LMFDB - Embedded newform 10000.2.a.bn.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [10000,2,Mod(1,10000)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(10000, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("10000.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 10000 = 2^{4} \cdot 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	10000.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:	$79.8504020213$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.8.6152203125.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} - 8x^{6} + 20x^{5} + 26x^{4} - 35x^{3} - 27x^{2} + 16x + 1 $ x^8 - 3x^7 - 8x^6 + 20x^5 + 26x^4 - 35x^3 - 27x^2 + 16*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 625)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.66501$ of defining polynomial
Character	$\chi$	$=$	10000.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.71538 q^{3} +3.42409 q^{7} -0.0574791 q^{9} +O(q^{10})$	$q-1.71538 q^{3} +3.42409 q^{7} -0.0574791 q^{9} +5.34111 q^{11} +3.52114 q^{13} -2.55787 q^{17} +2.02579 q^{19} -5.87362 q^{21} +7.57082 q^{23} +5.24473 q^{27} +4.74270 q^{29} -1.62421 q^{31} -9.16203 q^{33} +0.0134290 q^{37} -6.04008 q^{39} +9.67740 q^{41} +2.32645 q^{43} +6.94647 q^{47} +4.72443 q^{49} +4.38772 q^{51} -1.72246 q^{53} -3.47500 q^{57} +0.0221830 q^{59} +3.91768 q^{61} -0.196814 q^{63} +4.11832 q^{67} -12.9868 q^{69} -2.33894 q^{71} -1.51373 q^{73} +18.2885 q^{77} -0.426831 q^{79} -8.82426 q^{81} -6.04187 q^{83} -8.13552 q^{87} -6.09362 q^{89} +12.0567 q^{91} +2.78613 q^{93} +16.0018 q^{97} -0.307002 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 5 q^{3} + 10 q^{7} + 9 q^{9}+O(q^{10}) $ 8 * q + 5 * q^3 + 10 * q^7 + 9 * q^9	$ 8 q + 5 q^{3} + 10 q^{7} + 9 q^{9} - q^{11} - 10 q^{13} - 15 q^{17} + 10 q^{19} - 14 q^{21} + 30 q^{23} + 20 q^{27} + 10 q^{29} + 9 q^{31} - 5 q^{33} + 10 q^{37} - 8 q^{39} - 4 q^{41} + 30 q^{47} - 4 q^{49} + 14 q^{51} - 10 q^{53} + 10 q^{57} + 5 q^{59} + 6 q^{61} + 10 q^{67} + 3 q^{69} + 9 q^{71} - 5 q^{77} + 20 q^{79} + 8 q^{81} + 40 q^{83} + 40 q^{87} - 5 q^{89} - 6 q^{91} + 40 q^{93} + 22 q^{99}+O(q^{100}) $ 8 * q + 5 * q^3 + 10 * q^7 + 9 * q^9 - q^11 - 10 * q^13 - 15 * q^17 + 10 * q^19 - 14 * q^21 + 30 * q^23 + 20 * q^27 + 10 * q^29 + 9 * q^31 - 5 * q^33 + 10 * q^37 - 8 * q^39 - 4 * q^41 + 30 * q^47 - 4 * q^49 + 14 * q^51 - 10 * q^53 + 10 * q^57 + 5 * q^59 + 6 * q^61 + 10 * q^67 + 3 * q^69 + 9 * q^71 - 5 * q^77 + 20 * q^79 + 8 * q^81 + 40 * q^83 + 40 * q^87 - 5 * q^89 - 6 * q^91 + 40 * q^93 + 22 * q^99

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
$3$	−1.71538	−0.990374	−0.495187	−	0.868786i	$-0.664900\pi$
$3$	−1.71538	−0.990374	−0.495187	+	0.868786i	$0.664900\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	3.42409	1.29419	0.647093	−	0.762411i	$-0.275984\pi$
$7$	3.42409	1.29419	0.647093	+	0.762411i	$0.275984\pi$
$8$	0	0
$9$	−0.0574791	−0.0191597
$10$	0	0
$11$	5.34111	1.61041	0.805203	−	0.592999i	$-0.202056\pi$
$11$	5.34111	1.61041	0.805203	+	0.592999i	$0.202056\pi$
$12$	0	0
$13$	3.52114	0.976588	0.488294	−	0.872679i	$-0.337619\pi$
$13$	3.52114	0.976588	0.488294	+	0.872679i	$0.337619\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−2.55787	−0.620375	−0.310188	−	0.950675i	$-0.600392\pi$
$17$	−2.55787	−0.620375	−0.310188	+	0.950675i	$0.600392\pi$
$18$	0	0
$19$	2.02579	0.464748	0.232374	−	0.972626i	$-0.425351\pi$
$19$	2.02579	0.464748	0.232374	+	0.972626i	$0.425351\pi$
$20$	0	0
$21$	−5.87362	−1.28173
$22$	0	0
$23$	7.57082	1.57863	0.789313	−	0.613991i	$-0.210437\pi$
$23$	7.57082	1.57863	0.789313	+	0.613991i	$0.210437\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	5.24473	1.00935
$28$	0	0
$29$	4.74270	0.880697	0.440348	−	0.897827i	$-0.354855\pi$
$29$	4.74270	0.880697	0.440348	+	0.897827i	$0.354855\pi$
$30$	0	0
$31$	−1.62421	−0.291716	−0.145858	−	0.989306i	$-0.546594\pi$
$31$	−1.62421	−0.291716	−0.145858	+	0.989306i	$0.546594\pi$
$32$	0	0
$33$	−9.16203	−1.59490
$34$	0	0
$35$	0	0
$36$	0	0
$37$	0.0134290	0.00220771	0.00110385	−	0.999999i	$-0.499649\pi$
$37$	0.0134290	0.00220771	0.00110385	+	0.999999i	$0.499649\pi$
$38$	0	0
$39$	−6.04008	−0.967187
$40$	0	0
$41$	9.67740	1.51136	0.755678	−	0.654943i	$-0.227307\pi$
$41$	9.67740	1.51136	0.755678	+	0.654943i	$0.227307\pi$
$42$	0	0
$43$	2.32645	0.354780	0.177390	−	0.984141i	$-0.443235\pi$
$43$	2.32645	0.354780	0.177390	+	0.984141i	$0.443235\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	6.94647	1.01325	0.506624	−	0.862167i	$-0.330893\pi$
$47$	6.94647	1.01325	0.506624	+	0.862167i	$0.330893\pi$
$48$	0	0
$49$	4.72443	0.674918
$50$	0	0
$51$	4.38772	0.614403
$52$	0	0
$53$	−1.72246	−0.236598	−0.118299	−	0.992978i	$-0.537744\pi$
$53$	−1.72246	−0.236598	−0.118299	+	0.992978i	$0.537744\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−3.47500	−0.460275
$58$	0	0
$59$	0.0221830	0.00288798	0.00144399	−	0.999999i	$-0.499540\pi$
$59$	0.0221830	0.00288798	0.00144399	+	0.999999i	$0.499540\pi$
$60$	0	0
$61$	3.91768	0.501607	0.250804	−	0.968038i	$-0.419305\pi$
$61$	3.91768	0.501607	0.250804	+	0.968038i	$0.419305\pi$
$62$	0	0
$63$	−0.196814	−0.0247962
$64$	0	0
$65$	0	0
$66$	0	0
$67$	4.11832	0.503133	0.251566	−	0.967840i	$-0.419054\pi$
$67$	4.11832	0.503133	0.251566	+	0.967840i	$0.419054\pi$
$68$	0	0
$69$	−12.9868	−1.56343
$70$	0	0
$71$	−2.33894	−0.277581	−0.138790	−	0.990322i	$-0.544321\pi$
$71$	−2.33894	−0.277581	−0.138790	+	0.990322i	$0.544321\pi$
$72$	0	0
$73$	−1.51373	−0.177169	−0.0885843	−	0.996069i	$-0.528234\pi$
$73$	−1.51373	−0.177169	−0.0885843	+	0.996069i	$0.528234\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	18.2885	2.08417
$78$	0	0
$79$	−0.426831	−0.0480222	−0.0240111	−	0.999712i	$-0.507644\pi$
$79$	−0.426831	−0.0480222	−0.0240111	+	0.999712i	$0.507644\pi$
$80$	0	0
$81$	−8.82426	−0.980473
$82$	0	0
$83$	−6.04187	−0.663181	−0.331591	−	0.943423i	$-0.607585\pi$
$83$	−6.04187	−0.663181	−0.331591	+	0.943423i	$0.607585\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−8.13552	−0.872219
$88$	0	0
$89$	−6.09362	−0.645922	−0.322961	−	0.946412i	$-0.604678\pi$
$89$	−6.09362	−0.645922	−0.322961	+	0.946412i	$0.604678\pi$
$90$	0	0
$91$	12.0567	1.26389
$92$	0	0
$93$	2.78613	0.288908
$94$	0	0
$95$	0	0
$96$	0	0
$97$	16.0018	1.62474	0.812370	−	0.583143i	$-0.198177\pi$
$97$	16.0018	1.62474	0.812370	+	0.583143i	$0.198177\pi$
$98$	0	0
$99$	−0.307002	−0.0308549
$100$	0	0
$101$	1.44418	0.143701	0.0718505	−	0.997415i	$-0.477110\pi$
$101$	1.44418	0.143701	0.0718505	+	0.997415i	$0.477110\pi$
$102$	0	0
$103$	14.6657	1.44506	0.722529	−	0.691341i	$-0.242980\pi$
$103$	14.6657	1.44506	0.722529	+	0.691341i	$0.242980\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	12.2169	1.18106	0.590528	−	0.807017i	$-0.298919\pi$
$107$	12.2169	1.18106	0.590528	+	0.807017i	$0.298919\pi$
$108$	0	0
$109$	−15.3516	−1.47041	−0.735207	−	0.677843i	$-0.762915\pi$
$109$	−15.3516	−1.47041	−0.735207	+	0.677843i	$0.762915\pi$
$110$	0	0
$111$	−0.0230357	−0.00218646
$112$	0	0
$113$	−18.5544	−1.74545	−0.872727	−	0.488209i	$-0.837650\pi$
$113$	−18.5544	−1.74545	−0.872727	+	0.488209i	$0.837650\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−0.202392	−0.0187111
$118$	0	0
$119$	−8.75840	−0.802881
$120$	0	0
$121$	17.5275	1.59341
$122$	0	0
$123$	−16.6004	−1.49681
$124$	0	0
$125$	0	0
$126$	0	0
$127$	0.677902	0.0601541	0.0300771	−	0.999548i	$-0.490425\pi$
$127$	0.677902	0.0601541	0.0300771	+	0.999548i	$0.490425\pi$
$128$	0	0
$129$	−3.99074	−0.351365
$130$	0	0
$131$	−7.05058	−0.616012	−0.308006	−	0.951384i	$-0.599662\pi$
$131$	−7.05058	−0.616012	−0.308006	+	0.951384i	$0.599662\pi$
$132$	0	0
$133$	6.93650	0.601471
$134$	0	0
$135$	0	0
$136$	0	0
$137$	10.9186	0.932837	0.466418	−	0.884564i	$-0.345544\pi$
$137$	10.9186	0.932837	0.466418	+	0.884564i	$0.345544\pi$
$138$	0	0
$139$	−19.5102	−1.65483	−0.827416	−	0.561589i	$-0.810190\pi$
$139$	−19.5102	−1.65483	−0.827416	+	0.561589i	$0.810190\pi$
$140$	0	0
$141$	−11.9158	−1.00349
$142$	0	0
$143$	18.8068	1.57270
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−8.10418	−0.668421
$148$	0	0
$149$	−12.7945	−1.04817	−0.524085	−	0.851666i	$-0.675593\pi$
$149$	−12.7945	−1.04817	−0.524085	+	0.851666i	$0.675593\pi$
$150$	0	0
$151$	−2.15617	−0.175466	−0.0877331	−	0.996144i	$-0.527962\pi$
$151$	−2.15617	−0.175466	−0.0877331	+	0.996144i	$0.527962\pi$
$152$	0	0
$153$	0.147024	0.0118862
$154$	0	0
$155$	0	0
$156$	0	0
$157$	15.7474	1.25678	0.628389	−	0.777899i	$-0.283715\pi$
$157$	15.7474	1.25678	0.628389	+	0.777899i	$0.283715\pi$
$158$	0	0
$159$	2.95467	0.234321
$160$	0	0
$161$	25.9232	2.04304
$162$	0	0
$163$	7.39219	0.579001	0.289500	−	0.957178i	$-0.406511\pi$
$163$	7.39219	0.579001	0.289500	+	0.957178i	$0.406511\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−10.8165	−0.837007	−0.418504	−	0.908215i	$-0.637445\pi$
$167$	−10.8165	−0.837007	−0.418504	+	0.908215i	$0.637445\pi$
$168$	0	0
$169$	−0.601591	−0.0462762
$170$	0	0
$171$	−0.116441	−0.00890444
$172$	0	0
$173$	−14.9983	−1.14030	−0.570149	−	0.821541i	$-0.693115\pi$
$173$	−14.9983	−1.14030	−0.570149	+	0.821541i	$0.693115\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−0.0380522	−0.00286018
$178$	0	0
$179$	−7.39841	−0.552983	−0.276492	−	0.961016i	$-0.589172\pi$
$179$	−7.39841	−0.552983	−0.276492	+	0.961016i	$0.589172\pi$
$180$	0	0
$181$	−10.9177	−0.811503	−0.405752	−	0.913983i	$-0.632990\pi$
$181$	−10.9177	−0.811503	−0.405752	+	0.913983i	$0.632990\pi$
$182$	0	0
$183$	−6.72030	−0.496779
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−13.6619	−0.999056
$188$	0	0
$189$	17.9585	1.30629
$190$	0	0
$191$	1.75142	0.126728	0.0633642	−	0.997990i	$-0.479817\pi$
$191$	1.75142	0.126728	0.0633642	+	0.997990i	$0.479817\pi$
$192$	0	0
$193$	9.53146	0.686089	0.343045	−	0.939319i	$-0.388542\pi$
$193$	9.53146	0.686089	0.343045	+	0.939319i	$0.388542\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−23.9169	−1.70401	−0.852004	−	0.523535i	$-0.824613\pi$
$197$	−23.9169	−1.70401	−0.852004	+	0.523535i	$0.824613\pi$
$198$	0	0
$199$	−23.8281	−1.68913	−0.844566	−	0.535451i	$-0.820142\pi$
$199$	−23.8281	−1.68913	−0.844566	+	0.535451i	$0.820142\pi$
$200$	0	0
$201$	−7.06448	−0.498290
$202$	0	0
$203$	16.2394	1.13979
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−0.435164	−0.0302460
$208$	0	0
$209$	10.8200	0.748434
$210$	0	0
$211$	15.3923	1.05965	0.529826	−	0.848107i	$-0.322257\pi$
$211$	15.3923	1.05965	0.529826	+	0.848107i	$0.322257\pi$
$212$	0	0
$213$	4.01216	0.274909
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−5.56144	−0.377535
$218$	0	0
$219$	2.59662	0.175463
$220$	0	0
$221$	−9.00662	−0.605851
$222$	0	0
$223$	−19.5753	−1.31086	−0.655429	−	0.755257i	$-0.727512\pi$
$223$	−19.5753	−1.31086	−0.655429	+	0.755257i	$0.727512\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	1.34207	0.0890760	0.0445380	−	0.999008i	$-0.485818\pi$
$227$	1.34207	0.0890760	0.0445380	+	0.999008i	$0.485818\pi$
$228$	0	0
$229$	−22.3702	−1.47827	−0.739133	−	0.673560i	$-0.764764\pi$
$229$	−22.3702	−1.47827	−0.739133	+	0.673560i	$0.764764\pi$
$230$	0	0
$231$	−31.3717	−2.06410
$232$	0	0
$233$	−18.3651	−1.20314	−0.601568	−	0.798822i	$-0.705457\pi$
$233$	−18.3651	−1.20314	−0.601568	+	0.798822i	$0.705457\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0.732176	0.0475600
$238$	0	0
$239$	12.0037	0.776458	0.388229	−	0.921563i	$-0.373087\pi$
$239$	12.0037	0.776458	0.388229	+	0.921563i	$0.373087\pi$
$240$	0	0
$241$	10.1170	0.651692	0.325846	−	0.945423i	$-0.394351\pi$
$241$	10.1170	0.651692	0.325846	+	0.945423i	$0.394351\pi$
$242$	0	0
$243$	−0.597258	−0.0383141
$244$	0	0
$245$	0	0
$246$	0	0
$247$	7.13309	0.453867
$248$	0	0
$249$	10.3641	0.656797
$250$	0	0
$251$	−16.7258	−1.05573	−0.527863	−	0.849330i	$-0.677006\pi$
$251$	−16.7258	−1.05573	−0.527863	+	0.849330i	$0.677006\pi$
$252$	0	0
$253$	40.4366	2.54223
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−15.0170	−0.936732	−0.468366	−	0.883535i	$-0.655157\pi$
$257$	−15.0170	−0.936732	−0.468366	+	0.883535i	$0.655157\pi$
$258$	0	0
$259$	0.0459821	0.00285719
$260$	0	0
$261$	−0.272606	−0.0168739
$262$	0	0
$263$	6.17182	0.380571	0.190285	−	0.981729i	$-0.439059\pi$
$263$	6.17182	0.380571	0.190285	+	0.981729i	$0.439059\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	10.4529	0.639704
$268$	0	0
$269$	11.1052	0.677098	0.338549	−	0.940949i	$-0.390064\pi$
$269$	11.1052	0.677098	0.338549	+	0.940949i	$0.390064\pi$
$270$	0	0
$271$	1.16149	0.0705554	0.0352777	−	0.999378i	$-0.488768\pi$
$271$	1.16149	0.0705554	0.0352777	+	0.999378i	$0.488768\pi$
$272$	0	0
$273$	−20.6818	−1.25172
$274$	0	0
$275$	0	0
$276$	0	0
$277$	2.17486	0.130675	0.0653374	−	0.997863i	$-0.479188\pi$
$277$	2.17486	0.130675	0.0653374	+	0.997863i	$0.479188\pi$
$278$	0	0
$279$	0.0933580	0.00558920
$280$	0	0
$281$	−24.1177	−1.43874	−0.719370	−	0.694627i	$-0.755569\pi$
$281$	−24.1177	−1.43874	−0.719370	+	0.694627i	$0.755569\pi$
$282$	0	0
$283$	16.2144	0.963845	0.481923	−	0.876214i	$-0.339939\pi$
$283$	16.2144	0.963845	0.481923	+	0.876214i	$0.339939\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	33.1363	1.95598
$288$	0	0
$289$	−10.4573	−0.615135
$290$	0	0
$291$	−27.4492	−1.60910
$292$	0	0
$293$	−3.48929	−0.203846	−0.101923	−	0.994792i	$-0.532500\pi$
$293$	−3.48929	−0.203846	−0.101923	+	0.994792i	$0.532500\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	28.0127	1.62546
$298$	0	0
$299$	26.6579	1.54167
$300$	0	0
$301$	7.96599	0.459152
$302$	0	0
$303$	−2.47731	−0.142318
$304$	0	0
$305$	0	0
$306$	0	0
$307$	8.92690	0.509485	0.254742	−	0.967009i	$-0.418009\pi$
$307$	8.92690	0.509485	0.254742	+	0.967009i	$0.418009\pi$
$308$	0	0
$309$	−25.1573	−1.43115
$310$	0	0
$311$	27.1101	1.53727	0.768635	−	0.639687i	$-0.220936\pi$
$311$	27.1101	1.53727	0.768635	+	0.639687i	$0.220936\pi$
$312$	0	0
$313$	20.1073	1.13653	0.568267	−	0.822844i	$-0.307614\pi$
$313$	20.1073	1.13653	0.568267	+	0.822844i	$0.307614\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−9.67993	−0.543679	−0.271840	−	0.962343i	$-0.587632\pi$
$317$	−9.67993	−0.543679	−0.271840	+	0.962343i	$0.587632\pi$
$318$	0	0
$319$	25.3313	1.41828
$320$	0	0
$321$	−20.9567	−1.16969
$322$	0	0
$323$	−5.18171	−0.288318
$324$	0	0
$325$	0	0
$326$	0	0
$327$	26.3337	1.45626
$328$	0	0
$329$	23.7854	1.31133
$330$	0	0
$331$	−19.6759	−1.08148	−0.540742	−	0.841189i	$-0.681856\pi$
$331$	−19.6759	−1.08148	−0.540742	+	0.841189i	$0.681856\pi$
$332$	0	0
$333$	−0.000771885 0	−4.22990e−5 0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−18.8552	−1.02711	−0.513554	−	0.858057i	$-0.671671\pi$
$337$	−18.8552	−1.02711	−0.513554	+	0.858057i	$0.671671\pi$
$338$	0	0
$339$	31.8279	1.72865
$340$	0	0
$341$	−8.67508	−0.469782
$342$	0	0
$343$	−7.79178	−0.420717
$344$	0	0
$345$	0	0
$346$	0	0
$347$	25.9753	1.39442	0.697212	−	0.716865i	$-0.254423\pi$
$347$	25.9753	1.39442	0.697212	+	0.716865i	$0.254423\pi$
$348$	0	0
$349$	−26.1490	−1.39972	−0.699861	−	0.714279i	$-0.746755\pi$
$349$	−26.1490	−1.39972	−0.699861	+	0.714279i	$0.746755\pi$
$350$	0	0
$351$	18.4674	0.985718
$352$	0	0
$353$	5.32892	0.283630	0.141815	−	0.989893i	$-0.454706\pi$
$353$	5.32892	0.283630	0.141815	+	0.989893i	$0.454706\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	15.0240	0.795152
$358$	0	0
$359$	−9.89929	−0.522465	−0.261232	−	0.965276i	$-0.584129\pi$
$359$	−9.89929	−0.522465	−0.261232	+	0.965276i	$0.584129\pi$
$360$	0	0
$361$	−14.8962	−0.784009
$362$	0	0
$363$	−30.0663	−1.57807
$364$	0	0
$365$	0	0
$366$	0	0
$367$	16.4392	0.858118	0.429059	−	0.903276i	$-0.358845\pi$
$367$	16.4392	0.858118	0.429059	+	0.903276i	$0.358845\pi$
$368$	0	0
$369$	−0.556248	−0.0289571
$370$	0	0
$371$	−5.89787	−0.306202
$372$	0	0
$373$	−22.9933	−1.19055	−0.595273	−	0.803524i	$-0.702956\pi$
$373$	−22.9933	−1.19055	−0.595273	+	0.803524i	$0.702956\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	16.6997	0.860078
$378$	0	0
$379$	−16.4246	−0.843674	−0.421837	−	0.906672i	$-0.638615\pi$
$379$	−16.4246	−0.843674	−0.421837	+	0.906672i	$0.638615\pi$
$380$	0	0
$381$	−1.16286	−0.0595751
$382$	0	0
$383$	4.70503	0.240416	0.120208	−	0.992749i	$-0.461644\pi$
$383$	4.70503	0.240416	0.120208	+	0.992749i	$0.461644\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−0.133722	−0.00679748
$388$	0	0
$389$	3.53865	0.179417	0.0897083	−	0.995968i	$-0.471407\pi$
$389$	3.53865	0.179417	0.0897083	+	0.995968i	$0.471407\pi$
$390$	0	0
$391$	−19.3652	−0.979340
$392$	0	0
$393$	12.0944	0.610082
$394$	0	0
$395$	0	0
$396$	0	0
$397$	8.76374	0.439839	0.219920	−	0.975518i	$-0.429420\pi$
$397$	8.76374	0.439839	0.219920	+	0.975518i	$0.429420\pi$
$398$	0	0
$399$	−11.8987	−0.595681
$400$	0	0
$401$	22.7677	1.13697	0.568483	−	0.822695i	$-0.307531\pi$
$401$	22.7677	1.13697	0.568483	+	0.822695i	$0.307531\pi$
$402$	0	0
$403$	−5.71906	−0.284887
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0.0717256	0.00355531
$408$	0	0
$409$	28.0426	1.38662	0.693309	−	0.720640i	$-0.256152\pi$
$409$	28.0426	1.38662	0.693309	+	0.720640i	$0.256152\pi$
$410$	0	0
$411$	−18.7295	−0.923857
$412$	0	0
$413$	0.0759566	0.00373758
$414$	0	0
$415$	0	0
$416$	0	0
$417$	33.4673	1.63890
$418$	0	0
$419$	−34.0901	−1.66541	−0.832705	−	0.553717i	$-0.813209\pi$
$419$	−34.0901	−1.66541	−0.832705	+	0.553717i	$0.813209\pi$
$420$	0	0
$421$	−32.1390	−1.56636	−0.783180	−	0.621796i	$-0.786403\pi$
$421$	−32.1390	−1.56636	−0.783180	+	0.621796i	$0.786403\pi$
$422$	0	0
$423$	−0.399277	−0.0194135
$424$	0	0
$425$	0	0
$426$	0	0
$427$	13.4145	0.649173
$428$	0	0
$429$	−32.2608	−1.55756
$430$	0	0
$431$	17.7549	0.855222	0.427611	−	0.903963i	$-0.359355\pi$
$431$	17.7549	0.855222	0.427611	+	0.903963i	$0.359355\pi$
$432$	0	0
$433$	−3.06764	−0.147421	−0.0737107	−	0.997280i	$-0.523484\pi$
$433$	−3.06764	−0.147421	−0.0737107	+	0.997280i	$0.523484\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	15.3369	0.733664
$438$	0	0
$439$	−14.7475	−0.703861	−0.351931	−	0.936026i	$-0.614475\pi$
$439$	−14.7475	−0.703861	−0.351931	+	0.936026i	$0.614475\pi$
$440$	0	0
$441$	−0.271556	−0.0129312
$442$	0	0
$443$	−10.1857	−0.483935	−0.241968	−	0.970284i	$-0.577793\pi$
$443$	−10.1857	−0.483935	−0.241968	+	0.970284i	$0.577793\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	21.9475	1.03808
$448$	0	0
$449$	18.9484	0.894230	0.447115	−	0.894477i	$-0.352451\pi$
$449$	18.9484	0.894230	0.447115	+	0.894477i	$0.352451\pi$
$450$	0	0
$451$	51.6881	2.43390
$452$	0	0
$453$	3.69864	0.173777
$454$	0	0
$455$	0	0
$456$	0	0
$457$	30.4392	1.42389	0.711943	−	0.702237i	$-0.247815\pi$
$457$	30.4392	1.42389	0.711943	+	0.702237i	$0.247815\pi$
$458$	0	0
$459$	−13.4154	−0.626175
$460$	0	0
$461$	2.36972	0.110369	0.0551844	−	0.998476i	$-0.482425\pi$
$461$	2.36972	0.110369	0.0551844	+	0.998476i	$0.482425\pi$
$462$	0	0
$463$	0.320982	0.0149173	0.00745865	−	0.999972i	$-0.497626\pi$
$463$	0.320982	0.0149173	0.00745865	+	0.999972i	$0.497626\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	12.8071	0.592644	0.296322	−	0.955088i	$-0.404240\pi$
$467$	12.8071	0.592644	0.296322	+	0.955088i	$0.404240\pi$
$468$	0	0
$469$	14.1015	0.651148
$470$	0	0
$471$	−27.0127	−1.24468
$472$	0	0
$473$	12.4258	0.571341
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0.0990055	0.00453315
$478$	0	0
$479$	20.3446	0.929569	0.464784	−	0.885424i	$-0.346132\pi$
$479$	20.3446	0.929569	0.464784	+	0.885424i	$0.346132\pi$
$480$	0	0
$481$	0.0472852	0.00215602
$482$	0	0
$483$	−44.4681	−2.02337
$484$	0	0
$485$	0	0
$486$	0	0
$487$	15.4919	0.702003	0.351002	−	0.936375i	$-0.385841\pi$
$487$	15.4919	0.702003	0.351002	+	0.936375i	$0.385841\pi$
$488$	0	0
$489$	−12.6804	−0.573427
$490$	0	0
$491$	27.5085	1.24144	0.620722	−	0.784031i	$-0.286840\pi$
$491$	27.5085	1.24144	0.620722	+	0.784031i	$0.286840\pi$
$492$	0	0
$493$	−12.1312	−0.546362
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−8.00875	−0.359241
$498$	0	0
$499$	−4.91044	−0.219821	−0.109911	−	0.993941i	$-0.535057\pi$
$499$	−4.91044	−0.219821	−0.109911	+	0.993941i	$0.535057\pi$
$500$	0	0
$501$	18.5544	0.828950
$502$	0	0
$503$	41.0454	1.83012	0.915062	−	0.403314i	$-0.132142\pi$
$503$	41.0454	1.83012	0.915062	+	0.403314i	$0.132142\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	1.03196	0.0458308
$508$	0	0
$509$	41.4404	1.83681	0.918407	−	0.395637i	$-0.129476\pi$
$509$	41.4404	1.83681	0.918407	+	0.395637i	$0.129476\pi$
$510$	0	0
$511$	−5.18315	−0.229289
$512$	0	0
$513$	10.6247	0.469093
$514$	0	0
$515$	0	0
$516$	0	0
$517$	37.1019	1.63174
$518$	0	0
$519$	25.7277	1.12932
$520$	0	0
$521$	−1.62447	−0.0711691	−0.0355846	−	0.999367i	$-0.511329\pi$
$521$	−1.62447	−0.0711691	−0.0355846	+	0.999367i	$0.511329\pi$
$522$	0	0
$523$	−19.0009	−0.830853	−0.415427	−	0.909627i	$-0.636368\pi$
$523$	−19.0009	−0.830853	−0.415427	+	0.909627i	$0.636368\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	4.15451	0.180973
$528$	0	0
$529$	34.3174	1.49206
$530$	0	0
$531$	−0.00127506	−5.53328e−5 0
$532$	0	0
$533$	34.0755	1.47597
$534$	0	0
$535$	0	0
$536$	0	0
$537$	12.6911	0.547660
$538$	0	0
$539$	25.2337	1.08689
$540$	0	0
$541$	−33.5195	−1.44112	−0.720558	−	0.693394i	$-0.756114\pi$
$541$	−33.5195	−1.44112	−0.720558	+	0.693394i	$0.756114\pi$
$542$	0	0
$543$	18.7279	0.803691
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−17.4258	−0.745072	−0.372536	−	0.928018i	$-0.621512\pi$
$547$	−17.4258	−0.745072	−0.372536	+	0.928018i	$0.621512\pi$
$548$	0	0
$549$	−0.225185	−0.00961065
$550$	0	0
$551$	9.60771	0.409302
$552$	0	0
$553$	−1.46151	−0.0621497
$554$	0	0
$555$	0	0
$556$	0	0
$557$	16.2929	0.690351	0.345175	−	0.938538i	$-0.387819\pi$
$557$	16.2929	0.690351	0.345175	+	0.938538i	$0.387819\pi$
$558$	0	0
$559$	8.19175	0.346474
$560$	0	0
$561$	23.4353	0.989439
$562$	0	0
$563$	23.0999	0.973545	0.486773	−	0.873529i	$-0.338174\pi$
$563$	23.0999	0.973545	0.486773	+	0.873529i	$0.338174\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−30.2151	−1.26891
$568$	0	0
$569$	−9.28334	−0.389178	−0.194589	−	0.980885i	$-0.562337\pi$
$569$	−9.28334	−0.389178	−0.194589	+	0.980885i	$0.562337\pi$
$570$	0	0
$571$	−22.2143	−0.929640	−0.464820	−	0.885405i	$-0.653881\pi$
$571$	−22.2143	−0.929640	−0.464820	+	0.885405i	$0.653881\pi$
$572$	0	0
$573$	−3.00435	−0.125509
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−14.6093	−0.608192	−0.304096	−	0.952641i	$-0.598354\pi$
$577$	−14.6093	−0.608192	−0.304096	+	0.952641i	$0.598354\pi$
$578$	0	0
$579$	−16.3500	−0.679485
$580$	0	0
$581$	−20.6879	−0.858280
$582$	0	0
$583$	−9.19986	−0.381019
$584$	0	0
$585$	0	0
$586$	0	0
$587$	23.3435	0.963489	0.481744	−	0.876312i	$-0.340003\pi$
$587$	23.3435	0.963489	0.481744	+	0.876312i	$0.340003\pi$
$588$	0	0
$589$	−3.29030	−0.135575
$590$	0	0
$591$	41.0265	1.68760
$592$	0	0
$593$	41.9815	1.72397	0.861986	−	0.506932i	$-0.169220\pi$
$593$	41.9815	1.72397	0.861986	+	0.506932i	$0.169220\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	40.8743	1.67287
$598$	0	0
$599$	25.4160	1.03847	0.519236	−	0.854631i	$-0.326217\pi$
$599$	25.4160	1.03847	0.519236	+	0.854631i	$0.326217\pi$
$600$	0	0
$601$	37.1379	1.51489	0.757444	−	0.652900i	$-0.226448\pi$
$601$	37.1379	1.51489	0.757444	+	0.652900i	$0.226448\pi$
$602$	0	0
$603$	−0.236717	−0.00963988
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−18.9242	−0.768109	−0.384054	−	0.923310i	$-0.625473\pi$
$607$	−18.9242	−0.768109	−0.384054	+	0.923310i	$0.625473\pi$
$608$	0	0
$609$	−27.8568	−1.12881
$610$	0	0
$611$	24.4595	0.989525
$612$	0	0
$613$	−9.20317	−0.371713	−0.185856	−	0.982577i	$-0.559506\pi$
$613$	−9.20317	−0.371713	−0.185856	+	0.982577i	$0.559506\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	32.7375	1.31796	0.658981	−	0.752159i	$-0.270988\pi$
$617$	32.7375	1.31796	0.658981	+	0.752159i	$0.270988\pi$
$618$	0	0
$619$	−36.3952	−1.46285	−0.731424	−	0.681923i	$-0.761144\pi$
$619$	−36.3952	−1.46285	−0.731424	+	0.681923i	$0.761144\pi$
$620$	0	0
$621$	39.7069	1.59338
$622$	0	0
$623$	−20.8651	−0.835943
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−18.5604	−0.741229
$628$	0	0
$629$	−0.0343496	−0.00136961
$630$	0	0
$631$	−12.1083	−0.482024	−0.241012	−	0.970522i	$-0.577479\pi$
$631$	−12.1083	−0.482024	−0.241012	+	0.970522i	$0.577479\pi$
$632$	0	0
$633$	−26.4037	−1.04945
$634$	0	0
$635$	0	0
$636$	0	0
$637$	16.6354	0.659117
$638$	0	0
$639$	0.134440	0.00531837
$640$	0	0
$641$	9.75177	0.385172	0.192586	−	0.981280i	$-0.438313\pi$
$641$	9.75177	0.385172	0.192586	+	0.981280i	$0.438313\pi$
$642$	0	0
$643$	6.77862	0.267323	0.133661	−	0.991027i	$-0.457327\pi$
$643$	6.77862	0.267323	0.133661	+	0.991027i	$0.457327\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−40.1188	−1.57723	−0.788616	−	0.614886i	$-0.789202\pi$
$647$	−40.1188	−1.57723	−0.788616	+	0.614886i	$0.789202\pi$
$648$	0	0
$649$	0.118482	0.00465082
$650$	0	0
$651$	9.53997	0.373901
$652$	0	0
$653$	−6.14210	−0.240359	−0.120179	−	0.992752i	$-0.538347\pi$
$653$	−6.14210	−0.240359	−0.120179	+	0.992752i	$0.538347\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0.0870078	0.00339450
$658$	0	0
$659$	44.1645	1.72040	0.860202	−	0.509954i	$-0.170338\pi$
$659$	44.1645	1.72040	0.860202	+	0.509954i	$0.170338\pi$
$660$	0	0
$661$	27.7447	1.07915	0.539573	−	0.841939i	$-0.318586\pi$
$661$	27.7447	1.07915	0.539573	+	0.841939i	$0.318586\pi$
$662$	0	0
$663$	15.4498	0.600019
$664$	0	0
$665$	0	0
$666$	0	0
$667$	35.9061	1.39029
$668$	0	0
$669$	33.5790	1.29824
$670$	0	0
$671$	20.9248	0.807792
$672$	0	0
$673$	41.8324	1.61252	0.806259	−	0.591562i	$-0.201489\pi$
$673$	41.8324	1.61252	0.806259	+	0.591562i	$0.201489\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−22.6274	−0.869643	−0.434821	−	0.900517i	$-0.643188\pi$
$677$	−22.6274	−0.869643	−0.434821	+	0.900517i	$0.643188\pi$
$678$	0	0
$679$	54.7918	2.10272
$680$	0	0
$681$	−2.30215	−0.0882186
$682$	0	0
$683$	−49.0024	−1.87502	−0.937512	−	0.347953i	$-0.886877\pi$
$683$	−49.0024	−1.87502	−0.937512	+	0.347953i	$0.886877\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	38.3734	1.46404
$688$	0	0
$689$	−6.06502	−0.231059
$690$	0	0
$691$	−36.2583	−1.37933	−0.689665	−	0.724128i	$-0.742242\pi$
$691$	−36.2583	−1.37933	−0.689665	+	0.724128i	$0.742242\pi$
$692$	0	0
$693$	−1.05121	−0.0399320
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−24.7536	−0.937608
$698$	0	0
$699$	31.5030	1.19155
$700$	0	0
$701$	−8.32362	−0.314379	−0.157189	−	0.987568i	$-0.550243\pi$
$701$	−8.32362	−0.314379	−0.157189	+	0.987568i	$0.550243\pi$
$702$	0	0
$703$	0.0272043	0.00102603
$704$	0	0
$705$	0	0
$706$	0	0
$707$	4.94500	0.185976
$708$	0	0
$709$	37.3097	1.40119	0.700597	−	0.713557i	$-0.252917\pi$
$709$	37.3097	1.40119	0.700597	+	0.713557i	$0.252917\pi$
$710$	0	0
$711$	0.0245339	0.000920092 0
$712$	0	0
$713$	−12.2966	−0.460511
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−20.5910	−0.768983
$718$	0	0
$719$	7.66524	0.285865	0.142933	−	0.989732i	$-0.454347\pi$
$719$	7.66524	0.285865	0.142933	+	0.989732i	$0.454347\pi$
$720$	0	0
$721$	50.2169	1.87017
$722$	0	0
$723$	−17.3544	−0.645419
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−45.9089	−1.70267	−0.851334	−	0.524624i	$-0.824206\pi$
$727$	−45.9089	−1.70267	−0.851334	+	0.524624i	$0.824206\pi$
$728$	0	0
$729$	27.4973	1.01842
$730$	0	0
$731$	−5.95076	−0.220097
$732$	0	0
$733$	−23.7131	−0.875864	−0.437932	−	0.899008i	$-0.644289\pi$
$733$	−23.7131	−0.875864	−0.437932	+	0.899008i	$0.644289\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	21.9964	0.810249
$738$	0	0
$739$	3.53683	0.130104	0.0650522	−	0.997882i	$-0.479279\pi$
$739$	3.53683	0.130104	0.0650522	+	0.997882i	$0.479279\pi$
$740$	0	0
$741$	−12.2359	−0.449498
$742$	0	0
$743$	−15.7201	−0.576715	−0.288358	−	0.957523i	$-0.593109\pi$
$743$	−15.7201	−0.576715	−0.288358	+	0.957523i	$0.593109\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0.347281	0.0127063
$748$	0	0
$749$	41.8320	1.52851
$750$	0	0
$751$	14.1856	0.517642	0.258821	−	0.965925i	$-0.416666\pi$
$751$	14.1856	0.517642	0.258821	+	0.965925i	$0.416666\pi$
$752$	0	0
$753$	28.6911	1.04556
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−12.7388	−0.463000	−0.231500	−	0.972835i	$-0.574363\pi$
$757$	−12.7388	−0.463000	−0.231500	+	0.972835i	$0.574363\pi$
$758$	0	0
$759$	−69.3641	−2.51776
$760$	0	0
$761$	32.2299	1.16833	0.584167	−	0.811633i	$-0.301421\pi$
$761$	32.2299	1.16833	0.584167	+	0.811633i	$0.301421\pi$
$762$	0	0
$763$	−52.5652	−1.90299
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0.0781093	0.00282036
$768$	0	0
$769$	−11.3687	−0.409965	−0.204983	−	0.978766i	$-0.565714\pi$
$769$	−11.3687	−0.409965	−0.204983	+	0.978766i	$0.565714\pi$
$770$	0	0
$771$	25.7597	0.927715
$772$	0	0
$773$	−27.1994	−0.978295	−0.489147	−	0.872201i	$-0.662692\pi$
$773$	−27.1994	−0.978295	−0.489147	+	0.872201i	$0.662692\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−0.0788766	−0.00282968
$778$	0	0
$779$	19.6044	0.702400
$780$	0	0
$781$	−12.4925	−0.447018
$782$	0	0
$783$	24.8742	0.888931
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−25.7266	−0.917054	−0.458527	−	0.888680i	$-0.651623\pi$
$787$	−25.7266	−0.917054	−0.458527	+	0.888680i	$0.651623\pi$
$788$	0	0
$789$	−10.5870	−0.376908
$790$	0	0
$791$	−63.5321	−2.25894
$792$	0	0
$793$	13.7947	0.489864
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−17.6713	−0.625951	−0.312975	−	0.949761i	$-0.601326\pi$
$797$	−17.6713	−0.625951	−0.312975	+	0.949761i	$0.601326\pi$
$798$	0	0
$799$	−17.7682	−0.628593
$800$	0	0
$801$	0.350256	0.0123757
$802$	0	0
$803$	−8.08500	−0.285314
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−19.0497	−0.670580
$808$	0	0
$809$	−40.8576	−1.43648	−0.718238	−	0.695798i	$-0.755051\pi$
$809$	−40.8576	−1.43648	−0.718238	+	0.695798i	$0.755051\pi$
$810$	0	0
$811$	−16.6214	−0.583656	−0.291828	−	0.956471i	$-0.594263\pi$
$811$	−16.6214	−0.583656	−0.291828	+	0.956471i	$0.594263\pi$
$812$	0	0
$813$	−1.99239	−0.0698763
$814$	0	0
$815$	0	0
$816$	0	0
$817$	4.71290	0.164884
$818$	0	0
$819$	−0.693009	−0.0242157
$820$	0	0
$821$	−43.8983	−1.53206	−0.766031	−	0.642803i	$-0.777771\pi$
$821$	−43.8983	−1.53206	−0.766031	+	0.642803i	$0.777771\pi$
$822$	0	0
$823$	26.1962	0.913143	0.456571	−	0.889687i	$-0.349077\pi$
$823$	26.1962	0.913143	0.456571	+	0.889687i	$0.349077\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	22.3827	0.778321	0.389161	−	0.921170i	$-0.372765\pi$
$827$	22.3827	0.778321	0.389161	+	0.921170i	$0.372765\pi$
$828$	0	0
$829$	0.841282	0.0292189	0.0146095	−	0.999893i	$-0.495349\pi$
$829$	0.841282	0.0292189	0.0146095	+	0.999893i	$0.495349\pi$
$830$	0	0
$831$	−3.73071	−0.129417
$832$	0	0
$833$	−12.0845	−0.418702
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−8.51853	−0.294444
$838$	0	0
$839$	−35.5528	−1.22742	−0.613710	−	0.789532i	$-0.710323\pi$
$839$	−35.5528	−1.22742	−0.613710	+	0.789532i	$0.710323\pi$
$840$	0	0
$841$	−6.50681	−0.224373
$842$	0	0
$843$	41.3709	1.42489
$844$	0	0
$845$	0	0
$846$	0	0
$847$	60.0158	2.06217
$848$	0	0
$849$	−27.8138	−0.954567
$850$	0	0
$851$	0.101668	0.00348515
$852$	0	0
$853$	19.5406	0.669058	0.334529	−	0.942386i	$-0.391423\pi$
$853$	19.5406	0.669058	0.334529	+	0.942386i	$0.391423\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	0.570622	0.0194921	0.00974604	−	0.999953i	$-0.496898\pi$
$857$	0.570622	0.0194921	0.00974604	+	0.999953i	$0.496898\pi$
$858$	0	0
$859$	20.0983	0.685744	0.342872	−	0.939382i	$-0.388600\pi$
$859$	20.0983	0.685744	0.342872	+	0.939382i	$0.388600\pi$
$860$	0	0
$861$	−56.8413	−1.93715
$862$	0	0
$863$	13.9645	0.475358	0.237679	−	0.971344i	$-0.423613\pi$
$863$	13.9645	0.475358	0.237679	+	0.971344i	$0.423613\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	17.9382	0.609213
$868$	0	0
$869$	−2.27975	−0.0773353
$870$	0	0
$871$	14.5012	0.491353
$872$	0	0
$873$	−0.919771	−0.0311295
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−23.4512	−0.791891	−0.395946	−	0.918274i	$-0.629583\pi$
$877$	−23.4512	−0.791891	−0.395946	+	0.918274i	$0.629583\pi$
$878$	0	0
$879$	5.98545	0.201884
$880$	0	0
$881$	44.4714	1.49828	0.749139	−	0.662412i	$-0.230467\pi$
$881$	44.4714	1.49828	0.749139	+	0.662412i	$0.230467\pi$
$882$	0	0
$883$	−0.758169	−0.0255144	−0.0127572	−	0.999919i	$-0.504061\pi$
$883$	−0.758169	−0.0255144	−0.0127572	+	0.999919i	$0.504061\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	55.4695	1.86248	0.931242	−	0.364402i	$-0.118727\pi$
$887$	55.4695	1.86248	0.931242	+	0.364402i	$0.118727\pi$
$888$	0	0
$889$	2.32120	0.0778506
$890$	0	0
$891$	−47.1314	−1.57896
$892$	0	0
$893$	14.0721	0.470905
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−45.7284	−1.52683
$898$	0	0
$899$	−7.70312	−0.256914
$900$	0	0
$901$	4.40584	0.146780
$902$	0	0
$903$	−13.6647	−0.454732
$904$	0	0
$905$	0	0
$906$	0	0
$907$	15.8193	0.525271	0.262636	−	0.964895i	$-0.415408\pi$
$907$	15.8193	0.525271	0.262636	+	0.964895i	$0.415408\pi$
$908$	0	0
$909$	−0.0830100	−0.00275327
$910$	0	0
$911$	8.02411	0.265851	0.132925	−	0.991126i	$-0.457563\pi$
$911$	8.02411	0.265851	0.132925	+	0.991126i	$0.457563\pi$
$912$	0	0
$913$	−32.2703	−1.06799
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−24.1419	−0.797235
$918$	0	0
$919$	−22.8402	−0.753428	−0.376714	−	0.926330i	$-0.622946\pi$
$919$	−22.8402	−0.753428	−0.376714	+	0.926330i	$0.622946\pi$
$920$	0	0
$921$	−15.3130	−0.504581
$922$	0	0
$923$	−8.23572	−0.271082
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−0.842973	−0.0276869
$928$	0	0
$929$	−39.7278	−1.30343	−0.651713	−	0.758465i	$-0.725950\pi$
$929$	−39.7278	−1.30343	−0.651713	+	0.758465i	$0.725950\pi$
$930$	0	0
$931$	9.57070	0.313667
$932$	0	0
$933$	−46.5040	−1.52247
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−30.4367	−0.994325	−0.497162	−	0.867658i	$-0.665625\pi$
$937$	−30.4367	−0.994325	−0.497162	+	0.867658i	$0.665625\pi$
$938$	0	0
$939$	−34.4917	−1.12559
$940$	0	0
$941$	25.6117	0.834916	0.417458	−	0.908696i	$-0.362921\pi$
$941$	25.6117	0.834916	0.417458	+	0.908696i	$0.362921\pi$
$942$	0	0
$943$	73.2659	2.38587
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−58.0741	−1.88716	−0.943578	−	0.331151i	$-0.892563\pi$
$947$	−58.0741	−1.88716	−0.943578	+	0.331151i	$0.892563\pi$
$948$	0	0
$949$	−5.33005	−0.173021
$950$	0	0
$951$	16.6047	0.538446
$952$	0	0
$953$	45.6216	1.47783	0.738915	−	0.673799i	$-0.235339\pi$
$953$	45.6216	1.47783	0.738915	+	0.673799i	$0.235339\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−43.4527	−1.40463
$958$	0	0
$959$	37.3862	1.20726
$960$	0	0
$961$	−28.3620	−0.914902
$962$	0	0
$963$	−0.702219	−0.0226287
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−42.4449	−1.36494	−0.682468	−	0.730916i	$-0.739093\pi$
$967$	−42.4449	−1.36494	−0.682468	+	0.730916i	$0.739093\pi$
$968$	0	0
$969$	8.88860	0.285543
$970$	0	0
$971$	14.4332	0.463182	0.231591	−	0.972813i	$-0.425607\pi$
$971$	14.4332	0.463182	0.231591	+	0.972813i	$0.425607\pi$
$972$	0	0
$973$	−66.8047	−2.14166
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−16.5748	−0.530276	−0.265138	−	0.964211i	$-0.585417\pi$
$977$	−16.5748	−0.530276	−0.265138	+	0.964211i	$0.585417\pi$
$978$	0	0
$979$	−32.5467	−1.04020
$980$	0	0
$981$	0.882394	0.0281727
$982$	0	0
$983$	33.3174	1.06266	0.531330	−	0.847165i	$-0.321692\pi$
$983$	33.3174	1.06266	0.531330	+	0.847165i	$0.321692\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−40.8009	−1.29871
$988$	0	0
$989$	17.6131	0.560065
$990$	0	0
$991$	24.4962	0.778146	0.389073	−	0.921207i	$-0.372795\pi$
$991$	24.4962	0.778146	0.389073	+	0.921207i	$0.372795\pi$
$992$	0	0
$993$	33.7516	1.07107
$994$	0	0
$995$	0	0
$996$	0	0
$997$	59.8575	1.89571	0.947854	−	0.318706i	$-0.103248\pi$
$997$	59.8575	1.89571	0.947854	+	0.318706i	$0.103248\pi$
$998$	0	0
$999$	0.0704313	0.00222835

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	10000.2.a.bn.1.2		8
4.3	odd	2		625.2.a.e.1.6	✓	8
5.4	even	2		10000.2.a.be.1.7		8
12.11	even	2		5625.2.a.be.1.3		8
20.3	even	4		625.2.b.d.624.8		16
20.7	even	4		625.2.b.d.624.9		16
20.19	odd	2		625.2.a.g.1.3	yes	8
60.59	even	2		5625.2.a.s.1.6		8
100.3	even	20		625.2.e.k.249.4		32
100.11	odd	10		625.2.d.q.501.3		16
100.19	odd	10		625.2.d.n.251.3		16
100.23	even	20		625.2.e.j.124.4		32
100.27	even	20		625.2.e.j.124.5		32
100.31	odd	10		625.2.d.p.251.2		16
100.39	odd	10		625.2.d.m.501.2		16
100.47	even	20		625.2.e.k.249.5		32
100.59	odd	10		625.2.d.m.126.2		16
100.63	even	20		625.2.e.j.499.5		32
100.67	even	20		625.2.e.k.374.4		32
100.71	odd	10		625.2.d.p.376.2		16
100.79	odd	10		625.2.d.n.376.3		16
100.83	even	20		625.2.e.k.374.5		32
100.87	even	20		625.2.e.j.499.4		32
100.91	odd	10		625.2.d.q.126.3		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
625.2.a.e.1.6	✓	8	4.3	odd	2
625.2.a.g.1.3	yes	8	20.19	odd	2
625.2.b.d.624.8		16	20.3	even	4
625.2.b.d.624.9		16	20.7	even	4
625.2.d.m.126.2		16	100.59	odd	10
625.2.d.m.501.2		16	100.39	odd	10
625.2.d.n.251.3		16	100.19	odd	10
625.2.d.n.376.3		16	100.79	odd	10
625.2.d.p.251.2		16	100.31	odd	10
625.2.d.p.376.2		16	100.71	odd	10
625.2.d.q.126.3		16	100.91	odd	10
625.2.d.q.501.3		16	100.11	odd	10
625.2.e.j.124.4		32	100.23	even	20
625.2.e.j.124.5		32	100.27	even	20
625.2.e.j.499.4		32	100.87	even	20
625.2.e.j.499.5		32	100.63	even	20
625.2.e.k.249.4		32	100.3	even	20
625.2.e.k.249.5		32	100.47	even	20
625.2.e.k.374.4		32	100.67	even	20
625.2.e.k.374.5		32	100.83	even	20
5625.2.a.s.1.6		8	60.59	even	2
5625.2.a.be.1.3		8	12.11	even	2
10000.2.a.be.1.7		8	5.4	even	2
10000.2.a.bn.1.2		8	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 10000 = 2^{4} \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	10000.a (trivial)

\(f(q)\)	\(=\)	\(q-1.71538 q^{3} +3.42409 q^{7} -0.0574791 q^{9} +O(q^{10})\)	\(q-1.71538 q^{3} +3.42409 q^{7} -0.0574791 q^{9} +5.34111 q^{11} +3.52114 q^{13} -2.55787 q^{17} +2.02579 q^{19} -5.87362 q^{21} +7.57082 q^{23} +5.24473 q^{27} +4.74270 q^{29} -1.62421 q^{31} -9.16203 q^{33} +0.0134290 q^{37} -6.04008 q^{39} +9.67740 q^{41} +2.32645 q^{43} +6.94647 q^{47} +4.72443 q^{49} +4.38772 q^{51} -1.72246 q^{53} -3.47500 q^{57} +0.0221830 q^{59} +3.91768 q^{61} -0.196814 q^{63} +4.11832 q^{67} -12.9868 q^{69} -2.33894 q^{71} -1.51373 q^{73} +18.2885 q^{77} -0.426831 q^{79} -8.82426 q^{81} -6.04187 q^{83} -8.13552 q^{87} -6.09362 q^{89} +12.0567 q^{91} +2.78613 q^{93} +16.0018 q^{97} -0.307002 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 5 q^{3} + 10 q^{7} + 9 q^{9}+O(q^{10}) \) 8 * q + 5 * q^3 + 10 * q^7 + 9 * q^9	\( 8 q + 5 q^{3} + 10 q^{7} + 9 q^{9} - q^{11} - 10 q^{13} - 15 q^{17} + 10 q^{19} - 14 q^{21} + 30 q^{23} + 20 q^{27} + 10 q^{29} + 9 q^{31} - 5 q^{33} + 10 q^{37} - 8 q^{39} - 4 q^{41} + 30 q^{47} - 4 q^{49} + 14 q^{51} - 10 q^{53} + 10 q^{57} + 5 q^{59} + 6 q^{61} + 10 q^{67} + 3 q^{69} + 9 q^{71} - 5 q^{77} + 20 q^{79} + 8 q^{81} + 40 q^{83} + 40 q^{87} - 5 q^{89} - 6 q^{91} + 40 q^{93} + 22 q^{99}+O(q^{100}) \) 8 * q + 5 * q^3 + 10 * q^7 + 9 * q^9 - q^11 - 10 * q^13 - 15 * q^17 + 10 * q^19 - 14 * q^21 + 30 * q^23 + 20 * q^27 + 10 * q^29 + 9 * q^31 - 5 * q^33 + 10 * q^37 - 8 * q^39 - 4 * q^41 + 30 * q^47 - 4 * q^49 + 14 * q^51 - 10 * q^53 + 10 * q^57 + 5 * q^59 + 6 * q^61 + 10 * q^67 + 3 * q^69 + 9 * q^71 - 5 * q^77 + 20 * q^79 + 8 * q^81 + 40 * q^83 + 40 * q^87 - 5 * q^89 - 6 * q^91 + 40 * q^93 + 22 * q^99

Introduction

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