LMFDB - Embedded newform 4014.2.a.r.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4014,2,Mod(1,4014)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4014, 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("4014.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4014 = 2 \cdot 3^{2} \cdot 223 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4014.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:	$32.0519513713$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	5.5.356173.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 7x^{3} + 9x^{2} + 14x - 4 $ x^5 - 2x^4 - 7x^3 + 9x^2 + 14x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1338)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$0.253142$ of defining polynomial
Character	$\chi$	$=$	4014.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} -0.686428 q^{5} +2.87549 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} -0.686428 q^{5} +2.87549 q^{7} +1.00000 q^{8} -0.686428 q^{10} -5.51891 q^{11} +0.329853 q^{13} +2.87549 q^{14} +1.00000 q^{16} -4.44220 q^{17} -1.53786 q^{19} -0.686428 q^{20} -5.51891 q^{22} +3.17011 q^{23} -4.52882 q^{25} +0.329853 q^{26} +2.87549 q^{28} -7.03297 q^{29} -7.43855 q^{31} +1.00000 q^{32} -4.44220 q^{34} -1.97382 q^{35} -7.58613 q^{37} -1.53786 q^{38} -0.686428 q^{40} +2.51679 q^{41} +6.97369 q^{43} -5.51891 q^{44} +3.17011 q^{46} -6.00990 q^{47} +1.26844 q^{49} -4.52882 q^{50} +0.329853 q^{52} +0.840259 q^{53} +3.78834 q^{55} +2.87549 q^{56} -7.03297 q^{58} +3.14631 q^{59} -2.68741 q^{61} -7.43855 q^{62} +1.00000 q^{64} -0.226420 q^{65} -13.9815 q^{67} -4.44220 q^{68} -1.97382 q^{70} +6.67275 q^{71} +11.9575 q^{73} -7.58613 q^{74} -1.53786 q^{76} -15.8696 q^{77} +12.1543 q^{79} -0.686428 q^{80} +2.51679 q^{82} +5.22743 q^{83} +3.04925 q^{85} +6.97369 q^{86} -5.51891 q^{88} -8.29874 q^{89} +0.948489 q^{91} +3.17011 q^{92} -6.00990 q^{94} +1.05563 q^{95} +4.45112 q^{97} +1.26844 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 5 q^{2} + 5 q^{4} - 5 q^{5} - q^{7} + 5 q^{8}+O(q^{10}) $ 5 * q + 5 * q^2 + 5 * q^4 - 5 * q^5 - q^7 + 5 * q^8	$ 5 q + 5 q^{2} + 5 q^{4} - 5 q^{5} - q^{7} + 5 q^{8} - 5 q^{10} - 9 q^{11} - q^{14} + 5 q^{16} - 6 q^{17} - 4 q^{19} - 5 q^{20} - 9 q^{22} - 16 q^{23} + 8 q^{25} - q^{28} - 8 q^{29} - q^{31} + 5 q^{32} - 6 q^{34} - 22 q^{35} - 2 q^{37} - 4 q^{38} - 5 q^{40} - 4 q^{41} + 3 q^{43} - 9 q^{44} - 16 q^{46} - 18 q^{47} + 2 q^{49} + 8 q^{50} - 26 q^{53} + q^{55} - q^{56} - 8 q^{58} - 21 q^{59} - 20 q^{61} - q^{62} + 5 q^{64} + 3 q^{65} - 5 q^{67} - 6 q^{68} - 22 q^{70} - 17 q^{71} + 5 q^{73} - 2 q^{74} - 4 q^{76} - 2 q^{77} - 21 q^{79} - 5 q^{80} - 4 q^{82} - 11 q^{83} - 12 q^{85} + 3 q^{86} - 9 q^{88} + 5 q^{89} - 10 q^{91} - 16 q^{92} - 18 q^{94} - 10 q^{95} - 11 q^{97} + 2 q^{98}+O(q^{100}) $ 5 * q + 5 * q^2 + 5 * q^4 - 5 * q^5 - q^7 + 5 * q^8 - 5 * q^10 - 9 * q^11 - q^14 + 5 * q^16 - 6 * q^17 - 4 * q^19 - 5 * q^20 - 9 * q^22 - 16 * q^23 + 8 * q^25 - q^28 - 8 * q^29 - q^31 + 5 * q^32 - 6 * q^34 - 22 * q^35 - 2 * q^37 - 4 * q^38 - 5 * q^40 - 4 * q^41 + 3 * q^43 - 9 * q^44 - 16 * q^46 - 18 * q^47 + 2 * q^49 + 8 * q^50 - 26 * q^53 + q^55 - q^56 - 8 * q^58 - 21 * q^59 - 20 * q^61 - q^62 + 5 * q^64 + 3 * q^65 - 5 * q^67 - 6 * q^68 - 22 * q^70 - 17 * q^71 + 5 * q^73 - 2 * q^74 - 4 * q^76 - 2 * q^77 - 21 * q^79 - 5 * q^80 - 4 * q^82 - 11 * q^83 - 12 * q^85 + 3 * q^86 - 9 * q^88 + 5 * q^89 - 10 * q^91 - 16 * q^92 - 18 * q^94 - 10 * q^95 - 11 * q^97 + 2 * q^98

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−0.686428	−0.306980	−0.153490	−	0.988150i	$-0.549051\pi$
$5$	−0.686428	−0.306980	−0.153490	+	0.988150i	$0.549051\pi$
$6$	0	0
$7$	2.87549	1.08683	0.543416	−	0.839463i	$-0.317130\pi$
$7$	2.87549	1.08683	0.543416	+	0.839463i	$0.317130\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	−0.686428	−0.217068
$11$	−5.51891	−1.66402	−0.832008	−	0.554764i	$-0.812808\pi$
$11$	−5.51891	−1.66402	−0.832008	+	0.554764i	$0.812808\pi$
$12$	0	0
$13$	0.329853	0.0914848	0.0457424	−	0.998953i	$-0.485435\pi$
$13$	0.329853	0.0914848	0.0457424	+	0.998953i	$0.485435\pi$
$14$	2.87549	0.768507
$15$	0	0
$16$	1.00000	0.250000
$17$	−4.44220	−1.07739	−0.538696	−	0.842500i	$-0.681083\pi$
$17$	−4.44220	−1.07739	−0.538696	+	0.842500i	$0.681083\pi$
$18$	0	0
$19$	−1.53786	−0.352810	−0.176405	−	0.984318i	$-0.556447\pi$
$19$	−1.53786	−0.352810	−0.176405	+	0.984318i	$0.556447\pi$
$20$	−0.686428	−0.153490
$21$	0	0
$22$	−5.51891	−1.17664
$23$	3.17011	0.661014	0.330507	−	0.943804i	$-0.392780\pi$
$23$	3.17011	0.661014	0.330507	+	0.943804i	$0.392780\pi$
$24$	0	0
$25$	−4.52882	−0.905763
$26$	0.329853	0.0646895
$27$	0	0
$28$	2.87549	0.543416
$29$	−7.03297	−1.30599	−0.652995	−	0.757362i	$-0.726488\pi$
$29$	−7.03297	−1.30599	−0.652995	+	0.757362i	$0.726488\pi$
$30$	0	0
$31$	−7.43855	−1.33600	−0.668002	−	0.744160i	$-0.732850\pi$
$31$	−7.43855	−1.33600	−0.668002	+	0.744160i	$0.732850\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−4.44220	−0.761832
$35$	−1.97382	−0.333636
$36$	0	0
$37$	−7.58613	−1.24715	−0.623576	−	0.781762i	$-0.714321\pi$
$37$	−7.58613	−1.24715	−0.623576	+	0.781762i	$0.714321\pi$
$38$	−1.53786	−0.249474
$39$	0	0
$40$	−0.686428	−0.108534
$41$	2.51679	0.393056	0.196528	−	0.980498i	$-0.437033\pi$
$41$	2.51679	0.393056	0.196528	+	0.980498i	$0.437033\pi$
$42$	0	0
$43$	6.97369	1.06348	0.531739	−	0.846908i	$-0.321539\pi$
$43$	6.97369	1.06348	0.531739	+	0.846908i	$0.321539\pi$
$44$	−5.51891	−0.832008
$45$	0	0
$46$	3.17011	0.467408
$47$	−6.00990	−0.876634	−0.438317	−	0.898820i	$-0.644425\pi$
$47$	−6.00990	−0.876634	−0.438317	+	0.898820i	$0.644425\pi$
$48$	0	0
$49$	1.26844	0.181206
$50$	−4.52882	−0.640471
$51$	0	0
$52$	0.329853	0.0457424
$53$	0.840259	0.115419	0.0577093	−	0.998333i	$-0.481620\pi$
$53$	0.840259	0.115419	0.0577093	+	0.998333i	$0.481620\pi$
$54$	0	0
$55$	3.78834	0.510819
$56$	2.87549	0.384253
$57$	0	0
$58$	−7.03297	−0.923475
$59$	3.14631	0.409614	0.204807	−	0.978802i	$-0.434343\pi$
$59$	3.14631	0.409614	0.204807	+	0.978802i	$0.434343\pi$
$60$	0	0
$61$	−2.68741	−0.344088	−0.172044	−	0.985089i	$-0.555037\pi$
$61$	−2.68741	−0.344088	−0.172044	+	0.985089i	$0.555037\pi$
$62$	−7.43855	−0.944697
$63$	0	0
$64$	1.00000	0.125000
$65$	−0.226420	−0.0280840
$66$	0	0
$67$	−13.9815	−1.70811	−0.854057	−	0.520179i	$-0.825865\pi$
$67$	−13.9815	−1.70811	−0.854057	+	0.520179i	$0.825865\pi$
$68$	−4.44220	−0.538696
$69$	0	0
$70$	−1.97382	−0.235916
$71$	6.67275	0.791909	0.395955	−	0.918270i	$-0.370414\pi$
$71$	6.67275	0.791909	0.395955	+	0.918270i	$0.370414\pi$
$72$	0	0
$73$	11.9575	1.39952	0.699758	−	0.714380i	$-0.253291\pi$
$73$	11.9575	1.39952	0.699758	+	0.714380i	$0.253291\pi$
$74$	−7.58613	−0.881870
$75$	0	0
$76$	−1.53786	−0.176405
$77$	−15.8696	−1.80851
$78$	0	0
$79$	12.1543	1.36747	0.683733	−	0.729732i	$-0.260355\pi$
$79$	12.1543	1.36747	0.683733	+	0.729732i	$0.260355\pi$
$80$	−0.686428	−0.0767450
$81$	0	0
$82$	2.51679	0.277932
$83$	5.22743	0.573785	0.286892	−	0.957963i	$-0.407378\pi$
$83$	5.22743	0.573785	0.286892	+	0.957963i	$0.407378\pi$
$84$	0	0
$85$	3.04925	0.330738
$86$	6.97369	0.751992
$87$	0	0
$88$	−5.51891	−0.588318
$89$	−8.29874	−0.879665	−0.439833	−	0.898080i	$-0.644962\pi$
$89$	−8.29874	−0.879665	−0.439833	+	0.898080i	$0.644962\pi$
$90$	0	0
$91$	0.948489	0.0994286
$92$	3.17011	0.330507
$93$	0	0
$94$	−6.00990	−0.619874
$95$	1.05563	0.108306
$96$	0	0
$97$	4.45112	0.451943	0.225971	−	0.974134i	$-0.427444\pi$
$97$	4.45112	0.451943	0.225971	+	0.974134i	$0.427444\pi$
$98$	1.26844	0.128132
$99$	0	0
$100$	−4.52882	−0.452882
$101$	3.23632	0.322026	0.161013	−	0.986952i	$-0.448524\pi$
$101$	3.23632	0.322026	0.161013	+	0.986952i	$0.448524\pi$
$102$	0	0
$103$	−12.7274	−1.25407	−0.627034	−	0.778992i	$-0.715731\pi$
$103$	−12.7274	−1.25407	−0.627034	+	0.778992i	$0.715731\pi$
$104$	0.329853	0.0323447
$105$	0	0
$106$	0.840259	0.0816132
$107$	−4.52529	−0.437477	−0.218738	−	0.975784i	$-0.570194\pi$
$107$	−4.52529	−0.437477	−0.218738	+	0.975784i	$0.570194\pi$
$108$	0	0
$109$	−10.5371	−1.00927	−0.504637	−	0.863332i	$-0.668374\pi$
$109$	−10.5371	−1.00927	−0.504637	+	0.863332i	$0.668374\pi$
$110$	3.78834	0.361204
$111$	0	0
$112$	2.87549	0.271708
$113$	−2.93690	−0.276281	−0.138140	−	0.990413i	$-0.544112\pi$
$113$	−2.93690	−0.276281	−0.138140	+	0.990413i	$0.544112\pi$
$114$	0	0
$115$	−2.17605	−0.202918
$116$	−7.03297	−0.652995
$117$	0	0
$118$	3.14631	0.289641
$119$	−12.7735	−1.17095
$120$	0	0
$121$	19.4584	1.76895
$122$	−2.68741	−0.243307
$123$	0	0
$124$	−7.43855	−0.668002
$125$	6.54085	0.585031
$126$	0	0
$127$	4.04306	0.358764	0.179382	−	0.983780i	$-0.442590\pi$
$127$	4.04306	0.358764	0.179382	+	0.983780i	$0.442590\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−0.226420	−0.0198584
$131$	−6.64749	−0.580794	−0.290397	−	0.956906i	$-0.593787\pi$
$131$	−6.64749	−0.580794	−0.290397	+	0.956906i	$0.593787\pi$
$132$	0	0
$133$	−4.42211	−0.383446
$134$	−13.9815	−1.20782
$135$	0	0
$136$	−4.44220	−0.380916
$137$	−10.9737	−0.937545	−0.468773	−	0.883319i	$-0.655304\pi$
$137$	−10.9737	−0.937545	−0.468773	+	0.883319i	$0.655304\pi$
$138$	0	0
$139$	−15.2142	−1.29045	−0.645225	−	0.763992i	$-0.723237\pi$
$139$	−15.2142	−1.29045	−0.645225	+	0.763992i	$0.723237\pi$
$140$	−1.97382	−0.166818
$141$	0	0
$142$	6.67275	0.559964
$143$	−1.82043	−0.152232
$144$	0	0
$145$	4.82763	0.400913
$146$	11.9575	0.989607
$147$	0	0
$148$	−7.58613	−0.623576
$149$	−2.90167	−0.237714	−0.118857	−	0.992911i	$-0.537923\pi$
$149$	−2.90167	−0.237714	−0.118857	+	0.992911i	$0.537923\pi$
$150$	0	0
$151$	2.66399	0.216792	0.108396	−	0.994108i	$-0.465429\pi$
$151$	2.66399	0.216792	0.108396	+	0.994108i	$0.465429\pi$
$152$	−1.53786	−0.124737
$153$	0	0
$154$	−15.8696	−1.27881
$155$	5.10603	0.410126
$156$	0	0
$157$	15.2048	1.21347	0.606736	−	0.794903i	$-0.292478\pi$
$157$	15.2048	1.21347	0.606736	+	0.794903i	$0.292478\pi$
$158$	12.1543	0.966944
$159$	0	0
$160$	−0.686428	−0.0542669
$161$	9.11562	0.718412
$162$	0	0
$163$	12.4990	0.978996	0.489498	−	0.872004i	$-0.337180\pi$
$163$	12.4990	0.978996	0.489498	+	0.872004i	$0.337180\pi$
$164$	2.51679	0.196528
$165$	0	0
$166$	5.22743	0.405727
$167$	−6.88133	−0.532493	−0.266247	−	0.963905i	$-0.585784\pi$
$167$	−6.88133	−0.532493	−0.266247	+	0.963905i	$0.585784\pi$
$168$	0	0
$169$	−12.8912	−0.991631
$170$	3.04925	0.233867
$171$	0	0
$172$	6.97369	0.531739
$173$	3.65630	0.277984	0.138992	−	0.990294i	$-0.455614\pi$
$173$	3.65630	0.277984	0.138992	+	0.990294i	$0.455614\pi$
$174$	0	0
$175$	−13.0226	−0.984413
$176$	−5.51891	−0.416004
$177$	0	0
$178$	−8.29874	−0.622017
$179$	−11.6410	−0.870090	−0.435045	−	0.900409i	$-0.643267\pi$
$179$	−11.6410	−0.870090	−0.435045	+	0.900409i	$0.643267\pi$
$180$	0	0
$181$	−13.8301	−1.02798	−0.513990	−	0.857796i	$-0.671833\pi$
$181$	−13.8301	−1.02798	−0.513990	+	0.857796i	$0.671833\pi$
$182$	0.948489	0.0703067
$183$	0	0
$184$	3.17011	0.233704
$185$	5.20733	0.382851
$186$	0	0
$187$	24.5161	1.79280
$188$	−6.00990	−0.438317
$189$	0	0
$190$	1.05563	0.0765836
$191$	15.4332	1.11671	0.558354	−	0.829603i	$-0.311433\pi$
$191$	15.4332	1.11671	0.558354	+	0.829603i	$0.311433\pi$
$192$	0	0
$193$	−1.86292	−0.134096	−0.0670479	−	0.997750i	$-0.521358\pi$
$193$	−1.86292	−0.134096	−0.0670479	+	0.997750i	$0.521358\pi$
$194$	4.45112	0.319572
$195$	0	0
$196$	1.26844	0.0906028
$197$	18.0327	1.28477	0.642387	−	0.766381i	$-0.277944\pi$
$197$	18.0327	1.28477	0.642387	+	0.766381i	$0.277944\pi$
$198$	0	0
$199$	24.0118	1.70215	0.851077	−	0.525042i	$-0.175950\pi$
$199$	24.0118	1.70215	0.851077	+	0.525042i	$0.175950\pi$
$200$	−4.52882	−0.320236
$201$	0	0
$202$	3.23632	0.227707
$203$	−20.2232	−1.41939
$204$	0	0
$205$	−1.72759	−0.120660
$206$	−12.7274	−0.886760
$207$	0	0
$208$	0.329853	0.0228712
$209$	8.48734	0.587081
$210$	0	0
$211$	6.05163	0.416611	0.208306	−	0.978064i	$-0.433205\pi$
$211$	6.05163	0.416611	0.208306	+	0.978064i	$0.433205\pi$
$212$	0.840259	0.0577093
$213$	0	0
$214$	−4.52529	−0.309343
$215$	−4.78694	−0.326466
$216$	0	0
$217$	−21.3895	−1.45201
$218$	−10.5371	−0.713664
$219$	0	0
$220$	3.78834	0.255410
$221$	−1.46527	−0.0985650
$222$	0	0
$223$	−1.00000	−0.0669650
$223$	−1.00000	−0.0669650
$224$	2.87549	0.192127
$225$	0	0
$226$	−2.93690	−0.195360
$227$	−24.3730	−1.61769	−0.808846	−	0.588021i	$-0.799907\pi$
$227$	−24.3730	−1.61769	−0.808846	+	0.588021i	$0.799907\pi$
$228$	0	0
$229$	7.57496	0.500567	0.250284	−	0.968173i	$-0.419476\pi$
$229$	7.57496	0.500567	0.250284	+	0.968173i	$0.419476\pi$
$230$	−2.17605	−0.143485
$231$	0	0
$232$	−7.03297	−0.461737
$233$	−1.91430	−0.125410	−0.0627050	−	0.998032i	$-0.519973\pi$
$233$	−1.91430	−0.125410	−0.0627050	+	0.998032i	$0.519973\pi$
$234$	0	0
$235$	4.12537	0.269109
$236$	3.14631	0.204807
$237$	0	0
$238$	−12.7735	−0.827984
$239$	18.8018	1.21619	0.608094	−	0.793865i	$-0.291934\pi$
$239$	18.8018	1.21619	0.608094	+	0.793865i	$0.291934\pi$
$240$	0	0
$241$	−12.7202	−0.819378	−0.409689	−	0.912225i	$-0.634363\pi$
$241$	−12.7202	−0.819378	−0.409689	+	0.912225i	$0.634363\pi$
$242$	19.4584	1.25083
$243$	0	0
$244$	−2.68741	−0.172044
$245$	−0.870693	−0.0556265
$246$	0	0
$247$	−0.507269	−0.0322767
$248$	−7.43855	−0.472349
$249$	0	0
$250$	6.54085	0.413680
$251$	−8.40005	−0.530206	−0.265103	−	0.964220i	$-0.585406\pi$
$251$	−8.40005	−0.530206	−0.265103	+	0.964220i	$0.585406\pi$
$252$	0	0
$253$	−17.4956	−1.09994
$254$	4.04306	0.253684
$255$	0	0
$256$	1.00000	0.0625000
$257$	23.1611	1.44475	0.722375	−	0.691502i	$-0.243051\pi$
$257$	23.1611	1.44475	0.722375	+	0.691502i	$0.243051\pi$
$258$	0	0
$259$	−21.8138	−1.35545
$260$	−0.226420	−0.0140420
$261$	0	0
$262$	−6.64749	−0.410683
$263$	−6.61626	−0.407976	−0.203988	−	0.978973i	$-0.565390\pi$
$263$	−6.61626	−0.407976	−0.203988	+	0.978973i	$0.565390\pi$
$264$	0	0
$265$	−0.576778	−0.0354312
$266$	−4.42211	−0.271137
$267$	0	0
$268$	−13.9815	−0.854057
$269$	4.44005	0.270715	0.135357	−	0.990797i	$-0.456782\pi$
$269$	4.44005	0.270715	0.135357	+	0.990797i	$0.456782\pi$
$270$	0	0
$271$	−6.05643	−0.367902	−0.183951	−	0.982935i	$-0.558889\pi$
$271$	−6.05643	−0.367902	−0.183951	+	0.982935i	$0.558889\pi$
$272$	−4.44220	−0.269348
$273$	0	0
$274$	−10.9737	−0.662945
$275$	24.9942	1.50720
$276$	0	0
$277$	−25.7454	−1.54689	−0.773446	−	0.633862i	$-0.781469\pi$
$277$	−25.7454	−1.54689	−0.773446	+	0.633862i	$0.781469\pi$
$278$	−15.2142	−0.912487
$279$	0	0
$280$	−1.97382	−0.117958
$281$	14.7983	0.882790	0.441395	−	0.897313i	$-0.354484\pi$
$281$	14.7983	0.882790	0.441395	+	0.897313i	$0.354484\pi$
$282$	0	0
$283$	−4.17776	−0.248342	−0.124171	−	0.992261i	$-0.539627\pi$
$283$	−4.17776	−0.248342	−0.124171	+	0.992261i	$0.539627\pi$
$284$	6.67275	0.395955
$285$	0	0
$286$	−1.82043	−0.107644
$287$	7.23699	0.427186
$288$	0	0
$289$	2.73318	0.160775
$290$	4.82763	0.283488
$291$	0	0
$292$	11.9575	0.699758
$293$	12.6202	0.737281	0.368641	−	0.929572i	$-0.379823\pi$
$293$	12.6202	0.737281	0.368641	+	0.929572i	$0.379823\pi$
$294$	0	0
$295$	−2.15971	−0.125743
$296$	−7.58613	−0.440935
$297$	0	0
$298$	−2.90167	−0.168089
$299$	1.04567	0.0604727
$300$	0	0
$301$	20.0528	1.15582
$302$	2.66399	0.153295
$303$	0	0
$304$	−1.53786	−0.0882025
$305$	1.84472	0.105628
$306$	0	0
$307$	−32.7514	−1.86922	−0.934612	−	0.355669i	$-0.884253\pi$
$307$	−32.7514	−1.86922	−0.934612	+	0.355669i	$0.884253\pi$
$308$	−15.8696	−0.904253
$309$	0	0
$310$	5.10603	0.290003
$311$	5.09711	0.289031	0.144515	−	0.989503i	$-0.453838\pi$
$311$	5.09711	0.289031	0.144515	+	0.989503i	$0.453838\pi$
$312$	0	0
$313$	3.70513	0.209426	0.104713	−	0.994502i	$-0.466608\pi$
$313$	3.70513	0.209426	0.104713	+	0.994502i	$0.466608\pi$
$314$	15.2048	0.858055
$315$	0	0
$316$	12.1543	0.683733
$317$	−11.1944	−0.628743	−0.314371	−	0.949300i	$-0.601794\pi$
$317$	−11.1944	−0.628743	−0.314371	+	0.949300i	$0.601794\pi$
$318$	0	0
$319$	38.8144	2.17319
$320$	−0.686428	−0.0383725
$321$	0	0
$322$	9.11562	0.507994
$323$	6.83150	0.380115
$324$	0	0
$325$	−1.49384	−0.0828635
$326$	12.4990	0.692255
$327$	0	0
$328$	2.51679	0.138966
$329$	−17.2814	−0.952755
$330$	0	0
$331$	−30.7807	−1.69186	−0.845930	−	0.533293i	$-0.820954\pi$
$331$	−30.7807	−1.69186	−0.845930	+	0.533293i	$0.820954\pi$
$332$	5.22743	0.286892
$333$	0	0
$334$	−6.88133	−0.376530
$335$	9.59731	0.524357
$336$	0	0
$337$	−28.0781	−1.52951	−0.764755	−	0.644322i	$-0.777140\pi$
$337$	−28.0781	−1.52951	−0.764755	+	0.644322i	$0.777140\pi$
$338$	−12.8912	−0.701189
$339$	0	0
$340$	3.04925	0.165369
$341$	41.0527	2.22313
$342$	0	0
$343$	−16.4810	−0.889893
$344$	6.97369	0.375996
$345$	0	0
$346$	3.65630	0.196564
$347$	−4.81474	−0.258469	−0.129234	−	0.991614i	$-0.541252\pi$
$347$	−4.81474	−0.258469	−0.129234	+	0.991614i	$0.541252\pi$
$348$	0	0
$349$	20.7579	1.11114	0.555571	−	0.831469i	$-0.312500\pi$
$349$	20.7579	1.11114	0.555571	+	0.831469i	$0.312500\pi$
$350$	−13.0226	−0.696085
$351$	0	0
$352$	−5.51891	−0.294159
$353$	3.21876	0.171317	0.0856587	−	0.996325i	$-0.472701\pi$
$353$	3.21876	0.171317	0.0856587	+	0.996325i	$0.472701\pi$
$354$	0	0
$355$	−4.58036	−0.243100
$356$	−8.29874	−0.439833
$357$	0	0
$358$	−11.6410	−0.615246
$359$	−25.5107	−1.34641	−0.673203	−	0.739458i	$-0.735082\pi$
$359$	−25.5107	−1.34641	−0.673203	+	0.739458i	$0.735082\pi$
$360$	0	0
$361$	−16.6350	−0.875525
$362$	−13.8301	−0.726892
$363$	0	0
$364$	0.948489	0.0497143
$365$	−8.20794	−0.429623
$366$	0	0
$367$	−24.4579	−1.27669	−0.638346	−	0.769750i	$-0.720381\pi$
$367$	−24.4579	−1.27669	−0.638346	+	0.769750i	$0.720381\pi$
$368$	3.17011	0.165254
$369$	0	0
$370$	5.20733	0.270716
$371$	2.41616	0.125441
$372$	0	0
$373$	−9.79830	−0.507337	−0.253668	−	0.967291i	$-0.581637\pi$
$373$	−9.79830	−0.507337	−0.253668	+	0.967291i	$0.581637\pi$
$374$	24.5161	1.26770
$375$	0	0
$376$	−6.00990	−0.309937
$377$	−2.31985	−0.119478
$378$	0	0
$379$	23.3632	1.20009	0.600043	−	0.799968i	$-0.295150\pi$
$379$	23.3632	1.20009	0.600043	+	0.799968i	$0.295150\pi$
$380$	1.05563	0.0541528
$381$	0	0
$382$	15.4332	0.789632
$383$	−26.1424	−1.33582	−0.667908	−	0.744244i	$-0.732810\pi$
$383$	−26.1424	−1.33582	−0.667908	+	0.744244i	$0.732810\pi$
$384$	0	0
$385$	10.8933	0.555175
$386$	−1.86292	−0.0948201
$387$	0	0
$388$	4.45112	0.225971
$389$	−3.73549	−0.189397	−0.0946985	−	0.995506i	$-0.530189\pi$
$389$	−3.73549	−0.189397	−0.0946985	+	0.995506i	$0.530189\pi$
$390$	0	0
$391$	−14.0823	−0.712172
$392$	1.26844	0.0640659
$393$	0	0
$394$	18.0327	0.908472
$395$	−8.34305	−0.419785
$396$	0	0
$397$	17.6199	0.884316	0.442158	−	0.896937i	$-0.354213\pi$
$397$	17.6199	0.884316	0.442158	+	0.896937i	$0.354213\pi$
$398$	24.0118	1.20360
$399$	0	0
$400$	−4.52882	−0.226441
$401$	22.5232	1.12475	0.562376	−	0.826881i	$-0.309887\pi$
$401$	22.5232	1.12475	0.562376	+	0.826881i	$0.309887\pi$
$402$	0	0
$403$	−2.45363	−0.122224
$404$	3.23632	0.161013
$405$	0	0
$406$	−20.2232	−1.00366
$407$	41.8672	2.07528
$408$	0	0
$409$	−12.4542	−0.615821	−0.307910	−	0.951415i	$-0.599630\pi$
$409$	−12.4542	−0.615821	−0.307910	+	0.951415i	$0.599630\pi$
$410$	−1.72759	−0.0853197
$411$	0	0
$412$	−12.7274	−0.627034
$413$	9.04717	0.445182
$414$	0	0
$415$	−3.58825	−0.176140
$416$	0.329853	0.0161724
$417$	0	0
$418$	8.48734	0.415129
$419$	30.5170	1.49086	0.745428	−	0.666586i	$-0.232245\pi$
$419$	30.5170	1.49086	0.745428	+	0.666586i	$0.232245\pi$
$420$	0	0
$421$	5.17369	0.252150	0.126075	−	0.992021i	$-0.459762\pi$
$421$	5.17369	0.252150	0.126075	+	0.992021i	$0.459762\pi$
$422$	6.05163	0.294589
$423$	0	0
$424$	0.840259	0.0408066
$425$	20.1179	0.975863
$426$	0	0
$427$	−7.72762	−0.373966
$428$	−4.52529	−0.218738
$429$	0	0
$430$	−4.78694	−0.230847
$431$	11.8685	0.571686	0.285843	−	0.958277i	$-0.407726\pi$
$431$	11.8685	0.571686	0.285843	+	0.958277i	$0.407726\pi$
$432$	0	0
$433$	2.77072	0.133152	0.0665762	−	0.997781i	$-0.478792\pi$
$433$	2.77072	0.133152	0.0665762	+	0.997781i	$0.478792\pi$
$434$	−21.3895	−1.02673
$435$	0	0
$436$	−10.5371	−0.504637
$437$	−4.87520	−0.233212
$438$	0	0
$439$	−13.5404	−0.646248	−0.323124	−	0.946357i	$-0.604733\pi$
$439$	−13.5404	−0.646248	−0.323124	+	0.946357i	$0.604733\pi$
$440$	3.78834	0.180602
$441$	0	0
$442$	−1.46527	−0.0696960
$443$	−38.7298	−1.84011	−0.920054	−	0.391791i	$-0.871856\pi$
$443$	−38.7298	−1.84011	−0.920054	+	0.391791i	$0.871856\pi$
$444$	0	0
$445$	5.69649	0.270040
$446$	−1.00000	−0.0473514
$447$	0	0
$448$	2.87549	0.135854
$449$	35.2746	1.66471	0.832356	−	0.554241i	$-0.186992\pi$
$449$	35.2746	1.66471	0.832356	+	0.554241i	$0.186992\pi$
$450$	0	0
$451$	−13.8899	−0.654051
$452$	−2.93690	−0.138140
$453$	0	0
$454$	−24.3730	−1.14388
$455$	−0.651069	−0.0305226
$456$	0	0
$457$	−15.7179	−0.735251	−0.367626	−	0.929974i	$-0.619829\pi$
$457$	−15.7179	−0.735251	−0.367626	+	0.929974i	$0.619829\pi$
$458$	7.57496	0.353955
$459$	0	0
$460$	−2.17605	−0.101459
$461$	15.4964	0.721739	0.360869	−	0.932616i	$-0.382480\pi$
$461$	15.4964	0.721739	0.360869	+	0.932616i	$0.382480\pi$
$462$	0	0
$463$	34.0979	1.58466	0.792331	−	0.610091i	$-0.208867\pi$
$463$	34.0979	1.58466	0.792331	+	0.610091i	$0.208867\pi$
$464$	−7.03297	−0.326498
$465$	0	0
$466$	−1.91430	−0.0886783
$467$	−5.76270	−0.266666	−0.133333	−	0.991071i	$-0.542568\pi$
$467$	−5.76270	−0.266666	−0.133333	+	0.991071i	$0.542568\pi$
$468$	0	0
$469$	−40.2037	−1.85644
$470$	4.12537	0.190289
$471$	0	0
$472$	3.14631	0.144821
$473$	−38.4872	−1.76964
$474$	0	0
$475$	6.96470	0.319562
$476$	−12.7735	−0.585473
$477$	0	0
$478$	18.8018	0.859975
$479$	0.892156	0.0407637	0.0203818	−	0.999792i	$-0.493512\pi$
$479$	0.892156	0.0407637	0.0203818	+	0.999792i	$0.493512\pi$
$480$	0	0
$481$	−2.50231	−0.114095
$482$	−12.7202	−0.579387
$483$	0	0
$484$	19.4584	0.884473
$485$	−3.05538	−0.138737
$486$	0	0
$487$	4.83587	0.219134	0.109567	−	0.993979i	$-0.465054\pi$
$487$	4.83587	0.219134	0.109567	+	0.993979i	$0.465054\pi$
$488$	−2.68741	−0.121653
$489$	0	0
$490$	−0.870693	−0.0393339
$491$	14.5280	0.655640	0.327820	−	0.944740i	$-0.393686\pi$
$491$	14.5280	0.655640	0.327820	+	0.944740i	$0.393686\pi$
$492$	0	0
$493$	31.2419	1.40706
$494$	−0.507269	−0.0228231
$495$	0	0
$496$	−7.43855	−0.334001
$497$	19.1874	0.860673
$498$	0	0
$499$	42.7043	1.91171	0.955854	−	0.293842i	$-0.0949339\pi$
$499$	42.7043	1.91171	0.955854	+	0.293842i	$0.0949339\pi$
$500$	6.54085	0.292516
$501$	0	0
$502$	−8.40005	−0.374912
$503$	28.7298	1.28100	0.640499	−	0.767959i	$-0.278727\pi$
$503$	28.7298	1.28100	0.640499	+	0.767959i	$0.278727\pi$
$504$	0	0
$505$	−2.22150	−0.0988556
$506$	−17.4956	−0.777773
$507$	0	0
$508$	4.04306	0.179382
$509$	31.9400	1.41572	0.707858	−	0.706355i	$-0.249662\pi$
$509$	31.9400	1.41572	0.707858	+	0.706355i	$0.249662\pi$
$510$	0	0
$511$	34.3836	1.52104
$512$	1.00000	0.0441942
$513$	0	0
$514$	23.1611	1.02159
$515$	8.73644	0.384974
$516$	0	0
$517$	33.1681	1.45873
$518$	−21.8138	−0.958445
$519$	0	0
$520$	−0.226420	−0.00992919
$521$	−34.5500	−1.51366	−0.756832	−	0.653609i	$-0.773254\pi$
$521$	−34.5500	−1.51366	−0.756832	+	0.653609i	$0.773254\pi$
$522$	0	0
$523$	20.3878	0.891498	0.445749	−	0.895158i	$-0.352937\pi$
$523$	20.3878	0.891498	0.445749	+	0.895158i	$0.352937\pi$
$524$	−6.64749	−0.290397
$525$	0	0
$526$	−6.61626	−0.288483
$527$	33.0436	1.43940
$528$	0	0
$529$	−12.9504	−0.563060
$530$	−0.576778	−0.0250536
$531$	0	0
$532$	−4.42211	−0.191723
$533$	0.830169	0.0359586
$534$	0	0
$535$	3.10629	0.134297
$536$	−13.9815	−0.603910
$537$	0	0
$538$	4.44005	0.191424
$539$	−7.00041	−0.301529
$540$	0	0
$541$	−26.5035	−1.13948	−0.569738	−	0.821827i	$-0.692955\pi$
$541$	−26.5035	−1.13948	−0.569738	+	0.821827i	$0.692955\pi$
$542$	−6.05643	−0.260146
$543$	0	0
$544$	−4.44220	−0.190458
$545$	7.23298	0.309827
$546$	0	0
$547$	−0.362415	−0.0154958	−0.00774788	−	0.999970i	$-0.502466\pi$
$547$	−0.362415	−0.0154958	−0.00774788	+	0.999970i	$0.502466\pi$
$548$	−10.9737	−0.468773
$549$	0	0
$550$	24.9942	1.06575
$551$	10.8158	0.460766
$552$	0	0
$553$	34.9496	1.48621
$554$	−25.7454	−1.09382
$555$	0	0
$556$	−15.2142	−0.645225
$557$	17.2544	0.731092	0.365546	−	0.930793i	$-0.380882\pi$
$557$	17.2544	0.731092	0.365546	+	0.930793i	$0.380882\pi$
$558$	0	0
$559$	2.30029	0.0972920
$560$	−1.97382	−0.0834090
$561$	0	0
$562$	14.7983	0.624227
$563$	−38.1428	−1.60753	−0.803764	−	0.594948i	$-0.797173\pi$
$563$	−38.1428	−1.60753	−0.803764	+	0.594948i	$0.797173\pi$
$564$	0	0
$565$	2.01597	0.0848126
$566$	−4.17776	−0.175604
$567$	0	0
$568$	6.67275	0.279982
$569$	9.79988	0.410832	0.205416	−	0.978675i	$-0.434145\pi$
$569$	9.79988	0.410832	0.205416	+	0.978675i	$0.434145\pi$
$570$	0	0
$571$	28.2196	1.18095	0.590476	−	0.807055i	$-0.298940\pi$
$571$	28.2196	1.18095	0.590476	+	0.807055i	$0.298940\pi$
$572$	−1.82043	−0.0761160
$573$	0	0
$574$	7.23699	0.302066
$575$	−14.3569	−0.598722
$576$	0	0
$577$	28.2068	1.17427	0.587133	−	0.809491i	$-0.300257\pi$
$577$	28.2068	1.17427	0.587133	+	0.809491i	$0.300257\pi$
$578$	2.73318	0.113685
$579$	0	0
$580$	4.82763	0.200456
$581$	15.0314	0.623608
$582$	0	0
$583$	−4.63732	−0.192058
$584$	11.9575	0.494803
$585$	0	0
$586$	12.6202	0.521337
$587$	12.2413	0.505251	0.252626	−	0.967564i	$-0.418706\pi$
$587$	12.2413	0.505251	0.252626	+	0.967564i	$0.418706\pi$
$588$	0	0
$589$	11.4395	0.471355
$590$	−2.15971	−0.0889140
$591$	0	0
$592$	−7.58613	−0.311788
$593$	−28.6547	−1.17671	−0.588353	−	0.808604i	$-0.700223\pi$
$593$	−28.6547	−1.17671	−0.588353	+	0.808604i	$0.700223\pi$
$594$	0	0
$595$	8.76810	0.359457
$596$	−2.90167	−0.118857
$597$	0	0
$598$	1.04567	0.0427607
$599$	6.02231	0.246065	0.123032	−	0.992403i	$-0.460738\pi$
$599$	6.02231	0.246065	0.123032	+	0.992403i	$0.460738\pi$
$600$	0	0
$601$	33.1083	1.35052	0.675258	−	0.737582i	$-0.264032\pi$
$601$	33.1083	1.35052	0.675258	+	0.737582i	$0.264032\pi$
$602$	20.0528	0.817290
$603$	0	0
$604$	2.66399	0.108396
$605$	−13.3568	−0.543031
$606$	0	0
$607$	35.7642	1.45163	0.725813	−	0.687892i	$-0.241464\pi$
$607$	35.7642	1.45163	0.725813	+	0.687892i	$0.241464\pi$
$608$	−1.53786	−0.0623686
$609$	0	0
$610$	1.84472	0.0746903
$611$	−1.98238	−0.0801987
$612$	0	0
$613$	−23.7521	−0.959339	−0.479669	−	0.877449i	$-0.659243\pi$
$613$	−23.7521	−0.959339	−0.479669	+	0.877449i	$0.659243\pi$
$614$	−32.7514	−1.32174
$615$	0	0
$616$	−15.8696	−0.639404
$617$	37.5379	1.51122	0.755609	−	0.655023i	$-0.227341\pi$
$617$	37.5379	1.51122	0.755609	+	0.655023i	$0.227341\pi$
$618$	0	0
$619$	46.6282	1.87415	0.937073	−	0.349132i	$-0.113524\pi$
$619$	46.6282	1.87415	0.937073	+	0.349132i	$0.113524\pi$
$620$	5.10603	0.205063
$621$	0	0
$622$	5.09711	0.204376
$623$	−23.8630	−0.956049
$624$	0	0
$625$	18.1543	0.726170
$626$	3.70513	0.148087
$627$	0	0
$628$	15.2048	0.606736
$629$	33.6991	1.34367
$630$	0	0
$631$	28.3815	1.12985	0.564926	−	0.825142i	$-0.308905\pi$
$631$	28.3815	1.12985	0.564926	+	0.825142i	$0.308905\pi$
$632$	12.1543	0.483472
$633$	0	0
$634$	−11.1944	−0.444588
$635$	−2.77527	−0.110133
$636$	0	0
$637$	0.418399	0.0165776
$638$	38.8144	1.53668
$639$	0	0
$640$	−0.686428	−0.0271335
$641$	−11.2437	−0.444098	−0.222049	−	0.975036i	$-0.571274\pi$
$641$	−11.2437	−0.444098	−0.222049	+	0.975036i	$0.571274\pi$
$642$	0	0
$643$	−0.473139	−0.0186588	−0.00932940	−	0.999956i	$-0.502970\pi$
$643$	−0.473139	−0.0186588	−0.00932940	+	0.999956i	$0.502970\pi$
$644$	9.11562	0.359206
$645$	0	0
$646$	6.83150	0.268782
$647$	44.1533	1.73585	0.867923	−	0.496699i	$-0.165455\pi$
$647$	44.1533	1.73585	0.867923	+	0.496699i	$0.165455\pi$
$648$	0	0
$649$	−17.3642	−0.681604
$650$	−1.49384	−0.0585934
$651$	0	0
$652$	12.4990	0.489498
$653$	−17.7560	−0.694847	−0.347423	−	0.937708i	$-0.612943\pi$
$653$	−17.7560	−0.694847	−0.347423	+	0.937708i	$0.612943\pi$
$654$	0	0
$655$	4.56302	0.178292
$656$	2.51679	0.0982640
$657$	0	0
$658$	−17.2814	−0.673700
$659$	−28.1314	−1.09584	−0.547921	−	0.836530i	$-0.684581\pi$
$659$	−28.1314	−1.09584	−0.547921	+	0.836530i	$0.684581\pi$
$660$	0	0
$661$	13.7180	0.533567	0.266784	−	0.963756i	$-0.414039\pi$
$661$	13.7180	0.533567	0.266784	+	0.963756i	$0.414039\pi$
$662$	−30.7807	−1.19633
$663$	0	0
$664$	5.22743	0.202864
$665$	3.03546	0.117710
$666$	0	0
$667$	−22.2953	−0.863278
$668$	−6.88133	−0.266247
$669$	0	0
$670$	9.59731	0.370776
$671$	14.8316	0.572567
$672$	0	0
$673$	12.1670	0.469002	0.234501	−	0.972116i	$-0.424654\pi$
$673$	12.1670	0.469002	0.234501	+	0.972116i	$0.424654\pi$
$674$	−28.0781	−1.08153
$675$	0	0
$676$	−12.8912	−0.495815
$677$	−42.3323	−1.62696	−0.813481	−	0.581592i	$-0.802430\pi$
$677$	−42.3323	−1.62696	−0.813481	+	0.581592i	$0.802430\pi$
$678$	0	0
$679$	12.7992	0.491186
$680$	3.04925	0.116934
$681$	0	0
$682$	41.0527	1.57199
$683$	−32.6141	−1.24794	−0.623972	−	0.781446i	$-0.714482\pi$
$683$	−32.6141	−1.24794	−0.623972	+	0.781446i	$0.714482\pi$
$684$	0	0
$685$	7.53265	0.287808
$686$	−16.4810	−0.629249
$687$	0	0
$688$	6.97369	0.265869
$689$	0.277162	0.0105590
$690$	0	0
$691$	6.87327	0.261472	0.130736	−	0.991417i	$-0.458266\pi$
$691$	6.87327	0.261472	0.130736	+	0.991417i	$0.458266\pi$
$692$	3.65630	0.138992
$693$	0	0
$694$	−4.81474	−0.182765
$695$	10.4434	0.396143
$696$	0	0
$697$	−11.1801	−0.423475
$698$	20.7579	0.785697
$699$	0	0
$700$	−13.0226	−0.492207
$701$	30.3022	1.14450	0.572249	−	0.820080i	$-0.306071\pi$
$701$	30.3022	1.14450	0.572249	+	0.820080i	$0.306071\pi$
$702$	0	0
$703$	11.6664	0.440008
$704$	−5.51891	−0.208002
$705$	0	0
$706$	3.21876	0.121140
$707$	9.30601	0.349989
$708$	0	0
$709$	−45.6605	−1.71482	−0.857408	−	0.514638i	$-0.827926\pi$
$709$	−45.6605	−1.71482	−0.857408	+	0.514638i	$0.827926\pi$
$710$	−4.58036	−0.171898
$711$	0	0
$712$	−8.29874	−0.311009
$713$	−23.5810	−0.883117
$714$	0	0
$715$	1.24959	0.0467322
$716$	−11.6410	−0.435045
$717$	0	0
$718$	−25.5107	−0.952053
$719$	4.43820	0.165517	0.0827584	−	0.996570i	$-0.473627\pi$
$719$	4.43820	0.165517	0.0827584	+	0.996570i	$0.473627\pi$
$720$	0	0
$721$	−36.5975	−1.36296
$722$	−16.6350	−0.619090
$723$	0	0
$724$	−13.8301	−0.513990
$725$	31.8510	1.18292
$726$	0	0
$727$	3.18844	0.118253	0.0591263	−	0.998251i	$-0.481169\pi$
$727$	3.18844	0.118253	0.0591263	+	0.998251i	$0.481169\pi$
$728$	0.948489	0.0351533
$729$	0	0
$730$	−8.20794	−0.303790
$731$	−30.9785	−1.14578
$732$	0	0
$733$	7.01336	0.259045	0.129522	−	0.991577i	$-0.458656\pi$
$733$	7.01336	0.259045	0.129522	+	0.991577i	$0.458656\pi$
$734$	−24.4579	−0.902758
$735$	0	0
$736$	3.17011	0.116852
$737$	77.1628	2.84233
$738$	0	0
$739$	−24.9587	−0.918122	−0.459061	−	0.888405i	$-0.651814\pi$
$739$	−24.9587	−0.918122	−0.459061	+	0.888405i	$0.651814\pi$
$740$	5.20733	0.191425
$741$	0	0
$742$	2.41616	0.0886999
$743$	−28.8093	−1.05691	−0.528456	−	0.848961i	$-0.677229\pi$
$743$	−28.8093	−1.05691	−0.528456	+	0.848961i	$0.677229\pi$
$744$	0	0
$745$	1.99179	0.0729735
$746$	−9.79830	−0.358741
$747$	0	0
$748$	24.5161	0.896399
$749$	−13.0124	−0.475464
$750$	0	0
$751$	−51.0813	−1.86398	−0.931992	−	0.362479i	$-0.881931\pi$
$751$	−51.0813	−1.86398	−0.931992	+	0.362479i	$0.881931\pi$
$752$	−6.00990	−0.219159
$753$	0	0
$754$	−2.31985	−0.0844838
$755$	−1.82864	−0.0665509
$756$	0	0
$757$	15.0243	0.546068	0.273034	−	0.962004i	$-0.411973\pi$
$757$	15.0243	0.546068	0.273034	+	0.962004i	$0.411973\pi$
$758$	23.3632	0.848588
$759$	0	0
$760$	1.05563	0.0382918
$761$	−6.31216	−0.228816	−0.114408	−	0.993434i	$-0.536497\pi$
$761$	−6.31216	−0.228816	−0.114408	+	0.993434i	$0.536497\pi$
$762$	0	0
$763$	−30.2994	−1.09691
$764$	15.4332	0.558354
$765$	0	0
$766$	−26.1424	−0.944564
$767$	1.03782	0.0374735
$768$	0	0
$769$	24.2449	0.874293	0.437147	−	0.899390i	$-0.355989\pi$
$769$	24.2449	0.874293	0.437147	+	0.899390i	$0.355989\pi$
$770$	10.8933	0.392568
$771$	0	0
$772$	−1.86292	−0.0670479
$773$	−14.1903	−0.510390	−0.255195	−	0.966890i	$-0.582140\pi$
$773$	−14.1903	−0.510390	−0.255195	+	0.966890i	$0.582140\pi$
$774$	0	0
$775$	33.6878	1.21010
$776$	4.45112	0.159786
$777$	0	0
$778$	−3.73549	−0.133924
$779$	−3.87047	−0.138674
$780$	0	0
$781$	−36.8263	−1.31775
$782$	−14.0823	−0.503582
$783$	0	0
$784$	1.26844	0.0453014
$785$	−10.4370	−0.372512
$786$	0	0
$787$	−17.2608	−0.615280	−0.307640	−	0.951503i	$-0.599539\pi$
$787$	−17.2608	−0.615280	−0.307640	+	0.951503i	$0.599539\pi$
$788$	18.0327	0.642387
$789$	0	0
$790$	−8.34305	−0.296833
$791$	−8.44503	−0.300271
$792$	0	0
$793$	−0.886451	−0.0314788
$794$	17.6199	0.625306
$795$	0	0
$796$	24.0118	0.851077
$797$	13.2887	0.470710	0.235355	−	0.971909i	$-0.424375\pi$
$797$	13.2887	0.470710	0.235355	+	0.971909i	$0.424375\pi$
$798$	0	0
$799$	26.6972	0.944479
$800$	−4.52882	−0.160118
$801$	0	0
$802$	22.5232	0.795320
$803$	−65.9922	−2.32882
$804$	0	0
$805$	−6.25722	−0.220538
$806$	−2.45363	−0.0864254
$807$	0	0
$808$	3.23632	0.113853
$809$	9.08056	0.319255	0.159628	−	0.987177i	$-0.448971\pi$
$809$	9.08056	0.319255	0.159628	+	0.987177i	$0.448971\pi$
$810$	0	0
$811$	−49.9105	−1.75260	−0.876298	−	0.481770i	$-0.839994\pi$
$811$	−49.9105	−1.75260	−0.876298	+	0.481770i	$0.839994\pi$
$812$	−20.2232	−0.709697
$813$	0	0
$814$	41.8672	1.46745
$815$	−8.57965	−0.300532
$816$	0	0
$817$	−10.7246	−0.375206
$818$	−12.4542	−0.435451
$819$	0	0
$820$	−1.72759	−0.0603301
$821$	−9.08016	−0.316900	−0.158450	−	0.987367i	$-0.550650\pi$
$821$	−9.08016	−0.316900	−0.158450	+	0.987367i	$0.550650\pi$
$822$	0	0
$823$	19.7767	0.689371	0.344686	−	0.938718i	$-0.387986\pi$
$823$	19.7767	0.689371	0.344686	+	0.938718i	$0.387986\pi$
$824$	−12.7274	−0.443380
$825$	0	0
$826$	9.04717	0.314791
$827$	−37.8945	−1.31772	−0.658860	−	0.752266i	$-0.728961\pi$
$827$	−37.8945	−1.31772	−0.658860	+	0.752266i	$0.728961\pi$
$828$	0	0
$829$	−6.13822	−0.213189	−0.106595	−	0.994303i	$-0.533995\pi$
$829$	−6.13822	−0.213189	−0.106595	+	0.994303i	$0.533995\pi$
$830$	−3.58825	−0.124550
$831$	0	0
$832$	0.329853	0.0114356
$833$	−5.63467	−0.195230
$834$	0	0
$835$	4.72354	0.163465
$836$	8.48734	0.293541
$837$	0	0
$838$	30.5170	1.05419
$839$	−37.2877	−1.28731	−0.643657	−	0.765314i	$-0.722584\pi$
$839$	−37.2877	−1.28731	−0.643657	+	0.765314i	$0.722584\pi$
$840$	0	0
$841$	20.4627	0.705610
$842$	5.17369	0.178297
$843$	0	0
$844$	6.05163	0.208306
$845$	8.84888	0.304411
$846$	0	0
$847$	55.9525	1.92255
$848$	0.840259	0.0288546
$849$	0	0
$850$	20.1179	0.690039
$851$	−24.0489	−0.824385
$852$	0	0
$853$	−5.19329	−0.177815	−0.0889075	−	0.996040i	$-0.528338\pi$
$853$	−5.19329	−0.177815	−0.0889075	+	0.996040i	$0.528338\pi$
$854$	−7.72762	−0.264434
$855$	0	0
$856$	−4.52529	−0.154671
$857$	−33.5933	−1.14753	−0.573763	−	0.819022i	$-0.694517\pi$
$857$	−33.5933	−1.14753	−0.573763	+	0.819022i	$0.694517\pi$
$858$	0	0
$859$	10.4008	0.354872	0.177436	−	0.984132i	$-0.443220\pi$
$859$	10.4008	0.354872	0.177436	+	0.984132i	$0.443220\pi$
$860$	−4.78694	−0.163233
$861$	0	0
$862$	11.8685	0.404243
$863$	37.4393	1.27445	0.637224	−	0.770679i	$-0.280083\pi$
$863$	37.4393	1.27445	0.637224	+	0.770679i	$0.280083\pi$
$864$	0	0
$865$	−2.50979	−0.0853354
$866$	2.77072	0.0941530
$867$	0	0
$868$	−21.3895	−0.726006
$869$	−67.0785	−2.27548
$870$	0	0
$871$	−4.61185	−0.156266
$872$	−10.5371	−0.356832
$873$	0	0
$874$	−4.87520	−0.164906
$875$	18.8081	0.635831
$876$	0	0
$877$	49.1976	1.66129	0.830643	−	0.556805i	$-0.187973\pi$
$877$	49.1976	1.66129	0.830643	+	0.556805i	$0.187973\pi$
$878$	−13.5404	−0.456966
$879$	0	0
$880$	3.78834	0.127705
$881$	42.7892	1.44161	0.720803	−	0.693140i	$-0.243773\pi$
$881$	42.7892	1.44161	0.720803	+	0.693140i	$0.243773\pi$
$882$	0	0
$883$	−12.3874	−0.416870	−0.208435	−	0.978036i	$-0.566837\pi$
$883$	−12.3874	−0.416870	−0.208435	+	0.978036i	$0.566837\pi$
$884$	−1.46527	−0.0492825
$885$	0	0
$886$	−38.7298	−1.30115
$887$	1.39064	0.0466930	0.0233465	−	0.999727i	$-0.492568\pi$
$887$	1.39064	0.0466930	0.0233465	+	0.999727i	$0.492568\pi$
$888$	0	0
$889$	11.6258	0.389916
$890$	5.69649	0.190947
$891$	0	0
$892$	−1.00000	−0.0334825
$893$	9.24241	0.309285
$894$	0	0
$895$	7.99071	0.267100
$896$	2.87549	0.0960634
$897$	0	0
$898$	35.2746	1.17713
$899$	52.3151	1.74481
$900$	0	0
$901$	−3.73260	−0.124351
$902$	−13.8899	−0.462484
$903$	0	0
$904$	−2.93690	−0.0976799
$905$	9.49335	0.315570
$906$	0	0
$907$	−16.8327	−0.558921	−0.279460	−	0.960157i	$-0.590156\pi$
$907$	−16.8327	−0.558921	−0.279460	+	0.960157i	$0.590156\pi$
$908$	−24.3730	−0.808846
$909$	0	0
$910$	−0.651069	−0.0215827
$911$	−25.6813	−0.850860	−0.425430	−	0.904991i	$-0.639877\pi$
$911$	−25.6813	−0.850860	−0.425430	+	0.904991i	$0.639877\pi$
$912$	0	0
$913$	−28.8497	−0.954787
$914$	−15.7179	−0.519901
$915$	0	0
$916$	7.57496	0.250284
$917$	−19.1148	−0.631226
$918$	0	0
$919$	−34.0182	−1.12216	−0.561079	−	0.827762i	$-0.689614\pi$
$919$	−34.0182	−1.12216	−0.561079	+	0.827762i	$0.689614\pi$
$920$	−2.17605	−0.0717424
$921$	0	0
$922$	15.4964	0.510346
$923$	2.20103	0.0724476
$924$	0	0
$925$	34.3562	1.12963
$926$	34.0979	1.12053
$927$	0	0
$928$	−7.03297	−0.230869
$929$	16.0961	0.528096	0.264048	−	0.964510i	$-0.414942\pi$
$929$	16.0961	0.528096	0.264048	+	0.964510i	$0.414942\pi$
$930$	0	0
$931$	−1.95069	−0.0639312
$932$	−1.91430	−0.0627050
$933$	0	0
$934$	−5.76270	−0.188561
$935$	−16.8286	−0.550353
$936$	0	0
$937$	−11.1780	−0.365169	−0.182584	−	0.983190i	$-0.558446\pi$
$937$	−11.1780	−0.365169	−0.182584	+	0.983190i	$0.558446\pi$
$938$	−40.2037	−1.31270
$939$	0	0
$940$	4.12537	0.134555
$941$	55.8606	1.82101	0.910503	−	0.413504i	$-0.135695\pi$
$941$	55.8606	1.82101	0.910503	+	0.413504i	$0.135695\pi$
$942$	0	0
$943$	7.97849	0.259815
$944$	3.14631	0.102404
$945$	0	0
$946$	−38.4872	−1.25133
$947$	−33.1983	−1.07880	−0.539400	−	0.842050i	$-0.681349\pi$
$947$	−33.1983	−1.07880	−0.539400	+	0.842050i	$0.681349\pi$
$948$	0	0
$949$	3.94421	0.128034
$950$	6.96470	0.225965
$951$	0	0
$952$	−12.7735	−0.413992
$953$	52.1647	1.68978	0.844891	−	0.534939i	$-0.179665\pi$
$953$	52.1647	1.68978	0.844891	+	0.534939i	$0.179665\pi$
$954$	0	0
$955$	−10.5938	−0.342807
$956$	18.8018	0.608094
$957$	0	0
$958$	0.892156	0.0288243
$959$	−31.5547	−1.01896
$960$	0	0
$961$	24.3321	0.784905
$962$	−2.50231	−0.0806777
$963$	0	0
$964$	−12.7202	−0.409689
$965$	1.27876	0.0411648
$966$	0	0
$967$	−6.32108	−0.203272	−0.101636	−	0.994822i	$-0.532408\pi$
$967$	−6.32108	−0.203272	−0.101636	+	0.994822i	$0.532408\pi$
$968$	19.4584	0.625417
$969$	0	0
$970$	−3.05538	−0.0981022
$971$	−24.4643	−0.785098	−0.392549	−	0.919731i	$-0.628407\pi$
$971$	−24.4643	−0.785098	−0.392549	+	0.919731i	$0.628407\pi$
$972$	0	0
$973$	−43.7482	−1.40250
$974$	4.83587	0.154951
$975$	0	0
$976$	−2.68741	−0.0860220
$977$	23.1983	0.742181	0.371090	−	0.928597i	$-0.378984\pi$
$977$	23.1983	0.742181	0.371090	+	0.928597i	$0.378984\pi$
$978$	0	0
$979$	45.8001	1.46378
$980$	−0.870693	−0.0278133
$981$	0	0
$982$	14.5280	0.463608
$983$	44.9757	1.43450	0.717252	−	0.696814i	$-0.245400\pi$
$983$	44.9757	1.43450	0.717252	+	0.696814i	$0.245400\pi$
$984$	0	0
$985$	−12.3781	−0.394400
$986$	31.2419	0.994945
$987$	0	0
$988$	−0.507269	−0.0161384
$989$	22.1074	0.702974
$990$	0	0
$991$	−54.4523	−1.72973	−0.864867	−	0.502001i	$-0.832597\pi$
$991$	−54.4523	−1.72973	−0.864867	+	0.502001i	$0.832597\pi$
$992$	−7.43855	−0.236174
$993$	0	0
$994$	19.1874	0.608588
$995$	−16.4824	−0.522527
$996$	0	0
$997$	20.4748	0.648444	0.324222	−	0.945981i	$-0.394898\pi$
$997$	20.4748	0.648444	0.324222	+	0.945981i	$0.394898\pi$
$998$	42.7043	1.35178
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4014.2.a.r.1.4		5
3.2	odd	2		1338.2.a.h.1.2	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1338.2.a.h.1.2	✓	5	3.2	odd	2
4014.2.a.r.1.4		5	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:	\( N \)	\(=\)	\( 4014 = 2 \cdot 3^{2} \cdot 223 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4014.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -0.686428 q^{5} +2.87549 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} -0.686428 q^{5} +2.87549 q^{7} +1.00000 q^{8} -0.686428 q^{10} -5.51891 q^{11} +0.329853 q^{13} +2.87549 q^{14} +1.00000 q^{16} -4.44220 q^{17} -1.53786 q^{19} -0.686428 q^{20} -5.51891 q^{22} +3.17011 q^{23} -4.52882 q^{25} +0.329853 q^{26} +2.87549 q^{28} -7.03297 q^{29} -7.43855 q^{31} +1.00000 q^{32} -4.44220 q^{34} -1.97382 q^{35} -7.58613 q^{37} -1.53786 q^{38} -0.686428 q^{40} +2.51679 q^{41} +6.97369 q^{43} -5.51891 q^{44} +3.17011 q^{46} -6.00990 q^{47} +1.26844 q^{49} -4.52882 q^{50} +0.329853 q^{52} +0.840259 q^{53} +3.78834 q^{55} +2.87549 q^{56} -7.03297 q^{58} +3.14631 q^{59} -2.68741 q^{61} -7.43855 q^{62} +1.00000 q^{64} -0.226420 q^{65} -13.9815 q^{67} -4.44220 q^{68} -1.97382 q^{70} +6.67275 q^{71} +11.9575 q^{73} -7.58613 q^{74} -1.53786 q^{76} -15.8696 q^{77} +12.1543 q^{79} -0.686428 q^{80} +2.51679 q^{82} +5.22743 q^{83} +3.04925 q^{85} +6.97369 q^{86} -5.51891 q^{88} -8.29874 q^{89} +0.948489 q^{91} +3.17011 q^{92} -6.00990 q^{94} +1.05563 q^{95} +4.45112 q^{97} +1.26844 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 5 q^{2} + 5 q^{4} - 5 q^{5} - q^{7} + 5 q^{8}+O(q^{10}) \) 5 * q + 5 * q^2 + 5 * q^4 - 5 * q^5 - q^7 + 5 * q^8	\( 5 q + 5 q^{2} + 5 q^{4} - 5 q^{5} - q^{7} + 5 q^{8} - 5 q^{10} - 9 q^{11} - q^{14} + 5 q^{16} - 6 q^{17} - 4 q^{19} - 5 q^{20} - 9 q^{22} - 16 q^{23} + 8 q^{25} - q^{28} - 8 q^{29} - q^{31} + 5 q^{32} - 6 q^{34} - 22 q^{35} - 2 q^{37} - 4 q^{38} - 5 q^{40} - 4 q^{41} + 3 q^{43} - 9 q^{44} - 16 q^{46} - 18 q^{47} + 2 q^{49} + 8 q^{50} - 26 q^{53} + q^{55} - q^{56} - 8 q^{58} - 21 q^{59} - 20 q^{61} - q^{62} + 5 q^{64} + 3 q^{65} - 5 q^{67} - 6 q^{68} - 22 q^{70} - 17 q^{71} + 5 q^{73} - 2 q^{74} - 4 q^{76} - 2 q^{77} - 21 q^{79} - 5 q^{80} - 4 q^{82} - 11 q^{83} - 12 q^{85} + 3 q^{86} - 9 q^{88} + 5 q^{89} - 10 q^{91} - 16 q^{92} - 18 q^{94} - 10 q^{95} - 11 q^{97} + 2 q^{98}+O(q^{100}) \) 5 * q + 5 * q^2 + 5 * q^4 - 5 * q^5 - q^7 + 5 * q^8 - 5 * q^10 - 9 * q^11 - q^14 + 5 * q^16 - 6 * q^17 - 4 * q^19 - 5 * q^20 - 9 * q^22 - 16 * q^23 + 8 * q^25 - q^28 - 8 * q^29 - q^31 + 5 * q^32 - 6 * q^34 - 22 * q^35 - 2 * q^37 - 4 * q^38 - 5 * q^40 - 4 * q^41 + 3 * q^43 - 9 * q^44 - 16 * q^46 - 18 * q^47 + 2 * q^49 + 8 * q^50 - 26 * q^53 + q^55 - q^56 - 8 * q^58 - 21 * q^59 - 20 * q^61 - q^62 + 5 * q^64 + 3 * q^65 - 5 * q^67 - 6 * q^68 - 22 * q^70 - 17 * q^71 + 5 * q^73 - 2 * q^74 - 4 * q^76 - 2 * q^77 - 21 * q^79 - 5 * q^80 - 4 * q^82 - 11 * q^83 - 12 * q^85 + 3 * q^86 - 9 * q^88 + 5 * q^89 - 10 * q^91 - 16 * q^92 - 18 * q^94 - 10 * q^95 - 11 * q^97 + 2 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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