LMFDB - Embedded newform 5625.2.a.be.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5625,2,Mod(1,5625)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5625, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("5625.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5625 = 3^{2} \cdot 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5625.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:	$44.9158511370$
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			1.3
Root			$1.32675$ of defining polynomial
Character	$\chi$	$=$	5625.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.326751 q^{2} -1.89323 q^{4} -3.42409 q^{7} +1.27212 q^{8} +O(q^{10})$	$q-0.326751 q^{2} -1.89323 q^{4} -3.42409 q^{7} +1.27212 q^{8} +5.34111 q^{11} +3.52114 q^{13} +1.11883 q^{14} +3.37080 q^{16} +2.55787 q^{17} -2.02579 q^{19} -1.74522 q^{22} +7.57082 q^{23} -1.15054 q^{26} +6.48261 q^{28} -4.74270 q^{29} +1.62421 q^{31} -3.64565 q^{32} -0.835788 q^{34} +0.0134290 q^{37} +0.661929 q^{38} -9.67740 q^{41} -2.32645 q^{43} -10.1120 q^{44} -2.47377 q^{46} +6.94647 q^{47} +4.72443 q^{49} -6.66634 q^{52} +1.72246 q^{53} -4.35585 q^{56} +1.54968 q^{58} +0.0221830 q^{59} +3.91768 q^{61} -0.530712 q^{62} -5.55038 q^{64} -4.11832 q^{67} -4.84265 q^{68} -2.33894 q^{71} -1.51373 q^{73} -0.00438793 q^{74} +3.83530 q^{76} -18.2885 q^{77} +0.426831 q^{79} +3.16210 q^{82} -6.04187 q^{83} +0.760170 q^{86} +6.79453 q^{88} +6.09362 q^{89} -12.0567 q^{91} -14.3333 q^{92} -2.26977 q^{94} +16.0018 q^{97} -1.54371 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 5 q^{2} + 11 q^{4} - 10 q^{7} + 15 q^{8}+O(q^{10}) $ 8 * q + 5 * q^2 + 11 * q^4 - 10 * q^7 + 15 * q^8	$ 8 q + 5 q^{2} + 11 q^{4} - 10 q^{7} + 15 q^{8} - q^{11} - 10 q^{13} + 8 q^{14} + 13 q^{16} + 15 q^{17} - 10 q^{19} + 5 q^{22} + 30 q^{23} - 11 q^{26} + 5 q^{28} - 10 q^{29} - 9 q^{31} + 30 q^{32} + 7 q^{34} + 10 q^{37} + 20 q^{38} + 4 q^{41} + 18 q^{44} - 9 q^{46} + 30 q^{47} - 4 q^{49} - 5 q^{52} + 10 q^{53} + 30 q^{58} + 5 q^{59} + 6 q^{61} + 10 q^{62} - 9 q^{64} - 10 q^{67} + 40 q^{68} + 9 q^{71} + 18 q^{74} - 10 q^{76} + 5 q^{77} - 20 q^{79} + 45 q^{82} + 40 q^{83} + 24 q^{86} + 40 q^{88} + 5 q^{89} + 6 q^{91} + 15 q^{92} + 47 q^{94} - 30 q^{98}+O(q^{100}) $ 8 * q + 5 * q^2 + 11 * q^4 - 10 * q^7 + 15 * q^8 - q^11 - 10 * q^13 + 8 * q^14 + 13 * q^16 + 15 * q^17 - 10 * q^19 + 5 * q^22 + 30 * q^23 - 11 * q^26 + 5 * q^28 - 10 * q^29 - 9 * q^31 + 30 * q^32 + 7 * q^34 + 10 * q^37 + 20 * q^38 + 4 * q^41 + 18 * q^44 - 9 * q^46 + 30 * q^47 - 4 * q^49 - 5 * q^52 + 10 * q^53 + 30 * q^58 + 5 * q^59 + 6 * q^61 + 10 * q^62 - 9 * q^64 - 10 * q^67 + 40 * q^68 + 9 * q^71 + 18 * q^74 - 10 * q^76 + 5 * q^77 - 20 * q^79 + 45 * q^82 + 40 * q^83 + 24 * q^86 + 40 * q^88 + 5 * q^89 + 6 * q^91 + 15 * q^92 + 47 * q^94 - 30 * q^98

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 $

$2$	−0.326751	−0.231048	−0.115524	−	0.993305i	$-0.536855\pi$
$2$	−0.326751	−0.231048	−0.115524	+	0.993305i	$0.536855\pi$
$3$	0	0
$3$	0	0
$4$	−1.89323	−0.946617
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−3.42409	−1.29419	−0.647093	−	0.762411i	$-0.724016\pi$
$7$	−3.42409	−1.29419	−0.647093	+	0.762411i	$0.724016\pi$
$8$	1.27212	0.449762
$9$	0	0
$10$	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	$0.337619\pi$
$14$	1.11883	0.299019
$15$	0	0
$16$	3.37080	0.842700
$17$	2.55787	0.620375	0.310188	−	0.950675i	$-0.399608\pi$
$17$	2.55787	0.620375	0.310188	+	0.950675i	$0.399608\pi$
$18$	0	0
$19$	−2.02579	−0.464748	−0.232374	−	0.972626i	$-0.574649\pi$
$19$	−2.02579	−0.464748	−0.232374	+	0.972626i	$0.574649\pi$
$20$	0	0
$21$	0	0
$22$	−1.74522	−0.372081
$23$	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
$26$	−1.15054	−0.225639
$27$	0	0
$28$	6.48261	1.22510
$29$	−4.74270	−0.880697	−0.440348	−	0.897827i	$-0.645145\pi$
$29$	−4.74270	−0.880697	−0.440348	+	0.897827i	$0.645145\pi$
$30$	0	0
$31$	1.62421	0.291716	0.145858	−	0.989306i	$-0.453406\pi$
$31$	1.62421	0.291716	0.145858	+	0.989306i	$0.453406\pi$
$32$	−3.64565	−0.644466
$33$	0	0
$34$	−0.835788	−0.143336
$35$	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	$0.499649\pi$
$38$	0.661929	0.107379
$39$	0	0
$40$	0	0
$41$	−9.67740	−1.51136	−0.755678	−	0.654943i	$-0.772693\pi$
$41$	−9.67740	−1.51136	−0.755678	+	0.654943i	$0.772693\pi$
$42$	0	0
$43$	−2.32645	−0.354780	−0.177390	−	0.984141i	$-0.556765\pi$
$43$	−2.32645	−0.354780	−0.177390	+	0.984141i	$0.556765\pi$
$44$	−10.1120	−1.52444
$45$	0	0
$46$	−2.47377	−0.364738
$47$	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
$51$	0	0
$52$	−6.66634	−0.924454
$53$	1.72246	0.236598	0.118299	−	0.992978i	$-0.462256\pi$
$53$	1.72246	0.236598	0.118299	+	0.992978i	$0.462256\pi$
$54$	0	0
$55$	0	0
$56$	−4.35585	−0.582075
$57$	0	0
$58$	1.54968	0.203483
$59$	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	$0.419305\pi$
$62$	−0.530712	−0.0674004
$63$	0	0
$64$	−5.55038	−0.693798
$65$	0	0
$66$	0	0
$67$	−4.11832	−0.503133	−0.251566	−	0.967840i	$-0.580946\pi$
$67$	−4.11832	−0.503133	−0.251566	+	0.967840i	$0.580946\pi$
$68$	−4.84265	−0.587258
$69$	0	0
$70$	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	$0.528234\pi$
$74$	−0.00438793	−0.000510086 0
$75$	0	0
$76$	3.83530	0.439939
$77$	−18.2885	−2.08417
$78$	0	0
$79$	0.426831	0.0480222	0.0240111	−	0.999712i	$-0.492356\pi$
$79$	0.426831	0.0480222	0.0240111	+	0.999712i	$0.492356\pi$
$80$	0	0
$81$	0	0
$82$	3.16210	0.349196
$83$	−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
$86$	0.760170	0.0819712
$87$	0	0
$88$	6.79453	0.724299
$89$	6.09362	0.645922	0.322961	−	0.946412i	$-0.395322\pi$
$89$	6.09362	0.645922	0.322961	+	0.946412i	$0.395322\pi$
$90$	0	0
$91$	−12.0567	−1.26389
$92$	−14.3333	−1.49435
$93$	0	0
$94$	−2.26977	−0.234109
$95$	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	$0.198177\pi$
$98$	−1.54371	−0.155938
$99$	0	0
$100$	0	0
$101$	−1.44418	−0.143701	−0.0718505	−	0.997415i	$-0.522890\pi$
$101$	−1.44418	−0.143701	−0.0718505	+	0.997415i	$0.522890\pi$
$102$	0	0
$103$	−14.6657	−1.44506	−0.722529	−	0.691341i	$-0.757020\pi$
$103$	−14.6657	−1.44506	−0.722529	+	0.691341i	$0.757020\pi$
$104$	4.47930	0.439232
$105$	0	0
$106$	−0.562816	−0.0546655
$107$	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	0
$112$	−11.5419	−1.09061
$113$	18.5544	1.74545	0.872727	−	0.488209i	$-0.162350\pi$
$113$	18.5544	1.74545	0.872727	+	0.488209i	$0.162350\pi$
$114$	0	0
$115$	0	0
$116$	8.97904	0.833683
$117$	0	0
$118$	−0.00724831	−0.000667261 0
$119$	−8.75840	−0.802881
$120$	0	0
$121$	17.5275	1.59341
$122$	−1.28011	−0.115895
$123$	0	0
$124$	−3.07500	−0.276144
$125$	0	0
$126$	0	0
$127$	−0.677902	−0.0601541	−0.0300771	−	0.999548i	$-0.509575\pi$
$127$	−0.677902	−0.0601541	−0.0300771	+	0.999548i	$0.509575\pi$
$128$	9.10489	0.804766
$129$	0	0
$130$	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
$134$	1.34567	0.116248
$135$	0	0
$136$	3.25392	0.279021
$137$	−10.9186	−0.932837	−0.466418	−	0.884564i	$-0.654456\pi$
$137$	−10.9186	−0.932837	−0.466418	+	0.884564i	$0.654456\pi$
$138$	0	0
$139$	19.5102	1.65483	0.827416	−	0.561589i	$-0.189810\pi$
$139$	19.5102	1.65483	0.827416	+	0.561589i	$0.189810\pi$
$140$	0	0
$141$	0	0
$142$	0.764251	0.0641345
$143$	18.8068	1.57270
$144$	0	0
$145$	0	0
$146$	0.494613	0.0409345
$147$	0	0
$148$	−0.0254242	−0.00208985
$149$	12.7945	1.04817	0.524085	−	0.851666i	$-0.324407\pi$
$149$	12.7945	1.04817	0.524085	+	0.851666i	$0.324407\pi$
$150$	0	0
$151$	2.15617	0.175466	0.0877331	−	0.996144i	$-0.472038\pi$
$151$	2.15617	0.175466	0.0877331	+	0.996144i	$0.472038\pi$
$152$	−2.57705	−0.209026
$153$	0	0
$154$	5.97578	0.481542
$155$	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	$0.283715\pi$
$158$	−0.139467	−0.0110954
$159$	0	0
$160$	0	0
$161$	−25.9232	−2.04304
$162$	0	0
$163$	−7.39219	−0.579001	−0.289500	−	0.957178i	$-0.593489\pi$
$163$	−7.39219	−0.579001	−0.289500	+	0.957178i	$0.593489\pi$
$164$	18.3216	1.43068
$165$	0	0
$166$	1.97419	0.153227
$167$	−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	0
$172$	4.40451	0.335841
$173$	14.9983	1.14030	0.570149	−	0.821541i	$-0.306885\pi$
$173$	14.9983	1.14030	0.570149	+	0.821541i	$0.306885\pi$
$174$	0	0
$175$	0	0
$176$	18.0038	1.35709
$177$	0	0
$178$	−1.99110	−0.149239
$179$	−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	$0.632990\pi$
$182$	3.93954	0.292018
$183$	0	0
$184$	9.63098	0.710006
$185$	0	0
$186$	0	0
$187$	13.6619	0.999056
$188$	−13.1513	−0.959157
$189$	0	0
$190$	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	$0.388542\pi$
$194$	−5.22862	−0.375393
$195$	0	0
$196$	−8.94444	−0.638889
$197$	23.9169	1.70401	0.852004	−	0.523535i	$-0.175387\pi$
$197$	23.9169	1.70401	0.852004	+	0.523535i	$0.175387\pi$
$198$	0	0
$199$	23.8281	1.68913	0.844566	−	0.535451i	$-0.179858\pi$
$199$	23.8281	1.68913	0.844566	+	0.535451i	$0.179858\pi$
$200$	0	0
$201$	0	0
$202$	0.471886	0.0332018
$203$	16.2394	1.13979
$204$	0	0
$205$	0	0
$206$	4.79204	0.333878
$207$	0	0
$208$	11.8691	0.822971
$209$	−10.8200	−0.748434
$210$	0	0
$211$	−15.3923	−1.05965	−0.529826	−	0.848107i	$-0.677743\pi$
$211$	−15.3923	−1.05965	−0.529826	+	0.848107i	$0.677743\pi$
$212$	−3.26102	−0.223968
$213$	0	0
$214$	−3.99190	−0.272881
$215$	0	0
$216$	0	0
$217$	−5.56144	−0.377535
$218$	5.01614	0.339736
$219$	0	0
$220$	0	0
$221$	9.00662	0.605851
$222$	0	0
$223$	19.5753	1.31086	0.655429	−	0.755257i	$-0.272488\pi$
$223$	19.5753	1.31086	0.655429	+	0.755257i	$0.272488\pi$
$224$	12.4831	0.834059
$225$	0	0
$226$	−6.06268	−0.403283
$227$	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
$231$	0	0
$232$	−6.03327	−0.396104
$233$	18.3651	1.20314	0.601568	−	0.798822i	$-0.294543\pi$
$233$	18.3651	1.20314	0.601568	+	0.798822i	$0.294543\pi$
$234$	0	0
$235$	0	0
$236$	−0.0419976	−0.00273381
$237$	0	0
$238$	2.86182	0.185504
$239$	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	$0.394351\pi$
$242$	−5.72713	−0.368154
$243$	0	0
$244$	−7.41708	−0.474830
$245$	0	0
$246$	0	0
$247$	−7.13309	−0.453867
$248$	2.06618	0.131203
$249$	0	0
$250$	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
$254$	0.221505	0.0138985
$255$	0	0
$256$	8.12573	0.507858
$257$	15.0170	0.936732	0.468366	−	0.883535i	$-0.344843\pi$
$257$	15.0170	0.936732	0.468366	+	0.883535i	$0.344843\pi$
$258$	0	0
$259$	−0.0459821	−0.00285719
$260$	0	0
$261$	0	0
$262$	2.30379	0.142328
$263$	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
$266$	−2.26651	−0.138969
$267$	0	0
$268$	7.79694	0.476274
$269$	−11.1052	−0.677098	−0.338549	−	0.940949i	$-0.609936\pi$
$269$	−11.1052	−0.677098	−0.338549	+	0.940949i	$0.609936\pi$
$270$	0	0
$271$	−1.16149	−0.0705554	−0.0352777	−	0.999378i	$-0.511232\pi$
$271$	−1.16149	−0.0705554	−0.0352777	+	0.999378i	$0.511232\pi$
$272$	8.62208	0.522790
$273$	0	0
$274$	3.56766	0.215530
$275$	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	$0.479188\pi$
$278$	−6.37497	−0.382346
$279$	0	0
$280$	0	0
$281$	24.1177	1.43874	0.719370	−	0.694627i	$-0.244431\pi$
$281$	24.1177	1.43874	0.719370	+	0.694627i	$0.244431\pi$
$282$	0	0
$283$	−16.2144	−0.963845	−0.481923	−	0.876214i	$-0.660061\pi$
$283$	−16.2144	−0.963845	−0.481923	+	0.876214i	$0.660061\pi$
$284$	4.42816	0.262763
$285$	0	0
$286$	−6.14514	−0.363370
$287$	33.1363	1.95598
$288$	0	0
$289$	−10.4573	−0.615135
$290$	0	0
$291$	0	0
$292$	2.86584	0.167711
$293$	3.48929	0.203846	0.101923	−	0.994792i	$-0.467500\pi$
$293$	3.48929	0.203846	0.101923	+	0.994792i	$0.467500\pi$
$294$	0	0
$295$	0	0
$296$	0.0170832	0.000992943 0
$297$	0	0
$298$	−4.18063	−0.242178
$299$	26.6579	1.54167
$300$	0	0
$301$	7.96599	0.459152
$302$	−0.704529	−0.0405411
$303$	0	0
$304$	−6.82854	−0.391644
$305$	0	0
$306$	0	0
$307$	−8.92690	−0.509485	−0.254742	−	0.967009i	$-0.581991\pi$
$307$	−8.92690	−0.509485	−0.254742	+	0.967009i	$0.581991\pi$
$308$	34.6244	1.97291
$309$	0	0
$310$	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	$0.307614\pi$
$314$	−5.14548	−0.290376
$315$	0	0
$316$	−0.808091	−0.0454587
$317$	9.67993	0.543679	0.271840	−	0.962343i	$-0.412368\pi$
$317$	9.67993	0.543679	0.271840	+	0.962343i	$0.412368\pi$
$318$	0	0
$319$	−25.3313	−1.41828
$320$	0	0
$321$	0	0
$322$	8.47044	0.472039
$323$	−5.18171	−0.288318
$324$	0	0
$325$	0	0
$326$	2.41540	0.133777
$327$	0	0
$328$	−12.3108	−0.679750
$329$	−23.7854	−1.31133
$330$	0	0
$331$	19.6759	1.08148	0.540742	−	0.841189i	$-0.318144\pi$
$331$	19.6759	1.08148	0.540742	+	0.841189i	$0.318144\pi$
$332$	11.4387	0.627778
$333$	0	0
$334$	3.53431	0.193389
$335$	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	$0.671671\pi$
$338$	0.196571	0.0106920
$339$	0	0
$340$	0	0
$341$	8.67508	0.469782
$342$	0	0
$343$	7.79178	0.420717
$344$	−2.95952	−0.159567
$345$	0	0
$346$	−4.90071	−0.263464
$347$	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
$351$	0	0
$352$	−19.4718	−1.03785
$353$	−5.32892	−0.283630	−0.141815	−	0.989893i	$-0.545294\pi$
$353$	−5.32892	−0.283630	−0.141815	+	0.989893i	$0.545294\pi$
$354$	0	0
$355$	0	0
$356$	−11.5366	−0.611441
$357$	0	0
$358$	2.41744	0.127766
$359$	−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
$362$	3.56736	0.187496
$363$	0	0
$364$	22.8262	1.19642
$365$	0	0
$366$	0	0
$367$	−16.4392	−0.858118	−0.429059	−	0.903276i	$-0.641155\pi$
$367$	−16.4392	−0.858118	−0.429059	+	0.903276i	$0.641155\pi$
$368$	25.5197	1.33031
$369$	0	0
$370$	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	$0.702956\pi$
$374$	−4.46404	−0.230830
$375$	0	0
$376$	8.83674	0.455720
$377$	−16.6997	−0.860078
$378$	0	0
$379$	16.4246	0.843674	0.421837	−	0.906672i	$-0.361385\pi$
$379$	16.4246	0.843674	0.421837	+	0.906672i	$0.361385\pi$
$380$	0	0
$381$	0	0
$382$	−0.572279	−0.0292803
$383$	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
$386$	−3.11441	−0.158520
$387$	0	0
$388$	−30.2952	−1.53801
$389$	−3.53865	−0.179417	−0.0897083	−	0.995968i	$-0.528593\pi$
$389$	−3.53865	−0.179417	−0.0897083	+	0.995968i	$0.528593\pi$
$390$	0	0
$391$	19.3652	0.979340
$392$	6.01003	0.303552
$393$	0	0
$394$	−7.81487	−0.393707
$395$	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	$0.429420\pi$
$398$	−7.78587	−0.390270
$399$	0	0
$400$	0	0
$401$	−22.7677	−1.13697	−0.568483	−	0.822695i	$-0.692469\pi$
$401$	−22.7677	−1.13697	−0.568483	+	0.822695i	$0.692469\pi$
$402$	0	0
$403$	5.71906	0.284887
$404$	2.73416	0.136030
$405$	0	0
$406$	−5.30626	−0.263345
$407$	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
$411$	0	0
$412$	27.7657	1.36792
$413$	−0.0759566	−0.00373758
$414$	0	0
$415$	0	0
$416$	−12.8368	−0.629378
$417$	0	0
$418$	3.53544	0.172924
$419$	−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	$0.786403\pi$
$422$	5.02946	0.244830
$423$	0	0
$424$	2.19117	0.106413
$425$	0	0
$426$	0	0
$427$	−13.4145	−0.649173
$428$	−23.1295	−1.11801
$429$	0	0
$430$	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	$0.523484\pi$
$434$	1.81721	0.0872287
$435$	0	0
$436$	29.0641	1.39192
$437$	−15.3369	−0.733664
$438$	0	0
$439$	14.7475	0.703861	0.351931	−	0.936026i	$-0.385525\pi$
$439$	14.7475	0.703861	0.351931	+	0.936026i	$0.385525\pi$
$440$	0	0
$441$	0	0
$442$	−2.94292	−0.139981
$443$	−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
$446$	−6.39624	−0.302871
$447$	0	0
$448$	19.0050	0.897904
$449$	−18.9484	−0.894230	−0.447115	−	0.894477i	$-0.647549\pi$
$449$	−18.9484	−0.894230	−0.447115	+	0.894477i	$0.647549\pi$
$450$	0	0
$451$	−51.6881	−2.43390
$452$	−35.1279	−1.65228
$453$	0	0
$454$	−0.438522	−0.0205808
$455$	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	$0.247815\pi$
$458$	7.30949	0.341550
$459$	0	0
$460$	0	0
$461$	−2.36972	−0.110369	−0.0551844	−	0.998476i	$-0.517575\pi$
$461$	−2.36972	−0.110369	−0.0551844	+	0.998476i	$0.517575\pi$
$462$	0	0
$463$	−0.320982	−0.0149173	−0.00745865	−	0.999972i	$-0.502374\pi$
$463$	−0.320982	−0.0149173	−0.00745865	+	0.999972i	$0.502374\pi$
$464$	−15.9867	−0.742164
$465$	0	0
$466$	−6.00081	−0.277982
$467$	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
$471$	0	0
$472$	0.0282194	0.00129890
$473$	−12.4258	−0.571341
$474$	0	0
$475$	0	0
$476$	16.5817	0.760021
$477$	0	0
$478$	−3.92224	−0.179399
$479$	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
$482$	−3.30573	−0.150572
$483$	0	0
$484$	−33.1837	−1.50835
$485$	0	0
$486$	0	0
$487$	−15.4919	−0.702003	−0.351002	−	0.936375i	$-0.614159\pi$
$487$	−15.4919	−0.702003	−0.351002	+	0.936375i	$0.614159\pi$
$488$	4.98375	0.225604
$489$	0	0
$490$	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
$494$	2.33074	0.104865
$495$	0	0
$496$	5.47488	0.245829
$497$	8.00875	0.359241
$498$	0	0
$499$	4.91044	0.219821	0.109911	−	0.993941i	$-0.464943\pi$
$499$	4.91044	0.219821	0.109911	+	0.993941i	$0.464943\pi$
$500$	0	0
$501$	0	0
$502$	5.46518	0.243923
$503$	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
$506$	−13.2127	−0.587377
$507$	0	0
$508$	1.28343	0.0569429
$509$	−41.4404	−1.83681	−0.918407	−	0.395637i	$-0.870524\pi$
$509$	−41.4404	−1.83681	−0.918407	+	0.395637i	$0.870524\pi$
$510$	0	0
$511$	5.18315	0.229289
$512$	−20.8649	−0.922106
$513$	0	0
$514$	−4.90681	−0.216430
$515$	0	0
$516$	0	0
$517$	37.1019	1.63174
$518$	0.0150247	0.000660147 0
$519$	0	0
$520$	0	0
$521$	1.62447	0.0711691	0.0355846	−	0.999367i	$-0.488671\pi$
$521$	1.62447	0.0711691	0.0355846	+	0.999367i	$0.488671\pi$
$522$	0	0
$523$	19.0009	0.830853	0.415427	−	0.909627i	$-0.363632\pi$
$523$	19.0009	0.830853	0.415427	+	0.909627i	$0.363632\pi$
$524$	13.3484	0.583128
$525$	0	0
$526$	−2.01665	−0.0879301
$527$	4.15451	0.180973
$528$	0	0
$529$	34.3174	1.49206
$530$	0	0
$531$	0	0
$532$	−13.1324	−0.569362
$533$	−34.0755	−1.47597
$534$	0	0
$535$	0	0
$536$	−5.23899	−0.226290
$537$	0	0
$538$	3.62865	0.156442
$539$	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	$0.756114\pi$
$542$	0.379518	0.0163017
$543$	0	0
$544$	−9.32511	−0.399811
$545$	0	0
$546$	0	0
$547$	17.4258	0.745072	0.372536	−	0.928018i	$-0.378488\pi$
$547$	17.4258	0.745072	0.372536	+	0.928018i	$0.378488\pi$
$548$	20.6714	0.883039
$549$	0	0
$550$	0	0
$551$	9.60771	0.409302
$552$	0	0
$553$	−1.46151	−0.0621497
$554$	−0.710638	−0.0301921
$555$	0	0
$556$	−36.9373	−1.56649
$557$	−16.2929	−0.690351	−0.345175	−	0.938538i	$-0.612181\pi$
$557$	−16.2929	−0.690351	−0.345175	+	0.938538i	$0.612181\pi$
$558$	0	0
$559$	−8.19175	−0.346474
$560$	0	0
$561$	0	0
$562$	−7.88048	−0.332418
$563$	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
$566$	5.29807	0.222694
$567$	0	0
$568$	−2.97541	−0.124845
$569$	9.28334	0.389178	0.194589	−	0.980885i	$-0.437663\pi$
$569$	9.28334	0.389178	0.194589	+	0.980885i	$0.437663\pi$
$570$	0	0
$571$	22.2143	0.929640	0.464820	−	0.885405i	$-0.346119\pi$
$571$	22.2143	0.929640	0.464820	+	0.885405i	$0.346119\pi$
$572$	−35.6057	−1.48875
$573$	0	0
$574$	−10.8273	−0.451924
$575$	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	$0.598354\pi$
$578$	3.41693	0.142126
$579$	0	0
$580$	0	0
$581$	20.6879	0.858280
$582$	0	0
$583$	9.19986	0.381019
$584$	−1.92564	−0.0796837
$585$	0	0
$586$	−1.14013	−0.0470983
$587$	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
$591$	0	0
$592$	0.0452664	0.00186044
$593$	−41.9815	−1.72397	−0.861986	−	0.506932i	$-0.830780\pi$
$593$	−41.9815	−1.72397	−0.861986	+	0.506932i	$0.830780\pi$
$594$	0	0
$595$	0	0
$596$	−24.2231	−0.992215
$597$	0	0
$598$	−8.71050	−0.356199
$599$	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	$0.226448\pi$
$602$	−2.60289	−0.106086
$603$	0	0
$604$	−4.08213	−0.166099
$605$	0	0
$606$	0	0
$607$	18.9242	0.768109	0.384054	−	0.923310i	$-0.374527\pi$
$607$	18.9242	0.768109	0.384054	+	0.923310i	$0.374527\pi$
$608$	7.38532	0.299514
$609$	0	0
$610$	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	$0.559506\pi$
$614$	2.91687	0.117715
$615$	0	0
$616$	−23.2651	−0.937378
$617$	−32.7375	−1.31796	−0.658981	−	0.752159i	$-0.729012\pi$
$617$	−32.7375	−1.31796	−0.658981	+	0.752159i	$0.729012\pi$
$618$	0	0
$619$	36.3952	1.46285	0.731424	−	0.681923i	$-0.238856\pi$
$619$	36.3952	1.46285	0.731424	+	0.681923i	$0.238856\pi$
$620$	0	0
$621$	0	0
$622$	−8.85824	−0.355183
$623$	−20.8651	−0.835943
$624$	0	0
$625$	0	0
$626$	−6.57010	−0.262594
$627$	0	0
$628$	−29.8135	−1.18969
$629$	0.0343496	0.00136961
$630$	0	0
$631$	12.1083	0.482024	0.241012	−	0.970522i	$-0.422521\pi$
$631$	12.1083	0.482024	0.241012	+	0.970522i	$0.422521\pi$
$632$	0.542980	0.0215986
$633$	0	0
$634$	−3.16293	−0.125616
$635$	0	0
$636$	0	0
$637$	16.6354	0.659117
$638$	8.27703	0.327691
$639$	0	0
$640$	0	0
$641$	−9.75177	−0.385172	−0.192586	−	0.981280i	$-0.561687\pi$
$641$	−9.75177	−0.385172	−0.192586	+	0.981280i	$0.561687\pi$
$642$	0	0
$643$	−6.77862	−0.267323	−0.133661	−	0.991027i	$-0.542673\pi$
$643$	−6.77862	−0.267323	−0.133661	+	0.991027i	$0.542673\pi$
$644$	49.0787	1.93397
$645$	0	0
$646$	1.69313	0.0666153
$647$	−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
$651$	0	0
$652$	13.9951	0.548092
$653$	6.14210	0.240359	0.120179	−	0.992752i	$-0.461653\pi$
$653$	6.14210	0.240359	0.120179	+	0.992752i	$0.461653\pi$
$654$	0	0
$655$	0	0
$656$	−32.6206	−1.27362
$657$	0	0
$658$	7.77190	0.302980
$659$	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	$0.318586\pi$
$662$	−6.42911	−0.249875
$663$	0	0
$664$	−7.68597	−0.298274
$665$	0	0
$666$	0	0
$667$	−35.9061	−1.39029
$668$	20.4782	0.792325
$669$	0	0
$670$	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	$0.201489\pi$
$674$	6.16096	0.237311
$675$	0	0
$676$	1.13895	0.0438059
$677$	22.6274	0.869643	0.434821	−	0.900517i	$-0.356812\pi$
$677$	22.6274	0.869643	0.434821	+	0.900517i	$0.356812\pi$
$678$	0	0
$679$	−54.7918	−2.10272
$680$	0	0
$681$	0	0
$682$	−2.83459	−0.108542
$683$	−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
$686$	−2.54597	−0.0972057
$687$	0	0
$688$	−7.84200	−0.298973
$689$	6.06502	0.231059
$690$	0	0
$691$	36.2583	1.37933	0.689665	−	0.724128i	$-0.257758\pi$
$691$	36.2583	1.37933	0.689665	+	0.724128i	$0.257758\pi$
$692$	−28.3953	−1.07943
$693$	0	0
$694$	−8.48744	−0.322179
$695$	0	0
$696$	0	0
$697$	−24.7536	−0.937608
$698$	8.54421	0.323403
$699$	0	0
$700$	0	0
$701$	8.32362	0.314379	0.157189	−	0.987568i	$-0.449757\pi$
$701$	8.32362	0.314379	0.157189	+	0.987568i	$0.449757\pi$
$702$	0	0
$703$	−0.0272043	−0.00102603
$704$	−29.6452	−1.11730
$705$	0	0
$706$	1.74123	0.0655321
$707$	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	0
$712$	7.75180	0.290511
$713$	12.2966	0.460511
$714$	0	0
$715$	0	0
$716$	14.0069	0.523463
$717$	0	0
$718$	3.23460	0.120714
$719$	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
$722$	4.86734	0.181144
$723$	0	0
$724$	20.6697	0.768183
$725$	0	0
$726$	0	0
$727$	45.9089	1.70267	0.851334	−	0.524624i	$-0.175794\pi$
$727$	45.9089	1.70267	0.851334	+	0.524624i	$0.175794\pi$
$728$	−15.3376	−0.568448
$729$	0	0
$730$	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	$0.644289\pi$
$734$	5.37152	0.198266
$735$	0	0
$736$	−27.6006	−1.01737
$737$	−21.9964	−0.810249
$738$	0	0
$739$	−3.53683	−0.130104	−0.0650522	−	0.997882i	$-0.520721\pi$
$739$	−3.53683	−0.130104	−0.0650522	+	0.997882i	$0.520721\pi$
$740$	0	0
$741$	0	0
$742$	1.92714	0.0707474
$743$	−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
$746$	7.51307	0.275073
$747$	0	0
$748$	−25.8651	−0.945724
$749$	−41.8320	−1.52851
$750$	0	0
$751$	−14.1856	−0.517642	−0.258821	−	0.965925i	$-0.583334\pi$
$751$	−14.1856	−0.517642	−0.258821	+	0.965925i	$0.583334\pi$
$752$	23.4152	0.853864
$753$	0	0
$754$	5.45664	0.198719
$755$	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	$0.574363\pi$
$758$	−5.36675	−0.194929
$759$	0	0
$760$	0	0
$761$	−32.2299	−1.16833	−0.584167	−	0.811633i	$-0.698579\pi$
$761$	−32.2299	−1.16833	−0.584167	+	0.811633i	$0.698579\pi$
$762$	0	0
$763$	52.5652	1.90299
$764$	−3.31585	−0.119963
$765$	0	0
$766$	−1.53737	−0.0555476
$767$	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
$771$	0	0
$772$	−18.0453	−0.649464
$773$	27.1994	0.978295	0.489147	−	0.872201i	$-0.337308\pi$
$773$	27.1994	0.978295	0.489147	+	0.872201i	$0.337308\pi$
$774$	0	0
$775$	0	0
$776$	20.3562	0.730746
$777$	0	0
$778$	1.15626	0.0414538
$779$	19.6044	0.702400
$780$	0	0
$781$	−12.4925	−0.447018
$782$	−6.32760	−0.226275
$783$	0	0
$784$	15.9251	0.568754
$785$	0	0
$786$	0	0
$787$	25.7266	0.917054	0.458527	−	0.888680i	$-0.348377\pi$
$787$	25.7266	0.917054	0.458527	+	0.888680i	$0.348377\pi$
$788$	−45.2803	−1.61304
$789$	0	0
$790$	0	0
$791$	−63.5321	−2.25894
$792$	0	0
$793$	13.7947	0.489864
$794$	−2.86356	−0.101624
$795$	0	0
$796$	−45.1122	−1.59896
$797$	17.6713	0.625951	0.312975	−	0.949761i	$-0.398674\pi$
$797$	17.6713	0.625951	0.312975	+	0.949761i	$0.398674\pi$
$798$	0	0
$799$	17.7682	0.628593
$800$	0	0
$801$	0	0
$802$	7.43938	0.262694
$803$	−8.08500	−0.285314
$804$	0	0
$805$	0	0
$806$	−1.86871	−0.0658224
$807$	0	0
$808$	−1.83716	−0.0646312
$809$	40.8576	1.43648	0.718238	−	0.695798i	$-0.244949\pi$
$809$	40.8576	1.43648	0.718238	+	0.695798i	$0.244949\pi$
$810$	0	0
$811$	16.6214	0.583656	0.291828	−	0.956471i	$-0.405737\pi$
$811$	16.6214	0.583656	0.291828	+	0.956471i	$0.405737\pi$
$812$	−30.7451	−1.07894
$813$	0	0
$814$	−0.0234364	−0.000821447 0
$815$	0	0
$816$	0	0
$817$	4.71290	0.164884
$818$	−9.16296	−0.320375
$819$	0	0
$820$	0	0
$821$	43.8983	1.53206	0.766031	−	0.642803i	$-0.222229\pi$
$821$	43.8983	1.53206	0.766031	+	0.642803i	$0.222229\pi$
$822$	0	0
$823$	−26.1962	−0.913143	−0.456571	−	0.889687i	$-0.650923\pi$
$823$	−26.1962	−0.913143	−0.456571	+	0.889687i	$0.650923\pi$
$824$	−18.6565	−0.649932
$825$	0	0
$826$	0.0248189	0.000863561 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
$831$	0	0
$832$	−19.5437	−0.677554
$833$	12.0845	0.418702
$834$	0	0
$835$	0	0
$836$	20.4848	0.708480
$837$	0	0
$838$	11.1390	0.384790
$839$	−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
$842$	10.5015	0.361904
$843$	0	0
$844$	29.1413	1.00308
$845$	0	0
$846$	0	0
$847$	−60.0158	−2.06217
$848$	5.80608	0.199381
$849$	0	0
$850$	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	$0.391423\pi$
$854$	4.38320	0.149990
$855$	0	0
$856$	15.5414	0.531194
$857$	−0.570622	−0.0194921	−0.00974604	−	0.999953i	$-0.503102\pi$
$857$	−0.570622	−0.0194921	−0.00974604	+	0.999953i	$0.503102\pi$
$858$	0	0
$859$	−20.0983	−0.685744	−0.342872	−	0.939382i	$-0.611400\pi$
$859$	−20.0983	−0.685744	−0.342872	+	0.939382i	$0.611400\pi$
$860$	0	0
$861$	0	0
$862$	−5.80142	−0.197597
$863$	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
$866$	1.00236	0.0340614
$867$	0	0
$868$	10.5291	0.357381
$869$	2.27975	0.0773353
$870$	0	0
$871$	−14.5012	−0.491353
$872$	−19.5290	−0.661336
$873$	0	0
$874$	5.01135	0.169511
$875$	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	$0.629583\pi$
$878$	−4.81877	−0.162626
$879$	0	0
$880$	0	0
$881$	−44.4714	−1.49828	−0.749139	−	0.662412i	$-0.769533\pi$
$881$	−44.4714	−1.49828	−0.749139	+	0.662412i	$0.769533\pi$
$882$	0	0
$883$	0.758169	0.0255144	0.0127572	−	0.999919i	$-0.495939\pi$
$883$	0.758169	0.0255144	0.0127572	+	0.999919i	$0.495939\pi$
$884$	−17.0516	−0.573509
$885$	0	0
$886$	3.32818	0.111812
$887$	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
$891$	0	0
$892$	−37.0606	−1.24088
$893$	−14.0721	−0.470905
$894$	0	0
$895$	0	0
$896$	−31.1760	−1.04152
$897$	0	0
$898$	6.19141	0.206610
$899$	−7.70312	−0.256914
$900$	0	0
$901$	4.40584	0.146780
$902$	16.8891	0.562347
$903$	0	0
$904$	23.6034	0.785038
$905$	0	0
$906$	0	0
$907$	−15.8193	−0.525271	−0.262636	−	0.964895i	$-0.584592\pi$
$907$	−15.8193	−0.525271	−0.262636	+	0.964895i	$0.584592\pi$
$908$	−2.54084	−0.0843209
$909$	0	0
$910$	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
$914$	−9.94605	−0.328986
$915$	0	0
$916$	42.3520	1.39935
$917$	24.1419	0.797235
$918$	0	0
$919$	22.8402	0.753428	0.376714	−	0.926330i	$-0.377054\pi$
$919$	22.8402	0.753428	0.376714	+	0.926330i	$0.377054\pi$
$920$	0	0
$921$	0	0
$922$	0.774308	0.0255005
$923$	−8.23572	−0.271082
$924$	0	0
$925$	0	0
$926$	0.104881	0.00344661
$927$	0	0
$928$	17.2902	0.567579
$929$	39.7278	1.30343	0.651713	−	0.758465i	$-0.274050\pi$
$929$	39.7278	1.30343	0.651713	+	0.758465i	$0.274050\pi$
$930$	0	0
$931$	−9.57070	−0.313667
$932$	−34.7694	−1.13891
$933$	0	0
$934$	−4.18475	−0.136929
$935$	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	$0.665625\pi$
$938$	−4.60769	−0.150446
$939$	0	0
$940$	0	0
$941$	−25.6117	−0.834916	−0.417458	−	0.908696i	$-0.637079\pi$
$941$	−25.6117	−0.834916	−0.417458	+	0.908696i	$0.637079\pi$
$942$	0	0
$943$	−73.2659	−2.38587
$944$	0.0747744	0.00243370
$945$	0	0
$946$	4.06016	0.132007
$947$	−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
$951$	0	0
$952$	−11.1417	−0.361105
$953$	−45.6216	−1.47783	−0.738915	−	0.673799i	$-0.764661\pi$
$953$	−45.6216	−1.47783	−0.738915	+	0.673799i	$0.764661\pi$
$954$	0	0
$955$	0	0
$956$	−22.7259	−0.735008
$957$	0	0
$958$	−6.64762	−0.214775
$959$	37.3862	1.20726
$960$	0	0
$961$	−28.3620	−0.914902
$962$	−0.0154505	−0.000498144 0
$963$	0	0
$964$	−19.1538	−0.616903
$965$	0	0
$966$	0	0
$967$	42.4449	1.36494	0.682468	−	0.730916i	$-0.260907\pi$
$967$	42.4449	1.36494	0.682468	+	0.730916i	$0.260907\pi$
$968$	22.2971	0.716655
$969$	0	0
$970$	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
$974$	5.06198	0.162196
$975$	0	0
$976$	13.2057	0.422705
$977$	16.5748	0.530276	0.265138	−	0.964211i	$-0.414583\pi$
$977$	16.5748	0.530276	0.265138	+	0.964211i	$0.414583\pi$
$978$	0	0
$979$	32.5467	1.04020
$980$	0	0
$981$	0	0
$982$	−8.98845	−0.286833
$983$	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
$986$	3.96389	0.126236
$987$	0	0
$988$	13.5046	0.429639
$989$	−17.6131	−0.560065
$990$	0	0
$991$	−24.4962	−0.778146	−0.389073	−	0.921207i	$-0.627205\pi$
$991$	−24.4962	−0.778146	−0.389073	+	0.921207i	$0.627205\pi$
$992$	−5.92129	−0.188001
$993$	0	0
$994$	−2.61687	−0.0830020
$995$	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	$0.103248\pi$
$998$	−1.60449	−0.0507893
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5625.2.a.be.1.3		8
3.2	odd	2		625.2.a.e.1.6	✓	8
5.4	even	2		5625.2.a.s.1.6		8
12.11	even	2		10000.2.a.bn.1.2		8
15.2	even	4		625.2.b.d.624.9		16
15.8	even	4		625.2.b.d.624.8		16
15.14	odd	2		625.2.a.g.1.3	yes	8
60.59	even	2		10000.2.a.be.1.7		8
75.2	even	20		625.2.e.j.124.5		32
75.8	even	20		625.2.e.k.374.5		32
75.11	odd	10		625.2.d.q.501.3		16
75.14	odd	10		625.2.d.m.501.2		16
75.17	even	20		625.2.e.k.374.4		32
75.23	even	20		625.2.e.j.124.4		32
75.29	odd	10		625.2.d.n.376.3		16
75.38	even	20		625.2.e.j.499.5		32
75.41	odd	10		625.2.d.q.126.3		16
75.44	odd	10		625.2.d.n.251.3		16
75.47	even	20		625.2.e.k.249.5		32
75.53	even	20		625.2.e.k.249.4		32
75.56	odd	10		625.2.d.p.251.2		16
75.59	odd	10		625.2.d.m.126.2		16
75.62	even	20		625.2.e.j.499.4		32
75.71	odd	10		625.2.d.p.376.2		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
625.2.a.e.1.6	✓	8	3.2	odd	2
625.2.a.g.1.3	yes	8	15.14	odd	2
625.2.b.d.624.8		16	15.8	even	4
625.2.b.d.624.9		16	15.2	even	4
625.2.d.m.126.2		16	75.59	odd	10
625.2.d.m.501.2		16	75.14	odd	10
625.2.d.n.251.3		16	75.44	odd	10
625.2.d.n.376.3		16	75.29	odd	10
625.2.d.p.251.2		16	75.56	odd	10
625.2.d.p.376.2		16	75.71	odd	10
625.2.d.q.126.3		16	75.41	odd	10
625.2.d.q.501.3		16	75.11	odd	10
625.2.e.j.124.4		32	75.23	even	20
625.2.e.j.124.5		32	75.2	even	20
625.2.e.j.499.4		32	75.62	even	20
625.2.e.j.499.5		32	75.38	even	20
625.2.e.k.249.4		32	75.53	even	20
625.2.e.k.249.5		32	75.47	even	20
625.2.e.k.374.4		32	75.17	even	20
625.2.e.k.374.5		32	75.8	even	20
5625.2.a.s.1.6		8	5.4	even	2
5625.2.a.be.1.3		8	1.1	even	1	trivial
10000.2.a.be.1.7		8	60.59	even	2
10000.2.a.bn.1.2		8	12.11	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:	\( N \)	\(=\)	\( 5625 = 3^{2} \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5625.a (trivial)

\(f(q)\)	\(=\)	\(q-0.326751 q^{2} -1.89323 q^{4} -3.42409 q^{7} +1.27212 q^{8} +O(q^{10})\)	\(q-0.326751 q^{2} -1.89323 q^{4} -3.42409 q^{7} +1.27212 q^{8} +5.34111 q^{11} +3.52114 q^{13} +1.11883 q^{14} +3.37080 q^{16} +2.55787 q^{17} -2.02579 q^{19} -1.74522 q^{22} +7.57082 q^{23} -1.15054 q^{26} +6.48261 q^{28} -4.74270 q^{29} +1.62421 q^{31} -3.64565 q^{32} -0.835788 q^{34} +0.0134290 q^{37} +0.661929 q^{38} -9.67740 q^{41} -2.32645 q^{43} -10.1120 q^{44} -2.47377 q^{46} +6.94647 q^{47} +4.72443 q^{49} -6.66634 q^{52} +1.72246 q^{53} -4.35585 q^{56} +1.54968 q^{58} +0.0221830 q^{59} +3.91768 q^{61} -0.530712 q^{62} -5.55038 q^{64} -4.11832 q^{67} -4.84265 q^{68} -2.33894 q^{71} -1.51373 q^{73} -0.00438793 q^{74} +3.83530 q^{76} -18.2885 q^{77} +0.426831 q^{79} +3.16210 q^{82} -6.04187 q^{83} +0.760170 q^{86} +6.79453 q^{88} +6.09362 q^{89} -12.0567 q^{91} -14.3333 q^{92} -2.26977 q^{94} +16.0018 q^{97} -1.54371 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 5 q^{2} + 11 q^{4} - 10 q^{7} + 15 q^{8}+O(q^{10}) \) 8 * q + 5 * q^2 + 11 * q^4 - 10 * q^7 + 15 * q^8	\( 8 q + 5 q^{2} + 11 q^{4} - 10 q^{7} + 15 q^{8} - q^{11} - 10 q^{13} + 8 q^{14} + 13 q^{16} + 15 q^{17} - 10 q^{19} + 5 q^{22} + 30 q^{23} - 11 q^{26} + 5 q^{28} - 10 q^{29} - 9 q^{31} + 30 q^{32} + 7 q^{34} + 10 q^{37} + 20 q^{38} + 4 q^{41} + 18 q^{44} - 9 q^{46} + 30 q^{47} - 4 q^{49} - 5 q^{52} + 10 q^{53} + 30 q^{58} + 5 q^{59} + 6 q^{61} + 10 q^{62} - 9 q^{64} - 10 q^{67} + 40 q^{68} + 9 q^{71} + 18 q^{74} - 10 q^{76} + 5 q^{77} - 20 q^{79} + 45 q^{82} + 40 q^{83} + 24 q^{86} + 40 q^{88} + 5 q^{89} + 6 q^{91} + 15 q^{92} + 47 q^{94} - 30 q^{98}+O(q^{100}) \) 8 * q + 5 * q^2 + 11 * q^4 - 10 * q^7 + 15 * q^8 - q^11 - 10 * q^13 + 8 * q^14 + 13 * q^16 + 15 * q^17 - 10 * q^19 + 5 * q^22 + 30 * q^23 - 11 * q^26 + 5 * q^28 - 10 * q^29 - 9 * q^31 + 30 * q^32 + 7 * q^34 + 10 * q^37 + 20 * q^38 + 4 * q^41 + 18 * q^44 - 9 * q^46 + 30 * q^47 - 4 * q^49 - 5 * q^52 + 10 * q^53 + 30 * q^58 + 5 * q^59 + 6 * q^61 + 10 * q^62 - 9 * q^64 - 10 * q^67 + 40 * q^68 + 9 * q^71 + 18 * q^74 - 10 * q^76 + 5 * q^77 - 20 * q^79 + 45 * q^82 + 40 * q^83 + 24 * q^86 + 40 * q^88 + 5 * q^89 + 6 * q^91 + 15 * q^92 + 47 * q^94 - 30 * q^98

Introduction

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