LMFDB - Embedded newform 4925.2.a.r.1.17

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4925,2,Mod(1,4925)]

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

lf = mfeigenbasis(mf)

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")

//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);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$39.3263229955$
Analytic rank:	$1$
Dimension:	$49$
Twist minimal:	no (minimal twist has level 985)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.17
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)

$f(q)$	$=$	\(q-1.27373 q^{2} -3.21433 q^{3} -0.377621 q^{4} +4.09418 q^{6} +0.234977 q^{7} +3.02844 q^{8} +7.33194 q^{9} -0.577101 q^{11} +1.21380 q^{12} +3.89397 q^{13} -0.299296 q^{14} -3.10216 q^{16} -3.49246 q^{17} -9.33888 q^{18} +7.16189 q^{19} -0.755293 q^{21} +0.735069 q^{22} -4.85751 q^{23} -9.73441 q^{24} -4.95986 q^{26} -13.9243 q^{27} -0.0887321 q^{28} +8.44483 q^{29} -5.57555 q^{31} -2.10557 q^{32} +1.85499 q^{33} +4.44844 q^{34} -2.76869 q^{36} -5.18286 q^{37} -9.12229 q^{38} -12.5165 q^{39} +4.90259 q^{41} +0.962037 q^{42} -3.12584 q^{43} +0.217925 q^{44} +6.18714 q^{46} -6.99306 q^{47} +9.97138 q^{48} -6.94479 q^{49} +11.2259 q^{51} -1.47045 q^{52} +12.1609 q^{53} +17.7357 q^{54} +0.711612 q^{56} -23.0207 q^{57} -10.7564 q^{58} -14.3710 q^{59} -7.71724 q^{61} +7.10173 q^{62} +1.72283 q^{63} +8.88625 q^{64} -2.36276 q^{66} -2.57664 q^{67} +1.31883 q^{68} +15.6136 q^{69} +7.82253 q^{71} +22.2043 q^{72} -8.14734 q^{73} +6.60155 q^{74} -2.70448 q^{76} -0.135605 q^{77} +15.9426 q^{78} -0.388882 q^{79} +22.7615 q^{81} -6.24456 q^{82} +5.26069 q^{83} +0.285214 q^{84} +3.98147 q^{86} -27.1445 q^{87} -1.74771 q^{88} +8.36693 q^{89} +0.914993 q^{91} +1.83430 q^{92} +17.9217 q^{93} +8.90725 q^{94} +6.76801 q^{96} +8.83287 q^{97} +8.84576 q^{98} -4.23127 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1.27373	−0.900661	−0.450330	−	0.892862i	$-0.648694\pi$
$2$	−1.27373	−0.900661	−0.450330	+	0.892862i	$0.648694\pi$
$3$	−3.21433	−1.85580	−0.927898	−	0.372834i	$-0.878386\pi$
$3$	−3.21433	−1.85580	−0.927898	+	0.372834i	$0.878386\pi$
$4$	−0.377621	−0.188810
$5$	0	0
$5$	0	0
$6$	4.09418	1.67144
$7$	0.234977	0.0888128	0.0444064	−	0.999014i	$-0.485860\pi$
$7$	0.234977	0.0888128	0.0444064	+	0.999014i	$0.485860\pi$
$8$	3.02844	1.07071
$9$	7.33194	2.44398
$10$	0	0
$11$	−0.577101	−0.174002	−0.0870012	−	0.996208i	$-0.527728\pi$
$11$	−0.577101	−0.174002	−0.0870012	+	0.996208i	$0.527728\pi$
$12$	1.21380	0.350394
$13$	3.89397	1.07999	0.539997	−	0.841667i	$-0.318425\pi$
$13$	3.89397	1.07999	0.539997	+	0.841667i	$0.318425\pi$
$14$	−0.299296	−0.0799902
$15$	0	0
$16$	−3.10216	−0.775540
$17$	−3.49246	−0.847047	−0.423523	−	0.905885i	$-0.639207\pi$
$17$	−3.49246	−0.847047	−0.423523	+	0.905885i	$0.639207\pi$
$18$	−9.33888	−2.20120
$19$	7.16189	1.64305	0.821525	−	0.570173i	$-0.193124\pi$
$19$	7.16189	1.64305	0.821525	+	0.570173i	$0.193124\pi$
$20$	0	0
$21$	−0.755293	−0.164818
$22$	0.735069	0.156717
$23$	−4.85751	−1.01286	−0.506430	−	0.862281i	$-0.669035\pi$
$23$	−4.85751	−1.01286	−0.506430	+	0.862281i	$0.669035\pi$
$24$	−9.73441	−1.98703
$25$	0	0
$26$	−4.95986	−0.972708
$27$	−13.9243	−2.67973
$28$	−0.0887321	−0.0167688
$29$	8.44483	1.56817	0.784083	−	0.620656i	$-0.213134\pi$
$29$	8.44483	1.56817	0.784083	+	0.620656i	$0.213134\pi$
$30$	0	0
$31$	−5.57555	−1.00140	−0.500699	−	0.865621i	$-0.666924\pi$
$31$	−5.57555	−1.00140	−0.500699	+	0.865621i	$0.666924\pi$
$32$	−2.10557	−0.372216
$33$	1.85499	0.322913
$34$	4.44844	0.762901
$35$	0	0
$36$	−2.76869	−0.461449
$37$	−5.18286	−0.852057	−0.426028	−	0.904710i	$-0.640088\pi$
$37$	−5.18286	−0.852057	−0.426028	+	0.904710i	$0.640088\pi$
$38$	−9.12229	−1.47983
$39$	−12.5165	−2.00425
$40$	0	0
$41$	4.90259	0.765656	0.382828	−	0.923820i	$-0.374950\pi$
$41$	4.90259	0.765656	0.382828	+	0.923820i	$0.374950\pi$
$42$	0.962037	0.148446
$43$	−3.12584	−0.476686	−0.238343	−	0.971181i	$-0.576604\pi$
$43$	−3.12584	−0.476686	−0.238343	+	0.971181i	$0.576604\pi$
$44$	0.217925	0.0328535
$45$	0	0
$46$	6.18714	0.912243
$47$	−6.99306	−1.02004	−0.510021	−	0.860162i	$-0.670363\pi$
$47$	−6.99306	−1.02004	−0.510021	+	0.860162i	$0.670363\pi$
$48$	9.97138	1.43924
$49$	−6.94479	−0.992112
$50$	0	0
$51$	11.2259	1.57195
$52$	−1.47045	−0.203914
$53$	12.1609	1.67042	0.835212	−	0.549929i	$-0.185345\pi$
$53$	12.1609	1.67042	0.835212	+	0.549929i	$0.185345\pi$
$54$	17.7357	2.41353
$55$	0	0
$56$	0.711612	0.0950932
$57$	−23.0207	−3.04917
$58$	−10.7564	−1.41239
$59$	−14.3710	−1.87095	−0.935475	−	0.353393i	$-0.885028\pi$
$59$	−14.3710	−1.87095	−0.935475	+	0.353393i	$0.885028\pi$
$60$	0	0
$61$	−7.71724	−0.988092	−0.494046	−	0.869436i	$-0.664482\pi$
$61$	−7.71724	−0.988092	−0.494046	+	0.869436i	$0.664482\pi$
$62$	7.10173	0.901920
$63$	1.72283	0.217057
$64$	8.88625	1.11078
$65$	0	0
$66$	−2.36276	−0.290835
$67$	−2.57664	−0.314787	−0.157393	−	0.987536i	$-0.550309\pi$
$67$	−2.57664	−0.314787	−0.157393	+	0.987536i	$0.550309\pi$
$68$	1.31883	0.159931
$69$	15.6136	1.87966
$70$	0	0
$71$	7.82253	0.928363	0.464182	−	0.885740i	$-0.346348\pi$
$71$	7.82253	0.928363	0.464182	+	0.885740i	$0.346348\pi$
$72$	22.2043	2.61680
$73$	−8.14734	−0.953574	−0.476787	−	0.879019i	$-0.658199\pi$
$73$	−8.14734	−0.953574	−0.476787	+	0.879019i	$0.658199\pi$
$74$	6.60155	0.767414
$75$	0	0
$76$	−2.70448	−0.310225
$77$	−0.135605	−0.0154537
$78$	15.9426	1.80515
$79$	−0.388882	−0.0437526	−0.0218763	−	0.999761i	$-0.506964\pi$
$79$	−0.388882	−0.0437526	−0.0218763	+	0.999761i	$0.506964\pi$
$80$	0	0
$81$	22.7615	2.52906
$82$	−6.24456	−0.689597
$83$	5.26069	0.577436	0.288718	−	0.957414i	$-0.406771\pi$
$83$	5.26069	0.577436	0.288718	+	0.957414i	$0.406771\pi$
$84$	0.285214	0.0311195
$85$	0	0
$86$	3.98147	0.429333
$87$	−27.1445	−2.91020
$88$	−1.74771	−0.186307
$89$	8.36693	0.886893	0.443446	−	0.896301i	$-0.353756\pi$
$89$	8.36693	0.886893	0.443446	+	0.896301i	$0.353756\pi$
$90$	0	0
$91$	0.914993	0.0959173
$92$	1.83430	0.191239
$93$	17.9217	1.85839
$94$	8.90725	0.918712
$95$	0	0
$96$	6.76801	0.690757
$97$	8.83287	0.896842	0.448421	−	0.893822i	$-0.351987\pi$
$97$	8.83287	0.896842	0.448421	+	0.893822i	$0.351987\pi$
$98$	8.84576	0.893556
$99$	−4.23127	−0.425258
$100$	0	0
$101$	1.70238	0.169393	0.0846967	−	0.996407i	$-0.473008\pi$
$101$	1.70238	0.169393	0.0846967	+	0.996407i	$0.473008\pi$
$102$	−14.2988	−1.41579
$103$	−2.80970	−0.276848	−0.138424	−	0.990373i	$-0.544204\pi$
$103$	−2.80970	−0.276848	−0.138424	+	0.990373i	$0.544204\pi$
$104$	11.7927	1.15637
$105$	0	0
$106$	−15.4896	−1.50448
$107$	−6.50076	−0.628452	−0.314226	−	0.949348i	$-0.601745\pi$
$107$	−6.50076	−0.628452	−0.314226	+	0.949348i	$0.601745\pi$
$108$	5.25810	0.505961
$109$	11.9904	1.14847	0.574236	−	0.818690i	$-0.305299\pi$
$109$	11.9904	1.14847	0.574236	+	0.818690i	$0.305299\pi$
$110$	0	0
$111$	16.6594	1.58124
$112$	−0.728935	−0.0688779
$113$	−9.74277	−0.916522	−0.458261	−	0.888818i	$-0.651528\pi$
$113$	−9.74277	−0.916522	−0.458261	+	0.888818i	$0.651528\pi$
$114$	29.3221	2.74626
$115$	0	0
$116$	−3.18895	−0.296086
$117$	28.5504	2.63948
$118$	18.3048	1.68509
$119$	−0.820647	−0.0752286
$120$	0	0
$121$	−10.6670	−0.969723
$122$	9.82965	0.889935
$123$	−15.7586	−1.42090
$124$	2.10544	0.189075
$125$	0	0
$126$	−2.19442	−0.195494
$127$	10.3084	0.914723	0.457361	−	0.889281i	$-0.348795\pi$
$127$	10.3084	0.914723	0.457361	+	0.889281i	$0.348795\pi$
$128$	−7.10750	−0.628220
$129$	10.0475	0.884633
$130$	0	0
$131$	19.5189	1.70537	0.852686	−	0.522423i	$-0.174972\pi$
$131$	19.5189	1.70537	0.852686	+	0.522423i	$0.174972\pi$
$132$	−0.700485	−0.0609694
$133$	1.68288	0.145924
$134$	3.28193	0.283516
$135$	0	0
$136$	−10.5767	−0.906945
$137$	3.03314	0.259139	0.129569	−	0.991570i	$-0.458641\pi$
$137$	3.03314	0.259139	0.129569	+	0.991570i	$0.458641\pi$
$138$	−19.8875	−1.69294
$139$	2.29531	0.194686	0.0973428	−	0.995251i	$-0.468966\pi$
$139$	2.29531	0.194686	0.0973428	+	0.995251i	$0.468966\pi$
$140$	0	0
$141$	22.4780	1.89299
$142$	−9.96376	−0.836140
$143$	−2.24722	−0.187922
$144$	−22.7448	−1.89540
$145$	0	0
$146$	10.3775	0.858847
$147$	22.3229	1.84116
$148$	1.95716	0.160877
$149$	12.8336	1.05137	0.525685	−	0.850679i	$-0.323809\pi$
$149$	12.8336	1.05137	0.525685	+	0.850679i	$0.323809\pi$
$150$	0	0
$151$	−4.69271	−0.381887	−0.190944	−	0.981601i	$-0.561155\pi$
$151$	−4.69271	−0.381887	−0.190944	+	0.981601i	$0.561155\pi$
$152$	21.6893	1.75924
$153$	−25.6065	−2.07016
$154$	0.172724	0.0139185
$155$	0	0
$156$	4.72650	0.378423
$157$	3.54513	0.282933	0.141466	−	0.989943i	$-0.454818\pi$
$157$	3.54513	0.282933	0.141466	+	0.989943i	$0.454818\pi$
$158$	0.495329	0.0394063
$159$	−39.0891	−3.09997
$160$	0	0
$161$	−1.14140	−0.0899550
$162$	−28.9919	−2.27782
$163$	2.45347	0.192171	0.0960853	−	0.995373i	$-0.469368\pi$
$163$	2.45347	0.192171	0.0960853	+	0.995373i	$0.469368\pi$
$164$	−1.85132	−0.144564
$165$	0	0
$166$	−6.70068	−0.520074
$167$	−6.34224	−0.490777	−0.245389	−	0.969425i	$-0.578916\pi$
$167$	−6.34224	−0.490777	−0.245389	+	0.969425i	$0.578916\pi$
$168$	−2.28736	−0.176474
$169$	2.16303	0.166387
$170$	0	0
$171$	52.5105	4.01558
$172$	1.18038	0.0900034
$173$	−7.16447	−0.544705	−0.272352	−	0.962198i	$-0.587802\pi$
$173$	−7.16447	−0.544705	−0.272352	+	0.962198i	$0.587802\pi$
$174$	34.5747	2.62110
$175$	0	0
$176$	1.79026	0.134946
$177$	46.1933	3.47210
$178$	−10.6572	−0.798790
$179$	−7.39749	−0.552914	−0.276457	−	0.961026i	$-0.589160\pi$
$179$	−7.39749	−0.552914	−0.276457	+	0.961026i	$0.589160\pi$
$180$	0	0
$181$	−20.1924	−1.50089	−0.750445	−	0.660933i	$-0.770161\pi$
$181$	−20.1924	−1.50089	−0.750445	+	0.660933i	$0.770161\pi$
$182$	−1.16545	−0.0863889
$183$	24.8058	1.83370
$184$	−14.7107	−1.08448
$185$	0	0
$186$	−22.8273	−1.67378
$187$	2.01550	0.147388
$188$	2.64073	0.192595
$189$	−3.27188	−0.237994
$190$	0	0
$191$	−20.5634	−1.48791	−0.743957	−	0.668227i	$-0.767053\pi$
$191$	−20.5634	−1.48791	−0.743957	+	0.668227i	$0.767053\pi$
$192$	−28.5634	−2.06138
$193$	−2.68844	−0.193518	−0.0967591	−	0.995308i	$-0.530848\pi$
$193$	−2.68844	−0.193518	−0.0967591	+	0.995308i	$0.530848\pi$
$194$	−11.2507	−0.807750
$195$	0	0
$196$	2.62250	0.187321
$197$	1.00000	0.0712470
$197$	1.00000	0.0712470
$198$	5.38948	0.383014
$199$	−5.33760	−0.378373	−0.189186	−	0.981941i	$-0.560585\pi$
$199$	−5.33760	−0.378373	−0.189186	+	0.981941i	$0.560585\pi$
$200$	0	0
$201$	8.28218	0.584180
$202$	−2.16837	−0.152566
$203$	1.98434	0.139273
$204$	−4.23915	−0.296800
$205$	0	0
$206$	3.57878	0.249346
$207$	−35.6149	−2.47541
$208$	−12.0797	−0.837579
$209$	−4.13313	−0.285895
$210$	0	0
$211$	−27.4653	−1.89079	−0.945395	−	0.325927i	$-0.894324\pi$
$211$	−27.4653	−1.89079	−0.945395	+	0.325927i	$0.894324\pi$
$212$	−4.59220	−0.315393
$213$	−25.1442	−1.72285
$214$	8.28019	0.566022
$215$	0	0
$216$	−42.1689	−2.86923
$217$	−1.31012	−0.0889370
$218$	−15.2725	−1.03438
$219$	26.1883	1.76964
$220$	0	0
$221$	−13.5996	−0.914805
$222$	−21.2196	−1.42416
$223$	24.1738	1.61880	0.809399	−	0.587260i	$-0.199793\pi$
$223$	24.1738	1.61880	0.809399	+	0.587260i	$0.199793\pi$
$224$	−0.494760	−0.0330576
$225$	0	0
$226$	12.4096	0.825476
$227$	17.8340	1.18368	0.591842	−	0.806054i	$-0.298401\pi$
$227$	17.8340	1.18368	0.591842	+	0.806054i	$0.298401\pi$
$228$	8.69309	0.575714
$229$	−26.4544	−1.74816	−0.874079	−	0.485784i	$-0.838534\pi$
$229$	−26.4544	−1.74816	−0.874079	+	0.485784i	$0.838534\pi$
$230$	0	0
$231$	0.435880	0.0286788
$232$	25.5747	1.67906
$233$	30.1629	1.97604	0.988019	−	0.154335i	$-0.0493235\pi$
$233$	30.1629	1.97604	0.988019	+	0.154335i	$0.0493235\pi$
$234$	−36.3654	−2.37728
$235$	0	0
$236$	5.42680	0.353255
$237$	1.25000	0.0811959
$238$	1.04528	0.0677554
$239$	4.58904	0.296840	0.148420	−	0.988924i	$-0.452581\pi$
$239$	4.58904	0.296840	0.148420	+	0.988924i	$0.452581\pi$
$240$	0	0
$241$	23.5615	1.51773	0.758864	−	0.651249i	$-0.225755\pi$
$241$	23.5615	1.51773	0.758864	+	0.651249i	$0.225755\pi$
$242$	13.5868	0.873391
$243$	−31.3902	−2.01368
$244$	2.91419	0.186562
$245$	0	0
$246$	20.0721	1.27975
$247$	27.8882	1.77448
$248$	−16.8852	−1.07221
$249$	−16.9096	−1.07160
$250$	0	0
$251$	12.0057	0.757794	0.378897	−	0.925439i	$-0.376304\pi$
$251$	12.0057	0.757794	0.378897	+	0.925439i	$0.376304\pi$
$252$	−0.650578	−0.0409826
$253$	2.80327	0.176240
$254$	−13.1301	−0.823855
$255$	0	0
$256$	−8.71948	−0.544967
$257$	−14.1205	−0.880812	−0.440406	−	0.897799i	$-0.645165\pi$
$257$	−14.1205	−0.880812	−0.440406	+	0.897799i	$0.645165\pi$
$258$	−12.7978	−0.796754
$259$	−1.21785	−0.0756736
$260$	0	0
$261$	61.9170	3.83257
$262$	−24.8617	−1.53596
$263$	10.3347	0.637268	0.318634	−	0.947878i	$-0.396776\pi$
$263$	10.3347	0.637268	0.318634	+	0.947878i	$0.396776\pi$
$264$	5.61774	0.345748
$265$	0	0
$266$	−2.14352	−0.131428
$267$	−26.8941	−1.64589
$268$	0.972993	0.0594350
$269$	−15.0254	−0.916112	−0.458056	−	0.888923i	$-0.651454\pi$
$269$	−15.0254	−0.916112	−0.458056	+	0.888923i	$0.651454\pi$
$270$	0	0
$271$	−3.06893	−0.186424	−0.0932120	−	0.995646i	$-0.529713\pi$
$271$	−3.06893	−0.186424	−0.0932120	+	0.995646i	$0.529713\pi$
$272$	10.8342	0.656919
$273$	−2.94109	−0.178003
$274$	−3.86339	−0.233396
$275$	0	0
$276$	−5.89604	−0.354900
$277$	−32.3023	−1.94086	−0.970429	−	0.241385i	$-0.922398\pi$
$277$	−32.3023	−1.94086	−0.970429	+	0.241385i	$0.922398\pi$
$278$	−2.92360	−0.175346
$279$	−40.8796	−2.44740
$280$	0	0
$281$	−29.3552	−1.75118	−0.875592	−	0.483052i	$-0.839528\pi$
$281$	−29.3552	−1.75118	−0.875592	+	0.483052i	$0.839528\pi$
$282$	−28.6309	−1.70494
$283$	−15.3472	−0.912298	−0.456149	−	0.889903i	$-0.650772\pi$
$283$	−15.3472	−0.912298	−0.456149	+	0.889903i	$0.650772\pi$
$284$	−2.95395	−0.175285
$285$	0	0
$286$	2.86234	0.169254
$287$	1.15200	0.0680001
$288$	−15.4379	−0.909689
$289$	−4.80271	−0.282512
$290$	0	0
$291$	−28.3918	−1.66436
$292$	3.07661	0.180045
$293$	25.7169	1.50240	0.751199	−	0.660075i	$-0.229476\pi$
$293$	25.7169	1.50240	0.751199	+	0.660075i	$0.229476\pi$
$294$	−28.4332	−1.65826
$295$	0	0
$296$	−15.6960	−0.912310
$297$	8.03572	0.466280
$298$	−16.3465	−0.946928
$299$	−18.9150	−1.09388
$300$	0	0
$301$	−0.734500	−0.0423359
$302$	5.97723	0.343951
$303$	−5.47202	−0.314360
$304$	−22.2173	−1.27425
$305$	0	0
$306$	32.6157	1.86452
$307$	2.60692	0.148785	0.0743924	−	0.997229i	$-0.476298\pi$
$307$	2.60692	0.148785	0.0743924	+	0.997229i	$0.476298\pi$
$308$	0.0512074	0.00291781
$309$	9.03130	0.513773
$310$	0	0
$311$	−19.8431	−1.12520	−0.562599	−	0.826730i	$-0.690199\pi$
$311$	−19.8431	−1.12520	−0.562599	+	0.826730i	$0.690199\pi$
$312$	−37.9055	−2.14598
$313$	−18.1393	−1.02529	−0.512647	−	0.858599i	$-0.671335\pi$
$313$	−18.1393	−1.02529	−0.512647	+	0.858599i	$0.671335\pi$
$314$	−4.51553	−0.254826
$315$	0	0
$316$	0.146850	0.00826095
$317$	5.29542	0.297420	0.148710	−	0.988881i	$-0.452488\pi$
$317$	5.29542	0.297420	0.148710	+	0.988881i	$0.452488\pi$
$318$	49.7888	2.79202
$319$	−4.87352	−0.272865
$320$	0	0
$321$	20.8956	1.16628
$322$	1.45383	0.0810189
$323$	−25.0126	−1.39174
$324$	−8.59522	−0.477512
$325$	0	0
$326$	−3.12505	−0.173081
$327$	−38.5412	−2.13133
$328$	14.8472	0.819800
$329$	−1.64321	−0.0905929
$330$	0	0
$331$	29.2462	1.60752	0.803760	−	0.594954i	$-0.202830\pi$
$331$	29.2462	1.60752	0.803760	+	0.594954i	$0.202830\pi$
$332$	−1.98655	−0.109026
$333$	−38.0004	−2.08241
$334$	8.07828	0.442024
$335$	0	0
$336$	2.34304	0.127823
$337$	23.6870	1.29031	0.645156	−	0.764051i	$-0.276793\pi$
$337$	23.6870	1.29031	0.645156	+	0.764051i	$0.276793\pi$
$338$	−2.75511	−0.149858
$339$	31.3165	1.70088
$340$	0	0
$341$	3.21766	0.174246
$342$	−66.8840	−3.61667
$343$	−3.27670	−0.176925
$344$	−9.46642	−0.510395
$345$	0	0
$346$	9.12558	0.490594
$347$	−7.92141	−0.425244	−0.212622	−	0.977135i	$-0.568200\pi$
$347$	−7.92141	−0.425244	−0.212622	+	0.977135i	$0.568200\pi$
$348$	10.2503	0.549475
$349$	−25.6673	−1.37394	−0.686969	−	0.726687i	$-0.741059\pi$
$349$	−25.6673	−1.37394	−0.686969	+	0.726687i	$0.741059\pi$
$350$	0	0
$351$	−54.2208	−2.89409
$352$	1.21513	0.0647665
$353$	−13.2644	−0.705995	−0.352997	−	0.935624i	$-0.614838\pi$
$353$	−13.2644	−0.705995	−0.352997	+	0.935624i	$0.614838\pi$
$354$	−58.8376	−3.12718
$355$	0	0
$356$	−3.15953	−0.167455
$357$	2.63783	0.139609
$358$	9.42238	0.497988
$359$	−5.67509	−0.299520	−0.149760	−	0.988722i	$-0.547850\pi$
$359$	−5.67509	−0.299520	−0.149760	+	0.988722i	$0.547850\pi$
$360$	0	0
$361$	32.2926	1.69961
$362$	25.7196	1.35179
$363$	34.2871	1.79961
$364$	−0.345520	−0.0181102
$365$	0	0
$366$	−31.5958	−1.65154
$367$	−18.8576	−0.984361	−0.492180	−	0.870493i	$-0.663800\pi$
$367$	−18.8576	−0.984361	−0.492180	+	0.870493i	$0.663800\pi$
$368$	15.0688	0.785514
$369$	35.9455	1.87125
$370$	0	0
$371$	2.85752	0.148355
$372$	−6.76760	−0.350884
$373$	22.6052	1.17045	0.585226	−	0.810870i	$-0.301006\pi$
$373$	22.6052	1.17045	0.585226	+	0.810870i	$0.301006\pi$
$374$	−2.56720	−0.132747
$375$	0	0
$376$	−21.1781	−1.09217
$377$	32.8840	1.69361
$378$	4.16748	0.214352
$379$	−13.2151	−0.678815	−0.339408	−	0.940639i	$-0.610227\pi$
$379$	−13.2151	−0.678815	−0.339408	+	0.940639i	$0.610227\pi$
$380$	0	0
$381$	−33.1346	−1.69754
$382$	26.1921	1.34011
$383$	−9.06658	−0.463281	−0.231640	−	0.972801i	$-0.574409\pi$
$383$	−9.06658	−0.463281	−0.231640	+	0.972801i	$0.574409\pi$
$384$	22.8459	1.16585
$385$	0	0
$386$	3.42434	0.174294
$387$	−22.9185	−1.16501
$388$	−3.33548	−0.169333
$389$	24.1714	1.22554	0.612768	−	0.790263i	$-0.290056\pi$
$389$	24.1714	1.22554	0.612768	+	0.790263i	$0.290056\pi$
$390$	0	0
$391$	16.9647	0.857940
$392$	−21.0319	−1.06227
$393$	−62.7402	−3.16482
$394$	−1.27373	−0.0641694
$395$	0	0
$396$	1.59782	0.0802932
$397$	−10.7768	−0.540870	−0.270435	−	0.962738i	$-0.587167\pi$
$397$	−10.7768	−0.540870	−0.270435	+	0.962738i	$0.587167\pi$
$398$	6.79865	0.340785
$399$	−5.40933	−0.270805
$400$	0	0
$401$	0.512588	0.0255974	0.0127987	−	0.999918i	$-0.495926\pi$
$401$	0.512588	0.0255974	0.0127987	+	0.999918i	$0.495926\pi$
$402$	−10.5492	−0.526148
$403$	−21.7111	−1.08150
$404$	−0.642855	−0.0319832
$405$	0	0
$406$	−2.52750	−0.125438
$407$	2.99103	0.148260
$408$	33.9971	1.68311
$409$	29.1701	1.44237	0.721183	−	0.692744i	$-0.243598\pi$
$409$	29.1701	1.44237	0.721183	+	0.692744i	$0.243598\pi$
$410$	0	0
$411$	−9.74953	−0.480909
$412$	1.06100	0.0522717
$413$	−3.37686	−0.166164
$414$	45.3637	2.22950
$415$	0	0
$416$	−8.19905	−0.401991
$417$	−7.37789	−0.361297
$418$	5.26448	0.257494
$419$	−7.18784	−0.351149	−0.175574	−	0.984466i	$-0.556178\pi$
$419$	−7.18784	−0.351149	−0.175574	+	0.984466i	$0.556178\pi$
$420$	0	0
$421$	16.3859	0.798598	0.399299	−	0.916821i	$-0.369254\pi$
$421$	16.3859	0.798598	0.399299	+	0.916821i	$0.369254\pi$
$422$	34.9833	1.70296
$423$	−51.2727	−2.49296
$424$	36.8284	1.78855
$425$	0	0
$426$	32.0268	1.55171
$427$	−1.81337	−0.0877552
$428$	2.45482	0.118658
$429$	7.22330	0.348744
$430$	0	0
$431$	−24.6531	−1.18750	−0.593750	−	0.804650i	$-0.702353\pi$
$431$	−24.6531	−1.18750	−0.593750	+	0.804650i	$0.702353\pi$
$432$	43.1954	2.07824
$433$	16.5810	0.796831	0.398415	−	0.917205i	$-0.369560\pi$
$433$	16.5810	0.796831	0.398415	+	0.917205i	$0.369560\pi$
$434$	1.66874	0.0801021
$435$	0	0
$436$	−4.52783	−0.216844
$437$	−34.7889	−1.66418
$438$	−33.3567	−1.59384
$439$	14.3127	0.683110	0.341555	−	0.939862i	$-0.389046\pi$
$439$	14.3127	0.683110	0.341555	+	0.939862i	$0.389046\pi$
$440$	0	0
$441$	−50.9187	−2.42470
$442$	17.3221	0.823929
$443$	−1.91828	−0.0911403	−0.0455702	−	0.998961i	$-0.514510\pi$
$443$	−1.91828	−0.0911403	−0.0455702	+	0.998961i	$0.514510\pi$
$444$	−6.29095	−0.298555
$445$	0	0
$446$	−30.7908	−1.45799
$447$	−41.2515	−1.95113
$448$	2.08806	0.0986516
$449$	21.6287	1.02072	0.510360	−	0.859961i	$-0.329512\pi$
$449$	21.6287	1.02072	0.510360	+	0.859961i	$0.329512\pi$
$450$	0	0
$451$	−2.82929	−0.133226
$452$	3.67907	0.173049
$453$	15.0839	0.708705
$454$	−22.7157	−1.06610
$455$	0	0
$456$	−69.7168	−3.26479
$457$	−19.6638	−0.919835	−0.459917	−	0.887962i	$-0.652121\pi$
$457$	−19.6638	−0.919835	−0.459917	+	0.887962i	$0.652121\pi$
$458$	33.6957	1.57450
$459$	48.6301	2.26986
$460$	0	0
$461$	10.3420	0.481676	0.240838	−	0.970565i	$-0.422578\pi$
$461$	10.3420	0.481676	0.240838	+	0.970565i	$0.422578\pi$
$462$	−0.555192	−0.0258299
$463$	10.9518	0.508973	0.254486	−	0.967076i	$-0.418094\pi$
$463$	10.9518	0.508973	0.254486	+	0.967076i	$0.418094\pi$
$464$	−26.1972	−1.21618
$465$	0	0
$466$	−38.4193	−1.77974
$467$	−20.9307	−0.968556	−0.484278	−	0.874914i	$-0.660918\pi$
$467$	−20.9307	−0.968556	−0.484278	+	0.874914i	$0.660918\pi$
$468$	−10.7812	−0.498362
$469$	−0.605450	−0.0279571
$470$	0	0
$471$	−11.3952	−0.525065
$472$	−43.5218	−2.00325
$473$	1.80393	0.0829446
$474$	−1.59215	−0.0731300
$475$	0	0
$476$	0.309893	0.0142039
$477$	89.1627	4.08248
$478$	−5.84518	−0.267352
$479$	−20.8556	−0.952916	−0.476458	−	0.879197i	$-0.658080\pi$
$479$	−20.8556	−0.952916	−0.476458	+	0.879197i	$0.658080\pi$
$480$	0	0
$481$	−20.1819	−0.920216
$482$	−30.0109	−1.36696
$483$	3.66884	0.166938
$484$	4.02806	0.183094
$485$	0	0
$486$	39.9825	1.81364
$487$	27.2853	1.23642	0.618208	−	0.786015i	$-0.287859\pi$
$487$	27.2853	1.23642	0.618208	+	0.786015i	$0.287859\pi$
$488$	−23.3712	−1.05796
$489$	−7.88627	−0.356630
$490$	0	0
$491$	15.6305	0.705395	0.352697	−	0.935737i	$-0.385265\pi$
$491$	15.6305	0.705395	0.352697	+	0.935737i	$0.385265\pi$
$492$	5.95076	0.268281
$493$	−29.4933	−1.32831
$494$	−35.5219	−1.59821
$495$	0	0
$496$	17.2963	0.776625
$497$	1.83811	0.0824506
$498$	21.5382	0.965150
$499$	−22.7946	−1.02042	−0.510212	−	0.860048i	$-0.670433\pi$
$499$	−22.7946	−1.02042	−0.510212	+	0.860048i	$0.670433\pi$
$500$	0	0
$501$	20.3861	0.910783
$502$	−15.2920	−0.682515
$503$	21.7516	0.969858	0.484929	−	0.874554i	$-0.338845\pi$
$503$	21.7516	0.969858	0.484929	+	0.874554i	$0.338845\pi$
$504$	5.21750	0.232406
$505$	0	0
$506$	−3.57060	−0.158733
$507$	−6.95270	−0.308780
$508$	−3.89267	−0.172709
$509$	−36.9506	−1.63781	−0.818904	−	0.573930i	$-0.805418\pi$
$509$	−36.9506	−1.63781	−0.818904	+	0.573930i	$0.805418\pi$
$510$	0	0
$511$	−1.91444	−0.0846896
$512$	25.3212	1.11905
$513$	−99.7242	−4.40293
$514$	17.9856	0.793312
$515$	0	0
$516$	−3.79414	−0.167028
$517$	4.03570	0.177490
$518$	1.55121	0.0681562
$519$	23.0290	1.01086
$520$	0	0
$521$	0.0246447	0.00107970	0.000539852	−	1.00000i	$-0.499828\pi$
$521$	0.0246447	0.00107970	0.000539852		1.00000i	$0.499828\pi$
$522$	−78.8653	−3.45184
$523$	21.2265	0.928168	0.464084	−	0.885791i	$-0.346384\pi$
$523$	21.2265	0.928168	0.464084	+	0.885791i	$0.346384\pi$
$524$	−7.37074	−0.321992
$525$	0	0
$526$	−13.1636	−0.573962
$527$	19.4724	0.848231
$528$	−5.75449	−0.250432
$529$	0.595377	0.0258860
$530$	0	0
$531$	−105.368	−4.57256
$532$	−0.635489	−0.0275520
$533$	19.0906	0.826904
$534$	34.2557	1.48239
$535$	0	0
$536$	−7.80319	−0.337047
$537$	23.7780	1.02610
$538$	19.1382	0.825106
$539$	4.00784	0.172630
$540$	0	0
$541$	−25.2530	−1.08571	−0.542855	−	0.839827i	$-0.682657\pi$
$541$	−25.2530	−1.08571	−0.542855	+	0.839827i	$0.682657\pi$
$542$	3.90897	0.167905
$543$	64.9051	2.78535
$544$	7.35363	0.315284
$545$	0	0
$546$	3.74615	0.160320
$547$	12.9296	0.552830	0.276415	−	0.961038i	$-0.410854\pi$
$547$	12.9296	0.552830	0.276415	+	0.961038i	$0.410854\pi$
$548$	−1.14538	−0.0489281
$549$	−56.5823	−2.41488
$550$	0	0
$551$	60.4809	2.57657
$552$	47.2850	2.01258
$553$	−0.0913781	−0.00388579
$554$	41.1443	1.74806
$555$	0	0
$556$	−0.866757	−0.0367587
$557$	9.53791	0.404134	0.202067	−	0.979372i	$-0.435234\pi$
$557$	9.53791	0.404134	0.202067	+	0.979372i	$0.435234\pi$
$558$	52.0694	2.20427
$559$	−12.1719	−0.514818
$560$	0	0
$561$	−6.47850	−0.273522
$562$	37.3905	1.57722
$563$	−16.4517	−0.693356	−0.346678	−	0.937984i	$-0.612690\pi$
$563$	−16.4517	−0.693356	−0.346678	+	0.937984i	$0.612690\pi$
$564$	−8.48817	−0.357417
$565$	0	0
$566$	19.5482	0.821671
$567$	5.34842	0.224613
$568$	23.6900	0.994012
$569$	33.1531	1.38985	0.694926	−	0.719082i	$-0.255437\pi$
$569$	33.1531	1.38985	0.694926	+	0.719082i	$0.255437\pi$
$570$	0	0
$571$	−21.2096	−0.887595	−0.443798	−	0.896127i	$-0.646369\pi$
$571$	−21.2096	−0.887595	−0.443798	+	0.896127i	$0.646369\pi$
$572$	0.848596	0.0354816
$573$	66.0976	2.76127
$574$	−1.46733	−0.0612450
$575$	0	0
$576$	65.1534	2.71472
$577$	−3.39956	−0.141526	−0.0707628	−	0.997493i	$-0.522543\pi$
$577$	−3.39956	−0.141526	−0.0707628	+	0.997493i	$0.522543\pi$
$578$	6.11733	0.254448
$579$	8.64155	0.359130
$580$	0	0
$581$	1.23614	0.0512837
$582$	36.1634	1.49902
$583$	−7.01805	−0.290658
$584$	−24.6737	−1.02101
$585$	0	0
$586$	−32.7563	−1.35315
$587$	−37.5978	−1.55183	−0.775914	−	0.630839i	$-0.782711\pi$
$587$	−37.5978	−1.55183	−0.775914	+	0.630839i	$0.782711\pi$
$588$	−8.42958	−0.347630
$589$	−39.9315	−1.64535
$590$	0	0
$591$	−3.21433	−0.132220
$592$	16.0781	0.660804
$593$	17.6880	0.726360	0.363180	−	0.931719i	$-0.381691\pi$
$593$	17.6880	0.726360	0.363180	+	0.931719i	$0.381691\pi$
$594$	−10.2353	−0.419960
$595$	0	0
$596$	−4.84624	−0.198510
$597$	17.1568	0.702183
$598$	24.0925	0.985217
$599$	2.70489	0.110519	0.0552595	−	0.998472i	$-0.482401\pi$
$599$	2.70489	0.110519	0.0552595	+	0.998472i	$0.482401\pi$
$600$	0	0
$601$	−26.9989	−1.10131	−0.550654	−	0.834733i	$-0.685622\pi$
$601$	−26.9989	−1.10131	−0.550654	+	0.834733i	$0.685622\pi$
$602$	0.935552	0.0381302
$603$	−18.8918	−0.769332
$604$	1.77206	0.0721043
$605$	0	0
$606$	6.96986	0.283131
$607$	−6.45409	−0.261964	−0.130982	−	0.991385i	$-0.541813\pi$
$607$	−6.45409	−0.261964	−0.130982	+	0.991385i	$0.541813\pi$
$608$	−15.0799	−0.611570
$609$	−6.37833	−0.258463
$610$	0	0
$611$	−27.2308	−1.10164
$612$	9.66956	0.390869
$613$	9.63640	0.389211	0.194605	−	0.980882i	$-0.437657\pi$
$613$	9.63640	0.389211	0.194605	+	0.980882i	$0.437657\pi$
$614$	−3.32050	−0.134005
$615$	0	0
$616$	−0.410672	−0.0165465
$617$	8.69198	0.349926	0.174963	−	0.984575i	$-0.444019\pi$
$617$	8.69198	0.349926	0.174963	+	0.984575i	$0.444019\pi$
$618$	−11.5034	−0.462735
$619$	−22.1646	−0.890870	−0.445435	−	0.895314i	$-0.646951\pi$
$619$	−22.1646	−0.890870	−0.445435	+	0.895314i	$0.646951\pi$
$620$	0	0
$621$	67.6373	2.71419
$622$	25.2747	1.01342
$623$	1.96603	0.0787675
$624$	38.8283	1.55438
$625$	0	0
$626$	23.1045	0.923443
$627$	13.2853	0.530562
$628$	−1.33872	−0.0534206
$629$	18.1009	0.721732
$630$	0	0
$631$	−2.65768	−0.105800	−0.0529002	−	0.998600i	$-0.516847\pi$
$631$	−2.65768	−0.105800	−0.0529002	+	0.998600i	$0.516847\pi$
$632$	−1.17770	−0.0468466
$633$	88.2827	3.50892
$634$	−6.74492	−0.267875
$635$	0	0
$636$	14.7608	0.585306
$637$	−27.0428	−1.07148
$638$	6.20753	0.245759
$639$	57.3543	2.26890
$640$	0	0
$641$	27.4000	1.08224	0.541118	−	0.840946i	$-0.318001\pi$
$641$	27.4000	1.08224	0.541118	+	0.840946i	$0.318001\pi$
$642$	−26.6153	−1.05042
$643$	−26.5633	−1.04755	−0.523777	−	0.851856i	$-0.675477\pi$
$643$	−26.5633	−1.04755	−0.523777	+	0.851856i	$0.675477\pi$
$644$	0.431017	0.0169844
$645$	0	0
$646$	31.8592	1.25349
$647$	−25.6395	−1.00799	−0.503996	−	0.863706i	$-0.668137\pi$
$647$	−25.6395	−1.00799	−0.503996	+	0.863706i	$0.668137\pi$
$648$	68.9318	2.70790
$649$	8.29354	0.325550
$650$	0	0
$651$	4.21118	0.165049
$652$	−0.926482	−0.0362838
$653$	17.4067	0.681179	0.340589	−	0.940212i	$-0.389373\pi$
$653$	17.4067	0.681179	0.340589	+	0.940212i	$0.389373\pi$
$654$	49.0909	1.91961
$655$	0	0
$656$	−15.2086	−0.593797
$657$	−59.7358	−2.33052
$658$	2.09299	0.0815934
$659$	18.2733	0.711827	0.355913	−	0.934519i	$-0.384170\pi$
$659$	18.2733	0.711827	0.355913	+	0.934519i	$0.384170\pi$
$660$	0	0
$661$	13.4912	0.524748	0.262374	−	0.964966i	$-0.415495\pi$
$661$	13.4912	0.524748	0.262374	+	0.964966i	$0.415495\pi$
$662$	−37.2517	−1.44783
$663$	43.7135	1.69769
$664$	15.9317	0.618269
$665$	0	0
$666$	48.4021	1.87554
$667$	−41.0208	−1.58833
$668$	2.39496	0.0926639
$669$	−77.7026	−3.00416
$670$	0	0
$671$	4.45363	0.171930
$672$	1.59033	0.0613481
$673$	−24.5543	−0.946497	−0.473249	−	0.880929i	$-0.656919\pi$
$673$	−24.5543	−0.946497	−0.473249	+	0.880929i	$0.656919\pi$
$674$	−30.1707	−1.16213
$675$	0	0
$676$	−0.816805	−0.0314156
$677$	21.6592	0.832429	0.416215	−	0.909266i	$-0.363356\pi$
$677$	21.6592	0.832429	0.416215	+	0.909266i	$0.363356\pi$
$678$	−39.8887	−1.53191
$679$	2.07552	0.0796511
$680$	0	0
$681$	−57.3245	−2.19668
$682$	−4.09841	−0.156936
$683$	21.0697	0.806209	0.403105	−	0.915154i	$-0.367931\pi$
$683$	21.0697	0.806209	0.403105	+	0.915154i	$0.367931\pi$
$684$	−19.8291	−0.758183
$685$	0	0
$686$	4.17362	0.159349
$687$	85.0334	3.24423
$688$	9.69686	0.369689
$689$	47.3541	1.80405
$690$	0	0
$691$	22.7420	0.865146	0.432573	−	0.901599i	$-0.357606\pi$
$691$	22.7420	0.865146	0.432573	+	0.901599i	$0.357606\pi$
$692$	2.70545	0.102846
$693$	−0.994249	−0.0377684
$694$	10.0897	0.383000
$695$	0	0
$696$	−82.2055	−3.11599
$697$	−17.1221	−0.648547
$698$	32.6931	1.23745
$699$	−96.9536	−3.66712
$700$	0	0
$701$	−35.8276	−1.35319	−0.676596	−	0.736355i	$-0.736545\pi$
$701$	−35.8276	−1.35319	−0.676596	+	0.736355i	$0.736545\pi$
$702$	69.0625	2.60660
$703$	−37.1191	−1.39997
$704$	−5.12826	−0.193279
$705$	0	0
$706$	16.8953	0.635862
$707$	0.400020	0.0150443
$708$	−17.4436	−0.655569
$709$	−18.4033	−0.691152	−0.345576	−	0.938391i	$-0.612316\pi$
$709$	−18.4033	−0.691152	−0.345576	+	0.938391i	$0.612316\pi$
$710$	0	0
$711$	−2.85126	−0.106930
$712$	25.3387	0.949609
$713$	27.0833	1.01428
$714$	−3.35988	−0.125740
$715$	0	0
$716$	2.79345	0.104396
$717$	−14.7507	−0.550875
$718$	7.22852	0.269766
$719$	22.9617	0.856326	0.428163	−	0.903702i	$-0.359161\pi$
$719$	22.9617	0.856326	0.428163	+	0.903702i	$0.359161\pi$
$720$	0	0
$721$	−0.660213	−0.0245876
$722$	−41.1320	−1.53077
$723$	−75.7345	−2.81659
$724$	7.62507	0.283384
$725$	0	0
$726$	−43.6724	−1.62084
$727$	19.2162	0.712691	0.356345	−	0.934354i	$-0.384023\pi$
$727$	19.2162	0.712691	0.356345	+	0.934354i	$0.384023\pi$
$728$	2.77100	0.102700
$729$	32.6140	1.20792
$730$	0	0
$731$	10.9169	0.403776
$732$	−9.36718	−0.346221
$733$	0.407800	0.0150624	0.00753122	−	0.999972i	$-0.497603\pi$
$733$	0.407800	0.0150624	0.00753122	+	0.999972i	$0.497603\pi$
$734$	24.0195	0.886575
$735$	0	0
$736$	10.2278	0.377003
$737$	1.48698	0.0547736
$738$	−45.7847	−1.68536
$739$	−23.3824	−0.860135	−0.430068	−	0.902797i	$-0.641510\pi$
$739$	−23.3824	−0.860135	−0.430068	+	0.902797i	$0.641510\pi$
$740$	0	0
$741$	−89.6420	−3.29308
$742$	−3.63970	−0.133618
$743$	−49.3094	−1.80899	−0.904493	−	0.426488i	$-0.859751\pi$
$743$	−49.3094	−1.80899	−0.904493	+	0.426488i	$0.859751\pi$
$744$	54.2747	1.98981
$745$	0	0
$746$	−28.7928	−1.05418
$747$	38.5710	1.41124
$748$	−0.761096	−0.0278284
$749$	−1.52753	−0.0558146
$750$	0	0
$751$	0.947118	0.0345608	0.0172804	−	0.999851i	$-0.494499\pi$
$751$	0.947118	0.0345608	0.0172804	+	0.999851i	$0.494499\pi$
$752$	21.6936	0.791084
$753$	−38.5903	−1.40631
$754$	−41.8852	−1.52537
$755$	0	0
$756$	1.23553	0.0449358
$757$	−42.8580	−1.55770	−0.778850	−	0.627210i	$-0.784197\pi$
$757$	−42.8580	−1.55770	−0.778850	+	0.627210i	$0.784197\pi$
$758$	16.8325	0.611382
$759$	−9.01065	−0.327066
$760$	0	0
$761$	−14.6514	−0.531113	−0.265556	−	0.964095i	$-0.585556\pi$
$761$	−14.6514	−0.531113	−0.265556	+	0.964095i	$0.585556\pi$
$762$	42.2045	1.52891
$763$	2.81747	0.101999
$764$	7.76517	0.280934
$765$	0	0
$766$	11.5483	0.417259
$767$	−55.9604	−2.02061
$768$	28.0273	1.01135
$769$	−0.134376	−0.00484571	−0.00242285	−	0.999997i	$-0.500771\pi$
$769$	−0.134376	−0.00484571	−0.00242285	+	0.999997i	$0.500771\pi$
$770$	0	0
$771$	45.3879	1.63461
$772$	1.01521	0.0365383
$773$	−15.4519	−0.555767	−0.277884	−	0.960615i	$-0.589633\pi$
$773$	−15.4519	−0.555767	−0.277884	+	0.960615i	$0.589633\pi$
$774$	29.1919	1.04928
$775$	0	0
$776$	26.7498	0.960262
$777$	3.91458	0.140435
$778$	−30.7877	−1.10379
$779$	35.1118	1.25801
$780$	0	0
$781$	−4.51439	−0.161538
$782$	−21.6083	−0.772713
$783$	−117.588	−4.20226
$784$	21.5438	0.769423
$785$	0	0
$786$	79.9139	2.85043
$787$	−42.6277	−1.51951	−0.759757	−	0.650207i	$-0.774682\pi$
$787$	−42.6277	−1.51951	−0.759757	+	0.650207i	$0.774682\pi$
$788$	−0.377621	−0.0134522
$789$	−33.2193	−1.18264
$790$	0	0
$791$	−2.28932	−0.0813989
$792$	−12.8141	−0.455330
$793$	−30.0507	−1.06713
$794$	13.7266	0.487140
$795$	0	0
$796$	2.01559	0.0714407
$797$	11.6332	0.412069	0.206035	−	0.978545i	$-0.433944\pi$
$797$	11.6332	0.412069	0.206035	+	0.978545i	$0.433944\pi$
$798$	6.89000	0.243903
$799$	24.4230	0.864024
$800$	0	0
$801$	61.3458	2.16755
$802$	−0.652897	−0.0230546
$803$	4.70184	0.165924
$804$	−3.12752	−0.110299
$805$	0	0
$806$	27.6539	0.974069
$807$	48.2965	1.70012
$808$	5.15556	0.181372
$809$	−37.3413	−1.31285	−0.656425	−	0.754391i	$-0.727932\pi$
$809$	−37.3413	−1.31285	−0.656425	+	0.754391i	$0.727932\pi$
$810$	0	0
$811$	−9.00258	−0.316123	−0.158062	−	0.987429i	$-0.550524\pi$
$811$	−9.00258	−0.316123	−0.158062	+	0.987429i	$0.550524\pi$
$812$	−0.749328	−0.0262962
$813$	9.86455	0.345965
$814$	−3.80976	−0.133532
$815$	0	0
$816$	−34.8247	−1.21911
$817$	−22.3869	−0.783219
$818$	−37.1547	−1.29908
$819$	6.70867	0.234420
$820$	0	0
$821$	8.55499	0.298571	0.149286	−	0.988794i	$-0.452303\pi$
$821$	8.55499	0.298571	0.149286	+	0.988794i	$0.452303\pi$
$822$	12.4182	0.433136
$823$	27.1990	0.948097	0.474049	−	0.880499i	$-0.342792\pi$
$823$	27.1990	0.948097	0.474049	+	0.880499i	$0.342792\pi$
$824$	−8.50899	−0.296425
$825$	0	0
$826$	4.30119	0.149658
$827$	−8.38311	−0.291509	−0.145755	−	0.989321i	$-0.546561\pi$
$827$	−8.38311	−0.291509	−0.145755	+	0.989321i	$0.546561\pi$
$828$	13.4489	0.467383
$829$	37.2591	1.29406	0.647030	−	0.762464i	$-0.276011\pi$
$829$	37.2591	1.29406	0.647030	+	0.762464i	$0.276011\pi$
$830$	0	0
$831$	103.830	3.60184
$832$	34.6028	1.19964
$833$	24.2544	0.840365
$834$	9.39741	0.325406
$835$	0	0
$836$	1.56076	0.0539799
$837$	77.6356	2.68348
$838$	9.15534	0.316266
$839$	−12.5348	−0.432750	−0.216375	−	0.976310i	$-0.569423\pi$
$839$	−12.5348	−0.432750	−0.216375	+	0.976310i	$0.569423\pi$
$840$	0	0
$841$	42.3152	1.45914
$842$	−20.8711	−0.719265
$843$	94.3573	3.24984
$844$	10.3715	0.357001
$845$	0	0
$846$	65.3074	2.24531
$847$	−2.50649	−0.0861239
$848$	−37.7250	−1.29548
$849$	49.3311	1.69304
$850$	0	0
$851$	25.1758	0.863015
$852$	9.49498	0.325293
$853$	20.8885	0.715210	0.357605	−	0.933873i	$-0.383593\pi$
$853$	20.8885	0.715210	0.357605	+	0.933873i	$0.383593\pi$
$854$	2.30974	0.0790377
$855$	0	0
$856$	−19.6872	−0.672893
$857$	−30.0053	−1.02496	−0.512481	−	0.858699i	$-0.671273\pi$
$857$	−30.0053	−1.02496	−0.512481	+	0.858699i	$0.671273\pi$
$858$	−9.20051	−0.314100
$859$	44.8428	1.53002	0.765008	−	0.644020i	$-0.222735\pi$
$859$	44.8428	1.53002	0.765008	+	0.644020i	$0.222735\pi$
$860$	0	0
$861$	−3.70290	−0.126194
$862$	31.4013	1.06953
$863$	−10.1816	−0.346586	−0.173293	−	0.984870i	$-0.555441\pi$
$863$	−10.1816	−0.346586	−0.173293	+	0.984870i	$0.555441\pi$
$864$	29.3186	0.997439
$865$	0	0
$866$	−21.1196	−0.717674
$867$	15.4375	0.524285
$868$	0.494730	0.0167922
$869$	0.224424	0.00761306
$870$	0	0
$871$	−10.0334	−0.339968
$872$	36.3122	1.22969
$873$	64.7620	2.19186
$874$	44.3116	1.49886
$875$	0	0
$876$	−9.88924	−0.334126
$877$	−35.3951	−1.19521	−0.597604	−	0.801792i	$-0.703880\pi$
$877$	−35.3951	−1.19521	−0.597604	+	0.801792i	$0.703880\pi$
$878$	−18.2305	−0.615250
$879$	−82.6628	−2.78815
$880$	0	0
$881$	46.3047	1.56005	0.780023	−	0.625750i	$-0.215207\pi$
$881$	46.3047	1.56005	0.780023	+	0.625750i	$0.215207\pi$
$882$	64.8565	2.18383
$883$	−36.2361	−1.21944	−0.609722	−	0.792616i	$-0.708719\pi$
$883$	−36.2361	−1.21944	−0.609722	+	0.792616i	$0.708719\pi$
$884$	5.13548	0.172725
$885$	0	0
$886$	2.44337	0.0820865
$887$	26.9498	0.904887	0.452443	−	0.891793i	$-0.350552\pi$
$887$	26.9498	0.904887	0.452443	+	0.891793i	$0.350552\pi$
$888$	50.4521	1.69306
$889$	2.42223	0.0812391
$890$	0	0
$891$	−13.1357	−0.440062
$892$	−9.12853	−0.305646
$893$	−50.0835	−1.67598
$894$	52.5431	1.75731
$895$	0	0
$896$	−1.67010	−0.0557940
$897$	60.7991	2.03002
$898$	−27.5490	−0.919322
$899$	−47.0846	−1.57036
$900$	0	0
$901$	−42.4714	−1.41493
$902$	3.60374	0.119992
$903$	2.36093	0.0785667
$904$	−29.5054	−0.981334
$905$	0	0
$906$	−19.2128	−0.638303
$907$	−5.26836	−0.174933	−0.0874664	−	0.996167i	$-0.527877\pi$
$907$	−5.26836	−0.174933	−0.0874664	+	0.996167i	$0.527877\pi$
$908$	−6.73449	−0.223492
$909$	12.4818	0.413994
$910$	0	0
$911$	20.8539	0.690919	0.345459	−	0.938434i	$-0.387723\pi$
$911$	20.8539	0.690919	0.345459	+	0.938434i	$0.387723\pi$
$912$	71.4139	2.36475
$913$	−3.03595	−0.100475
$914$	25.0463	0.828459
$915$	0	0
$916$	9.98975	0.330070
$917$	4.58648	0.151459
$918$	−61.9414	−2.04437
$919$	−9.82871	−0.324219	−0.162110	−	0.986773i	$-0.551830\pi$
$919$	−9.82871	−0.324219	−0.162110	+	0.986773i	$0.551830\pi$
$920$	0	0
$921$	−8.37951	−0.276114
$922$	−13.1729	−0.433827
$923$	30.4607	1.00263
$924$	−0.164598	−0.00541486
$925$	0	0
$926$	−13.9496	−0.458412
$927$	−20.6005	−0.676610
$928$	−17.7812	−0.583697
$929$	14.8642	0.487677	0.243839	−	0.969816i	$-0.421593\pi$
$929$	14.8642	0.487677	0.243839	+	0.969816i	$0.421593\pi$
$930$	0	0
$931$	−49.7378	−1.63009
$932$	−11.3901	−0.373096
$933$	63.7823	2.08814
$934$	26.6599	0.872340
$935$	0	0
$936$	86.4630	2.82613
$937$	−56.7633	−1.85438	−0.927189	−	0.374594i	$-0.877782\pi$
$937$	−56.7633	−1.85438	−0.927189	+	0.374594i	$0.877782\pi$
$938$	0.771178	0.0251798
$939$	58.3058	1.90274
$940$	0	0
$941$	−10.6242	−0.346341	−0.173170	−	0.984892i	$-0.555401\pi$
$941$	−10.6242	−0.346341	−0.173170	+	0.984892i	$0.555401\pi$
$942$	14.5144	0.472905
$943$	−23.8144	−0.775503
$944$	44.5813	1.45100
$945$	0	0
$946$	−2.29771	−0.0747049
$947$	−35.9492	−1.16819	−0.584095	−	0.811685i	$-0.698551\pi$
$947$	−35.9492	−1.16819	−0.584095	+	0.811685i	$0.698551\pi$
$948$	−0.472024	−0.0153306
$949$	−31.7255	−1.02985
$950$	0	0
$951$	−17.0212	−0.551952
$952$	−2.48528	−0.0805484
$953$	54.1752	1.75491	0.877454	−	0.479661i	$-0.159240\pi$
$953$	54.1752	1.75491	0.877454	+	0.479661i	$0.159240\pi$
$954$	−113.569	−3.67693
$955$	0	0
$956$	−1.73292	−0.0560466
$957$	15.6651	0.506381
$958$	26.5643	0.858254
$959$	0.712717	0.0230148
$960$	0	0
$961$	0.0867807	0.00279938
$962$	25.7062	0.828803
$963$	−47.6632	−1.53592
$964$	−8.89731	−0.286563
$965$	0	0
$966$	−4.67310	−0.150355
$967$	5.91237	0.190129	0.0950645	−	0.995471i	$-0.469694\pi$
$967$	5.91237	0.190129	0.0950645	+	0.995471i	$0.469694\pi$
$968$	−32.3042	−1.03830
$969$	80.3989	2.58279
$970$	0	0
$971$	−25.0906	−0.805195	−0.402597	−	0.915377i	$-0.631893\pi$
$971$	−25.0906	−0.805195	−0.402597	+	0.915377i	$0.631893\pi$
$972$	11.8536	0.380204
$973$	0.539344	0.0172906
$974$	−34.7540	−1.11359
$975$	0	0
$976$	23.9401	0.766305
$977$	49.9680	1.59862	0.799309	−	0.600920i	$-0.205199\pi$
$977$	49.9680	1.59862	0.799309	+	0.600920i	$0.205199\pi$
$978$	10.0450	0.321202
$979$	−4.82856	−0.154322
$980$	0	0
$981$	87.9129	2.80684
$982$	−19.9090	−0.635321
$983$	−43.1014	−1.37472	−0.687361	−	0.726316i	$-0.741231\pi$
$983$	−43.1014	−1.37472	−0.687361	+	0.726316i	$0.741231\pi$
$984$	−47.7239	−1.52138
$985$	0	0
$986$	37.5663	1.19636
$987$	5.28181	0.168122
$988$	−10.5312	−0.335041
$989$	15.1838	0.482817
$990$	0	0
$991$	−39.3870	−1.25117	−0.625585	−	0.780156i	$-0.715140\pi$
$991$	−39.3870	−1.25117	−0.625585	+	0.780156i	$0.715140\pi$
$992$	11.7397	0.372737
$993$	−94.0072	−2.98323
$994$	−2.34125	−0.0742600
$995$	0	0
$996$	6.38542	0.202330
$997$	53.8563	1.70565	0.852823	−	0.522200i	$-0.174888\pi$
$997$	53.8563	1.70565	0.852823	+	0.522200i	$0.174888\pi$
$998$	29.0340	0.919056
$999$	72.1676	2.28328

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4925.2.a.r.1.17		49
5.2	odd	4		985.2.b.a.789.31	✓	98
5.3	odd	4		985.2.b.a.789.68	yes	98
5.4	even	2		4925.2.a.s.1.33		49

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
985.2.b.a.789.31	✓	98	5.2	odd	4
985.2.b.a.789.68	yes	98	5.3	odd	4
4925.2.a.r.1.17		49	1.1	even	1	trivial
4925.2.a.s.1.33		49	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-1.27373 q^{2} -3.21433 q^{3} -0.377621 q^{4} +4.09418 q^{6} +0.234977 q^{7} +3.02844 q^{8} +7.33194 q^{9} -0.577101 q^{11} +1.21380 q^{12} +3.89397 q^{13} -0.299296 q^{14} -3.10216 q^{16} -3.49246 q^{17} -9.33888 q^{18} +7.16189 q^{19} -0.755293 q^{21} +0.735069 q^{22} -4.85751 q^{23} -9.73441 q^{24} -4.95986 q^{26} -13.9243 q^{27} -0.0887321 q^{28} +8.44483 q^{29} -5.57555 q^{31} -2.10557 q^{32} +1.85499 q^{33} +4.44844 q^{34} -2.76869 q^{36} -5.18286 q^{37} -9.12229 q^{38} -12.5165 q^{39} +4.90259 q^{41} +0.962037 q^{42} -3.12584 q^{43} +0.217925 q^{44} +6.18714 q^{46} -6.99306 q^{47} +9.97138 q^{48} -6.94479 q^{49} +11.2259 q^{51} -1.47045 q^{52} +12.1609 q^{53} +17.7357 q^{54} +0.711612 q^{56} -23.0207 q^{57} -10.7564 q^{58} -14.3710 q^{59} -7.71724 q^{61} +7.10173 q^{62} +1.72283 q^{63} +8.88625 q^{64} -2.36276 q^{66} -2.57664 q^{67} +1.31883 q^{68} +15.6136 q^{69} +7.82253 q^{71} +22.2043 q^{72} -8.14734 q^{73} +6.60155 q^{74} -2.70448 q^{76} -0.135605 q^{77} +15.9426 q^{78} -0.388882 q^{79} +22.7615 q^{81} -6.24456 q^{82} +5.26069 q^{83} +0.285214 q^{84} +3.98147 q^{86} -27.1445 q^{87} -1.74771 q^{88} +8.36693 q^{89} +0.914993 q^{91} +1.83430 q^{92} +17.9217 q^{93} +8.90725 q^{94} +6.76801 q^{96} +8.83287 q^{97} +8.84576 q^{98} -4.23127 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 49 q - 5 q^{2} - 22 q^{3} + 49 q^{4} + 2 q^{6} - 32 q^{7} - 15 q^{8} + 51 q^{9} - 2 q^{11} - 44 q^{12} - 32 q^{13} - 8 q^{14} + 49 q^{16} - 14 q^{17} - 25 q^{18} + 4 q^{19} + 10 q^{21} - 38 q^{22} - 24 q^{23}+ \cdots + 22 q^{99}+O(q^{100}) \) 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

Related objects

Downloads

Learn more