LMFDB - Embedded newform 100.10.c.c.49.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [100,10,Mod(49,100)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(100, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 10, names="a")

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

chi := DirichletCharacter("100.49");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 100 = 2^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	100.c (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:	$51.5035836164$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{79})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 39x^{2} + 400 $ x^4 - 39*x^2 + 400
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{12}\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 20)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.3
Root			$4.44410 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	100.49
Dual form			100.10.c.c.49.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+12.2111i q^{3} -9622.57i q^{7} +19533.9 q^{9} +O(q^{10})$	$q+12.2111i q^{3} -9622.57i q^{7} +19533.9 q^{9} +55626.3 q^{11} +169777. i q^{13} -207499. i q^{17} -802445. q^{19} +117502. q^{21} -1.24189e6i q^{23} +478882. i q^{27} +4.28308e6 q^{29} -3.58713e6 q^{31} +679259. i q^{33} +2.89856e6i q^{37} -2.07317e6 q^{39} +2.51515e7 q^{41} -2.00204e7i q^{43} -3.73010e7i q^{47} -5.22402e7 q^{49} +2.53380e6 q^{51} -2.55155e7i q^{53} -9.79875e6i q^{57} +9.96495e7 q^{59} +2.00434e8 q^{61} -1.87966e8i q^{63} +8.09951e7i q^{67} +1.51648e7 q^{69} -4.31522e7 q^{71} -3.40820e8i q^{73} -5.35268e8i q^{77} -2.81089e8 q^{79} +3.78638e8 q^{81} -6.01017e8i q^{83} +5.23011e7i q^{87} -5.39917e8 q^{89} +1.63369e9 q^{91} -4.38028e7i q^{93} -4.23026e8i q^{97} +1.08660e9 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 69764 q^{9}+O(q^{10}) $ 4 * q - 69764 * q^9	$ 4 q - 69764 q^{9} + 205440 q^{11} - 274544 q^{19} + 5680624 q^{21} + 13787496 q^{29} + 583664 q^{31} + 951056 q^{39} + 59546904 q^{41} - 223875828 q^{49} - 54013392 q^{51} + 185861712 q^{59} + 391347848 q^{61} + 344148528 q^{69} + 622414032 q^{71} - 1084523552 q^{79} + 3762476756 q^{81} + 924583704 q^{89} + 3080075536 q^{91} - 2952090240 q^{99}+O(q^{100}) $ 4 * q - 69764 * q^9 + 205440 * q^11 - 274544 * q^19 + 5680624 * q^21 + 13787496 * q^29 + 583664 * q^31 + 951056 * q^39 + 59546904 * q^41 - 223875828 * q^49 - 54013392 * q^51 + 185861712 * q^59 + 391347848 * q^61 + 344148528 * q^69 + 622414032 * q^71 - 1084523552 * q^79 + 3762476756 * q^81 + 924583704 * q^89 + 3080075536 * q^91 - 2952090240 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/100\mathbb{Z}\right)^\times$.

$n$	$51$	$77$
$\chi(n)$	$1$	$-1$

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 $

$2$	0	0
$2$	0	0
$3$	12.2111i	0.0870381i	0.999053	+	0.0435191i	$0.0138569\pi$
$3$	12.2111i	0.0870381i	−0.999053	+	0.0435191i	$0.986143\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 9622.57i	− 1.51478i	−0.652962	−	0.757390i	$-0.726474\pi$
$7$	− 9622.57i	− 1.51478i	0.652962	−	0.757390i	$-0.273526\pi$
$8$	0	0
$9$	19533.9	0.992424
$10$	0	0
$11$	55626.3	1.14555	0.572774	−	0.819713i	$-0.305867\pi$
$11$	55626.3	1.14555	0.572774	+	0.819713i	$0.305867\pi$
$12$	0	0
$13$	169777.i	1.64867i	0.566102	+	0.824335i	$0.308451\pi$
$13$	169777.i	1.64867i	−0.566102	+	0.824335i	$0.691549\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 207499.i	− 0.602555i	−0.953536	−	0.301278i	$-0.902587\pi$
$17$	− 207499.i	− 0.602555i	0.953536	−	0.301278i	$-0.0974131\pi$
$18$	0	0
$19$	−802445.	−1.41262	−0.706308	−	0.707904i	$-0.749641\pi$
$19$	−802445.	−1.41262	−0.706308	+	0.707904i	$0.749641\pi$
$20$	0	0
$21$	117502.	0.131844
$22$	0	0
$23$	− 1.24189e6i	− 0.925351i	−0.886528	−	0.462675i	$-0.846890\pi$
$23$	− 1.24189e6i	− 0.925351i	0.886528	−	0.462675i	$-0.153110\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	478882.i	0.173417i
$28$	0	0
$29$	4.28308e6	1.12451	0.562257	−	0.826963i	$-0.309933\pi$
$29$	4.28308e6	1.12451	0.562257	+	0.826963i	$0.309933\pi$
$30$	0	0
$31$	−3.58713e6	−0.697620	−0.348810	−	0.937193i	$-0.613414\pi$
$31$	−3.58713e6	−0.697620	−0.348810	+	0.937193i	$0.613414\pi$
$32$	0	0
$33$	679259.i	0.0997064i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	2.89856e6i	0.254258i	0.991886	+	0.127129i	$0.0405763\pi$
$37$	2.89856e6i	0.254258i	−0.991886	+	0.127129i	$0.959424\pi$
$38$	0	0
$39$	−2.07317e6	−0.143497
$40$	0	0
$41$	2.51515e7	1.39007	0.695035	−	0.718975i	$-0.255389\pi$
$41$	2.51515e7	1.39007	0.695035	+	0.718975i	$0.255389\pi$
$42$	0	0
$43$	− 2.00204e7i	− 0.893029i	−0.894776	−	0.446515i	$-0.852665\pi$
$43$	− 2.00204e7i	− 0.893029i	0.894776	−	0.446515i	$-0.147335\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 3.73010e7i	− 1.11501i	−0.830172	−	0.557507i	$-0.811758\pi$
$47$	− 3.73010e7i	− 1.11501i	0.830172	−	0.557507i	$-0.188242\pi$
$48$	0	0
$49$	−5.22402e7	−1.29456
$50$	0	0
$51$	2.53380e6	0.0524453
$52$	0	0
$53$	− 2.55155e7i	− 0.444184i	−0.975026	−	0.222092i	$-0.928711\pi$
$53$	− 2.55155e7i	− 0.444184i	0.975026	−	0.222092i	$-0.0712886\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 9.79875e6i	− 0.122951i
$58$	0	0
$59$	9.96495e7	1.07063	0.535317	−	0.844651i	$-0.320192\pi$
$59$	9.96495e7	1.07063	0.535317	+	0.844651i	$0.320192\pi$
$60$	0	0
$61$	2.00434e8	1.85347	0.926737	−	0.375710i	$-0.122601\pi$
$61$	2.00434e8	1.85347	0.926737	+	0.375710i	$0.122601\pi$
$62$	0	0
$63$	− 1.87966e8i	− 1.50331i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	8.09951e7i	0.491046i	0.969391	+	0.245523i	$0.0789597\pi$
$67$	8.09951e7i	0.491046i	−0.969391	+	0.245523i	$0.921040\pi$
$68$	0	0
$69$	1.51648e7	0.0805408
$70$	0	0
$71$	−4.31522e7	−0.201530	−0.100765	−	0.994910i	$-0.532129\pi$
$71$	−4.31522e7	−0.201530	−0.100765	+	0.994910i	$0.532129\pi$
$72$	0	0
$73$	− 3.40820e8i	− 1.40466i	−0.711850	−	0.702332i	$-0.752142\pi$
$73$	− 3.40820e8i	− 1.40466i	0.711850	−	0.702332i	$-0.247858\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 5.35268e8i	− 1.73525i
$78$	0	0
$79$	−2.81089e8	−0.811935	−0.405967	−	0.913888i	$-0.633065\pi$
$79$	−2.81089e8	−0.811935	−0.405967	+	0.913888i	$0.633065\pi$
$80$	0	0
$81$	3.78638e8	0.977330
$82$	0	0
$83$	− 6.01017e8i	− 1.39007i	−0.718978	−	0.695033i	$-0.755390\pi$
$83$	− 6.01017e8i	− 1.39007i	0.718978	−	0.695033i	$-0.244610\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	5.23011e7i	0.0978756i
$88$	0	0
$89$	−5.39917e8	−0.912162	−0.456081	−	0.889938i	$-0.650747\pi$
$89$	−5.39917e8	−0.912162	−0.456081	+	0.889938i	$0.650747\pi$
$90$	0	0
$91$	1.63369e9	2.49737
$92$	0	0
$93$	− 4.38028e7i	− 0.0607195i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 4.23026e8i	− 0.485170i	−0.970130	−	0.242585i	$-0.922005\pi$
$97$	− 4.23026e8i	− 0.485170i	0.970130	−	0.242585i	$-0.0779953\pi$
$98$	0	0
$99$	1.08660e9	1.13687
$100$	0	0
$101$	−1.09286e9	−1.04501	−0.522503	−	0.852637i	$-0.675002\pi$
$101$	−1.09286e9	−1.04501	−0.522503	+	0.852637i	$0.675002\pi$
$102$	0	0
$103$	− 8.50494e7i	− 0.0744567i	−0.999307	−	0.0372284i	$-0.988147\pi$
$103$	− 8.50494e7i	− 0.0744567i	0.999307	−	0.0372284i	$-0.0118529\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 5.23881e8i	− 0.386372i	−0.981162	−	0.193186i	$-0.938118\pi$
$107$	− 5.23881e8i	− 0.386372i	0.981162	−	0.193186i	$-0.0618821\pi$
$108$	0	0
$109$	5.59417e8	0.379591	0.189796	−	0.981824i	$-0.439217\pi$
$109$	5.59417e8	0.379591	0.189796	+	0.981824i	$0.439217\pi$
$110$	0	0
$111$	−3.53947e7	−0.0221302
$112$	0	0
$113$	− 2.63801e9i	− 1.52203i	−0.648733	−	0.761016i	$-0.724701\pi$
$113$	− 2.63801e9i	− 1.52203i	0.648733	−	0.761016i	$-0.275299\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	3.31641e9i	1.63618i
$118$	0	0
$119$	−1.99668e9	−0.912739
$120$	0	0
$121$	7.36341e8	0.312281
$122$	0	0
$123$	3.07128e8i	0.120989i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 3.33268e9i	− 1.13678i	−0.822758	−	0.568391i	$-0.807566\pi$
$127$	− 3.33268e9i	− 1.13678i	0.822758	−	0.568391i	$-0.192434\pi$
$128$	0	0
$129$	2.44472e8	0.0777276
$130$	0	0
$131$	−3.78884e9	−1.12405	−0.562024	−	0.827121i	$-0.689977\pi$
$131$	−3.78884e9	−1.12405	−0.562024	+	0.827121i	$0.689977\pi$
$132$	0	0
$133$	7.72158e9i	2.13980i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	2.15185e9i	0.521879i	0.965355	+	0.260940i	$0.0840323\pi$
$137$	2.15185e9i	0.521879i	−0.965355	+	0.260940i	$0.915968\pi$
$138$	0	0
$139$	−2.75086e8	−0.0625032	−0.0312516	−	0.999512i	$-0.509949\pi$
$139$	−2.75086e8	−0.0625032	−0.0312516	+	0.999512i	$0.509949\pi$
$140$	0	0
$141$	4.55487e8	0.0970487
$142$	0	0
$143$	9.44408e9i	1.88863i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 6.37911e8i	− 0.112676i
$148$	0	0
$149$	6.16582e9	1.02483	0.512416	−	0.858737i	$-0.328751\pi$
$149$	6.16582e9	1.02483	0.512416	+	0.858737i	$0.328751\pi$
$150$	0	0
$151$	−3.28376e9	−0.514014	−0.257007	−	0.966410i	$-0.582736\pi$
$151$	−3.28376e9	−0.514014	−0.257007	+	0.966410i	$0.582736\pi$
$152$	0	0
$153$	− 4.05327e9i	− 0.597990i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	1.89255e9i	0.248599i	0.992245	+	0.124299i	$0.0396683\pi$
$157$	1.89255e9i	0.248599i	−0.992245	+	0.124299i	$0.960332\pi$
$158$	0	0
$159$	3.11573e8	0.0386610
$160$	0	0
$161$	−1.19501e10	−1.40170
$162$	0	0
$163$	4.14293e8i	0.0459688i	0.999736	+	0.0229844i	$0.00731681\pi$
$163$	4.14293e8i	0.0459688i	−0.999736	+	0.0229844i	$0.992683\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1.56225e10i	1.55427i	0.629332	+	0.777136i	$0.283328\pi$
$167$	1.56225e10i	1.55427i	−0.629332	+	0.777136i	$0.716672\pi$
$168$	0	0
$169$	−1.82198e10	−1.71812
$170$	0	0
$171$	−1.56749e10	−1.40191
$172$	0	0
$173$	1.16058e10i	0.985069i	0.870293	+	0.492535i	$0.163930\pi$
$173$	1.16058e10i	0.985069i	−0.870293	+	0.492535i	$0.836070\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	1.21683e9i	0.0931860i
$178$	0	0
$179$	2.45556e10	1.78777	0.893886	−	0.448294i	$-0.147968\pi$
$179$	2.45556e10	1.78777	0.893886	+	0.448294i	$0.147968\pi$
$180$	0	0
$181$	3.21032e9	0.222328	0.111164	−	0.993802i	$-0.464542\pi$
$181$	3.21032e9	0.222328	0.111164	+	0.993802i	$0.464542\pi$
$182$	0	0
$183$	2.44752e9i	0.161323i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 1.15424e10i	− 0.690256i
$188$	0	0
$189$	4.60807e9	0.262689
$190$	0	0
$191$	3.41548e10	1.85695	0.928477	−	0.371390i	$-0.121119\pi$
$191$	3.41548e10	1.85695	0.928477	+	0.371390i	$0.121119\pi$
$192$	0	0
$193$	2.50875e10i	1.30152i	0.759284	+	0.650759i	$0.225549\pi$
$193$	2.50875e10i	1.30152i	−0.759284	+	0.650759i	$0.774451\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 9.64109e9i	− 0.456067i	−0.973653	−	0.228033i	$-0.926771\pi$
$197$	− 9.64109e9i	− 0.456067i	0.973653	−	0.228033i	$-0.0732295\pi$
$198$	0	0
$199$	−3.34318e9	−0.151120	−0.0755598	−	0.997141i	$-0.524074\pi$
$199$	−3.34318e9	−0.151120	−0.0755598	+	0.997141i	$0.524074\pi$
$200$	0	0
$201$	−9.89040e8	−0.0427397
$202$	0	0
$203$	− 4.12142e10i	− 1.70339i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 2.42589e10i	− 0.918341i
$208$	0	0
$209$	−4.46371e10	−1.61822
$210$	0	0
$211$	−1.76087e10	−0.611585	−0.305792	−	0.952098i	$-0.598921\pi$
$211$	−1.76087e10	−0.611585	−0.305792	+	0.952098i	$0.598921\pi$
$212$	0	0
$213$	− 5.26936e8i	− 0.0175408i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	3.45174e10i	1.05674i
$218$	0	0
$219$	4.16179e9	0.122259
$220$	0	0
$221$	3.52287e10	0.993415
$222$	0	0
$223$	− 5.00503e10i	− 1.35530i	−0.735386	−	0.677649i	$-0.762999\pi$
$223$	− 5.00503e10i	− 1.35530i	0.735386	−	0.677649i	$-0.237001\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	8.68117e9i	0.217001i	0.994096	+	0.108501i	$0.0346049\pi$
$227$	8.68117e9i	0.217001i	−0.994096	+	0.108501i	$0.965395\pi$
$228$	0	0
$229$	2.96014e10	0.711299	0.355649	−	0.934619i	$-0.384260\pi$
$229$	2.96014e10	0.711299	0.355649	+	0.934619i	$0.384260\pi$
$230$	0	0
$231$	6.53622e9	0.151033
$232$	0	0
$233$	− 7.42617e10i	− 1.65068i	−0.564634	−	0.825341i	$-0.690983\pi$
$233$	− 7.42617e10i	− 1.65068i	0.564634	−	0.825341i	$-0.309017\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 3.43240e9i	− 0.0706693i
$238$	0	0
$239$	1.65577e10	0.328253	0.164126	−	0.986439i	$-0.447519\pi$
$239$	1.65577e10	0.328253	0.164126	+	0.986439i	$0.447519\pi$
$240$	0	0
$241$	3.61072e10	0.689473	0.344737	−	0.938699i	$-0.387968\pi$
$241$	3.61072e10	0.689473	0.344737	+	0.938699i	$0.387968\pi$
$242$	0	0
$243$	1.40494e10i	0.258482i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 1.36237e11i	− 2.32894i
$248$	0	0
$249$	7.33909e9	0.120989
$250$	0	0
$251$	2.32959e10	0.370465	0.185232	−	0.982695i	$-0.440696\pi$
$251$	2.32959e10	0.370465	0.185232	+	0.982695i	$0.440696\pi$
$252$	0	0
$253$	− 6.90815e10i	− 1.06003i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	4.57624e10i	0.654350i	0.944964	+	0.327175i	$0.106097\pi$
$257$	4.57624e10i	0.654350i	−0.944964	+	0.327175i	$0.893903\pi$
$258$	0	0
$259$	2.78916e10	0.385145
$260$	0	0
$261$	8.36651e10	1.11599
$262$	0	0
$263$	9.93911e10i	1.28099i	0.767961	+	0.640496i	$0.221271\pi$
$263$	9.93911e10i	1.28099i	−0.767961	+	0.640496i	$0.778729\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 6.59299e9i	− 0.0793929i
$268$	0	0
$269$	−1.57475e11	−1.83369	−0.916845	−	0.399242i	$-0.869273\pi$
$269$	−1.57475e11	−1.83369	−0.916845	+	0.399242i	$0.869273\pi$
$270$	0	0
$271$	2.79292e10	0.314555	0.157278	−	0.987554i	$-0.449728\pi$
$271$	2.79292e10	0.314555	0.157278	+	0.987554i	$0.449728\pi$
$272$	0	0
$273$	1.99492e10i	0.217367i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	1.68307e11i	1.71768i	0.512240	+	0.858842i	$0.328816\pi$
$277$	1.68307e11i	1.71768i	−0.512240	+	0.858842i	$0.671184\pi$
$278$	0	0
$279$	−7.00705e10	−0.692335
$280$	0	0
$281$	−8.51877e10	−0.815076	−0.407538	−	0.913188i	$-0.633613\pi$
$281$	−8.51877e10	−0.815076	−0.407538	+	0.913188i	$0.633613\pi$
$282$	0	0
$283$	− 2.02954e11i	− 1.88087i	−0.339970	−	0.940436i	$-0.610417\pi$
$283$	− 2.02954e11i	− 1.88087i	0.339970	−	0.940436i	$-0.389583\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 2.42022e11i	− 2.10565i
$288$	0	0
$289$	7.55318e10	0.636927
$290$	0	0
$291$	5.16561e9	0.0422283
$292$	0	0
$293$	1.31572e11i	1.04294i	0.853270	+	0.521470i	$0.174616\pi$
$293$	1.31572e11i	1.04294i	−0.853270	+	0.521470i	$0.825384\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	2.66384e10i	0.198657i
$298$	0	0
$299$	2.10844e11	1.52560
$300$	0	0
$301$	−1.92648e11	−1.35274
$302$	0	0
$303$	− 1.33450e10i	− 0.0909554i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	1.88785e11i	1.21296i	0.795099	+	0.606479i	$0.207419\pi$
$307$	1.88785e11i	1.21296i	−0.795099	+	0.606479i	$0.792581\pi$
$308$	0	0
$309$	1.03855e9	0.00648057
$310$	0	0
$311$	−7.10454e10	−0.430640	−0.215320	−	0.976544i	$-0.569079\pi$
$311$	−7.10454e10	−0.430640	−0.215320	+	0.976544i	$0.569079\pi$
$312$	0	0
$313$	3.35032e11i	1.97305i	0.163623	+	0.986523i	$0.447682\pi$
$313$	3.35032e11i	1.97305i	−0.163623	+	0.986523i	$0.552318\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1.98163e11i	1.10219i	0.834443	+	0.551094i	$0.185789\pi$
$317$	1.98163e11i	1.10219i	−0.834443	+	0.551094i	$0.814211\pi$
$318$	0	0
$319$	2.38252e11	1.28818
$320$	0	0
$321$	6.39717e9	0.0336291
$322$	0	0
$323$	1.66507e11i	0.851179i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	6.83110e9i	0.0330389i
$328$	0	0
$329$	−3.58931e11	−1.68900
$330$	0	0
$331$	−1.60576e11	−0.735284	−0.367642	−	0.929967i	$-0.619835\pi$
$331$	−1.60576e11	−0.735284	−0.367642	+	0.929967i	$0.619835\pi$
$332$	0	0
$333$	5.66202e10i	0.252332i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 8.77234e10i	− 0.370494i	−0.982692	−	0.185247i	$-0.940692\pi$
$337$	− 8.77234e10i	− 0.370494i	0.982692	−	0.185247i	$-0.0593085\pi$
$338$	0	0
$339$	3.22130e10	0.132475
$340$	0	0
$341$	−1.99539e11	−0.799157
$342$	0	0
$343$	1.14379e11i	0.446194i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 1.89987e11i	− 0.703464i	−0.936101	−	0.351732i	$-0.885593\pi$
$347$	− 1.89987e11i	− 0.703464i	0.936101	−	0.351732i	$-0.114407\pi$
$348$	0	0
$349$	−4.85678e11	−1.75240	−0.876201	−	0.481946i	$-0.839930\pi$
$349$	−4.85678e11	−1.75240	−0.876201	+	0.481946i	$0.839930\pi$
$350$	0	0
$351$	−8.13031e10	−0.285907
$352$	0	0
$353$	5.15816e10i	0.176811i	0.996085	+	0.0884054i	$0.0281771\pi$
$353$	5.15816e10i	0.176811i	−0.996085	+	0.0884054i	$0.971823\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 2.43816e10i	− 0.0794431i
$358$	0	0
$359$	1.79131e11	0.569176	0.284588	−	0.958650i	$-0.408143\pi$
$359$	1.79131e11	0.569176	0.284588	+	0.958650i	$0.408143\pi$
$360$	0	0
$361$	3.21231e11	0.995485
$362$	0	0
$363$	8.99154e9i	0.0271803i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	2.76314e10i	0.0795071i	0.999210	+	0.0397536i	$0.0126573\pi$
$367$	2.76314e10i	0.0795071i	−0.999210	+	0.0397536i	$0.987343\pi$
$368$	0	0
$369$	4.91307e11	1.37954
$370$	0	0
$371$	−2.45525e11	−0.672842
$372$	0	0
$373$	2.35196e11i	0.629129i	0.949236	+	0.314565i	$0.101858\pi$
$373$	2.35196e11i	0.629129i	−0.949236	+	0.314565i	$0.898142\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	7.27168e11i	1.85395i
$378$	0	0
$379$	3.11965e11	0.776658	0.388329	−	0.921521i	$-0.373052\pi$
$379$	3.11965e11	0.776658	0.388329	+	0.921521i	$0.373052\pi$
$380$	0	0
$381$	4.06958e10	0.0989434
$382$	0	0
$383$	− 4.78535e10i	− 0.113637i	−0.998385	−	0.0568185i	$-0.981904\pi$
$383$	− 4.78535e10i	− 0.113637i	0.998385	−	0.0568185i	$-0.0180956\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 3.91077e11i	− 0.886264i
$388$	0	0
$389$	−1.19528e11	−0.264665	−0.132332	−	0.991205i	$-0.542247\pi$
$389$	−1.19528e11	−0.264665	−0.132332	+	0.991205i	$0.542247\pi$
$390$	0	0
$391$	−2.57691e11	−0.557575
$392$	0	0
$393$	− 4.62659e10i	− 0.0978351i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 7.08892e10i	− 0.143226i	−0.997432	−	0.0716132i	$-0.977185\pi$
$397$	− 7.08892e10i	− 0.143226i	0.997432	−	0.0716132i	$-0.0228147\pi$
$398$	0	0
$399$	−9.42891e10	−0.186245
$400$	0	0
$401$	5.62486e11	1.08633	0.543165	−	0.839626i	$-0.317226\pi$
$401$	5.62486e11	1.08633	0.543165	+	0.839626i	$0.317226\pi$
$402$	0	0
$403$	− 6.09012e11i	− 1.15015i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1.61236e11i	0.291265i
$408$	0	0
$409$	−2.23341e11	−0.394651	−0.197325	−	0.980338i	$-0.563226\pi$
$409$	−2.23341e11	−0.394651	−0.197325	+	0.980338i	$0.563226\pi$
$410$	0	0
$411$	−2.62765e10	−0.0454234
$412$	0	0
$413$	− 9.58884e11i	− 1.62178i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 3.35911e9i	− 0.00544016i
$418$	0	0
$419$	−3.54170e11	−0.561370	−0.280685	−	0.959800i	$-0.590562\pi$
$419$	−3.54170e11	−0.561370	−0.280685	+	0.959800i	$0.590562\pi$
$420$	0	0
$421$	5.29294e11	0.821159	0.410580	−	0.911825i	$-0.365326\pi$
$421$	5.29294e11	0.821159	0.410580	+	0.911825i	$0.365326\pi$
$422$	0	0
$423$	− 7.28634e11i	− 1.10657i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 1.92869e12i	− 2.80761i
$428$	0	0
$429$	−1.15323e11	−0.164383
$430$	0	0
$431$	1.26760e12	1.76944	0.884718	−	0.466127i	$-0.154351\pi$
$431$	1.26760e12	1.76944	0.884718	+	0.466127i	$0.154351\pi$
$432$	0	0
$433$	3.28861e11i	0.449591i	0.974406	+	0.224795i	$0.0721713\pi$
$433$	3.28861e11i	0.449591i	−0.974406	+	0.224795i	$0.927829\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	9.96545e11i	1.30717i
$438$	0	0
$439$	−2.69396e11	−0.346178	−0.173089	−	0.984906i	$-0.555375\pi$
$439$	−2.69396e11	−0.346178	−0.173089	+	0.984906i	$0.555375\pi$
$440$	0	0
$441$	−1.02045e12	−1.28475
$442$	0	0
$443$	4.20444e11i	0.518671i	0.965787	+	0.259335i	$0.0835035\pi$
$443$	4.20444e11i	0.518671i	−0.965787	+	0.259335i	$0.916497\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	7.52915e10i	0.0891995i
$448$	0	0
$449$	−1.19773e11	−0.139075	−0.0695376	−	0.997579i	$-0.522152\pi$
$449$	−1.19773e11	−0.139075	−0.0695376	+	0.997579i	$0.522152\pi$
$450$	0	0
$451$	1.39909e12	1.59239
$452$	0	0
$453$	− 4.00984e10i	− 0.0447388i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 1.34943e12i	− 1.44719i	−0.690223	−	0.723597i	$-0.742488\pi$
$457$	− 1.34943e12i	− 1.44719i	0.690223	−	0.723597i	$-0.257512\pi$
$458$	0	0
$459$	9.93677e10	0.104493
$460$	0	0
$461$	2.09834e11	0.216383	0.108191	−	0.994130i	$-0.465494\pi$
$461$	2.09834e11	0.216383	0.108191	+	0.994130i	$0.465494\pi$
$462$	0	0
$463$	− 6.60348e11i	− 0.667819i	−0.942605	−	0.333909i	$-0.891632\pi$
$463$	− 6.60348e11i	− 0.667819i	0.942605	−	0.333909i	$-0.108368\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	8.28966e11i	0.806512i	0.915087	+	0.403256i	$0.132122\pi$
$467$	8.28966e11i	0.806512i	−0.915087	+	0.403256i	$0.867878\pi$
$468$	0	0
$469$	7.79381e11	0.743827
$470$	0	0
$471$	−2.31102e10	−0.0216376
$472$	0	0
$473$	− 1.11366e12i	− 1.02301i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 4.98417e11i	− 0.440819i
$478$	0	0
$479$	6.48710e11	0.563042	0.281521	−	0.959555i	$-0.409161\pi$
$479$	6.48710e11	0.563042	0.281521	+	0.959555i	$0.409161\pi$
$480$	0	0
$481$	−4.92110e11	−0.419188
$482$	0	0
$483$	− 1.45924e11i	− 0.122002i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	1.75979e12i	1.41768i	0.705367	+	0.708842i	$0.250782\pi$
$487$	1.75979e12i	1.41768i	−0.705367	+	0.708842i	$0.749218\pi$
$488$	0	0
$489$	−5.05898e9	−0.00400104
$490$	0	0
$491$	3.02135e11	0.234603	0.117302	−	0.993096i	$-0.462576\pi$
$491$	3.02135e11	0.234603	0.117302	+	0.993096i	$0.462576\pi$
$492$	0	0
$493$	− 8.88736e11i	− 0.677582i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	4.15235e11i	0.305274i
$498$	0	0
$499$	−5.11758e11	−0.369498	−0.184749	−	0.982786i	$-0.559147\pi$
$499$	−5.11758e11	−0.369498	−0.184749	+	0.982786i	$0.559147\pi$
$500$	0	0
$501$	−1.90768e11	−0.135281
$502$	0	0
$503$	1.69213e12i	1.17863i	0.807904	+	0.589314i	$0.200602\pi$
$503$	1.69213e12i	1.17863i	−0.807904	+	0.589314i	$0.799398\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 2.22483e11i	− 0.149542i
$508$	0	0
$509$	−7.10216e11	−0.468986	−0.234493	−	0.972118i	$-0.575343\pi$
$509$	−7.10216e11	−0.468986	−0.234493	+	0.972118i	$0.575343\pi$
$510$	0	0
$511$	−3.27956e12	−2.12776
$512$	0	0
$513$	− 3.84276e11i	− 0.244972i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 2.07492e12i	− 1.27730i
$518$	0	0
$519$	−1.41719e11	−0.0857386
$520$	0	0
$521$	−8.88794e11	−0.528483	−0.264242	−	0.964456i	$-0.585122\pi$
$521$	−8.88794e11	−0.528483	−0.264242	+	0.964456i	$0.585122\pi$
$522$	0	0
$523$	1.01046e12i	0.590555i	0.955412	+	0.295277i	$0.0954121\pi$
$523$	1.01046e12i	0.590555i	−0.955412	+	0.295277i	$0.904588\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	7.44327e11i	0.420355i
$528$	0	0
$529$	2.58873e11	0.143726
$530$	0	0
$531$	1.94654e12	1.06252
$532$	0	0
$533$	4.27015e12i	2.29177i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	2.99851e11i	0.155604i
$538$	0	0
$539$	−2.90593e12	−1.48298
$540$	0	0
$541$	3.39951e12	1.70619	0.853097	−	0.521753i	$-0.174722\pi$
$541$	3.39951e12	1.70619	0.853097	+	0.521753i	$0.174722\pi$
$542$	0	0
$543$	3.92015e10i	0.0193510i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	3.10308e12i	1.48201i	0.671501	+	0.741004i	$0.265650\pi$
$547$	3.10308e12i	1.48201i	−0.671501	+	0.741004i	$0.734350\pi$
$548$	0	0
$549$	3.91525e12	1.83943
$550$	0	0
$551$	−3.43693e12	−1.58851
$552$	0	0
$553$	2.70479e12i	1.22990i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	4.92558e11i	0.216825i	0.994106	+	0.108412i	$0.0345767\pi$
$557$	4.92558e11i	0.216825i	−0.994106	+	0.108412i	$0.965423\pi$
$558$	0	0
$559$	3.39901e12	1.47231
$560$	0	0
$561$	1.40946e11	0.0600786
$562$	0	0
$563$	1.67786e12i	0.703832i	0.936032	+	0.351916i	$0.114470\pi$
$563$	1.67786e12i	0.703832i	−0.936032	+	0.351916i	$0.885530\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 3.64347e12i	− 1.48044i
$568$	0	0
$569$	−3.40646e12	−1.36238	−0.681190	−	0.732107i	$-0.738537\pi$
$569$	−3.40646e12	−1.36238	−0.681190	+	0.732107i	$0.738537\pi$
$570$	0	0
$571$	−4.05234e12	−1.59530	−0.797651	−	0.603119i	$-0.793924\pi$
$571$	−4.05234e12	−1.59530	−0.797651	+	0.603119i	$0.793924\pi$
$572$	0	0
$573$	4.17068e11i	0.161626i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	4.55977e11i	0.171258i	0.996327	+	0.0856291i	$0.0272900\pi$
$577$	4.55977e11i	0.171258i	−0.996327	+	0.0856291i	$0.972710\pi$
$578$	0	0
$579$	−3.06347e11	−0.113282
$580$	0	0
$581$	−5.78333e12	−2.10565
$582$	0	0
$583$	− 1.41933e12i	− 0.508834i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	2.83858e12i	0.986801i	0.869802	+	0.493401i	$0.164246\pi$
$587$	2.83858e12i	0.986801i	−0.869802	+	0.493401i	$0.835754\pi$
$588$	0	0
$589$	2.87847e12	0.985470
$590$	0	0
$591$	1.17728e11	0.0396952
$592$	0	0
$593$	− 4.08687e12i	− 1.35720i	−0.734507	−	0.678601i	$-0.762586\pi$
$593$	− 4.08687e12i	− 1.35720i	0.734507	−	0.678601i	$-0.237414\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 4.08240e10i	− 0.0131532i
$598$	0	0
$599$	−4.47524e12	−1.42035	−0.710175	−	0.704025i	$-0.751384\pi$
$599$	−4.47524e12	−1.42035	−0.710175	+	0.704025i	$0.751384\pi$
$600$	0	0
$601$	−4.83907e11	−0.151296	−0.0756478	−	0.997135i	$-0.524102\pi$
$601$	−4.83907e11	−0.151296	−0.0756478	+	0.997135i	$0.524102\pi$
$602$	0	0
$603$	1.58215e12i	0.487326i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	4.96609e12i	1.48479i	0.669962	+	0.742395i	$0.266310\pi$
$607$	4.96609e12i	1.48479i	−0.669962	+	0.742395i	$0.733690\pi$
$608$	0	0
$609$	5.03271e11	0.148260
$610$	0	0
$611$	6.33285e12	1.83829
$612$	0	0
$613$	− 2.43460e12i	− 0.696396i	−0.937421	−	0.348198i	$-0.886794\pi$
$613$	− 2.43460e12i	− 0.696396i	0.937421	−	0.348198i	$-0.113206\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 1.82614e12i	− 0.507284i	−0.967298	−	0.253642i	$-0.918372\pi$
$617$	− 1.82614e12i	− 0.507284i	0.967298	−	0.253642i	$-0.0816285\pi$
$618$	0	0
$619$	1.13058e12	0.309523	0.154761	−	0.987952i	$-0.450539\pi$
$619$	1.13058e12	0.309523	0.154761	+	0.987952i	$0.450539\pi$
$620$	0	0
$621$	5.94716e11	0.160471
$622$	0	0
$623$	5.19539e12i	1.38173i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 5.45068e11i	− 0.140847i
$628$	0	0
$629$	6.01450e11	0.153205
$630$	0	0
$631$	1.96900e12	0.494440	0.247220	−	0.968959i	$-0.420483\pi$
$631$	1.96900e12	0.494440	0.247220	+	0.968959i	$0.420483\pi$
$632$	0	0
$633$	− 2.15022e11i	− 0.0532312i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 8.86918e12i	− 2.13430i
$638$	0	0
$639$	−8.42930e11	−0.200003
$640$	0	0
$641$	2.69141e12	0.629679	0.314840	−	0.949145i	$-0.398049\pi$
$641$	2.69141e12	0.629679	0.314840	+	0.949145i	$0.398049\pi$
$642$	0	0
$643$	− 2.27990e12i	− 0.525977i	−0.964799	−	0.262989i	$-0.915292\pi$
$643$	− 2.27990e12i	− 0.525977i	0.964799	−	0.262989i	$-0.0847081\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	4.86871e12i	1.09231i	0.837685	+	0.546153i	$0.183908\pi$
$647$	4.86871e12i	1.09231i	−0.837685	+	0.546153i	$0.816092\pi$
$648$	0	0
$649$	5.54314e12	1.22646
$650$	0	0
$651$	−4.21495e11	−0.0919768
$652$	0	0
$653$	2.30610e12i	0.496329i	0.968718	+	0.248164i	$0.0798273\pi$
$653$	2.30610e12i	0.496329i	−0.968718	+	0.248164i	$0.920173\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 6.65754e12i	− 1.39402i
$658$	0	0
$659$	−7.40069e12	−1.52858	−0.764289	−	0.644873i	$-0.776910\pi$
$659$	−7.40069e12	−1.52858	−0.764289	+	0.644873i	$0.776910\pi$
$660$	0	0
$661$	−2.00760e12	−0.409045	−0.204522	−	0.978862i	$-0.565564\pi$
$661$	−2.00760e12	−0.409045	−0.204522	+	0.978862i	$0.565564\pi$
$662$	0	0
$663$	4.30181e11i	0.0864650i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 5.31909e12i	− 1.04057i
$668$	0	0
$669$	6.11170e11	0.117963
$670$	0	0
$671$	1.11494e13	2.12324
$672$	0	0
$673$	− 1.01154e13i	− 1.90071i	−0.311167	−	0.950355i	$-0.600720\pi$
$673$	− 1.01154e13i	− 1.90071i	0.311167	−	0.950355i	$-0.399280\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	8.91166e12i	1.63046i	0.579139	+	0.815229i	$0.303389\pi$
$677$	8.91166e12i	1.63046i	−0.579139	+	0.815229i	$0.696611\pi$
$678$	0	0
$679$	−4.07059e12	−0.734926
$680$	0	0
$681$	−1.06007e11	−0.0188874
$682$	0	0
$683$	− 9.26863e12i	− 1.62976i	−0.579633	−	0.814878i	$-0.696804\pi$
$683$	− 9.26863e12i	− 1.62976i	0.579633	−	0.814878i	$-0.303196\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	3.61466e11i	0.0619101i
$688$	0	0
$689$	4.33195e12	0.732313
$690$	0	0
$691$	1.03490e11	0.0172682	0.00863409	−	0.999963i	$-0.497252\pi$
$691$	1.03490e11	0.0172682	0.00863409	+	0.999963i	$0.497252\pi$
$692$	0	0
$693$	− 1.04559e13i	− 1.72211i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 5.21893e12i	− 0.837594i
$698$	0	0
$699$	9.06818e11	0.143672
$700$	0	0
$701$	4.77133e12	0.746291	0.373145	−	0.927773i	$-0.378279\pi$
$701$	4.77133e12	0.746291	0.373145	+	0.927773i	$0.378279\pi$
$702$	0	0
$703$	− 2.32594e12i	− 0.359169i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	1.05161e13i	1.58295i
$708$	0	0
$709$	−7.58922e12	−1.12795	−0.563974	−	0.825793i	$-0.690728\pi$
$709$	−7.58922e12	−1.12795	−0.563974	+	0.825793i	$0.690728\pi$
$710$	0	0
$711$	−5.49075e12	−0.805784
$712$	0	0
$713$	4.45480e12i	0.645543i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	2.02188e11i	0.0285705i
$718$	0	0
$719$	−1.50766e12	−0.210389	−0.105194	−	0.994452i	$-0.533546\pi$
$719$	−1.50766e12	−0.210389	−0.105194	+	0.994452i	$0.533546\pi$
$720$	0	0
$721$	−8.18394e11	−0.112786
$722$	0	0
$723$	4.40909e11i	0.0600105i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 1.12985e13i	− 1.50009i	−0.661387	−	0.750045i	$-0.730032\pi$
$727$	− 1.12985e13i	− 1.50009i	0.661387	−	0.750045i	$-0.269968\pi$
$728$	0	0
$729$	7.28117e12	0.954833
$730$	0	0
$731$	−4.15423e12	−0.538100
$732$	0	0
$733$	1.07062e12i	0.136984i	0.997652	+	0.0684919i	$0.0218187\pi$
$733$	1.07062e12i	0.136984i	−0.997652	+	0.0684919i	$0.978181\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	4.50546e12i	0.562517i
$738$	0	0
$739$	1.44855e13	1.78662	0.893311	−	0.449439i	$-0.148376\pi$
$739$	1.44855e13	1.78662	0.893311	+	0.449439i	$0.148376\pi$
$740$	0	0
$741$	1.66360e12	0.202707
$742$	0	0
$743$	1.24339e13i	1.49678i	0.663261	+	0.748388i	$0.269172\pi$
$743$	1.24339e13i	1.49678i	−0.663261	+	0.748388i	$0.730828\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 1.17402e13i	− 1.37954i
$748$	0	0
$749$	−5.04108e12	−0.585269
$750$	0	0
$751$	8.79097e12	1.00846	0.504228	−	0.863570i	$-0.331777\pi$
$751$	8.79097e12	1.00846	0.504228	+	0.863570i	$0.331777\pi$
$752$	0	0
$753$	2.84468e11i	0.0322446i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 7.93877e12i	− 0.878662i	−0.898325	−	0.439331i	$-0.855216\pi$
$757$	− 7.93877e12i	− 0.878662i	0.898325	−	0.439331i	$-0.144784\pi$
$758$	0	0
$759$	8.43562e11	0.0922633
$760$	0	0
$761$	1.12763e13	1.21881	0.609405	−	0.792859i	$-0.291408\pi$
$761$	1.12763e13	1.21881	0.609405	+	0.792859i	$0.291408\pi$
$762$	0	0
$763$	− 5.38303e12i	− 0.574998i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1.69182e13i	1.76512i
$768$	0	0
$769$	−3.80743e12	−0.392612	−0.196306	−	0.980543i	$-0.562895\pi$
$769$	−3.80743e12	−0.392612	−0.196306	+	0.980543i	$0.562895\pi$
$770$	0	0
$771$	−5.58810e11	−0.0569534
$772$	0	0
$773$	− 3.89254e12i	− 0.392126i	−0.980591	−	0.196063i	$-0.937184\pi$
$773$	− 3.89254e12i	− 0.392126i	0.980591	−	0.196063i	$-0.0628156\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	3.40588e11i	0.0335223i
$778$	0	0
$779$	−2.01827e13	−1.96364
$780$	0	0
$781$	−2.40040e12	−0.230862
$782$	0	0
$783$	2.05109e12i	0.195010i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	2.70256e12i	0.251124i	0.992086	+	0.125562i	$0.0400734\pi$
$787$	2.70256e12i	0.251124i	−0.992086	+	0.125562i	$0.959927\pi$
$788$	0	0
$789$	−1.21368e12	−0.111495
$790$	0	0
$791$	−2.53844e13	−2.30554
$792$	0	0
$793$	3.40291e13i	3.05577i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	1.58748e12i	0.139363i	0.997569	+	0.0696813i	$0.0221982\pi$
$797$	1.58748e12i	0.139363i	−0.997569	+	0.0696813i	$0.977802\pi$
$798$	0	0
$799$	−7.73994e12	−0.671857
$800$	0	0
$801$	−1.05467e13	−0.905252
$802$	0	0
$803$	− 1.89586e13i	− 1.60911i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 1.92294e12i	− 0.159601i
$808$	0	0
$809$	4.55705e12	0.374038	0.187019	−	0.982356i	$-0.440117\pi$
$809$	4.55705e12	0.374038	0.187019	+	0.982356i	$0.440117\pi$
$810$	0	0
$811$	−2.77014e12	−0.224858	−0.112429	−	0.993660i	$-0.535863\pi$
$811$	−2.77014e12	−0.224858	−0.112429	+	0.993660i	$0.535863\pi$
$812$	0	0
$813$	3.41047e11i	0.0273783i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	1.60653e13i	1.26151i
$818$	0	0
$819$	3.19123e13	2.47846
$820$	0	0
$821$	−1.21734e13	−0.935124	−0.467562	−	0.883960i	$-0.654868\pi$
$821$	−1.21734e13	−0.935124	−0.467562	+	0.883960i	$0.654868\pi$
$822$	0	0
$823$	1.75355e13i	1.33235i	0.745796	+	0.666175i	$0.232069\pi$
$823$	1.75355e13i	1.33235i	−0.745796	+	0.666175i	$0.767931\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	9.15138e12i	0.680318i	0.940368	+	0.340159i	$0.110481\pi$
$827$	9.15138e12i	0.680318i	−0.940368	+	0.340159i	$0.889519\pi$
$828$	0	0
$829$	5.76678e12	0.424071	0.212035	−	0.977262i	$-0.431991\pi$
$829$	5.76678e12	0.424071	0.212035	+	0.977262i	$0.431991\pi$
$830$	0	0
$831$	−2.05522e12	−0.149504
$832$	0	0
$833$	1.08398e13i	0.780044i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 1.71781e12i	− 0.120979i
$838$	0	0
$839$	−8.69274e11	−0.0605658	−0.0302829	−	0.999541i	$-0.509641\pi$
$839$	−8.69274e11	−0.0605658	−0.0302829	+	0.999541i	$0.509641\pi$
$840$	0	0
$841$	3.83759e12	0.264531
$842$	0	0
$843$	− 1.04024e12i	− 0.0709427i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 7.08549e12i	− 0.473037i
$848$	0	0
$849$	2.47830e12	0.163708
$850$	0	0
$851$	3.59968e12	0.235278
$852$	0	0
$853$	− 9.19943e12i	− 0.594963i	−0.954728	−	0.297482i	$-0.903853\pi$
$853$	− 9.19943e12i	− 0.594963i	0.954728	−	0.297482i	$-0.0961467\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	6.57679e12i	0.416486i	0.978077	+	0.208243i	$0.0667745\pi$
$857$	6.57679e12i	0.416486i	−0.978077	+	0.208243i	$0.933225\pi$
$858$	0	0
$859$	1.80030e13	1.12817	0.564085	−	0.825717i	$-0.309229\pi$
$859$	1.80030e13	1.12817	0.564085	+	0.825717i	$0.309229\pi$
$860$	0	0
$861$	2.95536e12	0.183272
$862$	0	0
$863$	3.08737e13i	1.89470i	0.320200	+	0.947350i	$0.396250\pi$
$863$	3.08737e13i	1.89470i	−0.320200	+	0.947350i	$0.603750\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	9.22328e11i	0.0554370i
$868$	0	0
$869$	−1.56359e13	−0.930110
$870$	0	0
$871$	−1.37511e13	−0.809573
$872$	0	0
$873$	− 8.26334e12i	− 0.481495i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 2.77535e12i	− 0.158424i	−0.996858	−	0.0792118i	$-0.974760\pi$
$877$	− 2.77535e12i	− 0.158424i	0.996858	−	0.0792118i	$-0.0252403\pi$
$878$	0	0
$879$	−1.60664e12	−0.0907755
$880$	0	0
$881$	−1.64369e13	−0.919241	−0.459620	−	0.888116i	$-0.652014\pi$
$881$	−1.64369e13	−0.919241	−0.459620	+	0.888116i	$0.652014\pi$
$882$	0	0
$883$	− 5.38106e12i	− 0.297882i	−0.988846	−	0.148941i	$-0.952414\pi$
$883$	− 5.38106e12i	− 0.297882i	0.988846	−	0.148941i	$-0.0475865\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 1.61191e13i	− 0.874347i	−0.899377	−	0.437174i	$-0.855980\pi$
$887$	− 1.61191e13i	− 0.874347i	0.899377	−	0.437174i	$-0.144020\pi$
$888$	0	0
$889$	−3.20690e13	−1.72198
$890$	0	0
$891$	2.10622e13	1.11958
$892$	0	0
$893$	2.99320e13i	1.57509i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	2.57464e12i	0.132785i
$898$	0	0
$899$	−1.53639e13	−0.784483
$900$	0	0
$901$	−5.29446e12	−0.267645
$902$	0	0
$903$	− 2.35245e12i	− 0.117740i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 1.50156e13i	− 0.736734i	−0.929681	−	0.368367i	$-0.879917\pi$
$907$	− 1.50156e13i	− 0.736734i	0.929681	−	0.368367i	$-0.120083\pi$
$908$	0	0
$909$	−2.13478e13	−1.03709
$910$	0	0
$911$	−2.12904e12	−0.102412	−0.0512061	−	0.998688i	$-0.516307\pi$
$911$	−2.12904e12	−0.102412	−0.0512061	+	0.998688i	$0.516307\pi$
$912$	0	0
$913$	− 3.34324e13i	− 1.59239i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	3.64583e13i	1.70269i
$918$	0	0
$919$	2.82600e13	1.30693	0.653464	−	0.756957i	$-0.273315\pi$
$919$	2.82600e13	1.30693	0.653464	+	0.756957i	$0.273315\pi$
$920$	0	0
$921$	−2.30528e12	−0.105574
$922$	0	0
$923$	− 7.32625e12i	− 0.332257i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 1.66135e12i	− 0.0738927i
$928$	0	0
$929$	1.74749e11	0.00769739	0.00384869	−	0.999993i	$-0.498775\pi$
$929$	1.74749e11	0.00769739	0.00384869	+	0.999993i	$0.498775\pi$
$930$	0	0
$931$	4.19199e13	1.82872
$932$	0	0
$933$	− 8.67543e11i	− 0.0374821i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 6.04012e12i	− 0.255987i	−0.991775	−	0.127993i	$-0.959146\pi$
$937$	− 6.04012e12i	− 0.255987i	0.991775	−	0.127993i	$-0.0408536\pi$
$938$	0	0
$939$	−4.09112e12	−0.171730
$940$	0	0
$941$	2.13848e13	0.889104	0.444552	−	0.895753i	$-0.353363\pi$
$941$	2.13848e13	0.889104	0.444552	+	0.895753i	$0.353363\pi$
$942$	0	0
$943$	− 3.12353e13i	− 1.28630i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 4.58008e13i	− 1.85054i	−0.379312	−	0.925269i	$-0.623839\pi$
$947$	− 4.58008e13i	− 1.85054i	0.379312	−	0.925269i	$-0.376161\pi$
$948$	0	0
$949$	5.78634e13	2.31583
$950$	0	0
$951$	−2.41979e12	−0.0959323
$952$	0	0
$953$	− 1.57835e13i	− 0.619847i	−0.950761	−	0.309924i	$-0.899696\pi$
$953$	− 1.57835e13i	− 0.619847i	0.950761	−	0.309924i	$-0.100304\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	2.90932e12i	0.112121i
$958$	0	0
$959$	2.07063e13	0.790533
$960$	0	0
$961$	−1.35722e13	−0.513326
$962$	0	0
$963$	− 1.02334e13i	− 0.383445i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 1.61550e13i	− 0.594140i	−0.954856	−	0.297070i	$-0.903991\pi$
$967$	− 1.61550e13i	− 0.594140i	0.954856	−	0.297070i	$-0.0960095\pi$
$968$	0	0
$969$	−2.03324e12	−0.0740851
$970$	0	0
$971$	−5.05363e12	−0.182439	−0.0912194	−	0.995831i	$-0.529076\pi$
$971$	−5.05363e12	−0.182439	−0.0912194	+	0.995831i	$0.529076\pi$
$972$	0	0
$973$	2.64703e12i	0.0946786i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	3.22726e13i	1.13321i	0.823991	+	0.566603i	$0.191743\pi$
$977$	3.22726e13i	1.13321i	−0.823991	+	0.566603i	$0.808257\pi$
$978$	0	0
$979$	−3.00336e13	−1.04493
$980$	0	0
$981$	1.09276e13	0.376716
$982$	0	0
$983$	− 2.11820e13i	− 0.723561i	−0.932263	−	0.361781i	$-0.882169\pi$
$983$	− 2.11820e13i	− 0.723561i	0.932263	−	0.361781i	$-0.117831\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 4.38295e12i	− 0.147007i
$988$	0	0
$989$	−2.48631e13	−0.826365
$990$	0	0
$991$	−2.18946e13	−0.721116	−0.360558	−	0.932737i	$-0.617414\pi$
$991$	−2.18946e13	−0.721116	−0.360558	+	0.932737i	$0.617414\pi$
$992$	0	0
$993$	− 1.96081e12i	− 0.0639977i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	4.15332e13i	1.33127i	0.746277	+	0.665636i	$0.231840\pi$
$997$	4.15332e13i	1.33127i	−0.746277	+	0.665636i	$0.768160\pi$
$998$	0	0
$999$	−1.38807e12	−0.0440927

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	100.10.c.c.49.3		4
4.3	odd	2		400.10.c.l.49.2		4
5.2	odd	4		20.10.a.b.1.2	✓	2
5.3	odd	4		100.10.a.c.1.1		2
5.4	even	2	inner	100.10.c.c.49.2		4
15.2	even	4		180.10.a.e.1.2		2
20.3	even	4		400.10.a.l.1.2		2
20.7	even	4		80.10.a.j.1.1		2
20.19	odd	2		400.10.c.l.49.3		4
40.27	even	4		320.10.a.l.1.2		2
40.37	odd	4		320.10.a.t.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
20.10.a.b.1.2	✓	2	5.2	odd	4
80.10.a.j.1.1		2	20.7	even	4
100.10.a.c.1.1		2	5.3	odd	4
100.10.c.c.49.2		4	5.4	even	2	inner
100.10.c.c.49.3		4	1.1	even	1	trivial
180.10.a.e.1.2		2	15.2	even	4
320.10.a.l.1.2		2	40.27	even	4
320.10.a.t.1.1		2	40.37	odd	4
400.10.a.l.1.2		2	20.3	even	4
400.10.c.l.49.2		4	4.3	odd	2
400.10.c.l.49.3		4	20.19	odd	2

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:	\( N \)	\(=\)	\( 100 = 2^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	100.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+12.2111i q^{3} -9622.57i q^{7} +19533.9 q^{9} +O(q^{10})\)	\(q+12.2111i q^{3} -9622.57i q^{7} +19533.9 q^{9} +55626.3 q^{11} +169777. i q^{13} -207499. i q^{17} -802445. q^{19} +117502. q^{21} -1.24189e6i q^{23} +478882. i q^{27} +4.28308e6 q^{29} -3.58713e6 q^{31} +679259. i q^{33} +2.89856e6i q^{37} -2.07317e6 q^{39} +2.51515e7 q^{41} -2.00204e7i q^{43} -3.73010e7i q^{47} -5.22402e7 q^{49} +2.53380e6 q^{51} -2.55155e7i q^{53} -9.79875e6i q^{57} +9.96495e7 q^{59} +2.00434e8 q^{61} -1.87966e8i q^{63} +8.09951e7i q^{67} +1.51648e7 q^{69} -4.31522e7 q^{71} -3.40820e8i q^{73} -5.35268e8i q^{77} -2.81089e8 q^{79} +3.78638e8 q^{81} -6.01017e8i q^{83} +5.23011e7i q^{87} -5.39917e8 q^{89} +1.63369e9 q^{91} -4.38028e7i q^{93} -4.23026e8i q^{97} +1.08660e9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 69764 q^{9}+O(q^{10}) \) 4 * q - 69764 * q^9	\( 4 q - 69764 q^{9} + 205440 q^{11} - 274544 q^{19} + 5680624 q^{21} + 13787496 q^{29} + 583664 q^{31} + 951056 q^{39} + 59546904 q^{41} - 223875828 q^{49} - 54013392 q^{51} + 185861712 q^{59} + 391347848 q^{61} + 344148528 q^{69} + 622414032 q^{71} - 1084523552 q^{79} + 3762476756 q^{81} + 924583704 q^{89} + 3080075536 q^{91} - 2952090240 q^{99}+O(q^{100}) \) 4 * q - 69764 * q^9 + 205440 * q^11 - 274544 * q^19 + 5680624 * q^21 + 13787496 * q^29 + 583664 * q^31 + 951056 * q^39 + 59546904 * q^41 - 223875828 * q^49 - 54013392 * q^51 + 185861712 * q^59 + 391347848 * q^61 + 344148528 * q^69 + 622414032 * q^71 - 1084523552 * q^79 + 3762476756 * q^81 + 924583704 * q^89 + 3080075536 * q^91 - 2952090240 * q^99

Introduction

L-functions

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Database

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