LMFDB - Embedded newform 900.4.a.t.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [900,4,Mod(1,900)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(900, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("900.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	900.a (trivial)

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:	yes
Analytic conductor:	$53.1017190052$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{10}, \sqrt{34})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 22x^{2} + 36 $ x^4 - 22*x^2 + 36
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{7}\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 180)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.33434$ of defining polynomial
Character	$\chi$	$=$	900.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+23.3238 q^{7} +O(q^{10})$	$q+23.3238 q^{7} -63.2456 q^{11} +69.9714 q^{13} +92.1954 q^{17} +12.0000 q^{19} -184.391 q^{23} +189.737 q^{29} +136.000 q^{31} -116.619 q^{37} +126.491 q^{41} -186.590 q^{43} +184.391 q^{47} +201.000 q^{49} +276.586 q^{53} -316.228 q^{59} +794.000 q^{61} +326.533 q^{67} -379.473 q^{71} -466.476 q^{73} -1475.13 q^{77} +384.000 q^{79} +368.782 q^{83} -1011.93 q^{89} +1632.00 q^{91} -1072.90 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q + 48 q^{19} + 544 q^{31} + 804 q^{49} + 3176 q^{61} + 1536 q^{79} + 6528 q^{91}+O(q^{100}) $ 4 * q + 48 * q^19 + 544 * q^31 + 804 * q^49 + 3176 * q^61 + 1536 * q^79 + 6528 * q^91

Display 100 coefficients 10 coefficients

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	23.3238	1.25937	0.629684	−	0.776852i	$-0.283185\pi$
$7$	23.3238	1.25937	0.629684	+	0.776852i	$0.283185\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−63.2456	−1.73357	−0.866784	−	0.498683i	$-0.833817\pi$
$11$	−63.2456	−1.73357	−0.866784	+	0.498683i	$0.833817\pi$
$12$	0	0
$13$	69.9714	1.49281	0.746407	−	0.665490i	$-0.231777\pi$
$13$	69.9714	1.49281	0.746407	+	0.665490i	$0.231777\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	92.1954	1.31533	0.657667	−	0.753309i	$-0.271543\pi$
$17$	92.1954	1.31533	0.657667	+	0.753309i	$0.271543\pi$
$18$	0	0
$19$	12.0000	0.144894	0.0724471	−	0.997372i	$-0.476919\pi$
$19$	12.0000	0.144894	0.0724471	+	0.997372i	$0.476919\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−184.391	−1.67166	−0.835830	−	0.548989i	$-0.815013\pi$
$23$	−184.391	−1.67166	−0.835830	+	0.548989i	$0.815013\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	189.737	1.21494	0.607469	−	0.794343i	$-0.292185\pi$
$29$	189.737	1.21494	0.607469	+	0.794343i	$0.292185\pi$
$30$	0	0
$31$	136.000	0.787946	0.393973	−	0.919122i	$-0.371100\pi$
$31$	136.000	0.787946	0.393973	+	0.919122i	$0.371100\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−116.619	−0.518164	−0.259082	−	0.965855i	$-0.583420\pi$
$37$	−116.619	−0.518164	−0.259082	+	0.965855i	$0.583420\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	126.491	0.481819	0.240910	−	0.970548i	$-0.422554\pi$
$41$	126.491	0.481819	0.240910	+	0.970548i	$0.422554\pi$
$42$	0	0
$43$	−186.590	−0.661739	−0.330870	−	0.943677i	$-0.607342\pi$
$43$	−186.590	−0.661739	−0.330870	+	0.943677i	$0.607342\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	184.391	0.572259	0.286130	−	0.958191i	$-0.407631\pi$
$47$	184.391	0.572259	0.286130	+	0.958191i	$0.407631\pi$
$48$	0	0
$49$	201.000	0.586006
$50$	0	0
$51$	0	0
$52$	0	0
$53$	276.586	0.716831	0.358416	−	0.933562i	$-0.383317\pi$
$53$	276.586	0.716831	0.358416	+	0.933562i	$0.383317\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−316.228	−0.697786	−0.348893	−	0.937163i	$-0.613442\pi$
$59$	−316.228	−0.697786	−0.348893	+	0.937163i	$0.613442\pi$
$60$	0	0
$61$	794.000	1.66658	0.833289	−	0.552837i	$-0.186455\pi$
$61$	794.000	1.66658	0.833289	+	0.552837i	$0.186455\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	326.533	0.595409	0.297704	−	0.954658i	$-0.403779\pi$
$67$	326.533	0.595409	0.297704	+	0.954658i	$0.403779\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−379.473	−0.634299	−0.317149	−	0.948376i	$-0.602726\pi$
$71$	−379.473	−0.634299	−0.317149	+	0.948376i	$0.602726\pi$
$72$	0	0
$73$	−466.476	−0.747903	−0.373951	−	0.927448i	$-0.621997\pi$
$73$	−466.476	−0.747903	−0.373951	+	0.927448i	$0.621997\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−1475.13	−2.18320
$78$	0	0
$79$	384.000	0.546878	0.273439	−	0.961889i	$-0.411839\pi$
$79$	384.000	0.546878	0.273439	+	0.961889i	$0.411839\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	368.782	0.487700	0.243850	−	0.969813i	$-0.421590\pi$
$83$	368.782	0.487700	0.243850	+	0.969813i	$0.421590\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−1011.93	−1.20522	−0.602608	−	0.798037i	$-0.705872\pi$
$89$	−1011.93	−1.20522	−0.602608	+	0.798037i	$0.705872\pi$
$90$	0	0
$91$	1632.00	1.88000
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−1072.90	−1.12305	−0.561526	−	0.827459i	$-0.689785\pi$
$97$	−1072.90	−1.12305	−0.561526	+	0.827459i	$0.689785\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	1581.14	1.55771	0.778857	−	0.627201i	$-0.215800\pi$
$101$	1581.14	1.55771	0.778857	+	0.627201i	$0.215800\pi$
$102$	0	0
$103$	1516.05	1.45030	0.725149	−	0.688592i	$-0.241771\pi$
$103$	1516.05	1.45030	0.725149	+	0.688592i	$0.241771\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1106.35	0.999574	0.499787	−	0.866148i	$-0.333412\pi$
$107$	1106.35	0.999574	0.499787	+	0.866148i	$0.333412\pi$
$108$	0	0
$109$	1258.00	1.10545	0.552727	−	0.833362i	$-0.313587\pi$
$109$	1258.00	1.10545	0.552727	+	0.833362i	$0.313587\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	1382.93	1.15129	0.575643	−	0.817701i	$-0.304752\pi$
$113$	1382.93	1.15129	0.575643	+	0.817701i	$0.304752\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	2150.35	1.65649
$120$	0	0
$121$	2669.00	2.00526
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	1982.52	1.38520	0.692600	−	0.721321i	$-0.256465\pi$
$127$	1982.52	1.38520	0.692600	+	0.721321i	$0.256465\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−189.737	−0.126545	−0.0632724	−	0.997996i	$-0.520154\pi$
$131$	−189.737	−0.126545	−0.0632724	+	0.997996i	$0.520154\pi$
$132$	0	0
$133$	279.886	0.182475
$134$	0	0
$135$	0	0
$136$	0	0
$137$	276.586	0.172484	0.0862422	−	0.996274i	$-0.472514\pi$
$137$	276.586	0.172484	0.0862422	+	0.996274i	$0.472514\pi$
$138$	0	0
$139$	76.0000	0.0463758	0.0231879	−	0.999731i	$-0.492618\pi$
$139$	76.0000	0.0463758	0.0231879	+	0.999731i	$0.492618\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−4425.38	−2.58789
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−1075.17	−0.591152	−0.295576	−	0.955319i	$-0.595512\pi$
$149$	−1075.17	−0.591152	−0.295576	+	0.955319i	$0.595512\pi$
$150$	0	0
$151$	2536.00	1.36673	0.683367	−	0.730075i	$-0.260515\pi$
$151$	2536.00	1.36673	0.683367	+	0.730075i	$0.260515\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	676.390	0.343833	0.171917	−	0.985111i	$-0.445004\pi$
$157$	676.390	0.343833	0.171917	+	0.985111i	$0.445004\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−4300.70	−2.10523
$162$	0	0
$163$	−1352.78	−0.650049	−0.325024	−	0.945706i	$-0.605373\pi$
$163$	−1352.78	−0.650049	−0.325024	+	0.945706i	$0.605373\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1659.52	0.768966	0.384483	−	0.923132i	$-0.374380\pi$
$167$	1659.52	0.768966	0.384483	+	0.923132i	$0.374380\pi$
$168$	0	0
$169$	2699.00	1.22849
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−645.368	−0.283621	−0.141810	−	0.989894i	$-0.545292\pi$
$173$	−645.368	−0.283621	−0.141810	+	0.989894i	$0.545292\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	1960.61	0.818676	0.409338	−	0.912383i	$-0.365760\pi$
$179$	1960.61	0.818676	0.409338	+	0.912383i	$0.365760\pi$
$180$	0	0
$181$	3026.00	1.24266	0.621328	−	0.783550i	$-0.286593\pi$
$181$	3026.00	1.24266	0.621328	+	0.783550i	$0.286593\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−5830.95	−2.28022
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−1138.42	−0.431273	−0.215637	−	0.976474i	$-0.569183\pi$
$191$	−1138.42	−0.431273	−0.215637	+	0.976474i	$0.569183\pi$
$192$	0	0
$193$	−2845.50	−1.06126	−0.530632	−	0.847602i	$-0.678045\pi$
$193$	−2845.50	−1.06126	−0.530632	+	0.847602i	$0.678045\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	2673.67	0.966959	0.483480	−	0.875356i	$-0.339373\pi$
$197$	2673.67	0.966959	0.483480	+	0.875356i	$0.339373\pi$
$198$	0	0
$199$	−1152.00	−0.410367	−0.205184	−	0.978723i	$-0.565779\pi$
$199$	−1152.00	−0.410367	−0.205184	+	0.978723i	$0.565779\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	4425.38	1.53005
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−758.947	−0.251184
$210$	0	0
$211$	204.000	0.0665590	0.0332795	−	0.999446i	$-0.489405\pi$
$211$	204.000	0.0665590	0.0332795	+	0.999446i	$0.489405\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	3172.04	0.992313
$218$	0	0
$219$	0	0
$220$	0	0
$221$	6451.05	1.96355
$222$	0	0
$223$	−2495.65	−0.749421	−0.374711	−	0.927142i	$-0.622258\pi$
$223$	−2495.65	−0.749421	−0.374711	+	0.927142i	$0.622258\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−6269.29	−1.83307	−0.916536	−	0.399952i	$-0.869027\pi$
$227$	−6269.29	−1.83307	−0.916536	+	0.399952i	$0.869027\pi$
$228$	0	0
$229$	1502.00	0.433428	0.216714	−	0.976235i	$-0.430466\pi$
$229$	1502.00	0.433428	0.216714	+	0.976235i	$0.430466\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−1751.71	−0.492526	−0.246263	−	0.969203i	$-0.579203\pi$
$233$	−1751.71	−0.492526	−0.246263	+	0.969203i	$0.579203\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−4427.19	−1.19821	−0.599103	−	0.800672i	$-0.704476\pi$
$239$	−4427.19	−1.19821	−0.599103	+	0.800672i	$0.704476\pi$
$240$	0	0
$241$	994.000	0.265681	0.132841	−	0.991137i	$-0.457590\pi$
$241$	994.000	0.265681	0.132841	+	0.991137i	$0.457590\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	839.657	0.216300
$248$	0	0
$249$	0	0
$250$	0	0
$251$	5122.89	1.28826	0.644131	−	0.764915i	$-0.277219\pi$
$251$	5122.89	1.28826	0.644131	+	0.764915i	$0.277219\pi$
$252$	0	0
$253$	11661.9	2.89794
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−5255.14	−1.27551	−0.637756	−	0.770238i	$-0.720137\pi$
$257$	−5255.14	−1.27551	−0.637756	+	0.770238i	$0.720137\pi$
$258$	0	0
$259$	−2720.00	−0.652558
$260$	0	0
$261$	0	0
$262$	0	0
$263$	4240.99	0.994337	0.497169	−	0.867654i	$-0.334373\pi$
$263$	4240.99	0.994337	0.497169	+	0.867654i	$0.334373\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	6640.78	1.50519	0.752594	−	0.658485i	$-0.228802\pi$
$269$	6640.78	1.50519	0.752594	+	0.658485i	$0.228802\pi$
$270$	0	0
$271$	−4688.00	−1.05083	−0.525416	−	0.850845i	$-0.676090\pi$
$271$	−4688.00	−1.05083	−0.525416	+	0.850845i	$0.676090\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	5621.04	1.21926	0.609631	−	0.792686i	$-0.291318\pi$
$277$	5621.04	1.21926	0.609631	+	0.792686i	$0.291318\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−3035.79	−0.644483	−0.322242	−	0.946657i	$-0.604436\pi$
$281$	−3035.79	−0.644483	−0.322242	+	0.946657i	$0.604436\pi$
$282$	0	0
$283$	886.305	0.186167	0.0930836	−	0.995658i	$-0.470328\pi$
$283$	886.305	0.186167	0.0930836	+	0.995658i	$0.470328\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	2950.25	0.606787
$288$	0	0
$289$	3587.00	0.730104
$290$	0	0
$291$	0	0
$292$	0	0
$293$	1936.10	0.386036	0.193018	−	0.981195i	$-0.438172\pi$
$293$	1936.10	0.386036	0.193018	+	0.981195i	$0.438172\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−12902.1	−2.49548
$300$	0	0
$301$	−4352.00	−0.833372
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−839.657	−0.156097	−0.0780485	−	0.996950i	$-0.524869\pi$
$307$	−839.657	−0.156097	−0.0780485	+	0.996950i	$0.524869\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−4933.15	−0.899464	−0.449732	−	0.893163i	$-0.648481\pi$
$311$	−4933.15	−0.899464	−0.449732	+	0.893163i	$0.648481\pi$
$312$	0	0
$313$	−5084.59	−0.918205	−0.459102	−	0.888383i	$-0.651829\pi$
$313$	−5084.59	−0.918205	−0.459102	+	0.888383i	$0.651829\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	5070.75	0.898428	0.449214	−	0.893424i	$-0.351704\pi$
$317$	5070.75	0.898428	0.449214	+	0.893424i	$0.351704\pi$
$318$	0	0
$319$	−12000.0	−2.10618
$320$	0	0
$321$	0	0
$322$	0	0
$323$	1106.35	0.190584
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	4300.70	0.720684
$330$	0	0
$331$	−7508.00	−1.24676	−0.623379	−	0.781920i	$-0.714241\pi$
$331$	−7508.00	−1.24676	−0.623379	+	0.781920i	$0.714241\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−10822.2	−1.74933	−0.874667	−	0.484725i	$-0.838920\pi$
$337$	−10822.2	−1.74933	−0.874667	+	0.484725i	$0.838920\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−8601.40	−1.36596
$342$	0	0
$343$	−3311.98	−0.521371
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−6638.07	−1.02695	−0.513473	−	0.858106i	$-0.671641\pi$
$347$	−6638.07	−1.02695	−0.513473	+	0.858106i	$0.671641\pi$
$348$	0	0
$349$	6586.00	1.01014	0.505072	−	0.863077i	$-0.331466\pi$
$349$	6586.00	1.01014	0.505072	+	0.863077i	$0.331466\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−9496.13	−1.43181	−0.715904	−	0.698199i	$-0.753985\pi$
$353$	−9496.13	−1.43181	−0.715904	+	0.698199i	$0.753985\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	11384.2	1.67364	0.836818	−	0.547482i	$-0.184413\pi$
$359$	11384.2	1.67364	0.836818	+	0.547482i	$0.184413\pi$
$360$	0	0
$361$	−6715.00	−0.979006
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	23.3238	0.00331742	0.00165871	−	0.999999i	$-0.499472\pi$
$367$	23.3238	0.00331742	0.00165871	+	0.999999i	$0.499472\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	6451.05	0.902754
$372$	0	0
$373$	−9865.97	−1.36955	−0.684773	−	0.728757i	$-0.740099\pi$
$373$	−9865.97	−1.36955	−0.684773	+	0.728757i	$0.740099\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	13276.1	1.81368
$378$	0	0
$379$	−8788.00	−1.19105	−0.595527	−	0.803336i	$-0.703057\pi$
$379$	−8788.00	−1.19105	−0.595527	+	0.803336i	$0.703057\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−7560.03	−1.00861	−0.504307	−	0.863524i	$-0.668252\pi$
$383$	−7560.03	−1.00861	−0.504307	+	0.863524i	$0.668252\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	1454.65	0.189598	0.0947989	−	0.995496i	$-0.469779\pi$
$389$	1454.65	0.189598	0.0947989	+	0.995496i	$0.469779\pi$
$390$	0	0
$391$	−17000.0	−2.19879
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−2075.82	−0.262424	−0.131212	−	0.991354i	$-0.541887\pi$
$397$	−2075.82	−0.262424	−0.131212	+	0.991354i	$0.541887\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	5945.08	0.740357	0.370179	−	0.928961i	$-0.379296\pi$
$401$	5945.08	0.740357	0.370179	+	0.928961i	$0.379296\pi$
$402$	0	0
$403$	9516.11	1.17626
$404$	0	0
$405$	0	0
$406$	0	0
$407$	7375.64	0.898272
$408$	0	0
$409$	−3178.00	−0.384210	−0.192105	−	0.981374i	$-0.561531\pi$
$409$	−3178.00	−0.384210	−0.192105	+	0.981374i	$0.561531\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−7375.64	−0.878768
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	3352.01	0.390827	0.195414	−	0.980721i	$-0.437395\pi$
$419$	3352.01	0.390827	0.195414	+	0.980721i	$0.437395\pi$
$420$	0	0
$421$	−3362.00	−0.389202	−0.194601	−	0.980883i	$-0.562341\pi$
$421$	−3362.00	−0.389202	−0.194601	+	0.980883i	$0.562341\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	18519.1	2.09883
$428$	0	0
$429$	0	0
$430$	0	0
$431$	505.964	0.0565463	0.0282731	−	0.999600i	$-0.490999\pi$
$431$	505.964	0.0565463	0.0282731	+	0.999600i	$0.490999\pi$
$432$	0	0
$433$	−9842.65	−1.09240	−0.546198	−	0.837656i	$-0.683925\pi$
$433$	−9842.65	−1.09240	−0.546198	+	0.837656i	$0.683925\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−2212.69	−0.242214
$438$	0	0
$439$	−5952.00	−0.647092	−0.323546	−	0.946212i	$-0.604875\pi$
$439$	−5952.00	−0.647092	−0.323546	+	0.946212i	$0.604875\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	12169.8	1.30520	0.652601	−	0.757702i	$-0.273678\pi$
$443$	12169.8	1.30520	0.652601	+	0.757702i	$0.273678\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−8727.89	−0.917360	−0.458680	−	0.888602i	$-0.651677\pi$
$449$	−8727.89	−0.917360	−0.458680	+	0.888602i	$0.651677\pi$
$450$	0	0
$451$	−8000.00	−0.835267
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	1632.67	0.167118	0.0835590	−	0.996503i	$-0.473371\pi$
$457$	1632.67	0.167118	0.0835590	+	0.996503i	$0.473371\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−9676.57	−0.977620	−0.488810	−	0.872390i	$-0.662569\pi$
$461$	−9676.57	−0.977620	−0.488810	+	0.872390i	$0.662569\pi$
$462$	0	0
$463$	14344.1	1.43980	0.719902	−	0.694076i	$-0.244187\pi$
$463$	14344.1	1.43980	0.719902	+	0.694076i	$0.244187\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−15488.8	−1.53477	−0.767385	−	0.641186i	$-0.778443\pi$
$467$	−15488.8	−1.53477	−0.767385	+	0.641186i	$0.778443\pi$
$468$	0	0
$469$	7616.00	0.749838
$470$	0	0
$471$	0	0
$472$	0	0
$473$	11801.0	1.14717
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	4680.17	0.446435	0.223218	−	0.974769i	$-0.428344\pi$
$479$	4680.17	0.446435	0.223218	+	0.974769i	$0.428344\pi$
$480$	0	0
$481$	−8160.00	−0.773522
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−15883.5	−1.47793	−0.738964	−	0.673745i	$-0.764685\pi$
$487$	−15883.5	−1.47793	−0.738964	+	0.673745i	$0.764685\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−3225.52	−0.296468	−0.148234	−	0.988952i	$-0.547359\pi$
$491$	−3225.52	−0.296468	−0.148234	+	0.988952i	$0.547359\pi$
$492$	0	0
$493$	17492.9	1.59805
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−8850.76	−0.798815
$498$	0	0
$499$	−21732.0	−1.94962	−0.974808	−	0.223048i	$-0.928399\pi$
$499$	−21732.0	−1.94962	−0.974808	+	0.223048i	$0.928399\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−553.173	−0.0490353	−0.0245176	−	0.999699i	$-0.507805\pi$
$503$	−553.173	−0.0490353	−0.0245176	+	0.999699i	$0.507805\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−18783.9	−1.63572	−0.817862	−	0.575415i	$-0.804841\pi$
$509$	−18783.9	−1.63572	−0.817862	+	0.575415i	$0.804841\pi$
$510$	0	0
$511$	−10880.0	−0.941884
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−11661.9	−0.992050
$518$	0	0
$519$	0	0
$520$	0	0
$521$	9739.82	0.819019	0.409510	−	0.912306i	$-0.365700\pi$
$521$	9739.82	0.819019	0.409510	+	0.912306i	$0.365700\pi$
$522$	0	0
$523$	−15347.1	−1.28314	−0.641568	−	0.767066i	$-0.721716\pi$
$523$	−15347.1	−1.28314	−0.641568	+	0.767066i	$0.721716\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	12538.6	1.03641
$528$	0	0
$529$	21833.0	1.79444
$530$	0	0
$531$	0	0
$532$	0	0
$533$	8850.76	0.719267
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−12712.4	−1.01588
$540$	0	0
$541$	5914.00	0.469987	0.234993	−	0.971997i	$-0.424493\pi$
$541$	5914.00	0.469987	0.234993	+	0.971997i	$0.424493\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	20525.0	1.60436	0.802179	−	0.597084i	$-0.203674\pi$
$547$	20525.0	1.60436	0.802179	+	0.597084i	$0.203674\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	2276.84	0.176037
$552$	0	0
$553$	8956.34	0.688720
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−22956.7	−1.74633	−0.873164	−	0.487426i	$-0.837936\pi$
$557$	−22956.7	−1.74633	−0.873164	+	0.487426i	$0.837936\pi$
$558$	0	0
$559$	−13056.0	−0.987853
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−8113.20	−0.607337	−0.303668	−	0.952778i	$-0.598211\pi$
$563$	−8113.20	−0.607337	−0.303668	+	0.952778i	$0.598211\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	11510.7	0.848073	0.424036	−	0.905645i	$-0.360613\pi$
$569$	11510.7	0.848073	0.424036	+	0.905645i	$0.360613\pi$
$570$	0	0
$571$	4204.00	0.308112	0.154056	−	0.988062i	$-0.450766\pi$
$571$	4204.00	0.308112	0.154056	+	0.988062i	$0.450766\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	2892.15	0.208669	0.104334	−	0.994542i	$-0.466729\pi$
$577$	2892.15	0.208669	0.104334	+	0.994542i	$0.466729\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	8601.40	0.614193
$582$	0	0
$583$	−17492.9	−1.24268
$584$	0	0
$585$	0	0
$586$	0	0
$587$	1106.35	0.0777918	0.0388959	−	0.999243i	$-0.487616\pi$
$587$	1106.35	0.0777918	0.0388959	+	0.999243i	$0.487616\pi$
$588$	0	0
$589$	1632.00	0.114169
$590$	0	0
$591$	0	0
$592$	0	0
$593$	19084.5	1.32159	0.660797	−	0.750565i	$-0.270219\pi$
$593$	19084.5	1.32159	0.660797	+	0.750565i	$0.270219\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	20491.6	1.39777	0.698883	−	0.715236i	$-0.253681\pi$
$599$	20491.6	1.39777	0.698883	+	0.715236i	$0.253681\pi$
$600$	0	0
$601$	18326.0	1.24382	0.621908	−	0.783091i	$-0.286358\pi$
$601$	18326.0	1.24382	0.621908	+	0.783091i	$0.286358\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−6180.81	−0.413297	−0.206649	−	0.978415i	$-0.566256\pi$
$607$	−6180.81	−0.413297	−0.206649	+	0.978415i	$0.566256\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	12902.1	0.854276
$612$	0	0
$613$	−5714.33	−0.376509	−0.188254	−	0.982120i	$-0.560283\pi$
$613$	−5714.33	−0.376509	−0.188254	+	0.982120i	$0.560283\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	5623.92	0.366954	0.183477	−	0.983024i	$-0.441265\pi$
$617$	5623.92	0.366954	0.183477	+	0.983024i	$0.441265\pi$
$618$	0	0
$619$	7868.00	0.510891	0.255446	−	0.966823i	$-0.417778\pi$
$619$	7868.00	0.510891	0.255446	+	0.966823i	$0.417778\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−23602.0	−1.51781
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−10751.7	−0.681558
$630$	0	0
$631$	−4816.00	−0.303838	−0.151919	−	0.988393i	$-0.548545\pi$
$631$	−4816.00	−0.303838	−0.151919	+	0.988393i	$0.548545\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	14064.3	0.874798
$638$	0	0
$639$	0	0
$640$	0	0
$641$	16823.3	1.03663	0.518316	−	0.855189i	$-0.326559\pi$
$641$	16823.3	1.03663	0.518316	+	0.855189i	$0.326559\pi$
$642$	0	0
$643$	−699.714	−0.0429145	−0.0214573	−	0.999770i	$-0.506831\pi$
$643$	−699.714	−0.0429145	−0.0214573	+	0.999770i	$0.506831\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	18254.7	1.10922	0.554611	−	0.832110i	$-0.312867\pi$
$647$	18254.7	1.10922	0.554611	+	0.832110i	$0.312867\pi$
$648$	0	0
$649$	20000.0	1.20966
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−8205.39	−0.491733	−0.245867	−	0.969304i	$-0.579073\pi$
$653$	−8205.39	−0.491733	−0.245867	+	0.969304i	$0.579073\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	9929.55	0.586951	0.293475	−	0.955967i	$-0.405188\pi$
$659$	9929.55	0.586951	0.293475	+	0.955967i	$0.405188\pi$
$660$	0	0
$661$	8018.00	0.471806	0.235903	−	0.971777i	$-0.424195\pi$
$661$	8018.00	0.471806	0.235903	+	0.971777i	$0.424195\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−34985.7	−2.03096
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−50217.0	−2.88913
$672$	0	0
$673$	−3498.57	−0.200386	−0.100193	−	0.994968i	$-0.531946\pi$
$673$	−3498.57	−0.200386	−0.100193	+	0.994968i	$0.531946\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	8389.79	0.476286	0.238143	−	0.971230i	$-0.423461\pi$
$677$	8389.79	0.476286	0.238143	+	0.971230i	$0.423461\pi$
$678$	0	0
$679$	−25024.0	−1.41433
$680$	0	0
$681$	0	0
$682$	0	0
$683$	12538.6	0.702453	0.351227	−	0.936291i	$-0.385765\pi$
$683$	12538.6	0.702453	0.351227	+	0.936291i	$0.385765\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	19353.1	1.07010
$690$	0	0
$691$	35428.0	1.95043	0.975213	−	0.221267i	$-0.0710193\pi$
$691$	35428.0	1.95043	0.975213	+	0.221267i	$0.0710193\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	11661.9	0.633753
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−25867.4	−1.39372	−0.696861	−	0.717206i	$-0.745421\pi$
$701$	−25867.4	−1.39372	−0.696861	+	0.717206i	$0.745421\pi$
$702$	0	0
$703$	−1399.43	−0.0750789
$704$	0	0
$705$	0	0
$706$	0	0
$707$	36878.2	1.96173
$708$	0	0
$709$	27102.0	1.43560	0.717798	−	0.696252i	$-0.245150\pi$
$709$	27102.0	1.43560	0.717798	+	0.696252i	$0.245150\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−25077.2	−1.31718
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−35164.5	−1.82394	−0.911972	−	0.410253i	$-0.865441\pi$
$719$	−35164.5	−1.82394	−0.911972	+	0.410253i	$0.865441\pi$
$720$	0	0
$721$	35360.0	1.82646
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−8886.37	−0.453339	−0.226669	−	0.973972i	$-0.572784\pi$
$727$	−8886.37	−0.453339	−0.226669	+	0.973972i	$0.572784\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−17202.8	−0.870408
$732$	0	0
$733$	−18076.0	−0.910846	−0.455423	−	0.890275i	$-0.650512\pi$
$733$	−18076.0	−0.910846	−0.455423	+	0.890275i	$0.650512\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−20651.8	−1.03218
$738$	0	0
$739$	16876.0	0.840046	0.420023	−	0.907514i	$-0.362022\pi$
$739$	16876.0	0.840046	0.420023	+	0.907514i	$0.362022\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−40381.6	−1.99389	−0.996943	−	0.0781314i	$-0.975105\pi$
$743$	−40381.6	−1.99389	−0.996943	+	0.0781314i	$0.975105\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	25804.2	1.25883
$750$	0	0
$751$	−952.000	−0.0462570	−0.0231285	−	0.999733i	$-0.507363\pi$
$751$	−952.000	−0.0462570	−0.0231285	+	0.999733i	$0.507363\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	12618.2	0.605833	0.302916	−	0.953017i	$-0.402040\pi$
$757$	12618.2	0.605833	0.302916	+	0.953017i	$0.402040\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	11004.7	0.524206	0.262103	−	0.965040i	$-0.415584\pi$
$761$	11004.7	0.524206	0.262103	+	0.965040i	$0.415584\pi$
$762$	0	0
$763$	29341.3	1.39217
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−22126.9	−1.04166
$768$	0	0
$769$	−11442.0	−0.536553	−0.268276	−	0.963342i	$-0.586454\pi$
$769$	−11442.0	−0.536553	−0.268276	+	0.963342i	$0.586454\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	17978.1	0.836517	0.418259	−	0.908328i	$-0.362641\pi$
$773$	17978.1	0.836517	0.418259	+	0.908328i	$0.362641\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	1517.89	0.0698128
$780$	0	0
$781$	24000.0	1.09960
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	2472.32	0.111981	0.0559904	−	0.998431i	$-0.482168\pi$
$787$	2472.32	0.111981	0.0559904	+	0.998431i	$0.482168\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	32255.2	1.44989
$792$	0	0
$793$	55557.3	2.48789
$794$	0	0
$795$	0	0
$796$	0	0
$797$	11155.6	0.495801	0.247900	−	0.968786i	$-0.420259\pi$
$797$	11155.6	0.495801	0.247900	+	0.968786i	$0.420259\pi$
$798$	0	0
$799$	17000.0	0.752712
$800$	0	0
$801$	0	0
$802$	0	0
$803$	29502.5	1.29654
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−15558.4	−0.676149	−0.338074	−	0.941119i	$-0.609776\pi$
$809$	−15558.4	−0.676149	−0.338074	+	0.941119i	$0.609776\pi$
$810$	0	0
$811$	26116.0	1.13077	0.565386	−	0.824826i	$-0.308727\pi$
$811$	26116.0	1.13077	0.565386	+	0.824826i	$0.308727\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−2239.09	−0.0958821
$818$	0	0
$819$	0	0
$820$	0	0
$821$	13344.8	0.567280	0.283640	−	0.958931i	$-0.408458\pi$
$821$	13344.8	0.567280	0.283640	+	0.958931i	$0.408458\pi$
$822$	0	0
$823$	10425.7	0.441578	0.220789	−	0.975322i	$-0.429137\pi$
$823$	10425.7	0.441578	0.220789	+	0.975322i	$0.429137\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−31715.2	−1.33355	−0.666776	−	0.745259i	$-0.732326\pi$
$827$	−31715.2	−1.33355	−0.666776	+	0.745259i	$0.732326\pi$
$828$	0	0
$829$	27066.0	1.13395	0.566973	−	0.823736i	$-0.308114\pi$
$829$	27066.0	1.13395	0.566973	+	0.823736i	$0.308114\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	18531.3	0.770793
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	4047.72	0.166559	0.0832793	−	0.996526i	$-0.473461\pi$
$839$	4047.72	0.166559	0.0832793	+	0.996526i	$0.473461\pi$
$840$	0	0
$841$	11611.0	0.476075
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	62251.2	2.52536
$848$	0	0
$849$	0	0
$850$	0	0
$851$	21503.5	0.866193
$852$	0	0
$853$	−40047.0	−1.60748	−0.803741	−	0.594979i	$-0.797160\pi$
$853$	−40047.0	−1.60748	−0.803741	+	0.594979i	$0.797160\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	19637.6	0.782741	0.391370	−	0.920233i	$-0.372001\pi$
$857$	19637.6	0.782741	0.391370	+	0.920233i	$0.372001\pi$
$858$	0	0
$859$	−36196.0	−1.43771	−0.718854	−	0.695161i	$-0.755333\pi$
$859$	−36196.0	−1.43771	−0.718854	+	0.695161i	$0.755333\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	48126.0	1.89830	0.949148	−	0.314831i	$-0.101948\pi$
$863$	48126.0	1.89830	0.949148	+	0.314831i	$0.101948\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−24286.3	−0.948051
$870$	0	0
$871$	22848.0	0.888835
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−17236.3	−0.663658	−0.331829	−	0.943340i	$-0.607666\pi$
$877$	−17236.3	−0.663658	−0.331829	+	0.943340i	$0.607666\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	14420.0	0.551443	0.275722	−	0.961237i	$-0.411083\pi$
$881$	14420.0	0.551443	0.275722	+	0.961237i	$0.411083\pi$
$882$	0	0
$883$	−13014.7	−0.496012	−0.248006	−	0.968758i	$-0.579775\pi$
$883$	−13014.7	−0.496012	−0.248006	+	0.968758i	$0.579775\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−24892.8	−0.942297	−0.471148	−	0.882054i	$-0.656160\pi$
$887$	−24892.8	−0.942297	−0.471148	+	0.882054i	$0.656160\pi$
$888$	0	0
$889$	46240.0	1.74448
$890$	0	0
$891$	0	0
$892$	0	0
$893$	2212.69	0.0829170
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	25804.2	0.957306
$900$	0	0
$901$	25500.0	0.942873
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−51778.9	−1.89558	−0.947789	−	0.318899i	$-0.896687\pi$
$907$	−51778.9	−1.89558	−0.947789	+	0.318899i	$0.896687\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	33393.7	1.21447	0.607234	−	0.794523i	$-0.292279\pi$
$911$	33393.7	1.21447	0.607234	+	0.794523i	$0.292279\pi$
$912$	0	0
$913$	−23323.8	−0.845460
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−4425.38	−0.159366
$918$	0	0
$919$	−12408.0	−0.445378	−0.222689	−	0.974890i	$-0.571483\pi$
$919$	−12408.0	−0.445378	−0.222689	+	0.974890i	$0.571483\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−26552.3	−0.946890
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−50470.0	−1.78242	−0.891209	−	0.453594i	$-0.850142\pi$
$929$	−50470.0	−1.78242	−0.891209	+	0.453594i	$0.850142\pi$
$930$	0	0
$931$	2412.00	0.0849088
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	2705.56	0.0943296	0.0471648	−	0.998887i	$-0.484981\pi$
$937$	2705.56	0.0943296	0.0471648	+	0.998887i	$0.484981\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−22831.6	−0.790957	−0.395478	−	0.918475i	$-0.629421\pi$
$941$	−22831.6	−0.790957	−0.395478	+	0.918475i	$0.629421\pi$
$942$	0	0
$943$	−23323.8	−0.805438
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−16595.2	−0.569452	−0.284726	−	0.958609i	$-0.591903\pi$
$947$	−16595.2	−0.569452	−0.284726	+	0.958609i	$0.591903\pi$
$948$	0	0
$949$	−32640.0	−1.11648
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−34388.9	−1.16890	−0.584452	−	0.811428i	$-0.698691\pi$
$953$	−34388.9	−1.16890	−0.584452	+	0.811428i	$0.698691\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	6451.05	0.217221
$960$	0	0
$961$	−11295.0	−0.379141
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−16676.5	−0.554582	−0.277291	−	0.960786i	$-0.589437\pi$
$967$	−16676.5	−0.554582	−0.277291	+	0.960786i	$0.589437\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−27511.8	−0.909265	−0.454632	−	0.890679i	$-0.650229\pi$
$971$	−27511.8	−0.909265	−0.454632	+	0.890679i	$0.650229\pi$
$972$	0	0
$973$	1772.61	0.0584042
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−32360.6	−1.05968	−0.529840	−	0.848098i	$-0.677748\pi$
$977$	−32360.6	−1.05968	−0.529840	+	0.848098i	$0.677748\pi$
$978$	0	0
$979$	64000.0	2.08932
$980$	0	0
$981$	0	0
$982$	0	0
$983$	22311.3	0.723927	0.361963	−	0.932192i	$-0.382107\pi$
$983$	22311.3	0.723927	0.361963	+	0.932192i	$0.382107\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	34405.6	1.10620
$990$	0	0
$991$	−28016.0	−0.898040	−0.449020	−	0.893522i	$-0.648227\pi$
$991$	−28016.0	−0.898040	−0.449020	+	0.893522i	$0.648227\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−61924.7	−1.96708	−0.983538	−	0.180700i	$-0.942164\pi$
$997$	−61924.7	−1.96708	−0.983538	+	0.180700i	$0.942164\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	900.4.a.t.1.3		4
3.2	odd	2	inner	900.4.a.t.1.4		4
5.2	odd	4		180.4.d.c.109.4	yes	4
5.3	odd	4		180.4.d.c.109.3	yes	4
5.4	even	2	inner	900.4.a.t.1.1		4
15.2	even	4		180.4.d.c.109.1	✓	4
15.8	even	4		180.4.d.c.109.2	yes	4
15.14	odd	2	inner	900.4.a.t.1.2		4
20.3	even	4		720.4.f.k.289.3		4
20.7	even	4		720.4.f.k.289.4		4
60.23	odd	4		720.4.f.k.289.2		4
60.47	odd	4		720.4.f.k.289.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
180.4.d.c.109.1	✓	4	15.2	even	4
180.4.d.c.109.2	yes	4	15.8	even	4
180.4.d.c.109.3	yes	4	5.3	odd	4
180.4.d.c.109.4	yes	4	5.2	odd	4
720.4.f.k.289.1		4	60.47	odd	4
720.4.f.k.289.2		4	60.23	odd	4
720.4.f.k.289.3		4	20.3	even	4
720.4.f.k.289.4		4	20.7	even	4
900.4.a.t.1.1		4	5.4	even	2	inner
900.4.a.t.1.2		4	15.14	odd	2	inner
900.4.a.t.1.3		4	1.1	even	1	trivial
900.4.a.t.1.4		4	3.2	odd	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	900.a (trivial)

\(f(q)\)	\(=\)	\(q+23.3238 q^{7} +O(q^{10})\)	\(q+23.3238 q^{7} -63.2456 q^{11} +69.9714 q^{13} +92.1954 q^{17} +12.0000 q^{19} -184.391 q^{23} +189.737 q^{29} +136.000 q^{31} -116.619 q^{37} +126.491 q^{41} -186.590 q^{43} +184.391 q^{47} +201.000 q^{49} +276.586 q^{53} -316.228 q^{59} +794.000 q^{61} +326.533 q^{67} -379.473 q^{71} -466.476 q^{73} -1475.13 q^{77} +384.000 q^{79} +368.782 q^{83} -1011.93 q^{89} +1632.00 q^{91} -1072.90 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q + 48 q^{19} + 544 q^{31} + 804 q^{49} + 3176 q^{61} + 1536 q^{79} + 6528 q^{91}+O(q^{100}) \) 4 * q + 48 * q^19 + 544 * q^31 + 804 * q^49 + 3176 * q^61 + 1536 * q^79 + 6528 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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