LMFDB - Embedded newform 338.8.b.a.337.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [338,8,Mod(337,338)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(338, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("338.337");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 338 = 2 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	338.b (of order $2$, degree $1$, not 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:	$105.586138614$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 26)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	338.337
Dual form			338.8.b.a.337.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-8.00000i q^{2} -87.0000 q^{3} -64.0000 q^{4} -321.000i q^{5} +696.000i q^{6} -181.000i q^{7} +512.000i q^{8} +5382.00 q^{9} +O(q^{10})$	\(q-8.00000i q^{2} -87.0000 q^{3} -64.0000 q^{4} -321.000i q^{5} +696.000i q^{6} -181.000i q^{7} +512.000i q^{8} +5382.00 q^{9} -2568.00 q^{10} +7782.00i q^{11} +5568.00 q^{12} -1448.00 q^{14} +27927.0i q^{15} +4096.00 q^{16} -9069.00 q^{17} -43056.0i q^{18} +37150.0i q^{19} +20544.0i q^{20} +15747.0i q^{21} +62256.0 q^{22} -19008.0 q^{23} -44544.0i q^{24} -24916.0 q^{25} -277965. q^{27} +11584.0i q^{28} +174750. q^{29} +223416. q^{30} -29012.0i q^{31} -32768.0i q^{32} -677034. i q^{33} +72552.0i q^{34} -58101.0 q^{35} -344448. q^{36} +323669. i q^{37} +297200. q^{38} +164352. q^{40} -795312. i q^{41} +125976. q^{42} +314137. q^{43} -498048. i q^{44} -1.72762e6i q^{45} +152064. i q^{46} -447441. i q^{47} -356352. q^{48} +790782. q^{49} +199328. i q^{50} +789003. q^{51} -1.46923e6 q^{53} +2.22372e6i q^{54} +2.49802e6 q^{55} +92672.0 q^{56} -3.23205e6i q^{57} -1.39800e6i q^{58} +1.62777e6i q^{59} -1.78733e6i q^{60} -2.39961e6 q^{61} -232096. q^{62} -974142. i q^{63} -262144. q^{64} -5.41627e6 q^{66} +64066.0i q^{67} +580416. q^{68} +1.65370e6 q^{69} +464808. i q^{70} +322383. i q^{71} +2.75558e6i q^{72} -4.45478e6i q^{73} +2.58935e6 q^{74} +2.16769e6 q^{75} -2.37760e6i q^{76} +1.40854e6 q^{77} +753560. q^{79} -1.31482e6i q^{80} +1.24125e7 q^{81} -6.36250e6 q^{82} +1.21909e6i q^{83} -1.00781e6i q^{84} +2.91115e6i q^{85} -2.51310e6i q^{86} -1.52032e7 q^{87} -3.98438e6 q^{88} +3.39033e6i q^{89} -1.38210e7 q^{90} +1.21651e6 q^{92} +2.52404e6i q^{93} -3.57953e6 q^{94} +1.19252e7 q^{95} +2.85082e6i q^{96} -1.62877e6i q^{97} -6.32626e6i q^{98} +4.18827e7i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 174 q^{3} - 128 q^{4} + 10764 q^{9}+O(q^{10}) $ 2 * q - 174 * q^3 - 128 * q^4 + 10764 * q^9	$ 2 q - 174 q^{3} - 128 q^{4} + 10764 q^{9} - 5136 q^{10} + 11136 q^{12} - 2896 q^{14} + 8192 q^{16} - 18138 q^{17} + 124512 q^{22} - 38016 q^{23} - 49832 q^{25} - 555930 q^{27} + 349500 q^{29} + 446832 q^{30} - 116202 q^{35} - 688896 q^{36} + 594400 q^{38} + 328704 q^{40} + 251952 q^{42} + 628274 q^{43} - 712704 q^{48} + 1581564 q^{49} + 1578006 q^{51} - 2938464 q^{53} + 4996044 q^{55} + 185344 q^{56} - 4799216 q^{61} - 464192 q^{62} - 524288 q^{64} - 10832544 q^{66} + 1160832 q^{68} + 3307392 q^{69} + 5178704 q^{74} + 4335384 q^{75} + 2817084 q^{77} + 1507120 q^{79} + 24825042 q^{81} - 12724992 q^{82} - 30406500 q^{87} - 7968768 q^{88} - 27641952 q^{90} + 2433024 q^{92} - 7159056 q^{94} + 23850300 q^{95}+O(q^{100}) $ 2 * q - 174 * q^3 - 128 * q^4 + 10764 * q^9 - 5136 * q^10 + 11136 * q^12 - 2896 * q^14 + 8192 * q^16 - 18138 * q^17 + 124512 * q^22 - 38016 * q^23 - 49832 * q^25 - 555930 * q^27 + 349500 * q^29 + 446832 * q^30 - 116202 * q^35 - 688896 * q^36 + 594400 * q^38 + 328704 * q^40 + 251952 * q^42 + 628274 * q^43 - 712704 * q^48 + 1581564 * q^49 + 1578006 * q^51 - 2938464 * q^53 + 4996044 * q^55 + 185344 * q^56 - 4799216 * q^61 - 464192 * q^62 - 524288 * q^64 - 10832544 * q^66 + 1160832 * q^68 + 3307392 * q^69 + 5178704 * q^74 + 4335384 * q^75 + 2817084 * q^77 + 1507120 * q^79 + 24825042 * q^81 - 12724992 * q^82 - 30406500 * q^87 - 7968768 * q^88 - 27641952 * q^90 + 2433024 * q^92 - 7159056 * q^94 + 23850300 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/338\mathbb{Z}\right)^\times$.

$n$	$171$
$\chi(n)$	$-1$

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 $

$2$	− 8.00000i	− 0.707107i
$2$	− 8.00000i	− 0.707107i
$3$	−87.0000	−1.86035	−0.930175	−	0.367115i	$-0.880345\pi$
$3$	−87.0000	−1.86035	−0.930175	+	0.367115i	$0.880345\pi$
$4$	−64.0000	−0.500000
$5$	− 321.000i	− 1.14844i	−0.818699	−	0.574222i	$-0.805305\pi$
$5$	− 321.000i	− 1.14844i	0.818699	−	0.574222i	$-0.194695\pi$
$6$	696.000i	1.31547i
$7$	− 181.000i	− 0.199451i	−0.995015	−	0.0997253i	$-0.968204\pi$
$7$	− 181.000i	− 0.199451i	0.995015	−	0.0997253i	$-0.0317964\pi$
$8$	512.000i	0.353553i
$9$	5382.00	2.46091
$10$	−2568.00	−0.812073
$11$	7782.00i	1.76286i	0.472318	+	0.881428i	$0.343417\pi$
$11$	7782.00i	1.76286i	−0.472318	+	0.881428i	$0.656583\pi$
$12$	5568.00	0.930175
$13$	0	0
$13$	0	0
$14$	−1448.00	−0.141033
$15$	27927.0i	2.13651i
$16$	4096.00	0.250000
$17$	−9069.00	−0.447701	−0.223851	−	0.974623i	$-0.571863\pi$
$17$	−9069.00	−0.447701	−0.223851	+	0.974623i	$0.571863\pi$
$18$	− 43056.0i	− 1.74012i
$19$	37150.0i	1.24257i	0.783584	+	0.621286i	$0.213389\pi$
$19$	37150.0i	1.24257i	−0.783584	+	0.621286i	$0.786611\pi$
$20$	20544.0i	0.574222i
$21$	15747.0i	0.371048i
$22$	62256.0	1.24653
$23$	−19008.0	−0.325753	−0.162877	−	0.986646i	$-0.552077\pi$
$23$	−19008.0	−0.325753	−0.162877	+	0.986646i	$0.552077\pi$
$24$	− 44544.0i	− 0.657733i
$25$	−24916.0	−0.318925
$26$	0	0
$27$	−277965.	−2.71780
$28$	11584.0i	0.0997253i
$29$	174750.	1.33053	0.665264	−	0.746608i	$-0.268319\pi$
$29$	174750.	1.33053	0.665264	+	0.746608i	$0.268319\pi$
$30$	223416.	1.51074
$31$	− 29012.0i	− 0.174909i	−0.996169	−	0.0874544i	$-0.972127\pi$
$31$	− 29012.0i	− 0.174909i	0.996169	−	0.0874544i	$-0.0278732\pi$
$32$	− 32768.0i	− 0.176777i
$33$	− 677034.i	− 3.27953i
$34$	72552.0i	0.316572i
$35$	−58101.0	−0.229058
$36$	−344448.	−1.23045
$37$	323669.i	1.05050i	0.850949	+	0.525249i	$0.176028\pi$
$37$	323669.i	1.05050i	−0.850949	+	0.525249i	$0.823972\pi$
$38$	297200.	0.878630
$39$	0	0
$40$	164352.	0.406036
$41$	− 795312.i	− 1.80216i	−0.433650	−	0.901081i	$-0.642775\pi$
$41$	− 795312.i	− 1.80216i	0.433650	−	0.901081i	$-0.357225\pi$
$42$	125976.	0.262371
$43$	314137.	0.602531	0.301266	−	0.953540i	$-0.402591\pi$
$43$	314137.	0.602531	0.301266	+	0.953540i	$0.402591\pi$
$44$	− 498048.i	− 0.881428i
$45$	− 1.72762e6i	− 2.82621i
$46$	152064.i	0.230342i
$47$	− 447441.i	− 0.628627i	−0.949319	−	0.314314i	$-0.898226\pi$
$47$	− 447441.i	− 0.628627i	0.949319	−	0.314314i	$-0.101774\pi$
$48$	−356352.	−0.465088
$49$	790782.	0.960219
$50$	199328.i	0.225514i
$51$	789003.	0.832881
$52$	0	0
$53$	−1.46923e6	−1.35558	−0.677790	−	0.735256i	$-0.737062\pi$
$53$	−1.46923e6	−1.35558	−0.677790	+	0.735256i	$0.737062\pi$
$54$	2.22372e6i	1.92177i
$55$	2.49802e6	2.02454
$56$	92672.0	0.0705165
$57$	− 3.23205e6i	− 2.31162i
$58$	− 1.39800e6i	− 0.940826i
$59$	1.62777e6i	1.03184i	0.856638	+	0.515918i	$0.172549\pi$
$59$	1.62777e6i	1.03184i	−0.856638	+	0.515918i	$0.827451\pi$
$60$	− 1.78733e6i	− 1.06825i
$61$	−2.39961e6	−1.35359	−0.676793	−	0.736173i	$-0.736631\pi$
$61$	−2.39961e6	−1.35359	−0.676793	+	0.736173i	$0.736631\pi$
$62$	−232096.	−0.123679
$63$	− 974142.i	− 0.490829i
$64$	−262144.	−0.125000
$65$	0	0
$66$	−5.41627e6	−2.31898
$67$	64066.0i	0.0260235i	0.999915	+	0.0130118i	$0.00414189\pi$
$67$	64066.0i	0.0260235i	−0.999915	+	0.0130118i	$0.995858\pi$
$68$	580416.	0.223851
$69$	1.65370e6	0.606016
$70$	464808.i	0.161968i
$71$	322383.i	0.106898i	0.998571	+	0.0534488i	$0.0170214\pi$
$71$	322383.i	0.106898i	−0.998571	+	0.0534488i	$0.982979\pi$
$72$	2.75558e6i	0.870061i
$73$	− 4.45478e6i	− 1.34028i	−0.742233	−	0.670141i	$-0.766233\pi$
$73$	− 4.45478e6i	− 1.34028i	0.742233	−	0.670141i	$-0.233767\pi$
$74$	2.58935e6	0.742814
$75$	2.16769e6	0.593312
$76$	− 2.37760e6i	− 0.621286i
$77$	1.40854e6	0.351603
$78$	0	0
$79$	753560.	0.171958	0.0859791	−	0.996297i	$-0.472598\pi$
$79$	753560.	0.171958	0.0859791	+	0.996297i	$0.472598\pi$
$80$	− 1.31482e6i	− 0.287111i
$81$	1.24125e7	2.59515
$82$	−6.36250e6	−1.27432
$83$	1.21909e6i	0.234025i	0.993130	+	0.117013i	$0.0373318\pi$
$83$	1.21909e6i	0.234025i	−0.993130	+	0.117013i	$0.962668\pi$
$84$	− 1.00781e6i	− 0.185524i
$85$	2.91115e6i	0.514160i
$86$	− 2.51310e6i	− 0.426054i
$87$	−1.52032e7	−2.47525
$88$	−3.98438e6	−0.623264
$89$	3.39033e6i	0.509773i	0.966971	+	0.254887i	$0.0820381\pi$
$89$	3.39033e6i	0.509773i	−0.966971	+	0.254887i	$0.917962\pi$
$90$	−1.38210e7	−1.99843
$91$	0	0
$92$	1.21651e6	0.162877
$93$	2.52404e6i	0.325392i
$94$	−3.57953e6	−0.444507
$95$	1.19252e7	1.42702
$96$	2.85082e6i	0.328867i
$97$	− 1.62877e6i	− 0.181201i	−0.995887	−	0.0906003i	$-0.971121\pi$
$97$	− 1.62877e6i	− 0.181201i	0.995887	−	0.0906003i	$-0.0288786\pi$
$98$	− 6.32626e6i	− 0.678978i
$99$	4.18827e7i	4.33822i
$100$	1.59462e6	0.159462
$101$	1.53503e7	1.48249	0.741244	−	0.671236i	$-0.234236\pi$
$101$	1.53503e7	1.48249	0.741244	+	0.671236i	$0.234236\pi$
$102$	− 6.31202e6i	− 0.588936i
$103$	−6.87643e6	−0.620058	−0.310029	−	0.950727i	$-0.600339\pi$
$103$	−6.87643e6	−0.620058	−0.310029	+	0.950727i	$0.600339\pi$
$104$	0	0
$105$	5.05479e6	0.426128
$106$	1.17539e7i	0.958539i
$107$	−1.52027e7	−1.19971	−0.599857	−	0.800107i	$-0.704776\pi$
$107$	−1.52027e7	−1.19971	−0.599857	+	0.800107i	$0.704776\pi$
$108$	1.77898e7	1.35890
$109$	− 6.73260e6i	− 0.497955i	−0.968509	−	0.248978i	$-0.919905\pi$
$109$	− 6.73260e6i	− 0.497955i	0.968509	−	0.248978i	$-0.0800946\pi$
$110$	− 1.99842e7i	− 1.43157i
$111$	− 2.81592e7i	− 1.95429i
$112$	− 741376.i	− 0.0498627i
$113$	−1.15292e7	−0.751667	−0.375833	−	0.926687i	$-0.622644\pi$
$113$	−1.15292e7	−0.751667	−0.375833	+	0.926687i	$0.622644\pi$
$114$	−2.58564e7	−1.63456
$115$	6.10157e6i	0.374110i
$116$	−1.11840e7	−0.665264
$117$	0	0
$118$	1.30222e7	0.729619
$119$	1.64149e6i	0.0892943i
$120$	−1.42986e7	−0.755370
$121$	−4.10724e7	−2.10766
$122$	1.91969e7i	0.957130i
$123$	6.91921e7i	3.35266i
$124$	1.85677e6i	0.0874544i
$125$	− 1.70801e7i	− 0.782177i
$126$	−7.79314e6	−0.347069
$127$	−2.06699e7	−0.895418	−0.447709	−	0.894179i	$-0.647760\pi$
$127$	−2.06699e7	−0.895418	−0.447709	+	0.894179i	$0.647760\pi$
$128$	2.09715e6i	0.0883883i
$129$	−2.73299e7	−1.12092
$130$	0	0
$131$	−1.90949e7	−0.742107	−0.371054	−	0.928611i	$-0.621003\pi$
$131$	−1.90949e7	−0.742107	−0.371054	+	0.928611i	$0.621003\pi$
$132$	4.33302e7i	1.63977i
$133$	6.72415e6	0.247832
$134$	512528.	0.0184014
$135$	8.92268e7i	3.12124i
$136$	− 4.64333e6i	− 0.158286i
$137$	2.96901e7i	0.986482i	0.869893	+	0.493241i	$0.164188\pi$
$137$	2.96901e7i	0.986482i	−0.869893	+	0.493241i	$0.835812\pi$
$138$	− 1.32296e7i	− 0.428518i
$139$	1.55652e7	0.491591	0.245795	−	0.969322i	$-0.420951\pi$
$139$	1.55652e7	0.491591	0.245795	+	0.969322i	$0.420951\pi$
$140$	3.71846e6	0.114529
$141$	3.89274e7i	1.16947i
$142$	2.57906e6	0.0755880
$143$	0	0
$144$	2.20447e7	0.615226
$145$	− 5.60948e7i	− 1.52804i
$146$	−3.56383e7	−0.947723
$147$	−6.87980e7	−1.78635
$148$	− 2.07148e7i	− 0.525249i
$149$	2.49675e6i	0.0618334i	0.999522	+	0.0309167i	$0.00984266\pi$
$149$	2.49675e6i	0.0618334i	−0.999522	+	0.0309167i	$0.990157\pi$
$150$	− 1.73415e7i	− 0.419535i
$151$	2.39802e7i	0.566804i	0.959001	+	0.283402i	$0.0914630\pi$
$151$	2.39802e7i	0.566804i	−0.959001	+	0.283402i	$0.908537\pi$
$152$	−1.90208e7	−0.439315
$153$	−4.88094e7	−1.10175
$154$	− 1.12683e7i	− 0.248621i
$155$	−9.31285e6	−0.200873
$156$	0	0
$157$	1.70550e7	0.351725	0.175863	−	0.984415i	$-0.443729\pi$
$157$	1.70550e7	0.351725	0.175863	+	0.984415i	$0.443729\pi$
$158$	− 6.02848e6i	− 0.121593i
$159$	1.27823e8	2.52185
$160$	−1.05185e7	−0.203018
$161$	3.44045e6i	0.0649717i
$162$	− 9.93002e7i	− 1.83505i
$163$	− 7.34586e7i	− 1.32857i	−0.747477	−	0.664287i	$-0.768735\pi$
$163$	− 7.34586e7i	− 1.32857i	0.747477	−	0.664287i	$-0.231265\pi$
$164$	5.09000e7i	0.901081i
$165$	−2.17328e8	−3.76636
$166$	9.75274e6	0.165481
$167$	4.66860e7i	0.775674i	0.921728	+	0.387837i	$0.126778\pi$
$167$	4.66860e7i	0.775674i	−0.921728	+	0.387837i	$0.873222\pi$
$168$	−8.06246e6	−0.131185
$169$	0	0
$170$	2.32892e7	0.363566
$171$	1.99941e8i	3.05785i
$172$	−2.01048e7	−0.301266
$173$	7.80931e7	1.14670	0.573352	−	0.819309i	$-0.305643\pi$
$173$	7.80931e7	1.14670	0.573352	+	0.819309i	$0.305643\pi$
$174$	1.21626e8i	1.75027i
$175$	4.50980e6i	0.0636098i
$176$	3.18751e7i	0.440714i
$177$	− 1.41616e8i	− 1.91958i
$178$	2.71226e7	0.360464
$179$	5.56163e7	0.724797	0.362399	−	0.932023i	$-0.381958\pi$
$179$	5.56163e7	0.724797	0.362399	+	0.932023i	$0.381958\pi$
$180$	1.10568e8i	1.41311i
$181$	1.19435e8	1.49711	0.748557	−	0.663070i	$-0.230747\pi$
$181$	1.19435e8	1.49711	0.748557	+	0.663070i	$0.230747\pi$
$182$	0	0
$183$	2.08766e8	2.51815
$184$	− 9.73210e6i	− 0.115171i
$185$	1.03898e8	1.20644
$186$	2.01924e7	0.230087
$187$	− 7.05750e7i	− 0.789233i
$188$	2.86362e7i	0.314314i
$189$	5.03117e7i	0.542066i
$190$	− 9.54012e7i	− 1.00906i
$191$	1.05485e8	1.09540	0.547700	−	0.836675i	$-0.315504\pi$
$191$	1.05485e8	1.09540	0.547700	+	0.836675i	$0.315504\pi$
$192$	2.28065e7	0.232544
$193$	2.12059e7i	0.212327i	0.994349	+	0.106164i	$0.0338567\pi$
$193$	2.12059e7i	0.212327i	−0.994349	+	0.106164i	$0.966143\pi$
$194$	−1.30302e7	−0.128128
$195$	0	0
$196$	−5.06100e7	−0.480110
$197$	1.66535e8i	1.55194i	0.630771	+	0.775969i	$0.282739\pi$
$197$	1.66535e8i	1.55194i	−0.630771	+	0.775969i	$0.717261\pi$
$198$	3.35062e8	3.06759
$199$	1.26351e8	1.13656	0.568279	−	0.822836i	$-0.307609\pi$
$199$	1.26351e8	1.13656	0.568279	+	0.822836i	$0.307609\pi$
$200$	− 1.27570e7i	− 0.112757i
$201$	− 5.57374e6i	− 0.0484129i
$202$	− 1.22802e8i	− 1.04828i
$203$	− 3.16298e7i	− 0.265375i
$204$	−5.04962e7	−0.416441
$205$	−2.55295e8	−2.06968
$206$	5.50114e7i	0.438448i
$207$	−1.02301e8	−0.801648
$208$	0	0
$209$	−2.89101e8	−2.19047
$210$	− 4.04383e7i	− 0.301318i
$211$	1.08571e8	0.795655	0.397828	−	0.917460i	$-0.369764\pi$
$211$	1.08571e8	0.795655	0.397828	+	0.917460i	$0.369764\pi$
$212$	9.40308e7	0.677790
$213$	− 2.80473e7i	− 0.198867i
$214$	1.21622e8i	0.848326i
$215$	− 1.00838e8i	− 0.691974i
$216$	− 1.42318e8i	− 0.960886i
$217$	−5.25117e6	−0.0348857
$218$	−5.38608e7	−0.352108
$219$	3.87566e8i	2.49340i
$220$	−1.59873e8	−1.01227
$221$	0	0
$222$	−2.25274e8	−1.38189
$223$	1.25603e8i	0.758459i	0.925303	+	0.379229i	$0.123811\pi$
$223$	1.25603e8i	0.758459i	−0.925303	+	0.379229i	$0.876189\pi$
$224$	−5.93101e6	−0.0352582
$225$	−1.34098e8	−0.784844
$226$	9.22338e7i	0.531509i
$227$	− 1.90774e8i	− 1.08250i	−0.840861	−	0.541252i	$-0.817951\pi$
$227$	− 1.90774e8i	− 1.08250i	0.840861	−	0.541252i	$-0.182049\pi$
$228$	2.06851e8i	1.15581i
$229$	− 5.28911e7i	− 0.291044i	−0.989355	−	0.145522i	$-0.953514\pi$
$229$	− 5.28911e7i	− 0.291044i	0.989355	−	0.145522i	$-0.0464861\pi$
$230$	4.88125e7	0.264536
$231$	−1.22543e8	−0.654104
$232$	8.94720e7i	0.470413i
$233$	−1.51254e8	−0.783359	−0.391680	−	0.920102i	$-0.628106\pi$
$233$	−1.51254e8	−0.783359	−0.391680	+	0.920102i	$0.628106\pi$
$234$	0	0
$235$	−1.43629e8	−0.721944
$236$	− 1.04177e8i	− 0.515918i
$237$	−6.55597e7	−0.319903
$238$	1.31319e7	0.0631406
$239$	− 2.61917e8i	− 1.24100i	−0.784208	−	0.620498i	$-0.786930\pi$
$239$	− 2.61917e8i	− 1.24100i	0.784208	−	0.620498i	$-0.213070\pi$
$240$	1.14389e8i	0.534127i
$241$	− 1.31752e8i	− 0.606312i	−0.952941	−	0.303156i	$-0.901960\pi$
$241$	− 1.31752e8i	− 0.606312i	0.952941	−	0.303156i	$-0.0980404\pi$
$242$	3.28579e8i	1.49034i
$243$	−4.71980e8	−2.11009
$244$	1.53575e8	0.676793
$245$	− 2.53841e8i	− 1.10276i
$246$	5.53537e8	2.37069
$247$	0	0
$248$	1.48541e7	0.0618396
$249$	− 1.06061e8i	− 0.435370i
$250$	−1.36641e8	−0.553083
$251$	−2.47061e8	−0.986159	−0.493080	−	0.869984i	$-0.664129\pi$
$251$	−2.47061e8	−0.986159	−0.493080	+	0.869984i	$0.664129\pi$
$252$	6.23451e7i	0.245415i
$253$	− 1.47920e8i	− 0.574256i
$254$	1.65359e8i	0.633156i
$255$	− 2.53270e8i	− 0.956518i
$256$	1.67772e7	0.0625000
$257$	−2.27286e8	−0.835231	−0.417616	−	0.908624i	$-0.637134\pi$
$257$	−2.27286e8	−0.835231	−0.417616	+	0.908624i	$0.637134\pi$
$258$	2.18639e8i	0.792610i
$259$	5.85841e7	0.209522
$260$	0	0
$261$	9.40504e8	3.27430
$262$	1.52759e8i	0.524749i
$263$	−4.25872e8	−1.44356	−0.721779	−	0.692124i	$-0.756675\pi$
$263$	−4.25872e8	−1.44356	−0.721779	+	0.692124i	$0.756675\pi$
$264$	3.46641e8	1.15949
$265$	4.71623e8i	1.55681i
$266$	− 5.37932e7i	− 0.175243i
$267$	− 2.94959e8i	− 0.948357i
$268$	− 4.10022e6i	− 0.0130118i
$269$	−5.14154e8	−1.61050	−0.805250	−	0.592936i	$-0.797969\pi$
$269$	−5.14154e8	−1.61050	−0.805250	+	0.592936i	$0.797969\pi$
$270$	7.13814e8	2.20705
$271$	4.57096e7i	0.139513i	0.997564	+	0.0697565i	$0.0222222\pi$
$271$	4.57096e7i	0.139513i	−0.997564	+	0.0697565i	$0.977778\pi$
$272$	−3.71466e7	−0.111925
$273$	0	0
$274$	2.37521e8	0.697548
$275$	− 1.93896e8i	− 0.562218i
$276$	−1.05837e8	−0.303008
$277$	2.73964e8	0.774487	0.387244	−	0.921977i	$-0.373427\pi$
$277$	2.73964e8	0.774487	0.387244	+	0.921977i	$0.373427\pi$
$278$	− 1.24522e8i	− 0.347607i
$279$	− 1.56143e8i	− 0.430434i
$280$	− 2.97477e7i	− 0.0809842i
$281$	4.21707e8i	1.13381i	0.823784	+	0.566903i	$0.191859\pi$
$281$	4.21707e8i	1.13381i	−0.823784	+	0.566903i	$0.808141\pi$
$282$	3.11419e8	0.826938
$283$	−3.81957e8	−1.00176	−0.500878	−	0.865518i	$-0.666990\pi$
$283$	−3.81957e8	−1.00176	−0.500878	+	0.865518i	$0.666990\pi$
$284$	− 2.06325e7i	− 0.0534488i
$285$	−1.03749e9	−2.65477
$286$	0	0
$287$	−1.43951e8	−0.359443
$288$	− 1.76357e8i	− 0.435031i
$289$	−3.28092e8	−0.799564
$290$	−4.48758e8	−1.08049
$291$	1.41703e8i	0.337097i
$292$	2.85106e8i	0.670141i
$293$	− 4.04833e8i	− 0.940240i	−0.882602	−	0.470120i	$-0.844211\pi$
$293$	− 4.04833e8i	− 0.940240i	0.882602	−	0.470120i	$-0.155789\pi$
$294$	5.50384e8i	1.26314i
$295$	5.22514e8	1.18501
$296$	−1.65719e8	−0.371407
$297$	− 2.16312e9i	− 4.79108i
$298$	1.99740e7	0.0437228
$299$	0	0
$300$	−1.38732e8	−0.296656
$301$	− 5.68588e7i	− 0.120175i
$302$	1.91841e8	0.400791
$303$	−1.33547e9	−2.75795
$304$	1.52166e8i	0.310643i
$305$	7.70274e8i	1.55452i
$306$	3.90475e8i	0.779055i
$307$	− 4.75520e7i	− 0.0937960i	−0.998900	−	0.0468980i	$-0.985066\pi$
$307$	− 4.75520e7i	− 0.0937960i	0.998900	−	0.0468980i	$-0.0149336\pi$
$308$	−9.01467e7	−0.175801
$309$	5.98249e8	1.15353
$310$	7.45028e7i	0.142039i
$311$	3.02841e8	0.570892	0.285446	−	0.958395i	$-0.407858\pi$
$311$	3.02841e8	0.570892	0.285446	+	0.958395i	$0.407858\pi$
$312$	0	0
$313$	−6.31685e8	−1.16438	−0.582191	−	0.813052i	$-0.697804\pi$
$313$	−6.31685e8	−1.16438	−0.582191	+	0.813052i	$0.697804\pi$
$314$	− 1.36440e8i	− 0.248707i
$315$	−3.12700e8	−0.563690
$316$	−4.82278e7	−0.0859791
$317$	− 7.93332e8i	− 1.39877i	−0.714743	−	0.699387i	$-0.753456\pi$
$317$	− 7.93332e8i	− 1.39877i	0.714743	−	0.699387i	$-0.246544\pi$
$318$	− 1.02259e9i	− 1.78322i
$319$	1.35990e9i	2.34553i
$320$	8.41482e7i	0.143556i
$321$	1.32264e9	2.23189
$322$	2.75236e7	0.0459420
$323$	− 3.36913e8i	− 0.556300i
$324$	−7.94401e8	−1.29757
$325$	0	0
$326$	−5.87669e8	−0.939444
$327$	5.85737e8i	0.926372i
$328$	4.07200e8	0.637161
$329$	−8.09868e7	−0.125380
$330$	1.73862e9i	2.66322i
$331$	1.21628e9i	1.84346i	0.387829	+	0.921731i	$0.373225\pi$
$331$	1.21628e9i	1.84346i	−0.387829	+	0.921731i	$0.626775\pi$
$332$	− 7.80219e7i	− 0.117013i
$333$	1.74199e9i	2.58518i
$334$	3.73488e8	0.548484
$335$	2.05652e7	0.0298866
$336$	6.44997e7i	0.0927620i
$337$	1.51221e8	0.215232	0.107616	−	0.994193i	$-0.465678\pi$
$337$	1.51221e8	0.215232	0.107616	+	0.994193i	$0.465678\pi$
$338$	0	0
$339$	1.00304e9	1.39836
$340$	− 1.86314e8i	− 0.257080i
$341$	2.25771e8	0.308339
$342$	1.59953e9	2.16223
$343$	− 2.92193e8i	− 0.390967i
$344$	1.60838e8i	0.213027i
$345$	− 5.30836e8i	− 0.695975i
$346$	− 6.24745e8i	− 0.810842i
$347$	−5.97234e8	−0.767347	−0.383673	−	0.923469i	$-0.625341\pi$
$347$	−5.97234e8	−0.767347	−0.383673	+	0.923469i	$0.625341\pi$
$348$	9.73008e8	1.23762
$349$	1.19600e8i	0.150606i	0.997161	+	0.0753029i	$0.0239924\pi$
$349$	1.19600e8i	0.150606i	−0.997161	+	0.0753029i	$0.976008\pi$
$350$	3.60784e7	0.0449789
$351$	0	0
$352$	2.55001e8	0.311632
$353$	4.66414e8i	0.564366i	0.959361	+	0.282183i	$0.0910585\pi$
$353$	4.66414e8i	0.564366i	−0.959361	+	0.282183i	$0.908942\pi$
$354$	−1.13293e9	−1.35735
$355$	1.03485e8	0.122766
$356$	− 2.16981e8i	− 0.254887i
$357$	− 1.42810e8i	− 0.166119i
$358$	− 4.44931e8i	− 0.512509i
$359$	− 7.70102e8i	− 0.878451i	−0.898377	−	0.439225i	$-0.855253\pi$
$359$	− 7.70102e8i	− 0.878451i	0.898377	−	0.439225i	$-0.144747\pi$
$360$	8.84542e8	0.999217
$361$	−4.86251e8	−0.543983
$362$	− 9.55477e8i	− 1.05862i
$363$	3.57329e9	3.92099
$364$	0	0
$365$	−1.42999e9	−1.53924
$366$	− 1.67013e9i	− 1.78060i
$367$	−8.55319e8	−0.903227	−0.451613	−	0.892214i	$-0.649151\pi$
$367$	−8.55319e8	−0.903227	−0.451613	+	0.892214i	$0.649151\pi$
$368$	−7.78568e7	−0.0814384
$369$	− 4.28037e9i	− 4.43495i
$370$	− 8.31182e8i	− 0.853081i
$371$	2.65931e8i	0.270371i
$372$	− 1.61539e8i	− 0.162696i
$373$	−5.29609e8	−0.528414	−0.264207	−	0.964466i	$-0.585110\pi$
$373$	−5.29609e8	−0.528414	−0.264207	+	0.964466i	$0.585110\pi$
$374$	−5.64600e8	−0.558072
$375$	1.48597e9i	1.45512i
$376$	2.29090e8	0.222253
$377$	0	0
$378$	4.02493e8	0.383299
$379$	− 1.98358e9i	− 1.87159i	−0.352540	−	0.935797i	$-0.614682\pi$
$379$	− 1.98358e9i	− 1.87159i	0.352540	−	0.935797i	$-0.385318\pi$
$380$	−7.63210e8	−0.713512
$381$	1.79828e9	1.66579
$382$	− 8.43877e8i	− 0.774564i
$383$	− 8.98756e8i	− 0.817422i	−0.912664	−	0.408711i	$-0.865978\pi$
$383$	− 8.98756e8i	− 0.817422i	0.912664	−	0.408711i	$-0.134022\pi$
$384$	− 1.82452e8i	− 0.164433i
$385$	− 4.52142e8i	− 0.403796i
$386$	1.69647e8	0.150138
$387$	1.69069e9	1.48277
$388$	1.04242e8i	0.0906003i
$389$	−1.82475e9	−1.57174	−0.785868	−	0.618395i	$-0.787783\pi$
$389$	−1.82475e9	−1.57174	−0.785868	+	0.618395i	$0.787783\pi$
$390$	0	0
$391$	1.72384e8	0.145840
$392$	4.04880e8i	0.339489i
$393$	1.66125e9	1.38058
$394$	1.33228e9	1.09739
$395$	− 2.41893e8i	− 0.197485i
$396$	− 2.68049e9i	− 2.16911i
$397$	4.93083e8i	0.395506i	0.980252	+	0.197753i	$0.0633645\pi$
$397$	4.93083e8i	0.395506i	−0.980252	+	0.197753i	$0.936636\pi$
$398$	− 1.01081e9i	− 0.803668i
$399$	−5.85001e8	−0.461054
$400$	−1.02056e8	−0.0797312
$401$	− 5.68280e8i	− 0.440105i	−0.975488	−	0.220053i	$-0.929377\pi$
$401$	− 5.68280e8i	− 0.440105i	0.975488	−	0.220053i	$-0.0706229\pi$
$402$	−4.45899e7	−0.0342331
$403$	0	0
$404$	−9.82417e8	−0.741244
$405$	− 3.98442e9i	− 2.98039i
$406$	−2.53038e8	−0.187648
$407$	−2.51879e9	−1.85188
$408$	4.03970e8i	0.294468i
$409$	1.28472e9i	0.928489i	0.885707	+	0.464245i	$0.153674\pi$
$409$	1.28472e9i	0.928489i	−0.885707	+	0.464245i	$0.846326\pi$
$410$	2.04236e9i	1.46349i
$411$	− 2.58304e9i	− 1.83520i
$412$	4.40091e8	0.310029
$413$	2.94626e8	0.205801
$414$	8.18408e8i	0.566851i
$415$	3.91329e8	0.268765
$416$	0	0
$417$	−1.35418e9	−0.914532
$418$	2.31281e9i	1.54890i
$419$	2.74847e8	0.182533	0.0912667	−	0.995826i	$-0.470908\pi$
$419$	2.74847e8	0.182533	0.0912667	+	0.995826i	$0.470908\pi$
$420$	−3.23506e8	−0.213064
$421$	− 7.51368e8i	− 0.490756i	−0.969428	−	0.245378i	$-0.921088\pi$
$421$	− 7.51368e8i	− 0.490756i	0.969428	−	0.245378i	$-0.0789120\pi$
$422$	− 8.68568e8i	− 0.562613i
$423$	− 2.40813e9i	− 1.54699i
$424$	− 7.52247e8i	− 0.479270i
$425$	2.25963e8	0.142783
$426$	−2.24379e8	−0.140620
$427$	4.34329e8i	0.269974i
$428$	9.72974e8	0.599857
$429$	0	0
$430$	−8.06704e8	−0.489299
$431$	1.30756e8i	0.0786668i	0.999226	+	0.0393334i	$0.0125234\pi$
$431$	1.30756e8i	0.0786668i	−0.999226	+	0.0393334i	$0.987477\pi$
$432$	−1.13854e9	−0.679449
$433$	−1.66736e9	−0.987010	−0.493505	−	0.869743i	$-0.664284\pi$
$433$	−1.66736e9	−0.987010	−0.493505	+	0.869743i	$0.664284\pi$
$434$	4.20094e7i	0.0246679i
$435$	4.88024e9i	2.84269i
$436$	4.30887e8i	0.248978i
$437$	− 7.06147e8i	− 0.404772i
$438$	3.10053e9	1.76310
$439$	−2.31478e9	−1.30582	−0.652910	−	0.757436i	$-0.726452\pi$
$439$	−2.31478e9	−1.30582	−0.652910	+	0.757436i	$0.726452\pi$
$440$	1.27899e9i	0.715784i
$441$	4.25599e9	2.36301
$442$	0	0
$443$	−6.90047e8	−0.377108	−0.188554	−	0.982063i	$-0.560380\pi$
$443$	−6.90047e8	−0.377108	−0.188554	+	0.982063i	$0.560380\pi$
$444$	1.80219e9i	0.977147i
$445$	1.08830e9	0.585446
$446$	1.00482e9	0.536311
$447$	− 2.17217e8i	− 0.115032i
$448$	4.74481e7i	0.0249313i
$449$	2.63806e9i	1.37538i	0.726004	+	0.687690i	$0.241375\pi$
$449$	2.63806e9i	1.37538i	−0.726004	+	0.687690i	$0.758625\pi$
$450$	1.07278e9i	0.554968i
$451$	6.18912e9	3.17695
$452$	7.37870e8	0.375833
$453$	− 2.08627e9i	− 1.05445i
$454$	−1.52619e9	−0.765445
$455$	0	0
$456$	1.65481e9	0.817280
$457$	6.16222e8i	0.302016i	0.988533	+	0.151008i	$0.0482520\pi$
$457$	6.16222e8i	0.302016i	−0.988533	+	0.151008i	$0.951748\pi$
$458$	−4.23129e8	−0.205799
$459$	2.52086e9	1.21676
$460$	− 3.90500e8i	− 0.187055i
$461$	− 1.23621e9i	− 0.587679i	−0.955855	−	0.293839i	$-0.905067\pi$
$461$	− 1.23621e9i	− 0.587679i	0.955855	−	0.293839i	$-0.0949330\pi$
$462$	9.80345e8i	0.462522i
$463$	6.78469e7i	0.0317685i	0.999874	+	0.0158843i	$0.00505633\pi$
$463$	6.78469e7i	0.0317685i	−0.999874	+	0.0158843i	$0.994944\pi$
$464$	7.15776e8	0.332632
$465$	8.10218e8	0.373694
$466$	1.21003e9i	0.553919i
$467$	1.17502e9	0.533869	0.266934	−	0.963715i	$-0.413989\pi$
$467$	1.17502e9	0.533869	0.266934	+	0.963715i	$0.413989\pi$
$468$	0	0
$469$	1.15959e7	0.00519040
$470$	1.14903e9i	0.510491i
$471$	−1.48379e9	−0.654332
$472$	−8.33418e8	−0.364809
$473$	2.44461e9i	1.06218i
$474$	5.24478e8i	0.226205i
$475$	− 9.25629e8i	− 0.396287i
$476$	− 1.05055e8i	− 0.0446471i
$477$	−7.90741e9	−3.33595
$478$	−2.09533e9	−0.877517
$479$	3.96154e8i	0.164699i	0.996604	+	0.0823494i	$0.0262423\pi$
$479$	3.96154e8i	0.164699i	−0.996604	+	0.0823494i	$0.973758\pi$
$480$	9.15112e8	0.377685
$481$	0	0
$482$	−1.05401e9	−0.428728
$483$	− 2.99319e8i	− 0.120870i
$484$	2.62863e9	1.05383
$485$	−5.22836e8	−0.208099
$486$	3.77584e9i	1.49206i
$487$	3.03665e9i	1.19136i	0.803222	+	0.595680i	$0.203117\pi$
$487$	3.03665e9i	1.19136i	−0.803222	+	0.595680i	$0.796883\pi$
$488$	− 1.22860e9i	− 0.478565i
$489$	6.39090e9i	2.47162i
$490$	−2.03073e9	−0.779768
$491$	2.91974e9	1.11316	0.556582	−	0.830793i	$-0.312113\pi$
$491$	2.91974e9	1.11316	0.556582	+	0.830793i	$0.312113\pi$
$492$	− 4.42830e9i	− 1.67633i
$493$	−1.58481e9	−0.595679
$494$	0	0
$495$	1.34444e10	4.98221
$496$	− 1.18833e8i	− 0.0437272i
$497$	5.83513e7	0.0213208
$498$	−8.48488e8	−0.307853
$499$	1.62343e9i	0.584898i	0.956281	+	0.292449i	$0.0944702\pi$
$499$	1.62343e9i	0.584898i	−0.956281	+	0.292449i	$0.905530\pi$
$500$	1.09313e9i	0.391089i
$501$	− 4.06168e9i	− 1.44303i
$502$	1.97649e9i	0.697320i
$503$	4.75888e9	1.66731	0.833655	−	0.552285i	$-0.186244\pi$
$503$	4.75888e9	1.66731	0.833655	+	0.552285i	$0.186244\pi$
$504$	4.98761e8	0.173534
$505$	− 4.92744e9i	− 1.70256i
$506$	−1.18336e9	−0.406061
$507$	0	0
$508$	1.32288e9	0.447709
$509$	9.19375e8i	0.309016i	0.987992	+	0.154508i	$0.0493792\pi$
$509$	9.19375e8i	0.309016i	−0.987992	+	0.154508i	$0.950621\pi$
$510$	−2.02616e9	−0.676360
$511$	−8.06316e8	−0.267320
$512$	− 1.34218e8i	− 0.0441942i
$513$	− 1.03264e10i	− 3.37706i
$514$	1.81829e9i	0.590598i
$515$	2.20733e9i	0.712103i
$516$	1.74911e9	0.560460
$517$	3.48199e9	1.10818
$518$	− 4.68673e8i	− 0.148155i
$519$	−6.79410e9	−2.13327
$520$	0	0
$521$	−1.46089e9	−0.452569	−0.226284	−	0.974061i	$-0.572658\pi$
$521$	−1.46089e9	−0.452569	−0.226284	+	0.974061i	$0.572658\pi$
$522$	− 7.52404e9i	− 2.31528i
$523$	2.12856e9	0.650624	0.325312	−	0.945607i	$-0.394531\pi$
$523$	2.12856e9	0.650624	0.325312	+	0.945607i	$0.394531\pi$
$524$	1.22207e9	0.371054
$525$	− 3.92352e8i	− 0.118336i
$526$	3.40698e9i	1.02075i
$527$	2.63110e8i	0.0783069i
$528$	− 2.77313e9i	− 0.819883i
$529$	−3.04352e9	−0.893885
$530$	3.77299e9	1.10083
$531$	8.76066e9i	2.53925i
$532$	−4.30346e8	−0.123916
$533$	0	0
$534$	−2.35967e9	−0.670590
$535$	4.88007e9i	1.37781i
$536$	−3.28018e7	−0.00920070
$537$	−4.83862e9	−1.34838
$538$	4.11323e9i	1.13879i
$539$	6.15387e9i	1.69273i
$540$	− 5.71051e9i	− 1.56062i
$541$	− 1.72479e7i	− 0.00468324i	−0.999997	−	0.00234162i	$-0.999255\pi$
$541$	− 1.72479e7i	− 0.00468324i	0.999997	−	0.00234162i	$-0.000745361\pi$
$542$	3.65677e8	0.0986507
$543$	−1.03908e10	−2.78516
$544$	2.97173e8i	0.0791431i
$545$	−2.16117e9	−0.571874
$546$	0	0
$547$	7.51154e8	0.196234	0.0981168	−	0.995175i	$-0.468718\pi$
$547$	7.51154e8	0.196234	0.0981168	+	0.995175i	$0.468718\pi$
$548$	− 1.90016e9i	− 0.493241i
$549$	−1.29147e10	−3.33105
$550$	−1.55117e9	−0.397549
$551$	6.49196e9i	1.65328i
$552$	8.46692e8i	0.214259i
$553$	− 1.36394e8i	− 0.0342972i
$554$	− 2.19171e9i	− 0.547645i
$555$	−9.03910e9	−2.24440
$556$	−9.96175e8	−0.245795
$557$	3.00701e9i	0.737295i	0.929569	+	0.368647i	$0.120179\pi$
$557$	3.00701e9i	0.737295i	−0.929569	+	0.368647i	$0.879821\pi$
$558$	−1.24914e9	−0.304363
$559$	0	0
$560$	−2.37982e8	−0.0572645
$561$	6.14002e9i	1.46825i
$562$	3.37366e9	0.801722
$563$	−2.82880e9	−0.668070	−0.334035	−	0.942561i	$-0.608410\pi$
$563$	−2.82880e9	−0.668070	−0.334035	+	0.942561i	$0.608410\pi$
$564$	− 2.49135e9i	− 0.584734i
$565$	3.70088e9i	0.863248i
$566$	3.05566e9i	0.708349i
$567$	− 2.24667e9i	− 0.517604i
$568$	−1.65060e8	−0.0377940
$569$	−7.67290e9	−1.74609	−0.873045	−	0.487639i	$-0.837858\pi$
$569$	−7.67290e9	−1.74609	−0.873045	+	0.487639i	$0.837858\pi$
$570$	8.29990e9i	1.87720i
$571$	3.09363e9	0.695411	0.347706	−	0.937604i	$-0.386961\pi$
$571$	3.09363e9	0.695411	0.347706	+	0.937604i	$0.386961\pi$
$572$	0	0
$573$	−9.17717e9	−2.03783
$574$	1.15161e9i	0.254164i
$575$	4.73603e8	0.103891
$576$	−1.41086e9	−0.307613
$577$	− 3.71815e9i	− 0.805770i	−0.915251	−	0.402885i	$-0.868007\pi$
$577$	− 3.71815e9i	− 0.805770i	0.915251	−	0.402885i	$-0.131993\pi$
$578$	2.62474e9i	0.565377i
$579$	− 1.84491e9i	− 0.395003i
$580$	3.59006e9i	0.764019i
$581$	2.20656e8	0.0466765
$582$	1.13363e9	0.238363
$583$	− 1.14336e10i	− 2.38969i
$584$	2.28085e9	0.473862
$585$	0	0
$586$	−3.23866e9	−0.664850
$587$	2.74853e9i	0.560876i	0.959872	+	0.280438i	$0.0904796\pi$
$587$	2.74853e9i	0.560876i	−0.959872	+	0.280438i	$0.909520\pi$
$588$	4.40307e9	0.893173
$589$	1.07780e9	0.217337
$590$	− 4.18011e9i	− 0.837927i
$591$	− 1.44886e10i	− 2.88715i
$592$	1.32575e9i	0.262624i
$593$	− 9.11262e9i	− 1.79453i	−0.441488	−	0.897267i	$-0.645549\pi$
$593$	− 9.11262e9i	− 1.79453i	0.441488	−	0.897267i	$-0.354451\pi$
$594$	−1.73050e10	−3.38781
$595$	5.26918e8	0.102550
$596$	− 1.59792e8i	− 0.0309167i
$597$	−1.09925e10	−2.11440
$598$	0	0
$599$	−5.52493e9	−1.05035	−0.525174	−	0.850995i	$-0.676000\pi$
$599$	−5.52493e9	−1.05035	−0.525174	+	0.850995i	$0.676000\pi$
$600$	1.10986e9i	0.209767i
$601$	−1.78219e9	−0.334883	−0.167441	−	0.985882i	$-0.553551\pi$
$601$	−1.78219e9	−0.334883	−0.167441	+	0.985882i	$0.553551\pi$
$602$	−4.54870e8	−0.0849767
$603$	3.44803e8i	0.0640414i
$604$	− 1.53473e9i	− 0.283402i
$605$	1.31842e10i	2.42053i
$606$	1.06838e10i	1.95016i
$607$	9.53705e9	1.73083	0.865414	−	0.501058i	$-0.167056\pi$
$607$	9.53705e9	1.73083	0.865414	+	0.501058i	$0.167056\pi$
$608$	1.21733e9	0.219658
$609$	2.75179e9i	0.493690i
$610$	6.16219e9	1.09921
$611$	0	0
$612$	3.12380e9	0.550875
$613$	1.18627e9i	0.208004i	0.994577	+	0.104002i	$0.0331648\pi$
$613$	1.18627e9i	0.208004i	−0.994577	+	0.104002i	$0.966835\pi$
$614$	−3.80416e8	−0.0663238
$615$	2.22107e10	3.85034
$616$	7.21174e8i	0.124310i
$617$	− 1.32256e9i	− 0.226682i	−0.993556	−	0.113341i	$-0.963845\pi$
$617$	− 1.32256e9i	− 0.226682i	0.993556	−	0.113341i	$-0.0361552\pi$
$618$	− 4.78599e9i	− 0.815666i
$619$	− 3.59450e9i	− 0.609147i	−0.952489	−	0.304573i	$-0.901486\pi$
$619$	− 3.59450e9i	− 0.609147i	0.952489	−	0.304573i	$-0.0985138\pi$
$620$	5.96023e8	0.100437
$621$	5.28356e9	0.885332
$622$	− 2.42273e9i	− 0.403681i
$623$	6.13650e8	0.101675
$624$	0	0
$625$	−7.42927e9	−1.21721
$626$	5.05348e9i	0.823343i
$627$	2.51518e10	4.07505
$628$	−1.09152e9	−0.175863
$629$	− 2.93535e9i	− 0.470309i
$630$	2.50160e9i	0.398589i
$631$	7.49102e6i	0.00118697i	1.00000		0.000593483i	$0.000188911\pi$
$631$	7.49102e6i	0.00118697i	−1.00000		0.000593483i	$0.999811\pi$
$632$	3.85823e8i	0.0607964i
$633$	−9.44567e9	−1.48020
$634$	−6.34666e9	−0.989083
$635$	6.63505e9i	1.02834i
$636$	−8.18068e9	−1.26093
$637$	0	0
$638$	1.08792e10	1.65854
$639$	1.73507e9i	0.263065i
$640$	6.73186e8	0.101509
$641$	−4.06396e9	−0.609462	−0.304731	−	0.952438i	$-0.598567\pi$
$641$	−4.06396e9	−0.609462	−0.304731	+	0.952438i	$0.598567\pi$
$642$	− 1.05811e10i	− 1.57818i
$643$	− 1.56544e6i	0 0.000232219i	−1.00000		0.000116109i	$-0.999963\pi$
$643$	− 1.56544e6i	0 0.000232219i	1.00000		0.000116109i	$-3.69588e-5\pi$
$644$	− 2.20189e8i	− 0.0324859i
$645$	8.77290e9i	1.28731i
$646$	−2.69531e9	−0.393364
$647$	−1.31025e10	−1.90191	−0.950956	−	0.309325i	$-0.899897\pi$
$647$	−1.31025e10	−1.90191	−0.950956	+	0.309325i	$0.899897\pi$
$648$	6.35521e9i	0.917524i
$649$	−1.26673e10	−1.81898
$650$	0	0
$651$	4.56852e8	0.0648996
$652$	4.70135e9i	0.664287i
$653$	7.63326e9	1.07279	0.536394	−	0.843968i	$-0.319786\pi$
$653$	7.63326e9	1.07279	0.536394	+	0.843968i	$0.319786\pi$
$654$	4.68589e9	0.655044
$655$	6.12945e9i	0.852269i
$656$	− 3.25760e9i	− 0.450541i
$657$	− 2.39756e10i	− 3.29831i
$658$	6.47895e8i	0.0886571i
$659$	−9.25900e9	−1.26027	−0.630137	−	0.776484i	$-0.717001\pi$
$659$	−9.25900e9	−1.26027	−0.630137	+	0.776484i	$0.717001\pi$
$660$	1.39090e10	1.88318
$661$	4.79962e9i	0.646401i	0.946330	+	0.323201i	$0.104759\pi$
$661$	4.79962e9i	0.646401i	−0.946330	+	0.323201i	$0.895241\pi$
$662$	9.73021e9	1.30352
$663$	0	0
$664$	−6.24175e8	−0.0827405
$665$	− 2.15845e9i	− 0.284621i
$666$	1.39359e10	1.82799
$667$	−3.32165e9	−0.433424
$668$	− 2.98791e9i	− 0.387837i
$669$	− 1.09274e10i	− 1.41100i
$670$	− 1.64521e8i	− 0.0211330i
$671$	− 1.86737e10i	− 2.38618i
$672$	5.15998e8	0.0655927
$673$	1.08997e10	1.37836	0.689182	−	0.724589i	$-0.257970\pi$
$673$	1.08997e10	1.37836	0.689182	+	0.724589i	$0.257970\pi$
$674$	− 1.20977e9i	− 0.152192i
$675$	6.92578e9	0.866773
$676$	0	0
$677$	3.44099e9	0.426210	0.213105	−	0.977029i	$-0.431642\pi$
$677$	3.44099e9	0.426210	0.213105	+	0.977029i	$0.431642\pi$
$678$	− 8.02434e9i	− 0.988793i
$679$	−2.94808e8	−0.0361406
$680$	−1.49051e9	−0.181783
$681$	1.65974e10i	2.01384i
$682$	− 1.80617e9i	− 0.218029i
$683$	5.53553e9i	0.664794i	0.943140	+	0.332397i	$0.107857\pi$
$683$	5.53553e9i	0.664794i	−0.943140	+	0.332397i	$0.892143\pi$
$684$	− 1.27962e10i	− 1.52892i
$685$	9.53051e9	1.13292
$686$	−2.33754e9	−0.276455
$687$	4.60153e9i	0.541444i
$688$	1.28671e9	0.150633
$689$	0	0
$690$	−4.24669e9	−0.492129
$691$	− 4.21595e8i	− 0.0486097i	−0.999705	−	0.0243048i	$-0.992263\pi$
$691$	− 4.21595e8i	− 0.0486097i	0.999705	−	0.0243048i	$-0.00773723\pi$
$692$	−4.99796e9	−0.573352
$693$	7.58077e9	0.865261
$694$	4.77788e9i	0.542596i
$695$	− 4.99644e9i	− 0.564565i
$696$	− 7.78406e9i	− 0.875133i
$697$	7.21268e9i	0.806830i
$698$	9.56799e8	0.106494
$699$	1.31591e10	1.45732
$700$	− 2.88627e8i	− 0.0318049i
$701$	5.14995e9	0.564663	0.282332	−	0.959317i	$-0.408892\pi$
$701$	5.14995e9	0.564663	0.282332	+	0.959317i	$0.408892\pi$
$702$	0	0
$703$	−1.20243e10	−1.30532
$704$	− 2.04000e9i	− 0.220357i
$705$	1.24957e10	1.34307
$706$	3.73132e9	0.399067
$707$	− 2.77840e9i	− 0.295683i
$708$	9.06342e9i	0.959789i
$709$	1.05683e10i	1.11363i	0.830635	+	0.556817i	$0.187978\pi$
$709$	1.05683e10i	1.11363i	−0.830635	+	0.556817i	$0.812022\pi$
$710$	− 8.27880e8i	− 0.0868086i
$711$	4.05566e9	0.423173
$712$	−1.73585e9	−0.180232
$713$	5.51460e8i	0.0569772i
$714$	−1.14248e9	−0.117464
$715$	0	0
$716$	−3.55944e9	−0.362399
$717$	2.27868e10i	2.30869i
$718$	−6.16081e9	−0.621159
$719$	−1.53690e10	−1.54204	−0.771020	−	0.636811i	$-0.780253\pi$
$719$	−1.53690e10	−1.54204	−0.771020	+	0.636811i	$0.780253\pi$
$720$	− 7.07634e9i	− 0.706553i
$721$	1.24463e9i	0.123671i
$722$	3.89001e9i	0.384654i
$723$	1.14624e10i	1.12795i
$724$	−7.64381e9	−0.748557
$725$	−4.35407e9	−0.424339
$726$	− 2.85864e10i	− 2.77256i
$727$	−4.88599e9	−0.471609	−0.235804	−	0.971801i	$-0.575772\pi$
$727$	−4.88599e9	−0.471609	−0.235804	+	0.971801i	$0.575772\pi$
$728$	0	0
$729$	1.39161e10	1.33036
$730$	1.14399e10i	1.08841i
$731$	−2.84891e9	−0.269754
$732$	−1.33610e10	−1.25907
$733$	− 3.59889e9i	− 0.337524i	−0.985657	−	0.168762i	$-0.946023\pi$
$733$	− 3.59889e9i	− 0.337524i	0.985657	−	0.168762i	$-0.0539769\pi$
$734$	6.84255e9i	0.638678i
$735$	2.20842e10i	2.05152i
$736$	6.22854e8i	0.0575856i
$737$	−4.98562e8	−0.0458757
$738$	−3.42430e10	−3.13598
$739$	2.78886e9i	0.254198i	0.991890	+	0.127099i	$0.0405666\pi$
$739$	2.78886e9i	0.254198i	−0.991890	+	0.127099i	$0.959433\pi$
$740$	−6.64946e9	−0.603219
$741$	0	0
$742$	2.12745e9	0.191181
$743$	3.08130e9i	0.275597i	0.990460	+	0.137798i	$0.0440026\pi$
$743$	3.08130e9i	0.275597i	−0.990460	+	0.137798i	$0.955997\pi$
$744$	−1.29231e9	−0.115043
$745$	8.01457e8	0.0710122
$746$	4.23687e9i	0.373645i
$747$	6.56115e9i	0.575915i
$748$	4.51680e9i	0.394616i
$749$	2.75169e9i	0.239284i
$750$	1.18877e10	1.02893
$751$	6.41281e8	0.0552470	0.0276235	−	0.999618i	$-0.491206\pi$
$751$	6.41281e8	0.0552470	0.0276235	+	0.999618i	$0.491206\pi$
$752$	− 1.83272e9i	− 0.157157i
$753$	2.14943e10	1.83460
$754$	0	0
$755$	7.69763e9	0.650943
$756$	− 3.21995e9i	− 0.271033i
$757$	−1.60219e10	−1.34239	−0.671195	−	0.741280i	$-0.734219\pi$
$757$	−1.60219e10	−1.34239	−0.671195	+	0.741280i	$0.734219\pi$
$758$	−1.58686e10	−1.32342
$759$	1.28691e10i	1.06832i
$760$	6.10568e9i	0.504529i
$761$	− 5.73623e9i	− 0.471824i	−0.971774	−	0.235912i	$-0.924192\pi$
$761$	− 5.73623e9i	− 0.471824i	0.971774	−	0.235912i	$-0.0758077\pi$
$762$	− 1.43863e10i	− 1.17789i
$763$	−1.21860e9	−0.0993175
$764$	−6.75102e9	−0.547700
$765$	1.56678e10i	1.26530i
$766$	−7.19005e9	−0.578005
$767$	0	0
$768$	−1.45962e9	−0.116272
$769$	− 2.45874e10i	− 1.94971i	−0.222832	−	0.974857i	$-0.571530\pi$
$769$	− 2.45874e10i	− 1.94971i	0.222832	−	0.974857i	$-0.428470\pi$
$770$	−3.61714e9	−0.285527
$771$	1.97739e10	1.55382
$772$	− 1.35718e9i	− 0.106164i
$773$	1.31517e10i	1.02413i	0.858947	+	0.512065i	$0.171119\pi$
$773$	1.31517e10i	1.02413i	−0.858947	+	0.512065i	$0.828881\pi$
$774$	− 1.35255e10i	− 1.04848i
$775$	7.22863e8i	0.0557828i
$776$	8.33932e8	0.0640641
$777$	−5.09682e9	−0.389785
$778$	1.45980e10i	1.11138i
$779$	2.95458e10	2.23932
$780$	0	0
$781$	−2.50878e9	−0.188445
$782$	− 1.37907e9i	− 0.103125i
$783$	−4.85744e10	−3.61611
$784$	3.23904e9	0.240055
$785$	− 5.47466e9i	− 0.403937i
$786$	− 1.32900e10i	− 0.976218i
$787$	− 7.38863e9i	− 0.540322i	−0.962815	−	0.270161i	$-0.912923\pi$
$787$	− 7.38863e9i	− 0.540322i	0.962815	−	0.270161i	$-0.0870769\pi$
$788$	− 1.06583e10i	− 0.775969i
$789$	3.70509e10	2.68552
$790$	−1.93514e9	−0.139643
$791$	2.08679e9i	0.149920i
$792$	−2.14440e10	−1.53379
$793$	0	0
$794$	3.94467e9	0.279665
$795$	− 4.10312e10i	− 2.89621i
$796$	−8.08645e9	−0.568279
$797$	5.22399e9	0.365509	0.182754	−	0.983159i	$-0.441499\pi$
$797$	5.22399e9	0.365509	0.182754	+	0.983159i	$0.441499\pi$
$798$	4.68001e9i	0.326014i
$799$	4.05784e9i	0.281437i
$800$	8.16447e8i	0.0563785i
$801$	1.82468e10i	1.25450i
$802$	−4.54624e9	−0.311202
$803$	3.46671e10	2.36273
$804$	3.56719e8i	0.0242064i
$805$	1.10438e9	0.0746164
$806$	0	0
$807$	4.47314e10	2.99609
$808$	7.85934e9i	0.524139i
$809$	−7.92102e9	−0.525970	−0.262985	−	0.964800i	$-0.584707\pi$
$809$	−7.92102e9	−0.525970	−0.262985	+	0.964800i	$0.584707\pi$
$810$	−3.18754e10	−2.10745
$811$	8.16607e9i	0.537576i	0.963199	+	0.268788i	$0.0866232\pi$
$811$	8.16607e9i	0.537576i	−0.963199	+	0.268788i	$0.913377\pi$
$812$	2.02430e9i	0.132687i
$813$	− 3.97674e9i	− 0.259543i
$814$	2.01503e10i	1.30947i
$815$	−2.35802e10	−1.52579
$816$	3.23176e9	0.208220
$817$	1.16702e10i	0.748688i
$818$	1.02778e10	0.656541
$819$	0	0
$820$	1.63389e10	1.03484
$821$	− 2.63749e10i	− 1.66338i	−0.555244	−	0.831688i	$-0.687375\pi$
$821$	− 2.63749e10i	− 1.66338i	0.555244	−	0.831688i	$-0.312625\pi$
$822$	−2.06643e10	−1.29768
$823$	−2.04085e10	−1.27618	−0.638090	−	0.769962i	$-0.720275\pi$
$823$	−2.04085e10	−1.27618	−0.638090	+	0.769962i	$0.720275\pi$
$824$	− 3.52073e9i	− 0.219224i
$825$	1.68690e10i	1.04592i
$826$	− 2.35701e9i	− 0.145523i
$827$	− 2.55307e10i	− 1.56962i	−0.619738	−	0.784809i	$-0.712761\pi$
$827$	− 2.55307e10i	− 1.56962i	0.619738	−	0.784809i	$-0.287239\pi$
$828$	6.54727e9	0.400824
$829$	−8.48208e9	−0.517085	−0.258542	−	0.966000i	$-0.583242\pi$
$829$	−8.48208e9	−0.517085	−0.258542	+	0.966000i	$0.583242\pi$
$830$	− 3.13063e9i	− 0.190046i
$831$	−2.38349e10	−1.44082
$832$	0	0
$833$	−7.17160e9	−0.429891
$834$	1.08334e10i	0.646672i
$835$	1.49862e10	0.890819
$836$	1.85025e10	1.09524
$837$	8.06432e9i	0.475367i
$838$	− 2.19878e9i	− 0.129071i
$839$	− 2.29323e10i	− 1.34055i	−0.742115	−	0.670273i	$-0.766177\pi$
$839$	− 2.29323e10i	− 1.34055i	0.742115	−	0.670273i	$-0.233823\pi$
$840$	2.58805e9i	0.150659i
$841$	1.32877e10	0.770306
$842$	−6.01094e9	−0.347017
$843$	− 3.66885e10i	− 2.10928i
$844$	−6.94854e9	−0.397828
$845$	0	0
$846$	−1.92650e10	−1.09389
$847$	7.43410e9i	0.420374i
$848$	−6.01797e9	−0.338895
$849$	3.32303e10	1.86362
$850$	− 1.80771e9i	− 0.100963i
$851$	− 6.15230e9i	− 0.342203i
$852$	1.79503e9i	0.0994335i
$853$	2.47175e10i	1.36358i	0.731546	+	0.681792i	$0.238799\pi$
$853$	2.47175e10i	1.36358i	−0.731546	+	0.681792i	$0.761201\pi$
$854$	3.47463e9	0.190900
$855$	6.41812e10	3.51177
$856$	− 7.78379e9i	− 0.424163i
$857$	−1.19081e10	−0.646265	−0.323133	−	0.946354i	$-0.604736\pi$
$857$	−1.19081e10	−0.646265	−0.323133	+	0.946354i	$0.604736\pi$
$858$	0	0
$859$	−4.94214e9	−0.266035	−0.133018	−	0.991114i	$-0.542467\pi$
$859$	−4.94214e9	−0.266035	−0.133018	+	0.991114i	$0.542467\pi$
$860$	6.45363e9i	0.345987i
$861$	1.25238e10	0.668689
$862$	1.04605e9	0.0556259
$863$	2.05387e10i	1.08776i	0.839162	+	0.543881i	$0.183046\pi$
$863$	2.05387e10i	1.08776i	−0.839162	+	0.543881i	$0.816954\pi$
$864$	9.10836e9i	0.480443i
$865$	− 2.50679e10i	− 1.31693i
$866$	1.33389e10i	0.697921i
$867$	2.85440e10	1.48747
$868$	3.36075e8	0.0174428
$869$	5.86420e9i	0.303138i
$870$	3.90419e10	2.01008
$871$	0	0
$872$	3.44709e9	0.176054
$873$	− 8.76606e9i	− 0.445918i
$874$	−5.64918e9	−0.286217
$875$	−3.09150e9	−0.156006
$876$	− 2.48042e10i	− 1.24670i
$877$	1.42584e10i	0.713791i	0.934144	+	0.356895i	$0.116165\pi$
$877$	1.42584e10i	0.713791i	−0.934144	+	0.356895i	$0.883835\pi$
$878$	1.85182e10i	0.923354i
$879$	3.52204e10i	1.74918i
$880$	1.02319e10	0.506136
$881$	−1.78398e10	−0.878971	−0.439486	−	0.898250i	$-0.644839\pi$
$881$	−1.78398e10	−0.878971	−0.439486	+	0.898250i	$0.644839\pi$
$882$	− 3.40479e10i	− 1.67090i
$883$	−3.79954e10	−1.85724	−0.928622	−	0.371028i	$-0.879005\pi$
$883$	−3.79954e10	−1.85724	−0.928622	+	0.371028i	$0.879005\pi$
$884$	0	0
$885$	−4.54587e10	−2.20453
$886$	5.52038e9i	0.266656i
$887$	8.45195e7	0.00406653	0.00203327	−	0.999998i	$-0.499353\pi$
$887$	8.45195e7	0.00406653	0.00203327	+	0.999998i	$0.499353\pi$
$888$	1.44175e10	0.690947
$889$	3.74126e9i	0.178592i
$890$	− 8.70637e9i	− 0.413973i
$891$	9.65942e10i	4.57488i
$892$	− 8.03857e9i	− 0.379229i
$893$	1.66224e10	0.781114
$894$	−1.73774e9	−0.0813398
$895$	− 1.78528e10i	− 0.832390i
$896$	3.79585e8	0.0176291
$897$	0	0
$898$	2.11045e10	0.972541
$899$	− 5.06985e9i	− 0.232721i
$900$	8.58227e9	0.392422
$901$	1.33245e10	0.606894
$902$	− 4.95129e10i	− 2.24645i
$903$	4.94672e9i	0.223568i
$904$	− 5.90296e9i	− 0.265754i
$905$	− 3.83385e10i	− 1.71935i
$906$	−1.66902e10	−0.745611
$907$	−1.82024e10	−0.810033	−0.405017	−	0.914309i	$-0.632734\pi$
$907$	−1.82024e10	−0.810033	−0.405017	+	0.914309i	$0.632734\pi$
$908$	1.22096e10i	0.541252i
$909$	8.26151e10	3.64826
$910$	0	0
$911$	−3.66963e10	−1.60808	−0.804040	−	0.594575i	$-0.797320\pi$
$911$	−3.66963e10	−1.60808	−0.804040	+	0.594575i	$0.797320\pi$
$912$	− 1.32385e10i	− 0.577905i
$913$	−9.48697e9	−0.412553
$914$	4.92978e9	0.213558
$915$	− 6.70139e10i	− 2.89195i
$916$	3.38503e9i	0.145522i
$917$	3.45617e9i	0.148014i
$918$	− 2.01669e10i	− 0.860380i
$919$	1.33474e10	0.567275	0.283638	−	0.958932i	$-0.408459\pi$
$919$	1.33474e10	0.567275	0.283638	+	0.958932i	$0.408459\pi$
$920$	−3.12400e9	−0.132268
$921$	4.13702e9i	0.174493i
$922$	−9.88970e9	−0.415552
$923$	0	0
$924$	7.84276e9	0.327052
$925$	− 8.06454e9i	− 0.335030i
$926$	5.42775e8	0.0224637
$927$	−3.70089e10	−1.52591
$928$	− 5.72621e9i	− 0.235206i
$929$	− 2.71771e10i	− 1.11211i	−0.831146	−	0.556055i	$-0.812314\pi$
$929$	− 2.71771e10i	− 1.11211i	0.831146	−	0.556055i	$-0.187686\pi$
$930$	− 6.48174e9i	− 0.264242i
$931$	2.93776e10i	1.19314i
$932$	9.68025e9	0.391680
$933$	−2.63472e10	−1.06206
$934$	− 9.40012e9i	− 0.377502i
$935$	−2.26546e10	−0.906390
$936$	0	0
$937$	4.04333e10	1.60565	0.802825	−	0.596214i	$-0.203329\pi$
$937$	4.04333e10	1.60565	0.802825	+	0.596214i	$0.203329\pi$
$938$	− 9.27676e7i	− 0.00367017i
$939$	5.49566e10	2.16616
$940$	9.19223e9	0.360972
$941$	8.49843e9i	0.332487i	0.986085	+	0.166244i	$0.0531638\pi$
$941$	8.49843e9i	0.332487i	−0.986085	+	0.166244i	$0.946836\pi$
$942$	1.18703e10i	0.462683i
$943$	1.51173e10i	0.587061i
$944$	6.66735e9i	0.257959i
$945$	1.61500e10	0.622533
$946$	1.95569e10	0.751072
$947$	− 4.40082e9i	− 0.168387i	−0.996449	−	0.0841935i	$-0.973169\pi$
$947$	− 4.40082e9i	− 0.168387i	0.996449	−	0.0841935i	$-0.0268314\pi$
$948$	4.19582e9	0.159951
$949$	0	0
$950$	−7.40504e9	−0.280217
$951$	6.90199e10i	2.60221i
$952$	−8.40442e8	−0.0315703
$953$	1.73133e10	0.647970	0.323985	−	0.946062i	$-0.394977\pi$
$953$	1.73133e10	0.647970	0.323985	+	0.946062i	$0.394977\pi$
$954$	6.32593e10i	2.35887i
$955$	− 3.38606e10i	− 1.25801i
$956$	1.67627e10i	0.620498i
$957$	− 1.18312e11i	− 4.36351i
$958$	3.16924e9	0.116460
$959$	5.37390e9	0.196754
$960$	− 7.32090e9i	− 0.267064i
$961$	2.66709e10	0.969407
$962$	0	0
$963$	−8.18210e10	−2.95238
$964$	8.43211e9i	0.303156i
$965$	6.80708e9	0.243846
$966$	−2.39455e9	−0.0854681
$967$	− 1.40918e10i	− 0.501158i	−0.968096	−	0.250579i	$-0.919379\pi$
$967$	− 1.40918e10i	− 0.501158i	0.968096	−	0.250579i	$-0.0806210\pi$
$968$	− 2.10290e10i	− 0.745171i
$969$	2.93115e10i	1.03491i
$970$	4.18269e9i	0.147148i
$971$	−7.27843e9	−0.255135	−0.127568	−	0.991830i	$-0.540717\pi$
$971$	−7.27843e9	−0.255135	−0.127568	+	0.991830i	$0.540717\pi$
$972$	3.02067e10	1.05505
$973$	− 2.81731e9i	− 0.0980481i
$974$	2.42932e10	0.842419
$975$	0	0
$976$	−9.82879e9	−0.338397
$977$	− 2.43791e10i	− 0.836348i	−0.908367	−	0.418174i	$-0.862670\pi$
$977$	− 2.43791e10i	− 0.836348i	0.908367	−	0.418174i	$-0.137330\pi$
$978$	5.11272e10	1.74770
$979$	−2.63835e10	−0.898657
$980$	1.62458e10i	0.551379i
$981$	− 3.62349e10i	− 1.22542i
$982$	− 2.33579e10i	− 0.787126i
$983$	4.06556e10i	1.36516i	0.730811	+	0.682579i	$0.239142\pi$
$983$	4.06556e10i	1.36516i	−0.730811	+	0.682579i	$0.760858\pi$
$984$	−3.54264e10	−1.18534
$985$	5.34578e10	1.78231
$986$	1.26785e10i	0.421209i
$987$	7.04585e9	0.233251
$988$	0	0
$989$	−5.97112e9	−0.196277
$990$	− 1.07555e11i	− 3.52295i
$991$	−4.86636e10	−1.58835	−0.794175	−	0.607689i	$-0.792097\pi$
$991$	−4.86636e10	−1.58835	−0.794175	+	0.607689i	$0.792097\pi$
$992$	−9.50665e8	−0.0309198
$993$	− 1.05816e11i	− 3.42949i
$994$	− 4.66811e8i	− 0.0150761i
$995$	− 4.05586e10i	− 1.30527i
$996$	6.78790e9i	0.217685i
$997$	−1.76682e10	−0.564622	−0.282311	−	0.959323i	$-0.591101\pi$
$997$	−1.76682e10	−0.564622	−0.282311	+	0.959323i	$0.591101\pi$
$998$	1.29874e10	0.413586
$999$	− 8.99687e10i	− 2.85504i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	338.8.b.a.337.1		2
13.5	odd	4		338.8.a.a.1.1		1
13.8	odd	4		26.8.a.b.1.1	✓	1
13.12	even	2	inner	338.8.b.a.337.2		2
39.8	even	4		234.8.a.a.1.1		1
52.47	even	4		208.8.a.e.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
26.8.a.b.1.1	✓	1	13.8	odd	4
208.8.a.e.1.1		1	52.47	even	4
234.8.a.a.1.1		1	39.8	even	4
338.8.a.a.1.1		1	13.5	odd	4
338.8.b.a.337.1		2	1.1	even	1	trivial
338.8.b.a.337.2		2	13.12	even	2	inner

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:	\( N \)	\(=\)	\( 338 = 2 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	338.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-8.00000i q^{2} -87.0000 q^{3} -64.0000 q^{4} -321.000i q^{5} +696.000i q^{6} -181.000i q^{7} +512.000i q^{8} +5382.00 q^{9} +O(q^{10})\)	\(q-8.00000i q^{2} -87.0000 q^{3} -64.0000 q^{4} -321.000i q^{5} +696.000i q^{6} -181.000i q^{7} +512.000i q^{8} +5382.00 q^{9} -2568.00 q^{10} +7782.00i q^{11} +5568.00 q^{12} -1448.00 q^{14} +27927.0i q^{15} +4096.00 q^{16} -9069.00 q^{17} -43056.0i q^{18} +37150.0i q^{19} +20544.0i q^{20} +15747.0i q^{21} +62256.0 q^{22} -19008.0 q^{23} -44544.0i q^{24} -24916.0 q^{25} -277965. q^{27} +11584.0i q^{28} +174750. q^{29} +223416. q^{30} -29012.0i q^{31} -32768.0i q^{32} -677034. i q^{33} +72552.0i q^{34} -58101.0 q^{35} -344448. q^{36} +323669. i q^{37} +297200. q^{38} +164352. q^{40} -795312. i q^{41} +125976. q^{42} +314137. q^{43} -498048. i q^{44} -1.72762e6i q^{45} +152064. i q^{46} -447441. i q^{47} -356352. q^{48} +790782. q^{49} +199328. i q^{50} +789003. q^{51} -1.46923e6 q^{53} +2.22372e6i q^{54} +2.49802e6 q^{55} +92672.0 q^{56} -3.23205e6i q^{57} -1.39800e6i q^{58} +1.62777e6i q^{59} -1.78733e6i q^{60} -2.39961e6 q^{61} -232096. q^{62} -974142. i q^{63} -262144. q^{64} -5.41627e6 q^{66} +64066.0i q^{67} +580416. q^{68} +1.65370e6 q^{69} +464808. i q^{70} +322383. i q^{71} +2.75558e6i q^{72} -4.45478e6i q^{73} +2.58935e6 q^{74} +2.16769e6 q^{75} -2.37760e6i q^{76} +1.40854e6 q^{77} +753560. q^{79} -1.31482e6i q^{80} +1.24125e7 q^{81} -6.36250e6 q^{82} +1.21909e6i q^{83} -1.00781e6i q^{84} +2.91115e6i q^{85} -2.51310e6i q^{86} -1.52032e7 q^{87} -3.98438e6 q^{88} +3.39033e6i q^{89} -1.38210e7 q^{90} +1.21651e6 q^{92} +2.52404e6i q^{93} -3.57953e6 q^{94} +1.19252e7 q^{95} +2.85082e6i q^{96} -1.62877e6i q^{97} -6.32626e6i q^{98} +4.18827e7i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 174 q^{3} - 128 q^{4} + 10764 q^{9}+O(q^{10}) \) 2 * q - 174 * q^3 - 128 * q^4 + 10764 * q^9	\( 2 q - 174 q^{3} - 128 q^{4} + 10764 q^{9} - 5136 q^{10} + 11136 q^{12} - 2896 q^{14} + 8192 q^{16} - 18138 q^{17} + 124512 q^{22} - 38016 q^{23} - 49832 q^{25} - 555930 q^{27} + 349500 q^{29} + 446832 q^{30} - 116202 q^{35} - 688896 q^{36} + 594400 q^{38} + 328704 q^{40} + 251952 q^{42} + 628274 q^{43} - 712704 q^{48} + 1581564 q^{49} + 1578006 q^{51} - 2938464 q^{53} + 4996044 q^{55} + 185344 q^{56} - 4799216 q^{61} - 464192 q^{62} - 524288 q^{64} - 10832544 q^{66} + 1160832 q^{68} + 3307392 q^{69} + 5178704 q^{74} + 4335384 q^{75} + 2817084 q^{77} + 1507120 q^{79} + 24825042 q^{81} - 12724992 q^{82} - 30406500 q^{87} - 7968768 q^{88} - 27641952 q^{90} + 2433024 q^{92} - 7159056 q^{94} + 23850300 q^{95}+O(q^{100}) \) 2 * q - 174 * q^3 - 128 * q^4 + 10764 * q^9 - 5136 * q^10 + 11136 * q^12 - 2896 * q^14 + 8192 * q^16 - 18138 * q^17 + 124512 * q^22 - 38016 * q^23 - 49832 * q^25 - 555930 * q^27 + 349500 * q^29 + 446832 * q^30 - 116202 * q^35 - 688896 * q^36 + 594400 * q^38 + 328704 * q^40 + 251952 * q^42 + 628274 * q^43 - 712704 * q^48 + 1581564 * q^49 + 1578006 * q^51 - 2938464 * q^53 + 4996044 * q^55 + 185344 * q^56 - 4799216 * q^61 - 464192 * q^62 - 524288 * q^64 - 10832544 * q^66 + 1160832 * q^68 + 3307392 * q^69 + 5178704 * q^74 + 4335384 * q^75 + 2817084 * q^77 + 1507120 * q^79 + 24825042 * q^81 - 12724992 * q^82 - 30406500 * q^87 - 7968768 * q^88 - 27641952 * q^90 + 2433024 * q^92 - 7159056 * q^94 + 23850300 * q^95

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