LMFDB - Embedded newform 6025.2.a.p.1.39

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6025,2,Mod(1,6025)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6025, 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("6025.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6025 = 5^{2} \cdot 241 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6025.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:	$48.1098672178$
Analytic rank:	$1$
Dimension:	$46$
Twist minimal:	no (minimal twist has level 1205)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.39
Character	$\chi$	$=$	6025.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.88286 q^{2} +2.55671 q^{3} +1.54517 q^{4} +4.81392 q^{6} -3.51799 q^{7} -0.856390 q^{8} +3.53675 q^{9} +O(q^{10})$	\(q+1.88286 q^{2} +2.55671 q^{3} +1.54517 q^{4} +4.81392 q^{6} -3.51799 q^{7} -0.856390 q^{8} +3.53675 q^{9} -1.11714 q^{11} +3.95054 q^{12} -0.230541 q^{13} -6.62388 q^{14} -4.70279 q^{16} -4.35003 q^{17} +6.65921 q^{18} +3.14602 q^{19} -8.99446 q^{21} -2.10341 q^{22} -6.36510 q^{23} -2.18954 q^{24} -0.434076 q^{26} +1.37232 q^{27} -5.43587 q^{28} -0.0527009 q^{29} -4.55850 q^{31} -7.14193 q^{32} -2.85619 q^{33} -8.19050 q^{34} +5.46487 q^{36} -6.27282 q^{37} +5.92351 q^{38} -0.589425 q^{39} -8.79348 q^{41} -16.9353 q^{42} +2.95908 q^{43} -1.72616 q^{44} -11.9846 q^{46} +10.3334 q^{47} -12.0237 q^{48} +5.37623 q^{49} -11.1218 q^{51} -0.356224 q^{52} +6.03035 q^{53} +2.58389 q^{54} +3.01277 q^{56} +8.04344 q^{57} -0.0992284 q^{58} +6.43499 q^{59} -11.8995 q^{61} -8.58302 q^{62} -12.4423 q^{63} -4.04167 q^{64} -5.37781 q^{66} +2.08295 q^{67} -6.72152 q^{68} -16.2737 q^{69} -5.17552 q^{71} -3.02884 q^{72} +1.25992 q^{73} -11.8109 q^{74} +4.86112 q^{76} +3.93007 q^{77} -1.10981 q^{78} -0.749786 q^{79} -7.10164 q^{81} -16.5569 q^{82} +11.1374 q^{83} -13.8979 q^{84} +5.57154 q^{86} -0.134741 q^{87} +0.956703 q^{88} -8.15389 q^{89} +0.811039 q^{91} -9.83514 q^{92} -11.6547 q^{93} +19.4564 q^{94} -18.2598 q^{96} -7.78090 q^{97} +10.1227 q^{98} -3.95103 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 46 q + 34 q^{4} - 4 q^{6} + 34 q^{9}+O(q^{10}) $ 46 * q + 34 * q^4 - 4 * q^6 + 34 * q^9	$ 46 q + 34 q^{4} - 4 q^{6} + 34 q^{9} - 64 q^{11} - 66 q^{14} + 22 q^{16} - 14 q^{21} - 50 q^{24} - 60 q^{26} - 36 q^{29} - 36 q^{31} + 12 q^{34} - 34 q^{36} - 88 q^{39} - 76 q^{41} - 100 q^{44} - 12 q^{46} + 22 q^{49} - 112 q^{51} - 26 q^{54} - 120 q^{56} - 84 q^{59} - 78 q^{61} + 28 q^{64} - 2 q^{66} - 24 q^{69} - 172 q^{71} - 16 q^{74} - 18 q^{76} - 54 q^{79} - 42 q^{81} + 44 q^{84} - 80 q^{86} - 86 q^{89} - 88 q^{91} + 4 q^{94} - 122 q^{96} - 148 q^{99}+O(q^{100}) $ 46 * q + 34 * q^4 - 4 * q^6 + 34 * q^9 - 64 * q^11 - 66 * q^14 + 22 * q^16 - 14 * q^21 - 50 * q^24 - 60 * q^26 - 36 * q^29 - 36 * q^31 + 12 * q^34 - 34 * q^36 - 88 * q^39 - 76 * q^41 - 100 * q^44 - 12 * q^46 + 22 * q^49 - 112 * q^51 - 26 * q^54 - 120 * q^56 - 84 * q^59 - 78 * q^61 + 28 * q^64 - 2 * q^66 - 24 * q^69 - 172 * q^71 - 16 * q^74 - 18 * q^76 - 54 * q^79 - 42 * q^81 + 44 * q^84 - 80 * q^86 - 86 * q^89 - 88 * q^91 + 4 * q^94 - 122 * q^96 - 148 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.88286	1.33138	0.665692	−	0.746227i	$-0.268137\pi$
$2$	1.88286	1.33138	0.665692	+	0.746227i	$0.268137\pi$
$3$	2.55671	1.47612	0.738058	−	0.674737i	$-0.235743\pi$
$3$	2.55671	1.47612	0.738058	+	0.674737i	$0.235743\pi$
$4$	1.54517	0.772583
$5$	0	0
$5$	0	0
$6$	4.81392	1.96528
$7$	−3.51799	−1.32967	−0.664837	−	0.746988i	$-0.731499\pi$
$7$	−3.51799	−1.32967	−0.664837	+	0.746988i	$0.731499\pi$
$8$	−0.856390	−0.302780
$9$	3.53675	1.17892
$10$	0	0
$11$	−1.11714	−0.336829	−0.168414	−	0.985716i	$-0.553865\pi$
$11$	−1.11714	−0.336829	−0.168414	+	0.985716i	$0.553865\pi$
$12$	3.95054	1.14042
$13$	−0.230541	−0.0639405	−0.0319702	−	0.999489i	$-0.510178\pi$
$13$	−0.230541	−0.0639405	−0.0319702	+	0.999489i	$0.510178\pi$
$14$	−6.62388	−1.77031
$15$	0	0
$16$	−4.70279	−1.17570
$17$	−4.35003	−1.05504	−0.527519	−	0.849543i	$-0.676877\pi$
$17$	−4.35003	−1.05504	−0.527519	+	0.849543i	$0.676877\pi$
$18$	6.65921	1.56959
$19$	3.14602	0.721746	0.360873	−	0.932615i	$-0.382479\pi$
$19$	3.14602	0.721746	0.360873	+	0.932615i	$0.382479\pi$
$20$	0	0
$21$	−8.99446	−1.96275
$22$	−2.10341	−0.448449
$23$	−6.36510	−1.32722	−0.663608	−	0.748081i	$-0.730976\pi$
$23$	−6.36510	−1.32722	−0.663608	+	0.748081i	$0.730976\pi$
$24$	−2.18954	−0.446938
$25$	0	0
$26$	−0.434076	−0.0851293
$27$	1.37232	0.264103
$28$	−5.43587	−1.02728
$29$	−0.0527009	−0.00978631	−0.00489315	−	0.999988i	$-0.501558\pi$
$29$	−0.0527009	−0.00978631	−0.00489315	+	0.999988i	$0.501558\pi$
$30$	0	0
$31$	−4.55850	−0.818730	−0.409365	−	0.912371i	$-0.634250\pi$
$31$	−4.55850	−0.818730	−0.409365	+	0.912371i	$0.634250\pi$
$32$	−7.14193	−1.26253
$33$	−2.85619	−0.497199
$34$	−8.19050	−1.40466
$35$	0	0
$36$	5.46487	0.910811
$37$	−6.27282	−1.03125	−0.515623	−	0.856816i	$-0.672439\pi$
$37$	−6.27282	−1.03125	−0.515623	+	0.856816i	$0.672439\pi$
$38$	5.92351	0.960921
$39$	−0.589425	−0.0943835
$40$	0	0
$41$	−8.79348	−1.37331	−0.686655	−	0.726983i	$-0.740922\pi$
$41$	−8.79348	−1.37331	−0.686655	+	0.726983i	$0.740922\pi$
$42$	−16.9353	−2.61318
$43$	2.95908	0.451256	0.225628	−	0.974214i	$-0.427557\pi$
$43$	2.95908	0.451256	0.225628	+	0.974214i	$0.427557\pi$
$44$	−1.72616	−0.260228
$45$	0	0
$46$	−11.9846	−1.76703
$47$	10.3334	1.50729	0.753643	−	0.657284i	$-0.228295\pi$
$47$	10.3334	1.50729	0.753643	+	0.657284i	$0.228295\pi$
$48$	−12.0237	−1.73547
$49$	5.37623	0.768033
$50$	0	0
$51$	−11.1218	−1.55736
$52$	−0.356224	−0.0493993
$53$	6.03035	0.828332	0.414166	−	0.910201i	$-0.364073\pi$
$53$	6.03035	0.828332	0.414166	+	0.910201i	$0.364073\pi$
$54$	2.58389	0.351622
$55$	0	0
$56$	3.01277	0.402598
$57$	8.04344	1.06538
$58$	−0.0992284	−0.0130293
$59$	6.43499	0.837765	0.418882	−	0.908041i	$-0.362422\pi$
$59$	6.43499	0.837765	0.418882	+	0.908041i	$0.362422\pi$
$60$	0	0
$61$	−11.8995	−1.52358	−0.761789	−	0.647825i	$-0.775679\pi$
$61$	−11.8995	−1.52358	−0.761789	+	0.647825i	$0.775679\pi$
$62$	−8.58302	−1.09004
$63$	−12.4423	−1.56758
$64$	−4.04167	−0.505209
$65$	0	0
$66$	−5.37781	−0.661962
$67$	2.08295	0.254473	0.127236	−	0.991872i	$-0.459389\pi$
$67$	2.08295	0.254473	0.127236	+	0.991872i	$0.459389\pi$
$68$	−6.72152	−0.815104
$69$	−16.2737	−1.95912
$70$	0	0
$71$	−5.17552	−0.614222	−0.307111	−	0.951674i	$-0.599362\pi$
$71$	−5.17552	−0.614222	−0.307111	+	0.951674i	$0.599362\pi$
$72$	−3.02884	−0.356952
$73$	1.25992	0.147463	0.0737313	−	0.997278i	$-0.476509\pi$
$73$	1.25992	0.147463	0.0737313	+	0.997278i	$0.476509\pi$
$74$	−11.8109	−1.37298
$75$	0	0
$76$	4.86112	0.557608
$77$	3.93007	0.447873
$78$	−1.10981	−0.125661
$79$	−0.749786	−0.0843575	−0.0421788	−	0.999110i	$-0.513430\pi$
$79$	−0.749786	−0.0843575	−0.0421788	+	0.999110i	$0.513430\pi$
$80$	0	0
$81$	−7.10164	−0.789071
$82$	−16.5569	−1.82840
$83$	11.1374	1.22249	0.611244	−	0.791442i	$-0.290669\pi$
$83$	11.1374	1.22249	0.611244	+	0.791442i	$0.290669\pi$
$84$	−13.8979	−1.51639
$85$	0	0
$86$	5.57154	0.600795
$87$	−0.134741	−0.0144457
$88$	0.956703	0.101985
$89$	−8.15389	−0.864310	−0.432155	−	0.901799i	$-0.642247\pi$
$89$	−8.15389	−0.864310	−0.432155	+	0.901799i	$0.642247\pi$
$90$	0	0
$91$	0.811039	0.0850200
$92$	−9.83514	−1.02538
$93$	−11.6547	−1.20854
$94$	19.4564	2.00678
$95$	0	0
$96$	−18.2598	−1.86364
$97$	−7.78090	−0.790031	−0.395016	−	0.918674i	$-0.629261\pi$
$97$	−7.78090	−0.790031	−0.395016	+	0.918674i	$0.629261\pi$
$98$	10.1227	1.02255
$99$	−3.95103	−0.397094
$100$	0	0
$101$	6.17087	0.614025	0.307012	−	0.951706i	$-0.400671\pi$
$101$	6.17087	0.614025	0.307012	+	0.951706i	$0.400671\pi$
$102$	−20.9407	−2.07344
$103$	12.4124	1.22303	0.611513	−	0.791235i	$-0.290561\pi$
$103$	12.4124	1.22303	0.611513	+	0.791235i	$0.290561\pi$
$104$	0.197433	0.0193599
$105$	0	0
$106$	11.3543	1.10283
$107$	7.65910	0.740433	0.370216	−	0.928946i	$-0.379284\pi$
$107$	7.65910	0.740433	0.370216	+	0.928946i	$0.379284\pi$
$108$	2.12046	0.204041
$109$	−0.837327	−0.0802014	−0.0401007	−	0.999196i	$-0.512768\pi$
$109$	−0.837327	−0.0802014	−0.0401007	+	0.999196i	$0.512768\pi$
$110$	0	0
$111$	−16.0378	−1.52224
$112$	16.5444	1.56330
$113$	14.1036	1.32676	0.663378	−	0.748285i	$-0.269122\pi$
$113$	14.1036	1.32676	0.663378	+	0.748285i	$0.269122\pi$
$114$	15.1447	1.41843
$115$	0	0
$116$	−0.0814316	−0.00756073
$117$	−0.815365	−0.0753805
$118$	12.1162	1.11539
$119$	15.3033	1.40286
$120$	0	0
$121$	−9.75201	−0.886546
$122$	−22.4052	−2.02847
$123$	−22.4824	−2.02717
$124$	−7.04363	−0.632537
$125$	0	0
$126$	−23.4270	−2.08705
$127$	16.6119	1.47407	0.737033	−	0.675857i	$-0.236226\pi$
$127$	16.6119	1.47407	0.737033	+	0.675857i	$0.236226\pi$
$128$	6.67395	0.589900
$129$	7.56551	0.666106
$130$	0	0
$131$	−1.55100	−0.135512	−0.0677558	−	0.997702i	$-0.521584\pi$
$131$	−1.55100	−0.135512	−0.0677558	+	0.997702i	$0.521584\pi$
$132$	−4.41328	−0.384127
$133$	−11.0676	−0.959687
$134$	3.92190	0.338801
$135$	0	0
$136$	3.72532	0.319444
$137$	−1.06515	−0.0910016	−0.0455008	−	0.998964i	$-0.514488\pi$
$137$	−1.06515	−0.0910016	−0.0455008	+	0.998964i	$0.514488\pi$
$138$	−30.6411	−2.60835
$139$	10.1224	0.858570	0.429285	−	0.903169i	$-0.358766\pi$
$139$	10.1224	0.858570	0.429285	+	0.903169i	$0.358766\pi$
$140$	0	0
$141$	26.4196	2.22493
$142$	−9.74479	−0.817765
$143$	0.257545	0.0215370
$144$	−16.6326	−1.38605
$145$	0	0
$146$	2.37226	0.196329
$147$	13.7455	1.13371
$148$	−9.69255	−0.796723
$149$	−18.6972	−1.53173	−0.765867	−	0.642999i	$-0.777690\pi$
$149$	−18.6972	−1.53173	−0.765867	+	0.642999i	$0.777690\pi$
$150$	0	0
$151$	4.48916	0.365323	0.182661	−	0.983176i	$-0.441529\pi$
$151$	4.48916	0.365323	0.182661	+	0.983176i	$0.441529\pi$
$152$	−2.69422	−0.218530
$153$	−15.3850	−1.24380
$154$	7.39977	0.596291
$155$	0	0
$156$	−0.910759	−0.0729191
$157$	−2.73158	−0.218003	−0.109002	−	0.994042i	$-0.534765\pi$
$157$	−2.73158	−0.218003	−0.109002	+	0.994042i	$0.534765\pi$
$158$	−1.41174	−0.112312
$159$	15.4178	1.22271
$160$	0	0
$161$	22.3923	1.76476
$162$	−13.3714	−1.05056
$163$	−19.3913	−1.51884	−0.759421	−	0.650600i	$-0.774517\pi$
$163$	−19.3913	−1.51884	−0.759421	+	0.650600i	$0.774517\pi$
$164$	−13.5874	−1.06100
$165$	0	0
$166$	20.9702	1.62760
$167$	−14.3680	−1.11183	−0.555913	−	0.831241i	$-0.687631\pi$
$167$	−14.3680	−1.11183	−0.555913	+	0.831241i	$0.687631\pi$
$168$	7.70277	0.594281
$169$	−12.9469	−0.995912
$170$	0	0
$171$	11.1267	0.850879
$172$	4.57227	0.348632
$173$	−15.0237	−1.14223	−0.571114	−	0.820871i	$-0.693488\pi$
$173$	−15.0237	−1.14223	−0.571114	+	0.820871i	$0.693488\pi$
$174$	−0.253698	−0.0192328
$175$	0	0
$176$	5.25366	0.396009
$177$	16.4524	1.23664
$178$	−15.3526	−1.15073
$179$	−12.9071	−0.964719	−0.482360	−	0.875973i	$-0.660220\pi$
$179$	−12.9071	−0.964719	−0.482360	+	0.875973i	$0.660220\pi$
$180$	0	0
$181$	21.0476	1.56446	0.782229	−	0.622991i	$-0.214083\pi$
$181$	21.0476	1.56446	0.782229	+	0.622991i	$0.214083\pi$
$182$	1.52707	0.113194
$183$	−30.4236	−2.24898
$184$	5.45101	0.401854
$185$	0	0
$186$	−21.9443	−1.60903
$187$	4.85957	0.355367
$188$	15.9669	1.16450
$189$	−4.82780	−0.351171
$190$	0	0
$191$	−12.3891	−0.896442	−0.448221	−	0.893923i	$-0.647942\pi$
$191$	−12.3891	−0.896442	−0.448221	+	0.893923i	$0.647942\pi$
$192$	−10.3334	−0.745747
$193$	−12.3460	−0.888684	−0.444342	−	0.895857i	$-0.646562\pi$
$193$	−12.3460	−0.888684	−0.444342	+	0.895857i	$0.646562\pi$
$194$	−14.6504	−1.05183
$195$	0	0
$196$	8.30717	0.593369
$197$	7.55643	0.538373	0.269187	−	0.963088i	$-0.413245\pi$
$197$	7.55643	0.538373	0.269187	+	0.963088i	$0.413245\pi$
$198$	−7.43924	−0.528684
$199$	−19.3457	−1.37138	−0.685689	−	0.727895i	$-0.740499\pi$
$199$	−19.3457	−1.37138	−0.685689	+	0.727895i	$0.740499\pi$
$200$	0	0
$201$	5.32549	0.375631
$202$	11.6189	0.817503
$203$	0.185401	0.0130126
$204$	−17.1850	−1.20319
$205$	0	0
$206$	23.3707	1.62832
$207$	−22.5118	−1.56468
$208$	1.08419	0.0751747
$209$	−3.51453	−0.243105
$210$	0	0
$211$	24.8658	1.71183	0.855915	−	0.517116i	$-0.172994\pi$
$211$	24.8658	1.71183	0.855915	+	0.517116i	$0.172994\pi$
$212$	9.31788	0.639955
$213$	−13.2323	−0.906662
$214$	14.4210	0.985800
$215$	0	0
$216$	−1.17524	−0.0799650
$217$	16.0367	1.08864
$218$	−1.57657	−0.106779
$219$	3.22125	0.217672
$220$	0	0
$221$	1.00286	0.0674596
$222$	−30.1969	−2.02668
$223$	−1.62890	−0.109079	−0.0545396	−	0.998512i	$-0.517369\pi$
$223$	−1.62890	−0.109079	−0.0545396	+	0.998512i	$0.517369\pi$
$224$	25.1252	1.67875
$225$	0	0
$226$	26.5551	1.76642
$227$	11.5071	0.763751	0.381876	−	0.924214i	$-0.375278\pi$
$227$	11.5071	0.763751	0.381876	+	0.924214i	$0.375278\pi$
$228$	12.4285	0.823094
$229$	21.7531	1.43749	0.718743	−	0.695276i	$-0.244718\pi$
$229$	21.7531	1.43749	0.718743	+	0.695276i	$0.244718\pi$
$230$	0	0
$231$	10.0480	0.661112
$232$	0.0451325	0.00296309
$233$	6.93311	0.454203	0.227102	−	0.973871i	$-0.427075\pi$
$233$	6.93311	0.454203	0.227102	+	0.973871i	$0.427075\pi$
$234$	−1.53522	−0.100360
$235$	0	0
$236$	9.94313	0.647243
$237$	−1.91698	−0.124521
$238$	28.8141	1.86774
$239$	13.0493	0.844087	0.422044	−	0.906576i	$-0.361313\pi$
$239$	13.0493	0.844087	0.422044	+	0.906576i	$0.361313\pi$
$240$	0	0
$241$	1.00000	0.0644157
$241$	1.00000	0.0644157
$242$	−18.3617	−1.18033
$243$	−22.2738	−1.42886
$244$	−18.3867	−1.17709
$245$	0	0
$246$	−42.3312	−2.69894
$247$	−0.725285	−0.0461488
$248$	3.90385	0.247895
$249$	28.4751	1.80453
$250$	0	0
$251$	−28.8117	−1.81858	−0.909291	−	0.416162i	$-0.863375\pi$
$251$	−28.8117	−1.81858	−0.909291	+	0.416162i	$0.863375\pi$
$252$	−19.2253	−1.21108
$253$	7.11068	0.447045
$254$	31.2778	1.96255
$255$	0	0
$256$	20.6495	1.29059
$257$	−15.3709	−0.958809	−0.479404	−	0.877594i	$-0.659147\pi$
$257$	−15.3709	−0.958809	−0.479404	+	0.877594i	$0.659147\pi$
$258$	14.2448	0.886842
$259$	22.0677	1.37122
$260$	0	0
$261$	−0.186390	−0.0115372
$262$	−2.92032	−0.180418
$263$	−7.75814	−0.478387	−0.239194	−	0.970972i	$-0.576883\pi$
$263$	−7.75814	−0.478387	−0.239194	+	0.970972i	$0.576883\pi$
$264$	2.44601	0.150542
$265$	0	0
$266$	−20.8388	−1.27771
$267$	−20.8471	−1.27582
$268$	3.21850	0.196601
$269$	−26.5185	−1.61686	−0.808430	−	0.588592i	$-0.799682\pi$
$269$	−26.5185	−1.61686	−0.808430	+	0.588592i	$0.799682\pi$
$270$	0	0
$271$	−25.7841	−1.56627	−0.783137	−	0.621849i	$-0.786382\pi$
$271$	−25.7841	−1.56627	−0.783137	+	0.621849i	$0.786382\pi$
$272$	20.4573	1.24041
$273$	2.07359	0.125499
$274$	−2.00552	−0.121158
$275$	0	0
$276$	−25.1456	−1.51359
$277$	−16.8051	−1.00972	−0.504860	−	0.863201i	$-0.668456\pi$
$277$	−16.8051	−1.00972	−0.504860	+	0.863201i	$0.668456\pi$
$278$	19.0591	1.14309
$279$	−16.1223	−0.965215
$280$	0	0
$281$	7.13034	0.425360	0.212680	−	0.977122i	$-0.431781\pi$
$281$	7.13034	0.425360	0.212680	+	0.977122i	$0.431781\pi$
$282$	49.7444	2.96223
$283$	1.50077	0.0892114	0.0446057	−	0.999005i	$-0.485797\pi$
$283$	1.50077	0.0892114	0.0446057	+	0.999005i	$0.485797\pi$
$284$	−7.99704	−0.474537
$285$	0	0
$286$	0.484922	0.0286740
$287$	30.9354	1.82606
$288$	−25.2592	−1.48841
$289$	1.92276	0.113103
$290$	0	0
$291$	−19.8935	−1.16618
$292$	1.94679	0.113927
$293$	−27.0432	−1.57988	−0.789939	−	0.613186i	$-0.789888\pi$
$293$	−27.0432	−1.57988	−0.789939	+	0.613186i	$0.789888\pi$
$294$	25.8808	1.50940
$295$	0	0
$296$	5.37198	0.312240
$297$	−1.53307	−0.0889575
$298$	−35.2042	−2.03933
$299$	1.46741	0.0848628
$300$	0	0
$301$	−10.4100	−0.600023
$302$	8.45247	0.486385
$303$	15.7771	0.906371
$304$	−14.7951	−0.848555
$305$	0	0
$306$	−28.9678	−1.65598
$307$	1.85072	0.105626	0.0528130	−	0.998604i	$-0.483181\pi$
$307$	1.85072	0.105626	0.0528130	+	0.998604i	$0.483181\pi$
$308$	6.07261	0.346019
$309$	31.7347	1.80533
$310$	0	0
$311$	0.609280	0.0345491	0.0172745	−	0.999851i	$-0.494501\pi$
$311$	0.609280	0.0345491	0.0172745	+	0.999851i	$0.494501\pi$
$312$	0.504778	0.0285774
$313$	22.9454	1.29695	0.648476	−	0.761236i	$-0.275407\pi$
$313$	22.9454	1.29695	0.648476	+	0.761236i	$0.275407\pi$
$314$	−5.14318	−0.290246
$315$	0	0
$316$	−1.15854	−0.0651732
$317$	2.44078	0.137088	0.0685439	−	0.997648i	$-0.478165\pi$
$317$	2.44078	0.137088	0.0685439	+	0.997648i	$0.478165\pi$
$318$	29.0296	1.62790
$319$	0.0588740	0.00329631
$320$	0	0
$321$	19.5821	1.09296
$322$	42.1617	2.34958
$323$	−13.6853	−0.761469
$324$	−10.9732	−0.609623
$325$	0	0
$326$	−36.5111	−2.02216
$327$	−2.14080	−0.118387
$328$	7.53065	0.415810
$329$	−36.3529	−2.00420
$330$	0	0
$331$	6.91377	0.380015	0.190008	−	0.981783i	$-0.439149\pi$
$331$	6.91377	0.380015	0.190008	+	0.981783i	$0.439149\pi$
$332$	17.2091	0.944473
$333$	−22.1854	−1.21575
$334$	−27.0529	−1.48027
$335$	0	0
$336$	42.2991	2.30761
$337$	−14.0181	−0.763616	−0.381808	−	0.924242i	$-0.624698\pi$
$337$	−14.0181	−0.763616	−0.381808	+	0.924242i	$0.624698\pi$
$338$	−24.3771	−1.32594
$339$	36.0588	1.95845
$340$	0	0
$341$	5.09246	0.275772
$342$	20.9500	1.13285
$343$	5.71239	0.308440
$344$	−2.53413	−0.136631
$345$	0	0
$346$	−28.2874	−1.52074
$347$	25.2046	1.35305	0.676526	−	0.736419i	$-0.263485\pi$
$347$	25.2046	1.35305	0.676526	+	0.736419i	$0.263485\pi$
$348$	−0.208197	−0.0111605
$349$	0.543622	0.0290994	0.0145497	−	0.999894i	$-0.495369\pi$
$349$	0.543622	0.0290994	0.0145497	+	0.999894i	$0.495369\pi$
$350$	0	0
$351$	−0.316375	−0.0168869
$352$	7.97850	0.425256
$353$	7.06140	0.375840	0.187920	−	0.982184i	$-0.439825\pi$
$353$	7.06140	0.375840	0.187920	+	0.982184i	$0.439825\pi$
$354$	30.9776	1.64644
$355$	0	0
$356$	−12.5991	−0.667751
$357$	39.1262	2.07078
$358$	−24.3022	−1.28441
$359$	−32.0985	−1.69409	−0.847047	−	0.531519i	$-0.821622\pi$
$359$	−32.0985	−1.69409	−0.847047	+	0.531519i	$0.821622\pi$
$360$	0	0
$361$	−9.10258	−0.479083
$362$	39.6298	2.08289
$363$	−24.9330	−1.30864
$364$	1.25319	0.0656850
$365$	0	0
$366$	−57.2834	−2.99425
$367$	−15.2909	−0.798176	−0.399088	−	0.916913i	$-0.630673\pi$
$367$	−15.2909	−0.798176	−0.399088	+	0.916913i	$0.630673\pi$
$368$	29.9338	1.56041
$369$	−31.1004	−1.61902
$370$	0	0
$371$	−21.2147	−1.10141
$372$	−18.0085	−0.933698
$373$	15.6074	0.808122	0.404061	−	0.914732i	$-0.367598\pi$
$373$	15.6074	0.808122	0.404061	+	0.914732i	$0.367598\pi$
$374$	9.14990	0.473130
$375$	0	0
$376$	−8.84945	−0.456375
$377$	0.0121497	0.000625741 0
$378$	−9.09008	−0.467543
$379$	18.5818	0.954484	0.477242	−	0.878772i	$-0.341637\pi$
$379$	18.5818	0.954484	0.477242	+	0.878772i	$0.341637\pi$
$380$	0	0
$381$	42.4717	2.17589
$382$	−23.3269	−1.19351
$383$	31.3371	1.60125	0.800626	−	0.599165i	$-0.204501\pi$
$383$	31.3371	1.60125	0.800626	+	0.599165i	$0.204501\pi$
$384$	17.0633	0.870760
$385$	0	0
$386$	−23.2458	−1.18318
$387$	10.4655	0.531993
$388$	−12.0228	−0.610364
$389$	−24.3977	−1.23701	−0.618506	−	0.785780i	$-0.712262\pi$
$389$	−24.3977	−1.23701	−0.618506	+	0.785780i	$0.712262\pi$
$390$	0	0
$391$	27.6884	1.40026
$392$	−4.60415	−0.232545
$393$	−3.96545	−0.200031
$394$	14.2277	0.716781
$395$	0	0
$396$	−6.10500	−0.306788
$397$	9.32624	0.468071	0.234035	−	0.972228i	$-0.424807\pi$
$397$	9.32624	0.468071	0.234035	+	0.972228i	$0.424807\pi$
$398$	−36.4252	−1.82583
$399$	−28.2967	−1.41661
$400$	0	0
$401$	−10.9440	−0.546518	−0.273259	−	0.961940i	$-0.588102\pi$
$401$	−10.9440	−0.546518	−0.273259	+	0.961940i	$0.588102\pi$
$402$	10.0272	0.500109
$403$	1.05092	0.0523500
$404$	9.53502	0.474385
$405$	0	0
$406$	0.349084	0.0173248
$407$	7.00759	0.347353
$408$	9.52456	0.471536
$409$	29.3280	1.45017	0.725087	−	0.688657i	$-0.241799\pi$
$409$	29.3280	1.45017	0.725087	+	0.688657i	$0.241799\pi$
$410$	0	0
$411$	−2.72327	−0.134329
$412$	19.1791	0.944888
$413$	−22.6382	−1.11395
$414$	−42.3866	−2.08319
$415$	0	0
$416$	1.64650	0.0807265
$417$	25.8800	1.26735
$418$	−6.61737	−0.323666
$419$	28.6323	1.39878	0.699391	−	0.714740i	$-0.253455\pi$
$419$	28.6323	1.39878	0.699391	+	0.714740i	$0.253455\pi$
$420$	0	0
$421$	−0.0637058	−0.00310483	−0.00155241	−	0.999999i	$-0.500494\pi$
$421$	−0.0637058	−0.00310483	−0.00155241	+	0.999999i	$0.500494\pi$
$422$	46.8188	2.27910
$423$	36.5468	1.77697
$424$	−5.16433	−0.250802
$425$	0	0
$426$	−24.9146	−1.20712
$427$	41.8624	2.02586
$428$	11.8346	0.572046
$429$	0.658468	0.0317911
$430$	0	0
$431$	−25.7098	−1.23840	−0.619199	−	0.785234i	$-0.712543\pi$
$431$	−25.7098	−1.23840	−0.619199	+	0.785234i	$0.712543\pi$
$432$	−6.45374	−0.310506
$433$	22.4971	1.08114	0.540570	−	0.841299i	$-0.318209\pi$
$433$	22.4971	1.08114	0.540570	+	0.841299i	$0.318209\pi$
$434$	30.1949	1.44940
$435$	0	0
$436$	−1.29381	−0.0619622
$437$	−20.0247	−0.957912
$438$	6.06516	0.289805
$439$	−17.3064	−0.825989	−0.412995	−	0.910733i	$-0.635517\pi$
$439$	−17.3064	−0.825989	−0.412995	+	0.910733i	$0.635517\pi$
$440$	0	0
$441$	19.0144	0.905448
$442$	1.88824	0.0898146
$443$	35.4484	1.68421	0.842103	−	0.539317i	$-0.181318\pi$
$443$	35.4484	1.68421	0.842103	+	0.539317i	$0.181318\pi$
$444$	−24.7810	−1.17605
$445$	0	0
$446$	−3.06699	−0.145226
$447$	−47.8032	−2.26102
$448$	14.2185	0.671763
$449$	15.9564	0.753031	0.376516	−	0.926410i	$-0.377122\pi$
$449$	15.9564	0.753031	0.376516	+	0.926410i	$0.377122\pi$
$450$	0	0
$451$	9.82351	0.462571
$452$	21.7924	1.02503
$453$	11.4775	0.539259
$454$	21.6662	1.01685
$455$	0	0
$456$	−6.88832	−0.322575
$457$	−8.07516	−0.377740	−0.188870	−	0.982002i	$-0.560482\pi$
$457$	−8.07516	−0.377740	−0.188870	+	0.982002i	$0.560482\pi$
$458$	40.9581	1.91384
$459$	−5.96963	−0.278638
$460$	0	0
$461$	−27.1354	−1.26382	−0.631912	−	0.775040i	$-0.717730\pi$
$461$	−27.1354	−1.26382	−0.631912	+	0.775040i	$0.717730\pi$
$462$	18.9191	0.880194
$463$	−29.4190	−1.36721	−0.683607	−	0.729850i	$-0.739590\pi$
$463$	−29.4190	−1.36721	−0.683607	+	0.729850i	$0.739590\pi$
$464$	0.247841	0.0115057
$465$	0	0
$466$	13.0541	0.604719
$467$	−2.70008	−0.124945	−0.0624724	−	0.998047i	$-0.519899\pi$
$467$	−2.70008	−0.124945	−0.0624724	+	0.998047i	$0.519899\pi$
$468$	−1.25987	−0.0582377
$469$	−7.32779	−0.338366
$470$	0	0
$471$	−6.98384	−0.321798
$472$	−5.51086	−0.253658
$473$	−3.30570	−0.151996
$474$	−3.60941	−0.165786
$475$	0	0
$476$	23.6462	1.08382
$477$	21.3278	0.976535
$478$	24.5700	1.12380
$479$	−24.4078	−1.11522	−0.557609	−	0.830103i	$-0.688281\pi$
$479$	−24.4078	−1.11522	−0.557609	+	0.830103i	$0.688281\pi$
$480$	0	0
$481$	1.44614	0.0659383
$482$	1.88286	0.0857620
$483$	57.2507	2.60500
$484$	−15.0685	−0.684930
$485$	0	0
$486$	−41.9384	−1.90237
$487$	19.3677	0.877633	0.438817	−	0.898577i	$-0.355398\pi$
$487$	19.3677	0.877633	0.438817	+	0.898577i	$0.355398\pi$
$488$	10.1906	0.461308
$489$	−49.5778	−2.24199
$490$	0	0
$491$	14.3025	0.645462	0.322731	−	0.946491i	$-0.395399\pi$
$491$	14.3025	0.645462	0.322731	+	0.946491i	$0.395399\pi$
$492$	−34.7390	−1.56615
$493$	0.229250	0.0103249
$494$	−1.36561	−0.0614417
$495$	0	0
$496$	21.4377	0.962580
$497$	18.2074	0.816715
$498$	53.6146	2.40253
$499$	−22.6504	−1.01397	−0.506987	−	0.861954i	$-0.669241\pi$
$499$	−22.6504	−1.01397	−0.506987	+	0.861954i	$0.669241\pi$
$500$	0	0
$501$	−36.7347	−1.64118
$502$	−54.2485	−2.42123
$503$	33.2011	1.48036	0.740181	−	0.672407i	$-0.234740\pi$
$503$	33.2011	1.48036	0.740181	+	0.672407i	$0.234740\pi$
$504$	10.6554	0.474630
$505$	0	0
$506$	13.3884	0.595188
$507$	−33.1013	−1.47008
$508$	25.6681	1.13884
$509$	26.1038	1.15703	0.578515	−	0.815671i	$-0.303632\pi$
$509$	26.1038	1.15703	0.578515	+	0.815671i	$0.303632\pi$
$510$	0	0
$511$	−4.43239	−0.196077
$512$	25.5322	1.12837
$513$	4.31734	0.190615
$514$	−28.9412	−1.27654
$515$	0	0
$516$	11.6900	0.514622
$517$	−11.5438	−0.507698
$518$	41.5504	1.82562
$519$	−38.4111	−1.68606
$520$	0	0
$521$	11.1582	0.488849	0.244425	−	0.969668i	$-0.421401\pi$
$521$	11.1582	0.488849	0.244425	+	0.969668i	$0.421401\pi$
$522$	−0.350946	−0.0153605
$523$	−21.4491	−0.937902	−0.468951	−	0.883224i	$-0.655368\pi$
$523$	−21.4491	−0.937902	−0.468951	+	0.883224i	$0.655368\pi$
$524$	−2.39655	−0.104694
$525$	0	0
$526$	−14.6075	−0.636917
$527$	19.8296	0.863791
$528$	13.4321	0.584556
$529$	17.5145	0.761501
$530$	0	0
$531$	22.7590	0.987655
$532$	−17.1013	−0.741437
$533$	2.02725	0.0878102
$534$	−39.2522	−1.69861
$535$	0	0
$536$	−1.78382	−0.0770491
$537$	−32.9996	−1.42404
$538$	−49.9306	−2.15266
$539$	−6.00598	−0.258696
$540$	0	0
$541$	37.9845	1.63308	0.816541	−	0.577288i	$-0.195889\pi$
$541$	37.9845	1.63308	0.816541	+	0.577288i	$0.195889\pi$
$542$	−48.5479	−2.08531
$543$	53.8126	2.30932
$544$	31.0676	1.33201
$545$	0	0
$546$	3.90428	0.167088
$547$	−12.5760	−0.537711	−0.268856	−	0.963180i	$-0.586645\pi$
$547$	−12.5760	−0.537711	−0.268856	+	0.963180i	$0.586645\pi$
$548$	−1.64583	−0.0703063
$549$	−42.0857	−1.79617
$550$	0	0
$551$	−0.165798	−0.00706322
$552$	13.9366	0.593183
$553$	2.63774	0.112168
$554$	−31.6417	−1.34433
$555$	0	0
$556$	15.6408	0.663316
$557$	17.7488	0.752039	0.376020	−	0.926612i	$-0.377293\pi$
$557$	17.7488	0.752039	0.376020	+	0.926612i	$0.377293\pi$
$558$	−30.3560	−1.28507
$559$	−0.682189	−0.0288535
$560$	0	0
$561$	12.4245	0.524563
$562$	13.4254	0.566318
$563$	−19.5515	−0.823999	−0.411999	−	0.911184i	$-0.635169\pi$
$563$	−19.5515	−0.823999	−0.411999	+	0.911184i	$0.635169\pi$
$564$	40.8226	1.71894
$565$	0	0
$566$	2.82574	0.118775
$567$	24.9835	1.04921
$568$	4.43227	0.185974
$569$	41.0771	1.72204	0.861021	−	0.508570i	$-0.169826\pi$
$569$	41.0771	1.72204	0.861021	+	0.508570i	$0.169826\pi$
$570$	0	0
$571$	−34.8425	−1.45811	−0.729057	−	0.684453i	$-0.760041\pi$
$571$	−34.8425	−1.45811	−0.729057	+	0.684453i	$0.760041\pi$
$572$	0.397950	0.0166391
$573$	−31.6752	−1.32325
$574$	58.2470	2.43118
$575$	0	0
$576$	−14.2944	−0.595599
$577$	2.25604	0.0939202	0.0469601	−	0.998897i	$-0.485047\pi$
$577$	2.25604	0.0939202	0.0469601	+	0.998897i	$0.485047\pi$
$578$	3.62029	0.150584
$579$	−31.5651	−1.31180
$580$	0	0
$581$	−39.1812	−1.62551
$582$	−37.4567	−1.55263
$583$	−6.73671	−0.279006
$584$	−1.07898	−0.0446487
$585$	0	0
$586$	−50.9185	−2.10342
$587$	−27.6141	−1.13976	−0.569879	−	0.821729i	$-0.693010\pi$
$587$	−27.6141	−1.13976	−0.569879	+	0.821729i	$0.693010\pi$
$588$	21.2390	0.875882
$589$	−14.3411	−0.590915
$590$	0	0
$591$	19.3196	0.794701
$592$	29.4998	1.21243
$593$	32.2396	1.32392	0.661961	−	0.749538i	$-0.269724\pi$
$593$	32.2396	1.32392	0.661961	+	0.749538i	$0.269724\pi$
$594$	−2.88655	−0.118437
$595$	0	0
$596$	−28.8903	−1.18339
$597$	−49.4612	−2.02431
$598$	2.76294	0.112985
$599$	23.7981	0.972365	0.486183	−	0.873857i	$-0.338389\pi$
$599$	23.7981	0.972365	0.486183	+	0.873857i	$0.338389\pi$
$600$	0	0
$601$	8.59760	0.350703	0.175352	−	0.984506i	$-0.443894\pi$
$601$	8.59760	0.350703	0.175352	+	0.984506i	$0.443894\pi$
$602$	−19.6006	−0.798861
$603$	7.36687	0.300002
$604$	6.93650	0.282242
$605$	0	0
$606$	29.7061	1.20673
$607$	−13.7016	−0.556130	−0.278065	−	0.960562i	$-0.589693\pi$
$607$	−13.7016	−0.556130	−0.278065	+	0.960562i	$0.589693\pi$
$608$	−22.4686	−0.911223
$609$	0.474016	0.0192081
$610$	0	0
$611$	−2.38228	−0.0963766
$612$	−23.7723	−0.960940
$613$	44.7325	1.80673	0.903365	−	0.428873i	$-0.141089\pi$
$613$	44.7325	1.80673	0.903365	+	0.428873i	$0.141089\pi$
$614$	3.48464	0.140629
$615$	0	0
$616$	−3.36567	−0.135607
$617$	11.8716	0.477931	0.238965	−	0.971028i	$-0.423192\pi$
$617$	11.8716	0.477931	0.238965	+	0.971028i	$0.423192\pi$
$618$	59.7521	2.40358
$619$	−30.7196	−1.23472	−0.617362	−	0.786679i	$-0.711799\pi$
$619$	−30.7196	−1.23472	−0.617362	+	0.786679i	$0.711799\pi$
$620$	0	0
$621$	−8.73495	−0.350522
$622$	1.14719	0.0459981
$623$	28.6853	1.14925
$624$	2.77194	0.110967
$625$	0	0
$626$	43.2030	1.72674
$627$	−8.98562	−0.358851
$628$	−4.22074	−0.168426
$629$	27.2870	1.08800
$630$	0	0
$631$	−22.5309	−0.896943	−0.448471	−	0.893797i	$-0.648031\pi$
$631$	−22.5309	−0.896943	−0.448471	+	0.893797i	$0.648031\pi$
$632$	0.642109	0.0255417
$633$	63.5745	2.52686
$634$	4.59565	0.182517
$635$	0	0
$636$	23.8231	0.944647
$637$	−1.23944	−0.0491084
$638$	0.110852	0.00438866
$639$	−18.3045	−0.724117
$640$	0	0
$641$	31.9732	1.26287	0.631433	−	0.775430i	$-0.282467\pi$
$641$	31.9732	1.26287	0.631433	+	0.775430i	$0.282467\pi$
$642$	36.8703	1.45516
$643$	1.68365	0.0663968	0.0331984	−	0.999449i	$-0.489431\pi$
$643$	1.68365	0.0663968	0.0331984	+	0.999449i	$0.489431\pi$
$644$	34.5999	1.36343
$645$	0	0
$646$	−25.7675	−1.01381
$647$	4.78774	0.188225	0.0941127	−	0.995562i	$-0.469999\pi$
$647$	4.78774	0.188225	0.0941127	+	0.995562i	$0.469999\pi$
$648$	6.08177	0.238915
$649$	−7.18876	−0.282183
$650$	0	0
$651$	41.0012	1.60697
$652$	−29.9627	−1.17343
$653$	−17.3077	−0.677301	−0.338650	−	0.940912i	$-0.609970\pi$
$653$	−17.3077	−0.677301	−0.338650	+	0.940912i	$0.609970\pi$
$654$	−4.03083	−0.157618
$655$	0	0
$656$	41.3539	1.61460
$657$	4.45603	0.173846
$658$	−68.4474	−2.66836
$659$	−6.27035	−0.244258	−0.122129	−	0.992514i	$-0.538972\pi$
$659$	−6.27035	−0.244258	−0.122129	+	0.992514i	$0.538972\pi$
$660$	0	0
$661$	−6.20469	−0.241334	−0.120667	−	0.992693i	$-0.538503\pi$
$661$	−6.20469	−0.241334	−0.120667	+	0.992693i	$0.538503\pi$
$662$	13.0177	0.505946
$663$	2.56402	0.0995781
$664$	−9.53795	−0.370144
$665$	0	0
$666$	−41.7721	−1.61863
$667$	0.335446	0.0129885
$668$	−22.2009	−0.858977
$669$	−4.16462	−0.161013
$670$	0	0
$671$	13.2934	0.513185
$672$	64.2378	2.47803
$673$	32.2078	1.24152	0.620759	−	0.784001i	$-0.286824\pi$
$673$	32.2078	1.24152	0.620759	+	0.784001i	$0.286824\pi$
$674$	−26.3942	−1.01667
$675$	0	0
$676$	−20.0050	−0.769424
$677$	−16.1123	−0.619245	−0.309622	−	0.950860i	$-0.600203\pi$
$677$	−16.1123	−0.619245	−0.309622	+	0.950860i	$0.600203\pi$
$678$	67.8937	2.60744
$679$	27.3731	1.05048
$680$	0	0
$681$	29.4202	1.12739
$682$	9.58839	0.367158
$683$	−15.1853	−0.581048	−0.290524	−	0.956868i	$-0.593830\pi$
$683$	−15.1853	−0.581048	−0.290524	+	0.956868i	$0.593830\pi$
$684$	17.1926	0.657374
$685$	0	0
$686$	10.7556	0.410652
$687$	55.6163	2.12189
$688$	−13.9160	−0.530541
$689$	−1.39024	−0.0529639
$690$	0	0
$691$	15.7865	0.600547	0.300274	−	0.953853i	$-0.402922\pi$
$691$	15.7865	0.600547	0.300274	+	0.953853i	$0.402922\pi$
$692$	−23.2140	−0.882465
$693$	13.8997	0.528005
$694$	47.4567	1.80143
$695$	0	0
$696$	0.115391	0.00437387
$697$	38.2519	1.44889
$698$	1.02357	0.0387425
$699$	17.7259	0.670456
$700$	0	0
$701$	10.3713	0.391717	0.195859	−	0.980632i	$-0.437251\pi$
$701$	10.3713	0.391717	0.195859	+	0.980632i	$0.437251\pi$
$702$	−0.595691	−0.0224829
$703$	−19.7344	−0.744297
$704$	4.51509	0.170169
$705$	0	0
$706$	13.2956	0.500387
$707$	−21.7090	−0.816453
$708$	25.4217	0.955405
$709$	17.6528	0.662966	0.331483	−	0.943461i	$-0.392451\pi$
$709$	17.6528	0.662966	0.331483	+	0.943461i	$0.392451\pi$
$710$	0	0
$711$	−2.65181	−0.0994506
$712$	6.98291	0.261696
$713$	29.0153	1.08663
$714$	73.6692	2.75700
$715$	0	0
$716$	−19.9435	−0.745325
$717$	33.3632	1.24597
$718$	−60.4370	−2.25549
$719$	−20.1461	−0.751324	−0.375662	−	0.926757i	$-0.622585\pi$
$719$	−20.1461	−0.751324	−0.375662	+	0.926757i	$0.622585\pi$
$720$	0	0
$721$	−43.6665	−1.62623
$722$	−17.1389	−0.637843
$723$	2.55671	0.0950850
$724$	32.5221	1.20867
$725$	0	0
$726$	−46.9454	−1.74231
$727$	−49.4317	−1.83332	−0.916661	−	0.399666i	$-0.869126\pi$
$727$	−49.4317	−1.83332	−0.916661	+	0.399666i	$0.869126\pi$
$728$	−0.694566	−0.0257423
$729$	−35.6426	−1.32010
$730$	0	0
$731$	−12.8721	−0.476092
$732$	−47.0095	−1.73752
$733$	−31.2375	−1.15378	−0.576891	−	0.816821i	$-0.695734\pi$
$733$	−31.2375	−1.15378	−0.576891	+	0.816821i	$0.695734\pi$
$734$	−28.7905	−1.06268
$735$	0	0
$736$	45.4591	1.67564
$737$	−2.32694	−0.0857138
$738$	−58.5577	−2.15554
$739$	−0.931126	−0.0342520	−0.0171260	−	0.999853i	$-0.505452\pi$
$739$	−0.931126	−0.0342520	−0.0171260	+	0.999853i	$0.505452\pi$
$740$	0	0
$741$	−1.85434	−0.0681209
$742$	−39.9443	−1.46640
$743$	−50.7530	−1.86195	−0.930974	−	0.365085i	$-0.881040\pi$
$743$	−50.7530	−1.86195	−0.930974	+	0.365085i	$0.881040\pi$
$744$	9.98100	0.365921
$745$	0	0
$746$	29.3866	1.07592
$747$	39.3902	1.44121
$748$	7.50884	0.274551
$749$	−26.9446	−0.984534
$750$	0	0
$751$	−10.6990	−0.390412	−0.195206	−	0.980762i	$-0.562538\pi$
$751$	−10.6990	−0.390412	−0.195206	+	0.980762i	$0.562538\pi$
$752$	−48.5960	−1.77211
$753$	−73.6632	−2.68444
$754$	0.0228762	0.000833101 0
$755$	0	0
$756$	−7.45975	−0.271309
$757$	−3.26326	−0.118605	−0.0593027	−	0.998240i	$-0.518888\pi$
$757$	−3.26326	−0.118605	−0.0593027	+	0.998240i	$0.518888\pi$
$758$	34.9870	1.27078
$759$	18.1799	0.659890
$760$	0	0
$761$	−43.3419	−1.57114	−0.785572	−	0.618771i	$-0.787631\pi$
$761$	−43.3419	−1.57114	−0.785572	+	0.618771i	$0.787631\pi$
$762$	79.9683	2.89695
$763$	2.94571	0.106642
$764$	−19.1432	−0.692576
$765$	0	0
$766$	59.0034	2.13188
$767$	−1.48353	−0.0535671
$768$	52.7946	1.90506
$769$	26.6436	0.960793	0.480396	−	0.877051i	$-0.340493\pi$
$769$	26.6436	0.960793	0.480396	+	0.877051i	$0.340493\pi$
$770$	0	0
$771$	−39.2988	−1.41531
$772$	−19.0766	−0.686582
$773$	40.7749	1.46657	0.733285	−	0.679922i	$-0.237986\pi$
$773$	40.7749	1.46657	0.733285	+	0.679922i	$0.237986\pi$
$774$	19.7052	0.708287
$775$	0	0
$776$	6.66349	0.239205
$777$	56.4207	2.02408
$778$	−45.9375	−1.64694
$779$	−27.6644	−0.991181
$780$	0	0
$781$	5.78176	0.206888
$782$	52.1334	1.86429
$783$	−0.0723224	−0.00258459
$784$	−25.2833	−0.902976
$785$	0	0
$786$	−7.46640	−0.266318
$787$	−13.2146	−0.471051	−0.235525	−	0.971868i	$-0.575681\pi$
$787$	−13.2146	−0.471051	−0.235525	+	0.971868i	$0.575681\pi$
$788$	11.6759	0.415938
$789$	−19.8353	−0.706155
$790$	0	0
$791$	−49.6163	−1.76415
$792$	3.38362	0.120232
$793$	2.74333	0.0974183
$794$	17.5600	0.623182
$795$	0	0
$796$	−29.8923	−1.05950
$797$	3.31174	0.117308	0.0586539	−	0.998278i	$-0.481319\pi$
$797$	3.31174	0.117308	0.0586539	+	0.998278i	$0.481319\pi$
$798$	−53.2788	−1.88605
$799$	−44.9507	−1.59024
$800$	0	0
$801$	−28.8383	−1.01895
$802$	−20.6061	−0.727626
$803$	−1.40750	−0.0496697
$804$	8.22876	0.290206
$805$	0	0
$806$	1.97873	0.0696979
$807$	−67.8000	−2.38667
$808$	−5.28467	−0.185914
$809$	22.5759	0.793727	0.396864	−	0.917878i	$-0.370099\pi$
$809$	22.5759	0.793727	0.396864	+	0.917878i	$0.370099\pi$
$810$	0	0
$811$	17.1034	0.600581	0.300291	−	0.953848i	$-0.402916\pi$
$811$	17.1034	0.600581	0.300291	+	0.953848i	$0.402916\pi$
$812$	0.286475	0.0100533
$813$	−65.9225	−2.31200
$814$	13.1943	0.462461
$815$	0	0
$816$	52.3033	1.83098
$817$	9.30932	0.325692
$818$	55.2205	1.93074
$819$	2.86844	0.100232
$820$	0	0
$821$	−7.57786	−0.264469	−0.132235	−	0.991218i	$-0.542215\pi$
$821$	−7.57786	−0.264469	−0.132235	+	0.991218i	$0.542215\pi$
$822$	−5.12754	−0.178843
$823$	−43.7880	−1.52635	−0.763177	−	0.646190i	$-0.776361\pi$
$823$	−43.7880	−1.52635	−0.763177	+	0.646190i	$0.776361\pi$
$824$	−10.6298	−0.370307
$825$	0	0
$826$	−42.6246	−1.48310
$827$	−12.9806	−0.451379	−0.225690	−	0.974199i	$-0.572464\pi$
$827$	−12.9806	−0.451379	−0.225690	+	0.974199i	$0.572464\pi$
$828$	−34.7844	−1.20884
$829$	−2.03755	−0.0707669	−0.0353834	−	0.999374i	$-0.511265\pi$
$829$	−2.03755	−0.0707669	−0.0353834	+	0.999374i	$0.511265\pi$
$830$	0	0
$831$	−42.9657	−1.49046
$832$	0.931769	0.0323033
$833$	−23.3868	−0.810304
$834$	48.7284	1.68733
$835$	0	0
$836$	−5.43053	−0.187819
$837$	−6.25571	−0.216229
$838$	53.9107	1.86231
$839$	−27.3908	−0.945635	−0.472817	−	0.881160i	$-0.656763\pi$
$839$	−27.3908	−0.945635	−0.472817	+	0.881160i	$0.656763\pi$
$840$	0	0
$841$	−28.9972	−0.999904
$842$	−0.119949	−0.00413372
$843$	18.2302	0.627881
$844$	38.4217	1.32253
$845$	0	0
$846$	68.8125	2.36582
$847$	34.3074	1.17882
$848$	−28.3595	−0.973868
$849$	3.83703	0.131686
$850$	0	0
$851$	39.9271	1.36869
$852$	−20.4461	−0.700472
$853$	−26.0906	−0.893325	−0.446663	−	0.894702i	$-0.647388\pi$
$853$	−26.0906	−0.893325	−0.446663	+	0.894702i	$0.647388\pi$
$854$	78.8211	2.69720
$855$	0	0
$856$	−6.55917	−0.224188
$857$	−17.1819	−0.586922	−0.293461	−	0.955971i	$-0.594807\pi$
$857$	−17.1819	−0.586922	−0.293461	+	0.955971i	$0.594807\pi$
$858$	1.23980	0.0423262
$859$	2.77639	0.0947292	0.0473646	−	0.998878i	$-0.484918\pi$
$859$	2.77639	0.0947292	0.0473646	+	0.998878i	$0.484918\pi$
$860$	0	0
$861$	79.0926	2.69547
$862$	−48.4080	−1.64878
$863$	11.5989	0.394832	0.197416	−	0.980320i	$-0.436745\pi$
$863$	11.5989	0.394832	0.197416	+	0.980320i	$0.436745\pi$
$864$	−9.80101	−0.333437
$865$	0	0
$866$	42.3588	1.43941
$867$	4.91593	0.166954
$868$	24.7794	0.841068
$869$	0.837613	0.0284141
$870$	0	0
$871$	−0.480204	−0.0162711
$872$	0.717078	0.0242833
$873$	−27.5191	−0.931381
$874$	−37.7038	−1.27535
$875$	0	0
$876$	4.97736	0.168170
$877$	−19.8014	−0.668647	−0.334323	−	0.942458i	$-0.608508\pi$
$877$	−19.8014	−0.668647	−0.334323	+	0.942458i	$0.608508\pi$
$878$	−32.5855	−1.09971
$879$	−69.1414	−2.33208
$880$	0	0
$881$	50.1717	1.69033	0.845164	−	0.534508i	$-0.179503\pi$
$881$	50.1717	1.69033	0.845164	+	0.534508i	$0.179503\pi$
$882$	35.8015	1.20550
$883$	−13.0038	−0.437614	−0.218807	−	0.975768i	$-0.570216\pi$
$883$	−13.0038	−0.437614	−0.218807	+	0.975768i	$0.570216\pi$
$884$	1.54958	0.0521181
$885$	0	0
$886$	66.7444	2.24232
$887$	−14.2467	−0.478358	−0.239179	−	0.970975i	$-0.576878\pi$
$887$	−14.2467	−0.478358	−0.239179	+	0.970975i	$0.576878\pi$
$888$	13.7346	0.460902
$889$	−58.4403	−1.96003
$890$	0	0
$891$	7.93349	0.265782
$892$	−2.51692	−0.0842727
$893$	32.5091	1.08788
$894$	−90.0069	−3.01028
$895$	0	0
$896$	−23.4789	−0.784374
$897$	3.75175	0.125267
$898$	30.0438	1.00257
$899$	0.240237	0.00801234
$900$	0	0
$901$	−26.2322	−0.873921
$902$	18.4963	0.615859
$903$	−26.6154	−0.885704
$904$	−12.0782	−0.401715
$905$	0	0
$906$	21.6105	0.717960
$907$	15.5109	0.515032	0.257516	−	0.966274i	$-0.417096\pi$
$907$	15.5109	0.515032	0.257516	+	0.966274i	$0.417096\pi$
$908$	17.7803	0.590061
$909$	21.8248	0.723884
$910$	0	0
$911$	−56.4743	−1.87108	−0.935538	−	0.353225i	$-0.885085\pi$
$911$	−56.4743	−1.87108	−0.935538	+	0.353225i	$0.885085\pi$
$912$	−37.8267	−1.25257
$913$	−12.4420	−0.411769
$914$	−15.2044	−0.502917
$915$	0	0
$916$	33.6121	1.11058
$917$	5.45640	0.180186
$918$	−11.2400	−0.370975
$919$	50.4305	1.66355	0.831774	−	0.555115i	$-0.187326\pi$
$919$	50.4305	1.66355	0.831774	+	0.555115i	$0.187326\pi$
$920$	0	0
$921$	4.73174	0.155916
$922$	−51.0923	−1.68263
$923$	1.19317	0.0392736
$924$	15.5259	0.510764
$925$	0	0
$926$	−55.3918	−1.82029
$927$	43.8994	1.44185
$928$	0.376386	0.0123555
$929$	38.0255	1.24757	0.623787	−	0.781594i	$-0.285593\pi$
$929$	38.0255	1.24757	0.623787	+	0.781594i	$0.285593\pi$
$930$	0	0
$931$	16.9137	0.554325
$932$	10.7128	0.350909
$933$	1.55775	0.0509984
$934$	−5.08388	−0.166349
$935$	0	0
$936$	0.698271	0.0228237
$937$	−4.68279	−0.152980	−0.0764900	−	0.997070i	$-0.524371\pi$
$937$	−4.68279	−0.152980	−0.0764900	+	0.997070i	$0.524371\pi$
$938$	−13.7972	−0.450495
$939$	58.6647	1.91445
$940$	0	0
$941$	−51.3443	−1.67378	−0.836889	−	0.547373i	$-0.815628\pi$
$941$	−51.3443	−1.67378	−0.836889	+	0.547373i	$0.815628\pi$
$942$	−13.1496	−0.428437
$943$	55.9714	1.82268
$944$	−30.2624	−0.984959
$945$	0	0
$946$	−6.22417	−0.202365
$947$	29.5601	0.960575	0.480288	−	0.877111i	$-0.340532\pi$
$947$	29.5601	0.960575	0.480288	+	0.877111i	$0.340532\pi$
$948$	−2.96206	−0.0962032
$949$	−0.290463	−0.00942883
$950$	0	0
$951$	6.24036	0.202357
$952$	−13.1056	−0.424756
$953$	46.8102	1.51633	0.758166	−	0.652062i	$-0.226096\pi$
$953$	46.8102	1.51633	0.758166	+	0.652062i	$0.226096\pi$
$954$	40.1574	1.30014
$955$	0	0
$956$	20.1633	0.652127
$957$	0.150524	0.00486574
$958$	−45.9564	−1.48478
$959$	3.74717	0.121003
$960$	0	0
$961$	−10.2201	−0.329681
$962$	2.72288	0.0877892
$963$	27.0883	0.872909
$964$	1.54517	0.0497664
$965$	0	0
$966$	107.795	3.46825
$967$	37.6906	1.21205	0.606024	−	0.795446i	$-0.292763\pi$
$967$	37.6906	1.21205	0.606024	+	0.795446i	$0.292763\pi$
$968$	8.35152	0.268428
$969$	−34.9892	−1.12402
$970$	0	0
$971$	−53.4295	−1.71464	−0.857318	−	0.514788i	$-0.827871\pi$
$971$	−53.4295	−1.71464	−0.857318	+	0.514788i	$0.827871\pi$
$972$	−34.4167	−1.10392
$973$	−35.6104	−1.14162
$974$	36.4667	1.16847
$975$	0	0
$976$	55.9610	1.79127
$977$	9.03874	0.289175	0.144587	−	0.989492i	$-0.453814\pi$
$977$	9.03874	0.289175	0.144587	+	0.989492i	$0.453814\pi$
$978$	−93.3481	−2.98494
$979$	9.10900	0.291125
$980$	0	0
$981$	−2.96142	−0.0945508
$982$	26.9296	0.859357
$983$	−39.4162	−1.25718	−0.628590	−	0.777737i	$-0.716368\pi$
$983$	−39.4162	−1.25718	−0.628590	+	0.777737i	$0.716368\pi$
$984$	19.2537	0.613784
$985$	0	0
$986$	0.431647	0.0137464
$987$	−92.9437	−2.95843
$988$	−1.12069	−0.0356537
$989$	−18.8349	−0.598914
$990$	0	0
$991$	−50.5302	−1.60515	−0.802573	−	0.596554i	$-0.796536\pi$
$991$	−50.5302	−1.60515	−0.802573	+	0.596554i	$0.796536\pi$
$992$	32.5565	1.03367
$993$	17.6765	0.560946
$994$	34.2820	1.08736
$995$	0	0
$996$	43.9987	1.39415
$997$	−31.8629	−1.00911	−0.504553	−	0.863380i	$-0.668343\pi$
$997$	−31.8629	−1.00911	−0.504553	+	0.863380i	$0.668343\pi$
$998$	−42.6476	−1.34999
$999$	−8.60832	−0.272355

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6025.2.a.p.1.39		46
5.2	odd	4		1205.2.b.c.724.39	yes	46
5.3	odd	4		1205.2.b.c.724.8	✓	46
5.4	even	2	inner	6025.2.a.p.1.8		46

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1205.2.b.c.724.8	✓	46	5.3	odd	4
1205.2.b.c.724.39	yes	46	5.2	odd	4
6025.2.a.p.1.8		46	5.4	even	2	inner
6025.2.a.p.1.39		46	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 6025 = 5^{2} \cdot 241 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6025.a (trivial)

\(f(q)\)	\(=\)	\(q+1.88286 q^{2} +2.55671 q^{3} +1.54517 q^{4} +4.81392 q^{6} -3.51799 q^{7} -0.856390 q^{8} +3.53675 q^{9} +O(q^{10})\)	\(q+1.88286 q^{2} +2.55671 q^{3} +1.54517 q^{4} +4.81392 q^{6} -3.51799 q^{7} -0.856390 q^{8} +3.53675 q^{9} -1.11714 q^{11} +3.95054 q^{12} -0.230541 q^{13} -6.62388 q^{14} -4.70279 q^{16} -4.35003 q^{17} +6.65921 q^{18} +3.14602 q^{19} -8.99446 q^{21} -2.10341 q^{22} -6.36510 q^{23} -2.18954 q^{24} -0.434076 q^{26} +1.37232 q^{27} -5.43587 q^{28} -0.0527009 q^{29} -4.55850 q^{31} -7.14193 q^{32} -2.85619 q^{33} -8.19050 q^{34} +5.46487 q^{36} -6.27282 q^{37} +5.92351 q^{38} -0.589425 q^{39} -8.79348 q^{41} -16.9353 q^{42} +2.95908 q^{43} -1.72616 q^{44} -11.9846 q^{46} +10.3334 q^{47} -12.0237 q^{48} +5.37623 q^{49} -11.1218 q^{51} -0.356224 q^{52} +6.03035 q^{53} +2.58389 q^{54} +3.01277 q^{56} +8.04344 q^{57} -0.0992284 q^{58} +6.43499 q^{59} -11.8995 q^{61} -8.58302 q^{62} -12.4423 q^{63} -4.04167 q^{64} -5.37781 q^{66} +2.08295 q^{67} -6.72152 q^{68} -16.2737 q^{69} -5.17552 q^{71} -3.02884 q^{72} +1.25992 q^{73} -11.8109 q^{74} +4.86112 q^{76} +3.93007 q^{77} -1.10981 q^{78} -0.749786 q^{79} -7.10164 q^{81} -16.5569 q^{82} +11.1374 q^{83} -13.8979 q^{84} +5.57154 q^{86} -0.134741 q^{87} +0.956703 q^{88} -8.15389 q^{89} +0.811039 q^{91} -9.83514 q^{92} -11.6547 q^{93} +19.4564 q^{94} -18.2598 q^{96} -7.78090 q^{97} +10.1227 q^{98} -3.95103 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 46 q + 34 q^{4} - 4 q^{6} + 34 q^{9}+O(q^{10}) \) 46 * q + 34 * q^4 - 4 * q^6 + 34 * q^9	\( 46 q + 34 q^{4} - 4 q^{6} + 34 q^{9} - 64 q^{11} - 66 q^{14} + 22 q^{16} - 14 q^{21} - 50 q^{24} - 60 q^{26} - 36 q^{29} - 36 q^{31} + 12 q^{34} - 34 q^{36} - 88 q^{39} - 76 q^{41} - 100 q^{44} - 12 q^{46} + 22 q^{49} - 112 q^{51} - 26 q^{54} - 120 q^{56} - 84 q^{59} - 78 q^{61} + 28 q^{64} - 2 q^{66} - 24 q^{69} - 172 q^{71} - 16 q^{74} - 18 q^{76} - 54 q^{79} - 42 q^{81} + 44 q^{84} - 80 q^{86} - 86 q^{89} - 88 q^{91} + 4 q^{94} - 122 q^{96} - 148 q^{99}+O(q^{100}) \) 46 * q + 34 * q^4 - 4 * q^6 + 34 * q^9 - 64 * q^11 - 66 * q^14 + 22 * q^16 - 14 * q^21 - 50 * q^24 - 60 * q^26 - 36 * q^29 - 36 * q^31 + 12 * q^34 - 34 * q^36 - 88 * q^39 - 76 * q^41 - 100 * q^44 - 12 * q^46 + 22 * q^49 - 112 * q^51 - 26 * q^54 - 120 * q^56 - 84 * q^59 - 78 * q^61 + 28 * q^64 - 2 * q^66 - 24 * q^69 - 172 * q^71 - 16 * q^74 - 18 * q^76 - 54 * q^79 - 42 * q^81 + 44 * q^84 - 80 * q^86 - 86 * q^89 - 88 * q^91 + 4 * q^94 - 122 * q^96 - 148 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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