LMFDB - Embedded newform 6036.2.a.i.1.15

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			1.15
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)

$f(q)$	$=$	$q-1.00000 q^{3} +0.754498 q^{5} -4.03319 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +0.754498 q^{5} -4.03319 q^{7} +1.00000 q^{9} -3.82332 q^{11} -4.25616 q^{13} -0.754498 q^{15} -5.34396 q^{17} -3.06933 q^{19} +4.03319 q^{21} -3.68458 q^{23} -4.43073 q^{25} -1.00000 q^{27} -6.36746 q^{29} -10.0869 q^{31} +3.82332 q^{33} -3.04303 q^{35} +9.28645 q^{37} +4.25616 q^{39} +2.62840 q^{41} +7.26677 q^{43} +0.754498 q^{45} -10.3713 q^{47} +9.26663 q^{49} +5.34396 q^{51} +10.2864 q^{53} -2.88468 q^{55} +3.06933 q^{57} +6.42578 q^{59} +9.19171 q^{61} -4.03319 q^{63} -3.21126 q^{65} -6.71426 q^{67} +3.68458 q^{69} -0.934048 q^{71} -12.6801 q^{73} +4.43073 q^{75} +15.4202 q^{77} -0.170154 q^{79} +1.00000 q^{81} +10.4649 q^{83} -4.03201 q^{85} +6.36746 q^{87} +7.13287 q^{89} +17.1659 q^{91} +10.0869 q^{93} -2.31580 q^{95} -12.3455 q^{97} -3.82332 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	0.754498	0.337422	0.168711	−	0.985666i	$-0.446040\pi$
$5$	0.754498	0.337422	0.168711	+	0.985666i	$0.446040\pi$
$6$	0	0
$7$	−4.03319	−1.52440	−0.762201	−	0.647340i	$-0.775881\pi$
$7$	−4.03319	−1.52440	−0.762201	+	0.647340i	$0.775881\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−3.82332	−1.15277	−0.576387	−	0.817177i	$-0.695538\pi$
$11$	−3.82332	−1.15277	−0.576387	+	0.817177i	$0.695538\pi$
$12$	0	0
$13$	−4.25616	−1.18045	−0.590223	−	0.807240i	$-0.700960\pi$
$13$	−4.25616	−1.18045	−0.590223	+	0.807240i	$0.700960\pi$
$14$	0	0
$15$	−0.754498	−0.194810
$16$	0	0
$17$	−5.34396	−1.29610	−0.648051	−	0.761597i	$-0.724416\pi$
$17$	−5.34396	−1.29610	−0.648051	+	0.761597i	$0.724416\pi$
$18$	0	0
$19$	−3.06933	−0.704152	−0.352076	−	0.935971i	$-0.614524\pi$
$19$	−3.06933	−0.704152	−0.352076	+	0.935971i	$0.614524\pi$
$20$	0	0
$21$	4.03319	0.880114
$22$	0	0
$23$	−3.68458	−0.768288	−0.384144	−	0.923273i	$-0.625503\pi$
$23$	−3.68458	−0.768288	−0.384144	+	0.923273i	$0.625503\pi$
$24$	0	0
$25$	−4.43073	−0.886147
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−6.36746	−1.18241	−0.591204	−	0.806522i	$-0.701347\pi$
$29$	−6.36746	−1.18241	−0.591204	+	0.806522i	$0.701347\pi$
$30$	0	0
$31$	−10.0869	−1.81167	−0.905835	−	0.423631i	$-0.860755\pi$
$31$	−10.0869	−1.81167	−0.905835	+	0.423631i	$0.860755\pi$
$32$	0	0
$33$	3.82332	0.665554
$34$	0	0
$35$	−3.04303	−0.514366
$36$	0	0
$37$	9.28645	1.52668	0.763341	−	0.645996i	$-0.223558\pi$
$37$	9.28645	1.52668	0.763341	+	0.645996i	$0.223558\pi$
$38$	0	0
$39$	4.25616	0.681531
$40$	0	0
$41$	2.62840	0.410487	0.205244	−	0.978711i	$-0.434201\pi$
$41$	2.62840	0.410487	0.205244	+	0.978711i	$0.434201\pi$
$42$	0	0
$43$	7.26677	1.10817	0.554086	−	0.832459i	$-0.313068\pi$
$43$	7.26677	1.10817	0.554086	+	0.832459i	$0.313068\pi$
$44$	0	0
$45$	0.754498	0.112474
$46$	0	0
$47$	−10.3713	−1.51280	−0.756402	−	0.654107i	$-0.773044\pi$
$47$	−10.3713	−1.51280	−0.756402	+	0.654107i	$0.773044\pi$
$48$	0	0
$49$	9.26663	1.32380
$50$	0	0
$51$	5.34396	0.748305
$52$	0	0
$53$	10.2864	1.41295	0.706475	−	0.707738i	$-0.250284\pi$
$53$	10.2864	1.41295	0.706475	+	0.707738i	$0.250284\pi$
$54$	0	0
$55$	−2.88468	−0.388971
$56$	0	0
$57$	3.06933	0.406542
$58$	0	0
$59$	6.42578	0.836565	0.418282	−	0.908317i	$-0.362632\pi$
$59$	6.42578	0.836565	0.418282	+	0.908317i	$0.362632\pi$
$60$	0	0
$61$	9.19171	1.17688	0.588439	−	0.808541i	$-0.299743\pi$
$61$	9.19171	1.17688	0.588439	+	0.808541i	$0.299743\pi$
$62$	0	0
$63$	−4.03319	−0.508134
$64$	0	0
$65$	−3.21126	−0.398308
$66$	0	0
$67$	−6.71426	−0.820278	−0.410139	−	0.912023i	$-0.634520\pi$
$67$	−6.71426	−0.820278	−0.410139	+	0.912023i	$0.634520\pi$
$68$	0	0
$69$	3.68458	0.443571
$70$	0	0
$71$	−0.934048	−0.110851	−0.0554255	−	0.998463i	$-0.517652\pi$
$71$	−0.934048	−0.110851	−0.0554255	+	0.998463i	$0.517652\pi$
$72$	0	0
$73$	−12.6801	−1.48409	−0.742045	−	0.670351i	$-0.766144\pi$
$73$	−12.6801	−1.48409	−0.742045	+	0.670351i	$0.766144\pi$
$74$	0	0
$75$	4.43073	0.511617
$76$	0	0
$77$	15.4202	1.75729
$78$	0	0
$79$	−0.170154	−0.0191438	−0.00957192	−	0.999954i	$-0.503047\pi$
$79$	−0.170154	−0.0191438	−0.00957192	+	0.999954i	$0.503047\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	10.4649	1.14867	0.574334	−	0.818621i	$-0.305261\pi$
$83$	10.4649	1.14867	0.574334	+	0.818621i	$0.305261\pi$
$84$	0	0
$85$	−4.03201	−0.437333
$86$	0	0
$87$	6.36746	0.682663
$88$	0	0
$89$	7.13287	0.756083	0.378042	−	0.925789i	$-0.376598\pi$
$89$	7.13287	0.756083	0.378042	+	0.925789i	$0.376598\pi$
$90$	0	0
$91$	17.1659	1.79947
$92$	0	0
$93$	10.0869	1.04597
$94$	0	0
$95$	−2.31580	−0.237596
$96$	0	0
$97$	−12.3455	−1.25350	−0.626749	−	0.779221i	$-0.715615\pi$
$97$	−12.3455	−1.25350	−0.626749	+	0.779221i	$0.715615\pi$
$98$	0	0
$99$	−3.82332	−0.384258
$100$	0	0
$101$	5.74355	0.571505	0.285752	−	0.958303i	$-0.407756\pi$
$101$	5.74355	0.571505	0.285752	+	0.958303i	$0.407756\pi$
$102$	0	0
$103$	−6.83800	−0.673768	−0.336884	−	0.941546i	$-0.609373\pi$
$103$	−6.83800	−0.673768	−0.336884	+	0.941546i	$0.609373\pi$
$104$	0	0
$105$	3.04303	0.296970
$106$	0	0
$107$	−5.41431	−0.523421	−0.261711	−	0.965146i	$-0.584287\pi$
$107$	−5.41431	−0.523421	−0.261711	+	0.965146i	$0.584287\pi$
$108$	0	0
$109$	−15.8743	−1.52048	−0.760241	−	0.649642i	$-0.774919\pi$
$109$	−15.8743	−1.52048	−0.760241	+	0.649642i	$0.774919\pi$
$110$	0	0
$111$	−9.28645	−0.881430
$112$	0	0
$113$	−2.42307	−0.227943	−0.113971	−	0.993484i	$-0.536357\pi$
$113$	−2.42307	−0.227943	−0.113971	+	0.993484i	$0.536357\pi$
$114$	0	0
$115$	−2.78001	−0.259237
$116$	0	0
$117$	−4.25616	−0.393482
$118$	0	0
$119$	21.5532	1.97578
$120$	0	0
$121$	3.61776	0.328888
$122$	0	0
$123$	−2.62840	−0.236995
$124$	0	0
$125$	−7.11547	−0.636427
$126$	0	0
$127$	12.5823	1.11650	0.558250	−	0.829672i	$-0.311473\pi$
$127$	12.5823	1.11650	0.558250	+	0.829672i	$0.311473\pi$
$128$	0	0
$129$	−7.26677	−0.639803
$130$	0	0
$131$	1.11476	0.0973967	0.0486983	−	0.998814i	$-0.484493\pi$
$131$	1.11476	0.0973967	0.0486983	+	0.998814i	$0.484493\pi$
$132$	0	0
$133$	12.3792	1.07341
$134$	0	0
$135$	−0.754498	−0.0649368
$136$	0	0
$137$	2.03806	0.174123	0.0870617	−	0.996203i	$-0.472252\pi$
$137$	2.03806	0.174123	0.0870617	+	0.996203i	$0.472252\pi$
$138$	0	0
$139$	2.61705	0.221975	0.110988	−	0.993822i	$-0.464599\pi$
$139$	2.61705	0.221975	0.110988	+	0.993822i	$0.464599\pi$
$140$	0	0
$141$	10.3713	0.873417
$142$	0	0
$143$	16.2726	1.36079
$144$	0	0
$145$	−4.80423	−0.398970
$146$	0	0
$147$	−9.26663	−0.764299
$148$	0	0
$149$	−0.999867	−0.0819123	−0.0409561	−	0.999161i	$-0.513040\pi$
$149$	−0.999867	−0.0819123	−0.0409561	+	0.999161i	$0.513040\pi$
$150$	0	0
$151$	1.18736	0.0966257	0.0483128	−	0.998832i	$-0.484616\pi$
$151$	1.18736	0.0966257	0.0483128	+	0.998832i	$0.484616\pi$
$152$	0	0
$153$	−5.34396	−0.432034
$154$	0	0
$155$	−7.61058	−0.611296
$156$	0	0
$157$	2.87492	0.229443	0.114722	−	0.993398i	$-0.463402\pi$
$157$	2.87492	0.229443	0.114722	+	0.993398i	$0.463402\pi$
$158$	0	0
$159$	−10.2864	−0.815767
$160$	0	0
$161$	14.8606	1.17118
$162$	0	0
$163$	−15.3943	−1.20578	−0.602888	−	0.797826i	$-0.705984\pi$
$163$	−15.3943	−1.20578	−0.602888	+	0.797826i	$0.705984\pi$
$164$	0	0
$165$	2.88468	0.224572
$166$	0	0
$167$	−10.8494	−0.839554	−0.419777	−	0.907627i	$-0.637892\pi$
$167$	−10.8494	−0.839554	−0.419777	+	0.907627i	$0.637892\pi$
$168$	0	0
$169$	5.11487	0.393452
$170$	0	0
$171$	−3.06933	−0.234717
$172$	0	0
$173$	8.55402	0.650350	0.325175	−	0.945654i	$-0.394577\pi$
$173$	8.55402	0.650350	0.325175	+	0.945654i	$0.394577\pi$
$174$	0	0
$175$	17.8700	1.35084
$176$	0	0
$177$	−6.42578	−0.482991
$178$	0	0
$179$	−22.5055	−1.68214	−0.841070	−	0.540927i	$-0.818074\pi$
$179$	−22.5055	−1.68214	−0.841070	+	0.540927i	$0.818074\pi$
$180$	0	0
$181$	24.1911	1.79811	0.899057	−	0.437832i	$-0.144254\pi$
$181$	24.1911	1.79811	0.899057	+	0.437832i	$0.144254\pi$
$182$	0	0
$183$	−9.19171	−0.679471
$184$	0	0
$185$	7.00660	0.515135
$186$	0	0
$187$	20.4317	1.49411
$188$	0	0
$189$	4.03319	0.293371
$190$	0	0
$191$	−9.29327	−0.672438	−0.336219	−	0.941784i	$-0.609148\pi$
$191$	−9.29327	−0.672438	−0.336219	+	0.941784i	$0.609148\pi$
$192$	0	0
$193$	−11.8415	−0.852368	−0.426184	−	0.904637i	$-0.640142\pi$
$193$	−11.8415	−0.852368	−0.426184	+	0.904637i	$0.640142\pi$
$194$	0	0
$195$	3.21126	0.229963
$196$	0	0
$197$	10.2965	0.733598	0.366799	−	0.930300i	$-0.380454\pi$
$197$	10.2965	0.733598	0.366799	+	0.930300i	$0.380454\pi$
$198$	0	0
$199$	25.1760	1.78468	0.892341	−	0.451362i	$-0.149062\pi$
$199$	25.1760	1.78468	0.892341	+	0.451362i	$0.149062\pi$
$200$	0	0
$201$	6.71426	0.473587
$202$	0	0
$203$	25.6812	1.80246
$204$	0	0
$205$	1.98312	0.138507
$206$	0	0
$207$	−3.68458	−0.256096
$208$	0	0
$209$	11.7350	0.811728
$210$	0	0
$211$	−16.7238	−1.15132	−0.575658	−	0.817691i	$-0.695254\pi$
$211$	−16.7238	−1.15132	−0.575658	+	0.817691i	$0.695254\pi$
$212$	0	0
$213$	0.934048	0.0639999
$214$	0	0
$215$	5.48276	0.373921
$216$	0	0
$217$	40.6826	2.76171
$218$	0	0
$219$	12.6801	0.856839
$220$	0	0
$221$	22.7448	1.52998
$222$	0	0
$223$	6.83098	0.457436	0.228718	−	0.973493i	$-0.426547\pi$
$223$	6.83098	0.457436	0.228718	+	0.973493i	$0.426547\pi$
$224$	0	0
$225$	−4.43073	−0.295382
$226$	0	0
$227$	−27.7479	−1.84169	−0.920845	−	0.389929i	$-0.872500\pi$
$227$	−27.7479	−1.84169	−0.920845	+	0.389929i	$0.872500\pi$
$228$	0	0
$229$	−8.72641	−0.576657	−0.288329	−	0.957532i	$-0.593100\pi$
$229$	−8.72641	−0.576657	−0.288329	+	0.957532i	$0.593100\pi$
$230$	0	0
$231$	−15.4202	−1.01457
$232$	0	0
$233$	−12.2730	−0.804030	−0.402015	−	0.915633i	$-0.631690\pi$
$233$	−12.2730	−0.804030	−0.402015	+	0.915633i	$0.631690\pi$
$234$	0	0
$235$	−7.82509	−0.510452
$236$	0	0
$237$	0.170154	0.0110527
$238$	0	0
$239$	−18.0522	−1.16770	−0.583851	−	0.811861i	$-0.698455\pi$
$239$	−18.0522	−1.16770	−0.583851	+	0.811861i	$0.698455\pi$
$240$	0	0
$241$	25.8028	1.66211	0.831054	−	0.556192i	$-0.187738\pi$
$241$	25.8028	1.66211	0.831054	+	0.556192i	$0.187738\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	6.99165	0.446680
$246$	0	0
$247$	13.0635	0.831213
$248$	0	0
$249$	−10.4649	−0.663184
$250$	0	0
$251$	25.0414	1.58060	0.790299	−	0.612721i	$-0.209925\pi$
$251$	25.0414	1.58060	0.790299	+	0.612721i	$0.209925\pi$
$252$	0	0
$253$	14.0873	0.885663
$254$	0	0
$255$	4.03201	0.252494
$256$	0	0
$257$	14.6320	0.912716	0.456358	−	0.889796i	$-0.349154\pi$
$257$	14.6320	0.912716	0.456358	+	0.889796i	$0.349154\pi$
$258$	0	0
$259$	−37.4540	−2.32728
$260$	0	0
$261$	−6.36746	−0.394136
$262$	0	0
$263$	−27.1264	−1.67268	−0.836342	−	0.548208i	$-0.815310\pi$
$263$	−27.1264	−1.67268	−0.836342	+	0.548208i	$0.815310\pi$
$264$	0	0
$265$	7.76109	0.476760
$266$	0	0
$267$	−7.13287	−0.436525
$268$	0	0
$269$	−15.5230	−0.946457	−0.473228	−	0.880940i	$-0.656911\pi$
$269$	−15.5230	−0.946457	−0.473228	+	0.880940i	$0.656911\pi$
$270$	0	0
$271$	−3.76129	−0.228482	−0.114241	−	0.993453i	$-0.536444\pi$
$271$	−3.76129	−0.228482	−0.114241	+	0.993453i	$0.536444\pi$
$272$	0	0
$273$	−17.1659	−1.03893
$274$	0	0
$275$	16.9401	1.02153
$276$	0	0
$277$	−7.82723	−0.470293	−0.235146	−	0.971960i	$-0.575557\pi$
$277$	−7.82723	−0.470293	−0.235146	+	0.971960i	$0.575557\pi$
$278$	0	0
$279$	−10.0869	−0.603890
$280$	0	0
$281$	−11.0898	−0.661565	−0.330782	−	0.943707i	$-0.607313\pi$
$281$	−11.0898	−0.661565	−0.330782	+	0.943707i	$0.607313\pi$
$282$	0	0
$283$	−28.4603	−1.69179	−0.845895	−	0.533349i	$-0.820933\pi$
$283$	−28.4603	−1.69179	−0.845895	+	0.533349i	$0.820933\pi$
$284$	0	0
$285$	2.31580	0.137176
$286$	0	0
$287$	−10.6008	−0.625748
$288$	0	0
$289$	11.5580	0.679880
$290$	0	0
$291$	12.3455	0.723707
$292$	0	0
$293$	−31.1011	−1.81695	−0.908473	−	0.417943i	$-0.862751\pi$
$293$	−31.1011	−1.81695	−0.908473	+	0.417943i	$0.862751\pi$
$294$	0	0
$295$	4.84823	0.282275
$296$	0	0
$297$	3.82332	0.221851
$298$	0	0
$299$	15.6822	0.906922
$300$	0	0
$301$	−29.3083	−1.68930
$302$	0	0
$303$	−5.74355	−0.329959
$304$	0	0
$305$	6.93513	0.397104
$306$	0	0
$307$	22.1359	1.26336	0.631680	−	0.775229i	$-0.282366\pi$
$307$	22.1359	1.26336	0.631680	+	0.775229i	$0.282366\pi$
$308$	0	0
$309$	6.83800	0.389000
$310$	0	0
$311$	−20.5369	−1.16454	−0.582270	−	0.812995i	$-0.697835\pi$
$311$	−20.5369	−1.16454	−0.582270	+	0.812995i	$0.697835\pi$
$312$	0	0
$313$	16.4172	0.927957	0.463978	−	0.885847i	$-0.346422\pi$
$313$	16.4172	0.927957	0.463978	+	0.885847i	$0.346422\pi$
$314$	0	0
$315$	−3.04303	−0.171455
$316$	0	0
$317$	−9.49952	−0.533546	−0.266773	−	0.963759i	$-0.585957\pi$
$317$	−9.49952	−0.533546	−0.266773	+	0.963759i	$0.585957\pi$
$318$	0	0
$319$	24.3448	1.36305
$320$	0	0
$321$	5.41431	0.302197
$322$	0	0
$323$	16.4024	0.912653
$324$	0	0
$325$	18.8579	1.04605
$326$	0	0
$327$	15.8743	0.877850
$328$	0	0
$329$	41.8293	2.30612
$330$	0	0
$331$	−21.9003	−1.20375	−0.601874	−	0.798591i	$-0.705579\pi$
$331$	−21.9003	−1.20375	−0.601874	+	0.798591i	$0.705579\pi$
$332$	0	0
$333$	9.28645	0.508894
$334$	0	0
$335$	−5.06589	−0.276779
$336$	0	0
$337$	10.5657	0.575551	0.287776	−	0.957698i	$-0.407084\pi$
$337$	10.5657	0.575551	0.287776	+	0.957698i	$0.407084\pi$
$338$	0	0
$339$	2.42307	0.131603
$340$	0	0
$341$	38.5656	2.08845
$342$	0	0
$343$	−9.14174	−0.493608
$344$	0	0
$345$	2.78001	0.149671
$346$	0	0
$347$	−27.8517	−1.49516	−0.747578	−	0.664174i	$-0.768783\pi$
$347$	−27.8517	−1.49516	−0.747578	+	0.664174i	$0.768783\pi$
$348$	0	0
$349$	4.34629	0.232652	0.116326	−	0.993211i	$-0.462888\pi$
$349$	4.34629	0.232652	0.116326	+	0.993211i	$0.462888\pi$
$350$	0	0
$351$	4.25616	0.227177
$352$	0	0
$353$	12.6797	0.674874	0.337437	−	0.941348i	$-0.390440\pi$
$353$	12.6797	0.674874	0.337437	+	0.941348i	$0.390440\pi$
$354$	0	0
$355$	−0.704737	−0.0374035
$356$	0	0
$357$	−21.5532	−1.14072
$358$	0	0
$359$	28.9983	1.53047	0.765236	−	0.643750i	$-0.222622\pi$
$359$	28.9983	1.53047	0.765236	+	0.643750i	$0.222622\pi$
$360$	0	0
$361$	−9.57923	−0.504170
$362$	0	0
$363$	−3.61776	−0.189883
$364$	0	0
$365$	−9.56708	−0.500764
$366$	0	0
$367$	−10.5490	−0.550652	−0.275326	−	0.961351i	$-0.588786\pi$
$367$	−10.5490	−0.550652	−0.275326	+	0.961351i	$0.588786\pi$
$368$	0	0
$369$	2.62840	0.136829
$370$	0	0
$371$	−41.4872	−2.15391
$372$	0	0
$373$	4.24418	0.219755	0.109878	−	0.993945i	$-0.464954\pi$
$373$	4.24418	0.219755	0.109878	+	0.993945i	$0.464954\pi$
$374$	0	0
$375$	7.11547	0.367441
$376$	0	0
$377$	27.1009	1.39577
$378$	0	0
$379$	−1.87745	−0.0964381	−0.0482190	−	0.998837i	$-0.515355\pi$
$379$	−1.87745	−0.0964381	−0.0482190	+	0.998837i	$0.515355\pi$
$380$	0	0
$381$	−12.5823	−0.644612
$382$	0	0
$383$	31.6065	1.61502	0.807509	−	0.589855i	$-0.200815\pi$
$383$	31.6065	1.61502	0.807509	+	0.589855i	$0.200815\pi$
$384$	0	0
$385$	11.6345	0.592948
$386$	0	0
$387$	7.26677	0.369391
$388$	0	0
$389$	0.510967	0.0259070	0.0129535	−	0.999916i	$-0.495877\pi$
$389$	0.510967	0.0259070	0.0129535	+	0.999916i	$0.495877\pi$
$390$	0	0
$391$	19.6903	0.995780
$392$	0	0
$393$	−1.11476	−0.0562320
$394$	0	0
$395$	−0.128381	−0.00645955
$396$	0	0
$397$	−29.8037	−1.49580	−0.747902	−	0.663809i	$-0.768939\pi$
$397$	−29.8037	−1.49580	−0.747902	+	0.663809i	$0.768939\pi$
$398$	0	0
$399$	−12.3792	−0.619734
$400$	0	0
$401$	−15.9732	−0.797663	−0.398832	−	0.917024i	$-0.630584\pi$
$401$	−15.9732	−0.797663	−0.398832	+	0.917024i	$0.630584\pi$
$402$	0	0
$403$	42.9316	2.13858
$404$	0	0
$405$	0.754498	0.0374913
$406$	0	0
$407$	−35.5050	−1.75992
$408$	0	0
$409$	−5.71498	−0.282587	−0.141294	−	0.989968i	$-0.545126\pi$
$409$	−5.71498	−0.282587	−0.141294	+	0.989968i	$0.545126\pi$
$410$	0	0
$411$	−2.03806	−0.100530
$412$	0	0
$413$	−25.9164	−1.27526
$414$	0	0
$415$	7.89572	0.387586
$416$	0	0
$417$	−2.61705	−0.128158
$418$	0	0
$419$	11.8180	0.577349	0.288675	−	0.957427i	$-0.406785\pi$
$419$	11.8180	0.577349	0.288675	+	0.957427i	$0.406785\pi$
$420$	0	0
$421$	17.6267	0.859073	0.429537	−	0.903049i	$-0.358677\pi$
$421$	17.6267	0.859073	0.429537	+	0.903049i	$0.358677\pi$
$422$	0	0
$423$	−10.3713	−0.504268
$424$	0	0
$425$	23.6777	1.14854
$426$	0	0
$427$	−37.0719	−1.79404
$428$	0	0
$429$	−16.2726	−0.785651
$430$	0	0
$431$	30.5333	1.47074	0.735369	−	0.677667i	$-0.237009\pi$
$431$	30.5333	1.47074	0.735369	+	0.677667i	$0.237009\pi$
$432$	0	0
$433$	30.9147	1.48567	0.742833	−	0.669477i	$-0.233482\pi$
$433$	30.9147	1.48567	0.742833	+	0.669477i	$0.233482\pi$
$434$	0	0
$435$	4.80423	0.230345
$436$	0	0
$437$	11.3092	0.540992
$438$	0	0
$439$	−3.50027	−0.167059	−0.0835294	−	0.996505i	$-0.526619\pi$
$439$	−3.50027	−0.167059	−0.0835294	+	0.996505i	$0.526619\pi$
$440$	0	0
$441$	9.26663	0.441268
$442$	0	0
$443$	10.5864	0.502977	0.251489	−	0.967860i	$-0.419080\pi$
$443$	10.5864	0.502977	0.251489	+	0.967860i	$0.419080\pi$
$444$	0	0
$445$	5.38174	0.255119
$446$	0	0
$447$	0.999867	0.0472921
$448$	0	0
$449$	−19.7803	−0.933491	−0.466745	−	0.884392i	$-0.654574\pi$
$449$	−19.7803	−0.933491	−0.466745	+	0.884392i	$0.654574\pi$
$450$	0	0
$451$	−10.0492	−0.473199
$452$	0	0
$453$	−1.18736	−0.0557869
$454$	0	0
$455$	12.9516	0.607181
$456$	0	0
$457$	25.1346	1.17575	0.587875	−	0.808952i	$-0.299965\pi$
$457$	25.1346	1.17575	0.587875	+	0.808952i	$0.299965\pi$
$458$	0	0
$459$	5.34396	0.249435
$460$	0	0
$461$	6.56388	0.305711	0.152855	−	0.988249i	$-0.451153\pi$
$461$	6.56388	0.305711	0.152855	+	0.988249i	$0.451153\pi$
$462$	0	0
$463$	10.4817	0.487125	0.243562	−	0.969885i	$-0.421684\pi$
$463$	10.4817	0.487125	0.243562	+	0.969885i	$0.421684\pi$
$464$	0	0
$465$	7.61058	0.352932
$466$	0	0
$467$	−1.91751	−0.0887317	−0.0443659	−	0.999015i	$-0.514127\pi$
$467$	−1.91751	−0.0887317	−0.0443659	+	0.999015i	$0.514127\pi$
$468$	0	0
$469$	27.0799	1.25043
$470$	0	0
$471$	−2.87492	−0.132469
$472$	0	0
$473$	−27.7832	−1.27747
$474$	0	0
$475$	13.5994	0.623982
$476$	0	0
$477$	10.2864	0.470983
$478$	0	0
$479$	−13.4291	−0.613591	−0.306796	−	0.951775i	$-0.599257\pi$
$479$	−13.4291	−0.613591	−0.306796	+	0.951775i	$0.599257\pi$
$480$	0	0
$481$	−39.5246	−1.80217
$482$	0	0
$483$	−14.8606	−0.676182
$484$	0	0
$485$	−9.31467	−0.422957
$486$	0	0
$487$	−5.47244	−0.247980	−0.123990	−	0.992283i	$-0.539569\pi$
$487$	−5.47244	−0.247980	−0.123990	+	0.992283i	$0.539569\pi$
$488$	0	0
$489$	15.3943	0.696156
$490$	0	0
$491$	−25.7816	−1.16351	−0.581754	−	0.813365i	$-0.697634\pi$
$491$	−25.7816	−1.16351	−0.581754	+	0.813365i	$0.697634\pi$
$492$	0	0
$493$	34.0275	1.53252
$494$	0	0
$495$	−2.88468	−0.129657
$496$	0	0
$497$	3.76719	0.168982
$498$	0	0
$499$	33.2044	1.48643	0.743217	−	0.669051i	$-0.233299\pi$
$499$	33.2044	1.48643	0.743217	+	0.669051i	$0.233299\pi$
$500$	0	0
$501$	10.8494	0.484717
$502$	0	0
$503$	−1.00000	−0.0445878
$503$	−1.00000	−0.0445878
$504$	0	0
$505$	4.33350	0.192838
$506$	0	0
$507$	−5.11487	−0.227159
$508$	0	0
$509$	9.72689	0.431137	0.215568	−	0.976489i	$-0.430840\pi$
$509$	9.72689	0.431137	0.215568	+	0.976489i	$0.430840\pi$
$510$	0	0
$511$	51.1411	2.26235
$512$	0	0
$513$	3.06933	0.135514
$514$	0	0
$515$	−5.15926	−0.227344
$516$	0	0
$517$	39.6526	1.74392
$518$	0	0
$519$	−8.55402	−0.375480
$520$	0	0
$521$	37.5389	1.64461	0.822305	−	0.569047i	$-0.192688\pi$
$521$	37.5389	1.64461	0.822305	+	0.569047i	$0.192688\pi$
$522$	0	0
$523$	39.2063	1.71437	0.857185	−	0.515009i	$-0.172211\pi$
$523$	39.2063	1.71437	0.857185	+	0.515009i	$0.172211\pi$
$524$	0	0
$525$	−17.8700	−0.779910
$526$	0	0
$527$	53.9043	2.34811
$528$	0	0
$529$	−9.42386	−0.409733
$530$	0	0
$531$	6.42578	0.278855
$532$	0	0
$533$	−11.1869	−0.484558
$534$	0	0
$535$	−4.08508	−0.176614
$536$	0	0
$537$	22.5055	0.971184
$538$	0	0
$539$	−35.4293	−1.52605
$540$	0	0
$541$	−1.64741	−0.0708278	−0.0354139	−	0.999373i	$-0.511275\pi$
$541$	−1.64741	−0.0708278	−0.0354139	+	0.999373i	$0.511275\pi$
$542$	0	0
$543$	−24.1911	−1.03814
$544$	0	0
$545$	−11.9771	−0.513043
$546$	0	0
$547$	−33.6132	−1.43720	−0.718598	−	0.695425i	$-0.755216\pi$
$547$	−33.6132	−1.43720	−0.718598	+	0.695425i	$0.755216\pi$
$548$	0	0
$549$	9.19171	0.392293
$550$	0	0
$551$	19.5438	0.832595
$552$	0	0
$553$	0.686264	0.0291829
$554$	0	0
$555$	−7.00660	−0.297414
$556$	0	0
$557$	−23.9974	−1.01680	−0.508402	−	0.861120i	$-0.669763\pi$
$557$	−23.9974	−1.01680	−0.508402	+	0.861120i	$0.669763\pi$
$558$	0	0
$559$	−30.9285	−1.30814
$560$	0	0
$561$	−20.4317	−0.862626
$562$	0	0
$563$	−22.3031	−0.939965	−0.469983	−	0.882676i	$-0.655740\pi$
$563$	−22.3031	−0.939965	−0.469983	+	0.882676i	$0.655740\pi$
$564$	0	0
$565$	−1.82820	−0.0769128
$566$	0	0
$567$	−4.03319	−0.169378
$568$	0	0
$569$	−40.6422	−1.70381	−0.851905	−	0.523696i	$-0.824553\pi$
$569$	−40.6422	−1.70381	−0.851905	+	0.523696i	$0.824553\pi$
$570$	0	0
$571$	33.4244	1.39877	0.699385	−	0.714746i	$-0.253458\pi$
$571$	33.4244	1.39877	0.699385	+	0.714746i	$0.253458\pi$
$572$	0	0
$573$	9.29327	0.388232
$574$	0	0
$575$	16.3254	0.680816
$576$	0	0
$577$	−28.7613	−1.19735	−0.598674	−	0.800993i	$-0.704306\pi$
$577$	−28.7613	−1.19735	−0.598674	+	0.800993i	$0.704306\pi$
$578$	0	0
$579$	11.8415	0.492115
$580$	0	0
$581$	−42.2068	−1.75103
$582$	0	0
$583$	−39.3283	−1.62881
$584$	0	0
$585$	−3.21126	−0.132769
$586$	0	0
$587$	−11.3654	−0.469102	−0.234551	−	0.972104i	$-0.575362\pi$
$587$	−11.3654	−0.469102	−0.234551	+	0.972104i	$0.575362\pi$
$588$	0	0
$589$	30.9602	1.27569
$590$	0	0
$591$	−10.2965	−0.423543
$592$	0	0
$593$	−21.3119	−0.875177	−0.437588	−	0.899175i	$-0.644167\pi$
$593$	−21.3119	−0.875177	−0.437588	+	0.899175i	$0.644167\pi$
$594$	0	0
$595$	16.2619	0.666671
$596$	0	0
$597$	−25.1760	−1.03039
$598$	0	0
$599$	−7.58424	−0.309884	−0.154942	−	0.987924i	$-0.549519\pi$
$599$	−7.58424	−0.309884	−0.154942	+	0.987924i	$0.549519\pi$
$600$	0	0
$601$	35.3001	1.43992	0.719960	−	0.694015i	$-0.244160\pi$
$601$	35.3001	1.43992	0.719960	+	0.694015i	$0.244160\pi$
$602$	0	0
$603$	−6.71426	−0.273426
$604$	0	0
$605$	2.72959	0.110974
$606$	0	0
$607$	16.6902	0.677435	0.338718	−	0.940888i	$-0.390007\pi$
$607$	16.6902	0.677435	0.338718	+	0.940888i	$0.390007\pi$
$608$	0	0
$609$	−25.6812	−1.04065
$610$	0	0
$611$	44.1417	1.78578
$612$	0	0
$613$	−2.92515	−0.118146	−0.0590729	−	0.998254i	$-0.518814\pi$
$613$	−2.92515	−0.118146	−0.0590729	+	0.998254i	$0.518814\pi$
$614$	0	0
$615$	−1.98312	−0.0799672
$616$	0	0
$617$	−15.7400	−0.633670	−0.316835	−	0.948481i	$-0.602620\pi$
$617$	−15.7400	−0.633670	−0.316835	+	0.948481i	$0.602620\pi$
$618$	0	0
$619$	7.76427	0.312072	0.156036	−	0.987751i	$-0.450128\pi$
$619$	7.76427	0.312072	0.156036	+	0.987751i	$0.450128\pi$
$620$	0	0
$621$	3.68458	0.147857
$622$	0	0
$623$	−28.7682	−1.15258
$624$	0	0
$625$	16.7851	0.671403
$626$	0	0
$627$	−11.7350	−0.468651
$628$	0	0
$629$	−49.6264	−1.97874
$630$	0	0
$631$	14.0859	0.560749	0.280375	−	0.959891i	$-0.409541\pi$
$631$	14.0859	0.560749	0.280375	+	0.959891i	$0.409541\pi$
$632$	0	0
$633$	16.7238	0.664713
$634$	0	0
$635$	9.49333	0.376731
$636$	0	0
$637$	−39.4402	−1.56268
$638$	0	0
$639$	−0.934048	−0.0369504
$640$	0	0
$641$	−8.77631	−0.346643	−0.173322	−	0.984865i	$-0.555450\pi$
$641$	−8.77631	−0.346643	−0.173322	+	0.984865i	$0.555450\pi$
$642$	0	0
$643$	2.86752	0.113084	0.0565419	−	0.998400i	$-0.481993\pi$
$643$	2.86752	0.113084	0.0565419	+	0.998400i	$0.481993\pi$
$644$	0	0
$645$	−5.48276	−0.215883
$646$	0	0
$647$	−25.7123	−1.01085	−0.505427	−	0.862869i	$-0.668665\pi$
$647$	−25.7123	−1.01085	−0.505427	+	0.862869i	$0.668665\pi$
$648$	0	0
$649$	−24.5678	−0.964370
$650$	0	0
$651$	−40.6826	−1.59448
$652$	0	0
$653$	38.3735	1.50167	0.750835	−	0.660490i	$-0.229651\pi$
$653$	38.3735	1.50167	0.750835	+	0.660490i	$0.229651\pi$
$654$	0	0
$655$	0.841081	0.0328637
$656$	0	0
$657$	−12.6801	−0.494696
$658$	0	0
$659$	27.4948	1.07105	0.535523	−	0.844521i	$-0.320115\pi$
$659$	27.4948	1.07105	0.535523	+	0.844521i	$0.320115\pi$
$660$	0	0
$661$	31.4337	1.22263	0.611314	−	0.791388i	$-0.290641\pi$
$661$	31.4337	1.22263	0.611314	+	0.791388i	$0.290641\pi$
$662$	0	0
$663$	−22.7448	−0.883333
$664$	0	0
$665$	9.34006	0.362192
$666$	0	0
$667$	23.4614	0.908430
$668$	0	0
$669$	−6.83098	−0.264101
$670$	0	0
$671$	−35.1429	−1.35667
$672$	0	0
$673$	−28.9440	−1.11571	−0.557855	−	0.829939i	$-0.688375\pi$
$673$	−28.9440	−1.11571	−0.557855	+	0.829939i	$0.688375\pi$
$674$	0	0
$675$	4.43073	0.170539
$676$	0	0
$677$	−11.5512	−0.443950	−0.221975	−	0.975052i	$-0.571250\pi$
$677$	−11.5512	−0.443950	−0.221975	+	0.975052i	$0.571250\pi$
$678$	0	0
$679$	49.7918	1.91084
$680$	0	0
$681$	27.7479	1.06330
$682$	0	0
$683$	−27.8711	−1.06646	−0.533229	−	0.845971i	$-0.679022\pi$
$683$	−27.8711	−1.06646	−0.533229	+	0.845971i	$0.679022\pi$
$684$	0	0
$685$	1.53771	0.0587530
$686$	0	0
$687$	8.72641	0.332933
$688$	0	0
$689$	−43.7807	−1.66791
$690$	0	0
$691$	−3.75815	−0.142967	−0.0714834	−	0.997442i	$-0.522773\pi$
$691$	−3.75815	−0.142967	−0.0714834	+	0.997442i	$0.522773\pi$
$692$	0	0
$693$	15.4202	0.585764
$694$	0	0
$695$	1.97456	0.0748993
$696$	0	0
$697$	−14.0461	−0.532033
$698$	0	0
$699$	12.2730	0.464207
$700$	0	0
$701$	5.02223	0.189687	0.0948435	−	0.995492i	$-0.469765\pi$
$701$	5.02223	0.189687	0.0948435	+	0.995492i	$0.469765\pi$
$702$	0	0
$703$	−28.5031	−1.07502
$704$	0	0
$705$	7.82509	0.294710
$706$	0	0
$707$	−23.1648	−0.871204
$708$	0	0
$709$	36.4816	1.37009	0.685047	−	0.728499i	$-0.259782\pi$
$709$	36.4816	1.37009	0.685047	+	0.728499i	$0.259782\pi$
$710$	0	0
$711$	−0.170154	−0.00638128
$712$	0	0
$713$	37.1662	1.39188
$714$	0	0
$715$	12.2777	0.459159
$716$	0	0
$717$	18.0522	0.674173
$718$	0	0
$719$	43.9651	1.63962	0.819812	−	0.572633i	$-0.194078\pi$
$719$	43.9651	1.63962	0.819812	+	0.572633i	$0.194078\pi$
$720$	0	0
$721$	27.5790	1.02709
$722$	0	0
$723$	−25.8028	−0.959618
$724$	0	0
$725$	28.2125	1.04779
$726$	0	0
$727$	−9.53527	−0.353643	−0.176822	−	0.984243i	$-0.556582\pi$
$727$	−9.53527	−0.353643	−0.176822	+	0.984243i	$0.556582\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−38.8334	−1.43630
$732$	0	0
$733$	−33.9700	−1.25471	−0.627355	−	0.778733i	$-0.715863\pi$
$733$	−33.9700	−1.25471	−0.627355	+	0.778733i	$0.715863\pi$
$734$	0	0
$735$	−6.99165	−0.257891
$736$	0	0
$737$	25.6708	0.945594
$738$	0	0
$739$	−32.4665	−1.19430	−0.597150	−	0.802130i	$-0.703700\pi$
$739$	−32.4665	−1.19430	−0.597150	+	0.802130i	$0.703700\pi$
$740$	0	0
$741$	−13.0635	−0.479901
$742$	0	0
$743$	−34.4226	−1.26284	−0.631422	−	0.775440i	$-0.717528\pi$
$743$	−34.4226	−1.26284	−0.631422	+	0.775440i	$0.717528\pi$
$744$	0	0
$745$	−0.754397	−0.0276390
$746$	0	0
$747$	10.4649	0.382890
$748$	0	0
$749$	21.8369	0.797905
$750$	0	0
$751$	6.75620	0.246537	0.123269	−	0.992373i	$-0.460662\pi$
$751$	6.75620	0.246537	0.123269	+	0.992373i	$0.460662\pi$
$752$	0	0
$753$	−25.0414	−0.912559
$754$	0	0
$755$	0.895857	0.0326036
$756$	0	0
$757$	26.7652	0.972797	0.486398	−	0.873737i	$-0.338310\pi$
$757$	26.7652	0.972797	0.486398	+	0.873737i	$0.338310\pi$
$758$	0	0
$759$	−14.0873	−0.511338
$760$	0	0
$761$	2.22026	0.0804843	0.0402422	−	0.999190i	$-0.487187\pi$
$761$	2.22026	0.0804843	0.0402422	+	0.999190i	$0.487187\pi$
$762$	0	0
$763$	64.0240	2.31783
$764$	0	0
$765$	−4.03201	−0.145778
$766$	0	0
$767$	−27.3491	−0.987519
$768$	0	0
$769$	15.8742	0.572438	0.286219	−	0.958164i	$-0.407601\pi$
$769$	15.8742	0.572438	0.286219	+	0.958164i	$0.407601\pi$
$770$	0	0
$771$	−14.6320	−0.526957
$772$	0	0
$773$	41.1692	1.48075	0.740377	−	0.672192i	$-0.234647\pi$
$773$	41.1692	1.48075	0.740377	+	0.672192i	$0.234647\pi$
$774$	0	0
$775$	44.6926	1.60540
$776$	0	0
$777$	37.4540	1.34366
$778$	0	0
$779$	−8.06742	−0.289045
$780$	0	0
$781$	3.57116	0.127786
$782$	0	0
$783$	6.36746	0.227554
$784$	0	0
$785$	2.16912	0.0774192
$786$	0	0
$787$	−13.3361	−0.475380	−0.237690	−	0.971341i	$-0.576390\pi$
$787$	−13.3361	−0.475380	−0.237690	+	0.971341i	$0.576390\pi$
$788$	0	0
$789$	27.1264	0.965725
$790$	0	0
$791$	9.77269	0.347477
$792$	0	0
$793$	−39.1214	−1.38924
$794$	0	0
$795$	−7.76109	−0.275257
$796$	0	0
$797$	−39.2316	−1.38965	−0.694827	−	0.719177i	$-0.744519\pi$
$797$	−39.2316	−1.38965	−0.694827	+	0.719177i	$0.744519\pi$
$798$	0	0
$799$	55.4236	1.96075
$800$	0	0
$801$	7.13287	0.252028
$802$	0	0
$803$	48.4799	1.71082
$804$	0	0
$805$	11.2123	0.395182
$806$	0	0
$807$	15.5230	0.546437
$808$	0	0
$809$	43.5715	1.53189	0.765947	−	0.642904i	$-0.222271\pi$
$809$	43.5715	1.53189	0.765947	+	0.642904i	$0.222271\pi$
$810$	0	0
$811$	14.9480	0.524897	0.262448	−	0.964946i	$-0.415470\pi$
$811$	14.9480	0.524897	0.262448	+	0.964946i	$0.415470\pi$
$812$	0	0
$813$	3.76129	0.131914
$814$	0	0
$815$	−11.6150	−0.406855
$816$	0	0
$817$	−22.3041	−0.780322
$818$	0	0
$819$	17.1659	0.599825
$820$	0	0
$821$	−49.1473	−1.71525	−0.857626	−	0.514274i	$-0.828061\pi$
$821$	−49.1473	−1.71525	−0.857626	+	0.514274i	$0.828061\pi$
$822$	0	0
$823$	30.1302	1.05027	0.525137	−	0.851018i	$-0.324014\pi$
$823$	30.1302	1.05027	0.525137	+	0.851018i	$0.324014\pi$
$824$	0	0
$825$	−16.9401	−0.589779
$826$	0	0
$827$	28.8177	1.00209	0.501045	−	0.865421i	$-0.332949\pi$
$827$	28.8177	1.00209	0.501045	+	0.865421i	$0.332949\pi$
$828$	0	0
$829$	−26.4605	−0.919010	−0.459505	−	0.888175i	$-0.651973\pi$
$829$	−26.4605	−0.919010	−0.459505	+	0.888175i	$0.651973\pi$
$830$	0	0
$831$	7.82723	0.271524
$832$	0	0
$833$	−49.5205	−1.71578
$834$	0	0
$835$	−8.18586	−0.283283
$836$	0	0
$837$	10.0869	0.348656
$838$	0	0
$839$	−23.6556	−0.816680	−0.408340	−	0.912830i	$-0.633892\pi$
$839$	−23.6556	−0.816680	−0.408340	+	0.912830i	$0.633892\pi$
$840$	0	0
$841$	11.5445	0.398087
$842$	0	0
$843$	11.0898	0.381954
$844$	0	0
$845$	3.85916	0.132759
$846$	0	0
$847$	−14.5911	−0.501357
$848$	0	0
$849$	28.4603	0.976755
$850$	0	0
$851$	−34.2167	−1.17293
$852$	0	0
$853$	−29.0652	−0.995174	−0.497587	−	0.867414i	$-0.665780\pi$
$853$	−29.0652	−0.995174	−0.497587	+	0.867414i	$0.665780\pi$
$854$	0	0
$855$	−2.31580	−0.0791987
$856$	0	0
$857$	−6.11549	−0.208901	−0.104451	−	0.994530i	$-0.533308\pi$
$857$	−6.11549	−0.208901	−0.104451	+	0.994530i	$0.533308\pi$
$858$	0	0
$859$	−18.6271	−0.635548	−0.317774	−	0.948166i	$-0.602935\pi$
$859$	−18.6271	−0.635548	−0.317774	+	0.948166i	$0.602935\pi$
$860$	0	0
$861$	10.6008	0.361276
$862$	0	0
$863$	−55.7750	−1.89860	−0.949302	−	0.314366i	$-0.898208\pi$
$863$	−55.7750	−1.89860	−0.949302	+	0.314366i	$0.898208\pi$
$864$	0	0
$865$	6.45399	0.219442
$866$	0	0
$867$	−11.5580	−0.392529
$868$	0	0
$869$	0.650554	0.0220685
$870$	0	0
$871$	28.5769	0.968293
$872$	0	0
$873$	−12.3455	−0.417833
$874$	0	0
$875$	28.6980	0.970170
$876$	0	0
$877$	−47.5083	−1.60424	−0.802121	−	0.597162i	$-0.796295\pi$
$877$	−47.5083	−1.60424	−0.802121	+	0.597162i	$0.796295\pi$
$878$	0	0
$879$	31.1011	1.04901
$880$	0	0
$881$	2.62666	0.0884944	0.0442472	−	0.999021i	$-0.485911\pi$
$881$	2.62666	0.0884944	0.0442472	+	0.999021i	$0.485911\pi$
$882$	0	0
$883$	3.16601	0.106545	0.0532724	−	0.998580i	$-0.483035\pi$
$883$	3.16601	0.106545	0.0532724	+	0.998580i	$0.483035\pi$
$884$	0	0
$885$	−4.84823	−0.162972
$886$	0	0
$887$	−15.4145	−0.517569	−0.258785	−	0.965935i	$-0.583322\pi$
$887$	−15.4145	−0.517569	−0.258785	+	0.965935i	$0.583322\pi$
$888$	0	0
$889$	−50.7469	−1.70200
$890$	0	0
$891$	−3.82332	−0.128086
$892$	0	0
$893$	31.8328	1.06524
$894$	0	0
$895$	−16.9803	−0.567590
$896$	0	0
$897$	−15.6822	−0.523612
$898$	0	0
$899$	64.2282	2.14213
$900$	0	0
$901$	−54.9703	−1.83133
$902$	0	0
$903$	29.3083	0.975318
$904$	0	0
$905$	18.2522	0.606722
$906$	0	0
$907$	48.5123	1.61083	0.805413	−	0.592714i	$-0.201944\pi$
$907$	48.5123	1.61083	0.805413	+	0.592714i	$0.201944\pi$
$908$	0	0
$909$	5.74355	0.190502
$910$	0	0
$911$	3.30195	0.109398	0.0546992	−	0.998503i	$-0.482580\pi$
$911$	3.30195	0.109398	0.0546992	+	0.998503i	$0.482580\pi$
$912$	0	0
$913$	−40.0105	−1.32416
$914$	0	0
$915$	−6.93513	−0.229268
$916$	0	0
$917$	−4.49602	−0.148472
$918$	0	0
$919$	16.4454	0.542482	0.271241	−	0.962511i	$-0.412566\pi$
$919$	16.4454	0.542482	0.271241	+	0.962511i	$0.412566\pi$
$920$	0	0
$921$	−22.1359	−0.729402
$922$	0	0
$923$	3.97545	0.130854
$924$	0	0
$925$	−41.1458	−1.35286
$926$	0	0
$927$	−6.83800	−0.224589
$928$	0	0
$929$	4.67293	0.153314	0.0766569	−	0.997058i	$-0.475575\pi$
$929$	4.67293	0.153314	0.0766569	+	0.997058i	$0.475575\pi$
$930$	0	0
$931$	−28.4423	−0.932159
$932$	0	0
$933$	20.5369	0.672348
$934$	0	0
$935$	15.4156	0.504146
$936$	0	0
$937$	−36.3883	−1.18875	−0.594376	−	0.804187i	$-0.702601\pi$
$937$	−36.3883	−1.18875	−0.594376	+	0.804187i	$0.702601\pi$
$938$	0	0
$939$	−16.4172	−0.535756
$940$	0	0
$941$	41.0935	1.33961	0.669805	−	0.742537i	$-0.266378\pi$
$941$	41.0935	1.33961	0.669805	+	0.742537i	$0.266378\pi$
$942$	0	0
$943$	−9.68456	−0.315373
$944$	0	0
$945$	3.04303	0.0989899
$946$	0	0
$947$	−27.5567	−0.895473	−0.447737	−	0.894165i	$-0.647770\pi$
$947$	−27.5567	−0.895473	−0.447737	+	0.894165i	$0.647770\pi$
$948$	0	0
$949$	53.9683	1.75189
$950$	0	0
$951$	9.49952	0.308043
$952$	0	0
$953$	23.2999	0.754758	0.377379	−	0.926059i	$-0.376825\pi$
$953$	23.2999	0.754758	0.377379	+	0.926059i	$0.376825\pi$
$954$	0	0
$955$	−7.01175	−0.226895
$956$	0	0
$957$	−24.3448	−0.786956
$958$	0	0
$959$	−8.21989	−0.265434
$960$	0	0
$961$	70.7465	2.28215
$962$	0	0
$963$	−5.41431	−0.174474
$964$	0	0
$965$	−8.93436	−0.287607
$966$	0	0
$967$	46.7910	1.50470	0.752349	−	0.658765i	$-0.228921\pi$
$967$	46.7910	1.50470	0.752349	+	0.658765i	$0.228921\pi$
$968$	0	0
$969$	−16.4024	−0.526920
$970$	0	0
$971$	−38.9878	−1.25118	−0.625590	−	0.780152i	$-0.715142\pi$
$971$	−38.9878	−1.25118	−0.625590	+	0.780152i	$0.715142\pi$
$972$	0	0
$973$	−10.5551	−0.338380
$974$	0	0
$975$	−18.8579	−0.603936
$976$	0	0
$977$	−19.9510	−0.638289	−0.319144	−	0.947706i	$-0.603396\pi$
$977$	−19.9510	−0.638289	−0.319144	+	0.947706i	$0.603396\pi$
$978$	0	0
$979$	−27.2712	−0.871593
$980$	0	0
$981$	−15.8743	−0.506827
$982$	0	0
$983$	−29.4924	−0.940663	−0.470331	−	0.882490i	$-0.655866\pi$
$983$	−29.4924	−0.940663	−0.470331	+	0.882490i	$0.655866\pi$
$984$	0	0
$985$	7.76871	0.247532
$986$	0	0
$987$	−41.8293	−1.33144
$988$	0	0
$989$	−26.7750	−0.851396
$990$	0	0
$991$	−43.7780	−1.39065	−0.695327	−	0.718693i	$-0.744741\pi$
$991$	−43.7780	−1.39065	−0.695327	+	0.718693i	$0.744741\pi$
$992$	0	0
$993$	21.9003	0.694984
$994$	0	0
$995$	18.9953	0.602190
$996$	0	0
$997$	21.0399	0.666339	0.333170	−	0.942867i	$-0.391882\pi$
$997$	21.0399	0.666339	0.333170	+	0.942867i	$0.391882\pi$
$998$	0	0
$999$	−9.28645	−0.293810

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6036.2.a.i.1.15	✓	26

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6036.2.a.i.1.15	✓	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +0.754498 q^{5} -4.03319 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +0.754498 q^{5} -4.03319 q^{7} +1.00000 q^{9} -3.82332 q^{11} -4.25616 q^{13} -0.754498 q^{15} -5.34396 q^{17} -3.06933 q^{19} +4.03319 q^{21} -3.68458 q^{23} -4.43073 q^{25} -1.00000 q^{27} -6.36746 q^{29} -10.0869 q^{31} +3.82332 q^{33} -3.04303 q^{35} +9.28645 q^{37} +4.25616 q^{39} +2.62840 q^{41} +7.26677 q^{43} +0.754498 q^{45} -10.3713 q^{47} +9.26663 q^{49} +5.34396 q^{51} +10.2864 q^{53} -2.88468 q^{55} +3.06933 q^{57} +6.42578 q^{59} +9.19171 q^{61} -4.03319 q^{63} -3.21126 q^{65} -6.71426 q^{67} +3.68458 q^{69} -0.934048 q^{71} -12.6801 q^{73} +4.43073 q^{75} +15.4202 q^{77} -0.170154 q^{79} +1.00000 q^{81} +10.4649 q^{83} -4.03201 q^{85} +6.36746 q^{87} +7.13287 q^{89} +17.1659 q^{91} +10.0869 q^{93} -2.31580 q^{95} -12.3455 q^{97} -3.82332 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 26 q - 26 q^{3} + 6 q^{5} + 5 q^{7} + 26 q^{9}+O(q^{10}) \) 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

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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