LMFDB - Embedded newform 500.3.d.a.499.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [500,3,Mod(499,500)]

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, 1]))

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

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

chi := DirichletCharacter("500.499");

S:= CuspForms(chi, 3);

N := Newforms(S);

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

Embedding invariants

Embedding label			499.4
Root			$-1.90211$ of defining polynomial
Character	$\chi$	$=$	500.499

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000 q^{2} +5.48932 q^{3} +4.00000 q^{4} -10.9786 q^{6} +11.1220 q^{7} -8.00000 q^{8} +21.1327 q^{9} +O(q^{10})$	$q-2.00000 q^{2} +5.48932 q^{3} +4.00000 q^{4} -10.9786 q^{6} +11.1220 q^{7} -8.00000 q^{8} +21.1327 q^{9} +21.9573 q^{12} -22.2441 q^{14} +16.0000 q^{16} -42.2653 q^{18} +61.0524 q^{21} -43.4827 q^{23} -43.9146 q^{24} +66.6001 q^{27} +44.4881 q^{28} +44.2407 q^{29} -32.0000 q^{32} +84.5306 q^{36} -31.8800 q^{41} -122.105 q^{42} -61.7646 q^{43} +86.9654 q^{46} -90.5544 q^{47} +87.8292 q^{48} +74.6997 q^{49} -133.200 q^{54} -88.9763 q^{56} -88.4813 q^{58} -16.1647 q^{61} +235.038 q^{63} +64.0000 q^{64} +116.000 q^{67} -238.691 q^{69} -169.061 q^{72} +175.395 q^{81} +63.7600 q^{82} -25.2603 q^{83} +244.210 q^{84} +123.529 q^{86} +242.851 q^{87} +177.968 q^{89} -173.931 q^{92} +181.109 q^{94} -175.658 q^{96} -149.399 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{2} - 4 q^{3} + 16 q^{4} + 8 q^{6} + 4 q^{7} - 32 q^{8} + 38 q^{9}+O(q^{10}) $ 4 * q - 8 * q^2 - 4 * q^3 + 16 * q^4 + 8 * q^6 + 4 * q^7 - 32 * q^8 + 38 * q^9	$ 4 q - 8 q^{2} - 4 q^{3} + 16 q^{4} + 8 q^{6} + 4 q^{7} - 32 q^{8} + 38 q^{9} - 16 q^{12} - 8 q^{14} + 64 q^{16} - 76 q^{18} + 16 q^{21} - 44 q^{23} + 32 q^{24} + 8 q^{27} + 16 q^{28} + 22 q^{29} - 128 q^{32} + 152 q^{36} - 62 q^{41} - 32 q^{42} + 76 q^{43} + 88 q^{46} + 4 q^{47} - 64 q^{48} + 278 q^{49} - 16 q^{54} - 32 q^{56} - 44 q^{58} + 58 q^{61} + 648 q^{63} + 256 q^{64} + 464 q^{67} - 176 q^{69} - 304 q^{72} + 500 q^{81} + 124 q^{82} + 76 q^{83} + 64 q^{84} - 152 q^{86} + 468 q^{87} + 142 q^{89} - 176 q^{92} - 8 q^{94} + 128 q^{96} - 556 q^{98}+O(q^{100}) $ 4 * q - 8 * q^2 - 4 * q^3 + 16 * q^4 + 8 * q^6 + 4 * q^7 - 32 * q^8 + 38 * q^9 - 16 * q^12 - 8 * q^14 + 64 * q^16 - 76 * q^18 + 16 * q^21 - 44 * q^23 + 32 * q^24 + 8 * q^27 + 16 * q^28 + 22 * q^29 - 128 * q^32 + 152 * q^36 - 62 * q^41 - 32 * q^42 + 76 * q^43 + 88 * q^46 + 4 * q^47 - 64 * q^48 + 278 * q^49 - 16 * q^54 - 32 * q^56 - 44 * q^58 + 58 * q^61 + 648 * q^63 + 256 * q^64 + 464 * q^67 - 176 * q^69 - 304 * q^72 + 500 * q^81 + 124 * q^82 + 76 * q^83 + 64 * q^84 - 152 * q^86 + 468 * q^87 + 142 * q^89 - 176 * q^92 - 8 * q^94 + 128 * q^96 - 556 * q^98

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 $

$2$	−2.00000	−1.00000
$2$	−2.00000	−1.00000
$3$	5.48932	1.82977	0.914887	−	0.403710i	$-0.132280\pi$
$3$	5.48932	1.82977	0.914887	+	0.403710i	$0.132280\pi$
$4$	4.00000	1.00000
$5$	0	0
$5$	0	0
$6$	−10.9786	−1.82977
$7$	11.1220	1.58886	0.794431	−	0.607354i	$-0.207769\pi$
$7$	11.1220	1.58886	0.794431	+	0.607354i	$0.207769\pi$
$8$	−8.00000	−1.00000
$9$	21.1327	2.34807
$10$	0	0
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	21.9573	1.82977
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	−22.2441	−1.58886
$15$	0	0
$16$	16.0000	1.00000
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	−42.2653	−2.34807
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	0	0
$21$	61.0524	2.90726
$22$	0	0
$23$	−43.4827	−1.89055	−0.945276	−	0.326271i	$-0.894208\pi$
$23$	−43.4827	−1.89055	−0.945276	+	0.326271i	$0.894208\pi$
$24$	−43.9146	−1.82977
$25$	0	0
$26$	0	0
$27$	66.6001	2.46667
$28$	44.4881	1.58886
$29$	44.2407	1.52554	0.762770	−	0.646670i	$-0.223839\pi$
$29$	44.2407	1.52554	0.762770	+	0.646670i	$0.223839\pi$
$30$	0	0
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	−32.0000	−1.00000
$33$	0	0
$34$	0	0
$35$	0	0
$36$	84.5306	2.34807
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	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	$0.627104\pi$
$42$	−122.105	−2.90726
$43$	−61.7646	−1.43639	−0.718193	−	0.695844i	$-0.755030\pi$
$43$	−61.7646	−1.43639	−0.718193	+	0.695844i	$0.755030\pi$
$44$	0	0
$45$	0	0
$46$	86.9654	1.89055
$47$	−90.5544	−1.92669	−0.963345	−	0.268267i	$-0.913549\pi$
$47$	−90.5544	−1.92669	−0.963345	+	0.268267i	$0.913549\pi$
$48$	87.8292	1.82977
$49$	74.6997	1.52448
$50$	0	0
$51$	0	0
$52$	0	0
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	−133.200	−2.46667
$55$	0	0
$56$	−88.9763	−1.58886
$57$	0	0
$58$	−88.4813	−1.52554
$59$	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
$63$	235.038	3.73076
$64$	64.0000	1.00000
$65$	0	0
$66$	0	0
$67$	116.000	1.73134	0.865672	−	0.500612i	$-0.166892\pi$
$67$	116.000	1.73134	0.865672	+	0.500612i	$0.166892\pi$
$68$	0	0
$69$	−238.691	−3.45928
$70$	0	0
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	−169.061	−2.34807
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	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
$82$	63.7600	0.777561
$83$	−25.2603	−0.304341	−0.152171	−	0.988354i	$-0.548626\pi$
$83$	−25.2603	−0.304341	−0.152171	+	0.988354i	$0.548626\pi$
$84$	244.210	2.90726
$85$	0	0
$86$	123.529	1.43639
$87$	242.851	2.79139
$88$	0	0
$89$	177.968	1.99964	0.999821	−	0.0189175i	$-0.00602198\pi$
$89$	177.968	1.99964	0.999821	+	0.0189175i	$0.00602198\pi$
$90$	0	0
$91$	0	0
$92$	−173.931	−1.89055
$93$	0	0
$94$	181.109	1.92669
$95$	0	0
$96$	−175.658	−1.82977
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	−149.399	−1.52448
$99$	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
$103$	44.0000	0.427184	0.213592	−	0.976923i	$-0.431484\pi$
$103$	44.0000	0.427184	0.213592	+	0.976923i	$0.431484\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	127.559	1.19214	0.596069	−	0.802933i	$-0.296728\pi$
$107$	127.559	1.19214	0.596069	+	0.802933i	$0.296728\pi$
$108$	266.400	2.46667
$109$	−192.414	−1.76526	−0.882631	−	0.470067i	$-0.844230\pi$
$109$	−192.414	−1.76526	−0.882631	+	0.470067i	$0.844230\pi$
$110$	0	0
$111$	0	0
$112$	177.953	1.58886
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	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
$122$	32.3295	0.264996
$123$	−175.000	−1.42276
$124$	0	0
$125$	0	0
$126$	−470.076	−3.73076
$127$	−16.3903	−0.129058	−0.0645288	−	0.997916i	$-0.520554\pi$
$127$	−16.3903	−0.129058	−0.0645288	+	0.997916i	$0.520554\pi$
$128$	−128.000	−1.00000
$129$	−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	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	477.381	3.45928
$139$	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
$147$	410.050	2.78946
$148$	0	0
$149$	−161.819	−1.08603	−0.543017	−	0.839722i	$-0.682718\pi$
$149$	−161.819	−1.08603	−0.543017	+	0.839722i	$0.682718\pi$
$150$	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	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−483.616	−3.00383
$162$	−350.791	−2.16537
$163$	−298.284	−1.82996	−0.914982	−	0.403495i	$-0.867795\pi$
$163$	−298.284	−1.82996	−0.914982	+	0.403495i	$0.867795\pi$
$164$	−127.520	−0.777561
$165$	0	0
$166$	50.5207	0.304341
$167$	−292.316	−1.75040	−0.875198	−	0.483765i	$-0.839269\pi$
$167$	−292.316	−1.75040	−0.875198	+	0.483765i	$0.839269\pi$
$168$	−488.419	−2.90726
$169$	169.000	1.00000
$170$	0	0
$171$	0	0
$172$	−247.058	−1.43639
$173$	0	0	1.00000			$0$
$173$	0	0	−1.00000			$\pi$
$174$	−485.703	−2.79139
$175$	0	0
$176$	0	0
$177$	0	0
$178$	−355.936	−1.99964
$179$	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
$183$	−88.7335	−0.484883
$184$	347.862	1.89055
$185$	0	0
$186$	0	0
$187$	0	0
$188$	−362.218	−1.92669
$189$	740.728	3.91920
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	351.317	1.82977
$193$	0	0	1.00000			$0$
$193$	0	0	−1.00000			$\pi$
$194$	0	0
$195$	0	0
$196$	298.799	1.52448
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	0	0
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	0	0
$201$	636.761	3.16797
$202$	8.13695	0.0402819
$203$	492.046	2.42387
$204$	0	0
$205$	0	0
$206$	−88.0000	−0.427184
$207$	−918.905	−4.43916
$208$	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
$218$	384.827	1.76526
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−45.4131	−0.203646	−0.101823	−	0.994803i	$-0.532468\pi$
$223$	−45.4131	−0.203646	−0.101823	+	0.994803i	$0.532468\pi$
$224$	−355.905	−1.58886
$225$	0	0
$226$	0	0
$227$	−122.405	−0.539228	−0.269614	−	0.962968i	$-0.586896\pi$
$227$	−122.405	−0.539228	−0.269614	+	0.962968i	$0.586896\pi$
$228$	0	0
$229$	432.770	1.88982	0.944912	−	0.327326i	$-0.106147\pi$
$229$	432.770	1.88982	0.944912	+	0.327326i	$0.106147\pi$
$230$	0	0
$231$	0	0
$232$	−353.925	−1.52554
$233$	0	0	1.00000			$0$
$233$	0	0	−1.00000			$\pi$
$234$	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	$0.915536\pi$
$242$	−242.000	−1.00000
$243$	363.401	1.49548
$244$	−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			$\pi$
$252$	940.153	3.73076
$253$	0	0
$254$	32.7806	0.129058
$255$	0	0
$256$	256.000	1.00000
$257$	0	0	1.00000			$0$
$257$	0	0	−1.00000			$\pi$
$258$	678.091	2.62826
$259$	0	0
$260$	0	0
$261$	934.923	3.58208
$262$	0	0
$263$	−333.311	−1.26734	−0.633672	−	0.773602i	$-0.718453\pi$
$263$	−333.311	−1.26734	−0.633672	+	0.773602i	$0.718453\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	976.924	3.65889
$268$	464.000	1.73134
$269$	38.0000	0.141264	0.0706320	−	0.997502i	$-0.477498\pi$
$269$	38.0000	0.141264	0.0706320	+	0.997502i	$0.477498\pi$
$270$	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	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	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	$0.433035\pi$
$282$	994.165	3.52541
$283$	−316.000	−1.11661	−0.558304	−	0.829637i	$-0.688548\pi$
$283$	−316.000	−1.11661	−0.558304	+	0.829637i	$0.688548\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−354.570	−1.23544
$288$	−676.245	−2.34807
$289$	289.000	1.00000
$290$	0	0
$291$	0	0
$292$	0	0
$293$	0	0	1.00000			$0$
$293$	0	0	−1.00000			$\pi$
$294$	−820.101	−2.78946
$295$	0	0
$296$	0	0
$297$	0	0
$298$	323.638	1.08603
$299$	0	0
$300$	0	0
$301$	−686.948	−2.28222
$302$	0	0
$303$	−22.3332	−0.0737068
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−395.428	−1.28804	−0.644020	−	0.765008i	$-0.722735\pi$
$307$	−395.428	−1.28804	−0.644020	+	0.765008i	$0.722735\pi$
$308$	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	1.00000			$0$
$313$	0	0	−1.00000			$\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	700.211	2.18134
$322$	967.233	3.00383
$323$	0	0
$324$	701.581	2.16537
$325$	0	0
$326$	596.568	1.82996
$327$	−1056.22	−3.23003
$328$	255.040	0.777561
$329$	−1007.15	−3.06124
$330$	0	0
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	−101.041	−0.304341
$333$	0	0
$334$	584.632	1.75040
$335$	0	0
$336$	976.839	2.90726
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	−338.000	−1.00000
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	285.832	0.833331
$344$	494.117	1.43639
$345$	0	0
$346$	0	0
$347$	308.338	0.888583	0.444292	−	0.895882i	$-0.353455\pi$
$347$	308.338	0.888583	0.444292	+	0.895882i	$0.353455\pi$
$348$	971.405	2.79139
$349$	−392.272	−1.12399	−0.561994	−	0.827141i	$-0.689966\pi$
$349$	−392.272	−1.12399	−0.561994	+	0.827141i	$0.689966\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	0	0	1.00000			$0$
$353$	0	0	−1.00000			$\pi$
$354$	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
$362$	323.334	0.893189
$363$	664.208	1.82977
$364$	0	0
$365$	0	0
$366$	177.467	0.484883
$367$	514.755	1.40260	0.701301	−	0.712866i	$-0.252603\pi$
$367$	514.755	1.40260	0.701301	+	0.712866i	$0.252603\pi$
$368$	−695.723	−1.89055
$369$	−673.709	−1.82577
$370$	0	0
$371$	0	0
$372$	0	0
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0	0
$375$	0	0
$376$	724.435	1.92669
$377$	0	0
$378$	−1481.46	−3.91920
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	0	0
$381$	−89.9718	−0.236146
$382$	0	0
$383$	740.903	1.93447	0.967237	−	0.253877i	$-0.0817058\pi$
$383$	740.903	1.93447	0.967237	+	0.253877i	$0.0817058\pi$
$384$	−702.633	−1.82977
$385$	0	0
$386$	0	0
$387$	−1305.25	−3.37274
$388$	0	0
$389$	−776.968	−1.99735	−0.998674	−	0.0514870i	$-0.983604\pi$
$389$	−776.968	−1.99735	−0.998674	+	0.0514870i	$0.983604\pi$
$390$	0	0
$391$	0	0
$392$	−597.597	−1.52448
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	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	$0.496752\pi$
$402$	−1273.52	−3.16797
$403$	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	$-0.263061\pi$
$409$	554.200	1.35501	0.677506	+	0.735517i	$0.263061\pi$
$410$	0	0
$411$	0	0
$412$	176.000	0.427184
$413$	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
$423$	−1913.66	−4.52401
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−179.785	−0.421042
$428$	510.235	1.19214
$429$	0	0
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	1065.60	2.46667
$433$	0	0	1.00000			$0$
$433$	0	0	−1.00000			$\pi$
$434$	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
$443$	872.671	1.96991	0.984955	−	0.172808i	$-0.0552841\pi$
$443$	872.671	1.96991	0.984955	+	0.172808i	$0.0552841\pi$
$444$	0	0
$445$	0	0
$446$	90.8261	0.203646
$447$	−888.277	−1.98720
$448$	711.810	1.58886
$449$	398.000	0.886414	0.443207	−	0.896419i	$-0.353841\pi$
$449$	398.000	0.886414	0.443207	+	0.896419i	$0.353841\pi$
$450$	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	1.00000			$0$
$457$	0	0	−1.00000			$\pi$
$458$	−865.539	−1.88982
$459$	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
$463$	−310.536	−0.670705	−0.335352	−	0.942093i	$-0.608855\pi$
$463$	−310.536	−0.670705	−0.335352	+	0.942093i	$0.608855\pi$
$464$	707.851	1.52554
$465$	0	0
$466$	0	0
$467$	−443.814	−0.950350	−0.475175	−	0.879891i	$-0.657615\pi$
$467$	−443.814	−0.950350	−0.475175	+	0.879891i	$0.657615\pi$
$468$	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
$482$	930.260	1.93000
$483$	−2654.73	−5.49633
$484$	484.000	1.00000
$485$	0	0
$486$	−726.801	−1.49548
$487$	654.301	1.34353	0.671767	−	0.740763i	$-0.265536\pi$
$487$	654.301	1.34353	0.671767	+	0.740763i	$0.265536\pi$
$488$	129.318	0.264996
$489$	−1637.38	−3.34842
$490$	0	0
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	−699.998	−1.42276
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	277.324	0.556876
$499$	0	0	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	0	0
$501$	−1604.62	−3.20283
$502$	0	0
$503$	496.595	0.987266	0.493633	−	0.869670i	$-0.335669\pi$
$503$	496.595	0.987266	0.493633	+	0.869670i	$0.335669\pi$
$504$	−1880.31	−3.73076
$505$	0	0
$506$	0	0
$507$	927.695	1.82977
$508$	−65.5613	−0.129058
$509$	−982.000	−1.92927	−0.964637	−	0.263584i	$-0.915095\pi$
$509$	−982.000	−1.92927	−0.964637	+	0.263584i	$0.915095\pi$
$510$	0	0
$511$	0	0
$512$	−512.000	−1.00000
$513$	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	$0.143667\pi$
$522$	−1869.85	−3.58208
$523$	1033.18	1.97549	0.987744	−	0.156081i	$-0.0498861\pi$
$523$	1033.18	1.97549	0.987744	+	0.156081i	$0.0498861\pi$
$524$	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
$538$	−76.0000	−0.141264
$539$	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
$543$	−887.443	−1.63433
$544$	0	0
$545$	0	0
$546$	0	0
$547$	963.720	1.76183	0.880914	−	0.473276i	$-0.156929\pi$
$547$	963.720	1.76183	0.880914	+	0.473276i	$0.156929\pi$
$548$	0	0
$549$	−341.604	−0.622230
$550$	0	0
$551$	0	0
$552$	1909.52	3.45928
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	−234.724	−0.417658
$563$	1124.00	1.99645	0.998224	−	0.0595755i	$-0.0189747\pi$
$563$	1124.00	1.99645	0.998224	+	0.0595755i	$0.0189747\pi$
$564$	−1988.33	−3.52541
$565$	0	0
$566$	632.000	1.11661
$567$	1950.75	3.44048
$568$	0	0
$569$	1120.64	1.96950	0.984749	−	0.173979i	$-0.0556626\pi$
$569$	1120.64	1.96950	0.984749	+	0.173979i	$0.0556626\pi$
$570$	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	1.00000			$0$
$577$	0	0	−1.00000			$\pi$
$578$	−578.000	−1.00000
$579$	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
$587$	1076.00	1.83305	0.916525	−	0.399978i	$-0.130982\pi$
$587$	1076.00	1.83305	0.916525	+	0.399978i	$0.130982\pi$
$588$	1640.20	2.78946
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	0	0	1.00000			$0$
$593$	0	0	−1.00000			$\pi$
$594$	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	$0.986945\pi$
$602$	1373.90	2.28222
$603$	2451.39	4.06532
$604$	0	0
$605$	0	0
$606$	44.6663	0.0737068
$607$	−964.000	−1.58814	−0.794069	−	0.607827i	$-0.792041\pi$
$607$	−964.000	−1.58814	−0.794069	+	0.607827i	$0.792041\pi$
$608$	0	0
$609$	2701.00	4.43514
$610$	0	0
$611$	0	0
$612$	0	0
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	790.857	1.28804
$615$	0	0
$616$	0	0
$617$	0	0	1.00000			$0$
$617$	0	0	−1.00000			$\pi$
$618$	−483.060	−0.781651
$619$	0	0	1.00000			$0$
$619$	0	0	−1.00000			$\pi$
$620$	0	0
$621$	−2895.95	−4.66337
$622$	0	0
$623$	1979.37	3.17716
$624$	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	$0.752319\pi$
$642$	−1400.42	−2.18134
$643$	−1044.47	−1.62436	−0.812182	−	0.583404i	$-0.801720\pi$
$643$	−1044.47	−1.62436	−0.812182	+	0.583404i	$0.801720\pi$
$644$	−1934.47	−3.00383
$645$	0	0
$646$	0	0
$647$	956.000	1.47759	0.738794	−	0.673931i	$-0.235395\pi$
$647$	956.000	1.47759	0.738794	+	0.673931i	$0.235395\pi$
$648$	−1403.16	−2.16537
$649$	0	0
$650$	0	0
$651$	0	0
$652$	−1193.14	−1.82996
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	2112.44	3.23003
$655$	0	0
$656$	−510.080	−0.777561
$657$	0	0
$658$	2014.30	3.06124
$659$	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
$667$	−1923.70	−2.88411
$668$	−1169.26	−1.75040
$669$	−249.287	−0.372626
$670$	0	0
$671$	0	0
$672$	−1953.68	−2.90726
$673$	0	0	1.00000			$0$
$673$	0	0	−1.00000			$\pi$
$674$	0	0
$675$	0	0
$676$	676.000	1.00000
$677$	0	0	1.00000			$0$
$677$	0	0	−1.00000			$\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−671.919	−0.986665
$682$	0	0
$683$	−1358.47	−1.98898	−0.994489	−	0.104845i	$-0.966565\pi$
$683$	−1358.47	−1.98898	−0.994489	+	0.104845i	$0.966565\pi$
$684$	0	0
$685$	0	0
$686$	−571.665	−0.833331
$687$	2375.61	3.45795
$688$	−988.233	−1.43639
$689$	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
$698$	784.544	1.12399
$699$	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
$707$	−45.2497	−0.0640024
$708$	0	0
$709$	−1290.20	−1.81975	−0.909875	−	0.414882i	$-0.863823\pi$
$709$	−1290.20	−1.81975	−0.909875	+	0.414882i	$0.863823\pi$
$710$	0	0
$711$	0	0
$712$	−1423.75	−1.99964
$713$	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
$722$	−722.000	−1.00000
$723$	−2553.25	−3.53147
$724$	−646.669	−0.893189
$725$	0	0
$726$	−1328.42	−1.82977
$727$	226.832	0.312012	0.156006	−	0.987756i	$-0.450138\pi$
$727$	226.832	0.312012	0.156006	+	0.987756i	$0.450138\pi$
$728$	0	0
$729$	416.265	0.571008
$730$	0	0
$731$	0	0
$732$	−354.934	−0.484883
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	−1029.51	−1.40260
$735$	0	0
$736$	1391.45	1.89055
$737$	0	0
$738$	1347.42	1.82577
$739$	0	0	1.00000			$0$
$739$	0	0	−1.00000			$\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	764.000	1.02826	0.514132	−	0.857711i	$-0.328114\pi$
$743$	764.000	1.02826	0.514132	+	0.857711i	$0.328114\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−533.818	−0.714616
$748$	0	0
$749$	1418.71	1.89414
$750$	0	0
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	−1448.87	−1.92669
$753$	0	0
$754$	0	0
$755$	0	0
$756$	2962.91	3.91920
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	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	$0.350827\pi$
$762$	179.944	0.236146
$763$	−2140.03	−2.80476
$764$	0	0
$765$	0	0
$766$	−1481.81	−1.93447
$767$	0	0
$768$	1405.27	1.82977
$769$	−1129.25	−1.46846	−0.734231	−	0.678900i	$-0.762457\pi$
$769$	−1129.25	−1.46846	−0.734231	+	0.678900i	$0.762457\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	2610.50	3.37274
$775$	0	0
$776$	0	0
$777$	0	0
$778$	1553.94	1.99735
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	2946.43	3.76300
$784$	1195.19	1.52448
$785$	0	0
$786$	0	0
$787$	−1457.05	−1.85139	−0.925696	−	0.378267i	$-0.876520\pi$
$787$	−1457.05	−1.85139	−0.925696	+	0.378267i	$0.876520\pi$
$788$	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	1.00000			$0$
$797$	0	0	−1.00000			$\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	3760.94	4.69531
$802$	−16.3674	−0.0204083
$803$	0	0
$804$	2547.05	3.16797
$805$	0	0
$806$	0	0
$807$	208.594	0.258481
$808$	32.5478	0.0402819
$809$	−1617.89	−1.99987	−0.999934	−	0.0114648i	$-0.996351\pi$
$809$	−1617.89	−1.99987	−0.999934	+	0.0114648i	$0.996351\pi$
$810$	0	0
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	1968.19	2.42387
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	−1108.40	−1.35501
$819$	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
$823$	−1396.00	−1.69623	−0.848117	−	0.529810i	$-0.822263\pi$
$823$	−1396.00	−1.69623	−0.848117	+	0.529810i	$0.822263\pi$
$824$	−352.000	−0.427184
$825$	0	0
$826$	0	0
$827$	596.000	0.720677	0.360339	−	0.932822i	$-0.382661\pi$
$827$	596.000	0.720677	0.360339	+	0.932822i	$0.382661\pi$
$828$	−3675.62	−4.43916
$829$	−257.820	−0.311001	−0.155500	−	0.987836i	$-0.549699\pi$
$829$	−257.820	−0.311001	−0.155500	+	0.987836i	$0.549699\pi$
$830$	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
$842$	1093.30	1.29845
$843$	644.238	0.764221
$844$	0	0
$845$	0	0
$846$	3827.31	4.52401
$847$	1345.77	1.58886
$848$	0	0
$849$	−1734.63	−2.04314
$850$	0	0
$851$	0	0
$852$	0	0
$853$	0	0	1.00000			$0$
$853$	0	0	−1.00000			$\pi$
$854$	359.570	0.421042
$855$	0	0
$856$	−1020.47	−1.19214
$857$	0	0	1.00000			$0$
$857$	0	0	−1.00000			$\pi$
$858$	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
$863$	1000.23	1.15901	0.579506	−	0.814968i	$-0.303246\pi$
$863$	1000.23	1.15901	0.579506	+	0.814968i	$0.303246\pi$
$864$	−2131.20	−2.46667
$865$	0	0
$866$	0	0
$867$	1586.41	1.82977
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	1539.31	1.76526
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	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	$0.0704175\pi$
$882$	−3157.20	−3.57960
$883$	314.683	0.356379	0.178190	−	0.983996i	$-0.442976\pi$
$883$	314.683	0.356379	0.178190	+	0.983996i	$0.442976\pi$
$884$	0	0
$885$	0	0
$886$	−1745.34	−1.96991
$887$	1655.23	1.86610	0.933052	−	0.359741i	$-0.117135\pi$
$887$	1655.23	1.86610	0.933052	+	0.359741i	$0.117135\pi$
$888$	0	0
$889$	−182.294	−0.205055
$890$	0	0
$891$	0	0
$892$	−181.652	−0.203646
$893$	0	0
$894$	1776.55	1.98720
$895$	0	0
$896$	−1423.62	−1.58886
$897$	0	0
$898$	−796.000	−0.886414
$899$	0	0
$900$	0	0
$901$	0	0
$902$	0	0
$903$	−3770.88	−4.17594
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−1602.83	−1.76718	−0.883588	−	0.468266i	$-0.844879\pi$
$907$	−1602.83	−1.76718	−0.883588	+	0.468266i	$0.844879\pi$
$908$	−489.619	−0.539228
$909$	−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
$922$	1804.00	1.95661
$923$	0	0
$924$	0	0
$925$	0	0
$926$	621.073	0.670705
$927$	929.837	1.00306
$928$	−1415.70	−1.52554
$929$	−586.280	−0.631087	−0.315544	−	0.948911i	$-0.602187\pi$
$929$	−586.280	−0.631087	−0.315544	+	0.948911i	$0.602187\pi$
$930$	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	1.00000			$0$
$937$	0	0	−1.00000			$\pi$
$938$	−2580.31	−2.75087
$939$	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
$943$	1386.23	1.47002
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−8.79689	−0.00928921	−0.00464461	−	0.999989i	$-0.501478\pi$
$947$	−8.79689	−0.00928921	−0.00464461	+	0.999989i	$0.501478\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	0	0	1.00000			$0$
$953$	0	0	−1.00000			$\pi$
$954$	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
$963$	2695.66	2.79923
$964$	−1860.52	−1.93000
$965$	0	0
$966$	5309.45	5.49633
$967$	1749.25	1.80894	0.904470	−	0.426536i	$-0.140266\pi$
$967$	1749.25	1.80894	0.904470	+	0.426536i	$0.140266\pi$
$968$	−968.000	−1.00000
$969$	0	0
$970$	0	0
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	1453.60	1.49548
$973$	0	0
$974$	−1308.60	−1.34353
$975$	0	0
$976$	−258.636	−0.264996
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	3274.75	3.34842
$979$	0	0
$980$	0	0
$981$	−4066.21	−4.14496
$982$	0	0
$983$	284.000	0.288911	0.144456	−	0.989511i	$-0.453857\pi$
$983$	284.000	0.288911	0.144456	+	0.989511i	$0.453857\pi$
$984$	1400.00	1.42276
$985$	0	0
$986$	0	0
$987$	−5528.57	−5.60138
$988$	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	1.00000			$0$
$997$	0	0	−1.00000			$\pi$
$998$	0	0
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
500.3.b.a.251.1	✓	8	5.2	odd	4
500.3.b.a.251.1	✓	8	20.3	even	4
500.3.b.a.251.8	yes	8	5.3	odd	4
500.3.b.a.251.8	yes	8	20.7	even	4
500.3.d.a.499.4		4	1.1	even	1	trivial
500.3.d.a.499.4		4	20.19	odd	2	CM
500.3.d.b.499.1		4	4.3	odd	2
500.3.d.b.499.1		4	5.4	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:	\( N \)	\(=\)	\( 500 = 2^{2} \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	500.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} +5.48932 q^{3} +4.00000 q^{4} -10.9786 q^{6} +11.1220 q^{7} -8.00000 q^{8} +21.1327 q^{9} +O(q^{10})\)	\(q-2.00000 q^{2} +5.48932 q^{3} +4.00000 q^{4} -10.9786 q^{6} +11.1220 q^{7} -8.00000 q^{8} +21.1327 q^{9} +21.9573 q^{12} -22.2441 q^{14} +16.0000 q^{16} -42.2653 q^{18} +61.0524 q^{21} -43.4827 q^{23} -43.9146 q^{24} +66.6001 q^{27} +44.4881 q^{28} +44.2407 q^{29} -32.0000 q^{32} +84.5306 q^{36} -31.8800 q^{41} -122.105 q^{42} -61.7646 q^{43} +86.9654 q^{46} -90.5544 q^{47} +87.8292 q^{48} +74.6997 q^{49} -133.200 q^{54} -88.9763 q^{56} -88.4813 q^{58} -16.1647 q^{61} +235.038 q^{63} +64.0000 q^{64} +116.000 q^{67} -238.691 q^{69} -169.061 q^{72} +175.395 q^{81} +63.7600 q^{82} -25.2603 q^{83} +244.210 q^{84} +123.529 q^{86} +242.851 q^{87} +177.968 q^{89} -173.931 q^{92} +181.109 q^{94} -175.658 q^{96} -149.399 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{2} - 4 q^{3} + 16 q^{4} + 8 q^{6} + 4 q^{7} - 32 q^{8} + 38 q^{9}+O(q^{10}) \) 4 * q - 8 * q^2 - 4 * q^3 + 16 * q^4 + 8 * q^6 + 4 * q^7 - 32 * q^8 + 38 * q^9	\( 4 q - 8 q^{2} - 4 q^{3} + 16 q^{4} + 8 q^{6} + 4 q^{7} - 32 q^{8} + 38 q^{9} - 16 q^{12} - 8 q^{14} + 64 q^{16} - 76 q^{18} + 16 q^{21} - 44 q^{23} + 32 q^{24} + 8 q^{27} + 16 q^{28} + 22 q^{29} - 128 q^{32} + 152 q^{36} - 62 q^{41} - 32 q^{42} + 76 q^{43} + 88 q^{46} + 4 q^{47} - 64 q^{48} + 278 q^{49} - 16 q^{54} - 32 q^{56} - 44 q^{58} + 58 q^{61} + 648 q^{63} + 256 q^{64} + 464 q^{67} - 176 q^{69} - 304 q^{72} + 500 q^{81} + 124 q^{82} + 76 q^{83} + 64 q^{84} - 152 q^{86} + 468 q^{87} + 142 q^{89} - 176 q^{92} - 8 q^{94} + 128 q^{96} - 556 q^{98}+O(q^{100}) \) 4 * q - 8 * q^2 - 4 * q^3 + 16 * q^4 + 8 * q^6 + 4 * q^7 - 32 * q^8 + 38 * q^9 - 16 * q^12 - 8 * q^14 + 64 * q^16 - 76 * q^18 + 16 * q^21 - 44 * q^23 + 32 * q^24 + 8 * q^27 + 16 * q^28 + 22 * q^29 - 128 * q^32 + 152 * q^36 - 62 * q^41 - 32 * q^42 + 76 * q^43 + 88 * q^46 + 4 * q^47 - 64 * q^48 + 278 * q^49 - 16 * q^54 - 32 * q^56 - 44 * q^58 + 58 * q^61 + 648 * q^63 + 256 * q^64 + 464 * q^67 - 176 * q^69 - 304 * q^72 + 500 * q^81 + 124 * q^82 + 76 * q^83 + 64 * q^84 - 152 * q^86 + 468 * q^87 + 142 * q^89 - 176 * q^92 - 8 * q^94 + 128 * q^96 - 556 * q^98

Introduction

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