LMFDB - Embedded newform 200.10.c.d.49.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [200,10,Mod(49,200)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage: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")

magma://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:traces = [4,0,0,0,0,0,0,0,-38916] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$103.007167233$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{46})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 529 $ x^4 + 529
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{9}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 40)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.1
Root			$3.39116 - 3.39116i$ of defining polynomial
Character	$\chi$	$=$	200.49
Dual form			200.10.c.d.49.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-216.776i q^{3} +1662.09i q^{7} -27308.8 q^{9} +34372.0 q^{11} -162675. i q^{13} +91796.1i q^{17} +536227. q^{19} +360300. q^{21} -234899. i q^{23} +1.65309e6i q^{27} -7.25706e6 q^{29} +8.23051e6 q^{31} -7.45102e6i q^{33} -1.56850e7i q^{37} -3.52641e7 q^{39} -1.67318e7 q^{41} -9.36208e6i q^{43} -2.69410e7i q^{47} +3.75911e7 q^{49} +1.98992e7 q^{51} -3.41105e7i q^{53} -1.16241e8i q^{57} -1.12155e8 q^{59} +5.49531e6 q^{61} -4.53896e7i q^{63} -2.40686e8i q^{67} -5.09204e7 q^{69} -2.58502e8 q^{71} +3.53583e8i q^{73} +5.71292e7i q^{77} -3.66190e8 q^{79} -1.79169e8 q^{81} +9.74953e6i q^{83} +1.57316e9i q^{87} +1.14201e9 q^{89} +2.70380e8 q^{91} -1.78418e9i q^{93} +1.25555e9i q^{97} -9.38657e8 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 38916 q^{9} + 50240 q^{11} + 50960 q^{19} + 1279728 q^{21} - 14646680 q^{29} + 21354544 q^{31} - 73949616 q^{39} - 15591528 q^{41} + 142678668 q^{49} + 22944240 q^{51} - 119236528 q^{59} - 376327544 q^{61}+ \cdots - 2022585408 q^{99}+O(q^{100}) $ 4 * q - 38916 * q^9 + 50240 * q^11 + 50960 * q^19 + 1279728 * q^21 - 14646680 * q^29 + 21354544 * q^31 - 73949616 * q^39 - 15591528 * q^41 + 142678668 * q^49 + 22944240 * q^51 - 119236528 * q^59 - 376327544 * q^61 + 234896688 * q^69 - 337331184 * q^71 - 933892512 * q^79 - 700854444 * q^81 + 216669720 * q^89 + 621601232 * q^91 - 2022585408 * q^99

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
$3$	− 216.776i	− 1.54513i	−0.634935	−	0.772566i	$-0.718973\pi$
$3$	− 216.776i	− 1.54513i	0.634935	−	0.772566i	$-0.281027\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1662.09i	0.261645i	0.991406	+	0.130823i	$0.0417618\pi$
$7$	1662.09i	0.261645i	−0.991406	+	0.130823i	$0.958238\pi$
$8$	0	0
$9$	−27308.8	−1.38743
$10$	0	0
$11$	34372.0	0.707844	0.353922	−	0.935275i	$-0.384848\pi$
$11$	34372.0	0.707844	0.353922	+	0.935275i	$0.384848\pi$
$12$	0	0
$13$	− 162675.i	− 1.57971i	−0.613296	−	0.789853i	$-0.710157\pi$
$13$	− 162675.i	− 1.57971i	0.613296	−	0.789853i	$-0.289843\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	91796.1i	0.266566i	0.991078	+	0.133283i	$0.0425519\pi$
$17$	91796.1i	0.266566i	−0.991078	+	0.133283i	$0.957448\pi$
$18$	0	0
$19$	536227.	0.943969	0.471985	−	0.881607i	$-0.343538\pi$
$19$	536227.	0.943969	0.471985	+	0.881607i	$0.343538\pi$
$20$	0	0
$21$	360300.	0.404276
$22$	0	0
$23$	− 234899.i	− 0.175027i	−0.996163	−	0.0875136i	$-0.972108\pi$
$23$	− 234899.i	− 0.175027i	0.996163	−	0.0875136i	$-0.0278921\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.65309e6i	0.598631i
$28$	0	0
$29$	−7.25706e6	−1.90533	−0.952665	−	0.304023i	$-0.901670\pi$
$29$	−7.25706e6	−1.90533	−0.952665	+	0.304023i	$0.901670\pi$
$30$	0	0
$31$	8.23051e6	1.60066	0.800330	−	0.599559i	$-0.204657\pi$
$31$	8.23051e6	1.60066	0.800330	+	0.599559i	$0.204657\pi$
$32$	0	0
$33$	− 7.45102e6i	− 1.09371i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 1.56850e7i	− 1.37587i	−0.725771	−	0.687936i	$-0.758517\pi$
$37$	− 1.56850e7i	− 1.37587i	0.725771	−	0.687936i	$-0.241483\pi$
$38$	0	0
$39$	−3.52641e7	−2.44085
$40$	0	0
$41$	−1.67318e7	−0.924730	−0.462365	−	0.886690i	$-0.652999\pi$
$41$	−1.67318e7	−0.924730	−0.462365	+	0.886690i	$0.652999\pi$
$42$	0	0
$43$	− 9.36208e6i	− 0.417604i	−0.977958	−	0.208802i	$-0.933044\pi$
$43$	− 9.36208e6i	− 0.417604i	0.977958	−	0.208802i	$-0.0669564\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 2.69410e7i	− 0.805328i	−0.915348	−	0.402664i	$-0.868084\pi$
$47$	− 2.69410e7i	− 0.805328i	0.915348	−	0.402664i	$-0.131916\pi$
$48$	0	0
$49$	3.75911e7	0.931542
$50$	0	0
$51$	1.98992e7	0.411879
$52$	0	0
$53$	− 3.41105e7i	− 0.593809i	−0.954907	−	0.296905i	$-0.904046\pi$
$53$	− 3.41105e7i	− 0.593809i	0.954907	−	0.296905i	$-0.0959544\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 1.16241e8i	− 1.45856i
$58$	0	0
$59$	−1.12155e8	−1.20499	−0.602496	−	0.798122i	$-0.705827\pi$
$59$	−1.12155e8	−1.20499	−0.602496	+	0.798122i	$0.705827\pi$
$60$	0	0
$61$	5.49531e6	0.0508168	0.0254084	−	0.999677i	$-0.491911\pi$
$61$	5.49531e6	0.0508168	0.0254084	+	0.999677i	$0.491911\pi$
$62$	0	0
$63$	− 4.53896e7i	− 0.363014i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 2.40686e8i	− 1.45920i	−0.683876	−	0.729598i	$-0.739707\pi$
$67$	− 2.40686e8i	− 1.45920i	0.683876	−	0.729598i	$-0.260293\pi$
$68$	0	0
$69$	−5.09204e7	−0.270440
$70$	0	0
$71$	−2.58502e8	−1.20726	−0.603631	−	0.797264i	$-0.706280\pi$
$71$	−2.58502e8	−1.20726	−0.603631	+	0.797264i	$0.706280\pi$
$72$	0	0
$73$	3.53583e8i	1.45726i	0.684905	+	0.728632i	$0.259844\pi$
$73$	3.53583e8i	1.45726i	−0.684905	+	0.728632i	$0.740156\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	5.71292e7i	0.185204i
$78$	0	0
$79$	−3.66190e8	−1.05775	−0.528876	−	0.848699i	$-0.677386\pi$
$79$	−3.66190e8	−1.05775	−0.528876	+	0.848699i	$0.677386\pi$
$80$	0	0
$81$	−1.79169e8	−0.462467
$82$	0	0
$83$	9.74953e6i	0.0225493i	0.999936	+	0.0112746i	$0.00358890\pi$
$83$	9.74953e6i	0.0225493i	−0.999936	+	0.0112746i	$0.996411\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	1.57316e9i	2.94398i
$88$	0	0
$89$	1.14201e9	1.92937	0.964687	−	0.263397i	$-0.0848431\pi$
$89$	1.14201e9	1.92937	0.964687	+	0.263397i	$0.0848431\pi$
$90$	0	0
$91$	2.70380e8	0.413322
$92$	0	0
$93$	− 1.78418e9i	− 2.47323i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	1.25555e9i	1.44000i	0.693975	+	0.719999i	$0.255858\pi$
$97$	1.25555e9i	1.44000i	−0.693975	+	0.719999i	$0.744142\pi$
$98$	0	0
$99$	−9.38657e8	−0.982084
$100$	0	0
$101$	2.63563e8	0.252022	0.126011	−	0.992029i	$-0.459783\pi$
$101$	2.63563e8	0.252022	0.126011	+	0.992029i	$0.459783\pi$
$102$	0	0
$103$	2.09532e9i	1.83435i	0.398482	+	0.917176i	$0.369537\pi$
$103$	2.09532e9i	1.83435i	−0.398482	+	0.917176i	$0.630463\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 3.30509e8i	− 0.243757i	−0.992545	−	0.121878i	$-0.961108\pi$
$107$	− 3.30509e8i	− 0.243757i	0.992545	−	0.121878i	$-0.0388918\pi$
$108$	0	0
$109$	1.65293e9	1.12160	0.560798	−	0.827953i	$-0.310495\pi$
$109$	1.65293e9	1.12160	0.560798	+	0.827953i	$0.310495\pi$
$110$	0	0
$111$	−3.40014e9	−2.12590
$112$	0	0
$113$	− 1.15695e9i	− 0.667513i	−0.942659	−	0.333757i	$-0.891684\pi$
$113$	− 1.15695e9i	− 0.667513i	0.942659	−	0.333757i	$-0.108316\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	4.44246e9i	2.19173i
$118$	0	0
$119$	−1.52573e8	−0.0697456
$120$	0	0
$121$	−1.17652e9	−0.498957
$122$	0	0
$123$	3.62705e9i	1.42883i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	1.56129e9i	0.532559i	0.963896	+	0.266280i	$0.0857944\pi$
$127$	1.56129e9i	0.532559i	−0.963896	+	0.266280i	$0.914206\pi$
$128$	0	0
$129$	−2.02947e9	−0.645253
$130$	0	0
$131$	−4.75445e9	−1.41052	−0.705261	−	0.708948i	$-0.749170\pi$
$131$	−4.75445e9	−1.41052	−0.705261	+	0.708948i	$0.749170\pi$
$132$	0	0
$133$	8.91256e8i	0.246985i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	1.57706e9i	0.382478i	0.981543	+	0.191239i	$0.0612506\pi$
$137$	1.57706e9i	0.382478i	−0.981543	+	0.191239i	$0.938749\pi$
$138$	0	0
$139$	−3.40723e9	−0.774167	−0.387084	−	0.922045i	$-0.626518\pi$
$139$	−3.40723e9	−0.774167	−0.387084	+	0.922045i	$0.626518\pi$
$140$	0	0
$141$	−5.84015e9	−1.24434
$142$	0	0
$143$	− 5.59147e9i	− 1.11818i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 8.14884e9i	− 1.43935i
$148$	0	0
$149$	4.20713e8	0.0699274	0.0349637	−	0.999389i	$-0.488868\pi$
$149$	4.20713e8	0.0699274	0.0349637	+	0.999389i	$0.488868\pi$
$150$	0	0
$151$	−1.74377e9	−0.272957	−0.136478	−	0.990643i	$-0.543578\pi$
$151$	−1.74377e9	−0.272957	−0.136478	+	0.990643i	$0.543578\pi$
$152$	0	0
$153$	− 2.50684e9i	− 0.369841i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 6.13142e9i	− 0.805402i	−0.915332	−	0.402701i	$-0.868071\pi$
$157$	− 6.13142e9i	− 0.805402i	0.915332	−	0.402701i	$-0.131929\pi$
$158$	0	0
$159$	−7.39434e9	−0.917513
$160$	0	0
$161$	3.90422e8	0.0457950
$162$	0	0
$163$	− 1.07953e10i	− 1.19782i	−0.800816	−	0.598910i	$-0.795601\pi$
$163$	− 1.07953e10i	− 1.19782i	0.800816	−	0.598910i	$-0.204399\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1.55417e10i	1.54623i	0.634266	+	0.773115i	$0.281302\pi$
$167$	1.55417e10i	1.54623i	−0.634266	+	0.773115i	$0.718698\pi$
$168$	0	0
$169$	−1.58587e10	−1.49547
$170$	0	0
$171$	−1.46437e10	−1.30969
$172$	0	0
$173$	− 3.21498e9i	− 0.272880i	−0.990648	−	0.136440i	$-0.956434\pi$
$173$	− 3.21498e9i	− 0.272880i	0.990648	−	0.136440i	$-0.0435661\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	2.43125e10i	1.86187i
$178$	0	0
$179$	−2.10234e10	−1.53061	−0.765305	−	0.643667i	$-0.777412\pi$
$179$	−2.10234e10	−1.53061	−0.765305	+	0.643667i	$0.777412\pi$
$180$	0	0
$181$	−1.54514e10	−1.07008	−0.535038	−	0.844828i	$-0.679703\pi$
$181$	−1.54514e10	−1.07008	−0.535038	+	0.844828i	$0.679703\pi$
$182$	0	0
$183$	− 1.19125e9i	− 0.0785187i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	3.15521e9i	0.188687i
$188$	0	0
$189$	−2.74758e9	−0.156629
$190$	0	0
$191$	2.53988e10	1.38090	0.690451	−	0.723380i	$-0.257412\pi$
$191$	2.53988e10	1.38090	0.690451	+	0.723380i	$0.257412\pi$
$192$	0	0
$193$	1.70167e10i	0.882809i	0.897308	+	0.441404i	$0.145520\pi$
$193$	1.70167e10i	0.882809i	−0.897308	+	0.441404i	$0.854480\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	4.38218e9i	0.207297i	0.994614	+	0.103648i	$0.0330517\pi$
$197$	4.38218e9i	0.207297i	−0.994614	+	0.103648i	$0.966948\pi$
$198$	0	0
$199$	1.37237e10	0.620343	0.310171	−	0.950681i	$-0.399614\pi$
$199$	1.37237e10	0.620343	0.310171	+	0.950681i	$0.399614\pi$
$200$	0	0
$201$	−5.21749e10	−2.25465
$202$	0	0
$203$	− 1.20619e10i	− 0.498520i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	6.41480e9i	0.242838i
$208$	0	0
$209$	1.84312e10	0.668183
$210$	0	0
$211$	−1.21366e10	−0.421526	−0.210763	−	0.977537i	$-0.567595\pi$
$211$	−1.21366e10	−0.421526	−0.210763	+	0.977537i	$0.567595\pi$
$212$	0	0
$213$	5.60370e10i	1.86538i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	1.36798e10i	0.418805i
$218$	0	0
$219$	7.66483e10	2.25166
$220$	0	0
$221$	1.49330e10	0.421095
$222$	0	0
$223$	− 6.20297e10i	− 1.67968i	−0.542830	−	0.839842i	$-0.682647\pi$
$223$	− 6.20297e10i	− 1.67968i	0.542830	−	0.839842i	$-0.317353\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	3.51959e10i	0.879783i	0.898051	+	0.439891i	$0.144983\pi$
$227$	3.51959e10i	0.879783i	−0.898051	+	0.439891i	$0.855017\pi$
$228$	0	0
$229$	3.83973e10	0.922659	0.461329	−	0.887229i	$-0.347373\pi$
$229$	3.83973e10	0.922659	0.461329	+	0.887229i	$0.347373\pi$
$230$	0	0
$231$	1.23842e10	0.286164
$232$	0	0
$233$	2.59156e10i	0.576049i	0.957623	+	0.288024i	$0.0929984\pi$
$233$	2.59156e10i	0.576049i	−0.957623	+	0.288024i	$0.907002\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	7.93811e10i	1.63437i
$238$	0	0
$239$	−5.76903e10	−1.14370	−0.571850	−	0.820358i	$-0.693774\pi$
$239$	−5.76903e10	−1.14370	−0.571850	+	0.820358i	$0.693774\pi$
$240$	0	0
$241$	−4.59383e10	−0.877200	−0.438600	−	0.898682i	$-0.644525\pi$
$241$	−4.59383e10	−0.877200	−0.438600	+	0.898682i	$0.644525\pi$
$242$	0	0
$243$	7.13773e10i	1.31320i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 8.72309e10i	− 1.49119i
$248$	0	0
$249$	2.11346e9	0.0348416
$250$	0	0
$251$	−6.33143e10	−1.00686	−0.503431	−	0.864035i	$-0.667929\pi$
$251$	−6.33143e10	−1.00686	−0.503431	+	0.864035i	$0.667929\pi$
$252$	0	0
$253$	− 8.07393e9i	− 0.123892i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	4.21910e10i	0.603283i	0.953421	+	0.301642i	$0.0975346\pi$
$257$	4.21910e10i	0.603283i	−0.953421	+	0.301642i	$0.902465\pi$
$258$	0	0
$259$	2.60699e10	0.359990
$260$	0	0
$261$	1.98182e11	2.64351
$262$	0	0
$263$	− 5.01270e10i	− 0.646057i	−0.946389	−	0.323028i	$-0.895299\pi$
$263$	− 5.01270e10i	− 0.646057i	0.946389	−	0.323028i	$-0.104701\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 2.47561e11i	− 2.98114i
$268$	0	0
$269$	−7.21552e10	−0.840199	−0.420099	−	0.907478i	$-0.638005\pi$
$269$	−7.21552e10	−0.840199	−0.420099	+	0.907478i	$0.638005\pi$
$270$	0	0
$271$	5.23276e10	0.589344	0.294672	−	0.955598i	$-0.404790\pi$
$271$	5.23276e10	0.589344	0.294672	+	0.955598i	$0.404790\pi$
$272$	0	0
$273$	− 5.86119e10i	− 0.638637i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 5.78954e10i	− 0.590860i	−0.955364	−	0.295430i	$-0.904537\pi$
$277$	− 5.78954e10i	− 0.590860i	0.955364	−	0.295430i	$-0.0954630\pi$
$278$	0	0
$279$	−2.24765e11	−2.22081
$280$	0	0
$281$	7.31211e10	0.699623	0.349812	−	0.936820i	$-0.386246\pi$
$281$	7.31211e10	0.699623	0.349812	+	0.936820i	$0.386246\pi$
$282$	0	0
$283$	5.12368e10i	0.474835i	0.971408	+	0.237418i	$0.0763010\pi$
$283$	5.12368e10i	0.474835i	−0.971408	+	0.237418i	$0.923699\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 2.78097e10i	− 0.241951i
$288$	0	0
$289$	1.10161e11	0.928943
$290$	0	0
$291$	2.72173e11	2.22499
$292$	0	0
$293$	− 1.95934e11i	− 1.55312i	−0.630043	−	0.776561i	$-0.716963\pi$
$293$	− 1.95934e11i	− 1.55312i	0.630043	−	0.776561i	$-0.283037\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	5.68200e10i	0.423737i
$298$	0	0
$299$	−3.82122e10	−0.276491
$300$	0	0
$301$	1.55606e10	0.109264
$302$	0	0
$303$	− 5.71341e10i	− 0.389407i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 2.29564e11i	− 1.47496i	−0.675368	−	0.737481i	$-0.736015\pi$
$307$	− 2.29564e11i	− 1.47496i	0.675368	−	0.737481i	$-0.263985\pi$
$308$	0	0
$309$	4.54215e11	2.83431
$310$	0	0
$311$	4.99151e10	0.302559	0.151280	−	0.988491i	$-0.451661\pi$
$311$	4.99151e10	0.302559	0.151280	+	0.988491i	$0.451661\pi$
$312$	0	0
$313$	1.19272e11i	0.702408i	0.936299	+	0.351204i	$0.114228\pi$
$313$	1.19272e11i	0.702408i	−0.936299	+	0.351204i	$0.885772\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 2.85879e11i	− 1.59007i	−0.606565	−	0.795034i	$-0.707453\pi$
$317$	− 2.85879e11i	− 1.59007i	0.606565	−	0.795034i	$-0.292547\pi$
$318$	0	0
$319$	−2.49440e11	−1.34868
$320$	0	0
$321$	−7.16464e10	−0.376636
$322$	0	0
$323$	4.92236e10i	0.251630i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 3.58316e11i	− 1.73301i
$328$	0	0
$329$	4.47782e10	0.210710
$330$	0	0
$331$	−2.12047e11	−0.970973	−0.485486	−	0.874244i	$-0.661357\pi$
$331$	−2.12047e11	−0.970973	−0.485486	+	0.874244i	$0.661357\pi$
$332$	0	0
$333$	4.28340e11i	1.90893i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	2.00430e11i	0.846502i	0.906012	+	0.423251i	$0.139111\pi$
$337$	2.00430e11i	0.846502i	−0.906012	+	0.423251i	$0.860889\pi$
$338$	0	0
$339$	−2.50798e11	−1.03140
$340$	0	0
$341$	2.82899e11	1.13302
$342$	0	0
$343$	1.29551e11i	0.505378i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 9.86210e10i	− 0.365163i	−0.983191	−	0.182582i	$-0.941555\pi$
$347$	− 9.86210e10i	− 0.365163i	0.983191	−	0.182582i	$-0.0584453\pi$
$348$	0	0
$349$	4.79143e11	1.72882	0.864411	−	0.502785i	$-0.167691\pi$
$349$	4.79143e11	1.72882	0.864411	+	0.502785i	$0.167691\pi$
$350$	0	0
$351$	2.68917e11	0.945662
$352$	0	0
$353$	− 4.56755e11i	− 1.56566i	−0.622235	−	0.782830i	$-0.713775\pi$
$353$	− 4.56755e11i	− 1.56566i	0.622235	−	0.782830i	$-0.286225\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	3.30742e10i	0.107766i
$358$	0	0
$359$	1.54059e11	0.489509	0.244755	−	0.969585i	$-0.421293\pi$
$359$	1.54059e11	0.489509	0.244755	+	0.969585i	$0.421293\pi$
$360$	0	0
$361$	−3.51479e10	−0.108922
$362$	0	0
$363$	2.55040e11i	0.770954i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	3.42604e11i	0.985815i	0.870082	+	0.492908i	$0.164066\pi$
$367$	3.42604e11i	0.985815i	−0.870082	+	0.492908i	$0.835934\pi$
$368$	0	0
$369$	4.56925e11	1.28300
$370$	0	0
$371$	5.66947e10	0.155367
$372$	0	0
$373$	1.68130e11i	0.449734i	0.974389	+	0.224867i	$0.0721948\pi$
$373$	1.68130e11i	0.449734i	−0.974389	+	0.224867i	$0.927805\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	1.18054e12i	3.00986i
$378$	0	0
$379$	1.00434e11	0.250038	0.125019	−	0.992154i	$-0.460101\pi$
$379$	1.00434e11	0.250038	0.125019	+	0.992154i	$0.460101\pi$
$380$	0	0
$381$	3.38451e11	0.822874
$382$	0	0
$383$	− 5.07533e11i	− 1.20523i	−0.798032	−	0.602615i	$-0.794126\pi$
$383$	− 5.07533e11i	− 1.20523i	0.798032	−	0.602615i	$-0.205874\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	2.55667e11i	0.579396i
$388$	0	0
$389$	1.26566e11	0.280249	0.140125	−	0.990134i	$-0.455250\pi$
$389$	1.26566e11	0.280249	0.140125	+	0.990134i	$0.455250\pi$
$390$	0	0
$391$	2.15628e10	0.0466562
$392$	0	0
$393$	1.03065e12i	2.17944i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 4.70667e11i	− 0.950948i	−0.879730	−	0.475474i	$-0.842277\pi$
$397$	− 4.70667e11i	− 0.950948i	0.879730	−	0.475474i	$-0.157723\pi$
$398$	0	0
$399$	1.93203e11	0.381624
$400$	0	0
$401$	−9.20810e11	−1.77836	−0.889182	−	0.457554i	$-0.848726\pi$
$401$	−9.20810e11	−1.77836	−0.889182	+	0.457554i	$0.848726\pi$
$402$	0	0
$403$	− 1.33890e12i	− 2.52857i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 5.39126e11i	− 0.973902i
$408$	0	0
$409$	−8.23952e11	−1.45595	−0.727976	−	0.685603i	$-0.759539\pi$
$409$	−8.23952e11	−1.45595	−0.727976	+	0.685603i	$0.759539\pi$
$410$	0	0
$411$	3.41870e11	0.590979
$412$	0	0
$413$	− 1.86411e11i	− 0.315280i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	7.38605e11i	1.19619i
$418$	0	0
$419$	−7.01758e11	−1.11231	−0.556153	−	0.831080i	$-0.687723\pi$
$419$	−7.01758e11	−1.11231	−0.556153	+	0.831080i	$0.687723\pi$
$420$	0	0
$421$	1.22669e11	0.190311	0.0951555	−	0.995462i	$-0.469665\pi$
$421$	1.22669e11	0.190311	0.0951555	+	0.995462i	$0.469665\pi$
$422$	0	0
$423$	7.35725e11i	1.11734i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	9.13368e9i	0.0132960i
$428$	0	0
$429$	−1.21210e12	−1.72774
$430$	0	0
$431$	−4.82937e11	−0.674129	−0.337064	−	0.941482i	$-0.609434\pi$
$431$	−4.82937e11	−0.674129	−0.337064	+	0.941482i	$0.609434\pi$
$432$	0	0
$433$	− 7.71175e11i	− 1.05428i	−0.849777	−	0.527142i	$-0.823264\pi$
$433$	− 7.71175e11i	− 1.05428i	0.849777	−	0.527142i	$-0.176736\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 1.25959e11i	− 0.165220i
$438$	0	0
$439$	6.62301e11	0.851070	0.425535	−	0.904942i	$-0.360086\pi$
$439$	6.62301e11	0.851070	0.425535	+	0.904942i	$0.360086\pi$
$440$	0	0
$441$	−1.02657e12	−1.29245
$442$	0	0
$443$	3.76774e11i	0.464798i	0.972621	+	0.232399i	$0.0746574\pi$
$443$	3.76774e11i	0.464798i	−0.972621	+	0.232399i	$0.925343\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 9.12004e10i	− 0.108047i
$448$	0	0
$449$	−5.77774e11	−0.670888	−0.335444	−	0.942060i	$-0.608886\pi$
$449$	−5.77774e11	−0.670888	−0.335444	+	0.942060i	$0.608886\pi$
$450$	0	0
$451$	−5.75105e11	−0.654564
$452$	0	0
$453$	3.78008e11i	0.421754i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 8.28175e11i	− 0.888176i	−0.895983	−	0.444088i	$-0.853528\pi$
$457$	− 8.28175e11i	− 0.888176i	0.895983	−	0.444088i	$-0.146472\pi$
$458$	0	0
$459$	−1.51747e11	−0.159575
$460$	0	0
$461$	2.77554e11	0.286216	0.143108	−	0.989707i	$-0.454290\pi$
$461$	2.77554e11	0.286216	0.143108	+	0.989707i	$0.454290\pi$
$462$	0	0
$463$	1.01088e11i	0.102232i	0.998693	+	0.0511159i	$0.0162778\pi$
$463$	1.01088e11i	0.102232i	−0.998693	+	0.0511159i	$0.983722\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 1.30742e12i	− 1.27201i	−0.771686	−	0.636004i	$-0.780586\pi$
$467$	− 1.30742e12i	− 1.27201i	0.771686	−	0.636004i	$-0.219414\pi$
$468$	0	0
$469$	4.00041e11	0.381792
$470$	0	0
$471$	−1.32914e12	−1.24445
$472$	0	0
$473$	− 3.21793e11i	− 0.295598i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	9.31518e11i	0.823869i
$478$	0	0
$479$	1.38500e12	1.20210	0.601049	−	0.799212i	$-0.294750\pi$
$479$	1.38500e12	1.20210	0.601049	+	0.799212i	$0.294750\pi$
$480$	0	0
$481$	−2.55157e12	−2.17347
$482$	0	0
$483$	− 8.46341e10i	− 0.0707593i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	9.54305e11i	0.768789i	0.923169	+	0.384394i	$0.125590\pi$
$487$	9.54305e11i	0.768789i	−0.923169	+	0.384394i	$0.874410\pi$
$488$	0	0
$489$	−2.34017e12	−1.85079
$490$	0	0
$491$	−7.41251e11	−0.575571	−0.287785	−	0.957695i	$-0.592919\pi$
$491$	−7.41251e11	−0.575571	−0.287785	+	0.957695i	$0.592919\pi$
$492$	0	0
$493$	− 6.66171e11i	− 0.507895i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 4.29653e11i	− 0.315874i
$498$	0	0
$499$	−1.51393e12	−1.09308	−0.546542	−	0.837432i	$-0.684056\pi$
$499$	−1.51393e12	−1.09308	−0.546542	+	0.837432i	$0.684056\pi$
$500$	0	0
$501$	3.36906e12	2.38913
$502$	0	0
$503$	− 1.58191e12i	− 1.10186i	−0.834552	−	0.550929i	$-0.814274\pi$
$503$	− 1.58191e12i	− 1.10186i	0.834552	−	0.550929i	$-0.185726\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	3.43779e12i	2.31070i
$508$	0	0
$509$	2.29553e11	0.151584	0.0757918	−	0.997124i	$-0.475852\pi$
$509$	2.29553e11	0.151584	0.0757918	+	0.997124i	$0.475852\pi$
$510$	0	0
$511$	−5.87686e11	−0.381286
$512$	0	0
$513$	8.86432e11i	0.565089i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 9.26014e11i	− 0.570046i
$518$	0	0
$519$	−6.96931e11	−0.421635
$520$	0	0
$521$	−1.73274e12	−1.03030	−0.515149	−	0.857101i	$-0.672263\pi$
$521$	−1.73274e12	−1.03030	−0.515149	+	0.857101i	$0.672263\pi$
$522$	0	0
$523$	− 2.45691e12i	− 1.43592i	−0.696083	−	0.717961i	$-0.745075\pi$
$523$	− 2.45691e12i	− 1.43592i	0.696083	−	0.717961i	$-0.254925\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	7.55529e11i	0.426681i
$528$	0	0
$529$	1.74598e12	0.969366
$530$	0	0
$531$	3.06281e12	1.67184
$532$	0	0
$533$	2.72185e12i	1.46080i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	4.55737e12i	2.36499i
$538$	0	0
$539$	1.29208e12	0.659386
$540$	0	0
$541$	−2.10080e12	−1.05438	−0.527190	−	0.849748i	$-0.676754\pi$
$541$	−2.10080e12	−1.05438	−0.527190	+	0.849748i	$0.676754\pi$
$542$	0	0
$543$	3.34949e12i	1.65341i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	3.58621e12i	1.71275i	0.516357	+	0.856373i	$0.327288\pi$
$547$	3.58621e12i	1.71275i	−0.516357	+	0.856373i	$0.672712\pi$
$548$	0	0
$549$	−1.50070e11	−0.0705048
$550$	0	0
$551$	−3.89144e12	−1.79857
$552$	0	0
$553$	− 6.08639e11i	− 0.276756i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 3.15166e12i	− 1.38737i	−0.720280	−	0.693683i	$-0.755987\pi$
$557$	− 3.15166e12i	− 1.38737i	0.720280	−	0.693683i	$-0.244013\pi$
$558$	0	0
$559$	−1.52298e12	−0.659691
$560$	0	0
$561$	6.83975e11	0.291546
$562$	0	0
$563$	3.70986e12i	1.55621i	0.628131	+	0.778107i	$0.283820\pi$
$563$	3.70986e12i	1.55621i	−0.628131	+	0.778107i	$0.716180\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 2.97795e11i	− 0.121002i
$568$	0	0
$569$	1.52350e12	0.609308	0.304654	−	0.952463i	$-0.401459\pi$
$569$	1.52350e12	0.609308	0.304654	+	0.952463i	$0.401459\pi$
$570$	0	0
$571$	2.81914e12	1.10982	0.554911	−	0.831909i	$-0.312752\pi$
$571$	2.81914e12	1.10982	0.554911	+	0.831909i	$0.312752\pi$
$572$	0	0
$573$	− 5.50584e12i	− 2.13367i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 2.80815e11i	− 0.105470i	−0.998609	−	0.0527350i	$-0.983206\pi$
$577$	− 2.80815e11i	− 0.105470i	0.998609	−	0.0527350i	$-0.0167939\pi$
$578$	0	0
$579$	3.68880e12	1.36406
$580$	0	0
$581$	−1.62046e10	−0.00589990
$582$	0	0
$583$	− 1.17245e12i	− 0.420324i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	1.71929e12i	0.597691i	0.954302	+	0.298845i	$0.0966015\pi$
$587$	1.71929e12i	0.597691i	−0.954302	+	0.298845i	$0.903399\pi$
$588$	0	0
$589$	4.41343e12	1.51097
$590$	0	0
$591$	9.49952e11	0.320301
$592$	0	0
$593$	− 4.82940e12i	− 1.60379i	−0.597465	−	0.801895i	$-0.703825\pi$
$593$	− 4.82940e12i	− 1.60379i	0.597465	−	0.801895i	$-0.296175\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 2.97496e12i	− 0.958511i
$598$	0	0
$599$	−1.72589e12	−0.547763	−0.273881	−	0.961763i	$-0.588308\pi$
$599$	−1.72589e12	−0.547763	−0.273881	+	0.961763i	$0.588308\pi$
$600$	0	0
$601$	5.06692e12	1.58420	0.792098	−	0.610394i	$-0.208989\pi$
$601$	5.06692e12	1.58420	0.792098	+	0.610394i	$0.208989\pi$
$602$	0	0
$603$	6.57284e12i	2.02453i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 1.57706e12i	− 0.471520i	−0.971811	−	0.235760i	$-0.924242\pi$
$607$	− 1.57706e12i	− 0.471520i	0.971811	−	0.235760i	$-0.0757580\pi$
$608$	0	0
$609$	−2.61472e12	−0.770279
$610$	0	0
$611$	−4.38263e12	−1.27218
$612$	0	0
$613$	1.35557e12i	0.387747i	0.981026	+	0.193874i	$0.0621052\pi$
$613$	1.35557e12i	0.387747i	−0.981026	+	0.193874i	$0.937895\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 4.35935e12i	− 1.21098i	−0.795852	−	0.605492i	$-0.792977\pi$
$617$	− 4.35935e12i	− 1.21098i	0.795852	−	0.605492i	$-0.207023\pi$
$618$	0	0
$619$	−1.67780e12	−0.459338	−0.229669	−	0.973269i	$-0.573764\pi$
$619$	−1.67780e12	−0.459338	−0.229669	+	0.973269i	$0.573764\pi$
$620$	0	0
$621$	3.88309e11	0.104777
$622$	0	0
$623$	1.89813e12i	0.504811i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 3.99544e12i	− 1.03243i
$628$	0	0
$629$	1.43983e12	0.366760
$630$	0	0
$631$	4.12439e12	1.03568	0.517842	−	0.855476i	$-0.326736\pi$
$631$	4.12439e12	1.03568	0.517842	+	0.855476i	$0.326736\pi$
$632$	0	0
$633$	2.63091e12i	0.651313i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 6.11514e12i	− 1.47156i
$638$	0	0
$639$	7.05938e12	1.67499
$640$	0	0
$641$	2.54925e12	0.596420	0.298210	−	0.954500i	$-0.403610\pi$
$641$	2.54925e12	0.596420	0.298210	+	0.954500i	$0.403610\pi$
$642$	0	0
$643$	1.49409e11i	0.0344688i	0.999851	+	0.0172344i	$0.00548615\pi$
$643$	1.49409e11i	0.0344688i	−0.999851	+	0.0172344i	$0.994514\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 7.88552e11i	− 0.176913i	−0.996080	−	0.0884567i	$-0.971806\pi$
$647$	− 7.88552e11i	− 0.176913i	0.996080	−	0.0884567i	$-0.0281935\pi$
$648$	0	0
$649$	−3.85498e12	−0.852946
$650$	0	0
$651$	2.96546e12	0.647109
$652$	0	0
$653$	− 1.87557e12i	− 0.403668i	−0.979420	−	0.201834i	$-0.935310\pi$
$653$	− 1.87557e12i	− 0.403668i	0.979420	−	0.201834i	$-0.0646901\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 9.65592e12i	− 2.02185i
$658$	0	0
$659$	2.02453e11	0.0418158	0.0209079	−	0.999781i	$-0.493344\pi$
$659$	2.02453e11	0.0418158	0.0209079	+	0.999781i	$0.493344\pi$
$660$	0	0
$661$	2.41063e12	0.491160	0.245580	−	0.969376i	$-0.421022\pi$
$661$	2.41063e12	0.491160	0.245580	+	0.969376i	$0.421022\pi$
$662$	0	0
$663$	− 3.23711e12i	− 0.650648i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	1.70468e12i	0.333484i
$668$	0	0
$669$	−1.34465e13	−2.59533
$670$	0	0
$671$	1.88885e11	0.0359704
$672$	0	0
$673$	5.63582e11i	0.105898i	0.998597	+	0.0529491i	$0.0168621\pi$
$673$	5.63582e11i	0.105898i	−0.998597	+	0.0529491i	$0.983138\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	8.43518e11i	0.154328i	0.997018	+	0.0771641i	$0.0245865\pi$
$677$	8.43518e11i	0.154328i	−0.997018	+	0.0771641i	$0.975413\pi$
$678$	0	0
$679$	−2.08684e12	−0.376768
$680$	0	0
$681$	7.62962e12	1.35938
$682$	0	0
$683$	− 5.43927e12i	− 0.956418i	−0.878246	−	0.478209i	$-0.841286\pi$
$683$	− 5.43927e12i	− 0.956418i	0.878246	−	0.478209i	$-0.158714\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 8.32361e12i	− 1.42563i
$688$	0	0
$689$	−5.54894e12	−0.938044
$690$	0	0
$691$	−1.41900e12	−0.236773	−0.118386	−	0.992968i	$-0.537772\pi$
$691$	−1.41900e12	−0.236773	−0.118386	+	0.992968i	$0.537772\pi$
$692$	0	0
$693$	− 1.56013e12i	− 0.256957i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 1.53591e12i	− 0.246501i
$698$	0	0
$699$	5.61787e12	0.890071
$700$	0	0
$701$	9.10116e12	1.42353	0.711763	−	0.702419i	$-0.247897\pi$
$701$	9.10116e12	1.42353	0.711763	+	0.702419i	$0.247897\pi$
$702$	0	0
$703$	− 8.41075e12i	− 1.29878i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	4.38064e11i	0.0659402i
$708$	0	0
$709$	−3.00283e12	−0.446296	−0.223148	−	0.974785i	$-0.571633\pi$
$709$	−3.00283e12	−0.446296	−0.223148	+	0.974785i	$0.571633\pi$
$710$	0	0
$711$	1.00002e13	1.46756
$712$	0	0
$713$	− 1.93334e12i	− 0.280159i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	1.25059e13i	1.76717i
$718$	0	0
$719$	8.40313e12	1.17263	0.586315	−	0.810083i	$-0.300578\pi$
$719$	8.40313e12	1.17263	0.586315	+	0.810083i	$0.300578\pi$
$720$	0	0
$721$	−3.48260e12	−0.479949
$722$	0	0
$723$	9.95833e12i	1.35539i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	6.70907e12i	0.890753i	0.895343	+	0.445377i	$0.146930\pi$
$727$	6.70907e12i	0.890753i	−0.895343	+	0.445377i	$0.853070\pi$
$728$	0	0
$729$	1.19463e13	1.56660
$730$	0	0
$731$	8.59403e11	0.111319
$732$	0	0
$733$	− 1.50813e12i	− 0.192962i	−0.995335	−	0.0964810i	$-0.969241\pi$
$733$	− 1.50813e12i	− 0.192962i	0.995335	−	0.0964810i	$-0.0307587\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 8.27285e12i	− 1.03288i
$738$	0	0
$739$	1.48289e13	1.82898	0.914492	−	0.404603i	$-0.132590\pi$
$739$	1.48289e13	1.82898	0.914492	+	0.404603i	$0.132590\pi$
$740$	0	0
$741$	−1.89096e13	−2.30409
$742$	0	0
$743$	1.91312e12i	0.230299i	0.993348	+	0.115150i	$0.0367348\pi$
$743$	1.91312e12i	0.230299i	−0.993348	+	0.115150i	$0.963265\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 2.66248e11i	− 0.0312855i
$748$	0	0
$749$	5.49335e11	0.0637777
$750$	0	0
$751$	−6.50309e12	−0.746002	−0.373001	−	0.927831i	$-0.621671\pi$
$751$	−6.50309e12	−0.746002	−0.373001	+	0.927831i	$0.621671\pi$
$752$	0	0
$753$	1.37250e13i	1.55573i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 1.11069e11i	− 0.0122931i	−0.999981	−	0.00614653i	$-0.998043\pi$
$757$	− 1.11069e11i	− 0.0122931i	0.999981	−	0.00614653i	$-0.00195651\pi$
$758$	0	0
$759$	−1.75023e12	−0.191429
$760$	0	0
$761$	1.21294e13	1.31102	0.655510	−	0.755186i	$-0.272454\pi$
$761$	1.21294e13	1.31102	0.655510	+	0.755186i	$0.272454\pi$
$762$	0	0
$763$	2.74732e12i	0.293460i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1.82448e13i	1.90353i
$768$	0	0
$769$	1.19313e13	1.23033	0.615164	−	0.788399i	$-0.289090\pi$
$769$	1.19313e13	1.23033	0.615164	+	0.788399i	$0.289090\pi$
$770$	0	0
$771$	9.14600e12	0.932152
$772$	0	0
$773$	1.71471e13i	1.72736i	0.504038	+	0.863681i	$0.331847\pi$
$773$	1.71471e13i	1.72736i	−0.504038	+	0.863681i	$0.668153\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 5.65133e12i	− 0.556232i
$778$	0	0
$779$	−8.97204e12	−0.872917
$780$	0	0
$781$	−8.88523e12	−0.854553
$782$	0	0
$783$	− 1.19966e13i	− 1.14059i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 2.12129e11i	− 0.0197112i	−0.999951	−	0.00985561i	$-0.996863\pi$
$787$	− 2.12129e11i	− 0.0197112i	0.999951	−	0.00985561i	$-0.00313719\pi$
$788$	0	0
$789$	−1.08663e13	−0.998242
$790$	0	0
$791$	1.92294e12	0.174652
$792$	0	0
$793$	− 8.93950e11i	− 0.0802757i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	9.75044e12i	0.855976i	0.903784	+	0.427988i	$0.140777\pi$
$797$	9.75044e12i	0.855976i	−0.903784	+	0.427988i	$0.859223\pi$
$798$	0	0
$799$	2.47308e12	0.214673
$800$	0	0
$801$	−3.11870e13	−2.67687
$802$	0	0
$803$	1.21533e13i	1.03152i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	1.56415e13i	1.29822i
$808$	0	0
$809$	−1.00162e13	−0.822120	−0.411060	−	0.911608i	$-0.634841\pi$
$809$	−1.00162e13	−0.822120	−0.411060	+	0.911608i	$0.634841\pi$
$810$	0	0
$811$	1.09433e11	0.00888287	0.00444143	−	0.999990i	$-0.498586\pi$
$811$	1.09433e11	0.00888287	0.00444143	+	0.999990i	$0.498586\pi$
$812$	0	0
$813$	− 1.13434e13i	− 0.910614i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 5.02020e12i	− 0.394205i
$818$	0	0
$819$	−7.38376e12	−0.573456
$820$	0	0
$821$	−2.10914e13	−1.62017	−0.810086	−	0.586312i	$-0.800579\pi$
$821$	−2.10914e13	−1.62017	−0.810086	+	0.586312i	$0.800579\pi$
$822$	0	0
$823$	− 6.07029e12i	− 0.461222i	−0.973046	−	0.230611i	$-0.925927\pi$
$823$	− 6.07029e12i	− 0.461222i	0.973046	−	0.230611i	$-0.0740725\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 1.09879e13i	− 0.816846i	−0.912793	−	0.408423i	$-0.866079\pi$
$827$	− 1.09879e13i	− 0.816846i	0.912793	−	0.408423i	$-0.133921\pi$
$828$	0	0
$829$	1.34601e13	0.989811	0.494906	−	0.868947i	$-0.335203\pi$
$829$	1.34601e13	0.989811	0.494906	+	0.868947i	$0.335203\pi$
$830$	0	0
$831$	−1.25503e13	−0.912957
$832$	0	0
$833$	3.45072e12i	0.248317i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	1.36058e13i	0.958206i
$838$	0	0
$839$	4.34425e12	0.302681	0.151341	−	0.988482i	$-0.451641\pi$
$839$	4.34425e12	0.302681	0.151341	+	0.988482i	$0.451641\pi$
$840$	0	0
$841$	3.81578e13	2.63028
$842$	0	0
$843$	− 1.58509e13i	− 1.08101i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 1.95547e12i	− 0.130550i
$848$	0	0
$849$	1.11069e13	0.733683
$850$	0	0
$851$	−3.68440e12	−0.240815
$852$	0	0
$853$	1.33638e13i	0.864290i	0.901804	+	0.432145i	$0.142243\pi$
$853$	1.33638e13i	0.864290i	−0.901804	+	0.432145i	$0.857757\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	2.60497e13i	1.64964i	0.565397	+	0.824819i	$0.308723\pi$
$857$	2.60497e13i	1.64964i	−0.565397	+	0.824819i	$0.691277\pi$
$858$	0	0
$859$	−2.01187e12	−0.126076	−0.0630379	−	0.998011i	$-0.520079\pi$
$859$	−2.01187e12	−0.126076	−0.0630379	+	0.998011i	$0.520079\pi$
$860$	0	0
$861$	−6.02847e12	−0.373846
$862$	0	0
$863$	− 2.36870e13i	− 1.45366i	−0.686819	−	0.726828i	$-0.740994\pi$
$863$	− 2.36870e13i	− 1.45366i	0.686819	−	0.726828i	$-0.259006\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 2.38803e13i	− 1.43534i
$868$	0	0
$869$	−1.25867e13	−0.748723
$870$	0	0
$871$	−3.91536e13	−2.30510
$872$	0	0
$873$	− 3.42876e13i	− 1.99790i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	1.83122e13i	1.04531i	0.852545	+	0.522653i	$0.175058\pi$
$877$	1.83122e13i	1.04531i	−0.852545	+	0.522653i	$0.824942\pi$
$878$	0	0
$879$	−4.24738e13	−2.39978
$880$	0	0
$881$	3.34824e13	1.87251	0.936256	−	0.351320i	$-0.114267\pi$
$881$	3.34824e13	1.87251	0.936256	+	0.351320i	$0.114267\pi$
$882$	0	0
$883$	1.73556e13i	0.960763i	0.877060	+	0.480382i	$0.159502\pi$
$883$	1.73556e13i	0.960763i	−0.877060	+	0.480382i	$0.840498\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 1.34047e13i	− 0.727109i	−0.931573	−	0.363555i	$-0.881563\pi$
$887$	− 1.34047e13i	− 0.727109i	0.931573	−	0.363555i	$-0.118437\pi$
$888$	0	0
$889$	−2.59501e12	−0.139341
$890$	0	0
$891$	−6.15839e12	−0.327354
$892$	0	0
$893$	− 1.44465e13i	− 0.760204i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	8.28348e12i	0.427216i
$898$	0	0
$899$	−5.97294e13	−3.04979
$900$	0	0
$901$	3.13122e12	0.158289
$902$	0	0
$903$	− 3.37316e12i	− 0.168827i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	9.01114e12i	0.442127i	0.975259	+	0.221064i	$0.0709528\pi$
$907$	9.01114e12i	0.442127i	−0.975259	+	0.221064i	$0.929047\pi$
$908$	0	0
$909$	−7.19758e12	−0.349663
$910$	0	0
$911$	7.64836e12	0.367905	0.183952	−	0.982935i	$-0.441111\pi$
$911$	7.64836e12	0.367905	0.183952	+	0.982935i	$0.441111\pi$
$912$	0	0
$913$	3.35110e11i	0.0159613i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 7.90232e12i	− 0.369056i
$918$	0	0
$919$	−3.19451e13	−1.47735	−0.738676	−	0.674060i	$-0.764549\pi$
$919$	−3.19451e13	−1.47735	−0.738676	+	0.674060i	$0.764549\pi$
$920$	0	0
$921$	−4.97639e13	−2.27901
$922$	0	0
$923$	4.20519e13i	1.90712i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 5.72207e13i	− 2.54504i
$928$	0	0
$929$	1.36029e13	0.599186	0.299593	−	0.954067i	$-0.403149\pi$
$929$	1.36029e13	0.599186	0.299593	+	0.954067i	$0.403149\pi$
$930$	0	0
$931$	2.01574e13	0.879347
$932$	0	0
$933$	− 1.08204e13i	− 0.467494i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 1.10891e13i	− 0.469966i	−0.971999	−	0.234983i	$-0.924497\pi$
$937$	− 1.10891e13i	− 0.469966i	0.971999	−	0.234983i	$-0.0755035\pi$
$938$	0	0
$939$	2.58553e13	1.08531
$940$	0	0
$941$	−2.46259e13	−1.02385	−0.511927	−	0.859029i	$-0.671068\pi$
$941$	−2.46259e13	−1.02385	−0.511927	+	0.859029i	$0.671068\pi$
$942$	0	0
$943$	3.93028e12i	0.161853i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	4.82162e13i	1.94813i	0.226267	+	0.974065i	$0.427348\pi$
$947$	4.82162e13i	1.94813i	−0.226267	+	0.974065i	$0.572652\pi$
$948$	0	0
$949$	5.75192e13	2.30205
$950$	0	0
$951$	−6.19717e13	−2.45686
$952$	0	0
$953$	3.00142e12i	0.117872i	0.998262	+	0.0589358i	$0.0187707\pi$
$953$	3.00142e12i	0.117872i	−0.998262	+	0.0589358i	$0.981229\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	5.40725e13i	2.08388i
$958$	0	0
$959$	−2.62122e12	−0.100074
$960$	0	0
$961$	4.13017e13	1.56211
$962$	0	0
$963$	9.02581e12i	0.338195i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 1.36956e13i	− 0.503690i	−0.967768	−	0.251845i	$-0.918963\pi$
$967$	− 1.36956e13i	− 0.503690i	0.967768	−	0.251845i	$-0.0810373\pi$
$968$	0	0
$969$	1.06705e13	0.388801
$970$	0	0
$971$	2.83646e12	0.102398	0.0511988	−	0.998688i	$-0.483696\pi$
$971$	2.83646e12	0.102398	0.0511988	+	0.998688i	$0.483696\pi$
$972$	0	0
$973$	− 5.66311e12i	− 0.202557i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 9.66226e12i	− 0.339276i	−0.985506	−	0.169638i	$-0.945740\pi$
$977$	− 9.66226e12i	− 0.339276i	0.985506	−	0.169638i	$-0.0542599\pi$
$978$	0	0
$979$	3.92533e13	1.36570
$980$	0	0
$981$	−4.51396e13	−1.55614
$982$	0	0
$983$	1.05400e13i	0.360039i	0.983663	+	0.180019i	$0.0576160\pi$
$983$	1.05400e13i	0.360039i	−0.983663	+	0.180019i	$0.942384\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 9.70684e12i	− 0.325575i
$988$	0	0
$989$	−2.19914e12	−0.0730920
$990$	0	0
$991$	3.01597e13	0.993335	0.496668	−	0.867941i	$-0.334557\pi$
$991$	3.01597e13	0.993335	0.496668	+	0.867941i	$0.334557\pi$
$992$	0	0
$993$	4.59668e13i	1.50028i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	1.98318e13i	0.635675i	0.948145	+	0.317837i	$0.102957\pi$
$997$	1.98318e13i	0.635675i	−0.948145	+	0.317837i	$0.897043\pi$
$998$	0	0
$999$	2.59288e13	0.823640

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	200.10.c.d.49.1		4
4.3	odd	2		400.10.c.m.49.4		4
5.2	odd	4		200.10.a.c.1.1		2
5.3	odd	4		40.10.a.c.1.2	✓	2
5.4	even	2	inner	200.10.c.d.49.4		4
15.8	even	4		360.10.a.g.1.2		2
20.3	even	4		80.10.a.g.1.1		2
20.7	even	4		400.10.a.r.1.2		2
20.19	odd	2		400.10.c.m.49.1		4
40.3	even	4		320.10.a.q.1.2		2
40.13	odd	4		320.10.a.n.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.10.a.c.1.2	✓	2	5.3	odd	4
80.10.a.g.1.1		2	20.3	even	4
200.10.a.c.1.1		2	5.2	odd	4
200.10.c.d.49.1		4	1.1	even	1	trivial
200.10.c.d.49.4		4	5.4	even	2	inner
320.10.a.n.1.1		2	40.13	odd	4
320.10.a.q.1.2		2	40.3	even	4
360.10.a.g.1.2		2	15.8	even	4
400.10.a.r.1.2		2	20.7	even	4
400.10.c.m.49.1		4	20.19	odd	2
400.10.c.m.49.4		4	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 200 = 2^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	200.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-216.776i q^{3} +1662.09i q^{7} -27308.8 q^{9} +34372.0 q^{11} -162675. i q^{13} +91796.1i q^{17} +536227. q^{19} +360300. q^{21} -234899. i q^{23} +1.65309e6i q^{27} -7.25706e6 q^{29} +8.23051e6 q^{31} -7.45102e6i q^{33} -1.56850e7i q^{37} -3.52641e7 q^{39} -1.67318e7 q^{41} -9.36208e6i q^{43} -2.69410e7i q^{47} +3.75911e7 q^{49} +1.98992e7 q^{51} -3.41105e7i q^{53} -1.16241e8i q^{57} -1.12155e8 q^{59} +5.49531e6 q^{61} -4.53896e7i q^{63} -2.40686e8i q^{67} -5.09204e7 q^{69} -2.58502e8 q^{71} +3.53583e8i q^{73} +5.71292e7i q^{77} -3.66190e8 q^{79} -1.79169e8 q^{81} +9.74953e6i q^{83} +1.57316e9i q^{87} +1.14201e9 q^{89} +2.70380e8 q^{91} -1.78418e9i q^{93} +1.25555e9i q^{97} -9.38657e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 38916 q^{9} + 50240 q^{11} + 50960 q^{19} + 1279728 q^{21} - 14646680 q^{29} + 21354544 q^{31} - 73949616 q^{39} - 15591528 q^{41} + 142678668 q^{49} + 22944240 q^{51} - 119236528 q^{59} - 376327544 q^{61}+ \cdots - 2022585408 q^{99}+O(q^{100}) \) 4 * q - 38916 * q^9 + 50240 * q^11 + 50960 * q^19 + 1279728 * q^21 - 14646680 * q^29 + 21354544 * q^31 - 73949616 * q^39 - 15591528 * q^41 + 142678668 * q^49 + 22944240 * q^51 - 119236528 * q^59 - 376327544 * q^61 + 234896688 * q^69 - 337331184 * q^71 - 933892512 * q^79 - 700854444 * q^81 + 216669720 * q^89 + 621601232 * q^91 - 2022585408 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more