LMFDB - Embedded newform 6036.2.a.g.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:	$1$
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			1.15
Root			$3.64439$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} +2.64439 q^{5} -3.02199 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +2.64439 q^{5} -3.02199 q^{7} +1.00000 q^{9} +0.587142 q^{11} -2.23197 q^{13} +2.64439 q^{15} +1.24552 q^{17} -0.491597 q^{19} -3.02199 q^{21} -0.916648 q^{23} +1.99280 q^{25} +1.00000 q^{27} -5.33298 q^{29} -8.59992 q^{31} +0.587142 q^{33} -7.99132 q^{35} -11.8705 q^{37} -2.23197 q^{39} -2.54414 q^{41} +1.12078 q^{43} +2.64439 q^{45} -2.52693 q^{47} +2.13243 q^{49} +1.24552 q^{51} -0.789358 q^{53} +1.55263 q^{55} -0.491597 q^{57} +9.14016 q^{59} -9.78118 q^{61} -3.02199 q^{63} -5.90220 q^{65} -7.68153 q^{67} -0.916648 q^{69} +3.65743 q^{71} -6.58923 q^{73} +1.99280 q^{75} -1.77434 q^{77} +7.69343 q^{79} +1.00000 q^{81} +11.9522 q^{83} +3.29364 q^{85} -5.33298 q^{87} +3.28897 q^{89} +6.74500 q^{91} -8.59992 q^{93} -1.29997 q^{95} -8.50441 q^{97} +0.587142 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	2.64439	1.18261	0.591303	−	0.806449i	$-0.298614\pi$
$5$	2.64439	1.18261	0.591303	+	0.806449i	$0.298614\pi$
$6$	0	0
$7$	−3.02199	−1.14221	−0.571103	−	0.820879i	$-0.693484\pi$
$7$	−3.02199	−1.14221	−0.571103	+	0.820879i	$0.693484\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	0.587142	0.177030	0.0885150	−	0.996075i	$-0.471788\pi$
$11$	0.587142	0.177030	0.0885150	+	0.996075i	$0.471788\pi$
$12$	0	0
$13$	−2.23197	−0.619038	−0.309519	−	0.950893i	$-0.600168\pi$
$13$	−2.23197	−0.619038	−0.309519	+	0.950893i	$0.600168\pi$
$14$	0	0
$15$	2.64439	0.682778
$16$	0	0
$17$	1.24552	0.302083	0.151041	−	0.988527i	$-0.451737\pi$
$17$	1.24552	0.302083	0.151041	+	0.988527i	$0.451737\pi$
$18$	0	0
$19$	−0.491597	−0.112780	−0.0563900	−	0.998409i	$-0.517959\pi$
$19$	−0.491597	−0.112780	−0.0563900	+	0.998409i	$0.517959\pi$
$20$	0	0
$21$	−3.02199	−0.659452
$22$	0	0
$23$	−0.916648	−0.191134	−0.0955671	−	0.995423i	$-0.530466\pi$
$23$	−0.916648	−0.191134	−0.0955671	+	0.995423i	$0.530466\pi$
$24$	0	0
$25$	1.99280	0.398559
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−5.33298	−0.990309	−0.495155	−	0.868805i	$-0.664889\pi$
$29$	−5.33298	−0.990309	−0.495155	+	0.868805i	$0.664889\pi$
$30$	0	0
$31$	−8.59992	−1.54459	−0.772296	−	0.635263i	$-0.780892\pi$
$31$	−8.59992	−1.54459	−0.772296	+	0.635263i	$0.780892\pi$
$32$	0	0
$33$	0.587142	0.102208
$34$	0	0
$35$	−7.99132	−1.35078
$36$	0	0
$37$	−11.8705	−1.95150	−0.975748	−	0.218895i	$-0.929755\pi$
$37$	−11.8705	−1.95150	−0.975748	+	0.218895i	$0.929755\pi$
$38$	0	0
$39$	−2.23197	−0.357402
$40$	0	0
$41$	−2.54414	−0.397328	−0.198664	−	0.980068i	$-0.563660\pi$
$41$	−2.54414	−0.397328	−0.198664	+	0.980068i	$0.563660\pi$
$42$	0	0
$43$	1.12078	0.170918	0.0854590	−	0.996342i	$-0.472764\pi$
$43$	1.12078	0.170918	0.0854590	+	0.996342i	$0.472764\pi$
$44$	0	0
$45$	2.64439	0.394202
$46$	0	0
$47$	−2.52693	−0.368590	−0.184295	−	0.982871i	$-0.559000\pi$
$47$	−2.52693	−0.368590	−0.184295	+	0.982871i	$0.559000\pi$
$48$	0	0
$49$	2.13243	0.304632
$50$	0	0
$51$	1.24552	0.174408
$52$	0	0
$53$	−0.789358	−0.108427	−0.0542133	−	0.998529i	$-0.517265\pi$
$53$	−0.789358	−0.108427	−0.0542133	+	0.998529i	$0.517265\pi$
$54$	0	0
$55$	1.55263	0.209357
$56$	0	0
$57$	−0.491597	−0.0651136
$58$	0	0
$59$	9.14016	1.18995	0.594973	−	0.803745i	$-0.297163\pi$
$59$	9.14016	1.18995	0.594973	+	0.803745i	$0.297163\pi$
$60$	0	0
$61$	−9.78118	−1.25235	−0.626176	−	0.779682i	$-0.715381\pi$
$61$	−9.78118	−1.25235	−0.626176	+	0.779682i	$0.715381\pi$
$62$	0	0
$63$	−3.02199	−0.380735
$64$	0	0
$65$	−5.90220	−0.732078
$66$	0	0
$67$	−7.68153	−0.938449	−0.469224	−	0.883079i	$-0.655466\pi$
$67$	−7.68153	−0.938449	−0.469224	+	0.883079i	$0.655466\pi$
$68$	0	0
$69$	−0.916648	−0.110351
$70$	0	0
$71$	3.65743	0.434057	0.217028	−	0.976165i	$-0.430364\pi$
$71$	3.65743	0.434057	0.217028	+	0.976165i	$0.430364\pi$
$72$	0	0
$73$	−6.58923	−0.771210	−0.385605	−	0.922664i	$-0.626007\pi$
$73$	−6.58923	−0.771210	−0.385605	+	0.922664i	$0.626007\pi$
$74$	0	0
$75$	1.99280	0.230108
$76$	0	0
$77$	−1.77434	−0.202204
$78$	0	0
$79$	7.69343	0.865578	0.432789	−	0.901495i	$-0.357530\pi$
$79$	7.69343	0.865578	0.432789	+	0.901495i	$0.357530\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	11.9522	1.31192	0.655962	−	0.754794i	$-0.272263\pi$
$83$	11.9522	1.31192	0.655962	+	0.754794i	$0.272263\pi$
$84$	0	0
$85$	3.29364	0.357245
$86$	0	0
$87$	−5.33298	−0.571755
$88$	0	0
$89$	3.28897	0.348630	0.174315	−	0.984690i	$-0.444229\pi$
$89$	3.28897	0.348630	0.174315	+	0.984690i	$0.444229\pi$
$90$	0	0
$91$	6.74500	0.707068
$92$	0	0
$93$	−8.59992	−0.891770
$94$	0	0
$95$	−1.29997	−0.133375
$96$	0	0
$97$	−8.50441	−0.863492	−0.431746	−	0.901995i	$-0.642102\pi$
$97$	−8.50441	−0.863492	−0.431746	+	0.901995i	$0.642102\pi$
$98$	0	0
$99$	0.587142	0.0590100
$100$	0	0
$101$	−6.83549	−0.680157	−0.340079	−	0.940397i	$-0.610454\pi$
$101$	−6.83549	−0.680157	−0.340079	+	0.940397i	$0.610454\pi$
$102$	0	0
$103$	−1.17906	−0.116176	−0.0580881	−	0.998311i	$-0.518500\pi$
$103$	−1.17906	−0.116176	−0.0580881	+	0.998311i	$0.518500\pi$
$104$	0	0
$105$	−7.99132	−0.779873
$106$	0	0
$107$	15.7476	1.52238	0.761189	−	0.648530i	$-0.224616\pi$
$107$	15.7476	1.52238	0.761189	+	0.648530i	$0.224616\pi$
$108$	0	0
$109$	6.71332	0.643019	0.321510	−	0.946906i	$-0.395810\pi$
$109$	6.71332	0.643019	0.321510	+	0.946906i	$0.395810\pi$
$110$	0	0
$111$	−11.8705	−1.12670
$112$	0	0
$113$	10.7161	1.00808	0.504042	−	0.863679i	$-0.331846\pi$
$113$	10.7161	1.00808	0.504042	+	0.863679i	$0.331846\pi$
$114$	0	0
$115$	−2.42397	−0.226037
$116$	0	0
$117$	−2.23197	−0.206346
$118$	0	0
$119$	−3.76395	−0.345041
$120$	0	0
$121$	−10.6553	−0.968660
$122$	0	0
$123$	−2.54414	−0.229397
$124$	0	0
$125$	−7.95222	−0.711268
$126$	0	0
$127$	1.93781	0.171953	0.0859766	−	0.996297i	$-0.472599\pi$
$127$	1.93781	0.171953	0.0859766	+	0.996297i	$0.472599\pi$
$128$	0	0
$129$	1.12078	0.0986796
$130$	0	0
$131$	−8.74741	−0.764265	−0.382132	−	0.924108i	$-0.624810\pi$
$131$	−8.74741	−0.764265	−0.382132	+	0.924108i	$0.624810\pi$
$132$	0	0
$133$	1.48560	0.128818
$134$	0	0
$135$	2.64439	0.227593
$136$	0	0
$137$	13.4646	1.15036	0.575181	−	0.818026i	$-0.304932\pi$
$137$	13.4646	1.15036	0.575181	+	0.818026i	$0.304932\pi$
$138$	0	0
$139$	−6.12589	−0.519591	−0.259796	−	0.965664i	$-0.583655\pi$
$139$	−6.12589	−0.519591	−0.259796	+	0.965664i	$0.583655\pi$
$140$	0	0
$141$	−2.52693	−0.212806
$142$	0	0
$143$	−1.31048	−0.109588
$144$	0	0
$145$	−14.1025	−1.17115
$146$	0	0
$147$	2.13243	0.175880
$148$	0	0
$149$	6.53890	0.535687	0.267844	−	0.963462i	$-0.413689\pi$
$149$	6.53890	0.535687	0.267844	+	0.963462i	$0.413689\pi$
$150$	0	0
$151$	−16.6852	−1.35782	−0.678912	−	0.734219i	$-0.737548\pi$
$151$	−16.6852	−1.35782	−0.678912	+	0.734219i	$0.737548\pi$
$152$	0	0
$153$	1.24552	0.100694
$154$	0	0
$155$	−22.7415	−1.82664
$156$	0	0
$157$	−6.35568	−0.507238	−0.253619	−	0.967304i	$-0.581621\pi$
$157$	−6.35568	−0.507238	−0.253619	+	0.967304i	$0.581621\pi$
$158$	0	0
$159$	−0.789358	−0.0626002
$160$	0	0
$161$	2.77010	0.218315
$162$	0	0
$163$	12.1987	0.955477	0.477739	−	0.878502i	$-0.341457\pi$
$163$	12.1987	0.955477	0.477739	+	0.878502i	$0.341457\pi$
$164$	0	0
$165$	1.55263	0.120872
$166$	0	0
$167$	−14.8626	−1.15011	−0.575053	−	0.818116i	$-0.695018\pi$
$167$	−14.8626	−1.15011	−0.575053	+	0.818116i	$0.695018\pi$
$168$	0	0
$169$	−8.01830	−0.616792
$170$	0	0
$171$	−0.491597	−0.0375934
$172$	0	0
$173$	7.02508	0.534107	0.267054	−	0.963682i	$-0.413950\pi$
$173$	7.02508	0.534107	0.267054	+	0.963682i	$0.413950\pi$
$174$	0	0
$175$	−6.02221	−0.455236
$176$	0	0
$177$	9.14016	0.687016
$178$	0	0
$179$	5.16311	0.385909	0.192955	−	0.981208i	$-0.438193\pi$
$179$	5.16311	0.385909	0.192955	+	0.981208i	$0.438193\pi$
$180$	0	0
$181$	−1.26496	−0.0940238	−0.0470119	−	0.998894i	$-0.514970\pi$
$181$	−1.26496	−0.0940238	−0.0470119	+	0.998894i	$0.514970\pi$
$182$	0	0
$183$	−9.78118	−0.723046
$184$	0	0
$185$	−31.3902	−2.30785
$186$	0	0
$187$	0.731297	0.0534777
$188$	0	0
$189$	−3.02199	−0.219817
$190$	0	0
$191$	6.03083	0.436376	0.218188	−	0.975907i	$-0.429985\pi$
$191$	6.03083	0.436376	0.218188	+	0.975907i	$0.429985\pi$
$192$	0	0
$193$	−1.06344	−0.0765480	−0.0382740	−	0.999267i	$-0.512186\pi$
$193$	−1.06344	−0.0765480	−0.0382740	+	0.999267i	$0.512186\pi$
$194$	0	0
$195$	−5.90220	−0.422666
$196$	0	0
$197$	−22.3630	−1.59330	−0.796649	−	0.604443i	$-0.793396\pi$
$197$	−22.3630	−1.59330	−0.796649	+	0.604443i	$0.793396\pi$
$198$	0	0
$199$	−22.8245	−1.61799	−0.808993	−	0.587818i	$-0.799987\pi$
$199$	−22.8245	−1.61799	−0.808993	+	0.587818i	$0.799987\pi$
$200$	0	0
$201$	−7.68153	−0.541814
$202$	0	0
$203$	16.1162	1.13114
$204$	0	0
$205$	−6.72770	−0.469883
$206$	0	0
$207$	−0.916648	−0.0637114
$208$	0	0
$209$	−0.288637	−0.0199654
$210$	0	0
$211$	12.3297	0.848810	0.424405	−	0.905472i	$-0.360483\pi$
$211$	12.3297	0.848810	0.424405	+	0.905472i	$0.360483\pi$
$212$	0	0
$213$	3.65743	0.250603
$214$	0	0
$215$	2.96379	0.202129
$216$	0	0
$217$	25.9889	1.76424
$218$	0	0
$219$	−6.58923	−0.445259
$220$	0	0
$221$	−2.77997	−0.187001
$222$	0	0
$223$	12.5188	0.838318	0.419159	−	0.907913i	$-0.362325\pi$
$223$	12.5188	0.838318	0.419159	+	0.907913i	$0.362325\pi$
$224$	0	0
$225$	1.99280	0.132853
$226$	0	0
$227$	16.9465	1.12478	0.562389	−	0.826873i	$-0.309882\pi$
$227$	16.9465	1.12478	0.562389	+	0.826873i	$0.309882\pi$
$228$	0	0
$229$	−29.3819	−1.94161	−0.970807	−	0.239863i	$-0.922897\pi$
$229$	−29.3819	−1.94161	−0.970807	+	0.239863i	$0.922897\pi$
$230$	0	0
$231$	−1.77434	−0.116743
$232$	0	0
$233$	−28.4814	−1.86588	−0.932939	−	0.360035i	$-0.882765\pi$
$233$	−28.4814	−1.86588	−0.932939	+	0.360035i	$0.882765\pi$
$234$	0	0
$235$	−6.68218	−0.435898
$236$	0	0
$237$	7.69343	0.499742
$238$	0	0
$239$	22.9104	1.48195	0.740974	−	0.671533i	$-0.234364\pi$
$239$	22.9104	1.48195	0.740974	+	0.671533i	$0.234364\pi$
$240$	0	0
$241$	−7.46479	−0.480849	−0.240425	−	0.970668i	$-0.577287\pi$
$241$	−7.46479	−0.480849	−0.240425	+	0.970668i	$0.577287\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	5.63897	0.360260
$246$	0	0
$247$	1.09723	0.0698151
$248$	0	0
$249$	11.9522	0.757439
$250$	0	0
$251$	−6.63046	−0.418511	−0.209255	−	0.977861i	$-0.567104\pi$
$251$	−6.63046	−0.418511	−0.209255	+	0.977861i	$0.567104\pi$
$252$	0	0
$253$	−0.538202	−0.0338365
$254$	0	0
$255$	3.29364	0.206256
$256$	0	0
$257$	2.89579	0.180634	0.0903171	−	0.995913i	$-0.471212\pi$
$257$	2.89579	0.180634	0.0903171	+	0.995913i	$0.471212\pi$
$258$	0	0
$259$	35.8725	2.22901
$260$	0	0
$261$	−5.33298	−0.330103
$262$	0	0
$263$	−15.1587	−0.934727	−0.467364	−	0.884065i	$-0.654796\pi$
$263$	−15.1587	−0.934727	−0.467364	+	0.884065i	$0.654796\pi$
$264$	0	0
$265$	−2.08737	−0.128226
$266$	0	0
$267$	3.28897	0.201282
$268$	0	0
$269$	3.43706	0.209561	0.104781	−	0.994495i	$-0.466586\pi$
$269$	3.43706	0.209561	0.104781	+	0.994495i	$0.466586\pi$
$270$	0	0
$271$	27.7359	1.68484	0.842418	−	0.538824i	$-0.181131\pi$
$271$	27.7359	1.68484	0.842418	+	0.538824i	$0.181131\pi$
$272$	0	0
$273$	6.74500	0.408226
$274$	0	0
$275$	1.17005	0.0705569
$276$	0	0
$277$	20.9720	1.26009	0.630043	−	0.776561i	$-0.283037\pi$
$277$	20.9720	1.26009	0.630043	+	0.776561i	$0.283037\pi$
$278$	0	0
$279$	−8.59992	−0.514864
$280$	0	0
$281$	8.05468	0.480502	0.240251	−	0.970711i	$-0.422770\pi$
$281$	8.05468	0.480502	0.240251	+	0.970711i	$0.422770\pi$
$282$	0	0
$283$	−0.195610	−0.0116278	−0.00581392	−	0.999983i	$-0.501851\pi$
$283$	−0.195610	−0.0116278	−0.00581392	+	0.999983i	$0.501851\pi$
$284$	0	0
$285$	−1.29997	−0.0770038
$286$	0	0
$287$	7.68837	0.453830
$288$	0	0
$289$	−15.4487	−0.908746
$290$	0	0
$291$	−8.50441	−0.498538
$292$	0	0
$293$	−2.86744	−0.167518	−0.0837588	−	0.996486i	$-0.526693\pi$
$293$	−2.86744	−0.167518	−0.0837588	+	0.996486i	$0.526693\pi$
$294$	0	0
$295$	24.1701	1.40724
$296$	0	0
$297$	0.587142	0.0340694
$298$	0	0
$299$	2.04593	0.118319
$300$	0	0
$301$	−3.38700	−0.195223
$302$	0	0
$303$	−6.83549	−0.392689
$304$	0	0
$305$	−25.8653	−1.48104
$306$	0	0
$307$	3.76659	0.214971	0.107485	−	0.994207i	$-0.465720\pi$
$307$	3.76659	0.214971	0.107485	+	0.994207i	$0.465720\pi$
$308$	0	0
$309$	−1.17906	−0.0670744
$310$	0	0
$311$	27.9652	1.58576	0.792881	−	0.609377i	$-0.208580\pi$
$311$	27.9652	1.58576	0.792881	+	0.609377i	$0.208580\pi$
$312$	0	0
$313$	20.9426	1.18374	0.591872	−	0.806032i	$-0.298389\pi$
$313$	20.9426	1.18374	0.591872	+	0.806032i	$0.298389\pi$
$314$	0	0
$315$	−7.99132	−0.450260
$316$	0	0
$317$	−33.1900	−1.86413	−0.932067	−	0.362287i	$-0.881996\pi$
$317$	−33.1900	−1.86413	−0.932067	+	0.362287i	$0.881996\pi$
$318$	0	0
$319$	−3.13121	−0.175314
$320$	0	0
$321$	15.7476	0.878945
$322$	0	0
$323$	−0.612294	−0.0340689
$324$	0	0
$325$	−4.44787	−0.246723
$326$	0	0
$327$	6.71332	0.371247
$328$	0	0
$329$	7.63636	0.421006
$330$	0	0
$331$	−17.6285	−0.968950	−0.484475	−	0.874805i	$-0.660989\pi$
$331$	−17.6285	−0.968950	−0.484475	+	0.874805i	$0.660989\pi$
$332$	0	0
$333$	−11.8705	−0.650499
$334$	0	0
$335$	−20.3130	−1.10982
$336$	0	0
$337$	−4.53441	−0.247005	−0.123502	−	0.992344i	$-0.539413\pi$
$337$	−4.53441	−0.247005	−0.123502	+	0.992344i	$0.539413\pi$
$338$	0	0
$339$	10.7161	0.582017
$340$	0	0
$341$	−5.04937	−0.273439
$342$	0	0
$343$	14.7098	0.794252
$344$	0	0
$345$	−2.42397	−0.130502
$346$	0	0
$347$	2.66210	0.142909	0.0714545	−	0.997444i	$-0.477236\pi$
$347$	2.66210	0.142909	0.0714545	+	0.997444i	$0.477236\pi$
$348$	0	0
$349$	17.5601	0.939968	0.469984	−	0.882675i	$-0.344260\pi$
$349$	17.5601	0.939968	0.469984	+	0.882675i	$0.344260\pi$
$350$	0	0
$351$	−2.23197	−0.119134
$352$	0	0
$353$	0.525759	0.0279833	0.0139917	−	0.999902i	$-0.495546\pi$
$353$	0.525759	0.0279833	0.0139917	+	0.999902i	$0.495546\pi$
$354$	0	0
$355$	9.67166	0.513319
$356$	0	0
$357$	−3.76395	−0.199209
$358$	0	0
$359$	−10.8769	−0.574061	−0.287031	−	0.957921i	$-0.592668\pi$
$359$	−10.8769	−0.574061	−0.287031	+	0.957921i	$0.592668\pi$
$360$	0	0
$361$	−18.7583	−0.987281
$362$	0	0
$363$	−10.6553	−0.559256
$364$	0	0
$365$	−17.4245	−0.912039
$366$	0	0
$367$	−21.9209	−1.14426	−0.572132	−	0.820162i	$-0.693883\pi$
$367$	−21.9209	−1.14426	−0.572132	+	0.820162i	$0.693883\pi$
$368$	0	0
$369$	−2.54414	−0.132443
$370$	0	0
$371$	2.38543	0.123845
$372$	0	0
$373$	5.77269	0.298899	0.149449	−	0.988769i	$-0.452250\pi$
$373$	5.77269	0.298899	0.149449	+	0.988769i	$0.452250\pi$
$374$	0	0
$375$	−7.95222	−0.410651
$376$	0	0
$377$	11.9031	0.613039
$378$	0	0
$379$	−25.4249	−1.30599	−0.652994	−	0.757363i	$-0.726487\pi$
$379$	−25.4249	−1.30599	−0.652994	+	0.757363i	$0.726487\pi$
$380$	0	0
$381$	1.93781	0.0992772
$382$	0	0
$383$	−29.6411	−1.51459	−0.757294	−	0.653074i	$-0.773479\pi$
$383$	−29.6411	−1.51459	−0.757294	+	0.653074i	$0.773479\pi$
$384$	0	0
$385$	−4.69204	−0.239128
$386$	0	0
$387$	1.12078	0.0569727
$388$	0	0
$389$	−1.17069	−0.0593563	−0.0296781	−	0.999560i	$-0.509448\pi$
$389$	−1.17069	−0.0593563	−0.0296781	+	0.999560i	$0.509448\pi$
$390$	0	0
$391$	−1.14170	−0.0577384
$392$	0	0
$393$	−8.74741	−0.441248
$394$	0	0
$395$	20.3444	1.02364
$396$	0	0
$397$	−7.52841	−0.377840	−0.188920	−	0.981992i	$-0.560499\pi$
$397$	−7.52841	−0.377840	−0.188920	+	0.981992i	$0.560499\pi$
$398$	0	0
$399$	1.48560	0.0743731
$400$	0	0
$401$	10.8561	0.542128	0.271064	−	0.962561i	$-0.412625\pi$
$401$	10.8561	0.542128	0.271064	+	0.962561i	$0.412625\pi$
$402$	0	0
$403$	19.1948	0.956160
$404$	0	0
$405$	2.64439	0.131401
$406$	0	0
$407$	−6.96966	−0.345473
$408$	0	0
$409$	−18.0532	−0.892675	−0.446337	−	0.894865i	$-0.647272\pi$
$409$	−18.0532	−0.892675	−0.446337	+	0.894865i	$0.647272\pi$
$410$	0	0
$411$	13.4646	0.664162
$412$	0	0
$413$	−27.6215	−1.35916
$414$	0	0
$415$	31.6063	1.55149
$416$	0	0
$417$	−6.12589	−0.299986
$418$	0	0
$419$	0.721252	0.0352355	0.0176177	−	0.999845i	$-0.494392\pi$
$419$	0.721252	0.0352355	0.0176177	+	0.999845i	$0.494392\pi$
$420$	0	0
$421$	4.65373	0.226809	0.113404	−	0.993549i	$-0.463824\pi$
$421$	4.65373	0.226809	0.113404	+	0.993549i	$0.463824\pi$
$422$	0	0
$423$	−2.52693	−0.122863
$424$	0	0
$425$	2.48207	0.120398
$426$	0	0
$427$	29.5586	1.43044
$428$	0	0
$429$	−1.31048	−0.0632708
$430$	0	0
$431$	18.5254	0.892335	0.446167	−	0.894949i	$-0.352789\pi$
$431$	18.5254	0.892335	0.446167	+	0.894949i	$0.352789\pi$
$432$	0	0
$433$	12.2153	0.587028	0.293514	−	0.955955i	$-0.405175\pi$
$433$	12.2153	0.587028	0.293514	+	0.955955i	$0.405175\pi$
$434$	0	0
$435$	−14.1025	−0.676162
$436$	0	0
$437$	0.450621	0.0215561
$438$	0	0
$439$	5.86727	0.280029	0.140015	−	0.990149i	$-0.455285\pi$
$439$	5.86727	0.280029	0.140015	+	0.990149i	$0.455285\pi$
$440$	0	0
$441$	2.13243	0.101544
$442$	0	0
$443$	−26.1190	−1.24095	−0.620476	−	0.784225i	$-0.713061\pi$
$443$	−26.1190	−1.24095	−0.620476	+	0.784225i	$0.713061\pi$
$444$	0	0
$445$	8.69732	0.412292
$446$	0	0
$447$	6.53890	0.309279
$448$	0	0
$449$	33.6711	1.58904	0.794518	−	0.607241i	$-0.207724\pi$
$449$	33.6711	1.58904	0.794518	+	0.607241i	$0.207724\pi$
$450$	0	0
$451$	−1.49377	−0.0703389
$452$	0	0
$453$	−16.6852	−0.783940
$454$	0	0
$455$	17.8364	0.836184
$456$	0	0
$457$	19.7672	0.924670	0.462335	−	0.886705i	$-0.347012\pi$
$457$	19.7672	0.924670	0.462335	+	0.886705i	$0.347012\pi$
$458$	0	0
$459$	1.24552	0.0581359
$460$	0	0
$461$	24.9988	1.16431	0.582155	−	0.813078i	$-0.302210\pi$
$461$	24.9988	1.16431	0.582155	+	0.813078i	$0.302210\pi$
$462$	0	0
$463$	22.9788	1.06791	0.533957	−	0.845512i	$-0.320705\pi$
$463$	22.9788	1.06791	0.533957	+	0.845512i	$0.320705\pi$
$464$	0	0
$465$	−22.7415	−1.05461
$466$	0	0
$467$	13.4653	0.623098	0.311549	−	0.950230i	$-0.399152\pi$
$467$	13.4653	0.623098	0.311549	+	0.950230i	$0.399152\pi$
$468$	0	0
$469$	23.2135	1.07190
$470$	0	0
$471$	−6.35568	−0.292854
$472$	0	0
$473$	0.658060	0.0302576
$474$	0	0
$475$	−0.979653	−0.0449495
$476$	0	0
$477$	−0.789358	−0.0361422
$478$	0	0
$479$	−4.43406	−0.202598	−0.101299	−	0.994856i	$-0.532300\pi$
$479$	−4.43406	−0.202598	−0.101299	+	0.994856i	$0.532300\pi$
$480$	0	0
$481$	26.4946	1.20805
$482$	0	0
$483$	2.77010	0.126044
$484$	0	0
$485$	−22.4890	−1.02117
$486$	0	0
$487$	−16.2455	−0.736152	−0.368076	−	0.929796i	$-0.619983\pi$
$487$	−16.2455	−0.736152	−0.368076	+	0.929796i	$0.619983\pi$
$488$	0	0
$489$	12.1987	0.551645
$490$	0	0
$491$	−37.2553	−1.68131	−0.840655	−	0.541571i	$-0.817830\pi$
$491$	−37.2553	−1.68131	−0.840655	+	0.541571i	$0.817830\pi$
$492$	0	0
$493$	−6.64233	−0.299156
$494$	0	0
$495$	1.55263	0.0697856
$496$	0	0
$497$	−11.0527	−0.495782
$498$	0	0
$499$	18.9636	0.848928	0.424464	−	0.905445i	$-0.360463\pi$
$499$	18.9636	0.848928	0.424464	+	0.905445i	$0.360463\pi$
$500$	0	0
$501$	−14.8626	−0.664014
$502$	0	0
$503$	−1.00000	−0.0445878
$503$	−1.00000	−0.0445878
$504$	0	0
$505$	−18.0757	−0.804359
$506$	0	0
$507$	−8.01830	−0.356105
$508$	0	0
$509$	3.26730	0.144820	0.0724102	−	0.997375i	$-0.476931\pi$
$509$	3.26730	0.144820	0.0724102	+	0.997375i	$0.476931\pi$
$510$	0	0
$511$	19.9126	0.880881
$512$	0	0
$513$	−0.491597	−0.0217045
$514$	0	0
$515$	−3.11789	−0.137391
$516$	0	0
$517$	−1.48367	−0.0652515
$518$	0	0
$519$	7.02508	0.308367
$520$	0	0
$521$	−25.5062	−1.11745	−0.558723	−	0.829355i	$-0.688708\pi$
$521$	−25.5062	−1.11745	−0.558723	+	0.829355i	$0.688708\pi$
$522$	0	0
$523$	15.3011	0.669072	0.334536	−	0.942383i	$-0.391420\pi$
$523$	15.3011	0.669072	0.334536	+	0.942383i	$0.391420\pi$
$524$	0	0
$525$	−6.02221	−0.262831
$526$	0	0
$527$	−10.7114	−0.466595
$528$	0	0
$529$	−22.1598	−0.963468
$530$	0	0
$531$	9.14016	0.396649
$532$	0	0
$533$	5.67845	0.245961
$534$	0	0
$535$	41.6428	1.80037
$536$	0	0
$537$	5.16311	0.222805
$538$	0	0
$539$	1.25204	0.0539291
$540$	0	0
$541$	33.0023	1.41888	0.709441	−	0.704765i	$-0.248948\pi$
$541$	33.0023	1.41888	0.709441	+	0.704765i	$0.248948\pi$
$542$	0	0
$543$	−1.26496	−0.0542847
$544$	0	0
$545$	17.7526	0.760439
$546$	0	0
$547$	−6.64394	−0.284074	−0.142037	−	0.989861i	$-0.545365\pi$
$547$	−6.64394	−0.284074	−0.142037	+	0.989861i	$0.545365\pi$
$548$	0	0
$549$	−9.78118	−0.417451
$550$	0	0
$551$	2.62168	0.111687
$552$	0	0
$553$	−23.2495	−0.988668
$554$	0	0
$555$	−31.3902	−1.33244
$556$	0	0
$557$	29.2945	1.24125	0.620624	−	0.784108i	$-0.286879\pi$
$557$	29.2945	1.24125	0.620624	+	0.784108i	$0.286879\pi$
$558$	0	0
$559$	−2.50156	−0.105805
$560$	0	0
$561$	0.731297	0.0308754
$562$	0	0
$563$	−23.1598	−0.976071	−0.488035	−	0.872824i	$-0.662286\pi$
$563$	−23.1598	−0.976071	−0.488035	+	0.872824i	$0.662286\pi$
$564$	0	0
$565$	28.3375	1.19217
$566$	0	0
$567$	−3.02199	−0.126912
$568$	0	0
$569$	−3.99492	−0.167476	−0.0837379	−	0.996488i	$-0.526686\pi$
$569$	−3.99492	−0.167476	−0.0837379	+	0.996488i	$0.526686\pi$
$570$	0	0
$571$	−8.99135	−0.376276	−0.188138	−	0.982143i	$-0.560245\pi$
$571$	−8.99135	−0.376276	−0.188138	+	0.982143i	$0.560245\pi$
$572$	0	0
$573$	6.03083	0.251942
$574$	0	0
$575$	−1.82669	−0.0761783
$576$	0	0
$577$	−0.478841	−0.0199344	−0.00996722	−	0.999950i	$-0.503173\pi$
$577$	−0.478841	−0.0199344	−0.00996722	+	0.999950i	$0.503173\pi$
$578$	0	0
$579$	−1.06344	−0.0441950
$580$	0	0
$581$	−36.1194	−1.49849
$582$	0	0
$583$	−0.463465	−0.0191948
$584$	0	0
$585$	−5.90220	−0.244026
$586$	0	0
$587$	−27.1571	−1.12089	−0.560447	−	0.828191i	$-0.689371\pi$
$587$	−27.1571	−1.12089	−0.560447	+	0.828191i	$0.689371\pi$
$588$	0	0
$589$	4.22770	0.174199
$590$	0	0
$591$	−22.3630	−0.919891
$592$	0	0
$593$	31.2986	1.28528	0.642640	−	0.766169i	$-0.277839\pi$
$593$	31.2986	1.28528	0.642640	+	0.766169i	$0.277839\pi$
$594$	0	0
$595$	−9.95335	−0.408047
$596$	0	0
$597$	−22.8245	−0.934145
$598$	0	0
$599$	−41.2958	−1.68730	−0.843651	−	0.536892i	$-0.819598\pi$
$599$	−41.2958	−1.68730	−0.843651	+	0.536892i	$0.819598\pi$
$600$	0	0
$601$	−10.7160	−0.437115	−0.218558	−	0.975824i	$-0.570135\pi$
$601$	−10.7160	−0.437115	−0.218558	+	0.975824i	$0.570135\pi$
$602$	0	0
$603$	−7.68153	−0.312816
$604$	0	0
$605$	−28.1767	−1.14554
$606$	0	0
$607$	23.5255	0.954871	0.477436	−	0.878667i	$-0.341566\pi$
$607$	23.5255	0.954871	0.477436	+	0.878667i	$0.341566\pi$
$608$	0	0
$609$	16.1162	0.653062
$610$	0	0
$611$	5.64004	0.228171
$612$	0	0
$613$	22.0125	0.889075	0.444538	−	0.895760i	$-0.353368\pi$
$613$	22.0125	0.889075	0.444538	+	0.895760i	$0.353368\pi$
$614$	0	0
$615$	−6.72770	−0.271287
$616$	0	0
$617$	30.2743	1.21880	0.609400	−	0.792863i	$-0.291411\pi$
$617$	30.2743	1.21880	0.609400	+	0.792863i	$0.291411\pi$
$618$	0	0
$619$	−2.65262	−0.106618	−0.0533089	−	0.998578i	$-0.516977\pi$
$619$	−2.65262	−0.106618	−0.0533089	+	0.998578i	$0.516977\pi$
$620$	0	0
$621$	−0.916648	−0.0367838
$622$	0	0
$623$	−9.93924	−0.398207
$624$	0	0
$625$	−30.9927	−1.23971
$626$	0	0
$627$	−0.288637	−0.0115271
$628$	0	0
$629$	−14.7849	−0.589514
$630$	0	0
$631$	12.7539	0.507724	0.253862	−	0.967240i	$-0.418299\pi$
$631$	12.7539	0.507724	0.253862	+	0.967240i	$0.418299\pi$
$632$	0	0
$633$	12.3297	0.490061
$634$	0	0
$635$	5.12433	0.203353
$636$	0	0
$637$	−4.75952	−0.188579
$638$	0	0
$639$	3.65743	0.144686
$640$	0	0
$641$	−12.8796	−0.508715	−0.254358	−	0.967110i	$-0.581864\pi$
$641$	−12.8796	−0.508715	−0.254358	+	0.967110i	$0.581864\pi$
$642$	0	0
$643$	−28.0870	−1.10764	−0.553822	−	0.832635i	$-0.686831\pi$
$643$	−28.0870	−1.10764	−0.553822	+	0.832635i	$0.686831\pi$
$644$	0	0
$645$	2.96379	0.116699
$646$	0	0
$647$	1.25841	0.0494731	0.0247365	−	0.999694i	$-0.492125\pi$
$647$	1.25841	0.0494731	0.0247365	+	0.999694i	$0.492125\pi$
$648$	0	0
$649$	5.36657	0.210656
$650$	0	0
$651$	25.9889	1.01858
$652$	0	0
$653$	−45.8218	−1.79315	−0.896574	−	0.442895i	$-0.853952\pi$
$653$	−45.8218	−1.79315	−0.896574	+	0.442895i	$0.853952\pi$
$654$	0	0
$655$	−23.1316	−0.903825
$656$	0	0
$657$	−6.58923	−0.257070
$658$	0	0
$659$	−6.42765	−0.250386	−0.125193	−	0.992132i	$-0.539955\pi$
$659$	−6.42765	−0.250386	−0.125193	+	0.992132i	$0.539955\pi$
$660$	0	0
$661$	4.50232	0.175120	0.0875600	−	0.996159i	$-0.472093\pi$
$661$	4.50232	0.175120	0.0875600	+	0.996159i	$0.472093\pi$
$662$	0	0
$663$	−2.77997	−0.107965
$664$	0	0
$665$	3.92851	0.152341
$666$	0	0
$667$	4.88846	0.189282
$668$	0	0
$669$	12.5188	0.484003
$670$	0	0
$671$	−5.74294	−0.221704
$672$	0	0
$673$	43.8991	1.69219	0.846094	−	0.533034i	$-0.178948\pi$
$673$	43.8991	1.69219	0.846094	+	0.533034i	$0.178948\pi$
$674$	0	0
$675$	1.99280	0.0767028
$676$	0	0
$677$	−9.04130	−0.347486	−0.173743	−	0.984791i	$-0.555586\pi$
$677$	−9.04130	−0.347486	−0.173743	+	0.984791i	$0.555586\pi$
$678$	0	0
$679$	25.7003	0.986285
$680$	0	0
$681$	16.9465	0.649391
$682$	0	0
$683$	33.2015	1.27042	0.635210	−	0.772340i	$-0.280914\pi$
$683$	33.2015	1.27042	0.635210	+	0.772340i	$0.280914\pi$
$684$	0	0
$685$	35.6057	1.36043
$686$	0	0
$687$	−29.3819	−1.12099
$688$	0	0
$689$	1.76183	0.0671202
$690$	0	0
$691$	−12.2706	−0.466796	−0.233398	−	0.972381i	$-0.574984\pi$
$691$	−12.2706	−0.466796	−0.233398	+	0.972381i	$0.574984\pi$
$692$	0	0
$693$	−1.77434	−0.0674015
$694$	0	0
$695$	−16.1992	−0.614472
$696$	0	0
$697$	−3.16878	−0.120026
$698$	0	0
$699$	−28.4814	−1.07726
$700$	0	0
$701$	−16.5272	−0.624224	−0.312112	−	0.950045i	$-0.601036\pi$
$701$	−16.5272	−0.624224	−0.312112	+	0.950045i	$0.601036\pi$
$702$	0	0
$703$	5.83550	0.220090
$704$	0	0
$705$	−6.68218	−0.251666
$706$	0	0
$707$	20.6568	0.776879
$708$	0	0
$709$	−17.5665	−0.659722	−0.329861	−	0.944029i	$-0.607002\pi$
$709$	−17.5665	−0.659722	−0.329861	+	0.944029i	$0.607002\pi$
$710$	0	0
$711$	7.69343	0.288526
$712$	0	0
$713$	7.88310	0.295224
$714$	0	0
$715$	−3.46543	−0.129600
$716$	0	0
$717$	22.9104	0.855604
$718$	0	0
$719$	31.4665	1.17350	0.586751	−	0.809767i	$-0.300407\pi$
$719$	31.4665	1.17350	0.586751	+	0.809767i	$0.300407\pi$
$720$	0	0
$721$	3.56311	0.132697
$722$	0	0
$723$	−7.46479	−0.277618
$724$	0	0
$725$	−10.6275	−0.394697
$726$	0	0
$727$	17.8165	0.660778	0.330389	−	0.943845i	$-0.392820\pi$
$727$	17.8165	0.660778	0.330389	+	0.943845i	$0.392820\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	1.39596	0.0516314
$732$	0	0
$733$	−19.1822	−0.708509	−0.354255	−	0.935149i	$-0.615265\pi$
$733$	−19.1822	−0.708509	−0.354255	+	0.935149i	$0.615265\pi$
$734$	0	0
$735$	5.63897	0.207996
$736$	0	0
$737$	−4.51015	−0.166133
$738$	0	0
$739$	14.5735	0.536097	0.268048	−	0.963405i	$-0.413621\pi$
$739$	14.5735	0.536097	0.268048	+	0.963405i	$0.413621\pi$
$740$	0	0
$741$	1.09723	0.0403078
$742$	0	0
$743$	−7.55964	−0.277336	−0.138668	−	0.990339i	$-0.544282\pi$
$743$	−7.55964	−0.277336	−0.138668	+	0.990339i	$0.544282\pi$
$744$	0	0
$745$	17.2914	0.633508
$746$	0	0
$747$	11.9522	0.437308
$748$	0	0
$749$	−47.5891	−1.73887
$750$	0	0
$751$	−4.47286	−0.163217	−0.0816084	−	0.996664i	$-0.526006\pi$
$751$	−4.47286	−0.163217	−0.0816084	+	0.996664i	$0.526006\pi$
$752$	0	0
$753$	−6.63046	−0.241627
$754$	0	0
$755$	−44.1222	−1.60577
$756$	0	0
$757$	−37.9865	−1.38064	−0.690322	−	0.723503i	$-0.742531\pi$
$757$	−37.9865	−1.38064	−0.690322	+	0.723503i	$0.742531\pi$
$758$	0	0
$759$	−0.538202	−0.0195355
$760$	0	0
$761$	−11.9130	−0.431848	−0.215924	−	0.976410i	$-0.569276\pi$
$761$	−11.9130	−0.431848	−0.215924	+	0.976410i	$0.569276\pi$
$762$	0	0
$763$	−20.2876	−0.734460
$764$	0	0
$765$	3.29364	0.119082
$766$	0	0
$767$	−20.4006	−0.736622
$768$	0	0
$769$	−13.7398	−0.495468	−0.247734	−	0.968828i	$-0.579686\pi$
$769$	−13.7398	−0.495468	−0.247734	+	0.968828i	$0.579686\pi$
$770$	0	0
$771$	2.89579	0.104289
$772$	0	0
$773$	−30.1541	−1.08457	−0.542285	−	0.840195i	$-0.682441\pi$
$773$	−30.1541	−1.08457	−0.542285	+	0.840195i	$0.682441\pi$
$774$	0	0
$775$	−17.1379	−0.615611
$776$	0	0
$777$	35.8725	1.28692
$778$	0	0
$779$	1.25069	0.0448107
$780$	0	0
$781$	2.14743	0.0768410
$782$	0	0
$783$	−5.33298	−0.190585
$784$	0	0
$785$	−16.8069	−0.599863
$786$	0	0
$787$	−53.6781	−1.91342	−0.956709	−	0.291047i	$-0.905996\pi$
$787$	−53.6781	−1.91342	−0.956709	+	0.291047i	$0.905996\pi$
$788$	0	0
$789$	−15.1587	−0.539665
$790$	0	0
$791$	−32.3839	−1.15144
$792$	0	0
$793$	21.8313	0.775253
$794$	0	0
$795$	−2.08737	−0.0740314
$796$	0	0
$797$	28.5773	1.01226	0.506130	−	0.862457i	$-0.331076\pi$
$797$	28.5773	1.01226	0.506130	+	0.862457i	$0.331076\pi$
$798$	0	0
$799$	−3.14734	−0.111345
$800$	0	0
$801$	3.28897	0.116210
$802$	0	0
$803$	−3.86881	−0.136527
$804$	0	0
$805$	7.32523	0.258180
$806$	0	0
$807$	3.43706	0.120990
$808$	0	0
$809$	32.2686	1.13450	0.567251	−	0.823545i	$-0.308007\pi$
$809$	32.2686	1.13450	0.567251	+	0.823545i	$0.308007\pi$
$810$	0	0
$811$	−37.5741	−1.31940	−0.659702	−	0.751528i	$-0.729317\pi$
$811$	−37.5741	−1.31940	−0.659702	+	0.751528i	$0.729317\pi$
$812$	0	0
$813$	27.7359	0.972741
$814$	0	0
$815$	32.2582	1.12995
$816$	0	0
$817$	−0.550974	−0.0192762
$818$	0	0
$819$	6.74500	0.235689
$820$	0	0
$821$	10.9386	0.381761	0.190881	−	0.981613i	$-0.438866\pi$
$821$	10.9386	0.381761	0.190881	+	0.981613i	$0.438866\pi$
$822$	0	0
$823$	46.8692	1.63376	0.816878	−	0.576810i	$-0.195703\pi$
$823$	46.8692	1.63376	0.816878	+	0.576810i	$0.195703\pi$
$824$	0	0
$825$	1.17005	0.0407360
$826$	0	0
$827$	30.3247	1.05449	0.527247	−	0.849712i	$-0.323224\pi$
$827$	30.3247	1.05449	0.527247	+	0.849712i	$0.323224\pi$
$828$	0	0
$829$	31.7701	1.10342	0.551711	−	0.834035i	$-0.313975\pi$
$829$	31.7701	1.10342	0.551711	+	0.834035i	$0.313975\pi$
$830$	0	0
$831$	20.9720	0.727511
$832$	0	0
$833$	2.65598	0.0920243
$834$	0	0
$835$	−39.3026	−1.36012
$836$	0	0
$837$	−8.59992	−0.297257
$838$	0	0
$839$	−34.8586	−1.20345	−0.601726	−	0.798702i	$-0.705520\pi$
$839$	−34.8586	−1.20345	−0.601726	+	0.798702i	$0.705520\pi$
$840$	0	0
$841$	−0.559339	−0.0192876
$842$	0	0
$843$	8.05468	0.277418
$844$	0	0
$845$	−21.2035	−0.729423
$846$	0	0
$847$	32.2001	1.10641
$848$	0	0
$849$	−0.195610	−0.00671333
$850$	0	0
$851$	10.8811	0.372998
$852$	0	0
$853$	25.0784	0.858667	0.429333	−	0.903146i	$-0.358749\pi$
$853$	25.0784	0.858667	0.429333	+	0.903146i	$0.358749\pi$
$854$	0	0
$855$	−1.29997	−0.0444582
$856$	0	0
$857$	25.7574	0.879855	0.439927	−	0.898033i	$-0.355004\pi$
$857$	25.7574	0.879855	0.439927	+	0.898033i	$0.355004\pi$
$858$	0	0
$859$	−18.3344	−0.625561	−0.312780	−	0.949826i	$-0.601260\pi$
$859$	−18.3344	−0.625561	−0.312780	+	0.949826i	$0.601260\pi$
$860$	0	0
$861$	7.68837	0.262019
$862$	0	0
$863$	35.0214	1.19214	0.596072	−	0.802931i	$-0.296727\pi$
$863$	35.0214	1.19214	0.596072	+	0.802931i	$0.296727\pi$
$864$	0	0
$865$	18.5771	0.631639
$866$	0	0
$867$	−15.4487	−0.524665
$868$	0	0
$869$	4.51713	0.153233
$870$	0	0
$871$	17.1450	0.580935
$872$	0	0
$873$	−8.50441	−0.287831
$874$	0	0
$875$	24.0315	0.812414
$876$	0	0
$877$	−0.348285	−0.0117607	−0.00588037	−	0.999983i	$-0.501872\pi$
$877$	−0.348285	−0.0117607	−0.00588037	+	0.999983i	$0.501872\pi$
$878$	0	0
$879$	−2.86744	−0.0967163
$880$	0	0
$881$	−1.76588	−0.0594940	−0.0297470	−	0.999557i	$-0.509470\pi$
$881$	−1.76588	−0.0594940	−0.0297470	+	0.999557i	$0.509470\pi$
$882$	0	0
$883$	8.89614	0.299379	0.149690	−	0.988733i	$-0.452173\pi$
$883$	8.89614	0.299379	0.149690	+	0.988733i	$0.452173\pi$
$884$	0	0
$885$	24.1701	0.812470
$886$	0	0
$887$	−0.453290	−0.0152200	−0.00760999	−	0.999971i	$-0.502422\pi$
$887$	−0.453290	−0.0152200	−0.00760999	+	0.999971i	$0.502422\pi$
$888$	0	0
$889$	−5.85605	−0.196406
$890$	0	0
$891$	0.587142	0.0196700
$892$	0	0
$893$	1.24223	0.0415697
$894$	0	0
$895$	13.6533	0.456379
$896$	0	0
$897$	2.04593	0.0683117
$898$	0	0
$899$	45.8632	1.52962
$900$	0	0
$901$	−0.983161	−0.0327538
$902$	0	0
$903$	−3.38700	−0.112712
$904$	0	0
$905$	−3.34505	−0.111193
$906$	0	0
$907$	27.0906	0.899530	0.449765	−	0.893147i	$-0.351508\pi$
$907$	27.0906	0.899530	0.449765	+	0.893147i	$0.351508\pi$
$908$	0	0
$909$	−6.83549	−0.226719
$910$	0	0
$911$	−7.48478	−0.247982	−0.123991	−	0.992283i	$-0.539569\pi$
$911$	−7.48478	−0.247982	−0.123991	+	0.992283i	$0.539569\pi$
$912$	0	0
$913$	7.01763	0.232250
$914$	0	0
$915$	−25.8653	−0.855079
$916$	0	0
$917$	26.4346	0.872947
$918$	0	0
$919$	−1.29744	−0.0427984	−0.0213992	−	0.999771i	$-0.506812\pi$
$919$	−1.29744	−0.0427984	−0.0213992	+	0.999771i	$0.506812\pi$
$920$	0	0
$921$	3.76659	0.124113
$922$	0	0
$923$	−8.16328	−0.268697
$924$	0	0
$925$	−23.6555	−0.777787
$926$	0	0
$927$	−1.17906	−0.0387254
$928$	0	0
$929$	−46.7480	−1.53375	−0.766875	−	0.641796i	$-0.778190\pi$
$929$	−46.7480	−1.53375	−0.766875	+	0.641796i	$0.778190\pi$
$930$	0	0
$931$	−1.04829	−0.0343565
$932$	0	0
$933$	27.9652	0.915540
$934$	0	0
$935$	1.93383	0.0632431
$936$	0	0
$937$	−27.8406	−0.909514	−0.454757	−	0.890616i	$-0.650274\pi$
$937$	−27.8406	−0.909514	−0.454757	+	0.890616i	$0.650274\pi$
$938$	0	0
$939$	20.9426	0.683435
$940$	0	0
$941$	−40.3568	−1.31560	−0.657798	−	0.753195i	$-0.728512\pi$
$941$	−40.3568	−1.31560	−0.657798	+	0.753195i	$0.728512\pi$
$942$	0	0
$943$	2.33208	0.0759430
$944$	0	0
$945$	−7.99132	−0.259958
$946$	0	0
$947$	−5.07179	−0.164811	−0.0824055	−	0.996599i	$-0.526260\pi$
$947$	−5.07179	−0.164811	−0.0824055	+	0.996599i	$0.526260\pi$
$948$	0	0
$949$	14.7070	0.477408
$950$	0	0
$951$	−33.1900	−1.07626
$952$	0	0
$953$	−7.29876	−0.236430	−0.118215	−	0.992988i	$-0.537717\pi$
$953$	−7.29876	−0.236430	−0.118215	+	0.992988i	$0.537717\pi$
$954$	0	0
$955$	15.9479	0.516061
$956$	0	0
$957$	−3.13121	−0.101218
$958$	0	0
$959$	−40.6900	−1.31395
$960$	0	0
$961$	42.9587	1.38576
$962$	0	0
$963$	15.7476	0.507459
$964$	0	0
$965$	−2.81214	−0.0905261
$966$	0	0
$967$	41.1589	1.32358	0.661791	−	0.749688i	$-0.269797\pi$
$967$	41.1589	1.32358	0.661791	+	0.749688i	$0.269797\pi$
$968$	0	0
$969$	−0.612294	−0.0196697
$970$	0	0
$971$	29.1881	0.936691	0.468345	−	0.883545i	$-0.344850\pi$
$971$	29.1881	0.936691	0.468345	+	0.883545i	$0.344850\pi$
$972$	0	0
$973$	18.5124	0.593479
$974$	0	0
$975$	−4.44787	−0.142446
$976$	0	0
$977$	10.4025	0.332806	0.166403	−	0.986058i	$-0.446785\pi$
$977$	10.4025	0.332806	0.166403	+	0.986058i	$0.446785\pi$
$978$	0	0
$979$	1.93109	0.0617180
$980$	0	0
$981$	6.71332	0.214340
$982$	0	0
$983$	5.76886	0.183998	0.0919990	−	0.995759i	$-0.470674\pi$
$983$	5.76886	0.183998	0.0919990	+	0.995759i	$0.470674\pi$
$984$	0	0
$985$	−59.1365	−1.88424
$986$	0	0
$987$	7.63636	0.243068
$988$	0	0
$989$	−1.02736	−0.0326683
$990$	0	0
$991$	4.47663	0.142205	0.0711025	−	0.997469i	$-0.477348\pi$
$991$	4.47663	0.142205	0.0711025	+	0.997469i	$0.477348\pi$
$992$	0	0
$993$	−17.6285	−0.559423
$994$	0	0
$995$	−60.3569	−1.91344
$996$	0	0
$997$	43.3463	1.37279	0.686396	−	0.727228i	$-0.259192\pi$
$997$	43.3463	1.37279	0.686396	+	0.727228i	$0.259192\pi$
$998$	0	0
$999$	−11.8705	−0.375566

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6036.2.a.g.1.15	✓	15

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

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +2.64439 q^{5} -3.02199 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +2.64439 q^{5} -3.02199 q^{7} +1.00000 q^{9} +0.587142 q^{11} -2.23197 q^{13} +2.64439 q^{15} +1.24552 q^{17} -0.491597 q^{19} -3.02199 q^{21} -0.916648 q^{23} +1.99280 q^{25} +1.00000 q^{27} -5.33298 q^{29} -8.59992 q^{31} +0.587142 q^{33} -7.99132 q^{35} -11.8705 q^{37} -2.23197 q^{39} -2.54414 q^{41} +1.12078 q^{43} +2.64439 q^{45} -2.52693 q^{47} +2.13243 q^{49} +1.24552 q^{51} -0.789358 q^{53} +1.55263 q^{55} -0.491597 q^{57} +9.14016 q^{59} -9.78118 q^{61} -3.02199 q^{63} -5.90220 q^{65} -7.68153 q^{67} -0.916648 q^{69} +3.65743 q^{71} -6.58923 q^{73} +1.99280 q^{75} -1.77434 q^{77} +7.69343 q^{79} +1.00000 q^{81} +11.9522 q^{83} +3.29364 q^{85} -5.33298 q^{87} +3.28897 q^{89} +6.74500 q^{91} -8.59992 q^{93} -1.29997 q^{95} -8.50441 q^{97} +0.587142 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q + 15 q^{3} - 11 q^{5} - 4 q^{7} + 15 q^{9}+O(q^{10}) \) 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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