LMFDB - Embedded newform 200.10.c.a.49.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [200,10,Mod(49,200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(200, 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("200.49");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 200 = 2^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	200.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:	$103.007167233$
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 8)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	200.49
Dual form			200.10.c.a.49.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-68.0000i q^{3} +10248.0i q^{7} +15059.0 q^{9} +O(q^{10})$	$q-68.0000i q^{3} +10248.0i q^{7} +15059.0 q^{9} +3916.00 q^{11} +176594. i q^{13} +148370. i q^{17} -499796. q^{19} +696864. q^{21} +1.88977e6i q^{23} -2.36246e6i q^{27} +920898. q^{29} +1.37936e6 q^{31} -266288. i q^{33} +5.06497e6i q^{37} +1.20084e7 q^{39} -2.41008e7 q^{41} -2.57852e7i q^{43} -6.07902e7i q^{47} -6.46679e7 q^{49} +1.00892e7 q^{51} -2.94962e7i q^{53} +3.39861e7i q^{57} -5.18194e7 q^{59} +3.34269e7 q^{61} +1.54325e8i q^{63} +1.44856e8i q^{67} +1.28504e8 q^{69} +6.83971e7 q^{71} -1.68216e8i q^{73} +4.01312e7i q^{77} -2.35399e8 q^{79} +1.35759e8 q^{81} +6.46399e7i q^{83} -6.26211e7i q^{87} +7.87827e7 q^{89} -1.80974e9 q^{91} -9.37965e7i q^{93} -2.41136e7i q^{97} +5.89710e7 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 30118 q^{9}+O(q^{10}) $ 2 * q + 30118 * q^9	$ 2 q + 30118 q^{9} + 7832 q^{11} - 999592 q^{19} + 1393728 q^{21} + 1841796 q^{29} + 2758720 q^{31} + 24016784 q^{39} - 48201516 q^{41} - 129335794 q^{49} + 20178320 q^{51} - 103638776 q^{59} + 66853820 q^{61} + 257008448 q^{69} + 136794256 q^{71} - 470797472 q^{79} + 271518578 q^{81} + 157565388 q^{89} - 3619470624 q^{91} + 117942088 q^{99}+O(q^{100}) $ 2 * q + 30118 * q^9 + 7832 * q^11 - 999592 * q^19 + 1393728 * q^21 + 1841796 * q^29 + 2758720 * q^31 + 24016784 * q^39 - 48201516 * q^41 - 129335794 * q^49 + 20178320 * q^51 - 103638776 * q^59 + 66853820 * q^61 + 257008448 * q^69 + 136794256 * q^71 - 470797472 * q^79 + 271518578 * q^81 + 157565388 * q^89 - 3619470624 * q^91 + 117942088 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/200\mathbb{Z}\right)^\times$.

$n$	$101$	$151$	$177$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	− 68.0000i	− 0.484689i	−0.970190	−	0.242345i	$-0.922084\pi$
$3$	− 68.0000i	− 0.484689i	0.970190	−	0.242345i	$-0.0779164\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	10248.0i	1.61324i	0.591073	+	0.806618i	$0.298705\pi$
$7$	10248.0i	1.61324i	−0.591073	+	0.806618i	$0.701295\pi$
$8$	0	0
$9$	15059.0	0.765076
$10$	0	0
$11$	3916.00	0.0806447	0.0403223	−	0.999187i	$-0.487162\pi$
$11$	3916.00	0.0806447	0.0403223	+	0.999187i	$0.487162\pi$
$12$	0	0
$13$	176594.i	1.71487i	0.514593	+	0.857434i	$0.327943\pi$
$13$	176594.i	1.71487i	−0.514593	+	0.857434i	$0.672057\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	148370.i	0.430850i	0.976520	+	0.215425i	$0.0691137\pi$
$17$	148370.i	0.430850i	−0.976520	+	0.215425i	$0.930886\pi$
$18$	0	0
$19$	−499796.	−0.879836	−0.439918	−	0.898038i	$-0.644992\pi$
$19$	−499796.	−0.879836	−0.439918	+	0.898038i	$0.644992\pi$
$20$	0	0
$21$	696864.	0.781918
$22$	0	0
$23$	1.88977e6i	1.40810i	0.710151	+	0.704050i	$0.248627\pi$
$23$	1.88977e6i	1.40810i	−0.710151	+	0.704050i	$0.751373\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 2.36246e6i	− 0.855513i
$28$	0	0
$29$	920898.	0.241780	0.120890	−	0.992666i	$-0.461425\pi$
$29$	920898.	0.241780	0.120890	+	0.992666i	$0.461425\pi$
$30$	0	0
$31$	1.37936e6	0.268256	0.134128	−	0.990964i	$-0.457177\pi$
$31$	1.37936e6	0.268256	0.134128	+	0.990964i	$0.457177\pi$
$32$	0	0
$33$	− 266288.i	− 0.0390876i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	5.06497e6i	0.444292i	0.975013	+	0.222146i	$0.0713062\pi$
$37$	5.06497e6i	0.444292i	−0.975013	+	0.222146i	$0.928694\pi$
$38$	0	0
$39$	1.20084e7	0.831178
$40$	0	0
$41$	−2.41008e7	−1.33200	−0.665999	−	0.745953i	$-0.731994\pi$
$41$	−2.41008e7	−1.33200	−0.665999	+	0.745953i	$0.731994\pi$
$42$	0	0
$43$	− 2.57852e7i	− 1.15017i	−0.818093	−	0.575085i	$-0.804969\pi$
$43$	− 2.57852e7i	− 1.15017i	0.818093	−	0.575085i	$-0.195031\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 6.07902e7i	− 1.81716i	−0.417710	−	0.908580i	$-0.637167\pi$
$47$	− 6.07902e7i	− 1.81716i	0.417710	−	0.908580i	$-0.362833\pi$
$48$	0	0
$49$	−6.46679e7	−1.60253
$50$	0	0
$51$	1.00892e7	0.208828
$52$	0	0
$53$	− 2.94962e7i	− 0.513482i	−0.966480	−	0.256741i	$-0.917351\pi$
$53$	− 2.94962e7i	− 0.513482i	0.966480	−	0.256741i	$-0.0826487\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	3.39861e7i	0.426447i
$58$	0	0
$59$	−5.18194e7	−0.556747	−0.278374	−	0.960473i	$-0.589795\pi$
$59$	−5.18194e7	−0.556747	−0.278374	+	0.960473i	$0.589795\pi$
$60$	0	0
$61$	3.34269e7	0.309109	0.154555	−	0.987984i	$-0.450606\pi$
$61$	3.34269e7	0.309109	0.154555	+	0.987984i	$0.450606\pi$
$62$	0	0
$63$	1.54325e8i	1.23425i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	1.44856e8i	0.878214i	0.898435	+	0.439107i	$0.144705\pi$
$67$	1.44856e8i	0.878214i	−0.898435	+	0.439107i	$0.855295\pi$
$68$	0	0
$69$	1.28504e8	0.682490
$70$	0	0
$71$	6.83971e7	0.319430	0.159715	−	0.987163i	$-0.448943\pi$
$71$	6.83971e7	0.319430	0.159715	+	0.987163i	$0.448943\pi$
$72$	0	0
$73$	− 1.68216e8i	− 0.693290i	−0.937996	−	0.346645i	$-0.887321\pi$
$73$	− 1.68216e8i	− 0.693290i	0.937996	−	0.346645i	$-0.112679\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	4.01312e7i	0.130099i
$78$	0	0
$79$	−2.35399e8	−0.679958	−0.339979	−	0.940433i	$-0.610420\pi$
$79$	−2.35399e8	−0.679958	−0.339979	+	0.940433i	$0.610420\pi$
$80$	0	0
$81$	1.35759e8	0.350418
$82$	0	0
$83$	6.46399e7i	0.149503i	0.997202	+	0.0747513i	$0.0238163\pi$
$83$	6.46399e7i	0.149503i	−0.997202	+	0.0747513i	$0.976184\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 6.26211e7i	− 0.117188i
$88$	0	0
$89$	7.87827e7	0.133099	0.0665497	−	0.997783i	$-0.478801\pi$
$89$	7.87827e7	0.133099	0.0665497	+	0.997783i	$0.478801\pi$
$90$	0	0
$91$	−1.80974e9	−2.76649
$92$	0	0
$93$	− 9.37965e7i	− 0.130021i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 2.41136e7i	− 0.0276560i	−0.999904	−	0.0138280i	$-0.995598\pi$
$97$	− 2.41136e7i	− 0.0276560i	0.999904	−	0.0138280i	$-0.00440172\pi$
$98$	0	0
$99$	5.89710e7	0.0616993
$100$	0	0
$101$	−6.25963e8	−0.598553	−0.299277	−	0.954166i	$-0.596745\pi$
$101$	−6.25963e8	−0.598553	−0.299277	+	0.954166i	$0.596745\pi$
$102$	0	0
$103$	− 8.00618e8i	− 0.700902i	−0.936581	−	0.350451i	$-0.886028\pi$
$103$	− 8.00618e8i	− 0.700902i	0.936581	−	0.350451i	$-0.113972\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	2.45358e9i	1.80956i	0.425877	+	0.904781i	$0.359966\pi$
$107$	2.45358e9i	1.80956i	−0.425877	+	0.904781i	$0.640034\pi$
$108$	0	0
$109$	9.29043e8	0.630400	0.315200	−	0.949025i	$-0.397928\pi$
$109$	9.29043e8	0.630400	0.315200	+	0.949025i	$0.397928\pi$
$110$	0	0
$111$	3.44418e8	0.215344
$112$	0	0
$113$	1.65129e9i	0.952731i	0.879247	+	0.476366i	$0.158046\pi$
$113$	1.65129e9i	0.952731i	−0.879247	+	0.476366i	$0.841954\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	2.65933e9i	1.31201i
$118$	0	0
$119$	−1.52050e9	−0.695063
$120$	0	0
$121$	−2.34261e9	−0.993496
$122$	0	0
$123$	1.63885e9i	0.645605i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 1.83042e9i	− 0.624358i	−0.950023	−	0.312179i	$-0.898941\pi$
$127$	− 1.83042e9i	− 0.624358i	0.950023	−	0.312179i	$-0.101059\pi$
$128$	0	0
$129$	−1.75339e9	−0.557475
$130$	0	0
$131$	5.60254e8	0.166213	0.0831064	−	0.996541i	$-0.473516\pi$
$131$	5.60254e8	0.166213	0.0831064	+	0.996541i	$0.473516\pi$
$132$	0	0
$133$	− 5.12191e9i	− 1.41938i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	6.54232e9i	1.58668i	0.608778	+	0.793340i	$0.291660\pi$
$137$	6.54232e9i	1.58668i	−0.608778	+	0.793340i	$0.708340\pi$
$138$	0	0
$139$	−5.52722e9	−1.25586	−0.627929	−	0.778271i	$-0.716097\pi$
$139$	−5.52722e9	−1.25586	−0.627929	+	0.778271i	$0.716097\pi$
$140$	0	0
$141$	−4.13374e9	−0.880758
$142$	0	0
$143$	6.91542e8i	0.138295i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	4.39742e9i	0.776729i
$148$	0	0
$149$	3.25329e9	0.540734	0.270367	−	0.962757i	$-0.412855\pi$
$149$	3.25329e9	0.540734	0.270367	+	0.962757i	$0.412855\pi$
$150$	0	0
$151$	−8.54419e9	−1.33744	−0.668721	−	0.743514i	$-0.733158\pi$
$151$	−8.54419e9	−1.33744	−0.668721	+	0.743514i	$0.733158\pi$
$152$	0	0
$153$	2.23430e9i	0.329633i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 2.69871e8i	− 0.0354493i	−0.999843	−	0.0177246i	$-0.994358\pi$
$157$	− 2.69871e8i	− 0.0354493i	0.999843	−	0.0177246i	$-0.00564222\pi$
$158$	0	0
$159$	−2.00574e9	−0.248879
$160$	0	0
$161$	−1.93663e10	−2.27160
$162$	0	0
$163$	− 1.02903e10i	− 1.14178i	−0.821025	−	0.570892i	$-0.806597\pi$
$163$	− 1.02903e10i	− 1.14178i	0.821025	−	0.570892i	$-0.193403\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1.23946e8i	0.0123313i	0.999981	+	0.00616565i	$0.00196260\pi$
$167$	1.23946e8i	0.0123313i	−0.999981	+	0.00616565i	$0.998037\pi$
$168$	0	0
$169$	−2.05809e10	−1.94077
$170$	0	0
$171$	−7.52643e9	−0.673142
$172$	0	0
$173$	− 4.92770e9i	− 0.418250i	−0.977889	−	0.209125i	$-0.932938\pi$
$173$	− 4.92770e9i	− 0.418250i	0.977889	−	0.209125i	$-0.0670616\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	3.52372e9i	0.269849i
$178$	0	0
$179$	5.54564e9	0.403751	0.201875	−	0.979411i	$-0.435296\pi$
$179$	5.54564e9	0.403751	0.201875	+	0.979411i	$0.435296\pi$
$180$	0	0
$181$	1.16150e10	0.804391	0.402196	−	0.915554i	$-0.368247\pi$
$181$	1.16150e10	0.804391	0.402196	+	0.915554i	$0.368247\pi$
$182$	0	0
$183$	− 2.27303e9i	− 0.149822i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	5.81017e8i	0.0347457i
$188$	0	0
$189$	2.42104e10	1.38015
$190$	0	0
$191$	−2.72404e10	−1.48103	−0.740514	−	0.672041i	$-0.765418\pi$
$191$	−2.72404e10	−1.48103	−0.740514	+	0.672041i	$0.765418\pi$
$192$	0	0
$193$	2.88743e10i	1.49797i	0.662585	+	0.748987i	$0.269459\pi$
$193$	2.88743e10i	1.49797i	−0.662585	+	0.748987i	$0.730541\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	3.80169e10i	1.79837i	0.437571	+	0.899184i	$0.355839\pi$
$197$	3.80169e10i	1.79837i	−0.437571	+	0.899184i	$0.644161\pi$
$198$	0	0
$199$	−2.57386e10	−1.16344	−0.581722	−	0.813387i	$-0.697621\pi$
$199$	−2.57386e10	−1.16344	−0.581722	+	0.813387i	$0.697621\pi$
$200$	0	0
$201$	9.85022e9	0.425661
$202$	0	0
$203$	9.43736e9i	0.390048i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	2.84580e10i	1.07730i
$208$	0	0
$209$	−1.95720e9	−0.0709540
$210$	0	0
$211$	5.50064e9	0.191048	0.0955239	−	0.995427i	$-0.469547\pi$
$211$	5.50064e9	0.191048	0.0955239	+	0.995427i	$0.469547\pi$
$212$	0	0
$213$	− 4.65100e9i	− 0.154824i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	1.41357e10i	0.432761i
$218$	0	0
$219$	−1.14387e10	−0.336030
$220$	0	0
$221$	−2.62013e10	−0.738851
$222$	0	0
$223$	− 2.05983e10i	− 0.557774i	−0.960324	−	0.278887i	$-0.910034\pi$
$223$	− 2.05983e10i	− 0.557774i	0.960324	−	0.278887i	$-0.0899656\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 4.46842e10i	− 1.11696i	−0.829518	−	0.558480i	$-0.811385\pi$
$227$	− 4.46842e10i	− 1.11696i	0.829518	−	0.558480i	$-0.188615\pi$
$228$	0	0
$229$	5.69323e9	0.136804	0.0684020	−	0.997658i	$-0.478210\pi$
$229$	5.69323e9	0.136804	0.0684020	+	0.997658i	$0.478210\pi$
$230$	0	0
$231$	2.72892e9	0.0630575
$232$	0	0
$233$	− 1.74032e10i	− 0.386837i	−0.981116	−	0.193419i	$-0.938042\pi$
$233$	− 1.74032e10i	− 0.386837i	0.981116	−	0.193419i	$-0.0619575\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	1.60071e10i	0.329568i
$238$	0	0
$239$	3.35988e10	0.666090	0.333045	−	0.942911i	$-0.391924\pi$
$239$	3.35988e10	0.666090	0.333045	+	0.942911i	$0.391924\pi$
$240$	0	0
$241$	−6.08619e10	−1.16217	−0.581084	−	0.813844i	$-0.697371\pi$
$241$	−6.08619e10	−1.16217	−0.581084	+	0.813844i	$0.697371\pi$
$242$	0	0
$243$	− 5.57319e10i	− 1.02536i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 8.82610e10i	− 1.50880i
$248$	0	0
$249$	4.39551e9	0.0724623
$250$	0	0
$251$	8.74389e10	1.39051	0.695253	−	0.718765i	$-0.255292\pi$
$251$	8.74389e10	1.39051	0.695253	+	0.718765i	$0.255292\pi$
$252$	0	0
$253$	7.40033e9i	0.113556i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	7.11368e10i	1.01717i	0.861011	+	0.508587i	$0.169832\pi$
$257$	7.11368e10i	1.01717i	−0.861011	+	0.508587i	$0.830168\pi$
$258$	0	0
$259$	−5.19058e10	−0.716748
$260$	0	0
$261$	1.38678e10	0.184980
$262$	0	0
$263$	5.24741e10i	0.676308i	0.941091	+	0.338154i	$0.109802\pi$
$263$	5.24741e10i	0.676308i	−0.941091	+	0.338154i	$0.890198\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 5.35722e9i	− 0.0645118i
$268$	0	0
$269$	1.37810e11	1.60470	0.802350	−	0.596853i	$-0.203583\pi$
$269$	1.37810e11	1.60470	0.802350	+	0.596853i	$0.203583\pi$
$270$	0	0
$271$	1.18786e11	1.33784	0.668921	−	0.743333i	$-0.266756\pi$
$271$	1.18786e11	1.33784	0.668921	+	0.743333i	$0.266756\pi$
$272$	0	0
$273$	1.23062e11i	1.34089i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 6.20716e10i	− 0.633481i	−0.948512	−	0.316741i	$-0.897412\pi$
$277$	− 6.20716e10i	− 0.633481i	0.948512	−	0.316741i	$-0.102588\pi$
$278$	0	0
$279$	2.07718e10	0.205237
$280$	0	0
$281$	7.20927e10	0.689784	0.344892	−	0.938642i	$-0.387916\pi$
$281$	7.20927e10	0.689784	0.344892	+	0.938642i	$0.387916\pi$
$282$	0	0
$283$	− 3.90151e10i	− 0.361571i	−0.983523	−	0.180786i	$-0.942136\pi$
$283$	− 3.90151e10i	− 0.361571i	0.983523	−	0.180786i	$-0.0578640\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 2.46985e11i	− 2.14883i
$288$	0	0
$289$	9.65742e10	0.814368
$290$	0	0
$291$	−1.63972e9	−0.0134045
$292$	0	0
$293$	9.37214e10i	0.742907i	0.928451	+	0.371454i	$0.121141\pi$
$293$	9.37214e10i	0.742907i	−0.928451	+	0.371454i	$0.878859\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 9.25138e9i	− 0.0689926i
$298$	0	0
$299$	−3.33722e11	−2.41471
$300$	0	0
$301$	2.64247e11	1.85550
$302$	0	0
$303$	4.25655e10i	0.290112i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 1.63239e11i	− 1.04882i	−0.851465	−	0.524411i	$-0.824285\pi$
$307$	− 1.63239e11i	− 1.04882i	0.851465	−	0.524411i	$-0.175715\pi$
$308$	0	0
$309$	−5.44420e10	−0.339720
$310$	0	0
$311$	−5.54012e10	−0.335813	−0.167906	−	0.985803i	$-0.553701\pi$
$311$	−5.54012e10	−0.335813	−0.167906	+	0.985803i	$0.553701\pi$
$312$	0	0
$313$	− 2.35841e11i	− 1.38889i	−0.719544	−	0.694447i	$-0.755649\pi$
$313$	− 2.35841e11i	− 1.38889i	0.719544	−	0.694447i	$-0.244351\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 7.74070e10i	− 0.430540i	−0.976555	−	0.215270i	$-0.930937\pi$
$317$	− 7.74070e10i	− 0.430540i	0.976555	−	0.215270i	$-0.0690632\pi$
$318$	0	0
$319$	3.60624e9	0.0194983
$320$	0	0
$321$	1.66844e11	0.877075
$322$	0	0
$323$	− 7.41547e10i	− 0.379077i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 6.31749e10i	− 0.305548i
$328$	0	0
$329$	6.22978e11	2.93151
$330$	0	0
$331$	−7.90091e10	−0.361785	−0.180893	−	0.983503i	$-0.557899\pi$
$331$	−7.90091e10	−0.361785	−0.180893	+	0.983503i	$0.557899\pi$
$332$	0	0
$333$	7.62733e10i	0.339918i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	2.15728e11i	0.911114i	0.890207	+	0.455557i	$0.150560\pi$
$337$	2.15728e11i	0.911114i	−0.890207	+	0.455557i	$0.849440\pi$
$338$	0	0
$339$	1.12288e11	0.461778
$340$	0	0
$341$	5.40157e9	0.0216334
$342$	0	0
$343$	− 2.49173e11i	− 0.972024i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 2.38976e11i	− 0.884854i	−0.896804	−	0.442427i	$-0.854118\pi$
$347$	− 2.38976e11i	− 0.884854i	0.896804	−	0.442427i	$-0.145882\pi$
$348$	0	0
$349$	−5.14491e9	−0.0185637	−0.00928183	−	0.999957i	$-0.502955\pi$
$349$	−5.14491e9	−0.0185637	−0.00928183	+	0.999957i	$0.502955\pi$
$350$	0	0
$351$	4.17196e11	1.46709
$352$	0	0
$353$	− 5.71172e10i	− 0.195786i	−0.995197	−	0.0978928i	$-0.968790\pi$
$353$	− 5.71172e10i	− 0.195786i	0.995197	−	0.0978928i	$-0.0312102\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	1.03394e11i	0.336889i
$358$	0	0
$359$	−2.89030e11	−0.918371	−0.459185	−	0.888340i	$-0.651859\pi$
$359$	−2.89030e11	−0.918371	−0.459185	+	0.888340i	$0.651859\pi$
$360$	0	0
$361$	−7.28917e10	−0.225889
$362$	0	0
$363$	1.59298e11i	0.481537i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 1.68812e11i	− 0.485742i	−0.970059	−	0.242871i	$-0.921911\pi$
$367$	− 1.68812e11i	− 0.485742i	0.970059	−	0.242871i	$-0.0780892\pi$
$368$	0	0
$369$	−3.62933e11	−1.01908
$370$	0	0
$371$	3.02277e11	0.828367
$372$	0	0
$373$	6.06880e11i	1.62335i	0.584108	+	0.811676i	$0.301445\pi$
$373$	6.06880e11i	1.62335i	−0.584108	+	0.811676i	$0.698555\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	1.62625e11i	0.414621i
$378$	0	0
$379$	4.67636e11	1.16421	0.582105	−	0.813114i	$-0.302229\pi$
$379$	4.67636e11	1.16421	0.582105	+	0.813114i	$0.302229\pi$
$380$	0	0
$381$	−1.24468e11	−0.302619
$382$	0	0
$383$	3.90199e11i	0.926599i	0.886202	+	0.463300i	$0.153335\pi$
$383$	3.90199e11i	0.926599i	−0.886202	+	0.463300i	$0.846665\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 3.88299e11i	− 0.879969i
$388$	0	0
$389$	1.61508e11	0.357620	0.178810	−	0.983884i	$-0.442775\pi$
$389$	1.61508e11	0.357620	0.178810	+	0.983884i	$0.442775\pi$
$390$	0	0
$391$	−2.80385e11	−0.606679
$392$	0	0
$393$	− 3.80973e10i	− 0.0805615i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	8.18138e11i	1.65299i	0.562947	+	0.826493i	$0.309668\pi$
$397$	8.18138e11i	1.65299i	−0.562947	+	0.826493i	$0.690332\pi$
$398$	0	0
$399$	−3.48290e11	−0.687959
$400$	0	0
$401$	−1.08197e11	−0.208962	−0.104481	−	0.994527i	$-0.533318\pi$
$401$	−1.08197e11	−0.208962	−0.104481	+	0.994527i	$0.533318\pi$
$402$	0	0
$403$	2.43587e11i	0.460024i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1.98344e10i	0.0358298i
$408$	0	0
$409$	7.56200e11	1.33623	0.668115	−	0.744058i	$-0.267101\pi$
$409$	7.56200e11	1.33623	0.668115	+	0.744058i	$0.267101\pi$
$410$	0	0
$411$	4.44878e11	0.769047
$412$	0	0
$413$	− 5.31045e11i	− 0.898165i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	3.75851e11i	0.608701i
$418$	0	0
$419$	−6.64700e11	−1.05357	−0.526784	−	0.849999i	$-0.676602\pi$
$419$	−6.64700e11	−1.05357	−0.526784	+	0.849999i	$0.676602\pi$
$420$	0	0
$421$	−6.96689e11	−1.08086	−0.540430	−	0.841389i	$-0.681739\pi$
$421$	−6.96689e11	−1.08086	−0.540430	+	0.841389i	$0.681739\pi$
$422$	0	0
$423$	− 9.15440e11i	− 1.39027i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	3.42559e11i	0.498666i
$428$	0	0
$429$	4.70249e10	0.0670301
$430$	0	0
$431$	−1.37728e12	−1.92254	−0.961268	−	0.275614i	$-0.911119\pi$
$431$	−1.37728e12	−1.92254	−0.961268	+	0.275614i	$0.911119\pi$
$432$	0	0
$433$	2.26719e11i	0.309950i	0.987918	+	0.154975i	$0.0495297\pi$
$433$	2.26719e11i	0.309950i	−0.987918	+	0.154975i	$0.950470\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 9.44498e11i	− 1.23890i
$438$	0	0
$439$	−4.03484e9	−0.00518485	−0.00259242	−	0.999997i	$-0.500825\pi$
$439$	−4.03484e9	−0.00518485	−0.00259242	+	0.999997i	$0.500825\pi$
$440$	0	0
$441$	−9.73834e11	−1.22606
$442$	0	0
$443$	− 1.08930e12i	− 1.34379i	−0.740649	−	0.671893i	$-0.765482\pi$
$443$	− 1.08930e12i	− 1.34379i	0.740649	−	0.671893i	$-0.234518\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 2.21223e11i	− 0.262088i
$448$	0	0
$449$	−1.06107e12	−1.23207	−0.616035	−	0.787719i	$-0.711262\pi$
$449$	−1.06107e12	−1.23207	−0.616035	+	0.787719i	$0.711262\pi$
$450$	0	0
$451$	−9.43786e10	−0.107418
$452$	0	0
$453$	5.81005e11i	0.648243i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	7.52127e11i	0.806618i	0.915064	+	0.403309i	$0.132140\pi$
$457$	7.52127e11i	0.806618i	−0.915064	+	0.403309i	$0.867860\pi$
$458$	0	0
$459$	3.50518e11	0.368598
$460$	0	0
$461$	1.22335e12	1.26152	0.630761	−	0.775977i	$-0.282743\pi$
$461$	1.22335e12	1.26152	0.630761	+	0.775977i	$0.282743\pi$
$462$	0	0
$463$	− 1.00710e12i	− 1.01849i	−0.860621	−	0.509246i	$-0.829924\pi$
$463$	− 1.00710e12i	− 1.01849i	0.860621	−	0.509246i	$-0.170076\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	1.11362e12i	1.08345i	0.840556	+	0.541725i	$0.182229\pi$
$467$	1.11362e12i	1.08345i	−0.840556	+	0.541725i	$0.817771\pi$
$468$	0	0
$469$	−1.48449e12	−1.41677
$470$	0	0
$471$	−1.83512e10	−0.0171819
$472$	0	0
$473$	− 1.00975e11i	− 0.0927551i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 4.44183e11i	− 0.392853i
$478$	0	0
$479$	−7.94324e11	−0.689426	−0.344713	−	0.938708i	$-0.612024\pi$
$479$	−7.94324e11	−0.689426	−0.344713	+	0.938708i	$0.612024\pi$
$480$	0	0
$481$	−8.94443e11	−0.761903
$482$	0	0
$483$	1.31691e12i	1.10102i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	1.75226e12i	1.41162i	0.708401	+	0.705810i	$0.249417\pi$
$487$	1.75226e12i	1.41162i	−0.708401	+	0.705810i	$0.750583\pi$
$488$	0	0
$489$	−6.99741e11	−0.553411
$490$	0	0
$491$	−2.49979e12	−1.94105	−0.970526	−	0.240997i	$-0.922526\pi$
$491$	−2.49979e12	−1.94105	−0.970526	+	0.240997i	$0.922526\pi$
$492$	0	0
$493$	1.36634e11i	0.104171i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	7.00934e11i	0.515316i
$498$	0	0
$499$	3.03714e11	0.219286	0.109643	−	0.993971i	$-0.465029\pi$
$499$	3.03714e11	0.219286	0.109643	+	0.993971i	$0.465029\pi$
$500$	0	0
$501$	8.42834e9	0.00597685
$502$	0	0
$503$	1.56882e12i	1.09274i	0.837543	+	0.546372i	$0.183991\pi$
$503$	1.56882e12i	1.09274i	−0.837543	+	0.546372i	$0.816009\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	1.39950e12i	0.940672i
$508$	0	0
$509$	2.83572e12	1.87255	0.936274	−	0.351272i	$-0.114251\pi$
$509$	2.83572e12	1.87255	0.936274	+	0.351272i	$0.114251\pi$
$510$	0	0
$511$	1.72388e12	1.11844
$512$	0	0
$513$	1.18075e12i	0.752711i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 2.38055e11i	− 0.146544i
$518$	0	0
$519$	−3.35083e11	−0.202721
$520$	0	0
$521$	−1.43643e12	−0.854111	−0.427056	−	0.904225i	$-0.640449\pi$
$521$	−1.43643e12	−0.854111	−0.427056	+	0.904225i	$0.640449\pi$
$522$	0	0
$523$	9.78444e11i	0.571845i	0.958253	+	0.285923i	$0.0923000\pi$
$523$	9.78444e11i	0.571845i	−0.958253	+	0.285923i	$0.907700\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	2.04656e11i	0.115578i
$528$	0	0
$529$	−1.77007e12	−0.982743
$530$	0	0
$531$	−7.80348e11	−0.425954
$532$	0	0
$533$	− 4.25605e12i	− 2.28420i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 3.77104e11i	− 0.195694i
$538$	0	0
$539$	−2.53239e11	−0.129236
$540$	0	0
$541$	−2.85542e11	−0.143312	−0.0716560	−	0.997429i	$-0.522828\pi$
$541$	−2.85542e11	−0.143312	−0.0716560	+	0.997429i	$0.522828\pi$
$542$	0	0
$543$	− 7.89823e11i	− 0.389880i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 2.87296e12i	− 1.37210i	−0.727552	−	0.686052i	$-0.759342\pi$
$547$	− 2.87296e12i	− 1.37210i	0.727552	−	0.686052i	$-0.240658\pi$
$548$	0	0
$549$	5.03376e11	0.236492
$550$	0	0
$551$	−4.60261e11	−0.212727
$552$	0	0
$553$	− 2.41237e12i	− 1.09693i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	1.40595e12i	0.618899i	0.950916	+	0.309450i	$0.100145\pi$
$557$	1.40595e12i	0.618899i	−0.950916	+	0.309450i	$0.899855\pi$
$558$	0	0
$559$	4.55351e12	1.97239
$560$	0	0
$561$	3.95092e10	0.0168409
$562$	0	0
$563$	3.50500e11i	0.147028i	0.997294	+	0.0735139i	$0.0234213\pi$
$563$	3.50500e11i	0.147028i	−0.997294	+	0.0735139i	$0.976579\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	1.39126e12i	0.565308i
$568$	0	0
$569$	−3.17729e12	−1.27073	−0.635363	−	0.772214i	$-0.719149\pi$
$569$	−3.17729e12	−1.27073	−0.635363	+	0.772214i	$0.719149\pi$
$570$	0	0
$571$	−3.02125e12	−1.18939	−0.594695	−	0.803951i	$-0.702727\pi$
$571$	−3.02125e12	−1.18939	−0.594695	+	0.803951i	$0.702727\pi$
$572$	0	0
$573$	1.85235e12i	0.717838i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 3.01537e12i	− 1.13253i	−0.824224	−	0.566264i	$-0.808388\pi$
$577$	− 3.01537e12i	− 1.13253i	0.824224	−	0.566264i	$-0.191612\pi$
$578$	0	0
$579$	1.96345e12	0.726051
$580$	0	0
$581$	−6.62429e11	−0.241183
$582$	0	0
$583$	− 1.15507e11i	− 0.0414095i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	3.76756e12i	1.30975i	0.755736	+	0.654876i	$0.227279\pi$
$587$	3.76756e12i	1.30975i	−0.755736	+	0.654876i	$0.772721\pi$
$588$	0	0
$589$	−6.89399e11	−0.236022
$590$	0	0
$591$	2.58515e12	0.871649
$592$	0	0
$593$	4.38478e12i	1.45614i	0.685505	+	0.728068i	$0.259581\pi$
$593$	4.38478e12i	1.45614i	−0.685505	+	0.728068i	$0.740419\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	1.75022e12i	0.563909i
$598$	0	0
$599$	1.67903e11	0.0532890	0.0266445	−	0.999645i	$-0.491518\pi$
$599$	1.67903e11	0.0532890	0.0266445	+	0.999645i	$0.491518\pi$
$600$	0	0
$601$	2.30729e12	0.721384	0.360692	−	0.932685i	$-0.382541\pi$
$601$	2.30729e12	0.721384	0.360692	+	0.932685i	$0.382541\pi$
$602$	0	0
$603$	2.18139e12i	0.671901i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 1.50248e12i	− 0.449220i	−0.974449	−	0.224610i	$-0.927889\pi$
$607$	− 1.50248e12i	− 0.449220i	0.974449	−	0.224610i	$-0.0721107\pi$
$608$	0	0
$609$	6.41741e11	0.189052
$610$	0	0
$611$	1.07352e13	3.11619
$612$	0	0
$613$	− 3.94077e12i	− 1.12722i	−0.826041	−	0.563610i	$-0.809412\pi$
$613$	− 3.94077e12i	− 1.12722i	0.826041	−	0.563610i	$-0.190588\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 3.75098e12i	− 1.04198i	−0.853561	−	0.520992i	$-0.825562\pi$
$617$	− 3.75098e12i	− 1.04198i	0.853561	−	0.520992i	$-0.174438\pi$
$618$	0	0
$619$	−3.29947e12	−0.903308	−0.451654	−	0.892193i	$-0.649166\pi$
$619$	−3.29947e12	−0.903308	−0.451654	+	0.892193i	$0.649166\pi$
$620$	0	0
$621$	4.46449e12	1.20465
$622$	0	0
$623$	8.07365e11i	0.214721i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	1.33090e11i	0.0343907i
$628$	0	0
$629$	−7.51489e11	−0.191423
$630$	0	0
$631$	−7.22548e12	−1.81441	−0.907203	−	0.420693i	$-0.861787\pi$
$631$	−7.22548e12	−1.81441	−0.907203	+	0.420693i	$0.861787\pi$
$632$	0	0
$633$	− 3.74043e11i	− 0.0925988i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 1.14200e13i	− 2.74813i
$638$	0	0
$639$	1.02999e12	0.244388
$640$	0	0
$641$	4.43842e12	1.03841	0.519203	−	0.854651i	$-0.326229\pi$
$641$	4.43842e12	1.03841	0.519203	+	0.854651i	$0.326229\pi$
$642$	0	0
$643$	2.95078e11i	0.0680751i	0.999421	+	0.0340375i	$0.0108366\pi$
$643$	2.95078e11i	0.0680751i	−0.999421	+	0.0340375i	$0.989163\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	2.07880e12i	0.466383i	0.972431	+	0.233192i	$0.0749169\pi$
$647$	2.07880e12i	0.466383i	−0.972431	+	0.233192i	$0.925083\pi$
$648$	0	0
$649$	−2.02925e11	−0.0448987
$650$	0	0
$651$	9.61226e11	0.209754
$652$	0	0
$653$	1.80860e12i	0.389254i	0.980877	+	0.194627i	$0.0623496\pi$
$653$	1.80860e12i	0.389254i	−0.980877	+	0.194627i	$0.937650\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 2.53317e12i	− 0.530420i
$658$	0	0
$659$	6.58150e12	1.35938	0.679690	−	0.733500i	$-0.262114\pi$
$659$	6.58150e12	1.35938	0.679690	+	0.733500i	$0.262114\pi$
$660$	0	0
$661$	7.79747e12	1.58872	0.794360	−	0.607448i	$-0.207807\pi$
$661$	7.79747e12	1.58872	0.794360	+	0.607448i	$0.207807\pi$
$662$	0	0
$663$	1.78169e12i	0.358113i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	1.74028e12i	0.340450i
$668$	0	0
$669$	−1.40068e12	−0.270347
$670$	0	0
$671$	1.30900e11	0.0249280
$672$	0	0
$673$	− 7.86129e12i	− 1.47716i	−0.674168	−	0.738578i	$-0.735498\pi$
$673$	− 7.86129e12i	− 1.47716i	0.674168	−	0.738578i	$-0.264502\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 6.11388e12i	− 1.11858i	−0.828971	−	0.559291i	$-0.811073\pi$
$677$	− 6.11388e12i	− 1.11858i	0.828971	−	0.559291i	$-0.188927\pi$
$678$	0	0
$679$	2.47116e11	0.0446156
$680$	0	0
$681$	−3.03852e12	−0.541378
$682$	0	0
$683$	− 3.45133e11i	− 0.0606867i	−0.999540	−	0.0303434i	$-0.990340\pi$
$683$	− 3.45133e11i	− 0.0606867i	0.999540	−	0.0303434i	$-0.00966008\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 3.87140e11i	− 0.0663074i
$688$	0	0
$689$	5.20885e12	0.880553
$690$	0	0
$691$	5.18228e12	0.864708	0.432354	−	0.901704i	$-0.357683\pi$
$691$	5.18228e12	0.864708	0.432354	+	0.901704i	$0.357683\pi$
$692$	0	0
$693$	6.04335e11i	0.0995356i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 3.57583e12i	− 0.573891i
$698$	0	0
$699$	−1.18342e12	−0.187496
$700$	0	0
$701$	−9.33484e12	−1.46008	−0.730038	−	0.683406i	$-0.760498\pi$
$701$	−9.33484e12	−1.46008	−0.730038	+	0.683406i	$0.760498\pi$
$702$	0	0
$703$	− 2.53145e12i	− 0.390904i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 6.41487e12i	− 0.965607i
$708$	0	0
$709$	7.63912e11	0.113536	0.0567682	−	0.998387i	$-0.481920\pi$
$709$	7.63912e11	0.113536	0.0567682	+	0.998387i	$0.481920\pi$
$710$	0	0
$711$	−3.54487e12	−0.520220
$712$	0	0
$713$	2.60667e12i	0.377732i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 2.28472e12i	− 0.322846i
$718$	0	0
$719$	3.65176e12	0.509591	0.254796	−	0.966995i	$-0.417992\pi$
$719$	3.65176e12	0.509591	0.254796	+	0.966995i	$0.417992\pi$
$720$	0	0
$721$	8.20473e12	1.13072
$722$	0	0
$723$	4.13861e12i	0.563290i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	4.06692e12i	0.539959i	0.962866	+	0.269979i	$0.0870169\pi$
$727$	4.06692e12i	0.539959i	−0.962866	+	0.269979i	$0.912983\pi$
$728$	0	0
$729$	−1.11762e12	−0.146561
$730$	0	0
$731$	3.82575e12	0.495551
$732$	0	0
$733$	1.00125e13i	1.28107i	0.767927	+	0.640537i	$0.221288\pi$
$733$	1.00125e13i	1.28107i	−0.767927	+	0.640537i	$0.778712\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	5.67257e11i	0.0708233i
$738$	0	0
$739$	5.95622e12	0.734634	0.367317	−	0.930096i	$-0.380276\pi$
$739$	5.95622e12	0.734634	0.367317	+	0.930096i	$0.380276\pi$
$740$	0	0
$741$	−6.00175e12	−0.731300
$742$	0	0
$743$	− 8.85106e12i	− 1.06548i	−0.846279	−	0.532740i	$-0.821162\pi$
$743$	− 8.85106e12i	− 1.06548i	0.846279	−	0.532740i	$-0.178838\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	9.73412e11i	0.114381i
$748$	0	0
$749$	−2.51443e13	−2.91925
$750$	0	0
$751$	1.31039e13	1.50321	0.751606	−	0.659613i	$-0.229280\pi$
$751$	1.31039e13	1.50321	0.751606	+	0.659613i	$0.229280\pi$
$752$	0	0
$753$	− 5.94584e12i	− 0.673963i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	1.31072e13i	1.45071i	0.688378	+	0.725353i	$0.258323\pi$
$757$	1.31072e13i	1.45071i	−0.688378	+	0.725353i	$0.741677\pi$
$758$	0	0
$759$	5.03223e11	0.0550392
$760$	0	0
$761$	4.34902e12	0.470068	0.235034	−	0.971987i	$-0.424480\pi$
$761$	4.34902e12	0.470068	0.235034	+	0.971987i	$0.424480\pi$
$762$	0	0
$763$	9.52083e12i	1.01698i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 9.15099e12i	− 0.954749i
$768$	0	0
$769$	9.23095e12	0.951871	0.475935	−	0.879480i	$-0.342110\pi$
$769$	9.23095e12	0.951871	0.475935	+	0.879480i	$0.342110\pi$
$770$	0	0
$771$	4.83730e12	0.493013
$772$	0	0
$773$	4.89415e12i	0.493025i	0.969140	+	0.246513i	$0.0792847\pi$
$773$	4.89415e12i	0.493025i	−0.969140	+	0.246513i	$0.920715\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	3.52959e12i	0.347400i
$778$	0	0
$779$	1.20455e13	1.17194
$780$	0	0
$781$	2.67843e11	0.0257603
$782$	0	0
$783$	− 2.17558e12i	− 0.206846i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	1.57772e13i	1.46604i	0.680209	+	0.733018i	$0.261889\pi$
$787$	1.57772e13i	1.46604i	−0.680209	+	0.733018i	$0.738111\pi$
$788$	0	0
$789$	3.56824e12	0.327799
$790$	0	0
$791$	−1.69224e13	−1.53698
$792$	0	0
$793$	5.90299e12i	0.530082i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	1.39090e13i	1.22105i	0.791998	+	0.610524i	$0.209041\pi$
$797$	1.39090e13i	1.22105i	−0.791998	+	0.610524i	$0.790959\pi$
$798$	0	0
$799$	9.01945e12	0.782924
$800$	0	0
$801$	1.18639e12	0.101831
$802$	0	0
$803$	− 6.58735e11i	− 0.0559101i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 9.37105e12i	− 0.777781i
$808$	0	0
$809$	−3.81208e11	−0.0312891	−0.0156446	−	0.999878i	$-0.504980\pi$
$809$	−3.81208e11	−0.0312891	−0.0156446	+	0.999878i	$0.504980\pi$
$810$	0	0
$811$	−9.34915e12	−0.758889	−0.379445	−	0.925214i	$-0.623885\pi$
$811$	−9.34915e12	−0.758889	−0.379445	+	0.925214i	$0.623885\pi$
$812$	0	0
$813$	− 8.07748e12i	− 0.648438i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	1.28873e13i	1.01196i
$818$	0	0
$819$	−2.72528e13	−2.11657
$820$	0	0
$821$	−1.17330e13	−0.901290	−0.450645	−	0.892703i	$-0.648806\pi$
$821$	−1.17330e13	−0.901290	−0.450645	+	0.892703i	$0.648806\pi$
$822$	0	0
$823$	− 6.28533e12i	− 0.477561i	−0.971074	−	0.238781i	$-0.923252\pi$
$823$	− 6.28533e12i	− 0.477561i	0.971074	−	0.238781i	$-0.0767477\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1.80738e13i	1.34361i	0.740727	+	0.671806i	$0.234481\pi$
$827$	1.80738e13i	1.34361i	−0.740727	+	0.671806i	$0.765519\pi$
$828$	0	0
$829$	1.34766e13	0.991028	0.495514	−	0.868600i	$-0.334980\pi$
$829$	1.34766e13	0.991028	0.495514	+	0.868600i	$0.334980\pi$
$830$	0	0
$831$	−4.22087e12	−0.307041
$832$	0	0
$833$	− 9.59478e12i	− 0.690450i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 3.25868e12i	− 0.229497i
$838$	0	0
$839$	−2.14122e13	−1.49188	−0.745939	−	0.666014i	$-0.767999\pi$
$839$	−2.14122e13	−1.49188	−0.745939	+	0.666014i	$0.767999\pi$
$840$	0	0
$841$	−1.36591e13	−0.941542
$842$	0	0
$843$	− 4.90231e12i	− 0.334331i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 2.40071e13i	− 1.60274i
$848$	0	0
$849$	−2.65303e12	−0.175250
$850$	0	0
$851$	−9.57161e12	−0.625608
$852$	0	0
$853$	− 1.20087e13i	− 0.776649i	−0.921523	−	0.388324i	$-0.873054\pi$
$853$	− 1.20087e13i	− 0.776649i	0.921523	−	0.388324i	$-0.126946\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 2.89490e13i	− 1.83324i	−0.399759	−	0.916620i	$-0.630906\pi$
$857$	− 2.89490e13i	− 1.83324i	0.399759	−	0.916620i	$-0.369094\pi$
$858$	0	0
$859$	−2.31413e13	−1.45017	−0.725083	−	0.688662i	$-0.758199\pi$
$859$	−2.31413e13	−1.45017	−0.725083	+	0.688662i	$0.758199\pi$
$860$	0	0
$861$	−1.67950e13	−1.04151
$862$	0	0
$863$	6.84504e12i	0.420075i	0.977693	+	0.210038i	$0.0673587\pi$
$863$	6.84504e12i	0.420075i	−0.977693	+	0.210038i	$0.932641\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 6.56705e12i	− 0.394716i
$868$	0	0
$869$	−9.21821e11	−0.0548350
$870$	0	0
$871$	−2.55807e13	−1.50602
$872$	0	0
$873$	− 3.63126e11i	− 0.0211589i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	2.24289e13i	1.28029i	0.768253	+	0.640147i	$0.221126\pi$
$877$	2.24289e13i	1.28029i	−0.768253	+	0.640147i	$0.778874\pi$
$878$	0	0
$879$	6.37306e12	0.360079
$880$	0	0
$881$	1.24212e13	0.694658	0.347329	−	0.937743i	$-0.387089\pi$
$881$	1.24212e13	0.694658	0.347329	+	0.937743i	$0.387089\pi$
$882$	0	0
$883$	5.52878e12i	0.306060i	0.988222	+	0.153030i	$0.0489031\pi$
$883$	5.52878e12i	0.306060i	−0.988222	+	0.153030i	$0.951097\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	1.84924e13i	1.00308i	0.865133	+	0.501542i	$0.167234\pi$
$887$	1.84924e13i	1.00308i	−0.865133	+	0.501542i	$0.832766\pi$
$888$	0	0
$889$	1.87581e13	1.00724
$890$	0	0
$891$	5.31633e11	0.0282594
$892$	0	0
$893$	3.03827e13i	1.59880i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	2.26931e13i	1.17038i
$898$	0	0
$899$	1.27025e12	0.0648590
$900$	0	0
$901$	4.37635e12	0.221233
$902$	0	0
$903$	− 1.79688e13i	− 0.899339i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	2.72494e13i	1.33698i	0.743722	+	0.668489i	$0.233059\pi$
$907$	2.72494e13i	1.33698i	−0.743722	+	0.668489i	$0.766941\pi$
$908$	0	0
$909$	−9.42638e12	−0.457939
$910$	0	0
$911$	−7.31194e12	−0.351722	−0.175861	−	0.984415i	$-0.556271\pi$
$911$	−7.31194e12	−0.351722	−0.175861	+	0.984415i	$0.556271\pi$
$912$	0	0
$913$	2.53130e11i	0.0120566i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	5.74148e12i	0.268140i
$918$	0	0
$919$	6.17500e12	0.285573	0.142786	−	0.989754i	$-0.454394\pi$
$919$	6.17500e12	0.285573	0.142786	+	0.989754i	$0.454394\pi$
$920$	0	0
$921$	−1.11003e13	−0.508353
$922$	0	0
$923$	1.20785e13i	0.547780i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 1.20565e13i	− 0.536244i
$928$	0	0
$929$	−4.90710e12	−0.216149	−0.108075	−	0.994143i	$-0.534469\pi$
$929$	−4.90710e12	−0.216149	−0.108075	+	0.994143i	$0.534469\pi$
$930$	0	0
$931$	3.23208e13	1.40996
$932$	0	0
$933$	3.76728e12i	0.162765i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	2.66248e13i	1.12839i	0.825642	+	0.564195i	$0.190813\pi$
$937$	2.66248e13i	1.12839i	−0.825642	+	0.564195i	$0.809187\pi$
$938$	0	0
$939$	−1.60372e13	−0.673182
$940$	0	0
$941$	−3.78799e11	−0.0157491	−0.00787454	−	0.999969i	$-0.502507\pi$
$941$	−3.78799e11	−0.0157491	−0.00787454	+	0.999969i	$0.502507\pi$
$942$	0	0
$943$	− 4.55448e13i	− 1.87558i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 3.93752e13i	− 1.59092i	−0.606006	−	0.795460i	$-0.707229\pi$
$947$	− 3.93752e13i	− 1.59092i	0.606006	−	0.795460i	$-0.292771\pi$
$948$	0	0
$949$	2.97060e13	1.18890
$950$	0	0
$951$	−5.26368e12	−0.208678
$952$	0	0
$953$	3.77122e13i	1.48103i	0.672041	+	0.740514i	$0.265418\pi$
$953$	3.77122e13i	1.48103i	−0.672041	+	0.740514i	$0.734582\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 2.45224e11i	− 0.00945060i
$958$	0	0
$959$	−6.70457e13	−2.55969
$960$	0	0
$961$	−2.45370e13	−0.928039
$962$	0	0
$963$	3.69485e13i	1.38445i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 3.28952e12i	− 0.120980i	−0.998169	−	0.0604900i	$-0.980734\pi$
$967$	− 3.28952e12i	− 0.120980i	0.998169	−	0.0604900i	$-0.0192663\pi$
$968$	0	0
$969$	−5.04252e12	−0.183735
$970$	0	0
$971$	2.68793e11	0.00970357	0.00485179	−	0.999988i	$-0.498456\pi$
$971$	2.68793e11	0.00970357	0.00485179	+	0.999988i	$0.498456\pi$
$972$	0	0
$973$	− 5.66430e13i	− 2.02599i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	2.44634e13i	0.858998i	0.903067	+	0.429499i	$0.141310\pi$
$977$	2.44634e13i	0.858998i	−0.903067	+	0.429499i	$0.858690\pi$
$978$	0	0
$979$	3.08513e11	0.0107337
$980$	0	0
$981$	1.39905e13	0.482304
$982$	0	0
$983$	− 1.57199e13i	− 0.536983i	−0.963282	−	0.268491i	$-0.913475\pi$
$983$	− 1.57199e13i	− 0.536983i	0.963282	−	0.268491i	$-0.0865250\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 4.23625e13i	− 1.42087i
$988$	0	0
$989$	4.87280e13	1.61955
$990$	0	0
$991$	−1.11195e13	−0.366229	−0.183114	−	0.983092i	$-0.558618\pi$
$991$	−1.11195e13	−0.366229	−0.183114	+	0.983092i	$0.558618\pi$
$992$	0	0
$993$	5.37262e12i	0.175353i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	2.50821e13i	0.803962i	0.915648	+	0.401981i	$0.131678\pi$
$997$	2.50821e13i	0.803962i	−0.915648	+	0.401981i	$0.868322\pi$
$998$	0	0
$999$	1.19658e13	0.380098

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	200.10.c.a.49.1		2
4.3	odd	2		400.10.c.f.49.2		2
5.2	odd	4		200.10.a.a.1.1		1
5.3	odd	4		8.10.a.b.1.1	✓	1
5.4	even	2	inner	200.10.c.a.49.2		2
15.8	even	4		72.10.a.a.1.1		1
20.3	even	4		16.10.a.b.1.1		1
20.7	even	4		400.10.a.i.1.1		1
20.19	odd	2		400.10.c.f.49.1		2
35.13	even	4		392.10.a.a.1.1		1
40.3	even	4		64.10.a.g.1.1		1
40.13	odd	4		64.10.a.c.1.1		1
60.23	odd	4		144.10.a.b.1.1		1
80.3	even	4		256.10.b.k.129.1		2
80.13	odd	4		256.10.b.a.129.2		2
80.43	even	4		256.10.b.k.129.2		2
80.53	odd	4		256.10.b.a.129.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8.10.a.b.1.1	✓	1	5.3	odd	4
16.10.a.b.1.1		1	20.3	even	4
64.10.a.c.1.1		1	40.13	odd	4
64.10.a.g.1.1		1	40.3	even	4
72.10.a.a.1.1		1	15.8	even	4
144.10.a.b.1.1		1	60.23	odd	4
200.10.a.a.1.1		1	5.2	odd	4
200.10.c.a.49.1		2	1.1	even	1	trivial
200.10.c.a.49.2		2	5.4	even	2	inner
256.10.b.a.129.1		2	80.53	odd	4
256.10.b.a.129.2		2	80.13	odd	4
256.10.b.k.129.1		2	80.3	even	4
256.10.b.k.129.2		2	80.43	even	4
392.10.a.a.1.1		1	35.13	even	4
400.10.a.i.1.1		1	20.7	even	4
400.10.c.f.49.1		2	20.19	odd	2
400.10.c.f.49.2		2	4.3	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:	\( N \)	\(=\)	\( 200 = 2^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	200.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-68.0000i q^{3} +10248.0i q^{7} +15059.0 q^{9} +O(q^{10})\)	\(q-68.0000i q^{3} +10248.0i q^{7} +15059.0 q^{9} +3916.00 q^{11} +176594. i q^{13} +148370. i q^{17} -499796. q^{19} +696864. q^{21} +1.88977e6i q^{23} -2.36246e6i q^{27} +920898. q^{29} +1.37936e6 q^{31} -266288. i q^{33} +5.06497e6i q^{37} +1.20084e7 q^{39} -2.41008e7 q^{41} -2.57852e7i q^{43} -6.07902e7i q^{47} -6.46679e7 q^{49} +1.00892e7 q^{51} -2.94962e7i q^{53} +3.39861e7i q^{57} -5.18194e7 q^{59} +3.34269e7 q^{61} +1.54325e8i q^{63} +1.44856e8i q^{67} +1.28504e8 q^{69} +6.83971e7 q^{71} -1.68216e8i q^{73} +4.01312e7i q^{77} -2.35399e8 q^{79} +1.35759e8 q^{81} +6.46399e7i q^{83} -6.26211e7i q^{87} +7.87827e7 q^{89} -1.80974e9 q^{91} -9.37965e7i q^{93} -2.41136e7i q^{97} +5.89710e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 30118 q^{9}+O(q^{10}) \) 2 * q + 30118 * q^9	\( 2 q + 30118 q^{9} + 7832 q^{11} - 999592 q^{19} + 1393728 q^{21} + 1841796 q^{29} + 2758720 q^{31} + 24016784 q^{39} - 48201516 q^{41} - 129335794 q^{49} + 20178320 q^{51} - 103638776 q^{59} + 66853820 q^{61} + 257008448 q^{69} + 136794256 q^{71} - 470797472 q^{79} + 271518578 q^{81} + 157565388 q^{89} - 3619470624 q^{91} + 117942088 q^{99}+O(q^{100}) \) 2 * q + 30118 * q^9 + 7832 * q^11 - 999592 * q^19 + 1393728 * q^21 + 1841796 * q^29 + 2758720 * q^31 + 24016784 * q^39 - 48201516 * q^41 - 129335794 * q^49 + 20178320 * q^51 - 103638776 * q^59 + 66853820 * q^61 + 257008448 * q^69 + 136794256 * q^71 - 470797472 * q^79 + 271518578 * q^81 + 157565388 * q^89 - 3619470624 * q^91 + 117942088 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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