LMFDB - Embedded newform 400.9.b.c.351.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [400,9,Mod(351,400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

N = Newforms(chi, 9, names="a")

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

chi := DirichletCharacter("400.351");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 400 = 2^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	400.b (of order $2$, degree $1$, 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:	$162.951444024$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-35}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 9 $ x^2 - x + 9
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{4}\cdot 3 $
Twist minimal:	no (minimal twist has level 16)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			351.2
Root			$0.500000 - 2.95804i$ of defining polynomial
Character	$\chi$	$=$	400.351
Dual form			400.9.b.c.351.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+141.986i q^{3} -2555.75i q^{7} -13599.0 q^{9} -19168.1i q^{11} +27710.0 q^{13} -50370.0 q^{17} -108619. i q^{19} +362880. q^{21} +176347. i q^{23} -999297. i q^{27} +54978.0 q^{29} -1.17564e6i q^{31} +2.72160e6 q^{33} -793730. q^{37} +3.93443e6i q^{39} -75582.0 q^{41} +499648. i q^{43} +2.86755e6i q^{47} -767039. q^{49} -7.15183e6i q^{51} -1.11662e7 q^{53} +1.54224e7 q^{57} +2.18325e7i q^{59} -2.38266e7 q^{61} +3.47556e7i q^{63} +7.49473e6i q^{67} -2.50387e7 q^{69} +1.00824e7i q^{71} -6.51661e6 q^{73} -4.89888e7 q^{77} +4.87892e7i q^{79} +5.26630e7 q^{81} +7.34483e7i q^{83} +7.80610e6i q^{87} +8.67958e7 q^{89} -7.08197e7i q^{91} +1.66925e8 q^{93} +4.66703e7 q^{97} +2.60667e8i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 27198 q^{9} + 55420 q^{13} - 100740 q^{17} + 725760 q^{21} + 109956 q^{29} + 5443200 q^{33} - 1587460 q^{37} - 151164 q^{41} - 1534078 q^{49} - 22332420 q^{53} + 30844800 q^{57} - 47653244 q^{61}+ \cdots + 93340540 q^{97}+O(q^{100}) $ 2 * q - 27198 * q^9 + 55420 * q^13 - 100740 * q^17 + 725760 * q^21 + 109956 * q^29 + 5443200 * q^33 - 1587460 * q^37 - 151164 * q^41 - 1534078 * q^49 - 22332420 * q^53 + 30844800 * q^57 - 47653244 * q^61 - 50077440 * q^69 - 13033220 * q^73 - 97977600 * q^77 + 105326082 * q^81 + 173591556 * q^89 + 333849600 * q^93 + 93340540 * q^97

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
$3$	141.986i	1.75291i	0.481481	+	0.876456i	$0.340099\pi$
$3$	141.986i	1.75291i	−0.481481	+	0.876456i	$0.659901\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 2555.75i	− 1.06445i	−0.846603	−	0.532225i	$-0.821356\pi$
$7$	− 2555.75i	− 1.06445i	0.846603	−	0.532225i	$-0.178644\pi$
$8$	0	0
$9$	−13599.0	−2.07270
$10$	0	0
$11$	− 19168.1i	− 1.30921i	−0.755972	−	0.654603i	$-0.772836\pi$
$11$	− 19168.1i	− 1.30921i	0.755972	−	0.654603i	$-0.227164\pi$
$12$	0	0
$13$	27710.0	0.970204	0.485102	−	0.874458i	$-0.338782\pi$
$13$	27710.0	0.970204	0.485102	+	0.874458i	$0.338782\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−50370.0	−0.603082	−0.301541	−	0.953453i	$-0.597501\pi$
$17$	−50370.0	−0.603082	−0.301541	+	0.953453i	$0.597501\pi$
$18$	0	0
$19$	− 108619.i	− 0.833474i	−0.909027	−	0.416737i	$-0.863174\pi$
$19$	− 108619.i	− 0.833474i	0.909027	−	0.416737i	$-0.136826\pi$
$20$	0	0
$21$	362880.	1.86589
$22$	0	0
$23$	176347.i	0.630167i	0.949064	+	0.315083i	$0.102032\pi$
$23$	176347.i	0.630167i	−0.949064	+	0.315083i	$0.897968\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 999297.i	− 1.88035i
$28$	0	0
$29$	54978.0	0.0777315	0.0388657	−	0.999244i	$-0.487626\pi$
$29$	54978.0	0.0777315	0.0388657	+	0.999244i	$0.487626\pi$
$30$	0	0
$31$	− 1.17564e6i	− 1.27300i	−0.771276	−	0.636501i	$-0.780381\pi$
$31$	− 1.17564e6i	− 1.27300i	0.771276	−	0.636501i	$-0.219619\pi$
$32$	0	0
$33$	2.72160e6	2.29493
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−793730.	−0.423512	−0.211756	−	0.977323i	$-0.567918\pi$
$37$	−793730.	−0.423512	−0.211756	+	0.977323i	$0.567918\pi$
$38$	0	0
$39$	3.93443e6i	1.70068i
$40$	0	0
$41$	−75582.0	−0.0267475	−0.0133737	−	0.999911i	$-0.504257\pi$
$41$	−75582.0	−0.0267475	−0.0133737	+	0.999911i	$0.504257\pi$
$42$	0	0
$43$	499648.i	0.146147i	0.997327	+	0.0730736i	$0.0232808\pi$
$43$	499648.i	0.146147i	−0.997327	+	0.0730736i	$0.976719\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	2.86755e6i	0.587651i	0.955859	+	0.293825i	$0.0949284\pi$
$47$	2.86755e6i	0.587651i	−0.955859	+	0.293825i	$0.905072\pi$
$48$	0	0
$49$	−767039.	−0.133056
$50$	0	0
$51$	− 7.15183e6i	− 1.05715i
$52$	0	0
$53$	−1.11662e7	−1.41515	−0.707575	−	0.706639i	$-0.750211\pi$
$53$	−1.11662e7	−1.41515	−0.707575	+	0.706639i	$0.750211\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	1.54224e7	1.46101
$58$	0	0
$59$	2.18325e7i	1.80175i	0.434078	+	0.900875i	$0.357074\pi$
$59$	2.18325e7i	1.80175i	−0.434078	+	0.900875i	$0.642926\pi$
$60$	0	0
$61$	−2.38266e7	−1.72085	−0.860425	−	0.509577i	$-0.829802\pi$
$61$	−2.38266e7	−1.72085	−0.860425	+	0.509577i	$0.829802\pi$
$62$	0	0
$63$	3.47556e7i	2.20629i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	7.49473e6i	0.371926i	0.982557	+	0.185963i	$0.0595405\pi$
$67$	7.49473e6i	0.371926i	−0.982557	+	0.185963i	$0.940460\pi$
$68$	0	0
$69$	−2.50387e7	−1.10463
$70$	0	0
$71$	1.00824e7i	0.396763i	0.980125	+	0.198382i	$0.0635685\pi$
$71$	1.00824e7i	0.396763i	−0.980125	+	0.198382i	$0.936431\pi$
$72$	0	0
$73$	−6.51661e6	−0.229472	−0.114736	−	0.993396i	$-0.536602\pi$
$73$	−6.51661e6	−0.229472	−0.114736	+	0.993396i	$0.536602\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−4.89888e7	−1.39359
$78$	0	0
$79$	4.87892e7i	1.25261i	0.779579	+	0.626304i	$0.215433\pi$
$79$	4.87892e7i	1.25261i	−0.779579	+	0.626304i	$0.784567\pi$
$80$	0	0
$81$	5.26630e7	1.22339
$82$	0	0
$83$	7.34483e7i	1.54764i	0.633407	+	0.773819i	$0.281656\pi$
$83$	7.34483e7i	1.54764i	−0.633407	+	0.773819i	$0.718344\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	7.80610e6i	0.136256i
$88$	0	0
$89$	8.67958e7	1.38337	0.691685	−	0.722199i	$-0.256869\pi$
$89$	8.67958e7	1.38337	0.691685	+	0.722199i	$0.256869\pi$
$90$	0	0
$91$	− 7.08197e7i	− 1.03273i
$92$	0	0
$93$	1.66925e8	2.23146
$94$	0	0
$95$	0	0
$96$	0	0
$97$	4.66703e7	0.527173	0.263587	−	0.964636i	$-0.415095\pi$
$97$	4.66703e7	0.527173	0.263587	+	0.964636i	$0.415095\pi$
$98$	0	0
$99$	2.60667e8i	2.71360i
$100$	0	0
$101$	6.59910e7	0.634161	0.317080	−	0.948399i	$-0.397297\pi$
$101$	6.59910e7	0.634161	0.317080	+	0.948399i	$0.397297\pi$
$102$	0	0
$103$	1.64884e8i	1.46497i	0.680782	+	0.732486i	$0.261640\pi$
$103$	1.64884e8i	1.46497i	−0.680782	+	0.732486i	$0.738360\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1.27326e8i	0.971364i	0.874136	+	0.485682i	$0.161429\pi$
$107$	1.27326e8i	0.971364i	−0.874136	+	0.485682i	$0.838571\pi$
$108$	0	0
$109$	−1.56119e8	−1.10598	−0.552992	−	0.833186i	$-0.686514\pi$
$109$	−1.56119e8	−1.10598	−0.552992	+	0.833186i	$0.686514\pi$
$110$	0	0
$111$	− 1.12698e8i	− 0.742380i
$112$	0	0
$113$	−2.36346e8	−1.44955	−0.724776	−	0.688984i	$-0.758057\pi$
$113$	−2.36346e8	−1.44955	−0.724776	+	0.688984i	$0.758057\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−3.76828e8	−2.01094
$118$	0	0
$119$	1.28733e8i	0.641951i
$120$	0	0
$121$	−1.53057e8	−0.714023
$122$	0	0
$123$	− 1.07316e7i	− 0.0468860i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 3.67741e8i	− 1.41360i	−0.707412	−	0.706802i	$-0.750137\pi$
$127$	− 3.67741e8i	− 1.41360i	0.707412	−	0.706802i	$-0.249863\pi$
$128$	0	0
$129$	−7.09430e7	−0.256183
$130$	0	0
$131$	− 2.16350e8i	− 0.734636i	−0.930095	−	0.367318i	$-0.880276\pi$
$131$	− 2.16350e8i	− 0.734636i	0.930095	−	0.367318i	$-0.119724\pi$
$132$	0	0
$133$	−2.77603e8	−0.887193
$134$	0	0
$135$	0	0
$136$	0	0
$137$	3.86442e8	1.09699	0.548494	−	0.836155i	$-0.315201\pi$
$137$	3.86442e8	1.09699	0.548494	+	0.836155i	$0.315201\pi$
$138$	0	0
$139$	− 3.51077e8i	− 0.940465i	−0.882543	−	0.470232i	$-0.844170\pi$
$139$	− 3.51077e8i	− 0.940465i	0.882543	−	0.470232i	$-0.155830\pi$
$140$	0	0
$141$	−4.07151e8	−1.03010
$142$	0	0
$143$	− 5.31148e8i	− 1.27020i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 1.08909e8i	− 0.233235i
$148$	0	0
$149$	−4.54099e8	−0.921308	−0.460654	−	0.887580i	$-0.652385\pi$
$149$	−4.54099e8	−0.921308	−0.460654	+	0.887580i	$0.652385\pi$
$150$	0	0
$151$	6.60188e8i	1.26987i	0.772565	+	0.634936i	$0.218973\pi$
$151$	6.60188e8i	1.26987i	−0.772565	+	0.634936i	$0.781027\pi$
$152$	0	0
$153$	6.84982e8	1.25001
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−4.35318e8	−0.716486	−0.358243	−	0.933628i	$-0.616624\pi$
$157$	−4.35318e8	−0.716486	−0.358243	+	0.933628i	$0.616624\pi$
$158$	0	0
$159$	− 1.58544e9i	− 2.48063i
$160$	0	0
$161$	4.50697e8	0.670782
$162$	0	0
$163$	2.44065e8i	0.345744i	0.984944	+	0.172872i	$0.0553047\pi$
$163$	2.44065e8i	0.345744i	−0.984944	+	0.172872i	$0.944695\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	6.71351e8i	0.863145i	0.902078	+	0.431573i	$0.142041\pi$
$167$	6.71351e8i	0.863145i	−0.902078	+	0.431573i	$0.857959\pi$
$168$	0	0
$169$	−4.78866e7	−0.0587040
$170$	0	0
$171$	1.47711e9i	1.72754i
$172$	0	0
$173$	1.76764e9	1.97337	0.986685	−	0.162644i	$-0.0520022\pi$
$173$	1.76764e9	1.97337	0.986685	+	0.162644i	$0.0520022\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−3.09990e9	−3.15831
$178$	0	0
$179$	− 7.56967e8i	− 0.737335i	−0.929561	−	0.368668i	$-0.879814\pi$
$179$	− 7.56967e8i	− 0.737335i	0.929561	−	0.368668i	$-0.120186\pi$
$180$	0	0
$181$	6.27094e8	0.584277	0.292138	−	0.956376i	$-0.405633\pi$
$181$	6.27094e8	0.584277	0.292138	+	0.956376i	$0.405633\pi$
$182$	0	0
$183$	− 3.38304e9i	− 3.01650i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	9.65497e8i	0.789559i
$188$	0	0
$189$	−2.55395e9	−2.00154
$190$	0	0
$191$	− 1.07924e9i	− 0.810933i	−0.914110	−	0.405466i	$-0.867109\pi$
$191$	− 1.07924e9i	− 0.810933i	0.914110	−	0.405466i	$-0.132891\pi$
$192$	0	0
$193$	2.96757e7	0.0213881	0.0106940	−	0.999943i	$-0.496596\pi$
$193$	2.96757e7	0.0213881	0.0106940	+	0.999943i	$0.496596\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	1.12484e9	0.746837	0.373419	−	0.927663i	$-0.378186\pi$
$197$	1.12484e9	0.746837	0.373419	+	0.927663i	$0.378186\pi$
$198$	0	0
$199$	− 1.04718e9i	− 0.667742i	−0.942619	−	0.333871i	$-0.891645\pi$
$199$	− 1.04718e9i	− 0.667742i	0.942619	−	0.333871i	$-0.108355\pi$
$200$	0	0
$201$	−1.06415e9	−0.651954
$202$	0	0
$203$	− 1.40510e8i	− 0.0827413i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 2.39814e9i	− 1.30615i
$208$	0	0
$209$	−2.08202e9	−1.09119
$210$	0	0
$211$	2.77676e9i	1.40090i	0.713699	+	0.700452i	$0.247018\pi$
$211$	2.77676e9i	1.40090i	−0.713699	+	0.700452i	$0.752982\pi$
$212$	0	0
$213$	−1.43156e9	−0.695491
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−3.00465e9	−1.35505
$218$	0	0
$219$	− 9.25267e8i	− 0.402245i
$220$	0	0
$221$	−1.39575e9	−0.585113
$222$	0	0
$223$	2.27822e9i	0.921248i	0.887595	+	0.460624i	$0.152374\pi$
$223$	2.27822e9i	0.921248i	−0.887595	+	0.460624i	$0.847626\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 1.48863e9i	− 0.560639i	−0.959907	−	0.280319i	$-0.909560\pi$
$227$	− 1.48863e9i	− 0.560639i	0.959907	−	0.280319i	$-0.0904404\pi$
$228$	0	0
$229$	1.69447e9	0.616157	0.308079	−	0.951361i	$-0.400314\pi$
$229$	1.69447e9	0.616157	0.308079	+	0.951361i	$0.400314\pi$
$230$	0	0
$231$	− 6.95572e9i	− 2.44284i
$232$	0	0
$233$	−5.21423e8	−0.176916	−0.0884580	−	0.996080i	$-0.528194\pi$
$233$	−5.21423e8	−0.176916	−0.0884580	+	0.996080i	$0.528194\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−6.92738e9	−2.19571
$238$	0	0
$239$	4.40690e9i	1.35065i	0.737522	+	0.675323i	$0.235996\pi$
$239$	4.40690e9i	1.35065i	−0.737522	+	0.675323i	$0.764004\pi$
$240$	0	0
$241$	−1.62148e9	−0.480666	−0.240333	−	0.970691i	$-0.577257\pi$
$241$	−1.62148e9	−0.480666	−0.240333	+	0.970691i	$0.577257\pi$
$242$	0	0
$243$	9.21023e8i	0.264147i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 3.00984e9i	− 0.808640i
$248$	0	0
$249$	−1.04286e10	−2.71287
$250$	0	0
$251$	1.31321e8i	0.0330855i	0.999863	+	0.0165428i	$0.00526597\pi$
$251$	1.31321e8i	0.0330855i	−0.999863	+	0.0165428i	$0.994734\pi$
$252$	0	0
$253$	3.38023e9	0.825019
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−5.27789e9	−1.20984	−0.604920	−	0.796287i	$-0.706795\pi$
$257$	−5.27789e9	−1.20984	−0.604920	+	0.796287i	$0.706795\pi$
$258$	0	0
$259$	2.02857e9i	0.450808i
$260$	0	0
$261$	−7.47646e8	−0.161114
$262$	0	0
$263$	7.38745e9i	1.54409i	0.635570	+	0.772044i	$0.280765\pi$
$263$	7.38745e9i	1.54409i	−0.635570	+	0.772044i	$0.719235\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	1.23238e10i	2.42493i
$268$	0	0
$269$	3.46450e8	0.0661655	0.0330828	−	0.999453i	$-0.489468\pi$
$269$	3.46450e8	0.0661655	0.0330828	+	0.999453i	$0.489468\pi$
$270$	0	0
$271$	− 6.51715e6i	− 0.00120832i	−1.00000		0.000604158i	$-0.999808\pi$
$271$	− 6.51715e6i	− 0.00120832i	1.00000		0.000604158i	$-0.000192310\pi$
$272$	0	0
$273$	1.00554e10	1.81029
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−2.15061e9	−0.365293	−0.182647	−	0.983179i	$-0.558466\pi$
$277$	−2.15061e9	−0.365293	−0.182647	+	0.983179i	$0.558466\pi$
$278$	0	0
$279$	1.59876e10i	2.63855i
$280$	0	0
$281$	1.04256e10	1.67215	0.836074	−	0.548616i	$-0.184845\pi$
$281$	1.04256e10	1.67215	0.836074	+	0.548616i	$0.184845\pi$
$282$	0	0
$283$	1.28042e9i	0.199622i	0.995006	+	0.0998108i	$0.0318238\pi$
$283$	1.28042e9i	0.199622i	−0.995006	+	0.0998108i	$0.968176\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1.93168e8i	0.0284714i
$288$	0	0
$289$	−4.43862e9	−0.636292
$290$	0	0
$291$	6.62652e9i	0.924089i
$292$	0	0
$293$	2.13786e9	0.290074	0.145037	−	0.989426i	$-0.453670\pi$
$293$	2.13786e9	0.290074	0.145037	+	0.989426i	$0.453670\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−1.91546e10	−2.46177
$298$	0	0
$299$	4.88656e9i	0.611390i
$300$	0	0
$301$	1.27697e9	0.155567
$302$	0	0
$303$	9.36980e9i	1.11163i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	5.45140e9i	0.613698i	0.951758	+	0.306849i	$0.0992747\pi$
$307$	5.45140e9i	0.613698i	−0.951758	+	0.306849i	$0.900725\pi$
$308$	0	0
$309$	−2.34112e10	−2.56797
$310$	0	0
$311$	− 1.07550e10i	− 1.14965i	−0.818275	−	0.574827i	$-0.805069\pi$
$311$	− 1.07550e10i	− 1.14965i	0.818275	−	0.574827i	$-0.194931\pi$
$312$	0	0
$313$	2.99804e8	0.0312364	0.0156182	−	0.999878i	$-0.495028\pi$
$313$	2.99804e8	0.0312364	0.0156182	+	0.999878i	$0.495028\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−5.31172e9	−0.526015	−0.263007	−	0.964794i	$-0.584714\pi$
$317$	−5.31172e9	−0.526015	−0.263007	+	0.964794i	$0.584714\pi$
$318$	0	0
$319$	− 1.05382e9i	− 0.101767i
$320$	0	0
$321$	−1.80785e10	−1.70272
$322$	0	0
$323$	5.47115e9i	0.502653i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 2.21667e10i	− 1.93869i
$328$	0	0
$329$	7.32872e9	0.625525
$330$	0	0
$331$	1.01004e10i	0.841446i	0.907189	+	0.420723i	$0.138224\pi$
$331$	1.01004e10i	0.841446i	−0.907189	+	0.420723i	$0.861776\pi$
$332$	0	0
$333$	1.07939e10	0.877815
$334$	0	0
$335$	0	0
$336$	0	0
$337$	1.84359e10	1.42937	0.714684	−	0.699448i	$-0.246571\pi$
$337$	1.84359e10	1.42937	0.714684	+	0.699448i	$0.246571\pi$
$338$	0	0
$339$	− 3.35578e10i	− 2.54094i
$340$	0	0
$341$	−2.25348e10	−1.66662
$342$	0	0
$343$	− 1.27730e10i	− 0.922820i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	1.27822e10i	0.881636i	0.897597	+	0.440818i	$0.145312\pi$
$347$	1.27822e10i	0.881636i	−0.897597	+	0.440818i	$0.854688\pi$
$348$	0	0
$349$	−6.39381e8	−0.0430981	−0.0215490	−	0.999768i	$-0.506860\pi$
$349$	−6.39381e8	−0.0430981	−0.0215490	+	0.999768i	$0.506860\pi$
$350$	0	0
$351$	− 2.76905e10i	− 1.82433i
$352$	0	0
$353$	−2.59837e10	−1.67341	−0.836705	−	0.547653i	$-0.815521\pi$
$353$	−2.59837e10	−1.67341	−0.836705	+	0.547653i	$0.815521\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−1.82783e10	−1.12528
$358$	0	0
$359$	2.22541e10i	1.33978i	0.742461	+	0.669889i	$0.233658\pi$
$359$	2.22541e10i	1.33978i	−0.742461	+	0.669889i	$0.766342\pi$
$360$	0	0
$361$	5.18543e9	0.305320
$362$	0	0
$363$	− 2.17320e10i	− 1.25162i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 1.86229e10i	− 1.02656i	−0.858221	−	0.513280i	$-0.828430\pi$
$367$	− 1.86229e10i	− 1.02656i	0.858221	−	0.513280i	$-0.171570\pi$
$368$	0	0
$369$	1.02784e9	0.0554396
$370$	0	0
$371$	2.85380e10i	1.50636i
$372$	0	0
$373$	−1.19680e10	−0.618283	−0.309141	−	0.951016i	$-0.600042\pi$
$373$	−1.19680e10	−0.618283	−0.309141	+	0.951016i	$0.600042\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	1.52344e9	0.0754154
$378$	0	0
$379$	− 2.30787e10i	− 1.11855i	−0.828982	−	0.559275i	$-0.811080\pi$
$379$	− 2.30787e10i	− 1.11855i	0.828982	−	0.559275i	$-0.188920\pi$
$380$	0	0
$381$	5.22141e10	2.47792
$382$	0	0
$383$	− 1.43419e10i	− 0.666518i	−0.942835	−	0.333259i	$-0.891852\pi$
$383$	− 1.43419e10i	− 0.666518i	0.942835	−	0.333259i	$-0.108148\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 6.79472e9i	− 0.302920i
$388$	0	0
$389$	−2.73457e10	−1.19424	−0.597119	−	0.802152i	$-0.703688\pi$
$389$	−2.73457e10	−1.19424	−0.597119	+	0.802152i	$0.703688\pi$
$390$	0	0
$391$	− 8.88257e9i	− 0.380042i
$392$	0	0
$393$	3.07187e10	1.28775
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−3.99456e10	−1.60808	−0.804039	−	0.594576i	$-0.797320\pi$
$397$	−3.99456e10	−1.60808	−0.804039	+	0.594576i	$0.797320\pi$
$398$	0	0
$399$	− 3.94157e10i	− 1.55517i
$400$	0	0
$401$	−2.12767e10	−0.822863	−0.411431	−	0.911441i	$-0.634971\pi$
$401$	−2.12767e10	−0.822863	−0.411431	+	0.911441i	$0.634971\pi$
$402$	0	0
$403$	− 3.25771e10i	− 1.23507i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1.52143e10i	0.554465i
$408$	0	0
$409$	1.14283e10	0.408404	0.204202	−	0.978929i	$-0.434540\pi$
$409$	1.14283e10	0.408404	0.204202	+	0.978929i	$0.434540\pi$
$410$	0	0
$411$	5.48693e10i	1.92292i
$412$	0	0
$413$	5.57982e10	1.91788
$414$	0	0
$415$	0	0
$416$	0	0
$417$	4.98479e10	1.64855
$418$	0	0
$419$	− 1.10009e10i	− 0.356922i	−0.983947	−	0.178461i	$-0.942888\pi$
$419$	− 1.10009e10i	− 0.356922i	0.983947	−	0.178461i	$-0.0571119\pi$
$420$	0	0
$421$	2.28766e10	0.728220	0.364110	−	0.931356i	$-0.381373\pi$
$421$	2.28766e10	0.728220	0.364110	+	0.931356i	$0.381373\pi$
$422$	0	0
$423$	− 3.89958e10i	− 1.21802i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	6.08948e10i	1.83176i
$428$	0	0
$429$	7.54155e10	2.22655
$430$	0	0
$431$	9.55108e8i	0.0276786i	0.999904	+	0.0138393i	$0.00440532\pi$
$431$	9.55108e8i	0.0276786i	−0.999904	+	0.0138393i	$0.995595\pi$
$432$	0	0
$433$	−3.82225e10	−1.08735	−0.543673	−	0.839297i	$-0.682967\pi$
$433$	−3.82225e10	−1.08735	−0.543673	+	0.839297i	$0.682967\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	1.91546e10	0.525228
$438$	0	0
$439$	− 6.40288e10i	− 1.72392i	−0.506976	−	0.861960i	$-0.669237\pi$
$439$	− 6.40288e10i	− 1.72392i	0.506976	−	0.861960i	$-0.330763\pi$
$440$	0	0
$441$	1.04310e10	0.275785
$442$	0	0
$443$	7.47659e10i	1.94128i	0.240533	+	0.970641i	$0.422678\pi$
$443$	7.47659e10i	1.94128i	−0.240533	+	0.970641i	$0.577322\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 6.44756e10i	− 1.61497i
$448$	0	0
$449$	2.51987e10	0.620001	0.310000	−	0.950736i	$-0.399671\pi$
$449$	2.51987e10	0.620001	0.310000	+	0.950736i	$0.399671\pi$
$450$	0	0
$451$	1.44876e9i	0.0350180i
$452$	0	0
$453$	−9.37373e10	−2.22597
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−4.66828e9	−0.107027	−0.0535133	−	0.998567i	$-0.517042\pi$
$457$	−4.66828e9	−0.107027	−0.0535133	+	0.998567i	$0.517042\pi$
$458$	0	0
$459$	5.03346e10i	1.13401i
$460$	0	0
$461$	−3.88096e10	−0.859281	−0.429641	−	0.903000i	$-0.641360\pi$
$461$	−3.88096e10	−0.859281	−0.429641	+	0.903000i	$0.641360\pi$
$462$	0	0
$463$	− 3.23432e10i	− 0.703817i	−0.936034	−	0.351908i	$-0.885533\pi$
$463$	− 3.23432e10i	− 0.703817i	0.936034	−	0.351908i	$-0.114467\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 2.31902e10i	− 0.487570i	−0.969829	−	0.243785i	$-0.921611\pi$
$467$	− 2.31902e10i	− 0.487570i	0.969829	−	0.243785i	$-0.0783891\pi$
$468$	0	0
$469$	1.91546e10	0.395897
$470$	0	0
$471$	− 6.18090e10i	− 1.25594i
$472$	0	0
$473$	9.57731e9	0.191337
$474$	0	0
$475$	0	0
$476$	0	0
$477$	1.51849e11	2.93318
$478$	0	0
$479$	4.83542e9i	0.0918528i	0.998945	+	0.0459264i	$0.0146240\pi$
$479$	4.83542e9i	0.0918528i	−0.998945	+	0.0459264i	$0.985376\pi$
$480$	0	0
$481$	−2.19943e10	−0.410893
$482$	0	0
$483$	6.39926e10i	1.17582i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	3.03878e10i	0.540236i	0.962827	+	0.270118i	$0.0870628\pi$
$487$	3.03878e10i	0.540236i	−0.962827	+	0.270118i	$0.912937\pi$
$488$	0	0
$489$	−3.46538e10	−0.606059
$490$	0	0
$491$	5.56483e10i	0.957472i	0.877959	+	0.478736i	$0.158905\pi$
$491$	5.56483e10i	0.957472i	−0.877959	+	0.478736i	$0.841095\pi$
$492$	0	0
$493$	−2.76924e9	−0.0468784
$494$	0	0
$495$	0	0
$496$	0	0
$497$	2.57681e10	0.422335
$498$	0	0
$499$	7.88458e10i	1.27168i	0.771822	+	0.635838i	$0.219345\pi$
$499$	7.88458e10i	1.27168i	−0.771822	+	0.635838i	$0.780655\pi$
$500$	0	0
$501$	−9.53224e10	−1.51302
$502$	0	0
$503$	4.41092e10i	0.689061i	0.938775	+	0.344530i	$0.111962\pi$
$503$	4.41092e10i	0.689061i	−0.938775	+	0.344530i	$0.888038\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 6.79923e9i	− 0.102903i
$508$	0	0
$509$	1.05927e10	0.157811	0.0789055	−	0.996882i	$-0.474857\pi$
$509$	1.05927e10	0.157811	0.0789055	+	0.996882i	$0.474857\pi$
$510$	0	0
$511$	1.66548e10i	0.244262i
$512$	0	0
$513$	−1.08543e11	−1.56723
$514$	0	0
$515$	0	0
$516$	0	0
$517$	5.49654e10	0.769356
$518$	0	0
$519$	2.50979e11i	3.45914i
$520$	0	0
$521$	1.24958e11	1.69595	0.847973	−	0.530039i	$-0.177823\pi$
$521$	1.24958e11	1.69595	0.847973	+	0.530039i	$0.177823\pi$
$522$	0	0
$523$	2.80408e10i	0.374786i	0.982285	+	0.187393i	$0.0600038\pi$
$523$	2.80408e10i	0.374786i	−0.982285	+	0.187393i	$0.939996\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	5.92172e10i	0.767724i
$528$	0	0
$529$	4.72129e10	0.602890
$530$	0	0
$531$	− 2.96900e11i	− 3.73449i
$532$	0	0
$533$	−2.09438e9	−0.0259505
$534$	0	0
$535$	0	0
$536$	0	0
$537$	1.07479e11	1.29248
$538$	0	0
$539$	1.47027e10i	0.174197i
$540$	0	0
$541$	1.44659e11	1.68871	0.844356	−	0.535782i	$-0.179983\pi$
$541$	1.44659e11	1.68871	0.844356	+	0.535782i	$0.179983\pi$
$542$	0	0
$543$	8.90386e10i	1.02419i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	1.03774e10i	0.115915i	0.998319	+	0.0579573i	$0.0184587\pi$
$547$	1.03774e10i	0.115915i	−0.998319	+	0.0579573i	$0.981541\pi$
$548$	0	0
$549$	3.24018e11	3.56681
$550$	0	0
$551$	− 5.97167e9i	− 0.0647872i
$552$	0	0
$553$	1.24693e11	1.33334
$554$	0	0
$555$	0	0
$556$	0	0
$557$	5.47312e9	0.0568610	0.0284305	−	0.999596i	$-0.490949\pi$
$557$	5.47312e9	0.0568610	0.0284305	+	0.999596i	$0.490949\pi$
$558$	0	0
$559$	1.38453e10i	0.141793i
$560$	0	0
$561$	−1.37087e11	−1.38403
$562$	0	0
$563$	− 4.36118e10i	− 0.434081i	−0.976163	−	0.217040i	$-0.930360\pi$
$563$	− 4.36118e10i	− 0.434081i	0.976163	−	0.217040i	$-0.0696403\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 1.34593e11i	− 1.30224i
$568$	0	0
$569$	1.27822e10	0.121943	0.0609716	−	0.998140i	$-0.480580\pi$
$569$	1.27822e10	0.121943	0.0609716	+	0.998140i	$0.480580\pi$
$570$	0	0
$571$	7.59455e10i	0.714427i	0.934023	+	0.357213i	$0.116273\pi$
$571$	7.59455e10i	0.714427i	−0.934023	+	0.357213i	$0.883727\pi$
$572$	0	0
$573$	1.53237e11	1.42149
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−2.13827e10	−0.192912	−0.0964560	−	0.995337i	$-0.530751\pi$
$577$	−2.13827e10	−0.192912	−0.0964560	+	0.995337i	$0.530751\pi$
$578$	0	0
$579$	4.21353e9i	0.0374914i
$580$	0	0
$581$	1.87715e11	1.64738
$582$	0	0
$583$	2.14035e11i	1.85272i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	6.07298e10i	0.511504i	0.966742	+	0.255752i	$0.0823231\pi$
$587$	6.07298e10i	0.511504i	−0.966742	+	0.255752i	$0.917677\pi$
$588$	0	0
$589$	−1.27697e11	−1.06101
$590$	0	0
$591$	1.59711e11i	1.30914i
$592$	0	0
$593$	−1.15978e11	−0.937899	−0.468949	−	0.883225i	$-0.655367\pi$
$593$	−1.15978e11	−0.937899	−0.468949	+	0.883225i	$0.655367\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	1.48685e11	1.17049
$598$	0	0
$599$	− 2.40647e11i	− 1.86927i	−0.355607	−	0.934636i	$-0.615726\pi$
$599$	− 2.40647e11i	− 1.86927i	0.355607	−	0.934636i	$-0.384274\pi$
$600$	0	0
$601$	−1.92942e11	−1.47887	−0.739434	−	0.673229i	$-0.764907\pi$
$601$	−1.92942e11	−1.47887	−0.739434	+	0.673229i	$0.764907\pi$
$602$	0	0
$603$	− 1.01921e11i	− 0.770892i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	1.62042e11i	1.19364i	0.802376	+	0.596819i	$0.203569\pi$
$607$	1.62042e11i	1.19364i	−0.802376	+	0.596819i	$0.796431\pi$
$608$	0	0
$609$	1.99504e10	0.145038
$610$	0	0
$611$	7.94597e10i	0.570141i
$612$	0	0
$613$	−1.76424e11	−1.24944	−0.624722	−	0.780847i	$-0.714788\pi$
$613$	−1.76424e11	−1.24944	−0.624722	+	0.780847i	$0.714788\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	9.84986e10	0.679656	0.339828	−	0.940488i	$-0.389631\pi$
$617$	9.84986e10	0.679656	0.339828	+	0.940488i	$0.389631\pi$
$618$	0	0
$619$	1.28596e10i	0.0875923i	0.999040	+	0.0437961i	$0.0139452\pi$
$619$	1.28596e10i	0.0875923i	−0.999040	+	0.0437961i	$0.986055\pi$
$620$	0	0
$621$	1.76223e11	1.18494
$622$	0	0
$623$	− 2.21828e11i	− 1.47253i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 2.95618e11i	− 1.91276i
$628$	0	0
$629$	3.99802e10	0.255413
$630$	0	0
$631$	1.73463e11i	1.09418i	0.837073	+	0.547091i	$0.184265\pi$
$631$	1.73463e11i	1.09418i	−0.837073	+	0.547091i	$0.815735\pi$
$632$	0	0
$633$	−3.94261e11	−2.45566
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−2.12547e10	−0.129091
$638$	0	0
$639$	− 1.37111e11i	− 0.822372i
$640$	0	0
$641$	−1.13903e11	−0.674690	−0.337345	−	0.941381i	$-0.609529\pi$
$641$	−1.13903e11	−0.674690	−0.337345	+	0.941381i	$0.609529\pi$
$642$	0	0
$643$	− 7.04067e10i	− 0.411879i	−0.978565	−	0.205940i	$-0.933975\pi$
$643$	− 7.04067e10i	− 0.411879i	0.978565	−	0.205940i	$-0.0660250\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	1.99175e11i	1.13663i	0.822812	+	0.568314i	$0.192404\pi$
$647$	1.99175e11i	1.13663i	−0.822812	+	0.568314i	$0.807596\pi$
$648$	0	0
$649$	4.18487e11	2.35886
$650$	0	0
$651$	− 4.26617e11i	− 2.37528i
$652$	0	0
$653$	−6.49972e9	−0.0357472	−0.0178736	−	0.999840i	$-0.505690\pi$
$653$	−6.49972e9	−0.0357472	−0.0178736	+	0.999840i	$0.505690\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	8.86194e10	0.475628
$658$	0	0
$659$	− 2.20982e11i	− 1.17170i	−0.810421	−	0.585848i	$-0.800761\pi$
$659$	− 2.20982e11i	− 1.17170i	0.810421	−	0.585848i	$-0.199239\pi$
$660$	0	0
$661$	−2.69549e11	−1.41199	−0.705995	−	0.708217i	$-0.749500\pi$
$661$	−2.69549e11	−1.41199	−0.705995	+	0.708217i	$0.749500\pi$
$662$	0	0
$663$	− 1.98177e11i	− 1.02565i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	9.69518e9i	0.0489838i
$668$	0	0
$669$	−3.23476e11	−1.61487
$670$	0	0
$671$	4.56711e11i	2.25295i
$672$	0	0
$673$	−9.44470e10	−0.460392	−0.230196	−	0.973144i	$-0.573937\pi$
$673$	−9.44470e10	−0.460392	−0.230196	+	0.973144i	$0.573937\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−8.02735e10	−0.382136	−0.191068	−	0.981577i	$-0.561195\pi$
$677$	−8.02735e10	−0.382136	−0.191068	+	0.981577i	$0.561195\pi$
$678$	0	0
$679$	− 1.19277e11i	− 0.561150i
$680$	0	0
$681$	2.11364e11	0.982751
$682$	0	0
$683$	− 3.00783e11i	− 1.38220i	−0.722760	−	0.691099i	$-0.757127\pi$
$683$	− 3.00783e11i	− 1.38220i	0.722760	−	0.691099i	$-0.242873\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	2.40591e11i	1.08007i
$688$	0	0
$689$	−3.09416e11	−1.37298
$690$	0	0
$691$	3.06208e11i	1.34309i	0.740964	+	0.671544i	$0.234369\pi$
$691$	3.06208e11i	1.34309i	−0.740964	+	0.671544i	$0.765631\pi$
$692$	0	0
$693$	6.66199e11	2.88849
$694$	0	0
$695$	0	0
$696$	0	0
$697$	3.80707e9	0.0161309
$698$	0	0
$699$	− 7.40348e10i	− 0.310118i
$700$	0	0
$701$	−2.73603e11	−1.13305	−0.566524	−	0.824045i	$-0.691712\pi$
$701$	−2.73603e11	−1.13305	−0.566524	+	0.824045i	$0.691712\pi$
$702$	0	0
$703$	8.62143e10i	0.352987i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 1.68656e11i	− 0.675033i
$708$	0	0
$709$	−1.76662e11	−0.699129	−0.349564	−	0.936912i	$-0.613670\pi$
$709$	−1.76662e11	−0.699129	−0.349564	+	0.936912i	$0.613670\pi$
$710$	0	0
$711$	− 6.63484e11i	− 2.59628i
$712$	0	0
$713$	2.07321e11	0.802203
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−6.25718e11	−2.36756
$718$	0	0
$719$	− 2.25510e11i	− 0.843821i	−0.906637	−	0.421911i	$-0.861360\pi$
$719$	− 2.25510e11i	− 0.843821i	0.906637	−	0.421911i	$-0.138640\pi$
$720$	0	0
$721$	4.21402e11	1.55939
$722$	0	0
$723$	− 2.30227e11i	− 0.842565i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 2.87080e11i	− 1.02770i	−0.857881	−	0.513849i	$-0.828219\pi$
$727$	− 2.87080e11i	− 1.02770i	0.857881	−	0.513849i	$-0.171781\pi$
$728$	0	0
$729$	2.14750e11	0.760366
$730$	0	0
$731$	− 2.51673e10i	− 0.0881388i
$732$	0	0
$733$	2.94176e11	1.01904	0.509520	−	0.860459i	$-0.329823\pi$
$733$	2.94176e11	1.01904	0.509520	+	0.860459i	$0.329823\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1.43660e11	0.486928
$738$	0	0
$739$	− 9.69888e10i	− 0.325195i	−0.986692	−	0.162598i	$-0.948013\pi$
$739$	− 9.69888e10i	− 0.325195i	0.986692	−	0.162598i	$-0.0519872\pi$
$740$	0	0
$741$	4.27355e11	1.41748
$742$	0	0
$743$	1.01567e11i	0.333271i	0.986019	+	0.166635i	$0.0532902\pi$
$743$	1.01567e11i	0.333271i	−0.986019	+	0.166635i	$0.946710\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 9.98824e11i	− 3.20779i
$748$	0	0
$749$	3.25413e11	1.03397
$750$	0	0
$751$	4.17899e11i	1.31375i	0.754001	+	0.656873i	$0.228121\pi$
$751$	4.17899e11i	1.31375i	−0.754001	+	0.656873i	$0.771879\pi$
$752$	0	0
$753$	−1.86457e10	−0.0579960
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−1.82006e11	−0.554244	−0.277122	−	0.960835i	$-0.589381\pi$
$757$	−1.82006e11	−0.554244	−0.277122	+	0.960835i	$0.589381\pi$
$758$	0	0
$759$	4.79945e11i	1.44619i
$760$	0	0
$761$	−4.27419e11	−1.27443	−0.637213	−	0.770687i	$-0.719913\pi$
$761$	−4.27419e11	−1.27443	−0.637213	+	0.770687i	$0.719913\pi$
$762$	0	0
$763$	3.99000e11i	1.17727i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	6.04978e11i	1.74807i
$768$	0	0
$769$	−5.09969e11	−1.45827	−0.729136	−	0.684368i	$-0.760078\pi$
$769$	−5.09969e11	−1.45827	−0.729136	+	0.684368i	$0.760078\pi$
$770$	0	0
$771$	− 7.49386e11i	− 2.12074i
$772$	0	0
$773$	1.49408e11	0.418462	0.209231	−	0.977866i	$-0.432904\pi$
$773$	1.49408e11	0.418462	0.209231	+	0.977866i	$0.432904\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−2.88029e11	−0.790227
$778$	0	0
$779$	8.20966e9i	0.0222933i
$780$	0	0
$781$	1.93261e11	0.519445
$782$	0	0
$783$	− 5.49393e10i	− 0.146163i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	7.33252e11i	1.91141i	0.294323	+	0.955706i	$0.404906\pi$
$787$	7.33252e11i	1.91141i	−0.294323	+	0.955706i	$0.595094\pi$
$788$	0	0
$789$	−1.04891e12	−2.70665
$790$	0	0
$791$	6.04040e11i	1.54298i
$792$	0	0
$793$	−6.60236e11	−1.66958
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−3.02703e11	−0.750212	−0.375106	−	0.926982i	$-0.622394\pi$
$797$	−3.02703e11	−0.750212	−0.375106	+	0.926982i	$0.622394\pi$
$798$	0	0
$799$	− 1.44438e11i	− 0.354401i
$800$	0	0
$801$	−1.18034e12	−2.86732
$802$	0	0
$803$	1.24911e11i	0.300427i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	4.91911e10i	0.115982i
$808$	0	0
$809$	−5.84316e11	−1.36412	−0.682062	−	0.731295i	$-0.738916\pi$
$809$	−5.84316e11	−1.36412	−0.682062	+	0.731295i	$0.738916\pi$
$810$	0	0
$811$	1.21470e11i	0.280793i	0.990095	+	0.140396i	$0.0448377\pi$
$811$	1.21470e11i	0.280793i	−0.990095	+	0.140396i	$0.955162\pi$
$812$	0	0
$813$	9.25344e8	0.00211807
$814$	0	0
$815$	0	0
$816$	0	0
$817$	5.42714e10	0.121810
$818$	0	0
$819$	9.63078e11i	2.14055i
$820$	0	0
$821$	4.52470e11	0.995903	0.497952	−	0.867205i	$-0.334086\pi$
$821$	4.52470e11	0.995903	0.497952	+	0.867205i	$0.334086\pi$
$822$	0	0
$823$	3.06704e11i	0.668528i	0.942479	+	0.334264i	$0.108488\pi$
$823$	3.06704e11i	0.668528i	−0.942479	+	0.334264i	$0.891512\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	2.93276e11i	0.626982i	0.949591	+	0.313491i	$0.101499\pi$
$827$	2.93276e11i	0.626982i	−0.949591	+	0.313491i	$0.898501\pi$
$828$	0	0
$829$	3.35532e11	0.710421	0.355210	−	0.934786i	$-0.384409\pi$
$829$	3.35532e11	0.710421	0.355210	+	0.934786i	$0.384409\pi$
$830$	0	0
$831$	− 3.05356e11i	− 0.640327i
$832$	0	0
$833$	3.86358e10	0.0802434
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−1.17482e12	−2.39369
$838$	0	0
$839$	3.42844e11i	0.691908i	0.938252	+	0.345954i	$0.112445\pi$
$839$	3.42844e11i	0.691908i	−0.938252	+	0.345954i	$0.887555\pi$
$840$	0	0
$841$	−4.97224e11	−0.993958
$842$	0	0
$843$	1.48029e12i	2.93113i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	3.91175e11i	0.760042i
$848$	0	0
$849$	−1.81802e11	−0.349919
$850$	0	0
$851$	− 1.39972e11i	− 0.266883i
$852$	0	0
$853$	5.08662e11	0.960801	0.480400	−	0.877049i	$-0.340491\pi$
$853$	5.08662e11	0.960801	0.480400	+	0.877049i	$0.340491\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−6.06764e11	−1.12486	−0.562428	−	0.826846i	$-0.690133\pi$
$857$	−6.06764e11	−1.12486	−0.562428	+	0.826846i	$0.690133\pi$
$858$	0	0
$859$	9.49431e11i	1.74378i	0.489705	+	0.871888i	$0.337105\pi$
$859$	9.49431e11i	1.74378i	−0.489705	+	0.871888i	$0.662895\pi$
$860$	0	0
$861$	−2.74272e10	−0.0499078
$862$	0	0
$863$	− 2.99836e10i	− 0.0540556i	−0.999635	−	0.0270278i	$-0.991396\pi$
$863$	− 2.99836e10i	− 0.0540556i	0.999635	−	0.0270278i	$-0.00860426\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 6.30222e11i	− 1.11536i
$868$	0	0
$869$	9.35196e11	1.63992
$870$	0	0
$871$	2.07679e11i	0.360844i
$872$	0	0
$873$	−6.34669e11	−1.09267
$874$	0	0
$875$	0	0
$876$	0	0
$877$	8.80195e11	1.48792	0.743961	−	0.668223i	$-0.232945\pi$
$877$	8.80195e11	1.48792	0.743961	+	0.668223i	$0.232945\pi$
$878$	0	0
$879$	3.03546e11i	0.508474i
$880$	0	0
$881$	1.04085e12	1.72776	0.863879	−	0.503699i	$-0.168028\pi$
$881$	1.04085e12	1.72776	0.863879	+	0.503699i	$0.168028\pi$
$882$	0	0
$883$	− 7.60446e11i	− 1.25091i	−0.780261	−	0.625454i	$-0.784914\pi$
$883$	− 7.60446e11i	− 1.25091i	0.780261	−	0.625454i	$-0.215086\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	3.34097e11i	0.539732i	0.962898	+	0.269866i	$0.0869794\pi$
$887$	3.34097e11i	0.539732i	−0.962898	+	0.269866i	$0.913021\pi$
$888$	0	0
$889$	−9.39853e11	−1.50471
$890$	0	0
$891$	− 1.00945e12i	− 1.60167i
$892$	0	0
$893$	3.11471e11	0.489792
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−6.93823e11	−1.07171
$898$	0	0
$899$	− 6.46345e10i	− 0.0989523i
$900$	0	0
$901$	5.62442e11	0.853451
$902$	0	0
$903$	1.81312e11i	0.272695i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	7.65213e11i	1.13071i	0.824846	+	0.565357i	$0.191262\pi$
$907$	7.65213e11i	1.13071i	−0.824846	+	0.565357i	$0.808738\pi$
$908$	0	0
$909$	−8.97412e11	−1.31443
$910$	0	0
$911$	3.83541e11i	0.556851i	0.960458	+	0.278425i	$0.0898125\pi$
$911$	3.83541e11i	0.556851i	−0.960458	+	0.278425i	$0.910188\pi$
$912$	0	0
$913$	1.40786e12	2.02618
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−5.52937e11	−0.781984
$918$	0	0
$919$	− 6.82775e11i	− 0.957229i	−0.878025	−	0.478615i	$-0.841139\pi$
$919$	− 6.82775e11i	− 0.957229i	0.878025	−	0.478615i	$-0.158861\pi$
$920$	0	0
$921$	−7.74022e11	−1.07576
$922$	0	0
$923$	2.79384e11i	0.384941i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 2.24226e12i	− 3.03645i
$928$	0	0
$929$	2.94973e11	0.396021	0.198011	−	0.980200i	$-0.436552\pi$
$929$	2.94973e11	0.396021	0.198011	+	0.980200i	$0.436552\pi$
$930$	0	0
$931$	8.33152e10i	0.110898i
$932$	0	0
$933$	1.52705e12	2.01524
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−1.03941e12	−1.34843	−0.674217	−	0.738533i	$-0.735519\pi$
$937$	−1.03941e12	−1.34843	−0.674217	+	0.738533i	$0.735519\pi$
$938$	0	0
$939$	4.25680e10i	0.0547546i
$940$	0	0
$941$	1.26891e11	0.161836	0.0809178	−	0.996721i	$-0.474215\pi$
$941$	1.26891e11	0.161836	0.0809178	+	0.996721i	$0.474215\pi$
$942$	0	0
$943$	− 1.33286e10i	− 0.0168554i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 6.13064e11i	− 0.762265i	−0.924520	−	0.381133i	$-0.875534\pi$
$947$	− 6.13064e11i	− 0.762265i	0.924520	−	0.381133i	$-0.124466\pi$
$948$	0	0
$949$	−1.80575e11	−0.222635
$950$	0	0
$951$	− 7.54189e11i	− 0.922058i
$952$	0	0
$953$	−6.58227e11	−0.798002	−0.399001	−	0.916951i	$-0.630643\pi$
$953$	−6.58227e11	−0.798002	−0.399001	+	0.916951i	$0.630643\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	1.49628e11	0.178388
$958$	0	0
$959$	− 9.87647e11i	− 1.16769i
$960$	0	0
$961$	−5.29246e11	−0.620532
$962$	0	0
$963$	− 1.73151e12i	− 2.01335i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 5.41485e11i	− 0.619271i	−0.950855	−	0.309635i	$-0.899793\pi$
$967$	− 5.41485e11i	− 0.619271i	0.950855	−	0.309635i	$-0.100207\pi$
$968$	0	0
$969$	−7.76826e11	−0.881107
$970$	0	0
$971$	3.54981e11i	0.399327i	0.979865	+	0.199663i	$0.0639849\pi$
$971$	3.54981e11i	0.399327i	−0.979865	+	0.199663i	$0.936015\pi$
$972$	0	0
$973$	−8.97263e11	−1.00108
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−6.02238e11	−0.660982	−0.330491	−	0.943809i	$-0.607214\pi$
$977$	−6.02238e11	−0.660982	−0.330491	+	0.943809i	$0.607214\pi$
$978$	0	0
$979$	− 1.66371e12i	− 1.81112i
$980$	0	0
$981$	2.12306e12	2.29238
$982$	0	0
$983$	1.37465e12i	1.47223i	0.676854	+	0.736117i	$0.263343\pi$
$983$	1.37465e12i	1.47223i	−0.676854	+	0.736117i	$0.736657\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	1.04058e12i	1.09649i
$988$	0	0
$989$	−8.81113e10	−0.0920972
$990$	0	0
$991$	− 1.01081e12i	− 1.04803i	−0.851709	−	0.524015i	$-0.824433\pi$
$991$	− 1.01081e12i	− 1.04803i	0.851709	−	0.524015i	$-0.175567\pi$
$992$	0	0
$993$	−1.43411e12	−1.47498
$994$	0	0
$995$	0	0
$996$	0	0
$997$	3.28556e11	0.332528	0.166264	−	0.986081i	$-0.446830\pi$
$997$	3.28556e11	0.332528	0.166264	+	0.986081i	$0.446830\pi$
$998$	0	0
$999$	7.93172e11i	0.796353i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	400.9.b.c.351.2		2
4.3	odd	2	inner	400.9.b.c.351.1		2
5.2	odd	4		400.9.h.b.399.3		4
5.3	odd	4		400.9.h.b.399.1		4
5.4	even	2		16.9.c.a.15.1	✓	2
15.14	odd	2		144.9.g.g.127.2		2
20.3	even	4		400.9.h.b.399.4		4
20.7	even	4		400.9.h.b.399.2		4
20.19	odd	2		16.9.c.a.15.2	yes	2
40.19	odd	2		64.9.c.d.63.1		2
40.29	even	2		64.9.c.d.63.2		2
60.59	even	2		144.9.g.g.127.1		2
80.19	odd	4		256.9.d.f.127.2		4
80.29	even	4		256.9.d.f.127.4		4
80.59	odd	4		256.9.d.f.127.3		4
80.69	even	4		256.9.d.f.127.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.9.c.a.15.1	✓	2	5.4	even	2
16.9.c.a.15.2	yes	2	20.19	odd	2
64.9.c.d.63.1		2	40.19	odd	2
64.9.c.d.63.2		2	40.29	even	2
144.9.g.g.127.1		2	60.59	even	2
144.9.g.g.127.2		2	15.14	odd	2
256.9.d.f.127.1		4	80.69	even	4
256.9.d.f.127.2		4	80.19	odd	4
256.9.d.f.127.3		4	80.59	odd	4
256.9.d.f.127.4		4	80.29	even	4
400.9.b.c.351.1		2	4.3	odd	2	inner
400.9.b.c.351.2		2	1.1	even	1	trivial
400.9.h.b.399.1		4	5.3	odd	4
400.9.h.b.399.2		4	20.7	even	4
400.9.h.b.399.3		4	5.2	odd	4
400.9.h.b.399.4		4	20.3	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 400 = 2^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	400.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+141.986i q^{3} -2555.75i q^{7} -13599.0 q^{9} -19168.1i q^{11} +27710.0 q^{13} -50370.0 q^{17} -108619. i q^{19} +362880. q^{21} +176347. i q^{23} -999297. i q^{27} +54978.0 q^{29} -1.17564e6i q^{31} +2.72160e6 q^{33} -793730. q^{37} +3.93443e6i q^{39} -75582.0 q^{41} +499648. i q^{43} +2.86755e6i q^{47} -767039. q^{49} -7.15183e6i q^{51} -1.11662e7 q^{53} +1.54224e7 q^{57} +2.18325e7i q^{59} -2.38266e7 q^{61} +3.47556e7i q^{63} +7.49473e6i q^{67} -2.50387e7 q^{69} +1.00824e7i q^{71} -6.51661e6 q^{73} -4.89888e7 q^{77} +4.87892e7i q^{79} +5.26630e7 q^{81} +7.34483e7i q^{83} +7.80610e6i q^{87} +8.67958e7 q^{89} -7.08197e7i q^{91} +1.66925e8 q^{93} +4.66703e7 q^{97} +2.60667e8i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 27198 q^{9} + 55420 q^{13} - 100740 q^{17} + 725760 q^{21} + 109956 q^{29} + 5443200 q^{33} - 1587460 q^{37} - 151164 q^{41} - 1534078 q^{49} - 22332420 q^{53} + 30844800 q^{57} - 47653244 q^{61}+ \cdots + 93340540 q^{97}+O(q^{100}) \) 2 * q - 27198 * q^9 + 55420 * q^13 - 100740 * q^17 + 725760 * q^21 + 109956 * q^29 + 5443200 * q^33 - 1587460 * q^37 - 151164 * q^41 - 1534078 * q^49 - 22332420 * q^53 + 30844800 * q^57 - 47653244 * q^61 - 50077440 * q^69 - 13033220 * q^73 - 97977600 * q^77 + 105326082 * q^81 + 173591556 * q^89 + 333849600 * q^93 + 93340540 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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