LMFDB - Embedded newform 4024.2.a.e.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4024,2,Mod(1,4024)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4024, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("4024.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4024 = 2^{3} \cdot 503 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4024.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:	$32.1318017734$
Analytic rank:	$1$
Dimension:	$29$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Character	$\chi$	$=$	4024.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.05929 q^{3} +0.773308 q^{5} -3.93634 q^{7} +6.35923 q^{9} +O(q^{10})$	$q-3.05929 q^{3} +0.773308 q^{5} -3.93634 q^{7} +6.35923 q^{9} +0.488115 q^{11} +1.20103 q^{13} -2.36577 q^{15} +1.22510 q^{17} -5.48934 q^{19} +12.0424 q^{21} +3.02621 q^{23} -4.40199 q^{25} -10.2769 q^{27} +3.71159 q^{29} -6.08834 q^{31} -1.49328 q^{33} -3.04400 q^{35} +9.95392 q^{37} -3.67429 q^{39} +0.904055 q^{41} +0.153564 q^{43} +4.91765 q^{45} +12.7862 q^{47} +8.49474 q^{49} -3.74794 q^{51} -13.1913 q^{53} +0.377463 q^{55} +16.7935 q^{57} +11.1370 q^{59} -8.23214 q^{61} -25.0321 q^{63} +0.928766 q^{65} +5.79594 q^{67} -9.25804 q^{69} +10.5293 q^{71} +4.65869 q^{73} +13.4670 q^{75} -1.92138 q^{77} -14.5329 q^{79} +12.3621 q^{81} +17.2504 q^{83} +0.947383 q^{85} -11.3548 q^{87} +1.53848 q^{89} -4.72766 q^{91} +18.6260 q^{93} -4.24495 q^{95} +0.103224 q^{97} +3.10403 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 29 q - 7 q^{3} - 4 q^{5} - 13 q^{7} + 20 q^{9}+O(q^{10}) $ 29 * q - 7 * q^3 - 4 * q^5 - 13 * q^7 + 20 * q^9	$ 29 q - 7 q^{3} - 4 q^{5} - 13 q^{7} + 20 q^{9} - 27 q^{11} + 16 q^{13} - 14 q^{15} - 15 q^{17} - 14 q^{19} + q^{21} - 25 q^{23} + 21 q^{25} - 25 q^{27} - 13 q^{29} - 27 q^{31} - 9 q^{33} - 29 q^{35} + 35 q^{37} - 38 q^{39} - 30 q^{41} - 38 q^{43} + q^{45} - 35 q^{47} + 14 q^{49} - 21 q^{51} + 2 q^{53} - 25 q^{55} - 25 q^{57} - 40 q^{59} + 10 q^{61} - 56 q^{63} - 50 q^{65} - 31 q^{67} + 11 q^{69} - 65 q^{71} - 23 q^{73} - 32 q^{75} + 13 q^{77} - 44 q^{79} - 7 q^{81} - 41 q^{83} + 26 q^{85} - 25 q^{87} - 48 q^{89} - 44 q^{91} + 25 q^{93} - 75 q^{95} - 18 q^{97} - 80 q^{99}+O(q^{100}) $ 29 * q - 7 * q^3 - 4 * q^5 - 13 * q^7 + 20 * q^9 - 27 * q^11 + 16 * q^13 - 14 * q^15 - 15 * q^17 - 14 * q^19 + q^21 - 25 * q^23 + 21 * q^25 - 25 * q^27 - 13 * q^29 - 27 * q^31 - 9 * q^33 - 29 * q^35 + 35 * q^37 - 38 * q^39 - 30 * q^41 - 38 * q^43 + q^45 - 35 * q^47 + 14 * q^49 - 21 * q^51 + 2 * q^53 - 25 * q^55 - 25 * q^57 - 40 * q^59 + 10 * q^61 - 56 * q^63 - 50 * q^65 - 31 * q^67 + 11 * q^69 - 65 * q^71 - 23 * q^73 - 32 * q^75 + 13 * q^77 - 44 * q^79 - 7 * q^81 - 41 * q^83 + 26 * q^85 - 25 * q^87 - 48 * q^89 - 44 * q^91 + 25 * q^93 - 75 * q^95 - 18 * q^97 - 80 * 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
$3$	−3.05929	−1.76628	−0.883140	−	0.469110i	$-0.844575\pi$
$3$	−3.05929	−1.76628	−0.883140	+	0.469110i	$0.844575\pi$
$4$	0	0
$5$	0.773308	0.345834	0.172917	−	0.984936i	$-0.444681\pi$
$5$	0.773308	0.345834	0.172917	+	0.984936i	$0.444681\pi$
$6$	0	0
$7$	−3.93634	−1.48780	−0.743898	−	0.668294i	$-0.767025\pi$
$7$	−3.93634	−1.48780	−0.743898	+	0.668294i	$0.767025\pi$
$8$	0	0
$9$	6.35923	2.11974
$10$	0	0
$11$	0.488115	0.147172	0.0735861	−	0.997289i	$-0.476556\pi$
$11$	0.488115	0.147172	0.0735861	+	0.997289i	$0.476556\pi$
$12$	0	0
$13$	1.20103	0.333106	0.166553	−	0.986033i	$-0.446736\pi$
$13$	1.20103	0.333106	0.166553	+	0.986033i	$0.446736\pi$
$14$	0	0
$15$	−2.36577	−0.610839
$16$	0	0
$17$	1.22510	0.297131	0.148566	−	0.988903i	$-0.452534\pi$
$17$	1.22510	0.297131	0.148566	+	0.988903i	$0.452534\pi$
$18$	0	0
$19$	−5.48934	−1.25934	−0.629670	−	0.776862i	$-0.716810\pi$
$19$	−5.48934	−1.25934	−0.629670	+	0.776862i	$0.716810\pi$
$20$	0	0
$21$	12.0424	2.62786
$22$	0	0
$23$	3.02621	0.631008	0.315504	−	0.948924i	$-0.397826\pi$
$23$	3.02621	0.631008	0.315504	+	0.948924i	$0.397826\pi$
$24$	0	0
$25$	−4.40199	−0.880399
$26$	0	0
$27$	−10.2769	−1.97778
$28$	0	0
$29$	3.71159	0.689225	0.344613	−	0.938745i	$-0.388010\pi$
$29$	3.71159	0.689225	0.344613	+	0.938745i	$0.388010\pi$
$30$	0	0
$31$	−6.08834	−1.09350	−0.546749	−	0.837296i	$-0.684135\pi$
$31$	−6.08834	−1.09350	−0.546749	+	0.837296i	$0.684135\pi$
$32$	0	0
$33$	−1.49328	−0.259947
$34$	0	0
$35$	−3.04400	−0.514530
$36$	0	0
$37$	9.95392	1.63641	0.818207	−	0.574924i	$-0.194968\pi$
$37$	9.95392	1.63641	0.818207	+	0.574924i	$0.194968\pi$
$38$	0	0
$39$	−3.67429	−0.588358
$40$	0	0
$41$	0.904055	0.141190	0.0705948	−	0.997505i	$-0.477510\pi$
$41$	0.904055	0.141190	0.0705948	+	0.997505i	$0.477510\pi$
$42$	0	0
$43$	0.153564	0.0234183	0.0117091	−	0.999931i	$-0.496273\pi$
$43$	0.153564	0.0234183	0.0117091	+	0.999931i	$0.496273\pi$
$44$	0	0
$45$	4.91765	0.733079
$46$	0	0
$47$	12.7862	1.86506	0.932530	−	0.361093i	$-0.117596\pi$
$47$	12.7862	1.86506	0.932530	+	0.361093i	$0.117596\pi$
$48$	0	0
$49$	8.49474	1.21353
$50$	0	0
$51$	−3.74794	−0.524817
$52$	0	0
$53$	−13.1913	−1.81197	−0.905985	−	0.423309i	$-0.860868\pi$
$53$	−13.1913	−1.81197	−0.905985	+	0.423309i	$0.860868\pi$
$54$	0	0
$55$	0.377463	0.0508971
$56$	0	0
$57$	16.7935	2.22435
$58$	0	0
$59$	11.1370	1.44992	0.724958	−	0.688793i	$-0.241859\pi$
$59$	11.1370	1.44992	0.724958	+	0.688793i	$0.241859\pi$
$60$	0	0
$61$	−8.23214	−1.05402	−0.527009	−	0.849860i	$-0.676687\pi$
$61$	−8.23214	−1.05402	−0.527009	+	0.849860i	$0.676687\pi$
$62$	0	0
$63$	−25.0321	−3.15375
$64$	0	0
$65$	0.928766	0.115199
$66$	0	0
$67$	5.79594	0.708087	0.354043	−	0.935229i	$-0.384807\pi$
$67$	5.79594	0.708087	0.354043	+	0.935229i	$0.384807\pi$
$68$	0	0
$69$	−9.25804	−1.11454
$70$	0	0
$71$	10.5293	1.24960	0.624798	−	0.780786i	$-0.285181\pi$
$71$	10.5293	1.24960	0.624798	+	0.780786i	$0.285181\pi$
$72$	0	0
$73$	4.65869	0.545259	0.272629	−	0.962119i	$-0.412107\pi$
$73$	4.65869	0.545259	0.272629	+	0.962119i	$0.412107\pi$
$74$	0	0
$75$	13.4670	1.55503
$76$	0	0
$77$	−1.92138	−0.218962
$78$	0	0
$79$	−14.5329	−1.63508	−0.817538	−	0.575874i	$-0.804662\pi$
$79$	−14.5329	−1.63508	−0.817538	+	0.575874i	$0.804662\pi$
$80$	0	0
$81$	12.3621	1.37357
$82$	0	0
$83$	17.2504	1.89348	0.946738	−	0.322005i	$-0.104357\pi$
$83$	17.2504	1.89348	0.946738	+	0.322005i	$0.104357\pi$
$84$	0	0
$85$	0.947383	0.102758
$86$	0	0
$87$	−11.3548	−1.21736
$88$	0	0
$89$	1.53848	0.163078	0.0815390	−	0.996670i	$-0.474016\pi$
$89$	1.53848	0.163078	0.0815390	+	0.996670i	$0.474016\pi$
$90$	0	0
$91$	−4.72766	−0.495593
$92$	0	0
$93$	18.6260	1.93143
$94$	0	0
$95$	−4.24495	−0.435523
$96$	0	0
$97$	0.103224	0.0104808	0.00524042	−	0.999986i	$-0.498332\pi$
$97$	0.103224	0.0104808	0.00524042	+	0.999986i	$0.498332\pi$
$98$	0	0
$99$	3.10403	0.311967
$100$	0	0
$101$	4.38271	0.436095	0.218048	−	0.975938i	$-0.430031\pi$
$101$	4.38271	0.436095	0.218048	+	0.975938i	$0.430031\pi$
$102$	0	0
$103$	11.3796	1.12126	0.560630	−	0.828066i	$-0.310559\pi$
$103$	11.3796	1.12126	0.560630	+	0.828066i	$0.310559\pi$
$104$	0	0
$105$	9.31247	0.908804
$106$	0	0
$107$	−6.99164	−0.675908	−0.337954	−	0.941163i	$-0.609735\pi$
$107$	−6.99164	−0.675908	−0.337954	+	0.941163i	$0.609735\pi$
$108$	0	0
$109$	−8.86813	−0.849413	−0.424707	−	0.905331i	$-0.639623\pi$
$109$	−8.86813	−0.849413	−0.424707	+	0.905331i	$0.639623\pi$
$110$	0	0
$111$	−30.4519	−2.89036
$112$	0	0
$113$	−17.6949	−1.66460	−0.832298	−	0.554328i	$-0.812975\pi$
$113$	−17.6949	−1.66460	−0.832298	+	0.554328i	$0.812975\pi$
$114$	0	0
$115$	2.34019	0.218224
$116$	0	0
$117$	7.63763	0.706099
$118$	0	0
$119$	−4.82242	−0.442070
$120$	0	0
$121$	−10.7617	−0.978340
$122$	0	0
$123$	−2.76576	−0.249380
$124$	0	0
$125$	−7.27064	−0.650306
$126$	0	0
$127$	−4.25768	−0.377808	−0.188904	−	0.981996i	$-0.560494\pi$
$127$	−4.25768	−0.377808	−0.188904	+	0.981996i	$0.560494\pi$
$128$	0	0
$129$	−0.469796	−0.0413632
$130$	0	0
$131$	1.43659	0.125515	0.0627576	−	0.998029i	$-0.480010\pi$
$131$	1.43659	0.125515	0.0627576	+	0.998029i	$0.480010\pi$
$132$	0	0
$133$	21.6079	1.87364
$134$	0	0
$135$	−7.94718	−0.683984
$136$	0	0
$137$	17.7976	1.52055	0.760277	−	0.649599i	$-0.225063\pi$
$137$	17.7976	1.52055	0.760277	+	0.649599i	$0.225063\pi$
$138$	0	0
$139$	−16.1897	−1.37320	−0.686598	−	0.727037i	$-0.740897\pi$
$139$	−16.1897	−1.37320	−0.686598	+	0.727037i	$0.740897\pi$
$140$	0	0
$141$	−39.1167	−3.29422
$142$	0	0
$143$	0.586240	0.0490239
$144$	0	0
$145$	2.87020	0.238357
$146$	0	0
$147$	−25.9878	−2.14344
$148$	0	0
$149$	7.43524	0.609119	0.304559	−	0.952493i	$-0.401491\pi$
$149$	7.43524	0.609119	0.304559	+	0.952493i	$0.401491\pi$
$150$	0	0
$151$	−19.2005	−1.56251	−0.781256	−	0.624211i	$-0.785421\pi$
$151$	−19.2005	−1.56251	−0.781256	+	0.624211i	$0.785421\pi$
$152$	0	0
$153$	7.79072	0.629842
$154$	0	0
$155$	−4.70817	−0.378169
$156$	0	0
$157$	−12.6325	−1.00818	−0.504091	−	0.863650i	$-0.668172\pi$
$157$	−12.6325	−1.00818	−0.504091	+	0.863650i	$0.668172\pi$
$158$	0	0
$159$	40.3561	3.20045
$160$	0	0
$161$	−11.9122	−0.938811
$162$	0	0
$163$	−4.30607	−0.337277	−0.168639	−	0.985678i	$-0.553937\pi$
$163$	−4.30607	−0.337277	−0.168639	+	0.985678i	$0.553937\pi$
$164$	0	0
$165$	−1.15477	−0.0898985
$166$	0	0
$167$	−2.44801	−0.189432	−0.0947162	−	0.995504i	$-0.530194\pi$
$167$	−2.44801	−0.189432	−0.0947162	+	0.995504i	$0.530194\pi$
$168$	0	0
$169$	−11.5575	−0.889041
$170$	0	0
$171$	−34.9080	−2.66948
$172$	0	0
$173$	20.4598	1.55553	0.777765	−	0.628555i	$-0.216353\pi$
$173$	20.4598	1.55553	0.777765	+	0.628555i	$0.216353\pi$
$174$	0	0
$175$	17.3277	1.30985
$176$	0	0
$177$	−34.0713	−2.56096
$178$	0	0
$179$	16.0331	1.19837	0.599184	−	0.800611i	$-0.295492\pi$
$179$	16.0331	1.19837	0.599184	+	0.800611i	$0.295492\pi$
$180$	0	0
$181$	11.0477	0.821170	0.410585	−	0.911822i	$-0.365325\pi$
$181$	11.0477	0.821170	0.410585	+	0.911822i	$0.365325\pi$
$182$	0	0
$183$	25.1845	1.86169
$184$	0	0
$185$	7.69744	0.565927
$186$	0	0
$187$	0.597991	0.0437294
$188$	0	0
$189$	40.4531	2.94253
$190$	0	0
$191$	−14.9531	−1.08197	−0.540983	−	0.841033i	$-0.681948\pi$
$191$	−14.9531	−1.08197	−0.540983	+	0.841033i	$0.681948\pi$
$192$	0	0
$193$	4.33184	0.311812	0.155906	−	0.987772i	$-0.450170\pi$
$193$	4.33184	0.311812	0.155906	+	0.987772i	$0.450170\pi$
$194$	0	0
$195$	−2.84136	−0.203474
$196$	0	0
$197$	−5.63011	−0.401129	−0.200564	−	0.979681i	$-0.564278\pi$
$197$	−5.63011	−0.401129	−0.200564	+	0.979681i	$0.564278\pi$
$198$	0	0
$199$	−11.1931	−0.793456	−0.396728	−	0.917936i	$-0.629854\pi$
$199$	−11.1931	−0.793456	−0.396728	+	0.917936i	$0.629854\pi$
$200$	0	0
$201$	−17.7314	−1.25068
$202$	0	0
$203$	−14.6101	−1.02543
$204$	0	0
$205$	0.699113	0.0488282
$206$	0	0
$207$	19.2444	1.33758
$208$	0	0
$209$	−2.67943	−0.185340
$210$	0	0
$211$	−17.1277	−1.17912	−0.589558	−	0.807726i	$-0.700698\pi$
$211$	−17.1277	−1.17912	−0.589558	+	0.807726i	$0.700698\pi$
$212$	0	0
$213$	−32.2121	−2.20714
$214$	0	0
$215$	0.118752	0.00809884
$216$	0	0
$217$	23.9658	1.62690
$218$	0	0
$219$	−14.2523	−0.963079
$220$	0	0
$221$	1.47139	0.0989761
$222$	0	0
$223$	3.14940	0.210900	0.105450	−	0.994425i	$-0.466372\pi$
$223$	3.14940	0.210900	0.105450	+	0.994425i	$0.466372\pi$
$224$	0	0
$225$	−27.9933	−1.86622
$226$	0	0
$227$	−8.56998	−0.568810	−0.284405	−	0.958704i	$-0.591796\pi$
$227$	−8.56998	−0.568810	−0.284405	+	0.958704i	$0.591796\pi$
$228$	0	0
$229$	−24.4735	−1.61725	−0.808627	−	0.588321i	$-0.799789\pi$
$229$	−24.4735	−1.61725	−0.808627	+	0.588321i	$0.799789\pi$
$230$	0	0
$231$	5.87806	0.386748
$232$	0	0
$233$	−16.7489	−1.09726	−0.548629	−	0.836066i	$-0.684850\pi$
$233$	−16.7489	−1.09726	−0.548629	+	0.836066i	$0.684850\pi$
$234$	0	0
$235$	9.88767	0.645001
$236$	0	0
$237$	44.4602	2.88800
$238$	0	0
$239$	−28.2011	−1.82418	−0.912088	−	0.409995i	$-0.865530\pi$
$239$	−28.2011	−1.82418	−0.912088	+	0.409995i	$0.865530\pi$
$240$	0	0
$241$	−20.2240	−1.30274	−0.651371	−	0.758760i	$-0.725806\pi$
$241$	−20.2240	−1.30274	−0.651371	+	0.758760i	$0.725806\pi$
$242$	0	0
$243$	−6.98877	−0.448330
$244$	0	0
$245$	6.56905	0.419681
$246$	0	0
$247$	−6.59286	−0.419494
$248$	0	0
$249$	−52.7739	−3.34441
$250$	0	0
$251$	31.3104	1.97630	0.988148	−	0.153502i	$-0.0490552\pi$
$251$	31.3104	1.97630	0.988148	+	0.153502i	$0.0490552\pi$
$252$	0	0
$253$	1.47714	0.0928668
$254$	0	0
$255$	−2.89831	−0.181500
$256$	0	0
$257$	−16.7769	−1.04652	−0.523258	−	0.852174i	$-0.675284\pi$
$257$	−16.7769	−1.04652	−0.523258	+	0.852174i	$0.675284\pi$
$258$	0	0
$259$	−39.1820	−2.43465
$260$	0	0
$261$	23.6029	1.46098
$262$	0	0
$263$	−19.1869	−1.18312	−0.591558	−	0.806262i	$-0.701487\pi$
$263$	−19.1869	−1.18312	−0.591558	+	0.806262i	$0.701487\pi$
$264$	0	0
$265$	−10.2010	−0.626641
$266$	0	0
$267$	−4.70664	−0.288041
$268$	0	0
$269$	10.0973	0.615641	0.307820	−	0.951444i	$-0.400400\pi$
$269$	10.0973	0.615641	0.307820	+	0.951444i	$0.400400\pi$
$270$	0	0
$271$	24.0869	1.46318	0.731589	−	0.681746i	$-0.238779\pi$
$271$	24.0869	1.46318	0.731589	+	0.681746i	$0.238779\pi$
$272$	0	0
$273$	14.4633	0.875356
$274$	0	0
$275$	−2.14868	−0.129570
$276$	0	0
$277$	−1.79897	−0.108090	−0.0540449	−	0.998539i	$-0.517211\pi$
$277$	−1.79897	−0.108090	−0.0540449	+	0.998539i	$0.517211\pi$
$278$	0	0
$279$	−38.7172	−2.31794
$280$	0	0
$281$	−8.01290	−0.478010	−0.239005	−	0.971018i	$-0.576821\pi$
$281$	−8.01290	−0.478010	−0.239005	+	0.971018i	$0.576821\pi$
$282$	0	0
$283$	13.0403	0.775163	0.387581	−	0.921835i	$-0.373311\pi$
$283$	13.0403	0.775163	0.387581	+	0.921835i	$0.373311\pi$
$284$	0	0
$285$	12.9865	0.769255
$286$	0	0
$287$	−3.55866	−0.210061
$288$	0	0
$289$	−15.4991	−0.911713
$290$	0	0
$291$	−0.315793	−0.0185121
$292$	0	0
$293$	−1.41587	−0.0827159	−0.0413580	−	0.999144i	$-0.513168\pi$
$293$	−1.41587	−0.0827159	−0.0413580	+	0.999144i	$0.513168\pi$
$294$	0	0
$295$	8.61235	0.501430
$296$	0	0
$297$	−5.01628	−0.291074
$298$	0	0
$299$	3.63457	0.210193
$300$	0	0
$301$	−0.604479	−0.0348416
$302$	0	0
$303$	−13.4079	−0.770267
$304$	0	0
$305$	−6.36598	−0.364515
$306$	0	0
$307$	−23.3854	−1.33467	−0.667337	−	0.744756i	$-0.732566\pi$
$307$	−23.3854	−1.33467	−0.667337	+	0.744756i	$0.732566\pi$
$308$	0	0
$309$	−34.8133	−1.98046
$310$	0	0
$311$	−8.88023	−0.503552	−0.251776	−	0.967786i	$-0.581015\pi$
$311$	−8.88023	−0.503552	−0.251776	+	0.967786i	$0.581015\pi$
$312$	0	0
$313$	−10.3420	−0.584565	−0.292282	−	0.956332i	$-0.594415\pi$
$313$	−10.3420	−0.584565	−0.292282	+	0.956332i	$0.594415\pi$
$314$	0	0
$315$	−19.3575	−1.09067
$316$	0	0
$317$	18.6746	1.04887	0.524435	−	0.851451i	$-0.324277\pi$
$317$	18.6746	1.04887	0.524435	+	0.851451i	$0.324277\pi$
$318$	0	0
$319$	1.81168	0.101435
$320$	0	0
$321$	21.3894	1.19384
$322$	0	0
$323$	−6.72501	−0.374190
$324$	0	0
$325$	−5.28693	−0.293266
$326$	0	0
$327$	27.1302	1.50030
$328$	0	0
$329$	−50.3308	−2.77483
$330$	0	0
$331$	4.97030	0.273193	0.136596	−	0.990627i	$-0.456384\pi$
$331$	4.97030	0.273193	0.136596	+	0.990627i	$0.456384\pi$
$332$	0	0
$333$	63.2993	3.46878
$334$	0	0
$335$	4.48205	0.244880
$336$	0	0
$337$	0.0205908	0.00112165	0.000560827	−	1.00000i	$-0.499821\pi$
$337$	0.0205908	0.00112165	0.000560827		1.00000i	$0.499821\pi$
$338$	0	0
$339$	54.1338	2.94014
$340$	0	0
$341$	−2.97181	−0.160933
$342$	0	0
$343$	−5.88380	−0.317695
$344$	0	0
$345$	−7.15932	−0.385445
$346$	0	0
$347$	−12.4295	−0.667253	−0.333626	−	0.942705i	$-0.608272\pi$
$347$	−12.4295	−0.667253	−0.333626	+	0.942705i	$0.608272\pi$
$348$	0	0
$349$	−8.46707	−0.453232	−0.226616	−	0.973984i	$-0.572766\pi$
$349$	−8.46707	−0.453232	−0.226616	+	0.973984i	$0.572766\pi$
$350$	0	0
$351$	−12.3428	−0.658810
$352$	0	0
$353$	−13.2308	−0.704202	−0.352101	−	0.935962i	$-0.614533\pi$
$353$	−13.2308	−0.704202	−0.352101	+	0.935962i	$0.614533\pi$
$354$	0	0
$355$	8.14239	0.432153
$356$	0	0
$357$	14.7532	0.780820
$358$	0	0
$359$	−7.65605	−0.404071	−0.202036	−	0.979378i	$-0.564756\pi$
$359$	−7.65605	−0.404071	−0.202036	+	0.979378i	$0.564756\pi$
$360$	0	0
$361$	11.1328	0.585939
$362$	0	0
$363$	32.9233	1.72802
$364$	0	0
$365$	3.60260	0.188569
$366$	0	0
$367$	−14.6463	−0.764530	−0.382265	−	0.924053i	$-0.624856\pi$
$367$	−14.6463	−0.764530	−0.382265	+	0.924053i	$0.624856\pi$
$368$	0	0
$369$	5.74910	0.299286
$370$	0	0
$371$	51.9256	2.69584
$372$	0	0
$373$	−17.7581	−0.919481	−0.459740	−	0.888053i	$-0.652058\pi$
$373$	−17.7581	−0.919481	−0.459740	+	0.888053i	$0.652058\pi$
$374$	0	0
$375$	22.2430	1.14862
$376$	0	0
$377$	4.45773	0.229585
$378$	0	0
$379$	4.08381	0.209771	0.104886	−	0.994484i	$-0.466552\pi$
$379$	4.08381	0.209771	0.104886	+	0.994484i	$0.466552\pi$
$380$	0	0
$381$	13.0255	0.667315
$382$	0	0
$383$	−18.4872	−0.944650	−0.472325	−	0.881425i	$-0.656585\pi$
$383$	−18.4872	−0.944650	−0.472325	+	0.881425i	$0.656585\pi$
$384$	0	0
$385$	−1.48582	−0.0757245
$386$	0	0
$387$	0.976549	0.0496408
$388$	0	0
$389$	−5.88618	−0.298441	−0.149221	−	0.988804i	$-0.547676\pi$
$389$	−5.88618	−0.298441	−0.149221	+	0.988804i	$0.547676\pi$
$390$	0	0
$391$	3.70742	0.187492
$392$	0	0
$393$	−4.39493	−0.221695
$394$	0	0
$395$	−11.2384	−0.565465
$396$	0	0
$397$	37.2874	1.87140	0.935699	−	0.352798i	$-0.114770\pi$
$397$	37.2874	1.87140	0.935699	+	0.352798i	$0.114770\pi$
$398$	0	0
$399$	−66.1047	−3.30937
$400$	0	0
$401$	20.4727	1.02236	0.511178	−	0.859475i	$-0.329209\pi$
$401$	20.4727	1.02236	0.511178	+	0.859475i	$0.329209\pi$
$402$	0	0
$403$	−7.31228	−0.364251
$404$	0	0
$405$	9.55974	0.475028
$406$	0	0
$407$	4.85865	0.240834
$408$	0	0
$409$	−7.04647	−0.348425	−0.174213	−	0.984708i	$-0.555738\pi$
$409$	−7.04647	−0.348425	−0.174213	+	0.984708i	$0.555738\pi$
$410$	0	0
$411$	−54.4480	−2.68572
$412$	0	0
$413$	−43.8390	−2.15718
$414$	0	0
$415$	13.3399	0.654828
$416$	0	0
$417$	49.5290	2.42545
$418$	0	0
$419$	5.58187	0.272692	0.136346	−	0.990661i	$-0.456464\pi$
$419$	5.58187	0.272692	0.136346	+	0.990661i	$0.456464\pi$
$420$	0	0
$421$	8.34571	0.406745	0.203373	−	0.979101i	$-0.434810\pi$
$421$	8.34571	0.406745	0.203373	+	0.979101i	$0.434810\pi$
$422$	0	0
$423$	81.3104	3.95345
$424$	0	0
$425$	−5.39290	−0.261594
$426$	0	0
$427$	32.4045	1.56816
$428$	0	0
$429$	−1.79348	−0.0865899
$430$	0	0
$431$	−34.5612	−1.66475	−0.832377	−	0.554209i	$-0.813021\pi$
$431$	−34.5612	−1.66475	−0.832377	+	0.554209i	$0.813021\pi$
$432$	0	0
$433$	−17.4481	−0.838503	−0.419252	−	0.907870i	$-0.637708\pi$
$433$	−17.4481	−0.838503	−0.419252	+	0.907870i	$0.637708\pi$
$434$	0	0
$435$	−8.78077	−0.421006
$436$	0	0
$437$	−16.6119	−0.794655
$438$	0	0
$439$	6.96544	0.332442	0.166221	−	0.986088i	$-0.446843\pi$
$439$	6.96544	0.332442	0.166221	+	0.986088i	$0.446843\pi$
$440$	0	0
$441$	54.0200	2.57238
$442$	0	0
$443$	−16.7345	−0.795080	−0.397540	−	0.917585i	$-0.630136\pi$
$443$	−16.7345	−0.795080	−0.397540	+	0.917585i	$0.630136\pi$
$444$	0	0
$445$	1.18972	0.0563979
$446$	0	0
$447$	−22.7465	−1.07587
$448$	0	0
$449$	8.61450	0.406543	0.203272	−	0.979122i	$-0.434843\pi$
$449$	8.61450	0.406543	0.203272	+	0.979122i	$0.434843\pi$
$450$	0	0
$451$	0.441283	0.0207792
$452$	0	0
$453$	58.7397	2.75983
$454$	0	0
$455$	−3.65594	−0.171393
$456$	0	0
$457$	22.3211	1.04414	0.522068	−	0.852904i	$-0.325161\pi$
$457$	22.3211	1.04414	0.522068	+	0.852904i	$0.325161\pi$
$458$	0	0
$459$	−12.5902	−0.587661
$460$	0	0
$461$	−27.0353	−1.25916	−0.629579	−	0.776936i	$-0.716773\pi$
$461$	−27.0353	−1.25916	−0.629579	+	0.776936i	$0.716773\pi$
$462$	0	0
$463$	−32.3990	−1.50571	−0.752854	−	0.658187i	$-0.771324\pi$
$463$	−32.3990	−1.50571	−0.752854	+	0.658187i	$0.771324\pi$
$464$	0	0
$465$	14.4036	0.667952
$466$	0	0
$467$	0.0326884	0.00151264	0.000756319	−	1.00000i	$-0.499759\pi$
$467$	0.0326884	0.00151264	0.000756319		1.00000i	$0.499759\pi$
$468$	0	0
$469$	−22.8148	−1.05349
$470$	0	0
$471$	38.6464	1.78073
$472$	0	0
$473$	0.0749568	0.00344652
$474$	0	0
$475$	24.1640	1.10872
$476$	0	0
$477$	−83.8868	−3.84091
$478$	0	0
$479$	17.9000	0.817872	0.408936	−	0.912563i	$-0.365900\pi$
$479$	17.9000	0.817872	0.408936	+	0.912563i	$0.365900\pi$
$480$	0	0
$481$	11.9550	0.545099
$482$	0	0
$483$	36.4428	1.65820
$484$	0	0
$485$	0.0798242	0.00362463
$486$	0	0
$487$	6.68156	0.302770	0.151385	−	0.988475i	$-0.451627\pi$
$487$	6.68156	0.302770	0.151385	+	0.988475i	$0.451627\pi$
$488$	0	0
$489$	13.1735	0.595726
$490$	0	0
$491$	−21.3753	−0.964653	−0.482326	−	0.875992i	$-0.660208\pi$
$491$	−21.3753	−0.964653	−0.482326	+	0.875992i	$0.660208\pi$
$492$	0	0
$493$	4.54708	0.204790
$494$	0	0
$495$	2.40038	0.107889
$496$	0	0
$497$	−41.4468	−1.85914
$498$	0	0
$499$	5.65484	0.253145	0.126573	−	0.991957i	$-0.459602\pi$
$499$	5.65484	0.253145	0.126573	+	0.991957i	$0.459602\pi$
$500$	0	0
$501$	7.48916	0.334591
$502$	0	0
$503$	−1.00000	−0.0445878
$503$	−1.00000	−0.0445878
$504$	0	0
$505$	3.38918	0.150817
$506$	0	0
$507$	35.3578	1.57029
$508$	0	0
$509$	21.0049	0.931027	0.465513	−	0.885041i	$-0.345870\pi$
$509$	21.0049	0.931027	0.465513	+	0.885041i	$0.345870\pi$
$510$	0	0
$511$	−18.3382	−0.811233
$512$	0	0
$513$	56.4131	2.49070
$514$	0	0
$515$	8.79990	0.387770
$516$	0	0
$517$	6.24113	0.274485
$518$	0	0
$519$	−62.5924	−2.74750
$520$	0	0
$521$	1.76317	0.0772458	0.0386229	−	0.999254i	$-0.487703\pi$
$521$	1.76317	0.0772458	0.0386229	+	0.999254i	$0.487703\pi$
$522$	0	0
$523$	40.8784	1.78749	0.893744	−	0.448578i	$-0.148069\pi$
$523$	40.8784	1.78749	0.893744	+	0.448578i	$0.148069\pi$
$524$	0	0
$525$	−53.0105	−2.31357
$526$	0	0
$527$	−7.45885	−0.324913
$528$	0	0
$529$	−13.8421	−0.601828
$530$	0	0
$531$	70.8229	3.07345
$532$	0	0
$533$	1.08580	0.0470311
$534$	0	0
$535$	−5.40669	−0.233752
$536$	0	0
$537$	−49.0498	−2.11665
$538$	0	0
$539$	4.14641	0.178598
$540$	0	0
$541$	11.7905	0.506915	0.253458	−	0.967346i	$-0.418432\pi$
$541$	11.7905	0.506915	0.253458	+	0.967346i	$0.418432\pi$
$542$	0	0
$543$	−33.7981	−1.45042
$544$	0	0
$545$	−6.85780	−0.293756
$546$	0	0
$547$	−37.3877	−1.59858	−0.799290	−	0.600945i	$-0.794791\pi$
$547$	−37.3877	−1.59858	−0.799290	+	0.600945i	$0.794791\pi$
$548$	0	0
$549$	−52.3501	−2.23425
$550$	0	0
$551$	−20.3742	−0.867969
$552$	0	0
$553$	57.2063	2.43266
$554$	0	0
$555$	−23.5487	−0.999586
$556$	0	0
$557$	−16.3888	−0.694417	−0.347209	−	0.937788i	$-0.612870\pi$
$557$	−16.3888	−0.694417	−0.347209	+	0.937788i	$0.612870\pi$
$558$	0	0
$559$	0.184435	0.00780077
$560$	0	0
$561$	−1.82943	−0.0772384
$562$	0	0
$563$	−7.32113	−0.308549	−0.154274	−	0.988028i	$-0.549304\pi$
$563$	−7.32113	−0.308549	−0.154274	+	0.988028i	$0.549304\pi$
$564$	0	0
$565$	−13.6836	−0.575674
$566$	0	0
$567$	−48.6615	−2.04359
$568$	0	0
$569$	−16.9691	−0.711381	−0.355690	−	0.934604i	$-0.615754\pi$
$569$	−16.9691	−0.711381	−0.355690	+	0.934604i	$0.615754\pi$
$570$	0	0
$571$	−32.7259	−1.36953	−0.684767	−	0.728762i	$-0.740096\pi$
$571$	−32.7259	−1.36953	−0.684767	+	0.728762i	$0.740096\pi$
$572$	0	0
$573$	45.7457	1.91105
$574$	0	0
$575$	−13.3214	−0.555539
$576$	0	0
$577$	−4.02630	−0.167617	−0.0838085	−	0.996482i	$-0.526708\pi$
$577$	−4.02630	−0.167617	−0.0838085	+	0.996482i	$0.526708\pi$
$578$	0	0
$579$	−13.2523	−0.550748
$580$	0	0
$581$	−67.9033	−2.81710
$582$	0	0
$583$	−6.43889	−0.266672
$584$	0	0
$585$	5.90624	0.244193
$586$	0	0
$587$	5.35285	0.220936	0.110468	−	0.993880i	$-0.464765\pi$
$587$	5.35285	0.220936	0.110468	+	0.993880i	$0.464765\pi$
$588$	0	0
$589$	33.4210	1.37709
$590$	0	0
$591$	17.2241	0.708506
$592$	0	0
$593$	33.1494	1.36128	0.680642	−	0.732616i	$-0.261701\pi$
$593$	33.1494	1.36128	0.680642	+	0.732616i	$0.261701\pi$
$594$	0	0
$595$	−3.72922	−0.152883
$596$	0	0
$597$	34.2428	1.40146
$598$	0	0
$599$	39.2886	1.60529	0.802645	−	0.596457i	$-0.203425\pi$
$599$	39.2886	1.60529	0.802645	+	0.596457i	$0.203425\pi$
$600$	0	0
$601$	−37.5280	−1.53080	−0.765399	−	0.643556i	$-0.777458\pi$
$601$	−37.5280	−1.53080	−0.765399	+	0.643556i	$0.777458\pi$
$602$	0	0
$603$	36.8577	1.50096
$604$	0	0
$605$	−8.32214	−0.338343
$606$	0	0
$607$	−44.5418	−1.80790	−0.903948	−	0.427641i	$-0.859345\pi$
$607$	−44.5418	−1.80790	−0.903948	+	0.427641i	$0.859345\pi$
$608$	0	0
$609$	44.6964	1.81119
$610$	0	0
$611$	15.3566	0.621262
$612$	0	0
$613$	3.64962	0.147407	0.0737034	−	0.997280i	$-0.476518\pi$
$613$	3.64962	0.147407	0.0737034	+	0.997280i	$0.476518\pi$
$614$	0	0
$615$	−2.13879	−0.0862442
$616$	0	0
$617$	14.4523	0.581828	0.290914	−	0.956749i	$-0.406041\pi$
$617$	14.4523	0.581828	0.290914	+	0.956749i	$0.406041\pi$
$618$	0	0
$619$	−6.72409	−0.270264	−0.135132	−	0.990828i	$-0.543146\pi$
$619$	−6.72409	−0.270264	−0.135132	+	0.990828i	$0.543146\pi$
$620$	0	0
$621$	−31.0999	−1.24800
$622$	0	0
$623$	−6.05596	−0.242627
$624$	0	0
$625$	16.3875	0.655501
$626$	0	0
$627$	8.19713	0.327362
$628$	0	0
$629$	12.1946	0.486230
$630$	0	0
$631$	39.6883	1.57997	0.789983	−	0.613129i	$-0.210090\pi$
$631$	39.6883	1.57997	0.789983	+	0.613129i	$0.210090\pi$
$632$	0	0
$633$	52.3984	2.08265
$634$	0	0
$635$	−3.29250	−0.130659
$636$	0	0
$637$	10.2024	0.404235
$638$	0	0
$639$	66.9582	2.64883
$640$	0	0
$641$	−17.2655	−0.681947	−0.340973	−	0.940073i	$-0.610757\pi$
$641$	−17.2655	−0.681947	−0.340973	+	0.940073i	$0.610757\pi$
$642$	0	0
$643$	−24.2873	−0.957799	−0.478900	−	0.877870i	$-0.658964\pi$
$643$	−24.2873	−0.957799	−0.478900	+	0.877870i	$0.658964\pi$
$644$	0	0
$645$	−0.363297	−0.0143048
$646$	0	0
$647$	−6.79549	−0.267158	−0.133579	−	0.991038i	$-0.542647\pi$
$647$	−6.79549	−0.267158	−0.133579	+	0.991038i	$0.542647\pi$
$648$	0	0
$649$	5.43614	0.213387
$650$	0	0
$651$	−73.3181	−2.87356
$652$	0	0
$653$	43.3174	1.69514	0.847570	−	0.530684i	$-0.178065\pi$
$653$	43.3174	1.69514	0.847570	+	0.530684i	$0.178065\pi$
$654$	0	0
$655$	1.11092	0.0434074
$656$	0	0
$657$	29.6257	1.15581
$658$	0	0
$659$	32.9551	1.28375	0.641875	−	0.766809i	$-0.278157\pi$
$659$	32.9551	1.28375	0.641875	+	0.766809i	$0.278157\pi$
$660$	0	0
$661$	−7.87876	−0.306448	−0.153224	−	0.988191i	$-0.548966\pi$
$661$	−7.87876	−0.306448	−0.153224	+	0.988191i	$0.548966\pi$
$662$	0	0
$663$	−4.50139	−0.174820
$664$	0	0
$665$	16.7096	0.647969
$666$	0	0
$667$	11.2321	0.434907
$668$	0	0
$669$	−9.63493	−0.372508
$670$	0	0
$671$	−4.01823	−0.155122
$672$	0	0
$673$	−6.52047	−0.251346	−0.125673	−	0.992072i	$-0.540109\pi$
$673$	−6.52047	−0.251346	−0.125673	+	0.992072i	$0.540109\pi$
$674$	0	0
$675$	45.2387	1.74124
$676$	0	0
$677$	31.7503	1.22026	0.610132	−	0.792299i	$-0.291116\pi$
$677$	31.7503	1.22026	0.610132	+	0.792299i	$0.291116\pi$
$678$	0	0
$679$	−0.406326	−0.0155934
$680$	0	0
$681$	26.2180	1.00468
$682$	0	0
$683$	−2.33192	−0.0892283	−0.0446141	−	0.999004i	$-0.514206\pi$
$683$	−2.33192	−0.0892283	−0.0446141	+	0.999004i	$0.514206\pi$
$684$	0	0
$685$	13.7631	0.525859
$686$	0	0
$687$	74.8714	2.85652
$688$	0	0
$689$	−15.8432	−0.603578
$690$	0	0
$691$	−38.7882	−1.47557	−0.737786	−	0.675034i	$-0.764129\pi$
$691$	−38.7882	−1.47557	−0.737786	+	0.675034i	$0.764129\pi$
$692$	0	0
$693$	−12.2185	−0.464143
$694$	0	0
$695$	−12.5197	−0.474898
$696$	0	0
$697$	1.10756	0.0419519
$698$	0	0
$699$	51.2398	1.93807
$700$	0	0
$701$	3.46176	0.130749	0.0653745	−	0.997861i	$-0.479176\pi$
$701$	3.46176	0.130749	0.0653745	+	0.997861i	$0.479176\pi$
$702$	0	0
$703$	−54.6404	−2.06080
$704$	0	0
$705$	−30.2492	−1.13925
$706$	0	0
$707$	−17.2518	−0.648821
$708$	0	0
$709$	5.23449	0.196585	0.0982926	−	0.995158i	$-0.468662\pi$
$709$	5.23449	0.196585	0.0982926	+	0.995158i	$0.468662\pi$
$710$	0	0
$711$	−92.4180	−3.46594
$712$	0	0
$713$	−18.4246	−0.690007
$714$	0	0
$715$	0.453344	0.0169541
$716$	0	0
$717$	86.2751	3.22200
$718$	0	0
$719$	40.3254	1.50388	0.751941	−	0.659230i	$-0.229118\pi$
$719$	40.3254	1.50388	0.751941	+	0.659230i	$0.229118\pi$
$720$	0	0
$721$	−44.7937	−1.66821
$722$	0	0
$723$	61.8709	2.30101
$724$	0	0
$725$	−16.3384	−0.606793
$726$	0	0
$727$	−7.26666	−0.269505	−0.134753	−	0.990879i	$-0.543024\pi$
$727$	−7.26666	−0.269505	−0.134753	+	0.990879i	$0.543024\pi$
$728$	0	0
$729$	−15.7058	−0.581696
$730$	0	0
$731$	0.188132	0.00695831
$732$	0	0
$733$	30.4899	1.12617	0.563085	−	0.826399i	$-0.309615\pi$
$733$	30.4899	1.12617	0.563085	+	0.826399i	$0.309615\pi$
$734$	0	0
$735$	−20.0966	−0.741275
$736$	0	0
$737$	2.82908	0.104211
$738$	0	0
$739$	−27.9188	−1.02701	−0.513505	−	0.858087i	$-0.671653\pi$
$739$	−27.9188	−1.02701	−0.513505	+	0.858087i	$0.671653\pi$
$740$	0	0
$741$	20.1694	0.740943
$742$	0	0
$743$	51.7433	1.89828	0.949138	−	0.314860i	$-0.101958\pi$
$743$	51.7433	1.89828	0.949138	+	0.314860i	$0.101958\pi$
$744$	0	0
$745$	5.74973	0.210654
$746$	0	0
$747$	109.699	4.01368
$748$	0	0
$749$	27.5215	1.00561
$750$	0	0
$751$	13.5844	0.495700	0.247850	−	0.968798i	$-0.420276\pi$
$751$	13.5844	0.495700	0.247850	+	0.968798i	$0.420276\pi$
$752$	0	0
$753$	−95.7876	−3.49069
$754$	0	0
$755$	−14.8479	−0.540370
$756$	0	0
$757$	−17.4598	−0.634589	−0.317294	−	0.948327i	$-0.602774\pi$
$757$	−17.4598	−0.634589	−0.317294	+	0.948327i	$0.602774\pi$
$758$	0	0
$759$	−4.51899	−0.164029
$760$	0	0
$761$	35.2306	1.27711	0.638554	−	0.769577i	$-0.279533\pi$
$761$	35.2306	1.27711	0.638554	+	0.769577i	$0.279533\pi$
$762$	0	0
$763$	34.9079	1.26375
$764$	0	0
$765$	6.02463	0.217821
$766$	0	0
$767$	13.3759	0.482975
$768$	0	0
$769$	37.9736	1.36936	0.684682	−	0.728842i	$-0.259941\pi$
$769$	37.9736	1.36936	0.684682	+	0.728842i	$0.259941\pi$
$770$	0	0
$771$	51.3254	1.84844
$772$	0	0
$773$	−1.67535	−0.0602583	−0.0301291	−	0.999546i	$-0.509592\pi$
$773$	−1.67535	−0.0602583	−0.0301291	+	0.999546i	$0.509592\pi$
$774$	0	0
$775$	26.8009	0.962715
$776$	0	0
$777$	119.869	4.30027
$778$	0	0
$779$	−4.96266	−0.177806
$780$	0	0
$781$	5.13950	0.183906
$782$	0	0
$783$	−38.1435	−1.36314
$784$	0	0
$785$	−9.76880	−0.348664
$786$	0	0
$787$	−3.27997	−0.116918	−0.0584591	−	0.998290i	$-0.518619\pi$
$787$	−3.27997	−0.116918	−0.0584591	+	0.998290i	$0.518619\pi$
$788$	0	0
$789$	58.6983	2.08971
$790$	0	0
$791$	69.6531	2.47658
$792$	0	0
$793$	−9.88705	−0.351099
$794$	0	0
$795$	31.2077	1.10682
$796$	0	0
$797$	34.4315	1.21963	0.609813	−	0.792545i	$-0.291245\pi$
$797$	34.4315	1.21963	0.609813	+	0.792545i	$0.291245\pi$
$798$	0	0
$799$	15.6644	0.554168
$800$	0	0
$801$	9.78352	0.345684
$802$	0	0
$803$	2.27398	0.0802468
$804$	0	0
$805$	−9.21178	−0.324673
$806$	0	0
$807$	−30.8904	−1.08739
$808$	0	0
$809$	−33.3923	−1.17401	−0.587005	−	0.809583i	$-0.699693\pi$
$809$	−33.3923	−1.17401	−0.587005	+	0.809583i	$0.699693\pi$
$810$	0	0
$811$	25.4050	0.892092	0.446046	−	0.895010i	$-0.352832\pi$
$811$	25.4050	0.892092	0.446046	+	0.895010i	$0.352832\pi$
$812$	0	0
$813$	−73.6889	−2.58438
$814$	0	0
$815$	−3.32992	−0.116642
$816$	0	0
$817$	−0.842965	−0.0294916
$818$	0	0
$819$	−30.0643	−1.05053
$820$	0	0
$821$	−32.3219	−1.12804	−0.564021	−	0.825760i	$-0.690746\pi$
$821$	−32.3219	−1.12804	−0.564021	+	0.825760i	$0.690746\pi$
$822$	0	0
$823$	45.6137	1.58999	0.794997	−	0.606614i	$-0.207472\pi$
$823$	45.6137	1.58999	0.794997	+	0.606614i	$0.207472\pi$
$824$	0	0
$825$	6.57342	0.228857
$826$	0	0
$827$	0.893117	0.0310567	0.0155284	−	0.999879i	$-0.495057\pi$
$827$	0.893117	0.0310567	0.0155284	+	0.999879i	$0.495057\pi$
$828$	0	0
$829$	−21.2912	−0.739472	−0.369736	−	0.929137i	$-0.620552\pi$
$829$	−21.2912	−0.739472	−0.369736	+	0.929137i	$0.620552\pi$
$830$	0	0
$831$	5.50357	0.190917
$832$	0	0
$833$	10.4069	0.360579
$834$	0	0
$835$	−1.89306	−0.0655122
$836$	0	0
$837$	62.5690	2.16270
$838$	0	0
$839$	−16.8577	−0.581991	−0.290996	−	0.956724i	$-0.593986\pi$
$839$	−16.8577	−0.581991	−0.290996	+	0.956724i	$0.593986\pi$
$840$	0	0
$841$	−15.2241	−0.524969
$842$	0	0
$843$	24.5138	0.844299
$844$	0	0
$845$	−8.93753	−0.307460
$846$	0	0
$847$	42.3618	1.45557
$848$	0	0
$849$	−39.8939	−1.36915
$850$	0	0
$851$	30.1226	1.03259
$852$	0	0
$853$	−39.3839	−1.34848	−0.674239	−	0.738513i	$-0.735528\pi$
$853$	−39.3839	−1.34848	−0.674239	+	0.738513i	$0.735528\pi$
$854$	0	0
$855$	−26.9946	−0.923197
$856$	0	0
$857$	−17.8018	−0.608099	−0.304050	−	0.952656i	$-0.598339\pi$
$857$	−17.8018	−0.608099	−0.304050	+	0.952656i	$0.598339\pi$
$858$	0	0
$859$	2.15372	0.0734841	0.0367420	−	0.999325i	$-0.488302\pi$
$859$	2.15372	0.0734841	0.0367420	+	0.999325i	$0.488302\pi$
$860$	0	0
$861$	10.8870	0.371027
$862$	0	0
$863$	2.74862	0.0935641	0.0467821	−	0.998905i	$-0.485103\pi$
$863$	2.74862	0.0935641	0.0467821	+	0.998905i	$0.485103\pi$
$864$	0	0
$865$	15.8217	0.537955
$866$	0	0
$867$	47.4162	1.61034
$868$	0	0
$869$	−7.09371	−0.240638
$870$	0	0
$871$	6.96110	0.235868
$872$	0	0
$873$	0.656428	0.0222167
$874$	0	0
$875$	28.6197	0.967522
$876$	0	0
$877$	51.6730	1.74487	0.872436	−	0.488728i	$-0.162539\pi$
$877$	51.6730	1.74487	0.872436	+	0.488728i	$0.162539\pi$
$878$	0	0
$879$	4.33155	0.146099
$880$	0	0
$881$	−36.6925	−1.23620	−0.618101	−	0.786098i	$-0.712098\pi$
$881$	−36.6925	−1.23620	−0.618101	+	0.786098i	$0.712098\pi$
$882$	0	0
$883$	30.9564	1.04177	0.520883	−	0.853628i	$-0.325603\pi$
$883$	30.9564	1.04177	0.520883	+	0.853628i	$0.325603\pi$
$884$	0	0
$885$	−26.3476	−0.885666
$886$	0	0
$887$	8.84730	0.297063	0.148532	−	0.988908i	$-0.452545\pi$
$887$	8.84730	0.297063	0.148532	+	0.988908i	$0.452545\pi$
$888$	0	0
$889$	16.7597	0.562101
$890$	0	0
$891$	6.03414	0.202151
$892$	0	0
$893$	−70.1878	−2.34875
$894$	0	0
$895$	12.3985	0.414437
$896$	0	0
$897$	−11.1192	−0.371259
$898$	0	0
$899$	−22.5974	−0.753667
$900$	0	0
$901$	−16.1608	−0.538393
$902$	0	0
$903$	1.84928	0.0615400
$904$	0	0
$905$	8.54328	0.283988
$906$	0	0
$907$	11.4863	0.381395	0.190698	−	0.981649i	$-0.438925\pi$
$907$	11.4863	0.381395	0.190698	+	0.981649i	$0.438925\pi$
$908$	0	0
$909$	27.8706	0.924411
$910$	0	0
$911$	−41.4423	−1.37304	−0.686522	−	0.727109i	$-0.740863\pi$
$911$	−41.4423	−1.37304	−0.686522	+	0.727109i	$0.740863\pi$
$912$	0	0
$913$	8.42017	0.278667
$914$	0	0
$915$	19.4754	0.643836
$916$	0	0
$917$	−5.65489	−0.186741
$918$	0	0
$919$	−6.23059	−0.205528	−0.102764	−	0.994706i	$-0.532769\pi$
$919$	−6.23059	−0.205528	−0.102764	+	0.994706i	$0.532769\pi$
$920$	0	0
$921$	71.5426	2.35741
$922$	0	0
$923$	12.6460	0.416248
$924$	0	0
$925$	−43.8171	−1.44070
$926$	0	0
$927$	72.3652	2.37679
$928$	0	0
$929$	−15.1437	−0.496847	−0.248424	−	0.968652i	$-0.579912\pi$
$929$	−15.1437	−0.496847	−0.248424	+	0.968652i	$0.579912\pi$
$930$	0	0
$931$	−46.6305	−1.52825
$932$	0	0
$933$	27.1672	0.889413
$934$	0	0
$935$	0.462431	0.0151231
$936$	0	0
$937$	10.0741	0.329107	0.164553	−	0.986368i	$-0.447382\pi$
$937$	10.0741	0.329107	0.164553	+	0.986368i	$0.447382\pi$
$938$	0	0
$939$	31.6392	1.03250
$940$	0	0
$941$	−32.5870	−1.06231	−0.531153	−	0.847276i	$-0.678241\pi$
$941$	−32.5870	−1.06231	−0.531153	+	0.847276i	$0.678241\pi$
$942$	0	0
$943$	2.73586	0.0890919
$944$	0	0
$945$	31.2827	1.01763
$946$	0	0
$947$	−25.0570	−0.814244	−0.407122	−	0.913374i	$-0.633468\pi$
$947$	−25.0570	−0.814244	−0.407122	+	0.913374i	$0.633468\pi$
$948$	0	0
$949$	5.59523	0.181629
$950$	0	0
$951$	−57.1309	−1.85260
$952$	0	0
$953$	16.9661	0.549587	0.274794	−	0.961503i	$-0.411390\pi$
$953$	16.9661	0.549587	0.274794	+	0.961503i	$0.411390\pi$
$954$	0	0
$955$	−11.5633	−0.374181
$956$	0	0
$957$	−5.54245	−0.179162
$958$	0	0
$959$	−70.0574	−2.26227
$960$	0	0
$961$	6.06794	0.195740
$962$	0	0
$963$	−44.4615	−1.43275
$964$	0	0
$965$	3.34984	0.107835
$966$	0	0
$967$	22.8295	0.734148	0.367074	−	0.930192i	$-0.380360\pi$
$967$	22.8295	0.734148	0.367074	+	0.930192i	$0.380360\pi$
$968$	0	0
$969$	20.5737	0.660923
$970$	0	0
$971$	10.0301	0.321880	0.160940	−	0.986964i	$-0.448547\pi$
$971$	10.0301	0.321880	0.160940	+	0.986964i	$0.448547\pi$
$972$	0	0
$973$	63.7282	2.04303
$974$	0	0
$975$	16.1742	0.517990
$976$	0	0
$977$	−31.4831	−1.00723	−0.503617	−	0.863927i	$-0.667998\pi$
$977$	−31.4831	−1.00723	−0.503617	+	0.863927i	$0.667998\pi$
$978$	0	0
$979$	0.750952	0.0240005
$980$	0	0
$981$	−56.3945	−1.80054
$982$	0	0
$983$	−19.3223	−0.616286	−0.308143	−	0.951340i	$-0.599708\pi$
$983$	−19.3223	−0.616286	−0.308143	+	0.951340i	$0.599708\pi$
$984$	0	0
$985$	−4.35381	−0.138724
$986$	0	0
$987$	153.976	4.90112
$988$	0	0
$989$	0.464717	0.0147771
$990$	0	0
$991$	−33.7141	−1.07096	−0.535482	−	0.844547i	$-0.679870\pi$
$991$	−33.7141	−1.07096	−0.535482	+	0.844547i	$0.679870\pi$
$992$	0	0
$993$	−15.2056	−0.482534
$994$	0	0
$995$	−8.65569	−0.274404
$996$	0	0
$997$	−28.7104	−0.909268	−0.454634	−	0.890678i	$-0.650230\pi$
$997$	−28.7104	−0.909268	−0.454634	+	0.890678i	$0.650230\pi$
$998$	0	0
$999$	−102.295	−3.23647

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4024.2.a.e.1.2	✓	29
4.3	odd	2		8048.2.a.w.1.28		29

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4024.2.a.e.1.2	✓	29	1.1	even	1	trivial
8048.2.a.w.1.28		29	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4024 = 2^{3} \cdot 503 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4024.a (trivial)

\(f(q)\)	\(=\)	\(q-3.05929 q^{3} +0.773308 q^{5} -3.93634 q^{7} +6.35923 q^{9} +O(q^{10})\)	\(q-3.05929 q^{3} +0.773308 q^{5} -3.93634 q^{7} +6.35923 q^{9} +0.488115 q^{11} +1.20103 q^{13} -2.36577 q^{15} +1.22510 q^{17} -5.48934 q^{19} +12.0424 q^{21} +3.02621 q^{23} -4.40199 q^{25} -10.2769 q^{27} +3.71159 q^{29} -6.08834 q^{31} -1.49328 q^{33} -3.04400 q^{35} +9.95392 q^{37} -3.67429 q^{39} +0.904055 q^{41} +0.153564 q^{43} +4.91765 q^{45} +12.7862 q^{47} +8.49474 q^{49} -3.74794 q^{51} -13.1913 q^{53} +0.377463 q^{55} +16.7935 q^{57} +11.1370 q^{59} -8.23214 q^{61} -25.0321 q^{63} +0.928766 q^{65} +5.79594 q^{67} -9.25804 q^{69} +10.5293 q^{71} +4.65869 q^{73} +13.4670 q^{75} -1.92138 q^{77} -14.5329 q^{79} +12.3621 q^{81} +17.2504 q^{83} +0.947383 q^{85} -11.3548 q^{87} +1.53848 q^{89} -4.72766 q^{91} +18.6260 q^{93} -4.24495 q^{95} +0.103224 q^{97} +3.10403 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 29 q - 7 q^{3} - 4 q^{5} - 13 q^{7} + 20 q^{9}+O(q^{10}) \) 29 * q - 7 * q^3 - 4 * q^5 - 13 * q^7 + 20 * q^9	\( 29 q - 7 q^{3} - 4 q^{5} - 13 q^{7} + 20 q^{9} - 27 q^{11} + 16 q^{13} - 14 q^{15} - 15 q^{17} - 14 q^{19} + q^{21} - 25 q^{23} + 21 q^{25} - 25 q^{27} - 13 q^{29} - 27 q^{31} - 9 q^{33} - 29 q^{35} + 35 q^{37} - 38 q^{39} - 30 q^{41} - 38 q^{43} + q^{45} - 35 q^{47} + 14 q^{49} - 21 q^{51} + 2 q^{53} - 25 q^{55} - 25 q^{57} - 40 q^{59} + 10 q^{61} - 56 q^{63} - 50 q^{65} - 31 q^{67} + 11 q^{69} - 65 q^{71} - 23 q^{73} - 32 q^{75} + 13 q^{77} - 44 q^{79} - 7 q^{81} - 41 q^{83} + 26 q^{85} - 25 q^{87} - 48 q^{89} - 44 q^{91} + 25 q^{93} - 75 q^{95} - 18 q^{97} - 80 q^{99}+O(q^{100}) \) 29 * q - 7 * q^3 - 4 * q^5 - 13 * q^7 + 20 * q^9 - 27 * q^11 + 16 * q^13 - 14 * q^15 - 15 * q^17 - 14 * q^19 + q^21 - 25 * q^23 + 21 * q^25 - 25 * q^27 - 13 * q^29 - 27 * q^31 - 9 * q^33 - 29 * q^35 + 35 * q^37 - 38 * q^39 - 30 * q^41 - 38 * q^43 + q^45 - 35 * q^47 + 14 * q^49 - 21 * q^51 + 2 * q^53 - 25 * q^55 - 25 * q^57 - 40 * q^59 + 10 * q^61 - 56 * q^63 - 50 * q^65 - 31 * q^67 + 11 * q^69 - 65 * q^71 - 23 * q^73 - 32 * q^75 + 13 * q^77 - 44 * q^79 - 7 * q^81 - 41 * q^83 + 26 * q^85 - 25 * q^87 - 48 * q^89 - 44 * q^91 + 25 * q^93 - 75 * q^95 - 18 * q^97 - 80 * q^99

Introduction

L-functions

Modular forms

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