LMFDB - Embedded newform 6044.2.a.a.1.18

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			1.18
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)

$f(q)$	$=$	$q-1.59751 q^{3} +1.94100 q^{5} -3.58958 q^{7} -0.447960 q^{9} +O(q^{10})$	$q-1.59751 q^{3} +1.94100 q^{5} -3.58958 q^{7} -0.447960 q^{9} -4.13002 q^{11} +5.19584 q^{13} -3.10077 q^{15} +6.01864 q^{17} -1.62025 q^{19} +5.73439 q^{21} -3.90534 q^{23} -1.23251 q^{25} +5.50815 q^{27} -6.07764 q^{29} +8.17169 q^{31} +6.59775 q^{33} -6.96738 q^{35} +2.03167 q^{37} -8.30040 q^{39} +11.4818 q^{41} -7.37624 q^{43} -0.869491 q^{45} -3.40273 q^{47} +5.88509 q^{49} -9.61484 q^{51} +3.49595 q^{53} -8.01637 q^{55} +2.58837 q^{57} -0.828691 q^{59} -4.33801 q^{61} +1.60799 q^{63} +10.0851 q^{65} +0.611947 q^{67} +6.23881 q^{69} +3.21097 q^{71} +3.85902 q^{73} +1.96895 q^{75} +14.8250 q^{77} +16.5852 q^{79} -7.45545 q^{81} -2.02217 q^{83} +11.6822 q^{85} +9.70909 q^{87} +15.3887 q^{89} -18.6509 q^{91} -13.0544 q^{93} -3.14491 q^{95} -18.7836 q^{97} +1.85008 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−1.59751	−0.922323	−0.461162	−	0.887316i	$-0.652567\pi$
$3$	−1.59751	−0.922323	−0.461162	+	0.887316i	$0.652567\pi$
$4$	0	0
$5$	1.94100	0.868042	0.434021	−	0.900903i	$-0.357094\pi$
$5$	1.94100	0.868042	0.434021	+	0.900903i	$0.357094\pi$
$6$	0	0
$7$	−3.58958	−1.35673	−0.678367	−	0.734723i	$-0.737312\pi$
$7$	−3.58958	−1.35673	−0.678367	+	0.734723i	$0.737312\pi$
$8$	0	0
$9$	−0.447960	−0.149320
$10$	0	0
$11$	−4.13002	−1.24525	−0.622624	−	0.782521i	$-0.713933\pi$
$11$	−4.13002	−1.24525	−0.622624	+	0.782521i	$0.713933\pi$
$12$	0	0
$13$	5.19584	1.44107	0.720533	−	0.693421i	$-0.243897\pi$
$13$	5.19584	1.44107	0.720533	+	0.693421i	$0.243897\pi$
$14$	0	0
$15$	−3.10077	−0.800616
$16$	0	0
$17$	6.01864	1.45973	0.729867	−	0.683589i	$-0.239582\pi$
$17$	6.01864	1.45973	0.729867	+	0.683589i	$0.239582\pi$
$18$	0	0
$19$	−1.62025	−0.371711	−0.185856	−	0.982577i	$-0.559506\pi$
$19$	−1.62025	−0.371711	−0.185856	+	0.982577i	$0.559506\pi$
$20$	0	0
$21$	5.73439	1.25135
$22$	0	0
$23$	−3.90534	−0.814319	−0.407159	−	0.913357i	$-0.633481\pi$
$23$	−3.90534	−0.814319	−0.407159	+	0.913357i	$0.633481\pi$
$24$	0	0
$25$	−1.23251	−0.246502
$26$	0	0
$27$	5.50815	1.06004
$28$	0	0
$29$	−6.07764	−1.12859	−0.564295	−	0.825574i	$-0.690852\pi$
$29$	−6.07764	−1.12859	−0.564295	+	0.825574i	$0.690852\pi$
$30$	0	0
$31$	8.17169	1.46768	0.733839	−	0.679323i	$-0.237726\pi$
$31$	8.17169	1.46768	0.733839	+	0.679323i	$0.237726\pi$
$32$	0	0
$33$	6.59775	1.14852
$34$	0	0
$35$	−6.96738	−1.17770
$36$	0	0
$37$	2.03167	0.334004	0.167002	−	0.985957i	$-0.446591\pi$
$37$	2.03167	0.334004	0.167002	+	0.985957i	$0.446591\pi$
$38$	0	0
$39$	−8.30040	−1.32913
$40$	0	0
$41$	11.4818	1.79315	0.896576	−	0.442891i	$-0.146047\pi$
$41$	11.4818	1.79315	0.896576	+	0.442891i	$0.146047\pi$
$42$	0	0
$43$	−7.37624	−1.12487	−0.562433	−	0.826843i	$-0.690135\pi$
$43$	−7.37624	−1.12487	−0.562433	+	0.826843i	$0.690135\pi$
$44$	0	0
$45$	−0.869491	−0.129616
$46$	0	0
$47$	−3.40273	−0.496339	−0.248170	−	0.968717i	$-0.579829\pi$
$47$	−3.40273	−0.496339	−0.248170	+	0.968717i	$0.579829\pi$
$48$	0	0
$49$	5.88509	0.840726
$50$	0	0
$51$	−9.61484	−1.34635
$52$	0	0
$53$	3.49595	0.480205	0.240103	−	0.970748i	$-0.422819\pi$
$53$	3.49595	0.480205	0.240103	+	0.970748i	$0.422819\pi$
$54$	0	0
$55$	−8.01637	−1.08093
$56$	0	0
$57$	2.58837	0.342838
$58$	0	0
$59$	−0.828691	−0.107886	−0.0539432	−	0.998544i	$-0.517179\pi$
$59$	−0.828691	−0.107886	−0.0539432	+	0.998544i	$0.517179\pi$
$60$	0	0
$61$	−4.33801	−0.555425	−0.277712	−	0.960664i	$-0.589576\pi$
$61$	−4.33801	−0.555425	−0.277712	+	0.960664i	$0.589576\pi$
$62$	0	0
$63$	1.60799	0.202588
$64$	0	0
$65$	10.0851	1.25091
$66$	0	0
$67$	0.611947	0.0747612	0.0373806	−	0.999301i	$-0.488099\pi$
$67$	0.611947	0.0747612	0.0373806	+	0.999301i	$0.488099\pi$
$68$	0	0
$69$	6.23881	0.751065
$70$	0	0
$71$	3.21097	0.381072	0.190536	−	0.981680i	$-0.438977\pi$
$71$	3.21097	0.381072	0.190536	+	0.981680i	$0.438977\pi$
$72$	0	0
$73$	3.85902	0.451664	0.225832	−	0.974166i	$-0.427490\pi$
$73$	3.85902	0.451664	0.225832	+	0.974166i	$0.427490\pi$
$74$	0	0
$75$	1.96895	0.227355
$76$	0	0
$77$	14.8250	1.68947
$78$	0	0
$79$	16.5852	1.86598	0.932988	−	0.359908i	$-0.117192\pi$
$79$	16.5852	1.86598	0.932988	+	0.359908i	$0.117192\pi$
$80$	0	0
$81$	−7.45545	−0.828383
$82$	0	0
$83$	−2.02217	−0.221962	−0.110981	−	0.993823i	$-0.535399\pi$
$83$	−2.02217	−0.221962	−0.110981	+	0.993823i	$0.535399\pi$
$84$	0	0
$85$	11.6822	1.26711
$86$	0	0
$87$	9.70909	1.04092
$88$	0	0
$89$	15.3887	1.63120	0.815599	−	0.578618i	$-0.196408\pi$
$89$	15.3887	1.63120	0.815599	+	0.578618i	$0.196408\pi$
$90$	0	0
$91$	−18.6509	−1.95514
$92$	0	0
$93$	−13.0544	−1.35367
$94$	0	0
$95$	−3.14491	−0.322661
$96$	0	0
$97$	−18.7836	−1.90719	−0.953594	−	0.301095i	$-0.902648\pi$
$97$	−18.7836	−1.90719	−0.953594	+	0.301095i	$0.902648\pi$
$98$	0	0
$99$	1.85008	0.185940
$100$	0	0
$101$	−13.7111	−1.36431	−0.682153	−	0.731209i	$-0.738956\pi$
$101$	−13.7111	−1.36431	−0.682153	+	0.731209i	$0.738956\pi$
$102$	0	0
$103$	−9.04821	−0.891547	−0.445773	−	0.895146i	$-0.647071\pi$
$103$	−9.04821	−0.891547	−0.445773	+	0.895146i	$0.647071\pi$
$104$	0	0
$105$	11.1305	1.08622
$106$	0	0
$107$	4.73838	0.458076	0.229038	−	0.973417i	$-0.426442\pi$
$107$	4.73838	0.458076	0.229038	+	0.973417i	$0.426442\pi$
$108$	0	0
$109$	−15.4759	−1.48232	−0.741161	−	0.671328i	$-0.765724\pi$
$109$	−15.4759	−1.48232	−0.741161	+	0.671328i	$0.765724\pi$
$110$	0	0
$111$	−3.24561	−0.308060
$112$	0	0
$113$	−12.4237	−1.16872	−0.584361	−	0.811494i	$-0.698655\pi$
$113$	−12.4237	−1.16872	−0.584361	+	0.811494i	$0.698655\pi$
$114$	0	0
$115$	−7.58026	−0.706863
$116$	0	0
$117$	−2.32753	−0.215180
$118$	0	0
$119$	−21.6044	−1.98047
$120$	0	0
$121$	6.05705	0.550641
$122$	0	0
$123$	−18.3423	−1.65386
$124$	0	0
$125$	−12.0973	−1.08202
$126$	0	0
$127$	11.5435	1.02432	0.512160	−	0.858890i	$-0.328845\pi$
$127$	11.5435	1.02432	0.512160	+	0.858890i	$0.328845\pi$
$128$	0	0
$129$	11.7836	1.03749
$130$	0	0
$131$	−20.5059	−1.79161	−0.895805	−	0.444447i	$-0.853400\pi$
$131$	−20.5059	−1.79161	−0.895805	+	0.444447i	$0.853400\pi$
$132$	0	0
$133$	5.81602	0.504313
$134$	0	0
$135$	10.6913	0.920164
$136$	0	0
$137$	−9.80038	−0.837303	−0.418651	−	0.908147i	$-0.637497\pi$
$137$	−9.80038	−0.837303	−0.418651	+	0.908147i	$0.637497\pi$
$138$	0	0
$139$	13.4655	1.14213	0.571063	−	0.820906i	$-0.306531\pi$
$139$	13.4655	1.14213	0.571063	+	0.820906i	$0.306531\pi$
$140$	0	0
$141$	5.43590	0.457785
$142$	0	0
$143$	−21.4589	−1.79448
$144$	0	0
$145$	−11.7967	−0.979663
$146$	0	0
$147$	−9.40149	−0.775421
$148$	0	0
$149$	19.6202	1.60735	0.803674	−	0.595070i	$-0.202876\pi$
$149$	19.6202	1.60735	0.803674	+	0.595070i	$0.202876\pi$
$150$	0	0
$151$	−14.9842	−1.21939	−0.609697	−	0.792635i	$-0.708709\pi$
$151$	−14.9842	−1.21939	−0.609697	+	0.792635i	$0.708709\pi$
$152$	0	0
$153$	−2.69611	−0.217968
$154$	0	0
$155$	15.8613	1.27401
$156$	0	0
$157$	−9.40537	−0.750630	−0.375315	−	0.926897i	$-0.622466\pi$
$157$	−9.40537	−0.750630	−0.375315	+	0.926897i	$0.622466\pi$
$158$	0	0
$159$	−5.58481	−0.442904
$160$	0	0
$161$	14.0185	1.10481
$162$	0	0
$163$	10.0859	0.789987	0.394993	−	0.918684i	$-0.370747\pi$
$163$	10.0859	0.789987	0.394993	+	0.918684i	$0.370747\pi$
$164$	0	0
$165$	12.8062	0.996964
$166$	0	0
$167$	−7.02047	−0.543260	−0.271630	−	0.962402i	$-0.587563\pi$
$167$	−7.02047	−0.543260	−0.271630	+	0.962402i	$0.587563\pi$
$168$	0	0
$169$	13.9967	1.07667
$170$	0	0
$171$	0.725808	0.0555039
$172$	0	0
$173$	0.570895	0.0434043	0.0217022	−	0.999764i	$-0.493091\pi$
$173$	0.570895	0.0434043	0.0217022	+	0.999764i	$0.493091\pi$
$174$	0	0
$175$	4.42420	0.334438
$176$	0	0
$177$	1.32384	0.0995061
$178$	0	0
$179$	17.7859	1.32938	0.664691	−	0.747119i	$-0.268563\pi$
$179$	17.7859	1.32938	0.664691	+	0.747119i	$0.268563\pi$
$180$	0	0
$181$	4.62527	0.343794	0.171897	−	0.985115i	$-0.445010\pi$
$181$	4.62527	0.343794	0.171897	+	0.985115i	$0.445010\pi$
$182$	0	0
$183$	6.93001	0.512281
$184$	0	0
$185$	3.94347	0.289930
$186$	0	0
$187$	−24.8571	−1.81773
$188$	0	0
$189$	−19.7720	−1.43820
$190$	0	0
$191$	−10.1315	−0.733091	−0.366546	−	0.930400i	$-0.619460\pi$
$191$	−10.1315	−0.733091	−0.366546	+	0.930400i	$0.619460\pi$
$192$	0	0
$193$	−17.9253	−1.29029	−0.645147	−	0.764058i	$-0.723204\pi$
$193$	−17.9253	−1.29029	−0.645147	+	0.764058i	$0.723204\pi$
$194$	0	0
$195$	−16.1111	−1.15374
$196$	0	0
$197$	−10.3621	−0.738272	−0.369136	−	0.929375i	$-0.620346\pi$
$197$	−10.3621	−0.738272	−0.369136	+	0.929375i	$0.620346\pi$
$198$	0	0
$199$	−17.0359	−1.20764	−0.603822	−	0.797119i	$-0.706356\pi$
$199$	−17.0359	−1.20764	−0.603822	+	0.797119i	$0.706356\pi$
$200$	0	0
$201$	−0.977592	−0.0689540
$202$	0	0
$203$	21.8162	1.53119
$204$	0	0
$205$	22.2861	1.55653
$206$	0	0
$207$	1.74943	0.121594
$208$	0	0
$209$	6.69167	0.462872
$210$	0	0
$211$	−24.1414	−1.66196	−0.830980	−	0.556302i	$-0.812220\pi$
$211$	−24.1414	−1.66196	−0.830980	+	0.556302i	$0.812220\pi$
$212$	0	0
$213$	−5.12956	−0.351472
$214$	0	0
$215$	−14.3173	−0.976431
$216$	0	0
$217$	−29.3329	−1.99125
$218$	0	0
$219$	−6.16483	−0.416581
$220$	0	0
$221$	31.2719	2.10357
$222$	0	0
$223$	23.8001	1.59377	0.796887	−	0.604128i	$-0.206478\pi$
$223$	23.8001	1.59377	0.796887	+	0.604128i	$0.206478\pi$
$224$	0	0
$225$	0.552116	0.0368078
$226$	0	0
$227$	3.74839	0.248789	0.124395	−	0.992233i	$-0.460301\pi$
$227$	3.74839	0.248789	0.124395	+	0.992233i	$0.460301\pi$
$228$	0	0
$229$	2.59391	0.171410	0.0857052	−	0.996321i	$-0.472686\pi$
$229$	2.59391	0.171410	0.0857052	+	0.996321i	$0.472686\pi$
$230$	0	0
$231$	−23.6831	−1.55824
$232$	0	0
$233$	12.5912	0.824877	0.412439	−	0.910985i	$-0.364677\pi$
$233$	12.5912	0.824877	0.412439	+	0.910985i	$0.364677\pi$
$234$	0	0
$235$	−6.60470	−0.430843
$236$	0	0
$237$	−26.4950	−1.72103
$238$	0	0
$239$	−1.54406	−0.0998770	−0.0499385	−	0.998752i	$-0.515903\pi$
$239$	−1.54406	−0.0998770	−0.0499385	+	0.998752i	$0.515903\pi$
$240$	0	0
$241$	−20.3423	−1.31036	−0.655180	−	0.755473i	$-0.727407\pi$
$241$	−20.3423	−1.31036	−0.655180	+	0.755473i	$0.727407\pi$
$242$	0	0
$243$	−4.61430	−0.296007
$244$	0	0
$245$	11.4230	0.729786
$246$	0	0
$247$	−8.41856	−0.535660
$248$	0	0
$249$	3.23044	0.204721
$250$	0	0
$251$	13.3215	0.840844	0.420422	−	0.907329i	$-0.361882\pi$
$251$	13.3215	0.840844	0.420422	+	0.907329i	$0.361882\pi$
$252$	0	0
$253$	16.1291	1.01403
$254$	0	0
$255$	−18.6624	−1.16869
$256$	0	0
$257$	−3.87994	−0.242024	−0.121012	−	0.992651i	$-0.538614\pi$
$257$	−3.87994	−0.242024	−0.121012	+	0.992651i	$0.538614\pi$
$258$	0	0
$259$	−7.29284	−0.453155
$260$	0	0
$261$	2.72254	0.168521
$262$	0	0
$263$	−26.6562	−1.64369	−0.821845	−	0.569711i	$-0.807055\pi$
$263$	−26.6562	−1.64369	−0.821845	+	0.569711i	$0.807055\pi$
$264$	0	0
$265$	6.78564	0.416838
$266$	0	0
$267$	−24.5836	−1.50449
$268$	0	0
$269$	−26.0050	−1.58555	−0.792777	−	0.609512i	$-0.791365\pi$
$269$	−26.0050	−1.58555	−0.792777	+	0.609512i	$0.791365\pi$
$270$	0	0
$271$	−16.1828	−0.983035	−0.491517	−	0.870868i	$-0.663558\pi$
$271$	−16.1828	−0.983035	−0.491517	+	0.870868i	$0.663558\pi$
$272$	0	0
$273$	29.7950	1.80327
$274$	0	0
$275$	5.09030	0.306956
$276$	0	0
$277$	24.2405	1.45647	0.728237	−	0.685326i	$-0.240340\pi$
$277$	24.2405	1.45647	0.728237	+	0.685326i	$0.240340\pi$
$278$	0	0
$279$	−3.66059	−0.219154
$280$	0	0
$281$	2.40048	0.143201	0.0716004	−	0.997433i	$-0.477189\pi$
$281$	2.40048	0.143201	0.0716004	+	0.997433i	$0.477189\pi$
$282$	0	0
$283$	12.9101	0.767428	0.383714	−	0.923452i	$-0.374645\pi$
$283$	12.9101	0.767428	0.383714	+	0.923452i	$0.374645\pi$
$284$	0	0
$285$	5.02403	0.297598
$286$	0	0
$287$	−41.2147	−2.43283
$288$	0	0
$289$	19.2240	1.13082
$290$	0	0
$291$	30.0070	1.75904
$292$	0	0
$293$	22.3495	1.30567	0.652836	−	0.757500i	$-0.273579\pi$
$293$	22.3495	1.30567	0.652836	+	0.757500i	$0.273579\pi$
$294$	0	0
$295$	−1.60849	−0.0936500
$296$	0	0
$297$	−22.7488	−1.32002
$298$	0	0
$299$	−20.2915	−1.17349
$300$	0	0
$301$	26.4776	1.52614
$302$	0	0
$303$	21.9036	1.25833
$304$	0	0
$305$	−8.42008	−0.482132
$306$	0	0
$307$	−19.4669	−1.11103	−0.555516	−	0.831506i	$-0.687479\pi$
$307$	−19.4669	−1.11103	−0.555516	+	0.831506i	$0.687479\pi$
$308$	0	0
$309$	14.4546	0.822294
$310$	0	0
$311$	21.4507	1.21636	0.608179	−	0.793800i	$-0.291900\pi$
$311$	21.4507	1.21636	0.608179	+	0.793800i	$0.291900\pi$
$312$	0	0
$313$	−9.33517	−0.527655	−0.263828	−	0.964570i	$-0.584985\pi$
$313$	−9.33517	−0.527655	−0.263828	+	0.964570i	$0.584985\pi$
$314$	0	0
$315$	3.12111	0.175855
$316$	0	0
$317$	−11.4318	−0.642075	−0.321037	−	0.947067i	$-0.604032\pi$
$317$	−11.4318	−0.642075	−0.321037	+	0.947067i	$0.604032\pi$
$318$	0	0
$319$	25.1008	1.40537
$320$	0	0
$321$	−7.56961	−0.422494
$322$	0	0
$323$	−9.75171	−0.542599
$324$	0	0
$325$	−6.40393	−0.355226
$326$	0	0
$327$	24.7229	1.36718
$328$	0	0
$329$	12.2144	0.673400
$330$	0	0
$331$	12.5741	0.691135	0.345567	−	0.938394i	$-0.387686\pi$
$331$	12.5741	0.691135	0.345567	+	0.938394i	$0.387686\pi$
$332$	0	0
$333$	−0.910107	−0.0498735
$334$	0	0
$335$	1.18779	0.0648959
$336$	0	0
$337$	−25.7381	−1.40205	−0.701023	−	0.713139i	$-0.747273\pi$
$337$	−25.7381	−1.40205	−0.701023	+	0.713139i	$0.747273\pi$
$338$	0	0
$339$	19.8470	1.07794
$340$	0	0
$341$	−33.7492	−1.82762
$342$	0	0
$343$	4.00208	0.216092
$344$	0	0
$345$	12.1095	0.651956
$346$	0	0
$347$	19.2926	1.03568	0.517840	−	0.855477i	$-0.326736\pi$
$347$	19.2926	1.03568	0.517840	+	0.855477i	$0.326736\pi$
$348$	0	0
$349$	−23.7614	−1.27192	−0.635960	−	0.771722i	$-0.719396\pi$
$349$	−23.7614	−1.27192	−0.635960	+	0.771722i	$0.719396\pi$
$350$	0	0
$351$	28.6195	1.52759
$352$	0	0
$353$	−6.83958	−0.364034	−0.182017	−	0.983295i	$-0.558263\pi$
$353$	−6.83958	−0.364034	−0.182017	+	0.983295i	$0.558263\pi$
$354$	0	0
$355$	6.23250	0.330787
$356$	0	0
$357$	34.5132	1.82663
$358$	0	0
$359$	−25.6026	−1.35125	−0.675627	−	0.737244i	$-0.736127\pi$
$359$	−25.6026	−1.35125	−0.675627	+	0.737244i	$0.736127\pi$
$360$	0	0
$361$	−16.3748	−0.861831
$362$	0	0
$363$	−9.67620	−0.507869
$364$	0	0
$365$	7.49037	0.392064
$366$	0	0
$367$	4.62389	0.241365	0.120682	−	0.992691i	$-0.461492\pi$
$367$	4.62389	0.241365	0.120682	+	0.992691i	$0.461492\pi$
$368$	0	0
$369$	−5.14338	−0.267753
$370$	0	0
$371$	−12.5490	−0.651511
$372$	0	0
$373$	−29.7807	−1.54199	−0.770993	−	0.636844i	$-0.780240\pi$
$373$	−29.7807	−1.54199	−0.770993	+	0.636844i	$0.780240\pi$
$374$	0	0
$375$	19.3256	0.997969
$376$	0	0
$377$	−31.5784	−1.62637
$378$	0	0
$379$	−19.2060	−0.986544	−0.493272	−	0.869875i	$-0.664199\pi$
$379$	−19.2060	−0.986544	−0.493272	+	0.869875i	$0.664199\pi$
$380$	0	0
$381$	−18.4409	−0.944754
$382$	0	0
$383$	−27.0753	−1.38348	−0.691741	−	0.722145i	$-0.743156\pi$
$383$	−27.0753	−1.38348	−0.691741	+	0.722145i	$0.743156\pi$
$384$	0	0
$385$	28.7754	1.46653
$386$	0	0
$387$	3.30426	0.167965
$388$	0	0
$389$	−36.2241	−1.83664	−0.918319	−	0.395842i	$-0.870453\pi$
$389$	−36.2241	−1.83664	−0.918319	+	0.395842i	$0.870453\pi$
$390$	0	0
$391$	−23.5048	−1.18869
$392$	0	0
$393$	32.7584	1.65244
$394$	0	0
$395$	32.1918	1.61975
$396$	0	0
$397$	3.76652	0.189036	0.0945180	−	0.995523i	$-0.469869\pi$
$397$	3.76652	0.189036	0.0945180	+	0.995523i	$0.469869\pi$
$398$	0	0
$399$	−9.29116	−0.465140
$400$	0	0
$401$	−27.1120	−1.35391	−0.676954	−	0.736026i	$-0.736700\pi$
$401$	−27.1120	−1.35391	−0.676954	+	0.736026i	$0.736700\pi$
$402$	0	0
$403$	42.4588	2.11502
$404$	0	0
$405$	−14.4710	−0.719072
$406$	0	0
$407$	−8.39083	−0.415918
$408$	0	0
$409$	13.1519	0.650322	0.325161	−	0.945659i	$-0.394582\pi$
$409$	13.1519	0.650322	0.325161	+	0.945659i	$0.394582\pi$
$410$	0	0
$411$	15.6562	0.772264
$412$	0	0
$413$	2.97465	0.146373
$414$	0	0
$415$	−3.92503	−0.192672
$416$	0	0
$417$	−21.5112	−1.05341
$418$	0	0
$419$	−34.7632	−1.69829	−0.849146	−	0.528158i	$-0.822883\pi$
$419$	−34.7632	−1.69829	−0.849146	+	0.528158i	$0.822883\pi$
$420$	0	0
$421$	26.6405	1.29838	0.649188	−	0.760628i	$-0.275109\pi$
$421$	26.6405	1.29838	0.649188	+	0.760628i	$0.275109\pi$
$422$	0	0
$423$	1.52429	0.0741134
$424$	0	0
$425$	−7.41804	−0.359828
$426$	0	0
$427$	15.5716	0.753564
$428$	0	0
$429$	34.2808	1.65509
$430$	0	0
$431$	19.0325	0.916762	0.458381	−	0.888756i	$-0.348430\pi$
$431$	19.0325	0.916762	0.458381	+	0.888756i	$0.348430\pi$
$432$	0	0
$433$	23.6708	1.13755	0.568774	−	0.822494i	$-0.307418\pi$
$433$	23.6708	1.13755	0.568774	+	0.822494i	$0.307418\pi$
$434$	0	0
$435$	18.8454	0.903566
$436$	0	0
$437$	6.32762	0.302691
$438$	0	0
$439$	28.5431	1.36229	0.681144	−	0.732150i	$-0.261483\pi$
$439$	28.5431	1.36229	0.681144	+	0.732150i	$0.261483\pi$
$440$	0	0
$441$	−2.63628	−0.125537
$442$	0	0
$443$	−7.81765	−0.371428	−0.185714	−	0.982604i	$-0.559460\pi$
$443$	−7.81765	−0.371428	−0.185714	+	0.982604i	$0.559460\pi$
$444$	0	0
$445$	29.8695	1.41595
$446$	0	0
$447$	−31.3434	−1.48249
$448$	0	0
$449$	−17.0679	−0.805483	−0.402742	−	0.915314i	$-0.631943\pi$
$449$	−17.0679	−0.805483	−0.402742	+	0.915314i	$0.631943\pi$
$450$	0	0
$451$	−47.4199	−2.23292
$452$	0	0
$453$	23.9374	1.12468
$454$	0	0
$455$	−36.2014	−1.69715
$456$	0	0
$457$	−25.4375	−1.18992	−0.594958	−	0.803757i	$-0.702831\pi$
$457$	−25.4375	−1.18992	−0.594958	+	0.803757i	$0.702831\pi$
$458$	0	0
$459$	33.1516	1.54738
$460$	0	0
$461$	−34.9375	−1.62720	−0.813601	−	0.581423i	$-0.802496\pi$
$461$	−34.9375	−1.62720	−0.813601	+	0.581423i	$0.802496\pi$
$462$	0	0
$463$	4.45204	0.206904	0.103452	−	0.994634i	$-0.467011\pi$
$463$	4.45204	0.206904	0.103452	+	0.994634i	$0.467011\pi$
$464$	0	0
$465$	−25.3385	−1.17505
$466$	0	0
$467$	−15.1431	−0.700740	−0.350370	−	0.936611i	$-0.613944\pi$
$467$	−15.1431	−0.700740	−0.350370	+	0.936611i	$0.613944\pi$
$468$	0	0
$469$	−2.19663	−0.101431
$470$	0	0
$471$	15.0252	0.692324
$472$	0	0
$473$	30.4640	1.40074
$474$	0	0
$475$	1.99698	0.0916277
$476$	0	0
$477$	−1.56604	−0.0717043
$478$	0	0
$479$	−0.577563	−0.0263895	−0.0131948	−	0.999913i	$-0.504200\pi$
$479$	−0.577563	−0.0263895	−0.0131948	+	0.999913i	$0.504200\pi$
$480$	0	0
$481$	10.5562	0.481322
$482$	0	0
$483$	−22.3947	−1.01900
$484$	0	0
$485$	−36.4591	−1.65552
$486$	0	0
$487$	16.0575	0.727637	0.363818	−	0.931470i	$-0.381473\pi$
$487$	16.0575	0.727637	0.363818	+	0.931470i	$0.381473\pi$
$488$	0	0
$489$	−16.1123	−0.728623
$490$	0	0
$491$	14.0982	0.636245	0.318123	−	0.948050i	$-0.396948\pi$
$491$	14.0982	0.636245	0.318123	+	0.948050i	$0.396948\pi$
$492$	0	0
$493$	−36.5791	−1.64744
$494$	0	0
$495$	3.59101	0.161404
$496$	0	0
$497$	−11.5260	−0.517013
$498$	0	0
$499$	13.5785	0.607858	0.303929	−	0.952695i	$-0.401701\pi$
$499$	13.5785	0.607858	0.303929	+	0.952695i	$0.401701\pi$
$500$	0	0
$501$	11.2153	0.501062
$502$	0	0
$503$	36.7884	1.64032	0.820158	−	0.572137i	$-0.193886\pi$
$503$	36.7884	1.64032	0.820158	+	0.572137i	$0.193886\pi$
$504$	0	0
$505$	−26.6133	−1.18428
$506$	0	0
$507$	−22.3599	−0.993037
$508$	0	0
$509$	−33.3185	−1.47682	−0.738409	−	0.674353i	$-0.764423\pi$
$509$	−33.3185	−1.47682	−0.738409	+	0.674353i	$0.764423\pi$
$510$	0	0
$511$	−13.8523	−0.612788
$512$	0	0
$513$	−8.92459	−0.394030
$514$	0	0
$515$	−17.5626	−0.773900
$516$	0	0
$517$	14.0533	0.618065
$518$	0	0
$519$	−0.912011	−0.0400328
$520$	0	0
$521$	18.7030	0.819395	0.409697	−	0.912221i	$-0.365634\pi$
$521$	18.7030	0.819395	0.409697	+	0.912221i	$0.365634\pi$
$522$	0	0
$523$	−7.63498	−0.333854	−0.166927	−	0.985969i	$-0.553384\pi$
$523$	−7.63498	−0.333854	−0.166927	+	0.985969i	$0.553384\pi$
$524$	0	0
$525$	−7.06771	−0.308460
$526$	0	0
$527$	49.1824	2.14242
$528$	0	0
$529$	−7.74836	−0.336885
$530$	0	0
$531$	0.371221	0.0161096
$532$	0	0
$533$	59.6574	2.58405
$534$	0	0
$535$	9.19720	0.397630
$536$	0	0
$537$	−28.4132	−1.22612
$538$	0	0
$539$	−24.3055	−1.04691
$540$	0	0
$541$	18.4633	0.793798	0.396899	−	0.917862i	$-0.370086\pi$
$541$	18.4633	0.793798	0.396899	+	0.917862i	$0.370086\pi$
$542$	0	0
$543$	−7.38892	−0.317089
$544$	0	0
$545$	−30.0387	−1.28672
$546$	0	0
$547$	10.1934	0.435838	0.217919	−	0.975967i	$-0.430073\pi$
$547$	10.1934	0.435838	0.217919	+	0.975967i	$0.430073\pi$
$548$	0	0
$549$	1.94325	0.0829361
$550$	0	0
$551$	9.84730	0.419509
$552$	0	0
$553$	−59.5337	−2.53163
$554$	0	0
$555$	−6.29974	−0.267409
$556$	0	0
$557$	−25.2411	−1.06950	−0.534749	−	0.845011i	$-0.679594\pi$
$557$	−25.2411	−1.06950	−0.534749	+	0.845011i	$0.679594\pi$
$558$	0	0
$559$	−38.3257	−1.62101
$560$	0	0
$561$	39.7094	1.67653
$562$	0	0
$563$	−32.1521	−1.35505	−0.677524	−	0.735501i	$-0.736947\pi$
$563$	−32.1521	−1.35505	−0.677524	+	0.735501i	$0.736947\pi$
$564$	0	0
$565$	−24.1144	−1.01450
$566$	0	0
$567$	26.7619	1.12390
$568$	0	0
$569$	24.8277	1.04083	0.520417	−	0.853912i	$-0.325777\pi$
$569$	24.8277	1.04083	0.520417	+	0.853912i	$0.325777\pi$
$570$	0	0
$571$	−28.0388	−1.17339	−0.586694	−	0.809809i	$-0.699571\pi$
$571$	−28.0388	−1.17339	−0.586694	+	0.809809i	$0.699571\pi$
$572$	0	0
$573$	16.1852	0.676147
$574$	0	0
$575$	4.81337	0.200732
$576$	0	0
$577$	25.3908	1.05703	0.528516	−	0.848924i	$-0.322749\pi$
$577$	25.3908	1.05703	0.528516	+	0.848924i	$0.322749\pi$
$578$	0	0
$579$	28.6359	1.19007
$580$	0	0
$581$	7.25874	0.301143
$582$	0	0
$583$	−14.4383	−0.597974
$584$	0	0
$585$	−4.51773	−0.186785
$586$	0	0
$587$	−41.4557	−1.71106	−0.855530	−	0.517753i	$-0.826769\pi$
$587$	−41.4557	−1.71106	−0.855530	+	0.517753i	$0.826769\pi$
$588$	0	0
$589$	−13.2402	−0.545553
$590$	0	0
$591$	16.5536	0.680926
$592$	0	0
$593$	−9.83618	−0.403924	−0.201962	−	0.979393i	$-0.564732\pi$
$593$	−9.83618	−0.403924	−0.201962	+	0.979393i	$0.564732\pi$
$594$	0	0
$595$	−41.9341	−1.71913
$596$	0	0
$597$	27.2151	1.11384
$598$	0	0
$599$	37.5769	1.53535	0.767675	−	0.640840i	$-0.221414\pi$
$599$	37.5769	1.53535	0.767675	+	0.640840i	$0.221414\pi$
$600$	0	0
$601$	−47.7910	−1.94944	−0.974718	−	0.223439i	$-0.928272\pi$
$601$	−47.7910	−1.94944	−0.974718	+	0.223439i	$0.928272\pi$
$602$	0	0
$603$	−0.274128	−0.0111633
$604$	0	0
$605$	11.7567	0.477979
$606$	0	0
$607$	−0.596085	−0.0241944	−0.0120972	−	0.999927i	$-0.503851\pi$
$607$	−0.596085	−0.0241944	−0.0120972	+	0.999927i	$0.503851\pi$
$608$	0	0
$609$	−34.8516	−1.41226
$610$	0	0
$611$	−17.6800	−0.715257
$612$	0	0
$613$	0.464599	0.0187650	0.00938249	−	0.999956i	$-0.497013\pi$
$613$	0.464599	0.0187650	0.00938249	+	0.999956i	$0.497013\pi$
$614$	0	0
$615$	−35.6023	−1.43562
$616$	0	0
$617$	28.0887	1.13081	0.565405	−	0.824813i	$-0.308720\pi$
$617$	28.0887	1.13081	0.565405	+	0.824813i	$0.308720\pi$
$618$	0	0
$619$	41.4507	1.66604	0.833021	−	0.553241i	$-0.186609\pi$
$619$	41.4507	1.66604	0.833021	+	0.553241i	$0.186609\pi$
$620$	0	0
$621$	−21.5112	−0.863214
$622$	0	0
$623$	−55.2389	−2.21310
$624$	0	0
$625$	−17.3184	−0.692734
$626$	0	0
$627$	−10.6900	−0.426918
$628$	0	0
$629$	12.2279	0.487558
$630$	0	0
$631$	−25.8411	−1.02872	−0.514358	−	0.857576i	$-0.671970\pi$
$631$	−25.8411	−1.02872	−0.514358	+	0.857576i	$0.671970\pi$
$632$	0	0
$633$	38.5661	1.53286
$634$	0	0
$635$	22.4060	0.889153
$636$	0	0
$637$	30.5779	1.21154
$638$	0	0
$639$	−1.43839	−0.0569017
$640$	0	0
$641$	−48.6306	−1.92079	−0.960397	−	0.278634i	$-0.910118\pi$
$641$	−48.6306	−1.92079	−0.960397	+	0.278634i	$0.910118\pi$
$642$	0	0
$643$	−42.2290	−1.66535	−0.832675	−	0.553762i	$-0.813192\pi$
$643$	−42.2290	−1.66535	−0.832675	+	0.553762i	$0.813192\pi$
$644$	0	0
$645$	22.8720	0.900585
$646$	0	0
$647$	−32.1295	−1.26314	−0.631570	−	0.775318i	$-0.717589\pi$
$647$	−32.1295	−1.26314	−0.631570	+	0.775318i	$0.717589\pi$
$648$	0	0
$649$	3.42251	0.134345
$650$	0	0
$651$	46.8597	1.83658
$652$	0	0
$653$	29.2319	1.14393	0.571967	−	0.820277i	$-0.306181\pi$
$653$	29.2319	1.14393	0.571967	+	0.820277i	$0.306181\pi$
$654$	0	0
$655$	−39.8020	−1.55519
$656$	0	0
$657$	−1.72869	−0.0674426
$658$	0	0
$659$	28.2691	1.10121	0.550604	−	0.834767i	$-0.314398\pi$
$659$	28.2691	1.10121	0.550604	+	0.834767i	$0.314398\pi$
$660$	0	0
$661$	−12.9769	−0.504743	−0.252371	−	0.967630i	$-0.581210\pi$
$661$	−12.9769	−0.504743	−0.252371	+	0.967630i	$0.581210\pi$
$662$	0	0
$663$	−49.9571	−1.94017
$664$	0	0
$665$	11.2889	0.437765
$666$	0	0
$667$	23.7352	0.919031
$668$	0	0
$669$	−38.0210	−1.46998
$670$	0	0
$671$	17.9160	0.691641
$672$	0	0
$673$	−10.2783	−0.396198	−0.198099	−	0.980182i	$-0.563477\pi$
$673$	−10.2783	−0.396198	−0.198099	+	0.980182i	$0.563477\pi$
$674$	0	0
$675$	−6.78887	−0.261304
$676$	0	0
$677$	24.2035	0.930216	0.465108	−	0.885254i	$-0.346016\pi$
$677$	24.2035	0.930216	0.465108	+	0.885254i	$0.346016\pi$
$678$	0	0
$679$	67.4253	2.58755
$680$	0	0
$681$	−5.98809	−0.229464
$682$	0	0
$683$	−5.75554	−0.220230	−0.110115	−	0.993919i	$-0.535122\pi$
$683$	−5.75554	−0.220230	−0.110115	+	0.993919i	$0.535122\pi$
$684$	0	0
$685$	−19.0226	−0.726814
$686$	0	0
$687$	−4.14380	−0.158096
$688$	0	0
$689$	18.1644	0.692007
$690$	0	0
$691$	−39.0408	−1.48518	−0.742590	−	0.669746i	$-0.766403\pi$
$691$	−39.0408	−1.48518	−0.742590	+	0.669746i	$0.766403\pi$
$692$	0	0
$693$	−6.64102	−0.252272
$694$	0	0
$695$	26.1365	0.991414
$696$	0	0
$697$	69.1046	2.61752
$698$	0	0
$699$	−20.1146	−0.760803
$700$	0	0
$701$	38.6974	1.46158	0.730789	−	0.682603i	$-0.239152\pi$
$701$	38.6974	1.46158	0.730789	+	0.682603i	$0.239152\pi$
$702$	0	0
$703$	−3.29181	−0.124153
$704$	0	0
$705$	10.5511	0.397377
$706$	0	0
$707$	49.2171	1.85100
$708$	0	0
$709$	−41.9662	−1.57607	−0.788037	−	0.615627i	$-0.788903\pi$
$709$	−41.9662	−1.57607	−0.788037	+	0.615627i	$0.788903\pi$
$710$	0	0
$711$	−7.42949	−0.278628
$712$	0	0
$713$	−31.9132	−1.19516
$714$	0	0
$715$	−41.6517	−1.55769
$716$	0	0
$717$	2.46665	0.0921189
$718$	0	0
$719$	−9.94035	−0.370713	−0.185356	−	0.982671i	$-0.559344\pi$
$719$	−9.94035	−0.370713	−0.185356	+	0.982671i	$0.559344\pi$
$720$	0	0
$721$	32.4793	1.20959
$722$	0	0
$723$	32.4970	1.20858
$724$	0	0
$725$	7.49076	0.278200
$726$	0	0
$727$	48.5340	1.80003	0.900013	−	0.435864i	$-0.143557\pi$
$727$	48.5340	1.80003	0.900013	+	0.435864i	$0.143557\pi$
$728$	0	0
$729$	29.7377	1.10140
$730$	0	0
$731$	−44.3949	−1.64201
$732$	0	0
$733$	45.9485	1.69715	0.848573	−	0.529078i	$-0.177462\pi$
$733$	45.9485	1.69715	0.848573	+	0.529078i	$0.177462\pi$
$734$	0	0
$735$	−18.2483	−0.673099
$736$	0	0
$737$	−2.52735	−0.0930962
$738$	0	0
$739$	10.7206	0.394363	0.197182	−	0.980367i	$-0.436821\pi$
$739$	10.7206	0.394363	0.197182	+	0.980367i	$0.436821\pi$
$740$	0	0
$741$	13.4487	0.494052
$742$	0	0
$743$	4.32441	0.158647	0.0793236	−	0.996849i	$-0.474724\pi$
$743$	4.32441	0.158647	0.0793236	+	0.996849i	$0.474724\pi$
$744$	0	0
$745$	38.0828	1.39525
$746$	0	0
$747$	0.905851	0.0331434
$748$	0	0
$749$	−17.0088	−0.621488
$750$	0	0
$751$	−25.2763	−0.922344	−0.461172	−	0.887311i	$-0.652571\pi$
$751$	−25.2763	−0.922344	−0.461172	+	0.887311i	$0.652571\pi$
$752$	0	0
$753$	−21.2812	−0.775530
$754$	0	0
$755$	−29.0843	−1.05849
$756$	0	0
$757$	−18.5353	−0.673676	−0.336838	−	0.941563i	$-0.609358\pi$
$757$	−18.5353	−0.673676	−0.336838	+	0.941563i	$0.609358\pi$
$758$	0	0
$759$	−25.7664	−0.935262
$760$	0	0
$761$	−13.8759	−0.503002	−0.251501	−	0.967857i	$-0.580924\pi$
$761$	−13.8759	−0.503002	−0.251501	+	0.967857i	$0.580924\pi$
$762$	0	0
$763$	55.5519	2.01112
$764$	0	0
$765$	−5.23315	−0.189205
$766$	0	0
$767$	−4.30574	−0.155471
$768$	0	0
$769$	27.2268	0.981824	0.490912	−	0.871209i	$-0.336664\pi$
$769$	27.2268	0.981824	0.490912	+	0.871209i	$0.336664\pi$
$770$	0	0
$771$	6.19824	0.223224
$772$	0	0
$773$	−19.1780	−0.689786	−0.344893	−	0.938642i	$-0.612085\pi$
$773$	−19.1780	−0.689786	−0.344893	+	0.938642i	$0.612085\pi$
$774$	0	0
$775$	−10.0717	−0.361786
$776$	0	0
$777$	11.6504	0.417955
$778$	0	0
$779$	−18.6034	−0.666534
$780$	0	0
$781$	−13.2614	−0.474529
$782$	0	0
$783$	−33.4766	−1.19635
$784$	0	0
$785$	−18.2558	−0.651579
$786$	0	0
$787$	−47.3698	−1.68855	−0.844276	−	0.535909i	$-0.819969\pi$
$787$	−47.3698	−1.68855	−0.844276	+	0.535909i	$0.819969\pi$
$788$	0	0
$789$	42.5835	1.51601
$790$	0	0
$791$	44.5958	1.58564
$792$	0	0
$793$	−22.5396	−0.800404
$794$	0	0
$795$	−10.8401	−0.384460
$796$	0	0
$797$	−16.7527	−0.593411	−0.296706	−	0.954969i	$-0.595888\pi$
$797$	−16.7527	−0.593411	−0.296706	+	0.954969i	$0.595888\pi$
$798$	0	0
$799$	−20.4798	−0.724523
$800$	0	0
$801$	−6.89352	−0.243570
$802$	0	0
$803$	−15.9378	−0.562434
$804$	0	0
$805$	27.2100	0.959025
$806$	0	0
$807$	41.5433	1.46239
$808$	0	0
$809$	−18.4226	−0.647705	−0.323852	−	0.946108i	$-0.604978\pi$
$809$	−18.4226	−0.647705	−0.323852	+	0.946108i	$0.604978\pi$
$810$	0	0
$811$	16.3329	0.573527	0.286764	−	0.958001i	$-0.407421\pi$
$811$	16.3329	0.573527	0.286764	+	0.958001i	$0.407421\pi$
$812$	0	0
$813$	25.8522	0.906676
$814$	0	0
$815$	19.5767	0.685742
$816$	0	0
$817$	11.9514	0.418125
$818$	0	0
$819$	8.35484	0.291942
$820$	0	0
$821$	0.623164	0.0217486	0.0108743	−	0.999941i	$-0.496539\pi$
$821$	0.623164	0.0217486	0.0108743	+	0.999941i	$0.496539\pi$
$822$	0	0
$823$	−2.82032	−0.0983103	−0.0491552	−	0.998791i	$-0.515653\pi$
$823$	−2.82032	−0.0983103	−0.0491552	+	0.998791i	$0.515653\pi$
$824$	0	0
$825$	−8.13180	−0.283113
$826$	0	0
$827$	−56.9396	−1.97998	−0.989992	−	0.141124i	$-0.954928\pi$
$827$	−56.9396	−1.97998	−0.989992	+	0.141124i	$0.954928\pi$
$828$	0	0
$829$	12.1160	0.420805	0.210402	−	0.977615i	$-0.432523\pi$
$829$	12.1160	0.420805	0.210402	+	0.977615i	$0.432523\pi$
$830$	0	0
$831$	−38.7245	−1.34334
$832$	0	0
$833$	35.4202	1.22724
$834$	0	0
$835$	−13.6267	−0.471573
$836$	0	0
$837$	45.0109	1.55580
$838$	0	0
$839$	10.0482	0.346901	0.173450	−	0.984843i	$-0.444508\pi$
$839$	10.0482	0.346901	0.173450	+	0.984843i	$0.444508\pi$
$840$	0	0
$841$	7.93768	0.273713
$842$	0	0
$843$	−3.83480	−0.132077
$844$	0	0
$845$	27.1676	0.934595
$846$	0	0
$847$	−21.7423	−0.747073
$848$	0	0
$849$	−20.6241	−0.707816
$850$	0	0
$851$	−7.93435	−0.271986
$852$	0	0
$853$	29.1487	0.998034	0.499017	−	0.866592i	$-0.333695\pi$
$853$	29.1487	0.998034	0.499017	+	0.866592i	$0.333695\pi$
$854$	0	0
$855$	1.40879	0.0481798
$856$	0	0
$857$	32.5348	1.11137	0.555684	−	0.831394i	$-0.312457\pi$
$857$	32.5348	1.11137	0.555684	+	0.831394i	$0.312457\pi$
$858$	0	0
$859$	36.9928	1.26218	0.631089	−	0.775711i	$-0.282608\pi$
$859$	36.9928	1.26218	0.631089	+	0.775711i	$0.282608\pi$
$860$	0	0
$861$	65.8410	2.24385
$862$	0	0
$863$	46.9323	1.59759	0.798797	−	0.601601i	$-0.205470\pi$
$863$	46.9323	1.59759	0.798797	+	0.601601i	$0.205470\pi$
$864$	0	0
$865$	1.10811	0.0376768
$866$	0	0
$867$	−30.7105	−1.04298
$868$	0	0
$869$	−68.4970	−2.32360
$870$	0	0
$871$	3.17958	0.107736
$872$	0	0
$873$	8.41432	0.284781
$874$	0	0
$875$	43.4243	1.46801
$876$	0	0
$877$	41.9955	1.41809	0.709044	−	0.705165i	$-0.249127\pi$
$877$	41.9955	1.41809	0.709044	+	0.705165i	$0.249127\pi$
$878$	0	0
$879$	−35.7035	−1.20425
$880$	0	0
$881$	15.0343	0.506517	0.253259	−	0.967399i	$-0.418498\pi$
$881$	15.0343	0.506517	0.253259	+	0.967399i	$0.418498\pi$
$882$	0	0
$883$	23.0683	0.776309	0.388155	−	0.921594i	$-0.373113\pi$
$883$	23.0683	0.776309	0.388155	+	0.921594i	$0.373113\pi$
$884$	0	0
$885$	2.56958	0.0863756
$886$	0	0
$887$	0.866723	0.0291017	0.0145509	−	0.999894i	$-0.495368\pi$
$887$	0.866723	0.0291017	0.0145509	+	0.999894i	$0.495368\pi$
$888$	0	0
$889$	−41.4363	−1.38973
$890$	0	0
$891$	30.7911	1.03154
$892$	0	0
$893$	5.51328	0.184495
$894$	0	0
$895$	34.5225	1.15396
$896$	0	0
$897$	32.4159	1.08233
$898$	0	0
$899$	−49.6646	−1.65641
$900$	0	0
$901$	21.0408	0.700972
$902$	0	0
$903$	−42.2982	−1.40760
$904$	0	0
$905$	8.97766	0.298427
$906$	0	0
$907$	−41.1429	−1.36613	−0.683065	−	0.730358i	$-0.739353\pi$
$907$	−41.1429	−1.36613	−0.683065	+	0.730358i	$0.739353\pi$
$908$	0	0
$909$	6.14203	0.203718
$910$	0	0
$911$	43.3518	1.43631	0.718155	−	0.695884i	$-0.244987\pi$
$911$	43.3518	1.43631	0.718155	+	0.695884i	$0.244987\pi$
$912$	0	0
$913$	8.35159	0.276397
$914$	0	0
$915$	13.4512	0.444682
$916$	0	0
$917$	73.6076	2.43074
$918$	0	0
$919$	−5.96461	−0.196754	−0.0983772	−	0.995149i	$-0.531365\pi$
$919$	−5.96461	−0.196754	−0.0983772	+	0.995149i	$0.531365\pi$
$920$	0	0
$921$	31.0985	1.02473
$922$	0	0
$923$	16.6837	0.549150
$924$	0	0
$925$	−2.50406	−0.0823329
$926$	0	0
$927$	4.05324	0.133126
$928$	0	0
$929$	48.7720	1.60016	0.800079	−	0.599894i	$-0.204791\pi$
$929$	48.7720	1.60016	0.800079	+	0.599894i	$0.204791\pi$
$930$	0	0
$931$	−9.53532	−0.312507
$932$	0	0
$933$	−34.2677	−1.12187
$934$	0	0
$935$	−48.2476	−1.57787
$936$	0	0
$937$	2.85934	0.0934107	0.0467054	−	0.998909i	$-0.485128\pi$
$937$	2.85934	0.0934107	0.0467054	+	0.998909i	$0.485128\pi$
$938$	0	0
$939$	14.9130	0.486668
$940$	0	0
$941$	−39.1556	−1.27644	−0.638219	−	0.769855i	$-0.720328\pi$
$941$	−39.1556	−1.27644	−0.638219	+	0.769855i	$0.720328\pi$
$942$	0	0
$943$	−44.8402	−1.46020
$944$	0	0
$945$	−38.3774	−1.24842
$946$	0	0
$947$	−14.3420	−0.466054	−0.233027	−	0.972470i	$-0.574863\pi$
$947$	−14.3420	−0.466054	−0.233027	+	0.972470i	$0.574863\pi$
$948$	0	0
$949$	20.0508	0.650878
$950$	0	0
$951$	18.2624	0.592200
$952$	0	0
$953$	33.0667	1.07113	0.535567	−	0.844493i	$-0.320098\pi$
$953$	33.0667	1.07113	0.535567	+	0.844493i	$0.320098\pi$
$954$	0	0
$955$	−19.6653	−0.636354
$956$	0	0
$957$	−40.0987	−1.29621
$958$	0	0
$959$	35.1792	1.13600
$960$	0	0
$961$	35.7765	1.15408
$962$	0	0
$963$	−2.12260	−0.0684000
$964$	0	0
$965$	−34.7931	−1.12003
$966$	0	0
$967$	−14.2096	−0.456951	−0.228475	−	0.973550i	$-0.573374\pi$
$967$	−14.2096	−0.456951	−0.228475	+	0.973550i	$0.573374\pi$
$968$	0	0
$969$	15.5785	0.500452
$970$	0	0
$971$	7.71278	0.247515	0.123757	−	0.992313i	$-0.460506\pi$
$971$	7.71278	0.247515	0.123757	+	0.992313i	$0.460506\pi$
$972$	0	0
$973$	−48.3354	−1.54956
$974$	0	0
$975$	10.2303	0.327633
$976$	0	0
$977$	20.6237	0.659812	0.329906	−	0.944014i	$-0.392983\pi$
$977$	20.6237	0.659812	0.329906	+	0.944014i	$0.392983\pi$
$978$	0	0
$979$	−63.5555	−2.03124
$980$	0	0
$981$	6.93258	0.221340
$982$	0	0
$983$	22.9519	0.732052	0.366026	−	0.930605i	$-0.380718\pi$
$983$	22.9519	0.732052	0.366026	+	0.930605i	$0.380718\pi$
$984$	0	0
$985$	−20.1129	−0.640852
$986$	0	0
$987$	−19.5126	−0.621092
$988$	0	0
$989$	28.8067	0.915999
$990$	0	0
$991$	8.10042	0.257318	0.128659	−	0.991689i	$-0.458933\pi$
$991$	8.10042	0.257318	0.128659	+	0.991689i	$0.458933\pi$
$992$	0	0
$993$	−20.0872	−0.637450
$994$	0	0
$995$	−33.0668	−1.04829
$996$	0	0
$997$	−21.7782	−0.689722	−0.344861	−	0.938654i	$-0.612074\pi$
$997$	−21.7782	−0.689722	−0.344861	+	0.938654i	$0.612074\pi$
$998$	0	0
$999$	11.1907	0.354059

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6044.2.a.a.1.18	✓	63

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6044.2.a.a.1.18	✓	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)

\(f(q)\)	\(=\)	\(q-1.59751 q^{3} +1.94100 q^{5} -3.58958 q^{7} -0.447960 q^{9} +O(q^{10})\)	\(q-1.59751 q^{3} +1.94100 q^{5} -3.58958 q^{7} -0.447960 q^{9} -4.13002 q^{11} +5.19584 q^{13} -3.10077 q^{15} +6.01864 q^{17} -1.62025 q^{19} +5.73439 q^{21} -3.90534 q^{23} -1.23251 q^{25} +5.50815 q^{27} -6.07764 q^{29} +8.17169 q^{31} +6.59775 q^{33} -6.96738 q^{35} +2.03167 q^{37} -8.30040 q^{39} +11.4818 q^{41} -7.37624 q^{43} -0.869491 q^{45} -3.40273 q^{47} +5.88509 q^{49} -9.61484 q^{51} +3.49595 q^{53} -8.01637 q^{55} +2.58837 q^{57} -0.828691 q^{59} -4.33801 q^{61} +1.60799 q^{63} +10.0851 q^{65} +0.611947 q^{67} +6.23881 q^{69} +3.21097 q^{71} +3.85902 q^{73} +1.96895 q^{75} +14.8250 q^{77} +16.5852 q^{79} -7.45545 q^{81} -2.02217 q^{83} +11.6822 q^{85} +9.70909 q^{87} +15.3887 q^{89} -18.6509 q^{91} -13.0544 q^{93} -3.14491 q^{95} -18.7836 q^{97} +1.85008 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 63 q - 7 q^{3} - 7 q^{5} - 22 q^{7} + 62 q^{9}+O(q^{10}) \) 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

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