LMFDB - Embedded newform 650.6.b.o.599.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [650,6,Mod(599,650)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(650, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("650.599");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 650 = 2 \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	650.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:	$104.249482878$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 2075 x^{10} + 1596139 x^{8} + 560320345 x^{6} + 89526738200 x^{4} + 5863900282000 x^{2} + 131166627840000 $ x^12 + 2075x^10 + 1596139x^8 + 560320345x^6 + 89526738200x^4 + 5863900282000*x^2 + 131166627840000
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{8}\cdot 3^{4}\cdot 5^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			599.2
Root			$-20.7411i$ of defining polynomial
Character	$\chi$	$=$	650.599
Dual form			650.6.b.o.599.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-4.00000i q^{2} -18.7411i q^{3} -16.0000 q^{4} -74.9644 q^{6} +230.695i q^{7} +64.0000i q^{8} -108.229 q^{9} -544.656 q^{11} +299.858i q^{12} +169.000i q^{13} +922.779 q^{14} +256.000 q^{16} +554.057i q^{17} +432.917i q^{18} +2228.18 q^{19} +4323.47 q^{21} +2178.63i q^{22} -2915.01i q^{23} +1199.43 q^{24} +676.000 q^{26} -2525.75i q^{27} -3691.11i q^{28} +6864.22 q^{29} -6257.84 q^{31} -1024.00i q^{32} +10207.5i q^{33} +2216.23 q^{34} +1731.67 q^{36} -5695.55i q^{37} -8912.71i q^{38} +3167.25 q^{39} -11997.2 q^{41} -17293.9i q^{42} +15451.4i q^{43} +8714.50 q^{44} -11660.0 q^{46} +16278.3i q^{47} -4797.72i q^{48} -36413.0 q^{49} +10383.6 q^{51} -2704.00i q^{52} -11052.3i q^{53} -10103.0 q^{54} -14764.5 q^{56} -41758.5i q^{57} -27456.9i q^{58} +24145.9 q^{59} -10681.8 q^{61} +25031.4i q^{62} -24967.9i q^{63} -4096.00 q^{64} +40829.9 q^{66} -52777.4i q^{67} -8864.91i q^{68} -54630.5 q^{69} -44162.3 q^{71} -6926.67i q^{72} -41761.3i q^{73} -22782.2 q^{74} -35650.8 q^{76} -125649. i q^{77} -12669.0i q^{78} +59573.9 q^{79} -73635.1 q^{81} +47988.7i q^{82} +22640.6i q^{83} -69175.6 q^{84} +61805.8 q^{86} -128643. i q^{87} -34858.0i q^{88} -129864. q^{89} -38987.4 q^{91} +46640.1i q^{92} +117279. i q^{93} +65113.1 q^{94} -19190.9 q^{96} -63287.9i q^{97} +145652. i q^{98} +58947.7 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 192 q^{4} + 72 q^{6} - 1258 q^{9} + 740 q^{11} + 504 q^{14} + 3072 q^{16} + 3318 q^{19} + 8644 q^{21} - 1152 q^{24} + 8112 q^{26} + 11822 q^{29} + 2806 q^{31} - 6704 q^{34} + 20128 q^{36} - 3042 q^{39}+ \cdots - 772238 q^{99}+O(q^{100}) $ 12 * q - 192 * q^4 + 72 * q^6 - 1258 * q^9 + 740 * q^11 + 504 * q^14 + 3072 * q^16 + 3318 * q^19 + 8644 * q^21 - 1152 * q^24 + 8112 * q^26 + 11822 * q^29 + 2806 * q^31 - 6704 * q^34 + 20128 * q^36 - 3042 * q^39 + 2974 * q^41 - 11840 * q^44 + 12528 * q^46 - 57954 * q^49 + 14106 * q^51 - 60552 * q^54 - 8064 * q^56 - 44734 * q^59 + 13422 * q^61 - 49152 * q^64 + 71944 * q^66 - 268676 * q^69 + 30440 * q^71 - 84784 * q^74 - 53088 * q^76 - 190856 * q^79 - 16268 * q^81 - 138304 * q^84 + 128624 * q^86 - 584622 * q^89 - 21294 * q^91 - 57080 * q^94 + 18432 * q^96 - 772238 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/650\mathbb{Z}\right)^\times$.

$n$	$27$	$301$
$\chi(n)$	$-1$	$1$

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 $

$2$	− 4.00000i	− 0.707107i
$2$	− 4.00000i	− 0.707107i
$3$	− 18.7411i	− 1.20224i	−0.799158	−	0.601121i	$-0.794721\pi$
$3$	− 18.7411i	− 1.20224i	0.799158	−	0.601121i	$-0.205279\pi$
$4$	−16.0000	−0.500000
$5$	0	0
$5$	0	0
$6$	−74.9644	−0.850114
$7$	230.695i	1.77948i	0.456470	+	0.889739i	$0.349113\pi$
$7$	230.695i	1.77948i	−0.456470	+	0.889739i	$0.650887\pi$
$8$	64.0000i	0.353553i
$9$	−108.229	−0.445388
$10$	0	0
$11$	−544.656	−1.35719	−0.678595	−	0.734512i	$-0.737411\pi$
$11$	−544.656	−1.35719	−0.678595	+	0.734512i	$0.737411\pi$
$12$	299.858i	0.601121i
$13$	169.000i	0.277350i
$13$	169.000i	0.277350i
$14$	922.779	1.25828
$15$	0	0
$16$	256.000	0.250000
$17$	554.057i	0.464977i	0.972599	+	0.232489i	$0.0746869\pi$
$17$	554.057i	0.464977i	−0.972599	+	0.232489i	$0.925313\pi$
$18$	432.917i	0.314937i
$19$	2228.18	1.41601	0.708004	−	0.706208i	$-0.249596\pi$
$19$	2228.18	1.41601	0.708004	+	0.706208i	$0.249596\pi$
$20$	0	0
$21$	4323.47	2.13936
$22$	2178.63i	0.959679i
$23$	− 2915.01i	− 1.14900i	−0.818505	−	0.574500i	$-0.805197\pi$
$23$	− 2915.01i	− 1.14900i	0.818505	−	0.574500i	$-0.194803\pi$
$24$	1199.43	0.425057
$25$	0	0
$26$	676.000	0.196116
$27$	− 2525.75i	− 0.666779i
$28$	− 3691.11i	− 0.889739i
$29$	6864.22	1.51564	0.757820	−	0.652464i	$-0.226264\pi$
$29$	6864.22	1.51564	0.757820	+	0.652464i	$0.226264\pi$
$30$	0	0
$31$	−6257.84	−1.16955	−0.584777	−	0.811194i	$-0.698818\pi$
$31$	−6257.84	−1.16955	−0.584777	+	0.811194i	$0.698818\pi$
$32$	− 1024.00i	− 0.176777i
$33$	10207.5i	1.63167i
$34$	2216.23	0.328789
$35$	0	0
$36$	1731.67	0.222694
$37$	− 5695.55i	− 0.683961i	−0.939707	−	0.341980i	$-0.888902\pi$
$37$	− 5695.55i	− 0.683961i	0.939707	−	0.341980i	$-0.111098\pi$
$38$	− 8912.71i	− 1.00127i
$39$	3167.25	0.333442
$40$	0	0
$41$	−11997.2	−1.11460	−0.557300	−	0.830311i	$-0.688163\pi$
$41$	−11997.2	−1.11460	−0.557300	+	0.830311i	$0.688163\pi$
$42$	− 17293.9i	− 1.51276i
$43$	15451.4i	1.27438i	0.770708	+	0.637189i	$0.219903\pi$
$43$	15451.4i	1.27438i	−0.770708	+	0.637189i	$0.780097\pi$
$44$	8714.50	0.678595
$45$	0	0
$46$	−11660.0	−0.812466
$47$	16278.3i	1.07489i	0.843299	+	0.537445i	$0.180610\pi$
$47$	16278.3i	1.07489i	−0.843299	+	0.537445i	$0.819390\pi$
$48$	− 4797.72i	− 0.300561i
$49$	−36413.0	−2.16654
$50$	0	0
$51$	10383.6	0.559016
$52$	− 2704.00i	− 0.138675i
$53$	− 11052.3i	− 0.540462i	−0.962796	−	0.270231i	$-0.912900\pi$
$53$	− 11052.3i	− 0.540462i	0.962796	−	0.270231i	$-0.0871000\pi$
$54$	−10103.0	−0.471484
$55$	0	0
$56$	−14764.5	−0.629140
$57$	− 41758.5i	− 1.70239i
$58$	− 27456.9i	− 1.07172i
$59$	24145.9	0.903054	0.451527	−	0.892257i	$-0.350879\pi$
$59$	24145.9	0.903054	0.451527	+	0.892257i	$0.350879\pi$
$60$	0	0
$61$	−10681.8	−0.367552	−0.183776	−	0.982968i	$-0.558832\pi$
$61$	−10681.8	−0.367552	−0.183776	+	0.982968i	$0.558832\pi$
$62$	25031.4i	0.827000i
$63$	− 24967.9i	− 0.792557i
$64$	−4096.00	−0.125000
$65$	0	0
$66$	40829.9	1.15377
$67$	− 52777.4i	− 1.43635i	−0.695861	−	0.718176i	$-0.744977\pi$
$67$	− 52777.4i	− 1.43635i	0.695861	−	0.718176i	$-0.255023\pi$
$68$	− 8864.91i	− 0.232489i
$69$	−54630.5	−1.38138
$70$	0	0
$71$	−44162.3	−1.03970	−0.519848	−	0.854259i	$-0.674011\pi$
$71$	−44162.3	−1.03970	−0.519848	+	0.854259i	$0.674011\pi$
$72$	− 6926.67i	− 0.157468i
$73$	− 41761.3i	− 0.917205i	−0.888642	−	0.458602i	$-0.848350\pi$
$73$	− 41761.3i	− 0.917205i	0.888642	−	0.458602i	$-0.151650\pi$
$74$	−22782.2	−0.483633
$75$	0	0
$76$	−35650.8	−0.708004
$77$	− 125649.i	− 2.41509i
$78$	− 12669.0i	− 0.235779i
$79$	59573.9	1.07396	0.536980	−	0.843595i	$-0.319565\pi$
$79$	59573.9	1.07396	0.536980	+	0.843595i	$0.319565\pi$
$80$	0	0
$81$	−73635.1	−1.24702
$82$	47988.7i	0.788141i
$83$	22640.6i	0.360738i	0.983599	+	0.180369i	$0.0577292\pi$
$83$	22640.6i	0.360738i	−0.983599	+	0.180369i	$0.942271\pi$
$84$	−69175.6	−1.06968
$85$	0	0
$86$	61805.8	0.901121
$87$	− 128643.i	− 1.82217i
$88$	− 34858.0i	− 0.479839i
$89$	−129864.	−1.73786	−0.868930	−	0.494935i	$-0.835192\pi$
$89$	−129864.	−1.73786	−0.868930	+	0.494935i	$0.835192\pi$
$90$	0	0
$91$	−38987.4	−0.493538
$92$	46640.1i	0.574500i
$93$	117279.i	1.40609i
$94$	65113.1	0.760061
$95$	0	0
$96$	−19190.9	−0.212529
$97$	− 63287.9i	− 0.682953i	−0.939890	−	0.341477i	$-0.889073\pi$
$97$	− 63287.9i	− 0.682953i	0.939890	−	0.341477i	$-0.110927\pi$
$98$	145652.i	1.53197i
$99$	58947.7	0.604476
$100$	0	0
$101$	116458.	1.13596	0.567982	−	0.823041i	$-0.307724\pi$
$101$	116458.	1.13596	0.567982	+	0.823041i	$0.307724\pi$
$102$	− 41534.6i	− 0.395284i
$103$	− 48487.1i	− 0.450333i	−0.974320	−	0.225166i	$-0.927707\pi$
$103$	− 48487.1i	− 0.450333i	0.974320	−	0.225166i	$-0.0722926\pi$
$104$	−10816.0	−0.0980581
$105$	0	0
$106$	−44209.4	−0.382164
$107$	189220.i	1.59775i	0.601499	+	0.798874i	$0.294570\pi$
$107$	189220.i	1.59775i	−0.601499	+	0.798874i	$0.705430\pi$
$108$	40412.1i	0.333389i
$109$	−157628.	−1.27077	−0.635384	−	0.772197i	$-0.719158\pi$
$109$	−157628.	−1.27077	−0.635384	+	0.772197i	$0.719158\pi$
$110$	0	0
$111$	−106741.	−0.822287
$112$	59057.8i	0.444869i
$113$	− 200949.i	− 1.48044i	−0.672365	−	0.740219i	$-0.734722\pi$
$113$	− 200949.i	− 1.48044i	0.672365	−	0.740219i	$-0.265278\pi$
$114$	−167034.	−1.20377
$115$	0	0
$116$	−109827.	−0.757820
$117$	− 18290.7i	− 0.123528i
$118$	− 96583.7i	− 0.638556i
$119$	−127818.	−0.827417
$120$	0	0
$121$	135600.	0.841966
$122$	42727.1i	0.259898i
$123$	224840.i	1.34002i
$124$	100125.	0.584777
$125$	0	0
$126$	−99871.6	−0.560423
$127$	− 283395.i	− 1.55913i	−0.626321	−	0.779565i	$-0.715440\pi$
$127$	− 283395.i	− 1.55913i	0.626321	−	0.779565i	$-0.284560\pi$
$128$	16384.0i	0.0883883i
$129$	289577.	1.53211
$130$	0	0
$131$	−199352.	−1.01494	−0.507472	−	0.861668i	$-0.669420\pi$
$131$	−199352.	−1.01494	−0.507472	+	0.861668i	$0.669420\pi$
$132$	− 163319.i	− 0.815836i
$133$	514029.i	2.51975i
$134$	−211110.	−1.01565
$135$	0	0
$136$	−35459.6	−0.164394
$137$	265825.i	1.21003i	0.796216	+	0.605013i	$0.206832\pi$
$137$	265825.i	1.21003i	−0.796216	+	0.605013i	$0.793168\pi$
$138$	218522.i	0.976781i
$139$	−985.292	−0.00432541	−0.00216271	−	0.999998i	$-0.500688\pi$
$139$	−985.292	−0.00432541	−0.00216271	+	0.999998i	$0.500688\pi$
$140$	0	0
$141$	305073.	1.29228
$142$	176649.i	0.735176i
$143$	− 92046.9i	− 0.376417i
$144$	−27706.7	−0.111347
$145$	0	0
$146$	−167045.	−0.648562
$147$	682420.i	2.60471i
$148$	91128.7i	0.341980i
$149$	20636.5	0.0761502	0.0380751	−	0.999275i	$-0.487877\pi$
$149$	20636.5	0.0761502	0.0380751	+	0.999275i	$0.487877\pi$
$150$	0	0
$151$	31649.1	0.112959	0.0564793	−	0.998404i	$-0.482013\pi$
$151$	31649.1	0.112959	0.0564793	+	0.998404i	$0.482013\pi$
$152$	142603.i	0.500635i
$153$	− 59965.1i	− 0.207095i
$154$	−502597.	−1.70773
$155$	0	0
$156$	−50676.0	−0.166721
$157$	− 475091.i	− 1.53825i	−0.639097	−	0.769126i	$-0.720692\pi$
$157$	− 475091.i	− 1.53825i	0.639097	−	0.769126i	$-0.279308\pi$
$158$	− 238296.i	− 0.759405i
$159$	−207133.	−0.649766
$160$	0	0
$161$	672477.	2.04462
$162$	294541.i	0.881775i
$163$	− 368397.i	− 1.08604i	−0.839719	−	0.543022i	$-0.817280\pi$
$163$	− 368397.i	− 1.08604i	0.839719	−	0.543022i	$-0.182720\pi$
$164$	191955.	0.557300
$165$	0	0
$166$	90562.2	0.255080
$167$	− 446876.i	− 1.23993i	−0.784631	−	0.619963i	$-0.787148\pi$
$167$	− 446876.i	− 1.23993i	0.784631	−	0.619963i	$-0.212852\pi$
$168$	276702.i	0.756379i
$169$	−28561.0	−0.0769231
$170$	0	0
$171$	−241154.	−0.630673
$172$	− 247223.i	− 0.637189i
$173$	218403.i	0.554810i	0.960753	+	0.277405i	$0.0894743\pi$
$173$	218403.i	0.554810i	−0.960753	+	0.277405i	$0.910526\pi$
$174$	−514572.	−1.28847
$175$	0	0
$176$	−139432.	−0.339298
$177$	− 452521.i	− 1.08569i
$178$	519457.i	1.22885i
$179$	−338950.	−0.790684	−0.395342	−	0.918534i	$-0.629374\pi$
$179$	−338950.	−0.790684	−0.395342	+	0.918534i	$0.629374\pi$
$180$	0	0
$181$	−440716.	−0.999915	−0.499957	−	0.866050i	$-0.666651\pi$
$181$	−440716.	−0.999915	−0.499957	+	0.866050i	$0.666651\pi$
$182$	155950.i	0.348984i
$183$	200188.i	0.441887i
$184$	186560.	0.406233
$185$	0	0
$186$	469116.	0.994255
$187$	− 301771.i	− 0.631063i
$188$	− 260452.i	− 0.537445i
$189$	582678.	1.18652
$190$	0	0
$191$	643313.	1.27596	0.637982	−	0.770051i	$-0.279769\pi$
$191$	643313.	1.27596	0.637982	+	0.770051i	$0.279769\pi$
$192$	76763.6i	0.150280i
$193$	389679.i	0.753033i	0.926410	+	0.376517i	$0.122878\pi$
$193$	389679.i	0.753033i	−0.926410	+	0.376517i	$0.877122\pi$
$194$	−253151.	−0.482921
$195$	0	0
$196$	582608.	1.08327
$197$	− 326458.i	− 0.599323i	−0.954046	−	0.299662i	$-0.903126\pi$
$197$	− 326458.i	− 0.599323i	0.954046	−	0.299662i	$-0.0968738\pi$
$198$	− 235791.i	− 0.427429i
$199$	−433426.	−0.775858	−0.387929	−	0.921689i	$-0.626809\pi$
$199$	−433426.	−0.775858	−0.387929	+	0.921689i	$0.626809\pi$
$200$	0	0
$201$	−989107.	−1.72684
$202$	− 465830.i	− 0.803247i
$203$	1.58354e6i	2.69705i
$204$	−166138.	−0.279508
$205$	0	0
$206$	−193949.	−0.318433
$207$	315489.i	0.511751i
$208$	43264.0i	0.0693375i
$209$	−1.21359e6	−1.92179
$210$	0	0
$211$	−76110.5	−0.117690	−0.0588448	−	0.998267i	$-0.518742\pi$
$211$	−76110.5	−0.117690	−0.0588448	+	0.998267i	$0.518742\pi$
$212$	176838.i	0.270231i
$213$	827651.i	1.24997i
$214$	756881.	1.12978
$215$	0	0
$216$	161648.	0.235742
$217$	− 1.44365e6i	− 2.08120i
$218$	630510.i	0.898568i
$219$	−782652.	−1.10270
$220$	0	0
$221$	−93635.6	−0.128962
$222$	426963.i	0.581444i
$223$	− 141163.i	− 0.190090i	−0.995473	−	0.0950450i	$-0.969700\pi$
$223$	− 141163.i	− 0.190090i	0.995473	−	0.0950450i	$-0.0302995\pi$
$224$	236231.	0.314570
$225$	0	0
$226$	−803797.	−1.04683
$227$	968458.i	1.24743i	0.781652	+	0.623715i	$0.214377\pi$
$227$	968458.i	1.24743i	−0.781652	+	0.623715i	$0.785623\pi$
$228$	668136.i	0.851193i
$229$	−72887.3	−0.0918466	−0.0459233	−	0.998945i	$-0.514623\pi$
$229$	−72887.3	−0.0918466	−0.0459233	+	0.998945i	$0.514623\pi$
$230$	0	0
$231$	−2.35481e6	−2.90352
$232$	439310.i	0.535860i
$233$	613692.i	0.740561i	0.928920	+	0.370280i	$0.120738\pi$
$233$	613692.i	0.740561i	−0.928920	+	0.370280i	$0.879262\pi$
$234$	−73163.0	−0.0873477
$235$	0	0
$236$	−386335.	−0.451527
$237$	− 1.11648e6i	− 1.29116i
$238$	511272.i	0.585072i
$239$	1.21444e6	1.37525	0.687626	−	0.726065i	$-0.258653\pi$
$239$	1.21444e6	1.37525	0.687626	+	0.726065i	$0.258653\pi$
$240$	0	0
$241$	−1.78850e6	−1.98356	−0.991780	−	0.127957i	$-0.959158\pi$
$241$	−1.78850e6	−1.98356	−0.991780	+	0.127957i	$0.959158\pi$
$242$	− 542398.i	− 0.595360i
$243$	766246.i	0.832439i
$244$	170908.	0.183776
$245$	0	0
$246$	899361.	0.947537
$247$	376562.i	0.392730i
$248$	− 400502.i	− 0.413500i
$249$	424309.	0.433695
$250$	0	0
$251$	427715.	0.428519	0.214259	−	0.976777i	$-0.431266\pi$
$251$	427715.	0.428519	0.214259	+	0.976777i	$0.431266\pi$
$252$	399486.i	0.396279i
$253$	1.58768e6i	1.55941i
$254$	−1.13358e6	−1.10247
$255$	0	0
$256$	65536.0	0.0625000
$257$	271022.i	0.255960i	0.991777	+	0.127980i	$0.0408493\pi$
$257$	271022.i	0.255960i	−0.991777	+	0.127980i	$0.959151\pi$
$258$	− 1.15831e6i	− 1.08337i
$259$	1.31393e6	1.21709
$260$	0	0
$261$	−742909.	−0.675048
$262$	797407.i	0.717673i
$263$	− 1.31591e6i	− 1.17310i	−0.809912	−	0.586552i	$-0.800485\pi$
$263$	− 1.31591e6i	− 1.17310i	0.809912	−	0.586552i	$-0.199515\pi$
$264$	−653278.	−0.576883
$265$	0	0
$266$	2.05611e6	1.78174
$267$	2.43380e6i	2.08933i
$268$	844438.i	0.718176i
$269$	1.63605e6	1.37853	0.689266	−	0.724508i	$-0.257933\pi$
$269$	1.63605e6	1.37853	0.689266	+	0.724508i	$0.257933\pi$
$270$	0	0
$271$	−1.41289e6	−1.16865	−0.584326	−	0.811519i	$-0.698641\pi$
$271$	−1.41289e6	−1.16865	−0.584326	+	0.811519i	$0.698641\pi$
$272$	141839.i	0.116244i
$273$	730667.i	0.593353i
$274$	1.06330e6	0.855617
$275$	0	0
$276$	874088.	0.690689
$277$	− 755228.i	− 0.591397i	−0.955281	−	0.295698i	$-0.904448\pi$
$277$	− 755228.i	− 0.591397i	0.955281	−	0.295698i	$-0.0955523\pi$
$278$	3941.17i	0.00305853i
$279$	677282.	0.520905
$280$	0	0
$281$	−550427.	−0.415848	−0.207924	−	0.978145i	$-0.566671\pi$
$281$	−550427.	−0.415848	−0.207924	+	0.978145i	$0.566671\pi$
$282$	− 1.22029e6i	− 0.913778i
$283$	− 1.12011e6i	− 0.831367i	−0.909509	−	0.415683i	$-0.863542\pi$
$283$	− 1.12011e6i	− 0.831367i	0.909509	−	0.415683i	$-0.136458\pi$
$284$	706597.	0.519848
$285$	0	0
$286$	−368188.	−0.266167
$287$	− 2.76768e6i	− 1.98341i
$288$	110827.i	0.0787342i
$289$	1.11288e6	0.783796
$290$	0	0
$291$	−1.18608e6	−0.821076
$292$	668180.i	0.458602i
$293$	− 1.47320e6i	− 1.00252i	−0.865297	−	0.501260i	$-0.832870\pi$
$293$	− 1.47320e6i	− 1.00252i	0.865297	−	0.501260i	$-0.167130\pi$
$294$	2.72968e6	1.84181
$295$	0	0
$296$	364515.	0.241817
$297$	1.37567e6i	0.904946i
$298$	− 82546.1i	− 0.0538463i
$299$	492636.	0.318675
$300$	0	0
$301$	−3.56457e6	−2.26772
$302$	− 126596.i	− 0.0798737i
$303$	− 2.18254e6i	− 1.36570i
$304$	570414.	0.354002
$305$	0	0
$306$	−239861.	−0.146438
$307$	− 1.42452e6i	− 0.862623i	−0.902203	−	0.431312i	$-0.858051\pi$
$307$	− 1.42452e6i	− 0.862623i	0.902203	−	0.431312i	$-0.141949\pi$
$308$	2.01039e6i	1.20754i
$309$	−908703.	−0.541409
$310$	0	0
$311$	−1.79196e6	−1.05058	−0.525288	−	0.850924i	$-0.676043\pi$
$311$	−1.79196e6	−1.05058	−0.525288	+	0.850924i	$0.676043\pi$
$312$	202704.i	0.117890i
$313$	− 2.93685e6i	− 1.69442i	−0.531260	−	0.847209i	$-0.678281\pi$
$313$	− 2.93685e6i	− 1.69442i	0.531260	−	0.847209i	$-0.321719\pi$
$314$	−1.90036e6	−1.08771
$315$	0	0
$316$	−953182.	−0.536980
$317$	− 390977.i	− 0.218526i	−0.994013	−	0.109263i	$-0.965151\pi$
$317$	− 390977.i	− 0.218526i	0.994013	−	0.109263i	$-0.0348491\pi$
$318$	828533.i	0.459454i
$319$	−3.73864e6	−2.05701
$320$	0	0
$321$	3.54620e6	1.92088
$322$	− 2.68991e6i	− 1.44576i
$323$	1.23454e6i	0.658412i
$324$	1.17816e6	0.623509
$325$	0	0
$326$	−1.47359e6	−0.767949
$327$	2.95412e6i	1.52777i
$328$	− 767819.i	− 0.394071i
$329$	−3.75531e6	−1.91274
$330$	0	0
$331$	−2.78981e6	−1.39960	−0.699800	−	0.714339i	$-0.746728\pi$
$331$	−2.78981e6	−1.39960	−0.699800	+	0.714339i	$0.746728\pi$
$332$	− 362249.i	− 0.180369i
$333$	616425.i	0.304628i
$334$	−1.78750e6	−0.876760
$335$	0	0
$336$	1.10681e6	0.534841
$337$	− 2.51413e6i	− 1.20591i	−0.797776	−	0.602953i	$-0.793991\pi$
$337$	− 2.51413e6i	− 1.20591i	0.797776	−	0.602953i	$-0.206009\pi$
$338$	114244.i	0.0543928i
$339$	−3.76601e6	−1.77985
$340$	0	0
$341$	3.40837e6	1.58731
$342$	964616.i	0.445953i
$343$	− 4.52300e6i	− 2.07583i
$344$	−988893.	−0.450560
$345$	0	0
$346$	873614.	0.392310
$347$	− 310242.i	− 0.138317i	−0.997606	−	0.0691587i	$-0.977969\pi$
$347$	− 310242.i	− 0.138317i	0.997606	−	0.0691587i	$-0.0220315\pi$
$348$	2.05829e6i	0.911084i
$349$	1.17835e6	0.517859	0.258929	−	0.965896i	$-0.416630\pi$
$349$	1.17835e6	0.517859	0.258929	+	0.965896i	$0.416630\pi$
$350$	0	0
$351$	426852.	0.184931
$352$	557728.i	0.239920i
$353$	1.34357e6i	0.573885i	0.957948	+	0.286942i	$0.0926388\pi$
$353$	1.34357e6i	0.573885i	−0.957948	+	0.286942i	$0.907361\pi$
$354$	−1.81009e6	−0.767699
$355$	0	0
$356$	2.07783e6	0.868930
$357$	2.39545e6i	0.994756i
$358$	1.35580e6i	0.559098i
$359$	2.61851e6	1.07230	0.536152	−	0.844122i	$-0.319877\pi$
$359$	2.61851e6	1.07230	0.536152	+	0.844122i	$0.319877\pi$
$360$	0	0
$361$	2.48868e6	1.00508
$362$	1.76287e6i	0.707046i
$363$	− 2.54129e6i	− 1.01225i
$364$	623798.	0.246769
$365$	0	0
$366$	800753.	0.312461
$367$	− 119818.i	− 0.0464363i	−0.999730	−	0.0232182i	$-0.992609\pi$
$367$	− 119818.i	− 0.0464363i	0.999730	−	0.0232182i	$-0.00739124\pi$
$368$	− 746242.i	− 0.287250i
$369$	1.29844e6	0.496429
$370$	0	0
$371$	2.54972e6	0.961739
$372$	− 1.87646e6i	− 0.703044i
$373$	− 2.68734e6i	− 1.00012i	−0.865992	−	0.500058i	$-0.833312\pi$
$373$	− 2.68734e6i	− 1.00012i	0.865992	−	0.500058i	$-0.166688\pi$
$374$	−1.20708e6	−0.446229
$375$	0	0
$376$	−1.04181e6	−0.380031
$377$	1.16005e6i	0.420363i
$378$	− 2.33071e6i	− 0.838994i
$379$	−1.16326e6	−0.415988	−0.207994	−	0.978130i	$-0.566693\pi$
$379$	−1.16326e6	−0.415988	−0.207994	+	0.978130i	$0.566693\pi$
$380$	0	0
$381$	−5.31113e6	−1.87445
$382$	− 2.57325e6i	− 0.902243i
$383$	− 1.35360e6i	− 0.471514i	−0.971812	−	0.235757i	$-0.924243\pi$
$383$	− 1.35360e6i	− 0.471514i	0.971812	−	0.235757i	$-0.0757569\pi$
$384$	307054.	0.106264
$385$	0	0
$386$	1.55872e6	0.532475
$387$	− 1.67230e6i	− 0.567592i
$388$	1.01261e6i	0.341477i
$389$	−1.64100e6	−0.549839	−0.274920	−	0.961467i	$-0.588651\pi$
$389$	−1.64100e6	−0.549839	−0.274920	+	0.961467i	$0.588651\pi$
$390$	0	0
$391$	1.61508e6	0.534259
$392$	− 2.33043e6i	− 0.765987i
$393$	3.73607e6i	1.22021i
$394$	−1.30583e6	−0.423786
$395$	0	0
$396$	−943164.	−0.302238
$397$	1.51737e6i	0.483187i	0.970378	+	0.241593i	$0.0776700\pi$
$397$	1.51737e6i	0.483187i	−0.970378	+	0.241593i	$0.922330\pi$
$398$	1.73370e6i	0.548614i
$399$	9.63347e6	3.02936
$400$	0	0
$401$	−4.16517e6	−1.29352	−0.646758	−	0.762696i	$-0.723875\pi$
$401$	−4.16517e6	−1.29352	−0.646758	+	0.762696i	$0.723875\pi$
$402$	3.95643e6i	1.22106i
$403$	− 1.05758e6i	− 0.324376i
$404$	−1.86332e6	−0.567982
$405$	0	0
$406$	6.33415e6	1.90710
$407$	3.10212e6i	0.928265i
$408$	664553.i	0.197642i
$409$	−2.16258e6	−0.639240	−0.319620	−	0.947546i	$-0.603555\pi$
$409$	−2.16258e6	−0.639240	−0.319620	+	0.947546i	$0.603555\pi$
$410$	0	0
$411$	4.98186e6	1.45474
$412$	775794.i	0.225166i
$413$	5.57034e6i	1.60696i
$414$	1.26196e6	0.361862
$415$	0	0
$416$	173056.	0.0490290
$417$	18465.5i	0.00520020i
$418$	4.85436e6i	1.35891i
$419$	5.87037e6	1.63354	0.816772	−	0.576961i	$-0.195762\pi$
$419$	5.87037e6	1.63354	0.816772	+	0.576961i	$0.195762\pi$
$420$	0	0
$421$	−2.14426e6	−0.589621	−0.294810	−	0.955556i	$-0.595256\pi$
$421$	−2.14426e6	−0.589621	−0.294810	+	0.955556i	$0.595256\pi$
$422$	304442.i	0.0832192i
$423$	− 1.76178e6i	− 0.478742i
$424$	707350.	0.191082
$425$	0	0
$426$	3.31061e6	0.883860
$427$	− 2.46423e6i	− 0.654050i
$428$	− 3.02752e6i	− 0.798874i
$429$	−1.72506e6	−0.452545
$430$	0	0
$431$	2.56530e6	0.665189	0.332594	−	0.943070i	$-0.392076\pi$
$431$	2.56530e6	0.665189	0.332594	+	0.943070i	$0.392076\pi$
$432$	− 646593.i	− 0.166695i
$433$	− 6.07499e6i	− 1.55713i	−0.627561	−	0.778567i	$-0.715947\pi$
$433$	− 6.07499e6i	− 1.55713i	0.627561	−	0.778567i	$-0.284053\pi$
$434$	−5.77460e6	−1.47163
$435$	0	0
$436$	2.52204e6	0.635384
$437$	− 6.49515e6i	− 1.62699i
$438$	3.13061e6i	0.779729i
$439$	−7.17264e6	−1.77631	−0.888154	−	0.459547i	$-0.848012\pi$
$439$	−7.17264e6	−1.77631	−0.888154	+	0.459547i	$0.848012\pi$
$440$	0	0
$441$	3.94095e6	0.964950
$442$	374542.i	0.0911896i
$443$	− 1.55350e6i	− 0.376099i	−0.982160	−	0.188049i	$-0.939783\pi$
$443$	− 1.55350e6i	− 0.376099i	0.982160	−	0.188049i	$-0.0602165\pi$
$444$	1.70785e6	0.411143
$445$	0	0
$446$	−564653.	−0.134414
$447$	− 386751.i	− 0.0915510i
$448$	− 944925.i	− 0.222435i
$449$	3.57322e6	0.836456	0.418228	−	0.908342i	$-0.362651\pi$
$449$	3.57322e6	0.836456	0.418228	+	0.908342i	$0.362651\pi$
$450$	0	0
$451$	6.53433e6	1.51272
$452$	3.21519e6i	0.740219i
$453$	− 593139.i	− 0.135804i
$454$	3.87383e6	0.882066
$455$	0	0
$456$	2.67255e6	0.601884
$457$	8.13345e6i	1.82173i	0.412703	+	0.910866i	$0.364585\pi$
$457$	8.13345e6i	1.82173i	−0.412703	+	0.910866i	$0.635415\pi$
$458$	291549.i	0.0649454i
$459$	1.39941e6	0.310037
$460$	0	0
$461$	−3.69923e6	−0.810697	−0.405349	−	0.914162i	$-0.632850\pi$
$461$	−3.69923e6	−0.810697	−0.405349	+	0.914162i	$0.632850\pi$
$462$	9.41923e6i	2.05310i
$463$	− 1.34127e6i	− 0.290779i	−0.989375	−	0.145389i	$-0.953557\pi$
$463$	− 1.34127e6i	− 0.290779i	0.989375	−	0.145389i	$-0.0464435\pi$
$464$	1.75724e6	0.378910
$465$	0	0
$466$	2.45477e6	0.523655
$467$	− 2.41068e6i	− 0.511501i	−0.966743	−	0.255751i	$-0.917677\pi$
$467$	− 2.41068e6i	− 0.511501i	0.966743	−	0.255751i	$-0.0823226\pi$
$468$	292652.i	0.0617642i
$469$	1.21755e7	2.55596
$470$	0	0
$471$	−8.90373e6	−1.84935
$472$	1.54534e6i	0.319278i
$473$	− 8.41573e6i	− 1.72957i
$474$	−4.46592e6	−0.912989
$475$	0	0
$476$	2.04509e6	0.413708
$477$	1.19619e6i	0.240715i
$478$	− 4.85777e6i	− 0.972450i
$479$	−18669.1	−0.00371779	−0.00185890	−	0.999998i	$-0.500592\pi$
$479$	−18669.1	−0.00371779	−0.00185890	+	0.999998i	$0.500592\pi$
$480$	0	0
$481$	962547.	0.189697
$482$	7.15398e6i	1.40259i
$483$	− 1.26030e7i	− 2.45813i
$484$	−2.16959e6	−0.420983
$485$	0	0
$486$	3.06498e6	0.588623
$487$	4.53165e6i	0.865832i	0.901434	+	0.432916i	$0.142515\pi$
$487$	4.53165e6i	0.865832i	−0.901434	+	0.432916i	$0.857485\pi$
$488$	− 683633.i	− 0.129949i
$489$	−6.90417e6	−1.30569
$490$	0	0
$491$	3.84989e6	0.720684	0.360342	−	0.932820i	$-0.382660\pi$
$491$	3.84989e6	0.720684	0.360342	+	0.932820i	$0.382660\pi$
$492$	− 3.59744e6i	− 0.670010i
$493$	3.80317e6i	0.704739i
$494$	1.50625e6	0.277702
$495$	0	0
$496$	−1.60201e6	−0.292389
$497$	− 1.01880e7i	− 1.85011i
$498$	− 1.69724e6i	− 0.306668i
$499$	−2.18845e6	−0.393445	−0.196723	−	0.980459i	$-0.563030\pi$
$499$	−2.18845e6	−0.393445	−0.196723	+	0.980459i	$0.563030\pi$
$500$	0	0
$501$	−8.37495e6	−1.49069
$502$	− 1.71086e6i	− 0.303009i
$503$	1.37502e6i	0.242319i	0.992633	+	0.121160i	$0.0386613\pi$
$503$	1.37502e6i	0.242319i	−0.992633	+	0.121160i	$0.961339\pi$
$504$	1.59795e6	0.280211
$505$	0	0
$506$	6.35071e6	1.10267
$507$	535265.i	0.0924802i
$508$	4.53431e6i	0.779565i
$509$	−7.64261e6	−1.30752	−0.653759	−	0.756703i	$-0.726809\pi$
$509$	−7.64261e6	−1.30752	−0.653759	+	0.756703i	$0.726809\pi$
$510$	0	0
$511$	9.63410e6	1.63215
$512$	− 262144.i	− 0.0441942i
$513$	− 5.62783e6i	− 0.944164i
$514$	1.08409e6	0.180991
$515$	0	0
$516$	−4.63324e6	−0.766055
$517$	− 8.86606e6i	− 1.45883i
$518$	− 5.25573e6i	− 0.860614i
$519$	4.09312e6	0.667016
$520$	0	0
$521$	1.17487e7	1.89625	0.948123	−	0.317903i	$-0.102979\pi$
$521$	1.17487e7	1.89625	0.948123	+	0.317903i	$0.102979\pi$
$522$	2.97164e6i	0.477331i
$523$	5.40309e6i	0.863750i	0.901933	+	0.431875i	$0.142148\pi$
$523$	5.40309e6i	0.863750i	−0.901933	+	0.431875i	$0.857852\pi$
$524$	3.18963e6	0.507472
$525$	0	0
$526$	−5.26363e6	−0.829510
$527$	− 3.46720e6i	− 0.543817i
$528$	2.61311e6i	0.407918i
$529$	−2.06092e6	−0.320201
$530$	0	0
$531$	−2.61330e6	−0.402209
$532$	− 8.22446e6i	− 1.25988i
$533$	− 2.02752e6i	− 0.309134i
$534$	9.73520e6	1.47738
$535$	0	0
$536$	3.37775e6	0.507827
$537$	6.35230e6i	0.950594i
$538$	− 6.54422e6i	− 0.974769i
$539$	1.98326e7	2.94041
$540$	0	0
$541$	−1.33517e7	−1.96129	−0.980646	−	0.195789i	$-0.937273\pi$
$541$	−1.33517e7	−1.96129	−0.980646	+	0.195789i	$0.937273\pi$
$542$	5.65156e6i	0.826362i
$543$	8.25952e6i	1.20214i
$544$	567354.	0.0821972
$545$	0	0
$546$	2.92267e6	0.419564
$547$	− 2.40834e6i	− 0.344152i	−0.985084	−	0.172076i	$-0.944953\pi$
$547$	− 2.40834e6i	− 0.344152i	0.985084	−	0.172076i	$-0.0550475\pi$
$548$	− 4.25320e6i	− 0.605013i
$549$	1.15608e6	0.163703
$550$	0	0
$551$	1.52947e7	2.14616
$552$	− 3.49635e6i	− 0.488391i
$553$	1.37434e7i	1.91109i
$554$	−3.02091e6	−0.418180
$555$	0	0
$556$	15764.7	0.00216271
$557$	1.52042e6i	0.207647i	0.994596	+	0.103823i	$0.0331076\pi$
$557$	1.52042e6i	0.207647i	−0.994596	+	0.103823i	$0.966892\pi$
$558$	− 2.70913e6i	− 0.368336i
$559$	−2.61129e6	−0.353449
$560$	0	0
$561$	−5.65551e6	−0.758691
$562$	2.20171e6i	0.294049i
$563$	− 9.15496e6i	− 1.21727i	−0.793452	−	0.608633i	$-0.791718\pi$
$563$	− 9.15496e6i	− 1.21727i	0.793452	−	0.608633i	$-0.208282\pi$
$564$	−4.88117e6	−0.646139
$565$	0	0
$566$	−4.48042e6	−0.587865
$567$	− 1.69872e7i	− 2.21904i
$568$	− 2.82639e6i	− 0.367588i
$569$	−1.18973e7	−1.54052	−0.770262	−	0.637727i	$-0.779875\pi$
$569$	−1.18973e7	−1.54052	−0.770262	+	0.637727i	$0.779875\pi$
$570$	0	0
$571$	1.20642e7	1.54849	0.774245	−	0.632886i	$-0.218130\pi$
$571$	1.20642e7	1.54849	0.774245	+	0.632886i	$0.218130\pi$
$572$	1.47275e6i	0.188208i
$573$	− 1.20564e7i	− 1.53402i
$574$	−1.10707e7	−1.40248
$575$	0	0
$576$	443307.	0.0556735
$577$	1.38845e6i	0.173616i	0.996225	+	0.0868081i	$0.0276667\pi$
$577$	1.38845e6i	0.173616i	−0.996225	+	0.0868081i	$0.972333\pi$
$578$	− 4.45151e6i	− 0.554227i
$579$	7.30303e6	0.905329
$580$	0	0
$581$	−5.22305e6	−0.641925
$582$	4.74434e6i	0.580588i
$583$	6.01973e6i	0.733509i
$584$	2.67272e6	0.324281
$585$	0	0
$586$	−5.89280e6	−0.708888
$587$	− 5.80067e6i	− 0.694836i	−0.937710	−	0.347418i	$-0.887058\pi$
$587$	− 5.80067e6i	− 0.694836i	0.937710	−	0.347418i	$-0.112942\pi$
$588$	− 1.09187e7i	− 1.30235i
$589$	−1.39436e7	−1.65610
$590$	0	0
$591$	−6.11818e6	−0.720532
$592$	− 1.45806e6i	− 0.170990i
$593$	− 1.06130e7i	− 1.23937i	−0.784852	−	0.619684i	$-0.787261\pi$
$593$	− 1.06130e7i	− 1.23937i	0.784852	−	0.619684i	$-0.212739\pi$
$594$	5.50267e6	0.639893
$595$	0	0
$596$	−330184.	−0.0380751
$597$	8.12288e6i	0.932770i
$598$	− 1.97054e6i	− 0.225337i
$599$	4.68069e6	0.533020	0.266510	−	0.963832i	$-0.414130\pi$
$599$	4.68069e6	0.533020	0.266510	+	0.963832i	$0.414130\pi$
$600$	0	0
$601$	2.65345e6	0.299657	0.149828	−	0.988712i	$-0.452128\pi$
$601$	2.65345e6	0.299657	0.149828	+	0.988712i	$0.452128\pi$
$602$	1.42583e7i	1.60352i
$603$	5.71206e6i	0.639734i
$604$	−506386.	−0.0564793
$605$	0	0
$606$	−8.73018e6	−0.965698
$607$	6.58092e6i	0.724962i	0.931991	+	0.362481i	$0.118070\pi$
$607$	6.58092e6i	0.724962i	−0.931991	+	0.362481i	$0.881930\pi$
$608$	− 2.28165e6i	− 0.250317i
$609$	2.96773e7	3.24251
$610$	0	0
$611$	−2.75103e6	−0.298121
$612$	959442.i	0.103548i
$613$	− 8.28523e6i	− 0.890540i	−0.895396	−	0.445270i	$-0.853108\pi$
$613$	− 8.28523e6i	− 0.890540i	0.895396	−	0.445270i	$-0.146892\pi$
$614$	−5.69806e6	−0.609967
$615$	0	0
$616$	8.04155e6	0.853863
$617$	− 5.42747e6i	− 0.573964i	−0.957936	−	0.286982i	$-0.907348\pi$
$617$	− 5.42747e6i	− 0.573964i	0.957936	−	0.286982i	$-0.0926520\pi$
$618$	3.63481e6i	0.382834i
$619$	8.73793e6	0.916604	0.458302	−	0.888796i	$-0.348458\pi$
$619$	8.73793e6	0.916604	0.458302	+	0.888796i	$0.348458\pi$
$620$	0	0
$621$	−7.36259e6	−0.766129
$622$	7.16785e6i	0.742870i
$623$	− 2.99590e7i	− 3.09248i
$624$	810815.	0.0833605
$625$	0	0
$626$	−1.17474e7	−1.19813
$627$	2.27440e7i	2.31046i
$628$	7.60145e6i	0.769126i
$629$	3.15566e6	0.318026
$630$	0	0
$631$	−6.93247e6	−0.693130	−0.346565	−	0.938026i	$-0.612652\pi$
$631$	−6.93247e6	−0.693130	−0.346565	+	0.938026i	$0.612652\pi$
$632$	3.81273e6i	0.379702i
$633$	1.42639e6i	0.141492i
$634$	−1.56391e6	−0.154521
$635$	0	0
$636$	3.31413e6	0.324883
$637$	− 6.15380e6i	− 0.600890i
$638$	1.49546e7i	1.45453i
$639$	4.77966e6	0.463068
$640$	0	0
$641$	−9.65973e6	−0.928581	−0.464291	−	0.885683i	$-0.653691\pi$
$641$	−9.65973e6	−0.928581	−0.464291	+	0.885683i	$0.653691\pi$
$642$	− 1.41848e7i	− 1.35827i
$643$	588572.i	0.0561399i	0.999606	+	0.0280700i	$0.00893612\pi$
$643$	588572.i	0.0561399i	−0.999606	+	0.0280700i	$0.991064\pi$
$644$	−1.07596e7	−1.02231
$645$	0	0
$646$	4.93815e6	0.465568
$647$	7.53056e6i	0.707240i	0.935389	+	0.353620i	$0.115049\pi$
$647$	7.53056e6i	0.707240i	−0.935389	+	0.353620i	$0.884951\pi$
$648$	− 4.71265e6i	− 0.440887i
$649$	−1.31512e7	−1.22562
$650$	0	0
$651$	−2.70556e7	−2.50210
$652$	5.89436e6i	0.543022i
$653$	9.16834e6i	0.841410i	0.907197	+	0.420705i	$0.138217\pi$
$653$	9.16834e6i	0.841410i	−0.907197	+	0.420705i	$0.861783\pi$
$654$	1.18165e7	1.08030
$655$	0	0
$656$	−3.07127e6	−0.278650
$657$	4.51979e6i	0.408512i
$658$	1.50212e7i	1.35251i
$659$	1.25416e7	1.12496	0.562482	−	0.826809i	$-0.309847\pi$
$659$	1.25416e7	1.12496	0.562482	+	0.826809i	$0.309847\pi$
$660$	0	0
$661$	−2.80510e6	−0.249715	−0.124858	−	0.992175i	$-0.539847\pi$
$661$	−2.80510e6	−0.249715	−0.124858	+	0.992175i	$0.539847\pi$
$662$	1.11592e7i	0.989667i
$663$	1.75483e6i	0.155043i
$664$	−1.44900e6	−0.127540
$665$	0	0
$666$	2.46570e6	0.215404
$667$	− 2.00092e7i	− 1.74147i
$668$	7.15001e6i	0.619963i
$669$	−2.64556e6	−0.228534
$670$	0	0
$671$	5.81789e6	0.498838
$672$	− 4.42724e6i	− 0.378190i
$673$	1.88930e7i	1.60792i	0.594686	+	0.803958i	$0.297277\pi$
$673$	1.88930e7i	1.60792i	−0.594686	+	0.803958i	$0.702723\pi$
$674$	−1.00565e7	−0.852705
$675$	0	0
$676$	456976.	0.0384615
$677$	− 7.43678e6i	− 0.623610i	−0.950146	−	0.311805i	$-0.899066\pi$
$677$	− 7.43678e6i	− 0.623610i	0.950146	−	0.311805i	$-0.100934\pi$
$678$	1.50641e7i	1.25854i
$679$	1.46002e7	1.21530
$680$	0	0
$681$	1.81500e7	1.49971
$682$	− 1.36335e7i	− 1.12240i
$683$	1.01428e7i	0.831967i	0.909372	+	0.415984i	$0.136563\pi$
$683$	1.01428e7i	0.831967i	−0.909372	+	0.415984i	$0.863437\pi$
$684$	3.85846e6	0.315336
$685$	0	0
$686$	−1.80920e7	−1.46783
$687$	1.36599e6i	0.110422i
$688$	3.95557e6i	0.318594i
$689$	1.86785e6	0.149897
$690$	0	0
$691$	−2.15408e6	−0.171620	−0.0858098	−	0.996312i	$-0.527348\pi$
$691$	−2.15408e6	−0.171620	−0.0858098	+	0.996312i	$0.527348\pi$
$692$	− 3.49445e6i	− 0.277405i
$693$	1.35989e7i	1.07565i
$694$	−1.24097e6	−0.0978051
$695$	0	0
$696$	8.23316e6	0.644234
$697$	− 6.64711e6i	− 0.518264i
$698$	− 4.71341e6i	− 0.366181i
$699$	1.15013e7	0.890334
$700$	0	0
$701$	−6.90173e6	−0.530472	−0.265236	−	0.964183i	$-0.585450\pi$
$701$	−6.90173e6	−0.530472	−0.265236	+	0.964183i	$0.585450\pi$
$702$	− 1.70741e6i	− 0.130766i
$703$	− 1.26907e7i	− 0.968494i
$704$	2.23091e6	0.169649
$705$	0	0
$706$	5.37429e6	0.405798
$707$	2.68661e7i	2.02142i
$708$	7.24034e6i	0.542845i
$709$	−1.62641e7	−1.21511	−0.607555	−	0.794278i	$-0.707850\pi$
$709$	−1.62641e7	−1.21511	−0.607555	+	0.794278i	$0.707850\pi$
$710$	0	0
$711$	−6.44764e6	−0.478329
$712$	− 8.31131e6i	− 0.614426i
$713$	1.82417e7i	1.34382i
$714$	9.58180e6	0.703399
$715$	0	0
$716$	5.42320e6	0.395342
$717$	− 2.27600e7i	− 1.65339i
$718$	− 1.04740e7i	− 0.758233i
$719$	6.59643e6	0.475868	0.237934	−	0.971281i	$-0.423530\pi$
$719$	6.59643e6	0.475868	0.237934	+	0.971281i	$0.423530\pi$
$720$	0	0
$721$	1.11857e7	0.801357
$722$	− 9.95471e6i	− 0.710699i
$723$	3.35184e7i	2.38472i
$724$	7.05146e6	0.499957
$725$	0	0
$726$	−1.01651e7	−0.715767
$727$	2.19325e7i	1.53905i	0.638620	+	0.769523i	$0.279506\pi$
$727$	2.19325e7i	1.53905i	−0.638620	+	0.769523i	$0.720494\pi$
$728$	− 2.49519e6i	− 0.174492i
$729$	−3.53304e6	−0.246223
$730$	0	0
$731$	−8.56098e6	−0.592557
$732$	− 3.20301e6i	− 0.220943i
$733$	− 1.67003e7i	− 1.14806i	−0.818836	−	0.574028i	$-0.805380\pi$
$733$	− 1.67003e7i	− 1.14806i	0.818836	−	0.574028i	$-0.194620\pi$
$734$	−479273.	−0.0328355
$735$	0	0
$736$	−2.98497e6	−0.203116
$737$	2.87455e7i	1.94940i
$738$	− 5.19378e6i	− 0.351028i
$739$	1.69875e6	0.114425	0.0572123	−	0.998362i	$-0.481779\pi$
$739$	1.69875e6	0.114425	0.0572123	+	0.998362i	$0.481779\pi$
$740$	0	0
$741$	7.05719e6	0.472157
$742$	− 1.01989e7i	− 0.680052i
$743$	1.82334e7i	1.21170i	0.795578	+	0.605851i	$0.207167\pi$
$743$	1.82334e7i	1.21170i	−0.795578	+	0.605851i	$0.792833\pi$
$744$	−7.50585e6	−0.497127
$745$	0	0
$746$	−1.07493e7	−0.707188
$747$	− 2.45037e6i	− 0.160668i
$748$	4.82833e6i	0.315532i
$749$	−4.36521e7	−2.84315
$750$	0	0
$751$	−1.47109e6	−0.0951784	−0.0475892	−	0.998867i	$-0.515154\pi$
$751$	−1.47109e6	−0.0951784	−0.0475892	+	0.998867i	$0.515154\pi$
$752$	4.16724e6i	0.268722i
$753$	− 8.01585e6i	− 0.515184i
$754$	4.64021e6	0.297241
$755$	0	0
$756$	−9.32285e6	−0.593259
$757$	− 9.68714e6i	− 0.614407i	−0.951644	−	0.307203i	$-0.900607\pi$
$757$	− 9.68714e6i	− 0.614407i	0.951644	−	0.307203i	$-0.0993932\pi$
$758$	4.65306e6i	0.294148i
$759$	2.97548e7	1.87479
$760$	0	0
$761$	1.09169e7	0.683338	0.341669	−	0.939820i	$-0.389008\pi$
$761$	1.09169e7	0.683338	0.341669	+	0.939820i	$0.389008\pi$
$762$	2.12445e7i	1.32544i
$763$	− 3.63638e7i	− 2.26130i
$764$	−1.02930e7	−0.637982
$765$	0	0
$766$	−5.41441e6	−0.333411
$767$	4.08066e6i	0.250462i
$768$	− 1.22822e6i	− 0.0751402i
$769$	1.70886e7	1.04205	0.521027	−	0.853540i	$-0.325549\pi$
$769$	1.70886e7	1.04205	0.521027	+	0.853540i	$0.325549\pi$
$770$	0	0
$771$	5.07925e6	0.307726
$772$	− 6.23487e6i	− 0.376517i
$773$	5.92288e6i	0.356520i	0.983983	+	0.178260i	$0.0570468\pi$
$773$	5.92288e6i	0.356520i	−0.983983	+	0.178260i	$0.942953\pi$
$774$	−6.68919e6	−0.401348
$775$	0	0
$776$	4.05042e6	0.241461
$777$	− 2.46245e7i	− 1.46324i
$778$	6.56402e6i	0.388795i
$779$	−2.67318e7	−1.57828
$780$	0	0
$781$	2.40533e7	1.41107
$782$	− 6.46032e6i	− 0.377778i
$783$	− 1.73373e7i	− 1.01060i
$784$	−9.32173e6	−0.541635
$785$	0	0
$786$	1.49443e7	0.862818
$787$	− 2.75022e7i	− 1.58282i	−0.611288	−	0.791409i	$-0.709348\pi$
$787$	− 2.75022e7i	− 1.58282i	0.611288	−	0.791409i	$-0.290652\pi$
$788$	5.22332e6i	0.299662i
$789$	−2.46616e7	−1.41036
$790$	0	0
$791$	4.63579e7	2.63441
$792$	3.77265e6i	0.213715i
$793$	− 1.80522e6i	− 0.101941i
$794$	6.06947e6	0.341664
$795$	0	0
$796$	6.93481e6	0.387929
$797$	1.29045e7i	0.719606i	0.933028	+	0.359803i	$0.117156\pi$
$797$	1.29045e7i	0.719606i	−0.933028	+	0.359803i	$0.882844\pi$
$798$	− 3.85339e7i	− 2.14208i
$799$	−9.01909e6	−0.499799
$800$	0	0
$801$	1.40551e7	0.774021
$802$	1.66607e7i	0.914653i
$803$	2.27455e7i	1.24482i
$804$	1.58257e7	0.863422
$805$	0	0
$806$	−4.23030e6	−0.229369
$807$	− 3.06615e7i	− 1.65733i
$808$	7.45329e6i	0.401624i
$809$	−2.16172e7	−1.16126	−0.580628	−	0.814169i	$-0.697193\pi$
$809$	−2.16172e7	−1.16126	−0.580628	+	0.814169i	$0.697193\pi$
$810$	0	0
$811$	−1.68933e7	−0.901906	−0.450953	−	0.892548i	$-0.648916\pi$
$811$	−1.68933e7	−0.901906	−0.450953	+	0.892548i	$0.648916\pi$
$812$	− 2.53366e7i	− 1.34852i
$813$	2.64791e7i	1.40500i
$814$	1.24085e7	0.656382
$815$	0	0
$816$	2.65821e6	0.139754
$817$	3.44286e7i	1.80453i
$818$	8.65032e6i	0.452011i
$819$	4.21958e6	0.219816
$820$	0	0
$821$	−7.06381e6	−0.365747	−0.182874	−	0.983136i	$-0.558540\pi$
$821$	−7.06381e6	−0.365747	−0.182874	+	0.983136i	$0.558540\pi$
$822$	− 1.99274e7i	− 1.02866i
$823$	3.32953e7i	1.71350i	0.515735	+	0.856748i	$0.327519\pi$
$823$	3.32953e7i	1.71350i	−0.515735	+	0.856748i	$0.672481\pi$
$824$	3.10318e6	0.159217
$825$	0	0
$826$	2.22813e7	1.13630
$827$	1.49348e7i	0.759341i	0.925122	+	0.379670i	$0.123963\pi$
$827$	1.49348e7i	0.759341i	−0.925122	+	0.379670i	$0.876037\pi$
$828$	− 5.04782e6i	− 0.255875i
$829$	−1.10426e7	−0.558065	−0.279032	−	0.960282i	$-0.590014\pi$
$829$	−1.10426e7	−0.558065	−0.279032	+	0.960282i	$0.590014\pi$
$830$	0	0
$831$	−1.41538e7	−0.711002
$832$	− 692224.i	− 0.0346688i
$833$	− 2.01749e7i	− 1.00739i
$834$	73861.8	0.00367710
$835$	0	0
$836$	1.94175e7	0.960897
$837$	1.58058e7i	0.779834i
$838$	− 2.34815e7i	− 1.15509i
$839$	−2.86322e7	−1.40427	−0.702133	−	0.712046i	$-0.747769\pi$
$839$	−2.86322e7	−1.40427	−0.702133	+	0.712046i	$0.747769\pi$
$840$	0	0
$841$	2.66063e7	1.29717
$842$	8.57705e6i	0.416925i
$843$	1.03156e7i	0.499950i
$844$	1.21777e6	0.0588448
$845$	0	0
$846$	−7.04714e6	−0.338522
$847$	3.12821e7i	1.49826i
$848$	− 2.82940e6i	− 0.135115i
$849$	−2.09920e7	−0.999505
$850$	0	0
$851$	−1.66026e7	−0.785871
$852$	− 1.32424e7i	− 0.624983i
$853$	1.71429e7i	0.806700i	0.915046	+	0.403350i	$0.132154\pi$
$853$	1.71429e7i	0.806700i	−0.915046	+	0.403350i	$0.867846\pi$
$854$	−9.85691e6	−0.462483
$855$	0	0
$856$	−1.21101e7	−0.564889
$857$	− 132094.i	− 0.00614373i	−0.999995	−	0.00307186i	$-0.999022\pi$
$857$	− 132094.i	− 0.00614373i	0.999995	−	0.00307186i	$-0.000977806\pi$
$858$	6.90025e6i	0.319997i
$859$	−3.33727e7	−1.54315	−0.771577	−	0.636136i	$-0.780532\pi$
$859$	−3.33727e7	−1.54315	−0.771577	+	0.636136i	$0.780532\pi$
$860$	0	0
$861$	−5.18694e7	−2.38453
$862$	− 1.02612e7i	− 0.470359i
$863$	1.01919e7i	0.465829i	0.972497	+	0.232915i	$0.0748263\pi$
$863$	1.01919e7i	0.465829i	−0.972497	+	0.232915i	$0.925174\pi$
$864$	−2.58637e6	−0.117871
$865$	0	0
$866$	−2.43000e7	−1.10106
$867$	− 2.08566e7i	− 0.942313i
$868$	2.30984e7i	1.04060i
$869$	−3.24473e7	−1.45757
$870$	0	0
$871$	8.91938e6	0.398372
$872$	− 1.00882e7i	− 0.449284i
$873$	6.84960e6i	0.304179i
$874$	−2.59806e7	−1.15046
$875$	0	0
$876$	1.25224e7	0.551352
$877$	− 1.53857e6i	− 0.0675490i	−0.999429	−	0.0337745i	$-0.989247\pi$
$877$	− 1.53857e6i	− 0.0675490i	0.999429	−	0.0337745i	$-0.0107528\pi$
$878$	2.86906e7i	1.25604i
$879$	−2.76094e7	−1.20527
$880$	0	0
$881$	−5.38590e6	−0.233786	−0.116893	−	0.993145i	$-0.537293\pi$
$881$	−5.38590e6	−0.233786	−0.116893	+	0.993145i	$0.537293\pi$
$882$	− 1.57638e7i	− 0.682323i
$883$	− 3.24054e7i	− 1.39867i	−0.714793	−	0.699336i	$-0.753479\pi$
$883$	− 3.24054e7i	− 1.39867i	0.714793	−	0.699336i	$-0.246521\pi$
$884$	1.49817e6	0.0644808
$885$	0	0
$886$	−6.21400e6	−0.265942
$887$	3.98506e7i	1.70069i	0.526224	+	0.850346i	$0.323607\pi$
$887$	3.98506e7i	1.70069i	−0.526224	+	0.850346i	$0.676393\pi$
$888$	− 6.83142e6i	− 0.290722i
$889$	6.53776e7	2.77444
$890$	0	0
$891$	4.01058e7	1.69244
$892$	2.25861e6i	0.0950450i
$893$	3.62709e7i	1.52205i
$894$	−1.54701e6	−0.0647363
$895$	0	0
$896$	−3.77970e6	−0.157285
$897$	− 9.23255e6i	− 0.383125i
$898$	− 1.42929e7i	− 0.591464i
$899$	−4.29552e7	−1.77262
$900$	0	0
$901$	6.12363e6	0.251303
$902$	− 2.61373e7i	− 1.06966i
$903$	6.68039e7i	2.72636i
$904$	1.28608e7	0.523414
$905$	0	0
$906$	−2.37256e6	−0.0960276
$907$	494733.i	0.0199688i	0.999950	+	0.00998441i	$0.00317819\pi$
$907$	494733.i	0.0199688i	−0.999950	+	0.00998441i	$0.996822\pi$
$908$	− 1.54953e7i	− 0.623715i
$909$	−1.26041e7	−0.505944
$910$	0	0
$911$	1.03084e7	0.411526	0.205763	−	0.978602i	$-0.434032\pi$
$911$	1.03084e7	0.411526	0.205763	+	0.978602i	$0.434032\pi$
$912$	− 1.06902e7i	− 0.425597i
$913$	− 1.23313e7i	− 0.489590i
$914$	3.25338e7	1.28816
$915$	0	0
$916$	1.16620e6	0.0459233
$917$	− 4.59894e7i	− 1.80607i
$918$	− 5.59764e6i	− 0.219229i
$919$	−3.06006e7	−1.19520	−0.597602	−	0.801793i	$-0.703880\pi$
$919$	−3.06006e7	−1.19520	−0.597602	+	0.801793i	$0.703880\pi$
$920$	0	0
$921$	−2.66970e7	−1.03708
$922$	1.47969e7i	0.573250i
$923$	− 7.46344e6i	− 0.288360i
$924$	3.76769e7	1.45176
$925$	0	0
$926$	−5.36507e6	−0.205612
$927$	5.24772e6i	0.200573i
$928$	− 7.02896e6i	− 0.267930i
$929$	4.45717e7	1.69441	0.847207	−	0.531263i	$-0.178282\pi$
$929$	4.45717e7	1.69441	0.847207	+	0.531263i	$0.178282\pi$
$930$	0	0
$931$	−8.11347e7	−3.06784
$932$	− 9.81907e6i	− 0.370280i
$933$	3.35833e7i	1.26305i
$934$	−9.64271e6	−0.361686
$935$	0	0
$936$	1.17061e6	0.0436739
$937$	− 4.43114e7i	− 1.64879i	−0.566012	−	0.824397i	$-0.691515\pi$
$937$	− 4.43114e7i	− 1.64879i	0.566012	−	0.824397i	$-0.308485\pi$
$938$	− 4.87019e7i	− 1.80733i
$939$	−5.50398e7	−2.03710
$940$	0	0
$941$	1.48172e7	0.545498	0.272749	−	0.962085i	$-0.412067\pi$
$941$	1.48172e7	0.545498	0.272749	+	0.962085i	$0.412067\pi$
$942$	3.56149e7i	1.30769i
$943$	3.49718e7i	1.28068i
$944$	6.18136e6	0.225764
$945$	0	0
$946$	−3.36629e7	−1.22299
$947$	1.14497e7i	0.414876i	0.978248	+	0.207438i	$0.0665125\pi$
$947$	1.14497e7i	0.414876i	−0.978248	+	0.207438i	$0.933487\pi$
$948$	1.78637e7i	0.645581i
$949$	7.05765e6	0.254387
$950$	0	0
$951$	−7.32734e6	−0.262721
$952$	− 8.18035e6i	− 0.292536i
$953$	− 3.21943e7i	− 1.14828i	−0.818758	−	0.574139i	$-0.805337\pi$
$953$	− 3.21943e7i	− 1.14828i	0.818758	−	0.574139i	$-0.194663\pi$
$954$	4.78475e6	0.170211
$955$	0	0
$956$	−1.94311e7	−0.687626
$957$	7.00663e7i	2.47303i
$958$	74676.5i	0.00262888i
$959$	−6.13244e7	−2.15321
$960$	0	0
$961$	1.05315e7	0.367858
$962$	− 3.85019e6i	− 0.134136i
$963$	− 2.04792e7i	− 0.711617i
$964$	2.86159e7	0.991780
$965$	0	0
$966$	−5.04118e7	−1.73816
$967$	− 4.07255e7i	− 1.40055i	−0.713871	−	0.700277i	$-0.753060\pi$
$967$	− 4.07255e7i	− 1.40055i	0.713871	−	0.700277i	$-0.246940\pi$
$968$	8.67837e6i	0.297680i
$969$	2.31366e7	0.791571
$970$	0	0
$971$	1.63928e7	0.557961	0.278980	−	0.960297i	$-0.410004\pi$
$971$	1.63928e7	0.557961	0.278980	+	0.960297i	$0.410004\pi$
$972$	− 1.22599e7i	− 0.416220i
$973$	− 227301.i	− 0.00769698i
$974$	1.81266e7	0.612236
$975$	0	0
$976$	−2.73453e6	−0.0918880
$977$	− 4.03891e7i	− 1.35372i	−0.736114	−	0.676858i	$-0.763341\pi$
$977$	− 4.03891e7i	− 1.35372i	0.736114	−	0.676858i	$-0.236659\pi$
$978$	2.76167e7i	0.923261i
$979$	7.07314e7	2.35861
$980$	0	0
$981$	1.70599e7	0.565984
$982$	− 1.53996e7i	− 0.509601i
$983$	4.46121e7i	1.47254i	0.676685	+	0.736272i	$0.263416\pi$
$983$	4.46121e7i	1.47254i	−0.676685	+	0.736272i	$0.736584\pi$
$984$	−1.43898e7	−0.473769
$985$	0	0
$986$	1.52127e7	0.498325
$987$	7.03787e7i	2.29958i
$988$	− 6.02499e6i	− 0.196365i
$989$	4.50411e7	1.46426
$990$	0	0
$991$	−2.35070e7	−0.760351	−0.380175	−	0.924914i	$-0.624136\pi$
$991$	−2.35070e7	−0.760351	−0.380175	+	0.924914i	$0.624136\pi$
$992$	6.40803e6i	0.206750i
$993$	5.22841e7i	1.68266i
$994$	−4.07521e7	−1.30823
$995$	0	0
$996$	−6.78895e6	−0.216847
$997$	− 1.84154e6i	− 0.0586737i	−0.999570	−	0.0293369i	$-0.990660\pi$
$997$	− 1.84154e6i	− 0.0586737i	0.999570	−	0.0293369i	$-0.00933955\pi$
$998$	8.75378e6i	0.278208i
$999$	−1.43855e7	−0.456050

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	650.6.b.o.599.2		12
5.2	odd	4		650.6.a.t.1.2	yes	6
5.3	odd	4		650.6.a.s.1.5	✓	6
5.4	even	2	inner	650.6.b.o.599.11		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
650.6.a.s.1.5	✓	6	5.3	odd	4
650.6.a.t.1.2	yes	6	5.2	odd	4
650.6.b.o.599.2		12	1.1	even	1	trivial
650.6.b.o.599.11		12	5.4	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:	\( N \)	\(=\)	\( 650 = 2 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	650.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-4.00000i q^{2} -18.7411i q^{3} -16.0000 q^{4} -74.9644 q^{6} +230.695i q^{7} +64.0000i q^{8} -108.229 q^{9} -544.656 q^{11} +299.858i q^{12} +169.000i q^{13} +922.779 q^{14} +256.000 q^{16} +554.057i q^{17} +432.917i q^{18} +2228.18 q^{19} +4323.47 q^{21} +2178.63i q^{22} -2915.01i q^{23} +1199.43 q^{24} +676.000 q^{26} -2525.75i q^{27} -3691.11i q^{28} +6864.22 q^{29} -6257.84 q^{31} -1024.00i q^{32} +10207.5i q^{33} +2216.23 q^{34} +1731.67 q^{36} -5695.55i q^{37} -8912.71i q^{38} +3167.25 q^{39} -11997.2 q^{41} -17293.9i q^{42} +15451.4i q^{43} +8714.50 q^{44} -11660.0 q^{46} +16278.3i q^{47} -4797.72i q^{48} -36413.0 q^{49} +10383.6 q^{51} -2704.00i q^{52} -11052.3i q^{53} -10103.0 q^{54} -14764.5 q^{56} -41758.5i q^{57} -27456.9i q^{58} +24145.9 q^{59} -10681.8 q^{61} +25031.4i q^{62} -24967.9i q^{63} -4096.00 q^{64} +40829.9 q^{66} -52777.4i q^{67} -8864.91i q^{68} -54630.5 q^{69} -44162.3 q^{71} -6926.67i q^{72} -41761.3i q^{73} -22782.2 q^{74} -35650.8 q^{76} -125649. i q^{77} -12669.0i q^{78} +59573.9 q^{79} -73635.1 q^{81} +47988.7i q^{82} +22640.6i q^{83} -69175.6 q^{84} +61805.8 q^{86} -128643. i q^{87} -34858.0i q^{88} -129864. q^{89} -38987.4 q^{91} +46640.1i q^{92} +117279. i q^{93} +65113.1 q^{94} -19190.9 q^{96} -63287.9i q^{97} +145652. i q^{98} +58947.7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 192 q^{4} + 72 q^{6} - 1258 q^{9} + 740 q^{11} + 504 q^{14} + 3072 q^{16} + 3318 q^{19} + 8644 q^{21} - 1152 q^{24} + 8112 q^{26} + 11822 q^{29} + 2806 q^{31} - 6704 q^{34} + 20128 q^{36} - 3042 q^{39}+ \cdots - 772238 q^{99}+O(q^{100}) \) 12 * q - 192 * q^4 + 72 * q^6 - 1258 * q^9 + 740 * q^11 + 504 * q^14 + 3072 * q^16 + 3318 * q^19 + 8644 * q^21 - 1152 * q^24 + 8112 * q^26 + 11822 * q^29 + 2806 * q^31 - 6704 * q^34 + 20128 * q^36 - 3042 * q^39 + 2974 * q^41 - 11840 * q^44 + 12528 * q^46 - 57954 * q^49 + 14106 * q^51 - 60552 * q^54 - 8064 * q^56 - 44734 * q^59 + 13422 * q^61 - 49152 * q^64 + 71944 * q^66 - 268676 * q^69 + 30440 * q^71 - 84784 * q^74 - 53088 * q^76 - 190856 * q^79 - 16268 * q^81 - 138304 * q^84 + 128624 * q^86 - 584622 * q^89 - 21294 * q^91 - 57080 * q^94 + 18432 * q^96 - 772238 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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