LMFDB - Embedded newform 6036.2.a.g.1.12

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:	$1$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - 4 x^{14} - 27 x^{13} + 106 x^{12} + 266 x^{11} - 1004 x^{10} - 1105 x^{9} + 4076 x^{8} + \cdots - 44 $ x^15 - 4x^14 - 27x^13 + 106x^12 + 266x^11 - 1004x^10 - 1105x^9 + 4076x^8 + 1501x^7 - 7100x^6 - 134x^5 + 5356x^4 - 1041x^3 - 1381x^2 + 543x - 44
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 3 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Root			$1.40631$ of defining polynomial
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)

$f(q)$	$=$	$q+1.00000 q^{3} +0.406311 q^{5} -3.65793 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +0.406311 q^{5} -3.65793 q^{7} +1.00000 q^{9} +2.14503 q^{11} +3.44501 q^{13} +0.406311 q^{15} -3.83373 q^{17} -3.34349 q^{19} -3.65793 q^{21} -3.64859 q^{23} -4.83491 q^{25} +1.00000 q^{27} +9.24385 q^{29} +1.08272 q^{31} +2.14503 q^{33} -1.48626 q^{35} -2.54793 q^{37} +3.44501 q^{39} -8.54300 q^{41} +9.55542 q^{43} +0.406311 q^{45} +0.898817 q^{47} +6.38045 q^{49} -3.83373 q^{51} -1.91334 q^{53} +0.871548 q^{55} -3.34349 q^{57} +0.219503 q^{59} -11.3053 q^{61} -3.65793 q^{63} +1.39975 q^{65} -7.20894 q^{67} -3.64859 q^{69} -7.59769 q^{71} -13.9530 q^{73} -4.83491 q^{75} -7.84637 q^{77} -2.82794 q^{79} +1.00000 q^{81} +13.0368 q^{83} -1.55768 q^{85} +9.24385 q^{87} +5.24068 q^{89} -12.6016 q^{91} +1.08272 q^{93} -1.35849 q^{95} +3.08960 q^{97} +2.14503 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q + 15 q^{3} - 11 q^{5} - 4 q^{7} + 15 q^{9}+O(q^{10}) $ 15 * q + 15 * q^3 - 11 * q^5 - 4 * q^7 + 15 * q^9	$ 15 q + 15 q^{3} - 11 q^{5} - 4 q^{7} + 15 q^{9} - 5 q^{11} - 14 q^{13} - 11 q^{15} - 5 q^{17} + 3 q^{19} - 4 q^{21} - 32 q^{23} + 2 q^{25} + 15 q^{27} - 23 q^{29} - 13 q^{31} - 5 q^{33} - 16 q^{35} - 10 q^{37} - 14 q^{39} - 14 q^{41} + 4 q^{43} - 11 q^{45} - 20 q^{47} - 9 q^{49} - 5 q^{51} - 30 q^{53} - 10 q^{55} + 3 q^{57} - 14 q^{59} - 38 q^{61} - 4 q^{63} - 24 q^{65} - 8 q^{67} - 32 q^{69} - 41 q^{71} - 19 q^{73} + 2 q^{75} - 39 q^{77} - 27 q^{79} + 15 q^{81} - 17 q^{83} - 6 q^{85} - 23 q^{87} - 23 q^{89} + 4 q^{91} - 13 q^{93} - 30 q^{95} - 18 q^{97} - 5 q^{99}+O(q^{100}) $ 15 * q + 15 * q^3 - 11 * q^5 - 4 * q^7 + 15 * q^9 - 5 * q^11 - 14 * q^13 - 11 * q^15 - 5 * q^17 + 3 * q^19 - 4 * q^21 - 32 * q^23 + 2 * q^25 + 15 * q^27 - 23 * q^29 - 13 * q^31 - 5 * q^33 - 16 * q^35 - 10 * q^37 - 14 * q^39 - 14 * q^41 + 4 * q^43 - 11 * q^45 - 20 * q^47 - 9 * q^49 - 5 * q^51 - 30 * q^53 - 10 * q^55 + 3 * q^57 - 14 * q^59 - 38 * q^61 - 4 * q^63 - 24 * q^65 - 8 * q^67 - 32 * q^69 - 41 * q^71 - 19 * q^73 + 2 * q^75 - 39 * q^77 - 27 * q^79 + 15 * q^81 - 17 * q^83 - 6 * q^85 - 23 * q^87 - 23 * q^89 + 4 * q^91 - 13 * q^93 - 30 * q^95 - 18 * q^97 - 5 * 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
$5$	0.406311	0.181708	0.0908538	−	0.995864i	$-0.471040\pi$
$5$	0.406311	0.181708	0.0908538	+	0.995864i	$0.471040\pi$
$6$	0	0
$7$	−3.65793	−1.38257	−0.691284	−	0.722583i	$-0.742955\pi$
$7$	−3.65793	−1.38257	−0.691284	+	0.722583i	$0.742955\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	2.14503	0.646751	0.323375	−	0.946271i	$-0.395182\pi$
$11$	2.14503	0.646751	0.323375	+	0.946271i	$0.395182\pi$
$12$	0	0
$13$	3.44501	0.955475	0.477737	−	0.878503i	$-0.341457\pi$
$13$	3.44501	0.955475	0.477737	+	0.878503i	$0.341457\pi$
$14$	0	0
$15$	0.406311	0.104909
$16$	0	0
$17$	−3.83373	−0.929816	−0.464908	−	0.885359i	$-0.653913\pi$
$17$	−3.83373	−0.929816	−0.464908	+	0.885359i	$0.653913\pi$
$18$	0	0
$19$	−3.34349	−0.767049	−0.383524	−	0.923531i	$-0.625290\pi$
$19$	−3.34349	−0.767049	−0.383524	+	0.923531i	$0.625290\pi$
$20$	0	0
$21$	−3.65793	−0.798226
$22$	0	0
$23$	−3.64859	−0.760783	−0.380391	−	0.924826i	$-0.624211\pi$
$23$	−3.64859	−0.760783	−0.380391	+	0.924826i	$0.624211\pi$
$24$	0	0
$25$	−4.83491	−0.966982
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	9.24385	1.71654	0.858270	−	0.513199i	$-0.171540\pi$
$29$	9.24385	1.71654	0.858270	+	0.513199i	$0.171540\pi$
$30$	0	0
$31$	1.08272	0.194463	0.0972315	−	0.995262i	$-0.469001\pi$
$31$	1.08272	0.194463	0.0972315	+	0.995262i	$0.469001\pi$
$32$	0	0
$33$	2.14503	0.373402
$34$	0	0
$35$	−1.48626	−0.251223
$36$	0	0
$37$	−2.54793	−0.418876	−0.209438	−	0.977822i	$-0.567163\pi$
$37$	−2.54793	−0.418876	−0.209438	+	0.977822i	$0.567163\pi$
$38$	0	0
$39$	3.44501	0.551643
$40$	0	0
$41$	−8.54300	−1.33419	−0.667096	−	0.744972i	$-0.732463\pi$
$41$	−8.54300	−1.33419	−0.667096	+	0.744972i	$0.732463\pi$
$42$	0	0
$43$	9.55542	1.45719	0.728594	−	0.684946i	$-0.240174\pi$
$43$	9.55542	1.45719	0.728594	+	0.684946i	$0.240174\pi$
$44$	0	0
$45$	0.406311	0.0605692
$46$	0	0
$47$	0.898817	0.131106	0.0655530	−	0.997849i	$-0.479119\pi$
$47$	0.898817	0.131106	0.0655530	+	0.997849i	$0.479119\pi$
$48$	0	0
$49$	6.38045	0.911493
$50$	0	0
$51$	−3.83373	−0.536829
$52$	0	0
$53$	−1.91334	−0.262818	−0.131409	−	0.991328i	$-0.541950\pi$
$53$	−1.91334	−0.262818	−0.131409	+	0.991328i	$0.541950\pi$
$54$	0	0
$55$	0.871548	0.117520
$56$	0	0
$57$	−3.34349	−0.442856
$58$	0	0
$59$	0.219503	0.0285769	0.0142884	−	0.999898i	$-0.495452\pi$
$59$	0.219503	0.0285769	0.0142884	+	0.999898i	$0.495452\pi$
$60$	0	0
$61$	−11.3053	−1.44750	−0.723748	−	0.690064i	$-0.757582\pi$
$61$	−11.3053	−1.44750	−0.723748	+	0.690064i	$0.757582\pi$
$62$	0	0
$63$	−3.65793	−0.460856
$64$	0	0
$65$	1.39975	0.173617
$66$	0	0
$67$	−7.20894	−0.880712	−0.440356	−	0.897823i	$-0.645148\pi$
$67$	−7.20894	−0.880712	−0.440356	+	0.897823i	$0.645148\pi$
$68$	0	0
$69$	−3.64859	−0.439238
$70$	0	0
$71$	−7.59769	−0.901680	−0.450840	−	0.892605i	$-0.648876\pi$
$71$	−7.59769	−0.901680	−0.450840	+	0.892605i	$0.648876\pi$
$72$	0	0
$73$	−13.9530	−1.63308	−0.816538	−	0.577292i	$-0.804109\pi$
$73$	−13.9530	−1.63308	−0.816538	+	0.577292i	$0.804109\pi$
$74$	0	0
$75$	−4.83491	−0.558288
$76$	0	0
$77$	−7.84637	−0.894176
$78$	0	0
$79$	−2.82794	−0.318168	−0.159084	−	0.987265i	$-0.550854\pi$
$79$	−2.82794	−0.318168	−0.159084	+	0.987265i	$0.550854\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	13.0368	1.43097	0.715485	−	0.698628i	$-0.246206\pi$
$83$	13.0368	1.43097	0.715485	+	0.698628i	$0.246206\pi$
$84$	0	0
$85$	−1.55768	−0.168955
$86$	0	0
$87$	9.24385	0.991044
$88$	0	0
$89$	5.24068	0.555511	0.277756	−	0.960652i	$-0.410409\pi$
$89$	5.24068	0.555511	0.277756	+	0.960652i	$0.410409\pi$
$90$	0	0
$91$	−12.6016	−1.32101
$92$	0	0
$93$	1.08272	0.112273
$94$	0	0
$95$	−1.35849	−0.139379
$96$	0	0
$97$	3.08960	0.313701	0.156851	−	0.987622i	$-0.449866\pi$
$97$	3.08960	0.313701	0.156851	+	0.987622i	$0.449866\pi$
$98$	0	0
$99$	2.14503	0.215584
$100$	0	0
$101$	−8.72529	−0.868199	−0.434099	−	0.900865i	$-0.642933\pi$
$101$	−8.72529	−0.868199	−0.434099	+	0.900865i	$0.642933\pi$
$102$	0	0
$103$	12.0716	1.18945	0.594726	−	0.803928i	$-0.297260\pi$
$103$	12.0716	1.18945	0.594726	+	0.803928i	$0.297260\pi$
$104$	0	0
$105$	−1.48626	−0.145044
$106$	0	0
$107$	−13.3086	−1.28659	−0.643294	−	0.765619i	$-0.722433\pi$
$107$	−13.3086	−1.28659	−0.643294	+	0.765619i	$0.722433\pi$
$108$	0	0
$109$	−14.2247	−1.36248	−0.681240	−	0.732061i	$-0.738559\pi$
$109$	−14.2247	−1.36248	−0.681240	+	0.732061i	$0.738559\pi$
$110$	0	0
$111$	−2.54793	−0.241838
$112$	0	0
$113$	1.63423	0.153735	0.0768676	−	0.997041i	$-0.475508\pi$
$113$	1.63423	0.153735	0.0768676	+	0.997041i	$0.475508\pi$
$114$	0	0
$115$	−1.48246	−0.138240
$116$	0	0
$117$	3.44501	0.318492
$118$	0	0
$119$	14.0235	1.28553
$120$	0	0
$121$	−6.39885	−0.581714
$122$	0	0
$123$	−8.54300	−0.770296
$124$	0	0
$125$	−3.99603	−0.357416
$126$	0	0
$127$	−9.63369	−0.854851	−0.427426	−	0.904050i	$-0.640579\pi$
$127$	−9.63369	−0.854851	−0.427426	+	0.904050i	$0.640579\pi$
$128$	0	0
$129$	9.55542	0.841307
$130$	0	0
$131$	0.830899	0.0725959	0.0362980	−	0.999341i	$-0.488443\pi$
$131$	0.830899	0.0725959	0.0362980	+	0.999341i	$0.488443\pi$
$132$	0	0
$133$	12.2302	1.06050
$134$	0	0
$135$	0.406311	0.0349696
$136$	0	0
$137$	1.91974	0.164014	0.0820072	−	0.996632i	$-0.473867\pi$
$137$	1.91974	0.164014	0.0820072	+	0.996632i	$0.473867\pi$
$138$	0	0
$139$	18.9417	1.60662	0.803308	−	0.595563i	$-0.203071\pi$
$139$	18.9417	1.60662	0.803308	+	0.595563i	$0.203071\pi$
$140$	0	0
$141$	0.898817	0.0756941
$142$	0	0
$143$	7.38965	0.617954
$144$	0	0
$145$	3.75587	0.311908
$146$	0	0
$147$	6.38045	0.526251
$148$	0	0
$149$	−9.30578	−0.762359	−0.381179	−	0.924501i	$-0.624482\pi$
$149$	−9.30578	−0.762359	−0.381179	+	0.924501i	$0.624482\pi$
$150$	0	0
$151$	9.88578	0.804493	0.402247	−	0.915531i	$-0.368229\pi$
$151$	9.88578	0.804493	0.402247	+	0.915531i	$0.368229\pi$
$152$	0	0
$153$	−3.83373	−0.309939
$154$	0	0
$155$	0.439922	0.0353354
$156$	0	0
$157$	−14.3508	−1.14532	−0.572659	−	0.819794i	$-0.694088\pi$
$157$	−14.3508	−1.14532	−0.572659	+	0.819794i	$0.694088\pi$
$158$	0	0
$159$	−1.91334	−0.151738
$160$	0	0
$161$	13.3463	1.05183
$162$	0	0
$163$	−7.32959	−0.574097	−0.287049	−	0.957916i	$-0.592674\pi$
$163$	−7.32959	−0.574097	−0.287049	+	0.957916i	$0.592674\pi$
$164$	0	0
$165$	0.871548	0.0678499
$166$	0	0
$167$	16.5604	1.28148	0.640741	−	0.767757i	$-0.278627\pi$
$167$	16.5604	1.28148	0.640741	+	0.767757i	$0.278627\pi$
$168$	0	0
$169$	−1.13189	−0.0870684
$170$	0	0
$171$	−3.34349	−0.255683
$172$	0	0
$173$	−14.0530	−1.06843	−0.534214	−	0.845349i	$-0.679392\pi$
$173$	−14.0530	−1.06843	−0.534214	+	0.845349i	$0.679392\pi$
$174$	0	0
$175$	17.6858	1.33692
$176$	0	0
$177$	0.219503	0.0164989
$178$	0	0
$179$	−6.28478	−0.469746	−0.234873	−	0.972026i	$-0.575467\pi$
$179$	−6.28478	−0.469746	−0.234873	+	0.972026i	$0.575467\pi$
$180$	0	0
$181$	−15.6909	−1.16629	−0.583146	−	0.812367i	$-0.698178\pi$
$181$	−15.6909	−1.16629	−0.583146	+	0.812367i	$0.698178\pi$
$182$	0	0
$183$	−11.3053	−0.835713
$184$	0	0
$185$	−1.03525	−0.0761130
$186$	0	0
$187$	−8.22346	−0.601359
$188$	0	0
$189$	−3.65793	−0.266075
$190$	0	0
$191$	−14.9019	−1.07827	−0.539133	−	0.842221i	$-0.681248\pi$
$191$	−14.9019	−1.07827	−0.539133	+	0.842221i	$0.681248\pi$
$192$	0	0
$193$	−9.98140	−0.718477	−0.359238	−	0.933246i	$-0.616964\pi$
$193$	−9.98140	−0.718477	−0.359238	+	0.933246i	$0.616964\pi$
$194$	0	0
$195$	1.39975	0.100238
$196$	0	0
$197$	−0.547736	−0.0390246	−0.0195123	−	0.999810i	$-0.506211\pi$
$197$	−0.547736	−0.0390246	−0.0195123	+	0.999810i	$0.506211\pi$
$198$	0	0
$199$	4.41343	0.312860	0.156430	−	0.987689i	$-0.450001\pi$
$199$	4.41343	0.312860	0.156430	+	0.987689i	$0.450001\pi$
$200$	0	0
$201$	−7.20894	−0.508480
$202$	0	0
$203$	−33.8133	−2.37323
$204$	0	0
$205$	−3.47111	−0.242433
$206$	0	0
$207$	−3.64859	−0.253594
$208$	0	0
$209$	−7.17188	−0.496089
$210$	0	0
$211$	4.69009	0.322879	0.161440	−	0.986883i	$-0.448386\pi$
$211$	4.69009	0.322879	0.161440	+	0.986883i	$0.448386\pi$
$212$	0	0
$213$	−7.59769	−0.520585
$214$	0	0
$215$	3.88247	0.264782
$216$	0	0
$217$	−3.96053	−0.268858
$218$	0	0
$219$	−13.9530	−0.942857
$220$	0	0
$221$	−13.2072	−0.888415
$222$	0	0
$223$	5.27468	0.353218	0.176609	−	0.984281i	$-0.443487\pi$
$223$	5.27468	0.353218	0.176609	+	0.984281i	$0.443487\pi$
$224$	0	0
$225$	−4.83491	−0.322327
$226$	0	0
$227$	−21.5479	−1.43019	−0.715094	−	0.699029i	$-0.753616\pi$
$227$	−21.5479	−1.43019	−0.715094	+	0.699029i	$0.753616\pi$
$228$	0	0
$229$	24.2138	1.60009	0.800046	−	0.599938i	$-0.204808\pi$
$229$	24.2138	1.60009	0.800046	+	0.599938i	$0.204808\pi$
$230$	0	0
$231$	−7.84637	−0.516253
$232$	0	0
$233$	22.4540	1.47101	0.735505	−	0.677519i	$-0.236945\pi$
$233$	22.4540	1.47101	0.735505	+	0.677519i	$0.236945\pi$
$234$	0	0
$235$	0.365199	0.0238230
$236$	0	0
$237$	−2.82794	−0.183694
$238$	0	0
$239$	−6.86087	−0.443793	−0.221896	−	0.975070i	$-0.571225\pi$
$239$	−6.86087	−0.443793	−0.221896	+	0.975070i	$0.571225\pi$
$240$	0	0
$241$	0.0687447	0.00442823	0.00221412	−	0.999998i	$-0.499295\pi$
$241$	0.0687447	0.00442823	0.00221412	+	0.999998i	$0.499295\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	2.59244	0.165625
$246$	0	0
$247$	−11.5184	−0.732895
$248$	0	0
$249$	13.0368	0.826171
$250$	0	0
$251$	−3.14820	−0.198713	−0.0993564	−	0.995052i	$-0.531678\pi$
$251$	−3.14820	−0.198713	−0.0993564	+	0.995052i	$0.531678\pi$
$252$	0	0
$253$	−7.82632	−0.492037
$254$	0	0
$255$	−1.55768	−0.0975460
$256$	0	0
$257$	−17.2997	−1.07913	−0.539563	−	0.841945i	$-0.681411\pi$
$257$	−17.2997	−1.07913	−0.539563	+	0.841945i	$0.681411\pi$
$258$	0	0
$259$	9.32013	0.579125
$260$	0	0
$261$	9.24385	0.572180
$262$	0	0
$263$	7.74904	0.477827	0.238913	−	0.971041i	$-0.423209\pi$
$263$	7.74904	0.477827	0.238913	+	0.971041i	$0.423209\pi$
$264$	0	0
$265$	−0.777411	−0.0477560
$266$	0	0
$267$	5.24068	0.320724
$268$	0	0
$269$	−28.5783	−1.74245	−0.871224	−	0.490886i	$-0.836673\pi$
$269$	−28.5783	−1.74245	−0.871224	+	0.490886i	$0.836673\pi$
$270$	0	0
$271$	−20.5574	−1.24878	−0.624388	−	0.781115i	$-0.714651\pi$
$271$	−20.5574	−1.24878	−0.624388	+	0.781115i	$0.714651\pi$
$272$	0	0
$273$	−12.6016	−0.762684
$274$	0	0
$275$	−10.3710	−0.625396
$276$	0	0
$277$	−1.90200	−0.114280	−0.0571401	−	0.998366i	$-0.518198\pi$
$277$	−1.90200	−0.114280	−0.0571401	+	0.998366i	$0.518198\pi$
$278$	0	0
$279$	1.08272	0.0648210
$280$	0	0
$281$	−1.02847	−0.0613533	−0.0306767	−	0.999529i	$-0.509766\pi$
$281$	−1.02847	−0.0613533	−0.0306767	+	0.999529i	$0.509766\pi$
$282$	0	0
$283$	17.2609	1.02606	0.513028	−	0.858372i	$-0.328524\pi$
$283$	17.2609	1.02606	0.513028	+	0.858372i	$0.328524\pi$
$284$	0	0
$285$	−1.35849	−0.0804703
$286$	0	0
$287$	31.2497	1.84461
$288$	0	0
$289$	−2.30253	−0.135443
$290$	0	0
$291$	3.08960	0.181116
$292$	0	0
$293$	6.28937	0.367429	0.183715	−	0.982980i	$-0.441188\pi$
$293$	6.28937	0.367429	0.183715	+	0.982980i	$0.441188\pi$
$294$	0	0
$295$	0.0891864	0.00519263
$296$	0	0
$297$	2.14503	0.124467
$298$	0	0
$299$	−12.5694	−0.726909
$300$	0	0
$301$	−34.9530	−2.01466
$302$	0	0
$303$	−8.72529	−0.501255
$304$	0	0
$305$	−4.59347	−0.263021
$306$	0	0
$307$	−6.09689	−0.347968	−0.173984	−	0.984748i	$-0.555664\pi$
$307$	−6.09689	−0.347968	−0.173984	+	0.984748i	$0.555664\pi$
$308$	0	0
$309$	12.0716	0.686731
$310$	0	0
$311$	−23.3022	−1.32135	−0.660673	−	0.750674i	$-0.729729\pi$
$311$	−23.3022	−1.32135	−0.660673	+	0.750674i	$0.729729\pi$
$312$	0	0
$313$	−26.5747	−1.50209	−0.751046	−	0.660250i	$-0.770450\pi$
$313$	−26.5747	−1.50209	−0.751046	+	0.660250i	$0.770450\pi$
$314$	0	0
$315$	−1.48626	−0.0837410
$316$	0	0
$317$	10.2513	0.575772	0.287886	−	0.957665i	$-0.407048\pi$
$317$	10.2513	0.575772	0.287886	+	0.957665i	$0.407048\pi$
$318$	0	0
$319$	19.8283	1.11017
$320$	0	0
$321$	−13.3086	−0.742812
$322$	0	0
$323$	12.8180	0.713214
$324$	0	0
$325$	−16.6563	−0.923927
$326$	0	0
$327$	−14.2247	−0.786628
$328$	0	0
$329$	−3.28781	−0.181263
$330$	0	0
$331$	−7.48821	−0.411589	−0.205795	−	0.978595i	$-0.565978\pi$
$331$	−7.48821	−0.411589	−0.205795	+	0.978595i	$0.565978\pi$
$332$	0	0
$333$	−2.54793	−0.139625
$334$	0	0
$335$	−2.92907	−0.160032
$336$	0	0
$337$	12.4765	0.679638	0.339819	−	0.940491i	$-0.389634\pi$
$337$	12.4765	0.679638	0.339819	+	0.940491i	$0.389634\pi$
$338$	0	0
$339$	1.63423	0.0887591
$340$	0	0
$341$	2.32248	0.125769
$342$	0	0
$343$	2.26627	0.122367
$344$	0	0
$345$	−1.48246	−0.0798129
$346$	0	0
$347$	−13.0082	−0.698315	−0.349158	−	0.937064i	$-0.613532\pi$
$347$	−13.0082	−0.698315	−0.349158	+	0.937064i	$0.613532\pi$
$348$	0	0
$349$	−24.7891	−1.32693	−0.663464	−	0.748208i	$-0.730914\pi$
$349$	−24.7891	−1.32693	−0.663464	+	0.748208i	$0.730914\pi$
$350$	0	0
$351$	3.44501	0.183881
$352$	0	0
$353$	14.9658	0.796551	0.398276	−	0.917266i	$-0.369609\pi$
$353$	14.9658	0.796551	0.398276	+	0.917266i	$0.369609\pi$
$354$	0	0
$355$	−3.08702	−0.163842
$356$	0	0
$357$	14.0235	0.742203
$358$	0	0
$359$	−25.8342	−1.36348	−0.681738	−	0.731596i	$-0.738776\pi$
$359$	−25.8342	−1.36348	−0.681738	+	0.731596i	$0.738776\pi$
$360$	0	0
$361$	−7.82109	−0.411637
$362$	0	0
$363$	−6.39885	−0.335853
$364$	0	0
$365$	−5.66925	−0.296742
$366$	0	0
$367$	4.02532	0.210120	0.105060	−	0.994466i	$-0.466497\pi$
$367$	4.02532	0.210120	0.105060	+	0.994466i	$0.466497\pi$
$368$	0	0
$369$	−8.54300	−0.444731
$370$	0	0
$371$	6.99887	0.363363
$372$	0	0
$373$	−0.839469	−0.0434660	−0.0217330	−	0.999764i	$-0.506918\pi$
$373$	−0.839469	−0.0434660	−0.0217330	+	0.999764i	$0.506918\pi$
$374$	0	0
$375$	−3.99603	−0.206354
$376$	0	0
$377$	31.8452	1.64011
$378$	0	0
$379$	25.0804	1.28830	0.644148	−	0.764901i	$-0.277212\pi$
$379$	25.0804	1.28830	0.644148	+	0.764901i	$0.277212\pi$
$380$	0	0
$381$	−9.63369	−0.493549
$382$	0	0
$383$	16.7374	0.855243	0.427621	−	0.903958i	$-0.359352\pi$
$383$	16.7374	0.855243	0.427621	+	0.903958i	$0.359352\pi$
$384$	0	0
$385$	−3.18806	−0.162479
$386$	0	0
$387$	9.55542	0.485729
$388$	0	0
$389$	2.54922	0.129250	0.0646252	−	0.997910i	$-0.479415\pi$
$389$	2.54922	0.129250	0.0646252	+	0.997910i	$0.479415\pi$
$390$	0	0
$391$	13.9877	0.707388
$392$	0	0
$393$	0.830899	0.0419133
$394$	0	0
$395$	−1.14902	−0.0578135
$396$	0	0
$397$	20.7992	1.04388	0.521941	−	0.852982i	$-0.325208\pi$
$397$	20.7992	1.04388	0.521941	+	0.852982i	$0.325208\pi$
$398$	0	0
$399$	12.2302	0.612278
$400$	0	0
$401$	22.5961	1.12840	0.564198	−	0.825639i	$-0.309185\pi$
$401$	22.5961	1.12840	0.564198	+	0.825639i	$0.309185\pi$
$402$	0	0
$403$	3.73000	0.185805
$404$	0	0
$405$	0.406311	0.0201897
$406$	0	0
$407$	−5.46537	−0.270909
$408$	0	0
$409$	−14.2372	−0.703983	−0.351991	−	0.936003i	$-0.614495\pi$
$409$	−14.2372	−0.703983	−0.351991	+	0.936003i	$0.614495\pi$
$410$	0	0
$411$	1.91974	0.0946937
$412$	0	0
$413$	−0.802927	−0.0395094
$414$	0	0
$415$	5.29697	0.260018
$416$	0	0
$417$	18.9417	0.927581
$418$	0	0
$419$	−6.78220	−0.331332	−0.165666	−	0.986182i	$-0.552977\pi$
$419$	−6.78220	−0.331332	−0.165666	+	0.986182i	$0.552977\pi$
$420$	0	0
$421$	23.9079	1.16520	0.582600	−	0.812759i	$-0.302035\pi$
$421$	23.9079	1.16520	0.582600	+	0.812759i	$0.302035\pi$
$422$	0	0
$423$	0.898817	0.0437020
$424$	0	0
$425$	18.5357	0.899115
$426$	0	0
$427$	41.3540	2.00126
$428$	0	0
$429$	7.38965	0.356776
$430$	0	0
$431$	24.0546	1.15867	0.579334	−	0.815090i	$-0.303313\pi$
$431$	24.0546	1.15867	0.579334	+	0.815090i	$0.303313\pi$
$432$	0	0
$433$	22.2211	1.06788	0.533939	−	0.845523i	$-0.320711\pi$
$433$	22.2211	1.06788	0.533939	+	0.845523i	$0.320711\pi$
$434$	0	0
$435$	3.75587	0.180080
$436$	0	0
$437$	12.1990	0.583557
$438$	0	0
$439$	14.6703	0.700175	0.350087	−	0.936717i	$-0.386152\pi$
$439$	14.6703	0.700175	0.350087	+	0.936717i	$0.386152\pi$
$440$	0	0
$441$	6.38045	0.303831
$442$	0	0
$443$	−16.5212	−0.784944	−0.392472	−	0.919764i	$-0.628380\pi$
$443$	−16.5212	−0.784944	−0.392472	+	0.919764i	$0.628380\pi$
$444$	0	0
$445$	2.12934	0.100941
$446$	0	0
$447$	−9.30578	−0.440148
$448$	0	0
$449$	−2.49155	−0.117583	−0.0587917	−	0.998270i	$-0.518725\pi$
$449$	−2.49155	−0.117583	−0.0587917	+	0.998270i	$0.518725\pi$
$450$	0	0
$451$	−18.3250	−0.862890
$452$	0	0
$453$	9.88578	0.464474
$454$	0	0
$455$	−5.12017	−0.240037
$456$	0	0
$457$	13.1799	0.616532	0.308266	−	0.951300i	$-0.400251\pi$
$457$	13.1799	0.616532	0.308266	+	0.951300i	$0.400251\pi$
$458$	0	0
$459$	−3.83373	−0.178943
$460$	0	0
$461$	−29.1718	−1.35867	−0.679333	−	0.733830i	$-0.737731\pi$
$461$	−29.1718	−1.35867	−0.679333	+	0.733830i	$0.737731\pi$
$462$	0	0
$463$	9.74846	0.453049	0.226525	−	0.974005i	$-0.427264\pi$
$463$	9.74846	0.453049	0.226525	+	0.974005i	$0.427264\pi$
$464$	0	0
$465$	0.439922	0.0204009
$466$	0	0
$467$	3.62671	0.167824	0.0839120	−	0.996473i	$-0.473259\pi$
$467$	3.62671	0.167824	0.0839120	+	0.996473i	$0.473259\pi$
$468$	0	0
$469$	26.3698	1.21764
$470$	0	0
$471$	−14.3508	−0.661250
$472$	0	0
$473$	20.4966	0.942437
$474$	0	0
$475$	16.1655	0.741722
$476$	0	0
$477$	−1.91334	−0.0876059
$478$	0	0
$479$	−17.3619	−0.793286	−0.396643	−	0.917973i	$-0.629825\pi$
$479$	−17.3619	−0.793286	−0.396643	+	0.917973i	$0.629825\pi$
$480$	0	0
$481$	−8.77763	−0.400226
$482$	0	0
$483$	13.3463	0.607276
$484$	0	0
$485$	1.25534	0.0570020
$486$	0	0
$487$	12.0909	0.547891	0.273946	−	0.961745i	$-0.411671\pi$
$487$	12.0909	0.547891	0.273946	+	0.961745i	$0.411671\pi$
$488$	0	0
$489$	−7.32959	−0.331455
$490$	0	0
$491$	12.3346	0.556655	0.278327	−	0.960486i	$-0.410220\pi$
$491$	12.3346	0.556655	0.278327	+	0.960486i	$0.410220\pi$
$492$	0	0
$493$	−35.4384	−1.59606
$494$	0	0
$495$	0.871548	0.0391732
$496$	0	0
$497$	27.7918	1.24663
$498$	0	0
$499$	11.8077	0.528585	0.264293	−	0.964443i	$-0.414862\pi$
$499$	11.8077	0.528585	0.264293	+	0.964443i	$0.414862\pi$
$500$	0	0
$501$	16.5604	0.739865
$502$	0	0
$503$	−1.00000	−0.0445878
$503$	−1.00000	−0.0445878
$504$	0	0
$505$	−3.54518	−0.157758
$506$	0	0
$507$	−1.13189	−0.0502689
$508$	0	0
$509$	−39.3128	−1.74251	−0.871255	−	0.490830i	$-0.836693\pi$
$509$	−39.3128	−1.74251	−0.871255	+	0.490830i	$0.836693\pi$
$510$	0	0
$511$	51.0391	2.25784
$512$	0	0
$513$	−3.34349	−0.147619
$514$	0	0
$515$	4.90483	0.216133
$516$	0	0
$517$	1.92799	0.0847929
$518$	0	0
$519$	−14.0530	−0.616857
$520$	0	0
$521$	−19.6475	−0.860773	−0.430387	−	0.902645i	$-0.641623\pi$
$521$	−19.6475	−0.860773	−0.430387	+	0.902645i	$0.641623\pi$
$522$	0	0
$523$	15.6466	0.684180	0.342090	−	0.939667i	$-0.388865\pi$
$523$	15.6466	0.684180	0.342090	+	0.939667i	$0.388865\pi$
$524$	0	0
$525$	17.6858	0.771870
$526$	0	0
$527$	−4.15087	−0.180815
$528$	0	0
$529$	−9.68782	−0.421210
$530$	0	0
$531$	0.219503	0.00952562
$532$	0	0
$533$	−29.4307	−1.27479
$534$	0	0
$535$	−5.40741	−0.233783
$536$	0	0
$537$	−6.28478	−0.271208
$538$	0	0
$539$	13.6863	0.589509
$540$	0	0
$541$	33.2755	1.43062	0.715312	−	0.698805i	$-0.246285\pi$
$541$	33.2755	1.43062	0.715312	+	0.698805i	$0.246285\pi$
$542$	0	0
$543$	−15.6909	−0.673359
$544$	0	0
$545$	−5.77965	−0.247573
$546$	0	0
$547$	34.0810	1.45720	0.728599	−	0.684940i	$-0.240172\pi$
$547$	34.0810	1.45720	0.728599	+	0.684940i	$0.240172\pi$
$548$	0	0
$549$	−11.3053	−0.482499
$550$	0	0
$551$	−30.9067	−1.31667
$552$	0	0
$553$	10.3444	0.439889
$554$	0	0
$555$	−1.03525	−0.0439439
$556$	0	0
$557$	11.8757	0.503188	0.251594	−	0.967833i	$-0.419045\pi$
$557$	11.8757	0.503188	0.251594	+	0.967833i	$0.419045\pi$
$558$	0	0
$559$	32.9185	1.39231
$560$	0	0
$561$	−8.22346	−0.347195
$562$	0	0
$563$	−15.7880	−0.665386	−0.332693	−	0.943035i	$-0.607957\pi$
$563$	−15.7880	−0.665386	−0.332693	+	0.943035i	$0.607957\pi$
$564$	0	0
$565$	0.664004	0.0279349
$566$	0	0
$567$	−3.65793	−0.153619
$568$	0	0
$569$	11.6496	0.488378	0.244189	−	0.969728i	$-0.421478\pi$
$569$	11.6496	0.488378	0.244189	+	0.969728i	$0.421478\pi$
$570$	0	0
$571$	−31.6615	−1.32499	−0.662496	−	0.749066i	$-0.730503\pi$
$571$	−31.6615	−1.32499	−0.662496	+	0.749066i	$0.730503\pi$
$572$	0	0
$573$	−14.9019	−0.622537
$574$	0	0
$575$	17.6406	0.735664
$576$	0	0
$577$	4.00850	0.166876	0.0834380	−	0.996513i	$-0.473410\pi$
$577$	4.00850	0.166876	0.0834380	+	0.996513i	$0.473410\pi$
$578$	0	0
$579$	−9.98140	−0.414813
$580$	0	0
$581$	−47.6875	−1.97841
$582$	0	0
$583$	−4.10418	−0.169978
$584$	0	0
$585$	1.39975	0.0578723
$586$	0	0
$587$	−8.18405	−0.337792	−0.168896	−	0.985634i	$-0.554020\pi$
$587$	−8.18405	−0.337792	−0.168896	+	0.985634i	$0.554020\pi$
$588$	0	0
$589$	−3.62008	−0.149163
$590$	0	0
$591$	−0.547736	−0.0225308
$592$	0	0
$593$	−21.4944	−0.882669	−0.441334	−	0.897343i	$-0.645495\pi$
$593$	−21.4944	−0.882669	−0.441334	+	0.897343i	$0.645495\pi$
$594$	0	0
$595$	5.69790	0.233591
$596$	0	0
$597$	4.41343	0.180630
$598$	0	0
$599$	−33.2767	−1.35965	−0.679824	−	0.733375i	$-0.737944\pi$
$599$	−33.2767	−1.35965	−0.679824	+	0.733375i	$0.737944\pi$
$600$	0	0
$601$	30.8617	1.25888	0.629438	−	0.777051i	$-0.283285\pi$
$601$	30.8617	1.25888	0.629438	+	0.777051i	$0.283285\pi$
$602$	0	0
$603$	−7.20894	−0.293571
$604$	0	0
$605$	−2.59992	−0.105702
$606$	0	0
$607$	25.1748	1.02182	0.510908	−	0.859636i	$-0.329309\pi$
$607$	25.1748	1.02182	0.510908	+	0.859636i	$0.329309\pi$
$608$	0	0
$609$	−33.8133	−1.37019
$610$	0	0
$611$	3.09644	0.125268
$612$	0	0
$613$	−15.1119	−0.610364	−0.305182	−	0.952294i	$-0.598717\pi$
$613$	−15.1119	−0.610364	−0.305182	+	0.952294i	$0.598717\pi$
$614$	0	0
$615$	−3.47111	−0.139969
$616$	0	0
$617$	−15.8617	−0.638570	−0.319285	−	0.947659i	$-0.603443\pi$
$617$	−15.8617	−0.638570	−0.319285	+	0.947659i	$0.603443\pi$
$618$	0	0
$619$	40.0301	1.60895	0.804473	−	0.593989i	$-0.202448\pi$
$619$	40.0301	1.60895	0.804473	+	0.593989i	$0.202448\pi$
$620$	0	0
$621$	−3.64859	−0.146413
$622$	0	0
$623$	−19.1700	−0.768032
$624$	0	0
$625$	22.5509	0.902037
$626$	0	0
$627$	−7.17188	−0.286417
$628$	0	0
$629$	9.76805	0.389478
$630$	0	0
$631$	3.76994	0.150079	0.0750395	−	0.997181i	$-0.476092\pi$
$631$	3.76994	0.150079	0.0750395	+	0.997181i	$0.476092\pi$
$632$	0	0
$633$	4.69009	0.186414
$634$	0	0
$635$	−3.91427	−0.155333
$636$	0	0
$637$	21.9807	0.870908
$638$	0	0
$639$	−7.59769	−0.300560
$640$	0	0
$641$	22.3026	0.880901	0.440451	−	0.897777i	$-0.354819\pi$
$641$	22.3026	0.880901	0.440451	+	0.897777i	$0.354819\pi$
$642$	0	0
$643$	−19.0145	−0.749857	−0.374929	−	0.927054i	$-0.622333\pi$
$643$	−19.0145	−0.749857	−0.374929	+	0.927054i	$0.622333\pi$
$644$	0	0
$645$	3.88247	0.152872
$646$	0	0
$647$	−35.1435	−1.38163	−0.690816	−	0.723031i	$-0.742749\pi$
$647$	−35.1435	−1.38163	−0.690816	+	0.723031i	$0.742749\pi$
$648$	0	0
$649$	0.470840	0.0184821
$650$	0	0
$651$	−3.96053	−0.155225
$652$	0	0
$653$	−29.6822	−1.16155	−0.580777	−	0.814063i	$-0.697251\pi$
$653$	−29.6822	−1.16155	−0.580777	+	0.814063i	$0.697251\pi$
$654$	0	0
$655$	0.337603	0.0131912
$656$	0	0
$657$	−13.9530	−0.544359
$658$	0	0
$659$	−6.28017	−0.244641	−0.122320	−	0.992491i	$-0.539034\pi$
$659$	−6.28017	−0.244641	−0.122320	+	0.992491i	$0.539034\pi$
$660$	0	0
$661$	−13.2371	−0.514865	−0.257433	−	0.966296i	$-0.582877\pi$
$661$	−13.2371	−0.514865	−0.257433	+	0.966296i	$0.582877\pi$
$662$	0	0
$663$	−13.2072	−0.512927
$664$	0	0
$665$	4.96928	0.192700
$666$	0	0
$667$	−33.7270	−1.30591
$668$	0	0
$669$	5.27468	0.203931
$670$	0	0
$671$	−24.2502	−0.936169
$672$	0	0
$673$	32.6171	1.25730	0.628648	−	0.777690i	$-0.283609\pi$
$673$	32.6171	1.25730	0.628648	+	0.777690i	$0.283609\pi$
$674$	0	0
$675$	−4.83491	−0.186096
$676$	0	0
$677$	−3.86931	−0.148710	−0.0743549	−	0.997232i	$-0.523690\pi$
$677$	−3.86931	−0.148710	−0.0743549	+	0.997232i	$0.523690\pi$
$678$	0	0
$679$	−11.3015	−0.433713
$680$	0	0
$681$	−21.5479	−0.825719
$682$	0	0
$683$	−4.26938	−0.163363	−0.0816817	−	0.996658i	$-0.526029\pi$
$683$	−4.26938	−0.163363	−0.0816817	+	0.996658i	$0.526029\pi$
$684$	0	0
$685$	0.780010	0.0298027
$686$	0	0
$687$	24.2138	0.923814
$688$	0	0
$689$	−6.59149	−0.251116
$690$	0	0
$691$	−4.54311	−0.172828	−0.0864140	−	0.996259i	$-0.527541\pi$
$691$	−4.54311	−0.172828	−0.0864140	+	0.996259i	$0.527541\pi$
$692$	0	0
$693$	−7.84637	−0.298059
$694$	0	0
$695$	7.69623	0.291935
$696$	0	0
$697$	32.7515	1.24055
$698$	0	0
$699$	22.4540	0.849288
$700$	0	0
$701$	−1.64863	−0.0622679	−0.0311339	−	0.999515i	$-0.509912\pi$
$701$	−1.64863	−0.0622679	−0.0311339	+	0.999515i	$0.509912\pi$
$702$	0	0
$703$	8.51896	0.321298
$704$	0	0
$705$	0.365199	0.0137542
$706$	0	0
$707$	31.9165	1.20034
$708$	0	0
$709$	28.9639	1.08776	0.543880	−	0.839163i	$-0.316955\pi$
$709$	28.9639	1.08776	0.543880	+	0.839163i	$0.316955\pi$
$710$	0	0
$711$	−2.82794	−0.106056
$712$	0	0
$713$	−3.95041	−0.147944
$714$	0	0
$715$	3.00249	0.112287
$716$	0	0
$717$	−6.86087	−0.256224
$718$	0	0
$719$	−32.2589	−1.20305	−0.601527	−	0.798853i	$-0.705441\pi$
$719$	−32.2589	−1.20305	−0.601527	+	0.798853i	$0.705441\pi$
$720$	0	0
$721$	−44.1572	−1.64450
$722$	0	0
$723$	0.0687447	0.00255664
$724$	0	0
$725$	−44.6932	−1.65986
$726$	0	0
$727$	−36.8934	−1.36830	−0.684149	−	0.729342i	$-0.739826\pi$
$727$	−36.8934	−1.36830	−0.684149	+	0.729342i	$0.739826\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−36.6329	−1.35492
$732$	0	0
$733$	−17.4675	−0.645176	−0.322588	−	0.946539i	$-0.604553\pi$
$733$	−17.4675	−0.645176	−0.322588	+	0.946539i	$0.604553\pi$
$734$	0	0
$735$	2.59244	0.0956238
$736$	0	0
$737$	−15.4634	−0.569601
$738$	0	0
$739$	9.14408	0.336371	0.168185	−	0.985755i	$-0.446209\pi$
$739$	9.14408	0.336371	0.168185	+	0.985755i	$0.446209\pi$
$740$	0	0
$741$	−11.5184	−0.423137
$742$	0	0
$743$	−1.90908	−0.0700373	−0.0350187	−	0.999387i	$-0.511149\pi$
$743$	−1.90908	−0.0700373	−0.0350187	+	0.999387i	$0.511149\pi$
$744$	0	0
$745$	−3.78104	−0.138526
$746$	0	0
$747$	13.0368	0.476990
$748$	0	0
$749$	48.6818	1.77879
$750$	0	0
$751$	−2.87431	−0.104885	−0.0524426	−	0.998624i	$-0.516701\pi$
$751$	−2.87431	−0.104885	−0.0524426	+	0.998624i	$0.516701\pi$
$752$	0	0
$753$	−3.14820	−0.114727
$754$	0	0
$755$	4.01670	0.146183
$756$	0	0
$757$	10.3723	0.376986	0.188493	−	0.982075i	$-0.439640\pi$
$757$	10.3723	0.376986	0.188493	+	0.982075i	$0.439640\pi$
$758$	0	0
$759$	−7.82632	−0.284078
$760$	0	0
$761$	31.1830	1.13038	0.565191	−	0.824960i	$-0.308802\pi$
$761$	31.1830	1.13038	0.565191	+	0.824960i	$0.308802\pi$
$762$	0	0
$763$	52.0330	1.88372
$764$	0	0
$765$	−1.55768	−0.0563182
$766$	0	0
$767$	0.756191	0.0273045
$768$	0	0
$769$	8.37913	0.302159	0.151080	−	0.988522i	$-0.451725\pi$
$769$	8.37913	0.302159	0.151080	+	0.988522i	$0.451725\pi$
$770$	0	0
$771$	−17.2997	−0.623034
$772$	0	0
$773$	18.2922	0.657924	0.328962	−	0.944343i	$-0.393301\pi$
$773$	18.2922	0.657924	0.328962	+	0.944343i	$0.393301\pi$
$774$	0	0
$775$	−5.23488	−0.188042
$776$	0	0
$777$	9.32013	0.334358
$778$	0	0
$779$	28.5634	1.02339
$780$	0	0
$781$	−16.2973	−0.583162
$782$	0	0
$783$	9.24385	0.330348
$784$	0	0
$785$	−5.83088	−0.208113
$786$	0	0
$787$	12.8034	0.456392	0.228196	−	0.973615i	$-0.426717\pi$
$787$	12.8034	0.456392	0.228196	+	0.973615i	$0.426717\pi$
$788$	0	0
$789$	7.74904	0.275873
$790$	0	0
$791$	−5.97789	−0.212549
$792$	0	0
$793$	−38.9469	−1.38305
$794$	0	0
$795$	−0.777411	−0.0275719
$796$	0	0
$797$	11.4139	0.404302	0.202151	−	0.979354i	$-0.435207\pi$
$797$	11.4139	0.404302	0.202151	+	0.979354i	$0.435207\pi$
$798$	0	0
$799$	−3.44582	−0.121904
$800$	0	0
$801$	5.24068	0.185170
$802$	0	0
$803$	−29.9296	−1.05619
$804$	0	0
$805$	5.42273	0.191126
$806$	0	0
$807$	−28.5783	−1.00600
$808$	0	0
$809$	47.3479	1.66466	0.832332	−	0.554277i	$-0.187005\pi$
$809$	47.3479	1.66466	0.832332	+	0.554277i	$0.187005\pi$
$810$	0	0
$811$	23.2769	0.817361	0.408680	−	0.912678i	$-0.365989\pi$
$811$	23.2769	0.817361	0.408680	+	0.912678i	$0.365989\pi$
$812$	0	0
$813$	−20.5574	−0.720981
$814$	0	0
$815$	−2.97809	−0.104318
$816$	0	0
$817$	−31.9484	−1.11773
$818$	0	0
$819$	−12.6016	−0.440336
$820$	0	0
$821$	18.6905	0.652302	0.326151	−	0.945318i	$-0.394248\pi$
$821$	18.6905	0.652302	0.326151	+	0.945318i	$0.394248\pi$
$822$	0	0
$823$	−16.2117	−0.565105	−0.282553	−	0.959252i	$-0.591181\pi$
$823$	−16.2117	−0.565105	−0.282553	+	0.959252i	$0.591181\pi$
$824$	0	0
$825$	−10.3710	−0.361073
$826$	0	0
$827$	−4.92285	−0.171184	−0.0855921	−	0.996330i	$-0.527278\pi$
$827$	−4.92285	−0.171184	−0.0855921	+	0.996330i	$0.527278\pi$
$828$	0	0
$829$	−33.1013	−1.14966	−0.574828	−	0.818274i	$-0.694931\pi$
$829$	−33.1013	−1.14966	−0.574828	+	0.818274i	$0.694931\pi$
$830$	0	0
$831$	−1.90200	−0.0659797
$832$	0	0
$833$	−24.4609	−0.847520
$834$	0	0
$835$	6.72867	0.232855
$836$	0	0
$837$	1.08272	0.0374244
$838$	0	0
$839$	−28.8189	−0.994941	−0.497470	−	0.867481i	$-0.665738\pi$
$839$	−28.8189	−0.994941	−0.497470	+	0.867481i	$0.665738\pi$
$840$	0	0
$841$	56.4487	1.94651
$842$	0	0
$843$	−1.02847	−0.0354224
$844$	0	0
$845$	−0.459898	−0.0158210
$846$	0	0
$847$	23.4065	0.804258
$848$	0	0
$849$	17.2609	0.592394
$850$	0	0
$851$	9.29632	0.318674
$852$	0	0
$853$	−10.3896	−0.355733	−0.177867	−	0.984055i	$-0.556920\pi$
$853$	−10.3896	−0.355733	−0.177867	+	0.984055i	$0.556920\pi$
$854$	0	0
$855$	−1.35849	−0.0464595
$856$	0	0
$857$	50.3343	1.71939	0.859693	−	0.510811i	$-0.170655\pi$
$857$	50.3343	1.71939	0.859693	+	0.510811i	$0.170655\pi$
$858$	0	0
$859$	24.6385	0.840655	0.420327	−	0.907372i	$-0.361915\pi$
$859$	24.6385	0.840655	0.420327	+	0.907372i	$0.361915\pi$
$860$	0	0
$861$	31.2497	1.06499
$862$	0	0
$863$	31.3490	1.06713	0.533567	−	0.845758i	$-0.320851\pi$
$863$	31.3490	1.06713	0.533567	+	0.845758i	$0.320851\pi$
$864$	0	0
$865$	−5.70987	−0.194142
$866$	0	0
$867$	−2.30253	−0.0781981
$868$	0	0
$869$	−6.06601	−0.205775
$870$	0	0
$871$	−24.8349	−0.841498
$872$	0	0
$873$	3.08960	0.104567
$874$	0	0
$875$	14.6172	0.494151
$876$	0	0
$877$	10.5485	0.356197	0.178098	−	0.984013i	$-0.443005\pi$
$877$	10.5485	0.356197	0.178098	+	0.984013i	$0.443005\pi$
$878$	0	0
$879$	6.28937	0.212135
$880$	0	0
$881$	32.2167	1.08541	0.542704	−	0.839924i	$-0.317400\pi$
$881$	32.2167	1.08541	0.542704	+	0.839924i	$0.317400\pi$
$882$	0	0
$883$	24.4657	0.823336	0.411668	−	0.911334i	$-0.364946\pi$
$883$	24.4657	0.823336	0.411668	+	0.911334i	$0.364946\pi$
$884$	0	0
$885$	0.0891864	0.00299797
$886$	0	0
$887$	42.5837	1.42982	0.714910	−	0.699216i	$-0.246468\pi$
$887$	42.5837	1.42982	0.714910	+	0.699216i	$0.246468\pi$
$888$	0	0
$889$	35.2393	1.18189
$890$	0	0
$891$	2.14503	0.0718612
$892$	0	0
$893$	−3.00518	−0.100565
$894$	0	0
$895$	−2.55357	−0.0853565
$896$	0	0
$897$	−12.5694	−0.419681
$898$	0	0
$899$	10.0085	0.333803
$900$	0	0
$901$	7.33523	0.244372
$902$	0	0
$903$	−34.9530	−1.16316
$904$	0	0
$905$	−6.37536	−0.211924
$906$	0	0
$907$	−41.2043	−1.36817	−0.684083	−	0.729404i	$-0.739797\pi$
$907$	−41.2043	−1.36817	−0.684083	+	0.729404i	$0.739797\pi$
$908$	0	0
$909$	−8.72529	−0.289400
$910$	0	0
$911$	−39.4233	−1.30615	−0.653076	−	0.757293i	$-0.726522\pi$
$911$	−39.4233	−1.30615	−0.653076	+	0.757293i	$0.726522\pi$
$912$	0	0
$913$	27.9642	0.925481
$914$	0	0
$915$	−4.59347	−0.151855
$916$	0	0
$917$	−3.03937	−0.100369
$918$	0	0
$919$	−2.55207	−0.0841850	−0.0420925	−	0.999114i	$-0.513402\pi$
$919$	−2.55207	−0.0841850	−0.0420925	+	0.999114i	$0.513402\pi$
$920$	0	0
$921$	−6.09689	−0.200899
$922$	0	0
$923$	−26.1741	−0.861532
$924$	0	0
$925$	12.3190	0.405046
$926$	0	0
$927$	12.0716	0.396484
$928$	0	0
$929$	−40.9415	−1.34325	−0.671624	−	0.740892i	$-0.734403\pi$
$929$	−40.9415	−1.34325	−0.671624	+	0.740892i	$0.734403\pi$
$930$	0	0
$931$	−21.3330	−0.699159
$932$	0	0
$933$	−23.3022	−0.762879
$934$	0	0
$935$	−3.34128	−0.109271
$936$	0	0
$937$	49.5459	1.61859	0.809297	−	0.587400i	$-0.199848\pi$
$937$	49.5459	1.61859	0.809297	+	0.587400i	$0.199848\pi$
$938$	0	0
$939$	−26.5747	−0.867233
$940$	0	0
$941$	−40.6325	−1.32458	−0.662291	−	0.749246i	$-0.730416\pi$
$941$	−40.6325	−1.32458	−0.662291	+	0.749246i	$0.730416\pi$
$942$	0	0
$943$	31.1699	1.01503
$944$	0	0
$945$	−1.48626	−0.0483479
$946$	0	0
$947$	40.3218	1.31028	0.655141	−	0.755507i	$-0.272609\pi$
$947$	40.3218	1.31028	0.655141	+	0.755507i	$0.272609\pi$
$948$	0	0
$949$	−48.0683	−1.56036
$950$	0	0
$951$	10.2513	0.332422
$952$	0	0
$953$	−40.6134	−1.31560	−0.657798	−	0.753195i	$-0.728512\pi$
$953$	−40.6134	−1.31560	−0.657798	+	0.753195i	$0.728512\pi$
$954$	0	0
$955$	−6.05481	−0.195929
$956$	0	0
$957$	19.8283	0.640959
$958$	0	0
$959$	−7.02227	−0.226761
$960$	0	0
$961$	−29.8277	−0.962184
$962$	0	0
$963$	−13.3086	−0.428862
$964$	0	0
$965$	−4.05555	−0.130553
$966$	0	0
$967$	−11.1040	−0.357081	−0.178540	−	0.983933i	$-0.557138\pi$
$967$	−11.1040	−0.357081	−0.178540	+	0.983933i	$0.557138\pi$
$968$	0	0
$969$	12.8180	0.411774
$970$	0	0
$971$	40.5525	1.30139	0.650696	−	0.759339i	$-0.274477\pi$
$971$	40.5525	1.30139	0.650696	+	0.759339i	$0.274477\pi$
$972$	0	0
$973$	−69.2875	−2.22126
$974$	0	0
$975$	−16.6563	−0.533430
$976$	0	0
$977$	−1.56212	−0.0499767	−0.0249883	−	0.999688i	$-0.507955\pi$
$977$	−1.56212	−0.0499767	−0.0249883	+	0.999688i	$0.507955\pi$
$978$	0	0
$979$	11.2414	0.359277
$980$	0	0
$981$	−14.2247	−0.454160
$982$	0	0
$983$	25.6312	0.817509	0.408755	−	0.912644i	$-0.365963\pi$
$983$	25.6312	0.817509	0.408755	+	0.912644i	$0.365963\pi$
$984$	0	0
$985$	−0.222551	−0.00709106
$986$	0	0
$987$	−3.28781	−0.104652
$988$	0	0
$989$	−34.8638	−1.10860
$990$	0	0
$991$	−21.5294	−0.683903	−0.341951	−	0.939718i	$-0.611088\pi$
$991$	−21.5294	−0.683903	−0.341951	+	0.939718i	$0.611088\pi$
$992$	0	0
$993$	−7.48821	−0.237631
$994$	0	0
$995$	1.79322	0.0568490
$996$	0	0
$997$	46.6509	1.47745	0.738725	−	0.674007i	$-0.235428\pi$
$997$	46.6509	1.47745	0.738725	+	0.674007i	$0.235428\pi$
$998$	0	0
$999$	−2.54793	−0.0806128

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6036.2.a.g.1.12	✓	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6036.2.a.g.1.12	✓	15	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +0.406311 q^{5} -3.65793 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +0.406311 q^{5} -3.65793 q^{7} +1.00000 q^{9} +2.14503 q^{11} +3.44501 q^{13} +0.406311 q^{15} -3.83373 q^{17} -3.34349 q^{19} -3.65793 q^{21} -3.64859 q^{23} -4.83491 q^{25} +1.00000 q^{27} +9.24385 q^{29} +1.08272 q^{31} +2.14503 q^{33} -1.48626 q^{35} -2.54793 q^{37} +3.44501 q^{39} -8.54300 q^{41} +9.55542 q^{43} +0.406311 q^{45} +0.898817 q^{47} +6.38045 q^{49} -3.83373 q^{51} -1.91334 q^{53} +0.871548 q^{55} -3.34349 q^{57} +0.219503 q^{59} -11.3053 q^{61} -3.65793 q^{63} +1.39975 q^{65} -7.20894 q^{67} -3.64859 q^{69} -7.59769 q^{71} -13.9530 q^{73} -4.83491 q^{75} -7.84637 q^{77} -2.82794 q^{79} +1.00000 q^{81} +13.0368 q^{83} -1.55768 q^{85} +9.24385 q^{87} +5.24068 q^{89} -12.6016 q^{91} +1.08272 q^{93} -1.35849 q^{95} +3.08960 q^{97} +2.14503 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q + 15 q^{3} - 11 q^{5} - 4 q^{7} + 15 q^{9}+O(q^{10}) \) 15 * q + 15 * q^3 - 11 * q^5 - 4 * q^7 + 15 * q^9	\( 15 q + 15 q^{3} - 11 q^{5} - 4 q^{7} + 15 q^{9} - 5 q^{11} - 14 q^{13} - 11 q^{15} - 5 q^{17} + 3 q^{19} - 4 q^{21} - 32 q^{23} + 2 q^{25} + 15 q^{27} - 23 q^{29} - 13 q^{31} - 5 q^{33} - 16 q^{35} - 10 q^{37} - 14 q^{39} - 14 q^{41} + 4 q^{43} - 11 q^{45} - 20 q^{47} - 9 q^{49} - 5 q^{51} - 30 q^{53} - 10 q^{55} + 3 q^{57} - 14 q^{59} - 38 q^{61} - 4 q^{63} - 24 q^{65} - 8 q^{67} - 32 q^{69} - 41 q^{71} - 19 q^{73} + 2 q^{75} - 39 q^{77} - 27 q^{79} + 15 q^{81} - 17 q^{83} - 6 q^{85} - 23 q^{87} - 23 q^{89} + 4 q^{91} - 13 q^{93} - 30 q^{95} - 18 q^{97} - 5 q^{99}+O(q^{100}) \) 15 * q + 15 * q^3 - 11 * q^5 - 4 * q^7 + 15 * q^9 - 5 * q^11 - 14 * q^13 - 11 * q^15 - 5 * q^17 + 3 * q^19 - 4 * q^21 - 32 * q^23 + 2 * q^25 + 15 * q^27 - 23 * q^29 - 13 * q^31 - 5 * q^33 - 16 * q^35 - 10 * q^37 - 14 * q^39 - 14 * q^41 + 4 * q^43 - 11 * q^45 - 20 * q^47 - 9 * q^49 - 5 * q^51 - 30 * q^53 - 10 * q^55 + 3 * q^57 - 14 * q^59 - 38 * q^61 - 4 * q^63 - 24 * q^65 - 8 * q^67 - 32 * q^69 - 41 * q^71 - 19 * q^73 + 2 * q^75 - 39 * q^77 - 27 * q^79 + 15 * q^81 - 17 * q^83 - 6 * q^85 - 23 * q^87 - 23 * q^89 + 4 * q^91 - 13 * q^93 - 30 * q^95 - 18 * q^97 - 5 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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