LMFDB - Embedded newform 400.8.c.s.49.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [400,8,Mod(49,400)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("400.49"); S:= CuspForms(chi, 8); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(400, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 8, names="a")

Level:	$ N $	$=$	$ 400 = 2^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	400.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,0,0,4552] 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:	$124.954010194$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{649})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 325x^{2} + 26244 $ x^4 + 325*x^2 + 26244
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2}\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 25)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.1
Root			$12.2377i$ of defining polynomial
Character	$\chi$	$=$	400.49
Dual form			400.8.c.s.49.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-45.4755i q^{3} -1369.97i q^{7} +118.981 q^{9} +1012.43 q^{11} +3643.00i q^{13} +5532.37i q^{17} +23238.9 q^{19} -62300.0 q^{21} -96310.5i q^{23} -104866. i q^{27} +139729. q^{29} +208208. q^{31} -46041.0i q^{33} -448306. i q^{37} +165667. q^{39} -79718.5 q^{41} -227324. i q^{43} +719892. i q^{47} -1.05328e6 q^{49} +251587. q^{51} -495702. i q^{53} -1.05680e6i q^{57} -81006.9 q^{59} +3.15701e6 q^{61} -163000. i q^{63} +1.06340e6i q^{67} -4.37976e6 q^{69} +3.00099e6 q^{71} +405758. i q^{73} -1.38701e6i q^{77} -4.71733e6 q^{79} -4.50860e6 q^{81} -5.14106e6i q^{83} -6.35423e6i q^{87} +8.62652e6 q^{89} +4.99080e6 q^{91} -9.46835e6i q^{93} +6.99663e6i q^{97} +120460. q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4552 q^{9} - 8688 q^{11} + 36400 q^{19} - 133032 q^{21} - 111600 q^{29} + 603552 q^{31} + 177616 q^{39} - 216972 q^{41} - 1645172 q^{49} + 638992 q^{51} + 4135200 q^{59} + 1164088 q^{61} - 9582936 q^{69}+ \cdots - 22866944 q^{99}+O(q^{100}) $ 4 * q + 4552 * q^9 - 8688 * q^11 + 36400 * q^19 - 133032 * q^21 - 111600 * q^29 + 603552 * q^31 + 177616 * q^39 - 216972 * q^41 - 1645172 * q^49 + 638992 * q^51 + 4135200 * q^59 + 1164088 * q^61 - 9582936 * q^69 + 9456432 * q^71 - 14372400 * q^79 + 157124 * q^81 + 11981700 * q^89 - 11634528 * q^91 - 22866944 * q^99

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
$3$	− 45.4755i	− 0.972418i	−0.873843	−	0.486209i	$-0.838379\pi$
$3$	− 45.4755i	− 0.972418i	0.873843	−	0.486209i	$-0.161621\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 1369.97i	− 1.50962i	−0.655943	−	0.754811i	$-0.727729\pi$
$7$	− 1369.97i	− 1.50962i	0.655943	−	0.754811i	$-0.272271\pi$
$8$	0	0
$9$	118.981	0.0544037
$10$	0	0
$11$	1012.43	0.229347	0.114673	−	0.993403i	$-0.463418\pi$
$11$	1012.43	0.229347	0.114673	+	0.993403i	$0.463418\pi$
$12$	0	0
$13$	3643.00i	0.459894i	0.973203	+	0.229947i	$0.0738553\pi$
$13$	3643.00i	0.459894i	−0.973203	+	0.229947i	$0.926145\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	5532.37i	0.273111i	0.990632	+	0.136556i	$0.0436033\pi$
$17$	5532.37i	0.273111i	−0.990632	+	0.136556i	$0.956397\pi$
$18$	0	0
$19$	23238.9	0.777281	0.388640	−	0.921390i	$-0.372945\pi$
$19$	23238.9	0.777281	0.388640	+	0.921390i	$0.372945\pi$
$20$	0	0
$21$	−62300.0	−1.46798
$22$	0	0
$23$	− 96310.5i	− 1.65054i	−0.564738	−	0.825270i	$-0.691023\pi$
$23$	− 96310.5i	− 1.65054i	0.564738	−	0.825270i	$-0.308977\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 104866.i	− 1.02532i
$28$	0	0
$29$	139729.	1.06388	0.531940	−	0.846782i	$-0.321463\pi$
$29$	139729.	1.06388	0.531940	+	0.846782i	$0.321463\pi$
$30$	0	0
$31$	208208.	1.25525	0.627626	−	0.778515i	$-0.284027\pi$
$31$	208208.	1.25525	0.627626	+	0.778515i	$0.284027\pi$
$32$	0	0
$33$	− 46041.0i	− 0.223021i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 448306.i	− 1.45502i	−0.686098	−	0.727509i	$-0.740678\pi$
$37$	− 448306.i	− 1.45502i	0.686098	−	0.727509i	$-0.259322\pi$
$38$	0	0
$39$	165667.	0.447209
$40$	0	0
$41$	−79718.5	−0.180641	−0.0903203	−	0.995913i	$-0.528789\pi$
$41$	−79718.5	−0.180641	−0.0903203	+	0.995913i	$0.528789\pi$
$42$	0	0
$43$	− 227324.i	− 0.436020i	−0.975947	−	0.218010i	$-0.930043\pi$
$43$	− 227324.i	− 0.436020i	0.975947	−	0.218010i	$-0.0699565\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	719892.i	1.01140i	0.862708	+	0.505702i	$0.168766\pi$
$47$	719892.i	1.01140i	−0.862708	+	0.505702i	$0.831234\pi$
$48$	0	0
$49$	−1.05328e6	−1.27896
$50$	0	0
$51$	251587.	0.265578
$52$	0	0
$53$	− 495702.i	− 0.457357i	−0.973502	−	0.228678i	$-0.926560\pi$
$53$	− 495702.i	− 0.457357i	0.973502	−	0.228678i	$-0.0734404\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 1.05680e6i	− 0.755841i
$58$	0	0
$59$	−81006.9	−0.0513500	−0.0256750	−	0.999670i	$-0.508173\pi$
$59$	−81006.9	−0.0513500	−0.0256750	+	0.999670i	$0.508173\pi$
$60$	0	0
$61$	3.15701e6	1.78083	0.890414	−	0.455151i	$-0.150415\pi$
$61$	3.15701e6	1.78083	0.890414	+	0.455151i	$0.150415\pi$
$62$	0	0
$63$	− 163000.i	− 0.0821290i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	1.06340e6i	0.431950i	0.976399	+	0.215975i	$0.0692929\pi$
$67$	1.06340e6i	0.431950i	−0.976399	+	0.215975i	$0.930707\pi$
$68$	0	0
$69$	−4.37976e6	−1.60501
$70$	0	0
$71$	3.00099e6	0.995087	0.497543	−	0.867439i	$-0.334235\pi$
$71$	3.00099e6	0.995087	0.497543	+	0.867439i	$0.334235\pi$
$72$	0	0
$73$	405758.i	0.122078i	0.998135	+	0.0610389i	$0.0194414\pi$
$73$	405758.i	0.122078i	−0.998135	+	0.0610389i	$0.980559\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 1.38701e6i	− 0.346227i
$78$	0	0
$79$	−4.71733e6	−1.07647	−0.538235	−	0.842795i	$-0.680909\pi$
$79$	−4.71733e6	−1.07647	−0.538235	+	0.842795i	$0.680909\pi$
$80$	0	0
$81$	−4.50860e6	−0.942637
$82$	0	0
$83$	− 5.14106e6i	− 0.986914i	−0.869770	−	0.493457i	$-0.835733\pi$
$83$	− 5.14106e6i	− 0.986914i	0.869770	−	0.493457i	$-0.164267\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 6.35423e6i	− 1.03454i
$88$	0	0
$89$	8.62652e6	1.29709	0.648546	−	0.761175i	$-0.275377\pi$
$89$	8.62652e6	1.29709	0.648546	+	0.761175i	$0.275377\pi$
$90$	0	0
$91$	4.99080e6	0.694266
$92$	0	0
$93$	− 9.46835e6i	− 1.22063i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	6.99663e6i	0.778373i	0.921159	+	0.389187i	$0.127244\pi$
$97$	6.99663e6i	0.778373i	−0.921159	+	0.389187i	$0.872756\pi$
$98$	0	0
$99$	120460.	0.0124773
$100$	0	0
$101$	5.34837e6	0.516531	0.258266	−	0.966074i	$-0.416849\pi$
$101$	5.34837e6	0.516531	0.258266	+	0.966074i	$0.416849\pi$
$102$	0	0
$103$	− 7.09826e6i	− 0.640061i	−0.947407	−	0.320031i	$-0.896307\pi$
$103$	− 7.09826e6i	− 0.640061i	0.947407	−	0.320031i	$-0.103693\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 1.01014e7i	− 0.797145i	−0.917137	−	0.398573i	$-0.869506\pi$
$107$	− 1.01014e7i	− 0.797145i	0.917137	−	0.398573i	$-0.130494\pi$
$108$	0	0
$109$	1.20315e6	0.0889874	0.0444937	−	0.999010i	$-0.485833\pi$
$109$	1.20315e6	0.0889874	0.0444937	+	0.999010i	$0.485833\pi$
$110$	0	0
$111$	−2.03869e7	−1.41489
$112$	0	0
$113$	− 529610.i	− 0.0345288i	−0.999851	−	0.0172644i	$-0.994504\pi$
$113$	− 529610.i	− 0.0345288i	0.999851	−	0.0172644i	$-0.00549570\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	433448.i	0.0250199i
$118$	0	0
$119$	7.57918e6	0.412295
$120$	0	0
$121$	−1.84621e7	−0.947400
$122$	0	0
$123$	3.62524e6i	0.175658i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	1.90420e7i	0.824897i	0.910981	+	0.412448i	$0.135326\pi$
$127$	1.90420e7i	0.824897i	−0.910981	+	0.412448i	$0.864674\pi$
$128$	0	0
$129$	−1.03377e7	−0.423993
$130$	0	0
$131$	−4.51565e7	−1.75498	−0.877488	−	0.479599i	$-0.840782\pi$
$131$	−4.51565e7	−1.75498	−0.877488	+	0.479599i	$0.840782\pi$
$132$	0	0
$133$	− 3.18366e7i	− 1.17340i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	3.51625e7i	1.16831i	0.811642	+	0.584155i	$0.198574\pi$
$137$	3.51625e7i	1.16831i	−0.811642	+	0.584155i	$0.801426\pi$
$138$	0	0
$139$	−4.10740e7	−1.29722	−0.648612	−	0.761119i	$-0.724650\pi$
$139$	−4.10740e7	−1.29722	−0.648612	+	0.761119i	$0.724650\pi$
$140$	0	0
$141$	3.27374e7	0.983507
$142$	0	0
$143$	3.68830e6i	0.105475i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	4.78982e7i	1.24368i
$148$	0	0
$149$	−5.46024e7	−1.35226	−0.676129	−	0.736783i	$-0.736344\pi$
$149$	−5.46024e7	−1.35226	−0.676129	+	0.736783i	$0.736344\pi$
$150$	0	0
$151$	7.68654e6	0.181682	0.0908409	−	0.995865i	$-0.471045\pi$
$151$	7.68654e6	0.181682	0.0908409	+	0.995865i	$0.471045\pi$
$152$	0	0
$153$	658246.i	0.0148583i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	4.63213e6i	0.0955283i	0.998859	+	0.0477641i	$0.0152096\pi$
$157$	4.63213e6i	0.0955283i	−0.998859	+	0.0477641i	$0.984790\pi$
$158$	0	0
$159$	−2.25423e7	−0.444742
$160$	0	0
$161$	−1.31942e8	−2.49169
$162$	0	0
$163$	− 2.15892e7i	− 0.390462i	−0.980757	−	0.195231i	$-0.937454\pi$
$163$	− 2.15892e7i	− 0.390462i	0.980757	−	0.195231i	$-0.0625457\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	3.71330e7i	0.616953i	0.951232	+	0.308476i	$0.0998191\pi$
$167$	3.71330e7i	0.616953i	−0.951232	+	0.308476i	$0.900181\pi$
$168$	0	0
$169$	4.94771e7	0.788498
$170$	0	0
$171$	2.76498e6	0.0422869
$172$	0	0
$173$	6.07796e7i	0.892476i	0.894914	+	0.446238i	$0.147237\pi$
$173$	6.07796e7i	0.892476i	−0.894914	+	0.446238i	$0.852763\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	3.68383e6i	0.0499336i
$178$	0	0
$179$	3.05526e7	0.398164	0.199082	−	0.979983i	$-0.436204\pi$
$179$	3.05526e7	0.398164	0.199082	+	0.979983i	$0.436204\pi$
$180$	0	0
$181$	−1.18125e8	−1.48070	−0.740349	−	0.672223i	$-0.765340\pi$
$181$	−1.18125e8	−1.48070	−0.740349	+	0.672223i	$0.765340\pi$
$182$	0	0
$183$	− 1.43567e8i	− 1.73171i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	5.60116e6i	0.0626372i
$188$	0	0
$189$	−1.43663e8	−1.54785
$190$	0	0
$191$	−8.20283e7	−0.851818	−0.425909	−	0.904766i	$-0.640046\pi$
$191$	−8.20283e7	−0.851818	−0.425909	+	0.904766i	$0.640046\pi$
$192$	0	0
$193$	− 1.98988e8i	− 1.99240i	−0.0871152	−	0.996198i	$-0.527765\pi$
$193$	− 1.98988e8i	− 1.99240i	0.0871152	−	0.996198i	$-0.472235\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	8.48880e7i	0.791069i	0.918451	+	0.395535i	$0.129441\pi$
$197$	8.48880e7i	0.791069i	−0.918451	+	0.395535i	$0.870559\pi$
$198$	0	0
$199$	−4.12974e7	−0.371481	−0.185741	−	0.982599i	$-0.559468\pi$
$199$	−4.12974e7	−0.371481	−0.185741	+	0.982599i	$0.559468\pi$
$200$	0	0
$201$	4.83584e7	0.420036
$202$	0	0
$203$	− 1.91424e8i	− 1.60605i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 1.14591e7i	− 0.0897955i
$208$	0	0
$209$	2.35279e7	0.178267
$210$	0	0
$211$	1.55077e7	0.113647	0.0568236	−	0.998384i	$-0.481903\pi$
$211$	1.55077e7	0.113647	0.0568236	+	0.998384i	$0.481903\pi$
$212$	0	0
$213$	− 1.36472e8i	− 0.967640i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 2.85238e8i	− 1.89496i
$218$	0	0
$219$	1.84520e7	0.118711
$220$	0	0
$221$	−2.01544e7	−0.125602
$222$	0	0
$223$	− 4.16316e6i	− 0.0251395i	−0.999921	−	0.0125697i	$-0.995999\pi$
$223$	− 4.16316e6i	− 0.0251395i	0.999921	−	0.0125697i	$-0.00400118\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	6.41270e7i	0.363873i	0.983310	+	0.181937i	$0.0582366\pi$
$227$	6.41270e7i	0.363873i	−0.983310	+	0.181937i	$0.941763\pi$
$228$	0	0
$229$	−5.41678e7	−0.298069	−0.149035	−	0.988832i	$-0.547617\pi$
$229$	−5.41678e7	−0.298069	−0.149035	+	0.988832i	$0.547617\pi$
$230$	0	0
$231$	−6.30747e7	−0.336677
$232$	0	0
$233$	− 2.74855e8i	− 1.42350i	−0.702431	−	0.711752i	$-0.747902\pi$
$233$	− 2.74855e8i	− 1.42350i	0.702431	−	0.711752i	$-0.252098\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	2.14523e8i	1.04678i
$238$	0	0
$239$	−1.86000e8	−0.881291	−0.440646	−	0.897681i	$-0.645250\pi$
$239$	−1.86000e8	−0.881291	−0.440646	+	0.897681i	$0.645250\pi$
$240$	0	0
$241$	2.37791e8	1.09430	0.547150	−	0.837034i	$-0.315713\pi$
$241$	2.37791e8	1.09430	0.547150	+	0.837034i	$0.315713\pi$
$242$	0	0
$243$	− 2.43102e7i	− 0.108684i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	8.46593e7i	0.357467i
$248$	0	0
$249$	−2.33792e8	−0.959693
$250$	0	0
$251$	−3.28304e8	−1.31044	−0.655221	−	0.755438i	$-0.727424\pi$
$251$	−3.28304e8	−1.31044	−0.655221	+	0.755438i	$0.727424\pi$
$252$	0	0
$253$	− 9.75081e7i	− 0.378546i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	1.01567e8i	0.373238i	0.982432	+	0.186619i	$0.0597530\pi$
$257$	1.01567e8i	0.373238i	−0.982432	+	0.186619i	$0.940247\pi$
$258$	0	0
$259$	−6.14166e8	−2.19653
$260$	0	0
$261$	1.66250e7	0.0578790
$262$	0	0
$263$	− 1.98884e8i	− 0.674147i	−0.941478	−	0.337073i	$-0.890563\pi$
$263$	− 1.98884e8i	− 0.674147i	0.941478	−	0.337073i	$-0.109437\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 3.92295e8i	− 1.26132i
$268$	0	0
$269$	−2.44943e8	−0.767242	−0.383621	−	0.923491i	$-0.625323\pi$
$269$	−2.44943e8	−0.767242	−0.383621	+	0.923491i	$0.625323\pi$
$270$	0	0
$271$	3.91023e8	1.19347	0.596733	−	0.802440i	$-0.296465\pi$
$271$	3.91023e8	1.19347	0.596733	+	0.802440i	$0.296465\pi$
$272$	0	0
$273$	− 2.26959e8i	− 0.675116i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	5.08902e8i	1.43865i	0.694674	+	0.719325i	$0.255549\pi$
$277$	5.08902e8i	1.43865i	−0.694674	+	0.719325i	$0.744451\pi$
$278$	0	0
$279$	2.47727e7	0.0682904
$280$	0	0
$281$	−1.55563e8	−0.418248	−0.209124	−	0.977889i	$-0.567061\pi$
$281$	−1.55563e8	−0.418248	−0.209124	+	0.977889i	$0.567061\pi$
$282$	0	0
$283$	− 1.14570e8i	− 0.300481i	−0.988649	−	0.150241i	$-0.951995\pi$
$283$	− 1.14570e8i	− 0.300481i	0.988649	−	0.150241i	$-0.0480049\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1.09212e8i	0.272699i
$288$	0	0
$289$	3.79732e8	0.925410
$290$	0	0
$291$	3.18175e8	0.756904
$292$	0	0
$293$	8.42345e8i	1.95638i	0.207713	+	0.978190i	$0.433398\pi$
$293$	8.42345e8i	1.95638i	−0.207713	+	0.978190i	$0.566602\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 1.06170e8i	− 0.235154i
$298$	0	0
$299$	3.50859e8	0.759073
$300$	0	0
$301$	−3.11427e8	−0.658225
$302$	0	0
$303$	− 2.43220e8i	− 0.502284i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	2.38090e7i	0.0469631i	0.999724	+	0.0234816i	$0.00747510\pi$
$307$	2.38090e7i	0.0469631i	−0.999724	+	0.0234816i	$0.992525\pi$
$308$	0	0
$309$	−3.22797e8	−0.622407
$310$	0	0
$311$	−7.31704e8	−1.37935	−0.689674	−	0.724120i	$-0.742246\pi$
$311$	−7.31704e8	−1.37935	−0.689674	+	0.724120i	$0.742246\pi$
$312$	0	0
$313$	− 9.21577e8i	− 1.69874i	−0.527799	−	0.849369i	$-0.676983\pi$
$313$	− 9.21577e8i	− 1.69874i	0.527799	−	0.849369i	$-0.323017\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	7.85353e7i	0.138471i	0.997600	+	0.0692353i	$0.0220559\pi$
$317$	7.85353e7i	0.138471i	−0.997600	+	0.0692353i	$0.977944\pi$
$318$	0	0
$319$	1.41466e8	0.243997
$320$	0	0
$321$	−4.59365e8	−0.775158
$322$	0	0
$323$	1.28566e8i	0.212284i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 5.47140e7i	− 0.0865329i
$328$	0	0
$329$	9.86230e8	1.52684
$330$	0	0
$331$	−3.78878e8	−0.574250	−0.287125	−	0.957893i	$-0.592700\pi$
$331$	−3.78878e8	−0.574250	−0.287125	+	0.957893i	$0.592700\pi$
$332$	0	0
$333$	− 5.33398e7i	− 0.0791584i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 1.08031e9i	− 1.53760i	−0.639489	−	0.768800i	$-0.720854\pi$
$337$	− 1.08031e9i	− 1.53760i	0.639489	−	0.768800i	$-0.279146\pi$
$338$	0	0
$339$	−2.40843e7	−0.0335764
$340$	0	0
$341$	2.10797e8	0.287888
$342$	0	0
$343$	3.14726e8i	0.421117i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	8.77465e8i	1.12740i	0.825981	+	0.563698i	$0.190622\pi$
$347$	8.77465e8i	1.12740i	−0.825981	+	0.563698i	$0.809378\pi$
$348$	0	0
$349$	−3.31907e8	−0.417953	−0.208976	−	0.977921i	$-0.567013\pi$
$349$	−3.31907e8	−0.417953	−0.208976	+	0.977921i	$0.567013\pi$
$350$	0	0
$351$	3.82026e8	0.471539
$352$	0	0
$353$	9.88344e8i	1.19590i	0.801532	+	0.597952i	$0.204019\pi$
$353$	9.88344e8i	1.19590i	−0.801532	+	0.597952i	$0.795981\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 3.44667e8i	− 0.400923i
$358$	0	0
$359$	1.41167e9	1.61028	0.805140	−	0.593084i	$-0.202090\pi$
$359$	1.41167e9	1.61028	0.805140	+	0.593084i	$0.202090\pi$
$360$	0	0
$361$	−3.53826e8	−0.395835
$362$	0	0
$363$	8.39575e8i	0.921269i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	7.57010e8i	0.799412i	0.916643	+	0.399706i	$0.130888\pi$
$367$	7.57010e8i	0.799412i	−0.916643	+	0.399706i	$0.869112\pi$
$368$	0	0
$369$	−9.48497e6	−0.00982752
$370$	0	0
$371$	−6.79097e8	−0.690435
$372$	0	0
$373$	2.96259e8i	0.295591i	0.989018	+	0.147796i	$0.0472178\pi$
$373$	2.96259e8i	0.295591i	−0.989018	+	0.147796i	$0.952782\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	5.09032e8i	0.489272i
$378$	0	0
$379$	1.32923e9	1.25419	0.627093	−	0.778944i	$-0.284244\pi$
$379$	1.32923e9	1.25419	0.627093	+	0.778944i	$0.284244\pi$
$380$	0	0
$381$	8.65944e8	0.802144
$382$	0	0
$383$	8.35565e8i	0.759949i	0.924997	+	0.379974i	$0.124067\pi$
$383$	8.35565e8i	0.759949i	−0.924997	+	0.379974i	$0.875933\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 2.70472e7i	− 0.0237211i
$388$	0	0
$389$	−1.65397e8	−0.142464	−0.0712320	−	0.997460i	$-0.522693\pi$
$389$	−1.65397e8	−0.142464	−0.0712320	+	0.997460i	$0.522693\pi$
$390$	0	0
$391$	5.32825e8	0.450781
$392$	0	0
$393$	2.05351e9i	1.70657i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 2.01752e8i	− 0.161827i	−0.996721	−	0.0809134i	$-0.974216\pi$
$397$	− 2.01752e8i	− 0.161827i	0.996721	−	0.0809134i	$-0.0257837\pi$
$398$	0	0
$399$	−1.44778e9	−1.14103
$400$	0	0
$401$	−2.44937e8	−0.189692	−0.0948462	−	0.995492i	$-0.530236\pi$
$401$	−2.44937e8	−0.189692	−0.0948462	+	0.995492i	$0.530236\pi$
$402$	0	0
$403$	7.58502e8i	0.577283i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 4.53881e8i	− 0.333704i
$408$	0	0
$409$	−1.72583e8	−0.124728	−0.0623642	−	0.998053i	$-0.519864\pi$
$409$	−1.72583e8	−0.124728	−0.0623642	+	0.998053i	$0.519864\pi$
$410$	0	0
$411$	1.59903e9	1.13608
$412$	0	0
$413$	1.10977e8i	0.0775190i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	1.86786e9i	1.26144i
$418$	0	0
$419$	−2.64174e9	−1.75445	−0.877225	−	0.480079i	$-0.840608\pi$
$419$	−2.64174e9	−1.75445	−0.877225	+	0.480079i	$0.840608\pi$
$420$	0	0
$421$	1.11997e9	0.731509	0.365754	−	0.930711i	$-0.380811\pi$
$421$	1.11997e9	0.731509	0.365754	+	0.930711i	$0.380811\pi$
$422$	0	0
$423$	8.56533e7i	0.0550241i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 4.32501e9i	− 2.68838i
$428$	0	0
$429$	1.67727e8	0.102566
$430$	0	0
$431$	2.56747e9	1.54466	0.772332	−	0.635219i	$-0.219090\pi$
$431$	2.56747e9	1.54466	0.772332	+	0.635219i	$0.219090\pi$
$432$	0	0
$433$	9.65650e8i	0.571626i	0.958285	+	0.285813i	$0.0922637\pi$
$433$	9.65650e8i	0.571626i	−0.958285	+	0.285813i	$0.907736\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 2.23815e9i	− 1.28293i
$438$	0	0
$439$	2.67168e9	1.50716	0.753580	−	0.657356i	$-0.228325\pi$
$439$	2.67168e9	1.50716	0.753580	+	0.657356i	$0.228325\pi$
$440$	0	0
$441$	−1.25320e8	−0.0695799
$442$	0	0
$443$	− 9.25736e8i	− 0.505911i	−0.967478	−	0.252955i	$-0.918597\pi$
$443$	− 9.25736e8i	− 0.505911i	0.967478	−	0.252955i	$-0.0814026\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	2.48307e9i	1.31496i
$448$	0	0
$449$	3.43969e9	1.79332	0.896658	−	0.442724i	$-0.145988\pi$
$449$	3.43969e9	1.79332	0.896658	+	0.442724i	$0.145988\pi$
$450$	0	0
$451$	−8.07098e7	−0.0414294
$452$	0	0
$453$	− 3.49549e8i	− 0.176671i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	4.40856e8i	0.216068i	0.994147	+	0.108034i	$0.0344555\pi$
$457$	4.40856e8i	0.216068i	−0.994147	+	0.108034i	$0.965544\pi$
$458$	0	0
$459$	5.80155e8	0.280027
$460$	0	0
$461$	2.97064e9	1.41220	0.706100	−	0.708112i	$-0.250453\pi$
$461$	2.97064e9	1.41220	0.706100	+	0.708112i	$0.250453\pi$
$462$	0	0
$463$	1.59975e9i	0.749062i	0.927214	+	0.374531i	$0.122196\pi$
$463$	1.59975e9i	0.749062i	−0.927214	+	0.374531i	$0.877804\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 1.62694e9i	− 0.739201i	−0.929191	−	0.369600i	$-0.879495\pi$
$467$	− 1.62694e9i	− 0.739201i	0.929191	−	0.369600i	$-0.120505\pi$
$468$	0	0
$469$	1.45682e9	0.652080
$470$	0	0
$471$	2.10648e8	0.0928934
$472$	0	0
$473$	− 2.30151e8i	− 0.0999998i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 5.89790e7i	− 0.0248819i
$478$	0	0
$479$	−3.93755e8	−0.163701	−0.0818507	−	0.996645i	$-0.526083\pi$
$479$	−3.93755e8	−0.163701	−0.0818507	+	0.996645i	$0.526083\pi$
$480$	0	0
$481$	1.63318e9	0.669154
$482$	0	0
$483$	6.00015e9i	2.42296i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 2.90720e9i	− 1.14058i	−0.821445	−	0.570288i	$-0.806832\pi$
$487$	− 2.90720e9i	− 1.14058i	0.821445	−	0.570288i	$-0.193168\pi$
$488$	0	0
$489$	−9.81777e8	−0.379692
$490$	0	0
$491$	−1.49340e9	−0.569364	−0.284682	−	0.958622i	$-0.591888\pi$
$491$	−1.49340e9	−0.569364	−0.284682	+	0.958622i	$0.591888\pi$
$492$	0	0
$493$	7.73030e8i	0.290558i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 4.11127e9i	− 1.50220i
$498$	0	0
$499$	2.32305e9	0.836965	0.418482	−	0.908225i	$-0.362562\pi$
$499$	2.32305e9	0.836965	0.418482	+	0.908225i	$0.362562\pi$
$500$	0	0
$501$	1.68864e9	0.599936
$502$	0	0
$503$	− 4.58290e9i	− 1.60566i	−0.596210	−	0.802828i	$-0.703328\pi$
$503$	− 4.58290e9i	− 1.60566i	0.596210	−	0.802828i	$-0.296672\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 2.24999e9i	− 0.766749i
$508$	0	0
$509$	3.08836e9	1.03804	0.519022	−	0.854761i	$-0.326296\pi$
$509$	3.08836e9	1.03804	0.519022	+	0.854761i	$0.326296\pi$
$510$	0	0
$511$	5.55876e8	0.184291
$512$	0	0
$513$	− 2.43696e9i	− 0.796962i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	7.28843e8i	0.231962i
$518$	0	0
$519$	2.76398e9	0.867860
$520$	0	0
$521$	3.95517e9	1.22527	0.612637	−	0.790364i	$-0.290109\pi$
$521$	3.95517e9	1.22527	0.612637	+	0.790364i	$0.290109\pi$
$522$	0	0
$523$	− 2.97750e9i	− 0.910113i	−0.890463	−	0.455057i	$-0.849619\pi$
$523$	− 2.97750e9i	− 0.910113i	0.890463	−	0.455057i	$-0.150381\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1.15188e9i	0.342824i
$528$	0	0
$529$	−5.87088e9	−1.72428
$530$	0	0
$531$	−9.63828e6	−0.00279363
$532$	0	0
$533$	− 2.90415e8i	− 0.0830756i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 1.38939e9i	− 0.387182i
$538$	0	0
$539$	−1.06637e9	−0.293324
$540$	0	0
$541$	−4.75653e9	−1.29152	−0.645758	−	0.763542i	$-0.723459\pi$
$541$	−4.75653e9	−1.29152	−0.645758	+	0.763542i	$0.723459\pi$
$542$	0	0
$543$	5.37179e9i	1.43986i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	4.58098e9i	1.19675i	0.801217	+	0.598374i	$0.204186\pi$
$547$	4.58098e9i	1.19675i	−0.801217	+	0.598374i	$0.795814\pi$
$548$	0	0
$549$	3.75624e8	0.0968836
$550$	0	0
$551$	3.24714e9	0.826933
$552$	0	0
$553$	6.46260e9i	1.62506i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 1.98969e9i	− 0.487856i	−0.969793	−	0.243928i	$-0.921564\pi$
$557$	− 1.98969e9i	− 0.487856i	0.969793	−	0.243928i	$-0.0784361\pi$
$558$	0	0
$559$	8.28143e8	0.200523
$560$	0	0
$561$	2.54716e8	0.0609096
$562$	0	0
$563$	1.51071e9i	0.356781i	0.983960	+	0.178390i	$0.0570890\pi$
$563$	1.51071e9i	0.356781i	−0.983960	+	0.178390i	$0.942911\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	6.17665e9i	1.42302i
$568$	0	0
$569$	−1.75619e8	−0.0399650	−0.0199825	−	0.999800i	$-0.506361\pi$
$569$	−1.75619e8	−0.0399650	−0.0199825	+	0.999800i	$0.506361\pi$
$570$	0	0
$571$	2.21767e8	0.0498506	0.0249253	−	0.999689i	$-0.492065\pi$
$571$	2.21767e8	0.0498506	0.0249253	+	0.999689i	$0.492065\pi$
$572$	0	0
$573$	3.73028e9i	0.828323i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	7.51519e9i	1.62864i	0.580418	+	0.814319i	$0.302889\pi$
$577$	7.51519e9i	1.62864i	−0.580418	+	0.814319i	$0.697111\pi$
$578$	0	0
$579$	−9.04906e9	−1.93744
$580$	0	0
$581$	−7.04310e9	−1.48987
$582$	0	0
$583$	− 5.01866e8i	− 0.104893i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 5.57215e9i	− 1.13708i	−0.822657	−	0.568538i	$-0.807509\pi$
$587$	− 5.57215e9i	− 1.13708i	0.822657	−	0.568538i	$-0.192491\pi$
$588$	0	0
$589$	4.83852e9	0.975683
$590$	0	0
$591$	3.86032e9	0.769250
$592$	0	0
$593$	2.27962e9i	0.448922i	0.974483	+	0.224461i	$0.0720621\pi$
$593$	2.27962e9i	0.448922i	−0.974483	+	0.224461i	$0.927938\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	1.87802e9i	0.361235i
$598$	0	0
$599$	3.90822e9	0.742993	0.371497	−	0.928434i	$-0.378845\pi$
$599$	3.90822e9	0.742993	0.371497	+	0.928434i	$0.378845\pi$
$600$	0	0
$601$	−2.15505e9	−0.404946	−0.202473	−	0.979288i	$-0.564898\pi$
$601$	−2.15505e9	−0.404946	−0.202473	+	0.979288i	$0.564898\pi$
$602$	0	0
$603$	1.26524e8i	0.0234997i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	5.06857e9i	0.919867i	0.887953	+	0.459933i	$0.152127\pi$
$607$	5.06857e9i	0.919867i	−0.887953	+	0.459933i	$0.847873\pi$
$608$	0	0
$609$	−8.70510e9	−1.56176
$610$	0	0
$611$	−2.62257e9	−0.465138
$612$	0	0
$613$	− 6.56079e9i	− 1.15039i	−0.818017	−	0.575194i	$-0.804927\pi$
$613$	− 6.56079e9i	− 1.15039i	0.818017	−	0.575194i	$-0.195073\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 1.11814e8i	− 0.0191645i	−0.999954	−	0.00958223i	$-0.996950\pi$
$617$	− 1.11814e8i	− 0.0191645i	0.999954	−	0.00958223i	$-0.00305017\pi$
$618$	0	0
$619$	1.48440e9	0.251556	0.125778	−	0.992058i	$-0.459857\pi$
$619$	1.48440e9	0.251556	0.125778	+	0.992058i	$0.459857\pi$
$620$	0	0
$621$	−1.00997e10	−1.69233
$622$	0	0
$623$	− 1.18181e10i	− 1.95812i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 1.06994e9i	− 0.173350i
$628$	0	0
$629$	2.48019e9	0.397382
$630$	0	0
$631$	−6.22722e9	−0.986714	−0.493357	−	0.869827i	$-0.664230\pi$
$631$	−6.22722e9	−0.986714	−0.493357	+	0.869827i	$0.664230\pi$
$632$	0	0
$633$	− 7.05220e8i	− 0.110513i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 3.83708e9i	− 0.588184i
$638$	0	0
$639$	3.57061e8	0.0541364
$640$	0	0
$641$	6.72145e9	1.00800	0.503999	−	0.863704i	$-0.331861\pi$
$641$	6.72145e9	1.00800	0.503999	+	0.863704i	$0.331861\pi$
$642$	0	0
$643$	6.13000e9i	0.909332i	0.890662	+	0.454666i	$0.150241\pi$
$643$	6.13000e9i	0.909332i	−0.890662	+	0.454666i	$0.849759\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	1.10514e10i	1.60418i	0.597204	+	0.802089i	$0.296278\pi$
$647$	1.10514e10i	1.60418i	−0.597204	+	0.802089i	$0.703722\pi$
$648$	0	0
$649$	−8.20142e7	−0.0117769
$650$	0	0
$651$	−1.29714e10	−1.84269
$652$	0	0
$653$	− 1.90379e9i	− 0.267561i	−0.991011	−	0.133781i	$-0.957288\pi$
$653$	− 1.90379e9i	− 0.267561i	0.991011	−	0.133781i	$-0.0427117\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	4.82774e7i	0.00664149i
$658$	0	0
$659$	1.16276e9	0.158268	0.0791339	−	0.996864i	$-0.474785\pi$
$659$	1.16276e9	0.158268	0.0791339	+	0.996864i	$0.474785\pi$
$660$	0	0
$661$	−2.20847e9	−0.297431	−0.148716	−	0.988880i	$-0.547514\pi$
$661$	−2.20847e9	−0.297431	−0.148716	+	0.988880i	$0.547514\pi$
$662$	0	0
$663$	9.16532e8i	0.122138i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 1.34573e10i	− 1.75598i
$668$	0	0
$669$	−1.89322e8	−0.0244461
$670$	0	0
$671$	3.19627e9	0.408427
$672$	0	0
$673$	− 9.22930e9i	− 1.16712i	−0.812070	−	0.583561i	$-0.801659\pi$
$673$	− 9.22930e9i	− 1.16712i	0.812070	−	0.583561i	$-0.198341\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	5.75119e9i	0.712357i	0.934418	+	0.356178i	$0.115920\pi$
$677$	5.75119e9i	0.712357i	−0.934418	+	0.356178i	$0.884080\pi$
$678$	0	0
$679$	9.58518e9	1.17505
$680$	0	0
$681$	2.91620e9	0.353837
$682$	0	0
$683$	− 2.96482e9i	− 0.356063i	−0.984025	−	0.178031i	$-0.943027\pi$
$683$	− 2.96482e9i	− 0.356063i	0.984025	−	0.178031i	$-0.0569729\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	2.46331e9i	0.289848i
$688$	0	0
$689$	1.80584e9	0.210336
$690$	0	0
$691$	−1.35266e10	−1.55961	−0.779805	−	0.626023i	$-0.784682\pi$
$691$	−1.35266e10	−1.55961	−0.779805	+	0.626023i	$0.784682\pi$
$692$	0	0
$693$	− 1.65027e8i	− 0.0188360i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 4.41032e8i	− 0.0493350i
$698$	0	0
$699$	−1.24992e10	−1.38424
$700$	0	0
$701$	8.89669e9	0.975473	0.487737	−	0.872991i	$-0.337823\pi$
$701$	8.89669e9	0.975473	0.487737	+	0.872991i	$0.337823\pi$
$702$	0	0
$703$	− 1.04181e10i	− 1.13096i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 7.32710e9i	− 0.779766i
$708$	0	0
$709$	4.02759e9	0.424408	0.212204	−	0.977225i	$-0.431936\pi$
$709$	4.02759e9	0.424408	0.212204	+	0.977225i	$0.431936\pi$
$710$	0	0
$711$	−5.61272e8	−0.0585639
$712$	0	0
$713$	− 2.00526e10i	− 2.07184i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	8.45842e9i	0.856983i
$718$	0	0
$719$	−9.22714e9	−0.925798	−0.462899	−	0.886411i	$-0.653191\pi$
$719$	−9.22714e9	−0.925798	−0.462899	+	0.886411i	$0.653191\pi$
$720$	0	0
$721$	−9.72440e9	−0.966250
$722$	0	0
$723$	− 1.08137e10i	− 1.06412i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	9.12103e9i	0.880386i	0.897903	+	0.440193i	$0.145090\pi$
$727$	9.12103e9i	0.880386i	−0.897903	+	0.440193i	$0.854910\pi$
$728$	0	0
$729$	−1.09658e10	−1.04832
$730$	0	0
$731$	1.25764e9	0.119082
$732$	0	0
$733$	− 1.72029e10i	− 1.61338i	−0.590974	−	0.806691i	$-0.701256\pi$
$733$	− 1.72029e10i	− 1.61338i	0.590974	−	0.806691i	$-0.298744\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1.07662e9i	0.0990663i
$738$	0	0
$739$	−1.73905e10	−1.58510	−0.792552	−	0.609805i	$-0.791248\pi$
$739$	−1.73905e10	−1.58510	−0.792552	+	0.609805i	$0.791248\pi$
$740$	0	0
$741$	3.84992e9	0.347607
$742$	0	0
$743$	4.62464e9i	0.413635i	0.978380	+	0.206817i	$0.0663106\pi$
$743$	4.62464e9i	0.413635i	−0.978380	+	0.206817i	$0.933689\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 6.11688e8i	− 0.0536918i
$748$	0	0
$749$	−1.38386e10	−1.20339
$750$	0	0
$751$	1.36007e10	1.17171	0.585856	−	0.810415i	$-0.300758\pi$
$751$	1.36007e10	1.17171	0.585856	+	0.810415i	$0.300758\pi$
$752$	0	0
$753$	1.49298e10i	1.27430i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 1.78432e10i	− 1.49499i	−0.664269	−	0.747493i	$-0.731257\pi$
$757$	− 1.78432e10i	− 1.49499i	0.664269	−	0.747493i	$-0.268743\pi$
$758$	0	0
$759$	−4.43423e9	−0.368105
$760$	0	0
$761$	−1.41211e10	−1.16151	−0.580754	−	0.814079i	$-0.697242\pi$
$761$	−1.41211e10	−1.16151	−0.580754	+	0.814079i	$0.697242\pi$
$762$	0	0
$763$	− 1.64829e9i	− 0.134337i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 2.95108e8i	− 0.0236155i
$768$	0	0
$769$	6.98040e9	0.553526	0.276763	−	0.960938i	$-0.410738\pi$
$769$	6.98040e9	0.553526	0.276763	+	0.960938i	$0.410738\pi$
$770$	0	0
$771$	4.61880e9	0.362943
$772$	0	0
$773$	5.57792e9i	0.434354i	0.976132	+	0.217177i	$0.0696849\pi$
$773$	5.57792e9i	0.434354i	−0.976132	+	0.217177i	$0.930315\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	2.79295e10i	2.13594i
$778$	0	0
$779$	−1.85257e9	−0.140408
$780$	0	0
$781$	3.03831e9	0.228220
$782$	0	0
$783$	− 1.46527e10i	− 1.09082i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 1.26349e10i	− 0.923978i	−0.886886	−	0.461989i	$-0.847136\pi$
$787$	− 1.26349e10i	− 0.923978i	0.886886	−	0.461989i	$-0.152864\pi$
$788$	0	0
$789$	−9.04434e9	−0.655552
$790$	0	0
$791$	−7.25550e8	−0.0521254
$792$	0	0
$793$	1.15010e10i	0.818992i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	9.24597e9i	0.646916i	0.946242	+	0.323458i	$0.104846\pi$
$797$	9.24597e9i	0.646916i	−0.946242	+	0.323458i	$0.895154\pi$
$798$	0	0
$799$	−3.98271e9	−0.276226
$800$	0	0
$801$	1.02639e9	0.0705666
$802$	0	0
$803$	4.10803e8i	0.0279982i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	1.11389e10i	0.746080i
$808$	0	0
$809$	−3.77594e9	−0.250729	−0.125365	−	0.992111i	$-0.540010\pi$
$809$	−3.77594e9	−0.250729	−0.125365	+	0.992111i	$0.540010\pi$
$810$	0	0
$811$	8.24089e9	0.542502	0.271251	−	0.962509i	$-0.412563\pi$
$811$	8.24089e9	0.542502	0.271251	+	0.962509i	$0.412563\pi$
$812$	0	0
$813$	− 1.77820e10i	− 1.16055i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 5.28276e9i	− 0.338910i
$818$	0	0
$819$	5.93810e8	0.0377706
$820$	0	0
$821$	8.07649e9	0.509356	0.254678	−	0.967026i	$-0.418031\pi$
$821$	8.07649e9	0.509356	0.254678	+	0.967026i	$0.418031\pi$
$822$	0	0
$823$	− 1.76555e10i	− 1.10403i	−0.833834	−	0.552016i	$-0.813859\pi$
$823$	− 1.76555e10i	− 1.10403i	0.833834	−	0.552016i	$-0.186141\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 2.60329e9i	− 0.160049i	−0.996793	−	0.0800246i	$-0.974500\pi$
$827$	− 2.60329e9i	− 0.160049i	0.996793	−	0.0800246i	$-0.0254999\pi$
$828$	0	0
$829$	−9.16053e9	−0.558444	−0.279222	−	0.960227i	$-0.590077\pi$
$829$	−9.16053e9	−0.558444	−0.279222	+	0.960227i	$0.590077\pi$
$830$	0	0
$831$	2.31426e10	1.39897
$832$	0	0
$833$	− 5.82711e9i	− 0.349297i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 2.18338e10i	− 1.28704i
$838$	0	0
$839$	2.51395e10	1.46957	0.734784	−	0.678301i	$-0.237283\pi$
$839$	2.51395e10	1.46957	0.734784	+	0.678301i	$0.237283\pi$
$840$	0	0
$841$	2.27422e9	0.131840
$842$	0	0
$843$	7.07430e9i	0.406712i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	2.52926e10i	1.43022i
$848$	0	0
$849$	−5.21012e9	−0.292193
$850$	0	0
$851$	−4.31766e10	−2.40157
$852$	0	0
$853$	− 1.37385e10i	− 0.757908i	−0.925415	−	0.378954i	$-0.876284\pi$
$853$	− 1.37385e10i	− 0.757908i	0.925415	−	0.378954i	$-0.123716\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 2.41390e10i	− 1.31004i	−0.755610	−	0.655022i	$-0.772659\pi$
$857$	− 2.41390e10i	− 1.31004i	0.755610	−	0.655022i	$-0.227341\pi$
$858$	0	0
$859$	−2.62094e10	−1.41085	−0.705426	−	0.708783i	$-0.749244\pi$
$859$	−2.62094e10	−1.41085	−0.705426	+	0.708783i	$0.749244\pi$
$860$	0	0
$861$	4.96646e9	0.265177
$862$	0	0
$863$	− 1.61957e10i	− 0.857750i	−0.903364	−	0.428875i	$-0.858910\pi$
$863$	− 1.61957e10i	− 0.857750i	0.903364	−	0.428875i	$-0.141090\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 1.72685e10i	− 0.899885i
$868$	0	0
$869$	−4.77599e9	−0.246885
$870$	0	0
$871$	−3.87395e9	−0.198651
$872$	0	0
$873$	8.32465e8i	0.0423464i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 1.25717e10i	− 0.629356i	−0.949198	−	0.314678i	$-0.898103\pi$
$877$	− 1.25717e10i	− 0.629356i	0.949198	−	0.314678i	$-0.101897\pi$
$878$	0	0
$879$	3.83060e10	1.90242
$880$	0	0
$881$	−3.40786e10	−1.67906	−0.839530	−	0.543313i	$-0.817170\pi$
$881$	−3.40786e10	−1.67906	−0.839530	+	0.543313i	$0.817170\pi$
$882$	0	0
$883$	2.78949e10i	1.36352i	0.731574	+	0.681762i	$0.238786\pi$
$883$	2.78949e10i	1.36352i	−0.731574	+	0.681762i	$0.761214\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	3.29903e10i	1.58728i	0.608387	+	0.793641i	$0.291817\pi$
$887$	3.29903e10i	1.58728i	−0.608387	+	0.793641i	$0.708183\pi$
$888$	0	0
$889$	2.60870e10	1.24528
$890$	0	0
$891$	−4.56466e9	−0.216191
$892$	0	0
$893$	1.67295e10i	0.786144i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 1.59555e10i	− 0.738136i
$898$	0	0
$899$	2.90926e10	1.33544
$900$	0	0
$901$	2.74241e9	0.124909
$902$	0	0
$903$	1.41623e10i	0.640069i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 1.66008e10i	− 0.738759i	−0.929279	−	0.369380i	$-0.879570\pi$
$907$	− 1.66008e10i	− 0.738759i	0.929279	−	0.369380i	$-0.120430\pi$
$908$	0	0
$909$	6.36353e8	0.0281012
$910$	0	0
$911$	2.80189e9	0.122782	0.0613912	−	0.998114i	$-0.480446\pi$
$911$	2.80189e9	0.122782	0.0613912	+	0.998114i	$0.480446\pi$
$912$	0	0
$913$	− 5.20499e9i	− 0.226346i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	6.18631e10i	2.64935i
$918$	0	0
$919$	−7.97528e9	−0.338955	−0.169477	−	0.985534i	$-0.554208\pi$
$919$	−7.97528e9	−0.338955	−0.169477	+	0.985534i	$0.554208\pi$
$920$	0	0
$921$	1.08273e9	0.0456678
$922$	0	0
$923$	1.09326e10i	0.457635i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 8.44557e8i	− 0.0348217i
$928$	0	0
$929$	4.40021e10	1.80061	0.900303	−	0.435264i	$-0.143345\pi$
$929$	4.40021e10	1.80061	0.900303	+	0.435264i	$0.143345\pi$
$930$	0	0
$931$	−2.44769e10	−0.994107
$932$	0	0
$933$	3.32746e10i	1.34130i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 1.85516e10i	− 0.736702i	−0.929687	−	0.368351i	$-0.879922\pi$
$937$	− 1.85516e10i	− 0.736702i	0.929687	−	0.368351i	$-0.120078\pi$
$938$	0	0
$939$	−4.19092e10	−1.65188
$940$	0	0
$941$	−3.98382e8	−0.0155860	−0.00779302	−	0.999970i	$-0.502481\pi$
$941$	−3.98382e8	−0.0155860	−0.00779302	+	0.999970i	$0.502481\pi$
$942$	0	0
$943$	7.67772e9i	0.298155i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 1.22967e10i	− 0.470504i	−0.971934	−	0.235252i	$-0.924409\pi$
$947$	− 1.22967e10i	− 0.470504i	0.971934	−	0.235252i	$-0.0755915\pi$
$948$	0	0
$949$	−1.47818e9	−0.0561429
$950$	0	0
$951$	3.57143e9	0.134651
$952$	0	0
$953$	− 2.00329e10i	− 0.749754i	−0.927075	−	0.374877i	$-0.877685\pi$
$953$	− 2.00329e10i	− 0.749754i	0.927075	−	0.374877i	$-0.122315\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 6.43324e9i	− 0.237267i
$958$	0	0
$959$	4.81716e10	1.76370
$960$	0	0
$961$	1.58379e10	0.575659
$962$	0	0
$963$	− 1.20187e9i	− 0.0433676i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	2.01344e10i	0.716055i	0.933711	+	0.358027i	$0.116551\pi$
$967$	2.01344e10i	0.716055i	−0.933711	+	0.358027i	$0.883449\pi$
$968$	0	0
$969$	5.84661e9	0.206429
$970$	0	0
$971$	5.02424e10	1.76118	0.880589	−	0.473881i	$-0.157147\pi$
$971$	5.02424e10	1.76118	0.880589	+	0.473881i	$0.157147\pi$
$972$	0	0
$973$	5.62702e10i	1.95832i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	3.03613e10i	1.04157i	0.853687	+	0.520786i	$0.174361\pi$
$977$	3.03613e10i	1.04157i	−0.853687	+	0.520786i	$0.825639\pi$
$978$	0	0
$979$	8.73379e9	0.297484
$980$	0	0
$981$	1.43152e8	0.00484124
$982$	0	0
$983$	3.64309e10i	1.22330i	0.791129	+	0.611650i	$0.209494\pi$
$983$	3.64309e10i	1.22330i	−0.791129	+	0.611650i	$0.790506\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 4.48493e10i	− 1.48472i
$988$	0	0
$989$	−2.18937e10	−0.719668
$990$	0	0
$991$	2.79867e10	0.913470	0.456735	−	0.889603i	$-0.349019\pi$
$991$	2.79867e10	0.913470	0.456735	+	0.889603i	$0.349019\pi$
$992$	0	0
$993$	1.72296e10i	0.558411i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	3.06088e10i	0.978169i	0.872236	+	0.489084i	$0.162669\pi$
$997$	3.06088e10i	0.978169i	−0.872236	+	0.489084i	$0.837331\pi$
$998$	0	0
$999$	−4.70119e10	−1.49186

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	400.8.c.s.49.1		4
4.3	odd	2		25.8.b.b.24.1		4
5.2	odd	4		400.8.a.v.1.1		2
5.3	odd	4		400.8.a.bd.1.2		2
5.4	even	2	inner	400.8.c.s.49.4		4
12.11	even	2		225.8.b.l.199.4		4
20.3	even	4		25.8.a.c.1.1	✓	2
20.7	even	4		25.8.a.e.1.2	yes	2
20.19	odd	2		25.8.b.b.24.4		4
60.23	odd	4		225.8.a.v.1.2		2
60.47	odd	4		225.8.a.k.1.1		2
60.59	even	2		225.8.b.l.199.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
25.8.a.c.1.1	✓	2	20.3	even	4
25.8.a.e.1.2	yes	2	20.7	even	4
25.8.b.b.24.1		4	4.3	odd	2
25.8.b.b.24.4		4	20.19	odd	2
225.8.a.k.1.1		2	60.47	odd	4
225.8.a.v.1.2		2	60.23	odd	4
225.8.b.l.199.1		4	60.59	even	2
225.8.b.l.199.4		4	12.11	even	2
400.8.a.v.1.1		2	5.2	odd	4
400.8.a.bd.1.2		2	5.3	odd	4
400.8.c.s.49.1		4	1.1	even	1	trivial
400.8.c.s.49.4		4	5.4	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 400 = 2^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	400.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-45.4755i q^{3} -1369.97i q^{7} +118.981 q^{9} +1012.43 q^{11} +3643.00i q^{13} +5532.37i q^{17} +23238.9 q^{19} -62300.0 q^{21} -96310.5i q^{23} -104866. i q^{27} +139729. q^{29} +208208. q^{31} -46041.0i q^{33} -448306. i q^{37} +165667. q^{39} -79718.5 q^{41} -227324. i q^{43} +719892. i q^{47} -1.05328e6 q^{49} +251587. q^{51} -495702. i q^{53} -1.05680e6i q^{57} -81006.9 q^{59} +3.15701e6 q^{61} -163000. i q^{63} +1.06340e6i q^{67} -4.37976e6 q^{69} +3.00099e6 q^{71} +405758. i q^{73} -1.38701e6i q^{77} -4.71733e6 q^{79} -4.50860e6 q^{81} -5.14106e6i q^{83} -6.35423e6i q^{87} +8.62652e6 q^{89} +4.99080e6 q^{91} -9.46835e6i q^{93} +6.99663e6i q^{97} +120460. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4552 q^{9} - 8688 q^{11} + 36400 q^{19} - 133032 q^{21} - 111600 q^{29} + 603552 q^{31} + 177616 q^{39} - 216972 q^{41} - 1645172 q^{49} + 638992 q^{51} + 4135200 q^{59} + 1164088 q^{61} - 9582936 q^{69}+ \cdots - 22866944 q^{99}+O(q^{100}) \) 4 * q + 4552 * q^9 - 8688 * q^11 + 36400 * q^19 - 133032 * q^21 - 111600 * q^29 + 603552 * q^31 + 177616 * q^39 - 216972 * q^41 - 1645172 * q^49 + 638992 * q^51 + 4135200 * q^59 + 1164088 * q^61 - 9582936 * q^69 + 9456432 * q^71 - 14372400 * q^79 + 157124 * q^81 + 11981700 * q^89 - 11634528 * q^91 - 22866944 * q^99

Introduction

L-functions

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