LMFDB - Embedded newform 2450.4.a.ch.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2450,4,Mod(1,2450)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2450, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2450.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2450 = 2 \cdot 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2450.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:	$144.554679514$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.238585.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 67x - 189 $ x^3 - 67*x - 189
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 350)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$9.34000$ of defining polynomial
Character	$\chi$	$=$	2450.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000 q^{2} +8.34000 q^{3} +4.00000 q^{4} +16.6800 q^{6} +8.00000 q^{8} +42.5556 q^{9} +O(q^{10})$	\(q+2.00000 q^{2} +8.34000 q^{3} +4.00000 q^{4} +16.6800 q^{6} +8.00000 q^{8} +42.5556 q^{9} -26.2689 q^{11} +33.3600 q^{12} +38.1623 q^{13} +16.0000 q^{16} -2.39331 q^{17} +85.1111 q^{18} -99.7933 q^{19} -52.5377 q^{22} +101.778 q^{23} +66.7200 q^{24} +76.3245 q^{26} +129.733 q^{27} -223.936 q^{29} +267.327 q^{31} +32.0000 q^{32} -219.082 q^{33} -4.78661 q^{34} +170.222 q^{36} +363.287 q^{37} -199.587 q^{38} +318.273 q^{39} +94.9155 q^{41} +406.093 q^{43} -105.075 q^{44} +203.556 q^{46} +342.173 q^{47} +133.440 q^{48} -19.9602 q^{51} +152.649 q^{52} +543.653 q^{53} +259.467 q^{54} -832.276 q^{57} -447.871 q^{58} +393.158 q^{59} +504.309 q^{61} +534.653 q^{62} +64.0000 q^{64} -438.165 q^{66} +54.9465 q^{67} -9.57322 q^{68} +848.826 q^{69} +889.302 q^{71} +340.444 q^{72} -510.702 q^{73} +726.573 q^{74} -399.173 q^{76} +636.546 q^{78} +273.756 q^{79} -67.0246 q^{81} +189.831 q^{82} +525.188 q^{83} +812.187 q^{86} -1867.62 q^{87} -210.151 q^{88} -1655.87 q^{89} +407.111 q^{92} +2229.50 q^{93} +684.346 q^{94} +266.880 q^{96} +5.76230 q^{97} -1117.89 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 6 q^{2} - 3 q^{3} + 12 q^{4} - 6 q^{6} + 24 q^{8} + 56 q^{9}+O(q^{10}) $ 3 * q + 6 * q^2 - 3 * q^3 + 12 * q^4 - 6 * q^6 + 24 * q^8 + 56 * q^9	$ 3 q + 6 q^{2} - 3 q^{3} + 12 q^{4} - 6 q^{6} + 24 q^{8} + 56 q^{9} - 26 q^{11} - 12 q^{12} + 80 q^{13} + 48 q^{16} + 30 q^{17} + 112 q^{18} + 18 q^{19} - 52 q^{22} - 53 q^{23} - 24 q^{24} + 160 q^{26} + 324 q^{27} - 404 q^{29} + 615 q^{31} + 96 q^{32} - 314 q^{33} + 60 q^{34} + 224 q^{36} + 426 q^{37} + 36 q^{38} - 90 q^{39} + 101 q^{41} - 249 q^{43} - 104 q^{44} - 106 q^{46} + 402 q^{47} - 48 q^{48} - 339 q^{51} + 320 q^{52} + 390 q^{53} + 648 q^{54} - 1667 q^{57} - 808 q^{58} + 91 q^{59} + 647 q^{61} + 1230 q^{62} + 192 q^{64} - 628 q^{66} + 708 q^{67} + 120 q^{68} + 1548 q^{69} + 533 q^{71} + 448 q^{72} + 772 q^{73} + 852 q^{74} + 72 q^{76} - 180 q^{78} + 1421 q^{79} - 1393 q^{81} + 202 q^{82} - 520 q^{83} - 498 q^{86} - 833 q^{87} - 208 q^{88} + 194 q^{89} - 212 q^{92} + 148 q^{93} + 804 q^{94} - 96 q^{96} - 1027 q^{97} - 57 q^{99}+O(q^{100}) $ 3 * q + 6 * q^2 - 3 * q^3 + 12 * q^4 - 6 * q^6 + 24 * q^8 + 56 * q^9 - 26 * q^11 - 12 * q^12 + 80 * q^13 + 48 * q^16 + 30 * q^17 + 112 * q^18 + 18 * q^19 - 52 * q^22 - 53 * q^23 - 24 * q^24 + 160 * q^26 + 324 * q^27 - 404 * q^29 + 615 * q^31 + 96 * q^32 - 314 * q^33 + 60 * q^34 + 224 * q^36 + 426 * q^37 + 36 * q^38 - 90 * q^39 + 101 * q^41 - 249 * q^43 - 104 * q^44 - 106 * q^46 + 402 * q^47 - 48 * q^48 - 339 * q^51 + 320 * q^52 + 390 * q^53 + 648 * q^54 - 1667 * q^57 - 808 * q^58 + 91 * q^59 + 647 * q^61 + 1230 * q^62 + 192 * q^64 - 628 * q^66 + 708 * q^67 + 120 * q^68 + 1548 * q^69 + 533 * q^71 + 448 * q^72 + 772 * q^73 + 852 * q^74 + 72 * q^76 - 180 * q^78 + 1421 * q^79 - 1393 * q^81 + 202 * q^82 - 520 * q^83 - 498 * q^86 - 833 * q^87 - 208 * q^88 + 194 * q^89 - 212 * q^92 + 148 * q^93 + 804 * q^94 - 96 * q^96 - 1027 * q^97 - 57 * 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 $

$2$	2.00000	0.707107
$2$	2.00000	0.707107
$3$	8.34000	1.60503	0.802517	−	0.596630i	$-0.203494\pi$
$3$	8.34000	1.60503	0.802517	+	0.596630i	$0.203494\pi$
$4$	4.00000	0.500000
$5$	0	0
$5$	0	0
$6$	16.6800	1.13493
$7$	0	0
$7$	0	0
$8$	8.00000	0.353553
$9$	42.5556	1.57613
$10$	0	0
$11$	−26.2689	−0.720033	−0.360016	−	0.932946i	$-0.617229\pi$
$11$	−26.2689	−0.720033	−0.360016	+	0.932946i	$0.617229\pi$
$12$	33.3600	0.802517
$13$	38.1623	0.814177	0.407089	−	0.913389i	$-0.366544\pi$
$13$	38.1623	0.814177	0.407089	+	0.913389i	$0.366544\pi$
$14$	0	0
$15$	0	0
$16$	16.0000	0.250000
$17$	−2.39331	−0.0341448	−0.0170724	−	0.999854i	$-0.505435\pi$
$17$	−2.39331	−0.0341448	−0.0170724	+	0.999854i	$0.505435\pi$
$18$	85.1111	1.11449
$19$	−99.7933	−1.20496	−0.602478	−	0.798136i	$-0.705820\pi$
$19$	−99.7933	−1.20496	−0.602478	+	0.798136i	$0.705820\pi$
$20$	0	0
$21$	0	0
$22$	−52.5377	−0.509140
$23$	101.778	0.922702	0.461351	−	0.887218i	$-0.347365\pi$
$23$	101.778	0.922702	0.461351	+	0.887218i	$0.347365\pi$
$24$	66.7200	0.567465
$25$	0	0
$26$	76.3245	0.575710
$27$	129.733	0.924711
$28$	0	0
$29$	−223.936	−1.43392	−0.716962	−	0.697112i	$-0.754468\pi$
$29$	−223.936	−1.43392	−0.716962	+	0.697112i	$0.754468\pi$
$30$	0	0
$31$	267.327	1.54882	0.774408	−	0.632687i	$-0.218048\pi$
$31$	267.327	1.54882	0.774408	+	0.632687i	$0.218048\pi$
$32$	32.0000	0.176777
$33$	−219.082	−1.15568
$34$	−4.78661	−0.0241440
$35$	0	0
$36$	170.222	0.788066
$37$	363.287	1.61416	0.807081	−	0.590441i	$-0.201046\pi$
$37$	363.287	1.61416	0.807081	+	0.590441i	$0.201046\pi$
$38$	−199.587	−0.852032
$39$	318.273	1.30678
$40$	0	0
$41$	94.9155	0.361544	0.180772	−	0.983525i	$-0.442140\pi$
$41$	94.9155	0.361544	0.180772	+	0.983525i	$0.442140\pi$
$42$	0	0
$43$	406.093	1.44020	0.720101	−	0.693870i	$-0.244096\pi$
$43$	406.093	1.44020	0.720101	+	0.693870i	$0.244096\pi$
$44$	−105.075	−0.360016
$45$	0	0
$46$	203.556	0.652449
$47$	342.173	1.06194	0.530969	−	0.847391i	$-0.321828\pi$
$47$	342.173	1.06194	0.530969	+	0.847391i	$0.321828\pi$
$48$	133.440	0.401258
$49$	0	0
$50$	0	0
$51$	−19.9602	−0.0548036
$52$	152.649	0.407089
$53$	543.653	1.40899	0.704495	−	0.709709i	$-0.251173\pi$
$53$	543.653	1.40899	0.704495	+	0.709709i	$0.251173\pi$
$54$	259.467	0.653869
$55$	0	0
$56$	0	0
$57$	−832.276	−1.93399
$58$	−447.871	−1.01394
$59$	393.158	0.867539	0.433770	−	0.901024i	$-0.357183\pi$
$59$	393.158	0.867539	0.433770	+	0.901024i	$0.357183\pi$
$60$	0	0
$61$	504.309	1.05853	0.529263	−	0.848458i	$-0.322468\pi$
$61$	504.309	1.05853	0.529263	+	0.848458i	$0.322468\pi$
$62$	534.653	1.09518
$63$	0	0
$64$	64.0000	0.125000
$65$	0	0
$66$	−438.165	−0.817187
$67$	54.9465	0.100191	0.0500954	−	0.998744i	$-0.484047\pi$
$67$	54.9465	0.100191	0.0500954	+	0.998744i	$0.484047\pi$
$68$	−9.57322	−0.0170724
$69$	848.826	1.48097
$70$	0	0
$71$	889.302	1.48649	0.743245	−	0.669020i	$-0.233286\pi$
$71$	889.302	1.48649	0.743245	+	0.669020i	$0.233286\pi$
$72$	340.444	0.557247
$73$	−510.702	−0.818810	−0.409405	−	0.912353i	$-0.634264\pi$
$73$	−510.702	−0.818810	−0.409405	+	0.912353i	$0.634264\pi$
$74$	726.573	1.14138
$75$	0	0
$76$	−399.173	−0.602478
$77$	0	0
$78$	636.546	0.924034
$79$	273.756	0.389873	0.194936	−	0.980816i	$-0.437550\pi$
$79$	273.756	0.389873	0.194936	+	0.980816i	$0.437550\pi$
$80$	0	0
$81$	−67.0246	−0.0919405
$82$	189.831	0.255650
$83$	525.188	0.694541	0.347271	−	0.937765i	$-0.387109\pi$
$83$	525.188	0.694541	0.347271	+	0.937765i	$0.387109\pi$
$84$	0	0
$85$	0	0
$86$	812.187	1.01838
$87$	−1867.62	−2.30150
$88$	−210.151	−0.254570
$89$	−1655.87	−1.97216	−0.986079	−	0.166280i	$-0.946824\pi$
$89$	−1655.87	−1.97216	−0.986079	+	0.166280i	$0.946824\pi$
$90$	0	0
$91$	0	0
$92$	407.111	0.461351
$93$	2229.50	2.48590
$94$	684.346	0.750903
$95$	0	0
$96$	266.880	0.283732
$97$	5.76230	0.00603168	0.00301584	−	0.999995i	$-0.499040\pi$
$97$	5.76230	0.00603168	0.00301584	+	0.999995i	$0.499040\pi$
$98$	0	0
$99$	−1117.89	−1.13487
$100$	0	0
$101$	−1154.98	−1.13787	−0.568934	−	0.822383i	$-0.692644\pi$
$101$	−1154.98	−1.13787	−0.568934	+	0.822383i	$0.692644\pi$
$102$	−39.9203	−0.0387520
$103$	1635.81	1.56487	0.782433	−	0.622734i	$-0.213978\pi$
$103$	1635.81	1.56487	0.782433	+	0.622734i	$0.213978\pi$
$104$	305.298	0.287855
$105$	0	0
$106$	1087.31	0.996307
$107$	−663.857	−0.599790	−0.299895	−	0.953972i	$-0.596952\pi$
$107$	−663.857	−0.599790	−0.299895	+	0.953972i	$0.596952\pi$
$108$	518.933	0.462355
$109$	−1806.01	−1.58701	−0.793505	−	0.608564i	$-0.791746\pi$
$109$	−1806.01	−1.58701	−0.793505	+	0.608564i	$0.791746\pi$
$110$	0	0
$111$	3029.81	2.59078
$112$	0	0
$113$	448.406	0.373297	0.186648	−	0.982427i	$-0.440238\pi$
$113$	448.406	0.373297	0.186648	+	0.982427i	$0.440238\pi$
$114$	−1664.55	−1.36754
$115$	0	0
$116$	−895.742	−0.716962
$117$	1624.02	1.28325
$118$	786.316	0.613443
$119$	0	0
$120$	0	0
$121$	−640.947	−0.481553
$122$	1008.62	0.748491
$123$	791.595	0.580291
$124$	1069.31	0.774408
$125$	0	0
$126$	0	0
$127$	1780.09	1.24376	0.621881	−	0.783112i	$-0.286369\pi$
$127$	1780.09	1.24376	0.621881	+	0.783112i	$0.286369\pi$
$128$	128.000	0.0883883
$129$	3386.82	2.31157
$130$	0	0
$131$	−2940.88	−1.96142	−0.980709	−	0.195475i	$-0.937375\pi$
$131$	−2940.88	−1.96142	−0.980709	+	0.195475i	$0.937375\pi$
$132$	−876.329	−0.577838
$133$	0	0
$134$	109.893	0.0708456
$135$	0	0
$136$	−19.1464	−0.0120720
$137$	−1654.35	−1.03169	−0.515843	−	0.856683i	$-0.672521\pi$
$137$	−1654.35	−1.03169	−0.515843	+	0.856683i	$0.672521\pi$
$138$	1697.65	1.04720
$139$	−409.882	−0.250113	−0.125057	−	0.992150i	$-0.539911\pi$
$139$	−409.882	−0.250113	−0.125057	+	0.992150i	$0.539911\pi$
$140$	0	0
$141$	2853.72	1.70445
$142$	1778.60	1.05111
$143$	−1002.48	−0.586234
$144$	680.889	0.394033
$145$	0	0
$146$	−1021.40	−0.578986
$147$	0	0
$148$	1453.15	0.807081
$149$	−2979.18	−1.63802	−0.819008	−	0.573783i	$-0.805475\pi$
$149$	−2979.18	−1.63802	−0.819008	+	0.573783i	$0.805475\pi$
$150$	0	0
$151$	−1974.62	−1.06419	−0.532095	−	0.846685i	$-0.678595\pi$
$151$	−1974.62	−1.06419	−0.532095	+	0.846685i	$0.678595\pi$
$152$	−798.346	−0.426016
$153$	−101.848	−0.0538167
$154$	0	0
$155$	0	0
$156$	1273.09	0.653391
$157$	701.298	0.356495	0.178247	−	0.983986i	$-0.442957\pi$
$157$	701.298	0.356495	0.178247	+	0.983986i	$0.442957\pi$
$158$	547.512	0.275682
$159$	4534.07	2.26148
$160$	0	0
$161$	0	0
$162$	−134.049	−0.0650117
$163$	−894.125	−0.429652	−0.214826	−	0.976652i	$-0.568918\pi$
$163$	−894.125	−0.429652	−0.214826	+	0.976652i	$0.568918\pi$
$164$	379.662	0.180772
$165$	0	0
$166$	1050.38	0.491115
$167$	−1037.45	−0.480719	−0.240359	−	0.970684i	$-0.577265\pi$
$167$	−1037.45	−0.480719	−0.240359	+	0.970684i	$0.577265\pi$
$168$	0	0
$169$	−740.643	−0.337115
$170$	0	0
$171$	−4246.76	−1.89917
$172$	1624.37	0.720101
$173$	1837.08	0.807343	0.403672	−	0.914904i	$-0.367734\pi$
$173$	1837.08	0.807343	0.403672	+	0.914904i	$0.367734\pi$
$174$	−3735.24	−1.62740
$175$	0	0
$176$	−420.302	−0.180008
$177$	3278.94	1.39243
$178$	−3311.74	−1.39453
$179$	49.7945	0.0207923	0.0103961	−	0.999946i	$-0.496691\pi$
$179$	49.7945	0.0207923	0.0103961	+	0.999946i	$0.496691\pi$
$180$	0	0
$181$	−1000.59	−0.410903	−0.205452	−	0.978667i	$-0.565866\pi$
$181$	−1000.59	−0.410903	−0.205452	+	0.978667i	$0.565866\pi$
$182$	0	0
$183$	4205.93	1.69897
$184$	814.222	0.326224
$185$	0	0
$186$	4459.01	1.75780
$187$	62.8694	0.0245854
$188$	1368.69	0.530969
$189$	0	0
$190$	0	0
$191$	−997.807	−0.378004	−0.189002	−	0.981977i	$-0.560525\pi$
$191$	−997.807	−0.378004	−0.189002	+	0.981977i	$0.560525\pi$
$192$	533.760	0.200629
$193$	2563.06	0.955922	0.477961	−	0.878381i	$-0.341376\pi$
$193$	2563.06	0.955922	0.477961	+	0.878381i	$0.341376\pi$
$194$	11.5246	0.00426504
$195$	0	0
$196$	0	0
$197$	821.223	0.297004	0.148502	−	0.988912i	$-0.452555\pi$
$197$	821.223	0.297004	0.148502	+	0.988912i	$0.452555\pi$
$198$	−2235.77	−0.802472
$199$	3236.26	1.15283	0.576413	−	0.817158i	$-0.304452\pi$
$199$	3236.26	1.15283	0.576413	+	0.817158i	$0.304452\pi$
$200$	0	0
$201$	458.254	0.160810
$202$	−2309.96	−0.804595
$203$	0	0
$204$	−79.8407	−0.0274018
$205$	0	0
$206$	3271.62	1.10653
$207$	4331.21	1.45430
$208$	610.596	0.203544
$209$	2621.46	0.867607
$210$	0	0
$211$	−3164.93	−1.03262	−0.516309	−	0.856402i	$-0.672695\pi$
$211$	−3164.93	−1.03262	−0.516309	+	0.856402i	$0.672695\pi$
$212$	2174.61	0.704495
$213$	7416.78	2.38586
$214$	−1327.71	−0.424116
$215$	0	0
$216$	1037.87	0.326935
$217$	0	0
$218$	−3612.01	−1.12218
$219$	−4259.25	−1.31422
$220$	0	0
$221$	−91.3340	−0.0277999
$222$	6059.62	1.83196
$223$	−4697.63	−1.41066	−0.705329	−	0.708880i	$-0.749201\pi$
$223$	−4697.63	−1.41066	−0.705329	+	0.708880i	$0.749201\pi$
$224$	0	0
$225$	0	0
$226$	896.813	0.263961
$227$	743.942	0.217521	0.108760	−	0.994068i	$-0.465312\pi$
$227$	743.942	0.217521	0.108760	+	0.994068i	$0.465312\pi$
$228$	−3329.10	−0.966997
$229$	957.135	0.276198	0.138099	−	0.990418i	$-0.455901\pi$
$229$	957.135	0.276198	0.138099	+	0.990418i	$0.455901\pi$
$230$	0	0
$231$	0	0
$232$	−1791.48	−0.506969
$233$	857.919	0.241220	0.120610	−	0.992700i	$-0.461515\pi$
$233$	857.919	0.241220	0.120610	+	0.992700i	$0.461515\pi$
$234$	3248.03	0.907395
$235$	0	0
$236$	1572.63	0.433770
$237$	2283.12	0.625758
$238$	0	0
$239$	−1814.74	−0.491154	−0.245577	−	0.969377i	$-0.578977\pi$
$239$	−1814.74	−0.491154	−0.245577	+	0.969377i	$0.578977\pi$
$240$	0	0
$241$	3681.10	0.983901	0.491951	−	0.870623i	$-0.336284\pi$
$241$	3681.10	0.983901	0.491951	+	0.870623i	$0.336284\pi$
$242$	−1281.89	−0.340509
$243$	−4061.78	−1.07228
$244$	2017.24	0.529263
$245$	0	0
$246$	1583.19	0.410327
$247$	−3808.34	−0.981047
$248$	2138.61	0.547589
$249$	4380.07	1.11476
$250$	0	0
$251$	2588.41	0.650913	0.325457	−	0.945557i	$-0.394482\pi$
$251$	2588.41	0.650913	0.325457	+	0.945557i	$0.394482\pi$
$252$	0	0
$253$	−2673.59	−0.664375
$254$	3560.19	0.879472
$255$	0	0
$256$	256.000	0.0625000
$257$	1111.54	0.269790	0.134895	−	0.990860i	$-0.456930\pi$
$257$	1111.54	0.269790	0.134895	+	0.990860i	$0.456930\pi$
$258$	6773.63	1.63453
$259$	0	0
$260$	0	0
$261$	−9529.70	−2.26005
$262$	−5881.76	−1.38693
$263$	−2742.91	−0.643098	−0.321549	−	0.946893i	$-0.604204\pi$
$263$	−2742.91	−0.643098	−0.321549	+	0.946893i	$0.604204\pi$
$264$	−1752.66	−0.408593
$265$	0	0
$266$	0	0
$267$	−13810.0	−3.16538
$268$	219.786	0.0500954
$269$	1925.15	0.436350	0.218175	−	0.975910i	$-0.429990\pi$
$269$	1925.15	0.436350	0.218175	+	0.975910i	$0.429990\pi$
$270$	0	0
$271$	286.595	0.0642414	0.0321207	−	0.999484i	$-0.489774\pi$
$271$	286.595	0.0642414	0.0321207	+	0.999484i	$0.489774\pi$
$272$	−38.2929	−0.00853621
$273$	0	0
$274$	−3308.71	−0.729512
$275$	0	0
$276$	3395.31	0.740483
$277$	−1438.87	−0.312107	−0.156053	−	0.987749i	$-0.549877\pi$
$277$	−1438.87	−0.312107	−0.156053	+	0.987749i	$0.549877\pi$
$278$	−819.765	−0.176857
$279$	11376.2	2.44114
$280$	0	0
$281$	−2659.15	−0.564525	−0.282262	−	0.959337i	$-0.591085\pi$
$281$	−2659.15	−0.564525	−0.282262	+	0.959337i	$0.591085\pi$
$282$	5707.45	1.20522
$283$	333.145	0.0699768	0.0349884	−	0.999388i	$-0.488861\pi$
$283$	333.145	0.0699768	0.0349884	+	0.999388i	$0.488861\pi$
$284$	3557.21	0.743245
$285$	0	0
$286$	−2004.96	−0.414530
$287$	0	0
$288$	1361.78	0.278623
$289$	−4907.27	−0.998834
$290$	0	0
$291$	48.0576	0.00968105
$292$	−2042.81	−0.409405
$293$	7675.26	1.53035	0.765176	−	0.643821i	$-0.222652\pi$
$293$	7675.26	1.53035	0.765176	+	0.643821i	$0.222652\pi$
$294$	0	0
$295$	0	0
$296$	2906.29	0.570692
$297$	−3407.95	−0.665822
$298$	−5958.37	−1.15825
$299$	3884.07	0.751243
$300$	0	0
$301$	0	0
$302$	−3949.25	−0.752496
$303$	−9632.53	−1.82632
$304$	−1596.69	−0.301239
$305$	0	0
$306$	−203.697	−0.0380542
$307$	3833.02	0.712580	0.356290	−	0.934375i	$-0.384041\pi$
$307$	3833.02	0.712580	0.356290	+	0.934375i	$0.384041\pi$
$308$	0	0
$309$	13642.7	2.51166
$310$	0	0
$311$	−441.866	−0.0805656	−0.0402828	−	0.999188i	$-0.512826\pi$
$311$	−441.866	−0.0805656	−0.0402828	+	0.999188i	$0.512826\pi$
$312$	2546.18	0.462017
$313$	8326.03	1.50356	0.751781	−	0.659412i	$-0.229195\pi$
$313$	8326.03	1.50356	0.751781	+	0.659412i	$0.229195\pi$
$314$	1402.60	0.252080
$315$	0	0
$316$	1095.02	0.194936
$317$	−3468.11	−0.614475	−0.307237	−	0.951633i	$-0.599405\pi$
$317$	−3468.11	−0.614475	−0.307237	+	0.951633i	$0.599405\pi$
$318$	9068.13	1.59911
$319$	5882.53	1.03247
$320$	0	0
$321$	−5536.57	−0.962683
$322$	0	0
$323$	238.836	0.0411430
$324$	−268.098	−0.0459702
$325$	0	0
$326$	−1788.25	−0.303810
$327$	−15062.1	−2.54720
$328$	759.324	0.127825
$329$	0	0
$330$	0	0
$331$	−10968.7	−1.82144	−0.910720	−	0.413025i	$-0.864472\pi$
$331$	−10968.7	−1.82144	−0.910720	+	0.413025i	$0.864472\pi$
$332$	2100.75	0.347271
$333$	15459.9	2.54413
$334$	−2074.89	−0.339919
$335$	0	0
$336$	0	0
$337$	−4317.34	−0.697865	−0.348932	−	0.937148i	$-0.613456\pi$
$337$	−4317.34	−0.697865	−0.348932	+	0.937148i	$0.613456\pi$
$338$	−1481.29	−0.238377
$339$	3739.71	0.599153
$340$	0	0
$341$	−7022.37	−1.11520
$342$	−8493.52	−1.34291
$343$	0	0
$344$	3248.75	0.509188
$345$	0	0
$346$	3674.15	0.570878
$347$	−8407.43	−1.30068	−0.650338	−	0.759645i	$-0.725373\pi$
$347$	−8407.43	−1.30068	−0.650338	+	0.759645i	$0.725373\pi$
$348$	−7470.49	−1.15075
$349$	6083.47	0.933068	0.466534	−	0.884503i	$-0.345503\pi$
$349$	6083.47	0.933068	0.466534	+	0.884503i	$0.345503\pi$
$350$	0	0
$351$	4950.91	0.752878
$352$	−840.604	−0.127285
$353$	11635.2	1.75433	0.877166	−	0.480188i	$-0.159431\pi$
$353$	11635.2	1.75433	0.877166	+	0.480188i	$0.159431\pi$
$354$	6557.87	0.984596
$355$	0	0
$356$	−6623.48	−0.986079
$357$	0	0
$358$	99.5889	0.0147023
$359$	−1678.34	−0.246739	−0.123370	−	0.992361i	$-0.539370\pi$
$359$	−1678.34	−0.246739	−0.123370	+	0.992361i	$0.539370\pi$
$360$	0	0
$361$	3099.70	0.451917
$362$	−2001.19	−0.290552
$363$	−5345.49	−0.772908
$364$	0	0
$365$	0	0
$366$	8411.87	1.20135
$367$	−6817.74	−0.969709	−0.484854	−	0.874595i	$-0.661127\pi$
$367$	−6817.74	−0.969709	−0.484854	+	0.874595i	$0.661127\pi$
$368$	1628.44	0.230675
$369$	4039.18	0.569842
$370$	0	0
$371$	0	0
$372$	8918.02	1.24295
$373$	−13277.9	−1.84318	−0.921588	−	0.388170i	$-0.873108\pi$
$373$	−13277.9	−1.84318	−0.921588	+	0.388170i	$0.873108\pi$
$374$	125.739	0.0173845
$375$	0	0
$376$	2737.38	0.375452
$377$	−8545.88	−1.16747
$378$	0	0
$379$	11844.1	1.60525	0.802623	−	0.596487i	$-0.203437\pi$
$379$	11844.1	1.60525	0.802623	+	0.596487i	$0.203437\pi$
$380$	0	0
$381$	14846.0	1.99628
$382$	−1995.61	−0.267289
$383$	−6883.48	−0.918354	−0.459177	−	0.888345i	$-0.651856\pi$
$383$	−6883.48	−0.918354	−0.459177	+	0.888345i	$0.651856\pi$
$384$	1067.52	0.141866
$385$	0	0
$386$	5126.12	0.675939
$387$	17281.5	2.26995
$388$	23.0492	0.00301584
$389$	−4177.03	−0.544431	−0.272215	−	0.962236i	$-0.587756\pi$
$389$	−4177.03	−0.544431	−0.272215	+	0.962236i	$0.587756\pi$
$390$	0	0
$391$	−243.585	−0.0315055
$392$	0	0
$393$	−24526.9	−3.14814
$394$	1642.45	0.210013
$395$	0	0
$396$	−4471.55	−0.567433
$397$	5884.29	0.743889	0.371944	−	0.928255i	$-0.378691\pi$
$397$	5884.29	0.743889	0.371944	+	0.928255i	$0.378691\pi$
$398$	6472.52	0.815172
$399$	0	0
$400$	0	0
$401$	−9397.84	−1.17034	−0.585169	−	0.810911i	$-0.698972\pi$
$401$	−9397.84	−1.17034	−0.585169	+	0.810911i	$0.698972\pi$
$402$	916.508	0.113710
$403$	10201.8	1.26101
$404$	−4619.92	−0.568934
$405$	0	0
$406$	0	0
$407$	−9543.13	−1.16225
$408$	−159.681	−0.0193760
$409$	8324.14	1.00636	0.503181	−	0.864181i	$-0.332163\pi$
$409$	8324.14	1.00636	0.503181	+	0.864181i	$0.332163\pi$
$410$	0	0
$411$	−13797.3	−1.65589
$412$	6543.24	0.782433
$413$	0	0
$414$	8662.42	1.02834
$415$	0	0
$416$	1221.19	0.143928
$417$	−3418.42	−0.401440
$418$	5242.91	0.613491
$419$	7948.44	0.926746	0.463373	−	0.886163i	$-0.346639\pi$
$419$	7948.44	0.926746	0.463373	+	0.886163i	$0.346639\pi$
$420$	0	0
$421$	−7628.47	−0.883109	−0.441554	−	0.897235i	$-0.645573\pi$
$421$	−7628.47	−0.883109	−0.441554	+	0.897235i	$0.645573\pi$
$422$	−6329.85	−0.730172
$423$	14561.4	1.67375
$424$	4349.22	0.498153
$425$	0	0
$426$	14833.6	1.68706
$427$	0	0
$428$	−2655.43	−0.299895
$429$	−8360.67	−0.940926
$430$	0	0
$431$	6992.39	0.781465	0.390733	−	0.920504i	$-0.372222\pi$
$431$	6992.39	0.781465	0.390733	+	0.920504i	$0.372222\pi$
$432$	2075.73	0.231178
$433$	−10000.3	−1.10989	−0.554947	−	0.831886i	$-0.687261\pi$
$433$	−10000.3	−1.10989	−0.554947	+	0.831886i	$0.687261\pi$
$434$	0	0
$435$	0	0
$436$	−7224.02	−0.793505
$437$	−10156.7	−1.11181
$438$	−8518.51	−0.929292
$439$	−6730.83	−0.731765	−0.365882	−	0.930661i	$-0.619233\pi$
$439$	−6730.83	−0.731765	−0.365882	+	0.930661i	$0.619233\pi$
$440$	0	0
$441$	0	0
$442$	−182.668	−0.0196575
$443$	7006.11	0.751400	0.375700	−	0.926741i	$-0.377402\pi$
$443$	7006.11	0.751400	0.375700	+	0.926741i	$0.377402\pi$
$444$	12119.2	1.29539
$445$	0	0
$446$	−9395.26	−0.997486
$447$	−24846.4	−2.62907
$448$	0	0
$449$	5307.59	0.557863	0.278932	−	0.960311i	$-0.410020\pi$
$449$	5307.59	0.557863	0.278932	+	0.960311i	$0.410020\pi$
$450$	0	0
$451$	−2493.32	−0.260324
$452$	1793.63	0.186648
$453$	−16468.4	−1.70806
$454$	1487.88	0.153810
$455$	0	0
$456$	−6658.21	−0.683770
$457$	−11959.8	−1.22419	−0.612095	−	0.790784i	$-0.709673\pi$
$457$	−11959.8	−1.22419	−0.612095	+	0.790784i	$0.709673\pi$
$458$	1914.27	0.195301
$459$	−310.492	−0.0315741
$460$	0	0
$461$	3761.36	0.380009	0.190004	−	0.981783i	$-0.439150\pi$
$461$	3761.36	0.380009	0.190004	+	0.981783i	$0.439150\pi$
$462$	0	0
$463$	17527.2	1.75930	0.879652	−	0.475617i	$-0.157775\pi$
$463$	17527.2	1.75930	0.879652	+	0.475617i	$0.157775\pi$
$464$	−3582.97	−0.358481
$465$	0	0
$466$	1715.84	0.170568
$467$	−10196.3	−1.01034	−0.505168	−	0.863021i	$-0.668570\pi$
$467$	−10196.3	−1.01034	−0.505168	+	0.863021i	$0.668570\pi$
$468$	6496.06	0.641625
$469$	0	0
$470$	0	0
$471$	5848.83	0.572186
$472$	3145.26	0.306721
$473$	−10667.6	−1.03699
$474$	4566.24	0.442478
$475$	0	0
$476$	0	0
$477$	23135.5	2.22075
$478$	−3629.48	−0.347299
$479$	−17699.0	−1.68829	−0.844143	−	0.536119i	$-0.819890\pi$
$479$	−17699.0	−1.68829	−0.844143	+	0.536119i	$0.819890\pi$
$480$	0	0
$481$	13863.8	1.31421
$482$	7362.19	0.695723
$483$	0	0
$484$	−2563.79	−0.240776
$485$	0	0
$486$	−8123.57	−0.758215
$487$	−9879.80	−0.919295	−0.459647	−	0.888101i	$-0.652024\pi$
$487$	−9879.80	−0.919295	−0.459647	+	0.888101i	$0.652024\pi$
$488$	4034.47	0.374246
$489$	−7457.00	−0.689606
$490$	0	0
$491$	−13584.5	−1.24860	−0.624299	−	0.781185i	$-0.714615\pi$
$491$	−13584.5	−1.24860	−0.624299	+	0.781185i	$0.714615\pi$
$492$	3166.38	0.290145
$493$	535.946	0.0489611
$494$	−7616.67	−0.693705
$495$	0	0
$496$	4277.23	0.387204
$497$	0	0
$498$	8760.14	0.788255
$499$	11648.3	1.04499	0.522495	−	0.852642i	$-0.325001\pi$
$499$	11648.3	1.04499	0.522495	+	0.852642i	$0.325001\pi$
$500$	0	0
$501$	−8652.30	−0.771570
$502$	5176.83	0.460265
$503$	−14501.5	−1.28547	−0.642733	−	0.766091i	$-0.722199\pi$
$503$	−14501.5	−1.28547	−0.642733	+	0.766091i	$0.722199\pi$
$504$	0	0
$505$	0	0
$506$	−5347.17	−0.469784
$507$	−6176.96	−0.541081
$508$	7120.37	0.621881
$509$	2370.52	0.206428	0.103214	−	0.994659i	$-0.467087\pi$
$509$	2370.52	0.206428	0.103214	+	0.994659i	$0.467087\pi$
$510$	0	0
$511$	0	0
$512$	512.000	0.0441942
$513$	−12946.5	−1.11423
$514$	2223.09	0.190771
$515$	0	0
$516$	13547.3	1.15579
$517$	−8988.50	−0.764630
$518$	0	0
$519$	15321.2	1.29581
$520$	0	0
$521$	6074.36	0.510792	0.255396	−	0.966837i	$-0.417794\pi$
$521$	6074.36	0.510792	0.255396	+	0.966837i	$0.417794\pi$
$522$	−19059.4	−1.59810
$523$	1648.81	0.137853	0.0689266	−	0.997622i	$-0.478043\pi$
$523$	1648.81	0.137853	0.0689266	+	0.997622i	$0.478043\pi$
$524$	−11763.5	−0.980709
$525$	0	0
$526$	−5485.81	−0.454739
$527$	−639.795	−0.0528840
$528$	−3505.32	−0.288919
$529$	−1808.28	−0.148622
$530$	0	0
$531$	16731.1	1.36736
$532$	0	0
$533$	3622.19	0.294361
$534$	−27619.9	−2.23826
$535$	0	0
$536$	439.572	0.0354228
$537$	415.286	0.0333723
$538$	3850.29	0.308546
$539$	0	0
$540$	0	0
$541$	13758.5	1.09339	0.546697	−	0.837330i	$-0.315885\pi$
$541$	13758.5	1.09339	0.546697	+	0.837330i	$0.315885\pi$
$542$	573.190	0.0454255
$543$	−8344.94	−0.659513
$544$	−76.5858	−0.00603601
$545$	0	0
$546$	0	0
$547$	4368.90	0.341501	0.170750	−	0.985314i	$-0.445381\pi$
$547$	4368.90	0.341501	0.170750	+	0.985314i	$0.445381\pi$
$548$	−6617.42	−0.515843
$549$	21461.1	1.66838
$550$	0	0
$551$	22347.3	1.72781
$552$	6790.61	0.523601
$553$	0	0
$554$	−2877.75	−0.220693
$555$	0	0
$556$	−1639.53	−0.125057
$557$	10297.2	0.783313	0.391657	−	0.920111i	$-0.371902\pi$
$557$	10297.2	0.783313	0.391657	+	0.920111i	$0.371902\pi$
$558$	22752.5	1.72614
$559$	15497.4	1.17258
$560$	0	0
$561$	524.331	0.0394604
$562$	−5318.29	−0.399179
$563$	−14168.9	−1.06065	−0.530327	−	0.847793i	$-0.677931\pi$
$563$	−14168.9	−1.06065	−0.530327	+	0.847793i	$0.677931\pi$
$564$	11414.9	0.852223
$565$	0	0
$566$	666.290	0.0494811
$567$	0	0
$568$	7114.42	0.525553
$569$	−23967.1	−1.76582	−0.882912	−	0.469538i	$-0.844421\pi$
$569$	−23967.1	−1.76582	−0.882912	+	0.469538i	$0.844421\pi$
$570$	0	0
$571$	20705.2	1.51749	0.758745	−	0.651388i	$-0.225813\pi$
$571$	20705.2	1.51749	0.758745	+	0.651388i	$0.225813\pi$
$572$	−4009.92	−0.293117
$573$	−8321.71	−0.606709
$574$	0	0
$575$	0	0
$576$	2723.56	0.197016
$577$	1515.22	0.109323	0.0546616	−	0.998505i	$-0.482592\pi$
$577$	1515.22	0.109323	0.0546616	+	0.998505i	$0.482592\pi$
$578$	−9814.54	−0.706282
$579$	21375.9	1.53429
$580$	0	0
$581$	0	0
$582$	96.1151	0.00684553
$583$	−14281.2	−1.01452
$584$	−4085.62	−0.289493
$585$	0	0
$586$	15350.5	1.08212
$587$	25148.2	1.76828	0.884138	−	0.467226i	$-0.154746\pi$
$587$	25148.2	1.76828	0.884138	+	0.467226i	$0.154746\pi$
$588$	0	0
$589$	−26677.4	−1.86625
$590$	0	0
$591$	6849.00	0.476701
$592$	5812.59	0.403540
$593$	−15709.8	−1.08790	−0.543948	−	0.839119i	$-0.683071\pi$
$593$	−15709.8	−1.08790	−0.543948	+	0.839119i	$0.683071\pi$
$594$	−6815.89	−0.470807
$595$	0	0
$596$	−11916.7	−0.819008
$597$	26990.4	1.85033
$598$	7768.14	0.531209
$599$	−10487.8	−0.715393	−0.357696	−	0.933838i	$-0.616438\pi$
$599$	−10487.8	−0.715393	−0.357696	+	0.933838i	$0.616438\pi$
$600$	0	0
$601$	−19266.8	−1.30767	−0.653835	−	0.756637i	$-0.726841\pi$
$601$	−19266.8	−1.30767	−0.653835	+	0.756637i	$0.726841\pi$
$602$	0	0
$603$	2338.28	0.157914
$604$	−7898.50	−0.532095
$605$	0	0
$606$	−19265.1	−1.29140
$607$	−6981.75	−0.466854	−0.233427	−	0.972374i	$-0.574994\pi$
$607$	−6981.75	−0.466854	−0.233427	+	0.972374i	$0.574994\pi$
$608$	−3193.38	−0.213008
$609$	0	0
$610$	0	0
$611$	13058.1	0.864606
$612$	−407.394	−0.0269084
$613$	663.745	0.0437332	0.0218666	−	0.999761i	$-0.493039\pi$
$613$	663.745	0.0437332	0.0218666	+	0.999761i	$0.493039\pi$
$614$	7666.05	0.503870
$615$	0	0
$616$	0	0
$617$	7749.41	0.505640	0.252820	−	0.967513i	$-0.418642\pi$
$617$	7749.41	0.505640	0.252820	+	0.967513i	$0.418642\pi$
$618$	27285.3	1.77601
$619$	16431.9	1.06697	0.533483	−	0.845811i	$-0.320883\pi$
$619$	16431.9	1.06697	0.533483	+	0.845811i	$0.320883\pi$
$620$	0	0
$621$	13204.0	0.853232
$622$	−883.732	−0.0569685
$623$	0	0
$624$	5092.37	0.326695
$625$	0	0
$626$	16652.1	1.06318
$627$	21862.9	1.39254
$628$	2805.19	0.178247
$629$	−869.456	−0.0551153
$630$	0	0
$631$	14436.7	0.910804	0.455402	−	0.890286i	$-0.349496\pi$
$631$	14436.7	0.910804	0.455402	+	0.890286i	$0.349496\pi$
$632$	2190.05	0.137841
$633$	−26395.5	−1.65739
$634$	−6936.22	−0.434499
$635$	0	0
$636$	18136.3	1.13074
$637$	0	0
$638$	11765.1	0.730068
$639$	37844.7	2.34290
$640$	0	0
$641$	14889.2	0.917452	0.458726	−	0.888578i	$-0.348306\pi$
$641$	14889.2	0.917452	0.458726	+	0.888578i	$0.348306\pi$
$642$	−11073.1	−0.680720
$643$	26252.6	1.61011	0.805057	−	0.593198i	$-0.202135\pi$
$643$	26252.6	1.61011	0.805057	+	0.593198i	$0.202135\pi$
$644$	0	0
$645$	0	0
$646$	477.672	0.0290925
$647$	−7059.38	−0.428954	−0.214477	−	0.976729i	$-0.568805\pi$
$647$	−7059.38	−0.428954	−0.214477	+	0.976729i	$0.568805\pi$
$648$	−536.197	−0.0325059
$649$	−10327.8	−0.624657
$650$	0	0
$651$	0	0
$652$	−3576.50	−0.214826
$653$	−10720.3	−0.642446	−0.321223	−	0.947004i	$-0.604094\pi$
$653$	−10720.3	−0.642446	−0.321223	+	0.947004i	$0.604094\pi$
$654$	−30124.2	−1.80114
$655$	0	0
$656$	1518.65	0.0903861
$657$	−21733.2	−1.29055
$658$	0	0
$659$	5021.59	0.296834	0.148417	−	0.988925i	$-0.452582\pi$
$659$	5021.59	0.296834	0.148417	+	0.988925i	$0.452582\pi$
$660$	0	0
$661$	6338.74	0.372993	0.186497	−	0.982456i	$-0.440287\pi$
$661$	6338.74	0.372993	0.186497	+	0.982456i	$0.440287\pi$
$662$	−21937.5	−1.28795
$663$	−761.725	−0.0446198
$664$	4201.51	0.245557
$665$	0	0
$666$	30919.7	1.79897
$667$	−22791.7	−1.32308
$668$	−4149.79	−0.240359
$669$	−39178.2	−2.26415
$670$	0	0
$671$	−13247.6	−0.762174
$672$	0	0
$673$	−6498.81	−0.372230	−0.186115	−	0.982528i	$-0.559590\pi$
$673$	−6498.81	−0.372230	−0.186115	+	0.982528i	$0.559590\pi$
$674$	−8634.67	−0.493465
$675$	0	0
$676$	−2962.57	−0.168558
$677$	−892.599	−0.0506726	−0.0253363	−	0.999679i	$-0.508066\pi$
$677$	−892.599	−0.0506726	−0.0253363	+	0.999679i	$0.508066\pi$
$678$	7479.41	0.423665
$679$	0	0
$680$	0	0
$681$	6204.48	0.349128
$682$	−14044.7	−0.788564
$683$	20671.3	1.15808	0.579038	−	0.815301i	$-0.303428\pi$
$683$	20671.3	1.15808	0.579038	+	0.815301i	$0.303428\pi$
$684$	−16987.0	−0.949584
$685$	0	0
$686$	0	0
$687$	7982.50	0.443306
$688$	6497.49	0.360050
$689$	20747.0	1.14717
$690$	0	0
$691$	−11503.8	−0.633323	−0.316662	−	0.948539i	$-0.602562\pi$
$691$	−11503.8	−0.633323	−0.316662	+	0.948539i	$0.602562\pi$
$692$	7348.31	0.403672
$693$	0	0
$694$	−16814.9	−0.919717
$695$	0	0
$696$	−14941.0	−0.813701
$697$	−227.162	−0.0123449
$698$	12166.9	0.659778
$699$	7155.05	0.387165
$700$	0	0
$701$	9731.33	0.524318	0.262159	−	0.965025i	$-0.415565\pi$
$701$	9731.33	0.524318	0.262159	+	0.965025i	$0.415565\pi$
$702$	9901.83	0.532365
$703$	−36253.6	−1.94499
$704$	−1681.21	−0.0900041
$705$	0	0
$706$	23270.4	1.24050
$707$	0	0
$708$	13115.7	0.696214
$709$	4362.21	0.231067	0.115533	−	0.993304i	$-0.463142\pi$
$709$	4362.21	0.231067	0.115533	+	0.993304i	$0.463142\pi$
$710$	0	0
$711$	11649.8	0.614490
$712$	−13247.0	−0.697263
$713$	27207.9	1.42909
$714$	0	0
$715$	0	0
$716$	199.178	0.0103961
$717$	−15134.9	−0.788319
$718$	−3356.68	−0.174471
$719$	−25148.1	−1.30440	−0.652201	−	0.758046i	$-0.726154\pi$
$719$	−25148.1	−1.30440	−0.652201	+	0.758046i	$0.726154\pi$
$720$	0	0
$721$	0	0
$722$	6199.40	0.319554
$723$	30700.3	1.57919
$724$	−4002.37	−0.205452
$725$	0	0
$726$	−10691.0	−0.546528
$727$	−12965.7	−0.661446	−0.330723	−	0.943728i	$-0.607293\pi$
$727$	−12965.7	−0.661446	−0.330723	+	0.943728i	$0.607293\pi$
$728$	0	0
$729$	−32065.6	−1.62910
$730$	0	0
$731$	−971.906	−0.0491754
$732$	16823.7	0.849485
$733$	−9107.05	−0.458904	−0.229452	−	0.973320i	$-0.573693\pi$
$733$	−9107.05	−0.458904	−0.229452	+	0.973320i	$0.573693\pi$
$734$	−13635.5	−0.685688
$735$	0	0
$736$	3256.89	0.163112
$737$	−1443.38	−0.0721407
$738$	8078.37	0.402939
$739$	−6526.99	−0.324898	−0.162449	−	0.986717i	$-0.551939\pi$
$739$	−6526.99	−0.324898	−0.162449	+	0.986717i	$0.551939\pi$
$740$	0	0
$741$	−31761.5	−1.57461
$742$	0	0
$743$	35778.0	1.76658	0.883290	−	0.468827i	$-0.155323\pi$
$743$	35778.0	1.76658	0.883290	+	0.468827i	$0.155323\pi$
$744$	17836.0	0.878899
$745$	0	0
$746$	−26555.8	−1.30332
$747$	22349.7	1.09469
$748$	251.478	0.0122927
$749$	0	0
$750$	0	0
$751$	29588.4	1.43768	0.718838	−	0.695177i	$-0.244674\pi$
$751$	29588.4	1.43768	0.718838	+	0.695177i	$0.244674\pi$
$752$	5474.77	0.265484
$753$	21587.4	1.04474
$754$	−17091.8	−0.825525
$755$	0	0
$756$	0	0
$757$	7623.03	0.366002	0.183001	−	0.983113i	$-0.441419\pi$
$757$	7623.03	0.366002	0.183001	+	0.983113i	$0.441419\pi$
$758$	23688.1	1.13508
$759$	−22297.7	−1.06634
$760$	0	0
$761$	−25185.6	−1.19971	−0.599853	−	0.800110i	$-0.704774\pi$
$761$	−25185.6	−1.19971	−0.599853	+	0.800110i	$0.704774\pi$
$762$	29691.9	1.41158
$763$	0	0
$764$	−3991.23	−0.189002
$765$	0	0
$766$	−13767.0	−0.649374
$767$	15003.8	0.706331
$768$	2135.04	0.100315
$769$	7545.46	0.353831	0.176916	−	0.984226i	$-0.443388\pi$
$769$	7545.46	0.353831	0.176916	+	0.984226i	$0.443388\pi$
$770$	0	0
$771$	9270.27	0.433023
$772$	10252.2	0.477961
$773$	24203.4	1.12618	0.563089	−	0.826397i	$-0.309613\pi$
$773$	24203.4	1.12618	0.563089	+	0.826397i	$0.309613\pi$
$774$	34563.1	1.60509
$775$	0	0
$776$	46.0984	0.00213252
$777$	0	0
$778$	−8354.05	−0.384971
$779$	−9471.93	−0.435645
$780$	0	0
$781$	−23361.0	−1.07032
$782$	−487.171	−0.0222777
$783$	−29051.9	−1.32596
$784$	0	0
$785$	0	0
$786$	−49053.8	−2.22607
$787$	6262.86	0.283668	0.141834	−	0.989890i	$-0.454700\pi$
$787$	6262.86	0.283668	0.141834	+	0.989890i	$0.454700\pi$
$788$	3284.89	0.148502
$789$	−22875.8	−1.03219
$790$	0	0
$791$	0	0
$792$	−8943.09	−0.401236
$793$	19245.6	0.861828
$794$	11768.6	0.526009
$795$	0	0
$796$	12945.0	0.576413
$797$	41906.4	1.86248	0.931242	−	0.364402i	$-0.118726\pi$
$797$	41906.4	1.86248	0.931242	+	0.364402i	$0.118726\pi$
$798$	0	0
$799$	−818.925	−0.0362597
$800$	0	0
$801$	−70466.5	−3.10838
$802$	−18795.7	−0.827554
$803$	13415.6	0.589570
$804$	1833.02	0.0804048
$805$	0	0
$806$	20403.6	0.891669
$807$	16055.7	0.700357
$808$	−9239.84	−0.402297
$809$	34486.5	1.49874	0.749371	−	0.662150i	$-0.230356\pi$
$809$	34486.5	1.49874	0.749371	+	0.662150i	$0.230356\pi$
$810$	0	0
$811$	−15173.9	−0.657002	−0.328501	−	0.944504i	$-0.606543\pi$
$811$	−15173.9	−0.657002	−0.328501	+	0.944504i	$0.606543\pi$
$812$	0	0
$813$	2390.20	0.103110
$814$	−19086.3	−0.821834
$815$	0	0
$816$	−319.363	−0.0137009
$817$	−40525.4	−1.73538
$818$	16648.3	0.711605
$819$	0	0
$820$	0	0
$821$	−15214.7	−0.646767	−0.323383	−	0.946268i	$-0.604820\pi$
$821$	−15214.7	−0.646767	−0.323383	+	0.946268i	$0.604820\pi$
$822$	−27594.6	−1.17089
$823$	22509.7	0.953387	0.476693	−	0.879070i	$-0.341835\pi$
$823$	22509.7	0.953387	0.476693	+	0.879070i	$0.341835\pi$
$824$	13086.5	0.553264
$825$	0	0
$826$	0	0
$827$	−32024.2	−1.34654	−0.673272	−	0.739395i	$-0.735112\pi$
$827$	−32024.2	−1.34654	−0.673272	+	0.739395i	$0.735112\pi$
$828$	17324.8	0.727150
$829$	19083.7	0.799524	0.399762	−	0.916619i	$-0.369093\pi$
$829$	19083.7	0.799524	0.399762	+	0.916619i	$0.369093\pi$
$830$	0	0
$831$	−12000.2	−0.500942
$832$	2442.38	0.101772
$833$	0	0
$834$	−6836.84	−0.283861
$835$	0	0
$836$	10485.8	0.433804
$837$	34681.2	1.43221
$838$	15896.9	0.655309
$839$	−23198.0	−0.954568	−0.477284	−	0.878749i	$-0.658379\pi$
$839$	−23198.0	−0.954568	−0.477284	+	0.878749i	$0.658379\pi$
$840$	0	0
$841$	25758.1	1.05614
$842$	−15256.9	−0.624452
$843$	−22177.3	−0.906081
$844$	−12659.7	−0.516309
$845$	0	0
$846$	29122.7	1.18352
$847$	0	0
$848$	8698.45	0.352248
$849$	2778.43	0.112315
$850$	0	0
$851$	36974.5	1.48939
$852$	29667.1	1.19293
$853$	−26109.1	−1.04802	−0.524009	−	0.851713i	$-0.675564\pi$
$853$	−26109.1	−1.04802	−0.524009	+	0.851713i	$0.675564\pi$
$854$	0	0
$855$	0	0
$856$	−5310.86	−0.212058
$857$	20965.6	0.835672	0.417836	−	0.908522i	$-0.362789\pi$
$857$	20965.6	0.835672	0.417836	+	0.908522i	$0.362789\pi$
$858$	−16721.3	−0.665335
$859$	31346.3	1.24508	0.622540	−	0.782588i	$-0.286101\pi$
$859$	31346.3	1.24508	0.622540	+	0.782588i	$0.286101\pi$
$860$	0	0
$861$	0	0
$862$	13984.8	0.552579
$863$	−14508.0	−0.572257	−0.286129	−	0.958191i	$-0.592368\pi$
$863$	−14508.0	−0.572257	−0.286129	+	0.958191i	$0.592368\pi$
$864$	4151.47	0.163467
$865$	0	0
$866$	−20000.6	−0.784814
$867$	−40926.6	−1.60316
$868$	0	0
$869$	−7191.25	−0.280721
$870$	0	0
$871$	2096.88	0.0815731
$872$	−14448.0	−0.561092
$873$	245.218	0.00950672
$874$	−20313.5	−0.786171
$875$	0	0
$876$	−17037.0	−0.657109
$877$	105.690	0.00406942	0.00203471	−	0.999998i	$-0.499352\pi$
$877$	105.690	0.00406942	0.00203471	+	0.999998i	$0.499352\pi$
$878$	−13461.7	−0.517436
$879$	64011.6	2.45627
$880$	0	0
$881$	−22941.4	−0.877316	−0.438658	−	0.898654i	$-0.644546\pi$
$881$	−22941.4	−0.877316	−0.438658	+	0.898654i	$0.644546\pi$
$882$	0	0
$883$	−25169.9	−0.959267	−0.479634	−	0.877469i	$-0.659230\pi$
$883$	−25169.9	−0.959267	−0.479634	+	0.877469i	$0.659230\pi$
$884$	−365.336	−0.0139000
$885$	0	0
$886$	14012.2	0.531320
$887$	21260.2	0.804790	0.402395	−	0.915466i	$-0.368178\pi$
$887$	21260.2	0.804790	0.402395	+	0.915466i	$0.368178\pi$
$888$	24238.5	0.915980
$889$	0	0
$890$	0	0
$891$	1760.66	0.0662002
$892$	−18790.5	−0.705329
$893$	−34146.6	−1.27959
$894$	−49692.8	−1.85903
$895$	0	0
$896$	0	0
$897$	32393.1	1.20577
$898$	10615.2	0.394469
$899$	−59863.9	−2.22088
$900$	0	0
$901$	−1301.13	−0.0481097
$902$	−4986.65	−0.184077
$903$	0	0
$904$	3587.25	0.131980
$905$	0	0
$906$	−32936.7	−1.20778
$907$	−53385.7	−1.95440	−0.977202	−	0.212314i	$-0.931900\pi$
$907$	−53385.7	−1.95440	−0.977202	+	0.212314i	$0.931900\pi$
$908$	2975.77	0.108760
$909$	−49150.8	−1.79343
$910$	0	0
$911$	−3333.00	−0.121215	−0.0606076	−	0.998162i	$-0.519304\pi$
$911$	−3333.00	−0.121215	−0.0606076	+	0.998162i	$0.519304\pi$
$912$	−13316.4	−0.483498
$913$	−13796.1	−0.500092
$914$	−23919.5	−0.865633
$915$	0	0
$916$	3828.54	0.138099
$917$	0	0
$918$	−620.983	−0.0223262
$919$	−6223.24	−0.223380	−0.111690	−	0.993743i	$-0.535626\pi$
$919$	−6223.24	−0.223380	−0.111690	+	0.993743i	$0.535626\pi$
$920$	0	0
$921$	31967.4	1.14372
$922$	7522.72	0.268707
$923$	33937.8	1.21027
$924$	0	0
$925$	0	0
$926$	35054.4	1.24402
$927$	69612.8	2.46644
$928$	−7165.94	−0.253484
$929$	−3747.97	−0.132365	−0.0661825	−	0.997808i	$-0.521082\pi$
$929$	−3747.97	−0.132365	−0.0661825	+	0.997808i	$0.521082\pi$
$930$	0	0
$931$	0	0
$932$	3431.68	0.120610
$933$	−3685.16	−0.129311
$934$	−20392.5	−0.714416
$935$	0	0
$936$	12992.1	0.453698
$937$	31970.1	1.11464	0.557320	−	0.830298i	$-0.311830\pi$
$937$	31970.1	1.11464	0.557320	+	0.830298i	$0.311830\pi$
$938$	0	0
$939$	69439.1	2.41327
$940$	0	0
$941$	39612.1	1.37228	0.686141	−	0.727469i	$-0.259303\pi$
$941$	39612.1	1.37228	0.686141	+	0.727469i	$0.259303\pi$
$942$	11697.7	0.404597
$943$	9660.29	0.333598
$944$	6290.53	0.216885
$945$	0	0
$946$	−21335.2	−0.733264
$947$	−33485.8	−1.14904	−0.574520	−	0.818490i	$-0.694811\pi$
$947$	−33485.8	−1.14904	−0.574520	+	0.818490i	$0.694811\pi$
$948$	9132.49	0.312879
$949$	−19489.5	−0.666657
$950$	0	0
$951$	−28924.0	−0.986252
$952$	0	0
$953$	2871.26	0.0975963	0.0487981	−	0.998809i	$-0.484461\pi$
$953$	2871.26	0.0975963	0.0487981	+	0.998809i	$0.484461\pi$
$954$	46270.9	1.57031
$955$	0	0
$956$	−7258.97	−0.245577
$957$	49060.3	1.65715
$958$	−35398.0	−1.19380
$959$	0	0
$960$	0	0
$961$	41672.6	1.39883
$962$	27727.7	0.929289
$963$	−28250.8	−0.945348
$964$	14724.4	0.491951
$965$	0	0
$966$	0	0
$967$	−39768.7	−1.32252	−0.661259	−	0.750158i	$-0.729977\pi$
$967$	−39768.7	−1.32252	−0.661259	+	0.750158i	$0.729977\pi$
$968$	−5127.57	−0.170255
$969$	1991.89	0.0660359
$970$	0	0
$971$	28153.0	0.930456	0.465228	−	0.885191i	$-0.345972\pi$
$971$	28153.0	0.930456	0.465228	+	0.885191i	$0.345972\pi$
$972$	−16247.1	−0.536139
$973$	0	0
$974$	−19759.6	−0.650040
$975$	0	0
$976$	8068.94	0.264632
$977$	23068.3	0.755393	0.377696	−	0.925929i	$-0.376716\pi$
$977$	23068.3	0.755393	0.377696	+	0.925929i	$0.376716\pi$
$978$	−14914.0	−0.487625
$979$	43497.9	1.42002
$980$	0	0
$981$	−76855.6	−2.50134
$982$	−27169.1	−0.882893
$983$	2597.68	0.0842860	0.0421430	−	0.999112i	$-0.486581\pi$
$983$	2597.68	0.0842860	0.0421430	+	0.999112i	$0.486581\pi$
$984$	6332.76	0.205164
$985$	0	0
$986$	1071.89	0.0346207
$987$	0	0
$988$	−15233.3	−0.490524
$989$	41331.3	1.32888
$990$	0	0
$991$	−24438.0	−0.783348	−0.391674	−	0.920104i	$-0.628104\pi$
$991$	−24438.0	−0.783348	−0.391674	+	0.920104i	$0.628104\pi$
$992$	8554.45	0.273795
$993$	−91479.3	−2.92347
$994$	0	0
$995$	0	0
$996$	17520.3	0.557381
$997$	4218.63	0.134007	0.0670036	−	0.997753i	$-0.478656\pi$
$997$	4218.63	0.134007	0.0670036	+	0.997753i	$0.478656\pi$
$998$	23296.6	0.738920
$999$	47130.4	1.49263

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2450.4.a.ch.1.3		3
5.4	even	2		2450.4.a.cd.1.1		3
7.3	odd	6		350.4.e.i.51.3	✓	6
7.5	odd	6		350.4.e.i.151.3	yes	6
7.6	odd	2		2450.4.a.ci.1.1		3
35.3	even	12		350.4.j.h.149.4		12
35.12	even	12		350.4.j.h.249.4		12
35.17	even	12		350.4.j.h.149.3		12
35.19	odd	6		350.4.e.j.151.1	yes	6
35.24	odd	6		350.4.e.j.51.1	yes	6
35.33	even	12		350.4.j.h.249.3		12
35.34	odd	2		2450.4.a.cc.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
350.4.e.i.51.3	✓	6	7.3	odd	6
350.4.e.i.151.3	yes	6	7.5	odd	6
350.4.e.j.51.1	yes	6	35.24	odd	6
350.4.e.j.151.1	yes	6	35.19	odd	6
350.4.j.h.149.3		12	35.17	even	12
350.4.j.h.149.4		12	35.3	even	12
350.4.j.h.249.3		12	35.33	even	12
350.4.j.h.249.4		12	35.12	even	12
2450.4.a.cc.1.3		3	35.34	odd	2
2450.4.a.cd.1.1		3	5.4	even	2
2450.4.a.ch.1.3		3	1.1	even	1	trivial
2450.4.a.ci.1.1		3	7.6	odd	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:	\( N \)	\(=\)	\( 2450 = 2 \cdot 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2450.a (trivial)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} +8.34000 q^{3} +4.00000 q^{4} +16.6800 q^{6} +8.00000 q^{8} +42.5556 q^{9} +O(q^{10})\)	\(q+2.00000 q^{2} +8.34000 q^{3} +4.00000 q^{4} +16.6800 q^{6} +8.00000 q^{8} +42.5556 q^{9} -26.2689 q^{11} +33.3600 q^{12} +38.1623 q^{13} +16.0000 q^{16} -2.39331 q^{17} +85.1111 q^{18} -99.7933 q^{19} -52.5377 q^{22} +101.778 q^{23} +66.7200 q^{24} +76.3245 q^{26} +129.733 q^{27} -223.936 q^{29} +267.327 q^{31} +32.0000 q^{32} -219.082 q^{33} -4.78661 q^{34} +170.222 q^{36} +363.287 q^{37} -199.587 q^{38} +318.273 q^{39} +94.9155 q^{41} +406.093 q^{43} -105.075 q^{44} +203.556 q^{46} +342.173 q^{47} +133.440 q^{48} -19.9602 q^{51} +152.649 q^{52} +543.653 q^{53} +259.467 q^{54} -832.276 q^{57} -447.871 q^{58} +393.158 q^{59} +504.309 q^{61} +534.653 q^{62} +64.0000 q^{64} -438.165 q^{66} +54.9465 q^{67} -9.57322 q^{68} +848.826 q^{69} +889.302 q^{71} +340.444 q^{72} -510.702 q^{73} +726.573 q^{74} -399.173 q^{76} +636.546 q^{78} +273.756 q^{79} -67.0246 q^{81} +189.831 q^{82} +525.188 q^{83} +812.187 q^{86} -1867.62 q^{87} -210.151 q^{88} -1655.87 q^{89} +407.111 q^{92} +2229.50 q^{93} +684.346 q^{94} +266.880 q^{96} +5.76230 q^{97} -1117.89 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 6 q^{2} - 3 q^{3} + 12 q^{4} - 6 q^{6} + 24 q^{8} + 56 q^{9}+O(q^{10}) \) 3 * q + 6 * q^2 - 3 * q^3 + 12 * q^4 - 6 * q^6 + 24 * q^8 + 56 * q^9	\( 3 q + 6 q^{2} - 3 q^{3} + 12 q^{4} - 6 q^{6} + 24 q^{8} + 56 q^{9} - 26 q^{11} - 12 q^{12} + 80 q^{13} + 48 q^{16} + 30 q^{17} + 112 q^{18} + 18 q^{19} - 52 q^{22} - 53 q^{23} - 24 q^{24} + 160 q^{26} + 324 q^{27} - 404 q^{29} + 615 q^{31} + 96 q^{32} - 314 q^{33} + 60 q^{34} + 224 q^{36} + 426 q^{37} + 36 q^{38} - 90 q^{39} + 101 q^{41} - 249 q^{43} - 104 q^{44} - 106 q^{46} + 402 q^{47} - 48 q^{48} - 339 q^{51} + 320 q^{52} + 390 q^{53} + 648 q^{54} - 1667 q^{57} - 808 q^{58} + 91 q^{59} + 647 q^{61} + 1230 q^{62} + 192 q^{64} - 628 q^{66} + 708 q^{67} + 120 q^{68} + 1548 q^{69} + 533 q^{71} + 448 q^{72} + 772 q^{73} + 852 q^{74} + 72 q^{76} - 180 q^{78} + 1421 q^{79} - 1393 q^{81} + 202 q^{82} - 520 q^{83} - 498 q^{86} - 833 q^{87} - 208 q^{88} + 194 q^{89} - 212 q^{92} + 148 q^{93} + 804 q^{94} - 96 q^{96} - 1027 q^{97} - 57 q^{99}+O(q^{100}) \) 3 * q + 6 * q^2 - 3 * q^3 + 12 * q^4 - 6 * q^6 + 24 * q^8 + 56 * q^9 - 26 * q^11 - 12 * q^12 + 80 * q^13 + 48 * q^16 + 30 * q^17 + 112 * q^18 + 18 * q^19 - 52 * q^22 - 53 * q^23 - 24 * q^24 + 160 * q^26 + 324 * q^27 - 404 * q^29 + 615 * q^31 + 96 * q^32 - 314 * q^33 + 60 * q^34 + 224 * q^36 + 426 * q^37 + 36 * q^38 - 90 * q^39 + 101 * q^41 - 249 * q^43 - 104 * q^44 - 106 * q^46 + 402 * q^47 - 48 * q^48 - 339 * q^51 + 320 * q^52 + 390 * q^53 + 648 * q^54 - 1667 * q^57 - 808 * q^58 + 91 * q^59 + 647 * q^61 + 1230 * q^62 + 192 * q^64 - 628 * q^66 + 708 * q^67 + 120 * q^68 + 1548 * q^69 + 533 * q^71 + 448 * q^72 + 772 * q^73 + 852 * q^74 + 72 * q^76 - 180 * q^78 + 1421 * q^79 - 1393 * q^81 + 202 * q^82 - 520 * q^83 - 498 * q^86 - 833 * q^87 - 208 * q^88 + 194 * q^89 - 212 * q^92 + 148 * q^93 + 804 * q^94 - 96 * q^96 - 1027 * q^97 - 57 * q^99

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