LMFDB - Embedded newform 500.3.b.a.251.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [500,3,Mod(251,500)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(500, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("500.251");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 500 = 2^{2} \cdot 5^{3} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	500.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:	$13.6240132180$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{20})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{6} + x^{4} - x^{2} + 1 $ x^8 - x^6 + x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{8}\cdot 5^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			251.8
Root			$-0.951057 + 0.309017i$ of defining polynomial
Character	$\chi$	$=$	500.251
Dual form			500.3.b.a.251.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000i q^{2} +5.48932i q^{3} -4.00000 q^{4} -10.9786 q^{6} -11.1220i q^{7} -8.00000i q^{8} -21.1327 q^{9} +O(q^{10})$	$q+2.00000i q^{2} +5.48932i q^{3} -4.00000 q^{4} -10.9786 q^{6} -11.1220i q^{7} -8.00000i q^{8} -21.1327 q^{9} -21.9573i q^{12} +22.2441 q^{14} +16.0000 q^{16} -42.2653i q^{18} +61.0524 q^{21} -43.4827i q^{23} +43.9146 q^{24} -66.6001i q^{27} +44.4881i q^{28} -44.2407 q^{29} +32.0000i q^{32} +84.5306 q^{36} -31.8800 q^{41} +122.105i q^{42} -61.7646i q^{43} +86.9654 q^{46} +90.5544i q^{47} +87.8292i q^{48} -74.6997 q^{49} +133.200 q^{54} -88.9763 q^{56} -88.4813i q^{58} -16.1647 q^{61} +235.038i q^{63} -64.0000 q^{64} -116.000i q^{67} +238.691 q^{69} +169.061i q^{72} +175.395 q^{81} -63.7600i q^{82} -25.2603i q^{83} -244.210 q^{84} +123.529 q^{86} -242.851i q^{87} -177.968 q^{89} +173.931i q^{92} -181.109 q^{94} -175.658 q^{96} -149.399i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 32 q^{4} + 16 q^{6} - 76 q^{9}+O(q^{10}) $ 8 * q - 32 * q^4 + 16 * q^6 - 76 * q^9	$ 8 q - 32 q^{4} + 16 q^{6} - 76 q^{9} + 16 q^{14} + 128 q^{16} + 32 q^{21} - 64 q^{24} - 44 q^{29} + 304 q^{36} - 124 q^{41} + 176 q^{46} - 556 q^{49} + 32 q^{54} - 64 q^{56} + 116 q^{61} - 512 q^{64} + 352 q^{69} + 1000 q^{81} - 128 q^{84} - 304 q^{86} - 284 q^{89} + 16 q^{94} + 256 q^{96}+O(q^{100}) $ 8 * q - 32 * q^4 + 16 * q^6 - 76 * q^9 + 16 * q^14 + 128 * q^16 + 32 * q^21 - 64 * q^24 - 44 * q^29 + 304 * q^36 - 124 * q^41 + 176 * q^46 - 556 * q^49 + 32 * q^54 - 64 * q^56 + 116 * q^61 - 512 * q^64 + 352 * q^69 + 1000 * q^81 - 128 * q^84 - 304 * q^86 - 284 * q^89 + 16 * q^94 + 256 * q^96

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/500\mathbb{Z}\right)^\times$.

$n$	$251$	$377$
$\chi(n)$	$-1$	$1$

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 $

$2$	2.00000i	1.00000i
$2$	2.00000i	1.00000i
$3$	5.48932i	1.82977i	0.403710	+	0.914887i	$0.367720\pi$
$3$	5.48932i	1.82977i	−0.403710	+	0.914887i	$0.632280\pi$
$4$	−4.00000	−1.00000
$5$	0	0
$5$	0	0
$6$	−10.9786	−1.82977
$7$	− 11.1220i	− 1.58886i	−0.607354	−	0.794431i	$-0.707769\pi$
$7$	− 11.1220i	− 1.58886i	0.607354	−	0.794431i	$-0.292231\pi$
$8$	− 8.00000i	− 1.00000i
$9$	−21.1327	−2.34807
$10$	0	0
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	− 21.9573i	− 1.82977i
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	22.2441	1.58886
$15$	0	0
$16$	16.0000	1.00000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	− 42.2653i	− 2.34807i
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	0	0
$21$	61.0524	2.90726
$22$	0	0
$23$	− 43.4827i	− 1.89055i	−0.326271	−	0.945276i	$-0.605792\pi$
$23$	− 43.4827i	− 1.89055i	0.326271	−	0.945276i	$-0.394208\pi$
$24$	43.9146	1.82977
$25$	0	0
$26$	0	0
$27$	− 66.6001i	− 2.46667i
$28$	44.4881i	1.58886i
$29$	−44.2407	−1.52554	−0.762770	−	0.646670i	$-0.776161\pi$
$29$	−44.2407	−1.52554	−0.762770	+	0.646670i	$0.776161\pi$
$30$	0	0
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	32.0000i	1.00000i
$33$	0	0
$34$	0	0
$35$	0	0
$36$	84.5306	2.34807
$37$	0	0		−	1.00000i	$-0.5\pi$
$37$	0	0			1.00000i	$0.5\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−31.8800	−0.777561	−0.388780	−	0.921330i	$-0.627104\pi$
$41$	−31.8800	−0.777561	−0.388780	+	0.921330i	$0.627104\pi$
$42$	122.105i	2.90726i
$43$	− 61.7646i	− 1.43639i	−0.695844	−	0.718193i	$-0.744970\pi$
$43$	− 61.7646i	− 1.43639i	0.695844	−	0.718193i	$-0.255030\pi$
$44$	0	0
$45$	0	0
$46$	86.9654	1.89055
$47$	90.5544i	1.92669i	0.268267	+	0.963345i	$0.413549\pi$
$47$	90.5544i	1.92669i	−0.268267	+	0.963345i	$0.586451\pi$
$48$	87.8292i	1.82977i
$49$	−74.6997	−1.52448
$50$	0	0
$51$	0	0
$52$	0	0
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	133.200	2.46667
$55$	0	0
$56$	−88.9763	−1.58886
$57$	0	0
$58$	− 88.4813i	− 1.52554i
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	0	0
$61$	−16.1647	−0.264996	−0.132498	−	0.991183i	$-0.542300\pi$
$61$	−16.1647	−0.264996	−0.132498	+	0.991183i	$0.542300\pi$
$62$	0	0
$63$	235.038i	3.73076i
$64$	−64.0000	−1.00000
$65$	0	0
$66$	0	0
$67$	− 116.000i	− 1.73134i	−0.500612	−	0.865672i	$-0.666892\pi$
$67$	− 116.000i	− 1.73134i	0.500612	−	0.865672i	$-0.333108\pi$
$68$	0	0
$69$	238.691	3.45928
$70$	0	0
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	169.061i	2.34807i
$73$	0	0		−	1.00000i	$-0.5\pi$
$73$	0	0			1.00000i	$0.5\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	0	0
$81$	175.395	2.16537
$82$	− 63.7600i	− 0.777561i
$83$	− 25.2603i	− 0.304341i	−0.988354	−	0.152171i	$-0.951374\pi$
$83$	− 25.2603i	− 0.304341i	0.988354	−	0.152171i	$-0.0486264\pi$
$84$	−244.210	−2.90726
$85$	0	0
$86$	123.529	1.43639
$87$	− 242.851i	− 2.79139i
$88$	0	0
$89$	−177.968	−1.99964	−0.999821	−	0.0189175i	$-0.993978\pi$
$89$	−177.968	−1.99964	−0.999821	+	0.0189175i	$0.993978\pi$
$90$	0	0
$91$	0	0
$92$	173.931i	1.89055i
$93$	0	0
$94$	−181.109	−1.92669
$95$	0	0
$96$	−175.658	−1.82977
$97$	0	0		−	1.00000i	$-0.5\pi$
$97$	0	0			1.00000i	$0.5\pi$
$98$	− 149.399i	− 1.52448i
$99$	0	0
$100$	0	0
$101$	−4.06847	−0.0402819	−0.0201410	−	0.999797i	$-0.506411\pi$
$101$	−4.06847	−0.0402819	−0.0201410	+	0.999797i	$0.506411\pi$
$102$	0	0
$103$	44.0000i	0.427184i	0.976923	+	0.213592i	$0.0685164\pi$
$103$	44.0000i	0.427184i	−0.976923	+	0.213592i	$0.931484\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 127.559i	− 1.19214i	−0.802933	−	0.596069i	$-0.796728\pi$
$107$	− 127.559i	− 1.19214i	0.802933	−	0.596069i	$-0.203272\pi$
$108$	266.400i	2.46667i
$109$	192.414	1.76526	0.882631	−	0.470067i	$-0.155770\pi$
$109$	192.414	1.76526	0.882631	+	0.470067i	$0.155770\pi$
$110$	0	0
$111$	0	0
$112$	− 177.953i	− 1.58886i
$113$	0	0		−	1.00000i	$-0.5\pi$
$113$	0	0			1.00000i	$0.5\pi$
$114$	0	0
$115$	0	0
$116$	176.963	1.52554
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	121.000	1.00000
$122$	− 32.3295i	− 0.264996i
$123$	− 175.000i	− 1.42276i
$124$	0	0
$125$	0	0
$126$	−470.076	−3.73076
$127$	16.3903i	0.129058i	0.997916	+	0.0645288i	$0.0205544\pi$
$127$	16.3903i	0.129058i	−0.997916	+	0.0645288i	$0.979446\pi$
$128$	− 128.000i	− 1.00000i
$129$	339.046	2.62826
$130$	0	0
$131$	0	0	1.00000			$0$
$131$	0	0	−1.00000			$\pi$
$132$	0	0
$133$	0	0
$134$	232.000	1.73134
$135$	0	0
$136$	0	0
$137$	0	0		−	1.00000i	$-0.5\pi$
$137$	0	0			1.00000i	$0.5\pi$
$138$	477.381i	3.45928i
$139$	0	0	1.00000			$0$
$139$	0	0	−1.00000			$\pi$
$140$	0	0
$141$	−497.082	−3.52541
$142$	0	0
$143$	0	0
$144$	−338.122	−2.34807
$145$	0	0
$146$	0	0
$147$	− 410.050i	− 2.78946i
$148$	0	0
$149$	161.819	1.08603	0.543017	−	0.839722i	$-0.317282\pi$
$149$	161.819	1.08603	0.543017	+	0.839722i	$0.317282\pi$
$150$	0	0
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	0	0		−	1.00000i	$-0.5\pi$
$157$	0	0			1.00000i	$0.5\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−483.616	−3.00383
$162$	350.791i	2.16537i
$163$	− 298.284i	− 1.82996i	−0.403495	−	0.914982i	$-0.632205\pi$
$163$	− 298.284i	− 1.82996i	0.403495	−	0.914982i	$-0.367795\pi$
$164$	127.520	0.777561
$165$	0	0
$166$	50.5207	0.304341
$167$	292.316i	1.75040i	0.483765	+	0.875198i	$0.339269\pi$
$167$	292.316i	1.75040i	−0.483765	+	0.875198i	$0.660731\pi$
$168$	− 488.419i	− 2.90726i
$169$	−169.000	−1.00000
$170$	0	0
$171$	0	0
$172$	247.058i	1.43639i
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	485.703	2.79139
$175$	0	0
$176$	0	0
$177$	0	0
$178$	− 355.936i	− 1.99964i
$179$	0	0	1.00000			$0$
$179$	0	0	−1.00000			$\pi$
$180$	0	0
$181$	−161.667	−0.893189	−0.446594	−	0.894737i	$-0.647363\pi$
$181$	−161.667	−0.893189	−0.446594	+	0.894737i	$0.647363\pi$
$182$	0	0
$183$	− 88.7335i	− 0.484883i
$184$	−347.862	−1.89055
$185$	0	0
$186$	0	0
$187$	0	0
$188$	− 362.218i	− 1.92669i
$189$	−740.728	−3.91920
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	− 351.317i	− 1.82977i
$193$	0	0		−	1.00000i	$-0.5\pi$
$193$	0	0			1.00000i	$0.5\pi$
$194$	0	0
$195$	0	0
$196$	298.799	1.52448
$197$	0	0		−	1.00000i	$-0.5\pi$
$197$	0	0			1.00000i	$0.5\pi$
$198$	0	0
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	0	0
$201$	636.761	3.16797
$202$	− 8.13695i	− 0.0402819i
$203$	492.046i	2.42387i
$204$	0	0
$205$	0	0
$206$	−88.0000	−0.427184
$207$	918.905i	4.43916i
$208$	0	0
$209$	0	0
$210$	0	0
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	0	0
$213$	0	0
$214$	255.118	1.19214
$215$	0	0
$216$	−532.800	−2.46667
$217$	0	0
$218$	384.827i	1.76526i
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	− 45.4131i	− 0.203646i	−0.994803	−	0.101823i	$-0.967532\pi$
$223$	− 45.4131i	− 0.203646i	0.994803	−	0.101823i	$-0.0324675\pi$
$224$	355.905	1.58886
$225$	0	0
$226$	0	0
$227$	122.405i	0.539228i	0.962968	+	0.269614i	$0.0868961\pi$
$227$	122.405i	0.539228i	−0.962968	+	0.269614i	$0.913104\pi$
$228$	0	0
$229$	−432.770	−1.88982	−0.944912	−	0.327326i	$-0.893853\pi$
$229$	−432.770	−1.88982	−0.944912	+	0.327326i	$0.893853\pi$
$230$	0	0
$231$	0	0
$232$	353.925i	1.52554i
$233$	0	0		−	1.00000i	$-0.5\pi$
$233$	0	0			1.00000i	$0.5\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	0	0
$241$	−465.130	−1.93000	−0.965000	−	0.262248i	$-0.915536\pi$
$241$	−465.130	−1.93000	−0.965000	+	0.262248i	$0.915536\pi$
$242$	242.000i	1.00000i
$243$	363.401i	1.49548i
$244$	64.6590	0.264996
$245$	0	0
$246$	349.999	1.42276
$247$	0	0
$248$	0	0
$249$	138.662	0.556876
$250$	0	0
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	− 940.153i	− 3.73076i
$253$	0	0
$254$	−32.7806	−0.129058
$255$	0	0
$256$	256.000	1.00000
$257$	0	0		−	1.00000i	$-0.5\pi$
$257$	0	0			1.00000i	$0.5\pi$
$258$	678.091i	2.62826i
$259$	0	0
$260$	0	0
$261$	934.923	3.58208
$262$	0	0
$263$	− 333.311i	− 1.26734i	−0.773602	−	0.633672i	$-0.781547\pi$
$263$	− 333.311i	− 1.26734i	0.773602	−	0.633672i	$-0.218453\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 976.924i	− 3.65889i
$268$	464.000i	1.73134i
$269$	−38.0000	−0.141264	−0.0706320	−	0.997502i	$-0.522502\pi$
$269$	−38.0000	−0.141264	−0.0706320	+	0.997502i	$0.522502\pi$
$270$	0	0
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	−954.762	−3.45928
$277$	0	0		−	1.00000i	$-0.5\pi$
$277$	0	0			1.00000i	$0.5\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	117.362	0.417658	0.208829	−	0.977952i	$-0.433035\pi$
$281$	117.362	0.417658	0.208829	+	0.977952i	$0.433035\pi$
$282$	− 994.165i	− 3.52541i
$283$	− 316.000i	− 1.11661i	−0.829637	−	0.558304i	$-0.811452\pi$
$283$	− 316.000i	− 1.11661i	0.829637	−	0.558304i	$-0.188548\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	354.570i	1.23544i
$288$	− 676.245i	− 2.34807i
$289$	−289.000	−1.00000
$290$	0	0
$291$	0	0
$292$	0	0
$293$	0	0		−	1.00000i	$-0.5\pi$
$293$	0	0			1.00000i	$0.5\pi$
$294$	820.101	2.78946
$295$	0	0
$296$	0	0
$297$	0	0
$298$	323.638i	1.08603i
$299$	0	0
$300$	0	0
$301$	−686.948	−2.28222
$302$	0	0
$303$	− 22.3332i	− 0.0737068i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	395.428i	1.28804i	0.765008	+	0.644020i	$0.222735\pi$
$307$	395.428i	1.28804i	−0.765008	+	0.644020i	$0.777265\pi$
$308$	0	0
$309$	−241.530	−0.781651
$310$	0	0
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	0	0
$313$	0	0		−	1.00000i	$-0.5\pi$
$313$	0	0			1.00000i	$0.5\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	0	0		−	1.00000i	$-0.5\pi$
$317$	0	0			1.00000i	$0.5\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	700.211	2.18134
$322$	− 967.233i	− 3.00383i
$323$	0	0
$324$	−701.581	−2.16537
$325$	0	0
$326$	596.568	1.82996
$327$	1056.22i	3.23003i
$328$	255.040i	0.777561i
$329$	1007.15	3.06124
$330$	0	0
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	101.041i	0.304341i
$333$	0	0
$334$	−584.632	−1.75040
$335$	0	0
$336$	976.839	2.90726
$337$	0	0		−	1.00000i	$-0.5\pi$
$337$	0	0			1.00000i	$0.5\pi$
$338$	− 338.000i	− 1.00000i
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	285.832i	0.833331i
$344$	−494.117	−1.43639
$345$	0	0
$346$	0	0
$347$	− 308.338i	− 0.888583i	−0.895882	−	0.444292i	$-0.853455\pi$
$347$	− 308.338i	− 0.888583i	0.895882	−	0.444292i	$-0.146545\pi$
$348$	971.405i	2.79139i
$349$	392.272	1.12399	0.561994	−	0.827141i	$-0.310034\pi$
$349$	392.272	1.12399	0.561994	+	0.827141i	$0.310034\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	0	0		−	1.00000i	$-0.5\pi$
$353$	0	0			1.00000i	$0.5\pi$
$354$	0	0
$355$	0	0
$356$	711.873	1.99964
$357$	0	0
$358$	0	0
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	0	0
$361$	361.000	1.00000
$362$	− 323.334i	− 0.893189i
$363$	664.208i	1.82977i
$364$	0	0
$365$	0	0
$366$	177.467	0.484883
$367$	− 514.755i	− 1.40260i	−0.712866	−	0.701301i	$-0.752603\pi$
$367$	− 514.755i	− 1.40260i	0.712866	−	0.701301i	$-0.247397\pi$
$368$	− 695.723i	− 1.89055i
$369$	673.709	1.82577
$370$	0	0
$371$	0	0
$372$	0	0
$373$	0	0		−	1.00000i	$-0.5\pi$
$373$	0	0			1.00000i	$0.5\pi$
$374$	0	0
$375$	0	0
$376$	724.435	1.92669
$377$	0	0
$378$	− 1481.46i	− 3.91920i
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	0	0
$381$	−89.9718	−0.236146
$382$	0	0
$383$	740.903i	1.93447i	0.253877	+	0.967237i	$0.418294\pi$
$383$	740.903i	1.93447i	−0.253877	+	0.967237i	$0.581706\pi$
$384$	702.633	1.82977
$385$	0	0
$386$	0	0
$387$	1305.25i	3.37274i
$388$	0	0
$389$	776.968	1.99735	0.998674	−	0.0514870i	$-0.0163961\pi$
$389$	776.968	1.99735	0.998674	+	0.0514870i	$0.0163961\pi$
$390$	0	0
$391$	0	0
$392$	597.597i	1.52448i
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	0	0		−	1.00000i	$-0.5\pi$
$397$	0	0			1.00000i	$0.5\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	8.18372	0.0204083	0.0102041	−	0.999948i	$-0.496752\pi$
$401$	8.18372	0.0204083	0.0102041	+	0.999948i	$0.496752\pi$
$402$	1273.52i	3.16797i
$403$	0	0
$404$	16.2739	0.0402819
$405$	0	0
$406$	−984.093	−2.42387
$407$	0	0
$408$	0	0
$409$	−554.200	−1.35501	−0.677506	−	0.735517i	$-0.736939\pi$
$409$	−554.200	−1.35501	−0.677506	+	0.735517i	$0.736939\pi$
$410$	0	0
$411$	0	0
$412$	− 176.000i	− 0.427184i
$413$	0	0
$414$	−1837.81	−4.43916
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	0	0	1.00000			$0$
$419$	0	0	−1.00000			$\pi$
$420$	0	0
$421$	−546.650	−1.29845	−0.649227	−	0.760594i	$-0.724908\pi$
$421$	−546.650	−1.29845	−0.649227	+	0.760594i	$0.724908\pi$
$422$	0	0
$423$	− 1913.66i	− 4.52401i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	179.785i	0.421042i
$428$	510.235i	1.19214i
$429$	0	0
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	− 1065.60i	− 2.46667i
$433$	0	0		−	1.00000i	$-0.5\pi$
$433$	0	0			1.00000i	$0.5\pi$
$434$	0	0
$435$	0	0
$436$	−769.654	−1.76526
$437$	0	0
$438$	0	0
$439$	0	0	1.00000			$0$
$439$	0	0	−1.00000			$\pi$
$440$	0	0
$441$	1578.60	3.57960
$442$	0	0
$443$	872.671i	1.96991i	0.172808	+	0.984955i	$0.444716\pi$
$443$	872.671i	1.96991i	−0.172808	+	0.984955i	$0.555284\pi$
$444$	0	0
$445$	0	0
$446$	90.8261	0.203646
$447$	888.277i	1.98720i
$448$	711.810i	1.58886i
$449$	−398.000	−0.886414	−0.443207	−	0.896419i	$-0.646159\pi$
$449$	−398.000	−0.886414	−0.443207	+	0.896419i	$0.646159\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	−244.809	−0.539228
$455$	0	0
$456$	0	0
$457$	0	0		−	1.00000i	$-0.5\pi$
$457$	0	0			1.00000i	$0.5\pi$
$458$	− 865.539i	− 1.88982i
$459$	0	0
$460$	0	0
$461$	−901.999	−1.95661	−0.978307	−	0.207159i	$-0.933578\pi$
$461$	−901.999	−1.95661	−0.978307	+	0.207159i	$0.933578\pi$
$462$	0	0
$463$	− 310.536i	− 0.670705i	−0.942093	−	0.335352i	$-0.891145\pi$
$463$	− 310.536i	− 0.670705i	0.942093	−	0.335352i	$-0.108855\pi$
$464$	−707.851	−1.52554
$465$	0	0
$466$	0	0
$467$	443.814i	0.950350i	0.879891	+	0.475175i	$0.157615\pi$
$467$	443.814i	0.950350i	−0.879891	+	0.475175i	$0.842385\pi$
$468$	0	0
$469$	−1290.16	−2.75087
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	0	0
$481$	0	0
$482$	− 930.260i	− 1.93000i
$483$	− 2654.73i	− 5.49633i
$484$	−484.000	−1.00000
$485$	0	0
$486$	−726.801	−1.49548
$487$	− 654.301i	− 1.34353i	−0.740763	−	0.671767i	$-0.765536\pi$
$487$	− 654.301i	− 1.34353i	0.740763	−	0.671767i	$-0.234464\pi$
$488$	129.318i	0.264996i
$489$	1637.38	3.34842
$490$	0	0
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	699.998i	1.42276i
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	277.324i	0.556876i
$499$	0	0	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	0	0
$501$	−1604.62	−3.20283
$502$	0	0
$503$	496.595i	0.987266i	0.869670	+	0.493633i	$0.164331\pi$
$503$	496.595i	0.987266i	−0.869670	+	0.493633i	$0.835669\pi$
$504$	1880.31	3.73076
$505$	0	0
$506$	0	0
$507$	− 927.695i	− 1.82977i
$508$	− 65.5613i	− 0.129058i
$509$	982.000	1.92927	0.964637	−	0.263584i	$-0.0849045\pi$
$509$	982.000	1.92927	0.964637	+	0.263584i	$0.0849045\pi$
$510$	0	0
$511$	0	0
$512$	512.000i	1.00000i
$513$	0	0
$514$	0	0
$515$	0	0
$516$	−1356.18	−2.62826
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	937.657	1.79973	0.899863	−	0.436173i	$-0.143667\pi$
$521$	937.657	1.79973	0.899863	+	0.436173i	$0.143667\pi$
$522$	1869.85i	3.58208i
$523$	1033.18i	1.97549i	0.156081	+	0.987744i	$0.450114\pi$
$523$	1033.18i	1.97549i	−0.156081	+	0.987744i	$0.549886\pi$
$524$	0	0
$525$	0	0
$526$	666.623	1.26734
$527$	0	0
$528$	0	0
$529$	−1361.75	−2.57419
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	1953.85	3.65889
$535$	0	0
$536$	−928.000	−1.73134
$537$	0	0
$538$	− 76.0000i	− 0.141264i
$539$	0	0
$540$	0	0
$541$	1081.61	1.99927	0.999636	−	0.0269817i	$-0.00858959\pi$
$541$	1081.61	1.99927	0.999636	+	0.0269817i	$0.00858959\pi$
$542$	0	0
$543$	− 887.443i	− 1.63433i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 963.720i	− 1.76183i	−0.473276	−	0.880914i	$-0.656929\pi$
$547$	− 963.720i	− 1.76183i	0.473276	−	0.880914i	$-0.343071\pi$
$548$	0	0
$549$	341.604	0.622230
$550$	0	0
$551$	0	0
$552$	− 1909.52i	− 3.45928i
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	0	0		−	1.00000i	$-0.5\pi$
$557$	0	0			1.00000i	$0.5\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	234.724i	0.417658i
$563$	1124.00i	1.99645i	0.0595755	+	0.998224i	$0.481025\pi$
$563$	1124.00i	1.99645i	−0.0595755	+	0.998224i	$0.518975\pi$
$564$	1988.33	3.52541
$565$	0	0
$566$	632.000	1.11661
$567$	− 1950.75i	− 3.44048i
$568$	0	0
$569$	−1120.64	−1.96950	−0.984749	−	0.173979i	$-0.944337\pi$
$569$	−1120.64	−1.96950	−0.984749	+	0.173979i	$0.944337\pi$
$570$	0	0
$571$	0	0	1.00000			$0$
$571$	0	0	−1.00000			$\pi$
$572$	0	0
$573$	0	0
$574$	−709.141	−1.23544
$575$	0	0
$576$	1352.49	2.34807
$577$	0	0		−	1.00000i	$-0.5\pi$
$577$	0	0			1.00000i	$0.5\pi$
$578$	− 578.000i	− 1.00000i
$579$	0	0
$580$	0	0
$581$	−280.946	−0.483557
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 1076.00i	− 1.83305i	−0.399978	−	0.916525i	$-0.630982\pi$
$587$	− 1076.00i	− 1.83305i	0.399978	−	0.916525i	$-0.369018\pi$
$588$	1640.20i	2.78946i
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	0	0		−	1.00000i	$-0.5\pi$
$593$	0	0			1.00000i	$0.5\pi$
$594$	0	0
$595$	0	0
$596$	−647.276	−1.08603
$597$	0	0
$598$	0	0
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	0	0
$601$	−1200.99	−1.99832	−0.999159	−	0.0410035i	$-0.986945\pi$
$601$	−1200.99	−1.99832	−0.999159	+	0.0410035i	$0.986945\pi$
$602$	− 1373.90i	− 2.28222i
$603$	2451.39i	4.06532i
$604$	0	0
$605$	0	0
$606$	44.6663	0.0737068
$607$	964.000i	1.58814i	0.607827	+	0.794069i	$0.292041\pi$
$607$	964.000i	1.58814i	−0.607827	+	0.794069i	$0.707959\pi$
$608$	0	0
$609$	−2701.00	−4.43514
$610$	0	0
$611$	0	0
$612$	0	0
$613$	0	0		−	1.00000i	$-0.5\pi$
$613$	0	0			1.00000i	$0.5\pi$
$614$	−790.857	−1.28804
$615$	0	0
$616$	0	0
$617$	0	0		−	1.00000i	$-0.5\pi$
$617$	0	0			1.00000i	$0.5\pi$
$618$	− 483.060i	− 0.781651i
$619$	0	0	1.00000			$0$
$619$	0	0	−1.00000			$\pi$
$620$	0	0
$621$	−2895.95	−4.66337
$622$	0	0
$623$	1979.37i	3.17716i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−913.091	−1.42448	−0.712239	−	0.701937i	$-0.752319\pi$
$641$	−913.091	−1.42448	−0.712239	+	0.701937i	$0.752319\pi$
$642$	1400.42i	2.18134i
$643$	− 1044.47i	− 1.62436i	−0.583404	−	0.812182i	$-0.698280\pi$
$643$	− 1044.47i	− 1.62436i	0.583404	−	0.812182i	$-0.301720\pi$
$644$	1934.47	3.00383
$645$	0	0
$646$	0	0
$647$	− 956.000i	− 1.47759i	−0.673931	−	0.738794i	$-0.735395\pi$
$647$	− 956.000i	− 1.47759i	0.673931	−	0.738794i	$-0.264605\pi$
$648$	− 1403.16i	− 2.16537i
$649$	0	0
$650$	0	0
$651$	0	0
$652$	1193.14i	1.82996i
$653$	0	0		−	1.00000i	$-0.5\pi$
$653$	0	0			1.00000i	$0.5\pi$
$654$	−2112.44	−3.23003
$655$	0	0
$656$	−510.080	−0.777561
$657$	0	0
$658$	2014.30i	3.06124i
$659$	0	0	1.00000			$0$
$659$	0	0	−1.00000			$\pi$
$660$	0	0
$661$	1132.85	1.71384	0.856921	−	0.515447i	$-0.172374\pi$
$661$	1132.85	1.71384	0.856921	+	0.515447i	$0.172374\pi$
$662$	0	0
$663$	0	0
$664$	−202.083	−0.304341
$665$	0	0
$666$	0	0
$667$	1923.70i	2.88411i
$668$	− 1169.26i	− 1.75040i
$669$	249.287	0.372626
$670$	0	0
$671$	0	0
$672$	1953.68i	2.90726i
$673$	0	0		−	1.00000i	$-0.5\pi$
$673$	0	0			1.00000i	$0.5\pi$
$674$	0	0
$675$	0	0
$676$	676.000	1.00000
$677$	0	0		−	1.00000i	$-0.5\pi$
$677$	0	0			1.00000i	$0.5\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−671.919	−0.986665
$682$	0	0
$683$	− 1358.47i	− 1.98898i	−0.104845	−	0.994489i	$-0.533435\pi$
$683$	− 1358.47i	− 1.98898i	0.104845	−	0.994489i	$-0.466565\pi$
$684$	0	0
$685$	0	0
$686$	−571.665	−0.833331
$687$	− 2375.61i	− 3.45795i
$688$	− 988.233i	− 1.43639i
$689$	0	0
$690$	0	0
$691$	0	0	1.00000			$0$
$691$	0	0	−1.00000			$\pi$
$692$	0	0
$693$	0	0
$694$	616.677	0.888583
$695$	0	0
$696$	−1942.81	−2.79139
$697$	0	0
$698$	784.544i	1.12399i
$699$	0	0
$700$	0	0
$701$	902.000	1.28673	0.643367	−	0.765558i	$-0.277537\pi$
$701$	902.000	1.28673	0.643367	+	0.765558i	$0.277537\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	45.2497i	0.0640024i
$708$	0	0
$709$	1290.20	1.81975	0.909875	−	0.414882i	$-0.136177\pi$
$709$	1290.20	1.81975	0.909875	+	0.414882i	$0.136177\pi$
$710$	0	0
$711$	0	0
$712$	1423.75i	1.99964i
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	0	0	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	0	0
$721$	489.370	0.678737
$722$	722.000i	1.00000i
$723$	− 2553.25i	− 3.53147i
$724$	646.669	0.893189
$725$	0	0
$726$	−1328.42	−1.82977
$727$	− 226.832i	− 0.312012i	−0.987756	−	0.156006i	$-0.950138\pi$
$727$	− 226.832i	− 0.312012i	0.987756	−	0.156006i	$-0.0498619\pi$
$728$	0	0
$729$	−416.265	−0.571008
$730$	0	0
$731$	0	0
$732$	354.934i	0.484883i
$733$	0	0		−	1.00000i	$-0.5\pi$
$733$	0	0			1.00000i	$0.5\pi$
$734$	1029.51	1.40260
$735$	0	0
$736$	1391.45	1.89055
$737$	0	0
$738$	1347.42i	1.82577i
$739$	0	0	1.00000			$0$
$739$	0	0	−1.00000			$\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	764.000i	1.02826i	0.857711	+	0.514132i	$0.171886\pi$
$743$	764.000i	1.02826i	−0.857711	+	0.514132i	$0.828114\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	533.818i	0.714616i
$748$	0	0
$749$	−1418.71	−1.89414
$750$	0	0
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	1448.87i	1.92669i
$753$	0	0
$754$	0	0
$755$	0	0
$756$	2962.91	3.91920
$757$	0	0		−	1.00000i	$-0.5\pi$
$757$	0	0			1.00000i	$0.5\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	687.446	0.903346	0.451673	−	0.892184i	$-0.350827\pi$
$761$	687.446	0.903346	0.451673	+	0.892184i	$0.350827\pi$
$762$	− 179.944i	− 0.236146i
$763$	− 2140.03i	− 2.80476i
$764$	0	0
$765$	0	0
$766$	−1481.81	−1.93447
$767$	0	0
$768$	1405.27i	1.82977i
$769$	1129.25	1.46846	0.734231	−	0.678900i	$-0.237543\pi$
$769$	1129.25	1.46846	0.734231	+	0.678900i	$0.237543\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	0	0		−	1.00000i	$-0.5\pi$
$773$	0	0			1.00000i	$0.5\pi$
$774$	−2610.50	−3.37274
$775$	0	0
$776$	0	0
$777$	0	0
$778$	1553.94i	1.99735i
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	2946.43i	3.76300i
$784$	−1195.19	−1.52448
$785$	0	0
$786$	0	0
$787$	1457.05i	1.85139i	0.378267	+	0.925696i	$0.376520\pi$
$787$	1457.05i	1.85139i	−0.378267	+	0.925696i	$0.623480\pi$
$788$	0	0
$789$	1829.65	2.31895
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	0	0		−	1.00000i	$-0.5\pi$
$797$	0	0			1.00000i	$0.5\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	3760.94	4.69531
$802$	16.3674i	0.0204083i
$803$	0	0
$804$	−2547.05	−3.16797
$805$	0	0
$806$	0	0
$807$	− 208.594i	− 0.258481i
$808$	32.5478i	0.0402819i
$809$	1617.89	1.99987	0.999934	−	0.0114648i	$-0.00364945\pi$
$809$	1617.89	1.99987	0.999934	+	0.0114648i	$0.00364945\pi$
$810$	0	0
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	− 1968.19i	− 2.42387i
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	− 1108.40i	− 1.35501i
$819$	0	0
$820$	0	0
$821$	347.659	0.423458	0.211729	−	0.977328i	$-0.432091\pi$
$821$	347.659	0.423458	0.211729	+	0.977328i	$0.432091\pi$
$822$	0	0
$823$	− 1396.00i	− 1.69623i	−0.529810	−	0.848117i	$-0.677737\pi$
$823$	− 1396.00i	− 1.69623i	0.529810	−	0.848117i	$-0.322263\pi$
$824$	352.000	0.427184
$825$	0	0
$826$	0	0
$827$	− 596.000i	− 0.720677i	−0.932822	−	0.360339i	$-0.882661\pi$
$827$	− 596.000i	− 0.720677i	0.932822	−	0.360339i	$-0.117339\pi$
$828$	− 3675.62i	− 4.43916i
$829$	257.820	0.311001	0.155500	−	0.987836i	$-0.450301\pi$
$829$	257.820	0.311001	0.155500	+	0.987836i	$0.450301\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	0	0
$841$	1116.24	1.32727
$842$	− 1093.30i	− 1.29845i
$843$	644.238i	0.764221i
$844$	0	0
$845$	0	0
$846$	3827.31	4.52401
$847$	− 1345.77i	− 1.58886i
$848$	0	0
$849$	1734.63	2.04314
$850$	0	0
$851$	0	0
$852$	0	0
$853$	0	0		−	1.00000i	$-0.5\pi$
$853$	0	0			1.00000i	$0.5\pi$
$854$	−359.570	−0.421042
$855$	0	0
$856$	−1020.47	−1.19214
$857$	0	0		−	1.00000i	$-0.5\pi$
$857$	0	0			1.00000i	$0.5\pi$
$858$	0	0
$859$	0	0	1.00000			$0$
$859$	0	0	−1.00000			$\pi$
$860$	0	0
$861$	−1946.35	−2.26057
$862$	0	0
$863$	1000.23i	1.15901i	0.814968	+	0.579506i	$0.196754\pi$
$863$	1000.23i	1.15901i	−0.814968	+	0.579506i	$0.803246\pi$
$864$	2131.20	2.46667
$865$	0	0
$866$	0	0
$867$	− 1586.41i	− 1.82977i
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	− 1539.31i	− 1.76526i
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	0	0		−	1.00000i	$-0.5\pi$
$877$	0	0			1.00000i	$0.5\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	1719.06	1.95126	0.975630	−	0.219423i	$-0.0704175\pi$
$881$	1719.06	1.95126	0.975630	+	0.219423i	$0.0704175\pi$
$882$	3157.20i	3.57960i
$883$	314.683i	0.356379i	0.983996	+	0.178190i	$0.0570240\pi$
$883$	314.683i	0.356379i	−0.983996	+	0.178190i	$0.942976\pi$
$884$	0	0
$885$	0	0
$886$	−1745.34	−1.96991
$887$	− 1655.23i	− 1.86610i	−0.359741	−	0.933052i	$-0.617135\pi$
$887$	− 1655.23i	− 1.86610i	0.359741	−	0.933052i	$-0.382865\pi$
$888$	0	0
$889$	182.294	0.205055
$890$	0	0
$891$	0	0
$892$	181.652i	0.203646i
$893$	0	0
$894$	−1776.55	−1.98720
$895$	0	0
$896$	−1423.62	−1.58886
$897$	0	0
$898$	− 796.000i	− 0.886414i
$899$	0	0
$900$	0	0
$901$	0	0
$902$	0	0
$903$	− 3770.88i	− 4.17594i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	1602.83i	1.76718i	0.468266	+	0.883588i	$0.344879\pi$
$907$	1602.83i	1.76718i	−0.468266	+	0.883588i	$0.655121\pi$
$908$	− 489.619i	− 0.539228i
$909$	85.9776	0.0945849
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	1731.08	1.88982
$917$	0	0
$918$	0	0
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	0	0
$921$	−2170.63	−2.35682
$922$	− 1804.00i	− 1.95661i
$923$	0	0
$924$	0	0
$925$	0	0
$926$	621.073	0.670705
$927$	− 929.837i	− 1.00306i
$928$	− 1415.70i	− 1.52554i
$929$	586.280	0.631087	0.315544	−	0.948911i	$-0.397813\pi$
$929$	586.280	0.631087	0.315544	+	0.948911i	$0.397813\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	−887.627	−0.950350
$935$	0	0
$936$	0	0
$937$	0	0		−	1.00000i	$-0.5\pi$
$937$	0	0			1.00000i	$0.5\pi$
$938$	− 2580.31i	− 2.75087i
$939$	0	0
$940$	0	0
$941$	−118.000	−0.125399	−0.0626993	−	0.998032i	$-0.519971\pi$
$941$	−118.000	−0.125399	−0.0626993	+	0.998032i	$0.519971\pi$
$942$	0	0
$943$	1386.23i	1.47002i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	8.79689i	0.00928921i	0.999989	+	0.00464461i	$0.00147843\pi$
$947$	8.79689i	0.00928921i	−0.999989	+	0.00464461i	$0.998522\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	0	0		−	1.00000i	$-0.5\pi$
$953$	0	0			1.00000i	$0.5\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	961.000	1.00000
$962$	0	0
$963$	2695.66i	2.79923i
$964$	1860.52	1.93000
$965$	0	0
$966$	5309.45	5.49633
$967$	− 1749.25i	− 1.80894i	−0.426536	−	0.904470i	$-0.640266\pi$
$967$	− 1749.25i	− 1.80894i	0.426536	−	0.904470i	$-0.359734\pi$
$968$	− 968.000i	− 1.00000i
$969$	0	0
$970$	0	0
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	− 1453.60i	− 1.49548i
$973$	0	0
$974$	1308.60	1.34353
$975$	0	0
$976$	−258.636	−0.264996
$977$	0	0		−	1.00000i	$-0.5\pi$
$977$	0	0			1.00000i	$0.5\pi$
$978$	3274.75i	3.34842i
$979$	0	0
$980$	0	0
$981$	−4066.21	−4.14496
$982$	0	0
$983$	284.000i	0.288911i	0.989511	+	0.144456i	$0.0461431\pi$
$983$	284.000i	0.288911i	−0.989511	+	0.144456i	$0.953857\pi$
$984$	−1400.00	−1.42276
$985$	0	0
$986$	0	0
$987$	5528.57i	5.60138i
$988$	0	0
$989$	−2685.69	−2.71556
$990$	0	0
$991$	0	0	1.00000			$0$
$991$	0	0	−1.00000			$\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	−554.649	−0.556876
$997$	0	0		−	1.00000i	$-0.5\pi$
$997$	0	0			1.00000i	$0.5\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	500.3.b.a.251.8	yes	8
4.3	odd	2	inner	500.3.b.a.251.1	✓	8
5.2	odd	4		500.3.d.a.499.4		4
5.3	odd	4		500.3.d.b.499.1		4
5.4	even	2	inner	500.3.b.a.251.1	✓	8
20.3	even	4		500.3.d.a.499.4		4
20.7	even	4		500.3.d.b.499.1		4
20.19	odd	2	CM	500.3.b.a.251.8	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
500.3.b.a.251.1	✓	8	4.3	odd	2	inner
500.3.b.a.251.1	✓	8	5.4	even	2	inner
500.3.b.a.251.8	yes	8	1.1	even	1	trivial
500.3.b.a.251.8	yes	8	20.19	odd	2	CM
500.3.d.a.499.4		4	5.2	odd	4
500.3.d.a.499.4		4	20.3	even	4
500.3.d.b.499.1		4	5.3	odd	4
500.3.d.b.499.1		4	20.7	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:	\( N \)	\(=\)	\( 500 = 2^{2} \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	500.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.00000i q^{2} +5.48932i q^{3} -4.00000 q^{4} -10.9786 q^{6} -11.1220i q^{7} -8.00000i q^{8} -21.1327 q^{9} +O(q^{10})\)	\(q+2.00000i q^{2} +5.48932i q^{3} -4.00000 q^{4} -10.9786 q^{6} -11.1220i q^{7} -8.00000i q^{8} -21.1327 q^{9} -21.9573i q^{12} +22.2441 q^{14} +16.0000 q^{16} -42.2653i q^{18} +61.0524 q^{21} -43.4827i q^{23} +43.9146 q^{24} -66.6001i q^{27} +44.4881i q^{28} -44.2407 q^{29} +32.0000i q^{32} +84.5306 q^{36} -31.8800 q^{41} +122.105i q^{42} -61.7646i q^{43} +86.9654 q^{46} +90.5544i q^{47} +87.8292i q^{48} -74.6997 q^{49} +133.200 q^{54} -88.9763 q^{56} -88.4813i q^{58} -16.1647 q^{61} +235.038i q^{63} -64.0000 q^{64} -116.000i q^{67} +238.691 q^{69} +169.061i q^{72} +175.395 q^{81} -63.7600i q^{82} -25.2603i q^{83} -244.210 q^{84} +123.529 q^{86} -242.851i q^{87} -177.968 q^{89} +173.931i q^{92} -181.109 q^{94} -175.658 q^{96} -149.399i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 32 q^{4} + 16 q^{6} - 76 q^{9}+O(q^{10}) \) 8 * q - 32 * q^4 + 16 * q^6 - 76 * q^9	\( 8 q - 32 q^{4} + 16 q^{6} - 76 q^{9} + 16 q^{14} + 128 q^{16} + 32 q^{21} - 64 q^{24} - 44 q^{29} + 304 q^{36} - 124 q^{41} + 176 q^{46} - 556 q^{49} + 32 q^{54} - 64 q^{56} + 116 q^{61} - 512 q^{64} + 352 q^{69} + 1000 q^{81} - 128 q^{84} - 304 q^{86} - 284 q^{89} + 16 q^{94} + 256 q^{96}+O(q^{100}) \) 8 * q - 32 * q^4 + 16 * q^6 - 76 * q^9 + 16 * q^14 + 128 * q^16 + 32 * q^21 - 64 * q^24 - 44 * q^29 + 304 * q^36 - 124 * q^41 + 176 * q^46 - 556 * q^49 + 32 * q^54 - 64 * q^56 + 116 * q^61 - 512 * q^64 + 352 * q^69 + 1000 * q^81 - 128 * q^84 - 304 * q^86 - 284 * q^89 + 16 * q^94 + 256 * q^96

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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