LMFDB - Embedded newform 6036.2.a.g.1.8

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			1.8
Root			$0.357651$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} -0.642349 q^{5} -2.57693 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -0.642349 q^{5} -2.57693 q^{7} +1.00000 q^{9} -1.59395 q^{11} -3.01981 q^{13} -0.642349 q^{15} +5.85477 q^{17} +4.15803 q^{19} -2.57693 q^{21} +6.16672 q^{23} -4.58739 q^{25} +1.00000 q^{27} +0.244932 q^{29} -5.80673 q^{31} -1.59395 q^{33} +1.65529 q^{35} -2.35969 q^{37} -3.01981 q^{39} +3.00916 q^{41} -1.02072 q^{43} -0.642349 q^{45} -9.29799 q^{47} -0.359430 q^{49} +5.85477 q^{51} -8.77282 q^{53} +1.02388 q^{55} +4.15803 q^{57} +1.90637 q^{59} +3.09875 q^{61} -2.57693 q^{63} +1.93977 q^{65} +0.795114 q^{67} +6.16672 q^{69} +6.62085 q^{71} +5.75222 q^{73} -4.58739 q^{75} +4.10751 q^{77} -15.7483 q^{79} +1.00000 q^{81} +5.53292 q^{83} -3.76080 q^{85} +0.244932 q^{87} +6.54723 q^{89} +7.78184 q^{91} -5.80673 q^{93} -2.67091 q^{95} +12.6589 q^{97} -1.59395 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−0.642349	−0.287267	−0.143634	−	0.989631i	$-0.545879\pi$
$5$	−0.642349	−0.287267	−0.143634	+	0.989631i	$0.545879\pi$
$6$	0	0
$7$	−2.57693	−0.973988	−0.486994	−	0.873405i	$-0.661907\pi$
$7$	−2.57693	−0.973988	−0.486994	+	0.873405i	$0.661907\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.59395	−0.480595	−0.240298	−	0.970699i	$-0.577245\pi$
$11$	−1.59395	−0.480595	−0.240298	+	0.970699i	$0.577245\pi$
$12$	0	0
$13$	−3.01981	−0.837545	−0.418772	−	0.908091i	$-0.637540\pi$
$13$	−3.01981	−0.837545	−0.418772	+	0.908091i	$0.637540\pi$
$14$	0	0
$15$	−0.642349	−0.165854
$16$	0	0
$17$	5.85477	1.41999	0.709995	−	0.704207i	$-0.248697\pi$
$17$	5.85477	1.41999	0.709995	+	0.704207i	$0.248697\pi$
$18$	0	0
$19$	4.15803	0.953917	0.476958	−	0.878926i	$-0.341739\pi$
$19$	4.15803	0.953917	0.476958	+	0.878926i	$0.341739\pi$
$20$	0	0
$21$	−2.57693	−0.562332
$22$	0	0
$23$	6.16672	1.28585	0.642925	−	0.765929i	$-0.277721\pi$
$23$	6.16672	1.28585	0.642925	+	0.765929i	$0.277721\pi$
$24$	0	0
$25$	−4.58739	−0.917477
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	0.244932	0.0454827	0.0227414	−	0.999741i	$-0.492761\pi$
$29$	0.244932	0.0454827	0.0227414	+	0.999741i	$0.492761\pi$
$30$	0	0
$31$	−5.80673	−1.04292	−0.521460	−	0.853276i	$-0.674612\pi$
$31$	−5.80673	−1.04292	−0.521460	+	0.853276i	$0.674612\pi$
$32$	0	0
$33$	−1.59395	−0.277472
$34$	0	0
$35$	1.65529	0.279795
$36$	0	0
$37$	−2.35969	−0.387931	−0.193965	−	0.981008i	$-0.562135\pi$
$37$	−2.35969	−0.387931	−0.193965	+	0.981008i	$0.562135\pi$
$38$	0	0
$39$	−3.01981	−0.483557
$40$	0	0
$41$	3.00916	0.469952	0.234976	−	0.972001i	$-0.424499\pi$
$41$	3.00916	0.469952	0.234976	+	0.972001i	$0.424499\pi$
$42$	0	0
$43$	−1.02072	−0.155658	−0.0778290	−	0.996967i	$-0.524799\pi$
$43$	−1.02072	−0.155658	−0.0778290	+	0.996967i	$0.524799\pi$
$44$	0	0
$45$	−0.642349	−0.0957558
$46$	0	0
$47$	−9.29799	−1.35625	−0.678126	−	0.734946i	$-0.737207\pi$
$47$	−9.29799	−1.35625	−0.678126	+	0.734946i	$0.737207\pi$
$48$	0	0
$49$	−0.359430	−0.0513471
$50$	0	0
$51$	5.85477	0.819831
$52$	0	0
$53$	−8.77282	−1.20504	−0.602520	−	0.798104i	$-0.705836\pi$
$53$	−8.77282	−1.20504	−0.602520	+	0.798104i	$0.705836\pi$
$54$	0	0
$55$	1.02388	0.138059
$56$	0	0
$57$	4.15803	0.550744
$58$	0	0
$59$	1.90637	0.248188	0.124094	−	0.992270i	$-0.460398\pi$
$59$	1.90637	0.248188	0.124094	+	0.992270i	$0.460398\pi$
$60$	0	0
$61$	3.09875	0.396754	0.198377	−	0.980126i	$-0.436433\pi$
$61$	3.09875	0.396754	0.198377	+	0.980126i	$0.436433\pi$
$62$	0	0
$63$	−2.57693	−0.324663
$64$	0	0
$65$	1.93977	0.240599
$66$	0	0
$67$	0.795114	0.0971386	0.0485693	−	0.998820i	$-0.484534\pi$
$67$	0.795114	0.0971386	0.0485693	+	0.998820i	$0.484534\pi$
$68$	0	0
$69$	6.16672	0.742386
$70$	0	0
$71$	6.62085	0.785750	0.392875	−	0.919592i	$-0.371481\pi$
$71$	6.62085	0.785750	0.392875	+	0.919592i	$0.371481\pi$
$72$	0	0
$73$	5.75222	0.673247	0.336623	−	0.941639i	$-0.390715\pi$
$73$	5.75222	0.673247	0.336623	+	0.941639i	$0.390715\pi$
$74$	0	0
$75$	−4.58739	−0.529706
$76$	0	0
$77$	4.10751	0.468094
$78$	0	0
$79$	−15.7483	−1.77182	−0.885912	−	0.463854i	$-0.846466\pi$
$79$	−15.7483	−1.77182	−0.885912	+	0.463854i	$0.846466\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	5.53292	0.607316	0.303658	−	0.952781i	$-0.401792\pi$
$83$	5.53292	0.607316	0.303658	+	0.952781i	$0.401792\pi$
$84$	0	0
$85$	−3.76080	−0.407917
$86$	0	0
$87$	0.244932	0.0262595
$88$	0	0
$89$	6.54723	0.694005	0.347003	−	0.937864i	$-0.387200\pi$
$89$	6.54723	0.694005	0.347003	+	0.937864i	$0.387200\pi$
$90$	0	0
$91$	7.78184	0.815759
$92$	0	0
$93$	−5.80673	−0.602130
$94$	0	0
$95$	−2.67091	−0.274029
$96$	0	0
$97$	12.6589	1.28531	0.642657	−	0.766154i	$-0.277832\pi$
$97$	12.6589	1.28531	0.642657	+	0.766154i	$0.277832\pi$
$98$	0	0
$99$	−1.59395	−0.160198
$100$	0	0
$101$	−9.98911	−0.993953	−0.496977	−	0.867764i	$-0.665557\pi$
$101$	−9.98911	−0.993953	−0.496977	+	0.867764i	$0.665557\pi$
$102$	0	0
$103$	−6.44516	−0.635060	−0.317530	−	0.948248i	$-0.602853\pi$
$103$	−6.44516	−0.635060	−0.317530	+	0.948248i	$0.602853\pi$
$104$	0	0
$105$	1.65529	0.161540
$106$	0	0
$107$	−7.07502	−0.683968	−0.341984	−	0.939706i	$-0.611099\pi$
$107$	−7.07502	−0.683968	−0.341984	+	0.939706i	$0.611099\pi$
$108$	0	0
$109$	−12.6482	−1.21148	−0.605741	−	0.795662i	$-0.707123\pi$
$109$	−12.6482	−1.21148	−0.605741	+	0.795662i	$0.707123\pi$
$110$	0	0
$111$	−2.35969	−0.223972
$112$	0	0
$113$	−12.7692	−1.20122	−0.600612	−	0.799541i	$-0.705076\pi$
$113$	−12.7692	−1.20122	−0.600612	+	0.799541i	$0.705076\pi$
$114$	0	0
$115$	−3.96119	−0.369383
$116$	0	0
$117$	−3.01981	−0.279182
$118$	0	0
$119$	−15.0873	−1.38305
$120$	0	0
$121$	−8.45931	−0.769028
$122$	0	0
$123$	3.00916	0.271327
$124$	0	0
$125$	6.15845	0.550829
$126$	0	0
$127$	1.98151	0.175831	0.0879155	−	0.996128i	$-0.471979\pi$
$127$	1.98151	0.175831	0.0879155	+	0.996128i	$0.471979\pi$
$128$	0	0
$129$	−1.02072	−0.0898692
$130$	0	0
$131$	3.32004	0.290073	0.145037	−	0.989426i	$-0.453670\pi$
$131$	3.32004	0.290073	0.145037	+	0.989426i	$0.453670\pi$
$132$	0	0
$133$	−10.7149	−0.929104
$134$	0	0
$135$	−0.642349	−0.0552846
$136$	0	0
$137$	−19.6930	−1.68249	−0.841243	−	0.540657i	$-0.818176\pi$
$137$	−19.6930	−1.68249	−0.841243	+	0.540657i	$0.818176\pi$
$138$	0	0
$139$	−1.75068	−0.148491	−0.0742455	−	0.997240i	$-0.523655\pi$
$139$	−1.75068	−0.148491	−0.0742455	+	0.997240i	$0.523655\pi$
$140$	0	0
$141$	−9.29799	−0.783032
$142$	0	0
$143$	4.81344	0.402520
$144$	0	0
$145$	−0.157332	−0.0130657
$146$	0	0
$147$	−0.359430	−0.0296453
$148$	0	0
$149$	−19.8397	−1.62533	−0.812667	−	0.582729i	$-0.801985\pi$
$149$	−19.8397	−1.62533	−0.812667	+	0.582729i	$0.801985\pi$
$150$	0	0
$151$	13.2275	1.07644	0.538221	−	0.842804i	$-0.319097\pi$
$151$	13.2275	1.07644	0.538221	+	0.842804i	$0.319097\pi$
$152$	0	0
$153$	5.85477	0.473330
$154$	0	0
$155$	3.72995	0.299597
$156$	0	0
$157$	−2.79464	−0.223037	−0.111518	−	0.993762i	$-0.535571\pi$
$157$	−2.79464	−0.223037	−0.111518	+	0.993762i	$0.535571\pi$
$158$	0	0
$159$	−8.77282	−0.695730
$160$	0	0
$161$	−15.8912	−1.25240
$162$	0	0
$163$	0.834639	0.0653740	0.0326870	−	0.999466i	$-0.489594\pi$
$163$	0.834639	0.0653740	0.0326870	+	0.999466i	$0.489594\pi$
$164$	0	0
$165$	1.02388	0.0797086
$166$	0	0
$167$	−18.2561	−1.41270	−0.706351	−	0.707861i	$-0.749660\pi$
$167$	−18.2561	−1.41270	−0.706351	+	0.707861i	$0.749660\pi$
$168$	0	0
$169$	−3.88074	−0.298518
$170$	0	0
$171$	4.15803	0.317972
$172$	0	0
$173$	−6.01260	−0.457130	−0.228565	−	0.973529i	$-0.573403\pi$
$173$	−6.01260	−0.457130	−0.228565	+	0.973529i	$0.573403\pi$
$174$	0	0
$175$	11.8214	0.893612
$176$	0	0
$177$	1.90637	0.143291
$178$	0	0
$179$	2.51258	0.187799	0.0938996	−	0.995582i	$-0.470067\pi$
$179$	2.51258	0.187799	0.0938996	+	0.995582i	$0.470067\pi$
$180$	0	0
$181$	2.59082	0.192574	0.0962872	−	0.995354i	$-0.469303\pi$
$181$	2.59082	0.192574	0.0962872	+	0.995354i	$0.469303\pi$
$182$	0	0
$183$	3.09875	0.229066
$184$	0	0
$185$	1.51575	0.111440
$186$	0	0
$187$	−9.33223	−0.682440
$188$	0	0
$189$	−2.57693	−0.187444
$190$	0	0
$191$	−21.6020	−1.56307	−0.781533	−	0.623864i	$-0.785562\pi$
$191$	−21.6020	−1.56307	−0.781533	+	0.623864i	$0.785562\pi$
$192$	0	0
$193$	−15.6272	−1.12487	−0.562434	−	0.826843i	$-0.690135\pi$
$193$	−15.6272	−1.12487	−0.562434	+	0.826843i	$0.690135\pi$
$194$	0	0
$195$	1.93977	0.138910
$196$	0	0
$197$	−8.33545	−0.593876	−0.296938	−	0.954897i	$-0.595966\pi$
$197$	−8.33545	−0.593876	−0.296938	+	0.954897i	$0.595966\pi$
$198$	0	0
$199$	1.27093	0.0900942	0.0450471	−	0.998985i	$-0.485656\pi$
$199$	1.27093	0.0900942	0.0450471	+	0.998985i	$0.485656\pi$
$200$	0	0
$201$	0.795114	0.0560830
$202$	0	0
$203$	−0.631173	−0.0442996
$204$	0	0
$205$	−1.93293	−0.135002
$206$	0	0
$207$	6.16672	0.428617
$208$	0	0
$209$	−6.62771	−0.458448
$210$	0	0
$211$	−25.9437	−1.78604	−0.893019	−	0.450019i	$-0.851417\pi$
$211$	−25.9437	−1.78604	−0.893019	+	0.450019i	$0.851417\pi$
$212$	0	0
$213$	6.62085	0.453653
$214$	0	0
$215$	0.655657	0.0447155
$216$	0	0
$217$	14.9635	1.01579
$218$	0	0
$219$	5.75222	0.388699
$220$	0	0
$221$	−17.6803	−1.18930
$222$	0	0
$223$	26.9487	1.80462	0.902308	−	0.431091i	$-0.141871\pi$
$223$	26.9487	1.80462	0.902308	+	0.431091i	$0.141871\pi$
$224$	0	0
$225$	−4.58739	−0.305826
$226$	0	0
$227$	−6.41503	−0.425781	−0.212890	−	0.977076i	$-0.568288\pi$
$227$	−6.41503	−0.425781	−0.212890	+	0.977076i	$0.568288\pi$
$228$	0	0
$229$	21.6447	1.43032	0.715162	−	0.698959i	$-0.246353\pi$
$229$	21.6447	1.43032	0.715162	+	0.698959i	$0.246353\pi$
$230$	0	0
$231$	4.10751	0.270254
$232$	0	0
$233$	−7.27488	−0.476593	−0.238297	−	0.971192i	$-0.576589\pi$
$233$	−7.27488	−0.476593	−0.238297	+	0.971192i	$0.576589\pi$
$234$	0	0
$235$	5.97256	0.389607
$236$	0	0
$237$	−15.7483	−1.02296
$238$	0	0
$239$	−5.35849	−0.346612	−0.173306	−	0.984868i	$-0.555445\pi$
$239$	−5.35849	−0.346612	−0.173306	+	0.984868i	$0.555445\pi$
$240$	0	0
$241$	−14.9547	−0.963314	−0.481657	−	0.876360i	$-0.659965\pi$
$241$	−14.9547	−0.963314	−0.481657	+	0.876360i	$0.659965\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	0.230880	0.0147504
$246$	0	0
$247$	−12.5565	−0.798948
$248$	0	0
$249$	5.53292	0.350634
$250$	0	0
$251$	16.8397	1.06292	0.531458	−	0.847085i	$-0.321644\pi$
$251$	16.8397	1.06292	0.531458	+	0.847085i	$0.321644\pi$
$252$	0	0
$253$	−9.82948	−0.617974
$254$	0	0
$255$	−3.76080	−0.235511
$256$	0	0
$257$	2.48952	0.155292	0.0776461	−	0.996981i	$-0.475260\pi$
$257$	2.48952	0.155292	0.0776461	+	0.996981i	$0.475260\pi$
$258$	0	0
$259$	6.08076	0.377840
$260$	0	0
$261$	0.244932	0.0151609
$262$	0	0
$263$	18.4315	1.13653	0.568267	−	0.822844i	$-0.307614\pi$
$263$	18.4315	1.13653	0.568267	+	0.822844i	$0.307614\pi$
$264$	0	0
$265$	5.63521	0.346168
$266$	0	0
$267$	6.54723	0.400684
$268$	0	0
$269$	−7.78988	−0.474957	−0.237479	−	0.971393i	$-0.576321\pi$
$269$	−7.78988	−0.474957	−0.237479	+	0.971393i	$0.576321\pi$
$270$	0	0
$271$	−15.3470	−0.932262	−0.466131	−	0.884716i	$-0.654352\pi$
$271$	−15.3470	−0.932262	−0.466131	+	0.884716i	$0.654352\pi$
$272$	0	0
$273$	7.78184	0.470979
$274$	0	0
$275$	7.31209	0.440936
$276$	0	0
$277$	18.7422	1.12611	0.563055	−	0.826419i	$-0.309626\pi$
$277$	18.7422	1.12611	0.563055	+	0.826419i	$0.309626\pi$
$278$	0	0
$279$	−5.80673	−0.347640
$280$	0	0
$281$	−18.9025	−1.12763	−0.563813	−	0.825903i	$-0.690666\pi$
$281$	−18.9025	−1.12763	−0.563813	+	0.825903i	$0.690666\pi$
$282$	0	0
$283$	−9.54838	−0.567592	−0.283796	−	0.958885i	$-0.591594\pi$
$283$	−9.54838	−0.567592	−0.283796	+	0.958885i	$0.591594\pi$
$284$	0	0
$285$	−2.67091	−0.158211
$286$	0	0
$287$	−7.75440	−0.457728
$288$	0	0
$289$	17.2783	1.01637
$290$	0	0
$291$	12.6589	0.742077
$292$	0	0
$293$	30.6885	1.79284	0.896420	−	0.443205i	$-0.146159\pi$
$293$	30.6885	1.79284	0.896420	+	0.443205i	$0.146159\pi$
$294$	0	0
$295$	−1.22455	−0.0712963
$296$	0	0
$297$	−1.59395	−0.0924906
$298$	0	0
$299$	−18.6223	−1.07696
$300$	0	0
$301$	2.63032	0.151609
$302$	0	0
$303$	−9.98911	−0.573859
$304$	0	0
$305$	−1.99048	−0.113974
$306$	0	0
$307$	−9.47032	−0.540500	−0.270250	−	0.962790i	$-0.587106\pi$
$307$	−9.47032	−0.540500	−0.270250	+	0.962790i	$0.587106\pi$
$308$	0	0
$309$	−6.44516	−0.366652
$310$	0	0
$311$	−15.7391	−0.892483	−0.446242	−	0.894913i	$-0.647238\pi$
$311$	−15.7391	−0.892483	−0.446242	+	0.894913i	$0.647238\pi$
$312$	0	0
$313$	−10.6972	−0.604643	−0.302321	−	0.953206i	$-0.597762\pi$
$313$	−10.6972	−0.604643	−0.302321	+	0.953206i	$0.597762\pi$
$314$	0	0
$315$	1.65529	0.0932650
$316$	0	0
$317$	7.00074	0.393201	0.196600	−	0.980484i	$-0.437010\pi$
$317$	7.00074	0.393201	0.196600	+	0.980484i	$0.437010\pi$
$318$	0	0
$319$	−0.390411	−0.0218588
$320$	0	0
$321$	−7.07502	−0.394889
$322$	0	0
$323$	24.3443	1.35455
$324$	0	0
$325$	13.8530	0.768429
$326$	0	0
$327$	−12.6482	−0.699449
$328$	0	0
$329$	23.9603	1.32097
$330$	0	0
$331$	20.7091	1.13827	0.569137	−	0.822242i	$-0.307277\pi$
$331$	20.7091	1.13827	0.569137	+	0.822242i	$0.307277\pi$
$332$	0	0
$333$	−2.35969	−0.129310
$334$	0	0
$335$	−0.510741	−0.0279048
$336$	0	0
$337$	−11.7529	−0.640223	−0.320112	−	0.947380i	$-0.603720\pi$
$337$	−11.7529	−0.640223	−0.320112	+	0.947380i	$0.603720\pi$
$338$	0	0
$339$	−12.7692	−0.693527
$340$	0	0
$341$	9.25567	0.501223
$342$	0	0
$343$	18.9647	1.02400
$344$	0	0
$345$	−3.96119	−0.213263
$346$	0	0
$347$	5.39993	0.289883	0.144942	−	0.989440i	$-0.453701\pi$
$347$	5.39993	0.289883	0.144942	+	0.989440i	$0.453701\pi$
$348$	0	0
$349$	−6.03522	−0.323058	−0.161529	−	0.986868i	$-0.551643\pi$
$349$	−6.03522	−0.323058	−0.161529	+	0.986868i	$0.551643\pi$
$350$	0	0
$351$	−3.01981	−0.161186
$352$	0	0
$353$	−20.6927	−1.10136	−0.550679	−	0.834717i	$-0.685631\pi$
$353$	−20.6927	−1.10136	−0.550679	+	0.834717i	$0.685631\pi$
$354$	0	0
$355$	−4.25290	−0.225720
$356$	0	0
$357$	−15.0873	−0.798506
$358$	0	0
$359$	29.2602	1.54429	0.772147	−	0.635445i	$-0.219183\pi$
$359$	29.2602	1.54429	0.772147	+	0.635445i	$0.219183\pi$
$360$	0	0
$361$	−1.71081	−0.0900428
$362$	0	0
$363$	−8.45931	−0.443999
$364$	0	0
$365$	−3.69494	−0.193402
$366$	0	0
$367$	−14.9913	−0.782539	−0.391269	−	0.920276i	$-0.627964\pi$
$367$	−14.9913	−0.782539	−0.391269	+	0.920276i	$0.627964\pi$
$368$	0	0
$369$	3.00916	0.156651
$370$	0	0
$371$	22.6069	1.17369
$372$	0	0
$373$	−13.6212	−0.705277	−0.352638	−	0.935760i	$-0.614715\pi$
$373$	−13.6212	−0.705277	−0.352638	+	0.935760i	$0.614715\pi$
$374$	0	0
$375$	6.15845	0.318021
$376$	0	0
$377$	−0.739649	−0.0380938
$378$	0	0
$379$	20.5946	1.05788	0.528938	−	0.848661i	$-0.322591\pi$
$379$	20.5946	1.05788	0.528938	+	0.848661i	$0.322591\pi$
$380$	0	0
$381$	1.98151	0.101516
$382$	0	0
$383$	−35.7833	−1.82844	−0.914222	−	0.405214i	$-0.867197\pi$
$383$	−35.7833	−1.82844	−0.914222	+	0.405214i	$0.867197\pi$
$384$	0	0
$385$	−2.63846	−0.134468
$386$	0	0
$387$	−1.02072	−0.0518860
$388$	0	0
$389$	11.3770	0.576836	0.288418	−	0.957505i	$-0.406871\pi$
$389$	11.3770	0.576836	0.288418	+	0.957505i	$0.406871\pi$
$390$	0	0
$391$	36.1047	1.82589
$392$	0	0
$393$	3.32004	0.167474
$394$	0	0
$395$	10.1159	0.508987
$396$	0	0
$397$	−22.0959	−1.10896	−0.554482	−	0.832196i	$-0.687083\pi$
$397$	−22.0959	−1.10896	−0.554482	+	0.832196i	$0.687083\pi$
$398$	0	0
$399$	−10.7149	−0.536418
$400$	0	0
$401$	13.6270	0.680502	0.340251	−	0.940335i	$-0.389488\pi$
$401$	13.6270	0.680502	0.340251	+	0.940335i	$0.389488\pi$
$402$	0	0
$403$	17.5352	0.873492
$404$	0	0
$405$	−0.642349	−0.0319186
$406$	0	0
$407$	3.76124	0.186438
$408$	0	0
$409$	19.9098	0.984474	0.492237	−	0.870461i	$-0.336179\pi$
$409$	19.9098	0.984474	0.492237	+	0.870461i	$0.336179\pi$
$410$	0	0
$411$	−19.6930	−0.971384
$412$	0	0
$413$	−4.91258	−0.241732
$414$	0	0
$415$	−3.55406	−0.174462
$416$	0	0
$417$	−1.75068	−0.0857313
$418$	0	0
$419$	7.86876	0.384414	0.192207	−	0.981354i	$-0.438435\pi$
$419$	7.86876	0.384414	0.192207	+	0.981354i	$0.438435\pi$
$420$	0	0
$421$	−28.4676	−1.38743	−0.693713	−	0.720252i	$-0.744026\pi$
$421$	−28.4676	−1.38743	−0.693713	+	0.720252i	$0.744026\pi$
$422$	0	0
$423$	−9.29799	−0.452084
$424$	0	0
$425$	−26.8581	−1.30281
$426$	0	0
$427$	−7.98525	−0.386434
$428$	0	0
$429$	4.81344	0.232395
$430$	0	0
$431$	−32.9459	−1.58695	−0.793475	−	0.608603i	$-0.791730\pi$
$431$	−32.9459	−1.58695	−0.793475	+	0.608603i	$0.791730\pi$
$432$	0	0
$433$	17.8087	0.855830	0.427915	−	0.903819i	$-0.359248\pi$
$433$	17.8087	0.855830	0.427915	+	0.903819i	$0.359248\pi$
$434$	0	0
$435$	−0.157332	−0.00754349
$436$	0	0
$437$	25.6414	1.22659
$438$	0	0
$439$	30.6041	1.46066	0.730328	−	0.683096i	$-0.239367\pi$
$439$	30.6041	1.46066	0.730328	+	0.683096i	$0.239367\pi$
$440$	0	0
$441$	−0.359430	−0.0171157
$442$	0	0
$443$	−4.91149	−0.233352	−0.116676	−	0.993170i	$-0.537224\pi$
$443$	−4.91149	−0.233352	−0.116676	+	0.993170i	$0.537224\pi$
$444$	0	0
$445$	−4.20561	−0.199365
$446$	0	0
$447$	−19.8397	−0.938387
$448$	0	0
$449$	9.07590	0.428318	0.214159	−	0.976799i	$-0.431299\pi$
$449$	9.07590	0.428318	0.214159	+	0.976799i	$0.431299\pi$
$450$	0	0
$451$	−4.79647	−0.225857
$452$	0	0
$453$	13.2275	0.621484
$454$	0	0
$455$	−4.99866	−0.234341
$456$	0	0
$457$	−15.6041	−0.729929	−0.364964	−	0.931021i	$-0.618919\pi$
$457$	−15.6041	−0.729929	−0.364964	+	0.931021i	$0.618919\pi$
$458$	0	0
$459$	5.85477	0.273277
$460$	0	0
$461$	−19.1431	−0.891585	−0.445792	−	0.895136i	$-0.647078\pi$
$461$	−19.1431	−0.891585	−0.445792	+	0.895136i	$0.647078\pi$
$462$	0	0
$463$	9.87484	0.458923	0.229461	−	0.973318i	$-0.426304\pi$
$463$	9.87484	0.458923	0.229461	+	0.973318i	$0.426304\pi$
$464$	0	0
$465$	3.72995	0.172972
$466$	0	0
$467$	−13.8413	−0.640498	−0.320249	−	0.947333i	$-0.603767\pi$
$467$	−13.8413	−0.640498	−0.320249	+	0.947333i	$0.603767\pi$
$468$	0	0
$469$	−2.04895	−0.0946119
$470$	0	0
$471$	−2.79464	−0.128770
$472$	0	0
$473$	1.62698	0.0748085
$474$	0	0
$475$	−19.0745	−0.875197
$476$	0	0
$477$	−8.77282	−0.401680
$478$	0	0
$479$	−22.1039	−1.00995	−0.504977	−	0.863133i	$-0.668499\pi$
$479$	−22.1039	−1.00995	−0.504977	+	0.863133i	$0.668499\pi$
$480$	0	0
$481$	7.12582	0.324909
$482$	0	0
$483$	−15.8912	−0.723075
$484$	0	0
$485$	−8.13142	−0.369229
$486$	0	0
$487$	8.92524	0.404441	0.202221	−	0.979340i	$-0.435184\pi$
$487$	8.92524	0.404441	0.202221	+	0.979340i	$0.435184\pi$
$488$	0	0
$489$	0.834639	0.0377437
$490$	0	0
$491$	−25.3096	−1.14221	−0.571103	−	0.820878i	$-0.693484\pi$
$491$	−25.3096	−1.14221	−0.571103	+	0.820878i	$0.693484\pi$
$492$	0	0
$493$	1.43402	0.0645850
$494$	0	0
$495$	1.02388	0.0460198
$496$	0	0
$497$	−17.0615	−0.765311
$498$	0	0
$499$	−5.26088	−0.235509	−0.117755	−	0.993043i	$-0.537570\pi$
$499$	−5.26088	−0.235509	−0.117755	+	0.993043i	$0.537570\pi$
$500$	0	0
$501$	−18.2561	−0.815624
$502$	0	0
$503$	−1.00000	−0.0445878
$503$	−1.00000	−0.0445878
$504$	0	0
$505$	6.41650	0.285530
$506$	0	0
$507$	−3.88074	−0.172350
$508$	0	0
$509$	−17.3636	−0.769628	−0.384814	−	0.922994i	$-0.625734\pi$
$509$	−17.3636	−0.769628	−0.384814	+	0.922994i	$0.625734\pi$
$510$	0	0
$511$	−14.8231	−0.655734
$512$	0	0
$513$	4.15803	0.183581
$514$	0	0
$515$	4.14004	0.182432
$516$	0	0
$517$	14.8206	0.651808
$518$	0	0
$519$	−6.01260	−0.263924
$520$	0	0
$521$	32.6821	1.43183	0.715913	−	0.698189i	$-0.246011\pi$
$521$	32.6821	1.43183	0.715913	+	0.698189i	$0.246011\pi$
$522$	0	0
$523$	2.25399	0.0985602	0.0492801	−	0.998785i	$-0.484307\pi$
$523$	2.25399	0.0985602	0.0492801	+	0.998785i	$0.484307\pi$
$524$	0	0
$525$	11.8214	0.515927
$526$	0	0
$527$	−33.9971	−1.48094
$528$	0	0
$529$	15.0285	0.653412
$530$	0	0
$531$	1.90637	0.0827293
$532$	0	0
$533$	−9.08710	−0.393606
$534$	0	0
$535$	4.54463	0.196482
$536$	0	0
$537$	2.51258	0.108426
$538$	0	0
$539$	0.572915	0.0246772
$540$	0	0
$541$	26.8934	1.15624	0.578118	−	0.815953i	$-0.303787\pi$
$541$	26.8934	1.15624	0.578118	+	0.815953i	$0.303787\pi$
$542$	0	0
$543$	2.59082	0.111183
$544$	0	0
$545$	8.12458	0.348019
$546$	0	0
$547$	40.2434	1.72068	0.860342	−	0.509718i	$-0.170250\pi$
$547$	40.2434	1.72068	0.860342	+	0.509718i	$0.170250\pi$
$548$	0	0
$549$	3.09875	0.132251
$550$	0	0
$551$	1.01843	0.0433867
$552$	0	0
$553$	40.5823	1.72573
$554$	0	0
$555$	1.51575	0.0643398
$556$	0	0
$557$	39.4195	1.67026	0.835130	−	0.550053i	$-0.185393\pi$
$557$	39.4195	1.67026	0.835130	+	0.550053i	$0.185393\pi$
$558$	0	0
$559$	3.08238	0.130371
$560$	0	0
$561$	−9.33223	−0.394007
$562$	0	0
$563$	21.8100	0.919181	0.459590	−	0.888131i	$-0.347996\pi$
$563$	21.8100	0.919181	0.459590	+	0.888131i	$0.347996\pi$
$564$	0	0
$565$	8.20227	0.345072
$566$	0	0
$567$	−2.57693	−0.108221
$568$	0	0
$569$	2.32164	0.0973283	0.0486642	−	0.998815i	$-0.484504\pi$
$569$	2.32164	0.0973283	0.0486642	+	0.998815i	$0.484504\pi$
$570$	0	0
$571$	22.7309	0.951259	0.475629	−	0.879646i	$-0.342220\pi$
$571$	22.7309	0.951259	0.475629	+	0.879646i	$0.342220\pi$
$572$	0	0
$573$	−21.6020	−0.902436
$574$	0	0
$575$	−28.2892	−1.17974
$576$	0	0
$577$	20.7828	0.865200	0.432600	−	0.901586i	$-0.357596\pi$
$577$	20.7828	0.865200	0.432600	+	0.901586i	$0.357596\pi$
$578$	0	0
$579$	−15.6272	−0.649442
$580$	0	0
$581$	−14.2579	−0.591519
$582$	0	0
$583$	13.9835	0.579136
$584$	0	0
$585$	1.93977	0.0801998
$586$	0	0
$587$	16.3314	0.674070	0.337035	−	0.941492i	$-0.390576\pi$
$587$	16.3314	0.674070	0.337035	+	0.941492i	$0.390576\pi$
$588$	0	0
$589$	−24.1445	−0.994859
$590$	0	0
$591$	−8.33545	−0.342875
$592$	0	0
$593$	−22.4020	−0.919941	−0.459971	−	0.887934i	$-0.652140\pi$
$593$	−22.4020	−0.919941	−0.459971	+	0.887934i	$0.652140\pi$
$594$	0	0
$595$	9.69133	0.397306
$596$	0	0
$597$	1.27093	0.0520159
$598$	0	0
$599$	1.92173	0.0785197	0.0392599	−	0.999229i	$-0.487500\pi$
$599$	1.92173	0.0785197	0.0392599	+	0.999229i	$0.487500\pi$
$600$	0	0
$601$	−47.1530	−1.92341	−0.961705	−	0.274087i	$-0.911624\pi$
$601$	−47.1530	−1.92341	−0.961705	+	0.274087i	$0.911624\pi$
$602$	0	0
$603$	0.795114	0.0323795
$604$	0	0
$605$	5.43383	0.220917
$606$	0	0
$607$	11.4372	0.464221	0.232110	−	0.972689i	$-0.425437\pi$
$607$	11.4372	0.464221	0.232110	+	0.972689i	$0.425437\pi$
$608$	0	0
$609$	−0.631173	−0.0255764
$610$	0	0
$611$	28.0782	1.13592
$612$	0	0
$613$	42.2523	1.70656	0.853278	−	0.521457i	$-0.174611\pi$
$613$	42.2523	1.70656	0.853278	+	0.521457i	$0.174611\pi$
$614$	0	0
$615$	−1.93293	−0.0779434
$616$	0	0
$617$	−26.8177	−1.07964	−0.539819	−	0.841781i	$-0.681507\pi$
$617$	−26.8177	−1.07964	−0.539819	+	0.841781i	$0.681507\pi$
$618$	0	0
$619$	12.1200	0.487144	0.243572	−	0.969883i	$-0.421681\pi$
$619$	12.1200	0.487144	0.243572	+	0.969883i	$0.421681\pi$
$620$	0	0
$621$	6.16672	0.247462
$622$	0	0
$623$	−16.8718	−0.675953
$624$	0	0
$625$	18.9811	0.759242
$626$	0	0
$627$	−6.62771	−0.264685
$628$	0	0
$629$	−13.8154	−0.550857
$630$	0	0
$631$	20.5485	0.818021	0.409011	−	0.912530i	$-0.365874\pi$
$631$	20.5485	0.818021	0.409011	+	0.912530i	$0.365874\pi$
$632$	0	0
$633$	−25.9437	−1.03117
$634$	0	0
$635$	−1.27282	−0.0505105
$636$	0	0
$637$	1.08541	0.0430055
$638$	0	0
$639$	6.62085	0.261917
$640$	0	0
$641$	25.7509	1.01710	0.508550	−	0.861033i	$-0.330182\pi$
$641$	25.7509	1.01710	0.508550	+	0.861033i	$0.330182\pi$
$642$	0	0
$643$	29.6055	1.16753	0.583763	−	0.811924i	$-0.301580\pi$
$643$	29.6055	1.16753	0.583763	+	0.811924i	$0.301580\pi$
$644$	0	0
$645$	0.655657	0.0258165
$646$	0	0
$647$	−1.43032	−0.0562319	−0.0281159	−	0.999605i	$-0.508951\pi$
$647$	−1.43032	−0.0562319	−0.0281159	+	0.999605i	$0.508951\pi$
$648$	0	0
$649$	−3.03867	−0.119278
$650$	0	0
$651$	14.9635	0.586468
$652$	0	0
$653$	24.4989	0.958714	0.479357	−	0.877620i	$-0.340870\pi$
$653$	24.4989	0.958714	0.479357	+	0.877620i	$0.340870\pi$
$654$	0	0
$655$	−2.13263	−0.0833286
$656$	0	0
$657$	5.75222	0.224416
$658$	0	0
$659$	−17.2369	−0.671456	−0.335728	−	0.941959i	$-0.608982\pi$
$659$	−17.2369	−0.671456	−0.335728	+	0.941959i	$0.608982\pi$
$660$	0	0
$661$	−42.6049	−1.65714	−0.828569	−	0.559888i	$-0.810844\pi$
$661$	−42.6049	−1.65714	−0.828569	+	0.559888i	$0.810844\pi$
$662$	0	0
$663$	−17.6803	−0.686646
$664$	0	0
$665$	6.88274	0.266901
$666$	0	0
$667$	1.51043	0.0584840
$668$	0	0
$669$	26.9487	1.04190
$670$	0	0
$671$	−4.93926	−0.190678
$672$	0	0
$673$	−16.8197	−0.648354	−0.324177	−	0.945996i	$-0.605087\pi$
$673$	−16.8197	−0.648354	−0.324177	+	0.945996i	$0.605087\pi$
$674$	0	0
$675$	−4.58739	−0.176569
$676$	0	0
$677$	16.1519	0.620769	0.310385	−	0.950611i	$-0.399542\pi$
$677$	16.1519	0.620769	0.310385	+	0.950611i	$0.399542\pi$
$678$	0	0
$679$	−32.6211	−1.25188
$680$	0	0
$681$	−6.41503	−0.245825
$682$	0	0
$683$	−10.2592	−0.392557	−0.196279	−	0.980548i	$-0.562886\pi$
$683$	−10.2592	−0.392557	−0.196279	+	0.980548i	$0.562886\pi$
$684$	0	0
$685$	12.6498	0.483323
$686$	0	0
$687$	21.6447	0.825798
$688$	0	0
$689$	26.4923	1.00927
$690$	0	0
$691$	−11.9842	−0.455901	−0.227951	−	0.973673i	$-0.573202\pi$
$691$	−11.9842	−0.455901	−0.227951	+	0.973673i	$0.573202\pi$
$692$	0	0
$693$	4.10751	0.156031
$694$	0	0
$695$	1.12455	0.0426566
$696$	0	0
$697$	17.6179	0.667327
$698$	0	0
$699$	−7.27488	−0.275161
$700$	0	0
$701$	43.9885	1.66142	0.830712	−	0.556703i	$-0.187934\pi$
$701$	43.9885	1.66142	0.830712	+	0.556703i	$0.187934\pi$
$702$	0	0
$703$	−9.81166	−0.370054
$704$	0	0
$705$	5.97256	0.224940
$706$	0	0
$707$	25.7412	0.968099
$708$	0	0
$709$	−14.7967	−0.555701	−0.277850	−	0.960624i	$-0.589622\pi$
$709$	−14.7967	−0.555701	−0.277850	+	0.960624i	$0.589622\pi$
$710$	0	0
$711$	−15.7483	−0.590608
$712$	0	0
$713$	−35.8085	−1.34104
$714$	0	0
$715$	−3.09191	−0.115631
$716$	0	0
$717$	−5.35849	−0.200117
$718$	0	0
$719$	37.9691	1.41601	0.708004	−	0.706208i	$-0.249596\pi$
$719$	37.9691	1.41601	0.708004	+	0.706208i	$0.249596\pi$
$720$	0	0
$721$	16.6087	0.618541
$722$	0	0
$723$	−14.9547	−0.556170
$724$	0	0
$725$	−1.12360	−0.0417294
$726$	0	0
$727$	−18.1719	−0.673957	−0.336979	−	0.941512i	$-0.609405\pi$
$727$	−18.1719	−0.673957	−0.336979	+	0.941512i	$0.609405\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−5.97606	−0.221033
$732$	0	0
$733$	−34.3399	−1.26837	−0.634187	−	0.773180i	$-0.718665\pi$
$733$	−34.3399	−1.26837	−0.634187	+	0.773180i	$0.718665\pi$
$734$	0	0
$735$	0.230880	0.00851612
$736$	0	0
$737$	−1.26738	−0.0466844
$738$	0	0
$739$	−22.4903	−0.827319	−0.413660	−	0.910432i	$-0.635750\pi$
$739$	−22.4903	−0.827319	−0.413660	+	0.910432i	$0.635750\pi$
$740$	0	0
$741$	−12.5565	−0.461273
$742$	0	0
$743$	−26.8751	−0.985950	−0.492975	−	0.870043i	$-0.664091\pi$
$743$	−26.8751	−0.985950	−0.492975	+	0.870043i	$0.664091\pi$
$744$	0	0
$745$	12.7440	0.466905
$746$	0	0
$747$	5.53292	0.202439
$748$	0	0
$749$	18.2318	0.666177
$750$	0	0
$751$	52.2289	1.90586	0.952930	−	0.303190i	$-0.0980515\pi$
$751$	52.2289	1.90586	0.952930	+	0.303190i	$0.0980515\pi$
$752$	0	0
$753$	16.8397	0.613674
$754$	0	0
$755$	−8.49670	−0.309227
$756$	0	0
$757$	−34.3553	−1.24867	−0.624333	−	0.781159i	$-0.714629\pi$
$757$	−34.3553	−1.24867	−0.624333	+	0.781159i	$0.714629\pi$
$758$	0	0
$759$	−9.82948	−0.356788
$760$	0	0
$761$	36.8627	1.33627	0.668136	−	0.744039i	$-0.267092\pi$
$761$	36.8627	1.33627	0.668136	+	0.744039i	$0.267092\pi$
$762$	0	0
$763$	32.5936	1.17997
$764$	0	0
$765$	−3.76080	−0.135972
$766$	0	0
$767$	−5.75687	−0.207869
$768$	0	0
$769$	26.2610	0.946998	0.473499	−	0.880794i	$-0.342991\pi$
$769$	26.2610	0.946998	0.473499	+	0.880794i	$0.342991\pi$
$770$	0	0
$771$	2.48952	0.0896580
$772$	0	0
$773$	27.2625	0.980563	0.490282	−	0.871564i	$-0.336894\pi$
$773$	27.2625	0.980563	0.490282	+	0.871564i	$0.336894\pi$
$774$	0	0
$775$	26.6377	0.956856
$776$	0	0
$777$	6.08076	0.218146
$778$	0	0
$779$	12.5122	0.448295
$780$	0	0
$781$	−10.5533	−0.377628
$782$	0	0
$783$	0.244932	0.00875316
$784$	0	0
$785$	1.79514	0.0640711
$786$	0	0
$787$	10.7898	0.384614	0.192307	−	0.981335i	$-0.438403\pi$
$787$	10.7898	0.384614	0.192307	+	0.981335i	$0.438403\pi$
$788$	0	0
$789$	18.4315	0.656178
$790$	0	0
$791$	32.9053	1.16998
$792$	0	0
$793$	−9.35763	−0.332299
$794$	0	0
$795$	5.63521	0.199860
$796$	0	0
$797$	−45.2526	−1.60293	−0.801465	−	0.598042i	$-0.795946\pi$
$797$	−45.2526	−1.60293	−0.801465	+	0.598042i	$0.795946\pi$
$798$	0	0
$799$	−54.4376	−1.92586
$800$	0	0
$801$	6.54723	0.231335
$802$	0	0
$803$	−9.16878	−0.323559
$804$	0	0
$805$	10.2077	0.359775
$806$	0	0
$807$	−7.78988	−0.274217
$808$	0	0
$809$	−49.7421	−1.74884	−0.874420	−	0.485170i	$-0.838758\pi$
$809$	−49.7421	−1.74884	−0.874420	+	0.485170i	$0.838758\pi$
$810$	0	0
$811$	40.6208	1.42639	0.713194	−	0.700967i	$-0.247248\pi$
$811$	40.6208	1.42639	0.713194	+	0.700967i	$0.247248\pi$
$812$	0	0
$813$	−15.3470	−0.538242
$814$	0	0
$815$	−0.536130	−0.0187798
$816$	0	0
$817$	−4.24417	−0.148485
$818$	0	0
$819$	7.78184	0.271920
$820$	0	0
$821$	−25.9803	−0.906720	−0.453360	−	0.891328i	$-0.649775\pi$
$821$	−25.9803	−0.906720	−0.453360	+	0.891328i	$0.649775\pi$
$822$	0	0
$823$	41.4735	1.44567	0.722837	−	0.691018i	$-0.242838\pi$
$823$	41.4735	1.44567	0.722837	+	0.691018i	$0.242838\pi$
$824$	0	0
$825$	7.31209	0.254574
$826$	0	0
$827$	−1.17722	−0.0409358	−0.0204679	−	0.999791i	$-0.506516\pi$
$827$	−1.17722	−0.0409358	−0.0204679	+	0.999791i	$0.506516\pi$
$828$	0	0
$829$	2.51815	0.0874590	0.0437295	−	0.999043i	$-0.486076\pi$
$829$	2.51815	0.0874590	0.0437295	+	0.999043i	$0.486076\pi$
$830$	0	0
$831$	18.7422	0.650160
$832$	0	0
$833$	−2.10438	−0.0729124
$834$	0	0
$835$	11.7268	0.405823
$836$	0	0
$837$	−5.80673	−0.200710
$838$	0	0
$839$	−25.4576	−0.878895	−0.439447	−	0.898268i	$-0.644826\pi$
$839$	−25.4576	−0.878895	−0.439447	+	0.898268i	$0.644826\pi$
$840$	0	0
$841$	−28.9400	−0.997931
$842$	0	0
$843$	−18.9025	−0.651035
$844$	0	0
$845$	2.49279	0.0857546
$846$	0	0
$847$	21.7990	0.749024
$848$	0	0
$849$	−9.54838	−0.327699
$850$	0	0
$851$	−14.5516	−0.498821
$852$	0	0
$853$	38.7461	1.32664	0.663320	−	0.748336i	$-0.269147\pi$
$853$	38.7461	1.32664	0.663320	+	0.748336i	$0.269147\pi$
$854$	0	0
$855$	−2.67091	−0.0913430
$856$	0	0
$857$	4.95621	0.169301	0.0846505	−	0.996411i	$-0.473023\pi$
$857$	4.95621	0.169301	0.0846505	+	0.996411i	$0.473023\pi$
$858$	0	0
$859$	22.9446	0.782861	0.391430	−	0.920208i	$-0.371980\pi$
$859$	22.9446	0.782861	0.391430	+	0.920208i	$0.371980\pi$
$860$	0	0
$861$	−7.75440	−0.264269
$862$	0	0
$863$	−15.6756	−0.533605	−0.266802	−	0.963751i	$-0.585967\pi$
$863$	−15.6756	−0.533605	−0.266802	+	0.963751i	$0.585967\pi$
$864$	0	0
$865$	3.86219	0.131318
$866$	0	0
$867$	17.2783	0.586801
$868$	0	0
$869$	25.1021	0.851530
$870$	0	0
$871$	−2.40109	−0.0813580
$872$	0	0
$873$	12.6589	0.428438
$874$	0	0
$875$	−15.8699	−0.536501
$876$	0	0
$877$	−18.5255	−0.625562	−0.312781	−	0.949825i	$-0.601261\pi$
$877$	−18.5255	−0.625562	−0.312781	+	0.949825i	$0.601261\pi$
$878$	0	0
$879$	30.6885	1.03510
$880$	0	0
$881$	−26.9982	−0.909593	−0.454797	−	0.890595i	$-0.650288\pi$
$881$	−26.9982	−0.909593	−0.454797	+	0.890595i	$0.650288\pi$
$882$	0	0
$883$	−29.7939	−1.00264	−0.501322	−	0.865261i	$-0.667153\pi$
$883$	−29.7939	−1.00264	−0.501322	+	0.865261i	$0.667153\pi$
$884$	0	0
$885$	−1.22455	−0.0411629
$886$	0	0
$887$	−19.0104	−0.638306	−0.319153	−	0.947703i	$-0.603398\pi$
$887$	−19.0104	−0.638306	−0.319153	+	0.947703i	$0.603398\pi$
$888$	0	0
$889$	−5.10622	−0.171257
$890$	0	0
$891$	−1.59395	−0.0533995
$892$	0	0
$893$	−38.6613	−1.29375
$894$	0	0
$895$	−1.61396	−0.0539486
$896$	0	0
$897$	−18.6223	−0.621782
$898$	0	0
$899$	−1.42225	−0.0474349
$900$	0	0
$901$	−51.3628	−1.71114
$902$	0	0
$903$	2.63032	0.0875315
$904$	0	0
$905$	−1.66421	−0.0553203
$906$	0	0
$907$	15.0562	0.499932	0.249966	−	0.968255i	$-0.419581\pi$
$907$	15.0562	0.499932	0.249966	+	0.968255i	$0.419581\pi$
$908$	0	0
$909$	−9.98911	−0.331318
$910$	0	0
$911$	−5.37110	−0.177953	−0.0889763	−	0.996034i	$-0.528360\pi$
$911$	−5.37110	−0.177953	−0.0889763	+	0.996034i	$0.528360\pi$
$912$	0	0
$913$	−8.81922	−0.291873
$914$	0	0
$915$	−1.99048	−0.0658032
$916$	0	0
$917$	−8.55552	−0.282528
$918$	0	0
$919$	4.06392	0.134056	0.0670281	−	0.997751i	$-0.478648\pi$
$919$	4.06392	0.134056	0.0670281	+	0.997751i	$0.478648\pi$
$920$	0	0
$921$	−9.47032	−0.312058
$922$	0	0
$923$	−19.9937	−0.658101
$924$	0	0
$925$	10.8248	0.355918
$926$	0	0
$927$	−6.44516	−0.211687
$928$	0	0
$929$	21.5814	0.708061	0.354031	−	0.935234i	$-0.384811\pi$
$929$	21.5814	0.708061	0.354031	+	0.935234i	$0.384811\pi$
$930$	0	0
$931$	−1.49452	−0.0489809
$932$	0	0
$933$	−15.7391	−0.515275
$934$	0	0
$935$	5.99455	0.196043
$936$	0	0
$937$	17.1321	0.559683	0.279841	−	0.960046i	$-0.409718\pi$
$937$	17.1321	0.559683	0.279841	+	0.960046i	$0.409718\pi$
$938$	0	0
$939$	−10.6972	−0.349091
$940$	0	0
$941$	−52.4335	−1.70928	−0.854642	−	0.519218i	$-0.826223\pi$
$941$	−52.4335	−1.70928	−0.854642	+	0.519218i	$0.826223\pi$
$942$	0	0
$943$	18.5567	0.604288
$944$	0	0
$945$	1.65529	0.0538466
$946$	0	0
$947$	−26.4725	−0.860239	−0.430120	−	0.902772i	$-0.641529\pi$
$947$	−26.4725	−0.860239	−0.430120	+	0.902772i	$0.641529\pi$
$948$	0	0
$949$	−17.3706	−0.563874
$950$	0	0
$951$	7.00074	0.227014
$952$	0	0
$953$	−11.1278	−0.360465	−0.180232	−	0.983624i	$-0.557685\pi$
$953$	−11.1278	−0.360465	−0.180232	+	0.983624i	$0.557685\pi$
$954$	0	0
$955$	13.8760	0.449018
$956$	0	0
$957$	−0.390411	−0.0126202
$958$	0	0
$959$	50.7475	1.63872
$960$	0	0
$961$	2.71814	0.0876820
$962$	0	0
$963$	−7.07502	−0.227989
$964$	0	0
$965$	10.0381	0.323138
$966$	0	0
$967$	−7.69509	−0.247458	−0.123729	−	0.992316i	$-0.539485\pi$
$967$	−7.69509	−0.247458	−0.123729	+	0.992316i	$0.539485\pi$
$968$	0	0
$969$	24.3443	0.782051
$970$	0	0
$971$	49.7998	1.59815	0.799076	−	0.601231i	$-0.205323\pi$
$971$	49.7998	1.59815	0.799076	+	0.601231i	$0.205323\pi$
$972$	0	0
$973$	4.51139	0.144628
$974$	0	0
$975$	13.8530	0.443652
$976$	0	0
$977$	42.2241	1.35087	0.675435	−	0.737420i	$-0.263956\pi$
$977$	42.2241	1.35087	0.675435	+	0.737420i	$0.263956\pi$
$978$	0	0
$979$	−10.4360	−0.333536
$980$	0	0
$981$	−12.6482	−0.403827
$982$	0	0
$983$	27.4950	0.876953	0.438476	−	0.898743i	$-0.355518\pi$
$983$	27.4950	0.876953	0.438476	+	0.898743i	$0.355518\pi$
$984$	0	0
$985$	5.35427	0.170601
$986$	0	0
$987$	23.9603	0.762664
$988$	0	0
$989$	−6.29448	−0.200153
$990$	0	0
$991$	34.2405	1.08768	0.543842	−	0.839187i	$-0.316969\pi$
$991$	34.2405	1.08768	0.543842	+	0.839187i	$0.316969\pi$
$992$	0	0
$993$	20.7091	0.657183
$994$	0	0
$995$	−0.816384	−0.0258811
$996$	0	0
$997$	−56.8944	−1.80186	−0.900932	−	0.433959i	$-0.857116\pi$
$997$	−56.8944	−1.80186	−0.900932	+	0.433959i	$0.857116\pi$
$998$	0	0
$999$	−2.35969	−0.0746573

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6036.2.a.g.1.8	✓	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6036.2.a.g.1.8	✓	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -0.642349 q^{5} -2.57693 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -0.642349 q^{5} -2.57693 q^{7} +1.00000 q^{9} -1.59395 q^{11} -3.01981 q^{13} -0.642349 q^{15} +5.85477 q^{17} +4.15803 q^{19} -2.57693 q^{21} +6.16672 q^{23} -4.58739 q^{25} +1.00000 q^{27} +0.244932 q^{29} -5.80673 q^{31} -1.59395 q^{33} +1.65529 q^{35} -2.35969 q^{37} -3.01981 q^{39} +3.00916 q^{41} -1.02072 q^{43} -0.642349 q^{45} -9.29799 q^{47} -0.359430 q^{49} +5.85477 q^{51} -8.77282 q^{53} +1.02388 q^{55} +4.15803 q^{57} +1.90637 q^{59} +3.09875 q^{61} -2.57693 q^{63} +1.93977 q^{65} +0.795114 q^{67} +6.16672 q^{69} +6.62085 q^{71} +5.75222 q^{73} -4.58739 q^{75} +4.10751 q^{77} -15.7483 q^{79} +1.00000 q^{81} +5.53292 q^{83} -3.76080 q^{85} +0.244932 q^{87} +6.54723 q^{89} +7.78184 q^{91} -5.80673 q^{93} -2.67091 q^{95} +12.6589 q^{97} -1.59395 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q + 15 q^{3} - 11 q^{5} - 4 q^{7} + 15 q^{9}+O(q^{10}) \) 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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