LMFDB - Embedded newform 450.7.g.m.343.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [450,7,Mod(307,450)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(450, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 7, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("450.307");

S:= CuspForms(chi, 7);

N := Newforms(S);

Level:	$ N $	$=$	$ 450 = 2 \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	450.g (of order $4$, degree $2$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$103.524337629$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(i, \sqrt{129})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 65x^{2} + 1024 $ x^4 + 65*x^2 + 1024
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 10)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			343.2
Root			$-6.17891i$ of defining polynomial
Character	$\chi$	$=$	450.343
Dual form			450.7.g.m.307.2

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(4.00000 - 4.00000i) q^{2} -32.0000i q^{4} +(362.840 - 362.840i) q^{7} +(-128.000 - 128.000i) q^{8} +1321.26 q^{11} +(-254.789 - 254.789i) q^{13} -2902.72i q^{14} -1024.00 q^{16} +(4398.15 - 4398.15i) q^{17} +819.171i q^{19} +(5285.03 - 5285.03i) q^{22} +(11135.5 + 11135.5i) q^{23} -2038.31 q^{26} +(-11610.9 - 11610.9i) q^{28} -27495.1i q^{29} +15086.4 q^{31} +(-4096.00 + 4096.00i) q^{32} -35185.2i q^{34} +(-21550.0 + 21550.0i) q^{37} +(3276.68 + 3276.68i) q^{38} +45644.1 q^{41} +(78126.4 + 78126.4i) q^{43} -42280.3i q^{44} +89083.7 q^{46} +(-1995.12 + 1995.12i) q^{47} -145657. i q^{49} +(-8153.25 + 8153.25i) q^{52} +(106138. + 106138. i) q^{53} -92887.0 q^{56} +(-109980. - 109980. i) q^{58} +64466.6i q^{59} -320767. q^{61} +(60345.6 - 60345.6i) q^{62} +32768.0i q^{64} +(35044.1 - 35044.1i) q^{67} +(-140741. - 140741. i) q^{68} +251708. q^{71} +(-326907. - 326907. i) q^{73} +172400. i q^{74} +26213.5 q^{76} +(479405. - 479405. i) q^{77} +25722.3i q^{79} +(182577. - 182577. i) q^{82} +(-123104. - 123104. i) q^{83} +625011. q^{86} +(-169121. - 169121. i) q^{88} +198715. i q^{89} -184895. q^{91} +(356335. - 356335. i) q^{92} +15961.0i q^{94} +(738798. - 738798. i) q^{97} +(-582627. - 582627. i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 16 q^{2} + 202 q^{7} - 512 q^{8} + 2332 q^{11} - 792 q^{13} - 4096 q^{16} + 12368 q^{17} + 9328 q^{22} + 35342 q^{23} - 6336 q^{26} - 6464 q^{28} - 18932 q^{31} - 16384 q^{32} + 67812 q^{37} + 44000 q^{38}+ \cdots - 1321024 q^{98}+O(q^{100}) $ 4 * q + 16 * q^2 + 202 * q^7 - 512 * q^8 + 2332 * q^11 - 792 * q^13 - 4096 * q^16 + 12368 * q^17 + 9328 * q^22 + 35342 * q^23 - 6336 * q^26 - 6464 * q^28 - 18932 * q^31 - 16384 * q^32 + 67812 * q^37 + 44000 * q^38 - 110228 * q^41 + 293538 * q^43 + 282736 * q^46 + 195438 * q^47 - 25344 * q^52 - 121988 * q^53 - 51712 * q^56 - 229120 * q^58 - 243372 * q^61 - 75728 * q^62 + 773602 * q^67 - 395776 * q^68 + 1046132 * q^71 - 922372 * q^73 + 352000 * q^76 + 1040116 * q^77 - 440912 * q^82 - 1238058 * q^83 + 2348304 * q^86 - 298496 * q^88 - 221892 * q^91 + 1130944 * q^92 + 1937532 * q^97 - 1321024 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/450\mathbb{Z}\right)^\times$.

$n$	$101$	$127$
$\chi(n)$	$1$	$e\left(\frac{3}{4}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	4.00000	−	4.00000i	0.500000	−	0.500000i
$2$	4.00000	−	4.00000i	0.500000	−	0.500000i
$3$		0			0
$3$		0			0
$4$		−	32.0000i		−	0.500000i
$5$		0			0
$5$		0			0
$6$		0			0
$7$	362.840	−	362.840i	1.05784	−	1.05784i	0.0596214	−	0.998221i	$-0.481011\pi$
$7$	362.840	−	362.840i	1.05784	−	1.05784i	0.998221	−	0.0596214i	$-0.0189893\pi$
$8$	−128.000	−	128.000i	−0.250000	−	0.250000i
$9$		0			0
$10$		0			0
$11$	1321.26			0.992681			0.496340	−	0.868128i	$-0.334677\pi$
$11$	1321.26			0.992681			0.496340	+	0.868128i	$0.334677\pi$
$12$		0			0
$13$	−254.789	−	254.789i	−0.115971	−	0.115971i	0.646740	−	0.762711i	$-0.276132\pi$
$13$	−254.789	−	254.789i	−0.115971	−	0.115971i	−0.762711	+	0.646740i	$0.776132\pi$
$14$		−	2902.72i		−	1.05784i
$15$		0			0
$16$	−1024.00			−0.250000
$17$	4398.15	−	4398.15i	0.895206	−	0.895206i	−0.0998010	−	0.995007i	$-0.531821\pi$
$17$	4398.15	−	4398.15i	0.895206	−	0.895206i	0.995007	+	0.0998010i	$0.0318206\pi$
$18$		0			0
$19$			819.171i			0.119430i	0.998215	+	0.0597151i	$0.0190192\pi$
$19$			819.171i			0.119430i	−0.998215	+	0.0597151i	$0.980981\pi$
$20$		0			0
$21$		0			0
$22$	5285.03	−	5285.03i	0.496340	−	0.496340i
$23$	11135.5	+	11135.5i	0.915218	+	0.915218i	0.996677	−	0.0814587i	$-0.0259579\pi$
$23$	11135.5	+	11135.5i	0.915218	+	0.915218i	−0.0814587	+	0.996677i	$0.525958\pi$
$24$		0			0
$25$		0			0
$26$	−2038.31			−0.115971
$27$		0			0
$28$	−11610.9	−	11610.9i	−0.528921	−	0.528921i
$29$		−	27495.1i		−	1.12736i	−0.825995	−	0.563678i	$-0.809386\pi$
$29$		−	27495.1i		−	1.12736i	0.825995	−	0.563678i	$-0.190614\pi$
$30$		0			0
$31$	15086.4			0.506408			0.253204	−	0.967413i	$-0.418516\pi$
$31$	15086.4			0.506408			0.253204	+	0.967413i	$0.418516\pi$
$32$	−4096.00	+	4096.00i	−0.125000	+	0.125000i
$33$		0			0
$34$		−	35185.2i		−	0.895206i
$35$		0			0
$36$		0			0
$37$	−21550.0	+	21550.0i	−0.425444	+	0.425444i	−0.887073	−	0.461629i	$-0.847265\pi$
$37$	−21550.0	+	21550.0i	−0.425444	+	0.425444i	0.461629	+	0.887073i	$0.347265\pi$
$38$	3276.68	+	3276.68i	0.0597151	+	0.0597151i
$39$		0			0
$40$		0			0
$41$	45644.1			0.662267			0.331134	−	0.943584i	$-0.392569\pi$
$41$	45644.1			0.662267			0.331134	+	0.943584i	$0.392569\pi$
$42$		0			0
$43$	78126.4	+	78126.4i	0.982635	+	0.982635i	0.999852	−	0.0172164i	$-0.00548043\pi$
$43$	78126.4	+	78126.4i	0.982635	+	0.982635i	−0.0172164	+	0.999852i	$0.505480\pi$
$44$		−	42280.3i		−	0.496340i
$45$		0			0
$46$	89083.7			0.915218
$47$	−1995.12	+	1995.12i	−0.0192166	+	0.0192166i	−0.716650	−	0.697433i	$-0.754325\pi$
$47$	−1995.12	+	1995.12i	−0.0192166	+	0.0192166i	0.697433	+	0.716650i	$0.254325\pi$
$48$		0			0
$49$		−	145657.i		−	1.23806i
$50$		0			0
$51$		0			0
$52$	−8153.25	+	8153.25i	−0.0579857	+	0.0579857i
$53$	106138.	+	106138.i	0.712921	+	0.712921i	0.967145	−	0.254224i	$-0.0818202\pi$
$53$	106138.	+	106138.i	0.712921	+	0.712921i	−0.254224	+	0.967145i	$0.581820\pi$
$54$		0			0
$55$		0			0
$56$	−92887.0			−0.528921
$57$		0			0
$58$	−109980.	−	109980.i	−0.563678	−	0.563678i
$59$			64466.6i			0.313891i	0.987607	+	0.156945i	$0.0501647\pi$
$59$			64466.6i			0.313891i	−0.987607	+	0.156945i	$0.949835\pi$
$60$		0			0
$61$	−320767.			−1.41319			−0.706594	−	0.707620i	$-0.749769\pi$
$61$	−320767.			−1.41319			−0.706594	+	0.707620i	$0.749769\pi$
$62$	60345.6	−	60345.6i	0.253204	−	0.253204i
$63$		0			0
$64$			32768.0i			0.125000i
$65$		0			0
$66$		0			0
$67$	35044.1	−	35044.1i	0.116517	−	0.116517i	−0.646444	−	0.762961i	$-0.723745\pi$
$67$	35044.1	−	35044.1i	0.116517	−	0.116517i	0.762961	+	0.646444i	$0.223745\pi$
$68$	−140741.	−	140741.i	−0.447603	−	0.447603i
$69$		0			0
$70$		0			0
$71$	251708.			0.703271			0.351636	−	0.936137i	$-0.385626\pi$
$71$	251708.			0.703271			0.351636	+	0.936137i	$0.385626\pi$
$72$		0			0
$73$	−326907.	−	326907.i	−0.840342	−	0.840342i	0.148561	−	0.988903i	$-0.452536\pi$
$73$	−326907.	−	326907.i	−0.840342	−	0.840342i	−0.988903	+	0.148561i	$0.952536\pi$
$74$			172400.i			0.425444i
$75$		0			0
$76$	26213.5			0.0597151
$77$	479405.	−	479405.i	1.05010	−	1.05010i
$78$		0			0
$79$			25722.3i			0.0521710i	0.999660	+	0.0260855i	$0.00830421\pi$
$79$			25722.3i			0.0521710i	−0.999660	+	0.0260855i	$0.991696\pi$
$80$		0			0
$81$		0			0
$82$	182577.	−	182577.i	0.331134	−	0.331134i
$83$	−123104.	−	123104.i	−0.215298	−	0.215298i	0.591216	−	0.806513i	$-0.298648\pi$
$83$	−123104.	−	123104.i	−0.215298	−	0.215298i	−0.806513	+	0.591216i	$0.798648\pi$
$84$		0			0
$85$		0			0
$86$	625011.			0.982635
$87$		0			0
$88$	−169121.	−	169121.i	−0.248170	−	0.248170i
$89$			198715.i			0.281878i	0.990018	+	0.140939i	$0.0450121\pi$
$89$			198715.i			0.281878i	−0.990018	+	0.140939i	$0.954988\pi$
$90$		0			0
$91$	−184895.			−0.245359
$92$	356335.	−	356335.i	0.457609	−	0.457609i
$93$		0			0
$94$			15961.0i			0.0192166i
$95$		0			0
$96$		0			0
$97$	738798.	−	738798.i	0.809488	−	0.809488i	−0.175068	−	0.984556i	$-0.556015\pi$
$97$	738798.	−	738798.i	0.809488	−	0.809488i	0.984556	+	0.175068i	$0.0560145\pi$
$98$	−582627.	−	582627.i	−0.619031	−	0.619031i
$99$		0			0
$100$		0			0
$101$	122972.			0.119355			0.0596775	−	0.998218i	$-0.480993\pi$
$101$	122972.			0.119355			0.0596775	+	0.998218i	$0.480993\pi$
$102$		0			0
$103$	−419455.	−	419455.i	−0.383860	−	0.383860i	0.488631	−	0.872491i	$-0.337497\pi$
$103$	−419455.	−	419455.i	−0.383860	−	0.383860i	−0.872491	+	0.488631i	$0.837497\pi$
$104$			65226.0i			0.0579857i
$105$		0			0
$106$	849100.			0.712921
$107$	534643.	−	534643.i	0.436428	−	0.436428i	−0.454380	−	0.890808i	$-0.650139\pi$
$107$	534643.	−	534643.i	0.436428	−	0.436428i	0.890808	+	0.454380i	$0.150139\pi$
$108$		0			0
$109$		−	1.99464e6i		−	1.54023i	−0.637908	−	0.770113i	$-0.720200\pi$
$109$		−	1.99464e6i		−	1.54023i	0.637908	−	0.770113i	$-0.279800\pi$
$110$		0			0
$111$		0			0
$112$	−371548.	+	371548.i	−0.264461	+	0.264461i
$113$	−599768.	−	599768.i	−0.415669	−	0.415669i	0.468039	−	0.883708i	$-0.344961\pi$
$113$	−599768.	−	599768.i	−0.415669	−	0.415669i	−0.883708	+	0.468039i	$0.844961\pi$
$114$		0			0
$115$		0			0
$116$	−879842.			−0.563678
$117$		0			0
$118$	257866.	+	257866.i	0.156945	+	0.156945i
$119$		−	3.19165e6i		−	1.89397i
$120$		0			0
$121$	−25838.1			−0.0145849
$122$	−1.28307e6	+	1.28307e6i	−0.706594	+	0.706594i
$123$		0			0
$124$		−	482764.i		−	0.253204i
$125$		0			0
$126$		0			0
$127$	1.53367e6	−	1.53367e6i	0.748724	−	0.748724i	−0.225516	−	0.974240i	$-0.572407\pi$
$127$	1.53367e6	−	1.53367e6i	0.748724	−	0.748724i	0.974240	+	0.225516i	$0.0724067\pi$
$128$	131072.	+	131072.i	0.0625000	+	0.0625000i
$129$		0			0
$130$		0			0
$131$	−2.73248e6			−1.21547			−0.607734	−	0.794141i	$-0.707921\pi$
$131$	−2.73248e6			−1.21547			−0.607734	+	0.794141i	$0.707921\pi$
$132$		0			0
$133$	297228.	+	297228.i	0.126338	+	0.126338i
$134$		−	280353.i		−	0.116517i
$135$		0			0
$136$	−1.12593e6			−0.447603
$137$	1.14362e6	−	1.14362e6i	0.444753	−	0.444753i	−0.448853	−	0.893606i	$-0.648167\pi$
$137$	1.14362e6	−	1.14362e6i	0.444753	−	0.444753i	0.893606	+	0.448853i	$0.148167\pi$
$138$		0			0
$139$			1.81621e6i			0.676272i	0.941097	+	0.338136i	$0.109796\pi$
$139$			1.81621e6i			0.676272i	−0.941097	+	0.338136i	$0.890204\pi$
$140$		0			0
$141$		0			0
$142$	1.00683e6	−	1.00683e6i	0.351636	−	0.351636i
$143$	−336642.	−	336642.i	−0.115123	−	0.115123i
$144$		0			0
$145$		0			0
$146$	−2.61526e6			−0.840342
$147$		0			0
$148$	689600.	+	689600.i	0.212722	+	0.212722i
$149$		−	4.57708e6i		−	1.38366i	−0.722060	−	0.691830i	$-0.756805\pi$
$149$		−	4.57708e6i		−	1.38366i	0.722060	−	0.691830i	$-0.243195\pi$
$150$		0			0
$151$	−6.59081e6			−1.91429			−0.957145	−	0.289608i	$-0.906475\pi$
$151$	−6.59081e6			−1.91429			−0.957145	+	0.289608i	$0.906475\pi$
$152$	104854.	−	104854.i	0.0298575	−	0.0298575i
$153$		0			0
$154$		−	3.83524e6i		−	1.05010i
$155$		0			0
$156$		0			0
$157$	−2.59513e6	+	2.59513e6i	−0.670594	+	0.670594i	−0.957853	−	0.287259i	$-0.907256\pi$
$157$	−2.59513e6	+	2.59513e6i	−0.670594	+	0.670594i	0.287259	+	0.957853i	$0.407256\pi$
$158$	102889.	+	102889.i	0.0260855	+	0.0260855i
$159$		0			0
$160$		0			0
$161$	8.08078e6			1.93631
$162$		0			0
$163$	206143.	+	206143.i	0.0475999	+	0.0475999i	0.730506	−	0.682906i	$-0.239284\pi$
$163$	206143.	+	206143.i	0.0475999	+	0.0475999i	−0.682906	+	0.730506i	$0.739284\pi$
$164$		−	1.46061e6i		−	0.331134i
$165$		0			0
$166$	−984835.			−0.215298
$167$	−2.54852e6	+	2.54852e6i	−0.547191	+	0.547191i	−0.925627	−	0.378436i	$-0.876462\pi$
$167$	−2.54852e6	+	2.54852e6i	−0.547191	+	0.547191i	0.378436	+	0.925627i	$0.376462\pi$
$168$		0			0
$169$		−	4.69697e6i		−	0.973101i
$170$		0			0
$171$		0			0
$172$	2.50004e6	−	2.50004e6i	0.491318	−	0.491318i
$173$	4.48960e6	+	4.48960e6i	0.867100	+	0.867100i	0.992150	−	0.125050i	$-0.0399092\pi$
$173$	4.48960e6	+	4.48960e6i	0.867100	+	0.867100i	−0.125050	+	0.992150i	$0.539909\pi$
$174$		0			0
$175$		0			0
$176$	−1.35297e6			−0.248170
$177$		0			0
$178$	794860.	+	794860.i	0.140939	+	0.140939i
$179$			9.36798e6i			1.63338i	0.577078	+	0.816689i	$0.304193\pi$
$179$			9.36798e6i			1.63338i	−0.577078	+	0.816689i	$0.695807\pi$
$180$		0			0
$181$	7.30958e6			1.23270			0.616349	−	0.787473i	$-0.288611\pi$
$181$	7.30958e6			1.23270			0.616349	+	0.787473i	$0.288611\pi$
$182$	−739581.	+	739581.i	−0.122679	+	0.122679i
$183$		0			0
$184$		−	2.85068e6i		−	0.457609i
$185$		0			0
$186$		0			0
$187$	5.81109e6	−	5.81109e6i	0.888654	−	0.888654i
$188$	63844.0	+	63844.0i	0.00960830	+	0.00960830i
$189$		0			0
$190$		0			0
$191$	−8.54252e6			−1.22599			−0.612994	−	0.790088i	$-0.710035\pi$
$191$	−8.54252e6			−1.22599			−0.612994	+	0.790088i	$0.710035\pi$
$192$		0			0
$193$	−7.08708e6	−	7.08708e6i	−0.985815	−	0.985815i	0.0140856	−	0.999901i	$-0.495516\pi$
$193$	−7.08708e6	−	7.08708e6i	−0.985815	−	0.985815i	−0.999901	+	0.0140856i	$0.995516\pi$
$194$		−	5.91038e6i		−	0.809488i
$195$		0			0
$196$	−4.66101e6			−0.619031
$197$	5.09350e6	−	5.09350e6i	0.666220	−	0.666220i	−0.290619	−	0.956839i	$-0.593861\pi$
$197$	5.09350e6	−	5.09350e6i	0.666220	−	0.666220i	0.956839	+	0.290619i	$0.0938611\pi$
$198$		0			0
$199$		−	2.50957e6i		−	0.318449i	−0.987242	−	0.159224i	$-0.949101\pi$
$199$		−	2.50957e6i		−	0.318449i	0.987242	−	0.159224i	$-0.0508993\pi$
$200$		0			0
$201$		0			0
$202$	491887.	−	491887.i	0.0596775	−	0.0596775i
$203$	−9.97631e6	−	9.97631e6i	−1.19256	−	1.19256i
$204$		0			0
$205$		0			0
$206$	−3.35564e6			−0.383860
$207$		0			0
$208$	260904.	+	260904.i	0.0289928	+	0.0289928i
$209$			1.08234e6i			0.118556i
$210$		0			0
$211$	1.54519e6			0.164488			0.0822442	−	0.996612i	$-0.473791\pi$
$211$	1.54519e6			0.164488			0.0822442	+	0.996612i	$0.473791\pi$
$212$	3.39640e6	−	3.39640e6i	0.356460	−	0.356460i
$213$		0			0
$214$		−	4.27714e6i		−	0.436428i
$215$		0			0
$216$		0			0
$217$	5.47395e6	−	5.47395e6i	0.535700	−	0.535700i
$218$	−7.97855e6	−	7.97855e6i	−0.770113	−	0.770113i
$219$		0			0
$220$		0			0
$221$	−2.24120e6			−0.207637
$222$		0			0
$223$	−3.42480e6	−	3.42480e6i	−0.308831	−	0.308831i	0.535625	−	0.844456i	$-0.320076\pi$
$223$	−3.42480e6	−	3.42480e6i	−0.308831	−	0.308831i	−0.844456	+	0.535625i	$0.820076\pi$
$224$			2.97238e6i			0.264461i
$225$		0			0
$226$	−4.79814e6			−0.415669
$227$	1.45900e6	−	1.45900e6i	0.124732	−	0.124732i	−0.641985	−	0.766717i	$-0.721889\pi$
$227$	1.45900e6	−	1.45900e6i	0.124732	−	0.124732i	0.766717	+	0.641985i	$0.221889\pi$
$228$		0			0
$229$		−	1.46667e7i		−	1.22131i	−0.791895	−	0.610657i	$-0.790905\pi$
$229$		−	1.46667e7i		−	1.22131i	0.791895	−	0.610657i	$-0.209095\pi$
$230$		0			0
$231$		0			0
$232$	−3.51937e6	+	3.51937e6i	−0.281839	+	0.281839i
$233$	1.55030e7	+	1.55030e7i	1.22560	+	1.22560i	0.965613	+	0.259982i	$0.0837167\pi$
$233$	1.55030e7	+	1.55030e7i	1.22560	+	1.22560i	0.259982	+	0.965613i	$0.416283\pi$
$234$		0			0
$235$		0			0
$236$	2.06293e6			0.156945
$237$		0			0
$238$	−1.27666e7	−	1.27666e7i	−0.946987	−	0.946987i
$239$			1.47874e7i			1.08318i	0.840644	+	0.541588i	$0.182177\pi$
$239$			1.47874e7i			1.08318i	−0.840644	+	0.541588i	$0.817823\pi$
$240$		0			0
$241$	−2.05498e7			−1.46810			−0.734052	−	0.679094i	$-0.762373\pi$
$241$	−2.05498e7			−1.46810			−0.734052	+	0.679094i	$0.762373\pi$
$242$	−103352.	+	103352.i	−0.00729246	+	0.00729246i
$243$		0			0
$244$			1.02645e7i			0.706594i
$245$		0			0
$246$		0			0
$247$	208716.	−	208716.i	0.0138505	−	0.0138505i
$248$	−1.93106e6	−	1.93106e6i	−0.126602	−	0.126602i
$249$		0			0
$250$		0			0
$251$	−2.58613e6			−0.163542			−0.0817709	−	0.996651i	$-0.526058\pi$
$251$	−2.58613e6			−0.163542			−0.0817709	+	0.996651i	$0.526058\pi$
$252$		0			0
$253$	1.47128e7	+	1.47128e7i	0.908519	+	0.908519i
$254$		−	1.22694e7i		−	0.748724i
$255$		0			0
$256$	1.04858e6			0.0625000
$257$	−8.90392e6	+	8.90392e6i	−0.524544	+	0.524544i	−0.918940	−	0.394397i	$-0.870954\pi$
$257$	−8.90392e6	+	8.90392e6i	−0.524544	+	0.524544i	0.394397	+	0.918940i	$0.370954\pi$
$258$		0			0
$259$			1.56384e7i			0.900105i
$260$		0			0
$261$		0			0
$262$	−1.09299e7	+	1.09299e7i	−0.607734	+	0.607734i
$263$	1.24065e7	+	1.24065e7i	0.681996	+	0.681996i	0.960450	−	0.278454i	$-0.0898219\pi$
$263$	1.24065e7	+	1.24065e7i	0.681996	+	0.681996i	−0.278454	+	0.960450i	$0.589822\pi$
$264$		0			0
$265$		0			0
$266$	2.37782e6			0.126338
$267$		0			0
$268$	−1.12141e6	−	1.12141e6i	−0.0582587	−	0.0582587i
$269$		−	2.06412e7i		−	1.06042i	−0.847866	−	0.530211i	$-0.822113\pi$
$269$		−	2.06412e7i		−	1.06042i	0.847866	−	0.530211i	$-0.177887\pi$
$270$		0			0
$271$	−2.50344e7			−1.25785			−0.628926	−	0.777465i	$-0.716505\pi$
$271$	−2.50344e7			−1.25785			−0.628926	+	0.777465i	$0.716505\pi$
$272$	−4.50370e6	+	4.50370e6i	−0.223802	+	0.223802i
$273$		0			0
$274$		−	9.14894e6i		−	0.444753i
$275$		0			0
$276$		0			0
$277$	−1.30860e7	+	1.30860e7i	−0.615699	+	0.615699i	−0.944425	−	0.328726i	$-0.893381\pi$
$277$	−1.30860e7	+	1.30860e7i	−0.615699	+	0.615699i	0.328726	+	0.944425i	$0.393381\pi$
$278$	7.26483e6	+	7.26483e6i	0.338136	+	0.338136i
$279$		0			0
$280$		0			0
$281$	−7.83239e6			−0.353001			−0.176500	−	0.984301i	$-0.556478\pi$
$281$	−7.83239e6			−0.353001			−0.176500	+	0.984301i	$0.556478\pi$
$282$		0			0
$283$	3.44660e6	+	3.44660e6i	0.152066	+	0.152066i	0.779040	−	0.626974i	$-0.215707\pi$
$283$	3.44660e6	+	3.44660e6i	0.152066	+	0.152066i	−0.626974	+	0.779040i	$0.715707\pi$
$284$		−	8.05467e6i		−	0.351636i
$285$		0			0
$286$	−2.69314e6			−0.115123
$287$	1.65615e7	−	1.65615e7i	0.700575	−	0.700575i
$288$		0			0
$289$		−	1.45499e7i		−	0.602789i
$290$		0			0
$291$		0			0
$292$	−1.04610e7	+	1.04610e7i	−0.420171	+	0.420171i
$293$	2.34676e6	+	2.34676e6i	0.0932965	+	0.0932965i	0.752215	−	0.658918i	$-0.228986\pi$
$293$	2.34676e6	+	2.34676e6i	0.0932965	+	0.0932965i	−0.658918	+	0.752215i	$0.728986\pi$
$294$		0			0
$295$		0			0
$296$	5.51680e6			0.212722
$297$		0			0
$298$	−1.83083e7	−	1.83083e7i	−0.691830	−	0.691830i
$299$		−	5.67439e6i		−	0.212278i
$300$		0			0
$301$	5.66948e7			2.07895
$302$	−2.63632e7	+	2.63632e7i	−0.957145	+	0.957145i
$303$		0			0
$304$		−	838831.i		−	0.0298575i
$305$		0			0
$306$		0			0
$307$	3.16294e7	−	3.16294e7i	1.09314	−	1.09314i	0.0979492	−	0.995191i	$-0.468772\pi$
$307$	3.16294e7	−	3.16294e7i	1.09314	−	1.09314i	0.995191	−	0.0979492i	$-0.0312283\pi$
$308$	−1.53410e7	−	1.53410e7i	−0.525050	−	0.525050i
$309$		0			0
$310$		0			0
$311$	−2.07488e6			−0.0689781			−0.0344891	−	0.999405i	$-0.510980\pi$
$311$	−2.07488e6			−0.0689781			−0.0344891	+	0.999405i	$0.510980\pi$
$312$		0			0
$313$	3.33290e7	+	3.33290e7i	1.08690	+	1.08690i	0.995846	+	0.0910547i	$0.0290238\pi$
$313$	3.33290e7	+	3.33290e7i	1.08690	+	1.08690i	0.0910547	+	0.995846i	$0.470976\pi$
$314$			2.07610e7i			0.670594i
$315$		0			0
$316$	823114.			0.0260855
$317$	4.17645e6	−	4.17645e6i	0.131108	−	0.131108i	−0.638507	−	0.769616i	$-0.720448\pi$
$317$	4.17645e6	−	4.17645e6i	0.131108	−	0.131108i	0.769616	+	0.638507i	$0.220448\pi$
$318$		0			0
$319$		−	3.63281e7i		−	1.11910i
$320$		0			0
$321$		0			0
$322$	3.23231e7	−	3.23231e7i	0.968156	−	0.968156i
$323$	3.60284e6	+	3.60284e6i	0.106915	+	0.106915i
$324$		0			0
$325$		0			0
$326$	1.64914e6			0.0475999
$327$		0			0
$328$	−5.84245e6	−	5.84245e6i	−0.165567	−	0.165567i
$329$			1.44782e6i			0.0406563i
$330$		0			0
$331$	−1.41314e7			−0.389673			−0.194837	−	0.980836i	$-0.562418\pi$
$331$	−1.41314e7			−0.389673			−0.194837	+	0.980836i	$0.562418\pi$
$332$	−3.93934e6	+	3.93934e6i	−0.107649	+	0.107649i
$333$		0			0
$334$			2.03882e7i			0.547191i
$335$		0			0
$336$		0			0
$337$	−2.19464e7	+	2.19464e7i	−0.573420	+	0.573420i	−0.933083	−	0.359662i	$-0.882892\pi$
$337$	−2.19464e7	+	2.19464e7i	−0.573420	+	0.573420i	0.359662	+	0.933083i	$0.382892\pi$
$338$	−1.87879e7	−	1.87879e7i	−0.486551	−	0.486551i
$339$		0			0
$340$		0			0
$341$	1.99330e7			0.502701
$342$		0			0
$343$	−1.01623e7	−	1.01623e7i	−0.251831	−	0.251831i
$344$		−	2.00004e7i		−	0.491318i
$345$		0			0
$346$	3.59168e7			0.867100
$347$	3.70959e7	−	3.70959e7i	0.887846	−	0.887846i	−0.106470	−	0.994316i	$-0.533955\pi$
$347$	3.70959e7	−	3.70959e7i	0.887846	−	0.887846i	0.994316	+	0.106470i	$0.0339549\pi$
$348$		0			0
$349$			5.75499e7i			1.35384i	0.736056	+	0.676921i	$0.236686\pi$
$349$			5.75499e7i			1.35384i	−0.736056	+	0.676921i	$0.763314\pi$
$350$		0			0
$351$		0			0
$352$	−5.41187e6	+	5.41187e6i	−0.124085	+	0.124085i
$353$	−5.20285e7	−	5.20285e7i	−1.18282	−	1.18282i	−0.979011	−	0.203805i	$-0.934669\pi$
$353$	−5.20285e7	−	5.20285e7i	−1.18282	−	1.18282i	−0.203805	−	0.979011i	$-0.565331\pi$
$354$		0			0
$355$		0			0
$356$	6.35888e6			0.140939
$357$		0			0
$358$	3.74719e7	+	3.74719e7i	0.816689	+	0.816689i
$359$			3.95295e7i			0.854354i	0.904168	+	0.427177i	$0.140492\pi$
$359$			3.95295e7i			0.854354i	−0.904168	+	0.427177i	$0.859508\pi$
$360$		0			0
$361$	4.63748e7			0.985736
$362$	2.92383e7	−	2.92383e7i	0.616349	−	0.616349i
$363$		0			0
$364$			5.91665e6i			0.122679i
$365$		0			0
$366$		0			0
$367$	6.40678e7	−	6.40678e7i	1.29611	−	1.29611i	0.365168	−	0.930941i	$-0.381011\pi$
$367$	6.40678e7	−	6.40678e7i	1.29611	−	1.29611i	0.930941	−	0.365168i	$-0.118989\pi$
$368$	−1.14027e7	−	1.14027e7i	−0.228805	−	0.228805i
$369$		0			0
$370$		0			0
$371$	7.70219e7			1.50832
$372$		0			0
$373$	−1.23271e7	−	1.23271e7i	−0.237539	−	0.237539i	0.578291	−	0.815830i	$-0.303720\pi$
$373$	−1.23271e7	−	1.23271e7i	−0.237539	−	0.237539i	−0.815830	+	0.578291i	$0.803720\pi$
$374$		−	4.64887e7i		−	0.888654i
$375$		0			0
$376$	510752.			0.00960830
$377$	−7.00544e6	+	7.00544e6i	−0.130741	+	0.130741i
$378$		0			0
$379$			1.03307e8i			1.89764i	0.315817	+	0.948820i	$0.397721\pi$
$379$			1.03307e8i			1.89764i	−0.315817	+	0.948820i	$0.602279\pi$
$380$		0			0
$381$		0			0
$382$	−3.41701e7	+	3.41701e7i	−0.612994	+	0.612994i
$383$	4.46875e7	+	4.46875e7i	0.795408	+	0.795408i	0.982368	−	0.186959i	$-0.0598632\pi$
$383$	4.46875e7	+	4.46875e7i	0.795408	+	0.795408i	−0.186959	+	0.982368i	$0.559863\pi$
$384$		0			0
$385$		0			0
$386$	−5.66967e7			−0.985815
$387$		0			0
$388$	−2.36415e7	−	2.36415e7i	−0.404744	−	0.404744i
$389$			6.33567e7i			1.07633i	0.842841	+	0.538163i	$0.180882\pi$
$389$			6.33567e7i			1.07633i	−0.842841	+	0.538163i	$0.819118\pi$
$390$		0			0
$391$	9.79508e7			1.63862
$392$	−1.86441e7	+	1.86441e7i	−0.309515	+	0.309515i
$393$		0			0
$394$		−	4.07480e7i		−	0.666220i
$395$		0			0
$396$		0			0
$397$	−5.75784e7	+	5.75784e7i	−0.920213	+	0.920213i	−0.997044	−	0.0768313i	$-0.975520\pi$
$397$	−5.75784e7	+	5.75784e7i	−0.920213	+	0.920213i	0.0768313	+	0.997044i	$0.475520\pi$
$398$	−1.00383e7	−	1.00383e7i	−0.159224	−	0.159224i
$399$		0			0
$400$		0			0
$401$	3.99773e7			0.619984			0.309992	−	0.950739i	$-0.399674\pi$
$401$	3.99773e7			0.619984			0.309992	+	0.950739i	$0.399674\pi$
$402$		0			0
$403$	−3.84385e6	−	3.84385e6i	−0.0587288	−	0.0587288i
$404$		−	3.93509e6i		−	0.0596775i
$405$		0			0
$406$	−7.98105e7			−1.19256
$407$	−2.84731e7	+	2.84731e7i	−0.422330	+	0.422330i
$408$		0			0
$409$			1.10793e8i			1.61936i	0.586870	+	0.809681i	$0.300360\pi$
$409$			1.10793e8i			1.61936i	−0.586870	+	0.809681i	$0.699640\pi$
$410$		0			0
$411$		0			0
$412$	−1.34225e7	+	1.34225e7i	−0.191930	+	0.191930i
$413$	2.33910e7	+	2.33910e7i	0.332047	+	0.332047i
$414$		0			0
$415$		0			0
$416$	2.08723e6			0.0289928
$417$		0			0
$418$	4.32935e6	+	4.32935e6i	0.0592780	+	0.0592780i
$419$			1.22089e8i			1.65972i	0.557973	+	0.829859i	$0.311579\pi$
$419$			1.22089e8i			1.65972i	−0.557973	+	0.829859i	$0.688421\pi$
$420$		0			0
$421$	−4.21420e7			−0.564766			−0.282383	−	0.959302i	$-0.591125\pi$
$421$	−4.21420e7			−0.564766			−0.282383	+	0.959302i	$0.591125\pi$
$422$	6.18077e6	−	6.18077e6i	0.0822442	−	0.0822442i
$423$		0			0
$424$		−	2.71712e7i		−	0.356460i
$425$		0			0
$426$		0			0
$427$	−1.16387e8	+	1.16387e8i	−1.49493	+	1.49493i
$428$	−1.71086e7	−	1.71086e7i	−0.218214	−	0.218214i
$429$		0			0
$430$		0			0
$431$	3.91762e7			0.489317			0.244658	−	0.969609i	$-0.421324\pi$
$431$	3.91762e7			0.489317			0.244658	+	0.969609i	$0.421324\pi$
$432$		0			0
$433$	9.67310e7	+	9.67310e7i	1.19152	+	1.19152i	0.976640	+	0.214882i	$0.0689367\pi$
$433$	9.67310e7	+	9.67310e7i	1.19152	+	1.19152i	0.214882	+	0.976640i	$0.431063\pi$
$434$		−	4.37916e7i		−	0.535700i
$435$		0			0
$436$	−6.38284e7			−0.770113
$437$	−9.12185e6	+	9.12185e6i	−0.109305	+	0.109305i
$438$		0			0
$439$			7.95720e6i			0.0940517i	0.998894	+	0.0470258i	$0.0149743\pi$
$439$			7.95720e6i			0.0940517i	−0.998894	+	0.0470258i	$0.985026\pi$
$440$		0			0
$441$		0			0
$442$	−8.96480e6	+	8.96480e6i	−0.103818	+	0.103818i
$443$	8.63738e7	+	8.63738e7i	0.993507	+	0.993507i	0.999979	−	0.00647172i	$-0.00206003\pi$
$443$	8.63738e7	+	8.63738e7i	0.993507	+	0.993507i	−0.00647172	+	0.999979i	$0.502060\pi$
$444$		0			0
$445$		0			0
$446$	−2.73984e7			−0.308831
$447$		0			0
$448$	1.18895e7	+	1.18895e7i	0.132230	+	0.132230i
$449$		−	8.43204e7i		−	0.931523i	−0.884910	−	0.465762i	$-0.845780\pi$
$449$		−	8.43204e7i		−	0.931523i	0.884910	−	0.465762i	$-0.154220\pi$
$450$		0			0
$451$	6.03077e7			0.657420
$452$	−1.91926e7	+	1.91926e7i	−0.207835	+	0.207835i
$453$		0			0
$454$		−	1.16720e7i		−	0.124732i
$455$		0			0
$456$		0			0
$457$	1.32039e7	−	1.32039e7i	0.138341	−	0.138341i	−0.634545	−	0.772886i	$-0.718812\pi$
$457$	1.32039e7	−	1.32039e7i	0.138341	−	0.138341i	0.772886	+	0.634545i	$0.218812\pi$
$458$	−5.86670e7	−	5.86670e7i	−0.610657	−	0.610657i
$459$		0			0
$460$		0			0
$461$	−9.28082e7			−0.947291			−0.473645	−	0.880716i	$-0.657062\pi$
$461$	−9.28082e7			−0.947291			−0.473645	+	0.880716i	$0.657062\pi$
$462$		0			0
$463$	−4.15624e7	−	4.15624e7i	−0.418753	−	0.418753i	0.466021	−	0.884774i	$-0.345687\pi$
$463$	−4.15624e7	−	4.15624e7i	−0.418753	−	0.418753i	−0.884774	+	0.466021i	$0.845687\pi$
$464$			2.81549e7i			0.281839i
$465$		0			0
$466$	1.24024e8			1.22560
$467$	−5.74249e7	+	5.74249e7i	−0.563832	+	0.563832i	−0.930394	−	0.366562i	$-0.880535\pi$
$467$	−5.74249e7	+	5.74249e7i	−0.563832	+	0.563832i	0.366562	+	0.930394i	$0.380535\pi$
$468$		0			0
$469$		−	2.54308e7i		−	0.246514i
$470$		0			0
$471$		0			0
$472$	8.25172e6	−	8.25172e6i	0.0784727	−	0.0784727i
$473$	1.03225e8	+	1.03225e8i	0.975443	+	0.975443i
$474$		0			0
$475$		0			0
$476$	−1.02133e8			−0.946987
$477$		0			0
$478$	5.91498e7	+	5.91498e7i	0.541588	+	0.541588i
$479$			1.99190e7i			0.181243i	0.995885	+	0.0906215i	$0.0288854\pi$
$479$			1.99190e7i			0.181243i	−0.995885	+	0.0906215i	$0.971115\pi$
$480$		0			0
$481$	1.09814e7			0.0986786
$482$	−8.21992e7	+	8.21992e7i	−0.734052	+	0.734052i
$483$		0			0
$484$			826818.i			0.00729246i
$485$		0			0
$486$		0			0
$487$	4.50108e7	−	4.50108e7i	0.389700	−	0.389700i	−0.484881	−	0.874580i	$-0.661137\pi$
$487$	4.50108e7	−	4.50108e7i	0.389700	−	0.389700i	0.874580	+	0.484881i	$0.161137\pi$
$488$	4.10581e7	+	4.10581e7i	0.353297	+	0.353297i
$489$		0			0
$490$		0			0
$491$	1.28208e8			1.08310			0.541552	−	0.840667i	$-0.317837\pi$
$491$	1.28208e8			1.08310			0.541552	+	0.840667i	$0.317837\pi$
$492$		0			0
$493$	−1.20927e8	−	1.20927e8i	−1.00922	−	1.00922i
$494$		−	1.66973e6i		−	0.0138505i
$495$		0			0
$496$	−1.54485e7			−0.126602
$497$	9.13299e7	−	9.13299e7i	0.743950	−	0.743950i
$498$		0			0
$499$		−	5.72011e7i		−	0.460366i	−0.973147	−	0.230183i	$-0.926068\pi$
$499$		−	5.72011e7i		−	0.460366i	0.973147	−	0.230183i	$-0.0739324\pi$
$500$		0			0
$501$		0			0
$502$	−1.03445e7	+	1.03445e7i	−0.0817709	+	0.0817709i
$503$	−6.91106e6	−	6.91106e6i	−0.0543051	−	0.0543051i	0.679433	−	0.733738i	$-0.262226\pi$
$503$	−6.91106e6	−	6.91106e6i	−0.0543051	−	0.0543051i	−0.733738	+	0.679433i	$0.762226\pi$
$504$		0			0
$505$		0			0
$506$	1.17703e8			0.908519
$507$		0			0
$508$	−4.90775e7	−	4.90775e7i	−0.374362	−	0.374362i
$509$			1.61164e8i			1.22213i	0.791582	+	0.611063i	$0.209258\pi$
$509$			1.61164e8i			1.22213i	−0.791582	+	0.611063i	$0.790742\pi$
$510$		0			0
$511$	−2.37230e8			−1.77790
$512$	4.19430e6	−	4.19430e6i	0.0312500	−	0.0312500i
$513$		0			0
$514$			7.12313e7i			0.524544i
$515$		0			0
$516$		0			0
$517$	−2.63607e6	+	2.63607e6i	−0.0190759	+	0.0190759i
$518$	6.25536e7	+	6.25536e7i	0.450052	+	0.450052i
$519$		0			0
$520$		0			0
$521$	−9.62590e7			−0.680656			−0.340328	−	0.940307i	$-0.610538\pi$
$521$	−9.62590e7			−0.680656			−0.340328	+	0.940307i	$0.610538\pi$
$522$		0			0
$523$	1.16479e8	+	1.16479e8i	0.814221	+	0.814221i	0.985264	−	0.171042i	$-0.0547135\pi$
$523$	1.16479e8	+	1.16479e8i	0.814221	+	0.814221i	−0.171042	+	0.985264i	$0.554714\pi$
$524$			8.74394e7i			0.607734i
$525$		0			0
$526$	9.92520e7			0.681996
$527$	6.63522e7	−	6.63522e7i	0.453339	−	0.453339i
$528$		0			0
$529$			9.99610e7i			0.675248i
$530$		0			0
$531$		0			0
$532$	9.51130e6	−	9.51130e6i	0.0631691	−	0.0631691i
$533$	−1.16296e7	−	1.16296e7i	−0.0768040	−	0.0768040i
$534$		0			0
$535$		0			0
$536$	−8.97130e6			−0.0582587
$537$		0			0
$538$	−8.25650e7	−	8.25650e7i	−0.530211	−	0.530211i
$539$		−	1.92450e8i		−	1.22900i
$540$		0			0
$541$	8.42748e7			0.532238			0.266119	−	0.963940i	$-0.414259\pi$
$541$	8.42748e7			0.532238			0.266119	+	0.963940i	$0.414259\pi$
$542$	−1.00138e8	+	1.00138e8i	−0.628926	+	0.628926i
$543$		0			0
$544$			3.60296e7i			0.223802i
$545$		0			0
$546$		0			0
$547$	−1.60144e8	+	1.60144e8i	−0.978475	+	0.978475i	−0.999773	−	0.0212983i	$-0.993220\pi$
$547$	−1.60144e8	+	1.60144e8i	−0.978475	+	0.978475i	0.0212983	+	0.999773i	$0.493220\pi$
$548$	−3.65958e7	−	3.65958e7i	−0.222377	−	0.222377i
$549$		0			0
$550$		0			0
$551$	2.25232e7			0.134640
$552$		0			0
$553$	9.33308e6	+	9.33308e6i	0.0551887	+	0.0551887i
$554$			1.04688e8i			0.615699i
$555$		0			0
$556$	5.81187e7			0.338136
$557$	−1.89112e8	+	1.89112e8i	−1.09434	+	1.09434i	−0.0992858	+	0.995059i	$0.531656\pi$
$557$	−1.89112e8	+	1.89112e8i	−1.09434	+	1.09434i	−0.995059	+	0.0992858i	$0.968344\pi$
$558$		0			0
$559$		−	3.98115e7i		−	0.227915i
$560$		0			0
$561$		0			0
$562$	−3.13296e7	+	3.13296e7i	−0.176500	+	0.176500i
$563$	−3.67704e7	−	3.67704e7i	−0.206050	−	0.206050i	0.596536	−	0.802586i	$-0.296543\pi$
$563$	−3.67704e7	−	3.67704e7i	−0.206050	−	0.206050i	−0.802586	+	0.596536i	$0.796543\pi$
$564$		0			0
$565$		0			0
$566$	2.75728e7			0.152066
$567$		0			0
$568$	−3.22187e7	−	3.22187e7i	−0.175818	−	0.175818i
$569$		−	1.52471e8i		−	0.827658i	−0.910355	−	0.413829i	$-0.864191\pi$
$569$		−	1.52471e8i		−	0.827658i	0.910355	−	0.413829i	$-0.135809\pi$
$570$		0			0
$571$	−1.91478e8			−1.02851			−0.514257	−	0.857636i	$-0.671932\pi$
$571$	−1.91478e8			−1.02851			−0.514257	+	0.857636i	$0.671932\pi$
$572$	−1.07725e7	+	1.07725e7i	−0.0575613	+	0.0575613i
$573$		0			0
$574$		−	1.32492e8i		−	0.700575i
$575$		0			0
$576$		0			0
$577$	1.09464e8	−	1.09464e8i	0.569828	−	0.569828i	−0.362252	−	0.932080i	$-0.617992\pi$
$577$	1.09464e8	−	1.09464e8i	0.569828	−	0.569828i	0.932080	+	0.362252i	$0.117992\pi$
$578$	−5.81994e7	−	5.81994e7i	−0.301394	−	0.301394i
$579$		0			0
$580$		0			0
$581$	−8.93343e7			−0.455502
$582$		0			0
$583$	1.40235e8	+	1.40235e8i	0.707703	+	0.707703i
$584$			8.36883e7i			0.420171i
$585$		0			0
$586$	1.87741e7			0.0932965
$587$	1.33081e8	−	1.33081e8i	0.657961	−	0.657961i	−0.296936	−	0.954897i	$-0.595965\pi$
$587$	1.33081e8	−	1.33081e8i	0.657961	−	0.657961i	0.954897	+	0.296936i	$0.0959649\pi$
$588$		0			0
$589$			1.23583e7i			0.0604803i
$590$		0			0
$591$		0			0
$592$	2.20672e7	−	2.20672e7i	0.106361	−	0.106361i
$593$	−1.32615e8	−	1.32615e8i	−0.635956	−	0.635956i	0.313599	−	0.949555i	$-0.398465\pi$
$593$	−1.32615e8	−	1.32615e8i	−0.635956	−	0.635956i	−0.949555	+	0.313599i	$0.898465\pi$
$594$		0			0
$595$		0			0
$596$	−1.46466e8			−0.691830
$597$		0			0
$598$	−2.26975e7	−	2.26975e7i	−0.106139	−	0.106139i
$599$			9.86190e7i			0.458860i	0.973325	+	0.229430i	$0.0736862\pi$
$599$			9.86190e7i			0.458860i	−0.973325	+	0.229430i	$0.926314\pi$
$600$		0			0
$601$	1.55654e6			0.00717031			0.00358515	−	0.999994i	$-0.498859\pi$
$601$	1.55654e6			0.00717031			0.00358515	+	0.999994i	$0.498859\pi$
$602$	2.26779e8	−	2.26779e8i	1.03947	−	1.03947i
$603$		0			0
$604$			2.10906e8i			0.957145i
$605$		0			0
$606$		0			0
$607$	1.81568e8	−	1.81568e8i	0.811847	−	0.811847i	−0.173063	−	0.984911i	$-0.555367\pi$
$607$	1.81568e8	−	1.81568e8i	0.811847	−	0.811847i	0.984911	+	0.173063i	$0.0553665\pi$
$608$	−3.35533e6	−	3.35533e6i	−0.0149288	−	0.0149288i
$609$		0			0
$610$		0			0
$611$	1.01667e6			0.00445715
$612$		0			0
$613$	8.56979e7	+	8.56979e7i	0.372039	+	0.372039i	0.868220	−	0.496180i	$-0.165264\pi$
$613$	8.56979e7	+	8.56979e7i	0.372039	+	0.372039i	−0.496180	+	0.868220i	$0.665264\pi$
$614$		−	2.53035e8i		−	1.09314i
$615$		0			0
$616$	−1.22728e8			−0.525050
$617$	1.85185e8	−	1.85185e8i	0.788405	−	0.788405i	−0.192827	−	0.981233i	$-0.561766\pi$
$617$	1.85185e8	−	1.85185e8i	0.788405	−	0.788405i	0.981233	+	0.192827i	$0.0617658\pi$
$618$		0			0
$619$		−	2.30579e8i		−	0.972185i	−0.873907	−	0.486092i	$-0.838422\pi$
$619$		−	2.30579e8i		−	0.972185i	0.873907	−	0.486092i	$-0.161578\pi$
$620$		0			0
$621$		0			0
$622$	−8.29951e6	+	8.29951e6i	−0.0344891	+	0.0344891i
$623$	7.21018e7	+	7.21018e7i	0.298182	+	0.298182i
$624$		0			0
$625$		0			0
$626$	2.66632e8			1.08690
$627$		0			0
$628$	8.30441e7	+	8.30441e7i	0.335297	+	0.335297i
$629$			1.89560e8i			0.761720i
$630$		0			0
$631$	4.64011e8			1.84689			0.923444	−	0.383734i	$-0.125362\pi$
$631$	4.64011e8			1.84689			0.923444	+	0.383734i	$0.125362\pi$
$632$	3.29246e6	−	3.29246e6i	0.0130427	−	0.0130427i
$633$		0			0
$634$		−	3.34116e7i		−	0.131108i
$635$		0			0
$636$		0			0
$637$	−3.71117e7	+	3.71117e7i	−0.143580	+	0.143580i
$638$	−1.45312e8	−	1.45312e8i	−0.559552	−	0.559552i
$639$		0			0
$640$		0			0
$641$	−1.58114e8			−0.600337			−0.300168	−	0.953886i	$-0.597043\pi$
$641$	−1.58114e8			−0.600337			−0.300168	+	0.953886i	$0.597043\pi$
$642$		0			0
$643$	−7.86358e7	−	7.86358e7i	−0.295793	−	0.295793i	0.543571	−	0.839363i	$-0.317072\pi$
$643$	−7.86358e7	−	7.86358e7i	−0.295793	−	0.295793i	−0.839363	+	0.543571i	$0.817072\pi$
$644$		−	2.58585e8i		−	0.968156i
$645$		0			0
$646$	2.88227e7			0.106915
$647$	−3.62583e8	+	3.62583e8i	−1.33873	+	1.33873i	−0.441445	+	0.897288i	$0.645534\pi$
$647$	−3.62583e8	+	3.62583e8i	−1.33873	+	1.33873i	−0.897288	+	0.441445i	$0.854466\pi$
$648$		0			0
$649$			8.51770e7i			0.311593i
$650$		0			0
$651$		0			0
$652$	6.59658e6	−	6.59658e6i	0.0237999	−	0.0237999i
$653$	−2.28612e7	−	2.28612e7i	−0.0821031	−	0.0821031i	0.664863	−	0.746966i	$-0.268490\pi$
$653$	−2.28612e7	−	2.28612e7i	−0.0821031	−	0.0821031i	−0.746966	+	0.664863i	$0.768490\pi$
$654$		0			0
$655$		0			0
$656$	−4.67396e7			−0.165567
$657$		0			0
$658$	5.79129e6	+	5.79129e6i	0.0203281	+	0.0203281i
$659$		−	1.62416e8i		−	0.567510i	−0.958897	−	0.283755i	$-0.908420\pi$
$659$		−	1.62416e8i		−	0.567510i	0.958897	−	0.283755i	$-0.0915803\pi$
$660$		0			0
$661$	−5.53972e8			−1.91815			−0.959076	−	0.283148i	$-0.908621\pi$
$661$	−5.53972e8			−1.91815			−0.959076	+	0.283148i	$0.908621\pi$
$662$	−5.65255e7	+	5.65255e7i	−0.194837	+	0.194837i
$663$		0			0
$664$			3.15147e7i			0.107649i
$665$		0			0
$666$		0			0
$667$	3.06170e8	−	3.06170e8i	1.03178	−	1.03178i
$668$	8.15528e7	+	8.15528e7i	0.273596	+	0.273596i
$669$		0			0
$670$		0			0
$671$	−4.23816e8			−1.40284
$672$		0			0
$673$	−2.60781e8	−	2.60781e8i	−0.855521	−	0.855521i	0.135286	−	0.990807i	$-0.456805\pi$
$673$	−2.60781e8	−	2.60781e8i	−0.855521	−	0.855521i	−0.990807	+	0.135286i	$0.956805\pi$
$674$			1.75571e8i			0.573420i
$675$		0			0
$676$	−1.50303e8			−0.486551
$677$	8.65269e7	−	8.65269e7i	0.278859	−	0.278859i	−0.553794	−	0.832654i	$-0.686821\pi$
$677$	8.65269e7	−	8.65269e7i	0.278859	−	0.278859i	0.832654	+	0.553794i	$0.186821\pi$
$678$		0			0
$679$		−	5.36131e8i		−	1.71262i
$680$		0			0
$681$		0			0
$682$	7.97321e7	−	7.97321e7i	0.251351	−	0.251351i
$683$	−3.62755e8	−	3.62755e8i	−1.13855	−	1.13855i	−0.988710	−	0.149839i	$-0.952125\pi$
$683$	−3.62755e8	−	3.62755e8i	−1.13855	−	1.13855i	−0.149839	−	0.988710i	$-0.547875\pi$
$684$		0			0
$685$		0			0
$686$	−8.12984e7			−0.251831
$687$		0			0
$688$	−8.00014e7	−	8.00014e7i	−0.245659	−	0.245659i
$689$		−	5.40854e7i		−	0.165357i
$690$		0			0
$691$	5.53386e8			1.67724			0.838618	−	0.544720i	$-0.183364\pi$
$691$	5.53386e8			1.67724			0.838618	+	0.544720i	$0.183364\pi$
$692$	1.43667e8	−	1.43667e8i	0.433550	−	0.433550i
$693$		0			0
$694$		−	2.96767e8i		−	0.887846i
$695$		0			0
$696$		0			0
$697$	2.00750e8	−	2.00750e8i	0.592866	−	0.592866i
$698$	2.30199e8	+	2.30199e8i	0.676921	+	0.676921i
$699$		0			0
$700$		0			0
$701$	2.46611e8			0.715910			0.357955	−	0.933739i	$-0.383474\pi$
$701$	2.46611e8			0.715910			0.357955	+	0.933739i	$0.383474\pi$
$702$		0			0
$703$	−1.76531e7	−	1.76531e7i	−0.0508108	−	0.0508108i
$704$			4.32950e7i			0.124085i
$705$		0			0
$706$	−4.16228e8			−1.18282
$707$	4.46190e7	−	4.46190e7i	0.126259	−	0.126259i
$708$		0			0
$709$		−	2.37496e7i		−	0.0666374i	−0.999445	−	0.0333187i	$-0.989392\pi$
$709$		−	2.37496e7i		−	0.0666374i	0.999445	−	0.0333187i	$-0.0106076\pi$
$710$		0			0
$711$		0			0
$712$	2.54355e7	−	2.54355e7i	0.0704694	−	0.0704694i
$713$	1.67994e8	+	1.67994e8i	0.463473	+	0.463473i
$714$		0			0
$715$		0			0
$716$	2.99775e8			0.816689
$717$		0			0
$718$	1.58118e8	+	1.58118e8i	0.427177	+	0.427177i
$719$		−	5.91472e8i		−	1.59128i	−0.605767	−	0.795642i	$-0.707134\pi$
$719$		−	5.91472e8i		−	1.59128i	0.605767	−	0.795642i	$-0.292866\pi$
$720$		0			0
$721$	−3.04390e8			−0.812127
$722$	1.85499e8	−	1.85499e8i	0.492868	−	0.492868i
$723$		0			0
$724$		−	2.33906e8i		−	0.616349i
$725$		0			0
$726$		0			0
$727$	−1.16120e8	+	1.16120e8i	−0.302207	+	0.302207i	−0.841877	−	0.539670i	$-0.818549\pi$
$727$	−1.16120e8	+	1.16120e8i	−0.302207	+	0.302207i	0.539670	+	0.841877i	$0.318549\pi$
$728$	2.36666e7	+	2.36666e7i	0.0613397	+	0.0613397i
$729$		0			0
$730$		0			0
$731$	6.87223e8			1.75932
$732$		0			0
$733$	−9.94477e7	−	9.94477e7i	−0.252512	−	0.252512i	0.569488	−	0.822000i	$-0.307142\pi$
$733$	−9.94477e7	−	9.94477e7i	−0.252512	−	0.252512i	−0.822000	+	0.569488i	$0.807142\pi$
$734$		−	5.12543e8i		−	1.29611i
$735$		0			0
$736$	−9.12217e7			−0.228805
$737$	4.63024e7	−	4.63024e7i	0.115665	−	0.115665i
$738$		0			0
$739$		−	3.16873e8i		−	0.785150i	−0.919720	−	0.392575i	$-0.871584\pi$
$739$		−	3.16873e8i		−	0.785150i	0.919720	−	0.392575i	$-0.128416\pi$
$740$		0			0
$741$		0			0
$742$	3.08088e8	−	3.08088e8i	0.754158	−	0.754158i
$743$	−2.89147e8	−	2.89147e8i	−0.704939	−	0.704939i	0.260527	−	0.965467i	$-0.416104\pi$
$743$	−2.89147e8	−	2.89147e8i	−0.704939	−	0.704939i	−0.965467	+	0.260527i	$0.916104\pi$
$744$		0			0
$745$		0			0
$746$	−9.86170e7			−0.237539
$747$		0			0
$748$	−1.85955e8	−	1.85955e8i	−0.444327	−	0.444327i
$749$		−	3.87980e8i		−	0.923344i
$750$		0			0
$751$	−3.85591e8			−0.910346			−0.455173	−	0.890403i	$-0.650423\pi$
$751$	−3.85591e8			−0.910346			−0.455173	+	0.890403i	$0.650423\pi$
$752$	2.04301e6	−	2.04301e6i	0.00480415	−	0.00480415i
$753$		0			0
$754$			5.60435e7i			0.130741i
$755$		0			0
$756$		0			0
$757$	−2.20526e8	+	2.20526e8i	−0.508361	+	0.508361i	−0.914023	−	0.405662i	$-0.867041\pi$
$757$	−2.20526e8	+	2.20526e8i	−0.508361	+	0.508361i	0.405662	+	0.914023i	$0.367041\pi$
$758$	4.13230e8	+	4.13230e8i	0.948820	+	0.948820i
$759$		0			0
$760$		0			0
$761$	−2.54649e8			−0.577815			−0.288907	−	0.957357i	$-0.593292\pi$
$761$	−2.54649e8			−0.577815			−0.288907	+	0.957357i	$0.593292\pi$
$762$		0			0
$763$	−7.23734e8	−	7.23734e8i	−1.62932	−	1.62932i
$764$			2.73361e8i			0.612994i
$765$		0			0
$766$	3.57500e8			0.795408
$767$	1.64254e7	−	1.64254e7i	0.0364023	−	0.0364023i
$768$		0			0
$769$		−	4.95784e8i		−	1.09022i	−0.838365	−	0.545109i	$-0.816488\pi$
$769$		−	4.95784e8i		−	1.09022i	0.838365	−	0.545109i	$-0.183512\pi$
$770$		0			0
$771$		0			0
$772$	−2.26787e8	+	2.26787e8i	−0.492908	+	0.492908i
$773$	1.32543e8	+	1.32543e8i	0.286957	+	0.286957i	0.835876	−	0.548918i	$-0.184960\pi$
$773$	1.32543e8	+	1.32543e8i	0.286957	+	0.286957i	−0.548918	+	0.835876i	$0.684960\pi$
$774$		0			0
$775$		0			0
$776$	−1.89132e8			−0.404744
$777$		0			0
$778$	2.53427e8	+	2.53427e8i	0.538163	+	0.538163i
$779$			3.73904e7i			0.0790947i
$780$		0			0
$781$	3.32572e8			0.698124
$782$	3.91803e8	−	3.91803e8i	0.819309	−	0.819309i
$783$		0			0
$784$			1.49152e8i			0.309515i
$785$		0			0
$786$		0			0
$787$	−1.81673e8	+	1.81673e8i	−0.372705	+	0.372705i	−0.868462	−	0.495757i	$-0.834891\pi$
$787$	−1.81673e8	+	1.81673e8i	−0.372705	+	0.372705i	0.495757	+	0.868462i	$0.334891\pi$
$788$	−1.62992e8	−	1.62992e8i	−0.333110	−	0.333110i
$789$		0			0
$790$		0			0
$791$	−4.35239e8			−0.879425
$792$		0			0
$793$	8.17278e7	+	8.17278e7i	0.163889	+	0.163889i
$794$			4.60627e8i			0.920213i
$795$		0			0
$796$	−8.03061e7			−0.159224
$797$	−1.28320e7	+	1.28320e7i	−0.0253466	+	0.0253466i	−0.719666	−	0.694320i	$-0.755705\pi$
$797$	−1.28320e7	+	1.28320e7i	−0.0253466	+	0.0253466i	0.694320	+	0.719666i	$0.255705\pi$
$798$		0			0
$799$			1.75497e7i			0.0344056i
$800$		0			0
$801$		0			0
$802$	1.59909e8	−	1.59909e8i	0.309992	−	0.309992i
$803$	−4.31929e8	−	4.31929e8i	−0.834191	−	0.834191i
$804$		0			0
$805$		0			0
$806$	−3.07508e7			−0.0587288
$807$		0			0
$808$	−1.57404e7	−	1.57404e7i	−0.0298388	−	0.0298388i
$809$			2.71170e8i			0.512150i	0.966657	+	0.256075i	$0.0824293\pi$
$809$			2.71170e8i			0.512150i	−0.966657	+	0.256075i	$0.917571\pi$
$810$		0			0
$811$	−5.93403e8			−1.11247			−0.556234	−	0.831026i	$-0.687754\pi$
$811$	−5.93403e8			−1.11247			−0.556234	+	0.831026i	$0.687754\pi$
$812$	−3.19242e8	+	3.19242e8i	−0.596282	+	0.596282i
$813$		0			0
$814$			2.27785e8i			0.422330i
$815$		0			0
$816$		0			0
$817$	−6.39989e7	+	6.39989e7i	−0.117356	+	0.117356i
$818$	4.43174e8	+	4.43174e8i	0.809681	+	0.809681i
$819$		0			0
$820$		0			0
$821$	2.50203e7			0.0452130			0.0226065	−	0.999744i	$-0.492804\pi$
$821$	2.50203e7			0.0452130			0.0226065	+	0.999744i	$0.492804\pi$
$822$		0			0
$823$	5.91490e6	+	5.91490e6i	0.0106108	+	0.0106108i	0.712392	−	0.701781i	$-0.247612\pi$
$823$	5.91490e6	+	5.91490e6i	0.0106108	+	0.0106108i	−0.701781	+	0.712392i	$0.747612\pi$
$824$			1.07380e8i			0.191930i
$825$		0			0
$826$	1.87128e8			0.332047
$827$	3.92300e8	−	3.92300e8i	0.693588	−	0.693588i	−0.269432	−	0.963019i	$-0.586836\pi$
$827$	3.92300e8	−	3.92300e8i	0.693588	−	0.693588i	0.963019	+	0.269432i	$0.0868359\pi$
$828$		0			0
$829$			6.16308e8i			1.08177i	0.841097	+	0.540884i	$0.181910\pi$
$829$			6.16308e8i			1.08177i	−0.841097	+	0.540884i	$0.818090\pi$
$830$		0			0
$831$		0			0
$832$	8.34893e6	−	8.34893e6i	0.0144964	−	0.0144964i
$833$	−6.40620e8	−	6.40620e8i	−1.10832	−	1.10832i
$834$		0			0
$835$		0			0
$836$	3.46348e7			0.0592780
$837$		0			0
$838$	4.88356e8	+	4.88356e8i	0.829859	+	0.829859i
$839$			5.89741e8i			0.998564i	0.866440	+	0.499282i	$0.166403\pi$
$839$			5.89741e8i			0.998564i	−0.866440	+	0.499282i	$0.833597\pi$
$840$		0			0
$841$	−1.61155e8			−0.270930
$842$	−1.68568e8	+	1.68568e8i	−0.282383	+	0.282383i
$843$		0			0
$844$		−	4.94461e7i		−	0.0822442i
$845$		0			0
$846$		0			0
$847$	−9.37509e6	+	9.37509e6i	−0.0154285	+	0.0154285i
$848$	−1.08685e8	−	1.08685e8i	−0.178230	−	0.178230i
$849$		0			0
$850$		0			0
$851$	−4.79938e8			−0.778747
$852$		0			0
$853$	5.34794e8	+	5.34794e8i	0.861666	+	0.861666i	0.991532	−	0.129865i	$-0.0414545\pi$
$853$	5.34794e8	+	5.34794e8i	0.861666	+	0.861666i	−0.129865	+	0.991532i	$0.541454\pi$
$854$			9.31096e8i			1.49493i
$855$		0			0
$856$	−1.36869e8			−0.218214
$857$	−3.90046e7	+	3.90046e7i	−0.0619689	+	0.0619689i	−0.737412	−	0.675443i	$-0.763952\pi$
$857$	−3.90046e7	+	3.90046e7i	−0.0619689	+	0.0619689i	0.675443	+	0.737412i	$0.263952\pi$
$858$		0			0
$859$			8.15365e8i			1.28639i	0.765702	+	0.643195i	$0.222392\pi$
$859$			8.15365e8i			1.28639i	−0.765702	+	0.643195i	$0.777608\pi$
$860$		0			0
$861$		0			0
$862$	1.56705e8	−	1.56705e8i	0.244658	−	0.244658i
$863$	5.03491e8	+	5.03491e8i	0.783357	+	0.783357i	0.980396	−	0.197039i	$-0.0631325\pi$
$863$	5.03491e8	+	5.03491e8i	0.783357	+	0.783357i	−0.197039	+	0.980396i	$0.563132\pi$
$864$		0			0
$865$		0			0
$866$	7.73848e8			1.19152
$867$		0			0
$868$	−1.75166e8	−	1.75166e8i	−0.267850	−	0.267850i
$869$			3.39858e7i			0.0517891i
$870$		0			0
$871$	−1.78577e7			−0.0270254
$872$	−2.55314e8	+	2.55314e8i	−0.385056	+	0.385056i
$873$		0			0
$874$			7.29748e7i			0.109305i
$875$		0			0
$876$		0			0
$877$	4.25203e8	−	4.25203e8i	0.630373	−	0.630373i	−0.317788	−	0.948162i	$-0.602940\pi$
$877$	4.25203e8	−	4.25203e8i	0.630373	−	0.630373i	0.948162	+	0.317788i	$0.102940\pi$
$878$	3.18288e7	+	3.18288e7i	0.0470258	+	0.0470258i
$879$		0			0
$880$		0			0
$881$	−4.01941e8			−0.587806			−0.293903	−	0.955835i	$-0.594954\pi$
$881$	−4.01941e8			−0.587806			−0.293903	+	0.955835i	$0.594954\pi$
$882$		0			0
$883$	−4.01338e8	−	4.01338e8i	−0.582946	−	0.582946i	0.352766	−	0.935712i	$-0.385241\pi$
$883$	−4.01338e8	−	4.01338e8i	−0.582946	−	0.582946i	−0.935712	+	0.352766i	$0.885241\pi$
$884$			7.17184e7i			0.103818i
$885$		0			0
$886$	6.90991e8			0.993507
$887$	5.18983e6	−	5.18983e6i	0.00743674	−	0.00743674i	−0.703379	−	0.710815i	$-0.748326\pi$
$887$	5.18983e6	−	5.18983e6i	0.00743674	−	0.00743674i	0.710815	+	0.703379i	$0.248326\pi$
$888$		0			0
$889$		−	1.11296e9i		−	1.58406i
$890$		0			0
$891$		0			0
$892$	−1.09594e8	+	1.09594e8i	−0.154415	+	0.154415i
$893$	−1.63435e6	−	1.63435e6i	−0.00229504	−	0.00229504i
$894$		0			0
$895$		0			0
$896$	9.51163e7			0.132230
$897$		0			0
$898$	−3.37282e8	−	3.37282e8i	−0.465762	−	0.465762i
$899$		−	4.14801e8i		−	0.570901i
$900$		0			0
$901$	9.33617e8			1.27642
$902$	2.41231e8	−	2.41231e8i	0.328710	−	0.328710i
$903$		0			0
$904$			1.53541e8i			0.207835i
$905$		0			0
$906$		0			0
$907$	−3.76104e8	+	3.76104e8i	−0.504065	+	0.504065i	−0.912698	−	0.408634i	$-0.866005\pi$
$907$	−3.76104e8	+	3.76104e8i	−0.504065	+	0.504065i	0.408634	+	0.912698i	$0.366005\pi$
$908$	−4.66879e7	−	4.66879e7i	−0.0623659	−	0.0623659i
$909$		0			0
$910$		0			0
$911$	9.07902e8			1.20084			0.600418	−	0.799686i	$-0.295001\pi$
$911$	9.07902e8			1.20084			0.600418	+	0.799686i	$0.295001\pi$
$912$		0			0
$913$	−1.62653e8	−	1.62653e8i	−0.213722	−	0.213722i
$914$		−	1.05631e8i		−	0.138341i
$915$		0			0
$916$	−4.69336e8			−0.610657
$917$	−9.91453e8	+	9.91453e8i	−1.28577	+	1.28577i
$918$		0			0
$919$		−	2.04129e8i		−	0.263002i	−0.991316	−	0.131501i	$-0.958020\pi$
$919$		−	2.04129e8i		−	0.263002i	0.991316	−	0.131501i	$-0.0419796\pi$
$920$		0			0
$921$		0			0
$922$	−3.71233e8	+	3.71233e8i	−0.473645	+	0.473645i
$923$	−6.41326e7	−	6.41326e7i	−0.0815593	−	0.0815593i
$924$		0			0
$925$		0			0
$926$	−3.32499e8			−0.418753
$927$		0			0
$928$	1.12620e8	+	1.12620e8i	0.140919	+	0.140919i
$929$		−	1.33077e9i		−	1.65980i	−0.557914	−	0.829898i	$-0.688398\pi$
$929$		−	1.33077e9i		−	1.65980i	0.557914	−	0.829898i	$-0.311602\pi$
$930$		0			0
$931$	1.19318e8			0.147862
$932$	4.96095e8	−	4.96095e8i	0.612798	−	0.612798i
$933$		0			0
$934$			4.59399e8i			0.563832i
$935$		0			0
$936$		0			0
$937$	9.92661e7	−	9.92661e7i	0.120665	−	0.120665i	−0.644196	−	0.764861i	$-0.722808\pi$
$937$	9.92661e7	−	9.92661e7i	0.120665	−	0.120665i	0.764861	+	0.644196i	$0.222808\pi$
$938$	−1.01723e8	−	1.01723e8i	−0.123257	−	0.123257i
$939$		0			0
$940$		0			0
$941$	3.77955e7			0.0453598			0.0226799	−	0.999743i	$-0.492780\pi$
$941$	3.77955e7			0.0453598			0.0226799	+	0.999743i	$0.492780\pi$
$942$		0			0
$943$	5.08268e8	+	5.08268e8i	0.606119	+	0.606119i
$944$		−	6.60138e7i		−	0.0784727i
$945$		0			0
$946$	8.25801e8			0.975443
$947$	−1.09711e7	+	1.09711e7i	−0.0129181	+	0.0129181i	−0.713536	−	0.700618i	$-0.752908\pi$
$947$	−1.09711e7	+	1.09711e7i	−0.0129181	+	0.0129181i	0.700618	+	0.713536i	$0.252908\pi$
$948$		0			0
$949$			1.66585e8i			0.194911i
$950$		0			0
$951$		0			0
$952$	−4.08531e8	+	4.08531e8i	−0.473494	+	0.473494i
$953$	1.06083e9	+	1.06083e9i	1.22565	+	1.22565i	0.965593	+	0.260059i	$0.0837420\pi$
$953$	1.06083e9	+	1.06083e9i	1.22565	+	1.22565i	0.260059	+	0.965593i	$0.416258\pi$
$954$		0			0
$955$		0			0
$956$	4.73198e8			0.541588
$957$		0			0
$958$	7.96761e7	+	7.96761e7i	0.0906215	+	0.0906215i
$959$		−	8.29900e8i		−	0.940958i
$960$		0			0
$961$	−6.59905e8			−0.743551
$962$	4.39256e7	−	4.39256e7i	0.0493393	−	0.0493393i
$963$		0			0
$964$			6.57594e8i			0.734052i
$965$		0			0
$966$		0			0
$967$	−3.42662e8	+	3.42662e8i	−0.378954	+	0.378954i	−0.870725	−	0.491771i	$-0.836350\pi$
$967$	−3.42662e8	+	3.42662e8i	−0.378954	+	0.378954i	0.491771	+	0.870725i	$0.336350\pi$
$968$	3.30727e6	+	3.30727e6i	0.00364623	+	0.00364623i
$969$		0			0
$970$		0			0
$971$	−3.06097e8			−0.334350			−0.167175	−	0.985927i	$-0.553464\pi$
$971$	−3.06097e8			−0.334350			−0.167175	+	0.985927i	$0.553464\pi$
$972$		0			0
$973$	6.58993e8	+	6.58993e8i	0.715389	+	0.715389i
$974$		−	3.60087e8i		−	0.389700i
$975$		0			0
$976$	3.28465e8			0.353297
$977$	−1.01093e9	+	1.01093e9i	−1.08402	+	1.08402i	−0.0878855	+	0.996131i	$0.528011\pi$
$977$	−1.01093e9	+	1.01093e9i	−1.08402	+	1.08402i	−0.996131	+	0.0878855i	$0.971989\pi$
$978$		0			0
$979$			2.62554e8i			0.279815i
$980$		0			0
$981$		0			0
$982$	5.12831e8	−	5.12831e8i	0.541552	−	0.541552i
$983$	3.66759e8	+	3.66759e8i	0.386118	+	0.386118i	0.873300	−	0.487182i	$-0.161975\pi$
$983$	3.66759e8	+	3.66759e8i	0.386118	+	0.386118i	−0.487182	+	0.873300i	$0.661975\pi$
$984$		0			0
$985$		0			0
$986$	−9.67419e8			−1.00922
$987$		0			0
$988$	−6.67891e6	−	6.67891e6i	−0.00692524	−	0.00692524i
$989$			1.73995e9i			1.79865i
$990$		0			0
$991$	−7.65601e8			−0.786649			−0.393325	−	0.919400i	$-0.628675\pi$
$991$	−7.65601e8			−0.786649			−0.393325	+	0.919400i	$0.628675\pi$
$992$	−6.17939e7	+	6.17939e7i	−0.0633010	+	0.0633010i
$993$		0			0
$994$		−	7.30639e8i		−	0.743950i
$995$		0			0
$996$		0			0
$997$	−3.07440e8	+	3.07440e8i	−0.310223	+	0.310223i	−0.844996	−	0.534773i	$-0.820397\pi$
$997$	−3.07440e8	+	3.07440e8i	−0.310223	+	0.310223i	0.534773	+	0.844996i	$0.320397\pi$
$998$	−2.28805e8	−	2.28805e8i	−0.230183	−	0.230183i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	450.7.g.m.343.2		4
3.2	odd	2		50.7.c.d.43.1		4
5.2	odd	4	inner	450.7.g.m.307.2		4
5.3	odd	4		90.7.g.b.37.2		4
5.4	even	2		90.7.g.b.73.2		4
15.2	even	4		50.7.c.d.7.1		4
15.8	even	4		10.7.c.b.7.2	yes	4
15.14	odd	2		10.7.c.b.3.2	✓	4
60.23	odd	4		80.7.p.c.17.1		4
60.59	even	2		80.7.p.c.33.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
10.7.c.b.3.2	✓	4	15.14	odd	2
10.7.c.b.7.2	yes	4	15.8	even	4
50.7.c.d.7.1		4	15.2	even	4
50.7.c.d.43.1		4	3.2	odd	2
80.7.p.c.17.1		4	60.23	odd	4
80.7.p.c.33.1		4	60.59	even	2
90.7.g.b.37.2		4	5.3	odd	4
90.7.g.b.73.2		4	5.4	even	2
450.7.g.m.307.2		4	5.2	odd	4	inner
450.7.g.m.343.2		4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	450.g (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(4.00000 - 4.00000i) q^{2} -32.0000i q^{4} +(362.840 - 362.840i) q^{7} +(-128.000 - 128.000i) q^{8} +1321.26 q^{11} +(-254.789 - 254.789i) q^{13} -2902.72i q^{14} -1024.00 q^{16} +(4398.15 - 4398.15i) q^{17} +819.171i q^{19} +(5285.03 - 5285.03i) q^{22} +(11135.5 + 11135.5i) q^{23} -2038.31 q^{26} +(-11610.9 - 11610.9i) q^{28} -27495.1i q^{29} +15086.4 q^{31} +(-4096.00 + 4096.00i) q^{32} -35185.2i q^{34} +(-21550.0 + 21550.0i) q^{37} +(3276.68 + 3276.68i) q^{38} +45644.1 q^{41} +(78126.4 + 78126.4i) q^{43} -42280.3i q^{44} +89083.7 q^{46} +(-1995.12 + 1995.12i) q^{47} -145657. i q^{49} +(-8153.25 + 8153.25i) q^{52} +(106138. + 106138. i) q^{53} -92887.0 q^{56} +(-109980. - 109980. i) q^{58} +64466.6i q^{59} -320767. q^{61} +(60345.6 - 60345.6i) q^{62} +32768.0i q^{64} +(35044.1 - 35044.1i) q^{67} +(-140741. - 140741. i) q^{68} +251708. q^{71} +(-326907. - 326907. i) q^{73} +172400. i q^{74} +26213.5 q^{76} +(479405. - 479405. i) q^{77} +25722.3i q^{79} +(182577. - 182577. i) q^{82} +(-123104. - 123104. i) q^{83} +625011. q^{86} +(-169121. - 169121. i) q^{88} +198715. i q^{89} -184895. q^{91} +(356335. - 356335. i) q^{92} +15961.0i q^{94} +(738798. - 738798. i) q^{97} +(-582627. - 582627. i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 16 q^{2} + 202 q^{7} - 512 q^{8} + 2332 q^{11} - 792 q^{13} - 4096 q^{16} + 12368 q^{17} + 9328 q^{22} + 35342 q^{23} - 6336 q^{26} - 6464 q^{28} - 18932 q^{31} - 16384 q^{32} + 67812 q^{37} + 44000 q^{38}+ \cdots - 1321024 q^{98}+O(q^{100}) \) 4 * q + 16 * q^2 + 202 * q^7 - 512 * q^8 + 2332 * q^11 - 792 * q^13 - 4096 * q^16 + 12368 * q^17 + 9328 * q^22 + 35342 * q^23 - 6336 * q^26 - 6464 * q^28 - 18932 * q^31 - 16384 * q^32 + 67812 * q^37 + 44000 * q^38 - 110228 * q^41 + 293538 * q^43 + 282736 * q^46 + 195438 * q^47 - 25344 * q^52 - 121988 * q^53 - 51712 * q^56 - 229120 * q^58 - 243372 * q^61 - 75728 * q^62 + 773602 * q^67 - 395776 * q^68 + 1046132 * q^71 - 922372 * q^73 + 352000 * q^76 + 1040116 * q^77 - 440912 * q^82 - 1238058 * q^83 + 2348304 * q^86 - 298496 * q^88 - 221892 * q^91 + 1130944 * q^92 + 1937532 * q^97 - 1321024 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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