LMFDB - Embedded newform 400.6.c.f.49.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [400,6,Mod(49,400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("400.49");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 400 = 2^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	400.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$64.1535279252$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 4)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	400.49
Dual form			400.6.c.f.49.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-12.0000i q^{3} +88.0000i q^{7} +99.0000 q^{9} +O(q^{10})$	$q-12.0000i q^{3} +88.0000i q^{7} +99.0000 q^{9} -540.000 q^{11} +418.000i q^{13} +594.000i q^{17} +836.000 q^{19} +1056.00 q^{21} -4104.00i q^{23} -4104.00i q^{27} +594.000 q^{29} -4256.00 q^{31} +6480.00i q^{33} -298.000i q^{37} +5016.00 q^{39} +17226.0 q^{41} -12100.0i q^{43} +1296.00i q^{47} +9063.00 q^{49} +7128.00 q^{51} -19494.0i q^{53} -10032.0i q^{57} -7668.00 q^{59} -34738.0 q^{61} +8712.00i q^{63} -21812.0i q^{67} -49248.0 q^{69} +46872.0 q^{71} -67562.0i q^{73} -47520.0i q^{77} -76912.0 q^{79} -25191.0 q^{81} +67716.0i q^{83} -7128.00i q^{87} -29754.0 q^{89} -36784.0 q^{91} +51072.0i q^{93} -122398. i q^{97} -53460.0 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 198 q^{9}+O(q^{10}) $ 2 * q + 198 * q^9	$ 2 q + 198 q^{9} - 1080 q^{11} + 1672 q^{19} + 2112 q^{21} + 1188 q^{29} - 8512 q^{31} + 10032 q^{39} + 34452 q^{41} + 18126 q^{49} + 14256 q^{51} - 15336 q^{59} - 69476 q^{61} - 98496 q^{69} + 93744 q^{71} - 153824 q^{79} - 50382 q^{81} - 59508 q^{89} - 73568 q^{91} - 106920 q^{99}+O(q^{100}) $ 2 * q + 198 * q^9 - 1080 * q^11 + 1672 * q^19 + 2112 * q^21 + 1188 * q^29 - 8512 * q^31 + 10032 * q^39 + 34452 * q^41 + 18126 * q^49 + 14256 * q^51 - 15336 * q^59 - 69476 * q^61 - 98496 * q^69 + 93744 * q^71 - 153824 * q^79 - 50382 * q^81 - 59508 * q^89 - 73568 * q^91 - 106920 * q^99

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	− 12.0000i	− 0.769800i	−0.922958	−	0.384900i	$-0.874236\pi$
$3$	− 12.0000i	− 0.769800i	0.922958	−	0.384900i	$-0.125764\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	88.0000i	0.678793i	0.940643	+	0.339397i	$0.110223\pi$
$7$	88.0000i	0.678793i	−0.940643	+	0.339397i	$0.889777\pi$
$8$	0	0
$9$	99.0000	0.407407
$10$	0	0
$11$	−540.000	−1.34559	−0.672794	−	0.739830i	$-0.734906\pi$
$11$	−540.000	−1.34559	−0.672794	+	0.739830i	$0.734906\pi$
$12$	0	0
$13$	418.000i	0.685990i	0.939337	+	0.342995i	$0.111441\pi$
$13$	418.000i	0.685990i	−0.939337	+	0.342995i	$0.888559\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	594.000i	0.498499i	0.968439	+	0.249249i	$0.0801839\pi$
$17$	594.000i	0.498499i	−0.968439	+	0.249249i	$0.919816\pi$
$18$	0	0
$19$	836.000	0.531279	0.265639	−	0.964072i	$-0.414417\pi$
$19$	836.000	0.531279	0.265639	+	0.964072i	$0.414417\pi$
$20$	0	0
$21$	1056.00	0.522535
$22$	0	0
$23$	− 4104.00i	− 1.61766i	−0.588041	−	0.808831i	$-0.700101\pi$
$23$	− 4104.00i	− 1.61766i	0.588041	−	0.808831i	$-0.299899\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 4104.00i	− 1.08342i
$28$	0	0
$29$	594.000	0.131157	0.0655785	−	0.997847i	$-0.479111\pi$
$29$	594.000	0.131157	0.0655785	+	0.997847i	$0.479111\pi$
$30$	0	0
$31$	−4256.00	−0.795422	−0.397711	−	0.917511i	$-0.630195\pi$
$31$	−4256.00	−0.795422	−0.397711	+	0.917511i	$0.630195\pi$
$32$	0	0
$33$	6480.00i	1.03583i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 298.000i	− 0.0357859i	−0.999840	−	0.0178930i	$-0.994304\pi$
$37$	− 298.000i	− 0.0357859i	0.999840	−	0.0178930i	$-0.00569581\pi$
$38$	0	0
$39$	5016.00	0.528075
$40$	0	0
$41$	17226.0	1.60039	0.800193	−	0.599742i	$-0.204730\pi$
$41$	17226.0	1.60039	0.800193	+	0.599742i	$0.204730\pi$
$42$	0	0
$43$	− 12100.0i	− 0.997963i	−0.866613	−	0.498981i	$-0.833708\pi$
$43$	− 12100.0i	− 0.997963i	0.866613	−	0.498981i	$-0.166292\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	1296.00i	0.0855777i	0.999084	+	0.0427888i	$0.0136243\pi$
$47$	1296.00i	0.0855777i	−0.999084	+	0.0427888i	$0.986376\pi$
$48$	0	0
$49$	9063.00	0.539240
$50$	0	0
$51$	7128.00	0.383745
$52$	0	0
$53$	− 19494.0i	− 0.953260i	−0.879104	−	0.476630i	$-0.841858\pi$
$53$	− 19494.0i	− 0.953260i	0.879104	−	0.476630i	$-0.158142\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 10032.0i	− 0.408978i
$58$	0	0
$59$	−7668.00	−0.286782	−0.143391	−	0.989666i	$-0.545801\pi$
$59$	−7668.00	−0.286782	−0.143391	+	0.989666i	$0.545801\pi$
$60$	0	0
$61$	−34738.0	−1.19531	−0.597655	−	0.801754i	$-0.703901\pi$
$61$	−34738.0	−1.19531	−0.597655	+	0.801754i	$0.703901\pi$
$62$	0	0
$63$	8712.00i	0.276545i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 21812.0i	− 0.593620i	−0.954937	−	0.296810i	$-0.904077\pi$
$67$	− 21812.0i	− 0.593620i	0.954937	−	0.296810i	$-0.0959228\pi$
$68$	0	0
$69$	−49248.0	−1.24528
$70$	0	0
$71$	46872.0	1.10349	0.551744	−	0.834014i	$-0.313963\pi$
$71$	46872.0	1.10349	0.551744	+	0.834014i	$0.313963\pi$
$72$	0	0
$73$	− 67562.0i	− 1.48387i	−0.670473	−	0.741934i	$-0.733909\pi$
$73$	− 67562.0i	− 1.48387i	0.670473	−	0.741934i	$-0.266091\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 47520.0i	− 0.913376i
$78$	0	0
$79$	−76912.0	−1.38652	−0.693260	−	0.720687i	$-0.743826\pi$
$79$	−76912.0	−1.38652	−0.693260	+	0.720687i	$0.743826\pi$
$80$	0	0
$81$	−25191.0	−0.426612
$82$	0	0
$83$	67716.0i	1.07894i	0.842006	+	0.539468i	$0.181375\pi$
$83$	67716.0i	1.07894i	−0.842006	+	0.539468i	$0.818625\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 7128.00i	− 0.100965i
$88$	0	0
$89$	−29754.0	−0.398172	−0.199086	−	0.979982i	$-0.563797\pi$
$89$	−29754.0	−0.398172	−0.199086	+	0.979982i	$0.563797\pi$
$90$	0	0
$91$	−36784.0	−0.465646
$92$	0	0
$93$	51072.0i	0.612316i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 122398.i	− 1.32082i	−0.750903	−	0.660412i	$-0.770382\pi$
$97$	− 122398.i	− 1.32082i	0.750903	−	0.660412i	$-0.229618\pi$
$98$	0	0
$99$	−53460.0	−0.548202
$100$	0	0
$101$	11286.0	0.110087	0.0550436	−	0.998484i	$-0.482470\pi$
$101$	11286.0	0.110087	0.0550436	+	0.998484i	$0.482470\pi$
$102$	0	0
$103$	− 27256.0i	− 0.253145i	−0.991957	−	0.126572i	$-0.959602\pi$
$103$	− 27256.0i	− 0.253145i	0.991957	−	0.126572i	$-0.0403976\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 122364.i	− 1.03322i	−0.856220	−	0.516612i	$-0.827193\pi$
$107$	− 122364.i	− 1.03322i	0.856220	−	0.516612i	$-0.172807\pi$
$108$	0	0
$109$	−99902.0	−0.805393	−0.402697	−	0.915334i	$-0.631927\pi$
$109$	−99902.0	−0.805393	−0.402697	+	0.915334i	$0.631927\pi$
$110$	0	0
$111$	−3576.00	−0.0275480
$112$	0	0
$113$	29646.0i	0.218409i	0.994019	+	0.109204i	$0.0348303\pi$
$113$	29646.0i	0.218409i	−0.994019	+	0.109204i	$0.965170\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	41382.0i	0.279477i
$118$	0	0
$119$	−52272.0	−0.338378
$120$	0	0
$121$	130549.	0.810607
$122$	0	0
$123$	− 206712.i	− 1.23198i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 336512.i	− 1.85136i	−0.378305	−	0.925681i	$-0.623493\pi$
$127$	− 336512.i	− 1.85136i	0.378305	−	0.925681i	$-0.376507\pi$
$128$	0	0
$129$	−145200.	−0.768232
$130$	0	0
$131$	−100980.	−0.514111	−0.257056	−	0.966397i	$-0.582752\pi$
$131$	−100980.	−0.514111	−0.257056	+	0.966397i	$0.582752\pi$
$132$	0	0
$133$	73568.0i	0.360628i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 317142.i	− 1.44362i	−0.692092	−	0.721809i	$-0.743311\pi$
$137$	− 317142.i	− 1.44362i	0.692092	−	0.721809i	$-0.256689\pi$
$138$	0	0
$139$	−148324.	−0.651140	−0.325570	−	0.945518i	$-0.605556\pi$
$139$	−148324.	−0.651140	−0.325570	+	0.945518i	$0.605556\pi$
$140$	0	0
$141$	15552.0	0.0658777
$142$	0	0
$143$	− 225720.i	− 0.923060i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 108756.i	− 0.415107i
$148$	0	0
$149$	−196614.	−0.725519	−0.362759	−	0.931883i	$-0.618165\pi$
$149$	−196614.	−0.725519	−0.362759	+	0.931883i	$0.618165\pi$
$150$	0	0
$151$	−74360.0	−0.265398	−0.132699	−	0.991156i	$-0.542364\pi$
$151$	−74360.0	−0.265398	−0.132699	+	0.991156i	$0.542364\pi$
$152$	0	0
$153$	58806.0i	0.203092i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	120878.i	0.391380i	0.980666	+	0.195690i	$0.0626946\pi$
$157$	120878.i	0.391380i	−0.980666	+	0.195690i	$0.937305\pi$
$158$	0	0
$159$	−233928.	−0.733820
$160$	0	0
$161$	361152.	1.09806
$162$	0	0
$163$	− 111340.i	− 0.328233i	−0.986441	−	0.164116i	$-0.947523\pi$
$163$	− 111340.i	− 0.328233i	0.986441	−	0.164116i	$-0.0524773\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	491832.i	1.36466i	0.731043	+	0.682332i	$0.239034\pi$
$167$	491832.i	1.36466i	−0.731043	+	0.682332i	$0.760966\pi$
$168$	0	0
$169$	196569.	0.529417
$170$	0	0
$171$	82764.0	0.216447
$172$	0	0
$173$	− 707454.i	− 1.79714i	−0.438826	−	0.898572i	$-0.644605\pi$
$173$	− 707454.i	− 1.79714i	0.438826	−	0.898572i	$-0.355395\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	92016.0i	0.220765i
$178$	0	0
$179$	493668.	1.15160	0.575801	−	0.817590i	$-0.304690\pi$
$179$	493668.	1.15160	0.575801	+	0.817590i	$0.304690\pi$
$180$	0	0
$181$	−559450.	−1.26930	−0.634651	−	0.772799i	$-0.718856\pi$
$181$	−559450.	−1.26930	−0.634651	+	0.772799i	$0.718856\pi$
$182$	0	0
$183$	416856.i	0.920149i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 320760.i	− 0.670774i
$188$	0	0
$189$	361152.	0.735420
$190$	0	0
$191$	724032.	1.43607	0.718033	−	0.696009i	$-0.245043\pi$
$191$	724032.	1.43607	0.718033	+	0.696009i	$0.245043\pi$
$192$	0	0
$193$	− 7106.00i	− 0.0137319i	−0.999976	−	0.00686597i	$-0.997814\pi$
$193$	− 7106.00i	− 0.0137319i	0.999976	−	0.00686597i	$-0.00218552\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 530442.i	− 0.973806i	−0.873456	−	0.486903i	$-0.838127\pi$
$197$	− 530442.i	− 0.973806i	0.873456	−	0.486903i	$-0.161873\pi$
$198$	0	0
$199$	56168.0	0.100544	0.0502720	−	0.998736i	$-0.483991\pi$
$199$	56168.0	0.100544	0.0502720	+	0.998736i	$0.483991\pi$
$200$	0	0
$201$	−261744.	−0.456969
$202$	0	0
$203$	52272.0i	0.0890285i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 406296.i	− 0.659047i
$208$	0	0
$209$	−451440.	−0.714882
$210$	0	0
$211$	339196.	0.524499	0.262249	−	0.965000i	$-0.415536\pi$
$211$	339196.	0.524499	0.262249	+	0.965000i	$0.415536\pi$
$212$	0	0
$213$	− 562464.i	− 0.849465i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 374528.i	− 0.539927i
$218$	0	0
$219$	−810744.	−1.14228
$220$	0	0
$221$	−248292.	−0.341965
$222$	0	0
$223$	779360.i	1.04948i	0.851261	+	0.524742i	$0.175838\pi$
$223$	779360.i	1.04948i	−0.851261	+	0.524742i	$0.824162\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	744876.i	0.959443i	0.877421	+	0.479722i	$0.159262\pi$
$227$	744876.i	0.959443i	−0.877421	+	0.479722i	$0.840738\pi$
$228$	0	0
$229$	272746.	0.343692	0.171846	−	0.985124i	$-0.445027\pi$
$229$	272746.	0.343692	0.171846	+	0.985124i	$0.445027\pi$
$230$	0	0
$231$	−570240.	−0.703117
$232$	0	0
$233$	153846.i	0.185651i	0.995682	+	0.0928253i	$0.0295898\pi$
$233$	153846.i	0.185651i	−0.995682	+	0.0928253i	$0.970410\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	922944.i	1.06734i
$238$	0	0
$239$	1.15474e6	1.30764	0.653820	−	0.756650i	$-0.273166\pi$
$239$	1.15474e6	1.30764	0.653820	+	0.756650i	$0.273166\pi$
$240$	0	0
$241$	657074.	0.728738	0.364369	−	0.931255i	$-0.381285\pi$
$241$	657074.	0.728738	0.364369	+	0.931255i	$0.381285\pi$
$242$	0	0
$243$	− 694980.i	− 0.755017i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	349448.i	0.364452i
$248$	0	0
$249$	812592.	0.830566
$250$	0	0
$251$	−1.34190e6	−1.34442	−0.672211	−	0.740359i	$-0.734655\pi$
$251$	−1.34190e6	−1.34442	−0.672211	+	0.740359i	$0.734655\pi$
$252$	0	0
$253$	2.21616e6i	2.17671i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	132354.i	0.124998i	0.998045	+	0.0624992i	$0.0199071\pi$
$257$	132354.i	0.124998i	−0.998045	+	0.0624992i	$0.980093\pi$
$258$	0	0
$259$	26224.0	0.0242912
$260$	0	0
$261$	58806.0	0.0534343
$262$	0	0
$263$	943272.i	0.840906i	0.907314	+	0.420453i	$0.138129\pi$
$263$	943272.i	0.840906i	−0.907314	+	0.420453i	$0.861871\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	357048.i	0.306513i
$268$	0	0
$269$	−967518.	−0.815227	−0.407613	−	0.913155i	$-0.633639\pi$
$269$	−967518.	−0.815227	−0.407613	+	0.913155i	$0.633639\pi$
$270$	0	0
$271$	518320.	0.428721	0.214360	−	0.976755i	$-0.431233\pi$
$271$	518320.	0.428721	0.214360	+	0.976755i	$0.431233\pi$
$272$	0	0
$273$	441408.i	0.358454i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	2.22273e6i	1.74055i	0.492566	+	0.870275i	$0.336059\pi$
$277$	2.22273e6i	1.74055i	−0.492566	+	0.870275i	$0.663941\pi$
$278$	0	0
$279$	−421344.	−0.324061
$280$	0	0
$281$	−196614.	−0.148542	−0.0742709	−	0.997238i	$-0.523663\pi$
$281$	−196614.	−0.148542	−0.0742709	+	0.997238i	$0.523663\pi$
$282$	0	0
$283$	− 1.55228e6i	− 1.15213i	−0.817403	−	0.576067i	$-0.804587\pi$
$283$	− 1.55228e6i	− 1.15213i	0.817403	−	0.576067i	$-0.195413\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1.51589e6i	1.08633i
$288$	0	0
$289$	1.06702e6	0.751499
$290$	0	0
$291$	−1.46878e6	−1.01677
$292$	0	0
$293$	1.07217e6i	0.729616i	0.931083	+	0.364808i	$0.118865\pi$
$293$	1.07217e6i	0.729616i	−0.931083	+	0.364808i	$0.881135\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	2.21616e6i	1.45784i
$298$	0	0
$299$	1.71547e6	1.10970
$300$	0	0
$301$	1.06480e6	0.677410
$302$	0	0
$303$	− 135432.i	− 0.0847451i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 1.58589e6i	− 0.960346i	−0.877174	−	0.480173i	$-0.840574\pi$
$307$	− 1.58589e6i	− 0.960346i	0.877174	−	0.480173i	$-0.159426\pi$
$308$	0	0
$309$	−327072.	−0.194871
$310$	0	0
$311$	730728.	0.428405	0.214203	−	0.976789i	$-0.431285\pi$
$311$	730728.	0.428405	0.214203	+	0.976789i	$0.431285\pi$
$312$	0	0
$313$	− 584858.i	− 0.337435i	−0.985664	−	0.168717i	$-0.946038\pi$
$313$	− 584858.i	− 0.337435i	0.985664	−	0.168717i	$-0.0539625\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 2.48287e6i	− 1.38773i	−0.720105	−	0.693865i	$-0.755906\pi$
$317$	− 2.48287e6i	− 1.38773i	0.720105	−	0.693865i	$-0.244094\pi$
$318$	0	0
$319$	−320760.	−0.176483
$320$	0	0
$321$	−1.46837e6	−0.795376
$322$	0	0
$323$	496584.i	0.264842i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	1.19882e6i	0.619992i
$328$	0	0
$329$	−114048.	−0.0580895
$330$	0	0
$331$	−377948.	−0.189610	−0.0948052	−	0.995496i	$-0.530223\pi$
$331$	−377948.	−0.189610	−0.0948052	+	0.995496i	$0.530223\pi$
$332$	0	0
$333$	− 29502.0i	− 0.0145794i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	639122.i	0.306555i	0.988183	+	0.153278i	$0.0489829\pi$
$337$	639122.i	0.306555i	−0.988183	+	0.153278i	$0.951017\pi$
$338$	0	0
$339$	355752.	0.168131
$340$	0	0
$341$	2.29824e6	1.07031
$342$	0	0
$343$	2.27656e6i	1.04483i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	2.90466e6i	1.29501i	0.762063	+	0.647503i	$0.224187\pi$
$347$	2.90466e6i	1.29501i	−0.762063	+	0.647503i	$0.775813\pi$
$348$	0	0
$349$	3.99157e6	1.75420	0.877102	−	0.480304i	$-0.159474\pi$
$349$	3.99157e6	1.75420	0.877102	+	0.480304i	$0.159474\pi$
$350$	0	0
$351$	1.71547e6	0.743217
$352$	0	0
$353$	− 1.42922e6i	− 0.610466i	−0.952278	−	0.305233i	$-0.901266\pi$
$353$	− 1.42922e6i	− 0.610466i	0.952278	−	0.305233i	$-0.0987344\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	627264.i	0.260483i
$358$	0	0
$359$	1.16186e6	0.475794	0.237897	−	0.971290i	$-0.423542\pi$
$359$	1.16186e6	0.475794	0.237897	+	0.971290i	$0.423542\pi$
$360$	0	0
$361$	−1.77720e6	−0.717743
$362$	0	0
$363$	− 1.56659e6i	− 0.624005i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	1.08923e6i	0.422139i	0.977471	+	0.211069i	$0.0676946\pi$
$367$	1.08923e6i	0.422139i	−0.977471	+	0.211069i	$0.932305\pi$
$368$	0	0
$369$	1.70537e6	0.652009
$370$	0	0
$371$	1.71547e6	0.647066
$372$	0	0
$373$	− 3.50577e6i	− 1.30470i	−0.757918	−	0.652350i	$-0.773783\pi$
$373$	− 3.50577e6i	− 1.30470i	0.757918	−	0.652350i	$-0.226217\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	248292.i	0.0899724i
$378$	0	0
$379$	4.04385e6	1.44610	0.723048	−	0.690798i	$-0.242740\pi$
$379$	4.04385e6	1.44610	0.723048	+	0.690798i	$0.242740\pi$
$380$	0	0
$381$	−4.03814e6	−1.42518
$382$	0	0
$383$	5.18746e6i	1.80700i	0.428591	+	0.903499i	$0.359010\pi$
$383$	5.18746e6i	1.80700i	−0.428591	+	0.903499i	$0.640990\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 1.19790e6i	− 0.406577i
$388$	0	0
$389$	950346.	0.318425	0.159213	−	0.987244i	$-0.449104\pi$
$389$	950346.	0.318425	0.159213	+	0.987244i	$0.449104\pi$
$390$	0	0
$391$	2.43778e6	0.806403
$392$	0	0
$393$	1.21176e6i	0.395763i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 520738.i	− 0.165822i	−0.996557	−	0.0829112i	$-0.973578\pi$
$397$	− 520738.i	− 0.165822i	0.996557	−	0.0829112i	$-0.0264218\pi$
$398$	0	0
$399$	882816.	0.277612
$400$	0	0
$401$	764370.	0.237379	0.118690	−	0.992931i	$-0.462131\pi$
$401$	764370.	0.237379	0.118690	+	0.992931i	$0.462131\pi$
$402$	0	0
$403$	− 1.77901e6i	− 0.545651i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	160920.i	0.0481531i
$408$	0	0
$409$	−2.64051e6	−0.780511	−0.390255	−	0.920707i	$-0.627613\pi$
$409$	−2.64051e6	−0.780511	−0.390255	+	0.920707i	$0.627613\pi$
$410$	0	0
$411$	−3.80570e6	−1.11130
$412$	0	0
$413$	− 674784.i	− 0.194666i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	1.77989e6i	0.501248i
$418$	0	0
$419$	−4.98020e6	−1.38584	−0.692918	−	0.721016i	$-0.743675\pi$
$419$	−4.98020e6	−1.38584	−0.692918	+	0.721016i	$0.743675\pi$
$420$	0	0
$421$	−237994.	−0.0654426	−0.0327213	−	0.999465i	$-0.510417\pi$
$421$	−237994.	−0.0654426	−0.0327213	+	0.999465i	$0.510417\pi$
$422$	0	0
$423$	128304.i	0.0348650i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 3.05694e6i	− 0.811368i
$428$	0	0
$429$	−2.70864e6	−0.710572
$430$	0	0
$431$	3.88238e6	1.00671	0.503356	−	0.864079i	$-0.332098\pi$
$431$	3.88238e6	1.00671	0.503356	+	0.864079i	$0.332098\pi$
$432$	0	0
$433$	66958.0i	0.0171626i	0.999963	+	0.00858129i	$0.00273154\pi$
$433$	66958.0i	0.0171626i	−0.999963	+	0.00858129i	$0.997268\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 3.43094e6i	− 0.859429i
$438$	0	0
$439$	−6.50135e6	−1.61006	−0.805031	−	0.593233i	$-0.797851\pi$
$439$	−6.50135e6	−1.61006	−0.805031	+	0.593233i	$0.797851\pi$
$440$	0	0
$441$	897237.	0.219690
$442$	0	0
$443$	− 4.60760e6i	− 1.11549i	−0.830012	−	0.557745i	$-0.811667\pi$
$443$	− 4.60760e6i	− 1.11549i	0.830012	−	0.557745i	$-0.188333\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	2.35937e6i	0.558505i
$448$	0	0
$449$	−3.77671e6	−0.884092	−0.442046	−	0.896992i	$-0.645747\pi$
$449$	−3.77671e6	−0.884092	−0.442046	+	0.896992i	$0.645747\pi$
$450$	0	0
$451$	−9.30204e6	−2.15346
$452$	0	0
$453$	892320.i	0.204303i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 3.18069e6i	− 0.712412i	−0.934407	−	0.356206i	$-0.884070\pi$
$457$	− 3.18069e6i	− 0.712412i	0.934407	−	0.356206i	$-0.115930\pi$
$458$	0	0
$459$	2.43778e6	0.540085
$460$	0	0
$461$	6.68547e6	1.46514	0.732571	−	0.680691i	$-0.238320\pi$
$461$	6.68547e6	1.46514	0.732571	+	0.680691i	$0.238320\pi$
$462$	0	0
$463$	− 4.35122e6i	− 0.943318i	−0.881781	−	0.471659i	$-0.843655\pi$
$463$	− 4.35122e6i	− 0.943318i	0.881781	−	0.471659i	$-0.156345\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 7.07994e6i	− 1.50223i	−0.660170	−	0.751117i	$-0.729516\pi$
$467$	− 7.07994e6i	− 1.50223i	0.660170	−	0.751117i	$-0.270484\pi$
$468$	0	0
$469$	1.91946e6	0.402945
$470$	0	0
$471$	1.45054e6	0.301284
$472$	0	0
$473$	6.53400e6i	1.34285i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 1.92991e6i	− 0.388365i
$478$	0	0
$479$	3.22186e6	0.641604	0.320802	−	0.947146i	$-0.396048\pi$
$479$	3.22186e6	0.641604	0.320802	+	0.947146i	$0.396048\pi$
$480$	0	0
$481$	124564.	0.0245488
$482$	0	0
$483$	− 4.33382e6i	− 0.845286i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 2.29710e6i	− 0.438891i	−0.975625	−	0.219446i	$-0.929575\pi$
$487$	− 2.29710e6i	− 0.438891i	0.975625	−	0.219446i	$-0.0704248\pi$
$488$	0	0
$489$	−1.33608e6	−0.252674
$490$	0	0
$491$	−2.82150e6	−0.528173	−0.264087	−	0.964499i	$-0.585070\pi$
$491$	−2.82150e6	−0.528173	−0.264087	+	0.964499i	$0.585070\pi$
$492$	0	0
$493$	352836.i	0.0653816i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	4.12474e6i	0.749040i
$498$	0	0
$499$	−4.13628e6	−0.743634	−0.371817	−	0.928306i	$-0.621265\pi$
$499$	−4.13628e6	−0.743634	−0.371817	+	0.928306i	$0.621265\pi$
$500$	0	0
$501$	5.90198e6	1.05052
$502$	0	0
$503$	8.33263e6i	1.46846i	0.678901	+	0.734230i	$0.262457\pi$
$503$	8.33263e6i	1.46846i	−0.678901	+	0.734230i	$0.737543\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 2.35883e6i	− 0.407546i
$508$	0	0
$509$	−4.34101e6	−0.742670	−0.371335	−	0.928499i	$-0.621100\pi$
$509$	−4.34101e6	−0.742670	−0.371335	+	0.928499i	$0.621100\pi$
$510$	0	0
$511$	5.94546e6	1.00724
$512$	0	0
$513$	− 3.43094e6i	− 0.575599i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 699840.i	− 0.115152i
$518$	0	0
$519$	−8.48945e6	−1.38344
$520$	0	0
$521$	−6.74185e6	−1.08814	−0.544070	−	0.839040i	$-0.683117\pi$
$521$	−6.74185e6	−1.08814	−0.544070	+	0.839040i	$0.683117\pi$
$522$	0	0
$523$	− 7.72196e6i	− 1.23445i	−0.786787	−	0.617224i	$-0.788257\pi$
$523$	− 7.72196e6i	− 1.23445i	0.786787	−	0.617224i	$-0.211743\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 2.52806e6i	− 0.396517i
$528$	0	0
$529$	−1.04065e7	−1.61683
$530$	0	0
$531$	−759132.	−0.116837
$532$	0	0
$533$	7.20047e6i	1.09785i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 5.92402e6i	− 0.886504i
$538$	0	0
$539$	−4.89402e6	−0.725594
$540$	0	0
$541$	−682066.	−0.100192	−0.0500960	−	0.998744i	$-0.515953\pi$
$541$	−682066.	−0.100192	−0.0500960	+	0.998744i	$0.515953\pi$
$542$	0	0
$543$	6.71340e6i	0.977109i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 2.15772e6i	− 0.308337i	−0.988045	−	0.154169i	$-0.950730\pi$
$547$	− 2.15772e6i	− 0.308337i	0.988045	−	0.154169i	$-0.0492699\pi$
$548$	0	0
$549$	−3.43906e6	−0.486978
$550$	0	0
$551$	496584.	0.0696809
$552$	0	0
$553$	− 6.76826e6i	− 0.941161i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 2.67597e6i	− 0.365463i	−0.983163	−	0.182731i	$-0.941506\pi$
$557$	− 2.67597e6i	− 0.365463i	0.983163	−	0.182731i	$-0.0584939\pi$
$558$	0	0
$559$	5.05780e6	0.684592
$560$	0	0
$561$	−3.84912e6	−0.516362
$562$	0	0
$563$	− 3.55331e6i	− 0.472457i	−0.971698	−	0.236228i	$-0.924089\pi$
$563$	− 3.55331e6i	− 0.472457i	0.971698	−	0.236228i	$-0.0759113\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 2.21681e6i	− 0.289581i
$568$	0	0
$569$	1.29225e7	1.67327	0.836633	−	0.547764i	$-0.184521\pi$
$569$	1.29225e7	1.67327	0.836633	+	0.547764i	$0.184521\pi$
$570$	0	0
$571$	6.08357e6	0.780851	0.390426	−	0.920634i	$-0.372328\pi$
$571$	6.08357e6	0.780851	0.390426	+	0.920634i	$0.372328\pi$
$572$	0	0
$573$	− 8.68838e6i	− 1.10548i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 1.58241e7i	− 1.97869i	−0.145579	−	0.989347i	$-0.546505\pi$
$577$	− 1.58241e7i	− 1.97869i	0.145579	−	0.989347i	$-0.453495\pi$
$578$	0	0
$579$	−85272.0	−0.0105709
$580$	0	0
$581$	−5.95901e6	−0.732375
$582$	0	0
$583$	1.05268e7i	1.28269i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 4.60220e6i	− 0.551278i	−0.961261	−	0.275639i	$-0.911111\pi$
$587$	− 4.60220e6i	− 0.551278i	0.961261	−	0.275639i	$-0.0888894\pi$
$588$	0	0
$589$	−3.55802e6	−0.422590
$590$	0	0
$591$	−6.36530e6	−0.749636
$592$	0	0
$593$	− 8.61122e6i	− 1.00561i	−0.864401	−	0.502803i	$-0.832302\pi$
$593$	− 8.61122e6i	− 1.00561i	0.864401	−	0.502803i	$-0.167698\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 674016.i	− 0.0773988i
$598$	0	0
$599$	−7.98228e6	−0.908992	−0.454496	−	0.890749i	$-0.650181\pi$
$599$	−7.98228e6	−0.908992	−0.454496	+	0.890749i	$0.650181\pi$
$600$	0	0
$601$	1.01740e7	1.14896	0.574481	−	0.818518i	$-0.305204\pi$
$601$	1.01740e7	1.14896	0.574481	+	0.818518i	$0.305204\pi$
$602$	0	0
$603$	− 2.15939e6i	− 0.241845i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	9.95843e6i	1.09703i	0.836140	+	0.548516i	$0.184807\pi$
$607$	9.95843e6i	1.09703i	−0.836140	+	0.548516i	$0.815193\pi$
$608$	0	0
$609$	627264.	0.0685342
$610$	0	0
$611$	−541728.	−0.0587054
$612$	0	0
$613$	− 4.19586e6i	− 0.450993i	−0.974244	−	0.225497i	$-0.927600\pi$
$613$	− 4.19586e6i	− 0.450993i	0.974244	−	0.225497i	$-0.0724005\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	9.12551e6i	0.965038i	0.875885	+	0.482519i	$0.160278\pi$
$617$	9.12551e6i	0.965038i	−0.875885	+	0.482519i	$0.839722\pi$
$618$	0	0
$619$	6.45734e6	0.677372	0.338686	−	0.940900i	$-0.390018\pi$
$619$	6.45734e6	0.677372	0.338686	+	0.940900i	$0.390018\pi$
$620$	0	0
$621$	−1.68428e7	−1.75261
$622$	0	0
$623$	− 2.61835e6i	− 0.270276i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	5.41728e6i	0.550316i
$628$	0	0
$629$	177012.	0.0178392
$630$	0	0
$631$	1.40514e7	1.40490	0.702450	−	0.711733i	$-0.252090\pi$
$631$	1.40514e7	1.40490	0.702450	+	0.711733i	$0.252090\pi$
$632$	0	0
$633$	− 4.07035e6i	− 0.403759i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	3.78833e6i	0.369913i
$638$	0	0
$639$	4.64033e6	0.449569
$640$	0	0
$641$	8.47168e6	0.814375	0.407188	−	0.913345i	$-0.366510\pi$
$641$	8.47168e6	0.814375	0.407188	+	0.913345i	$0.366510\pi$
$642$	0	0
$643$	488564.i	0.0466009i	0.999729	+	0.0233004i	$0.00741743\pi$
$643$	488564.i	0.0466009i	−0.999729	+	0.0233004i	$0.992583\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 2.48119e6i	− 0.233023i	−0.993189	−	0.116512i	$-0.962829\pi$
$647$	− 2.48119e6i	− 0.233023i	0.993189	−	0.116512i	$-0.0371713\pi$
$648$	0	0
$649$	4.14072e6	0.385891
$650$	0	0
$651$	−4.49434e6	−0.415636
$652$	0	0
$653$	5.29130e6i	0.485601i	0.970076	+	0.242800i	$0.0780660\pi$
$653$	5.29130e6i	0.485601i	−0.970076	+	0.242800i	$0.921934\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 6.68864e6i	− 0.604539i
$658$	0	0
$659$	4.72468e6	0.423798	0.211899	−	0.977292i	$-0.432035\pi$
$659$	4.72468e6	0.423798	0.211899	+	0.977292i	$0.432035\pi$
$660$	0	0
$661$	−6.17420e6	−0.549639	−0.274819	−	0.961496i	$-0.588618\pi$
$661$	−6.17420e6	−0.549639	−0.274819	+	0.961496i	$0.588618\pi$
$662$	0	0
$663$	2.97950e6i	0.263245i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 2.43778e6i	− 0.212168i
$668$	0	0
$669$	9.35232e6	0.807893
$670$	0	0
$671$	1.87585e7	1.60839
$672$	0	0
$673$	9.40925e6i	0.800787i	0.916343	+	0.400394i	$0.131127\pi$
$673$	9.40925e6i	0.800787i	−0.916343	+	0.400394i	$0.868873\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	1.50086e7i	1.25854i	0.777185	+	0.629272i	$0.216647\pi$
$677$	1.50086e7i	1.25854i	−0.777185	+	0.629272i	$0.783353\pi$
$678$	0	0
$679$	1.07710e7	0.896567
$680$	0	0
$681$	8.93851e6	0.738580
$682$	0	0
$683$	− 1.29707e7i	− 1.06393i	−0.846768	−	0.531963i	$-0.821455\pi$
$683$	− 1.29707e7i	− 1.06393i	0.846768	−	0.531963i	$-0.178545\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 3.27295e6i	− 0.264574i
$688$	0	0
$689$	8.14849e6	0.653927
$690$	0	0
$691$	−2.26556e7	−1.80501	−0.902506	−	0.430677i	$-0.858275\pi$
$691$	−2.26556e7	−1.80501	−0.902506	+	0.430677i	$0.858275\pi$
$692$	0	0
$693$	− 4.70448e6i	− 0.372116i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	1.02322e7i	0.797791i
$698$	0	0
$699$	1.84615e6	0.142914
$700$	0	0
$701$	1.90169e7	1.46166	0.730828	−	0.682562i	$-0.239134\pi$
$701$	1.90169e7	1.46166	0.730828	+	0.682562i	$0.239134\pi$
$702$	0	0
$703$	− 249128.i	− 0.0190123i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	993168.i	0.0747264i
$708$	0	0
$709$	−1.51311e7	−1.13046	−0.565231	−	0.824933i	$-0.691213\pi$
$709$	−1.51311e7	−1.13046	−0.565231	+	0.824933i	$0.691213\pi$
$710$	0	0
$711$	−7.61429e6	−0.564879
$712$	0	0
$713$	1.74666e7i	1.28672i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 1.38568e7i	− 1.00662i
$718$	0	0
$719$	−1.50323e7	−1.08443	−0.542217	−	0.840238i	$-0.682415\pi$
$719$	−1.50323e7	−1.08443	−0.542217	+	0.840238i	$0.682415\pi$
$720$	0	0
$721$	2.39853e6	0.171833
$722$	0	0
$723$	− 7.88489e6i	− 0.560983i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	7.41230e6i	0.520136i	0.965590	+	0.260068i	$0.0837449\pi$
$727$	7.41230e6i	0.520136i	−0.965590	+	0.260068i	$0.916255\pi$
$728$	0	0
$729$	−1.44612e7	−1.00782
$730$	0	0
$731$	7.18740e6	0.497483
$732$	0	0
$733$	2.77928e6i	0.191061i	0.995426	+	0.0955306i	$0.0304548\pi$
$733$	2.77928e6i	0.191061i	−0.995426	+	0.0955306i	$0.969545\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1.17785e7i	0.798768i
$738$	0	0
$739$	−1.21046e7	−0.815342	−0.407671	−	0.913129i	$-0.633659\pi$
$739$	−1.21046e7	−0.815342	−0.407671	+	0.913129i	$0.633659\pi$
$740$	0	0
$741$	4.19338e6	0.280555
$742$	0	0
$743$	4.46926e6i	0.297005i	0.988912	+	0.148502i	$0.0474452\pi$
$743$	4.46926e6i	0.297005i	−0.988912	+	0.148502i	$0.952555\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	6.70388e6i	0.439567i
$748$	0	0
$749$	1.07680e7	0.701345
$750$	0	0
$751$	−2.88463e7	−1.86634	−0.933168	−	0.359442i	$-0.882967\pi$
$751$	−2.88463e7	−1.86634	−0.933168	+	0.359442i	$0.882967\pi$
$752$	0	0
$753$	1.61028e7i	1.03494i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	9.60868e6i	0.609430i	0.952444	+	0.304715i	$0.0985612\pi$
$757$	9.60868e6i	0.609430i	−0.952444	+	0.304715i	$0.901439\pi$
$758$	0	0
$759$	2.65939e7	1.67563
$760$	0	0
$761$	4.54588e6	0.284549	0.142274	−	0.989827i	$-0.454558\pi$
$761$	4.54588e6	0.284549	0.142274	+	0.989827i	$0.454558\pi$
$762$	0	0
$763$	− 8.79138e6i	− 0.546696i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 3.20522e6i	− 0.196730i
$768$	0	0
$769$	2.15923e7	1.31669	0.658345	−	0.752716i	$-0.271257\pi$
$769$	2.15923e7	1.31669	0.658345	+	0.752716i	$0.271257\pi$
$770$	0	0
$771$	1.58825e6	0.0962238
$772$	0	0
$773$	1.48400e7i	0.893276i	0.894715	+	0.446638i	$0.147379\pi$
$773$	1.48400e7i	0.893276i	−0.894715	+	0.446638i	$0.852621\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 314688.i	− 0.0186994i
$778$	0	0
$779$	1.44009e7	0.850251
$780$	0	0
$781$	−2.53109e7	−1.48484
$782$	0	0
$783$	− 2.43778e6i	− 0.142098i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	2.48785e7i	1.43182i	0.698194	+	0.715909i	$0.253987\pi$
$787$	2.48785e7i	1.43182i	−0.698194	+	0.715909i	$0.746013\pi$
$788$	0	0
$789$	1.13193e7	0.647330
$790$	0	0
$791$	−2.60885e6	−0.148254
$792$	0	0
$793$	− 1.45205e7i	− 0.819970i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	3.16080e7i	1.76259i	0.472568	+	0.881294i	$0.343327\pi$
$797$	3.16080e7i	1.76259i	−0.472568	+	0.881294i	$0.656673\pi$
$798$	0	0
$799$	−769824.	−0.0426604
$800$	0	0
$801$	−2.94565e6	−0.162218
$802$	0	0
$803$	3.64835e7i	1.99668i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	1.16102e7i	0.627562i
$808$	0	0
$809$	3.10009e6	0.166534	0.0832669	−	0.996527i	$-0.473465\pi$
$809$	3.10009e6	0.166534	0.0832669	+	0.996527i	$0.473465\pi$
$810$	0	0
$811$	−1.87180e6	−0.0999328	−0.0499664	−	0.998751i	$-0.515911\pi$
$811$	−1.87180e6	−0.0999328	−0.0499664	+	0.998751i	$0.515911\pi$
$812$	0	0
$813$	− 6.21984e6i	− 0.330030i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 1.01156e7i	− 0.530196i
$818$	0	0
$819$	−3.64162e6	−0.189707
$820$	0	0
$821$	−2.00184e7	−1.03650	−0.518252	−	0.855228i	$-0.673417\pi$
$821$	−2.00184e7	−1.03650	−0.518252	+	0.855228i	$0.673417\pi$
$822$	0	0
$823$	1.53118e7i	0.787999i	0.919111	+	0.394000i	$0.128909\pi$
$823$	1.53118e7i	0.787999i	−0.919111	+	0.394000i	$0.871091\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 9.59310e6i	− 0.487748i	−0.969807	−	0.243874i	$-0.921582\pi$
$827$	− 9.59310e6i	− 0.487748i	0.969807	−	0.243874i	$-0.0784183\pi$
$828$	0	0
$829$	−2.52209e7	−1.27460	−0.637302	−	0.770615i	$-0.719949\pi$
$829$	−2.52209e7	−1.27460	−0.637302	+	0.770615i	$0.719949\pi$
$830$	0	0
$831$	2.66727e7	1.33988
$832$	0	0
$833$	5.38342e6i	0.268810i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	1.74666e7i	0.861778i
$838$	0	0
$839$	−1.77623e7	−0.871154	−0.435577	−	0.900151i	$-0.643456\pi$
$839$	−1.77623e7	−0.871154	−0.435577	+	0.900151i	$0.643456\pi$
$840$	0	0
$841$	−2.01583e7	−0.982798
$842$	0	0
$843$	2.35937e6i	0.114348i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	1.14883e7i	0.550234i
$848$	0	0
$849$	−1.86273e7	−0.886913
$850$	0	0
$851$	−1.22299e6	−0.0578895
$852$	0	0
$853$	486970.i	0.0229155i	0.999934	+	0.0114578i	$0.00364720\pi$
$853$	486970.i	0.0229155i	−0.999934	+	0.0114578i	$0.996353\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 1.92634e6i	− 0.0895945i	−0.998996	−	0.0447972i	$-0.985736\pi$
$857$	− 1.92634e6i	− 0.0895945i	0.998996	−	0.0447972i	$-0.0142642\pi$
$858$	0	0
$859$	2.23538e7	1.03364	0.516820	−	0.856094i	$-0.327116\pi$
$859$	2.23538e7	1.03364	0.516820	+	0.856094i	$0.327116\pi$
$860$	0	0
$861$	1.81907e7	0.836258
$862$	0	0
$863$	1.85838e7i	0.849390i	0.905337	+	0.424695i	$0.139619\pi$
$863$	1.85838e7i	0.849390i	−0.905337	+	0.424695i	$0.860381\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 1.28043e7i	− 0.578504i
$868$	0	0
$869$	4.15325e7	1.86569
$870$	0	0
$871$	9.11742e6	0.407217
$872$	0	0
$873$	− 1.21174e7i	− 0.538114i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 2.91048e7i	− 1.27781i	−0.769286	−	0.638905i	$-0.779388\pi$
$877$	− 2.91048e7i	− 1.27781i	0.769286	−	0.638905i	$-0.220612\pi$
$878$	0	0
$879$	1.28660e7	0.561659
$880$	0	0
$881$	−3.14696e6	−0.136600	−0.0683001	−	0.997665i	$-0.521758\pi$
$881$	−3.14696e6	−0.136600	−0.0683001	+	0.997665i	$0.521758\pi$
$882$	0	0
$883$	1.59995e7i	0.690566i	0.938499	+	0.345283i	$0.112217\pi$
$883$	1.59995e7i	0.690566i	−0.938499	+	0.345283i	$0.887783\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	3.45874e7i	1.47608i	0.674758	+	0.738039i	$0.264248\pi$
$887$	3.45874e7i	1.47608i	−0.674758	+	0.738039i	$0.735752\pi$
$888$	0	0
$889$	2.96131e7	1.25669
$890$	0	0
$891$	1.36031e7	0.574044
$892$	0	0
$893$	1.08346e6i	0.0454656i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 2.05857e7i	− 0.854248i
$898$	0	0
$899$	−2.52806e6	−0.104325
$900$	0	0
$901$	1.15794e7	0.475199
$902$	0	0
$903$	− 1.27776e7i	− 0.521471i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 1.74396e7i	− 0.703914i	−0.936016	−	0.351957i	$-0.885516\pi$
$907$	− 1.74396e7i	− 0.703914i	0.936016	−	0.351957i	$-0.114484\pi$
$908$	0	0
$909$	1.11731e6	0.0448503
$910$	0	0
$911$	2.59589e6	0.103631	0.0518155	−	0.998657i	$-0.483499\pi$
$911$	2.59589e6	0.103631	0.0518155	+	0.998657i	$0.483499\pi$
$912$	0	0
$913$	− 3.65666e7i	− 1.45180i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 8.88624e6i	− 0.348975i
$918$	0	0
$919$	−1.76411e7	−0.689028	−0.344514	−	0.938781i	$-0.611956\pi$
$919$	−1.76411e7	−0.689028	−0.344514	+	0.938781i	$0.611956\pi$
$920$	0	0
$921$	−1.90307e7	−0.739275
$922$	0	0
$923$	1.95925e7i	0.756982i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 2.69834e6i	− 0.103133i
$928$	0	0
$929$	−3.96785e7	−1.50840	−0.754199	−	0.656646i	$-0.771975\pi$
$929$	−3.96785e7	−1.50840	−0.754199	+	0.656646i	$0.771975\pi$
$930$	0	0
$931$	7.57667e6	0.286486
$932$	0	0
$933$	− 8.76874e6i	− 0.329787i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	3.93413e7i	1.46386i	0.681380	+	0.731930i	$0.261380\pi$
$937$	3.93413e7i	1.46386i	−0.681380	+	0.731930i	$0.738620\pi$
$938$	0	0
$939$	−7.01830e6	−0.259757
$940$	0	0
$941$	4.62506e7	1.70272	0.851361	−	0.524581i	$-0.175778\pi$
$941$	4.62506e7	1.70272	0.851361	+	0.524581i	$0.175778\pi$
$942$	0	0
$943$	− 7.06955e7i	− 2.58888i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	3.79025e7i	1.37339i	0.726947	+	0.686693i	$0.240938\pi$
$947$	3.79025e7i	1.37339i	−0.726947	+	0.686693i	$0.759062\pi$
$948$	0	0
$949$	2.82409e7	1.01792
$950$	0	0
$951$	−2.97944e7	−1.06828
$952$	0	0
$953$	2.66462e7i	0.950394i	0.879879	+	0.475197i	$0.157623\pi$
$953$	2.66462e7i	0.950394i	−0.879879	+	0.475197i	$0.842377\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	3.84912e6i	0.135857i
$958$	0	0
$959$	2.79085e7	0.979918
$960$	0	0
$961$	−1.05156e7	−0.367304
$962$	0	0
$963$	− 1.21140e7i	− 0.420943i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 4.09790e7i	− 1.40927i	−0.709568	−	0.704637i	$-0.751110\pi$
$967$	− 4.09790e7i	− 1.40927i	0.709568	−	0.704637i	$-0.248890\pi$
$968$	0	0
$969$	5.95901e6	0.203875
$970$	0	0
$971$	2.72034e7	0.925922	0.462961	−	0.886379i	$-0.346787\pi$
$971$	2.72034e7	0.925922	0.462961	+	0.886379i	$0.346787\pi$
$972$	0	0
$973$	− 1.30525e7i	− 0.441990i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	2.53555e7i	0.849839i	0.905231	+	0.424919i	$0.139698\pi$
$977$	2.53555e7i	0.849839i	−0.905231	+	0.424919i	$0.860302\pi$
$978$	0	0
$979$	1.60672e7	0.535775
$980$	0	0
$981$	−9.89030e6	−0.328123
$982$	0	0
$983$	1.19139e7i	0.393252i	0.980479	+	0.196626i	$0.0629984\pi$
$983$	1.19139e7i	0.393252i	−0.980479	+	0.196626i	$0.937002\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	1.36858e6i	0.0447173i
$988$	0	0
$989$	−4.96584e7	−1.61437
$990$	0	0
$991$	−2.91931e7	−0.944268	−0.472134	−	0.881527i	$-0.656516\pi$
$991$	−2.91931e7	−0.944268	−0.472134	+	0.881527i	$0.656516\pi$
$992$	0	0
$993$	4.53538e6i	0.145962i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 1.73001e7i	− 0.551201i	−0.961272	−	0.275601i	$-0.911123\pi$
$997$	− 1.73001e7i	− 0.551201i	0.961272	−	0.275601i	$-0.0888767\pi$
$998$	0	0
$999$	−1.22299e6	−0.0387713

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	400.6.c.f.49.1		2
4.3	odd	2		100.6.c.b.49.2		2
5.2	odd	4		400.6.a.d.1.1		1
5.3	odd	4		16.6.a.b.1.1		1
5.4	even	2	inner	400.6.c.f.49.2		2
12.11	even	2		900.6.d.a.649.1		2
15.8	even	4		144.6.a.c.1.1		1
20.3	even	4		4.6.a.a.1.1	✓	1
20.7	even	4		100.6.a.b.1.1		1
20.19	odd	2		100.6.c.b.49.1		2
35.13	even	4		784.6.a.d.1.1		1
40.3	even	4		64.6.a.f.1.1		1
40.13	odd	4		64.6.a.b.1.1		1
60.23	odd	4		36.6.a.a.1.1		1
60.47	odd	4		900.6.a.h.1.1		1
60.59	even	2		900.6.d.a.649.2		2
80.3	even	4		256.6.b.g.129.1		2
80.13	odd	4		256.6.b.c.129.2		2
80.43	even	4		256.6.b.g.129.2		2
80.53	odd	4		256.6.b.c.129.1		2
120.53	even	4		576.6.a.bd.1.1		1
120.83	odd	4		576.6.a.bc.1.1		1
140.3	odd	12		196.6.e.d.177.1		2
140.23	even	12		196.6.e.g.165.1		2
140.83	odd	4		196.6.a.e.1.1		1
140.103	odd	12		196.6.e.d.165.1		2
140.123	even	12		196.6.e.g.177.1		2
180.23	odd	12		324.6.e.d.217.1		2
180.43	even	12		324.6.e.a.109.1		2
180.83	odd	12		324.6.e.d.109.1		2
180.103	even	12		324.6.e.a.217.1		2
220.43	odd	4		484.6.a.a.1.1		1
260.83	odd	4		676.6.d.a.337.1		2
260.103	even	4		676.6.a.a.1.1		1
260.203	odd	4		676.6.d.a.337.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4.6.a.a.1.1	✓	1	20.3	even	4
16.6.a.b.1.1		1	5.3	odd	4
36.6.a.a.1.1		1	60.23	odd	4
64.6.a.b.1.1		1	40.13	odd	4
64.6.a.f.1.1		1	40.3	even	4
100.6.a.b.1.1		1	20.7	even	4
100.6.c.b.49.1		2	20.19	odd	2
100.6.c.b.49.2		2	4.3	odd	2
144.6.a.c.1.1		1	15.8	even	4
196.6.a.e.1.1		1	140.83	odd	4
196.6.e.d.165.1		2	140.103	odd	12
196.6.e.d.177.1		2	140.3	odd	12
196.6.e.g.165.1		2	140.23	even	12
196.6.e.g.177.1		2	140.123	even	12
256.6.b.c.129.1		2	80.53	odd	4
256.6.b.c.129.2		2	80.13	odd	4
256.6.b.g.129.1		2	80.3	even	4
256.6.b.g.129.2		2	80.43	even	4
324.6.e.a.109.1		2	180.43	even	12
324.6.e.a.217.1		2	180.103	even	12
324.6.e.d.109.1		2	180.83	odd	12
324.6.e.d.217.1		2	180.23	odd	12
400.6.a.d.1.1		1	5.2	odd	4
400.6.c.f.49.1		2	1.1	even	1	trivial
400.6.c.f.49.2		2	5.4	even	2	inner
484.6.a.a.1.1		1	220.43	odd	4
576.6.a.bc.1.1		1	120.83	odd	4
576.6.a.bd.1.1		1	120.53	even	4
676.6.a.a.1.1		1	260.103	even	4
676.6.d.a.337.1		2	260.83	odd	4
676.6.d.a.337.2		2	260.203	odd	4
784.6.a.d.1.1		1	35.13	even	4
900.6.a.h.1.1		1	60.47	odd	4
900.6.d.a.649.1		2	12.11	even	2
900.6.d.a.649.2		2	60.59	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 400 = 2^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	400.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-12.0000i q^{3} +88.0000i q^{7} +99.0000 q^{9} +O(q^{10})\)	\(q-12.0000i q^{3} +88.0000i q^{7} +99.0000 q^{9} -540.000 q^{11} +418.000i q^{13} +594.000i q^{17} +836.000 q^{19} +1056.00 q^{21} -4104.00i q^{23} -4104.00i q^{27} +594.000 q^{29} -4256.00 q^{31} +6480.00i q^{33} -298.000i q^{37} +5016.00 q^{39} +17226.0 q^{41} -12100.0i q^{43} +1296.00i q^{47} +9063.00 q^{49} +7128.00 q^{51} -19494.0i q^{53} -10032.0i q^{57} -7668.00 q^{59} -34738.0 q^{61} +8712.00i q^{63} -21812.0i q^{67} -49248.0 q^{69} +46872.0 q^{71} -67562.0i q^{73} -47520.0i q^{77} -76912.0 q^{79} -25191.0 q^{81} +67716.0i q^{83} -7128.00i q^{87} -29754.0 q^{89} -36784.0 q^{91} +51072.0i q^{93} -122398. i q^{97} -53460.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 198 q^{9}+O(q^{10}) \) 2 * q + 198 * q^9	\( 2 q + 198 q^{9} - 1080 q^{11} + 1672 q^{19} + 2112 q^{21} + 1188 q^{29} - 8512 q^{31} + 10032 q^{39} + 34452 q^{41} + 18126 q^{49} + 14256 q^{51} - 15336 q^{59} - 69476 q^{61} - 98496 q^{69} + 93744 q^{71} - 153824 q^{79} - 50382 q^{81} - 59508 q^{89} - 73568 q^{91} - 106920 q^{99}+O(q^{100}) \) 2 * q + 198 * q^9 - 1080 * q^11 + 1672 * q^19 + 2112 * q^21 + 1188 * q^29 - 8512 * q^31 + 10032 * q^39 + 34452 * q^41 + 18126 * q^49 + 14256 * q^51 - 15336 * q^59 - 69476 * q^61 - 98496 * q^69 + 93744 * q^71 - 153824 * q^79 - 50382 * q^81 - 59508 * q^89 - 73568 * q^91 - 106920 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more