LMFDB - Embedded newform 425.4.a.i.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [425,4,Mod(1,425)] 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("425.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(425, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 425 = 5^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	425.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,-2,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$25.0758117524$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 20x^{3} + 38x^{2} + 69x - 126 $ x^5 - 2x^4 - 20x^3 + 38x^2 + 69x - 126
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 85)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$4.05155$ of defining polynomial
Character	$\chi$	$=$	425.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.290753 q^{2} +7.70584 q^{3} -7.91546 q^{4} -2.24049 q^{6} -22.4661 q^{7} +4.62746 q^{8} +32.3800 q^{9} +39.6109 q^{11} -60.9953 q^{12} +6.70527 q^{13} +6.53208 q^{14} +61.9783 q^{16} -17.0000 q^{17} -9.41457 q^{18} +42.9490 q^{19} -173.120 q^{21} -11.5170 q^{22} +115.726 q^{23} +35.6585 q^{24} -1.94958 q^{26} +41.4574 q^{27} +177.830 q^{28} +184.009 q^{29} +201.848 q^{31} -55.0401 q^{32} +305.235 q^{33} +4.94280 q^{34} -256.303 q^{36} +189.885 q^{37} -12.4876 q^{38} +51.6697 q^{39} +297.987 q^{41} +50.3352 q^{42} -428.676 q^{43} -313.539 q^{44} -33.6476 q^{46} +311.134 q^{47} +477.595 q^{48} +161.725 q^{49} -130.999 q^{51} -53.0753 q^{52} +307.329 q^{53} -12.0538 q^{54} -103.961 q^{56} +330.959 q^{57} -53.5011 q^{58} +704.985 q^{59} -929.140 q^{61} -58.6879 q^{62} -727.452 q^{63} -479.823 q^{64} -88.7480 q^{66} -587.978 q^{67} +134.563 q^{68} +891.764 q^{69} +507.339 q^{71} +149.837 q^{72} +13.3237 q^{73} -55.2094 q^{74} -339.962 q^{76} -889.903 q^{77} -15.0231 q^{78} -143.573 q^{79} -554.796 q^{81} -86.6407 q^{82} -1017.25 q^{83} +1370.33 q^{84} +124.639 q^{86} +1417.94 q^{87} +183.298 q^{88} +772.337 q^{89} -150.641 q^{91} -916.023 q^{92} +1555.41 q^{93} -90.4630 q^{94} -424.130 q^{96} +1700.56 q^{97} -47.0221 q^{98} +1282.60 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 2 q^{2} + q^{3} + 34 q^{4} - 5 q^{6} - 10 q^{7} - 30 q^{8} - 30 q^{9} + 126 q^{11} - 15 q^{12} - 83 q^{13} + 90 q^{14} + 322 q^{16} - 85 q^{17} + 97 q^{18} + 55 q^{19} + 6 q^{21} + 240 q^{22} + 2 q^{23}+ \cdots - 858 q^{99}+O(q^{100}) $ 5 * q - 2 * q^2 + q^3 + 34 * q^4 - 5 * q^6 - 10 * q^7 - 30 * q^8 - 30 * q^9 + 126 * q^11 - 15 * q^12 - 83 * q^13 + 90 * q^14 + 322 * q^16 - 85 * q^17 + 97 * q^18 + 55 * q^19 + 6 * q^21 + 240 * q^22 + 2 * q^23 + 155 * q^24 - 395 * q^26 + 163 * q^27 + 1062 * q^28 + 195 * q^29 + 97 * q^31 - 182 * q^32 + 352 * q^33 + 34 * q^34 - 1437 * q^36 + 476 * q^37 + 149 * q^38 + 373 * q^39 + 298 * q^41 + 18 * q^42 - 168 * q^43 + 1136 * q^44 - 2106 * q^46 - 1095 * q^47 + 1193 * q^48 + 737 * q^49 - 17 * q^51 - 281 * q^52 + 1035 * q^53 - 291 * q^54 - 350 * q^56 + 15 * q^57 + 2199 * q^58 + 1163 * q^59 - 351 * q^61 + 1241 * q^62 - 1584 * q^63 + 2042 * q^64 + 356 * q^66 - 1064 * q^67 - 578 * q^68 + 1298 * q^69 + 1393 * q^71 + 269 * q^72 + 163 * q^73 + 1530 * q^74 - 393 * q^76 + 732 * q^77 + 67 * q^78 + 750 * q^79 + 85 * q^81 - 1144 * q^82 - 270 * q^83 + 2422 * q^84 + 3502 * q^86 + 907 * q^87 + 3216 * q^88 + 1179 * q^89 + 282 * q^91 - 758 * q^92 + 3923 * q^93 + 3473 * q^94 - 2045 * q^96 + 2745 * q^97 - 502 * q^98 - 858 * 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 $

$2$	−0.290753	−0.102797	−0.0513983	−	0.998678i	$-0.516368\pi$
$2$	−0.290753	−0.102797	−0.0513983	+	0.998678i	$0.516368\pi$
$3$	7.70584	1.48299	0.741495	−	0.670958i	$-0.234117\pi$
$3$	7.70584	1.48299	0.741495	+	0.670958i	$0.234117\pi$
$4$	−7.91546	−0.989433
$5$	0	0
$5$	0	0
$6$	−2.24049	−0.152446
$7$	−22.4661	−1.21306	−0.606528	−	0.795062i	$-0.707438\pi$
$7$	−22.4661	−1.21306	−0.606528	+	0.795062i	$0.707438\pi$
$8$	4.62746	0.204507
$9$	32.3800	1.19926
$10$	0	0
$11$	39.6109	1.08574	0.542870	−	0.839817i	$-0.317338\pi$
$11$	39.6109	1.08574	0.542870	+	0.839817i	$0.317338\pi$
$12$	−60.9953	−1.46732
$13$	6.70527	0.143054	0.0715272	−	0.997439i	$-0.477213\pi$
$13$	6.70527	0.143054	0.0715272	+	0.997439i	$0.477213\pi$
$14$	6.53208	0.124698
$15$	0	0
$16$	61.9783	0.968410
$17$	−17.0000	−0.242536
$17$	−17.0000	−0.242536
$18$	−9.41457	−0.123280
$19$	42.9490	0.518589	0.259294	−	0.965798i	$-0.416510\pi$
$19$	42.9490	0.518589	0.259294	+	0.965798i	$0.416510\pi$
$20$	0	0
$21$	−173.120	−1.79895
$22$	−11.5170	−0.111610
$23$	115.726	1.04915	0.524576	−	0.851364i	$-0.324224\pi$
$23$	115.726	1.04915	0.524576	+	0.851364i	$0.324224\pi$
$24$	35.6585	0.303282
$25$	0	0
$26$	−1.94958	−0.0147055
$27$	41.4574	0.295499
$28$	177.830	1.20024
$29$	184.009	1.17826	0.589131	−	0.808037i	$-0.299470\pi$
$29$	184.009	1.17826	0.589131	+	0.808037i	$0.299470\pi$
$30$	0	0
$31$	201.848	1.16945	0.584726	−	0.811231i	$-0.301202\pi$
$31$	201.848	1.16945	0.584726	+	0.811231i	$0.301202\pi$
$32$	−55.0401	−0.304056
$33$	305.235	1.61014
$34$	4.94280	0.0249318
$35$	0	0
$36$	−256.303	−1.18659
$37$	189.885	0.843698	0.421849	−	0.906666i	$-0.361381\pi$
$37$	189.885	0.843698	0.421849	+	0.906666i	$0.361381\pi$
$38$	−12.4876	−0.0533092
$39$	51.6697	0.212148
$40$	0	0
$41$	297.987	1.13507	0.567534	−	0.823350i	$-0.307897\pi$
$41$	297.987	1.13507	0.567534	+	0.823350i	$0.307897\pi$
$42$	50.3352	0.184926
$43$	−428.676	−1.52029	−0.760144	−	0.649754i	$-0.774872\pi$
$43$	−428.676	−1.52029	−0.760144	+	0.649754i	$0.774872\pi$
$44$	−313.539	−1.07427
$45$	0	0
$46$	−33.6476	−0.107849
$47$	311.134	0.965607	0.482803	−	0.875729i	$-0.339619\pi$
$47$	311.134	0.965607	0.482803	+	0.875729i	$0.339619\pi$
$48$	477.595	1.43614
$49$	161.725	0.471503
$50$	0	0
$51$	−130.999	−0.359678
$52$	−53.0753	−0.141543
$53$	307.329	0.796507	0.398253	−	0.917275i	$-0.369617\pi$
$53$	307.329	0.796507	0.398253	+	0.917275i	$0.369617\pi$
$54$	−12.0538	−0.0303763
$55$	0	0
$56$	−103.961	−0.248078
$57$	330.959	0.769062
$58$	−53.5011	−0.121121
$59$	704.985	1.55561	0.777807	−	0.628503i	$-0.216332\pi$
$59$	704.985	1.55561	0.777807	+	0.628503i	$0.216332\pi$
$60$	0	0
$61$	−929.140	−1.95023	−0.975117	−	0.221693i	$-0.928842\pi$
$61$	−929.140	−1.95023	−0.975117	+	0.221693i	$0.928842\pi$
$62$	−58.6879	−0.120216
$63$	−727.452	−1.45477
$64$	−479.823	−0.937154
$65$	0	0
$66$	−88.7480	−0.165517
$67$	−587.978	−1.07213	−0.536067	−	0.844175i	$-0.680091\pi$
$67$	−587.978	−1.07213	−0.536067	+	0.844175i	$0.680091\pi$
$68$	134.563	0.239973
$69$	891.764	1.55588
$70$	0	0
$71$	507.339	0.848030	0.424015	−	0.905655i	$-0.360620\pi$
$71$	507.339	0.848030	0.424015	+	0.905655i	$0.360620\pi$
$72$	149.837	0.245257
$73$	13.3237	0.0213619	0.0106810	−	0.999943i	$-0.496600\pi$
$73$	13.3237	0.0213619	0.0106810	+	0.999943i	$0.496600\pi$
$74$	−55.2094	−0.0867293
$75$	0	0
$76$	−339.962	−0.513109
$77$	−889.903	−1.31706
$78$	−15.0231	−0.0218081
$79$	−143.573	−0.204472	−0.102236	−	0.994760i	$-0.532600\pi$
$79$	−143.573	−0.204472	−0.102236	+	0.994760i	$0.532600\pi$
$80$	0	0
$81$	−554.796	−0.761037
$82$	−86.6407	−0.116681
$83$	−1017.25	−1.34528	−0.672638	−	0.739972i	$-0.734839\pi$
$83$	−1017.25	−1.34528	−0.672638	+	0.739972i	$0.734839\pi$
$84$	1370.33	1.77994
$85$	0	0
$86$	124.639	0.156280
$87$	1417.94	1.74735
$88$	183.298	0.222041
$89$	772.337	0.919860	0.459930	−	0.887955i	$-0.347874\pi$
$89$	772.337	0.919860	0.459930	+	0.887955i	$0.347874\pi$
$90$	0	0
$91$	−150.641	−0.173533
$92$	−916.023	−1.03806
$93$	1555.41	1.73429
$94$	−90.4630	−0.0992611
$95$	0	0
$96$	−424.130	−0.450912
$97$	1700.56	1.78006	0.890031	−	0.455899i	$-0.150682\pi$
$97$	1700.56	1.78006	0.890031	+	0.455899i	$0.150682\pi$
$98$	−47.0221	−0.0484689
$99$	1282.60	1.30208
$100$	0	0
$101$	721.370	0.710683	0.355342	−	0.934736i	$-0.384365\pi$
$101$	721.370	0.710683	0.355342	+	0.934736i	$0.384365\pi$
$102$	38.0884	0.0369737
$103$	95.7866	0.0916324	0.0458162	−	0.998950i	$-0.485411\pi$
$103$	95.7866	0.0916324	0.0458162	+	0.998950i	$0.485411\pi$
$104$	31.0284	0.0292556
$105$	0	0
$106$	−89.3567	−0.0818782
$107$	−361.596	−0.326699	−0.163349	−	0.986568i	$-0.552230\pi$
$107$	−361.596	−0.326699	−0.163349	+	0.986568i	$0.552230\pi$
$108$	−328.154	−0.292377
$109$	−1717.63	−1.50935	−0.754676	−	0.656098i	$-0.772206\pi$
$109$	−1717.63	−1.50935	−0.754676	+	0.656098i	$0.772206\pi$
$110$	0	0
$111$	1463.22	1.25120
$112$	−1392.41	−1.17473
$113$	1863.75	1.55156	0.775782	−	0.631001i	$-0.217355\pi$
$113$	1863.75	1.55156	0.775782	+	0.631001i	$0.217355\pi$
$114$	−96.2271	−0.0790570
$115$	0	0
$116$	−1456.52	−1.16581
$117$	217.117	0.171559
$118$	−204.976	−0.159912
$119$	381.924	0.294209
$120$	0	0
$121$	238.024	0.178831
$122$	270.150	0.200477
$123$	2296.24	1.68330
$124$	−1597.72	−1.15709
$125$	0	0
$126$	211.509	0.149545
$127$	1431.63	1.00029	0.500143	−	0.865943i	$-0.333281\pi$
$127$	1431.63	1.00029	0.500143	+	0.865943i	$0.333281\pi$
$128$	579.830	0.400393
$129$	−3303.31	−2.25457
$130$	0	0
$131$	199.420	0.133003	0.0665016	−	0.997786i	$-0.478816\pi$
$131$	199.420	0.133003	0.0665016	+	0.997786i	$0.478816\pi$
$132$	−2416.08	−1.59313
$133$	−964.897	−0.629077
$134$	170.956	0.110212
$135$	0	0
$136$	−78.6669	−0.0496002
$137$	−1625.54	−1.01372	−0.506860	−	0.862029i	$-0.669194\pi$
$137$	−1625.54	−1.01372	−0.506860	+	0.862029i	$0.669194\pi$
$138$	−259.283	−0.159939
$139$	−832.842	−0.508207	−0.254103	−	0.967177i	$-0.581780\pi$
$139$	−832.842	−0.508207	−0.254103	+	0.967177i	$0.581780\pi$
$140$	0	0
$141$	2397.55	1.43198
$142$	−147.510	−0.0871746
$143$	265.602	0.155320
$144$	2006.86	1.16137
$145$	0	0
$146$	−3.87390	−0.00219593
$147$	1246.23	0.699234
$148$	−1503.02	−0.834783
$149$	891.789	0.490324	0.245162	−	0.969482i	$-0.421159\pi$
$149$	891.789	0.490324	0.245162	+	0.969482i	$0.421159\pi$
$150$	0	0
$151$	1764.98	0.951208	0.475604	−	0.879660i	$-0.342230\pi$
$151$	1764.98	0.951208	0.475604	+	0.879660i	$0.342230\pi$
$152$	198.745	0.106055
$153$	−550.460	−0.290863
$154$	258.742	0.135390
$155$	0	0
$156$	−408.990	−0.209906
$157$	−2692.39	−1.36864	−0.684318	−	0.729184i	$-0.739900\pi$
$157$	−2692.39	−1.36864	−0.684318	+	0.729184i	$0.739900\pi$
$158$	41.7443	0.0210190
$159$	2368.23	1.18121
$160$	0	0
$161$	−2599.90	−1.27268
$162$	161.308	0.0782320
$163$	1704.29	0.818959	0.409479	−	0.912319i	$-0.365710\pi$
$163$	1704.29	0.818959	0.409479	+	0.912319i	$0.365710\pi$
$164$	−2358.71	−1.12307
$165$	0	0
$166$	295.769	0.138290
$167$	2228.69	1.03270	0.516351	−	0.856377i	$-0.327290\pi$
$167$	2228.69	1.03270	0.516351	+	0.856377i	$0.327290\pi$
$168$	−801.107	−0.367897
$169$	−2152.04	−0.979535
$170$	0	0
$171$	1390.69	0.621922
$172$	3393.17	1.50422
$173$	−2754.28	−1.21043	−0.605213	−	0.796064i	$-0.706912\pi$
$173$	−2754.28	−1.21043	−0.605213	+	0.796064i	$0.706912\pi$
$174$	−412.271	−0.179622
$175$	0	0
$176$	2455.02	1.05144
$177$	5432.50	2.30696
$178$	−224.559	−0.0945585
$179$	−2107.61	−0.880055	−0.440028	−	0.897984i	$-0.645031\pi$
$179$	−2107.61	−0.880055	−0.440028	+	0.897984i	$0.645031\pi$
$180$	0	0
$181$	3481.93	1.42989	0.714944	−	0.699181i	$-0.246452\pi$
$181$	3481.93	1.42989	0.714944	+	0.699181i	$0.246452\pi$
$182$	43.7993	0.0178386
$183$	−7159.81	−2.89218
$184$	535.517	0.214559
$185$	0	0
$186$	−452.240	−0.178279
$187$	−673.386	−0.263331
$188$	−2462.77	−0.955403
$189$	−931.386	−0.358457
$190$	0	0
$191$	−455.488	−0.172555	−0.0862774	−	0.996271i	$-0.527497\pi$
$191$	−455.488	−0.172555	−0.0862774	+	0.996271i	$0.527497\pi$
$192$	−3697.44	−1.38979
$193$	−1120.95	−0.418073	−0.209036	−	0.977908i	$-0.567033\pi$
$193$	−1120.95	−0.418073	−0.209036	+	0.977908i	$0.567033\pi$
$194$	−494.443	−0.182984
$195$	0	0
$196$	−1280.13	−0.466520
$197$	−4226.11	−1.52842	−0.764208	−	0.644970i	$-0.776870\pi$
$197$	−4226.11	−1.52842	−0.764208	+	0.644970i	$0.776870\pi$
$198$	−372.920	−0.133850
$199$	−3705.46	−1.31996	−0.659982	−	0.751281i	$-0.729436\pi$
$199$	−3705.46	−1.31996	−0.659982	+	0.751281i	$0.729436\pi$
$200$	0	0
$201$	−4530.87	−1.58996
$202$	−209.740	−0.0730558
$203$	−4133.96	−1.42930
$204$	1036.92	0.355877
$205$	0	0
$206$	−27.8502	−0.00941950
$207$	3747.20	1.25820
$208$	415.581	0.138535
$209$	1701.25	0.563053
$210$	0	0
$211$	224.116	0.0731222	0.0365611	−	0.999331i	$-0.488360\pi$
$211$	224.116	0.0731222	0.0365611	+	0.999331i	$0.488360\pi$
$212$	−2432.65	−0.788090
$213$	3909.48	1.25762
$214$	105.135	0.0335835
$215$	0	0
$216$	191.843	0.0604316
$217$	−4534.74	−1.41861
$218$	499.406	0.155156
$219$	102.670	0.0316795
$220$	0	0
$221$	−113.990	−0.0346958
$222$	−425.435	−0.128619
$223$	883.204	0.265218	0.132609	−	0.991168i	$-0.457664\pi$
$223$	883.204	0.265218	0.132609	+	0.991168i	$0.457664\pi$
$224$	1236.54	0.368837
$225$	0	0
$226$	−541.890	−0.159496
$227$	−2795.44	−0.817357	−0.408678	−	0.912678i	$-0.634010\pi$
$227$	−2795.44	−0.817357	−0.408678	+	0.912678i	$0.634010\pi$
$228$	−2619.69	−0.760935
$229$	−5951.09	−1.71729	−0.858644	−	0.512572i	$-0.828693\pi$
$229$	−5951.09	−1.71729	−0.858644	+	0.512572i	$0.828693\pi$
$230$	0	0
$231$	−6857.45	−1.95319
$232$	851.495	0.240963
$233$	1995.41	0.561047	0.280523	−	0.959847i	$-0.409492\pi$
$233$	1995.41	0.561047	0.280523	+	0.959847i	$0.409492\pi$
$234$	−63.1272	−0.0176357
$235$	0	0
$236$	−5580.28	−1.53918
$237$	−1106.35	−0.303229
$238$	−111.045	−0.0302437
$239$	−3216.85	−0.870630	−0.435315	−	0.900278i	$-0.643363\pi$
$239$	−3216.85	−0.870630	−0.435315	+	0.900278i	$0.643363\pi$
$240$	0	0
$241$	3958.14	1.05795	0.528976	−	0.848637i	$-0.322576\pi$
$241$	3958.14	1.05795	0.528976	+	0.848637i	$0.322576\pi$
$242$	−69.2062	−0.0183832
$243$	−5394.52	−1.42411
$244$	7354.58	1.92962
$245$	0	0
$246$	−667.639	−0.173037
$247$	287.985	0.0741864
$248$	934.045	0.239161
$249$	−7838.78	−1.99503
$250$	0	0
$251$	−7420.61	−1.86608	−0.933038	−	0.359779i	$-0.882852\pi$
$251$	−7420.61	−1.86608	−0.933038	+	0.359779i	$0.882852\pi$
$252$	5758.12	1.43939
$253$	4584.00	1.13911
$254$	−416.249	−0.102826
$255$	0	0
$256$	3670.00	0.895995
$257$	262.594	0.0637360	0.0318680	−	0.999492i	$-0.489854\pi$
$257$	262.594	0.0637360	0.0318680	+	0.999492i	$0.489854\pi$
$258$	960.445	0.231762
$259$	−4265.96	−1.02345
$260$	0	0
$261$	5958.21	1.41304
$262$	−57.9819	−0.0136723
$263$	4074.12	0.955213	0.477606	−	0.878574i	$-0.341504\pi$
$263$	4074.12	0.955213	0.477606	+	0.878574i	$0.341504\pi$
$264$	1412.47	0.329285
$265$	0	0
$266$	280.547	0.0646670
$267$	5951.51	1.36414
$268$	4654.12	1.06080
$269$	5030.54	1.14021	0.570107	−	0.821570i	$-0.306902\pi$
$269$	5030.54	1.14021	0.570107	+	0.821570i	$0.306902\pi$
$270$	0	0
$271$	4277.34	0.958783	0.479392	−	0.877601i	$-0.340857\pi$
$271$	4277.34	0.958783	0.479392	+	0.877601i	$0.340857\pi$
$272$	−1053.63	−0.234874
$273$	−1160.82	−0.257347
$274$	472.631	0.104207
$275$	0	0
$276$	−7058.72	−1.53944
$277$	6194.21	1.34359	0.671795	−	0.740737i	$-0.265524\pi$
$277$	6194.21	1.34359	0.671795	+	0.740737i	$0.265524\pi$
$278$	242.151	0.0522419
$279$	6535.84	1.40248
$280$	0	0
$281$	8161.19	1.73258	0.866291	−	0.499539i	$-0.166497\pi$
$281$	8161.19	1.73258	0.866291	+	0.499539i	$0.166497\pi$
$282$	−697.093	−0.147203
$283$	−7482.95	−1.57178	−0.785892	−	0.618363i	$-0.787796\pi$
$283$	−7482.95	−1.57178	−0.785892	+	0.618363i	$0.787796\pi$
$284$	−4015.83	−0.839069
$285$	0	0
$286$	−77.2245	−0.0159664
$287$	−6694.62	−1.37690
$288$	−1782.20	−0.364642
$289$	289.000	0.0588235
$290$	0	0
$291$	13104.3	2.63981
$292$	−105.463	−0.0211362
$293$	162.179	0.0323365	0.0161682	−	0.999869i	$-0.494853\pi$
$293$	162.179	0.0323365	0.0161682	+	0.999869i	$0.494853\pi$
$294$	−362.345	−0.0718789
$295$	0	0
$296$	878.684	0.172542
$297$	1642.17	0.320835
$298$	−259.290	−0.0504036
$299$	775.972	0.150086
$300$	0	0
$301$	9630.67	1.84419
$302$	−513.174	−0.0977809
$303$	5558.76	1.05394
$304$	2661.91	0.502207
$305$	0	0
$306$	160.048	0.0298997
$307$	−6507.52	−1.20978	−0.604892	−	0.796308i	$-0.706784\pi$
$307$	−6507.52	−1.20978	−0.604892	+	0.796308i	$0.706784\pi$
$308$	7043.99	1.30314
$309$	738.116	0.135890
$310$	0	0
$311$	−492.092	−0.0897234	−0.0448617	−	0.998993i	$-0.514285\pi$
$311$	−492.092	−0.0897234	−0.0448617	+	0.998993i	$0.514285\pi$
$312$	239.100	0.0433858
$313$	7298.76	1.31805	0.659026	−	0.752120i	$-0.270969\pi$
$313$	7298.76	1.31805	0.659026	+	0.752120i	$0.270969\pi$
$314$	782.819	0.140691
$315$	0	0
$316$	1136.45	0.202311
$317$	7469.73	1.32348	0.661738	−	0.749735i	$-0.269819\pi$
$317$	7469.73	1.32348	0.661738	+	0.749735i	$0.269819\pi$
$318$	−688.568	−0.121424
$319$	7288.76	1.27929
$320$	0	0
$321$	−2786.40	−0.484491
$322$	755.929	0.130827
$323$	−730.134	−0.125776
$324$	4391.47	0.752995
$325$	0	0
$326$	−495.527	−0.0841862
$327$	−13235.8	−2.23835
$328$	1378.93	0.232129
$329$	−6989.96	−1.17133
$330$	0	0
$331$	−8196.55	−1.36110	−0.680548	−	0.732703i	$-0.738258\pi$
$331$	−8196.55	−1.36110	−0.680548	+	0.732703i	$0.738258\pi$
$332$	8052.02	1.33106
$333$	6148.46	1.01181
$334$	−647.998	−0.106158
$335$	0	0
$336$	−10729.7	−1.74212
$337$	−4439.22	−0.717565	−0.358783	−	0.933421i	$-0.616808\pi$
$337$	−4439.22	−0.717565	−0.358783	+	0.933421i	$0.616808\pi$
$338$	625.711	0.100693
$339$	14361.8	2.30095
$340$	0	0
$341$	7995.39	1.26972
$342$	−404.347	−0.0639315
$343$	4072.53	0.641096
$344$	−1983.68	−0.310910
$345$	0	0
$346$	800.813	0.124428
$347$	8604.97	1.33124	0.665618	−	0.746292i	$-0.268168\pi$
$347$	8604.97	1.33124	0.665618	+	0.746292i	$0.268168\pi$
$348$	−11223.7	−1.72889
$349$	2072.68	0.317903	0.158951	−	0.987286i	$-0.449189\pi$
$349$	2072.68	0.317903	0.158951	+	0.987286i	$0.449189\pi$
$350$	0	0
$351$	277.983	0.0422725
$352$	−2180.19	−0.330126
$353$	970.542	0.146336	0.0731682	−	0.997320i	$-0.476689\pi$
$353$	970.542	0.146336	0.0731682	+	0.997320i	$0.476689\pi$
$354$	−1579.51	−0.237148
$355$	0	0
$356$	−6113.41	−0.910140
$357$	2943.04	0.436309
$358$	612.792	0.0904667
$359$	535.411	0.0787129	0.0393564	−	0.999225i	$-0.487469\pi$
$359$	535.411	0.0787129	0.0393564	+	0.999225i	$0.487469\pi$
$360$	0	0
$361$	−5014.38	−0.731066
$362$	−1012.38	−0.146988
$363$	1834.18	0.265205
$364$	1192.39	0.171699
$365$	0	0
$366$	2081.73	0.297306
$367$	5821.92	0.828070	0.414035	−	0.910261i	$-0.364119\pi$
$367$	5821.92	0.828070	0.414035	+	0.910261i	$0.364119\pi$
$368$	7172.48	1.01601
$369$	9648.83	1.36124
$370$	0	0
$371$	−6904.48	−0.966206
$372$	−12311.8	−1.71596
$373$	6328.48	0.878488	0.439244	−	0.898368i	$-0.355246\pi$
$373$	6328.48	0.878488	0.439244	+	0.898368i	$0.355246\pi$
$374$	195.789	0.0270695
$375$	0	0
$376$	1439.76	0.197473
$377$	1233.83	0.168556
$378$	270.803	0.0368481
$379$	−1168.28	−0.158339	−0.0791695	−	0.996861i	$-0.525227\pi$
$379$	−1168.28	−0.158339	−0.0791695	+	0.996861i	$0.525227\pi$
$380$	0	0
$381$	11031.9	1.48341
$382$	132.434	0.0177380
$383$	−6305.10	−0.841189	−0.420595	−	0.907249i	$-0.638179\pi$
$383$	−6305.10	−0.841189	−0.420595	+	0.907249i	$0.638179\pi$
$384$	4468.08	0.593778
$385$	0	0
$386$	325.920	0.0429764
$387$	−13880.5	−1.82322
$388$	−13460.7	−1.76125
$389$	−68.6812	−0.00895186	−0.00447593	−	0.999990i	$-0.501425\pi$
$389$	−68.6812	−0.00895186	−0.00447593	+	0.999990i	$0.501425\pi$
$390$	0	0
$391$	−1967.34	−0.254457
$392$	748.379	0.0964256
$393$	1536.70	0.197242
$394$	1228.75	0.157116
$395$	0	0
$396$	−10152.4	−1.28832
$397$	5941.54	0.751127	0.375563	−	0.926797i	$-0.377449\pi$
$397$	5941.54	0.751127	0.375563	+	0.926797i	$0.377449\pi$
$398$	1077.37	0.135688
$399$	−7435.35	−0.932914
$400$	0	0
$401$	−12044.6	−1.49994	−0.749971	−	0.661471i	$-0.769932\pi$
$401$	−12044.6	−1.49994	−0.749971	+	0.661471i	$0.769932\pi$
$402$	1317.36	0.163443
$403$	1353.45	0.167295
$404$	−5709.98	−0.703173
$405$	0	0
$406$	1201.96	0.146927
$407$	7521.50	0.916037
$408$	−606.195	−0.0735566
$409$	−7908.90	−0.956161	−0.478080	−	0.878316i	$-0.658667\pi$
$409$	−7908.90	−0.956161	−0.478080	+	0.878316i	$0.658667\pi$
$410$	0	0
$411$	−12526.2	−1.50334
$412$	−758.195	−0.0906641
$413$	−15838.3	−1.88705
$414$	−1089.51	−0.129339
$415$	0	0
$416$	−369.058	−0.0434966
$417$	−6417.75	−0.753665
$418$	−494.643	−0.0578799
$419$	6026.48	0.702656	0.351328	−	0.936252i	$-0.385730\pi$
$419$	6026.48	0.702656	0.351328	+	0.936252i	$0.385730\pi$
$420$	0	0
$421$	−12244.9	−1.41753	−0.708765	−	0.705445i	$-0.750747\pi$
$421$	−12244.9	−1.41753	−0.708765	+	0.705445i	$0.750747\pi$
$422$	−65.1624	−0.00751672
$423$	10074.5	1.15801
$424$	1422.15	0.162891
$425$	0	0
$426$	−1136.69	−0.129279
$427$	20874.2	2.36574
$428$	2862.20	0.323246
$429$	2046.69	0.230338
$430$	0	0
$431$	−14325.4	−1.60100	−0.800498	−	0.599336i	$-0.795431\pi$
$431$	−14325.4	−1.60100	−0.800498	+	0.599336i	$0.795431\pi$
$432$	2569.46	0.286164
$433$	−4880.86	−0.541707	−0.270853	−	0.962621i	$-0.587306\pi$
$433$	−4880.86	−0.541707	−0.270853	+	0.962621i	$0.587306\pi$
$434$	1318.49	0.145828
$435$	0	0
$436$	13595.9	1.49340
$437$	4970.31	0.544078
$438$	−29.8516	−0.00325654
$439$	12177.0	1.32387	0.661933	−	0.749563i	$-0.269736\pi$
$439$	12177.0	1.32387	0.661933	+	0.749563i	$0.269736\pi$
$440$	0	0
$441$	5236.67	0.565454
$442$	33.1428	0.00356661
$443$	7619.52	0.817188	0.408594	−	0.912716i	$-0.366019\pi$
$443$	7619.52	0.817188	0.408594	+	0.912716i	$0.366019\pi$
$444$	−11582.1	−1.23797
$445$	0	0
$446$	−256.794	−0.0272636
$447$	6871.99	0.727145
$448$	10779.7	1.13682
$449$	−7330.68	−0.770504	−0.385252	−	0.922811i	$-0.625885\pi$
$449$	−7330.68	−0.770504	−0.385252	+	0.922811i	$0.625885\pi$
$450$	0	0
$451$	11803.6	1.23239
$452$	−14752.4	−1.53517
$453$	13600.7	1.41063
$454$	812.782	0.0840215
$455$	0	0
$456$	1531.50	0.157279
$457$	5644.06	0.577720	0.288860	−	0.957371i	$-0.406724\pi$
$457$	5644.06	0.577720	0.288860	+	0.957371i	$0.406724\pi$
$458$	1730.30	0.176531
$459$	−704.776	−0.0716691
$460$	0	0
$461$	−273.212	−0.0276025	−0.0138013	−	0.999905i	$-0.504393\pi$
$461$	−273.212	−0.0276025	−0.0138013	+	0.999905i	$0.504393\pi$
$462$	1993.82	0.200781
$463$	2215.54	0.222387	0.111193	−	0.993799i	$-0.464533\pi$
$463$	2215.54	0.222387	0.111193	+	0.993799i	$0.464533\pi$
$464$	11404.6	1.14104
$465$	0	0
$466$	−580.172	−0.0576737
$467$	−12288.4	−1.21764	−0.608821	−	0.793308i	$-0.708357\pi$
$467$	−12288.4	−1.21764	−0.608821	+	0.793308i	$0.708357\pi$
$468$	−1718.58	−0.169746
$469$	13209.6	1.30056
$470$	0	0
$471$	−20747.1	−2.02967
$472$	3262.29	0.318134
$473$	−16980.2	−1.65064
$474$	321.675	0.0311709
$475$	0	0
$476$	−3023.10	−0.291100
$477$	9951.30	0.955218
$478$	935.308	0.0894979
$479$	849.329	0.0810164	0.0405082	−	0.999179i	$-0.487102\pi$
$479$	849.329	0.0810164	0.0405082	+	0.999179i	$0.487102\pi$
$480$	0	0
$481$	1273.23	0.120695
$482$	−1150.84	−0.108754
$483$	−20034.5	−1.88737
$484$	−1884.07	−0.176941
$485$	0	0
$486$	1568.47	0.146394
$487$	11460.6	1.06639	0.533194	−	0.845993i	$-0.320992\pi$
$487$	11460.6	1.06639	0.533194	+	0.845993i	$0.320992\pi$
$488$	−4299.56	−0.398836
$489$	13133.0	1.21451
$490$	0	0
$491$	8975.92	0.825006	0.412503	−	0.910956i	$-0.364655\pi$
$491$	8975.92	0.825006	0.412503	+	0.910956i	$0.364655\pi$
$492$	−18175.8	−1.66551
$493$	−3128.15	−0.285771
$494$	−83.7324	−0.00762611
$495$	0	0
$496$	12510.2	1.13251
$497$	−11397.9	−1.02871
$498$	2279.15	0.205082
$499$	−17449.9	−1.56546	−0.782730	−	0.622362i	$-0.786173\pi$
$499$	−17449.9	−1.56546	−0.782730	+	0.622362i	$0.786173\pi$
$500$	0	0
$501$	17173.9	1.53149
$502$	2157.56	0.191826
$503$	9536.30	0.845334	0.422667	−	0.906285i	$-0.361094\pi$
$503$	9536.30	0.845334	0.422667	+	0.906285i	$0.361094\pi$
$504$	−3366.26	−0.297510
$505$	0	0
$506$	−1332.81	−0.117096
$507$	−16583.3	−1.45264
$508$	−11332.0	−0.989715
$509$	11124.1	0.968699	0.484349	−	0.874875i	$-0.339056\pi$
$509$	11124.1	0.968699	0.484349	+	0.874875i	$0.339056\pi$
$510$	0	0
$511$	−299.331	−0.0259132
$512$	−5705.70	−0.492498
$513$	1780.56	0.153243
$514$	−76.3499	−0.00655185
$515$	0	0
$516$	26147.2	2.23075
$517$	12324.3	1.04840
$518$	1240.34	0.105207
$519$	−21224.0	−1.79505
$520$	0	0
$521$	−4282.84	−0.360143	−0.180071	−	0.983654i	$-0.557633\pi$
$521$	−4282.84	−0.360143	−0.180071	+	0.983654i	$0.557633\pi$
$522$	−1732.37	−0.145256
$523$	−14815.1	−1.23866	−0.619332	−	0.785129i	$-0.712596\pi$
$523$	−14815.1	−1.23866	−0.619332	+	0.785129i	$0.712596\pi$
$524$	−1578.50	−0.131598
$525$	0	0
$526$	−1184.56	−0.0981927
$527$	−3431.42	−0.283634
$528$	18918.0	1.55928
$529$	1225.44	0.100718
$530$	0	0
$531$	22827.4	1.86558
$532$	7637.61	0.622429
$533$	1998.09	0.162377
$534$	−1730.42	−0.140229
$535$	0	0
$536$	−2720.85	−0.219259
$537$	−16240.9	−1.30511
$538$	−1462.64	−0.117210
$539$	6406.09	0.511929
$540$	0	0
$541$	−2526.68	−0.200795	−0.100398	−	0.994947i	$-0.532012\pi$
$541$	−2526.68	−0.200795	−0.100398	+	0.994947i	$0.532012\pi$
$542$	−1243.65	−0.0985596
$543$	26831.2	2.12051
$544$	935.681	0.0737445
$545$	0	0
$546$	337.511	0.0264544
$547$	−5034.13	−0.393499	−0.196749	−	0.980454i	$-0.563039\pi$
$547$	−5034.13	−0.393499	−0.196749	+	0.980454i	$0.563039\pi$
$548$	12866.9	1.00301
$549$	−30085.6	−2.33883
$550$	0	0
$551$	7903.01	0.611034
$552$	4126.61	0.318188
$553$	3225.53	0.248035
$554$	−1800.98	−0.138116
$555$	0	0
$556$	6592.33	0.502836
$557$	−177.914	−0.0135340	−0.00676702	−	0.999977i	$-0.502154\pi$
$557$	−177.914	−0.0135340	−0.00676702	+	0.999977i	$0.502154\pi$
$558$	−1900.31	−0.144170
$559$	−2874.38	−0.217484
$560$	0	0
$561$	−5189.00	−0.390517
$562$	−2372.89	−0.178104
$563$	−14626.4	−1.09490	−0.547451	−	0.836838i	$-0.684402\pi$
$563$	−14626.4	−1.09490	−0.547451	+	0.836838i	$0.684402\pi$
$564$	−18977.7	−1.41685
$565$	0	0
$566$	2175.69	0.161574
$567$	12464.1	0.923179
$568$	2347.70	0.173428
$569$	1106.84	0.0815485	0.0407742	−	0.999168i	$-0.487018\pi$
$569$	1106.84	0.0815485	0.0407742	+	0.999168i	$0.487018\pi$
$570$	0	0
$571$	17900.2	1.31191	0.655956	−	0.754800i	$-0.272266\pi$
$571$	17900.2	1.31191	0.655956	+	0.754800i	$0.272266\pi$
$572$	−2102.36	−0.153679
$573$	−3509.92	−0.255897
$574$	1946.48	0.141541
$575$	0	0
$576$	−15536.7	−1.12389
$577$	2108.65	0.152139	0.0760694	−	0.997103i	$-0.475763\pi$
$577$	2108.65	0.152139	0.0760694	+	0.997103i	$0.475763\pi$
$578$	−84.0275	−0.00604686
$579$	−8637.89	−0.619997
$580$	0	0
$581$	22853.7	1.63189
$582$	−3810.10	−0.271364
$583$	12173.6	0.864799
$584$	61.6549	0.00436866
$585$	0	0
$586$	−47.1539	−0.00332408
$587$	12504.8	0.879267	0.439634	−	0.898177i	$-0.355108\pi$
$587$	12504.8	0.879267	0.439634	+	0.898177i	$0.355108\pi$
$588$	−9864.49	−0.691845
$589$	8669.19	0.606465
$590$	0	0
$591$	−32565.7	−2.26662
$592$	11768.7	0.817046
$593$	−14161.6	−0.980686	−0.490343	−	0.871529i	$-0.663129\pi$
$593$	−14161.6	−0.980686	−0.490343	+	0.871529i	$0.663129\pi$
$594$	−477.464	−0.0329808
$595$	0	0
$596$	−7058.92	−0.485142
$597$	−28553.7	−1.95749
$598$	−225.616	−0.0154283
$599$	23580.2	1.60845	0.804225	−	0.594325i	$-0.202581\pi$
$599$	23580.2	1.60845	0.804225	+	0.594325i	$0.202581\pi$
$600$	0	0
$601$	7861.34	0.533562	0.266781	−	0.963757i	$-0.414040\pi$
$601$	7861.34	0.533562	0.266781	+	0.963757i	$0.414040\pi$
$602$	−2800.14	−0.189577
$603$	−19038.7	−1.28577
$604$	−13970.7	−0.941156
$605$	0	0
$606$	−1616.23	−0.108341
$607$	12847.1	0.859058	0.429529	−	0.903053i	$-0.358680\pi$
$607$	12847.1	0.859058	0.429529	+	0.903053i	$0.358680\pi$
$608$	−2363.92	−0.157680
$609$	−31855.7	−2.11963
$610$	0	0
$611$	2086.24	0.138134
$612$	4357.15	0.287789
$613$	−18323.0	−1.20728	−0.603638	−	0.797258i	$-0.706283\pi$
$613$	−18323.0	−1.20728	−0.603638	+	0.797258i	$0.706283\pi$
$614$	1892.08	0.124362
$615$	0	0
$616$	−4117.99	−0.269348
$617$	4105.27	0.267864	0.133932	−	0.990991i	$-0.457240\pi$
$617$	4105.27	0.267864	0.133932	+	0.990991i	$0.457240\pi$
$618$	−214.609	−0.0139690
$619$	−8193.31	−0.532014	−0.266007	−	0.963971i	$-0.585704\pi$
$619$	−8193.31	−0.532014	−0.266007	+	0.963971i	$0.585704\pi$
$620$	0	0
$621$	4797.69	0.310023
$622$	143.077	0.00922326
$623$	−17351.4	−1.11584
$624$	3202.40	0.205447
$625$	0	0
$626$	−2122.13	−0.135491
$627$	13109.6	0.835001
$628$	21311.5	1.35417
$629$	−3228.04	−0.204627
$630$	0	0
$631$	11714.4	0.739052	0.369526	−	0.929220i	$-0.379520\pi$
$631$	11714.4	0.739052	0.369526	+	0.929220i	$0.379520\pi$
$632$	−664.380	−0.0418158
$633$	1727.00	0.108440
$634$	−2171.85	−0.136049
$635$	0	0
$636$	−18745.6	−1.16873
$637$	1084.41	0.0674505
$638$	−2119.23	−0.131506
$639$	16427.7	1.01701
$640$	0	0
$641$	911.726	0.0561794	0.0280897	−	0.999605i	$-0.491058\pi$
$641$	911.726	0.0561794	0.0280897	+	0.999605i	$0.491058\pi$
$642$	810.153	0.0498040
$643$	12522.8	0.768040	0.384020	−	0.923325i	$-0.374539\pi$
$643$	12522.8	0.768040	0.384020	+	0.923325i	$0.374539\pi$
$644$	20579.4	1.25923
$645$	0	0
$646$	212.288	0.0129294
$647$	3887.31	0.236207	0.118104	−	0.993001i	$-0.462319\pi$
$647$	3887.31	0.236207	0.118104	+	0.993001i	$0.462319\pi$
$648$	−2567.30	−0.155637
$649$	27925.1	1.68899
$650$	0	0
$651$	−34944.0	−2.10378
$652$	−13490.2	−0.810305
$653$	1025.73	0.0614698	0.0307349	−	0.999528i	$-0.490215\pi$
$653$	1025.73	0.0614698	0.0307349	+	0.999528i	$0.490215\pi$
$654$	3848.35	0.230095
$655$	0	0
$656$	18468.7	1.09921
$657$	431.421	0.0256185
$658$	2032.35	0.120409
$659$	−18150.7	−1.07291	−0.536456	−	0.843928i	$-0.680237\pi$
$659$	−18150.7	−1.07291	−0.536456	+	0.843928i	$0.680237\pi$
$660$	0	0
$661$	−20665.5	−1.21603	−0.608015	−	0.793926i	$-0.708034\pi$
$661$	−20665.5	−1.21603	−0.608015	+	0.793926i	$0.708034\pi$
$662$	2383.17	0.139916
$663$	−878.386	−0.0514535
$664$	−4707.30	−0.275118
$665$	0	0
$666$	−1787.68	−0.104011
$667$	21294.6	1.23618
$668$	−17641.1	−1.02179
$669$	6805.83	0.393316
$670$	0	0
$671$	−36804.1	−2.11745
$672$	9528.54	0.546982
$673$	−14571.3	−0.834597	−0.417298	−	0.908770i	$-0.637023\pi$
$673$	−14571.3	−0.834597	−0.417298	+	0.908770i	$0.637023\pi$
$674$	1290.71	0.0737633
$675$	0	0
$676$	17034.4	0.969185
$677$	−18919.4	−1.07405	−0.537026	−	0.843566i	$-0.680452\pi$
$677$	−18919.4	−1.07405	−0.537026	+	0.843566i	$0.680452\pi$
$678$	−4175.72	−0.236530
$679$	−38205.0	−2.15931
$680$	0	0
$681$	−21541.2	−1.21213
$682$	−2324.68	−0.130523
$683$	4059.13	0.227406	0.113703	−	0.993515i	$-0.463729\pi$
$683$	4059.13	0.227406	0.113703	+	0.993515i	$0.463729\pi$
$684$	−11008.0	−0.615350
$685$	0	0
$686$	−1184.10	−0.0659025
$687$	−45858.2	−2.54672
$688$	−26568.6	−1.47226
$689$	2060.72	0.113944
$690$	0	0
$691$	702.880	0.0386958	0.0193479	−	0.999813i	$-0.493841\pi$
$691$	702.880	0.0386958	0.0193479	+	0.999813i	$0.493841\pi$
$692$	21801.4	1.19764
$693$	−28815.0	−1.57950
$694$	−2501.92	−0.136847
$695$	0	0
$696$	6561.49	0.357345
$697$	−5065.79	−0.275295
$698$	−602.638	−0.0326793
$699$	15376.3	0.832027
$700$	0	0
$701$	17518.0	0.943860	0.471930	−	0.881636i	$-0.343558\pi$
$701$	17518.0	0.943860	0.471930	+	0.881636i	$0.343558\pi$
$702$	−80.8243	−0.00434547
$703$	8155.36	0.437532
$704$	−19006.2	−1.01751
$705$	0	0
$706$	−282.188	−0.0150429
$707$	−16206.4	−0.862098
$708$	−43000.8	−2.28258
$709$	−18773.1	−0.994412	−0.497206	−	0.867633i	$-0.665641\pi$
$709$	−18773.1	−0.994412	−0.497206	+	0.867633i	$0.665641\pi$
$710$	0	0
$711$	−4648.90	−0.245214
$712$	3573.96	0.188118
$713$	23359.0	1.22693
$714$	−855.698	−0.0448511
$715$	0	0
$716$	16682.7	0.870756
$717$	−24788.5	−1.29114
$718$	−155.672	−0.00809141
$719$	−5559.32	−0.288356	−0.144178	−	0.989552i	$-0.546054\pi$
$719$	−5559.32	−0.288356	−0.144178	+	0.989552i	$0.546054\pi$
$720$	0	0
$721$	−2151.95	−0.111155
$722$	1457.94	0.0751511
$723$	30500.8	1.56893
$724$	−27561.1	−1.41478
$725$	0	0
$726$	−533.292	−0.0272622
$727$	30185.8	1.53993	0.769965	−	0.638086i	$-0.220274\pi$
$727$	30185.8	1.53993	0.769965	+	0.638086i	$0.220274\pi$
$728$	−697.087	−0.0354887
$729$	−26589.8	−1.35090
$730$	0	0
$731$	7287.48	0.368724
$732$	56673.2	2.86161
$733$	5727.12	0.288589	0.144295	−	0.989535i	$-0.453909\pi$
$733$	5727.12	0.288589	0.144295	+	0.989535i	$0.453909\pi$
$734$	−1692.74	−0.0851228
$735$	0	0
$736$	−6369.55	−0.319001
$737$	−23290.4	−1.16406
$738$	−2805.42	−0.139931
$739$	20981.4	1.04440	0.522202	−	0.852822i	$-0.325111\pi$
$739$	20981.4	1.04440	0.522202	+	0.852822i	$0.325111\pi$
$740$	0	0
$741$	2219.17	0.110018
$742$	2007.50	0.0993227
$743$	−10189.0	−0.503092	−0.251546	−	0.967845i	$-0.580939\pi$
$743$	−10189.0	−0.503092	−0.251546	+	0.967845i	$0.580939\pi$
$744$	7197.60	0.354673
$745$	0	0
$746$	−1840.02	−0.0903056
$747$	−32938.6	−1.61333
$748$	5330.16	0.260548
$749$	8123.64	0.396304
$750$	0	0
$751$	20640.3	1.00289	0.501447	−	0.865188i	$-0.332801\pi$
$751$	20640.3	1.00289	0.501447	+	0.865188i	$0.332801\pi$
$752$	19283.5	0.935103
$753$	−57182.1	−2.76737
$754$	−358.739	−0.0173269
$755$	0	0
$756$	7372.35	0.354669
$757$	−22987.1	−1.10368	−0.551838	−	0.833952i	$-0.686073\pi$
$757$	−22987.1	−1.10368	−0.551838	+	0.833952i	$0.686073\pi$
$758$	339.681	0.0162767
$759$	35323.6	1.68928
$760$	0	0
$761$	2412.34	0.114911	0.0574554	−	0.998348i	$-0.481701\pi$
$761$	2412.34	0.114911	0.0574554	+	0.998348i	$0.481701\pi$
$762$	−3207.55	−0.152490
$763$	38588.5	1.83093
$764$	3605.40	0.170731
$765$	0	0
$766$	1833.22	0.0864714
$767$	4727.11	0.222537
$768$	28280.4	1.32875
$769$	20079.6	0.941597	0.470799	−	0.882241i	$-0.343966\pi$
$769$	20079.6	0.941597	0.470799	+	0.882241i	$0.343966\pi$
$770$	0	0
$771$	2023.51	0.0945199
$772$	8872.87	0.413655
$773$	−4641.90	−0.215986	−0.107993	−	0.994152i	$-0.534442\pi$
$773$	−4641.90	−0.215986	−0.107993	+	0.994152i	$0.534442\pi$
$774$	4035.80	0.187421
$775$	0	0
$776$	7869.30	0.364035
$777$	−32872.8	−1.51777
$778$	19.9692	0.000920221 0
$779$	12798.3	0.588634
$780$	0	0
$781$	20096.2	0.920740
$782$	572.009	0.0261573
$783$	7628.53	0.348176
$784$	10023.5	0.456608
$785$	0	0
$786$	−446.800	−0.0202758
$787$	13570.7	0.614668	0.307334	−	0.951602i	$-0.400563\pi$
$787$	13570.7	0.614668	0.307334	+	0.951602i	$0.400563\pi$
$788$	33451.6	1.51226
$789$	31394.5	1.41657
$790$	0	0
$791$	−41871.2	−1.88213
$792$	5935.19	0.266285
$793$	−6230.14	−0.278989
$794$	−1727.52	−0.0772133
$795$	0	0
$796$	29330.4	1.30602
$797$	31882.9	1.41700	0.708500	−	0.705710i	$-0.249372\pi$
$797$	31882.9	1.41700	0.708500	+	0.705710i	$0.249372\pi$
$798$	2161.85	0.0959004
$799$	−5289.27	−0.234194
$800$	0	0
$801$	25008.3	1.10315
$802$	3501.99	0.154189
$803$	527.763	0.0231935
$804$	35863.9	1.57316
$805$	0	0
$806$	−393.518	−0.0171974
$807$	38764.6	1.69093
$808$	3338.11	0.145340
$809$	9675.40	0.420481	0.210240	−	0.977650i	$-0.432575\pi$
$809$	9675.40	0.420481	0.210240	+	0.977650i	$0.432575\pi$
$810$	0	0
$811$	−11925.8	−0.516364	−0.258182	−	0.966096i	$-0.583123\pi$
$811$	−11925.8	−0.516364	−0.258182	+	0.966096i	$0.583123\pi$
$812$	32722.2	1.41419
$813$	32960.5	1.42187
$814$	−2186.90	−0.0941655
$815$	0	0
$816$	−8119.11	−0.348316
$817$	−18411.2	−0.788405
$818$	2299.53	0.0982901
$819$	−4877.76	−0.208111
$820$	0	0
$821$	−24823.9	−1.05525	−0.527624	−	0.849478i	$-0.676917\pi$
$821$	−24823.9	−1.05525	−0.527624	+	0.849478i	$0.676917\pi$
$822$	3642.02	0.154538
$823$	14094.9	0.596985	0.298492	−	0.954412i	$-0.403516\pi$
$823$	14094.9	0.596985	0.298492	+	0.954412i	$0.403516\pi$
$824$	443.249	0.0187395
$825$	0	0
$826$	4605.02	0.193982
$827$	−47083.2	−1.97974	−0.989870	−	0.141978i	$-0.954654\pi$
$827$	−47083.2	−1.97974	−0.989870	+	0.141978i	$0.954654\pi$
$828$	−29660.8	−1.24491
$829$	−23982.5	−1.00476	−0.502380	−	0.864647i	$-0.667542\pi$
$829$	−23982.5	−1.00476	−0.502380	+	0.864647i	$0.667542\pi$
$830$	0	0
$831$	47731.6	1.99253
$832$	−3217.34	−0.134064
$833$	−2749.33	−0.114356
$834$	1865.98	0.0774742
$835$	0	0
$836$	−13466.2	−0.557103
$837$	8368.10	0.345572
$838$	−1752.22	−0.0722307
$839$	16754.7	0.689437	0.344719	−	0.938706i	$-0.387974\pi$
$839$	16754.7	0.689437	0.344719	+	0.938706i	$0.387974\pi$
$840$	0	0
$841$	9470.31	0.388302
$842$	3560.24	0.145717
$843$	62888.8	2.56940
$844$	−1773.98	−0.0723495
$845$	0	0
$846$	−2929.19	−0.119040
$847$	−5347.48	−0.216932
$848$	19047.7	0.771345
$849$	−57662.4	−2.33094
$850$	0	0
$851$	21974.5	0.885167
$852$	−30945.3	−1.24433
$853$	−46054.7	−1.84863	−0.924316	−	0.381627i	$-0.875364\pi$
$853$	−46054.7	−1.84863	−0.924316	+	0.381627i	$0.875364\pi$
$854$	−6069.22	−0.243190
$855$	0	0
$856$	−1673.27	−0.0668122
$857$	−11332.1	−0.451688	−0.225844	−	0.974163i	$-0.572514\pi$
$857$	−11332.1	−0.451688	−0.225844	+	0.974163i	$0.572514\pi$
$858$	−595.079	−0.0236779
$859$	41498.6	1.64833	0.824165	−	0.566350i	$-0.191645\pi$
$859$	41498.6	1.64833	0.824165	+	0.566350i	$0.191645\pi$
$860$	0	0
$861$	−51587.6	−2.04193
$862$	4165.14	0.164577
$863$	−20533.4	−0.809926	−0.404963	−	0.914333i	$-0.632716\pi$
$863$	−20533.4	−0.809926	−0.404963	+	0.914333i	$0.632716\pi$
$864$	−2281.82	−0.0898484
$865$	0	0
$866$	1419.12	0.0556856
$867$	2226.99	0.0872347
$868$	35894.6	1.40362
$869$	−5687.07	−0.222003
$870$	0	0
$871$	−3942.55	−0.153373
$872$	−7948.28	−0.308673
$873$	55064.2	2.13476
$874$	−1445.13	−0.0559294
$875$	0	0
$876$	−812.682	−0.0313447
$877$	28184.2	1.08519	0.542596	−	0.839994i	$-0.317442\pi$
$877$	28184.2	1.08519	0.542596	+	0.839994i	$0.317442\pi$
$878$	−3540.50	−0.136089
$879$	1249.72	0.0479546
$880$	0	0
$881$	−33164.1	−1.26825	−0.634124	−	0.773232i	$-0.718639\pi$
$881$	−33164.1	−1.26825	−0.634124	+	0.773232i	$0.718639\pi$
$882$	−1522.58	−0.0581267
$883$	27.8443	0.00106119	0.000530597	−	1.00000i	$-0.499831\pi$
$883$	27.8443	0.00106119	0.000530597		1.00000i	$0.499831\pi$
$884$	902.280	0.0343292
$885$	0	0
$886$	−2215.40	−0.0840041
$887$	−13237.4	−0.501091	−0.250546	−	0.968105i	$-0.580610\pi$
$887$	−13237.4	−0.501091	−0.250546	+	0.968105i	$0.580610\pi$
$888$	6771.00	0.255878
$889$	−32163.0	−1.21340
$890$	0	0
$891$	−21976.0	−0.826288
$892$	−6990.97	−0.262416
$893$	13362.9	0.500753
$894$	−1998.05	−0.0747480
$895$	0	0
$896$	−13026.5	−0.485698
$897$	5979.52	0.222576
$898$	2131.41	0.0792052
$899$	37141.9	1.37792
$900$	0	0
$901$	−5224.59	−0.193181
$902$	−3431.92	−0.126685
$903$	74212.4	2.73492
$904$	8624.43	0.317306
$905$	0	0
$906$	−3954.44	−0.145008
$907$	8903.26	0.325940	0.162970	−	0.986631i	$-0.447893\pi$
$907$	8903.26	0.325940	0.162970	+	0.986631i	$0.447893\pi$
$908$	22127.2	0.808720
$909$	23358.0	0.852293
$910$	0	0
$911$	41123.0	1.49557	0.747786	−	0.663940i	$-0.231117\pi$
$911$	41123.0	1.49557	0.747786	+	0.663940i	$0.231117\pi$
$912$	20512.2	0.744767
$913$	−40294.3	−1.46062
$914$	−1641.03	−0.0593876
$915$	0	0
$916$	47105.6	1.69914
$917$	−4480.19	−0.161340
$918$	204.915	0.00736734
$919$	−10153.2	−0.364442	−0.182221	−	0.983258i	$-0.558329\pi$
$919$	−10153.2	−0.364442	−0.182221	+	0.983258i	$0.558329\pi$
$920$	0	0
$921$	−50145.9	−1.79410
$922$	79.4371	0.00283744
$923$	3401.85	0.121314
$924$	54279.9	1.93255
$925$	0	0
$926$	−644.175	−0.0228606
$927$	3101.57	0.109891
$928$	−10127.9	−0.358258
$929$	−54460.5	−1.92335	−0.961674	−	0.274196i	$-0.911588\pi$
$929$	−54460.5	−1.92335	−0.961674	+	0.274196i	$0.911588\pi$
$930$	0	0
$931$	6945.95	0.244516
$932$	−15794.6	−0.555118
$933$	−3791.98	−0.133059
$934$	3572.88	0.125169
$935$	0	0
$936$	1004.70	0.0350851
$937$	−42095.7	−1.46767	−0.733834	−	0.679328i	$-0.762271\pi$
$937$	−42095.7	−1.46767	−0.733834	+	0.679328i	$0.762271\pi$
$938$	−3840.72	−0.133693
$939$	56243.1	1.95466
$940$	0	0
$941$	−31820.4	−1.10236	−0.551178	−	0.834388i	$-0.685821\pi$
$941$	−31820.4	−1.10236	−0.551178	+	0.834388i	$0.685821\pi$
$942$	6032.28	0.208644
$943$	34484.8	1.19086
$944$	43693.7	1.50647
$945$	0	0
$946$	4937.05	0.169680
$947$	−46199.7	−1.58531	−0.792656	−	0.609670i	$-0.791302\pi$
$947$	−46199.7	−1.58531	−0.792656	+	0.609670i	$0.791302\pi$
$948$	8757.29	0.300025
$949$	89.3389	0.00305592
$950$	0	0
$951$	57560.6	1.96270
$952$	1767.34	0.0601678
$953$	−18232.9	−0.619750	−0.309875	−	0.950777i	$-0.600287\pi$
$953$	−18232.9	−0.619750	−0.309875	+	0.950777i	$0.600287\pi$
$954$	−2893.37	−0.0981931
$955$	0	0
$956$	25462.9	0.861430
$957$	56166.1	1.89717
$958$	−246.945	−0.00832821
$959$	36519.6	1.22970
$960$	0	0
$961$	10951.7	0.367617
$962$	−370.194	−0.0124070
$963$	−11708.5	−0.391796
$964$	−31330.5	−1.04677
$965$	0	0
$966$	5825.07	0.194015
$967$	−12670.2	−0.421352	−0.210676	−	0.977556i	$-0.567567\pi$
$967$	−12670.2	−0.421352	−0.210676	+	0.977556i	$0.567567\pi$
$968$	1101.45	0.0365722
$969$	−5626.30	−0.186525
$970$	0	0
$971$	−44212.3	−1.46122	−0.730608	−	0.682797i	$-0.760763\pi$
$971$	−44212.3	−1.46122	−0.730608	+	0.682797i	$0.760763\pi$
$972$	42700.1	1.40906
$973$	18710.7	0.616483
$974$	−3332.21	−0.109621
$975$	0	0
$976$	−57586.5	−1.88863
$977$	4923.23	0.161216	0.0806081	−	0.996746i	$-0.474314\pi$
$977$	4923.23	0.161216	0.0806081	+	0.996746i	$0.474314\pi$
$978$	−3818.45	−0.124847
$979$	30593.0	0.998729
$980$	0	0
$981$	−55616.9	−1.81010
$982$	−2609.77	−0.0848078
$983$	−40398.0	−1.31078	−0.655390	−	0.755291i	$-0.727496\pi$
$983$	−40398.0	−1.31078	−0.655390	+	0.755291i	$0.727496\pi$
$984$	10625.8	0.344246
$985$	0	0
$986$	909.519	0.0293763
$987$	−53863.5	−1.73708
$988$	−2279.53	−0.0734025
$989$	−49608.8	−1.59501
$990$	0	0
$991$	−30649.9	−0.982469	−0.491235	−	0.871027i	$-0.663454\pi$
$991$	−30649.9	−0.982469	−0.491235	+	0.871027i	$0.663454\pi$
$992$	−11109.7	−0.355579
$993$	−63161.3	−2.01849
$994$	3313.98	0.105748
$995$	0	0
$996$	62047.6	1.97395
$997$	36971.9	1.17444	0.587218	−	0.809429i	$-0.300223\pi$
$997$	36971.9	1.17444	0.587218	+	0.809429i	$0.300223\pi$
$998$	5073.60	0.160924
$999$	7872.12	0.249312

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	425.4.a.i.1.3		5
5.2	odd	4		425.4.b.i.324.5		10
5.3	odd	4		425.4.b.i.324.6		10
5.4	even	2		85.4.a.g.1.3	✓	5
15.14	odd	2		765.4.a.m.1.3		5
20.19	odd	2		1360.4.a.w.1.5		5
85.84	even	2		1445.4.a.l.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
85.4.a.g.1.3	✓	5	5.4	even	2
425.4.a.i.1.3		5	1.1	even	1	trivial
425.4.b.i.324.5		10	5.2	odd	4
425.4.b.i.324.6		10	5.3	odd	4
765.4.a.m.1.3		5	15.14	odd	2
1360.4.a.w.1.5		5	20.19	odd	2
1445.4.a.l.1.3		5	85.84	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 425 = 5^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	425.a (trivial)

\(f(q)\)	\(=\)	\(q-0.290753 q^{2} +7.70584 q^{3} -7.91546 q^{4} -2.24049 q^{6} -22.4661 q^{7} +4.62746 q^{8} +32.3800 q^{9} +39.6109 q^{11} -60.9953 q^{12} +6.70527 q^{13} +6.53208 q^{14} +61.9783 q^{16} -17.0000 q^{17} -9.41457 q^{18} +42.9490 q^{19} -173.120 q^{21} -11.5170 q^{22} +115.726 q^{23} +35.6585 q^{24} -1.94958 q^{26} +41.4574 q^{27} +177.830 q^{28} +184.009 q^{29} +201.848 q^{31} -55.0401 q^{32} +305.235 q^{33} +4.94280 q^{34} -256.303 q^{36} +189.885 q^{37} -12.4876 q^{38} +51.6697 q^{39} +297.987 q^{41} +50.3352 q^{42} -428.676 q^{43} -313.539 q^{44} -33.6476 q^{46} +311.134 q^{47} +477.595 q^{48} +161.725 q^{49} -130.999 q^{51} -53.0753 q^{52} +307.329 q^{53} -12.0538 q^{54} -103.961 q^{56} +330.959 q^{57} -53.5011 q^{58} +704.985 q^{59} -929.140 q^{61} -58.6879 q^{62} -727.452 q^{63} -479.823 q^{64} -88.7480 q^{66} -587.978 q^{67} +134.563 q^{68} +891.764 q^{69} +507.339 q^{71} +149.837 q^{72} +13.3237 q^{73} -55.2094 q^{74} -339.962 q^{76} -889.903 q^{77} -15.0231 q^{78} -143.573 q^{79} -554.796 q^{81} -86.6407 q^{82} -1017.25 q^{83} +1370.33 q^{84} +124.639 q^{86} +1417.94 q^{87} +183.298 q^{88} +772.337 q^{89} -150.641 q^{91} -916.023 q^{92} +1555.41 q^{93} -90.4630 q^{94} -424.130 q^{96} +1700.56 q^{97} -47.0221 q^{98} +1282.60 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 2 q^{2} + q^{3} + 34 q^{4} - 5 q^{6} - 10 q^{7} - 30 q^{8} - 30 q^{9} + 126 q^{11} - 15 q^{12} - 83 q^{13} + 90 q^{14} + 322 q^{16} - 85 q^{17} + 97 q^{18} + 55 q^{19} + 6 q^{21} + 240 q^{22} + 2 q^{23}+ \cdots - 858 q^{99}+O(q^{100}) \) 5 * q - 2 * q^2 + q^3 + 34 * q^4 - 5 * q^6 - 10 * q^7 - 30 * q^8 - 30 * q^9 + 126 * q^11 - 15 * q^12 - 83 * q^13 + 90 * q^14 + 322 * q^16 - 85 * q^17 + 97 * q^18 + 55 * q^19 + 6 * q^21 + 240 * q^22 + 2 * q^23 + 155 * q^24 - 395 * q^26 + 163 * q^27 + 1062 * q^28 + 195 * q^29 + 97 * q^31 - 182 * q^32 + 352 * q^33 + 34 * q^34 - 1437 * q^36 + 476 * q^37 + 149 * q^38 + 373 * q^39 + 298 * q^41 + 18 * q^42 - 168 * q^43 + 1136 * q^44 - 2106 * q^46 - 1095 * q^47 + 1193 * q^48 + 737 * q^49 - 17 * q^51 - 281 * q^52 + 1035 * q^53 - 291 * q^54 - 350 * q^56 + 15 * q^57 + 2199 * q^58 + 1163 * q^59 - 351 * q^61 + 1241 * q^62 - 1584 * q^63 + 2042 * q^64 + 356 * q^66 - 1064 * q^67 - 578 * q^68 + 1298 * q^69 + 1393 * q^71 + 269 * q^72 + 163 * q^73 + 1530 * q^74 - 393 * q^76 + 732 * q^77 + 67 * q^78 + 750 * q^79 + 85 * q^81 - 1144 * q^82 - 270 * q^83 + 2422 * q^84 + 3502 * q^86 + 907 * q^87 + 3216 * q^88 + 1179 * q^89 + 282 * q^91 - 758 * q^92 + 3923 * q^93 + 3473 * q^94 - 2045 * q^96 + 2745 * q^97 - 502 * q^98 - 858 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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