LMFDB - Embedded newform 6044.2.a.a.1.8

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			1.8
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)

$f(q)$	$=$	$q-2.59590 q^{3} -0.0339717 q^{5} +0.331169 q^{7} +3.73871 q^{9} +O(q^{10})$	$q-2.59590 q^{3} -0.0339717 q^{5} +0.331169 q^{7} +3.73871 q^{9} +6.14371 q^{11} +4.21115 q^{13} +0.0881871 q^{15} -5.01151 q^{17} -1.42119 q^{19} -0.859682 q^{21} -0.363482 q^{23} -4.99885 q^{25} -1.91763 q^{27} -3.36350 q^{29} -10.2633 q^{31} -15.9485 q^{33} -0.0112503 q^{35} +6.64801 q^{37} -10.9317 q^{39} +7.66964 q^{41} -0.970784 q^{43} -0.127010 q^{45} -4.08347 q^{47} -6.89033 q^{49} +13.0094 q^{51} -13.7386 q^{53} -0.208712 q^{55} +3.68927 q^{57} +3.94093 q^{59} -5.44308 q^{61} +1.23815 q^{63} -0.143060 q^{65} +12.3571 q^{67} +0.943564 q^{69} -9.00961 q^{71} +9.68285 q^{73} +12.9765 q^{75} +2.03461 q^{77} -13.4904 q^{79} -6.23816 q^{81} +6.37716 q^{83} +0.170249 q^{85} +8.73131 q^{87} +16.8946 q^{89} +1.39460 q^{91} +26.6426 q^{93} +0.0482801 q^{95} +12.5513 q^{97} +22.9696 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−2.59590	−1.49875	−0.749373	−	0.662148i	$-0.769645\pi$
$3$	−2.59590	−1.49875	−0.749373	+	0.662148i	$0.769645\pi$
$4$	0	0
$5$	−0.0339717	−0.0151926	−0.00759629	−	0.999971i	$-0.502418\pi$
$5$	−0.0339717	−0.0151926	−0.00759629	+	0.999971i	$0.502418\pi$
$6$	0	0
$7$	0.331169	0.125170	0.0625850	−	0.998040i	$-0.480066\pi$
$7$	0.331169	0.125170	0.0625850	+	0.998040i	$0.480066\pi$
$8$	0	0
$9$	3.73871	1.24624
$10$	0	0
$11$	6.14371	1.85240	0.926199	−	0.377034i	$-0.123056\pi$
$11$	6.14371	1.85240	0.926199	+	0.377034i	$0.123056\pi$
$12$	0	0
$13$	4.21115	1.16796	0.583981	−	0.811767i	$-0.301494\pi$
$13$	4.21115	1.16796	0.583981	+	0.811767i	$0.301494\pi$
$14$	0	0
$15$	0.0881871	0.0227698
$16$	0	0
$17$	−5.01151	−1.21547	−0.607735	−	0.794140i	$-0.707922\pi$
$17$	−5.01151	−1.21547	−0.607735	+	0.794140i	$0.707922\pi$
$18$	0	0
$19$	−1.42119	−0.326043	−0.163021	−	0.986623i	$-0.552124\pi$
$19$	−1.42119	−0.326043	−0.163021	+	0.986623i	$0.552124\pi$
$20$	0	0
$21$	−0.859682	−0.187598
$22$	0	0
$23$	−0.363482	−0.0757913	−0.0378956	−	0.999282i	$-0.512065\pi$
$23$	−0.363482	−0.0757913	−0.0378956	+	0.999282i	$0.512065\pi$
$24$	0	0
$25$	−4.99885	−0.999769
$26$	0	0
$27$	−1.91763	−0.369048
$28$	0	0
$29$	−3.36350	−0.624586	−0.312293	−	0.949986i	$-0.601097\pi$
$29$	−3.36350	−0.624586	−0.312293	+	0.949986i	$0.601097\pi$
$30$	0	0
$31$	−10.2633	−1.84335	−0.921675	−	0.387964i	$-0.873179\pi$
$31$	−10.2633	−1.84335	−0.921675	+	0.387964i	$0.873179\pi$
$32$	0	0
$33$	−15.9485	−2.77627
$34$	0	0
$35$	−0.0112503	−0.00190166
$36$	0	0
$37$	6.64801	1.09293	0.546463	−	0.837483i	$-0.315974\pi$
$37$	6.64801	1.09293	0.546463	+	0.837483i	$0.315974\pi$
$38$	0	0
$39$	−10.9317	−1.75048
$40$	0	0
$41$	7.66964	1.19780	0.598898	−	0.800825i	$-0.295605\pi$
$41$	7.66964	1.19780	0.598898	+	0.800825i	$0.295605\pi$
$42$	0	0
$43$	−0.970784	−0.148043	−0.0740216	−	0.997257i	$-0.523583\pi$
$43$	−0.970784	−0.148043	−0.0740216	+	0.997257i	$0.523583\pi$
$44$	0	0
$45$	−0.127010	−0.0189336
$46$	0	0
$47$	−4.08347	−0.595635	−0.297817	−	0.954623i	$-0.596259\pi$
$47$	−4.08347	−0.595635	−0.297817	+	0.954623i	$0.596259\pi$
$48$	0	0
$49$	−6.89033	−0.984332
$50$	0	0
$51$	13.0094	1.82168
$52$	0	0
$53$	−13.7386	−1.88714	−0.943568	−	0.331180i	$-0.892553\pi$
$53$	−13.7386	−1.88714	−0.943568	+	0.331180i	$0.892553\pi$
$54$	0	0
$55$	−0.208712	−0.0281427
$56$	0	0
$57$	3.68927	0.488655
$58$	0	0
$59$	3.94093	0.513066	0.256533	−	0.966536i	$-0.417420\pi$
$59$	3.94093	0.513066	0.256533	+	0.966536i	$0.417420\pi$
$60$	0	0
$61$	−5.44308	−0.696915	−0.348458	−	0.937325i	$-0.613294\pi$
$61$	−5.44308	−0.696915	−0.348458	+	0.937325i	$0.613294\pi$
$62$	0	0
$63$	1.23815	0.155992
$64$	0	0
$65$	−0.143060	−0.0177444
$66$	0	0
$67$	12.3571	1.50966	0.754831	−	0.655919i	$-0.227719\pi$
$67$	12.3571	1.50966	0.754831	+	0.655919i	$0.227719\pi$
$68$	0	0
$69$	0.943564	0.113592
$70$	0	0
$71$	−9.00961	−1.06924	−0.534622	−	0.845091i	$-0.679546\pi$
$71$	−9.00961	−1.06924	−0.534622	+	0.845091i	$0.679546\pi$
$72$	0	0
$73$	9.68285	1.13329	0.566646	−	0.823961i	$-0.308241\pi$
$73$	9.68285	1.13329	0.566646	+	0.823961i	$0.308241\pi$
$74$	0	0
$75$	12.9765	1.49840
$76$	0	0
$77$	2.03461	0.231865
$78$	0	0
$79$	−13.4904	−1.51778	−0.758892	−	0.651217i	$-0.774259\pi$
$79$	−13.4904	−1.51778	−0.758892	+	0.651217i	$0.774259\pi$
$80$	0	0
$81$	−6.23816	−0.693129
$82$	0	0
$83$	6.37716	0.699985	0.349992	−	0.936753i	$-0.386184\pi$
$83$	6.37716	0.699985	0.349992	+	0.936753i	$0.386184\pi$
$84$	0	0
$85$	0.170249	0.0184661
$86$	0	0
$87$	8.73131	0.936095
$88$	0	0
$89$	16.8946	1.79083	0.895414	−	0.445235i	$-0.146880\pi$
$89$	16.8946	1.79083	0.895414	+	0.445235i	$0.146880\pi$
$90$	0	0
$91$	1.39460	0.146194
$92$	0	0
$93$	26.6426	2.76271
$94$	0	0
$95$	0.0482801	0.00495343
$96$	0	0
$97$	12.5513	1.27439	0.637193	−	0.770704i	$-0.280095\pi$
$97$	12.5513	1.27439	0.637193	+	0.770704i	$0.280095\pi$
$98$	0	0
$99$	22.9696	2.30853
$100$	0	0
$101$	−9.11066	−0.906544	−0.453272	−	0.891372i	$-0.649743\pi$
$101$	−9.11066	−0.906544	−0.453272	+	0.891372i	$0.649743\pi$
$102$	0	0
$103$	−9.75480	−0.961169	−0.480584	−	0.876948i	$-0.659575\pi$
$103$	−9.75480	−0.961169	−0.480584	+	0.876948i	$0.659575\pi$
$104$	0	0
$105$	0.0292048	0.00285010
$106$	0	0
$107$	−19.2900	−1.86484	−0.932418	−	0.361381i	$-0.882305\pi$
$107$	−19.2900	−1.86484	−0.932418	+	0.361381i	$0.882305\pi$
$108$	0	0
$109$	10.3240	0.988864	0.494432	−	0.869216i	$-0.335376\pi$
$109$	10.3240	0.988864	0.494432	+	0.869216i	$0.335376\pi$
$110$	0	0
$111$	−17.2576	−1.63802
$112$	0	0
$113$	9.76770	0.918868	0.459434	−	0.888212i	$-0.348052\pi$
$113$	9.76770	0.918868	0.459434	+	0.888212i	$0.348052\pi$
$114$	0	0
$115$	0.0123481	0.00115147
$116$	0	0
$117$	15.7443	1.45556
$118$	0	0
$119$	−1.65966	−0.152140
$120$	0	0
$121$	26.7452	2.43138
$122$	0	0
$123$	−19.9096	−1.79519
$124$	0	0
$125$	0.339677	0.0303817
$126$	0	0
$127$	−11.9360	−1.05915	−0.529573	−	0.848264i	$-0.677648\pi$
$127$	−11.9360	−1.05915	−0.529573	+	0.848264i	$0.677648\pi$
$128$	0	0
$129$	2.52006	0.221879
$130$	0	0
$131$	−4.91269	−0.429223	−0.214612	−	0.976699i	$-0.568849\pi$
$131$	−4.91269	−0.429223	−0.214612	+	0.976699i	$0.568849\pi$
$132$	0	0
$133$	−0.470653	−0.0408108
$134$	0	0
$135$	0.0651451	0.00560680
$136$	0	0
$137$	−1.94085	−0.165818	−0.0829089	−	0.996557i	$-0.526421\pi$
$137$	−1.94085	−0.165818	−0.0829089	+	0.996557i	$0.526421\pi$
$138$	0	0
$139$	10.1524	0.861118	0.430559	−	0.902562i	$-0.358316\pi$
$139$	10.1524	0.861118	0.430559	+	0.902562i	$0.358316\pi$
$140$	0	0
$141$	10.6003	0.892705
$142$	0	0
$143$	25.8721	2.16353
$144$	0	0
$145$	0.114264	0.00948907
$146$	0	0
$147$	17.8866	1.47526
$148$	0	0
$149$	15.7398	1.28946	0.644728	−	0.764412i	$-0.276971\pi$
$149$	15.7398	1.28946	0.644728	+	0.764412i	$0.276971\pi$
$150$	0	0
$151$	−23.4188	−1.90579	−0.952896	−	0.303297i	$-0.901912\pi$
$151$	−23.4188	−1.90579	−0.952896	+	0.303297i	$0.901912\pi$
$152$	0	0
$153$	−18.7366	−1.51477
$154$	0	0
$155$	0.348662	0.0280052
$156$	0	0
$157$	−1.44901	−0.115644	−0.0578219	−	0.998327i	$-0.518416\pi$
$157$	−1.44901	−0.115644	−0.0578219	+	0.998327i	$0.518416\pi$
$158$	0	0
$159$	35.6640	2.82834
$160$	0	0
$161$	−0.120374	−0.00948679
$162$	0	0
$163$	9.84686	0.771265	0.385633	−	0.922652i	$-0.373983\pi$
$163$	9.84686	0.771265	0.385633	+	0.922652i	$0.373983\pi$
$164$	0	0
$165$	0.541796	0.0421788
$166$	0	0
$167$	4.49152	0.347564	0.173782	−	0.984784i	$-0.444401\pi$
$167$	4.49152	0.347564	0.173782	+	0.984784i	$0.444401\pi$
$168$	0	0
$169$	4.73374	0.364134
$170$	0	0
$171$	−5.31341	−0.406327
$172$	0	0
$173$	−12.6262	−0.959952	−0.479976	−	0.877282i	$-0.659355\pi$
$173$	−12.6262	−0.959952	−0.479976	+	0.877282i	$0.659355\pi$
$174$	0	0
$175$	−1.65546	−0.125141
$176$	0	0
$177$	−10.2303	−0.768955
$178$	0	0
$179$	−17.8696	−1.33564	−0.667820	−	0.744323i	$-0.732772\pi$
$179$	−17.8696	−1.33564	−0.667820	+	0.744323i	$0.732772\pi$
$180$	0	0
$181$	−10.7454	−0.798696	−0.399348	−	0.916799i	$-0.630763\pi$
$181$	−10.7454	−0.798696	−0.399348	+	0.916799i	$0.630763\pi$
$182$	0	0
$183$	14.1297	1.04450
$184$	0	0
$185$	−0.225844	−0.0166044
$186$	0	0
$187$	−30.7893	−2.25154
$188$	0	0
$189$	−0.635060	−0.0461938
$190$	0	0
$191$	−9.07146	−0.656388	−0.328194	−	0.944610i	$-0.606440\pi$
$191$	−9.07146	−0.656388	−0.328194	+	0.944610i	$0.606440\pi$
$192$	0	0
$193$	−13.6163	−0.980120	−0.490060	−	0.871689i	$-0.663025\pi$
$193$	−13.6163	−0.980120	−0.490060	+	0.871689i	$0.663025\pi$
$194$	0	0
$195$	0.371369	0.0265943
$196$	0	0
$197$	−17.6726	−1.25912	−0.629560	−	0.776952i	$-0.716765\pi$
$197$	−17.6726	−1.25912	−0.629560	+	0.776952i	$0.716765\pi$
$198$	0	0
$199$	−27.4642	−1.94688	−0.973442	−	0.228935i	$-0.926476\pi$
$199$	−27.4642	−1.94688	−0.973442	+	0.228935i	$0.926476\pi$
$200$	0	0
$201$	−32.0779	−2.26260
$202$	0	0
$203$	−1.11388	−0.0781794
$204$	0	0
$205$	−0.260550	−0.0181976
$206$	0	0
$207$	−1.35896	−0.0944539
$208$	0	0
$209$	−8.73137	−0.603961
$210$	0	0
$211$	16.5222	1.13744	0.568718	−	0.822533i	$-0.307440\pi$
$211$	16.5222	1.13744	0.568718	+	0.822533i	$0.307440\pi$
$212$	0	0
$213$	23.3881	1.60252
$214$	0	0
$215$	0.0329792	0.00224916
$216$	0	0
$217$	−3.39889	−0.230732
$218$	0	0
$219$	−25.1357	−1.69852
$220$	0	0
$221$	−21.1042	−1.41962
$222$	0	0
$223$	20.7762	1.39128	0.695639	−	0.718391i	$-0.255121\pi$
$223$	20.7762	1.39128	0.695639	+	0.718391i	$0.255121\pi$
$224$	0	0
$225$	−18.6893	−1.24595
$226$	0	0
$227$	−6.27029	−0.416174	−0.208087	−	0.978110i	$-0.566724\pi$
$227$	−6.27029	−0.416174	−0.208087	+	0.978110i	$0.566724\pi$
$228$	0	0
$229$	25.3259	1.67358	0.836792	−	0.547521i	$-0.184428\pi$
$229$	25.3259	1.67358	0.836792	+	0.547521i	$0.184428\pi$
$230$	0	0
$231$	−5.28164	−0.347506
$232$	0	0
$233$	−11.8189	−0.774280	−0.387140	−	0.922021i	$-0.626537\pi$
$233$	−11.8189	−0.774280	−0.387140	+	0.922021i	$0.626537\pi$
$234$	0	0
$235$	0.138722	0.00904923
$236$	0	0
$237$	35.0197	2.27477
$238$	0	0
$239$	−0.823412	−0.0532621	−0.0266310	−	0.999645i	$-0.508478\pi$
$239$	−0.823412	−0.0532621	−0.0266310	+	0.999645i	$0.508478\pi$
$240$	0	0
$241$	−6.78170	−0.436848	−0.218424	−	0.975854i	$-0.570092\pi$
$241$	−6.78170	−0.436848	−0.218424	+	0.975854i	$0.570092\pi$
$242$	0	0
$243$	21.9465	1.40787
$244$	0	0
$245$	0.234076	0.0149546
$246$	0	0
$247$	−5.98483	−0.380805
$248$	0	0
$249$	−16.5545	−1.04910
$250$	0	0
$251$	7.00733	0.442299	0.221149	−	0.975240i	$-0.429019\pi$
$251$	7.00733	0.442299	0.221149	+	0.975240i	$0.429019\pi$
$252$	0	0
$253$	−2.23313	−0.140396
$254$	0	0
$255$	−0.441951	−0.0276760
$256$	0	0
$257$	0.615426	0.0383892	0.0191946	−	0.999816i	$-0.493890\pi$
$257$	0.615426	0.0383892	0.0191946	+	0.999816i	$0.493890\pi$
$258$	0	0
$259$	2.20161	0.136801
$260$	0	0
$261$	−12.5752	−0.778382
$262$	0	0
$263$	2.15975	0.133176	0.0665878	−	0.997781i	$-0.478789\pi$
$263$	2.15975	0.133176	0.0665878	+	0.997781i	$0.478789\pi$
$264$	0	0
$265$	0.466721	0.0286705
$266$	0	0
$267$	−43.8568	−2.68400
$268$	0	0
$269$	15.9787	0.974240	0.487120	−	0.873335i	$-0.338047\pi$
$269$	15.9787	0.974240	0.487120	+	0.873335i	$0.338047\pi$
$270$	0	0
$271$	−12.4670	−0.757318	−0.378659	−	0.925536i	$-0.623615\pi$
$271$	−12.4670	−0.757318	−0.378659	+	0.925536i	$0.623615\pi$
$272$	0	0
$273$	−3.62025	−0.219107
$274$	0	0
$275$	−30.7115	−1.85197
$276$	0	0
$277$	−9.12106	−0.548032	−0.274016	−	0.961725i	$-0.588352\pi$
$277$	−9.12106	−0.548032	−0.274016	+	0.961725i	$0.588352\pi$
$278$	0	0
$279$	−38.3717	−2.29725
$280$	0	0
$281$	22.0885	1.31769	0.658845	−	0.752279i	$-0.271045\pi$
$281$	22.0885	1.31769	0.658845	+	0.752279i	$0.271045\pi$
$282$	0	0
$283$	−17.3321	−1.03029	−0.515143	−	0.857104i	$-0.672261\pi$
$283$	−17.3321	−1.03029	−0.515143	+	0.857104i	$0.672261\pi$
$284$	0	0
$285$	−0.125330	−0.00742393
$286$	0	0
$287$	2.53994	0.149928
$288$	0	0
$289$	8.11524	0.477367
$290$	0	0
$291$	−32.5818	−1.90998
$292$	0	0
$293$	0.151595	0.00885627	0.00442813	−	0.999990i	$-0.498590\pi$
$293$	0.151595	0.00885627	0.00442813	+	0.999990i	$0.498590\pi$
$294$	0	0
$295$	−0.133880	−0.00779479
$296$	0	0
$297$	−11.7814	−0.683625
$298$	0	0
$299$	−1.53068	−0.0885213
$300$	0	0
$301$	−0.321493	−0.0185306
$302$	0	0
$303$	23.6504	1.35868
$304$	0	0
$305$	0.184910	0.0105879
$306$	0	0
$307$	−24.9944	−1.42650	−0.713252	−	0.700907i	$-0.752779\pi$
$307$	−24.9944	−1.42650	−0.713252	+	0.700907i	$0.752779\pi$
$308$	0	0
$309$	25.3225	1.44055
$310$	0	0
$311$	−5.69043	−0.322674	−0.161337	−	0.986899i	$-0.551581\pi$
$311$	−5.69043	−0.322674	−0.161337	+	0.986899i	$0.551581\pi$
$312$	0	0
$313$	−13.2610	−0.749556	−0.374778	−	0.927115i	$-0.622281\pi$
$313$	−13.2610	−0.749556	−0.374778	+	0.927115i	$0.622281\pi$
$314$	0	0
$315$	−0.0420618	−0.00236992
$316$	0	0
$317$	15.2088	0.854213	0.427107	−	0.904201i	$-0.359533\pi$
$317$	15.2088	0.854213	0.427107	+	0.904201i	$0.359533\pi$
$318$	0	0
$319$	−20.6644	−1.15698
$320$	0	0
$321$	50.0750	2.79492
$322$	0	0
$323$	7.12230	0.396295
$324$	0	0
$325$	−21.0509	−1.16769
$326$	0	0
$327$	−26.8002	−1.48206
$328$	0	0
$329$	−1.35232	−0.0745556
$330$	0	0
$331$	−10.9118	−0.599767	−0.299883	−	0.953976i	$-0.596948\pi$
$331$	−10.9118	−0.599767	−0.299883	+	0.953976i	$0.596948\pi$
$332$	0	0
$333$	24.8550	1.36205
$334$	0	0
$335$	−0.419792	−0.0229357
$336$	0	0
$337$	−18.1557	−0.989002	−0.494501	−	0.869177i	$-0.664649\pi$
$337$	−18.1557	−0.989002	−0.494501	+	0.869177i	$0.664649\pi$
$338$	0	0
$339$	−25.3560	−1.37715
$340$	0	0
$341$	−63.0550	−3.41462
$342$	0	0
$343$	−4.60004	−0.248379
$344$	0	0
$345$	−0.0320544	−0.00172575
$346$	0	0
$347$	10.1921	0.547140	0.273570	−	0.961852i	$-0.411796\pi$
$347$	10.1921	0.547140	0.273570	+	0.961852i	$0.411796\pi$
$348$	0	0
$349$	−10.8576	−0.581196	−0.290598	−	0.956845i	$-0.593854\pi$
$349$	−10.8576	−0.581196	−0.290598	+	0.956845i	$0.593854\pi$
$350$	0	0
$351$	−8.07543	−0.431034
$352$	0	0
$353$	23.2216	1.23596	0.617981	−	0.786193i	$-0.287951\pi$
$353$	23.2216	1.23596	0.617981	+	0.786193i	$0.287951\pi$
$354$	0	0
$355$	0.306071	0.0162446
$356$	0	0
$357$	4.30831	0.228020
$358$	0	0
$359$	−16.1798	−0.853938	−0.426969	−	0.904266i	$-0.640419\pi$
$359$	−16.1798	−0.853938	−0.426969	+	0.904266i	$0.640419\pi$
$360$	0	0
$361$	−16.9802	−0.893696
$362$	0	0
$363$	−69.4280	−3.64402
$364$	0	0
$365$	−0.328942	−0.0172176
$366$	0	0
$367$	24.5884	1.28350	0.641752	−	0.766912i	$-0.278208\pi$
$367$	24.5884	1.28350	0.641752	+	0.766912i	$0.278208\pi$
$368$	0	0
$369$	28.6746	1.49274
$370$	0	0
$371$	−4.54978	−0.236213
$372$	0	0
$373$	6.90225	0.357385	0.178692	−	0.983905i	$-0.442813\pi$
$373$	6.90225	0.357385	0.178692	+	0.983905i	$0.442813\pi$
$374$	0	0
$375$	−0.881770	−0.0455344
$376$	0	0
$377$	−14.1642	−0.729492
$378$	0	0
$379$	−28.1246	−1.44466	−0.722331	−	0.691547i	$-0.756929\pi$
$379$	−28.1246	−1.44466	−0.722331	+	0.691547i	$0.756929\pi$
$380$	0	0
$381$	30.9846	1.58739
$382$	0	0
$383$	15.6264	0.798470	0.399235	−	0.916849i	$-0.369276\pi$
$383$	15.6264	0.798470	0.399235	+	0.916849i	$0.369276\pi$
$384$	0	0
$385$	−0.0691189	−0.00352263
$386$	0	0
$387$	−3.62949	−0.184497
$388$	0	0
$389$	1.24363	0.0630546	0.0315273	−	0.999503i	$-0.489963\pi$
$389$	1.24363	0.0630546	0.0315273	+	0.999503i	$0.489963\pi$
$390$	0	0
$391$	1.82159	0.0921220
$392$	0	0
$393$	12.7529	0.643297
$394$	0	0
$395$	0.458290	0.0230591
$396$	0	0
$397$	18.1537	0.911107	0.455553	−	0.890208i	$-0.349441\pi$
$397$	18.1537	0.911107	0.455553	+	0.890208i	$0.349441\pi$
$398$	0	0
$399$	1.22177	0.0611650
$400$	0	0
$401$	−8.28280	−0.413623	−0.206812	−	0.978381i	$-0.566309\pi$
$401$	−8.28280	−0.413623	−0.206812	+	0.978381i	$0.566309\pi$
$402$	0	0
$403$	−43.2204	−2.15296
$404$	0	0
$405$	0.211921	0.0105304
$406$	0	0
$407$	40.8434	2.02453
$408$	0	0
$409$	8.80928	0.435591	0.217795	−	0.975994i	$-0.430113\pi$
$409$	8.80928	0.435591	0.217795	+	0.975994i	$0.430113\pi$
$410$	0	0
$411$	5.03825	0.248519
$412$	0	0
$413$	1.30511	0.0642204
$414$	0	0
$415$	−0.216643	−0.0106346
$416$	0	0
$417$	−26.3547	−1.29060
$418$	0	0
$419$	8.94011	0.436753	0.218377	−	0.975865i	$-0.429924\pi$
$419$	8.94011	0.436753	0.218377	+	0.975865i	$0.429924\pi$
$420$	0	0
$421$	−37.2027	−1.81315	−0.906574	−	0.422048i	$-0.861311\pi$
$421$	−37.2027	−1.81315	−0.906574	+	0.422048i	$0.861311\pi$
$422$	0	0
$423$	−15.2669	−0.742303
$424$	0	0
$425$	25.0518	1.21519
$426$	0	0
$427$	−1.80258	−0.0872329
$428$	0	0
$429$	−67.1614	−3.24258
$430$	0	0
$431$	−29.5506	−1.42340	−0.711701	−	0.702483i	$-0.752075\pi$
$431$	−29.5506	−1.42340	−0.711701	+	0.702483i	$0.752075\pi$
$432$	0	0
$433$	24.3352	1.16947	0.584736	−	0.811223i	$-0.301198\pi$
$433$	24.3352	1.16947	0.584736	+	0.811223i	$0.301198\pi$
$434$	0	0
$435$	−0.296617	−0.0142217
$436$	0	0
$437$	0.516576	0.0247112
$438$	0	0
$439$	−27.6644	−1.32035	−0.660175	−	0.751112i	$-0.729518\pi$
$439$	−27.6644	−1.32035	−0.660175	+	0.751112i	$0.729518\pi$
$440$	0	0
$441$	−25.7610	−1.22671
$442$	0	0
$443$	−19.4249	−0.922904	−0.461452	−	0.887165i	$-0.652671\pi$
$443$	−19.4249	−0.922904	−0.461452	+	0.887165i	$0.652671\pi$
$444$	0	0
$445$	−0.573939	−0.0272073
$446$	0	0
$447$	−40.8590	−1.93257
$448$	0	0
$449$	8.82497	0.416476	0.208238	−	0.978078i	$-0.433227\pi$
$449$	8.82497	0.416476	0.208238	+	0.978078i	$0.433227\pi$
$450$	0	0
$451$	47.1200	2.21880
$452$	0	0
$453$	60.7928	2.85630
$454$	0	0
$455$	−0.0473768	−0.00222106
$456$	0	0
$457$	8.67396	0.405751	0.202875	−	0.979205i	$-0.434971\pi$
$457$	8.67396	0.405751	0.202875	+	0.979205i	$0.434971\pi$
$458$	0	0
$459$	9.61023	0.448567
$460$	0	0
$461$	−18.9882	−0.884367	−0.442184	−	0.896925i	$-0.645796\pi$
$461$	−18.9882	−0.884367	−0.442184	+	0.896925i	$0.645796\pi$
$462$	0	0
$463$	−5.32301	−0.247381	−0.123691	−	0.992321i	$-0.539473\pi$
$463$	−5.32301	−0.247381	−0.123691	+	0.992321i	$0.539473\pi$
$464$	0	0
$465$	−0.905094	−0.0419727
$466$	0	0
$467$	19.1704	0.887100	0.443550	−	0.896250i	$-0.353719\pi$
$467$	19.1704	0.887100	0.443550	+	0.896250i	$0.353719\pi$
$468$	0	0
$469$	4.09229	0.188964
$470$	0	0
$471$	3.76150	0.173321
$472$	0	0
$473$	−5.96422	−0.274235
$474$	0	0
$475$	7.10430	0.325967
$476$	0	0
$477$	−51.3645	−2.35182
$478$	0	0
$479$	6.29309	0.287539	0.143769	−	0.989611i	$-0.454078\pi$
$479$	6.29309	0.287539	0.143769	+	0.989611i	$0.454078\pi$
$480$	0	0
$481$	27.9957	1.27650
$482$	0	0
$483$	0.312479	0.0142183
$484$	0	0
$485$	−0.426387	−0.0193612
$486$	0	0
$487$	22.5516	1.02191	0.510956	−	0.859607i	$-0.329291\pi$
$487$	22.5516	1.02191	0.510956	+	0.859607i	$0.329291\pi$
$488$	0	0
$489$	−25.5615	−1.15593
$490$	0	0
$491$	−21.5720	−0.973531	−0.486766	−	0.873533i	$-0.661823\pi$
$491$	−21.5720	−0.973531	−0.486766	+	0.873533i	$0.661823\pi$
$492$	0	0
$493$	16.8562	0.759165
$494$	0	0
$495$	−0.780315	−0.0350725
$496$	0	0
$497$	−2.98370	−0.133837
$498$	0	0
$499$	5.43374	0.243248	0.121624	−	0.992576i	$-0.461190\pi$
$499$	5.43374	0.243248	0.121624	+	0.992576i	$0.461190\pi$
$500$	0	0
$501$	−11.6595	−0.520910
$502$	0	0
$503$	1.45796	0.0650071	0.0325035	−	0.999472i	$-0.489652\pi$
$503$	1.45796	0.0650071	0.0325035	+	0.999472i	$0.489652\pi$
$504$	0	0
$505$	0.309504	0.0137727
$506$	0	0
$507$	−12.2883	−0.545745
$508$	0	0
$509$	−29.6818	−1.31562	−0.657812	−	0.753182i	$-0.728518\pi$
$509$	−29.6818	−1.31562	−0.657812	+	0.753182i	$0.728518\pi$
$510$	0	0
$511$	3.20666	0.141854
$512$	0	0
$513$	2.72531	0.120326
$514$	0	0
$515$	0.331387	0.0146026
$516$	0	0
$517$	−25.0876	−1.10335
$518$	0	0
$519$	32.7764	1.43872
$520$	0	0
$521$	−26.6859	−1.16913	−0.584565	−	0.811347i	$-0.698735\pi$
$521$	−26.6859	−1.16913	−0.584565	+	0.811347i	$0.698735\pi$
$522$	0	0
$523$	22.5546	0.986245	0.493122	−	0.869960i	$-0.335855\pi$
$523$	22.5546	0.986245	0.493122	+	0.869960i	$0.335855\pi$
$524$	0	0
$525$	4.29742	0.187555
$526$	0	0
$527$	51.4348	2.24054
$528$	0	0
$529$	−22.8679	−0.994256
$530$	0	0
$531$	14.7340	0.639402
$532$	0	0
$533$	32.2980	1.39898
$534$	0	0
$535$	0.655314	0.0283317
$536$	0	0
$537$	46.3878	2.00178
$538$	0	0
$539$	−42.3322	−1.82338
$540$	0	0
$541$	22.9472	0.986579	0.493289	−	0.869865i	$-0.335794\pi$
$541$	22.9472	0.986579	0.493289	+	0.869865i	$0.335794\pi$
$542$	0	0
$543$	27.8939	1.19704
$544$	0	0
$545$	−0.350725	−0.0150234
$546$	0	0
$547$	4.29149	0.183491	0.0917455	−	0.995782i	$-0.470755\pi$
$547$	4.29149	0.183491	0.0917455	+	0.995782i	$0.470755\pi$
$548$	0	0
$549$	−20.3501	−0.868522
$550$	0	0
$551$	4.78016	0.203642
$552$	0	0
$553$	−4.46758	−0.189981
$554$	0	0
$555$	0.586269	0.0248857
$556$	0	0
$557$	−11.0570	−0.468499	−0.234249	−	0.972177i	$-0.575263\pi$
$557$	−11.0570	−0.468499	−0.234249	+	0.972177i	$0.575263\pi$
$558$	0	0
$559$	−4.08811	−0.172909
$560$	0	0
$561$	79.9260	3.37448
$562$	0	0
$563$	25.5127	1.07523	0.537616	−	0.843190i	$-0.319325\pi$
$563$	25.5127	1.07523	0.537616	+	0.843190i	$0.319325\pi$
$564$	0	0
$565$	−0.331825	−0.0139600
$566$	0	0
$567$	−2.06588	−0.0867589
$568$	0	0
$569$	16.2694	0.682047	0.341023	−	0.940055i	$-0.389226\pi$
$569$	16.2694	0.682047	0.341023	+	0.940055i	$0.389226\pi$
$570$	0	0
$571$	−24.8674	−1.04067	−0.520334	−	0.853963i	$-0.674193\pi$
$571$	−24.8674	−1.04067	−0.520334	+	0.853963i	$0.674193\pi$
$572$	0	0
$573$	23.5486	0.983758
$574$	0	0
$575$	1.81699	0.0757738
$576$	0	0
$577$	44.4533	1.85062	0.925308	−	0.379217i	$-0.123807\pi$
$577$	44.4533	1.85062	0.925308	+	0.379217i	$0.123807\pi$
$578$	0	0
$579$	35.3465	1.46895
$580$	0	0
$581$	2.11192	0.0876171
$582$	0	0
$583$	−84.4057	−3.49573
$584$	0	0
$585$	−0.534859	−0.0221137
$586$	0	0
$587$	9.14093	0.377287	0.188643	−	0.982046i	$-0.439591\pi$
$587$	9.14093	0.377287	0.188643	+	0.982046i	$0.439591\pi$
$588$	0	0
$589$	14.5861	0.601011
$590$	0	0
$591$	45.8763	1.88710
$592$	0	0
$593$	−22.9989	−0.944450	−0.472225	−	0.881478i	$-0.656549\pi$
$593$	−22.9989	−0.944450	−0.472225	+	0.881478i	$0.656549\pi$
$594$	0	0
$595$	0.0563812	0.00231141
$596$	0	0
$597$	71.2943	2.91788
$598$	0	0
$599$	−16.9287	−0.691688	−0.345844	−	0.938292i	$-0.612407\pi$
$599$	−16.9287	−0.691688	−0.345844	+	0.938292i	$0.612407\pi$
$600$	0	0
$601$	−26.1010	−1.06468	−0.532342	−	0.846530i	$-0.678688\pi$
$601$	−26.1010	−1.06468	−0.532342	+	0.846530i	$0.678688\pi$
$602$	0	0
$603$	46.1997	1.88140
$604$	0	0
$605$	−0.908579	−0.0369390
$606$	0	0
$607$	−2.45147	−0.0995021	−0.0497511	−	0.998762i	$-0.515843\pi$
$607$	−2.45147	−0.0995021	−0.0497511	+	0.998762i	$0.515843\pi$
$608$	0	0
$609$	2.89154	0.117171
$610$	0	0
$611$	−17.1961	−0.695679
$612$	0	0
$613$	−3.13297	−0.126539	−0.0632697	−	0.997996i	$-0.520153\pi$
$613$	−3.13297	−0.126539	−0.0632697	+	0.997996i	$0.520153\pi$
$614$	0	0
$615$	0.676363	0.0272736
$616$	0	0
$617$	43.6378	1.75679	0.878396	−	0.477934i	$-0.158614\pi$
$617$	43.6378	1.75679	0.878396	+	0.477934i	$0.158614\pi$
$618$	0	0
$619$	−20.4286	−0.821093	−0.410547	−	0.911840i	$-0.634662\pi$
$619$	−20.4286	−0.821093	−0.410547	+	0.911840i	$0.634662\pi$
$620$	0	0
$621$	0.697025	0.0279706
$622$	0	0
$623$	5.59497	0.224158
$624$	0	0
$625$	24.9827	0.999308
$626$	0	0
$627$	22.6658	0.905184
$628$	0	0
$629$	−33.3166	−1.32842
$630$	0	0
$631$	−39.2728	−1.56343	−0.781713	−	0.623638i	$-0.785654\pi$
$631$	−39.2728	−1.56343	−0.781713	+	0.623638i	$0.785654\pi$
$632$	0	0
$633$	−42.8900	−1.70473
$634$	0	0
$635$	0.405485	0.0160912
$636$	0	0
$637$	−29.0162	−1.14966
$638$	0	0
$639$	−33.6844	−1.33253
$640$	0	0
$641$	5.43695	0.214746	0.107373	−	0.994219i	$-0.465756\pi$
$641$	5.43695	0.214746	0.107373	+	0.994219i	$0.465756\pi$
$642$	0	0
$643$	−36.4666	−1.43810	−0.719051	−	0.694958i	$-0.755423\pi$
$643$	−36.4666	−1.43810	−0.719051	+	0.694958i	$0.755423\pi$
$644$	0	0
$645$	−0.0856107	−0.00337092
$646$	0	0
$647$	29.0860	1.14349	0.571745	−	0.820431i	$-0.306267\pi$
$647$	29.0860	1.14349	0.571745	+	0.820431i	$0.306267\pi$
$648$	0	0
$649$	24.2119	0.950402
$650$	0	0
$651$	8.82320	0.345809
$652$	0	0
$653$	−35.8275	−1.40204	−0.701020	−	0.713141i	$-0.747272\pi$
$653$	−35.8275	−1.40204	−0.701020	+	0.713141i	$0.747272\pi$
$654$	0	0
$655$	0.166892	0.00652101
$656$	0	0
$657$	36.2014	1.41235
$658$	0	0
$659$	14.2240	0.554088	0.277044	−	0.960857i	$-0.410645\pi$
$659$	14.2240	0.554088	0.277044	+	0.960857i	$0.410645\pi$
$660$	0	0
$661$	30.3039	1.17868	0.589342	−	0.807883i	$-0.299387\pi$
$661$	30.3039	1.17868	0.589342	+	0.807883i	$0.299387\pi$
$662$	0	0
$663$	54.7845	2.12765
$664$	0	0
$665$	0.0159889	0.000620021 0
$666$	0	0
$667$	1.22257	0.0473381
$668$	0	0
$669$	−53.9330	−2.08517
$670$	0	0
$671$	−33.4407	−1.29096
$672$	0	0
$673$	−8.08898	−0.311807	−0.155904	−	0.987772i	$-0.549829\pi$
$673$	−8.08898	−0.311807	−0.155904	+	0.987772i	$0.549829\pi$
$674$	0	0
$675$	9.58594	0.368963
$676$	0	0
$677$	−35.3760	−1.35961	−0.679805	−	0.733393i	$-0.737936\pi$
$677$	−35.3760	−1.35961	−0.679805	+	0.733393i	$0.737936\pi$
$678$	0	0
$679$	4.15658	0.159515
$680$	0	0
$681$	16.2771	0.623738
$682$	0	0
$683$	−34.1005	−1.30482	−0.652409	−	0.757867i	$-0.726242\pi$
$683$	−34.1005	−1.30482	−0.652409	+	0.757867i	$0.726242\pi$
$684$	0	0
$685$	0.0659338	0.00251920
$686$	0	0
$687$	−65.7436	−2.50828
$688$	0	0
$689$	−57.8550	−2.20410
$690$	0	0
$691$	−34.3006	−1.30486	−0.652428	−	0.757850i	$-0.726250\pi$
$691$	−34.3006	−1.30486	−0.652428	+	0.757850i	$0.726250\pi$
$692$	0	0
$693$	7.60681	0.288959
$694$	0	0
$695$	−0.344895	−0.0130826
$696$	0	0
$697$	−38.4365	−1.45589
$698$	0	0
$699$	30.6806	1.16045
$700$	0	0
$701$	5.71243	0.215756	0.107878	−	0.994164i	$-0.465594\pi$
$701$	5.71243	0.215756	0.107878	+	0.994164i	$0.465594\pi$
$702$	0	0
$703$	−9.44806	−0.356340
$704$	0	0
$705$	−0.360109	−0.0135625
$706$	0	0
$707$	−3.01716	−0.113472
$708$	0	0
$709$	25.3485	0.951982	0.475991	−	0.879450i	$-0.342089\pi$
$709$	25.3485	0.951982	0.475991	+	0.879450i	$0.342089\pi$
$710$	0	0
$711$	−50.4366	−1.89152
$712$	0	0
$713$	3.73054	0.139710
$714$	0	0
$715$	−0.878917	−0.0328696
$716$	0	0
$717$	2.13750	0.0798263
$718$	0	0
$719$	37.2404	1.38883	0.694417	−	0.719573i	$-0.255662\pi$
$719$	37.2404	1.38883	0.694417	+	0.719573i	$0.255662\pi$
$720$	0	0
$721$	−3.23048	−0.120310
$722$	0	0
$723$	17.6046	0.654723
$724$	0	0
$725$	16.8136	0.624441
$726$	0	0
$727$	23.0016	0.853081	0.426541	−	0.904468i	$-0.359732\pi$
$727$	23.0016	0.853081	0.426541	+	0.904468i	$0.359732\pi$
$728$	0	0
$729$	−38.2566	−1.41691
$730$	0	0
$731$	4.86510	0.179942
$732$	0	0
$733$	−12.2025	−0.450710	−0.225355	−	0.974277i	$-0.572354\pi$
$733$	−12.2025	−0.450710	−0.225355	+	0.974277i	$0.572354\pi$
$734$	0	0
$735$	−0.607638	−0.0224131
$736$	0	0
$737$	75.9186	2.79650
$738$	0	0
$739$	−41.0041	−1.50836	−0.754181	−	0.656667i	$-0.771966\pi$
$739$	−41.0041	−1.50836	−0.754181	+	0.656667i	$0.771966\pi$
$740$	0	0
$741$	15.5360	0.570730
$742$	0	0
$743$	3.20160	0.117455	0.0587276	−	0.998274i	$-0.481296\pi$
$743$	3.20160	0.117455	0.0587276	+	0.998274i	$0.481296\pi$
$744$	0	0
$745$	−0.534707	−0.0195902
$746$	0	0
$747$	23.8424	0.872348
$748$	0	0
$749$	−6.38825	−0.233422
$750$	0	0
$751$	−35.4092	−1.29210	−0.646051	−	0.763294i	$-0.723581\pi$
$751$	−35.4092	−1.29210	−0.646051	+	0.763294i	$0.723581\pi$
$752$	0	0
$753$	−18.1904	−0.662894
$754$	0	0
$755$	0.795574	0.0289539
$756$	0	0
$757$	0.834068	0.0303147	0.0151574	−	0.999885i	$-0.495175\pi$
$757$	0.834068	0.0303147	0.0151574	+	0.999885i	$0.495175\pi$
$758$	0	0
$759$	5.79699	0.210417
$760$	0	0
$761$	−27.5986	−1.00045	−0.500224	−	0.865896i	$-0.666749\pi$
$761$	−27.5986	−1.00045	−0.500224	+	0.865896i	$0.666749\pi$
$762$	0	0
$763$	3.41900	0.123776
$764$	0	0
$765$	0.636514	0.0230132
$766$	0	0
$767$	16.5958	0.599241
$768$	0	0
$769$	−29.5786	−1.06663	−0.533315	−	0.845917i	$-0.679054\pi$
$769$	−29.5786	−1.06663	−0.533315	+	0.845917i	$0.679054\pi$
$770$	0	0
$771$	−1.59759	−0.0575356
$772$	0	0
$773$	−25.2995	−0.909960	−0.454980	−	0.890502i	$-0.650354\pi$
$773$	−25.2995	−0.909960	−0.454980	+	0.890502i	$0.650354\pi$
$774$	0	0
$775$	51.3048	1.84292
$776$	0	0
$777$	−5.71517	−0.205031
$778$	0	0
$779$	−10.9000	−0.390533
$780$	0	0
$781$	−55.3525	−1.98067
$782$	0	0
$783$	6.44995	0.230502
$784$	0	0
$785$	0.0492254	0.00175693
$786$	0	0
$787$	−38.4903	−1.37203	−0.686015	−	0.727587i	$-0.740642\pi$
$787$	−38.4903	−1.37203	−0.686015	+	0.727587i	$0.740642\pi$
$788$	0	0
$789$	−5.60649	−0.199596
$790$	0	0
$791$	3.23476	0.115015
$792$	0	0
$793$	−22.9216	−0.813970
$794$	0	0
$795$	−1.21156	−0.0429697
$796$	0	0
$797$	−33.0690	−1.17137	−0.585683	−	0.810540i	$-0.699174\pi$
$797$	−33.0690	−1.17137	−0.585683	+	0.810540i	$0.699174\pi$
$798$	0	0
$799$	20.4643	0.723976
$800$	0	0
$801$	63.1642	2.23180
$802$	0	0
$803$	59.4886	2.09931
$804$	0	0
$805$	0.00408930	0.000144129 0
$806$	0	0
$807$	−41.4792	−1.46014
$808$	0	0
$809$	38.0367	1.33730	0.668649	−	0.743578i	$-0.266873\pi$
$809$	38.0367	1.33730	0.668649	+	0.743578i	$0.266873\pi$
$810$	0	0
$811$	14.1435	0.496646	0.248323	−	0.968677i	$-0.420121\pi$
$811$	14.1435	0.496646	0.248323	+	0.968677i	$0.420121\pi$
$812$	0	0
$813$	32.3632	1.13503
$814$	0	0
$815$	−0.334514	−0.0117175
$816$	0	0
$817$	1.37967	0.0482684
$818$	0	0
$819$	5.21401	0.182192
$820$	0	0
$821$	−35.2476	−1.23015	−0.615074	−	0.788469i	$-0.710874\pi$
$821$	−35.2476	−1.23015	−0.615074	+	0.788469i	$0.710874\pi$
$822$	0	0
$823$	43.9330	1.53141	0.765703	−	0.643194i	$-0.222391\pi$
$823$	43.9330	1.53141	0.765703	+	0.643194i	$0.222391\pi$
$824$	0	0
$825$	79.7240	2.77563
$826$	0	0
$827$	2.35080	0.0817452	0.0408726	−	0.999164i	$-0.486986\pi$
$827$	2.35080	0.0817452	0.0408726	+	0.999164i	$0.486986\pi$
$828$	0	0
$829$	7.66185	0.266107	0.133054	−	0.991109i	$-0.457522\pi$
$829$	7.66185	0.266107	0.133054	+	0.991109i	$0.457522\pi$
$830$	0	0
$831$	23.6774	0.821360
$832$	0	0
$833$	34.5310	1.19643
$834$	0	0
$835$	−0.152584	−0.00528039
$836$	0	0
$837$	19.6813	0.680285
$838$	0	0
$839$	−35.9587	−1.24143	−0.620717	−	0.784035i	$-0.713158\pi$
$839$	−35.9587	−1.24143	−0.620717	+	0.784035i	$0.713158\pi$
$840$	0	0
$841$	−17.6869	−0.609893
$842$	0	0
$843$	−57.3396	−1.97488
$844$	0	0
$845$	−0.160813	−0.00553214
$846$	0	0
$847$	8.85717	0.304336
$848$	0	0
$849$	44.9924	1.54414
$850$	0	0
$851$	−2.41643	−0.0828342
$852$	0	0
$853$	−0.996636	−0.0341242	−0.0170621	−	0.999854i	$-0.505431\pi$
$853$	−0.996636	−0.0341242	−0.0170621	+	0.999854i	$0.505431\pi$
$854$	0	0
$855$	0.180505	0.00617316
$856$	0	0
$857$	−20.8701	−0.712910	−0.356455	−	0.934313i	$-0.616015\pi$
$857$	−20.8701	−0.712910	−0.356455	+	0.934313i	$0.616015\pi$
$858$	0	0
$859$	−39.0187	−1.33130	−0.665651	−	0.746264i	$-0.731846\pi$
$859$	−39.0187	−1.33130	−0.665651	+	0.746264i	$0.731846\pi$
$860$	0	0
$861$	−6.59345	−0.224704
$862$	0	0
$863$	19.0296	0.647774	0.323887	−	0.946096i	$-0.395010\pi$
$863$	19.0296	0.647774	0.323887	+	0.946096i	$0.395010\pi$
$864$	0	0
$865$	0.428933	0.0145842
$866$	0	0
$867$	−21.0664	−0.715452
$868$	0	0
$869$	−82.8809	−2.81154
$870$	0	0
$871$	52.0376	1.76323
$872$	0	0
$873$	46.9256	1.58819
$874$	0	0
$875$	0.112490	0.00380287
$876$	0	0
$877$	−40.8777	−1.38034	−0.690170	−	0.723647i	$-0.742464\pi$
$877$	−40.8777	−1.38034	−0.690170	+	0.723647i	$0.742464\pi$
$878$	0	0
$879$	−0.393526	−0.0132733
$880$	0	0
$881$	41.0933	1.38447	0.692233	−	0.721674i	$-0.256627\pi$
$881$	41.0933	1.38447	0.692233	+	0.721674i	$0.256627\pi$
$882$	0	0
$883$	−26.7542	−0.900350	−0.450175	−	0.892940i	$-0.648638\pi$
$883$	−26.7542	−0.900350	−0.450175	+	0.892940i	$0.648638\pi$
$884$	0	0
$885$	0.347539	0.0116824
$886$	0	0
$887$	25.3686	0.851793	0.425896	−	0.904772i	$-0.359959\pi$
$887$	25.3686	0.851793	0.425896	+	0.904772i	$0.359959\pi$
$888$	0	0
$889$	−3.95282	−0.132573
$890$	0	0
$891$	−38.3254	−1.28395
$892$	0	0
$893$	5.80337	0.194202
$894$	0	0
$895$	0.607061	0.0202918
$896$	0	0
$897$	3.97349	0.132671
$898$	0	0
$899$	34.5207	1.15133
$900$	0	0
$901$	68.8509	2.29376
$902$	0	0
$903$	0.834566	0.0277726
$904$	0	0
$905$	0.365038	0.0121343
$906$	0	0
$907$	−4.87293	−0.161803	−0.0809015	−	0.996722i	$-0.525780\pi$
$907$	−4.87293	−0.161803	−0.0809015	+	0.996722i	$0.525780\pi$
$908$	0	0
$909$	−34.0621	−1.12977
$910$	0	0
$911$	−3.73625	−0.123788	−0.0618938	−	0.998083i	$-0.519714\pi$
$911$	−3.73625	−0.123788	−0.0618938	+	0.998083i	$0.519714\pi$
$912$	0	0
$913$	39.1795	1.29665
$914$	0	0
$915$	−0.480010	−0.0158686
$916$	0	0
$917$	−1.62693	−0.0537259
$918$	0	0
$919$	−50.2804	−1.65860	−0.829299	−	0.558805i	$-0.811260\pi$
$919$	−50.2804	−1.65860	−0.829299	+	0.558805i	$0.811260\pi$
$920$	0	0
$921$	64.8830	2.13797
$922$	0	0
$923$	−37.9408	−1.24884
$924$	0	0
$925$	−33.2324	−1.09267
$926$	0	0
$927$	−36.4704	−1.19785
$928$	0	0
$929$	−47.2459	−1.55009	−0.775044	−	0.631908i	$-0.782272\pi$
$929$	−47.2459	−1.55009	−0.775044	+	0.631908i	$0.782272\pi$
$930$	0	0
$931$	9.79245	0.320934
$932$	0	0
$933$	14.7718	0.483607
$934$	0	0
$935$	1.04596	0.0342066
$936$	0	0
$937$	58.2437	1.90274	0.951369	−	0.308052i	$-0.0996772\pi$
$937$	58.2437	1.90274	0.951369	+	0.308052i	$0.0996772\pi$
$938$	0	0
$939$	34.4243	1.12339
$940$	0	0
$941$	30.8779	1.00659	0.503295	−	0.864115i	$-0.332121\pi$
$941$	30.8779	1.00659	0.503295	+	0.864115i	$0.332121\pi$
$942$	0	0
$943$	−2.78778	−0.0907825
$944$	0	0
$945$	0.0215740	0.000701803 0
$946$	0	0
$947$	−47.9777	−1.55907	−0.779533	−	0.626362i	$-0.784543\pi$
$947$	−47.9777	−1.55907	−0.779533	+	0.626362i	$0.784543\pi$
$948$	0	0
$949$	40.7759	1.32364
$950$	0	0
$951$	−39.4807	−1.28025
$952$	0	0
$953$	0.799072	0.0258845	0.0129422	−	0.999916i	$-0.495880\pi$
$953$	0.799072	0.0258845	0.0129422	+	0.999916i	$0.495880\pi$
$954$	0	0
$955$	0.308173	0.00997223
$956$	0	0
$957$	53.6427	1.73402
$958$	0	0
$959$	−0.642748	−0.0207554
$960$	0	0
$961$	74.3360	2.39794
$962$	0	0
$963$	−72.1199	−2.32403
$964$	0	0
$965$	0.462567	0.0148906
$966$	0	0
$967$	−18.1463	−0.583546	−0.291773	−	0.956488i	$-0.594245\pi$
$967$	−18.1463	−0.583546	−0.291773	+	0.956488i	$0.594245\pi$
$968$	0	0
$969$	−18.4888	−0.593946
$970$	0	0
$971$	−1.82548	−0.0585825	−0.0292913	−	0.999571i	$-0.509325\pi$
$971$	−1.82548	−0.0585825	−0.0292913	+	0.999571i	$0.509325\pi$
$972$	0	0
$973$	3.36217	0.107786
$974$	0	0
$975$	54.6460	1.75007
$976$	0	0
$977$	−6.29536	−0.201407	−0.100703	−	0.994917i	$-0.532109\pi$
$977$	−6.29536	−0.201407	−0.100703	+	0.994917i	$0.532109\pi$
$978$	0	0
$979$	103.796	3.31733
$980$	0	0
$981$	38.5987	1.23236
$982$	0	0
$983$	−52.5032	−1.67459	−0.837296	−	0.546750i	$-0.815865\pi$
$983$	−52.5032	−1.67459	−0.837296	+	0.546750i	$0.815865\pi$
$984$	0	0
$985$	0.600367	0.0191293
$986$	0	0
$987$	3.51048	0.111740
$988$	0	0
$989$	0.352863	0.0112204
$990$	0	0
$991$	−15.8944	−0.504901	−0.252450	−	0.967610i	$-0.581236\pi$
$991$	−15.8944	−0.504901	−0.252450	+	0.967610i	$0.581236\pi$
$992$	0	0
$993$	28.3260	0.898898
$994$	0	0
$995$	0.933003	0.0295782
$996$	0	0
$997$	23.4832	0.743720	0.371860	−	0.928289i	$-0.378720\pi$
$997$	23.4832	0.743720	0.371860	+	0.928289i	$0.378720\pi$
$998$	0	0
$999$	−12.7484	−0.403342

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6044.2.a.a.1.8	✓	63

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6044.2.a.a.1.8	✓	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)

\(f(q)\)	\(=\)	\(q-2.59590 q^{3} -0.0339717 q^{5} +0.331169 q^{7} +3.73871 q^{9} +O(q^{10})\)	\(q-2.59590 q^{3} -0.0339717 q^{5} +0.331169 q^{7} +3.73871 q^{9} +6.14371 q^{11} +4.21115 q^{13} +0.0881871 q^{15} -5.01151 q^{17} -1.42119 q^{19} -0.859682 q^{21} -0.363482 q^{23} -4.99885 q^{25} -1.91763 q^{27} -3.36350 q^{29} -10.2633 q^{31} -15.9485 q^{33} -0.0112503 q^{35} +6.64801 q^{37} -10.9317 q^{39} +7.66964 q^{41} -0.970784 q^{43} -0.127010 q^{45} -4.08347 q^{47} -6.89033 q^{49} +13.0094 q^{51} -13.7386 q^{53} -0.208712 q^{55} +3.68927 q^{57} +3.94093 q^{59} -5.44308 q^{61} +1.23815 q^{63} -0.143060 q^{65} +12.3571 q^{67} +0.943564 q^{69} -9.00961 q^{71} +9.68285 q^{73} +12.9765 q^{75} +2.03461 q^{77} -13.4904 q^{79} -6.23816 q^{81} +6.37716 q^{83} +0.170249 q^{85} +8.73131 q^{87} +16.8946 q^{89} +1.39460 q^{91} +26.6426 q^{93} +0.0482801 q^{95} +12.5513 q^{97} +22.9696 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 63 q - 7 q^{3} - 7 q^{5} - 22 q^{7} + 62 q^{9}+O(q^{10}) \) 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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