LMFDB - Embedded newform 5225.2.a.bb.1.11

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5225,2,Mod(1,5225)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("5225.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5225, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 5225 = 5^{2} \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5225.a (trivial)

Newform invariants

comment:select newform

sage:traces = [22,0,0,32,0,-12,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$41.7218350561$
Analytic rank:	$0$
Dimension:	$22$
Twist minimal:	no (minimal twist has level 1045)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Character	$\chi$	$=$	5225.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.104144 q^{2} -0.696995 q^{3} -1.98915 q^{4} +0.0725876 q^{6} +3.21657 q^{7} +0.415445 q^{8} -2.51420 q^{9} -1.00000 q^{11} +1.38643 q^{12} +4.65405 q^{13} -0.334986 q^{14} +3.93504 q^{16} -0.00602394 q^{17} +0.261838 q^{18} -1.00000 q^{19} -2.24194 q^{21} +0.104144 q^{22} -1.65858 q^{23} -0.289563 q^{24} -0.484690 q^{26} +3.84337 q^{27} -6.39826 q^{28} +3.49988 q^{29} -3.34225 q^{31} -1.24070 q^{32} +0.696995 q^{33} +0.000627356 q^{34} +5.00113 q^{36} +6.13742 q^{37} +0.104144 q^{38} -3.24385 q^{39} -9.38032 q^{41} +0.233483 q^{42} +8.61740 q^{43} +1.98915 q^{44} +0.172731 q^{46} +0.893231 q^{47} -2.74270 q^{48} +3.34635 q^{49} +0.00419866 q^{51} -9.25763 q^{52} +1.78431 q^{53} -0.400263 q^{54} +1.33631 q^{56} +0.696995 q^{57} -0.364491 q^{58} +2.57022 q^{59} -3.68861 q^{61} +0.348074 q^{62} -8.08710 q^{63} -7.74087 q^{64} -0.0725876 q^{66} +5.98644 q^{67} +0.0119826 q^{68} +1.15602 q^{69} +0.852858 q^{71} -1.04451 q^{72} +4.72786 q^{73} -0.639174 q^{74} +1.98915 q^{76} -3.21657 q^{77} +0.337827 q^{78} -4.94098 q^{79} +4.86379 q^{81} +0.976902 q^{82} -15.0339 q^{83} +4.45956 q^{84} -0.897448 q^{86} -2.43940 q^{87} -0.415445 q^{88} +5.11449 q^{89} +14.9701 q^{91} +3.29917 q^{92} +2.32953 q^{93} -0.0930244 q^{94} +0.864762 q^{96} -16.5462 q^{97} -0.348501 q^{98} +2.51420 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 22 q + 32 q^{4} - 12 q^{6} + 34 q^{9} - 22 q^{11} - 8 q^{14} + 40 q^{16} - 22 q^{19} - 22 q^{21} - 22 q^{24} + 16 q^{26} - 10 q^{29} + 76 q^{31} + 56 q^{34} + 104 q^{36} - 8 q^{39} + 6 q^{41} - 32 q^{44}+ \cdots - 34 q^{99}+O(q^{100}) $ 22 * q + 32 * q^4 - 12 * q^6 + 34 * q^9 - 22 * q^11 - 8 * q^14 + 40 * q^16 - 22 * q^19 - 22 * q^21 - 22 * q^24 + 16 * q^26 - 10 * q^29 + 76 * q^31 + 56 * q^34 + 104 * q^36 - 8 * q^39 + 6 * q^41 - 32 * q^44 + 88 * q^46 + 28 * q^49 + 8 * q^51 + 38 * q^54 + 44 * q^56 + 40 * q^59 - 6 * q^61 + 140 * q^64 + 12 * q^66 + 74 * q^69 + 62 * q^71 - 26 * q^74 - 32 * q^76 + 102 * q^79 + 94 * q^81 - 38 * q^84 + 28 * q^86 + 54 * q^89 + 88 * q^91 + 36 * q^94 + 2 * q^96 - 34 * q^99

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 $

$2$	−0.104144	−0.0736407	−0.0368204	−	0.999322i	$-0.511723\pi$
$2$	−0.104144	−0.0736407	−0.0368204	+	0.999322i	$0.511723\pi$
$3$	−0.696995	−0.402410	−0.201205	−	0.979549i	$-0.564486\pi$
$3$	−0.696995	−0.402410	−0.201205	+	0.979549i	$0.564486\pi$
$4$	−1.98915	−0.994577
$5$	0	0
$5$	0	0
$6$	0.0725876	0.0296338
$7$	3.21657	1.21575	0.607875	−	0.794033i	$-0.292022\pi$
$7$	3.21657	1.21575	0.607875	+	0.794033i	$0.292022\pi$
$8$	0.415445	0.146882
$9$	−2.51420	−0.838066
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	1.38643	0.400228
$13$	4.65405	1.29080	0.645401	−	0.763844i	$-0.276690\pi$
$13$	4.65405	1.29080	0.645401	+	0.763844i	$0.276690\pi$
$14$	−0.334986	−0.0895287
$15$	0	0
$16$	3.93504	0.983761
$17$	−0.00602394	−0.00146102	−0.000730510	−	1.00000i	$-0.500233\pi$
$17$	−0.00602394	−0.00146102	−0.000730510		1.00000i	$0.500233\pi$
$18$	0.261838	0.0617158
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−2.24194	−0.489230
$22$	0.104144	0.0222035
$23$	−1.65858	−0.345838	−0.172919	−	0.984936i	$-0.555320\pi$
$23$	−1.65858	−0.345838	−0.172919	+	0.984936i	$0.555320\pi$
$24$	−0.289563	−0.0591069
$25$	0	0
$26$	−0.484690	−0.0950556
$27$	3.84337	0.739657
$28$	−6.39826	−1.20916
$29$	3.49988	0.649912	0.324956	−	0.945729i	$-0.394650\pi$
$29$	3.49988	0.649912	0.324956	+	0.945729i	$0.394650\pi$
$30$	0	0
$31$	−3.34225	−0.600285	−0.300143	−	0.953894i	$-0.597034\pi$
$31$	−3.34225	−0.600285	−0.300143	+	0.953894i	$0.597034\pi$
$32$	−1.24070	−0.219327
$33$	0.696995	0.121331
$34$	0.000627356 0	0.000107591 0
$35$	0	0
$36$	5.00113	0.833521
$37$	6.13742	1.00899	0.504493	−	0.863416i	$-0.331680\pi$
$37$	6.13742	1.00899	0.504493	+	0.863416i	$0.331680\pi$
$38$	0.104144	0.0168943
$39$	−3.24385	−0.519432
$40$	0	0
$41$	−9.38032	−1.46496	−0.732480	−	0.680788i	$-0.761637\pi$
$41$	−9.38032	−1.46496	−0.732480	+	0.680788i	$0.761637\pi$
$42$	0.233483	0.0360273
$43$	8.61740	1.31414	0.657071	−	0.753829i	$-0.271795\pi$
$43$	8.61740	1.31414	0.657071	+	0.753829i	$0.271795\pi$
$44$	1.98915	0.299876
$45$	0	0
$46$	0.172731	0.0254677
$47$	0.893231	0.130291	0.0651456	−	0.997876i	$-0.479249\pi$
$47$	0.893231	0.130291	0.0651456	+	0.997876i	$0.479249\pi$
$48$	−2.74270	−0.395875
$49$	3.34635	0.478049
$50$	0	0
$51$	0.00419866	0.000587930 0
$52$	−9.25763	−1.28380
$53$	1.78431	0.245093	0.122547	−	0.992463i	$-0.460894\pi$
$53$	1.78431	0.245093	0.122547	+	0.992463i	$0.460894\pi$
$54$	−0.400263	−0.0544688
$55$	0	0
$56$	1.33631	0.178572
$57$	0.696995	0.0923192
$58$	−0.364491	−0.0478600
$59$	2.57022	0.334614	0.167307	−	0.985905i	$-0.446493\pi$
$59$	2.57022	0.334614	0.167307	+	0.985905i	$0.446493\pi$
$60$	0	0
$61$	−3.68861	−0.472278	−0.236139	−	0.971719i	$-0.575882\pi$
$61$	−3.68861	−0.472278	−0.236139	+	0.971719i	$0.575882\pi$
$62$	0.348074	0.0442054
$63$	−8.08710	−1.01888
$64$	−7.74087	−0.967609
$65$	0	0
$66$	−0.0725876	−0.00893492
$67$	5.98644	0.731361	0.365680	−	0.930741i	$-0.380836\pi$
$67$	5.98644	0.731361	0.365680	+	0.930741i	$0.380836\pi$
$68$	0.0119826	0.00145310
$69$	1.15602	0.139169
$70$	0	0
$71$	0.852858	0.101216	0.0506078	−	0.998719i	$-0.483884\pi$
$71$	0.852858	0.101216	0.0506078	+	0.998719i	$0.483884\pi$
$72$	−1.04451	−0.123097
$73$	4.72786	0.553354	0.276677	−	0.960963i	$-0.410767\pi$
$73$	4.72786	0.553354	0.276677	+	0.960963i	$0.410767\pi$
$74$	−0.639174	−0.0743024
$75$	0	0
$76$	1.98915	0.228172
$77$	−3.21657	−0.366563
$78$	0.337827	0.0382513
$79$	−4.94098	−0.555903	−0.277952	−	0.960595i	$-0.589656\pi$
$79$	−4.94098	−0.555903	−0.277952	+	0.960595i	$0.589656\pi$
$80$	0	0
$81$	4.86379	0.540421
$82$	0.976902	0.107881
$83$	−15.0339	−1.65019	−0.825093	−	0.564997i	$-0.808877\pi$
$83$	−15.0339	−1.65019	−0.825093	+	0.564997i	$0.808877\pi$
$84$	4.45956	0.486577
$85$	0	0
$86$	−0.897448	−0.0967743
$87$	−2.43940	−0.261531
$88$	−0.415445	−0.0442866
$89$	5.11449	0.542135	0.271067	−	0.962560i	$-0.412623\pi$
$89$	5.11449	0.542135	0.271067	+	0.962560i	$0.412623\pi$
$90$	0	0
$91$	14.9701	1.56929
$92$	3.29917	0.343962
$93$	2.32953	0.241561
$94$	−0.0930244	−0.00959474
$95$	0	0
$96$	0.864762	0.0882594
$97$	−16.5462	−1.68001	−0.840005	−	0.542578i	$-0.817448\pi$
$97$	−16.5462	−1.68001	−0.840005	+	0.542578i	$0.817448\pi$
$98$	−0.348501	−0.0352039
$99$	2.51420	0.252686
$100$	0	0
$101$	5.62840	0.560047	0.280023	−	0.959993i	$-0.409658\pi$
$101$	5.62840	0.560047	0.280023	+	0.959993i	$0.409658\pi$
$102$	−0.000437264 0	−4.32956e−5 0
$103$	−7.56366	−0.745270	−0.372635	−	0.927978i	$-0.621546\pi$
$103$	−7.56366	−0.745270	−0.372635	+	0.927978i	$0.621546\pi$
$104$	1.93350	0.189596
$105$	0	0
$106$	−0.185824	−0.0180488
$107$	5.01113	0.484444	0.242222	−	0.970221i	$-0.422124\pi$
$107$	5.01113	0.484444	0.242222	+	0.970221i	$0.422124\pi$
$108$	−7.64505	−0.735645
$109$	3.73155	0.357418	0.178709	−	0.983902i	$-0.442808\pi$
$109$	3.73155	0.357418	0.178709	+	0.983902i	$0.442808\pi$
$110$	0	0
$111$	−4.27775	−0.406026
$112$	12.6574	1.19601
$113$	−16.9702	−1.59642	−0.798209	−	0.602380i	$-0.794219\pi$
$113$	−16.9702	−1.59642	−0.798209	+	0.602380i	$0.794219\pi$
$114$	−0.0725876	−0.00679846
$115$	0	0
$116$	−6.96181	−0.646388
$117$	−11.7012	−1.08178
$118$	−0.267672	−0.0246412
$119$	−0.0193765	−0.00177624
$120$	0	0
$121$	1.00000	0.0909091
$122$	0.384146	0.0347789
$123$	6.53804	0.589515
$124$	6.64824	0.597030
$125$	0	0
$126$	0.842221	0.0750310
$127$	0.870581	0.0772516	0.0386258	−	0.999254i	$-0.487702\pi$
$127$	0.870581	0.0772516	0.0386258	+	0.999254i	$0.487702\pi$
$128$	3.28756	0.290582
$129$	−6.00628	−0.528824
$130$	0	0
$131$	−0.332706	−0.0290687	−0.0145343	−	0.999894i	$-0.504627\pi$
$131$	−0.332706	−0.0290687	−0.0145343	+	0.999894i	$0.504627\pi$
$132$	−1.38643	−0.120673
$133$	−3.21657	−0.278912
$134$	−0.623450	−0.0538579
$135$	0	0
$136$	−0.00250262	−0.000214598 0
$137$	−13.0314	−1.11335	−0.556673	−	0.830732i	$-0.687922\pi$
$137$	−13.0314	−1.11335	−0.556673	+	0.830732i	$0.687922\pi$
$138$	−0.120392	−0.0102485
$139$	12.6827	1.07573	0.537867	−	0.843030i	$-0.319230\pi$
$139$	12.6827	1.07573	0.537867	+	0.843030i	$0.319230\pi$
$140$	0	0
$141$	−0.622578	−0.0524305
$142$	−0.0888197	−0.00745359
$143$	−4.65405	−0.389192
$144$	−9.89348	−0.824456
$145$	0	0
$146$	−0.492376	−0.0407494
$147$	−2.33239	−0.192372
$148$	−12.2083	−1.00351
$149$	17.2444	1.41272	0.706358	−	0.707855i	$-0.250337\pi$
$149$	17.2444	1.41272	0.706358	+	0.707855i	$0.250337\pi$
$150$	0	0
$151$	22.5478	1.83491	0.917456	−	0.397836i	$-0.130239\pi$
$151$	22.5478	1.83491	0.917456	+	0.397836i	$0.130239\pi$
$152$	−0.415445	−0.0336971
$153$	0.0151454	0.00122443
$154$	0.334986	0.0269939
$155$	0	0
$156$	6.45252	0.516615
$157$	12.1662	0.970972	0.485486	−	0.874244i	$-0.338643\pi$
$157$	12.1662	0.970972	0.485486	+	0.874244i	$0.338643\pi$
$158$	0.514572	0.0409371
$159$	−1.24365	−0.0986280
$160$	0	0
$161$	−5.33494	−0.420452
$162$	−0.506533	−0.0397970
$163$	21.8086	1.70818	0.854090	−	0.520125i	$-0.174115\pi$
$163$	21.8086	1.70818	0.854090	+	0.520125i	$0.174115\pi$
$164$	18.6589	1.45702
$165$	0	0
$166$	1.56569	0.121521
$167$	−15.5829	−1.20584	−0.602919	−	0.797802i	$-0.705996\pi$
$167$	−15.5829	−1.20584	−0.602919	+	0.797802i	$0.705996\pi$
$168$	−0.931402	−0.0718592
$169$	8.66022	0.666170
$170$	0	0
$171$	2.51420	0.192266
$172$	−17.1413	−1.30701
$173$	10.4213	0.792315	0.396157	−	0.918183i	$-0.370344\pi$
$173$	10.4213	0.792315	0.396157	+	0.918183i	$0.370344\pi$
$174$	0.254048	0.0192594
$175$	0	0
$176$	−3.93504	−0.296615
$177$	−1.79143	−0.134652
$178$	−0.532642	−0.0399232
$179$	20.5356	1.53490	0.767452	−	0.641107i	$-0.221524\pi$
$179$	20.5356	1.53490	0.767452	+	0.641107i	$0.221524\pi$
$180$	0	0
$181$	0.0667973	0.00496500	0.00248250	−	0.999997i	$-0.499210\pi$
$181$	0.0667973	0.00496500	0.00248250	+	0.999997i	$0.499210\pi$
$182$	−1.55904	−0.115564
$183$	2.57094	0.190050
$184$	−0.689049	−0.0507974
$185$	0	0
$186$	−0.242606	−0.0177887
$187$	0.00602394	0.000440514 0
$188$	−1.77677	−0.129585
$189$	12.3625	0.899238
$190$	0	0
$191$	−2.09641	−0.151691	−0.0758456	−	0.997120i	$-0.524166\pi$
$191$	−2.09641	−0.151691	−0.0758456	+	0.997120i	$0.524166\pi$
$192$	5.39535	0.389376
$193$	25.1960	1.81365	0.906825	−	0.421507i	$-0.138499\pi$
$193$	25.1960	1.81365	0.906825	+	0.421507i	$0.138499\pi$
$194$	1.72318	0.123717
$195$	0	0
$196$	−6.65640	−0.475457
$197$	5.70885	0.406739	0.203369	−	0.979102i	$-0.434811\pi$
$197$	5.70885	0.406739	0.203369	+	0.979102i	$0.434811\pi$
$198$	−0.261838	−0.0186080
$199$	−10.2003	−0.723081	−0.361541	−	0.932356i	$-0.617749\pi$
$199$	−10.2003	−0.723081	−0.361541	+	0.932356i	$0.617749\pi$
$200$	0	0
$201$	−4.17252	−0.294307
$202$	−0.586162	−0.0412422
$203$	11.2576	0.790131
$204$	−0.00835178	−0.000584741 0
$205$	0	0
$206$	0.787708	0.0548822
$207$	4.17000	0.289835
$208$	18.3139	1.26984
$209$	1.00000	0.0691714
$210$	0	0
$211$	1.03962	0.0715706	0.0357853	−	0.999360i	$-0.488607\pi$
$211$	1.03962	0.0715706	0.0357853	+	0.999360i	$0.488607\pi$
$212$	−3.54926	−0.243764
$213$	−0.594437	−0.0407302
$214$	−0.521877	−0.0356748
$215$	0	0
$216$	1.59671	0.108642
$217$	−10.7506	−0.729797
$218$	−0.388618	−0.0263205
$219$	−3.29529	−0.222675
$220$	0	0
$221$	−0.0280358	−0.00188589
$222$	0.445501	0.0299001
$223$	6.42634	0.430339	0.215170	−	0.976577i	$-0.430970\pi$
$223$	6.42634	0.430339	0.215170	+	0.976577i	$0.430970\pi$
$224$	−3.99080	−0.266647
$225$	0	0
$226$	1.76734	0.117561
$227$	6.25815	0.415368	0.207684	−	0.978196i	$-0.433407\pi$
$227$	6.25815	0.415368	0.207684	+	0.978196i	$0.433407\pi$
$228$	−1.38643	−0.0918186
$229$	26.2671	1.73578	0.867890	−	0.496757i	$-0.165476\pi$
$229$	26.2671	1.73578	0.867890	+	0.496757i	$0.165476\pi$
$230$	0	0
$231$	2.24194	0.147509
$232$	1.45401	0.0954605
$233$	2.94990	0.193255	0.0966273	−	0.995321i	$-0.469195\pi$
$233$	2.94990	0.193255	0.0966273	+	0.995321i	$0.469195\pi$
$234$	1.21861	0.0796629
$235$	0	0
$236$	−5.11257	−0.332800
$237$	3.44384	0.223701
$238$	0.00201794	0.000130803 0
$239$	18.5942	1.20276	0.601378	−	0.798965i	$-0.294619\pi$
$239$	18.5942	1.20276	0.601378	+	0.798965i	$0.294619\pi$
$240$	0	0
$241$	−8.68183	−0.559246	−0.279623	−	0.960110i	$-0.590209\pi$
$241$	−8.68183	−0.559246	−0.279623	+	0.960110i	$0.590209\pi$
$242$	−0.104144	−0.00669461
$243$	−14.9201	−0.957127
$244$	7.33722	0.469717
$245$	0	0
$246$	−0.680896	−0.0434123
$247$	−4.65405	−0.296130
$248$	−1.38852	−0.0881711
$249$	10.4786	0.664052
$250$	0	0
$251$	−27.7431	−1.75113	−0.875565	−	0.483100i	$-0.839511\pi$
$251$	−27.7431	−1.75113	−0.875565	+	0.483100i	$0.839511\pi$
$252$	16.0865	1.01335
$253$	1.65858	0.104274
$254$	−0.0906655	−0.00568886
$255$	0	0
$256$	15.1394	0.946210
$257$	−17.7003	−1.10412	−0.552059	−	0.833805i	$-0.686158\pi$
$257$	−17.7003	−1.10412	−0.552059	+	0.833805i	$0.686158\pi$
$258$	0.625517	0.0389430
$259$	19.7415	1.22667
$260$	0	0
$261$	−8.79940	−0.544669
$262$	0.0346492	0.00214064
$263$	22.4026	1.38140	0.690702	−	0.723139i	$-0.257302\pi$
$263$	22.4026	1.38140	0.690702	+	0.723139i	$0.257302\pi$
$264$	0.289563	0.0178214
$265$	0	0
$266$	0.334986	0.0205393
$267$	−3.56477	−0.218161
$268$	−11.9080	−0.727394
$269$	16.1724	0.986046	0.493023	−	0.870016i	$-0.335892\pi$
$269$	16.1724	0.986046	0.493023	+	0.870016i	$0.335892\pi$
$270$	0	0
$271$	2.09973	0.127550	0.0637748	−	0.997964i	$-0.479686\pi$
$271$	2.09973	0.127550	0.0637748	+	0.997964i	$0.479686\pi$
$272$	−0.0237045	−0.00143729
$273$	−10.4341	−0.631500
$274$	1.35714	0.0819876
$275$	0	0
$276$	−2.29950	−0.138414
$277$	29.9790	1.80127	0.900633	−	0.434580i	$-0.143103\pi$
$277$	29.9790	1.80127	0.900633	+	0.434580i	$0.143103\pi$
$278$	−1.32082	−0.0792178
$279$	8.40307	0.503078
$280$	0	0
$281$	−0.469805	−0.0280262	−0.0140131	−	0.999902i	$-0.504461\pi$
$281$	−0.469805	−0.0280262	−0.0140131	+	0.999902i	$0.504461\pi$
$282$	0.0648376	0.00386102
$283$	−13.2122	−0.785386	−0.392693	−	0.919670i	$-0.628457\pi$
$283$	−13.2122	−0.785386	−0.392693	+	0.919670i	$0.628457\pi$
$284$	−1.69647	−0.100667
$285$	0	0
$286$	0.484690	0.0286603
$287$	−30.1725	−1.78103
$288$	3.11937	0.183810
$289$	−17.0000	−0.999998
$290$	0	0
$291$	11.5326	0.676054
$292$	−9.40443	−0.550353
$293$	5.47146	0.319646	0.159823	−	0.987146i	$-0.448908\pi$
$293$	5.47146	0.319646	0.159823	+	0.987146i	$0.448908\pi$
$294$	0.242903	0.0141664
$295$	0	0
$296$	2.54976	0.148202
$297$	−3.84337	−0.223015
$298$	−1.79590	−0.104033
$299$	−7.71912	−0.446408
$300$	0	0
$301$	27.7185	1.59767
$302$	−2.34821	−0.135124
$303$	−3.92297	−0.225369
$304$	−3.93504	−0.225690
$305$	0	0
$306$	−0.00157730	−9.01681e−5 0
$307$	30.8787	1.76234	0.881169	−	0.472801i	$-0.156757\pi$
$307$	30.8787	1.76234	0.881169	+	0.472801i	$0.156757\pi$
$308$	6.39826	0.364575
$309$	5.27183	0.299904
$310$	0	0
$311$	24.2528	1.37525	0.687625	−	0.726066i	$-0.258653\pi$
$311$	24.2528	1.37525	0.687625	+	0.726066i	$0.258653\pi$
$312$	−1.34764	−0.0762953
$313$	−12.2922	−0.694796	−0.347398	−	0.937718i	$-0.612935\pi$
$313$	−12.2922	−0.694796	−0.347398	+	0.937718i	$0.612935\pi$
$314$	−1.26704	−0.0715031
$315$	0	0
$316$	9.82837	0.552889
$317$	−20.0634	−1.12687	−0.563436	−	0.826160i	$-0.690521\pi$
$317$	−20.0634	−1.12687	−0.563436	+	0.826160i	$0.690521\pi$
$318$	0.129519	0.00726304
$319$	−3.49988	−0.195956
$320$	0	0
$321$	−3.49273	−0.194945
$322$	0.555601	0.0309624
$323$	0.00602394	0.000335181 0
$324$	−9.67482	−0.537490
$325$	0	0
$326$	−2.27123	−0.125792
$327$	−2.60087	−0.143829
$328$	−3.89701	−0.215176
$329$	2.87314	0.158402
$330$	0	0
$331$	−28.3147	−1.55632	−0.778159	−	0.628067i	$-0.783846\pi$
$331$	−28.3147	−1.55632	−0.778159	+	0.628067i	$0.783846\pi$
$332$	29.9048	1.64124
$333$	−15.4307	−0.845596
$334$	1.62286	0.0887988
$335$	0	0
$336$	−8.82211	−0.481286
$337$	35.6067	1.93962	0.969810	−	0.243863i	$-0.0784146\pi$
$337$	35.6067	1.93962	0.969810	+	0.243863i	$0.0784146\pi$
$338$	−0.901907	−0.0490573
$339$	11.8281	0.642415
$340$	0	0
$341$	3.34225	0.180993
$342$	−0.261838	−0.0141586
$343$	−11.7522	−0.634562
$344$	3.58006	0.193024
$345$	0	0
$346$	−1.08531	−0.0583466
$347$	8.36165	0.448877	0.224438	−	0.974488i	$-0.427945\pi$
$347$	8.36165	0.448877	0.224438	+	0.974488i	$0.427945\pi$
$348$	4.85235	0.260113
$349$	−26.3753	−1.41184	−0.705919	−	0.708293i	$-0.749466\pi$
$349$	−26.3753	−1.41184	−0.705919	+	0.708293i	$0.749466\pi$
$350$	0	0
$351$	17.8872	0.954750
$352$	1.24070	0.0661296
$353$	13.3537	0.710745	0.355372	−	0.934725i	$-0.384354\pi$
$353$	13.3537	0.710745	0.355372	+	0.934725i	$0.384354\pi$
$354$	0.186566	0.00991589
$355$	0	0
$356$	−10.1735	−0.539195
$357$	0.0135053	0.000714776 0
$358$	−2.13865	−0.113031
$359$	14.7872	0.780439	0.390220	−	0.920722i	$-0.372399\pi$
$359$	14.7872	0.780439	0.390220	+	0.920722i	$0.372399\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−0.00695651	−0.000365626 0
$363$	−0.696995	−0.0365827
$364$	−29.7778	−1.56078
$365$	0	0
$366$	−0.267748	−0.0139954
$367$	26.2183	1.36858	0.684291	−	0.729209i	$-0.260112\pi$
$367$	26.2183	1.36858	0.684291	+	0.729209i	$0.260112\pi$
$368$	−6.52658	−0.340221
$369$	23.5840	1.22773
$370$	0	0
$371$	5.73935	0.297972
$372$	−4.63379	−0.240251
$373$	−1.30489	−0.0675648	−0.0337824	−	0.999429i	$-0.510755\pi$
$373$	−1.30489	−0.0675648	−0.0337824	+	0.999429i	$0.510755\pi$
$374$	−0.000627356 0	−3.24398e−5 0
$375$	0	0
$376$	0.371089	0.0191374
$377$	16.2887	0.838908
$378$	−1.28747	−0.0662205
$379$	−10.0189	−0.514634	−0.257317	−	0.966327i	$-0.582838\pi$
$379$	−10.0189	−0.514634	−0.257317	+	0.966327i	$0.582838\pi$
$380$	0	0
$381$	−0.606791	−0.0310868
$382$	0.218328	0.0111706
$383$	5.44686	0.278322	0.139161	−	0.990270i	$-0.455560\pi$
$383$	5.44686	0.278322	0.139161	+	0.990270i	$0.455560\pi$
$384$	−2.29142	−0.116933
$385$	0	0
$386$	−2.62401	−0.133559
$387$	−21.6659	−1.10134
$388$	32.9129	1.67090
$389$	−7.59394	−0.385028	−0.192514	−	0.981294i	$-0.561664\pi$
$389$	−7.59394	−0.385028	−0.192514	+	0.981294i	$0.561664\pi$
$390$	0	0
$391$	0.00999119	0.000505276 0
$392$	1.39022	0.0702169
$393$	0.231894	0.0116975
$394$	−0.594541	−0.0299525
$395$	0	0
$396$	−5.00113	−0.251316
$397$	−6.27453	−0.314910	−0.157455	−	0.987526i	$-0.550329\pi$
$397$	−6.27453	−0.314910	−0.157455	+	0.987526i	$0.550329\pi$
$398$	1.06230	0.0532482
$399$	2.24194	0.112237
$400$	0	0
$401$	17.8048	0.889131	0.444565	−	0.895746i	$-0.353358\pi$
$401$	17.8048	0.889131	0.444565	+	0.895746i	$0.353358\pi$
$402$	0.434542	0.0216730
$403$	−15.5550	−0.774849
$404$	−11.1958	−0.557010
$405$	0	0
$406$	−1.17241	−0.0581858
$407$	−6.13742	−0.304221
$408$	0.00174431	8.63564e−5 0
$409$	8.47192	0.418909	0.209455	−	0.977818i	$-0.432831\pi$
$409$	8.47192	0.418909	0.209455	+	0.977818i	$0.432831\pi$
$410$	0	0
$411$	9.08281	0.448022
$412$	15.0453	0.741228
$413$	8.26731	0.406807
$414$	−0.434279	−0.0213436
$415$	0	0
$416$	−5.77429	−0.283108
$417$	−8.83978	−0.432886
$418$	−0.104144	−0.00509384
$419$	10.4558	0.510799	0.255399	−	0.966836i	$-0.417793\pi$
$419$	10.4558	0.510799	0.255399	+	0.966836i	$0.417793\pi$
$420$	0	0
$421$	32.4814	1.58305	0.791523	−	0.611139i	$-0.209288\pi$
$421$	32.4814	1.58305	0.791523	+	0.611139i	$0.209288\pi$
$422$	−0.108270	−0.00527051
$423$	−2.24576	−0.109193
$424$	0.741281	0.0359998
$425$	0	0
$426$	0.0619069	0.00299940
$427$	−11.8647	−0.574173
$428$	−9.96790	−0.481817
$429$	3.24385	0.156615
$430$	0	0
$431$	−2.28726	−0.110174	−0.0550868	−	0.998482i	$-0.517544\pi$
$431$	−2.28726	−0.110174	−0.0550868	+	0.998482i	$0.517544\pi$
$432$	15.1238	0.727645
$433$	36.5224	1.75515	0.877576	−	0.479437i	$-0.159159\pi$
$433$	36.5224	1.75515	0.877576	+	0.479437i	$0.159159\pi$
$434$	1.11961	0.0537428
$435$	0	0
$436$	−7.42264	−0.355480
$437$	1.65858	0.0793406
$438$	0.343184	0.0163980
$439$	−23.0323	−1.09927	−0.549636	−	0.835404i	$-0.685233\pi$
$439$	−23.0323	−1.09927	−0.549636	+	0.835404i	$0.685233\pi$
$440$	0	0
$441$	−8.41338	−0.400637
$442$	0.00291975	0.000138878 0
$443$	3.63563	0.172734	0.0863671	−	0.996263i	$-0.472474\pi$
$443$	3.63563	0.172734	0.0863671	+	0.996263i	$0.472474\pi$
$444$	8.50910	0.403824
$445$	0	0
$446$	−0.669263	−0.0316905
$447$	−12.0193	−0.568492
$448$	−24.8991	−1.17637
$449$	25.5726	1.20685	0.603423	−	0.797422i	$-0.293803\pi$
$449$	25.5726	1.20685	0.603423	+	0.797422i	$0.293803\pi$
$450$	0	0
$451$	9.38032	0.441702
$452$	33.7563	1.58776
$453$	−15.7157	−0.738388
$454$	−0.651747	−0.0305880
$455$	0	0
$456$	0.289563	0.0135600
$457$	6.45150	0.301789	0.150894	−	0.988550i	$-0.451785\pi$
$457$	6.45150	0.301789	0.150894	+	0.988550i	$0.451785\pi$
$458$	−2.73555	−0.127824
$459$	−0.0231522	−0.00108065
$460$	0	0
$461$	−23.1293	−1.07724	−0.538619	−	0.842550i	$-0.681054\pi$
$461$	−23.1293	−1.07724	−0.538619	+	0.842550i	$0.681054\pi$
$462$	−0.233483	−0.0108626
$463$	36.4434	1.69367	0.846834	−	0.531858i	$-0.178506\pi$
$463$	36.4434	1.69367	0.846834	+	0.531858i	$0.178506\pi$
$464$	13.7722	0.639358
$465$	0	0
$466$	−0.307214	−0.0142314
$467$	−36.0540	−1.66838	−0.834191	−	0.551476i	$-0.814065\pi$
$467$	−36.0540	−1.66838	−0.834191	+	0.551476i	$0.814065\pi$
$468$	23.2755	1.07591
$469$	19.2558	0.889152
$470$	0	0
$471$	−8.47981	−0.390729
$472$	1.06779	0.0491488
$473$	−8.61740	−0.396228
$474$	−0.358654	−0.0164735
$475$	0	0
$476$	0.0385428	0.00176660
$477$	−4.48610	−0.205404
$478$	−1.93646	−0.0885718
$479$	17.2250	0.787032	0.393516	−	0.919318i	$-0.371259\pi$
$479$	17.2250	0.787032	0.393516	+	0.919318i	$0.371259\pi$
$480$	0	0
$481$	28.5639	1.30240
$482$	0.904158	0.0411832
$483$	3.71843	0.169194
$484$	−1.98915	−0.0904161
$485$	0	0
$486$	1.55384	0.0704835
$487$	−31.6437	−1.43391	−0.716957	−	0.697117i	$-0.754466\pi$
$487$	−31.6437	−1.43391	−0.716957	+	0.697117i	$0.754466\pi$
$488$	−1.53242	−0.0693692
$489$	−15.2005	−0.687389
$490$	0	0
$491$	38.9066	1.75583	0.877916	−	0.478814i	$-0.158933\pi$
$491$	38.9066	1.75583	0.877916	+	0.478814i	$0.158933\pi$
$492$	−13.0052	−0.586318
$493$	−0.0210831	−0.000949536 0
$494$	0.484690	0.0218073
$495$	0	0
$496$	−13.1519	−0.590537
$497$	2.74328	0.123053
$498$	−1.09128	−0.0489013
$499$	16.4194	0.735033	0.367516	−	0.930017i	$-0.380208\pi$
$499$	16.4194	0.735033	0.367516	+	0.930017i	$0.380208\pi$
$500$	0	0
$501$	10.8612	0.485242
$502$	2.88927	0.128954
$503$	4.73308	0.211038	0.105519	−	0.994417i	$-0.466350\pi$
$503$	4.73308	0.211038	0.105519	+	0.994417i	$0.466350\pi$
$504$	−3.35975	−0.149655
$505$	0	0
$506$	−0.172731	−0.00767881
$507$	−6.03613	−0.268074
$508$	−1.73172	−0.0768326
$509$	−32.8750	−1.45716	−0.728580	−	0.684961i	$-0.759819\pi$
$509$	−32.8750	−1.45716	−0.728580	+	0.684961i	$0.759819\pi$
$510$	0	0
$511$	15.2075	0.672740
$512$	−8.15180	−0.360262
$513$	−3.84337	−0.169689
$514$	1.84338	0.0813080
$515$	0	0
$516$	11.9474	0.525956
$517$	−0.893231	−0.0392843
$518$	−2.05595	−0.0903332
$519$	−7.26357	−0.318835
$520$	0	0
$521$	7.50334	0.328727	0.164364	−	0.986400i	$-0.447443\pi$
$521$	7.50334	0.328727	0.164364	+	0.986400i	$0.447443\pi$
$522$	0.916402	0.0401098
$523$	11.7845	0.515300	0.257650	−	0.966238i	$-0.417052\pi$
$523$	11.7845	0.515300	0.257650	+	0.966238i	$0.417052\pi$
$524$	0.661803	0.0289110
$525$	0	0
$526$	−2.33309	−0.101728
$527$	0.0201335	0.000877029 0
$528$	2.74270	0.119361
$529$	−20.2491	−0.880396
$530$	0	0
$531$	−6.46204	−0.280429
$532$	6.39826	0.277400
$533$	−43.6565	−1.89097
$534$	0.371249	0.0160655
$535$	0	0
$536$	2.48704	0.107424
$537$	−14.3132	−0.617661
$538$	−1.68425	−0.0726131
$539$	−3.34635	−0.144137
$540$	0	0
$541$	18.8061	0.808535	0.404268	−	0.914641i	$-0.367526\pi$
$541$	18.8061	0.808535	0.404268	+	0.914641i	$0.367526\pi$
$542$	−0.218674	−0.00939285
$543$	−0.0465574	−0.00199797
$544$	0.00747391	0.000320441 0
$545$	0	0
$546$	1.08664	0.0465041
$547$	11.1690	0.477550	0.238775	−	0.971075i	$-0.423254\pi$
$547$	11.1690	0.477550	0.238775	+	0.971075i	$0.423254\pi$
$548$	25.9214	1.10731
$549$	9.27390	0.395800
$550$	0	0
$551$	−3.49988	−0.149100
$552$	0.480264	0.0204414
$553$	−15.8930	−0.675840
$554$	−3.12213	−0.132647
$555$	0	0
$556$	−25.2279	−1.06990
$557$	26.3044	1.11455	0.557276	−	0.830327i	$-0.311846\pi$
$557$	26.3044	1.11455	0.557276	+	0.830327i	$0.311846\pi$
$558$	−0.875127	−0.0370471
$559$	40.1058	1.69630
$560$	0	0
$561$	−0.00419866	−0.000177267 0
$562$	0.0489273	0.00206387
$563$	27.3566	1.15294	0.576472	−	0.817117i	$-0.304429\pi$
$563$	27.3566	1.15294	0.576472	+	0.817117i	$0.304429\pi$
$564$	1.23840	0.0521462
$565$	0	0
$566$	1.37597	0.0578364
$567$	15.6447	0.657017
$568$	0.354316	0.0148668
$569$	−17.7921	−0.745882	−0.372941	−	0.927855i	$-0.621651\pi$
$569$	−17.7921	−0.745882	−0.372941	+	0.927855i	$0.621651\pi$
$570$	0	0
$571$	25.9440	1.08572	0.542862	−	0.839822i	$-0.317341\pi$
$571$	25.9440	1.08572	0.542862	+	0.839822i	$0.317341\pi$
$572$	9.25763	0.387081
$573$	1.46119	0.0610421
$574$	3.14228	0.131156
$575$	0	0
$576$	19.4621	0.810920
$577$	−13.7994	−0.574476	−0.287238	−	0.957859i	$-0.592737\pi$
$577$	−13.7994	−0.574476	−0.287238	+	0.957859i	$0.592737\pi$
$578$	1.77044	0.0736406
$579$	−17.5615	−0.729831
$580$	0	0
$581$	−48.3577	−2.00621
$582$	−1.20105	−0.0497851
$583$	−1.78431	−0.0738984
$584$	1.96417	0.0812777
$585$	0	0
$586$	−0.569818	−0.0235390
$587$	31.3230	1.29284	0.646420	−	0.762982i	$-0.276266\pi$
$587$	31.3230	1.29284	0.646420	+	0.762982i	$0.276266\pi$
$588$	4.63947	0.191329
$589$	3.34225	0.137715
$590$	0	0
$591$	−3.97904	−0.163676
$592$	24.1510	0.992600
$593$	−33.8912	−1.39174	−0.695872	−	0.718166i	$-0.744982\pi$
$593$	−33.8912	−1.39174	−0.695872	+	0.718166i	$0.744982\pi$
$594$	0.400263	0.0164230
$595$	0	0
$596$	−34.3018	−1.40506
$597$	7.10957	0.290975
$598$	0.803897	0.0328738
$599$	−20.5165	−0.838280	−0.419140	−	0.907922i	$-0.637668\pi$
$599$	−20.5165	−0.838280	−0.419140	+	0.907922i	$0.637668\pi$
$600$	0	0
$601$	23.2037	0.946501	0.473250	−	0.880928i	$-0.343081\pi$
$601$	23.2037	0.946501	0.473250	+	0.880928i	$0.343081\pi$
$602$	−2.88671	−0.117653
$603$	−15.0511	−0.612928
$604$	−44.8510	−1.82496
$605$	0	0
$606$	0.408552	0.0165963
$607$	2.28113	0.0925881	0.0462941	−	0.998928i	$-0.485259\pi$
$607$	2.28113	0.0925881	0.0462941	+	0.998928i	$0.485259\pi$
$608$	1.24070	0.0503170
$609$	−7.84652	−0.317957
$610$	0	0
$611$	4.15715	0.168180
$612$	−0.0301265	−0.00121779
$613$	2.06670	0.0834732	0.0417366	−	0.999129i	$-0.486711\pi$
$613$	2.06670	0.0834732	0.0417366	+	0.999129i	$0.486711\pi$
$614$	−3.21582	−0.129780
$615$	0	0
$616$	−1.33631	−0.0538415
$617$	39.7201	1.59907	0.799536	−	0.600618i	$-0.205079\pi$
$617$	39.7201	1.59907	0.799536	+	0.600618i	$0.205079\pi$
$618$	−0.549028	−0.0220852
$619$	3.94724	0.158653	0.0793266	−	0.996849i	$-0.474723\pi$
$619$	3.94724	0.158653	0.0793266	+	0.996849i	$0.474723\pi$
$620$	0	0
$621$	−6.37453	−0.255801
$622$	−2.52578	−0.101274
$623$	16.4511	0.659101
$624$	−12.7647	−0.510997
$625$	0	0
$626$	1.28016	0.0511653
$627$	−0.696995	−0.0278353
$628$	−24.2005	−0.965707
$629$	−0.0369715	−0.00147415
$630$	0	0
$631$	−12.3504	−0.491663	−0.245832	−	0.969313i	$-0.579061\pi$
$631$	−12.3504	−0.491663	−0.245832	+	0.969313i	$0.579061\pi$
$632$	−2.05271	−0.0816523
$633$	−0.724612	−0.0288007
$634$	2.08947	0.0829836
$635$	0	0
$636$	2.47382	0.0980932
$637$	15.5741	0.617067
$638$	0.364491	0.0144303
$639$	−2.14425	−0.0848253
$640$	0	0
$641$	44.1646	1.74440	0.872198	−	0.489153i	$-0.162694\pi$
$641$	44.1646	1.74440	0.872198	+	0.489153i	$0.162694\pi$
$642$	0.363746	0.0143559
$643$	−7.27326	−0.286829	−0.143415	−	0.989663i	$-0.545808\pi$
$643$	−7.27326	−0.286829	−0.143415	+	0.989663i	$0.545808\pi$
$644$	10.6120	0.418172
$645$	0	0
$646$	−0.000627356 0	−2.46830e−5 0
$647$	−11.7738	−0.462878	−0.231439	−	0.972849i	$-0.574343\pi$
$647$	−11.7738	−0.462878	−0.231439	+	0.972849i	$0.574343\pi$
$648$	2.02064	0.0793781
$649$	−2.57022	−0.100890
$650$	0	0
$651$	7.49310	0.293678
$652$	−43.3806	−1.69892
$653$	−31.7958	−1.24427	−0.622134	−	0.782911i	$-0.713734\pi$
$653$	−31.7958	−1.24427	−0.622134	+	0.782911i	$0.713734\pi$
$654$	0.270865	0.0105916
$655$	0	0
$656$	−36.9120	−1.44117
$657$	−11.8868	−0.463747
$658$	−0.299220	−0.0116648
$659$	−23.9090	−0.931361	−0.465680	−	0.884953i	$-0.654190\pi$
$659$	−23.9090	−0.931361	−0.465680	+	0.884953i	$0.654190\pi$
$660$	0	0
$661$	−39.9343	−1.55327	−0.776633	−	0.629953i	$-0.783074\pi$
$661$	−39.9343	−1.55327	−0.776633	+	0.629953i	$0.783074\pi$
$662$	2.94880	0.114608
$663$	0.0195408	0.000758901 0
$664$	−6.24577	−0.242383
$665$	0	0
$666$	1.60701	0.0622703
$667$	−5.80484	−0.224764
$668$	30.9967	1.19930
$669$	−4.47913	−0.173173
$670$	0	0
$671$	3.68861	0.142397
$672$	2.78157	0.107301
$673$	−15.3378	−0.591228	−0.295614	−	0.955308i	$-0.595524\pi$
$673$	−15.3378	−0.591228	−0.295614	+	0.955308i	$0.595524\pi$
$674$	−3.70821	−0.142835
$675$	0	0
$676$	−17.2265	−0.662558
$677$	−20.2183	−0.777054	−0.388527	−	0.921437i	$-0.627016\pi$
$677$	−20.2183	−0.777054	−0.388527	+	0.921437i	$0.627016\pi$
$678$	−1.23182	−0.0473079
$679$	−53.2220	−2.04247
$680$	0	0
$681$	−4.36190	−0.167148
$682$	−0.348074	−0.0133284
$683$	−9.75882	−0.373411	−0.186705	−	0.982416i	$-0.559781\pi$
$683$	−9.75882	−0.373411	−0.186705	+	0.982416i	$0.559781\pi$
$684$	−5.00113	−0.191223
$685$	0	0
$686$	1.22392	0.0467296
$687$	−18.3080	−0.698495
$688$	33.9098	1.29280
$689$	8.30425	0.316367
$690$	0	0
$691$	−3.10119	−0.117975	−0.0589873	−	0.998259i	$-0.518787\pi$
$691$	−3.10119	−0.117975	−0.0589873	+	0.998259i	$0.518787\pi$
$692$	−20.7295	−0.788018
$693$	8.08710	0.307204
$694$	−0.870813	−0.0330556
$695$	0	0
$696$	−1.01344	−0.0384143
$697$	0.0565066	0.00214034
$698$	2.74682	0.103969
$699$	−2.05607	−0.0777676
$700$	0	0
$701$	1.49983	0.0566478	0.0283239	−	0.999599i	$-0.490983\pi$
$701$	1.49983	0.0566478	0.0283239	+	0.999599i	$0.490983\pi$
$702$	−1.86284	−0.0703085
$703$	−6.13742	−0.231477
$704$	7.74087	0.291745
$705$	0	0
$706$	−1.39070	−0.0523397
$707$	18.1042	0.680877
$708$	3.56343	0.133922
$709$	−33.3647	−1.25304	−0.626518	−	0.779407i	$-0.715520\pi$
$709$	−33.3647	−1.25304	−0.626518	+	0.779407i	$0.715520\pi$
$710$	0	0
$711$	12.4226	0.465884
$712$	2.12479	0.0796299
$713$	5.54338	0.207601
$714$	−0.00140649	−5.26366e−5 0
$715$	0	0
$716$	−40.8485	−1.52658
$717$	−12.9600	−0.484001
$718$	−1.53999	−0.0574721
$719$	−11.9782	−0.446711	−0.223355	−	0.974737i	$-0.571701\pi$
$719$	−11.9782	−0.446711	−0.223355	+	0.974737i	$0.571701\pi$
$720$	0	0
$721$	−24.3291	−0.906062
$722$	−0.104144	−0.00387583
$723$	6.05119	0.225046
$724$	−0.132870	−0.00493808
$725$	0	0
$726$	0.0725876	0.00269398
$727$	−4.56220	−0.169203	−0.0846014	−	0.996415i	$-0.526962\pi$
$727$	−4.56220	−0.169203	−0.0846014	+	0.996415i	$0.526962\pi$
$728$	6.21926	0.230501
$729$	−4.19210	−0.155263
$730$	0	0
$731$	−0.0519107	−0.00191999
$732$	−5.11400	−0.189019
$733$	22.2267	0.820963	0.410482	−	0.911869i	$-0.365361\pi$
$733$	22.2267	0.820963	0.410482	+	0.911869i	$0.365361\pi$
$734$	−2.73047	−0.100783
$735$	0	0
$736$	2.05780	0.0758515
$737$	−5.98644	−0.220513
$738$	−2.45612	−0.0904112
$739$	14.4390	0.531149	0.265574	−	0.964090i	$-0.414438\pi$
$739$	14.4390	0.531149	0.265574	+	0.964090i	$0.414438\pi$
$740$	0	0
$741$	3.24385	0.119166
$742$	−0.597717	−0.0219429
$743$	−43.0239	−1.57839	−0.789197	−	0.614141i	$-0.789503\pi$
$743$	−43.0239	−1.57839	−0.789197	+	0.614141i	$0.789503\pi$
$744$	0.967792	0.0354810
$745$	0	0
$746$	0.135896	0.00497552
$747$	37.7982	1.38297
$748$	−0.0119826	−0.000438126 0
$749$	16.1187	0.588963
$750$	0	0
$751$	8.50342	0.310294	0.155147	−	0.987891i	$-0.450415\pi$
$751$	8.50342	0.310294	0.155147	+	0.987891i	$0.450415\pi$
$752$	3.51490	0.128175
$753$	19.3368	0.704673
$754$	−1.69636	−0.0617778
$755$	0	0
$756$	−24.5909	−0.894361
$757$	−18.2012	−0.661533	−0.330766	−	0.943713i	$-0.607307\pi$
$757$	−18.2012	−0.661533	−0.330766	+	0.943713i	$0.607307\pi$
$758$	1.04340	0.0378980
$759$	−1.15602	−0.0419609
$760$	0	0
$761$	45.3295	1.64319	0.821597	−	0.570069i	$-0.193083\pi$
$761$	45.3295	1.64319	0.821597	+	0.570069i	$0.193083\pi$
$762$	0.0631934	0.00228926
$763$	12.0028	0.434531
$764$	4.17009	0.150869
$765$	0	0
$766$	−0.567256	−0.0204958
$767$	11.9619	0.431921
$768$	−10.5521	−0.380765
$769$	−27.6341	−0.996510	−0.498255	−	0.867031i	$-0.666026\pi$
$769$	−27.6341	−0.996510	−0.498255	+	0.867031i	$0.666026\pi$
$770$	0	0
$771$	12.3371	0.444308
$772$	−50.1188	−1.80382
$773$	−27.7758	−0.999027	−0.499513	−	0.866306i	$-0.666488\pi$
$773$	−27.7758	−0.999027	−0.499513	+	0.866306i	$0.666488\pi$
$774$	2.25636	0.0811033
$775$	0	0
$776$	−6.87404	−0.246764
$777$	−13.7597	−0.493626
$778$	0.790861	0.0283538
$779$	9.38032	0.336085
$780$	0	0
$781$	−0.852858	−0.0305176
$782$	−0.00104052	−3.72089e−5 0
$783$	13.4513	0.480712
$784$	13.1680	0.470286
$785$	0	0
$786$	−0.0241503	−0.000861414 0
$787$	1.98342	0.0707012	0.0353506	−	0.999375i	$-0.488745\pi$
$787$	1.98342	0.0707012	0.0353506	+	0.999375i	$0.488745\pi$
$788$	−11.3558	−0.404533
$789$	−15.6145	−0.555891
$790$	0	0
$791$	−54.5858	−1.94085
$792$	1.04451	0.0371151
$793$	−17.1670	−0.609618
$794$	0.653453	0.0231902
$795$	0	0
$796$	20.2900	0.719160
$797$	30.9790	1.09733	0.548666	−	0.836041i	$-0.315136\pi$
$797$	30.9790	1.09733	0.548666	+	0.836041i	$0.315136\pi$
$798$	−0.233483	−0.00826523
$799$	−0.00538078	−0.000190358 0
$800$	0	0
$801$	−12.8588	−0.454345
$802$	−1.85426	−0.0654762
$803$	−4.72786	−0.166842
$804$	8.29979	0.292711
$805$	0	0
$806$	1.61995	0.0570605
$807$	−11.2721	−0.396795
$808$	2.33829	0.0822608
$809$	33.4595	1.17637	0.588186	−	0.808725i	$-0.299842\pi$
$809$	33.4595	1.17637	0.588186	+	0.808725i	$0.299842\pi$
$810$	0	0
$811$	−33.7987	−1.18683	−0.593416	−	0.804896i	$-0.702221\pi$
$811$	−33.7987	−1.18683	−0.593416	+	0.804896i	$0.702221\pi$
$812$	−22.3932	−0.785846
$813$	−1.46350	−0.0513273
$814$	0.639174	0.0224030
$815$	0	0
$816$	0.0165219	0.000578382 0
$817$	−8.61740	−0.301485
$818$	−0.882297	−0.0308488
$819$	−37.6378	−1.31517
$820$	0	0
$821$	−17.6687	−0.616641	−0.308320	−	0.951283i	$-0.599767\pi$
$821$	−17.6687	−0.616641	−0.308320	+	0.951283i	$0.599767\pi$
$822$	−0.945917	−0.0329927
$823$	−46.0491	−1.60517	−0.802585	−	0.596538i	$-0.796543\pi$
$823$	−46.0491	−1.60517	−0.802585	+	0.596538i	$0.796543\pi$
$824$	−3.14229	−0.109467
$825$	0	0
$826$	−0.860988	−0.0299576
$827$	−23.9121	−0.831503	−0.415752	−	0.909478i	$-0.636481\pi$
$827$	−23.9121	−0.831503	−0.415752	+	0.909478i	$0.636481\pi$
$828$	−8.29476	−0.288263
$829$	−10.2688	−0.356649	−0.178324	−	0.983972i	$-0.557068\pi$
$829$	−10.2688	−0.356649	−0.178324	+	0.983972i	$0.557068\pi$
$830$	0	0
$831$	−20.8952	−0.724848
$832$	−36.0264	−1.24899
$833$	−0.0201582	−0.000698440 0
$834$	0.920608	0.0318780
$835$	0	0
$836$	−1.98915	−0.0687963
$837$	−12.8455	−0.444005
$838$	−1.08890	−0.0376156
$839$	−40.5454	−1.39978	−0.699890	−	0.714250i	$-0.746768\pi$
$839$	−40.5454	−1.39978	−0.699890	+	0.714250i	$0.746768\pi$
$840$	0	0
$841$	−16.7508	−0.577614
$842$	−3.38273	−0.116577
$843$	0.327452	0.0112780
$844$	−2.06797	−0.0711825
$845$	0	0
$846$	0.233882	0.00804102
$847$	3.21657	0.110523
$848$	7.02132	0.241113
$849$	9.20886	0.316047
$850$	0	0
$851$	−10.1794	−0.348945
$852$	1.18243	0.0405093
$853$	−1.99694	−0.0683740	−0.0341870	−	0.999415i	$-0.510884\pi$
$853$	−1.99694	−0.0683740	−0.0341870	+	0.999415i	$0.510884\pi$
$854$	1.23563	0.0422825
$855$	0	0
$856$	2.08185	0.0711561
$857$	25.1830	0.860235	0.430118	−	0.902773i	$-0.358472\pi$
$857$	25.1830	0.860235	0.430118	+	0.902773i	$0.358472\pi$
$858$	−0.337827	−0.0115332
$859$	−0.915472	−0.0312355	−0.0156178	−	0.999878i	$-0.504971\pi$
$859$	−0.915472	−0.0312355	−0.0156178	+	0.999878i	$0.504971\pi$
$860$	0	0
$861$	21.0301	0.716703
$862$	0.238204	0.00811326
$863$	−49.8812	−1.69797	−0.848987	−	0.528413i	$-0.822787\pi$
$863$	−49.8812	−1.69797	−0.848987	+	0.528413i	$0.822787\pi$
$864$	−4.76847	−0.162227
$865$	0	0
$866$	−3.80357	−0.129251
$867$	11.8489	0.402409
$868$	21.3846	0.725839
$869$	4.94098	0.167611
$870$	0	0
$871$	27.8612	0.944042
$872$	1.55026	0.0524983
$873$	41.6004	1.40796
$874$	−0.172731	−0.00584270
$875$	0	0
$876$	6.55484	0.221468
$877$	18.6769	0.630673	0.315337	−	0.948980i	$-0.397883\pi$
$877$	18.6769	0.630673	0.315337	+	0.948980i	$0.397883\pi$
$878$	2.39867	0.0809511
$879$	−3.81358	−0.128629
$880$	0	0
$881$	−46.9216	−1.58083	−0.790414	−	0.612573i	$-0.790134\pi$
$881$	−46.9216	−1.58083	−0.790414	+	0.612573i	$0.790134\pi$
$882$	0.876200	0.0295032
$883$	−44.8499	−1.50932	−0.754659	−	0.656117i	$-0.772198\pi$
$883$	−44.8499	−1.50932	−0.754659	+	0.656117i	$0.772198\pi$
$884$	0.0557674	0.00187566
$885$	0	0
$886$	−0.378628	−0.0127203
$887$	14.5451	0.488376	0.244188	−	0.969728i	$-0.421479\pi$
$887$	14.5451	0.488376	0.244188	+	0.969728i	$0.421479\pi$
$888$	−1.77717	−0.0596380
$889$	2.80029	0.0939186
$890$	0	0
$891$	−4.86379	−0.162943
$892$	−12.7830	−0.428006
$893$	−0.893231	−0.0298908
$894$	1.25173	0.0418641
$895$	0	0
$896$	10.5747	0.353276
$897$	5.38018	0.179639
$898$	−2.66322	−0.0888730
$899$	−11.6975	−0.390133
$900$	0	0
$901$	−0.0107486	−0.000358086 0
$902$	−0.976902	−0.0325273
$903$	−19.3197	−0.642918
$904$	−7.05017	−0.234485
$905$	0	0
$906$	1.63669	0.0543754
$907$	56.2751	1.86858	0.934292	−	0.356510i	$-0.116033\pi$
$907$	56.2751	1.86858	0.934292	+	0.356510i	$0.116033\pi$
$908$	−12.4484	−0.413115
$909$	−14.1509	−0.469356
$910$	0	0
$911$	11.8656	0.393123	0.196562	−	0.980491i	$-0.437022\pi$
$911$	11.8656	0.393123	0.196562	+	0.980491i	$0.437022\pi$
$912$	2.74270	0.0908200
$913$	15.0339	0.497550
$914$	−0.671883	−0.0222239
$915$	0	0
$916$	−52.2493	−1.72637
$917$	−1.07017	−0.0353402
$918$	0.00241116	7.95801e−5 0
$919$	17.4567	0.575844	0.287922	−	0.957654i	$-0.407036\pi$
$919$	17.4567	0.575844	0.287922	+	0.957654i	$0.407036\pi$
$920$	0	0
$921$	−21.5223	−0.709183
$922$	2.40877	0.0793286
$923$	3.96924	0.130649
$924$	−4.45956	−0.146709
$925$	0	0
$926$	−3.79535	−0.124723
$927$	19.0165	0.624585
$928$	−4.34231	−0.142543
$929$	6.75452	0.221609	0.110804	−	0.993842i	$-0.464657\pi$
$929$	6.75452	0.221609	0.110804	+	0.993842i	$0.464657\pi$
$930$	0	0
$931$	−3.34635	−0.109672
$932$	−5.86781	−0.192206
$933$	−16.9041	−0.553415
$934$	3.75480	0.122861
$935$	0	0
$936$	−4.86121	−0.158894
$937$	−45.1286	−1.47429	−0.737143	−	0.675736i	$-0.763826\pi$
$937$	−45.1286	−1.47429	−0.737143	+	0.675736i	$0.763826\pi$
$938$	−2.00537	−0.0654778
$939$	8.56760	0.279593
$940$	0	0
$941$	26.7484	0.871972	0.435986	−	0.899954i	$-0.356400\pi$
$941$	26.7484	0.871972	0.435986	+	0.899954i	$0.356400\pi$
$942$	0.883119	0.0287736
$943$	15.5580	0.506639
$944$	10.1139	0.329180
$945$	0	0
$946$	0.897448	0.0291786
$947$	−16.2900	−0.529355	−0.264677	−	0.964337i	$-0.585265\pi$
$947$	−16.2900	−0.529355	−0.264677	+	0.964337i	$0.585265\pi$
$948$	−6.85032	−0.222488
$949$	22.0037	0.714270
$950$	0	0
$951$	13.9841	0.453465
$952$	−0.00804986	−0.000260897 0
$953$	4.10009	0.132815	0.0664075	−	0.997793i	$-0.478846\pi$
$953$	4.10009	0.132815	0.0664075	+	0.997793i	$0.478846\pi$
$954$	0.467199	0.0151261
$955$	0	0
$956$	−36.9866	−1.19623
$957$	2.43940	0.0788547
$958$	−1.79388	−0.0579576
$959$	−41.9164	−1.35355
$960$	0	0
$961$	−19.8294	−0.639658
$962$	−2.97475	−0.0959097
$963$	−12.5990	−0.405996
$964$	17.2695	0.556213
$965$	0	0
$966$	−0.387251	−0.0124596
$967$	−34.0890	−1.09623	−0.548115	−	0.836403i	$-0.684654\pi$
$967$	−34.0890	−1.09623	−0.548115	+	0.836403i	$0.684654\pi$
$968$	0.415445	0.0133529
$969$	−0.00419866	−0.000134880 0
$970$	0	0
$971$	40.8253	1.31015	0.655073	−	0.755565i	$-0.272638\pi$
$971$	40.8253	1.31015	0.655073	+	0.755565i	$0.272638\pi$
$972$	29.6785	0.951937
$973$	40.7949	1.30782
$974$	3.29550	0.105594
$975$	0	0
$976$	−14.5148	−0.464609
$977$	−14.7093	−0.470591	−0.235296	−	0.971924i	$-0.575606\pi$
$977$	−14.7093	−0.470591	−0.235296	+	0.971924i	$0.575606\pi$
$978$	1.58303	0.0506198
$979$	−5.11449	−0.163460
$980$	0	0
$981$	−9.38187	−0.299540
$982$	−4.05188	−0.129301
$983$	31.0823	0.991372	0.495686	−	0.868502i	$-0.334917\pi$
$983$	31.0823	0.991372	0.495686	+	0.868502i	$0.334917\pi$
$984$	2.71620	0.0865892
$985$	0	0
$986$	0.00219567	6.99245e−5 0
$987$	−2.00257	−0.0637424
$988$	9.25763	0.294524
$989$	−14.2926	−0.454479
$990$	0	0
$991$	45.1569	1.43446	0.717228	−	0.696838i	$-0.245411\pi$
$991$	45.1569	1.43446	0.717228	+	0.696838i	$0.245411\pi$
$992$	4.14673	0.131659
$993$	19.7352	0.626279
$994$	−0.285695	−0.00906170
$995$	0	0
$996$	−20.8435	−0.660451
$997$	−46.8604	−1.48409	−0.742043	−	0.670353i	$-0.766143\pi$
$997$	−46.8604	−1.48409	−0.742043	+	0.670353i	$0.766143\pi$
$998$	−1.70998	−0.0541283
$999$	23.5884	0.746303

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5225.2.a.bb.1.11		22
5.2	odd	4		1045.2.b.d.419.11	✓	22
5.3	odd	4		1045.2.b.d.419.12	yes	22
5.4	even	2	inner	5225.2.a.bb.1.12		22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1045.2.b.d.419.11	✓	22	5.2	odd	4
1045.2.b.d.419.12	yes	22	5.3	odd	4
5225.2.a.bb.1.11		22	1.1	even	1	trivial
5225.2.a.bb.1.12		22	5.4	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 5225 = 5^{2} \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5225.a (trivial)

\(f(q)\)	\(=\)	\(q-0.104144 q^{2} -0.696995 q^{3} -1.98915 q^{4} +0.0725876 q^{6} +3.21657 q^{7} +0.415445 q^{8} -2.51420 q^{9} -1.00000 q^{11} +1.38643 q^{12} +4.65405 q^{13} -0.334986 q^{14} +3.93504 q^{16} -0.00602394 q^{17} +0.261838 q^{18} -1.00000 q^{19} -2.24194 q^{21} +0.104144 q^{22} -1.65858 q^{23} -0.289563 q^{24} -0.484690 q^{26} +3.84337 q^{27} -6.39826 q^{28} +3.49988 q^{29} -3.34225 q^{31} -1.24070 q^{32} +0.696995 q^{33} +0.000627356 q^{34} +5.00113 q^{36} +6.13742 q^{37} +0.104144 q^{38} -3.24385 q^{39} -9.38032 q^{41} +0.233483 q^{42} +8.61740 q^{43} +1.98915 q^{44} +0.172731 q^{46} +0.893231 q^{47} -2.74270 q^{48} +3.34635 q^{49} +0.00419866 q^{51} -9.25763 q^{52} +1.78431 q^{53} -0.400263 q^{54} +1.33631 q^{56} +0.696995 q^{57} -0.364491 q^{58} +2.57022 q^{59} -3.68861 q^{61} +0.348074 q^{62} -8.08710 q^{63} -7.74087 q^{64} -0.0725876 q^{66} +5.98644 q^{67} +0.0119826 q^{68} +1.15602 q^{69} +0.852858 q^{71} -1.04451 q^{72} +4.72786 q^{73} -0.639174 q^{74} +1.98915 q^{76} -3.21657 q^{77} +0.337827 q^{78} -4.94098 q^{79} +4.86379 q^{81} +0.976902 q^{82} -15.0339 q^{83} +4.45956 q^{84} -0.897448 q^{86} -2.43940 q^{87} -0.415445 q^{88} +5.11449 q^{89} +14.9701 q^{91} +3.29917 q^{92} +2.32953 q^{93} -0.0930244 q^{94} +0.864762 q^{96} -16.5462 q^{97} -0.348501 q^{98} +2.51420 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q + 32 q^{4} - 12 q^{6} + 34 q^{9} - 22 q^{11} - 8 q^{14} + 40 q^{16} - 22 q^{19} - 22 q^{21} - 22 q^{24} + 16 q^{26} - 10 q^{29} + 76 q^{31} + 56 q^{34} + 104 q^{36} - 8 q^{39} + 6 q^{41} - 32 q^{44}+ \cdots - 34 q^{99}+O(q^{100}) \) 22 * q + 32 * q^4 - 12 * q^6 + 34 * q^9 - 22 * q^11 - 8 * q^14 + 40 * q^16 - 22 * q^19 - 22 * q^21 - 22 * q^24 + 16 * q^26 - 10 * q^29 + 76 * q^31 + 56 * q^34 + 104 * q^36 - 8 * q^39 + 6 * q^41 - 32 * q^44 + 88 * q^46 + 28 * q^49 + 8 * q^51 + 38 * q^54 + 44 * q^56 + 40 * q^59 - 6 * q^61 + 140 * q^64 + 12 * q^66 + 74 * q^69 + 62 * q^71 - 26 * q^74 - 32 * q^76 + 102 * q^79 + 94 * q^81 - 38 * q^84 + 28 * q^86 + 54 * q^89 + 88 * q^91 + 36 * q^94 + 2 * q^96 - 34 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more