LMFDB - Embedded newform 400.10.c.e.49.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [400,10,Mod(49,400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(400, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("400.49");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 400 = 2^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	400.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:	$206.014334466$
Analytic rank:	$0$
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:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 5)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	400.49
Dual form			400.10.c.e.49.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+114.000i q^{3} +4242.00i q^{7} +6687.00 q^{9} +O(q^{10})$	$q+114.000i q^{3} +4242.00i q^{7} +6687.00 q^{9} +46208.0 q^{11} -115934. i q^{13} -494842. i q^{17} -1.00874e6 q^{19} -483588. q^{21} +532554. i q^{23} +3.00618e6i q^{27} -4.19639e6 q^{29} +3.36503e6 q^{31} +5.26771e6i q^{33} +1.49314e7i q^{37} +1.32165e7 q^{39} +1.10563e7 q^{41} +6.39679e6i q^{43} -3.55592e7i q^{47} +2.23590e7 q^{49} +5.64120e7 q^{51} +3.97386e7i q^{53} -1.14996e8i q^{57} -8.51856e7 q^{59} +4.57486e7 q^{61} +2.83663e7i q^{63} -4.52862e7i q^{67} -6.07112e7 q^{69} +1.89967e8 q^{71} +4.12171e8i q^{73} +1.96014e8i q^{77} +9.50408e7 q^{79} -2.11084e8 q^{81} -2.61706e8i q^{83} -4.78388e8i q^{87} +1.99386e7 q^{89} +4.91792e8 q^{91} +3.83613e8i q^{93} +1.95034e7i q^{97} +3.08993e8 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 13374 q^{9}+O(q^{10}) $ 2 * q + 13374 * q^9	$ 2 q + 13374 q^{9} + 92416 q^{11} - 2017480 q^{19} - 967176 q^{21} - 8392780 q^{29} + 6730056 q^{31} + 26432952 q^{39} + 22112524 q^{41} + 44718086 q^{49} + 112823976 q^{51} - 170371240 q^{59} + 91497284 q^{61} - 121422312 q^{69} + 379934936 q^{71} + 190081680 q^{79} - 422168598 q^{81} + 39877260 q^{89} + 983584056 q^{91} + 617985792 q^{99}+O(q^{100}) $ 2 * q + 13374 * q^9 + 92416 * q^11 - 2017480 * q^19 - 967176 * q^21 - 8392780 * q^29 + 6730056 * q^31 + 26432952 * q^39 + 22112524 * q^41 + 44718086 * q^49 + 112823976 * q^51 - 170371240 * q^59 + 91497284 * q^61 - 121422312 * q^69 + 379934936 * q^71 + 190081680 * q^79 - 422168598 * q^81 + 39877260 * q^89 + 983584056 * q^91 + 617985792 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/400\mathbb{Z}\right)^\times$.

$n$	$101$	$177$	$351$
$\chi(n)$	$1$	$-1$	$1$

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
$3$	114.000i	0.812567i	0.913747	+	0.406284i	$0.133175\pi$
$3$	114.000i	0.812567i	−0.913747	+	0.406284i	$0.866825\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	4242.00i	0.667774i	0.942613	+	0.333887i	$0.108360\pi$
$7$	4242.00i	0.667774i	−0.942613	+	0.333887i	$0.891640\pi$
$8$	0	0
$9$	6687.00	0.339735
$10$	0	0
$11$	46208.0	0.951590	0.475795	−	0.879556i	$-0.342160\pi$
$11$	46208.0	0.951590	0.475795	+	0.879556i	$0.342160\pi$
$12$	0	0
$13$	− 115934.i	− 1.12581i	−0.826521	−	0.562906i	$-0.809683\pi$
$13$	− 115934.i	− 1.12581i	0.826521	−	0.562906i	$-0.190317\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 494842.i	− 1.43697i	−0.695545	−	0.718483i	$-0.744837\pi$
$17$	− 494842.i	− 1.43697i	0.695545	−	0.718483i	$-0.255163\pi$
$18$	0	0
$19$	−1.00874e6	−1.77578	−0.887888	−	0.460060i	$-0.847828\pi$
$19$	−1.00874e6	−1.77578	−0.887888	+	0.460060i	$0.847828\pi$
$20$	0	0
$21$	−483588.	−0.542611
$22$	0	0
$23$	532554.i	0.396815i	0.980120	+	0.198408i	$0.0635770\pi$
$23$	532554.i	0.396815i	−0.980120	+	0.198408i	$0.936423\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	3.00618e6i	1.08862i
$28$	0	0
$29$	−4.19639e6	−1.10175	−0.550877	−	0.834586i	$-0.685707\pi$
$29$	−4.19639e6	−1.10175	−0.550877	+	0.834586i	$0.685707\pi$
$30$	0	0
$31$	3.36503e6	0.654427	0.327213	−	0.944950i	$-0.393890\pi$
$31$	3.36503e6	0.654427	0.327213	+	0.944950i	$0.393890\pi$
$32$	0	0
$33$	5.26771e6i	0.773231i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	1.49314e7i	1.30976i	0.755733	+	0.654880i	$0.227281\pi$
$37$	1.49314e7i	1.30976i	−0.755733	+	0.654880i	$0.772719\pi$
$38$	0	0
$39$	1.32165e7	0.914797
$40$	0	0
$41$	1.10563e7	0.611056	0.305528	−	0.952183i	$-0.401167\pi$
$41$	1.10563e7	0.611056	0.305528	+	0.952183i	$0.401167\pi$
$42$	0	0
$43$	6.39679e6i	0.285335i	0.989771	+	0.142667i	$0.0455679\pi$
$43$	6.39679e6i	0.285335i	−0.989771	+	0.142667i	$0.954432\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 3.55592e7i	− 1.06295i	−0.847075	−	0.531473i	$-0.821639\pi$
$47$	− 3.55592e7i	− 1.06295i	0.847075	−	0.531473i	$-0.178361\pi$
$48$	0	0
$49$	2.23590e7	0.554078
$50$	0	0
$51$	5.64120e7	1.16763
$52$	0	0
$53$	3.97386e7i	0.691785i	0.938274	+	0.345892i	$0.112424\pi$
$53$	3.97386e7i	0.691785i	−0.938274	+	0.345892i	$0.887576\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 1.14996e8i	− 1.44294i
$58$	0	0
$59$	−8.51856e7	−0.915234	−0.457617	−	0.889149i	$-0.651297\pi$
$59$	−8.51856e7	−0.915234	−0.457617	+	0.889149i	$0.651297\pi$
$60$	0	0
$61$	4.57486e7	0.423052	0.211526	−	0.977372i	$-0.432157\pi$
$61$	4.57486e7	0.423052	0.211526	+	0.977372i	$0.432157\pi$
$62$	0	0
$63$	2.83663e7i	0.226866i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 4.52862e7i	− 0.274555i	−0.990533	−	0.137277i	$-0.956165\pi$
$67$	− 4.52862e7i	− 0.274555i	0.990533	−	0.137277i	$-0.0438352\pi$
$68$	0	0
$69$	−6.07112e7	−0.322439
$70$	0	0
$71$	1.89967e8	0.887190	0.443595	−	0.896227i	$-0.353703\pi$
$71$	1.89967e8	0.887190	0.443595	+	0.896227i	$0.353703\pi$
$72$	0	0
$73$	4.12171e8i	1.69873i	0.527805	+	0.849365i	$0.323015\pi$
$73$	4.12171e8i	1.69873i	−0.527805	+	0.849365i	$0.676985\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1.96014e8i	0.635447i
$78$	0	0
$79$	9.50408e7	0.274529	0.137265	−	0.990534i	$-0.456169\pi$
$79$	9.50408e7	0.274529	0.137265	+	0.990534i	$0.456169\pi$
$80$	0	0
$81$	−2.11084e8	−0.544845
$82$	0	0
$83$	− 2.61706e8i	− 0.605289i	−0.953104	−	0.302645i	$-0.902131\pi$
$83$	− 2.61706e8i	− 0.605289i	0.953104	−	0.302645i	$-0.0978695\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 4.78388e8i	− 0.895249i
$88$	0	0
$89$	1.99386e7	0.0336853	0.0168426	−	0.999858i	$-0.494639\pi$
$89$	1.99386e7	0.0336853	0.0168426	+	0.999858i	$0.494639\pi$
$90$	0	0
$91$	4.91792e8	0.751788
$92$	0	0
$93$	3.83613e8i	0.531766i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	1.95034e7i	0.0223685i	0.999937	+	0.0111842i	$0.00356013\pi$
$97$	1.95034e7i	0.0223685i	−0.999937	+	0.0111842i	$0.996440\pi$
$98$	0	0
$99$	3.08993e8	0.323288
$100$	0	0
$101$	−2.16672e8	−0.207184	−0.103592	−	0.994620i	$-0.533034\pi$
$101$	−2.16672e8	−0.207184	−0.103592	+	0.994620i	$0.533034\pi$
$102$	0	0
$103$	7.48234e8i	0.655043i	0.944844	+	0.327522i	$0.106213\pi$
$103$	7.48234e8i	0.655043i	−0.944844	+	0.327522i	$0.893787\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1.05711e9i	0.779637i	0.920892	+	0.389819i	$0.127462\pi$
$107$	1.05711e9i	0.779637i	−0.920892	+	0.389819i	$0.872538\pi$
$108$	0	0
$109$	2.49659e9	1.69406	0.847029	−	0.531547i	$-0.178389\pi$
$109$	2.49659e9	1.69406	0.847029	+	0.531547i	$0.178389\pi$
$110$	0	0
$111$	−1.70217e9	−1.06427
$112$	0	0
$113$	1.11566e9i	0.643695i	0.946791	+	0.321848i	$0.104304\pi$
$113$	1.11566e9i	0.643695i	−0.946791	+	0.321848i	$0.895696\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 7.75251e8i	− 0.382477i
$118$	0	0
$119$	2.09912e9	0.959568
$120$	0	0
$121$	−2.22768e8	−0.0944756
$122$	0	0
$123$	1.26041e9i	0.496524i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	3.12054e8i	0.106442i	0.998583	+	0.0532210i	$0.0169488\pi$
$127$	3.12054e8i	0.106442i	−0.998583	+	0.0532210i	$0.983051\pi$
$128$	0	0
$129$	−7.29235e8	−0.231853
$130$	0	0
$131$	2.81701e9	0.835733	0.417867	−	0.908508i	$-0.362778\pi$
$131$	2.81701e9	0.835733	0.417867	+	0.908508i	$0.362778\pi$
$132$	0	0
$133$	− 4.27908e9i	− 1.18582i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	7.15311e9i	1.73481i	0.497600	+	0.867406i	$0.334215\pi$
$137$	7.15311e9i	1.73481i	−0.497600	+	0.867406i	$0.665785\pi$
$138$	0	0
$139$	2.52323e9	0.573310	0.286655	−	0.958034i	$-0.407457\pi$
$139$	2.52323e9	0.573310	0.286655	+	0.958034i	$0.407457\pi$
$140$	0	0
$141$	4.05374e9	0.863715
$142$	0	0
$143$	− 5.35708e9i	− 1.07131i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	2.54893e9i	0.450225i
$148$	0	0
$149$	−1.46531e9	−0.243551	−0.121775	−	0.992558i	$-0.538859\pi$
$149$	−1.46531e9	−0.243551	−0.121775	+	0.992558i	$0.538859\pi$
$150$	0	0
$151$	−3.65515e9	−0.572149	−0.286075	−	0.958207i	$-0.592350\pi$
$151$	−3.65515e9	−0.572149	−0.286075	+	0.958207i	$0.592350\pi$
$152$	0	0
$153$	− 3.30901e9i	− 0.488187i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 5.55191e9i	− 0.729279i	−0.931149	−	0.364639i	$-0.881192\pi$
$157$	− 5.55191e9i	− 0.729279i	0.931149	−	0.364639i	$-0.118808\pi$
$158$	0	0
$159$	−4.53020e9	−0.562122
$160$	0	0
$161$	−2.25909e9	−0.264983
$162$	0	0
$163$	1.67637e10i	1.86006i	0.367485	+	0.930029i	$0.380219\pi$
$163$	1.67637e10i	1.86006i	−0.367485	+	0.930029i	$0.619781\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1.39549e10i	1.38836i	0.719799	+	0.694182i	$0.244234\pi$
$167$	1.39549e10i	1.38836i	−0.719799	+	0.694182i	$0.755766\pi$
$168$	0	0
$169$	−2.83619e9	−0.267452
$170$	0	0
$171$	−6.74544e9	−0.603293
$172$	0	0
$173$	1.57000e10i	1.33258i	0.745694	+	0.666289i	$0.232118\pi$
$173$	1.57000e10i	1.33258i	−0.745694	+	0.666289i	$0.767882\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 9.71116e9i	− 0.743689i
$178$	0	0
$179$	−2.51902e10	−1.83397	−0.916985	−	0.398921i	$-0.869385\pi$
$179$	−2.51902e10	−1.83397	−0.916985	+	0.398921i	$0.869385\pi$
$180$	0	0
$181$	1.04482e10	0.723579	0.361789	−	0.932260i	$-0.382166\pi$
$181$	1.04482e10	0.723579	0.361789	+	0.932260i	$0.382166\pi$
$182$	0	0
$183$	5.21535e9i	0.343758i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 2.28657e10i	− 1.36740i
$188$	0	0
$189$	−1.27522e10	−0.726955
$190$	0	0
$191$	9.68625e9	0.526630	0.263315	−	0.964710i	$-0.415184\pi$
$191$	9.68625e9	0.526630	0.263315	+	0.964710i	$0.415184\pi$
$192$	0	0
$193$	− 2.04431e10i	− 1.06057i	−0.847820	−	0.530285i	$-0.822085\pi$
$193$	− 2.04431e10i	− 1.06057i	0.847820	−	0.530285i	$-0.177915\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	2.52431e10i	1.19411i	0.802200	+	0.597056i	$0.203663\pi$
$197$	2.52431e10i	1.19411i	−0.802200	+	0.597056i	$0.796337\pi$
$198$	0	0
$199$	8.06736e9	0.364664	0.182332	−	0.983237i	$-0.441635\pi$
$199$	8.06736e9	0.364664	0.182332	+	0.983237i	$0.441635\pi$
$200$	0	0
$201$	5.16262e9	0.223094
$202$	0	0
$203$	− 1.78011e10i	− 0.735723i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	3.56119e9i	0.134812i
$208$	0	0
$209$	−4.66119e10	−1.68981
$210$	0	0
$211$	8.04589e9	0.279449	0.139725	−	0.990190i	$-0.455378\pi$
$211$	8.04589e9	0.279449	0.139725	+	0.990190i	$0.455378\pi$
$212$	0	0
$213$	2.16563e10i	0.720901i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	1.42744e10i	0.437009i
$218$	0	0
$219$	−4.69875e10	−1.38033
$220$	0	0
$221$	−5.73690e10	−1.61775
$222$	0	0
$223$	− 3.18618e10i	− 0.862777i	−0.902166	−	0.431388i	$-0.858024\pi$
$223$	− 3.18618e10i	− 0.862777i	0.902166	−	0.431388i	$-0.141976\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	3.02621e10i	0.756454i	0.925713	+	0.378227i	$0.123466\pi$
$227$	3.02621e10i	0.756454i	−0.925713	+	0.378227i	$0.876534\pi$
$228$	0	0
$229$	−2.06101e10	−0.495245	−0.247623	−	0.968857i	$-0.579649\pi$
$229$	−2.06101e10	−0.495245	−0.247623	+	0.968857i	$0.579649\pi$
$230$	0	0
$231$	−2.23456e10	−0.516344
$232$	0	0
$233$	3.61735e9i	0.0804061i	0.999192	+	0.0402031i	$0.0128005\pi$
$233$	3.61735e9i	0.0804061i	−0.999192	+	0.0402031i	$0.987200\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	1.08347e10i	0.223073i
$238$	0	0
$239$	2.21916e9	0.0439945	0.0219973	−	0.999758i	$-0.492997\pi$
$239$	2.21916e9	0.0439945	0.0219973	+	0.999758i	$0.492997\pi$
$240$	0	0
$241$	2.57977e10	0.492611	0.246306	−	0.969192i	$-0.420783\pi$
$241$	2.57977e10	0.492611	0.246306	+	0.969192i	$0.420783\pi$
$242$	0	0
$243$	3.51070e10i	0.645901i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	1.16947e11i	1.99919i
$248$	0	0
$249$	2.98345e10	0.491838
$250$	0	0
$251$	8.91848e10	1.41827	0.709135	−	0.705072i	$-0.249086\pi$
$251$	8.91848e10	1.41827	0.709135	+	0.705072i	$0.249086\pi$
$252$	0	0
$253$	2.46083e10i	0.377606i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	1.10058e11i	1.57371i	0.617141	+	0.786853i	$0.288291\pi$
$257$	1.10058e11i	1.57371i	−0.617141	+	0.786853i	$0.711709\pi$
$258$	0	0
$259$	−6.33388e10	−0.874623
$260$	0	0
$261$	−2.80613e10	−0.374304
$262$	0	0
$263$	− 2.06374e10i	− 0.265983i	−0.991117	−	0.132992i	$-0.957542\pi$
$263$	− 2.06374e10i	− 0.265983i	0.991117	−	0.132992i	$-0.0424584\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	2.27300e9i	0.0273716i
$268$	0	0
$269$	2.64890e10	0.308446	0.154223	−	0.988036i	$-0.450713\pi$
$269$	2.64890e10	0.308446	0.154223	+	0.988036i	$0.450713\pi$
$270$	0	0
$271$	−3.44004e10	−0.387438	−0.193719	−	0.981057i	$-0.562055\pi$
$271$	−3.44004e10	−0.387438	−0.193719	+	0.981057i	$0.562055\pi$
$272$	0	0
$273$	5.60643e10i	0.610878i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	1.06023e11i	1.08203i	0.841012	+	0.541017i	$0.181960\pi$
$277$	1.06023e11i	1.08203i	−0.841012	+	0.541017i	$0.818040\pi$
$278$	0	0
$279$	2.25019e10	0.222332
$280$	0	0
$281$	1.00299e11	0.959662	0.479831	−	0.877361i	$-0.340698\pi$
$281$	1.00299e11	0.959662	0.479831	+	0.877361i	$0.340698\pi$
$282$	0	0
$283$	2.49296e9i	0.0231035i	0.999933	+	0.0115517i	$0.00367711\pi$
$283$	2.49296e9i	0.0231035i	−0.999933	+	0.0115517i	$0.996323\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	4.69007e10i	0.408047i
$288$	0	0
$289$	−1.26281e11	−1.06487
$290$	0	0
$291$	−2.22338e9	−0.0181759
$292$	0	0
$293$	− 8.47068e10i	− 0.671451i	−0.941960	−	0.335725i	$-0.891019\pi$
$293$	− 8.47068e10i	− 0.671451i	0.941960	−	0.335725i	$-0.108981\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	1.38910e11i	1.03592i
$298$	0	0
$299$	6.17411e10	0.446739
$300$	0	0
$301$	−2.71352e10	−0.190539
$302$	0	0
$303$	− 2.47006e10i	− 0.168351i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	6.35507e9i	0.0408317i	0.999792	+	0.0204159i	$0.00649902\pi$
$307$	6.35507e9i	0.0408317i	−0.999792	+	0.0204159i	$0.993501\pi$
$308$	0	0
$309$	−8.52987e10	−0.532267
$310$	0	0
$311$	−2.71253e11	−1.64419	−0.822096	−	0.569350i	$-0.807195\pi$
$311$	−2.71253e11	−1.64419	−0.822096	+	0.569350i	$0.807195\pi$
$312$	0	0
$313$	1.63499e11i	0.962865i	0.876483	+	0.481432i	$0.159883\pi$
$313$	1.63499e11i	0.962865i	−0.876483	+	0.481432i	$0.840117\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 2.53606e11i	− 1.41056i	−0.708927	−	0.705282i	$-0.750820\pi$
$317$	− 2.53606e11i	− 1.41056i	0.708927	−	0.705282i	$-0.249180\pi$
$318$	0	0
$319$	−1.93907e11	−1.04842
$320$	0	0
$321$	−1.20510e11	−0.633508
$322$	0	0
$323$	4.99167e11i	2.55173i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	2.84611e11i	1.37654i
$328$	0	0
$329$	1.50842e11	0.709808
$330$	0	0
$331$	2.40064e11	1.09926	0.549631	−	0.835408i	$-0.314768\pi$
$331$	2.40064e11	1.09926	0.549631	+	0.835408i	$0.314768\pi$
$332$	0	0
$333$	9.98460e10i	0.444971i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	5.13812e10i	0.217005i	0.994096	+	0.108502i	$0.0346055\pi$
$337$	5.13812e10i	0.217005i	−0.994096	+	0.108502i	$0.965394\pi$
$338$	0	0
$339$	−1.27186e11	−0.523046
$340$	0	0
$341$	1.55491e11	0.622746
$342$	0	0
$343$	2.66027e11i	1.03777i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	2.76560e11i	1.02401i	0.858981	+	0.512007i	$0.171098\pi$
$347$	2.76560e11i	1.02401i	−0.858981	+	0.512007i	$0.828902\pi$
$348$	0	0
$349$	−4.66592e11	−1.68354	−0.841770	−	0.539837i	$-0.818486\pi$
$349$	−4.66592e11	−1.68354	−0.841770	+	0.539837i	$0.818486\pi$
$350$	0	0
$351$	3.48518e11	1.22559
$352$	0	0
$353$	− 1.00998e11i	− 0.346198i	−0.984904	−	0.173099i	$-0.944622\pi$
$353$	− 1.00998e11i	− 0.346198i	0.984904	−	0.173099i	$-0.0553781\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	2.39300e11i	0.779714i
$358$	0	0
$359$	−1.66447e11	−0.528874	−0.264437	−	0.964403i	$-0.585186\pi$
$359$	−1.66447e11	−0.528874	−0.264437	+	0.964403i	$0.585186\pi$
$360$	0	0
$361$	6.94869e11	2.15338
$362$	0	0
$363$	− 2.53956e10i	− 0.0767677i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 2.97381e11i	− 0.855688i	−0.903853	−	0.427844i	$-0.859273\pi$
$367$	− 2.97381e11i	− 0.855688i	0.903853	−	0.427844i	$-0.140727\pi$
$368$	0	0
$369$	7.39332e10	0.207597
$370$	0	0
$371$	−1.68571e11	−0.461956
$372$	0	0
$373$	1.95714e11i	0.523517i	0.965133	+	0.261759i	$0.0843024\pi$
$373$	1.95714e11i	0.523517i	−0.965133	+	0.261759i	$0.915698\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	4.86504e11i	1.24037i
$378$	0	0
$379$	−1.67009e11	−0.415781	−0.207890	−	0.978152i	$-0.566660\pi$
$379$	−1.67009e11	−0.415781	−0.207890	+	0.978152i	$0.566660\pi$
$380$	0	0
$381$	−3.55741e10	−0.0864912
$382$	0	0
$383$	− 7.20782e10i	− 0.171163i	−0.996331	−	0.0855814i	$-0.972725\pi$
$383$	− 7.20782e10i	− 0.171163i	0.996331	−	0.0855814i	$-0.0272748\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	4.27754e10i	0.0969381i
$388$	0	0
$389$	−7.92249e11	−1.75424	−0.877119	−	0.480272i	$-0.840538\pi$
$389$	−7.92249e11	−1.75424	−0.877119	+	0.480272i	$0.840538\pi$
$390$	0	0
$391$	2.63530e11	0.570210
$392$	0	0
$393$	3.21139e11i	0.679089i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	1.47288e11i	0.297584i	0.988869	+	0.148792i	$0.0475384\pi$
$397$	1.47288e11i	0.297584i	−0.988869	+	0.148792i	$0.952462\pi$
$398$	0	0
$399$	4.87815e11	0.963555
$400$	0	0
$401$	−3.22101e11	−0.622075	−0.311037	−	0.950398i	$-0.600676\pi$
$401$	−3.22101e11	−0.622075	−0.311037	+	0.950398i	$0.600676\pi$
$402$	0	0
$403$	− 3.90121e11i	− 0.736761i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	6.89948e11i	1.24635i
$408$	0	0
$409$	3.96423e11	0.700493	0.350247	−	0.936658i	$-0.386098\pi$
$409$	3.96423e11	0.700493	0.350247	+	0.936658i	$0.386098\pi$
$410$	0	0
$411$	−8.15455e11	−1.40965
$412$	0	0
$413$	− 3.61357e11i	− 0.611170i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	2.87648e11i	0.465853i
$418$	0	0
$419$	−1.23162e12	−1.95215	−0.976074	−	0.217441i	$-0.930229\pi$
$419$	−1.23162e12	−1.95215	−0.976074	+	0.217441i	$0.930229\pi$
$420$	0	0
$421$	−1.17500e12	−1.82293	−0.911465	−	0.411379i	$-0.865047\pi$
$421$	−1.17500e12	−1.82293	−0.911465	+	0.411379i	$0.865047\pi$
$422$	0	0
$423$	− 2.37784e11i	− 0.361120i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	1.94066e11i	0.282503i
$428$	0	0
$429$	6.10707e11	0.870513
$430$	0	0
$431$	2.32327e11	0.324304	0.162152	−	0.986766i	$-0.448157\pi$
$431$	2.32327e11	0.324304	0.162152	+	0.986766i	$0.448157\pi$
$432$	0	0
$433$	− 1.51554e11i	− 0.207191i	−0.994620	−	0.103596i	$-0.966965\pi$
$433$	− 1.51554e11i	− 0.207191i	0.994620	−	0.103596i	$-0.0330347\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 5.37209e11i	− 0.704655i
$438$	0	0
$439$	−5.21385e11	−0.669990	−0.334995	−	0.942220i	$-0.608735\pi$
$439$	−5.21385e11	−0.669990	−0.334995	+	0.942220i	$0.608735\pi$
$440$	0	0
$441$	1.49515e11	0.188240
$442$	0	0
$443$	− 1.10169e12i	− 1.35908i	−0.733639	−	0.679539i	$-0.762180\pi$
$443$	− 1.10169e12i	− 1.35908i	0.733639	−	0.679539i	$-0.237820\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 1.67045e11i	− 0.197901i
$448$	0	0
$449$	−5.85672e11	−0.680058	−0.340029	−	0.940415i	$-0.610437\pi$
$449$	−5.85672e11	−0.680058	−0.340029	+	0.940415i	$0.610437\pi$
$450$	0	0
$451$	5.10888e11	0.581475
$452$	0	0
$453$	− 4.16687e11i	− 0.464909i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	5.70804e11i	0.612159i	0.952006	+	0.306079i	$0.0990173\pi$
$457$	5.70804e11i	0.612159i	−0.952006	+	0.306079i	$0.900983\pi$
$458$	0	0
$459$	1.48758e12	1.56432
$460$	0	0
$461$	−7.60279e11	−0.784005	−0.392002	−	0.919964i	$-0.628218\pi$
$461$	−7.60279e11	−0.784005	−0.392002	+	0.919964i	$0.628218\pi$
$462$	0	0
$463$	7.14289e11i	0.722370i	0.932494	+	0.361185i	$0.117628\pi$
$463$	7.14289e11i	0.722370i	−0.932494	+	0.361185i	$0.882372\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	7.82521e11i	0.761325i	0.924714	+	0.380662i	$0.124304\pi$
$467$	7.82521e11i	0.761325i	−0.924714	+	0.380662i	$0.875696\pi$
$468$	0	0
$469$	1.92104e11	0.183340
$470$	0	0
$471$	6.32917e11	0.592588
$472$	0	0
$473$	2.95583e11i	0.271522i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	2.65732e11i	0.235023i
$478$	0	0
$479$	−6.95738e11	−0.603860	−0.301930	−	0.953330i	$-0.597631\pi$
$479$	−6.95738e11	−0.603860	−0.301930	+	0.953330i	$0.597631\pi$
$480$	0	0
$481$	1.73105e12	1.47454
$482$	0	0
$483$	− 2.57537e11i	− 0.215316i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 1.65196e12i	− 1.33082i	−0.746480	−	0.665408i	$-0.768258\pi$
$487$	− 1.65196e12i	− 1.33082i	0.746480	−	0.665408i	$-0.231742\pi$
$488$	0	0
$489$	−1.91107e12	−1.51142
$490$	0	0
$491$	−1.19989e12	−0.931694	−0.465847	−	0.884865i	$-0.654250\pi$
$491$	−1.19989e12	−0.931694	−0.465847	+	0.884865i	$0.654250\pi$
$492$	0	0
$493$	2.07655e12i	1.58318i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	8.05842e11i	0.592442i
$498$	0	0
$499$	1.43146e12	1.03354	0.516768	−	0.856125i	$-0.327135\pi$
$499$	1.43146e12	1.03354	0.516768	+	0.856125i	$0.327135\pi$
$500$	0	0
$501$	−1.59086e12	−1.12814
$502$	0	0
$503$	− 1.41833e12i	− 0.987919i	−0.869485	−	0.493959i	$-0.835549\pi$
$503$	− 1.41833e12i	− 0.987919i	0.869485	−	0.493959i	$-0.164451\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 3.23326e11i	− 0.217323i
$508$	0	0
$509$	1.16463e12	0.769059	0.384529	−	0.923113i	$-0.374364\pi$
$509$	1.16463e12	0.769059	0.384529	+	0.923113i	$0.374364\pi$
$510$	0	0
$511$	−1.74843e12	−1.13437
$512$	0	0
$513$	− 3.03245e12i	− 1.93315i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 1.64312e12i	− 1.01149i
$518$	0	0
$519$	−1.78980e12	−1.08281
$520$	0	0
$521$	−8.36946e11	−0.497654	−0.248827	−	0.968548i	$-0.580045\pi$
$521$	−8.36946e11	−0.497654	−0.248827	+	0.968548i	$0.580045\pi$
$522$	0	0
$523$	2.60288e12i	1.52124i	0.649199	+	0.760618i	$0.275104\pi$
$523$	2.60288e12i	1.52124i	−0.649199	+	0.760618i	$0.724896\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 1.66516e12i	− 0.940389i
$528$	0	0
$529$	1.51754e12	0.842538
$530$	0	0
$531$	−5.69636e11	−0.310937
$532$	0	0
$533$	− 1.28180e12i	− 0.687934i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 2.87168e12i	− 1.49022i
$538$	0	0
$539$	1.03317e12	0.527255
$540$	0	0
$541$	3.07406e12	1.54286	0.771428	−	0.636317i	$-0.219543\pi$
$541$	3.07406e12	1.54286	0.771428	+	0.636317i	$0.219543\pi$
$542$	0	0
$543$	1.19109e12i	0.587956i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	1.32972e12i	0.635062i	0.948248	+	0.317531i	$0.102854\pi$
$547$	1.32972e12i	0.635062i	−0.948248	+	0.317531i	$0.897146\pi$
$548$	0	0
$549$	3.05921e11	0.143726
$550$	0	0
$551$	4.23307e12	1.95647
$552$	0	0
$553$	4.03163e11i	0.183323i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 2.49418e12i	− 1.09794i	−0.835841	−	0.548971i	$-0.815020\pi$
$557$	− 2.49418e12i	− 1.09794i	0.835841	−	0.548971i	$-0.184980\pi$
$558$	0	0
$559$	7.41606e11	0.321233
$560$	0	0
$561$	2.60669e12	1.11111
$562$	0	0
$563$	− 1.06689e12i	− 0.447541i	−0.974642	−	0.223770i	$-0.928163\pi$
$563$	− 1.06689e12i	− 0.447541i	0.974642	−	0.223770i	$-0.0718365\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 8.95420e11i	− 0.363834i
$568$	0	0
$569$	4.57278e12	1.82884	0.914420	−	0.404768i	$-0.132648\pi$
$569$	4.57278e12	1.82884	0.914420	+	0.404768i	$0.132648\pi$
$570$	0	0
$571$	2.32305e12	0.914525	0.457263	−	0.889332i	$-0.348830\pi$
$571$	2.32305e12	0.914525	0.457263	+	0.889332i	$0.348830\pi$
$572$	0	0
$573$	1.10423e12i	0.427922i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 1.01144e12i	− 0.379881i	−0.981796	−	0.189940i	$-0.939171\pi$
$577$	− 1.01144e12i	− 0.379881i	0.981796	−	0.189940i	$-0.0608295\pi$
$578$	0	0
$579$	2.33051e12	0.861784
$580$	0	0
$581$	1.11016e12	0.404196
$582$	0	0
$583$	1.83624e12i	0.658296i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 1.10143e12i	− 0.382899i	−0.981502	−	0.191449i	$-0.938681\pi$
$587$	− 1.10143e12i	− 0.382899i	0.981502	−	0.191449i	$-0.0613188\pi$
$588$	0	0
$589$	−3.39444e12	−1.16211
$590$	0	0
$591$	−2.87772e12	−0.970296
$592$	0	0
$593$	9.84366e11i	0.326897i	0.986552	+	0.163448i	$0.0522617\pi$
$593$	9.84366e11i	0.326897i	−0.986552	+	0.163448i	$0.947738\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	9.19679e11i	0.296314i
$598$	0	0
$599$	4.51397e12	1.43264	0.716321	−	0.697771i	$-0.245825\pi$
$599$	4.51397e12	1.43264	0.716321	+	0.697771i	$0.245825\pi$
$600$	0	0
$601$	−1.44212e12	−0.450885	−0.225443	−	0.974256i	$-0.572383\pi$
$601$	−1.44212e12	−0.450885	−0.225443	+	0.974256i	$0.572383\pi$
$602$	0	0
$603$	− 3.02829e11i	− 0.0932758i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 4.12105e12i	− 1.23214i	−0.787692	−	0.616069i	$-0.788724\pi$
$607$	− 4.12105e12i	− 1.23214i	0.787692	−	0.616069i	$-0.211276\pi$
$608$	0	0
$609$	2.02932e12	0.597824
$610$	0	0
$611$	−4.12252e12	−1.19668
$612$	0	0
$613$	1.38470e12i	0.396080i	0.980194	+	0.198040i	$0.0634576\pi$
$613$	1.38470e12i	0.396080i	−0.980194	+	0.198040i	$0.936542\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 6.22714e11i	− 0.172984i	−0.996253	−	0.0864918i	$-0.972434\pi$
$617$	− 6.22714e11i	− 0.172984i	0.996253	−	0.0864918i	$-0.0275657\pi$
$618$	0	0
$619$	1.89438e12	0.518631	0.259315	−	0.965793i	$-0.416503\pi$
$619$	1.89438e12	0.518631	0.259315	+	0.965793i	$0.416503\pi$
$620$	0	0
$621$	−1.60095e12	−0.431983
$622$	0	0
$623$	8.45797e10i	0.0224942i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 5.31375e12i	− 1.37308i
$628$	0	0
$629$	7.38866e12	1.88208
$630$	0	0
$631$	1.10160e12	0.276624	0.138312	−	0.990389i	$-0.455832\pi$
$631$	1.10160e12	0.276624	0.138312	+	0.990389i	$0.455832\pi$
$632$	0	0
$633$	9.17231e11i	0.227071i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 2.59217e12i	− 0.623787i
$638$	0	0
$639$	1.27031e12	0.301409
$640$	0	0
$641$	−4.85252e12	−1.13529	−0.567644	−	0.823274i	$-0.692145\pi$
$641$	−4.85252e12	−1.13529	−0.567644	+	0.823274i	$0.692145\pi$
$642$	0	0
$643$	− 1.57487e12i	− 0.363325i	−0.983361	−	0.181662i	$-0.941852\pi$
$643$	− 1.57487e12i	− 0.363325i	0.983361	−	0.181662i	$-0.0581478\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	6.93025e12i	1.55482i	0.628995	+	0.777410i	$0.283467\pi$
$647$	6.93025e12i	1.55482i	−0.628995	+	0.777410i	$0.716533\pi$
$648$	0	0
$649$	−3.93626e12	−0.870928
$650$	0	0
$651$	−1.62729e12	−0.355099
$652$	0	0
$653$	− 8.15499e12i	− 1.75515i	−0.479440	−	0.877575i	$-0.659160\pi$
$653$	− 8.15499e12i	− 1.75515i	0.479440	−	0.877575i	$-0.340840\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	2.75619e12i	0.577118i
$658$	0	0
$659$	−2.92553e12	−0.604255	−0.302127	−	0.953268i	$-0.597697\pi$
$659$	−2.92553e12	−0.604255	−0.302127	+	0.953268i	$0.597697\pi$
$660$	0	0
$661$	−1.62921e12	−0.331947	−0.165974	−	0.986130i	$-0.553077\pi$
$661$	−1.62921e12	−0.331947	−0.165974	+	0.986130i	$0.553077\pi$
$662$	0	0
$663$	− 6.54007e12i	− 1.31453i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 2.23480e12i	− 0.437193i
$668$	0	0
$669$	3.63225e12	0.701064
$670$	0	0
$671$	2.11395e12	0.402572
$672$	0	0
$673$	− 2.87521e12i	− 0.540258i	−0.962824	−	0.270129i	$-0.912934\pi$
$673$	− 2.87521e12i	− 0.540258i	0.962824	−	0.270129i	$-0.0870664\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 3.19620e12i	− 0.584769i	−0.956301	−	0.292385i	$-0.905551\pi$
$677$	− 3.19620e12i	− 0.584769i	0.956301	−	0.292385i	$-0.0944487\pi$
$678$	0	0
$679$	−8.27332e10	−0.0149371
$680$	0	0
$681$	−3.44988e12	−0.614670
$682$	0	0
$683$	5.47901e11i	0.0963406i	0.998839	+	0.0481703i	$0.0153390\pi$
$683$	5.47901e11i	0.0963406i	−0.998839	+	0.0481703i	$0.984661\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 2.34955e12i	− 0.402420i
$688$	0	0
$689$	4.60705e12	0.778819
$690$	0	0
$691$	−5.90996e11	−0.0986128	−0.0493064	−	0.998784i	$-0.515701\pi$
$691$	−5.90996e11	−0.0986128	−0.0493064	+	0.998784i	$0.515701\pi$
$692$	0	0
$693$	1.31075e12i	0.215884i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 5.47110e12i	− 0.878066i
$698$	0	0
$699$	−4.12378e11	−0.0653354
$700$	0	0
$701$	7.12738e12	1.11481	0.557403	−	0.830242i	$-0.311798\pi$
$701$	7.12738e12	1.11481	0.557403	+	0.830242i	$0.311798\pi$
$702$	0	0
$703$	− 1.50619e13i	− 2.32584i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 9.19122e11i	− 0.138352i
$708$	0	0
$709$	−9.09269e11	−0.135140	−0.0675701	−	0.997715i	$-0.521525\pi$
$709$	−9.09269e11	−0.135140	−0.0675701	+	0.997715i	$0.521525\pi$
$710$	0	0
$711$	6.35538e11	0.0932671
$712$	0	0
$713$	1.79206e12i	0.259687i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	2.52985e11i	0.0357485i
$718$	0	0
$719$	1.08017e13	1.50734	0.753669	−	0.657254i	$-0.228282\pi$
$719$	1.08017e13	1.50734	0.753669	+	0.657254i	$0.228282\pi$
$720$	0	0
$721$	−3.17401e12	−0.437421
$722$	0	0
$723$	2.94094e12i	0.400280i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 9.77691e12i	− 1.29807i	−0.760760	−	0.649033i	$-0.775174\pi$
$727$	− 9.77691e12i	− 1.29807i	0.760760	−	0.649033i	$-0.224826\pi$
$728$	0	0
$729$	−8.15697e12	−1.06968
$730$	0	0
$731$	3.16540e12	0.410016
$732$	0	0
$733$	3.52295e11i	0.0450753i	0.999746	+	0.0225377i	$0.00717457\pi$
$733$	3.52295e11i	0.0450753i	−0.999746	+	0.0225377i	$0.992825\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 2.09258e12i	− 0.261264i
$738$	0	0
$739$	−2.72124e12	−0.335635	−0.167818	−	0.985818i	$-0.553672\pi$
$739$	−2.72124e12	−0.335635	−0.167818	+	0.985818i	$0.553672\pi$
$740$	0	0
$741$	−1.33320e13	−1.62447
$742$	0	0
$743$	5.85311e12i	0.704590i	0.935889	+	0.352295i	$0.114599\pi$
$743$	5.85311e12i	0.704590i	−0.935889	+	0.352295i	$0.885401\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 1.75003e12i	− 0.205638i
$748$	0	0
$749$	−4.48426e12	−0.520622
$750$	0	0
$751$	5.13723e12	0.589317	0.294659	−	0.955603i	$-0.404794\pi$
$751$	5.13723e12	0.589317	0.294659	+	0.955603i	$0.404794\pi$
$752$	0	0
$753$	1.01671e13i	1.15244i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	1.45713e13i	1.61275i	0.591407	+	0.806373i	$0.298573\pi$
$757$	1.45713e13i	1.61275i	−0.591407	+	0.806373i	$0.701427\pi$
$758$	0	0
$759$	−2.80534e12	−0.306830
$760$	0	0
$761$	1.03936e13	1.12340	0.561700	−	0.827341i	$-0.310148\pi$
$761$	1.03936e13	1.12340	0.561700	+	0.827341i	$0.310148\pi$
$762$	0	0
$763$	1.05905e13i	1.13125i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	9.87591e12i	1.03038i
$768$	0	0
$769$	3.91664e12	0.403873	0.201936	−	0.979399i	$-0.435277\pi$
$769$	3.91664e12	0.403873	0.201936	+	0.979399i	$0.435277\pi$
$770$	0	0
$771$	−1.25466e13	−1.27874
$772$	0	0
$773$	− 6.36894e12i	− 0.641592i	−0.947148	−	0.320796i	$-0.896050\pi$
$773$	− 6.36894e12i	− 0.641592i	0.947148	−	0.320796i	$-0.103950\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 7.22063e12i	− 0.710690i
$778$	0	0
$779$	−1.11529e13	−1.08510
$780$	0	0
$781$	8.77802e12	0.844242
$782$	0	0
$783$	− 1.26151e13i	− 1.19940i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	6.55343e11i	0.0608951i	0.999536	+	0.0304475i	$0.00969325\pi$
$787$	6.55343e11i	0.0608951i	−0.999536	+	0.0304475i	$0.990307\pi$
$788$	0	0
$789$	2.35267e12	0.216129
$790$	0	0
$791$	−4.73265e12	−0.429843
$792$	0	0
$793$	− 5.30382e12i	− 0.476277i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 3.83799e12i	− 0.336931i	−0.985708	−	0.168466i	$-0.946119\pi$
$797$	− 3.83799e12i	− 0.336931i	0.985708	−	0.168466i	$-0.0538812\pi$
$798$	0	0
$799$	−1.75962e13	−1.52742
$800$	0	0
$801$	1.33330e11	0.0114441
$802$	0	0
$803$	1.90456e13i	1.61650i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	3.01974e12i	0.250633i
$808$	0	0
$809$	−2.67581e12	−0.219627	−0.109814	−	0.993952i	$-0.535025\pi$
$809$	−2.67581e12	−0.219627	−0.109814	+	0.993952i	$0.535025\pi$
$810$	0	0
$811$	2.32586e13	1.88795	0.943973	−	0.330023i	$-0.107057\pi$
$811$	2.32586e13	1.88795	0.943973	+	0.330023i	$0.107057\pi$
$812$	0	0
$813$	− 3.92165e12i	− 0.314819i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 6.45270e12i	− 0.506690i
$818$	0	0
$819$	3.28861e12	0.255408
$820$	0	0
$821$	1.65740e13	1.27316	0.636580	−	0.771211i	$-0.280349\pi$
$821$	1.65740e13	1.27316	0.636580	+	0.771211i	$0.280349\pi$
$822$	0	0
$823$	− 1.37332e13i	− 1.04345i	−0.853113	−	0.521726i	$-0.825288\pi$
$823$	− 1.37332e13i	− 1.04345i	0.853113	−	0.521726i	$-0.174712\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1.80036e13i	1.33840i	0.743085	+	0.669198i	$0.233362\pi$
$827$	1.80036e13i	1.33840i	−0.743085	+	0.669198i	$0.766638\pi$
$828$	0	0
$829$	−2.11116e13	−1.55248	−0.776240	−	0.630438i	$-0.782875\pi$
$829$	−2.11116e13	−1.55248	−0.776240	+	0.630438i	$0.782875\pi$
$830$	0	0
$831$	−1.20866e13	−0.879225
$832$	0	0
$833$	− 1.10642e13i	− 0.796191i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	1.01159e13i	0.712425i
$838$	0	0
$839$	−3.47988e12	−0.242458	−0.121229	−	0.992625i	$-0.538683\pi$
$839$	−3.47988e12	−0.242458	−0.121229	+	0.992625i	$0.538683\pi$
$840$	0	0
$841$	3.10254e12	0.213863
$842$	0	0
$843$	1.14341e13i	0.779789i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 9.44984e11i	− 0.0630883i
$848$	0	0
$849$	−2.84198e11	−0.0187731
$850$	0	0
$851$	−7.95175e12	−0.519733
$852$	0	0
$853$	1.12686e13i	0.728784i	0.931246	+	0.364392i	$0.118723\pi$
$853$	1.12686e13i	0.728784i	−0.931246	+	0.364392i	$0.881277\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 3.75227e12i	− 0.237618i	−0.992917	−	0.118809i	$-0.962092\pi$
$857$	− 3.75227e12i	− 0.237618i	0.992917	−	0.118809i	$-0.0379077\pi$
$858$	0	0
$859$	1.57014e13	0.983944	0.491972	−	0.870611i	$-0.336276\pi$
$859$	1.57014e13	0.983944	0.491972	+	0.870611i	$0.336276\pi$
$860$	0	0
$861$	−5.34668e12	−0.331566
$862$	0	0
$863$	1.39316e13i	0.854975i	0.904021	+	0.427487i	$0.140601\pi$
$863$	1.39316e13i	0.854975i	−0.904021	+	0.427487i	$0.859399\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 1.43960e13i	− 0.865279i
$868$	0	0
$869$	4.39165e12	0.261239
$870$	0	0
$871$	−5.25021e12	−0.309097
$872$	0	0
$873$	1.30419e11i	0.00759935i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	1.59832e13i	0.912357i	0.889888	+	0.456179i	$0.150782\pi$
$877$	1.59832e13i	0.912357i	−0.889888	+	0.456179i	$0.849218\pi$
$878$	0	0
$879$	9.65658e12	0.545599
$880$	0	0
$881$	−3.16227e13	−1.76851	−0.884255	−	0.467005i	$-0.845333\pi$
$881$	−3.16227e13	−1.76851	−0.884255	+	0.467005i	$0.845333\pi$
$882$	0	0
$883$	1.48526e13i	0.822202i	0.911590	+	0.411101i	$0.134856\pi$
$883$	1.48526e13i	0.822202i	−0.911590	+	0.411101i	$0.865144\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 9.39325e12i	− 0.509518i	−0.967005	−	0.254759i	$-0.918004\pi$
$887$	− 9.39325e12i	− 0.509518i	0.967005	−	0.254759i	$-0.0819961\pi$
$888$	0	0
$889$	−1.32373e12	−0.0710792
$890$	0	0
$891$	−9.75378e12	−0.518470
$892$	0	0
$893$	3.58699e13i	1.88755i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	7.03849e12i	0.363006i
$898$	0	0
$899$	−1.41210e13	−0.721018
$900$	0	0
$901$	1.96643e13	0.994071
$902$	0	0
$903$	− 3.09341e12i	− 0.154826i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	2.64429e13i	1.29741i	0.761041	+	0.648704i	$0.224689\pi$
$907$	2.64429e13i	1.29741i	−0.761041	+	0.648704i	$0.775311\pi$
$908$	0	0
$909$	−1.44888e12	−0.0703876
$910$	0	0
$911$	3.41645e13	1.64339	0.821697	−	0.569924i	$-0.193028\pi$
$911$	3.41645e13	1.64339	0.821697	+	0.569924i	$0.193028\pi$
$912$	0	0
$913$	− 1.20929e13i	− 0.575987i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	1.19498e13i	0.558081i
$918$	0	0
$919$	3.14366e13	1.45384	0.726920	−	0.686723i	$-0.240951\pi$
$919$	3.14366e13	1.45384	0.726920	+	0.686723i	$0.240951\pi$
$920$	0	0
$921$	−7.24478e11	−0.0331785
$922$	0	0
$923$	− 2.20237e13i	− 0.998809i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	5.00344e12i	0.222541i
$928$	0	0
$929$	8.77007e12	0.386307	0.193153	−	0.981169i	$-0.438128\pi$
$929$	8.77007e12	0.386307	0.193153	+	0.981169i	$0.438128\pi$
$930$	0	0
$931$	−2.25545e13	−0.983918
$932$	0	0
$933$	− 3.09228e13i	− 1.33602i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	1.66294e13i	0.704770i	0.935855	+	0.352385i	$0.114629\pi$
$937$	1.66294e13i	0.704770i	−0.935855	+	0.352385i	$0.885371\pi$
$938$	0	0
$939$	−1.86389e13	−0.782392
$940$	0	0
$941$	−7.56052e12	−0.314339	−0.157169	−	0.987572i	$-0.550237\pi$
$941$	−7.56052e12	−0.314339	−0.157169	+	0.987572i	$0.550237\pi$
$942$	0	0
$943$	5.88806e12i	0.242476i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	3.40925e13i	1.37747i	0.725011	+	0.688737i	$0.241835\pi$
$947$	3.40925e13i	1.37747i	−0.725011	+	0.688737i	$0.758165\pi$
$948$	0	0
$949$	4.77846e13	1.91245
$950$	0	0
$951$	2.89111e13	1.14618
$952$	0	0
$953$	3.45885e13i	1.35836i	0.733973	+	0.679178i	$0.237664\pi$
$953$	3.45885e13i	1.35836i	−0.733973	+	0.679178i	$0.762336\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 2.21054e13i	− 0.851911i
$958$	0	0
$959$	−3.03435e13	−1.15846
$960$	0	0
$961$	−1.51162e13	−0.571726
$962$	0	0
$963$	7.06889e12i	0.264870i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	4.59891e13i	1.69136i	0.533691	+	0.845680i	$0.320805\pi$
$967$	4.59891e13i	1.69136i	−0.533691	+	0.845680i	$0.679195\pi$
$968$	0	0
$969$	−5.69050e13	−2.07345
$970$	0	0
$971$	−2.72663e13	−0.984327	−0.492164	−	0.870503i	$-0.663794\pi$
$971$	−2.72663e13	−0.984327	−0.492164	+	0.870503i	$0.663794\pi$
$972$	0	0
$973$	1.07035e13i	0.382842i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	4.92228e13i	1.72839i	0.503160	+	0.864193i	$0.332171\pi$
$977$	4.92228e13i	1.72839i	−0.503160	+	0.864193i	$0.667829\pi$
$978$	0	0
$979$	9.21324e11	0.0320546
$980$	0	0
$981$	1.66947e13	0.575530
$982$	0	0
$983$	− 3.98337e13i	− 1.36069i	−0.732891	−	0.680346i	$-0.761830\pi$
$983$	− 3.98337e13i	− 1.36069i	0.732891	−	0.680346i	$-0.238170\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	1.71960e13i	0.576766i
$988$	0	0
$989$	−3.40664e12	−0.113225
$990$	0	0
$991$	2.56388e13	0.844434	0.422217	−	0.906495i	$-0.361252\pi$
$991$	2.56388e13	0.844434	0.422217	+	0.906495i	$0.361252\pi$
$992$	0	0
$993$	2.73673e13i	0.893224i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	4.97241e13i	1.59382i	0.604099	+	0.796909i	$0.293533\pi$
$997$	4.97241e13i	1.59382i	−0.604099	+	0.796909i	$0.706467\pi$
$998$	0	0
$999$	−4.48863e13	−1.42584

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	400.10.c.e.49.2		2
4.3	odd	2		25.10.b.a.24.2		2
5.2	odd	4		80.10.a.d.1.1		1
5.3	odd	4		400.10.a.c.1.1		1
5.4	even	2	inner	400.10.c.e.49.1		2
12.11	even	2		225.10.b.d.199.1		2
20.3	even	4		25.10.a.a.1.1		1
20.7	even	4		5.10.a.a.1.1	✓	1
20.19	odd	2		25.10.b.a.24.1		2
40.27	even	4		320.10.a.h.1.1		1
40.37	odd	4		320.10.a.c.1.1		1
60.23	odd	4		225.10.a.b.1.1		1
60.47	odd	4		45.10.a.c.1.1		1
60.59	even	2		225.10.b.d.199.2		2
140.27	odd	4		245.10.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5.10.a.a.1.1	✓	1	20.7	even	4
25.10.a.a.1.1		1	20.3	even	4
25.10.b.a.24.1		2	20.19	odd	2
25.10.b.a.24.2		2	4.3	odd	2
45.10.a.c.1.1		1	60.47	odd	4
80.10.a.d.1.1		1	5.2	odd	4
225.10.a.b.1.1		1	60.23	odd	4
225.10.b.d.199.1		2	12.11	even	2
225.10.b.d.199.2		2	60.59	even	2
245.10.a.a.1.1		1	140.27	odd	4
320.10.a.c.1.1		1	40.37	odd	4
320.10.a.h.1.1		1	40.27	even	4
400.10.a.c.1.1		1	5.3	odd	4
400.10.c.e.49.1		2	5.4	even	2	inner
400.10.c.e.49.2		2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 400 = 2^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	400.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+114.000i q^{3} +4242.00i q^{7} +6687.00 q^{9} +O(q^{10})\)	\(q+114.000i q^{3} +4242.00i q^{7} +6687.00 q^{9} +46208.0 q^{11} -115934. i q^{13} -494842. i q^{17} -1.00874e6 q^{19} -483588. q^{21} +532554. i q^{23} +3.00618e6i q^{27} -4.19639e6 q^{29} +3.36503e6 q^{31} +5.26771e6i q^{33} +1.49314e7i q^{37} +1.32165e7 q^{39} +1.10563e7 q^{41} +6.39679e6i q^{43} -3.55592e7i q^{47} +2.23590e7 q^{49} +5.64120e7 q^{51} +3.97386e7i q^{53} -1.14996e8i q^{57} -8.51856e7 q^{59} +4.57486e7 q^{61} +2.83663e7i q^{63} -4.52862e7i q^{67} -6.07112e7 q^{69} +1.89967e8 q^{71} +4.12171e8i q^{73} +1.96014e8i q^{77} +9.50408e7 q^{79} -2.11084e8 q^{81} -2.61706e8i q^{83} -4.78388e8i q^{87} +1.99386e7 q^{89} +4.91792e8 q^{91} +3.83613e8i q^{93} +1.95034e7i q^{97} +3.08993e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 13374 q^{9}+O(q^{10}) \) 2 * q + 13374 * q^9	\( 2 q + 13374 q^{9} + 92416 q^{11} - 2017480 q^{19} - 967176 q^{21} - 8392780 q^{29} + 6730056 q^{31} + 26432952 q^{39} + 22112524 q^{41} + 44718086 q^{49} + 112823976 q^{51} - 170371240 q^{59} + 91497284 q^{61} - 121422312 q^{69} + 379934936 q^{71} + 190081680 q^{79} - 422168598 q^{81} + 39877260 q^{89} + 983584056 q^{91} + 617985792 q^{99}+O(q^{100}) \) 2 * q + 13374 * q^9 + 92416 * q^11 - 2017480 * q^19 - 967176 * q^21 - 8392780 * q^29 + 6730056 * q^31 + 26432952 * q^39 + 22112524 * q^41 + 44718086 * q^49 + 112823976 * q^51 - 170371240 * q^59 + 91497284 * q^61 - 121422312 * q^69 + 379934936 * q^71 + 190081680 * q^79 - 422168598 * q^81 + 39877260 * q^89 + 983584056 * q^91 + 617985792 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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