LMFDB - Embedded newform 3750.2.a.o.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3750,2,Mod(1,3750)] 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("3750.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3750, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3750 = 2 \cdot 3 \cdot 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3750.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,4,-4,4,0,-4,-4,4,4,0,-10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$29.9439007580$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{20})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 5x^{2} + 5 $ x^4 - 5*x^2 + 5
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 150)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.17557$ of defining polynomial
Character	$\chi$	$=$	3750.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +2.07768 q^{7} +1.00000 q^{8} +1.00000 q^{9} +0.520147 q^{11} -1.00000 q^{12} -2.18619 q^{13} +2.07768 q^{14} +1.00000 q^{16} -2.69572 q^{17} +1.00000 q^{18} -6.87129 q^{19} -2.07768 q^{21} +0.520147 q^{22} -3.86067 q^{23} -1.00000 q^{24} -2.18619 q^{26} -1.00000 q^{27} +2.07768 q^{28} -9.03373 q^{29} -7.40507 q^{31} +1.00000 q^{32} -0.520147 q^{33} -2.69572 q^{34} +1.00000 q^{36} +5.30719 q^{37} -6.87129 q^{38} +2.18619 q^{39} -3.22358 q^{41} -2.07768 q^{42} -9.53920 q^{43} +0.520147 q^{44} -3.86067 q^{46} +9.26574 q^{47} -1.00000 q^{48} -2.68323 q^{49} +2.69572 q^{51} -2.18619 q^{52} +2.43841 q^{53} -1.00000 q^{54} +2.07768 q^{56} +6.87129 q^{57} -9.03373 q^{58} +8.64875 q^{59} +12.5882 q^{61} -7.40507 q^{62} +2.07768 q^{63} +1.00000 q^{64} -0.520147 q^{66} -6.69385 q^{67} -2.69572 q^{68} +3.86067 q^{69} -8.16901 q^{71} +1.00000 q^{72} -3.84348 q^{73} +5.30719 q^{74} -6.87129 q^{76} +1.08070 q^{77} +2.18619 q^{78} +3.51243 q^{79} +1.00000 q^{81} -3.22358 q^{82} +13.1311 q^{83} -2.07768 q^{84} -9.53920 q^{86} +9.03373 q^{87} +0.520147 q^{88} -9.85880 q^{89} -4.54222 q^{91} -3.86067 q^{92} +7.40507 q^{93} +9.26574 q^{94} -1.00000 q^{96} +10.7598 q^{97} -2.68323 q^{98} +0.520147 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{2} - 4 q^{3} + 4 q^{4} - 4 q^{6} - 4 q^{7} + 4 q^{8} + 4 q^{9} - 10 q^{11} - 4 q^{12} + 2 q^{13} - 4 q^{14} + 4 q^{16} + 6 q^{17} + 4 q^{18} - 6 q^{19} + 4 q^{21} - 10 q^{22} - 4 q^{24} + 2 q^{26}+ \cdots - 10 q^{99}+O(q^{100}) $ 4 * q + 4 * q^2 - 4 * q^3 + 4 * q^4 - 4 * q^6 - 4 * q^7 + 4 * q^8 + 4 * q^9 - 10 * q^11 - 4 * q^12 + 2 * q^13 - 4 * q^14 + 4 * q^16 + 6 * q^17 + 4 * q^18 - 6 * q^19 + 4 * q^21 - 10 * q^22 - 4 * q^24 + 2 * q^26 - 4 * q^27 - 4 * q^28 - 14 * q^29 - 18 * q^31 + 4 * q^32 + 10 * q^33 + 6 * q^34 + 4 * q^36 + 2 * q^37 - 6 * q^38 - 2 * q^39 - 14 * q^41 + 4 * q^42 - 14 * q^43 - 10 * q^44 + 10 * q^47 - 4 * q^48 - 4 * q^49 - 6 * q^51 + 2 * q^52 + 14 * q^53 - 4 * q^54 - 4 * q^56 + 6 * q^57 - 14 * q^58 - 20 * q^59 - 18 * q^62 - 4 * q^63 + 4 * q^64 + 10 * q^66 - 14 * q^67 + 6 * q^68 - 30 * q^71 + 4 * q^72 - 8 * q^73 + 2 * q^74 - 6 * q^76 + 20 * q^77 - 2 * q^78 - 28 * q^79 + 4 * q^81 - 14 * q^82 + 12 * q^83 + 4 * q^84 - 14 * q^86 + 14 * q^87 - 10 * q^88 - 28 * q^89 - 22 * q^91 + 18 * q^93 + 10 * q^94 - 4 * q^96 - 8 * q^97 - 4 * q^98 - 10 * 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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−1.00000	−0.408248
$7$	2.07768	0.785291	0.392645	−	0.919690i	$-0.371560\pi$
$7$	2.07768	0.785291	0.392645	+	0.919690i	$0.371560\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	0	0
$11$	0.520147	0.156830	0.0784151	−	0.996921i	$-0.475014\pi$
$11$	0.520147	0.156830	0.0784151	+	0.996921i	$0.475014\pi$
$12$	−1.00000	−0.288675
$13$	−2.18619	−0.606341	−0.303170	−	0.952936i	$-0.598045\pi$
$13$	−2.18619	−0.606341	−0.303170	+	0.952936i	$0.598045\pi$
$14$	2.07768	0.555284
$15$	0	0
$16$	1.00000	0.250000
$17$	−2.69572	−0.653808	−0.326904	−	0.945058i	$-0.606005\pi$
$17$	−2.69572	−0.653808	−0.326904	+	0.945058i	$0.606005\pi$
$18$	1.00000	0.235702
$19$	−6.87129	−1.57638	−0.788191	−	0.615431i	$-0.788982\pi$
$19$	−6.87129	−1.57638	−0.788191	+	0.615431i	$0.788982\pi$
$20$	0	0
$21$	−2.07768	−0.453388
$22$	0.520147	0.110896
$23$	−3.86067	−0.805005	−0.402502	−	0.915419i	$-0.631859\pi$
$23$	−3.86067	−0.805005	−0.402502	+	0.915419i	$0.631859\pi$
$24$	−1.00000	−0.204124
$25$	0	0
$26$	−2.18619	−0.428748
$27$	−1.00000	−0.192450
$28$	2.07768	0.392645
$29$	−9.03373	−1.67752	−0.838761	−	0.544500i	$-0.816719\pi$
$29$	−9.03373	−1.67752	−0.838761	+	0.544500i	$0.816719\pi$
$30$	0	0
$31$	−7.40507	−1.32999	−0.664995	−	0.746848i	$-0.731566\pi$
$31$	−7.40507	−1.32999	−0.664995	+	0.746848i	$0.731566\pi$
$32$	1.00000	0.176777
$33$	−0.520147	−0.0905460
$34$	−2.69572	−0.462312
$35$	0	0
$36$	1.00000	0.166667
$37$	5.30719	0.872496	0.436248	−	0.899826i	$-0.356307\pi$
$37$	5.30719	0.872496	0.436248	+	0.899826i	$0.356307\pi$
$38$	−6.87129	−1.11467
$39$	2.18619	0.350071
$40$	0	0
$41$	−3.22358	−0.503439	−0.251719	−	0.967800i	$-0.580996\pi$
$41$	−3.22358	−0.503439	−0.251719	+	0.967800i	$0.580996\pi$
$42$	−2.07768	−0.320594
$43$	−9.53920	−1.45471	−0.727357	−	0.686259i	$-0.759252\pi$
$43$	−9.53920	−1.45471	−0.727357	+	0.686259i	$0.759252\pi$
$44$	0.520147	0.0784151
$45$	0	0
$46$	−3.86067	−0.569224
$47$	9.26574	1.35155	0.675774	−	0.737109i	$-0.263810\pi$
$47$	9.26574	1.35155	0.675774	+	0.737109i	$0.263810\pi$
$48$	−1.00000	−0.144338
$49$	−2.68323	−0.383319
$50$	0	0
$51$	2.69572	0.377476
$52$	−2.18619	−0.303170
$53$	2.43841	0.334941	0.167470	−	0.985877i	$-0.446440\pi$
$53$	2.43841	0.334941	0.167470	+	0.985877i	$0.446440\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	2.07768	0.277642
$57$	6.87129	0.910124
$58$	−9.03373	−1.18619
$59$	8.64875	1.12597	0.562985	−	0.826467i	$-0.309653\pi$
$59$	8.64875	1.12597	0.562985	+	0.826467i	$0.309653\pi$
$60$	0	0
$61$	12.5882	1.61176	0.805880	−	0.592079i	$-0.201693\pi$
$61$	12.5882	1.61176	0.805880	+	0.592079i	$0.201693\pi$
$62$	−7.40507	−0.940445
$63$	2.07768	0.261764
$64$	1.00000	0.125000
$65$	0	0
$66$	−0.520147	−0.0640257
$67$	−6.69385	−0.817784	−0.408892	−	0.912583i	$-0.634085\pi$
$67$	−6.69385	−0.817784	−0.408892	+	0.912583i	$0.634085\pi$
$68$	−2.69572	−0.326904
$69$	3.86067	0.464770
$70$	0	0
$71$	−8.16901	−0.969483	−0.484741	−	0.874658i	$-0.661086\pi$
$71$	−8.16901	−0.969483	−0.484741	+	0.874658i	$0.661086\pi$
$72$	1.00000	0.117851
$73$	−3.84348	−0.449845	−0.224923	−	0.974377i	$-0.572213\pi$
$73$	−3.84348	−0.449845	−0.224923	+	0.974377i	$0.572213\pi$
$74$	5.30719	0.616948
$75$	0	0
$76$	−6.87129	−0.788191
$77$	1.08070	0.123157
$78$	2.18619	0.247538
$79$	3.51243	0.395179	0.197590	−	0.980285i	$-0.436689\pi$
$79$	3.51243	0.395179	0.197590	+	0.980285i	$0.436689\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−3.22358	−0.355985
$83$	13.1311	1.44133	0.720663	−	0.693285i	$-0.243837\pi$
$83$	13.1311	1.44133	0.720663	+	0.693285i	$0.243837\pi$
$84$	−2.07768	−0.226694
$85$	0	0
$86$	−9.53920	−1.02864
$87$	9.03373	0.968517
$88$	0.520147	0.0554479
$89$	−9.85880	−1.04503	−0.522515	−	0.852630i	$-0.675006\pi$
$89$	−9.85880	−1.04503	−0.522515	+	0.852630i	$0.675006\pi$
$90$	0	0
$91$	−4.54222	−0.476154
$92$	−3.86067	−0.402502
$93$	7.40507	0.767870
$94$	9.26574	0.955688
$95$	0	0
$96$	−1.00000	−0.102062
$97$	10.7598	1.09249	0.546244	−	0.837626i	$-0.316057\pi$
$97$	10.7598	1.09249	0.546244	+	0.837626i	$0.316057\pi$
$98$	−2.68323	−0.271047
$99$	0.520147	0.0522767
$100$	0	0
$101$	−17.6400	−1.75524	−0.877622	−	0.479353i	$-0.840871\pi$
$101$	−17.6400	−1.75524	−0.877622	+	0.479353i	$0.840871\pi$
$102$	2.69572	0.266916
$103$	−8.69758	−0.856998	−0.428499	−	0.903542i	$-0.640957\pi$
$103$	−8.69758	−0.856998	−0.428499	+	0.903542i	$0.640957\pi$
$104$	−2.18619	−0.214374
$105$	0	0
$106$	2.43841	0.236839
$107$	−5.40977	−0.522983	−0.261491	−	0.965206i	$-0.584214\pi$
$107$	−5.40977	−0.522983	−0.261491	+	0.965206i	$0.584214\pi$
$108$	−1.00000	−0.0962250
$109$	2.88377	0.276215	0.138108	−	0.990417i	$-0.455898\pi$
$109$	2.88377	0.276215	0.138108	+	0.990417i	$0.455898\pi$
$110$	0	0
$111$	−5.30719	−0.503736
$112$	2.07768	0.196323
$113$	−6.36955	−0.599197	−0.299598	−	0.954065i	$-0.596853\pi$
$113$	−6.36955	−0.599197	−0.299598	+	0.954065i	$0.596853\pi$
$114$	6.87129	0.643555
$115$	0	0
$116$	−9.03373	−0.838761
$117$	−2.18619	−0.202114
$118$	8.64875	0.796182
$119$	−5.60085	−0.513429
$120$	0	0
$121$	−10.7294	−0.975404
$122$	12.5882	1.13969
$123$	3.22358	0.290661
$124$	−7.40507	−0.664995
$125$	0	0
$126$	2.07768	0.185095
$127$	−14.9673	−1.32813	−0.664067	−	0.747673i	$-0.731171\pi$
$127$	−14.9673	−1.32813	−0.664067	+	0.747673i	$0.731171\pi$
$128$	1.00000	0.0883883
$129$	9.53920	0.839880
$130$	0	0
$131$	13.8551	1.21053	0.605265	−	0.796024i	$-0.293067\pi$
$131$	13.8551	1.21053	0.605265	+	0.796024i	$0.293067\pi$
$132$	−0.520147	−0.0452730
$133$	−14.2764	−1.23792
$134$	−6.69385	−0.578261
$135$	0	0
$136$	−2.69572	−0.231156
$137$	19.8939	1.69965	0.849825	−	0.527065i	$-0.176708\pi$
$137$	19.8939	1.69965	0.849825	+	0.527065i	$0.176708\pi$
$138$	3.86067	0.328642
$139$	5.64365	0.478688	0.239344	−	0.970935i	$-0.423068\pi$
$139$	5.64365	0.478688	0.239344	+	0.970935i	$0.423068\pi$
$140$	0	0
$141$	−9.26574	−0.780316
$142$	−8.16901	−0.685528
$143$	−1.13714	−0.0950925
$144$	1.00000	0.0833333
$145$	0	0
$146$	−3.84348	−0.318089
$147$	2.68323	0.221309
$148$	5.30719	0.436248
$149$	−15.0956	−1.23668	−0.618339	−	0.785911i	$-0.712194\pi$
$149$	−15.0956	−1.23668	−0.618339	+	0.785911i	$0.712194\pi$
$150$	0	0
$151$	−20.9353	−1.70369	−0.851847	−	0.523791i	$-0.824517\pi$
$151$	−20.9353	−1.70369	−0.851847	+	0.523791i	$0.824517\pi$
$152$	−6.87129	−0.557335
$153$	−2.69572	−0.217936
$154$	1.08070	0.0870854
$155$	0	0
$156$	2.18619	0.175035
$157$	5.71998	0.456504	0.228252	−	0.973602i	$-0.426699\pi$
$157$	5.71998	0.456504	0.228252	+	0.973602i	$0.426699\pi$
$158$	3.51243	0.279434
$159$	−2.43841	−0.193378
$160$	0	0
$161$	−8.02124	−0.632163
$162$	1.00000	0.0785674
$163$	11.8950	0.931691	0.465846	−	0.884866i	$-0.345750\pi$
$163$	11.8950	0.931691	0.465846	+	0.884866i	$0.345750\pi$
$164$	−3.22358	−0.251719
$165$	0	0
$166$	13.1311	1.01917
$167$	8.94427	0.692129	0.346064	−	0.938211i	$-0.387518\pi$
$167$	8.94427	0.692129	0.346064	+	0.938211i	$0.387518\pi$
$168$	−2.07768	−0.160297
$169$	−8.22056	−0.632351
$170$	0	0
$171$	−6.87129	−0.525461
$172$	−9.53920	−0.727357
$173$	16.3391	1.24224	0.621118	−	0.783717i	$-0.286679\pi$
$173$	16.3391	1.24224	0.621118	+	0.783717i	$0.286679\pi$
$174$	9.03373	0.684845
$175$	0	0
$176$	0.520147	0.0392076
$177$	−8.64875	−0.650080
$178$	−9.85880	−0.738948
$179$	3.67853	0.274946	0.137473	−	0.990505i	$-0.456102\pi$
$179$	3.67853	0.274946	0.137473	+	0.990505i	$0.456102\pi$
$180$	0	0
$181$	−23.0493	−1.71324	−0.856619	−	0.515950i	$-0.827439\pi$
$181$	−23.0493	−1.71324	−0.856619	+	0.515950i	$0.827439\pi$
$182$	−4.54222	−0.336691
$183$	−12.5882	−0.930550
$184$	−3.86067	−0.284612
$185$	0	0
$186$	7.40507	0.542966
$187$	−1.40217	−0.102537
$188$	9.26574	0.675774
$189$	−2.07768	−0.151129
$190$	0	0
$191$	−1.78768	−0.129352	−0.0646761	−	0.997906i	$-0.520601\pi$
$191$	−1.78768	−0.129352	−0.0646761	+	0.997906i	$0.520601\pi$
$192$	−1.00000	−0.0721688
$193$	−6.60138	−0.475178	−0.237589	−	0.971366i	$-0.576357\pi$
$193$	−6.60138	−0.475178	−0.237589	+	0.971366i	$0.576357\pi$
$194$	10.7598	0.772506
$195$	0	0
$196$	−2.68323	−0.191659
$197$	12.6575	0.901810	0.450905	−	0.892572i	$-0.351101\pi$
$197$	12.6575	0.901810	0.450905	+	0.892572i	$0.351101\pi$
$198$	0.520147	0.0369652
$199$	6.24148	0.442447	0.221223	−	0.975223i	$-0.428995\pi$
$199$	6.24148	0.442447	0.221223	+	0.975223i	$0.428995\pi$
$200$	0	0
$201$	6.69385	0.472148
$202$	−17.6400	−1.24115
$203$	−18.7692	−1.31734
$204$	2.69572	0.188738
$205$	0	0
$206$	−8.69758	−0.605989
$207$	−3.86067	−0.268335
$208$	−2.18619	−0.151585
$209$	−3.57408	−0.247224
$210$	0	0
$211$	−22.4779	−1.54745	−0.773723	−	0.633524i	$-0.781608\pi$
$211$	−22.4779	−1.54745	−0.773723	+	0.633524i	$0.781608\pi$
$212$	2.43841	0.167470
$213$	8.16901	0.559731
$214$	−5.40977	−0.369805
$215$	0	0
$216$	−1.00000	−0.0680414
$217$	−15.3854	−1.04443
$218$	2.88377	0.195314
$219$	3.84348	0.259718
$220$	0	0
$221$	5.89336	0.396430
$222$	−5.30719	−0.356195
$223$	−17.9299	−1.20068	−0.600338	−	0.799746i	$-0.704967\pi$
$223$	−17.9299	−1.20068	−0.600338	+	0.799746i	$0.704967\pi$
$224$	2.07768	0.138821
$225$	0	0
$226$	−6.36955	−0.423696
$227$	−8.85765	−0.587903	−0.293951	−	0.955820i	$-0.594970\pi$
$227$	−8.85765	−0.587903	−0.293951	+	0.955820i	$0.594970\pi$
$228$	6.87129	0.455062
$229$	22.0626	1.45794	0.728968	−	0.684548i	$-0.240000\pi$
$229$	22.0626	1.45794	0.728968	+	0.684548i	$0.240000\pi$
$230$	0	0
$231$	−1.08070	−0.0711049
$232$	−9.03373	−0.593093
$233$	11.5427	0.756190	0.378095	−	0.925767i	$-0.376579\pi$
$233$	11.5427	0.756190	0.378095	+	0.925767i	$0.376579\pi$
$234$	−2.18619	−0.142916
$235$	0	0
$236$	8.64875	0.562985
$237$	−3.51243	−0.228157
$238$	−5.60085	−0.363049
$239$	−0.884927	−0.0572412	−0.0286206	−	0.999590i	$-0.509111\pi$
$239$	−0.884927	−0.0572412	−0.0286206	+	0.999590i	$0.509111\pi$
$240$	0	0
$241$	12.6736	0.816382	0.408191	−	0.912897i	$-0.366160\pi$
$241$	12.6736	0.816382	0.408191	+	0.912897i	$0.366160\pi$
$242$	−10.7294	−0.689715
$243$	−1.00000	−0.0641500
$244$	12.5882	0.805880
$245$	0	0
$246$	3.22358	0.205528
$247$	15.0220	0.955824
$248$	−7.40507	−0.470223
$249$	−13.1311	−0.832150
$250$	0	0
$251$	18.0966	1.14225	0.571124	−	0.820864i	$-0.306507\pi$
$251$	18.0966	1.14225	0.571124	+	0.820864i	$0.306507\pi$
$252$	2.07768	0.130882
$253$	−2.00811	−0.126249
$254$	−14.9673	−0.939133
$255$	0	0
$256$	1.00000	0.0625000
$257$	1.28053	0.0798774	0.0399387	−	0.999202i	$-0.487284\pi$
$257$	1.28053	0.0798774	0.0399387	+	0.999202i	$0.487284\pi$
$258$	9.53920	0.593885
$259$	11.0267	0.685163
$260$	0	0
$261$	−9.03373	−0.559174
$262$	13.8551	0.855974
$263$	−7.11425	−0.438683	−0.219342	−	0.975648i	$-0.570391\pi$
$263$	−7.11425	−0.438683	−0.219342	+	0.975648i	$0.570391\pi$
$264$	−0.520147	−0.0320128
$265$	0	0
$266$	−14.2764	−0.875340
$267$	9.85880	0.603349
$268$	−6.69385	−0.408892
$269$	−9.42986	−0.574949	−0.287474	−	0.957788i	$-0.592816\pi$
$269$	−9.42986	−0.574949	−0.287474	+	0.957788i	$0.592816\pi$
$270$	0	0
$271$	1.37489	0.0835187	0.0417594	−	0.999128i	$-0.486704\pi$
$271$	1.37489	0.0835187	0.0417594	+	0.999128i	$0.486704\pi$
$272$	−2.69572	−0.163452
$273$	4.54222	0.274907
$274$	19.8939	1.20183
$275$	0	0
$276$	3.86067	0.232385
$277$	19.5967	1.17745	0.588725	−	0.808333i	$-0.299630\pi$
$277$	19.5967	1.17745	0.588725	+	0.808333i	$0.299630\pi$
$278$	5.64365	0.338484
$279$	−7.40507	−0.443330
$280$	0	0
$281$	−2.83807	−0.169305	−0.0846525	−	0.996411i	$-0.526978\pi$
$281$	−2.83807	−0.169305	−0.0846525	+	0.996411i	$0.526978\pi$
$282$	−9.26574	−0.551767
$283$	−2.23356	−0.132771	−0.0663857	−	0.997794i	$-0.521147\pi$
$283$	−2.23356	−0.132771	−0.0663857	+	0.997794i	$0.521147\pi$
$284$	−8.16901	−0.484741
$285$	0	0
$286$	−1.13714	−0.0672406
$287$	−6.69758	−0.395346
$288$	1.00000	0.0589256
$289$	−9.73311	−0.572536
$290$	0	0
$291$	−10.7598	−0.630748
$292$	−3.84348	−0.224923
$293$	24.0260	1.40361	0.701807	−	0.712367i	$-0.252377\pi$
$293$	24.0260	1.40361	0.701807	+	0.712367i	$0.252377\pi$
$294$	2.68323	0.156489
$295$	0	0
$296$	5.30719	0.308474
$297$	−0.520147	−0.0301820
$298$	−15.0956	−0.874464
$299$	8.44016	0.488107
$300$	0	0
$301$	−19.8194	−1.14237
$302$	−20.9353	−1.20469
$303$	17.6400	1.01339
$304$	−6.87129	−0.394095
$305$	0	0
$306$	−2.69572	−0.154104
$307$	17.1204	0.977111	0.488556	−	0.872533i	$-0.337524\pi$
$307$	17.1204	0.977111	0.488556	+	0.872533i	$0.337524\pi$
$308$	1.08070	0.0615786
$309$	8.69758	0.494788
$310$	0	0
$311$	−20.1527	−1.14276	−0.571379	−	0.820687i	$-0.693591\pi$
$311$	−20.1527	−1.14276	−0.571379	+	0.820687i	$0.693591\pi$
$312$	2.18619	0.123769
$313$	−25.5043	−1.44159	−0.720795	−	0.693149i	$-0.756223\pi$
$313$	−25.5043	−1.44159	−0.720795	+	0.693149i	$0.756223\pi$
$314$	5.71998	0.322797
$315$	0	0
$316$	3.51243	0.197590
$317$	−31.5182	−1.77024	−0.885120	−	0.465362i	$-0.845924\pi$
$317$	−31.5182	−1.77024	−0.885120	+	0.465362i	$0.845924\pi$
$318$	−2.43841	−0.136739
$319$	−4.69887	−0.263086
$320$	0	0
$321$	5.40977	0.301944
$322$	−8.02124	−0.447006
$323$	18.5231	1.03065
$324$	1.00000	0.0555556
$325$	0	0
$326$	11.8950	0.658805
$327$	−2.88377	−0.159473
$328$	−3.22358	−0.177993
$329$	19.2513	1.06136
$330$	0	0
$331$	−7.30062	−0.401279	−0.200639	−	0.979665i	$-0.564302\pi$
$331$	−7.30062	−0.401279	−0.200639	+	0.979665i	$0.564302\pi$
$332$	13.1311	0.720663
$333$	5.30719	0.290832
$334$	8.94427	0.489409
$335$	0	0
$336$	−2.07768	−0.113347
$337$	−26.7448	−1.45688	−0.728440	−	0.685109i	$-0.759754\pi$
$337$	−26.7448	−1.45688	−0.728440	+	0.685109i	$0.759754\pi$
$338$	−8.22056	−0.447140
$339$	6.36955	0.345946
$340$	0	0
$341$	−3.85173	−0.208583
$342$	−6.87129	−0.371557
$343$	−20.1187	−1.08631
$344$	−9.53920	−0.514319
$345$	0	0
$346$	16.3391	0.878393
$347$	−4.87129	−0.261504	−0.130752	−	0.991415i	$-0.541739\pi$
$347$	−4.87129	−0.261504	−0.130752	+	0.991415i	$0.541739\pi$
$348$	9.03373	0.484259
$349$	16.7173	0.894855	0.447428	−	0.894320i	$-0.352340\pi$
$349$	16.7173	0.894855	0.447428	+	0.894320i	$0.352340\pi$
$350$	0	0
$351$	2.18619	0.116690
$352$	0.520147	0.0277239
$353$	27.0805	1.44135	0.720674	−	0.693274i	$-0.243832\pi$
$353$	27.0805	1.44135	0.720674	+	0.693274i	$0.243832\pi$
$354$	−8.64875	−0.459676
$355$	0	0
$356$	−9.85880	−0.522515
$357$	5.60085	0.296428
$358$	3.67853	0.194416
$359$	−8.48817	−0.447988	−0.223994	−	0.974590i	$-0.571910\pi$
$359$	−8.48817	−0.447988	−0.223994	+	0.974590i	$0.571910\pi$
$360$	0	0
$361$	28.2146	1.48498
$362$	−23.0493	−1.21144
$363$	10.7294	0.563150
$364$	−4.54222	−0.238077
$365$	0	0
$366$	−12.5882	−0.657998
$367$	12.7477	0.665423	0.332712	−	0.943029i	$-0.392036\pi$
$367$	12.7477	0.665423	0.332712	+	0.943029i	$0.392036\pi$
$368$	−3.86067	−0.201251
$369$	−3.22358	−0.167813
$370$	0	0
$371$	5.06624	0.263026
$372$	7.40507	0.383935
$373$	10.9405	0.566480	0.283240	−	0.959049i	$-0.408591\pi$
$373$	10.9405	0.566480	0.283240	+	0.959049i	$0.408591\pi$
$374$	−1.40217	−0.0725045
$375$	0	0
$376$	9.26574	0.477844
$377$	19.7495	1.01715
$378$	−2.07768	−0.106865
$379$	−14.2301	−0.730954	−0.365477	−	0.930820i	$-0.619094\pi$
$379$	−14.2301	−0.730954	−0.365477	+	0.930820i	$0.619094\pi$
$380$	0	0
$381$	14.9673	0.766799
$382$	−1.78768	−0.0914658
$383$	−7.16419	−0.366073	−0.183037	−	0.983106i	$-0.558593\pi$
$383$	−7.16419	−0.366073	−0.183037	+	0.983106i	$0.558593\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	−6.60138	−0.336001
$387$	−9.53920	−0.484905
$388$	10.7598	0.546244
$389$	−0.918972	−0.0465937	−0.0232969	−	0.999729i	$-0.507416\pi$
$389$	−0.918972	−0.0465937	−0.0232969	+	0.999729i	$0.507416\pi$
$390$	0	0
$391$	10.4073	0.526318
$392$	−2.68323	−0.135524
$393$	−13.8551	−0.698899
$394$	12.6575	0.637676
$395$	0	0
$396$	0.520147	0.0261384
$397$	21.8019	1.09421	0.547104	−	0.837065i	$-0.315730\pi$
$397$	21.8019	1.09421	0.547104	+	0.837065i	$0.315730\pi$
$398$	6.24148	0.312857
$399$	14.2764	0.714712
$400$	0	0
$401$	22.8035	1.13875	0.569375	−	0.822078i	$-0.307185\pi$
$401$	22.8035	1.13875	0.569375	+	0.822078i	$0.307185\pi$
$402$	6.69385	0.333859
$403$	16.1889	0.806427
$404$	−17.6400	−0.877622
$405$	0	0
$406$	−18.7692	−0.931501
$407$	2.76052	0.136834
$408$	2.69572	0.133458
$409$	5.00147	0.247307	0.123653	−	0.992325i	$-0.460539\pi$
$409$	5.00147	0.247307	0.123653	+	0.992325i	$0.460539\pi$
$410$	0	0
$411$	−19.8939	−0.981293
$412$	−8.69758	−0.428499
$413$	17.9694	0.884214
$414$	−3.86067	−0.189741
$415$	0	0
$416$	−2.18619	−0.107187
$417$	−5.64365	−0.276371
$418$	−3.57408	−0.174814
$419$	−22.8254	−1.11509	−0.557546	−	0.830146i	$-0.688257\pi$
$419$	−22.8254	−1.11509	−0.557546	+	0.830146i	$0.688257\pi$
$420$	0	0
$421$	15.7134	0.765826	0.382913	−	0.923784i	$-0.374921\pi$
$421$	15.7134	0.765826	0.382913	+	0.923784i	$0.374921\pi$
$422$	−22.4779	−1.09421
$423$	9.26574	0.450516
$424$	2.43841	0.118419
$425$	0	0
$426$	8.16901	0.395790
$427$	26.1544	1.26570
$428$	−5.40977	−0.261491
$429$	1.13714	0.0549017
$430$	0	0
$431$	−9.55671	−0.460331	−0.230165	−	0.973152i	$-0.573927\pi$
$431$	−9.55671	−0.460331	−0.230165	+	0.973152i	$0.573927\pi$
$432$	−1.00000	−0.0481125
$433$	8.12215	0.390325	0.195163	−	0.980771i	$-0.437476\pi$
$433$	8.12215	0.390325	0.195163	+	0.980771i	$0.437476\pi$
$434$	−15.3854	−0.738523
$435$	0	0
$436$	2.88377	0.138108
$437$	26.5278	1.26899
$438$	3.84348	0.183649
$439$	9.21952	0.440024	0.220012	−	0.975497i	$-0.429390\pi$
$439$	9.21952	0.440024	0.220012	+	0.975497i	$0.429390\pi$
$440$	0	0
$441$	−2.68323	−0.127773
$442$	5.89336	0.280318
$443$	9.63232	0.457645	0.228823	−	0.973468i	$-0.426512\pi$
$443$	9.63232	0.457645	0.228823	+	0.973468i	$0.426512\pi$
$444$	−5.30719	−0.251868
$445$	0	0
$446$	−17.9299	−0.849006
$447$	15.0956	0.713997
$448$	2.07768	0.0981613
$449$	−21.3096	−1.00566	−0.502831	−	0.864385i	$-0.667708\pi$
$449$	−21.3096	−1.00566	−0.502831	+	0.864385i	$0.667708\pi$
$450$	0	0
$451$	−1.67674	−0.0789544
$452$	−6.36955	−0.299598
$453$	20.9353	0.983628
$454$	−8.85765	−0.415710
$455$	0	0
$456$	6.87129	0.321778
$457$	−13.4662	−0.629923	−0.314961	−	0.949104i	$-0.601992\pi$
$457$	−13.4662	−0.629923	−0.314961	+	0.949104i	$0.601992\pi$
$458$	22.0626	1.03092
$459$	2.69572	0.125825
$460$	0	0
$461$	32.8357	1.52931	0.764654	−	0.644441i	$-0.222910\pi$
$461$	32.8357	1.52931	0.764654	+	0.644441i	$0.222910\pi$
$462$	−1.08070	−0.0502788
$463$	−22.9306	−1.06568	−0.532838	−	0.846217i	$-0.678875\pi$
$463$	−22.9306	−1.06568	−0.532838	+	0.846217i	$0.678875\pi$
$464$	−9.03373	−0.419380
$465$	0	0
$466$	11.5427	0.534707
$467$	20.8340	0.964083	0.482042	−	0.876148i	$-0.339895\pi$
$467$	20.8340	0.964083	0.482042	+	0.876148i	$0.339895\pi$
$468$	−2.18619	−0.101057
$469$	−13.9077	−0.642198
$470$	0	0
$471$	−5.71998	−0.263563
$472$	8.64875	0.398091
$473$	−4.96179	−0.228143
$474$	−3.51243	−0.161331
$475$	0	0
$476$	−5.60085	−0.256714
$477$	2.43841	0.111647
$478$	−0.884927	−0.0404756
$479$	13.5227	0.617866	0.308933	−	0.951084i	$-0.400028\pi$
$479$	13.5227	0.617866	0.308933	+	0.951084i	$0.400028\pi$
$480$	0	0
$481$	−11.6025	−0.529030
$482$	12.6736	0.577269
$483$	8.02124	0.364979
$484$	−10.7294	−0.487702
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	26.6228	1.20639	0.603197	−	0.797592i	$-0.293893\pi$
$487$	26.6228	1.20639	0.603197	+	0.797592i	$0.293893\pi$
$488$	12.5882	0.569843
$489$	−11.8950	−0.537912
$490$	0	0
$491$	−12.0227	−0.542575	−0.271288	−	0.962498i	$-0.587449\pi$
$491$	−12.0227	−0.542575	−0.271288	+	0.962498i	$0.587449\pi$
$492$	3.22358	0.145330
$493$	24.3524	1.09678
$494$	15.0220	0.675870
$495$	0	0
$496$	−7.40507	−0.332498
$497$	−16.9726	−0.761326
$498$	−13.1311	−0.588419
$499$	−27.3600	−1.22480	−0.612400	−	0.790548i	$-0.709796\pi$
$499$	−27.3600	−1.22480	−0.612400	+	0.790548i	$0.709796\pi$
$500$	0	0
$501$	−8.94427	−0.399601
$502$	18.0966	0.807692
$503$	−9.16257	−0.408539	−0.204269	−	0.978915i	$-0.565482\pi$
$503$	−9.16257	−0.408539	−0.204269	+	0.978915i	$0.565482\pi$
$504$	2.07768	0.0925474
$505$	0	0
$506$	−2.00811	−0.0892716
$507$	8.22056	0.365088
$508$	−14.9673	−0.664067
$509$	−2.34831	−0.104087	−0.0520434	−	0.998645i	$-0.516573\pi$
$509$	−2.34831	−0.104087	−0.0520434	+	0.998645i	$0.516573\pi$
$510$	0	0
$511$	−7.98554	−0.353259
$512$	1.00000	0.0441942
$513$	6.87129	0.303375
$514$	1.28053	0.0564818
$515$	0	0
$516$	9.53920	0.419940
$517$	4.81955	0.211963
$518$	11.0267	0.484483
$519$	−16.3391	−0.717205
$520$	0	0
$521$	−1.53432	−0.0672196	−0.0336098	−	0.999435i	$-0.510700\pi$
$521$	−1.53432	−0.0672196	−0.0336098	+	0.999435i	$0.510700\pi$
$522$	−9.03373	−0.395396
$523$	−41.5860	−1.81843	−0.909215	−	0.416328i	$-0.863317\pi$
$523$	−41.5860	−1.81843	−0.909215	+	0.416328i	$0.863317\pi$
$524$	13.8551	0.605265
$525$	0	0
$526$	−7.11425	−0.310196
$527$	19.9620	0.869558
$528$	−0.520147	−0.0226365
$529$	−8.09525	−0.351968
$530$	0	0
$531$	8.64875	0.375324
$532$	−14.2764	−0.618959
$533$	7.04737	0.305255
$534$	9.85880	0.426632
$535$	0	0
$536$	−6.69385	−0.289130
$537$	−3.67853	−0.158740
$538$	−9.42986	−0.406550
$539$	−1.39567	−0.0601160
$540$	0	0
$541$	19.0419	0.818677	0.409338	−	0.912383i	$-0.365760\pi$
$541$	19.0419	0.818677	0.409338	+	0.912383i	$0.365760\pi$
$542$	1.37489	0.0590566
$543$	23.0493	0.989138
$544$	−2.69572	−0.115578
$545$	0	0
$546$	4.54222	0.194389
$547$	−27.3208	−1.16816	−0.584078	−	0.811698i	$-0.698544\pi$
$547$	−27.3208	−1.16816	−0.584078	+	0.811698i	$0.698544\pi$
$548$	19.8939	0.849825
$549$	12.5882	0.537253
$550$	0	0
$551$	62.0734	2.64441
$552$	3.86067	0.164321
$553$	7.29772	0.310331
$554$	19.5967	0.832583
$555$	0	0
$556$	5.64365	0.239344
$557$	9.89921	0.419443	0.209721	−	0.977761i	$-0.432744\pi$
$557$	9.89921	0.419443	0.209721	+	0.977761i	$0.432744\pi$
$558$	−7.40507	−0.313482
$559$	20.8545	0.882052
$560$	0	0
$561$	1.40217	0.0591996
$562$	−2.83807	−0.119717
$563$	−2.01184	−0.0847891	−0.0423946	−	0.999101i	$-0.513499\pi$
$563$	−2.01184	−0.0847891	−0.0423946	+	0.999101i	$0.513499\pi$
$564$	−9.26574	−0.390158
$565$	0	0
$566$	−2.23356	−0.0938836
$567$	2.07768	0.0872545
$568$	−8.16901	−0.342764
$569$	23.6051	0.989578	0.494789	−	0.869013i	$-0.335245\pi$
$569$	23.6051	0.989578	0.494789	+	0.869013i	$0.335245\pi$
$570$	0	0
$571$	21.2334	0.888592	0.444296	−	0.895880i	$-0.353454\pi$
$571$	21.2334	0.888592	0.444296	+	0.895880i	$0.353454\pi$
$572$	−1.13714	−0.0475463
$573$	1.78768	0.0746815
$574$	−6.69758	−0.279552
$575$	0	0
$576$	1.00000	0.0416667
$577$	−45.3356	−1.88735	−0.943673	−	0.330880i	$-0.892654\pi$
$577$	−45.3356	−1.88735	−0.943673	+	0.330880i	$0.892654\pi$
$578$	−9.73311	−0.404844
$579$	6.60138	0.274344
$580$	0	0
$581$	27.2823	1.13186
$582$	−10.7598	−0.446006
$583$	1.26833	0.0525289
$584$	−3.84348	−0.159044
$585$	0	0
$586$	24.0260	0.992505
$587$	16.3234	0.673739	0.336869	−	0.941551i	$-0.390632\pi$
$587$	16.3234	0.673739	0.336869	+	0.941551i	$0.390632\pi$
$588$	2.68323	0.110655
$589$	50.8824	2.09657
$590$	0	0
$591$	−12.6575	−0.520660
$592$	5.30719	0.218124
$593$	32.8357	1.34840	0.674199	−	0.738549i	$-0.264489\pi$
$593$	32.8357	1.34840	0.674199	+	0.738549i	$0.264489\pi$
$594$	−0.520147	−0.0213419
$595$	0	0
$596$	−15.0956	−0.618339
$597$	−6.24148	−0.255447
$598$	8.44016	0.345144
$599$	−13.1905	−0.538947	−0.269474	−	0.963008i	$-0.586850\pi$
$599$	−13.1905	−0.538947	−0.269474	+	0.963008i	$0.586850\pi$
$600$	0	0
$601$	−19.1992	−0.783152	−0.391576	−	0.920146i	$-0.628070\pi$
$601$	−19.1992	−0.783152	−0.391576	+	0.920146i	$0.628070\pi$
$602$	−19.8194	−0.807780
$603$	−6.69385	−0.272595
$604$	−20.9353	−0.851847
$605$	0	0
$606$	17.6400	0.716576
$607$	−6.54268	−0.265559	−0.132779	−	0.991146i	$-0.542390\pi$
$607$	−6.54268	−0.265559	−0.132779	+	0.991146i	$0.542390\pi$
$608$	−6.87129	−0.278668
$609$	18.7692	0.760568
$610$	0	0
$611$	−20.2567	−0.819498
$612$	−2.69572	−0.108968
$613$	9.33202	0.376917	0.188458	−	0.982081i	$-0.439651\pi$
$613$	9.33202	0.376917	0.188458	+	0.982081i	$0.439651\pi$
$614$	17.1204	0.690922
$615$	0	0
$616$	1.08070	0.0435427
$617$	33.0016	1.32860	0.664298	−	0.747468i	$-0.268731\pi$
$617$	33.0016	1.32860	0.664298	+	0.747468i	$0.268731\pi$
$618$	8.69758	0.349868
$619$	28.1291	1.13060	0.565302	−	0.824884i	$-0.308760\pi$
$619$	28.1291	1.13060	0.565302	+	0.824884i	$0.308760\pi$
$620$	0	0
$621$	3.86067	0.154923
$622$	−20.1527	−0.808052
$623$	−20.4835	−0.820653
$624$	2.18619	0.0875177
$625$	0	0
$626$	−25.5043	−1.01936
$627$	3.57408	0.142735
$628$	5.71998	0.228252
$629$	−14.3067	−0.570445
$630$	0	0
$631$	−2.39685	−0.0954170	−0.0477085	−	0.998861i	$-0.515192\pi$
$631$	−2.39685	−0.0954170	−0.0477085	+	0.998861i	$0.515192\pi$
$632$	3.51243	0.139717
$633$	22.4779	0.893418
$634$	−31.5182	−1.25175
$635$	0	0
$636$	−2.43841	−0.0966891
$637$	5.86606	0.232422
$638$	−4.69887	−0.186030
$639$	−8.16901	−0.323161
$640$	0	0
$641$	12.1756	0.480907	0.240453	−	0.970661i	$-0.422704\pi$
$641$	12.1756	0.480907	0.240453	+	0.970661i	$0.422704\pi$
$642$	5.40977	0.213507
$643$	−27.0249	−1.06576	−0.532878	−	0.846192i	$-0.678890\pi$
$643$	−27.0249	−1.06576	−0.532878	+	0.846192i	$0.678890\pi$
$644$	−8.02124	−0.316081
$645$	0	0
$646$	18.5231	0.728780
$647$	−33.9198	−1.33352	−0.666762	−	0.745271i	$-0.732320\pi$
$647$	−33.9198	−1.33352	−0.666762	+	0.745271i	$0.732320\pi$
$648$	1.00000	0.0392837
$649$	4.49862	0.176586
$650$	0	0
$651$	15.3854	0.603001
$652$	11.8950	0.465846
$653$	−14.8530	−0.581244	−0.290622	−	0.956838i	$-0.593862\pi$
$653$	−14.8530	−0.581244	−0.290622	+	0.956838i	$0.593862\pi$
$654$	−2.88377	−0.112765
$655$	0	0
$656$	−3.22358	−0.125860
$657$	−3.84348	−0.149948
$658$	19.2513	0.750493
$659$	4.71934	0.183839	0.0919196	−	0.995766i	$-0.470700\pi$
$659$	4.71934	0.183839	0.0919196	+	0.995766i	$0.470700\pi$
$660$	0	0
$661$	3.49686	0.136012	0.0680060	−	0.997685i	$-0.478336\pi$
$661$	3.49686	0.136012	0.0680060	+	0.997685i	$0.478336\pi$
$662$	−7.30062	−0.283747
$663$	−5.89336	−0.228879
$664$	13.1311	0.509586
$665$	0	0
$666$	5.30719	0.205649
$667$	34.8762	1.35041
$668$	8.94427	0.346064
$669$	17.9299	0.693211
$670$	0	0
$671$	6.54774	0.252773
$672$	−2.07768	−0.0801484
$673$	24.6344	0.949586	0.474793	−	0.880098i	$-0.342523\pi$
$673$	24.6344	0.949586	0.474793	+	0.880098i	$0.342523\pi$
$674$	−26.7448	−1.03017
$675$	0	0
$676$	−8.22056	−0.316176
$677$	29.8252	1.14628	0.573139	−	0.819458i	$-0.305726\pi$
$677$	29.8252	1.14628	0.573139	+	0.819458i	$0.305726\pi$
$678$	6.36955	0.244621
$679$	22.3554	0.857921
$680$	0	0
$681$	8.85765	0.339426
$682$	−3.85173	−0.147490
$683$	−26.0533	−0.996902	−0.498451	−	0.866918i	$-0.666098\pi$
$683$	−26.0533	−0.996902	−0.498451	+	0.866918i	$0.666098\pi$
$684$	−6.87129	−0.262730
$685$	0	0
$686$	−20.1187	−0.768135
$687$	−22.0626	−0.841740
$688$	−9.53920	−0.363679
$689$	−5.33082	−0.203088
$690$	0	0
$691$	34.4035	1.30877	0.654385	−	0.756161i	$-0.272928\pi$
$691$	34.4035	1.30877	0.654385	+	0.756161i	$0.272928\pi$
$692$	16.3391	0.621118
$693$	1.08070	0.0410524
$694$	−4.87129	−0.184912
$695$	0	0
$696$	9.03373	0.342423
$697$	8.68987	0.329152
$698$	16.7173	0.632758
$699$	−11.5427	−0.436587
$700$	0	0
$701$	13.2937	0.502095	0.251047	−	0.967975i	$-0.419225\pi$
$701$	13.2937	0.502095	0.251047	+	0.967975i	$0.419225\pi$
$702$	2.18619	0.0825125
$703$	−36.4672	−1.37539
$704$	0.520147	0.0196038
$705$	0	0
$706$	27.0805	1.01919
$707$	−36.6503	−1.37838
$708$	−8.64875	−0.325040
$709$	−7.27532	−0.273230	−0.136615	−	0.990624i	$-0.543622\pi$
$709$	−7.27532	−0.273230	−0.136615	+	0.990624i	$0.543622\pi$
$710$	0	0
$711$	3.51243	0.131726
$712$	−9.85880	−0.369474
$713$	28.5885	1.07065
$714$	5.60085	0.209606
$715$	0	0
$716$	3.67853	0.137473
$717$	0.884927	0.0330482
$718$	−8.48817	−0.316776
$719$	−32.1466	−1.19886	−0.599432	−	0.800425i	$-0.704607\pi$
$719$	−32.1466	−1.19886	−0.599432	+	0.800425i	$0.704607\pi$
$720$	0	0
$721$	−18.0708	−0.672993
$722$	28.2146	1.05004
$723$	−12.6736	−0.471338
$724$	−23.0493	−0.856619
$725$	0	0
$726$	10.7294	0.398207
$727$	3.12558	0.115921	0.0579607	−	0.998319i	$-0.481540\pi$
$727$	3.12558	0.115921	0.0579607	+	0.998319i	$0.481540\pi$
$728$	−4.54222	−0.168346
$729$	1.00000	0.0370370
$730$	0	0
$731$	25.7150	0.951103
$732$	−12.5882	−0.465275
$733$	−46.3584	−1.71229	−0.856143	−	0.516739i	$-0.827146\pi$
$733$	−46.3584	−1.71229	−0.856143	+	0.516739i	$0.827146\pi$
$734$	12.7477	0.470525
$735$	0	0
$736$	−3.86067	−0.142306
$737$	−3.48179	−0.128253
$738$	−3.22358	−0.118662
$739$	−48.0286	−1.76676	−0.883381	−	0.468656i	$-0.844738\pi$
$739$	−48.0286	−1.76676	−0.883381	+	0.468656i	$0.844738\pi$
$740$	0	0
$741$	−15.0220	−0.551845
$742$	5.06624	0.185987
$743$	−2.88963	−0.106010	−0.0530051	−	0.998594i	$-0.516880\pi$
$743$	−2.88963	−0.106010	−0.0530051	+	0.998594i	$0.516880\pi$
$744$	7.40507	0.271483
$745$	0	0
$746$	10.9405	0.400562
$747$	13.1311	0.480442
$748$	−1.40217	−0.0512684
$749$	−11.2398	−0.410693
$750$	0	0
$751$	−39.4965	−1.44125	−0.720624	−	0.693326i	$-0.756144\pi$
$751$	−39.4965	−1.44125	−0.720624	+	0.693326i	$0.756144\pi$
$752$	9.26574	0.337887
$753$	−18.0966	−0.659478
$754$	19.7495	0.719233
$755$	0	0
$756$	−2.07768	−0.0755646
$757$	25.4654	0.925555	0.462777	−	0.886475i	$-0.346853\pi$
$757$	25.4654	0.925555	0.462777	+	0.886475i	$0.346853\pi$
$758$	−14.2301	−0.516862
$759$	2.00811	0.0728899
$760$	0	0
$761$	41.6705	1.51056	0.755278	−	0.655405i	$-0.227502\pi$
$761$	41.6705	1.51056	0.755278	+	0.655405i	$0.227502\pi$
$762$	14.9673	0.542208
$763$	5.99157	0.216909
$764$	−1.78768	−0.0646761
$765$	0	0
$766$	−7.16419	−0.258853
$767$	−18.9078	−0.682722
$768$	−1.00000	−0.0360844
$769$	2.07247	0.0747353	0.0373677	−	0.999302i	$-0.488103\pi$
$769$	2.07247	0.0747353	0.0373677	+	0.999302i	$0.488103\pi$
$770$	0	0
$771$	−1.28053	−0.0461172
$772$	−6.60138	−0.237589
$773$	−7.98959	−0.287366	−0.143683	−	0.989624i	$-0.545895\pi$
$773$	−7.98959	−0.287366	−0.143683	+	0.989624i	$0.545895\pi$
$774$	−9.53920	−0.342879
$775$	0	0
$776$	10.7598	0.386253
$777$	−11.0267	−0.395579
$778$	−0.918972	−0.0329467
$779$	22.1502	0.793612
$780$	0	0
$781$	−4.24908	−0.152044
$782$	10.4073	0.372163
$783$	9.03373	0.322839
$784$	−2.68323	−0.0958297
$785$	0	0
$786$	−13.8551	−0.494197
$787$	−17.9103	−0.638433	−0.319217	−	0.947682i	$-0.603420\pi$
$787$	−17.9103	−0.638433	−0.319217	+	0.947682i	$0.603420\pi$
$788$	12.6575	0.450905
$789$	7.11425	0.253274
$790$	0	0
$791$	−13.2339	−0.470544
$792$	0.520147	0.0184826
$793$	−27.5203	−0.977276
$794$	21.8019	0.773721
$795$	0	0
$796$	6.24148	0.221223
$797$	−26.2543	−0.929973	−0.464987	−	0.885318i	$-0.653941\pi$
$797$	−26.2543	−0.929973	−0.464987	+	0.885318i	$0.653941\pi$
$798$	14.2764	0.505378
$799$	−24.9778	−0.883652
$800$	0	0
$801$	−9.85880	−0.348344
$802$	22.8035	0.805219
$803$	−1.99917	−0.0705493
$804$	6.69385	0.236074
$805$	0	0
$806$	16.1889	0.570230
$807$	9.42986	0.331947
$808$	−17.6400	−0.620573
$809$	29.5316	1.03828	0.519138	−	0.854690i	$-0.326253\pi$
$809$	29.5316	1.03828	0.519138	+	0.854690i	$0.326253\pi$
$810$	0	0
$811$	29.4744	1.03499	0.517493	−	0.855687i	$-0.326865\pi$
$811$	29.4744	1.03499	0.517493	+	0.855687i	$0.326865\pi$
$812$	−18.7692	−0.658671
$813$	−1.37489	−0.0482196
$814$	2.76052	0.0967561
$815$	0	0
$816$	2.69572	0.0943690
$817$	65.5466	2.29318
$818$	5.00147	0.174872
$819$	−4.54222	−0.158718
$820$	0	0
$821$	44.8174	1.56414	0.782069	−	0.623192i	$-0.214165\pi$
$821$	44.8174	1.56414	0.782069	+	0.623192i	$0.214165\pi$
$822$	−19.8939	−0.693879
$823$	−21.7635	−0.758628	−0.379314	−	0.925268i	$-0.623840\pi$
$823$	−21.7635	−0.758628	−0.379314	+	0.925268i	$0.623840\pi$
$824$	−8.69758	−0.302995
$825$	0	0
$826$	17.9694	0.625234
$827$	19.1072	0.664424	0.332212	−	0.943205i	$-0.392205\pi$
$827$	19.1072	0.664424	0.332212	+	0.943205i	$0.392205\pi$
$828$	−3.86067	−0.134167
$829$	−36.3234	−1.26156	−0.630781	−	0.775961i	$-0.717265\pi$
$829$	−36.3234	−1.26156	−0.630781	+	0.775961i	$0.717265\pi$
$830$	0	0
$831$	−19.5967	−0.679801
$832$	−2.18619	−0.0757926
$833$	7.23323	0.250617
$834$	−5.64365	−0.195424
$835$	0	0
$836$	−3.57408	−0.123612
$837$	7.40507	0.255957
$838$	−22.8254	−0.788489
$839$	19.8589	0.685604	0.342802	−	0.939408i	$-0.388624\pi$
$839$	19.8589	0.685604	0.342802	+	0.939408i	$0.388624\pi$
$840$	0	0
$841$	52.6083	1.81408
$842$	15.7134	0.541521
$843$	2.83807	0.0977483
$844$	−22.4779	−0.773723
$845$	0	0
$846$	9.26574	0.318563
$847$	−22.2924	−0.765976
$848$	2.43841	0.0837352
$849$	2.23356	0.0766556
$850$	0	0
$851$	−20.4893	−0.702363
$852$	8.16901	0.279866
$853$	−26.0342	−0.891394	−0.445697	−	0.895184i	$-0.647044\pi$
$853$	−26.0342	−0.891394	−0.445697	+	0.895184i	$0.647044\pi$
$854$	26.1544	0.894985
$855$	0	0
$856$	−5.40977	−0.184902
$857$	−34.8614	−1.19084	−0.595421	−	0.803414i	$-0.703015\pi$
$857$	−34.8614	−1.19084	−0.595421	+	0.803414i	$0.703015\pi$
$858$	1.13714	0.0388214
$859$	−15.1199	−0.515883	−0.257942	−	0.966161i	$-0.583044\pi$
$859$	−15.1199	−0.515883	−0.257942	+	0.966161i	$0.583044\pi$
$860$	0	0
$861$	6.69758	0.228253
$862$	−9.55671	−0.325503
$863$	−22.9029	−0.779624	−0.389812	−	0.920895i	$-0.627460\pi$
$863$	−22.9029	−0.779624	−0.389812	+	0.920895i	$0.627460\pi$
$864$	−1.00000	−0.0340207
$865$	0	0
$866$	8.12215	0.276002
$867$	9.73311	0.330554
$868$	−15.3854	−0.522215
$869$	1.82698	0.0619761
$870$	0	0
$871$	14.6340	0.495856
$872$	2.88377	0.0976569
$873$	10.7598	0.364163
$874$	26.5278	0.897315
$875$	0	0
$876$	3.84348	0.129859
$877$	16.4875	0.556744	0.278372	−	0.960473i	$-0.410205\pi$
$877$	16.4875	0.556744	0.278372	+	0.960473i	$0.410205\pi$
$878$	9.21952	0.311144
$879$	−24.0260	−0.810377
$880$	0	0
$881$	23.1913	0.781334	0.390667	−	0.920532i	$-0.372244\pi$
$881$	23.1913	0.781334	0.390667	+	0.920532i	$0.372244\pi$
$882$	−2.68323	−0.0903491
$883$	−24.4716	−0.823536	−0.411768	−	0.911289i	$-0.635089\pi$
$883$	−24.4716	−0.823536	−0.411768	+	0.911289i	$0.635089\pi$
$884$	5.89336	0.198215
$885$	0	0
$886$	9.63232	0.323604
$887$	−3.94954	−0.132613	−0.0663064	−	0.997799i	$-0.521121\pi$
$887$	−3.94954	−0.132613	−0.0663064	+	0.997799i	$0.521121\pi$
$888$	−5.30719	−0.178098
$889$	−31.0973	−1.04297
$890$	0	0
$891$	0.520147	0.0174256
$892$	−17.9299	−0.600338
$893$	−63.6676	−2.13055
$894$	15.0956	0.504872
$895$	0	0
$896$	2.07768	0.0694105
$897$	−8.44016	−0.281809
$898$	−21.3096	−0.711110
$899$	66.8954	2.23109
$900$	0	0
$901$	−6.57325	−0.218987
$902$	−1.67674	−0.0558292
$903$	19.8194	0.659549
$904$	−6.36955	−0.211848
$905$	0	0
$906$	20.9353	0.695530
$907$	52.8637	1.75531	0.877656	−	0.479291i	$-0.159106\pi$
$907$	52.8637	1.75531	0.877656	+	0.479291i	$0.159106\pi$
$908$	−8.85765	−0.293951
$909$	−17.6400	−0.585082
$910$	0	0
$911$	11.5092	0.381317	0.190659	−	0.981656i	$-0.438938\pi$
$911$	11.5092	0.381317	0.190659	+	0.981656i	$0.438938\pi$
$912$	6.87129	0.227531
$913$	6.83011	0.226044
$914$	−13.4662	−0.445423
$915$	0	0
$916$	22.0626	0.728968
$917$	28.7866	0.950617
$918$	2.69572	0.0889719
$919$	31.0263	1.02346	0.511732	−	0.859145i	$-0.329004\pi$
$919$	31.0263	1.02346	0.511732	+	0.859145i	$0.329004\pi$
$920$	0	0
$921$	−17.1204	−0.564135
$922$	32.8357	1.08138
$923$	17.8590	0.587837
$924$	−1.08070	−0.0355524
$925$	0	0
$926$	−22.9306	−0.753547
$927$	−8.69758	−0.285666
$928$	−9.03373	−0.296547
$929$	−11.9550	−0.392230	−0.196115	−	0.980581i	$-0.562833\pi$
$929$	−11.9550	−0.392230	−0.196115	+	0.980581i	$0.562833\pi$
$930$	0	0
$931$	18.4373	0.604257
$932$	11.5427	0.378095
$933$	20.1527	0.659771
$934$	20.8340	0.681710
$935$	0	0
$936$	−2.18619	−0.0714579
$937$	14.6931	0.480003	0.240001	−	0.970773i	$-0.422852\pi$
$937$	14.6931	0.480003	0.240001	+	0.970773i	$0.422852\pi$
$938$	−13.9077	−0.454103
$939$	25.5043	0.832302
$940$	0	0
$941$	50.5426	1.64764	0.823820	−	0.566851i	$-0.191838\pi$
$941$	50.5426	1.64764	0.823820	+	0.566851i	$0.191838\pi$
$942$	−5.71998	−0.186367
$943$	12.4452	0.405271
$944$	8.64875	0.281493
$945$	0	0
$946$	−4.96179	−0.161322
$947$	−39.6824	−1.28950	−0.644752	−	0.764391i	$-0.723040\pi$
$947$	−39.6824	−1.28950	−0.644752	+	0.764391i	$0.723040\pi$
$948$	−3.51243	−0.114078
$949$	8.40259	0.272759
$950$	0	0
$951$	31.5182	1.02205
$952$	−5.60085	−0.181525
$953$	50.6244	1.63989	0.819943	−	0.572445i	$-0.194005\pi$
$953$	50.6244	1.63989	0.819943	+	0.572445i	$0.194005\pi$
$954$	2.43841	0.0789463
$955$	0	0
$956$	−0.884927	−0.0286206
$957$	4.69887	0.151893
$958$	13.5227	0.436897
$959$	41.3332	1.33472
$960$	0	0
$961$	23.8351	0.768875
$962$	−11.6025	−0.374081
$963$	−5.40977	−0.174328
$964$	12.6736	0.408191
$965$	0	0
$966$	8.02124	0.258079
$967$	−11.0521	−0.355413	−0.177706	−	0.984084i	$-0.556868\pi$
$967$	−11.0521	−0.355413	−0.177706	+	0.984084i	$0.556868\pi$
$968$	−10.7294	−0.344857
$969$	−18.5231	−0.595046
$970$	0	0
$971$	24.6590	0.791344	0.395672	−	0.918392i	$-0.370512\pi$
$971$	24.6590	0.791344	0.395672	+	0.918392i	$0.370512\pi$
$972$	−1.00000	−0.0320750
$973$	11.7257	0.375909
$974$	26.6228	0.853050
$975$	0	0
$976$	12.5882	0.402940
$977$	8.11694	0.259684	0.129842	−	0.991535i	$-0.458553\pi$
$977$	8.11694	0.259684	0.129842	+	0.991535i	$0.458553\pi$
$978$	−11.8950	−0.380361
$979$	−5.12803	−0.163892
$980$	0	0
$981$	2.88377	0.0920718
$982$	−12.0227	−0.383659
$983$	26.7026	0.851680	0.425840	−	0.904799i	$-0.359979\pi$
$983$	26.7026	0.851680	0.425840	+	0.904799i	$0.359979\pi$
$984$	3.22358	0.102764
$985$	0	0
$986$	24.3524	0.775538
$987$	−19.2513	−0.612775
$988$	15.0220	0.477912
$989$	36.8277	1.17105
$990$	0	0
$991$	−52.9146	−1.68089	−0.840444	−	0.541899i	$-0.817706\pi$
$991$	−52.9146	−1.68089	−0.840444	+	0.541899i	$0.817706\pi$
$992$	−7.40507	−0.235111
$993$	7.30062	0.231678
$994$	−16.9726	−0.538338
$995$	0	0
$996$	−13.1311	−0.416075
$997$	−28.2703	−0.895330	−0.447665	−	0.894201i	$-0.647744\pi$
$997$	−28.2703	−0.895330	−0.447665	+	0.894201i	$0.647744\pi$
$998$	−27.3600	−0.866065
$999$	−5.30719	−0.167912

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3750.2.a.o.1.4		4
5.2	odd	4		3750.2.c.e.1249.8		8
5.3	odd	4		3750.2.c.e.1249.1		8
5.4	even	2		3750.2.a.m.1.1		4
25.2	odd	20		150.2.h.a.79.2	yes	8
25.9	even	10		750.2.g.e.151.1		8
25.11	even	5		750.2.g.c.601.2		8
25.12	odd	20		750.2.h.c.349.1		8
25.13	odd	20		150.2.h.a.19.2	✓	8
25.14	even	10		750.2.g.e.601.1		8
25.16	even	5		750.2.g.c.151.2		8
25.23	odd	20		750.2.h.c.649.1		8
75.2	even	20		450.2.l.a.379.1		8
75.38	even	20		450.2.l.a.19.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
150.2.h.a.19.2	✓	8	25.13	odd	20
150.2.h.a.79.2	yes	8	25.2	odd	20
450.2.l.a.19.1		8	75.38	even	20
450.2.l.a.379.1		8	75.2	even	20
750.2.g.c.151.2		8	25.16	even	5
750.2.g.c.601.2		8	25.11	even	5
750.2.g.e.151.1		8	25.9	even	10
750.2.g.e.601.1		8	25.14	even	10
750.2.h.c.349.1		8	25.12	odd	20
750.2.h.c.649.1		8	25.23	odd	20
3750.2.a.m.1.1		4	5.4	even	2
3750.2.a.o.1.4		4	1.1	even	1	trivial
3750.2.c.e.1249.1		8	5.3	odd	4
3750.2.c.e.1249.8		8	5.2	odd	4

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:	\( N \)	\(=\)	\( 3750 = 2 \cdot 3 \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3750.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +2.07768 q^{7} +1.00000 q^{8} +1.00000 q^{9} +0.520147 q^{11} -1.00000 q^{12} -2.18619 q^{13} +2.07768 q^{14} +1.00000 q^{16} -2.69572 q^{17} +1.00000 q^{18} -6.87129 q^{19} -2.07768 q^{21} +0.520147 q^{22} -3.86067 q^{23} -1.00000 q^{24} -2.18619 q^{26} -1.00000 q^{27} +2.07768 q^{28} -9.03373 q^{29} -7.40507 q^{31} +1.00000 q^{32} -0.520147 q^{33} -2.69572 q^{34} +1.00000 q^{36} +5.30719 q^{37} -6.87129 q^{38} +2.18619 q^{39} -3.22358 q^{41} -2.07768 q^{42} -9.53920 q^{43} +0.520147 q^{44} -3.86067 q^{46} +9.26574 q^{47} -1.00000 q^{48} -2.68323 q^{49} +2.69572 q^{51} -2.18619 q^{52} +2.43841 q^{53} -1.00000 q^{54} +2.07768 q^{56} +6.87129 q^{57} -9.03373 q^{58} +8.64875 q^{59} +12.5882 q^{61} -7.40507 q^{62} +2.07768 q^{63} +1.00000 q^{64} -0.520147 q^{66} -6.69385 q^{67} -2.69572 q^{68} +3.86067 q^{69} -8.16901 q^{71} +1.00000 q^{72} -3.84348 q^{73} +5.30719 q^{74} -6.87129 q^{76} +1.08070 q^{77} +2.18619 q^{78} +3.51243 q^{79} +1.00000 q^{81} -3.22358 q^{82} +13.1311 q^{83} -2.07768 q^{84} -9.53920 q^{86} +9.03373 q^{87} +0.520147 q^{88} -9.85880 q^{89} -4.54222 q^{91} -3.86067 q^{92} +7.40507 q^{93} +9.26574 q^{94} -1.00000 q^{96} +10.7598 q^{97} -2.68323 q^{98} +0.520147 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{2} - 4 q^{3} + 4 q^{4} - 4 q^{6} - 4 q^{7} + 4 q^{8} + 4 q^{9} - 10 q^{11} - 4 q^{12} + 2 q^{13} - 4 q^{14} + 4 q^{16} + 6 q^{17} + 4 q^{18} - 6 q^{19} + 4 q^{21} - 10 q^{22} - 4 q^{24} + 2 q^{26}+ \cdots - 10 q^{99}+O(q^{100}) \) 4 * q + 4 * q^2 - 4 * q^3 + 4 * q^4 - 4 * q^6 - 4 * q^7 + 4 * q^8 + 4 * q^9 - 10 * q^11 - 4 * q^12 + 2 * q^13 - 4 * q^14 + 4 * q^16 + 6 * q^17 + 4 * q^18 - 6 * q^19 + 4 * q^21 - 10 * q^22 - 4 * q^24 + 2 * q^26 - 4 * q^27 - 4 * q^28 - 14 * q^29 - 18 * q^31 + 4 * q^32 + 10 * q^33 + 6 * q^34 + 4 * q^36 + 2 * q^37 - 6 * q^38 - 2 * q^39 - 14 * q^41 + 4 * q^42 - 14 * q^43 - 10 * q^44 + 10 * q^47 - 4 * q^48 - 4 * q^49 - 6 * q^51 + 2 * q^52 + 14 * q^53 - 4 * q^54 - 4 * q^56 + 6 * q^57 - 14 * q^58 - 20 * q^59 - 18 * q^62 - 4 * q^63 + 4 * q^64 + 10 * q^66 - 14 * q^67 + 6 * q^68 - 30 * q^71 + 4 * q^72 - 8 * q^73 + 2 * q^74 - 6 * q^76 + 20 * q^77 - 2 * q^78 - 28 * q^79 + 4 * q^81 - 14 * q^82 + 12 * q^83 + 4 * q^84 - 14 * q^86 + 14 * q^87 - 10 * q^88 - 28 * q^89 - 22 * q^91 + 18 * q^93 + 10 * q^94 - 4 * q^96 - 8 * q^97 - 4 * q^98 - 10 * q^99

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