LMFDB - Embedded newform 4925.2.a.s.1.10

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4925,2,Mod(1,4925)] 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("4925.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4925, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4925 = 5^{2} \cdot 197 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4925.a (trivial)

Newform invariants

comment:select newform

sage:traces = [49,5,22] 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:	$39.3263229955$
Analytic rank:	$0$
Dimension:	$49$
Twist minimal:	no (minimal twist has level 985)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Character	$\chi$	$=$	4925.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.80911 q^{2} -2.24718 q^{3} +1.27287 q^{4} +4.06540 q^{6} +2.84950 q^{7} +1.31546 q^{8} +2.04984 q^{9} -0.745776 q^{11} -2.86037 q^{12} +4.06489 q^{13} -5.15504 q^{14} -4.92554 q^{16} -6.80009 q^{17} -3.70837 q^{18} -7.39798 q^{19} -6.40334 q^{21} +1.34919 q^{22} -1.49894 q^{23} -2.95608 q^{24} -7.35382 q^{26} +2.13519 q^{27} +3.62703 q^{28} -5.21854 q^{29} +2.04295 q^{31} +6.27992 q^{32} +1.67590 q^{33} +12.3021 q^{34} +2.60917 q^{36} +6.87951 q^{37} +13.3837 q^{38} -9.13456 q^{39} +6.86920 q^{41} +11.5843 q^{42} -5.24725 q^{43} -0.949275 q^{44} +2.71175 q^{46} +2.06962 q^{47} +11.0686 q^{48} +1.11963 q^{49} +15.2811 q^{51} +5.17407 q^{52} -4.36958 q^{53} -3.86279 q^{54} +3.74840 q^{56} +16.6246 q^{57} +9.44090 q^{58} -6.79294 q^{59} -10.4968 q^{61} -3.69592 q^{62} +5.84100 q^{63} -1.50995 q^{64} -3.03188 q^{66} +11.1601 q^{67} -8.65562 q^{68} +3.36840 q^{69} -3.12691 q^{71} +2.69648 q^{72} +15.0523 q^{73} -12.4458 q^{74} -9.41665 q^{76} -2.12509 q^{77} +16.5254 q^{78} +15.7598 q^{79} -10.9477 q^{81} -12.4271 q^{82} +17.6697 q^{83} -8.15061 q^{84} +9.49284 q^{86} +11.7270 q^{87} -0.981038 q^{88} -9.96196 q^{89} +11.5829 q^{91} -1.90796 q^{92} -4.59089 q^{93} -3.74416 q^{94} -14.1121 q^{96} -17.1696 q^{97} -2.02553 q^{98} -1.52872 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 49 q + 5 q^{2} + 22 q^{3} + 49 q^{4} + 2 q^{6} + 32 q^{7} + 15 q^{8} + 51 q^{9} - 2 q^{11} + 44 q^{12} + 32 q^{13} - 8 q^{14} + 49 q^{16} + 14 q^{17} + 25 q^{18} + 4 q^{19} + 10 q^{21} + 38 q^{22} + 24 q^{23}+ \cdots + 22 q^{99}+O(q^{100}) $ 49 * q + 5 * q^2 + 22 * q^3 + 49 * q^4 + 2 * q^6 + 32 * q^7 + 15 * q^8 + 51 * q^9 - 2 * q^11 + 44 * q^12 + 32 * q^13 - 8 * q^14 + 49 * q^16 + 14 * q^17 + 25 * q^18 + 4 * q^19 + 10 * q^21 + 38 * q^22 + 24 * q^23 - 2 * q^26 + 88 * q^27 + 80 * q^28 + 6 * q^29 - 2 * q^31 + 35 * q^32 + 48 * q^33 - 12 * q^34 + 43 * q^36 + 60 * q^37 + 34 * q^38 - 2 * q^41 + 36 * q^42 + 80 * q^43 - 8 * q^44 - 22 * q^46 + 48 * q^47 + 88 * q^48 + 65 * q^49 + 16 * q^51 + 96 * q^52 + 18 * q^53 + 42 * q^54 - 22 * q^56 + 40 * q^57 + 52 * q^58 + 8 * q^59 + 16 * q^61 + 38 * q^62 + 82 * q^63 + 55 * q^64 + 42 * q^66 + 146 * q^67 - 68 * q^68 - 6 * q^69 + 75 * q^72 + 78 * q^73 - 24 * q^74 - 4 * q^76 + 32 * q^77 - 38 * q^78 + 10 * q^79 + 41 * q^81 + 96 * q^82 + 64 * q^83 + 42 * q^84 - 28 * q^86 + 56 * q^87 + 72 * q^88 + 4 * q^89 + 60 * q^92 - 28 * q^93 - 8 * q^94 - 34 * q^96 + 124 * q^97 - 19 * q^98 + 22 * 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 $

$2$	−1.80911	−1.27923	−0.639616	−	0.768695i	$-0.720907\pi$
$2$	−1.80911	−1.27923	−0.639616	+	0.768695i	$0.720907\pi$
$3$	−2.24718	−1.29741	−0.648706	−	0.761039i	$-0.724690\pi$
$3$	−2.24718	−1.29741	−0.648706	+	0.761039i	$0.724690\pi$
$4$	1.27287	0.636434
$5$	0	0
$5$	0	0
$6$	4.06540	1.65969
$7$	2.84950	1.07701	0.538504	−	0.842623i	$-0.318990\pi$
$7$	2.84950	1.07701	0.538504	+	0.842623i	$0.318990\pi$
$8$	1.31546	0.465085
$9$	2.04984	0.683279
$10$	0	0
$11$	−0.745776	−0.224860	−0.112430	−	0.993660i	$-0.535863\pi$
$11$	−0.745776	−0.224860	−0.112430	+	0.993660i	$0.535863\pi$
$12$	−2.86037	−0.825717
$13$	4.06489	1.12740	0.563699	−	0.825980i	$-0.309378\pi$
$13$	4.06489	1.12740	0.563699	+	0.825980i	$0.309378\pi$
$14$	−5.15504	−1.37774
$15$	0	0
$16$	−4.92554	−1.23139
$17$	−6.80009	−1.64926	−0.824632	−	0.565669i	$-0.808618\pi$
$17$	−6.80009	−1.64926	−0.824632	+	0.565669i	$0.808618\pi$
$18$	−3.70837	−0.874072
$19$	−7.39798	−1.69721	−0.848606	−	0.529025i	$-0.822558\pi$
$19$	−7.39798	−1.69721	−0.848606	+	0.529025i	$0.822558\pi$
$20$	0	0
$21$	−6.40334	−1.39732
$22$	1.34919	0.287648
$23$	−1.49894	−0.312551	−0.156276	−	0.987713i	$-0.549949\pi$
$23$	−1.49894	−0.312551	−0.156276	+	0.987713i	$0.549949\pi$
$24$	−2.95608	−0.603407
$25$	0	0
$26$	−7.35382	−1.44220
$27$	2.13519	0.410918
$28$	3.62703	0.685445
$29$	−5.21854	−0.969059	−0.484529	−	0.874775i	$-0.661009\pi$
$29$	−5.21854	−0.969059	−0.484529	+	0.874775i	$0.661009\pi$
$30$	0	0
$31$	2.04295	0.366925	0.183462	−	0.983027i	$-0.441269\pi$
$31$	2.04295	0.366925	0.183462	+	0.983027i	$0.441269\pi$
$32$	6.27992	1.11014
$33$	1.67590	0.291736
$34$	12.3021	2.10979
$35$	0	0
$36$	2.60917	0.434862
$37$	6.87951	1.13098	0.565492	−	0.824754i	$-0.308686\pi$
$37$	6.87951	1.13098	0.565492	+	0.824754i	$0.308686\pi$
$38$	13.3837	2.17113
$39$	−9.13456	−1.46270
$40$	0	0
$41$	6.86920	1.07279	0.536394	−	0.843968i	$-0.319786\pi$
$41$	6.86920	1.07279	0.536394	+	0.843968i	$0.319786\pi$
$42$	11.5843	1.78750
$43$	−5.24725	−0.800198	−0.400099	−	0.916472i	$-0.631024\pi$
$43$	−5.24725	−0.800198	−0.400099	+	0.916472i	$0.631024\pi$
$44$	−0.949275	−0.143109
$45$	0	0
$46$	2.71175	0.399826
$47$	2.06962	0.301885	0.150943	−	0.988543i	$-0.451769\pi$
$47$	2.06962	0.301885	0.150943	+	0.988543i	$0.451769\pi$
$48$	11.0686	1.59762
$49$	1.11963	0.159947
$50$	0	0
$51$	15.2811	2.13978
$52$	5.17407	0.717515
$53$	−4.36958	−0.600208	−0.300104	−	0.953906i	$-0.597021\pi$
$53$	−4.36958	−0.600208	−0.300104	+	0.953906i	$0.597021\pi$
$54$	−3.86279	−0.525660
$55$	0	0
$56$	3.74840	0.500901
$57$	16.6246	2.20198
$58$	9.44090	1.23965
$59$	−6.79294	−0.884365	−0.442183	−	0.896925i	$-0.645796\pi$
$59$	−6.79294	−0.884365	−0.442183	+	0.896925i	$0.645796\pi$
$60$	0	0
$61$	−10.4968	−1.34398	−0.671992	−	0.740559i	$-0.734561\pi$
$61$	−10.4968	−1.34398	−0.671992	+	0.740559i	$0.734561\pi$
$62$	−3.69592	−0.469382
$63$	5.84100	0.735897
$64$	−1.50995	−0.188744
$65$	0	0
$66$	−3.03188	−0.373198
$67$	11.1601	1.36342	0.681710	−	0.731622i	$-0.261237\pi$
$67$	11.1601	1.36342	0.681710	+	0.731622i	$0.261237\pi$
$68$	−8.65562	−1.04965
$69$	3.36840	0.405508
$70$	0	0
$71$	−3.12691	−0.371096	−0.185548	−	0.982635i	$-0.559406\pi$
$71$	−3.12691	−0.371096	−0.185548	+	0.982635i	$0.559406\pi$
$72$	2.69648	0.317783
$73$	15.0523	1.76174	0.880870	−	0.473358i	$-0.156958\pi$
$73$	15.0523	1.76174	0.880870	+	0.473358i	$0.156958\pi$
$74$	−12.4458	−1.44679
$75$	0	0
$76$	−9.41665	−1.08016
$77$	−2.12509	−0.242176
$78$	16.5254	1.87113
$79$	15.7598	1.77311	0.886556	−	0.462621i	$-0.153091\pi$
$79$	15.7598	1.77311	0.886556	+	0.462621i	$0.153091\pi$
$80$	0	0
$81$	−10.9477	−1.21641
$82$	−12.4271	−1.37234
$83$	17.6697	1.93950	0.969751	−	0.244098i	$-0.0784919\pi$
$83$	17.6697	1.93950	0.969751	+	0.244098i	$0.0784919\pi$
$84$	−8.15061	−0.889305
$85$	0	0
$86$	9.49284	1.02364
$87$	11.7270	1.25727
$88$	−0.981038	−0.104579
$89$	−9.96196	−1.05597	−0.527983	−	0.849255i	$-0.677051\pi$
$89$	−9.96196	−1.05597	−0.527983	+	0.849255i	$0.677051\pi$
$90$	0	0
$91$	11.5829	1.21422
$92$	−1.90796	−0.198918
$93$	−4.59089	−0.476053
$94$	−3.74416	−0.386181
$95$	0	0
$96$	−14.1121	−1.44031
$97$	−17.1696	−1.74331	−0.871653	−	0.490123i	$-0.836952\pi$
$97$	−17.1696	−1.74331	−0.871653	+	0.490123i	$0.836952\pi$
$98$	−2.02553	−0.204610
$99$	−1.52872	−0.153642
$100$	0	0
$101$	6.61755	0.658471	0.329235	−	0.944248i	$-0.393209\pi$
$101$	6.61755	0.658471	0.329235	+	0.944248i	$0.393209\pi$
$102$	−27.6451	−2.73727
$103$	−16.2135	−1.59756	−0.798780	−	0.601623i	$-0.794521\pi$
$103$	−16.2135	−1.59756	−0.798780	+	0.601623i	$0.794521\pi$
$104$	5.34720	0.524336
$105$	0	0
$106$	7.90504	0.767806
$107$	−8.27804	−0.800269	−0.400134	−	0.916457i	$-0.631037\pi$
$107$	−8.27804	−0.800269	−0.400134	+	0.916457i	$0.631037\pi$
$108$	2.71782	0.261522
$109$	6.12234	0.586414	0.293207	−	0.956049i	$-0.405277\pi$
$109$	6.12234	0.586414	0.293207	+	0.956049i	$0.405277\pi$
$110$	0	0
$111$	−15.4595	−1.46735
$112$	−14.0353	−1.32621
$113$	−9.92629	−0.933787	−0.466893	−	0.884314i	$-0.654627\pi$
$113$	−9.92629	−0.933787	−0.466893	+	0.884314i	$0.654627\pi$
$114$	−30.0757	−2.81685
$115$	0	0
$116$	−6.64252	−0.616742
$117$	8.33236	0.770327
$118$	12.2892	1.13131
$119$	−19.3768	−1.77627
$120$	0	0
$121$	−10.4438	−0.949438
$122$	18.9899	1.71927
$123$	−15.4363	−1.39185
$124$	2.60041	0.233523
$125$	0	0
$126$	−10.5670	−0.941383
$127$	−7.74417	−0.687184	−0.343592	−	0.939119i	$-0.611644\pi$
$127$	−7.74417	−0.687184	−0.343592	+	0.939119i	$0.611644\pi$
$128$	−9.82816	−0.868695
$129$	11.7915	1.03819
$130$	0	0
$131$	−21.2818	−1.85940	−0.929700	−	0.368318i	$-0.879934\pi$
$131$	−21.2818	−1.85940	−0.929700	+	0.368318i	$0.879934\pi$
$132$	2.13319	0.185671
$133$	−21.0805	−1.82791
$134$	−20.1898	−1.74413
$135$	0	0
$136$	−8.94524	−0.767048
$137$	12.6452	1.08035	0.540174	−	0.841553i	$-0.318358\pi$
$137$	12.6452	1.08035	0.540174	+	0.841553i	$0.318358\pi$
$138$	−6.09380	−0.518739
$139$	17.3823	1.47434	0.737172	−	0.675706i	$-0.236161\pi$
$139$	17.3823	1.47434	0.737172	+	0.675706i	$0.236161\pi$
$140$	0	0
$141$	−4.65082	−0.391669
$142$	5.65692	0.474718
$143$	−3.03150	−0.253507
$144$	−10.0966	−0.841380
$145$	0	0
$146$	−27.2313	−2.25367
$147$	−2.51602	−0.207518
$148$	8.75671	0.719797
$149$	7.34550	0.601767	0.300884	−	0.953661i	$-0.402718\pi$
$149$	7.34550	0.601767	0.300884	+	0.953661i	$0.402718\pi$
$150$	0	0
$151$	2.98039	0.242541	0.121270	−	0.992620i	$-0.461303\pi$
$151$	2.98039	0.242541	0.121270	+	0.992620i	$0.461303\pi$
$152$	−9.73174	−0.789348
$153$	−13.9391	−1.12691
$154$	3.84451	0.309799
$155$	0	0
$156$	−11.6271	−0.930912
$157$	18.2772	1.45868	0.729342	−	0.684150i	$-0.239826\pi$
$157$	18.2772	1.45868	0.729342	+	0.684150i	$0.239826\pi$
$158$	−28.5111	−2.26822
$159$	9.81925	0.778718
$160$	0	0
$161$	−4.27123	−0.336620
$162$	19.8055	1.55607
$163$	12.7146	0.995885	0.497943	−	0.867210i	$-0.334089\pi$
$163$	12.7146	0.995885	0.497943	+	0.867210i	$0.334089\pi$
$164$	8.74358	0.682759
$165$	0	0
$166$	−31.9664	−2.48107
$167$	1.77532	0.137378	0.0686892	−	0.997638i	$-0.478118\pi$
$167$	1.77532	0.137378	0.0686892	+	0.997638i	$0.478118\pi$
$168$	−8.42334	−0.649875
$169$	3.52334	0.271026
$170$	0	0
$171$	−15.1646	−1.15967
$172$	−6.67906	−0.509274
$173$	−6.59088	−0.501095	−0.250547	−	0.968104i	$-0.580611\pi$
$173$	−6.59088	−0.501095	−0.250547	+	0.968104i	$0.580611\pi$
$174$	−21.2154	−1.60834
$175$	0	0
$176$	3.67335	0.276889
$177$	15.2650	1.14739
$178$	18.0223	1.35082
$179$	10.1511	0.758732	0.379366	−	0.925247i	$-0.376142\pi$
$179$	10.1511	0.758732	0.379366	+	0.925247i	$0.376142\pi$
$180$	0	0
$181$	24.6663	1.83343	0.916717	−	0.399536i	$-0.130829\pi$
$181$	24.6663	1.83343	0.916717	+	0.399536i	$0.130829\pi$
$182$	−20.9547	−1.55327
$183$	23.5883	1.74370
$184$	−1.97180	−0.145363
$185$	0	0
$186$	8.30541	0.608982
$187$	5.07135	0.370854
$188$	2.63435	0.192130
$189$	6.08423	0.442562
$190$	0	0
$191$	0.954691	0.0690790	0.0345395	−	0.999403i	$-0.489004\pi$
$191$	0.954691	0.0690790	0.0345395	+	0.999403i	$0.489004\pi$
$192$	3.39314	0.244879
$193$	−4.52820	−0.325947	−0.162973	−	0.986630i	$-0.552108\pi$
$193$	−4.52820	−0.325947	−0.162973	+	0.986630i	$0.552108\pi$
$194$	31.0616	2.23009
$195$	0	0
$196$	1.42514	0.101796
$197$	−1.00000	−0.0712470
$197$	−1.00000	−0.0712470
$198$	2.76562	0.196544
$199$	12.3248	0.873682	0.436841	−	0.899539i	$-0.356097\pi$
$199$	12.3248	0.873682	0.436841	+	0.899539i	$0.356097\pi$
$200$	0	0
$201$	−25.0787	−1.76892
$202$	−11.9719	−0.842337
$203$	−14.8702	−1.04368
$204$	19.4508	1.36183
$205$	0	0
$206$	29.3319	2.04365
$207$	−3.07259	−0.213560
$208$	−20.0218	−1.38826
$209$	5.51723	0.381635
$210$	0	0
$211$	−3.96349	−0.272858	−0.136429	−	0.990650i	$-0.543563\pi$
$211$	−3.96349	−0.272858	−0.136429	+	0.990650i	$0.543563\pi$
$212$	−5.56190	−0.381993
$213$	7.02674	0.481465
$214$	14.9759	1.02373
$215$	0	0
$216$	2.80876	0.191112
$217$	5.82138	0.395181
$218$	−11.0760	−0.750160
$219$	−33.8253	−2.28570
$220$	0	0
$221$	−27.6416	−1.85938
$222$	27.9679	1.87708
$223$	0.0193322	0.00129458	0.000647291	−	1.00000i	$-0.499794\pi$
$223$	0.0193322	0.00129458	0.000647291		1.00000i	$0.499794\pi$
$224$	17.8946	1.19563
$225$	0	0
$226$	17.9577	1.19453
$227$	4.56798	0.303187	0.151594	−	0.988443i	$-0.451559\pi$
$227$	4.56798	0.303187	0.151594	+	0.988443i	$0.451559\pi$
$228$	21.1609	1.40142
$229$	−3.24750	−0.214601	−0.107300	−	0.994227i	$-0.534221\pi$
$229$	−3.24750	−0.214601	−0.107300	+	0.994227i	$0.534221\pi$
$230$	0	0
$231$	4.77546	0.314202
$232$	−6.86478	−0.450695
$233$	−3.73326	−0.244574	−0.122287	−	0.992495i	$-0.539023\pi$
$233$	−3.73326	−0.244574	−0.122287	+	0.992495i	$0.539023\pi$
$234$	−15.0741	−0.985427
$235$	0	0
$236$	−8.64651	−0.562840
$237$	−35.4151	−2.30046
$238$	35.0548	2.27226
$239$	1.47211	0.0952231	0.0476115	−	0.998866i	$-0.484839\pi$
$239$	1.47211	0.0952231	0.0476115	+	0.998866i	$0.484839\pi$
$240$	0	0
$241$	1.03890	0.0669213	0.0334606	−	0.999440i	$-0.489347\pi$
$241$	1.03890	0.0669213	0.0334606	+	0.999440i	$0.489347\pi$
$242$	18.8940	1.21455
$243$	18.1959	1.16727
$244$	−13.3611	−0.855357
$245$	0	0
$246$	27.9260	1.78050
$247$	−30.0720	−1.91343
$248$	2.68742	0.170651
$249$	−39.7071	−2.51633
$250$	0	0
$251$	7.15367	0.451536	0.225768	−	0.974181i	$-0.427511\pi$
$251$	7.15367	0.451536	0.225768	+	0.974181i	$0.427511\pi$
$252$	7.43482	0.468350
$253$	1.11788	0.0702803
$254$	14.0100	0.879068
$255$	0	0
$256$	20.8001	1.30001
$257$	16.2257	1.01213	0.506066	−	0.862495i	$-0.331099\pi$
$257$	16.2257	1.01213	0.506066	+	0.862495i	$0.331099\pi$
$258$	−21.3322	−1.32808
$259$	19.6031	1.21808
$260$	0	0
$261$	−10.6972	−0.662137
$262$	38.5011	2.37860
$263$	21.8595	1.34792	0.673958	−	0.738770i	$-0.264593\pi$
$263$	21.8595	1.34792	0.673958	+	0.738770i	$0.264593\pi$
$264$	2.20457	0.135682
$265$	0	0
$266$	38.1369	2.33832
$267$	22.3864	1.37002
$268$	14.2053	0.867727
$269$	−4.27811	−0.260841	−0.130421	−	0.991459i	$-0.541633\pi$
$269$	−4.27811	−0.260841	−0.130421	+	0.991459i	$0.541633\pi$
$270$	0	0
$271$	7.02587	0.426791	0.213396	−	0.976966i	$-0.431548\pi$
$271$	7.02587	0.426791	0.213396	+	0.976966i	$0.431548\pi$
$272$	33.4941	2.03088
$273$	−26.0289	−1.57534
$274$	−22.8764	−1.38202
$275$	0	0
$276$	4.28753	0.258079
$277$	−20.1351	−1.20980	−0.604899	−	0.796302i	$-0.706787\pi$
$277$	−20.1351	−1.20980	−0.604899	+	0.796302i	$0.706787\pi$
$278$	−31.4464	−1.88603
$279$	4.18771	0.250712
$280$	0	0
$281$	−3.44937	−0.205772	−0.102886	−	0.994693i	$-0.532808\pi$
$281$	−3.44937	−0.205772	−0.102886	+	0.994693i	$0.532808\pi$
$282$	8.41383	0.501036
$283$	−5.98364	−0.355690	−0.177845	−	0.984058i	$-0.556913\pi$
$283$	−5.98364	−0.355690	−0.177845	+	0.984058i	$0.556913\pi$
$284$	−3.98014	−0.236178
$285$	0	0
$286$	5.48431	0.324294
$287$	19.5738	1.15540
$288$	12.8728	0.758537
$289$	29.2413	1.72007
$290$	0	0
$291$	38.5832	2.26179
$292$	19.1596	1.12123
$293$	−15.4879	−0.904813	−0.452406	−	0.891812i	$-0.649434\pi$
$293$	−15.4879	−0.904813	−0.452406	+	0.891812i	$0.649434\pi$
$294$	4.55175	0.265463
$295$	0	0
$296$	9.04971	0.526004
$297$	−1.59238	−0.0923990
$298$	−13.2888	−0.769800
$299$	−6.09304	−0.352370
$300$	0	0
$301$	−14.9520	−0.861820
$302$	−5.39184	−0.310266
$303$	−14.8708	−0.854308
$304$	36.4390	2.08992
$305$	0	0
$306$	25.2173	1.44158
$307$	12.5020	0.713526	0.356763	−	0.934195i	$-0.383880\pi$
$307$	12.5020	0.713526	0.356763	+	0.934195i	$0.383880\pi$
$308$	−2.70495	−0.154129
$309$	36.4346	2.07269
$310$	0	0
$311$	−15.4593	−0.876616	−0.438308	−	0.898825i	$-0.644422\pi$
$311$	−15.4593	−0.876616	−0.438308	+	0.898825i	$0.644422\pi$
$312$	−12.0161	−0.680280
$313$	−19.8301	−1.12086	−0.560430	−	0.828202i	$-0.689364\pi$
$313$	−19.8301	−1.12086	−0.560430	+	0.828202i	$0.689364\pi$
$314$	−33.0655	−1.86599
$315$	0	0
$316$	20.0601	1.12847
$317$	−17.4047	−0.977546	−0.488773	−	0.872411i	$-0.662555\pi$
$317$	−17.4047	−0.977546	−0.488773	+	0.872411i	$0.662555\pi$
$318$	−17.7641	−0.996160
$319$	3.89186	0.217903
$320$	0	0
$321$	18.6023	1.03828
$322$	7.72712	0.430616
$323$	50.3069	2.79915
$324$	−13.9350	−0.774164
$325$	0	0
$326$	−23.0021	−1.27397
$327$	−13.7580	−0.760821
$328$	9.03615	0.498938
$329$	5.89738	0.325133
$330$	0	0
$331$	−28.2178	−1.55099	−0.775496	−	0.631353i	$-0.782500\pi$
$331$	−28.2178	−1.55099	−0.775496	+	0.631353i	$0.782500\pi$
$332$	22.4912	1.23436
$333$	14.1019	0.772777
$334$	−3.21174	−0.175739
$335$	0	0
$336$	31.5399	1.72064
$337$	−23.1981	−1.26368	−0.631841	−	0.775098i	$-0.717701\pi$
$337$	−23.1981	−1.26368	−0.631841	+	0.775098i	$0.717701\pi$
$338$	−6.37410	−0.346705
$339$	22.3062	1.21151
$340$	0	0
$341$	−1.52358	−0.0825067
$342$	27.4344	1.48349
$343$	−16.7561	−0.904744
$344$	−6.90255	−0.372160
$345$	0	0
$346$	11.9236	0.641017
$347$	28.0414	1.50534	0.752671	−	0.658396i	$-0.228765\pi$
$347$	28.0414	1.50534	0.752671	+	0.658396i	$0.228765\pi$
$348$	14.9270	0.800169
$349$	−0.609875	−0.0326459	−0.0163229	−	0.999867i	$-0.505196\pi$
$349$	−0.609875	−0.0326459	−0.0163229	+	0.999867i	$0.505196\pi$
$350$	0	0
$351$	8.67933	0.463268
$352$	−4.68341	−0.249627
$353$	8.46810	0.450711	0.225356	−	0.974277i	$-0.427646\pi$
$353$	8.46810	0.450711	0.225356	+	0.974277i	$0.427646\pi$
$354$	−27.6160	−1.46777
$355$	0	0
$356$	−12.6803	−0.672052
$357$	43.5433	2.30456
$358$	−18.3645	−0.970594
$359$	−33.9630	−1.79250	−0.896248	−	0.443553i	$-0.853718\pi$
$359$	−33.9630	−1.79250	−0.896248	+	0.443553i	$0.853718\pi$
$360$	0	0
$361$	35.7300	1.88053
$362$	−44.6241	−2.34539
$363$	23.4692	1.23181
$364$	14.7435	0.772769
$365$	0	0
$366$	−42.6738	−2.23060
$367$	23.1703	1.20948	0.604741	−	0.796422i	$-0.293277\pi$
$367$	23.1703	1.20948	0.604741	+	0.796422i	$0.293277\pi$
$368$	7.38311	0.384871
$369$	14.0807	0.733013
$370$	0	0
$371$	−12.4511	−0.646429
$372$	−5.84359	−0.302976
$373$	−0.768561	−0.0397946	−0.0198973	−	0.999802i	$-0.506334\pi$
$373$	−0.768561	−0.0397946	−0.0198973	+	0.999802i	$0.506334\pi$
$374$	−9.17461	−0.474408
$375$	0	0
$376$	2.72250	0.140402
$377$	−21.2128	−1.09252
$378$	−11.0070	−0.566140
$379$	3.18413	0.163558	0.0817789	−	0.996650i	$-0.473940\pi$
$379$	3.18413	0.163558	0.0817789	+	0.996650i	$0.473940\pi$
$380$	0	0
$381$	17.4026	0.891561
$382$	−1.72714	−0.0883680
$383$	−19.7260	−1.00795	−0.503977	−	0.863717i	$-0.668130\pi$
$383$	−19.7260	−1.00795	−0.503977	+	0.863717i	$0.668130\pi$
$384$	22.0857	1.12706
$385$	0	0
$386$	8.19199	0.416961
$387$	−10.7560	−0.546758
$388$	−21.8546	−1.10950
$389$	4.51662	0.229002	0.114501	−	0.993423i	$-0.463473\pi$
$389$	4.51662	0.229002	0.114501	+	0.993423i	$0.463473\pi$
$390$	0	0
$391$	10.1930	0.515480
$392$	1.47283	0.0743891
$393$	47.8241	2.41241
$394$	1.80911	0.0911415
$395$	0	0
$396$	−1.94586	−0.0977830
$397$	21.9779	1.10304	0.551519	−	0.834162i	$-0.314048\pi$
$397$	21.9779	1.10304	0.551519	+	0.834162i	$0.314048\pi$
$398$	−22.2969	−1.11764
$399$	47.3718	2.37156
$400$	0	0
$401$	−7.89978	−0.394496	−0.197248	−	0.980354i	$-0.563200\pi$
$401$	−7.89978	−0.394496	−0.197248	+	0.980354i	$0.563200\pi$
$402$	45.3701	2.26286
$403$	8.30437	0.413670
$404$	8.42327	0.419073
$405$	0	0
$406$	26.9018	1.33511
$407$	−5.13057	−0.254313
$408$	20.1016	0.995178
$409$	−0.0919677	−0.00454751	−0.00227376	−	0.999997i	$-0.500724\pi$
$409$	−0.0919677	−0.00454751	−0.00227376	+	0.999997i	$0.500724\pi$
$410$	0	0
$411$	−28.4160	−1.40166
$412$	−20.6376	−1.01674
$413$	−19.3565	−0.952469
$414$	5.55864	0.273192
$415$	0	0
$416$	25.5272	1.25157
$417$	−39.0611	−1.91283
$418$	−9.98126	−0.488200
$419$	20.9183	1.02193	0.510963	−	0.859602i	$-0.329289\pi$
$419$	20.9183	1.02193	0.510963	+	0.859602i	$0.329289\pi$
$420$	0	0
$421$	3.95842	0.192922	0.0964608	−	0.995337i	$-0.469248\pi$
$421$	3.95842	0.192922	0.0964608	+	0.995337i	$0.469248\pi$
$422$	7.17038	0.349048
$423$	4.24238	0.206272
$424$	−5.74801	−0.279148
$425$	0	0
$426$	−12.7121	−0.615905
$427$	−29.9107	−1.44748
$428$	−10.5369	−0.509318
$429$	6.81234	0.328903
$430$	0	0
$431$	10.9486	0.527375	0.263688	−	0.964608i	$-0.415061\pi$
$431$	10.9486	0.527375	0.263688	+	0.964608i	$0.415061\pi$
$432$	−10.5170	−0.505999
$433$	30.9442	1.48709	0.743543	−	0.668689i	$-0.233144\pi$
$433$	30.9442	1.48709	0.743543	+	0.668689i	$0.233144\pi$
$434$	−10.5315	−0.505528
$435$	0	0
$436$	7.79293	0.373214
$437$	11.0891	0.530466
$438$	61.1936	2.92395
$439$	28.3938	1.35516	0.677582	−	0.735447i	$-0.263028\pi$
$439$	28.3938	1.35516	0.677582	+	0.735447i	$0.263028\pi$
$440$	0	0
$441$	2.29506	0.109289
$442$	50.0067	2.37857
$443$	8.40244	0.399212	0.199606	−	0.979876i	$-0.436034\pi$
$443$	8.40244	0.399212	0.199606	+	0.979876i	$0.436034\pi$
$444$	−19.6779	−0.933873
$445$	0	0
$446$	−0.0349741	−0.00165607
$447$	−16.5067	−0.780740
$448$	−4.30261	−0.203279
$449$	1.94020	0.0915636	0.0457818	−	0.998951i	$-0.485422\pi$
$449$	1.94020	0.0915636	0.0457818	+	0.998951i	$0.485422\pi$
$450$	0	0
$451$	−5.12288	−0.241227
$452$	−12.6349	−0.594294
$453$	−6.69748	−0.314675
$454$	−8.26396	−0.387847
$455$	0	0
$456$	21.8690	1.02411
$457$	32.9131	1.53961	0.769804	−	0.638280i	$-0.220354\pi$
$457$	32.9131	1.53961	0.769804	+	0.638280i	$0.220354\pi$
$458$	5.87507	0.274524
$459$	−14.5195	−0.677713
$460$	0	0
$461$	22.0106	1.02513	0.512567	−	0.858647i	$-0.328695\pi$
$461$	22.0106	1.02513	0.512567	+	0.858647i	$0.328695\pi$
$462$	−8.63932	−0.401937
$463$	−1.14649	−0.0532819	−0.0266410	−	0.999645i	$-0.508481\pi$
$463$	−1.14649	−0.0532819	−0.0266410	+	0.999645i	$0.508481\pi$
$464$	25.7042	1.19329
$465$	0	0
$466$	6.75387	0.312867
$467$	11.6655	0.539814	0.269907	−	0.962886i	$-0.413007\pi$
$467$	11.6655	0.539814	0.269907	+	0.962886i	$0.413007\pi$
$468$	10.6060	0.490262
$469$	31.8006	1.46842
$470$	0	0
$471$	−41.0723	−1.89251
$472$	−8.93583	−0.411305
$473$	3.91327	0.179933
$474$	64.0697	2.94282
$475$	0	0
$476$	−24.6642	−1.13048
$477$	−8.95693	−0.410109
$478$	−2.66321	−0.121812
$479$	−8.60694	−0.393261	−0.196631	−	0.980478i	$-0.563000\pi$
$479$	−8.60694	−0.393261	−0.196631	+	0.980478i	$0.563000\pi$
$480$	0	0
$481$	27.9645	1.27507
$482$	−1.87948	−0.0856078
$483$	9.59825	0.436735
$484$	−13.2936	−0.604255
$485$	0	0
$486$	−32.9183	−1.49320
$487$	1.99510	0.0904067	0.0452033	−	0.998978i	$-0.485606\pi$
$487$	1.99510	0.0904067	0.0452033	+	0.998978i	$0.485606\pi$
$488$	−13.8082	−0.625067
$489$	−28.5721	−1.29207
$490$	0	0
$491$	−38.9159	−1.75625	−0.878124	−	0.478433i	$-0.841205\pi$
$491$	−38.9159	−1.75625	−0.878124	+	0.478433i	$0.841205\pi$
$492$	−19.6484	−0.885820
$493$	35.4866	1.59823
$494$	54.4034	2.44772
$495$	0	0
$496$	−10.0626	−0.451826
$497$	−8.91012	−0.399674
$498$	71.8343	3.21897
$499$	−3.41314	−0.152793	−0.0763966	−	0.997078i	$-0.524342\pi$
$499$	−3.41314	−0.152793	−0.0763966	+	0.997078i	$0.524342\pi$
$500$	0	0
$501$	−3.98947	−0.178236
$502$	−12.9418	−0.577619
$503$	7.16856	0.319630	0.159815	−	0.987147i	$-0.448910\pi$
$503$	7.16856	0.319630	0.159815	+	0.987147i	$0.448910\pi$
$504$	7.68360	0.342255
$505$	0	0
$506$	−2.02236	−0.0899048
$507$	−7.91760	−0.351633
$508$	−9.85731	−0.437347
$509$	6.96089	0.308536	0.154268	−	0.988029i	$-0.450698\pi$
$509$	6.96089	0.308536	0.154268	+	0.988029i	$0.450698\pi$
$510$	0	0
$511$	42.8915	1.89741
$512$	−17.9733	−0.794315
$513$	−15.7961	−0.697415
$514$	−29.3540	−1.29475
$515$	0	0
$516$	15.0091	0.660738
$517$	−1.54347	−0.0678819
$518$	−35.4642	−1.55821
$519$	14.8109	0.650127
$520$	0	0
$521$	−16.0653	−0.703834	−0.351917	−	0.936031i	$-0.614470\pi$
$521$	−16.0653	−0.703834	−0.351917	+	0.936031i	$0.614470\pi$
$522$	19.3523	0.847027
$523$	28.1079	1.22907	0.614537	−	0.788888i	$-0.289343\pi$
$523$	28.1079	1.22907	0.614537	+	0.788888i	$0.289343\pi$
$524$	−27.0889	−1.18339
$525$	0	0
$526$	−39.5462	−1.72430
$527$	−13.8923	−0.605156
$528$	−8.25470	−0.359240
$529$	−20.7532	−0.902312
$530$	0	0
$531$	−13.9244	−0.604268
$532$	−26.8327	−1.16335
$533$	27.9225	1.20946
$534$	−40.4993	−1.75258
$535$	0	0
$536$	14.6806	0.634106
$537$	−22.8115	−0.984388
$538$	7.73956	0.333676
$539$	−0.834994	−0.0359657
$540$	0	0
$541$	−0.688559	−0.0296034	−0.0148017	−	0.999890i	$-0.504712\pi$
$541$	−0.688559	−0.0296034	−0.0148017	+	0.999890i	$0.504712\pi$
$542$	−12.7106	−0.545965
$543$	−55.4298	−2.37872
$544$	−42.7040	−1.83092
$545$	0	0
$546$	47.0891	2.01523
$547$	16.4885	0.704996	0.352498	−	0.935813i	$-0.385332\pi$
$547$	16.4885	0.704996	0.352498	+	0.935813i	$0.385332\pi$
$548$	16.0956	0.687571
$549$	−21.5168	−0.918315
$550$	0	0
$551$	38.6066	1.64470
$552$	4.43100	0.188596
$553$	44.9074	1.90966
$554$	36.4265	1.54761
$555$	0	0
$556$	22.1253	0.938322
$557$	19.8953	0.842990	0.421495	−	0.906831i	$-0.361505\pi$
$557$	19.8953	0.842990	0.421495	+	0.906831i	$0.361505\pi$
$558$	−7.57602	−0.320719
$559$	−21.3295	−0.902142
$560$	0	0
$561$	−11.3962	−0.481150
$562$	6.24028	0.263230
$563$	5.23265	0.220530	0.110265	−	0.993902i	$-0.464830\pi$
$563$	5.23265	0.220530	0.110265	+	0.993902i	$0.464830\pi$
$564$	−5.91988	−0.249272
$565$	0	0
$566$	10.8250	0.455010
$567$	−31.1954	−1.31008
$568$	−4.11332	−0.172591
$569$	29.3627	1.23095	0.615475	−	0.788157i	$-0.288964\pi$
$569$	29.3627	1.23095	0.615475	+	0.788157i	$0.288964\pi$
$570$	0	0
$571$	28.4996	1.19267	0.596336	−	0.802735i	$-0.296622\pi$
$571$	28.4996	1.19267	0.596336	+	0.802735i	$0.296622\pi$
$572$	−3.85870	−0.161340
$573$	−2.14537	−0.0896239
$574$	−35.4110	−1.47803
$575$	0	0
$576$	−3.09516	−0.128965
$577$	−3.78549	−0.157592	−0.0787959	−	0.996891i	$-0.525108\pi$
$577$	−3.78549	−0.157592	−0.0787959	+	0.996891i	$0.525108\pi$
$578$	−52.9006	−2.20037
$579$	10.1757	0.422887
$580$	0	0
$581$	50.3497	2.08886
$582$	−69.8011	−2.89335
$583$	3.25873	0.134963
$584$	19.8007	0.819359
$585$	0	0
$586$	28.0193	1.15747
$587$	27.2568	1.12501	0.562504	−	0.826795i	$-0.309838\pi$
$587$	27.2568	1.12501	0.562504	+	0.826795i	$0.309838\pi$
$588$	−3.20256	−0.132071
$589$	−15.1137	−0.622749
$590$	0	0
$591$	2.24718	0.0924368
$592$	−33.8853	−1.39268
$593$	−30.4143	−1.24897	−0.624484	−	0.781038i	$-0.714691\pi$
$593$	−30.4143	−1.24897	−0.624484	+	0.781038i	$0.714691\pi$
$594$	2.88078	0.118200
$595$	0	0
$596$	9.34986	0.382985
$597$	−27.6961	−1.13353
$598$	11.0230	0.450763
$599$	1.43537	0.0586475	0.0293237	−	0.999570i	$-0.490665\pi$
$599$	1.43537	0.0586475	0.0293237	+	0.999570i	$0.490665\pi$
$600$	0	0
$601$	−34.3674	−1.40188	−0.700939	−	0.713222i	$-0.747235\pi$
$601$	−34.3674	−1.40188	−0.700939	+	0.713222i	$0.747235\pi$
$602$	27.0498	1.10247
$603$	22.8763	0.931596
$604$	3.79364	0.154361
$605$	0	0
$606$	26.9030	1.09286
$607$	30.4221	1.23479	0.617397	−	0.786652i	$-0.288187\pi$
$607$	30.4221	1.23479	0.617397	+	0.786652i	$0.288187\pi$
$608$	−46.4587	−1.88415
$609$	33.4161	1.35409
$610$	0	0
$611$	8.41278	0.340345
$612$	−17.7426	−0.717202
$613$	18.4853	0.746615	0.373308	−	0.927708i	$-0.378224\pi$
$613$	18.4853	0.746615	0.373308	+	0.927708i	$0.378224\pi$
$614$	−22.6174	−0.912766
$615$	0	0
$616$	−2.79546	−0.112632
$617$	38.3447	1.54370	0.771850	−	0.635804i	$-0.219331\pi$
$617$	38.3447	1.54370	0.771850	+	0.635804i	$0.219331\pi$
$618$	−65.9142	−2.65146
$619$	9.89770	0.397822	0.198911	−	0.980018i	$-0.436260\pi$
$619$	9.89770	0.397822	0.198911	+	0.980018i	$0.436260\pi$
$620$	0	0
$621$	−3.20053	−0.128433
$622$	27.9675	1.12140
$623$	−28.3866	−1.13728
$624$	44.9927	1.80115
$625$	0	0
$626$	35.8747	1.43384
$627$	−12.3982	−0.495138
$628$	23.2645	0.928356
$629$	−46.7813	−1.86529
$630$	0	0
$631$	2.16277	0.0860986	0.0430493	−	0.999073i	$-0.486293\pi$
$631$	2.16277	0.0860986	0.0430493	+	0.999073i	$0.486293\pi$
$632$	20.7313	0.824648
$633$	8.90669	0.354009
$634$	31.4870	1.25051
$635$	0	0
$636$	12.4986	0.495602
$637$	4.55118	0.180324
$638$	−7.04080	−0.278748
$639$	−6.40965	−0.253562
$640$	0	0
$641$	6.80536	0.268796	0.134398	−	0.990927i	$-0.457090\pi$
$641$	6.80536	0.268796	0.134398	+	0.990927i	$0.457090\pi$
$642$	−33.6535	−1.32820
$643$	29.2180	1.15224	0.576122	−	0.817363i	$-0.304565\pi$
$643$	29.2180	1.15224	0.576122	+	0.817363i	$0.304565\pi$
$644$	−5.43672	−0.214237
$645$	0	0
$646$	−91.0106	−3.58076
$647$	−42.1419	−1.65677	−0.828385	−	0.560160i	$-0.810740\pi$
$647$	−42.1419	−1.65677	−0.828385	+	0.560160i	$0.810740\pi$
$648$	−14.4012	−0.565734
$649$	5.06601	0.198858
$650$	0	0
$651$	−13.0817	−0.512713
$652$	16.1840	0.633815
$653$	−42.3133	−1.65585	−0.827925	−	0.560839i	$-0.810479\pi$
$653$	−42.3133	−1.65585	−0.827925	+	0.560839i	$0.810479\pi$
$654$	24.8897	0.973266
$655$	0	0
$656$	−33.8345	−1.32102
$657$	30.8548	1.20376
$658$	−10.6690	−0.415920
$659$	−18.9681	−0.738894	−0.369447	−	0.929252i	$-0.620453\pi$
$659$	−18.9681	−0.738894	−0.369447	+	0.929252i	$0.620453\pi$
$660$	0	0
$661$	−22.4832	−0.874495	−0.437247	−	0.899341i	$-0.644047\pi$
$661$	−22.4832	−0.874495	−0.437247	+	0.899341i	$0.644047\pi$
$662$	51.0490	1.98408
$663$	62.1158	2.41238
$664$	23.2438	0.902033
$665$	0	0
$666$	−25.5118	−0.988561
$667$	7.82230	0.302881
$668$	2.25975	0.0874323
$669$	−0.0434431	−0.00167961
$670$	0	0
$671$	7.82830	0.302208
$672$	−40.2125	−1.55123
$673$	9.83248	0.379014	0.189507	−	0.981879i	$-0.439311\pi$
$673$	9.83248	0.379014	0.189507	+	0.981879i	$0.439311\pi$
$674$	41.9679	1.61654
$675$	0	0
$676$	4.48475	0.172490
$677$	37.2861	1.43302	0.716512	−	0.697575i	$-0.245738\pi$
$677$	37.2861	1.43302	0.716512	+	0.697575i	$0.245738\pi$
$678$	−40.3543	−1.54980
$679$	−48.9247	−1.87756
$680$	0	0
$681$	−10.2651	−0.393359
$682$	2.75633	0.105545
$683$	39.5287	1.51252	0.756261	−	0.654270i	$-0.227024\pi$
$683$	39.5287	1.51252	0.756261	+	0.654270i	$0.227024\pi$
$684$	−19.3026	−0.738053
$685$	0	0
$686$	30.3136	1.15738
$687$	7.29773	0.278426
$688$	25.8456	0.985353
$689$	−17.7619	−0.676674
$690$	0	0
$691$	−0.510033	−0.0194026	−0.00970129	−	0.999953i	$-0.503088\pi$
$691$	−0.510033	−0.0194026	−0.00970129	+	0.999953i	$0.503088\pi$
$692$	−8.38932	−0.318914
$693$	−4.35608	−0.165474
$694$	−50.7299	−1.92568
$695$	0	0
$696$	15.4264	0.584737
$697$	−46.7112	−1.76931
$698$	1.10333	0.0417616
$699$	8.38932	0.317313
$700$	0	0
$701$	31.1167	1.17526	0.587631	−	0.809129i	$-0.300060\pi$
$701$	31.1167	1.17526	0.587631	+	0.809129i	$0.300060\pi$
$702$	−15.7018	−0.592628
$703$	−50.8944	−1.91952
$704$	1.12609	0.0424410
$705$	0	0
$706$	−15.3197	−0.576564
$707$	18.8567	0.709178
$708$	19.4303	0.730236
$709$	−13.2510	−0.497653	−0.248827	−	0.968548i	$-0.580045\pi$
$709$	−13.2510	−0.497653	−0.248827	+	0.968548i	$0.580045\pi$
$710$	0	0
$711$	32.3049	1.21153
$712$	−13.1046	−0.491114
$713$	−3.06227	−0.114683
$714$	−78.7745	−2.94806
$715$	0	0
$716$	12.9211	0.482883
$717$	−3.30811	−0.123544
$718$	61.4426	2.29302
$719$	10.2589	0.382591	0.191296	−	0.981532i	$-0.438731\pi$
$719$	10.2589	0.382591	0.191296	+	0.981532i	$0.438731\pi$
$720$	0	0
$721$	−46.2002	−1.72059
$722$	−64.6395	−2.40563
$723$	−2.33459	−0.0868245
$724$	31.3970	1.16686
$725$	0	0
$726$	−42.4583	−1.57577
$727$	41.1951	1.52784	0.763922	−	0.645309i	$-0.223271\pi$
$727$	41.1951	1.52784	0.763922	+	0.645309i	$0.223271\pi$
$728$	15.2368	0.564714
$729$	−8.04643	−0.298016
$730$	0	0
$731$	35.6818	1.31974
$732$	30.0248	1.10975
$733$	8.80769	0.325320	0.162660	−	0.986682i	$-0.447993\pi$
$733$	8.80769	0.325320	0.162660	+	0.986682i	$0.447993\pi$
$734$	−41.9176	−1.54721
$735$	0	0
$736$	−9.41324	−0.346977
$737$	−8.32292	−0.306579
$738$	−25.4735	−0.937694
$739$	10.8085	0.397596	0.198798	−	0.980040i	$-0.436296\pi$
$739$	10.8085	0.397596	0.198798	+	0.980040i	$0.436296\pi$
$740$	0	0
$741$	67.5772	2.48251
$742$	22.5254	0.826933
$743$	46.3861	1.70174	0.850870	−	0.525376i	$-0.176075\pi$
$743$	46.3861	1.70174	0.850870	+	0.525376i	$0.176075\pi$
$744$	−6.03913	−0.221405
$745$	0	0
$746$	1.39041	0.0509065
$747$	36.2200	1.32522
$748$	6.45515	0.236024
$749$	−23.5883	−0.861896
$750$	0	0
$751$	28.1339	1.02662	0.513311	−	0.858203i	$-0.328419\pi$
$751$	28.1339	1.02662	0.513311	+	0.858203i	$0.328419\pi$
$752$	−10.1940	−0.371737
$753$	−16.0756	−0.585828
$754$	38.3762	1.39758
$755$	0	0
$756$	7.74442	0.281662
$757$	32.4601	1.17978	0.589891	−	0.807483i	$-0.299171\pi$
$757$	32.4601	1.17978	0.589891	+	0.807483i	$0.299171\pi$
$758$	−5.76044	−0.209228
$759$	−2.51207	−0.0911825
$760$	0	0
$761$	−5.92476	−0.214772	−0.107386	−	0.994217i	$-0.534248\pi$
$761$	−5.92476	−0.214772	−0.107386	+	0.994217i	$0.534248\pi$
$762$	−31.4831	−1.14051
$763$	17.4456	0.631573
$764$	1.21520	0.0439642
$765$	0	0
$766$	35.6865	1.28941
$767$	−27.6126	−0.997032
$768$	−46.7417	−1.68664
$769$	12.5599	0.452923	0.226461	−	0.974020i	$-0.427284\pi$
$769$	12.5599	0.452923	0.226461	+	0.974020i	$0.427284\pi$
$770$	0	0
$771$	−36.4621	−1.31315
$772$	−5.76380	−0.207444
$773$	14.4118	0.518357	0.259179	−	0.965829i	$-0.416548\pi$
$773$	14.4118	0.518357	0.259179	+	0.965829i	$0.416548\pi$
$774$	19.4588	0.699431
$775$	0	0
$776$	−22.5859	−0.810786
$777$	−44.0519	−1.58035
$778$	−8.17105	−0.292946
$779$	−50.8181	−1.82075
$780$	0	0
$781$	2.33198	0.0834446
$782$	−18.4401	−0.659418
$783$	−11.1426	−0.398204
$784$	−5.51479	−0.196957
$785$	0	0
$786$	−86.5190	−3.08603
$787$	−3.51279	−0.125217	−0.0626087	−	0.998038i	$-0.519942\pi$
$787$	−3.51279	−0.125217	−0.0626087	+	0.998038i	$0.519942\pi$
$788$	−1.27287	−0.0453441
$789$	−49.1224	−1.74880
$790$	0	0
$791$	−28.2849	−1.00570
$792$	−2.01097	−0.0714566
$793$	−42.6685	−1.51520
$794$	−39.7604	−1.41104
$795$	0	0
$796$	15.6878	0.556041
$797$	10.4129	0.368843	0.184421	−	0.982847i	$-0.440959\pi$
$797$	10.4129	0.368843	0.184421	+	0.982847i	$0.440959\pi$
$798$	−85.7006	−3.03377
$799$	−14.0736	−0.497888
$800$	0	0
$801$	−20.4204	−0.721519
$802$	14.2916	0.504652
$803$	−11.2257	−0.396145
$804$	−31.9219	−1.12580
$805$	0	0
$806$	−15.0235	−0.529180
$807$	9.61371	0.338418
$808$	8.70511	0.306245
$809$	−22.6403	−0.795990	−0.397995	−	0.917388i	$-0.630294\pi$
$809$	−22.6403	−0.795990	−0.397995	+	0.917388i	$0.630294\pi$
$810$	0	0
$811$	14.2638	0.500870	0.250435	−	0.968133i	$-0.419426\pi$
$811$	14.2638	0.500870	0.250435	+	0.968133i	$0.419426\pi$
$812$	−18.9278	−0.664237
$813$	−15.7884	−0.553724
$814$	9.28175	0.325325
$815$	0	0
$816$	−75.2675	−2.63489
$817$	38.8190	1.35811
$818$	0.166379	0.00581732
$819$	23.7430	0.829649
$820$	0	0
$821$	46.6504	1.62811	0.814054	−	0.580789i	$-0.197256\pi$
$821$	46.6504	1.62811	0.814054	+	0.580789i	$0.197256\pi$
$822$	51.4076	1.79304
$823$	1.29527	0.0451502	0.0225751	−	0.999745i	$-0.492814\pi$
$823$	1.29527	0.0451502	0.0225751	+	0.999745i	$0.492814\pi$
$824$	−21.3282	−0.743001
$825$	0	0
$826$	35.0179	1.21843
$827$	18.4609	0.641948	0.320974	−	0.947088i	$-0.395990\pi$
$827$	18.4609	0.641948	0.320974	+	0.947088i	$0.395990\pi$
$828$	−3.91100	−0.135917
$829$	−35.6828	−1.23932	−0.619658	−	0.784872i	$-0.712729\pi$
$829$	−35.6828	−1.23932	−0.619658	+	0.784872i	$0.712729\pi$
$830$	0	0
$831$	45.2472	1.56961
$832$	−6.13780	−0.212790
$833$	−7.61360	−0.263795
$834$	70.6657	2.44695
$835$	0	0
$836$	7.02271	0.242886
$837$	4.36210	0.150776
$838$	−37.8435	−1.30728
$839$	−15.4370	−0.532943	−0.266472	−	0.963843i	$-0.585858\pi$
$839$	−15.4370	−0.532943	−0.266472	+	0.963843i	$0.585858\pi$
$840$	0	0
$841$	−1.76682	−0.0609248
$842$	−7.16120	−0.246791
$843$	7.75137	0.266971
$844$	−5.04500	−0.173656
$845$	0	0
$846$	−7.67492	−0.263869
$847$	−29.7596	−1.02255
$848$	21.5226	0.739088
$849$	13.4463	0.461477
$850$	0	0
$851$	−10.3120	−0.353491
$852$	8.94412	0.306420
$853$	9.33840	0.319741	0.159870	−	0.987138i	$-0.448892\pi$
$853$	9.33840	0.319741	0.159870	+	0.987138i	$0.448892\pi$
$854$	54.1117	1.85166
$855$	0	0
$856$	−10.8894	−0.372193
$857$	−2.80292	−0.0957460	−0.0478730	−	0.998853i	$-0.515244\pi$
$857$	−2.80292	−0.0957460	−0.0478730	+	0.998853i	$0.515244\pi$
$858$	−12.3242	−0.420743
$859$	−37.2809	−1.27201	−0.636003	−	0.771686i	$-0.719414\pi$
$859$	−37.2809	−1.27201	−0.636003	+	0.771686i	$0.719414\pi$
$860$	0	0
$861$	−43.9858	−1.49903
$862$	−19.8072	−0.674635
$863$	43.5426	1.48221	0.741104	−	0.671390i	$-0.234303\pi$
$863$	43.5426	1.48221	0.741104	+	0.671390i	$0.234303\pi$
$864$	13.4088	0.456178
$865$	0	0
$866$	−55.9814	−1.90233
$867$	−65.7105	−2.23164
$868$	7.40985	0.251507
$869$	−11.7533	−0.398702
$870$	0	0
$871$	45.3645	1.53712
$872$	8.05369	0.272732
$873$	−35.1948	−1.19116
$874$	−20.0615	−0.678589
$875$	0	0
$876$	−43.0552	−1.45470
$877$	−7.12426	−0.240569	−0.120285	−	0.992739i	$-0.538381\pi$
$877$	−7.12426	−0.240569	−0.120285	+	0.992739i	$0.538381\pi$
$878$	−51.3675	−1.73357
$879$	34.8042	1.17392
$880$	0	0
$881$	3.36338	0.113315	0.0566576	−	0.998394i	$-0.481956\pi$
$881$	3.36338	0.113315	0.0566576	+	0.998394i	$0.481956\pi$
$882$	−4.15201	−0.139805
$883$	49.2397	1.65705	0.828523	−	0.559955i	$-0.189181\pi$
$883$	49.2397	1.65705	0.828523	+	0.559955i	$0.189181\pi$
$884$	−35.1842	−1.18337
$885$	0	0
$886$	−15.2009	−0.510685
$887$	−39.4318	−1.32399	−0.661995	−	0.749508i	$-0.730290\pi$
$887$	−39.4318	−1.32399	−0.661995	+	0.749508i	$0.730290\pi$
$888$	−20.3364	−0.682444
$889$	−22.0670	−0.740103
$890$	0	0
$891$	8.16452	0.273522
$892$	0.0246074	0.000823916 0
$893$	−15.3110	−0.512363
$894$	29.8624	0.998748
$895$	0	0
$896$	−28.0053	−0.935592
$897$	13.6922	0.457169
$898$	−3.51003	−0.117131
$899$	−10.6612	−0.355572
$900$	0	0
$901$	29.7136	0.989902
$902$	9.26784	0.308585
$903$	33.6000	1.11814
$904$	−13.0576	−0.434290
$905$	0	0
$906$	12.1165	0.402542
$907$	8.54232	0.283643	0.141822	−	0.989892i	$-0.454704\pi$
$907$	8.54232	0.283643	0.141822	+	0.989892i	$0.454704\pi$
$908$	5.81443	0.192959
$909$	13.5649	0.449919
$910$	0	0
$911$	2.86873	0.0950451	0.0475226	−	0.998870i	$-0.484867\pi$
$911$	2.86873	0.0950451	0.0475226	+	0.998870i	$0.484867\pi$
$912$	−81.8852	−2.71149
$913$	−13.1776	−0.436116
$914$	−59.5432	−1.96952
$915$	0	0
$916$	−4.13364	−0.136579
$917$	−60.6424	−2.00259
$918$	26.2673	0.866952
$919$	52.6798	1.73775	0.868873	−	0.495035i	$-0.164845\pi$
$919$	52.6798	1.73775	0.868873	+	0.495035i	$0.164845\pi$
$920$	0	0
$921$	−28.0943	−0.925738
$922$	−39.8195	−1.31138
$923$	−12.7106	−0.418373
$924$	6.07853	0.199969
$925$	0	0
$926$	2.07412	0.0681599
$927$	−33.2349	−1.09158
$928$	−32.7720	−1.07579
$929$	16.4388	0.539339	0.269670	−	0.962953i	$-0.413085\pi$
$929$	16.4388	0.539339	0.269670	+	0.962953i	$0.413085\pi$
$930$	0	0
$931$	−8.28300	−0.271465
$932$	−4.75195	−0.155655
$933$	34.7399	1.13733
$934$	−21.1041	−0.690548
$935$	0	0
$936$	10.9609	0.358268
$937$	−27.3664	−0.894023	−0.447011	−	0.894528i	$-0.647512\pi$
$937$	−27.3664	−0.894023	−0.447011	+	0.894528i	$0.647512\pi$
$938$	−57.5307	−1.87844
$939$	44.5618	1.45422
$940$	0	0
$941$	28.6192	0.932959	0.466479	−	0.884532i	$-0.345522\pi$
$941$	28.6192	0.932959	0.466479	+	0.884532i	$0.345522\pi$
$942$	74.3043	2.42096
$943$	−10.2965	−0.335301
$944$	33.4589	1.08899
$945$	0	0
$946$	−7.07953	−0.230175
$947$	−54.5673	−1.77320	−0.886600	−	0.462537i	$-0.846939\pi$
$947$	−54.5673	−1.77320	−0.886600	+	0.462537i	$0.846939\pi$
$948$	−45.0788	−1.46409
$949$	61.1860	1.98618
$950$	0	0
$951$	39.1116	1.26828
$952$	−25.4894	−0.826118
$953$	−14.3729	−0.465583	−0.232791	−	0.972527i	$-0.574786\pi$
$953$	−14.3729	−0.465583	−0.232791	+	0.972527i	$0.574786\pi$
$954$	16.2040	0.524625
$955$	0	0
$956$	1.87381	0.0606032
$957$	−8.74573	−0.282709
$958$	15.5709	0.503072
$959$	36.0323	1.16354
$960$	0	0
$961$	−26.8264	−0.865366
$962$	−50.5907	−1.63111
$963$	−16.9686	−0.546806
$964$	1.32238	0.0425910
$965$	0	0
$966$	−17.3643	−0.558686
$967$	55.8404	1.79571	0.897853	−	0.440295i	$-0.145126\pi$
$967$	55.8404	1.79571	0.897853	+	0.440295i	$0.145126\pi$
$968$	−13.7384	−0.441569
$969$	−113.049	−3.63165
$970$	0	0
$971$	1.97942	0.0635227	0.0317613	−	0.999495i	$-0.489888\pi$
$971$	1.97942	0.0635227	0.0317613	+	0.999495i	$0.489888\pi$
$972$	23.1609	0.742888
$973$	49.5307	1.58788
$974$	−3.60935	−0.115651
$975$	0	0
$976$	51.7027	1.65496
$977$	−42.9153	−1.37298	−0.686492	−	0.727138i	$-0.740850\pi$
$977$	−42.9153	−1.37298	−0.686492	+	0.727138i	$0.740850\pi$
$978$	51.6899	1.65286
$979$	7.42939	0.237444
$980$	0	0
$981$	12.5498	0.400684
$982$	70.4030	2.24665
$983$	−13.5082	−0.430844	−0.215422	−	0.976521i	$-0.569113\pi$
$983$	−13.5082	−0.430844	−0.215422	+	0.976521i	$0.569113\pi$
$984$	−20.3059	−0.647328
$985$	0	0
$986$	−64.1990	−2.04451
$987$	−13.2525	−0.421831
$988$	−38.2776	−1.21777
$989$	7.86533	0.250103
$990$	0	0
$991$	52.5064	1.66792	0.833961	−	0.551824i	$-0.186068\pi$
$991$	52.5064	1.66792	0.833961	+	0.551824i	$0.186068\pi$
$992$	12.8296	0.407339
$993$	63.4106	2.01228
$994$	16.1194	0.511275
$995$	0	0
$996$	−50.5419	−1.60148
$997$	−10.1737	−0.322205	−0.161102	−	0.986938i	$-0.551505\pi$
$997$	−10.1737	−0.322205	−0.161102	+	0.986938i	$0.551505\pi$
$998$	6.17474	0.195458
$999$	14.6891	0.464742

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4925.2.a.s.1.10		49
5.2	odd	4		985.2.b.a.789.21	✓	98
5.3	odd	4		985.2.b.a.789.78	yes	98
5.4	even	2		4925.2.a.r.1.40		49

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
985.2.b.a.789.21	✓	98	5.2	odd	4
985.2.b.a.789.78	yes	98	5.3	odd	4
4925.2.a.r.1.40		49	5.4	even	2
4925.2.a.s.1.10		49	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4925 = 5^{2} \cdot 197 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4925.a (trivial)

\(f(q)\)	\(=\)	\(q-1.80911 q^{2} -2.24718 q^{3} +1.27287 q^{4} +4.06540 q^{6} +2.84950 q^{7} +1.31546 q^{8} +2.04984 q^{9} -0.745776 q^{11} -2.86037 q^{12} +4.06489 q^{13} -5.15504 q^{14} -4.92554 q^{16} -6.80009 q^{17} -3.70837 q^{18} -7.39798 q^{19} -6.40334 q^{21} +1.34919 q^{22} -1.49894 q^{23} -2.95608 q^{24} -7.35382 q^{26} +2.13519 q^{27} +3.62703 q^{28} -5.21854 q^{29} +2.04295 q^{31} +6.27992 q^{32} +1.67590 q^{33} +12.3021 q^{34} +2.60917 q^{36} +6.87951 q^{37} +13.3837 q^{38} -9.13456 q^{39} +6.86920 q^{41} +11.5843 q^{42} -5.24725 q^{43} -0.949275 q^{44} +2.71175 q^{46} +2.06962 q^{47} +11.0686 q^{48} +1.11963 q^{49} +15.2811 q^{51} +5.17407 q^{52} -4.36958 q^{53} -3.86279 q^{54} +3.74840 q^{56} +16.6246 q^{57} +9.44090 q^{58} -6.79294 q^{59} -10.4968 q^{61} -3.69592 q^{62} +5.84100 q^{63} -1.50995 q^{64} -3.03188 q^{66} +11.1601 q^{67} -8.65562 q^{68} +3.36840 q^{69} -3.12691 q^{71} +2.69648 q^{72} +15.0523 q^{73} -12.4458 q^{74} -9.41665 q^{76} -2.12509 q^{77} +16.5254 q^{78} +15.7598 q^{79} -10.9477 q^{81} -12.4271 q^{82} +17.6697 q^{83} -8.15061 q^{84} +9.49284 q^{86} +11.7270 q^{87} -0.981038 q^{88} -9.96196 q^{89} +11.5829 q^{91} -1.90796 q^{92} -4.59089 q^{93} -3.74416 q^{94} -14.1121 q^{96} -17.1696 q^{97} -2.02553 q^{98} -1.52872 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 49 q + 5 q^{2} + 22 q^{3} + 49 q^{4} + 2 q^{6} + 32 q^{7} + 15 q^{8} + 51 q^{9} - 2 q^{11} + 44 q^{12} + 32 q^{13} - 8 q^{14} + 49 q^{16} + 14 q^{17} + 25 q^{18} + 4 q^{19} + 10 q^{21} + 38 q^{22} + 24 q^{23}+ \cdots + 22 q^{99}+O(q^{100}) \) 49 * q + 5 * q^2 + 22 * q^3 + 49 * q^4 + 2 * q^6 + 32 * q^7 + 15 * q^8 + 51 * q^9 - 2 * q^11 + 44 * q^12 + 32 * q^13 - 8 * q^14 + 49 * q^16 + 14 * q^17 + 25 * q^18 + 4 * q^19 + 10 * q^21 + 38 * q^22 + 24 * q^23 - 2 * q^26 + 88 * q^27 + 80 * q^28 + 6 * q^29 - 2 * q^31 + 35 * q^32 + 48 * q^33 - 12 * q^34 + 43 * q^36 + 60 * q^37 + 34 * q^38 - 2 * q^41 + 36 * q^42 + 80 * q^43 - 8 * q^44 - 22 * q^46 + 48 * q^47 + 88 * q^48 + 65 * q^49 + 16 * q^51 + 96 * q^52 + 18 * q^53 + 42 * q^54 - 22 * q^56 + 40 * q^57 + 52 * q^58 + 8 * q^59 + 16 * q^61 + 38 * q^62 + 82 * q^63 + 55 * q^64 + 42 * q^66 + 146 * q^67 - 68 * q^68 - 6 * q^69 + 75 * q^72 + 78 * q^73 - 24 * q^74 - 4 * q^76 + 32 * q^77 - 38 * q^78 + 10 * q^79 + 41 * q^81 + 96 * q^82 + 64 * q^83 + 42 * q^84 - 28 * q^86 + 56 * q^87 + 72 * q^88 + 4 * q^89 + 60 * q^92 - 28 * q^93 - 8 * q^94 - 34 * q^96 + 124 * q^97 - 19 * q^98 + 22 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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