LMFDB - Embedded newform 6044.2.a.a.1.19

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6044,2,Mod(1,6044)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6044, 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("6044.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6044 = 2^{2} \cdot 1511 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6044.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$48.2615829817$
Analytic rank:	$1$
Dimension:	$63$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.19
Character	$\chi$	$=$	6044.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.58770 q^{3} -0.675402 q^{5} -1.82211 q^{7} -0.479212 q^{9} +O(q^{10})$	$q-1.58770 q^{3} -0.675402 q^{5} -1.82211 q^{7} -0.479212 q^{9} -0.0892749 q^{11} +1.89681 q^{13} +1.07234 q^{15} -2.72739 q^{17} -0.950237 q^{19} +2.89297 q^{21} +0.452609 q^{23} -4.54383 q^{25} +5.52394 q^{27} +4.76295 q^{29} +7.32378 q^{31} +0.141742 q^{33} +1.23066 q^{35} +10.4886 q^{37} -3.01156 q^{39} -6.42533 q^{41} -0.288879 q^{43} +0.323661 q^{45} -2.76537 q^{47} -3.67991 q^{49} +4.33028 q^{51} +8.22321 q^{53} +0.0602965 q^{55} +1.50869 q^{57} +5.51634 q^{59} -1.26310 q^{61} +0.873178 q^{63} -1.28111 q^{65} +0.673526 q^{67} -0.718606 q^{69} -7.15399 q^{71} -5.05339 q^{73} +7.21424 q^{75} +0.162669 q^{77} -0.224643 q^{79} -7.33272 q^{81} -2.39356 q^{83} +1.84209 q^{85} -7.56213 q^{87} +12.4780 q^{89} -3.45620 q^{91} -11.6280 q^{93} +0.641792 q^{95} +4.64727 q^{97} +0.0427817 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 63 q - 7 q^{3} - 7 q^{5} - 22 q^{7} + 62 q^{9}+O(q^{10}) $ 63 * q - 7 * q^3 - 7 * q^5 - 22 * q^7 + 62 * q^9	$ 63 q - 7 q^{3} - 7 q^{5} - 22 q^{7} + 62 q^{9} - 21 q^{11} - 19 q^{13} - 30 q^{15} - 5 q^{17} - 59 q^{19} - 30 q^{21} - 24 q^{23} + 60 q^{25} - 34 q^{27} - 28 q^{29} - 48 q^{31} - q^{33} - 44 q^{35} - 29 q^{37} - 75 q^{39} - 3 q^{41} - 88 q^{43} - 21 q^{45} - 21 q^{47} + 63 q^{49} - 85 q^{51} - 24 q^{53} - 85 q^{55} - 35 q^{59} - 78 q^{61} - 74 q^{63} - 13 q^{65} - 68 q^{67} - 43 q^{69} - 59 q^{71} - q^{73} - 45 q^{75} - 33 q^{77} - 140 q^{79} + 51 q^{81} - 27 q^{83} - 84 q^{85} - 61 q^{87} - 2 q^{89} - 92 q^{91} - 51 q^{93} - 51 q^{95} - 10 q^{97} - 115 q^{99}+O(q^{100}) $ 63 * q - 7 * q^3 - 7 * q^5 - 22 * q^7 + 62 * q^9 - 21 * q^11 - 19 * q^13 - 30 * q^15 - 5 * q^17 - 59 * q^19 - 30 * q^21 - 24 * q^23 + 60 * q^25 - 34 * q^27 - 28 * q^29 - 48 * q^31 - q^33 - 44 * q^35 - 29 * q^37 - 75 * q^39 - 3 * q^41 - 88 * q^43 - 21 * q^45 - 21 * q^47 + 63 * q^49 - 85 * q^51 - 24 * q^53 - 85 * q^55 - 35 * q^59 - 78 * q^61 - 74 * q^63 - 13 * q^65 - 68 * q^67 - 43 * q^69 - 59 * q^71 - q^73 - 45 * q^75 - 33 * q^77 - 140 * q^79 + 51 * q^81 - 27 * q^83 - 84 * q^85 - 61 * q^87 - 2 * q^89 - 92 * q^91 - 51 * q^93 - 51 * q^95 - 10 * q^97 - 115 * 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 $

$2$	0	0
$2$	0	0
$3$	−1.58770	−0.916658	−0.458329	−	0.888783i	$-0.651552\pi$
$3$	−1.58770	−0.916658	−0.458329	+	0.888783i	$0.651552\pi$
$4$	0	0
$5$	−0.675402	−0.302049	−0.151025	−	0.988530i	$-0.548257\pi$
$5$	−0.675402	−0.302049	−0.151025	+	0.988530i	$0.548257\pi$
$6$	0	0
$7$	−1.82211	−0.688694	−0.344347	−	0.938843i	$-0.611900\pi$
$7$	−1.82211	−0.688694	−0.344347	+	0.938843i	$0.611900\pi$
$8$	0	0
$9$	−0.479212	−0.159737
$10$	0	0
$11$	−0.0892749	−0.0269174	−0.0134587	−	0.999909i	$-0.504284\pi$
$11$	−0.0892749	−0.0269174	−0.0134587	+	0.999909i	$0.504284\pi$
$12$	0	0
$13$	1.89681	0.526080	0.263040	−	0.964785i	$-0.415275\pi$
$13$	1.89681	0.526080	0.263040	+	0.964785i	$0.415275\pi$
$14$	0	0
$15$	1.07234	0.276876
$16$	0	0
$17$	−2.72739	−0.661490	−0.330745	−	0.943720i	$-0.607300\pi$
$17$	−2.72739	−0.661490	−0.330745	+	0.943720i	$0.607300\pi$
$18$	0	0
$19$	−0.950237	−0.217999	−0.109000	−	0.994042i	$-0.534765\pi$
$19$	−0.950237	−0.217999	−0.109000	+	0.994042i	$0.534765\pi$
$20$	0	0
$21$	2.89297	0.631297
$22$	0	0
$23$	0.452609	0.0943754	0.0471877	−	0.998886i	$-0.484974\pi$
$23$	0.452609	0.0943754	0.0471877	+	0.998886i	$0.484974\pi$
$24$	0	0
$25$	−4.54383	−0.908766
$26$	0	0
$27$	5.52394	1.06308
$28$	0	0
$29$	4.76295	0.884458	0.442229	−	0.896902i	$-0.354188\pi$
$29$	4.76295	0.884458	0.442229	+	0.896902i	$0.354188\pi$
$30$	0	0
$31$	7.32378	1.31539	0.657695	−	0.753284i	$-0.271532\pi$
$31$	7.32378	1.31539	0.657695	+	0.753284i	$0.271532\pi$
$32$	0	0
$33$	0.141742	0.0246741
$34$	0	0
$35$	1.23066	0.208019
$36$	0	0
$37$	10.4886	1.72431	0.862155	−	0.506645i	$-0.169114\pi$
$37$	10.4886	1.72431	0.862155	+	0.506645i	$0.169114\pi$
$38$	0	0
$39$	−3.01156	−0.482236
$40$	0	0
$41$	−6.42533	−1.00347	−0.501734	−	0.865022i	$-0.667304\pi$
$41$	−6.42533	−1.00347	−0.501734	+	0.865022i	$0.667304\pi$
$42$	0	0
$43$	−0.288879	−0.0440536	−0.0220268	−	0.999757i	$-0.507012\pi$
$43$	−0.288879	−0.0440536	−0.0220268	+	0.999757i	$0.507012\pi$
$44$	0	0
$45$	0.323661	0.0482485
$46$	0	0
$47$	−2.76537	−0.403371	−0.201685	−	0.979450i	$-0.564642\pi$
$47$	−2.76537	−0.403371	−0.201685	+	0.979450i	$0.564642\pi$
$48$	0	0
$49$	−3.67991	−0.525701
$50$	0	0
$51$	4.33028	0.606360
$52$	0	0
$53$	8.22321	1.12955	0.564773	−	0.825247i	$-0.308964\pi$
$53$	8.22321	1.12955	0.564773	+	0.825247i	$0.308964\pi$
$54$	0	0
$55$	0.0602965	0.00813038
$56$	0	0
$57$	1.50869	0.199831
$58$	0	0
$59$	5.51634	0.718167	0.359083	−	0.933305i	$-0.383089\pi$
$59$	5.51634	0.718167	0.359083	+	0.933305i	$0.383089\pi$
$60$	0	0
$61$	−1.26310	−0.161723	−0.0808614	−	0.996725i	$-0.525767\pi$
$61$	−1.26310	−0.161723	−0.0808614	+	0.996725i	$0.525767\pi$
$62$	0	0
$63$	0.873178	0.110010
$64$	0	0
$65$	−1.28111	−0.158902
$66$	0	0
$67$	0.673526	0.0822843	0.0411421	−	0.999153i	$-0.486900\pi$
$67$	0.673526	0.0822843	0.0411421	+	0.999153i	$0.486900\pi$
$68$	0	0
$69$	−0.718606	−0.0865100
$70$	0	0
$71$	−7.15399	−0.849022	−0.424511	−	0.905423i	$-0.639554\pi$
$71$	−7.15399	−0.849022	−0.424511	+	0.905423i	$0.639554\pi$
$72$	0	0
$73$	−5.05339	−0.591454	−0.295727	−	0.955272i	$-0.595562\pi$
$73$	−5.05339	−0.591454	−0.295727	+	0.955272i	$0.595562\pi$
$74$	0	0
$75$	7.21424	0.833028
$76$	0	0
$77$	0.162669	0.0185378
$78$	0	0
$79$	−0.224643	−0.0252744	−0.0126372	−	0.999920i	$-0.504023\pi$
$79$	−0.224643	−0.0252744	−0.0126372	+	0.999920i	$0.504023\pi$
$80$	0	0
$81$	−7.33272	−0.814747
$82$	0	0
$83$	−2.39356	−0.262727	−0.131364	−	0.991334i	$-0.541936\pi$
$83$	−2.39356	−0.262727	−0.131364	+	0.991334i	$0.541936\pi$
$84$	0	0
$85$	1.84209	0.199802
$86$	0	0
$87$	−7.56213	−0.810746
$88$	0	0
$89$	12.4780	1.32266	0.661331	−	0.750094i	$-0.269992\pi$
$89$	12.4780	1.32266	0.661331	+	0.750094i	$0.269992\pi$
$90$	0	0
$91$	−3.45620	−0.362308
$92$	0	0
$93$	−11.6280	−1.20576
$94$	0	0
$95$	0.641792	0.0658465
$96$	0	0
$97$	4.64727	0.471859	0.235929	−	0.971770i	$-0.424187\pi$
$97$	4.64727	0.471859	0.235929	+	0.971770i	$0.424187\pi$
$98$	0	0
$99$	0.0427817	0.00429972
$100$	0	0
$101$	12.4195	1.23579	0.617893	−	0.786263i	$-0.287987\pi$
$101$	12.4195	1.23579	0.617893	+	0.786263i	$0.287987\pi$
$102$	0	0
$103$	−16.1363	−1.58996	−0.794978	−	0.606639i	$-0.792517\pi$
$103$	−16.1363	−1.58996	−0.794978	+	0.606639i	$0.792517\pi$
$104$	0	0
$105$	−1.95391	−0.190683
$106$	0	0
$107$	2.44513	0.236379	0.118190	−	0.992991i	$-0.462291\pi$
$107$	2.44513	0.236379	0.118190	+	0.992991i	$0.462291\pi$
$108$	0	0
$109$	3.02094	0.289354	0.144677	−	0.989479i	$-0.453786\pi$
$109$	3.02094	0.289354	0.144677	+	0.989479i	$0.453786\pi$
$110$	0	0
$111$	−16.6527	−1.58060
$112$	0	0
$113$	9.02305	0.848818	0.424409	−	0.905471i	$-0.360482\pi$
$113$	9.02305	0.848818	0.424409	+	0.905471i	$0.360482\pi$
$114$	0	0
$115$	−0.305693	−0.0285060
$116$	0	0
$117$	−0.908974	−0.0840347
$118$	0	0
$119$	4.96961	0.455564
$120$	0	0
$121$	−10.9920	−0.999275
$122$	0	0
$123$	10.2015	0.919837
$124$	0	0
$125$	6.44592	0.576541
$126$	0	0
$127$	−11.8929	−1.05532	−0.527661	−	0.849455i	$-0.676931\pi$
$127$	−11.8929	−1.05532	−0.527661	+	0.849455i	$0.676931\pi$
$128$	0	0
$129$	0.458653	0.0403821
$130$	0	0
$131$	−15.6705	−1.36914	−0.684569	−	0.728948i	$-0.740010\pi$
$131$	−15.6705	−1.36914	−0.684569	+	0.728948i	$0.740010\pi$
$132$	0	0
$133$	1.73144	0.150135
$134$	0	0
$135$	−3.73088	−0.321103
$136$	0	0
$137$	−0.172875	−0.0147697	−0.00738487	−	0.999973i	$-0.502351\pi$
$137$	−0.172875	−0.0147697	−0.00738487	+	0.999973i	$0.502351\pi$
$138$	0	0
$139$	−10.7203	−0.909288	−0.454644	−	0.890673i	$-0.650233\pi$
$139$	−10.7203	−0.909288	−0.454644	+	0.890673i	$0.650233\pi$
$140$	0	0
$141$	4.39058	0.369753
$142$	0	0
$143$	−0.169337	−0.0141607
$144$	0	0
$145$	−3.21691	−0.267150
$146$	0	0
$147$	5.84259	0.481888
$148$	0	0
$149$	−0.135175	−0.0110740	−0.00553700	−	0.999985i	$-0.501762\pi$
$149$	−0.135175	−0.0110740	−0.00553700	+	0.999985i	$0.501762\pi$
$150$	0	0
$151$	10.2251	0.832103	0.416052	−	0.909341i	$-0.363414\pi$
$151$	10.2251	0.832103	0.416052	+	0.909341i	$0.363414\pi$
$152$	0	0
$153$	1.30700	0.105665
$154$	0	0
$155$	−4.94650	−0.397312
$156$	0	0
$157$	9.10458	0.726624	0.363312	−	0.931667i	$-0.381646\pi$
$157$	9.10458	0.726624	0.363312	+	0.931667i	$0.381646\pi$
$158$	0	0
$159$	−13.0560	−1.03541
$160$	0	0
$161$	−0.824703	−0.0649957
$162$	0	0
$163$	−13.6309	−1.06765	−0.533827	−	0.845594i	$-0.679246\pi$
$163$	−13.6309	−1.06765	−0.533827	+	0.845594i	$0.679246\pi$
$164$	0	0
$165$	−0.0957327	−0.00745278
$166$	0	0
$167$	2.00325	0.155016	0.0775079	−	0.996992i	$-0.475304\pi$
$167$	2.00325	0.155016	0.0775079	+	0.996992i	$0.475304\pi$
$168$	0	0
$169$	−9.40212	−0.723240
$170$	0	0
$171$	0.455365	0.0348226
$172$	0	0
$173$	8.94727	0.680249	0.340124	−	0.940380i	$-0.389531\pi$
$173$	8.94727	0.680249	0.340124	+	0.940380i	$0.389531\pi$
$174$	0	0
$175$	8.27937	0.625862
$176$	0	0
$177$	−8.75829	−0.658314
$178$	0	0
$179$	−12.8035	−0.956977	−0.478489	−	0.878094i	$-0.658815\pi$
$179$	−12.8035	−0.956977	−0.478489	+	0.878094i	$0.658815\pi$
$180$	0	0
$181$	20.5704	1.52898	0.764492	−	0.644633i	$-0.222990\pi$
$181$	20.5704	1.52898	0.764492	+	0.644633i	$0.222990\pi$
$182$	0	0
$183$	2.00542	0.148245
$184$	0	0
$185$	−7.08400	−0.520826
$186$	0	0
$187$	0.243488	0.0178056
$188$	0	0
$189$	−10.0652	−0.732138
$190$	0	0
$191$	25.1215	1.81772	0.908862	−	0.417096i	$-0.136952\pi$
$191$	25.1215	1.81772	0.908862	+	0.417096i	$0.136952\pi$
$192$	0	0
$193$	−22.6811	−1.63262	−0.816310	−	0.577615i	$-0.803984\pi$
$193$	−22.6811	−1.63262	−0.816310	+	0.577615i	$0.803984\pi$
$194$	0	0
$195$	2.03401	0.145659
$196$	0	0
$197$	−23.7721	−1.69369	−0.846845	−	0.531840i	$-0.821501\pi$
$197$	−23.7721	−1.69369	−0.846845	+	0.531840i	$0.821501\pi$
$198$	0	0
$199$	17.1249	1.21395	0.606976	−	0.794720i	$-0.292382\pi$
$199$	17.1249	1.21395	0.606976	+	0.794720i	$0.292382\pi$
$200$	0	0
$201$	−1.06936	−0.0754266
$202$	0	0
$203$	−8.67863	−0.609120
$204$	0	0
$205$	4.33968	0.303096
$206$	0	0
$207$	−0.216896	−0.0150753
$208$	0	0
$209$	0.0848323	0.00586797
$210$	0	0
$211$	−13.2391	−0.911420	−0.455710	−	0.890128i	$-0.650615\pi$
$211$	−13.2391	−0.911420	−0.455710	+	0.890128i	$0.650615\pi$
$212$	0	0
$213$	11.3584	0.778263
$214$	0	0
$215$	0.195109	0.0133063
$216$	0	0
$217$	−13.3448	−0.905901
$218$	0	0
$219$	8.02326	0.542161
$220$	0	0
$221$	−5.17334	−0.347996
$222$	0	0
$223$	−0.448661	−0.0300445	−0.0150223	−	0.999887i	$-0.504782\pi$
$223$	−0.448661	−0.0300445	−0.0150223	+	0.999887i	$0.504782\pi$
$224$	0	0
$225$	2.17746	0.145164
$226$	0	0
$227$	−18.3016	−1.21472	−0.607359	−	0.794427i	$-0.707771\pi$
$227$	−18.3016	−1.21472	−0.607359	+	0.794427i	$0.707771\pi$
$228$	0	0
$229$	5.88328	0.388778	0.194389	−	0.980925i	$-0.437728\pi$
$229$	5.88328	0.388778	0.194389	+	0.980925i	$0.437728\pi$
$230$	0	0
$231$	−0.258269	−0.0169929
$232$	0	0
$233$	14.3305	0.938825	0.469412	−	0.882979i	$-0.344466\pi$
$233$	14.3305	0.938825	0.469412	+	0.882979i	$0.344466\pi$
$234$	0	0
$235$	1.86774	0.121838
$236$	0	0
$237$	0.356666	0.0231680
$238$	0	0
$239$	1.56559	0.101270	0.0506348	−	0.998717i	$-0.483876\pi$
$239$	1.56559	0.101270	0.0506348	+	0.998717i	$0.483876\pi$
$240$	0	0
$241$	18.6317	1.20017	0.600087	−	0.799935i	$-0.295133\pi$
$241$	18.6317	1.20017	0.600087	+	0.799935i	$0.295133\pi$
$242$	0	0
$243$	−4.92968	−0.316239
$244$	0	0
$245$	2.48542	0.158788
$246$	0	0
$247$	−1.80242	−0.114685
$248$	0	0
$249$	3.80025	0.240831
$250$	0	0
$251$	−19.1757	−1.21036	−0.605181	−	0.796088i	$-0.706899\pi$
$251$	−19.1757	−1.21036	−0.605181	+	0.796088i	$0.706899\pi$
$252$	0	0
$253$	−0.0404066	−0.00254034
$254$	0	0
$255$	−2.92468	−0.183150
$256$	0	0
$257$	7.59180	0.473563	0.236782	−	0.971563i	$-0.423907\pi$
$257$	7.59180	0.473563	0.236782	+	0.971563i	$0.423907\pi$
$258$	0	0
$259$	−19.1113	−1.18752
$260$	0	0
$261$	−2.28246	−0.141281
$262$	0	0
$263$	−27.2357	−1.67943	−0.839713	−	0.543031i	$-0.817277\pi$
$263$	−27.2357	−1.67943	−0.839713	+	0.543031i	$0.817277\pi$
$264$	0	0
$265$	−5.55398	−0.341178
$266$	0	0
$267$	−19.8113	−1.21243
$268$	0	0
$269$	−26.9339	−1.64219	−0.821095	−	0.570791i	$-0.806637\pi$
$269$	−26.9339	−1.64219	−0.821095	+	0.570791i	$0.806637\pi$
$270$	0	0
$271$	−19.3389	−1.17475	−0.587377	−	0.809314i	$-0.699839\pi$
$271$	−19.3389	−1.17475	−0.587377	+	0.809314i	$0.699839\pi$
$272$	0	0
$273$	5.48740	0.332113
$274$	0	0
$275$	0.405650	0.0244616
$276$	0	0
$277$	23.5193	1.41314	0.706569	−	0.707644i	$-0.250242\pi$
$277$	23.5193	1.41314	0.706569	+	0.707644i	$0.250242\pi$
$278$	0	0
$279$	−3.50965	−0.210117
$280$	0	0
$281$	−23.8870	−1.42498	−0.712490	−	0.701682i	$-0.752433\pi$
$281$	−23.8870	−1.42498	−0.712490	+	0.701682i	$0.752433\pi$
$282$	0	0
$283$	−20.1080	−1.19530	−0.597649	−	0.801758i	$-0.703898\pi$
$283$	−20.1080	−1.19530	−0.597649	+	0.801758i	$0.703898\pi$
$284$	0	0
$285$	−1.01897	−0.0603587
$286$	0	0
$287$	11.7077	0.691082
$288$	0	0
$289$	−9.56134	−0.562432
$290$	0	0
$291$	−7.37846	−0.432533
$292$	0	0
$293$	−11.8757	−0.693786	−0.346893	−	0.937905i	$-0.612763\pi$
$293$	−11.8757	−0.693786	−0.346893	+	0.937905i	$0.612763\pi$
$294$	0	0
$295$	−3.72575	−0.216922
$296$	0	0
$297$	−0.493150	−0.0286154
$298$	0	0
$299$	0.858512	0.0496490
$300$	0	0
$301$	0.526370	0.0303394
$302$	0	0
$303$	−19.7184	−1.13279
$304$	0	0
$305$	0.853098	0.0488482
$306$	0	0
$307$	8.78194	0.501212	0.250606	−	0.968089i	$-0.419370\pi$
$307$	8.78194	0.501212	0.250606	+	0.968089i	$0.419370\pi$
$308$	0	0
$309$	25.6196	1.45745
$310$	0	0
$311$	−21.3638	−1.21143	−0.605714	−	0.795682i	$-0.707113\pi$
$311$	−21.3638	−1.21143	−0.605714	+	0.795682i	$0.707113\pi$
$312$	0	0
$313$	13.8088	0.780519	0.390260	−	0.920705i	$-0.372385\pi$
$313$	13.8088	0.780519	0.390260	+	0.920705i	$0.372385\pi$
$314$	0	0
$315$	−0.589747	−0.0332285
$316$	0	0
$317$	0.371584	0.0208702	0.0104351	−	0.999946i	$-0.496678\pi$
$317$	0.371584	0.0208702	0.0104351	+	0.999946i	$0.496678\pi$
$318$	0	0
$319$	−0.425212	−0.0238073
$320$	0	0
$321$	−3.88213	−0.216679
$322$	0	0
$323$	2.59167	0.144204
$324$	0	0
$325$	−8.61878	−0.478084
$326$	0	0
$327$	−4.79634	−0.265238
$328$	0	0
$329$	5.03881	0.277799
$330$	0	0
$331$	−13.3678	−0.734759	−0.367379	−	0.930071i	$-0.619745\pi$
$331$	−13.3678	−0.734759	−0.367379	+	0.930071i	$0.619745\pi$
$332$	0	0
$333$	−5.02625	−0.275437
$334$	0	0
$335$	−0.454901	−0.0248539
$336$	0	0
$337$	−26.3143	−1.43343	−0.716715	−	0.697367i	$-0.754355\pi$
$337$	−26.3143	−1.43343	−0.716715	+	0.697367i	$0.754355\pi$
$338$	0	0
$339$	−14.3259	−0.778076
$340$	0	0
$341$	−0.653830	−0.0354069
$342$	0	0
$343$	19.4600	1.05074
$344$	0	0
$345$	0.485348	0.0261303
$346$	0	0
$347$	−31.7007	−1.70178	−0.850892	−	0.525341i	$-0.823938\pi$
$347$	−31.7007	−1.70178	−0.850892	+	0.525341i	$0.823938\pi$
$348$	0	0
$349$	−23.0153	−1.23198	−0.615989	−	0.787754i	$-0.711244\pi$
$349$	−23.0153	−1.23198	−0.615989	+	0.787754i	$0.711244\pi$
$350$	0	0
$351$	10.4779	0.559267
$352$	0	0
$353$	−1.60368	−0.0853552	−0.0426776	−	0.999089i	$-0.513589\pi$
$353$	−1.60368	−0.0853552	−0.0426776	+	0.999089i	$0.513589\pi$
$354$	0	0
$355$	4.83182	0.256446
$356$	0	0
$357$	−7.89025	−0.417596
$358$	0	0
$359$	22.5067	1.18786	0.593928	−	0.804518i	$-0.297576\pi$
$359$	22.5067	1.18786	0.593928	+	0.804518i	$0.297576\pi$
$360$	0	0
$361$	−18.0971	−0.952476
$362$	0	0
$363$	17.4520	0.915994
$364$	0	0
$365$	3.41307	0.178648
$366$	0	0
$367$	−28.6784	−1.49700	−0.748500	−	0.663134i	$-0.769226\pi$
$367$	−28.6784	−1.49700	−0.748500	+	0.663134i	$0.769226\pi$
$368$	0	0
$369$	3.07910	0.160291
$370$	0	0
$371$	−14.9836	−0.777911
$372$	0	0
$373$	29.8023	1.54311	0.771553	−	0.636165i	$-0.219480\pi$
$373$	29.8023	1.54311	0.771553	+	0.636165i	$0.219480\pi$
$374$	0	0
$375$	−10.2342	−0.528491
$376$	0	0
$377$	9.03441	0.465296
$378$	0	0
$379$	−1.18516	−0.0608776	−0.0304388	−	0.999537i	$-0.509690\pi$
$379$	−1.18516	−0.0608776	−0.0304388	+	0.999537i	$0.509690\pi$
$380$	0	0
$381$	18.8823	0.967370
$382$	0	0
$383$	38.1176	1.94772	0.973860	−	0.227151i	$-0.0729411\pi$
$383$	38.1176	1.94772	0.973860	+	0.227151i	$0.0729411\pi$
$384$	0	0
$385$	−0.109867	−0.00559934
$386$	0	0
$387$	0.138434	0.00703701
$388$	0	0
$389$	9.62100	0.487804	0.243902	−	0.969800i	$-0.421572\pi$
$389$	9.62100	0.487804	0.243902	+	0.969800i	$0.421572\pi$
$390$	0	0
$391$	−1.23444	−0.0624284
$392$	0	0
$393$	24.8800	1.25503
$394$	0	0
$395$	0.151725	0.00763410
$396$	0	0
$397$	−12.5908	−0.631912	−0.315956	−	0.948774i	$-0.602325\pi$
$397$	−12.5908	−0.631912	−0.315956	+	0.948774i	$0.602325\pi$
$398$	0	0
$399$	−2.74900	−0.137622
$400$	0	0
$401$	−15.5879	−0.778420	−0.389210	−	0.921149i	$-0.627252\pi$
$401$	−15.5879	−0.778420	−0.389210	+	0.921149i	$0.627252\pi$
$402$	0	0
$403$	13.8918	0.692001
$404$	0	0
$405$	4.95253	0.246093
$406$	0	0
$407$	−0.936366	−0.0464140
$408$	0	0
$409$	−2.81666	−0.139275	−0.0696375	−	0.997572i	$-0.522184\pi$
$409$	−2.81666	−0.139275	−0.0696375	+	0.997572i	$0.522184\pi$
$410$	0	0
$411$	0.274474	0.0135388
$412$	0	0
$413$	−10.0514	−0.494597
$414$	0	0
$415$	1.61662	0.0793565
$416$	0	0
$417$	17.0207	0.833506
$418$	0	0
$419$	28.3010	1.38259	0.691297	−	0.722570i	$-0.257040\pi$
$419$	28.3010	1.38259	0.691297	+	0.722570i	$0.257040\pi$
$420$	0	0
$421$	−14.4509	−0.704296	−0.352148	−	0.935944i	$-0.614549\pi$
$421$	−14.4509	−0.704296	−0.352148	+	0.935944i	$0.614549\pi$
$422$	0	0
$423$	1.32520	0.0644334
$424$	0	0
$425$	12.3928	0.601140
$426$	0	0
$427$	2.30150	0.111378
$428$	0	0
$429$	0.268857	0.0129805
$430$	0	0
$431$	−29.2251	−1.40772	−0.703861	−	0.710338i	$-0.748542\pi$
$431$	−29.2251	−1.40772	−0.703861	+	0.710338i	$0.748542\pi$
$432$	0	0
$433$	32.7032	1.57161	0.785807	−	0.618472i	$-0.212248\pi$
$433$	32.7032	1.57161	0.785807	+	0.618472i	$0.212248\pi$
$434$	0	0
$435$	5.10748	0.244885
$436$	0	0
$437$	−0.430085	−0.0205738
$438$	0	0
$439$	−30.5966	−1.46029	−0.730147	−	0.683290i	$-0.760548\pi$
$439$	−30.5966	−1.46029	−0.730147	+	0.683290i	$0.760548\pi$
$440$	0	0
$441$	1.76346	0.0839742
$442$	0	0
$443$	21.8386	1.03758	0.518791	−	0.854901i	$-0.326382\pi$
$443$	21.8386	1.03758	0.518791	+	0.854901i	$0.326382\pi$
$444$	0	0
$445$	−8.42765	−0.399509
$446$	0	0
$447$	0.214618	0.0101511
$448$	0	0
$449$	−13.2634	−0.625937	−0.312969	−	0.949764i	$-0.601323\pi$
$449$	−13.2634	−0.625937	−0.312969	+	0.949764i	$0.601323\pi$
$450$	0	0
$451$	0.573621	0.0270108
$452$	0	0
$453$	−16.2343	−0.762755
$454$	0	0
$455$	2.33432	0.109435
$456$	0	0
$457$	23.2927	1.08959	0.544794	−	0.838570i	$-0.316608\pi$
$457$	23.2927	1.08959	0.544794	+	0.838570i	$0.316608\pi$
$458$	0	0
$459$	−15.0660	−0.703218
$460$	0	0
$461$	−16.4752	−0.767325	−0.383662	−	0.923473i	$-0.625337\pi$
$461$	−16.4752	−0.767325	−0.383662	+	0.923473i	$0.625337\pi$
$462$	0	0
$463$	34.5528	1.60580	0.802902	−	0.596110i	$-0.203288\pi$
$463$	34.5528	1.60580	0.802902	+	0.596110i	$0.203288\pi$
$464$	0	0
$465$	7.85355	0.364200
$466$	0	0
$467$	−35.5206	−1.64370	−0.821848	−	0.569706i	$-0.807057\pi$
$467$	−35.5206	−1.64370	−0.821848	+	0.569706i	$0.807057\pi$
$468$	0	0
$469$	−1.22724	−0.0566687
$470$	0	0
$471$	−14.4553	−0.666066
$472$	0	0
$473$	0.0257896	0.00118581
$474$	0	0
$475$	4.31772	0.198110
$476$	0	0
$477$	−3.94067	−0.180431
$478$	0	0
$479$	1.66173	0.0759266	0.0379633	−	0.999279i	$-0.487913\pi$
$479$	1.66173	0.0759266	0.0379633	+	0.999279i	$0.487913\pi$
$480$	0	0
$481$	19.8948	0.907125
$482$	0	0
$483$	1.30938	0.0595789
$484$	0	0
$485$	−3.13877	−0.142524
$486$	0	0
$487$	−17.9924	−0.815315	−0.407657	−	0.913135i	$-0.633654\pi$
$487$	−17.9924	−0.815315	−0.407657	+	0.913135i	$0.633654\pi$
$488$	0	0
$489$	21.6417	0.978673
$490$	0	0
$491$	40.8236	1.84234	0.921172	−	0.389156i	$-0.127233\pi$
$491$	40.8236	1.84234	0.921172	+	0.389156i	$0.127233\pi$
$492$	0	0
$493$	−12.9904	−0.585060
$494$	0	0
$495$	−0.0288948	−0.00129873
$496$	0	0
$497$	13.0354	0.584716
$498$	0	0
$499$	−36.9180	−1.65268	−0.826338	−	0.563174i	$-0.809580\pi$
$499$	−36.9180	−1.65268	−0.826338	+	0.563174i	$0.809580\pi$
$500$	0	0
$501$	−3.18055	−0.142097
$502$	0	0
$503$	−0.470722	−0.0209884	−0.0104942	−	0.999945i	$-0.503340\pi$
$503$	−0.470722	−0.0209884	−0.0104942	+	0.999945i	$0.503340\pi$
$504$	0	0
$505$	−8.38815	−0.373268
$506$	0	0
$507$	14.9277	0.662964
$508$	0	0
$509$	25.6957	1.13894	0.569471	−	0.822012i	$-0.307148\pi$
$509$	25.6957	1.13894	0.569471	+	0.822012i	$0.307148\pi$
$510$	0	0
$511$	9.20784	0.407331
$512$	0	0
$513$	−5.24905	−0.231751
$514$	0	0
$515$	10.8985	0.480244
$516$	0	0
$517$	0.246878	0.0108577
$518$	0	0
$519$	−14.2056	−0.623556
$520$	0	0
$521$	−11.4503	−0.501646	−0.250823	−	0.968033i	$-0.580701\pi$
$521$	−11.4503	−0.501646	−0.250823	+	0.968033i	$0.580701\pi$
$522$	0	0
$523$	−15.7055	−0.686753	−0.343376	−	0.939198i	$-0.611571\pi$
$523$	−15.7055	−0.686753	−0.343376	+	0.939198i	$0.611571\pi$
$524$	0	0
$525$	−13.1451	−0.573701
$526$	0	0
$527$	−19.9748	−0.870117
$528$	0	0
$529$	−22.7951	−0.991093
$530$	0	0
$531$	−2.64350	−0.114718
$532$	0	0
$533$	−12.1876	−0.527904
$534$	0	0
$535$	−1.65144	−0.0713982
$536$	0	0
$537$	20.3281	0.877221
$538$	0	0
$539$	0.328524	0.0141505
$540$	0	0
$541$	−3.66316	−0.157492	−0.0787458	−	0.996895i	$-0.525092\pi$
$541$	−3.66316	−0.157492	−0.0787458	+	0.996895i	$0.525092\pi$
$542$	0	0
$543$	−32.6596	−1.40156
$544$	0	0
$545$	−2.04035	−0.0873990
$546$	0	0
$547$	−16.1587	−0.690898	−0.345449	−	0.938438i	$-0.612273\pi$
$547$	−16.1587	−0.690898	−0.345449	+	0.938438i	$0.612273\pi$
$548$	0	0
$549$	0.605291	0.0258332
$550$	0	0
$551$	−4.52593	−0.192811
$552$	0	0
$553$	0.409326	0.0174063
$554$	0	0
$555$	11.2473	0.477420
$556$	0	0
$557$	−23.1742	−0.981923	−0.490962	−	0.871181i	$-0.663355\pi$
$557$	−23.1742	−0.981923	−0.490962	+	0.871181i	$0.663355\pi$
$558$	0	0
$559$	−0.547948	−0.0231757
$560$	0	0
$561$	−0.386585	−0.0163216
$562$	0	0
$563$	18.9513	0.798703	0.399351	−	0.916798i	$-0.369235\pi$
$563$	18.9513	0.798703	0.399351	+	0.916798i	$0.369235\pi$
$564$	0	0
$565$	−6.09419	−0.256384
$566$	0	0
$567$	13.3610	0.561111
$568$	0	0
$569$	−21.9531	−0.920320	−0.460160	−	0.887836i	$-0.652208\pi$
$569$	−21.9531	−0.920320	−0.460160	+	0.887836i	$0.652208\pi$
$570$	0	0
$571$	−20.9637	−0.877303	−0.438652	−	0.898657i	$-0.644544\pi$
$571$	−20.9637	−0.877303	−0.438652	+	0.898657i	$0.644544\pi$
$572$	0	0
$573$	−39.8853	−1.66623
$574$	0	0
$575$	−2.05658	−0.0857652
$576$	0	0
$577$	−27.2559	−1.13468	−0.567339	−	0.823484i	$-0.692027\pi$
$577$	−27.2559	−1.13468	−0.567339	+	0.823484i	$0.692027\pi$
$578$	0	0
$579$	36.0107	1.49655
$580$	0	0
$581$	4.36133	0.180939
$582$	0	0
$583$	−0.734127	−0.0304044
$584$	0	0
$585$	0.613923	0.0253826
$586$	0	0
$587$	15.0611	0.621639	0.310820	−	0.950469i	$-0.399396\pi$
$587$	15.0611	0.621639	0.310820	+	0.950469i	$0.399396\pi$
$588$	0	0
$589$	−6.95933	−0.286754
$590$	0	0
$591$	37.7429	1.55254
$592$	0	0
$593$	5.74352	0.235858	0.117929	−	0.993022i	$-0.462374\pi$
$593$	5.74352	0.235858	0.117929	+	0.993022i	$0.462374\pi$
$594$	0	0
$595$	−3.35649	−0.137603
$596$	0	0
$597$	−27.1892	−1.11278
$598$	0	0
$599$	17.5387	0.716610	0.358305	−	0.933605i	$-0.383355\pi$
$599$	17.5387	0.716610	0.358305	+	0.933605i	$0.383355\pi$
$600$	0	0
$601$	42.3076	1.72576	0.862882	−	0.505405i	$-0.168657\pi$
$601$	42.3076	1.72576	0.862882	+	0.505405i	$0.168657\pi$
$602$	0	0
$603$	−0.322762	−0.0131439
$604$	0	0
$605$	7.42404	0.301830
$606$	0	0
$607$	−45.8865	−1.86248	−0.931238	−	0.364412i	$-0.881270\pi$
$607$	−45.8865	−1.86248	−0.931238	+	0.364412i	$0.881270\pi$
$608$	0	0
$609$	13.7790	0.558355
$610$	0	0
$611$	−5.24538	−0.212205
$612$	0	0
$613$	−23.0133	−0.929499	−0.464749	−	0.885442i	$-0.653856\pi$
$613$	−23.0133	−0.929499	−0.464749	+	0.885442i	$0.653856\pi$
$614$	0	0
$615$	−6.89011	−0.277836
$616$	0	0
$617$	41.9067	1.68710	0.843550	−	0.537051i	$-0.180462\pi$
$617$	41.9067	1.68710	0.843550	+	0.537051i	$0.180462\pi$
$618$	0	0
$619$	−23.2286	−0.933637	−0.466818	−	0.884353i	$-0.654600\pi$
$619$	−23.2286	−0.933637	−0.466818	+	0.884353i	$0.654600\pi$
$620$	0	0
$621$	2.50018	0.100329
$622$	0	0
$623$	−22.7363	−0.910909
$624$	0	0
$625$	18.3656	0.734623
$626$	0	0
$627$	−0.134688	−0.00537893
$628$	0	0
$629$	−28.6064	−1.14061
$630$	0	0
$631$	14.7607	0.587615	0.293807	−	0.955865i	$-0.405078\pi$
$631$	14.7607	0.587615	0.293807	+	0.955865i	$0.405078\pi$
$632$	0	0
$633$	21.0198	0.835461
$634$	0	0
$635$	8.03248	0.318759
$636$	0	0
$637$	−6.98008	−0.276561
$638$	0	0
$639$	3.42828	0.135621
$640$	0	0
$641$	−29.0842	−1.14876	−0.574379	−	0.818589i	$-0.694756\pi$
$641$	−29.0842	−1.14876	−0.574379	+	0.818589i	$0.694756\pi$
$642$	0	0
$643$	25.6979	1.01343	0.506714	−	0.862114i	$-0.330860\pi$
$643$	25.6979	1.01343	0.506714	+	0.862114i	$0.330860\pi$
$644$	0	0
$645$	−0.309775	−0.0121974
$646$	0	0
$647$	2.52377	0.0992196	0.0496098	−	0.998769i	$-0.484202\pi$
$647$	2.52377	0.0992196	0.0496098	+	0.998769i	$0.484202\pi$
$648$	0	0
$649$	−0.492471	−0.0193312
$650$	0	0
$651$	21.1874	0.830401
$652$	0	0
$653$	29.2430	1.14437	0.572184	−	0.820125i	$-0.306096\pi$
$653$	29.2430	1.14437	0.572184	+	0.820125i	$0.306096\pi$
$654$	0	0
$655$	10.5839	0.413547
$656$	0	0
$657$	2.42164	0.0944774
$658$	0	0
$659$	−23.4016	−0.911597	−0.455799	−	0.890083i	$-0.650646\pi$
$659$	−23.4016	−0.911597	−0.455799	+	0.890083i	$0.650646\pi$
$660$	0	0
$661$	−5.53757	−0.215386	−0.107693	−	0.994184i	$-0.534346\pi$
$661$	−5.53757	−0.215386	−0.107693	+	0.994184i	$0.534346\pi$
$662$	0	0
$663$	8.21371	0.318994
$664$	0	0
$665$	−1.16942	−0.0453480
$666$	0	0
$667$	2.15575	0.0834711
$668$	0	0
$669$	0.712338	0.0275406
$670$	0	0
$671$	0.112763	0.00435316
$672$	0	0
$673$	15.5015	0.597539	0.298769	−	0.954325i	$-0.403424\pi$
$673$	15.5015	0.597539	0.298769	+	0.954325i	$0.403424\pi$
$674$	0	0
$675$	−25.0999	−0.966094
$676$	0	0
$677$	37.3580	1.43578	0.717892	−	0.696155i	$-0.245107\pi$
$677$	37.3580	1.43578	0.717892	+	0.696155i	$0.245107\pi$
$678$	0	0
$679$	−8.46784	−0.324966
$680$	0	0
$681$	29.0574	1.11348
$682$	0	0
$683$	21.1231	0.808252	0.404126	−	0.914703i	$-0.367576\pi$
$683$	21.1231	0.808252	0.404126	+	0.914703i	$0.367576\pi$
$684$	0	0
$685$	0.116760	0.00446118
$686$	0	0
$687$	−9.34087	−0.356376
$688$	0	0
$689$	15.5979	0.594231
$690$	0	0
$691$	−24.5953	−0.935648	−0.467824	−	0.883822i	$-0.654962\pi$
$691$	−24.5953	−0.935648	−0.467824	+	0.883822i	$0.654962\pi$
$692$	0	0
$693$	−0.0779530	−0.00296119
$694$	0	0
$695$	7.24054	0.274649
$696$	0	0
$697$	17.5244	0.663783
$698$	0	0
$699$	−22.7526	−0.860582
$700$	0	0
$701$	−36.5608	−1.38088	−0.690441	−	0.723388i	$-0.742584\pi$
$701$	−36.5608	−1.38088	−0.690441	+	0.723388i	$0.742584\pi$
$702$	0	0
$703$	−9.96662	−0.375898
$704$	0	0
$705$	−2.96540	−0.111684
$706$	0	0
$707$	−22.6297	−0.851077
$708$	0	0
$709$	3.44144	0.129246	0.0646231	−	0.997910i	$-0.479415\pi$
$709$	3.44144	0.129246	0.0646231	+	0.997910i	$0.479415\pi$
$710$	0	0
$711$	0.107652	0.00403726
$712$	0	0
$713$	3.31481	0.124140
$714$	0	0
$715$	0.114371	0.00427723
$716$	0	0
$717$	−2.48568	−0.0928295
$718$	0	0
$719$	1.13671	0.0423922	0.0211961	−	0.999775i	$-0.493253\pi$
$719$	1.13671	0.0423922	0.0211961	+	0.999775i	$0.493253\pi$
$720$	0	0
$721$	29.4021	1.09499
$722$	0	0
$723$	−29.5816	−1.10015
$724$	0	0
$725$	−21.6420	−0.803765
$726$	0	0
$727$	30.6595	1.13710	0.568549	−	0.822650i	$-0.307505\pi$
$727$	30.6595	1.13710	0.568549	+	0.822650i	$0.307505\pi$
$728$	0	0
$729$	29.8250	1.10463
$730$	0	0
$731$	0.787886	0.0291410
$732$	0	0
$733$	1.01538	0.0375038	0.0187519	−	0.999824i	$-0.494031\pi$
$733$	1.01538	0.0375038	0.0187519	+	0.999824i	$0.494031\pi$
$734$	0	0
$735$	−3.94610	−0.145554
$736$	0	0
$737$	−0.0601290	−0.00221488
$738$	0	0
$739$	−14.2486	−0.524141	−0.262071	−	0.965049i	$-0.584405\pi$
$739$	−14.2486	−0.524141	−0.262071	+	0.965049i	$0.584405\pi$
$740$	0	0
$741$	2.86170	0.105127
$742$	0	0
$743$	−13.9188	−0.510630	−0.255315	−	0.966858i	$-0.582179\pi$
$743$	−13.9188	−0.510630	−0.255315	+	0.966858i	$0.582179\pi$
$744$	0	0
$745$	0.0912978	0.00334489
$746$	0	0
$747$	1.14702	0.0419674
$748$	0	0
$749$	−4.45530	−0.162793
$750$	0	0
$751$	33.3106	1.21552	0.607760	−	0.794121i	$-0.292068\pi$
$751$	33.3106	1.21552	0.607760	+	0.794121i	$0.292068\pi$
$752$	0	0
$753$	30.4453	1.10949
$754$	0	0
$755$	−6.90603	−0.251336
$756$	0	0
$757$	50.2129	1.82502	0.912509	−	0.409057i	$-0.134142\pi$
$757$	50.2129	1.82502	0.912509	+	0.409057i	$0.134142\pi$
$758$	0	0
$759$	0.0641535	0.00232863
$760$	0	0
$761$	−20.2137	−0.732746	−0.366373	−	0.930468i	$-0.619401\pi$
$761$	−20.2137	−0.732746	−0.366373	+	0.930468i	$0.619401\pi$
$762$	0	0
$763$	−5.50449	−0.199276
$764$	0	0
$765$	−0.882750	−0.0319159
$766$	0	0
$767$	10.4634	0.377813
$768$	0	0
$769$	−41.9685	−1.51342	−0.756711	−	0.653749i	$-0.773195\pi$
$769$	−41.9685	−1.51342	−0.756711	+	0.653749i	$0.773195\pi$
$770$	0	0
$771$	−12.0535	−0.434096
$772$	0	0
$773$	−24.6200	−0.885520	−0.442760	−	0.896640i	$-0.646001\pi$
$773$	−24.6200	−0.885520	−0.442760	+	0.896640i	$0.646001\pi$
$774$	0	0
$775$	−33.2780	−1.19538
$776$	0	0
$777$	30.3431	1.08855
$778$	0	0
$779$	6.10558	0.218755
$780$	0	0
$781$	0.638672	0.0228535
$782$	0	0
$783$	26.3103	0.940252
$784$	0	0
$785$	−6.14925	−0.219476
$786$	0	0
$787$	−10.7874	−0.384529	−0.192264	−	0.981343i	$-0.561583\pi$
$787$	−10.7874	−0.384529	−0.192264	+	0.981343i	$0.561583\pi$
$788$	0	0
$789$	43.2421	1.53946
$790$	0	0
$791$	−16.4410	−0.584575
$792$	0	0
$793$	−2.39585	−0.0850792
$794$	0	0
$795$	8.81804	0.312744
$796$	0	0
$797$	30.8599	1.09312	0.546558	−	0.837422i	$-0.315938\pi$
$797$	30.8599	1.09312	0.546558	+	0.837422i	$0.315938\pi$
$798$	0	0
$799$	7.54225	0.266826
$800$	0	0
$801$	−5.97960	−0.211279
$802$	0	0
$803$	0.451141	0.0159204
$804$	0	0
$805$	0.557006	0.0196319
$806$	0	0
$807$	42.7630	1.50533
$808$	0	0
$809$	−8.01506	−0.281795	−0.140897	−	0.990024i	$-0.544999\pi$
$809$	−8.01506	−0.281795	−0.140897	+	0.990024i	$0.544999\pi$
$810$	0	0
$811$	−53.2463	−1.86973	−0.934865	−	0.355004i	$-0.884479\pi$
$811$	−53.2463	−1.86973	−0.934865	+	0.355004i	$0.884479\pi$
$812$	0	0
$813$	30.7043	1.07685
$814$	0	0
$815$	9.20633	0.322484
$816$	0	0
$817$	0.274503	0.00960365
$818$	0	0
$819$	1.65625	0.0578741
$820$	0	0
$821$	41.2683	1.44027	0.720137	−	0.693832i	$-0.244079\pi$
$821$	41.2683	1.44027	0.720137	+	0.693832i	$0.244079\pi$
$822$	0	0
$823$	−45.3250	−1.57993	−0.789966	−	0.613151i	$-0.789902\pi$
$823$	−45.3250	−1.57993	−0.789966	+	0.613151i	$0.789902\pi$
$824$	0	0
$825$	−0.644051	−0.0224230
$826$	0	0
$827$	6.30338	0.219190	0.109595	−	0.993976i	$-0.465045\pi$
$827$	6.30338	0.219190	0.109595	+	0.993976i	$0.465045\pi$
$828$	0	0
$829$	−42.6317	−1.48066	−0.740330	−	0.672244i	$-0.765331\pi$
$829$	−42.6317	−1.48066	−0.740330	+	0.672244i	$0.765331\pi$
$830$	0	0
$831$	−37.3416	−1.29537
$832$	0	0
$833$	10.0366	0.347746
$834$	0	0
$835$	−1.35300	−0.0468224
$836$	0	0
$837$	40.4561	1.39837
$838$	0	0
$839$	−21.8594	−0.754670	−0.377335	−	0.926077i	$-0.623159\pi$
$839$	−21.8594	−0.754670	−0.377335	+	0.926077i	$0.623159\pi$
$840$	0	0
$841$	−6.31430	−0.217735
$842$	0	0
$843$	37.9254	1.30622
$844$	0	0
$845$	6.35021	0.218454
$846$	0	0
$847$	20.0287	0.688195
$848$	0	0
$849$	31.9255	1.09568
$850$	0	0
$851$	4.74722	0.162732
$852$	0	0
$853$	−51.1302	−1.75066	−0.875332	−	0.483523i	$-0.839357\pi$
$853$	−51.1302	−1.75066	−0.875332	+	0.483523i	$0.839357\pi$
$854$	0	0
$855$	−0.307555	−0.0105181
$856$	0	0
$857$	−24.1550	−0.825119	−0.412560	−	0.910931i	$-0.635365\pi$
$857$	−24.1550	−0.825119	−0.412560	+	0.910931i	$0.635365\pi$
$858$	0	0
$859$	−21.1700	−0.722312	−0.361156	−	0.932505i	$-0.617618\pi$
$859$	−21.1700	−0.722312	−0.361156	+	0.932505i	$0.617618\pi$
$860$	0	0
$861$	−18.5883	−0.633486
$862$	0	0
$863$	−35.9252	−1.22291	−0.611453	−	0.791280i	$-0.709415\pi$
$863$	−35.9252	−1.22291	−0.611453	+	0.791280i	$0.709415\pi$
$864$	0	0
$865$	−6.04301	−0.205468
$866$	0	0
$867$	15.1805	0.515558
$868$	0	0
$869$	0.0200550	0.000680320 0
$870$	0	0
$871$	1.27755	0.0432881
$872$	0	0
$873$	−2.22703	−0.0753735
$874$	0	0
$875$	−11.7452	−0.397060
$876$	0	0
$877$	20.9374	0.707007	0.353503	−	0.935433i	$-0.384990\pi$
$877$	20.9374	0.707007	0.353503	+	0.935433i	$0.384990\pi$
$878$	0	0
$879$	18.8550	0.635965
$880$	0	0
$881$	−3.43105	−0.115595	−0.0577975	−	0.998328i	$-0.518408\pi$
$881$	−3.43105	−0.115595	−0.0577975	+	0.998328i	$0.518408\pi$
$882$	0	0
$883$	45.4118	1.52823	0.764115	−	0.645081i	$-0.223176\pi$
$883$	45.4118	1.52823	0.764115	+	0.645081i	$0.223176\pi$
$884$	0	0
$885$	5.91537	0.198843
$886$	0	0
$887$	15.2862	0.513259	0.256630	−	0.966510i	$-0.417388\pi$
$887$	15.2862	0.513259	0.256630	+	0.966510i	$0.417388\pi$
$888$	0	0
$889$	21.6702	0.726794
$890$	0	0
$891$	0.654628	0.0219309
$892$	0	0
$893$	2.62776	0.0879345
$894$	0	0
$895$	8.64750	0.289054
$896$	0	0
$897$	−1.36306	−0.0455112
$898$	0	0
$899$	34.8828	1.16341
$900$	0	0
$901$	−22.4279	−0.747182
$902$	0	0
$903$	−0.835716	−0.0278109
$904$	0	0
$905$	−13.8933	−0.461828
$906$	0	0
$907$	−17.9678	−0.596609	−0.298305	−	0.954471i	$-0.596421\pi$
$907$	−17.9678	−0.596609	−0.298305	+	0.954471i	$0.596421\pi$
$908$	0	0
$909$	−5.95157	−0.197401
$910$	0	0
$911$	8.81443	0.292035	0.146018	−	0.989282i	$-0.453354\pi$
$911$	8.81443	0.292035	0.146018	+	0.989282i	$0.453354\pi$
$912$	0	0
$913$	0.213685	0.00707194
$914$	0	0
$915$	−1.35446	−0.0447771
$916$	0	0
$917$	28.5534	0.942917
$918$	0	0
$919$	31.0109	1.02296	0.511478	−	0.859296i	$-0.329098\pi$
$919$	31.0109	1.02296	0.511478	+	0.859296i	$0.329098\pi$
$920$	0	0
$921$	−13.9431	−0.459440
$922$	0	0
$923$	−13.5697	−0.446654
$924$	0	0
$925$	−47.6583	−1.56700
$926$	0	0
$927$	7.73271	0.253975
$928$	0	0
$929$	−6.05504	−0.198659	−0.0993297	−	0.995055i	$-0.531670\pi$
$929$	−6.05504	−0.198659	−0.0993297	+	0.995055i	$0.531670\pi$
$930$	0	0
$931$	3.49678	0.114602
$932$	0	0
$933$	33.9192	1.11047
$934$	0	0
$935$	−0.164452	−0.00537816
$936$	0	0
$937$	−1.23885	−0.0404716	−0.0202358	−	0.999795i	$-0.506442\pi$
$937$	−1.23885	−0.0404716	−0.0202358	+	0.999795i	$0.506442\pi$
$938$	0	0
$939$	−21.9242	−0.715470
$940$	0	0
$941$	29.8465	0.972968	0.486484	−	0.873690i	$-0.338279\pi$
$941$	29.8465	0.972968	0.486484	+	0.873690i	$0.338279\pi$
$942$	0	0
$943$	−2.90816	−0.0947027
$944$	0	0
$945$	6.79808	0.221142
$946$	0	0
$947$	9.97110	0.324017	0.162009	−	0.986789i	$-0.448203\pi$
$947$	9.97110	0.324017	0.162009	+	0.986789i	$0.448203\pi$
$948$	0	0
$949$	−9.58531	−0.311152
$950$	0	0
$951$	−0.589963	−0.0191309
$952$	0	0
$953$	−41.4868	−1.34389	−0.671944	−	0.740602i	$-0.734540\pi$
$953$	−41.4868	−1.34389	−0.671944	+	0.740602i	$0.734540\pi$
$954$	0	0
$955$	−16.9671	−0.549042
$956$	0	0
$957$	0.675109	0.0218232
$958$	0	0
$959$	0.314998	0.0101718
$960$	0	0
$961$	22.6378	0.730251
$962$	0	0
$963$	−1.17174	−0.0377587
$964$	0	0
$965$	15.3188	0.493131
$966$	0	0
$967$	−3.02957	−0.0974246	−0.0487123	−	0.998813i	$-0.515512\pi$
$967$	−3.02957	−0.0974246	−0.0487123	+	0.998813i	$0.515512\pi$
$968$	0	0
$969$	−4.11479	−0.132186
$970$	0	0
$971$	34.9180	1.12057	0.560286	−	0.828299i	$-0.310691\pi$
$971$	34.9180	1.12057	0.560286	+	0.828299i	$0.310691\pi$
$972$	0	0
$973$	19.5337	0.626221
$974$	0	0
$975$	13.6840	0.438240
$976$	0	0
$977$	7.08792	0.226763	0.113381	−	0.993552i	$-0.463832\pi$
$977$	7.08792	0.226763	0.113381	+	0.993552i	$0.463832\pi$
$978$	0	0
$979$	−1.11397	−0.0356026
$980$	0	0
$981$	−1.44767	−0.0462206
$982$	0	0
$983$	51.8102	1.65249	0.826245	−	0.563311i	$-0.190473\pi$
$983$	51.8102	1.65249	0.826245	+	0.563311i	$0.190473\pi$
$984$	0	0
$985$	16.0557	0.511577
$986$	0	0
$987$	−8.00012	−0.254647
$988$	0	0
$989$	−0.130749	−0.00415758
$990$	0	0
$991$	−40.9830	−1.30187	−0.650935	−	0.759134i	$-0.725623\pi$
$991$	−40.9830	−1.30187	−0.650935	+	0.759134i	$0.725623\pi$
$992$	0	0
$993$	21.2240	0.673523
$994$	0	0
$995$	−11.5662	−0.366673
$996$	0	0
$997$	44.4942	1.40915	0.704573	−	0.709631i	$-0.251138\pi$
$997$	44.4942	1.40915	0.704573	+	0.709631i	$0.251138\pi$
$998$	0	0
$999$	57.9382	1.83308

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6044.2.a.a.1.19	✓	63

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6044.2.a.a.1.19	✓	63	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:	\( N \)	\(=\)	\( 6044 = 2^{2} \cdot 1511 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6044.a (trivial)

\(f(q)\)	\(=\)	\(q-1.58770 q^{3} -0.675402 q^{5} -1.82211 q^{7} -0.479212 q^{9} +O(q^{10})\)	\(q-1.58770 q^{3} -0.675402 q^{5} -1.82211 q^{7} -0.479212 q^{9} -0.0892749 q^{11} +1.89681 q^{13} +1.07234 q^{15} -2.72739 q^{17} -0.950237 q^{19} +2.89297 q^{21} +0.452609 q^{23} -4.54383 q^{25} +5.52394 q^{27} +4.76295 q^{29} +7.32378 q^{31} +0.141742 q^{33} +1.23066 q^{35} +10.4886 q^{37} -3.01156 q^{39} -6.42533 q^{41} -0.288879 q^{43} +0.323661 q^{45} -2.76537 q^{47} -3.67991 q^{49} +4.33028 q^{51} +8.22321 q^{53} +0.0602965 q^{55} +1.50869 q^{57} +5.51634 q^{59} -1.26310 q^{61} +0.873178 q^{63} -1.28111 q^{65} +0.673526 q^{67} -0.718606 q^{69} -7.15399 q^{71} -5.05339 q^{73} +7.21424 q^{75} +0.162669 q^{77} -0.224643 q^{79} -7.33272 q^{81} -2.39356 q^{83} +1.84209 q^{85} -7.56213 q^{87} +12.4780 q^{89} -3.45620 q^{91} -11.6280 q^{93} +0.641792 q^{95} +4.64727 q^{97} +0.0427817 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 63 q - 7 q^{3} - 7 q^{5} - 22 q^{7} + 62 q^{9}+O(q^{10}) \) 63 * q - 7 * q^3 - 7 * q^5 - 22 * q^7 + 62 * q^9	\( 63 q - 7 q^{3} - 7 q^{5} - 22 q^{7} + 62 q^{9} - 21 q^{11} - 19 q^{13} - 30 q^{15} - 5 q^{17} - 59 q^{19} - 30 q^{21} - 24 q^{23} + 60 q^{25} - 34 q^{27} - 28 q^{29} - 48 q^{31} - q^{33} - 44 q^{35} - 29 q^{37} - 75 q^{39} - 3 q^{41} - 88 q^{43} - 21 q^{45} - 21 q^{47} + 63 q^{49} - 85 q^{51} - 24 q^{53} - 85 q^{55} - 35 q^{59} - 78 q^{61} - 74 q^{63} - 13 q^{65} - 68 q^{67} - 43 q^{69} - 59 q^{71} - q^{73} - 45 q^{75} - 33 q^{77} - 140 q^{79} + 51 q^{81} - 27 q^{83} - 84 q^{85} - 61 q^{87} - 2 q^{89} - 92 q^{91} - 51 q^{93} - 51 q^{95} - 10 q^{97} - 115 q^{99}+O(q^{100}) \) 63 * q - 7 * q^3 - 7 * q^5 - 22 * q^7 + 62 * q^9 - 21 * q^11 - 19 * q^13 - 30 * q^15 - 5 * q^17 - 59 * q^19 - 30 * q^21 - 24 * q^23 + 60 * q^25 - 34 * q^27 - 28 * q^29 - 48 * q^31 - q^33 - 44 * q^35 - 29 * q^37 - 75 * q^39 - 3 * q^41 - 88 * q^43 - 21 * q^45 - 21 * q^47 + 63 * q^49 - 85 * q^51 - 24 * q^53 - 85 * q^55 - 35 * q^59 - 78 * q^61 - 74 * q^63 - 13 * q^65 - 68 * q^67 - 43 * q^69 - 59 * q^71 - q^73 - 45 * q^75 - 33 * q^77 - 140 * q^79 + 51 * q^81 - 27 * q^83 - 84 * q^85 - 61 * q^87 - 2 * q^89 - 92 * q^91 - 51 * q^93 - 51 * q^95 - 10 * q^97 - 115 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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