LMFDB - Embedded newform 6036.2.a.i.1.25

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6036,2,Mod(1,6036)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6036, 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("6036.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6036 = 2^{2} \cdot 3 \cdot 503 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6036.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:	$48.1977026600$
Analytic rank:	$0$
Dimension:	$26$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.25
Character	$\chi$	$=$	6036.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +4.20338 q^{5} -4.85430 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +4.20338 q^{5} -4.85430 q^{7} +1.00000 q^{9} -2.29477 q^{11} +4.87495 q^{13} -4.20338 q^{15} -5.90737 q^{17} -6.37966 q^{19} +4.85430 q^{21} -7.01785 q^{23} +12.6684 q^{25} -1.00000 q^{27} -3.54478 q^{29} +10.5787 q^{31} +2.29477 q^{33} -20.4045 q^{35} -6.92120 q^{37} -4.87495 q^{39} +7.28161 q^{41} -0.286483 q^{43} +4.20338 q^{45} +10.1863 q^{47} +16.5642 q^{49} +5.90737 q^{51} -5.30299 q^{53} -9.64580 q^{55} +6.37966 q^{57} +6.05413 q^{59} +4.89088 q^{61} -4.85430 q^{63} +20.4913 q^{65} +5.18407 q^{67} +7.01785 q^{69} +6.89990 q^{71} +6.70542 q^{73} -12.6684 q^{75} +11.1395 q^{77} +8.84232 q^{79} +1.00000 q^{81} -1.41587 q^{83} -24.8309 q^{85} +3.54478 q^{87} +4.76102 q^{89} -23.6645 q^{91} -10.5787 q^{93} -26.8161 q^{95} -14.4392 q^{97} -2.29477 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 26 q - 26 q^{3} + 6 q^{5} + 5 q^{7} + 26 q^{9}+O(q^{10}) $ 26 * q - 26 * q^3 + 6 * q^5 + 5 * q^7 + 26 * q^9	$ 26 q - 26 q^{3} + 6 q^{5} + 5 q^{7} + 26 q^{9} - 11 q^{11} + 13 q^{13} - 6 q^{15} + 12 q^{17} - q^{19} - 5 q^{21} - 22 q^{23} + 48 q^{25} - 26 q^{27} + 6 q^{29} + 19 q^{31} + 11 q^{33} - 21 q^{35} + 20 q^{37} - 13 q^{39} + 25 q^{41} + 4 q^{43} + 6 q^{45} + 8 q^{47} + 67 q^{49} - 12 q^{51} - 5 q^{53} + 20 q^{55} + q^{57} - 18 q^{59} + 43 q^{61} + 5 q^{63} + 41 q^{65} + 5 q^{67} + 22 q^{69} - q^{71} + 22 q^{73} - 48 q^{75} + 23 q^{77} + 16 q^{79} + 26 q^{81} - 19 q^{83} + 29 q^{85} - 6 q^{87} + 49 q^{89} - 13 q^{91} - 19 q^{93} - 26 q^{95} + 25 q^{97} - 11 q^{99}+O(q^{100}) $ 26 * q - 26 * q^3 + 6 * q^5 + 5 * q^7 + 26 * q^9 - 11 * q^11 + 13 * q^13 - 6 * q^15 + 12 * q^17 - q^19 - 5 * q^21 - 22 * q^23 + 48 * q^25 - 26 * q^27 + 6 * q^29 + 19 * q^31 + 11 * q^33 - 21 * q^35 + 20 * q^37 - 13 * q^39 + 25 * q^41 + 4 * q^43 + 6 * q^45 + 8 * q^47 + 67 * q^49 - 12 * q^51 - 5 * q^53 + 20 * q^55 + q^57 - 18 * q^59 + 43 * q^61 + 5 * q^63 + 41 * q^65 + 5 * q^67 + 22 * q^69 - q^71 + 22 * q^73 - 48 * q^75 + 23 * q^77 + 16 * q^79 + 26 * q^81 - 19 * q^83 + 29 * q^85 - 6 * q^87 + 49 * q^89 - 13 * q^91 - 19 * q^93 - 26 * q^95 + 25 * q^97 - 11 * 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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	4.20338	1.87981	0.939904	−	0.341438i	$-0.110914\pi$
$5$	4.20338	1.87981	0.939904	+	0.341438i	$0.110914\pi$
$6$	0	0
$7$	−4.85430	−1.83475	−0.917377	−	0.398020i	$-0.869697\pi$
$7$	−4.85430	−1.83475	−0.917377	+	0.398020i	$0.869697\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.29477	−0.691900	−0.345950	−	0.938253i	$-0.612443\pi$
$11$	−2.29477	−0.691900	−0.345950	+	0.938253i	$0.612443\pi$
$12$	0	0
$13$	4.87495	1.35207	0.676034	−	0.736870i	$-0.263697\pi$
$13$	4.87495	1.35207	0.676034	+	0.736870i	$0.263697\pi$
$14$	0	0
$15$	−4.20338	−1.08531
$16$	0	0
$17$	−5.90737	−1.43275	−0.716374	−	0.697716i	$-0.754200\pi$
$17$	−5.90737	−1.43275	−0.716374	+	0.697716i	$0.754200\pi$
$18$	0	0
$19$	−6.37966	−1.46360	−0.731798	−	0.681522i	$-0.761318\pi$
$19$	−6.37966	−1.46360	−0.731798	+	0.681522i	$0.761318\pi$
$20$	0	0
$21$	4.85430	1.05930
$22$	0	0
$23$	−7.01785	−1.46332	−0.731661	−	0.681668i	$-0.761255\pi$
$23$	−7.01785	−1.46332	−0.731661	+	0.681668i	$0.761255\pi$
$24$	0	0
$25$	12.6684	2.53368
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−3.54478	−0.658250	−0.329125	−	0.944286i	$-0.606754\pi$
$29$	−3.54478	−0.658250	−0.329125	+	0.944286i	$0.606754\pi$
$30$	0	0
$31$	10.5787	1.89998	0.949992	−	0.312274i	$-0.101091\pi$
$31$	10.5787	1.89998	0.949992	+	0.312274i	$0.101091\pi$
$32$	0	0
$33$	2.29477	0.399469
$34$	0	0
$35$	−20.4045	−3.44898
$36$	0	0
$37$	−6.92120	−1.13784	−0.568919	−	0.822394i	$-0.692638\pi$
$37$	−6.92120	−1.13784	−0.568919	+	0.822394i	$0.692638\pi$
$38$	0	0
$39$	−4.87495	−0.780617
$40$	0	0
$41$	7.28161	1.13720	0.568598	−	0.822615i	$-0.307486\pi$
$41$	7.28161	1.13720	0.568598	+	0.822615i	$0.307486\pi$
$42$	0	0
$43$	−0.286483	−0.0436882	−0.0218441	−	0.999761i	$-0.506954\pi$
$43$	−0.286483	−0.0436882	−0.0218441	+	0.999761i	$0.506954\pi$
$44$	0	0
$45$	4.20338	0.626603
$46$	0	0
$47$	10.1863	1.48583	0.742914	−	0.669387i	$-0.233443\pi$
$47$	10.1863	1.48583	0.742914	+	0.669387i	$0.233443\pi$
$48$	0	0
$49$	16.5642	2.36632
$50$	0	0
$51$	5.90737	0.827198
$52$	0	0
$53$	−5.30299	−0.728422	−0.364211	−	0.931316i	$-0.618661\pi$
$53$	−5.30299	−0.728422	−0.364211	+	0.931316i	$0.618661\pi$
$54$	0	0
$55$	−9.64580	−1.30064
$56$	0	0
$57$	6.37966	0.845007
$58$	0	0
$59$	6.05413	0.788181	0.394090	−	0.919072i	$-0.371060\pi$
$59$	6.05413	0.788181	0.394090	+	0.919072i	$0.371060\pi$
$60$	0	0
$61$	4.89088	0.626213	0.313107	−	0.949718i	$-0.398630\pi$
$61$	4.89088	0.626213	0.313107	+	0.949718i	$0.398630\pi$
$62$	0	0
$63$	−4.85430	−0.611584
$64$	0	0
$65$	20.4913	2.54163
$66$	0	0
$67$	5.18407	0.633334	0.316667	−	0.948537i	$-0.397436\pi$
$67$	5.18407	0.633334	0.316667	+	0.948537i	$0.397436\pi$
$68$	0	0
$69$	7.01785	0.844850
$70$	0	0
$71$	6.89990	0.818867	0.409434	−	0.912340i	$-0.365726\pi$
$71$	6.89990	0.818867	0.409434	+	0.912340i	$0.365726\pi$
$72$	0	0
$73$	6.70542	0.784810	0.392405	−	0.919792i	$-0.371643\pi$
$73$	6.70542	0.784810	0.392405	+	0.919792i	$0.371643\pi$
$74$	0	0
$75$	−12.6684	−1.46282
$76$	0	0
$77$	11.1395	1.26947
$78$	0	0
$79$	8.84232	0.994839	0.497419	−	0.867510i	$-0.334281\pi$
$79$	8.84232	0.994839	0.497419	+	0.867510i	$0.334281\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−1.41587	−0.155412	−0.0777061	−	0.996976i	$-0.524760\pi$
$83$	−1.41587	−0.155412	−0.0777061	+	0.996976i	$0.524760\pi$
$84$	0	0
$85$	−24.8309	−2.69329
$86$	0	0
$87$	3.54478	0.380041
$88$	0	0
$89$	4.76102	0.504668	0.252334	−	0.967640i	$-0.418802\pi$
$89$	4.76102	0.504668	0.252334	+	0.967640i	$0.418802\pi$
$90$	0	0
$91$	−23.6645	−2.48071
$92$	0	0
$93$	−10.5787	−1.09696
$94$	0	0
$95$	−26.8161	−2.75128
$96$	0	0
$97$	−14.4392	−1.46608	−0.733040	−	0.680186i	$-0.761899\pi$
$97$	−14.4392	−1.46608	−0.733040	+	0.680186i	$0.761899\pi$
$98$	0	0
$99$	−2.29477	−0.230633
$100$	0	0
$101$	1.30060	0.129414	0.0647072	−	0.997904i	$-0.479389\pi$
$101$	1.30060	0.129414	0.0647072	+	0.997904i	$0.479389\pi$
$102$	0	0
$103$	18.6871	1.84129	0.920645	−	0.390400i	$-0.127663\pi$
$103$	18.6871	1.84129	0.920645	+	0.390400i	$0.127663\pi$
$104$	0	0
$105$	20.4045	1.99127
$106$	0	0
$107$	3.67107	0.354896	0.177448	−	0.984130i	$-0.443216\pi$
$107$	3.67107	0.354896	0.177448	+	0.984130i	$0.443216\pi$
$108$	0	0
$109$	8.31311	0.796252	0.398126	−	0.917331i	$-0.369661\pi$
$109$	8.31311	0.796252	0.398126	+	0.917331i	$0.369661\pi$
$110$	0	0
$111$	6.92120	0.656931
$112$	0	0
$113$	−0.573625	−0.0539621	−0.0269811	−	0.999636i	$-0.508589\pi$
$113$	−0.573625	−0.0539621	−0.0269811	+	0.999636i	$0.508589\pi$
$114$	0	0
$115$	−29.4987	−2.75077
$116$	0	0
$117$	4.87495	0.450690
$118$	0	0
$119$	28.6762	2.62874
$120$	0	0
$121$	−5.73402	−0.521275
$122$	0	0
$123$	−7.28161	−0.656561
$124$	0	0
$125$	32.2332	2.88302
$126$	0	0
$127$	−5.51341	−0.489236	−0.244618	−	0.969620i	$-0.578662\pi$
$127$	−5.51341	−0.489236	−0.244618	+	0.969620i	$0.578662\pi$
$128$	0	0
$129$	0.286483	0.0252234
$130$	0	0
$131$	−3.58966	−0.313630	−0.156815	−	0.987628i	$-0.550123\pi$
$131$	−3.58966	−0.313630	−0.156815	+	0.987628i	$0.550123\pi$
$132$	0	0
$133$	30.9688	2.68534
$134$	0	0
$135$	−4.20338	−0.361769
$136$	0	0
$137$	22.0824	1.88662	0.943312	−	0.331907i	$-0.107692\pi$
$137$	22.0824	1.88662	0.943312	+	0.331907i	$0.107692\pi$
$138$	0	0
$139$	2.49725	0.211814	0.105907	−	0.994376i	$-0.466225\pi$
$139$	2.49725	0.211814	0.105907	+	0.994376i	$0.466225\pi$
$140$	0	0
$141$	−10.1863	−0.857843
$142$	0	0
$143$	−11.1869	−0.935496
$144$	0	0
$145$	−14.9001	−1.23738
$146$	0	0
$147$	−16.5642	−1.36620
$148$	0	0
$149$	6.74731	0.552761	0.276381	−	0.961048i	$-0.410865\pi$
$149$	6.74731	0.552761	0.276381	+	0.961048i	$0.410865\pi$
$150$	0	0
$151$	22.6346	1.84198	0.920991	−	0.389584i	$-0.127381\pi$
$151$	22.6346	1.84198	0.920991	+	0.389584i	$0.127381\pi$
$152$	0	0
$153$	−5.90737	−0.477583
$154$	0	0
$155$	44.4661	3.57161
$156$	0	0
$157$	−18.3228	−1.46232	−0.731160	−	0.682206i	$-0.761021\pi$
$157$	−18.3228	−1.46232	−0.731160	+	0.682206i	$0.761021\pi$
$158$	0	0
$159$	5.30299	0.420555
$160$	0	0
$161$	34.0668	2.68484
$162$	0	0
$163$	17.6153	1.37973	0.689867	−	0.723936i	$-0.257669\pi$
$163$	17.6153	1.37973	0.689867	+	0.723936i	$0.257669\pi$
$164$	0	0
$165$	9.64580	0.750924
$166$	0	0
$167$	4.65158	0.359950	0.179975	−	0.983671i	$-0.442398\pi$
$167$	4.65158	0.359950	0.179975	+	0.983671i	$0.442398\pi$
$168$	0	0
$169$	10.7652	0.828090
$170$	0	0
$171$	−6.37966	−0.487865
$172$	0	0
$173$	−8.43275	−0.641130	−0.320565	−	0.947227i	$-0.603873\pi$
$173$	−8.43275	−0.641130	−0.320565	+	0.947227i	$0.603873\pi$
$174$	0	0
$175$	−61.4962	−4.64868
$176$	0	0
$177$	−6.05413	−0.455057
$178$	0	0
$179$	−13.5775	−1.01483	−0.507417	−	0.861701i	$-0.669399\pi$
$179$	−13.5775	−1.01483	−0.507417	+	0.861701i	$0.669399\pi$
$180$	0	0
$181$	−4.64377	−0.345169	−0.172584	−	0.984995i	$-0.555212\pi$
$181$	−4.64377	−0.345169	−0.172584	+	0.984995i	$0.555212\pi$
$182$	0	0
$183$	−4.89088	−0.361545
$184$	0	0
$185$	−29.0924	−2.13892
$186$	0	0
$187$	13.5561	0.991319
$188$	0	0
$189$	4.85430	0.353098
$190$	0	0
$191$	17.7981	1.28783	0.643914	−	0.765098i	$-0.277309\pi$
$191$	17.7981	1.28783	0.643914	+	0.765098i	$0.277309\pi$
$192$	0	0
$193$	0.996260	0.0717123	0.0358562	−	0.999357i	$-0.488584\pi$
$193$	0.996260	0.0717123	0.0358562	+	0.999357i	$0.488584\pi$
$194$	0	0
$195$	−20.4913	−1.46741
$196$	0	0
$197$	5.17054	0.368386	0.184193	−	0.982890i	$-0.441033\pi$
$197$	5.17054	0.368386	0.184193	+	0.982890i	$0.441033\pi$
$198$	0	0
$199$	−19.4428	−1.37826	−0.689130	−	0.724638i	$-0.742007\pi$
$199$	−19.4428	−1.37826	−0.689130	+	0.724638i	$0.742007\pi$
$200$	0	0
$201$	−5.18407	−0.365656
$202$	0	0
$203$	17.2075	1.20773
$204$	0	0
$205$	30.6074	2.13771
$206$	0	0
$207$	−7.01785	−0.487774
$208$	0	0
$209$	14.6399	1.01266
$210$	0	0
$211$	12.0636	0.830495	0.415247	−	0.909709i	$-0.363695\pi$
$211$	12.0636	0.830495	0.415247	+	0.909709i	$0.363695\pi$
$212$	0	0
$213$	−6.89990	−0.472773
$214$	0	0
$215$	−1.20420	−0.0821255
$216$	0	0
$217$	−51.3520	−3.48600
$218$	0	0
$219$	−6.70542	−0.453110
$220$	0	0
$221$	−28.7982	−1.93718
$222$	0	0
$223$	8.37340	0.560724	0.280362	−	0.959894i	$-0.409545\pi$
$223$	8.37340	0.560724	0.280362	+	0.959894i	$0.409545\pi$
$224$	0	0
$225$	12.6684	0.844560
$226$	0	0
$227$	−19.8272	−1.31598	−0.657989	−	0.753027i	$-0.728593\pi$
$227$	−19.8272	−1.31598	−0.657989	+	0.753027i	$0.728593\pi$
$228$	0	0
$229$	2.58173	0.170605	0.0853027	−	0.996355i	$-0.472814\pi$
$229$	2.58173	0.170605	0.0853027	+	0.996355i	$0.472814\pi$
$230$	0	0
$231$	−11.1395	−0.732926
$232$	0	0
$233$	−9.86101	−0.646016	−0.323008	−	0.946396i	$-0.604694\pi$
$233$	−9.86101	−0.646016	−0.323008	+	0.946396i	$0.604694\pi$
$234$	0	0
$235$	42.8170	2.79307
$236$	0	0
$237$	−8.84232	−0.574370
$238$	0	0
$239$	−25.4514	−1.64631	−0.823156	−	0.567815i	$-0.807789\pi$
$239$	−25.4514	−1.64631	−0.823156	+	0.567815i	$0.807789\pi$
$240$	0	0
$241$	−11.5744	−0.745571	−0.372785	−	0.927918i	$-0.621597\pi$
$241$	−11.5744	−0.745571	−0.372785	+	0.927918i	$0.621597\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	69.6258	4.44823
$246$	0	0
$247$	−31.1006	−1.97888
$248$	0	0
$249$	1.41587	0.0897273
$250$	0	0
$251$	−2.18518	−0.137928	−0.0689638	−	0.997619i	$-0.521969\pi$
$251$	−2.18518	−0.137928	−0.0689638	+	0.997619i	$0.521969\pi$
$252$	0	0
$253$	16.1044	1.01247
$254$	0	0
$255$	24.8309	1.55497
$256$	0	0
$257$	18.6738	1.16484	0.582421	−	0.812887i	$-0.302106\pi$
$257$	18.6738	1.16484	0.582421	+	0.812887i	$0.302106\pi$
$258$	0	0
$259$	33.5976	2.08765
$260$	0	0
$261$	−3.54478	−0.219417
$262$	0	0
$263$	−12.7689	−0.787366	−0.393683	−	0.919246i	$-0.628799\pi$
$263$	−12.7689	−0.787366	−0.393683	+	0.919246i	$0.628799\pi$
$264$	0	0
$265$	−22.2905	−1.36929
$266$	0	0
$267$	−4.76102	−0.291370
$268$	0	0
$269$	3.81283	0.232473	0.116236	−	0.993222i	$-0.462917\pi$
$269$	3.81283	0.232473	0.116236	+	0.993222i	$0.462917\pi$
$270$	0	0
$271$	−13.8637	−0.842161	−0.421080	−	0.907023i	$-0.638349\pi$
$271$	−13.8637	−0.842161	−0.421080	+	0.907023i	$0.638349\pi$
$272$	0	0
$273$	23.6645	1.43224
$274$	0	0
$275$	−29.0711	−1.75305
$276$	0	0
$277$	−2.57537	−0.154739	−0.0773694	−	0.997002i	$-0.524652\pi$
$277$	−2.57537	−0.154739	−0.0773694	+	0.997002i	$0.524652\pi$
$278$	0	0
$279$	10.5787	0.633328
$280$	0	0
$281$	5.49420	0.327756	0.163878	−	0.986481i	$-0.447600\pi$
$281$	5.49420	0.327756	0.163878	+	0.986481i	$0.447600\pi$
$282$	0	0
$283$	5.70200	0.338949	0.169474	−	0.985535i	$-0.445793\pi$
$283$	5.70200	0.338949	0.169474	+	0.985535i	$0.445793\pi$
$284$	0	0
$285$	26.8161	1.58845
$286$	0	0
$287$	−35.3471	−2.08648
$288$	0	0
$289$	17.8971	1.05277
$290$	0	0
$291$	14.4392	0.846442
$292$	0	0
$293$	−2.08811	−0.121989	−0.0609943	−	0.998138i	$-0.519427\pi$
$293$	−2.08811	−0.121989	−0.0609943	+	0.998138i	$0.519427\pi$
$294$	0	0
$295$	25.4478	1.48163
$296$	0	0
$297$	2.29477	0.133156
$298$	0	0
$299$	−34.2117	−1.97851
$300$	0	0
$301$	1.39067	0.0801572
$302$	0	0
$303$	−1.30060	−0.0747175
$304$	0	0
$305$	20.5582	1.17716
$306$	0	0
$307$	22.2877	1.27202	0.636012	−	0.771679i	$-0.280583\pi$
$307$	22.2877	1.27202	0.636012	+	0.771679i	$0.280583\pi$
$308$	0	0
$309$	−18.6871	−1.06307
$310$	0	0
$311$	10.2272	0.579929	0.289964	−	0.957037i	$-0.406357\pi$
$311$	10.2272	0.579929	0.289964	+	0.957037i	$0.406357\pi$
$312$	0	0
$313$	−8.58534	−0.485272	−0.242636	−	0.970117i	$-0.578012\pi$
$313$	−8.58534	−0.485272	−0.242636	+	0.970117i	$0.578012\pi$
$314$	0	0
$315$	−20.4045	−1.14966
$316$	0	0
$317$	−30.3507	−1.70466	−0.852332	−	0.523002i	$-0.824812\pi$
$317$	−30.3507	−1.70466	−0.852332	+	0.523002i	$0.824812\pi$
$318$	0	0
$319$	8.13447	0.455443
$320$	0	0
$321$	−3.67107	−0.204899
$322$	0	0
$323$	37.6871	2.09696
$324$	0	0
$325$	61.7578	3.42571
$326$	0	0
$327$	−8.31311	−0.459716
$328$	0	0
$329$	−49.4475	−2.72613
$330$	0	0
$331$	7.14727	0.392849	0.196425	−	0.980519i	$-0.437067\pi$
$331$	7.14727	0.392849	0.196425	+	0.980519i	$0.437067\pi$
$332$	0	0
$333$	−6.92120	−0.379279
$334$	0	0
$335$	21.7906	1.19055
$336$	0	0
$337$	−32.9243	−1.79350	−0.896750	−	0.442538i	$-0.854078\pi$
$337$	−32.9243	−1.79350	−0.896750	+	0.442538i	$0.854078\pi$
$338$	0	0
$339$	0.573625	0.0311550
$340$	0	0
$341$	−24.2756	−1.31460
$342$	0	0
$343$	−46.4277	−2.50686
$344$	0	0
$345$	29.4987	1.58816
$346$	0	0
$347$	18.5012	0.993199	0.496599	−	0.867980i	$-0.334582\pi$
$347$	18.5012	0.993199	0.496599	+	0.867980i	$0.334582\pi$
$348$	0	0
$349$	−6.52219	−0.349125	−0.174562	−	0.984646i	$-0.555851\pi$
$349$	−6.52219	−0.349125	−0.174562	+	0.984646i	$0.555851\pi$
$350$	0	0
$351$	−4.87495	−0.260206
$352$	0	0
$353$	33.1013	1.76181	0.880903	−	0.473296i	$-0.156936\pi$
$353$	33.1013	1.76181	0.880903	+	0.473296i	$0.156936\pi$
$354$	0	0
$355$	29.0029	1.53931
$356$	0	0
$357$	−28.6762	−1.51770
$358$	0	0
$359$	−10.4551	−0.551799	−0.275900	−	0.961186i	$-0.588976\pi$
$359$	−10.4551	−0.551799	−0.275900	+	0.961186i	$0.588976\pi$
$360$	0	0
$361$	21.7001	1.14211
$362$	0	0
$363$	5.73402	0.300958
$364$	0	0
$365$	28.1854	1.47529
$366$	0	0
$367$	−12.5353	−0.654335	−0.327168	−	0.944966i	$-0.606094\pi$
$367$	−12.5353	−0.654335	−0.327168	+	0.944966i	$0.606094\pi$
$368$	0	0
$369$	7.28161	0.379066
$370$	0	0
$371$	25.7423	1.33647
$372$	0	0
$373$	15.0282	0.778128	0.389064	−	0.921211i	$-0.372798\pi$
$373$	15.0282	0.778128	0.389064	+	0.921211i	$0.372798\pi$
$374$	0	0
$375$	−32.2332	−1.66451
$376$	0	0
$377$	−17.2807	−0.889999
$378$	0	0
$379$	1.27420	0.0654510	0.0327255	−	0.999464i	$-0.489581\pi$
$379$	1.27420	0.0654510	0.0327255	+	0.999464i	$0.489581\pi$
$380$	0	0
$381$	5.51341	0.282460
$382$	0	0
$383$	17.3490	0.886494	0.443247	−	0.896399i	$-0.353826\pi$
$383$	17.3490	0.886494	0.443247	+	0.896399i	$0.353826\pi$
$384$	0	0
$385$	46.8236	2.38635
$386$	0	0
$387$	−0.286483	−0.0145627
$388$	0	0
$389$	−23.6260	−1.19789	−0.598943	−	0.800791i	$-0.704413\pi$
$389$	−23.6260	−1.19789	−0.598943	+	0.800791i	$0.704413\pi$
$390$	0	0
$391$	41.4571	2.09657
$392$	0	0
$393$	3.58966	0.181075
$394$	0	0
$395$	37.1676	1.87011
$396$	0	0
$397$	−8.74972	−0.439136	−0.219568	−	0.975597i	$-0.570465\pi$
$397$	−8.74972	−0.439136	−0.219568	+	0.975597i	$0.570465\pi$
$398$	0	0
$399$	−30.9688	−1.55038
$400$	0	0
$401$	36.0776	1.80163	0.900815	−	0.434204i	$-0.142970\pi$
$401$	36.0776	1.80163	0.900815	+	0.434204i	$0.142970\pi$
$402$	0	0
$403$	51.5705	2.56891
$404$	0	0
$405$	4.20338	0.208868
$406$	0	0
$407$	15.8826	0.787270
$408$	0	0
$409$	35.8707	1.77369	0.886847	−	0.462064i	$-0.152891\pi$
$409$	35.8707	1.77369	0.886847	+	0.462064i	$0.152891\pi$
$410$	0	0
$411$	−22.0824	−1.08924
$412$	0	0
$413$	−29.3886	−1.44612
$414$	0	0
$415$	−5.95145	−0.292145
$416$	0	0
$417$	−2.49725	−0.122291
$418$	0	0
$419$	28.5823	1.39633	0.698167	−	0.715935i	$-0.253999\pi$
$419$	28.5823	1.39633	0.698167	+	0.715935i	$0.253999\pi$
$420$	0	0
$421$	35.8817	1.74877	0.874385	−	0.485234i	$-0.161265\pi$
$421$	35.8817	1.74877	0.874385	+	0.485234i	$0.161265\pi$
$422$	0	0
$423$	10.1863	0.495276
$424$	0	0
$425$	−74.8370	−3.63013
$426$	0	0
$427$	−23.7418	−1.14895
$428$	0	0
$429$	11.1869	0.540109
$430$	0	0
$431$	−12.6839	−0.610963	−0.305481	−	0.952198i	$-0.598817\pi$
$431$	−12.6839	−0.610963	−0.305481	+	0.952198i	$0.598817\pi$
$432$	0	0
$433$	−31.0612	−1.49271	−0.746354	−	0.665549i	$-0.768197\pi$
$433$	−31.0612	−1.49271	−0.746354	+	0.665549i	$0.768197\pi$
$434$	0	0
$435$	14.9001	0.714404
$436$	0	0
$437$	44.7715	2.14171
$438$	0	0
$439$	14.7780	0.705315	0.352657	−	0.935753i	$-0.385278\pi$
$439$	14.7780	0.705315	0.352657	+	0.935753i	$0.385278\pi$
$440$	0	0
$441$	16.5642	0.788773
$442$	0	0
$443$	−14.4542	−0.686740	−0.343370	−	0.939200i	$-0.611569\pi$
$443$	−14.4542	−0.686740	−0.343370	+	0.939200i	$0.611569\pi$
$444$	0	0
$445$	20.0124	0.948678
$446$	0	0
$447$	−6.74731	−0.319137
$448$	0	0
$449$	40.1656	1.89553	0.947766	−	0.318966i	$-0.103336\pi$
$449$	40.1656	1.89553	0.947766	+	0.318966i	$0.103336\pi$
$450$	0	0
$451$	−16.7096	−0.786826
$452$	0	0
$453$	−22.6346	−1.06347
$454$	0	0
$455$	−99.4708	−4.66327
$456$	0	0
$457$	1.85518	0.0867815	0.0433908	−	0.999058i	$-0.486184\pi$
$457$	1.85518	0.0867815	0.0433908	+	0.999058i	$0.486184\pi$
$458$	0	0
$459$	5.90737	0.275733
$460$	0	0
$461$	−22.6343	−1.05419	−0.527093	−	0.849808i	$-0.676718\pi$
$461$	−22.6343	−1.05419	−0.527093	+	0.849808i	$0.676718\pi$
$462$	0	0
$463$	33.9939	1.57983	0.789915	−	0.613217i	$-0.210125\pi$
$463$	33.9939	1.57983	0.789915	+	0.613217i	$0.210125\pi$
$464$	0	0
$465$	−44.4661	−2.06207
$466$	0	0
$467$	−4.11620	−0.190475	−0.0952374	−	0.995455i	$-0.530361\pi$
$467$	−4.11620	−0.190475	−0.0952374	+	0.995455i	$0.530361\pi$
$468$	0	0
$469$	−25.1650	−1.16201
$470$	0	0
$471$	18.3228	0.844271
$472$	0	0
$473$	0.657413	0.0302279
$474$	0	0
$475$	−80.8201	−3.70828
$476$	0	0
$477$	−5.30299	−0.242807
$478$	0	0
$479$	−32.0537	−1.46457	−0.732285	−	0.680999i	$-0.761546\pi$
$479$	−32.0537	−1.46457	−0.732285	+	0.680999i	$0.761546\pi$
$480$	0	0
$481$	−33.7405	−1.53844
$482$	0	0
$483$	−34.0668	−1.55009
$484$	0	0
$485$	−60.6935	−2.75595
$486$	0	0
$487$	18.2243	0.825821	0.412910	−	0.910772i	$-0.364512\pi$
$487$	18.2243	0.825821	0.412910	+	0.910772i	$0.364512\pi$
$488$	0	0
$489$	−17.6153	−0.796590
$490$	0	0
$491$	3.62906	0.163777	0.0818885	−	0.996641i	$-0.473905\pi$
$491$	3.62906	0.163777	0.0818885	+	0.996641i	$0.473905\pi$
$492$	0	0
$493$	20.9404	0.943107
$494$	0	0
$495$	−9.64580	−0.433546
$496$	0	0
$497$	−33.4942	−1.50242
$498$	0	0
$499$	−17.5606	−0.786119	−0.393059	−	0.919513i	$-0.628583\pi$
$499$	−17.5606	−0.786119	−0.393059	+	0.919513i	$0.628583\pi$
$500$	0	0
$501$	−4.65158	−0.207817
$502$	0	0
$503$	−1.00000	−0.0445878
$503$	−1.00000	−0.0445878
$504$	0	0
$505$	5.46691	0.243274
$506$	0	0
$507$	−10.7652	−0.478098
$508$	0	0
$509$	−18.7771	−0.832280	−0.416140	−	0.909301i	$-0.636617\pi$
$509$	−18.7771	−0.832280	−0.416140	+	0.909301i	$0.636617\pi$
$510$	0	0
$511$	−32.5501	−1.43993
$512$	0	0
$513$	6.37966	0.281669
$514$	0	0
$515$	78.5488	3.46127
$516$	0	0
$517$	−23.3753	−1.02804
$518$	0	0
$519$	8.43275	0.370157
$520$	0	0
$521$	6.77978	0.297027	0.148514	−	0.988910i	$-0.452551\pi$
$521$	6.77978	0.297027	0.148514	+	0.988910i	$0.452551\pi$
$522$	0	0
$523$	25.2712	1.10503	0.552516	−	0.833502i	$-0.313668\pi$
$523$	25.2712	1.10503	0.552516	+	0.833502i	$0.313668\pi$
$524$	0	0
$525$	61.4962	2.68391
$526$	0	0
$527$	−62.4921	−2.72220
$528$	0	0
$529$	26.2502	1.14131
$530$	0	0
$531$	6.05413	0.262727
$532$	0	0
$533$	35.4975	1.53757
$534$	0	0
$535$	15.4309	0.667136
$536$	0	0
$537$	13.5775	0.585914
$538$	0	0
$539$	−38.0112	−1.63726
$540$	0	0
$541$	1.91720	0.0824270	0.0412135	−	0.999150i	$-0.486878\pi$
$541$	1.91720	0.0824270	0.0412135	+	0.999150i	$0.486878\pi$
$542$	0	0
$543$	4.64377	0.199283
$544$	0	0
$545$	34.9432	1.49680
$546$	0	0
$547$	5.44869	0.232969	0.116485	−	0.993192i	$-0.462837\pi$
$547$	5.44869	0.232969	0.116485	+	0.993192i	$0.462837\pi$
$548$	0	0
$549$	4.89088	0.208738
$550$	0	0
$551$	22.6145	0.963411
$552$	0	0
$553$	−42.9233	−1.82528
$554$	0	0
$555$	29.0924	1.23490
$556$	0	0
$557$	−13.7680	−0.583368	−0.291684	−	0.956515i	$-0.594216\pi$
$557$	−13.7680	−0.583368	−0.291684	+	0.956515i	$0.594216\pi$
$558$	0	0
$559$	−1.39659	−0.0590695
$560$	0	0
$561$	−13.5561	−0.572338
$562$	0	0
$563$	−23.1208	−0.974425	−0.487213	−	0.873283i	$-0.661986\pi$
$563$	−23.1208	−0.974425	−0.487213	+	0.873283i	$0.661986\pi$
$564$	0	0
$565$	−2.41116	−0.101438
$566$	0	0
$567$	−4.85430	−0.203861
$568$	0	0
$569$	29.3450	1.23021	0.615103	−	0.788447i	$-0.289114\pi$
$569$	29.3450	1.23021	0.615103	+	0.788447i	$0.289114\pi$
$570$	0	0
$571$	9.71654	0.406625	0.203312	−	0.979114i	$-0.434829\pi$
$571$	9.71654	0.406625	0.203312	+	0.979114i	$0.434829\pi$
$572$	0	0
$573$	−17.7981	−0.743528
$574$	0	0
$575$	−88.9049	−3.70759
$576$	0	0
$577$	1.33390	0.0555310	0.0277655	−	0.999614i	$-0.491161\pi$
$577$	1.33390	0.0555310	0.0277655	+	0.999614i	$0.491161\pi$
$578$	0	0
$579$	−0.996260	−0.0414031
$580$	0	0
$581$	6.87307	0.285143
$582$	0	0
$583$	12.1692	0.503995
$584$	0	0
$585$	20.4913	0.847210
$586$	0	0
$587$	0.501195	0.0206865	0.0103433	−	0.999947i	$-0.496708\pi$
$587$	0.501195	0.0206865	0.0103433	+	0.999947i	$0.496708\pi$
$588$	0	0
$589$	−67.4883	−2.78081
$590$	0	0
$591$	−5.17054	−0.212688
$592$	0	0
$593$	21.6302	0.888245	0.444123	−	0.895966i	$-0.353515\pi$
$593$	21.6302	0.888245	0.444123	+	0.895966i	$0.353515\pi$
$594$	0	0
$595$	120.537	4.94153
$596$	0	0
$597$	19.4428	0.795739
$598$	0	0
$599$	35.0337	1.43144	0.715718	−	0.698389i	$-0.246099\pi$
$599$	35.0337	1.43144	0.715718	+	0.698389i	$0.246099\pi$
$600$	0	0
$601$	30.7061	1.25253	0.626264	−	0.779611i	$-0.284583\pi$
$601$	30.7061	1.25253	0.626264	+	0.779611i	$0.284583\pi$
$602$	0	0
$603$	5.18407	0.211111
$604$	0	0
$605$	−24.1023	−0.979896
$606$	0	0
$607$	25.5060	1.03526	0.517628	−	0.855606i	$-0.326815\pi$
$607$	25.5060	1.03526	0.517628	+	0.855606i	$0.326815\pi$
$608$	0	0
$609$	−17.2075	−0.697281
$610$	0	0
$611$	49.6578	2.00894
$612$	0	0
$613$	44.6266	1.80245	0.901227	−	0.433348i	$-0.142668\pi$
$613$	44.6266	1.80245	0.901227	+	0.433348i	$0.142668\pi$
$614$	0	0
$615$	−30.6074	−1.23421
$616$	0	0
$617$	−19.3710	−0.779846	−0.389923	−	0.920848i	$-0.627498\pi$
$617$	−19.3710	−0.779846	−0.389923	+	0.920848i	$0.627498\pi$
$618$	0	0
$619$	−44.9388	−1.80624	−0.903121	−	0.429387i	$-0.858730\pi$
$619$	−44.9388	−1.80624	−0.903121	+	0.429387i	$0.858730\pi$
$620$	0	0
$621$	7.01785	0.281617
$622$	0	0
$623$	−23.1114	−0.925941
$624$	0	0
$625$	72.1463	2.88585
$626$	0	0
$627$	−14.6399	−0.584660
$628$	0	0
$629$	40.8861	1.63024
$630$	0	0
$631$	14.7107	0.585622	0.292811	−	0.956170i	$-0.405409\pi$
$631$	14.7107	0.585622	0.292811	+	0.956170i	$0.405409\pi$
$632$	0	0
$633$	−12.0636	−0.479486
$634$	0	0
$635$	−23.1749	−0.919669
$636$	0	0
$637$	80.7499	3.19943
$638$	0	0
$639$	6.89990	0.272956
$640$	0	0
$641$	−20.6791	−0.816776	−0.408388	−	0.912808i	$-0.633909\pi$
$641$	−20.6791	−0.816776	−0.408388	+	0.912808i	$0.633909\pi$
$642$	0	0
$643$	2.94019	0.115950	0.0579749	−	0.998318i	$-0.481536\pi$
$643$	2.94019	0.115950	0.0579749	+	0.998318i	$0.481536\pi$
$644$	0	0
$645$	1.20420	0.0474152
$646$	0	0
$647$	−30.3700	−1.19397	−0.596983	−	0.802254i	$-0.703634\pi$
$647$	−30.3700	−1.19397	−0.596983	+	0.802254i	$0.703634\pi$
$648$	0	0
$649$	−13.8929	−0.545342
$650$	0	0
$651$	51.3520	2.01264
$652$	0	0
$653$	29.0608	1.13724	0.568618	−	0.822602i	$-0.307478\pi$
$653$	29.0608	1.13724	0.568618	+	0.822602i	$0.307478\pi$
$654$	0	0
$655$	−15.0887	−0.589565
$656$	0	0
$657$	6.70542	0.261603
$658$	0	0
$659$	12.8396	0.500158	0.250079	−	0.968225i	$-0.419543\pi$
$659$	12.8396	0.500158	0.250079	+	0.968225i	$0.419543\pi$
$660$	0	0
$661$	13.8726	0.539580	0.269790	−	0.962919i	$-0.413046\pi$
$661$	13.8726	0.539580	0.269790	+	0.962919i	$0.413046\pi$
$662$	0	0
$663$	28.7982	1.11843
$664$	0	0
$665$	130.174	5.04792
$666$	0	0
$667$	24.8768	0.963232
$668$	0	0
$669$	−8.37340	−0.323734
$670$	0	0
$671$	−11.2235	−0.433277
$672$	0	0
$673$	0.901074	0.0347339	0.0173669	−	0.999849i	$-0.494472\pi$
$673$	0.901074	0.0347339	0.0173669	+	0.999849i	$0.494472\pi$
$674$	0	0
$675$	−12.6684	−0.487607
$676$	0	0
$677$	18.0633	0.694229	0.347114	−	0.937823i	$-0.387162\pi$
$677$	18.0633	0.694229	0.347114	+	0.937823i	$0.387162\pi$
$678$	0	0
$679$	70.0923	2.68990
$680$	0	0
$681$	19.8272	0.759781
$682$	0	0
$683$	12.4242	0.475398	0.237699	−	0.971339i	$-0.423607\pi$
$683$	12.4242	0.475398	0.237699	+	0.971339i	$0.423607\pi$
$684$	0	0
$685$	92.8206	3.54649
$686$	0	0
$687$	−2.58173	−0.0984990
$688$	0	0
$689$	−25.8518	−0.984877
$690$	0	0
$691$	17.0184	0.647409	0.323705	−	0.946158i	$-0.395072\pi$
$691$	17.0184	0.647409	0.323705	+	0.946158i	$0.395072\pi$
$692$	0	0
$693$	11.1395	0.423155
$694$	0	0
$695$	10.4969	0.398169
$696$	0	0
$697$	−43.0152	−1.62932
$698$	0	0
$699$	9.86101	0.372978
$700$	0	0
$701$	34.9301	1.31929	0.659647	−	0.751576i	$-0.270706\pi$
$701$	34.9301	1.31929	0.659647	+	0.751576i	$0.270706\pi$
$702$	0	0
$703$	44.1549	1.66533
$704$	0	0
$705$	−42.8170	−1.61258
$706$	0	0
$707$	−6.31350	−0.237444
$708$	0	0
$709$	−32.5708	−1.22322	−0.611610	−	0.791159i	$-0.709478\pi$
$709$	−32.5708	−1.22322	−0.611610	+	0.791159i	$0.709478\pi$
$710$	0	0
$711$	8.84232	0.331613
$712$	0	0
$713$	−74.2395	−2.78029
$714$	0	0
$715$	−47.0228	−1.75855
$716$	0	0
$717$	25.4514	0.950499
$718$	0	0
$719$	−4.90065	−0.182763	−0.0913817	−	0.995816i	$-0.529128\pi$
$719$	−4.90065	−0.182763	−0.0913817	+	0.995816i	$0.529128\pi$
$720$	0	0
$721$	−90.7126	−3.37831
$722$	0	0
$723$	11.5744	0.430456
$724$	0	0
$725$	−44.9067	−1.66779
$726$	0	0
$727$	27.3922	1.01592	0.507960	−	0.861381i	$-0.330400\pi$
$727$	27.3922	1.01592	0.507960	+	0.861381i	$0.330400\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	1.69236	0.0625943
$732$	0	0
$733$	−21.0855	−0.778811	−0.389406	−	0.921066i	$-0.627320\pi$
$733$	−21.0855	−0.778811	−0.389406	+	0.921066i	$0.627320\pi$
$734$	0	0
$735$	−69.6258	−2.56819
$736$	0	0
$737$	−11.8962	−0.438204
$738$	0	0
$739$	−39.4441	−1.45098	−0.725488	−	0.688235i	$-0.758386\pi$
$739$	−39.4441	−1.45098	−0.725488	+	0.688235i	$0.758386\pi$
$740$	0	0
$741$	31.1006	1.14251
$742$	0	0
$743$	8.61959	0.316222	0.158111	−	0.987421i	$-0.449460\pi$
$743$	8.61959	0.316222	0.158111	+	0.987421i	$0.449460\pi$
$744$	0	0
$745$	28.3615	1.03909
$746$	0	0
$747$	−1.41587	−0.0518041
$748$	0	0
$749$	−17.8205	−0.651147
$750$	0	0
$751$	−3.56093	−0.129940	−0.0649700	−	0.997887i	$-0.520695\pi$
$751$	−3.56093	−0.129940	−0.0649700	+	0.997887i	$0.520695\pi$
$752$	0	0
$753$	2.18518	0.0796326
$754$	0	0
$755$	95.1420	3.46257
$756$	0	0
$757$	−2.90445	−0.105564	−0.0527820	−	0.998606i	$-0.516809\pi$
$757$	−2.90445	−0.105564	−0.0527820	+	0.998606i	$0.516809\pi$
$758$	0	0
$759$	−16.1044	−0.584551
$760$	0	0
$761$	−24.4839	−0.887542	−0.443771	−	0.896140i	$-0.646360\pi$
$761$	−24.4839	−0.887542	−0.443771	+	0.896140i	$0.646360\pi$
$762$	0	0
$763$	−40.3543	−1.46093
$764$	0	0
$765$	−24.8309	−0.897764
$766$	0	0
$767$	29.5136	1.06568
$768$	0	0
$769$	14.0319	0.506003	0.253001	−	0.967466i	$-0.418582\pi$
$769$	14.0319	0.506003	0.253001	+	0.967466i	$0.418582\pi$
$770$	0	0
$771$	−18.6738	−0.672522
$772$	0	0
$773$	−23.4430	−0.843187	−0.421594	−	0.906785i	$-0.638529\pi$
$773$	−23.4430	−0.843187	−0.421594	+	0.906785i	$0.638529\pi$
$774$	0	0
$775$	134.015	4.81395
$776$	0	0
$777$	−33.5976	−1.20531
$778$	0	0
$779$	−46.4542	−1.66440
$780$	0	0
$781$	−15.8337	−0.566574
$782$	0	0
$783$	3.54478	0.126680
$784$	0	0
$785$	−77.0177	−2.74888
$786$	0	0
$787$	8.61203	0.306986	0.153493	−	0.988150i	$-0.450948\pi$
$787$	8.61203	0.306986	0.153493	+	0.988150i	$0.450948\pi$
$788$	0	0
$789$	12.7689	0.454586
$790$	0	0
$791$	2.78455	0.0990072
$792$	0	0
$793$	23.8428	0.846684
$794$	0	0
$795$	22.2905	0.790562
$796$	0	0
$797$	16.0844	0.569738	0.284869	−	0.958566i	$-0.408050\pi$
$797$	16.0844	0.569738	0.284869	+	0.958566i	$0.408050\pi$
$798$	0	0
$799$	−60.1744	−2.12882
$800$	0	0
$801$	4.76102	0.168223
$802$	0	0
$803$	−15.3874	−0.543010
$804$	0	0
$805$	143.195	5.04698
$806$	0	0
$807$	−3.81283	−0.134218
$808$	0	0
$809$	−11.6765	−0.410524	−0.205262	−	0.978707i	$-0.565805\pi$
$809$	−11.6765	−0.410524	−0.205262	+	0.978707i	$0.565805\pi$
$810$	0	0
$811$	35.8519	1.25893	0.629466	−	0.777028i	$-0.283274\pi$
$811$	35.8519	1.25893	0.629466	+	0.777028i	$0.283274\pi$
$812$	0	0
$813$	13.8637	0.486222
$814$	0	0
$815$	74.0436	2.59364
$816$	0	0
$817$	1.82766	0.0639419
$818$	0	0
$819$	−23.6645	−0.826904
$820$	0	0
$821$	−40.0652	−1.39829	−0.699143	−	0.714982i	$-0.746435\pi$
$821$	−40.0652	−1.39829	−0.699143	+	0.714982i	$0.746435\pi$
$822$	0	0
$823$	3.97867	0.138688	0.0693439	−	0.997593i	$-0.477909\pi$
$823$	3.97867	0.138688	0.0693439	+	0.997593i	$0.477909\pi$
$824$	0	0
$825$	29.0711	1.01213
$826$	0	0
$827$	−17.7428	−0.616976	−0.308488	−	0.951228i	$-0.599823\pi$
$827$	−17.7428	−0.616976	−0.308488	+	0.951228i	$0.599823\pi$
$828$	0	0
$829$	−43.8889	−1.52433	−0.762163	−	0.647386i	$-0.775862\pi$
$829$	−43.8889	−1.52433	−0.762163	+	0.647386i	$0.775862\pi$
$830$	0	0
$831$	2.57537	0.0893385
$832$	0	0
$833$	−97.8512	−3.39034
$834$	0	0
$835$	19.5523	0.676637
$836$	0	0
$837$	−10.5787	−0.365652
$838$	0	0
$839$	20.4178	0.704900	0.352450	−	0.935831i	$-0.385349\pi$
$839$	20.4178	0.704900	0.352450	+	0.935831i	$0.385349\pi$
$840$	0	0
$841$	−16.4345	−0.566707
$842$	0	0
$843$	−5.49420	−0.189230
$844$	0	0
$845$	45.2501	1.55665
$846$	0	0
$847$	27.8347	0.956411
$848$	0	0
$849$	−5.70200	−0.195692
$850$	0	0
$851$	48.5719	1.66502
$852$	0	0
$853$	9.31856	0.319061	0.159531	−	0.987193i	$-0.449002\pi$
$853$	9.31856	0.319061	0.159531	+	0.987193i	$0.449002\pi$
$854$	0	0
$855$	−26.8161	−0.917093
$856$	0	0
$857$	−38.0916	−1.30118	−0.650592	−	0.759428i	$-0.725479\pi$
$857$	−38.0916	−1.30118	−0.650592	+	0.759428i	$0.725479\pi$
$858$	0	0
$859$	33.2157	1.13330	0.566652	−	0.823957i	$-0.308238\pi$
$859$	33.2157	1.13330	0.566652	+	0.823957i	$0.308238\pi$
$860$	0	0
$861$	35.3471	1.20463
$862$	0	0
$863$	28.7897	0.980012	0.490006	−	0.871719i	$-0.336995\pi$
$863$	28.7897	0.980012	0.490006	+	0.871719i	$0.336995\pi$
$864$	0	0
$865$	−35.4460	−1.20520
$866$	0	0
$867$	−17.8971	−0.607816
$868$	0	0
$869$	−20.2911	−0.688329
$870$	0	0
$871$	25.2721	0.856312
$872$	0	0
$873$	−14.4392	−0.488693
$874$	0	0
$875$	−156.470	−5.28964
$876$	0	0
$877$	45.7262	1.54407	0.772033	−	0.635583i	$-0.219240\pi$
$877$	45.7262	1.54407	0.772033	+	0.635583i	$0.219240\pi$
$878$	0	0
$879$	2.08811	0.0704302
$880$	0	0
$881$	−23.1112	−0.778635	−0.389318	−	0.921104i	$-0.627289\pi$
$881$	−23.1112	−0.778635	−0.389318	+	0.921104i	$0.627289\pi$
$882$	0	0
$883$	−39.5316	−1.33034	−0.665172	−	0.746690i	$-0.731642\pi$
$883$	−39.5316	−1.33034	−0.665172	+	0.746690i	$0.731642\pi$
$884$	0	0
$885$	−25.4478	−0.855419
$886$	0	0
$887$	−38.0736	−1.27839	−0.639193	−	0.769047i	$-0.720731\pi$
$887$	−38.0736	−1.27839	−0.639193	+	0.769047i	$0.720731\pi$
$888$	0	0
$889$	26.7637	0.897627
$890$	0	0
$891$	−2.29477	−0.0768778
$892$	0	0
$893$	−64.9853	−2.17465
$894$	0	0
$895$	−57.0716	−1.90769
$896$	0	0
$897$	34.2117	1.14230
$898$	0	0
$899$	−37.4991	−1.25066
$900$	0	0
$901$	31.3268	1.04365
$902$	0	0
$903$	−1.39067	−0.0462788
$904$	0	0
$905$	−19.5195	−0.648851
$906$	0	0
$907$	44.3595	1.47293	0.736466	−	0.676475i	$-0.236493\pi$
$907$	44.3595	1.47293	0.736466	+	0.676475i	$0.236493\pi$
$908$	0	0
$909$	1.30060	0.0431381
$910$	0	0
$911$	−34.3923	−1.13947	−0.569734	−	0.821829i	$-0.692954\pi$
$911$	−34.3923	−1.13947	−0.569734	+	0.821829i	$0.692954\pi$
$912$	0	0
$913$	3.24911	0.107530
$914$	0	0
$915$	−20.5582	−0.679634
$916$	0	0
$917$	17.4253	0.575434
$918$	0	0
$919$	−41.3984	−1.36561	−0.682804	−	0.730602i	$-0.739240\pi$
$919$	−41.3984	−1.36561	−0.682804	+	0.730602i	$0.739240\pi$
$920$	0	0
$921$	−22.2877	−0.734403
$922$	0	0
$923$	33.6367	1.10717
$924$	0	0
$925$	−87.6805	−2.88292
$926$	0	0
$927$	18.6871	0.613763
$928$	0	0
$929$	−16.0918	−0.527956	−0.263978	−	0.964529i	$-0.585035\pi$
$929$	−16.0918	−0.527956	−0.263978	+	0.964529i	$0.585035\pi$
$930$	0	0
$931$	−105.674	−3.46334
$932$	0	0
$933$	−10.2272	−0.334822
$934$	0	0
$935$	56.9813	1.86349
$936$	0	0
$937$	4.27201	0.139561	0.0697803	−	0.997562i	$-0.477770\pi$
$937$	4.27201	0.139561	0.0697803	+	0.997562i	$0.477770\pi$
$938$	0	0
$939$	8.58534	0.280172
$940$	0	0
$941$	0.693819	0.0226178	0.0113089	−	0.999936i	$-0.496400\pi$
$941$	0.693819	0.0226178	0.0113089	+	0.999936i	$0.496400\pi$
$942$	0	0
$943$	−51.1013	−1.66409
$944$	0	0
$945$	20.4045	0.663757
$946$	0	0
$947$	15.5960	0.506803	0.253402	−	0.967361i	$-0.418451\pi$
$947$	15.5960	0.506803	0.253402	+	0.967361i	$0.418451\pi$
$948$	0	0
$949$	32.6886	1.06112
$950$	0	0
$951$	30.3507	0.984188
$952$	0	0
$953$	−26.1799	−0.848051	−0.424026	−	0.905650i	$-0.639383\pi$
$953$	−26.1799	−0.848051	−0.424026	+	0.905650i	$0.639383\pi$
$954$	0	0
$955$	74.8123	2.42087
$956$	0	0
$957$	−8.13447	−0.262950
$958$	0	0
$959$	−107.194	−3.46149
$960$	0	0
$961$	80.9081	2.60994
$962$	0	0
$963$	3.67107	0.118299
$964$	0	0
$965$	4.18766	0.134805
$966$	0	0
$967$	32.5634	1.04717	0.523585	−	0.851974i	$-0.324594\pi$
$967$	32.5634	1.04717	0.523585	+	0.851974i	$0.324594\pi$
$968$	0	0
$969$	−37.6871	−1.21068
$970$	0	0
$971$	−5.19389	−0.166680	−0.0833400	−	0.996521i	$-0.526559\pi$
$971$	−5.19389	−0.166680	−0.0833400	+	0.996521i	$0.526559\pi$
$972$	0	0
$973$	−12.1224	−0.388626
$974$	0	0
$975$	−61.7578	−1.97783
$976$	0	0
$977$	15.6344	0.500188	0.250094	−	0.968222i	$-0.419538\pi$
$977$	15.6344	0.500188	0.250094	+	0.968222i	$0.419538\pi$
$978$	0	0
$979$	−10.9255	−0.349179
$980$	0	0
$981$	8.31311	0.265417
$982$	0	0
$983$	1.14902	0.0366481	0.0183241	−	0.999832i	$-0.494167\pi$
$983$	1.14902	0.0366481	0.0183241	+	0.999832i	$0.494167\pi$
$984$	0	0
$985$	21.7338	0.692495
$986$	0	0
$987$	49.4475	1.57393
$988$	0	0
$989$	2.01049	0.0639300
$990$	0	0
$991$	19.4026	0.616343	0.308171	−	0.951331i	$-0.400283\pi$
$991$	19.4026	0.616343	0.308171	+	0.951331i	$0.400283\pi$
$992$	0	0
$993$	−7.14727	−0.226812
$994$	0	0
$995$	−81.7253	−2.59087
$996$	0	0
$997$	−22.3616	−0.708199	−0.354099	−	0.935208i	$-0.615212\pi$
$997$	−22.3616	−0.708199	−0.354099	+	0.935208i	$0.615212\pi$
$998$	0	0
$999$	6.92120	0.218977

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6036.2.a.i.1.25	✓	26

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6036.2.a.i.1.25	✓	26	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:	\( N \)	\(=\)	\( 6036 = 2^{2} \cdot 3 \cdot 503 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6036.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +4.20338 q^{5} -4.85430 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +4.20338 q^{5} -4.85430 q^{7} +1.00000 q^{9} -2.29477 q^{11} +4.87495 q^{13} -4.20338 q^{15} -5.90737 q^{17} -6.37966 q^{19} +4.85430 q^{21} -7.01785 q^{23} +12.6684 q^{25} -1.00000 q^{27} -3.54478 q^{29} +10.5787 q^{31} +2.29477 q^{33} -20.4045 q^{35} -6.92120 q^{37} -4.87495 q^{39} +7.28161 q^{41} -0.286483 q^{43} +4.20338 q^{45} +10.1863 q^{47} +16.5642 q^{49} +5.90737 q^{51} -5.30299 q^{53} -9.64580 q^{55} +6.37966 q^{57} +6.05413 q^{59} +4.89088 q^{61} -4.85430 q^{63} +20.4913 q^{65} +5.18407 q^{67} +7.01785 q^{69} +6.89990 q^{71} +6.70542 q^{73} -12.6684 q^{75} +11.1395 q^{77} +8.84232 q^{79} +1.00000 q^{81} -1.41587 q^{83} -24.8309 q^{85} +3.54478 q^{87} +4.76102 q^{89} -23.6645 q^{91} -10.5787 q^{93} -26.8161 q^{95} -14.4392 q^{97} -2.29477 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 26 q - 26 q^{3} + 6 q^{5} + 5 q^{7} + 26 q^{9}+O(q^{10}) \) 26 * q - 26 * q^3 + 6 * q^5 + 5 * q^7 + 26 * q^9	\( 26 q - 26 q^{3} + 6 q^{5} + 5 q^{7} + 26 q^{9} - 11 q^{11} + 13 q^{13} - 6 q^{15} + 12 q^{17} - q^{19} - 5 q^{21} - 22 q^{23} + 48 q^{25} - 26 q^{27} + 6 q^{29} + 19 q^{31} + 11 q^{33} - 21 q^{35} + 20 q^{37} - 13 q^{39} + 25 q^{41} + 4 q^{43} + 6 q^{45} + 8 q^{47} + 67 q^{49} - 12 q^{51} - 5 q^{53} + 20 q^{55} + q^{57} - 18 q^{59} + 43 q^{61} + 5 q^{63} + 41 q^{65} + 5 q^{67} + 22 q^{69} - q^{71} + 22 q^{73} - 48 q^{75} + 23 q^{77} + 16 q^{79} + 26 q^{81} - 19 q^{83} + 29 q^{85} - 6 q^{87} + 49 q^{89} - 13 q^{91} - 19 q^{93} - 26 q^{95} + 25 q^{97} - 11 q^{99}+O(q^{100}) \) 26 * q - 26 * q^3 + 6 * q^5 + 5 * q^7 + 26 * q^9 - 11 * q^11 + 13 * q^13 - 6 * q^15 + 12 * q^17 - q^19 - 5 * q^21 - 22 * q^23 + 48 * q^25 - 26 * q^27 + 6 * q^29 + 19 * q^31 + 11 * q^33 - 21 * q^35 + 20 * q^37 - 13 * q^39 + 25 * q^41 + 4 * q^43 + 6 * q^45 + 8 * q^47 + 67 * q^49 - 12 * q^51 - 5 * q^53 + 20 * q^55 + q^57 - 18 * q^59 + 43 * q^61 + 5 * q^63 + 41 * q^65 + 5 * q^67 + 22 * q^69 - q^71 + 22 * q^73 - 48 * q^75 + 23 * q^77 + 16 * q^79 + 26 * q^81 - 19 * q^83 + 29 * q^85 - 6 * q^87 + 49 * q^89 - 13 * q^91 - 19 * q^93 - 26 * q^95 + 25 * q^97 - 11 * q^99

Introduction

L-functions

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Database

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