LMFDB - Embedded newform 6036.2.a.g.1.13

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			1.13
Root			$3.48442$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} +2.48442 q^{5} +1.05465 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +2.48442 q^{5} +1.05465 q^{7} +1.00000 q^{9} -4.01743 q^{11} -4.37086 q^{13} +2.48442 q^{15} -5.30462 q^{17} +4.18305 q^{19} +1.05465 q^{21} -5.97856 q^{23} +1.17232 q^{25} +1.00000 q^{27} -2.50134 q^{29} -0.980193 q^{31} -4.01743 q^{33} +2.62020 q^{35} +0.119159 q^{37} -4.37086 q^{39} -2.22539 q^{41} -0.812728 q^{43} +2.48442 q^{45} -4.81548 q^{47} -5.88770 q^{49} -5.30462 q^{51} +0.831360 q^{53} -9.98098 q^{55} +4.18305 q^{57} +4.34939 q^{59} +3.11748 q^{61} +1.05465 q^{63} -10.8590 q^{65} -2.10873 q^{67} -5.97856 q^{69} -11.8866 q^{71} +13.6096 q^{73} +1.17232 q^{75} -4.23700 q^{77} -4.21360 q^{79} +1.00000 q^{81} -9.78346 q^{83} -13.1789 q^{85} -2.50134 q^{87} +15.2828 q^{89} -4.60974 q^{91} -0.980193 q^{93} +10.3924 q^{95} -11.5419 q^{97} -4.01743 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	2.48442	1.11106	0.555532	−	0.831495i	$-0.312515\pi$
$5$	2.48442	1.11106	0.555532	+	0.831495i	$0.312515\pi$
$6$	0	0
$7$	1.05465	0.398622	0.199311	−	0.979936i	$-0.436130\pi$
$7$	1.05465	0.398622	0.199311	+	0.979936i	$0.436130\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−4.01743	−1.21130	−0.605651	−	0.795730i	$-0.707087\pi$
$11$	−4.01743	−1.21130	−0.605651	+	0.795730i	$0.707087\pi$
$12$	0	0
$13$	−4.37086	−1.21226	−0.606129	−	0.795367i	$-0.707278\pi$
$13$	−4.37086	−1.21226	−0.606129	+	0.795367i	$0.707278\pi$
$14$	0	0
$15$	2.48442	0.641473
$16$	0	0
$17$	−5.30462	−1.28656	−0.643280	−	0.765631i	$-0.722427\pi$
$17$	−5.30462	−1.28656	−0.643280	+	0.765631i	$0.722427\pi$
$18$	0	0
$19$	4.18305	0.959656	0.479828	−	0.877362i	$-0.340699\pi$
$19$	4.18305	0.959656	0.479828	+	0.877362i	$0.340699\pi$
$20$	0	0
$21$	1.05465	0.230144
$22$	0	0
$23$	−5.97856	−1.24662	−0.623308	−	0.781977i	$-0.714212\pi$
$23$	−5.97856	−1.24662	−0.623308	+	0.781977i	$0.714212\pi$
$24$	0	0
$25$	1.17232	0.234464
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−2.50134	−0.464488	−0.232244	−	0.972658i	$-0.574607\pi$
$29$	−2.50134	−0.464488	−0.232244	+	0.972658i	$0.574607\pi$
$30$	0	0
$31$	−0.980193	−0.176048	−0.0880239	−	0.996118i	$-0.528055\pi$
$31$	−0.980193	−0.176048	−0.0880239	+	0.996118i	$0.528055\pi$
$32$	0	0
$33$	−4.01743	−0.699346
$34$	0	0
$35$	2.62020	0.442895
$36$	0	0
$37$	0.119159	0.0195896	0.00979482	−	0.999952i	$-0.496882\pi$
$37$	0.119159	0.0195896	0.00979482	+	0.999952i	$0.496882\pi$
$38$	0	0
$39$	−4.37086	−0.699897
$40$	0	0
$41$	−2.22539	−0.347548	−0.173774	−	0.984786i	$-0.555596\pi$
$41$	−2.22539	−0.347548	−0.173774	+	0.984786i	$0.555596\pi$
$42$	0	0
$43$	−0.812728	−0.123940	−0.0619699	−	0.998078i	$-0.519738\pi$
$43$	−0.812728	−0.123940	−0.0619699	+	0.998078i	$0.519738\pi$
$44$	0	0
$45$	2.48442	0.370355
$46$	0	0
$47$	−4.81548	−0.702409	−0.351205	−	0.936299i	$-0.614228\pi$
$47$	−4.81548	−0.702409	−0.351205	+	0.936299i	$0.614228\pi$
$48$	0	0
$49$	−5.88770	−0.841101
$50$	0	0
$51$	−5.30462	−0.742795
$52$	0	0
$53$	0.831360	0.114196	0.0570980	−	0.998369i	$-0.481815\pi$
$53$	0.831360	0.114196	0.0570980	+	0.998369i	$0.481815\pi$
$54$	0	0
$55$	−9.98098	−1.34583
$56$	0	0
$57$	4.18305	0.554058
$58$	0	0
$59$	4.34939	0.566243	0.283121	−	0.959084i	$-0.408630\pi$
$59$	4.34939	0.566243	0.283121	+	0.959084i	$0.408630\pi$
$60$	0	0
$61$	3.11748	0.399153	0.199576	−	0.979882i	$-0.436043\pi$
$61$	3.11748	0.399153	0.199576	+	0.979882i	$0.436043\pi$
$62$	0	0
$63$	1.05465	0.132874
$64$	0	0
$65$	−10.8590	−1.34690
$66$	0	0
$67$	−2.10873	−0.257622	−0.128811	−	0.991669i	$-0.541116\pi$
$67$	−2.10873	−0.257622	−0.128811	+	0.991669i	$0.541116\pi$
$68$	0	0
$69$	−5.97856	−0.719734
$70$	0	0
$71$	−11.8866	−1.41068	−0.705342	−	0.708867i	$-0.749207\pi$
$71$	−11.8866	−1.41068	−0.705342	+	0.708867i	$0.749207\pi$
$72$	0	0
$73$	13.6096	1.59288	0.796440	−	0.604718i	$-0.206714\pi$
$73$	13.6096	1.59288	0.796440	+	0.604718i	$0.206714\pi$
$74$	0	0
$75$	1.17232	0.135368
$76$	0	0
$77$	−4.23700	−0.482851
$78$	0	0
$79$	−4.21360	−0.474067	−0.237033	−	0.971502i	$-0.576175\pi$
$79$	−4.21360	−0.474067	−0.237033	+	0.971502i	$0.576175\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−9.78346	−1.07387	−0.536937	−	0.843622i	$-0.680419\pi$
$83$	−9.78346	−1.07387	−0.536937	+	0.843622i	$0.680419\pi$
$84$	0	0
$85$	−13.1789	−1.42945
$86$	0	0
$87$	−2.50134	−0.268172
$88$	0	0
$89$	15.2828	1.61998	0.809989	−	0.586445i	$-0.199473\pi$
$89$	15.2828	1.61998	0.809989	+	0.586445i	$0.199473\pi$
$90$	0	0
$91$	−4.60974	−0.483232
$92$	0	0
$93$	−0.980193	−0.101641
$94$	0	0
$95$	10.3924	1.06624
$96$	0	0
$97$	−11.5419	−1.17190	−0.585949	−	0.810348i	$-0.699278\pi$
$97$	−11.5419	−1.17190	−0.585949	+	0.810348i	$0.699278\pi$
$98$	0	0
$99$	−4.01743	−0.403767
$100$	0	0
$101$	−19.7358	−1.96379	−0.981893	−	0.189439i	$-0.939333\pi$
$101$	−19.7358	−1.96379	−0.981893	+	0.189439i	$0.939333\pi$
$102$	0	0
$103$	8.48413	0.835966	0.417983	−	0.908455i	$-0.362737\pi$
$103$	8.48413	0.835966	0.417983	+	0.908455i	$0.362737\pi$
$104$	0	0
$105$	2.62020	0.255705
$106$	0	0
$107$	−12.5953	−1.21764	−0.608818	−	0.793310i	$-0.708356\pi$
$107$	−12.5953	−1.21764	−0.608818	+	0.793310i	$0.708356\pi$
$108$	0	0
$109$	12.8417	1.23001	0.615004	−	0.788524i	$-0.289154\pi$
$109$	12.8417	1.23001	0.615004	+	0.788524i	$0.289154\pi$
$110$	0	0
$111$	0.119159	0.0113101
$112$	0	0
$113$	−10.8335	−1.01913	−0.509564	−	0.860433i	$-0.670193\pi$
$113$	−10.8335	−1.01913	−0.509564	+	0.860433i	$0.670193\pi$
$114$	0	0
$115$	−14.8532	−1.38507
$116$	0	0
$117$	−4.37086	−0.404086
$118$	0	0
$119$	−5.59454	−0.512851
$120$	0	0
$121$	5.13978	0.467252
$122$	0	0
$123$	−2.22539	−0.200657
$124$	0	0
$125$	−9.50955	−0.850560
$126$	0	0
$127$	−1.98049	−0.175740	−0.0878701	−	0.996132i	$-0.528006\pi$
$127$	−1.98049	−0.175740	−0.0878701	+	0.996132i	$0.528006\pi$
$128$	0	0
$129$	−0.812728	−0.0715567
$130$	0	0
$131$	19.0312	1.66277	0.831383	−	0.555700i	$-0.187550\pi$
$131$	19.0312	1.66277	0.831383	+	0.555700i	$0.187550\pi$
$132$	0	0
$133$	4.41167	0.382540
$134$	0	0
$135$	2.48442	0.213824
$136$	0	0
$137$	15.9477	1.36250	0.681251	−	0.732050i	$-0.261436\pi$
$137$	15.9477	1.36250	0.681251	+	0.732050i	$0.261436\pi$
$138$	0	0
$139$	−12.5090	−1.06100	−0.530498	−	0.847686i	$-0.677995\pi$
$139$	−12.5090	−1.06100	−0.530498	+	0.847686i	$0.677995\pi$
$140$	0	0
$141$	−4.81548	−0.405536
$142$	0	0
$143$	17.5596	1.46841
$144$	0	0
$145$	−6.21438	−0.516076
$146$	0	0
$147$	−5.88770	−0.485610
$148$	0	0
$149$	−21.4384	−1.75630	−0.878151	−	0.478383i	$-0.841223\pi$
$149$	−21.4384	−1.75630	−0.878151	+	0.478383i	$0.841223\pi$
$150$	0	0
$151$	−9.67697	−0.787500	−0.393750	−	0.919217i	$-0.628822\pi$
$151$	−9.67697	−0.787500	−0.393750	+	0.919217i	$0.628822\pi$
$152$	0	0
$153$	−5.30462	−0.428853
$154$	0	0
$155$	−2.43521	−0.195600
$156$	0	0
$157$	−12.0099	−0.958497	−0.479249	−	0.877679i	$-0.659091\pi$
$157$	−12.0099	−0.958497	−0.479249	+	0.877679i	$0.659091\pi$
$158$	0	0
$159$	0.831360	0.0659311
$160$	0	0
$161$	−6.30531	−0.496928
$162$	0	0
$163$	−2.49487	−0.195413	−0.0977065	−	0.995215i	$-0.531151\pi$
$163$	−2.49487	−0.195413	−0.0977065	+	0.995215i	$0.531151\pi$
$164$	0	0
$165$	−9.98098	−0.777018
$166$	0	0
$167$	12.9914	1.00531	0.502654	−	0.864488i	$-0.332357\pi$
$167$	12.9914	1.00531	0.502654	+	0.864488i	$0.332357\pi$
$168$	0	0
$169$	6.10439	0.469568
$170$	0	0
$171$	4.18305	0.319885
$172$	0	0
$173$	11.1085	0.844566	0.422283	−	0.906464i	$-0.361229\pi$
$173$	11.1085	0.844566	0.422283	+	0.906464i	$0.361229\pi$
$174$	0	0
$175$	1.23639	0.0934625
$176$	0	0
$177$	4.34939	0.326920
$178$	0	0
$179$	0.128562	0.00960917	0.00480459	−	0.999988i	$-0.498471\pi$
$179$	0.128562	0.00960917	0.00480459	+	0.999988i	$0.498471\pi$
$180$	0	0
$181$	−1.48913	−0.110686	−0.0553432	−	0.998467i	$-0.517625\pi$
$181$	−1.48913	−0.110686	−0.0553432	+	0.998467i	$0.517625\pi$
$182$	0	0
$183$	3.11748	0.230451
$184$	0	0
$185$	0.296041	0.0217654
$186$	0	0
$187$	21.3110	1.55841
$188$	0	0
$189$	1.05465	0.0767148
$190$	0	0
$191$	24.1225	1.74544	0.872720	−	0.488220i	$-0.162354\pi$
$191$	24.1225	1.74544	0.872720	+	0.488220i	$0.162354\pi$
$192$	0	0
$193$	−3.41216	−0.245613	−0.122806	−	0.992431i	$-0.539189\pi$
$193$	−3.41216	−0.245613	−0.122806	+	0.992431i	$0.539189\pi$
$194$	0	0
$195$	−10.8590	−0.777631
$196$	0	0
$197$	4.66110	0.332090	0.166045	−	0.986118i	$-0.446900\pi$
$197$	4.66110	0.332090	0.166045	+	0.986118i	$0.446900\pi$
$198$	0	0
$199$	10.4261	0.739088	0.369544	−	0.929213i	$-0.379514\pi$
$199$	10.4261	0.739088	0.369544	+	0.929213i	$0.379514\pi$
$200$	0	0
$201$	−2.10873	−0.148738
$202$	0	0
$203$	−2.63805	−0.185155
$204$	0	0
$205$	−5.52880	−0.386148
$206$	0	0
$207$	−5.97856	−0.415539
$208$	0	0
$209$	−16.8051	−1.16243
$210$	0	0
$211$	−15.9315	−1.09677	−0.548384	−	0.836226i	$-0.684757\pi$
$211$	−15.9315	−1.09677	−0.548384	+	0.836226i	$0.684757\pi$
$212$	0	0
$213$	−11.8866	−0.814459
$214$	0	0
$215$	−2.01915	−0.137705
$216$	0	0
$217$	−1.03376	−0.0701765
$218$	0	0
$219$	13.6096	0.919649
$220$	0	0
$221$	23.1857	1.55964
$222$	0	0
$223$	13.5783	0.909270	0.454635	−	0.890678i	$-0.349770\pi$
$223$	13.5783	0.909270	0.454635	+	0.890678i	$0.349770\pi$
$224$	0	0
$225$	1.17232	0.0781547
$226$	0	0
$227$	1.76364	0.117057	0.0585283	−	0.998286i	$-0.481359\pi$
$227$	1.76364	0.117057	0.0585283	+	0.998286i	$0.481359\pi$
$228$	0	0
$229$	−3.21524	−0.212469	−0.106234	−	0.994341i	$-0.533879\pi$
$229$	−3.21524	−0.212469	−0.106234	+	0.994341i	$0.533879\pi$
$230$	0	0
$231$	−4.23700	−0.278774
$232$	0	0
$233$	25.8981	1.69664	0.848321	−	0.529482i	$-0.177614\pi$
$233$	25.8981	1.69664	0.848321	+	0.529482i	$0.177614\pi$
$234$	0	0
$235$	−11.9636	−0.780422
$236$	0	0
$237$	−4.21360	−0.273703
$238$	0	0
$239$	−2.26966	−0.146812	−0.0734062	−	0.997302i	$-0.523387\pi$
$239$	−2.26966	−0.146812	−0.0734062	+	0.997302i	$0.523387\pi$
$240$	0	0
$241$	5.61835	0.361910	0.180955	−	0.983491i	$-0.442081\pi$
$241$	5.61835	0.361910	0.180955	+	0.983491i	$0.442081\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−14.6275	−0.934517
$246$	0	0
$247$	−18.2835	−1.16335
$248$	0	0
$249$	−9.78346	−0.620002
$250$	0	0
$251$	14.3387	0.905048	0.452524	−	0.891752i	$-0.350524\pi$
$251$	14.3387	0.905048	0.452524	+	0.891752i	$0.350524\pi$
$252$	0	0
$253$	24.0185	1.51003
$254$	0	0
$255$	−13.1789	−0.825293
$256$	0	0
$257$	1.56506	0.0976257	0.0488128	−	0.998808i	$-0.484456\pi$
$257$	1.56506	0.0976257	0.0488128	+	0.998808i	$0.484456\pi$
$258$	0	0
$259$	0.125672	0.00780886
$260$	0	0
$261$	−2.50134	−0.154829
$262$	0	0
$263$	26.0080	1.60372	0.801862	−	0.597510i	$-0.203843\pi$
$263$	26.0080	1.60372	0.801862	+	0.597510i	$0.203843\pi$
$264$	0	0
$265$	2.06544	0.126879
$266$	0	0
$267$	15.2828	0.935295
$268$	0	0
$269$	−25.3580	−1.54610	−0.773052	−	0.634343i	$-0.781271\pi$
$269$	−25.3580	−1.54610	−0.773052	+	0.634343i	$0.781271\pi$
$270$	0	0
$271$	5.93004	0.360224	0.180112	−	0.983646i	$-0.442354\pi$
$271$	5.93004	0.360224	0.180112	+	0.983646i	$0.442354\pi$
$272$	0	0
$273$	−4.60974	−0.278994
$274$	0	0
$275$	−4.70972	−0.284007
$276$	0	0
$277$	−3.14081	−0.188713	−0.0943566	−	0.995538i	$-0.530079\pi$
$277$	−3.14081	−0.188713	−0.0943566	+	0.995538i	$0.530079\pi$
$278$	0	0
$279$	−0.980193	−0.0586826
$280$	0	0
$281$	−22.5471	−1.34505	−0.672524	−	0.740076i	$-0.734790\pi$
$281$	−22.5471	−1.34505	−0.672524	+	0.740076i	$0.734790\pi$
$282$	0	0
$283$	31.0880	1.84799	0.923995	−	0.382404i	$-0.124904\pi$
$283$	31.0880	1.84799	0.923995	+	0.382404i	$0.124904\pi$
$284$	0	0
$285$	10.3924	0.615594
$286$	0	0
$287$	−2.34702	−0.138540
$288$	0	0
$289$	11.1390	0.655234
$290$	0	0
$291$	−11.5419	−0.676596
$292$	0	0
$293$	12.6151	0.736983	0.368492	−	0.929631i	$-0.379874\pi$
$293$	12.6151	0.736983	0.368492	+	0.929631i	$0.379874\pi$
$294$	0	0
$295$	10.8057	0.629132
$296$	0	0
$297$	−4.01743	−0.233115
$298$	0	0
$299$	26.1314	1.51122
$300$	0	0
$301$	−0.857147	−0.0494051
$302$	0	0
$303$	−19.7358	−1.13379
$304$	0	0
$305$	7.74512	0.443484
$306$	0	0
$307$	12.4460	0.710332	0.355166	−	0.934803i	$-0.384424\pi$
$307$	12.4460	0.710332	0.355166	+	0.934803i	$0.384424\pi$
$308$	0	0
$309$	8.48413	0.482645
$310$	0	0
$311$	17.8323	1.01118	0.505588	−	0.862775i	$-0.331276\pi$
$311$	17.8323	1.01118	0.505588	+	0.862775i	$0.331276\pi$
$312$	0	0
$313$	−26.8191	−1.51591	−0.757953	−	0.652309i	$-0.773800\pi$
$313$	−26.8191	−1.51591	−0.757953	+	0.652309i	$0.773800\pi$
$314$	0	0
$315$	2.62020	0.147632
$316$	0	0
$317$	−18.9847	−1.06629	−0.533144	−	0.846024i	$-0.678990\pi$
$317$	−18.9847	−1.06629	−0.533144	+	0.846024i	$0.678990\pi$
$318$	0	0
$319$	10.0490	0.562635
$320$	0	0
$321$	−12.5953	−0.703003
$322$	0	0
$323$	−22.1895	−1.23465
$324$	0	0
$325$	−5.12404	−0.284231
$326$	0	0
$327$	12.8417	0.710145
$328$	0	0
$329$	−5.07866	−0.279996
$330$	0	0
$331$	14.3796	0.790372	0.395186	−	0.918601i	$-0.370680\pi$
$331$	14.3796	0.790372	0.395186	+	0.918601i	$0.370680\pi$
$332$	0	0
$333$	0.119159	0.00652988
$334$	0	0
$335$	−5.23896	−0.286235
$336$	0	0
$337$	0.924650	0.0503689	0.0251845	−	0.999683i	$-0.491983\pi$
$337$	0.924650	0.0503689	0.0251845	+	0.999683i	$0.491983\pi$
$338$	0	0
$339$	−10.8335	−0.588393
$340$	0	0
$341$	3.93786	0.213247
$342$	0	0
$343$	−13.5921	−0.733903
$344$	0	0
$345$	−14.8532	−0.799671
$346$	0	0
$347$	1.79552	0.0963884	0.0481942	−	0.998838i	$-0.484653\pi$
$347$	1.79552	0.0963884	0.0481942	+	0.998838i	$0.484653\pi$
$348$	0	0
$349$	9.74494	0.521635	0.260817	−	0.965388i	$-0.416008\pi$
$349$	9.74494	0.521635	0.260817	+	0.965388i	$0.416008\pi$
$350$	0	0
$351$	−4.37086	−0.233299
$352$	0	0
$353$	−27.0143	−1.43783	−0.718913	−	0.695100i	$-0.755360\pi$
$353$	−27.0143	−1.43783	−0.718913	+	0.695100i	$0.755360\pi$
$354$	0	0
$355$	−29.5313	−1.56736
$356$	0	0
$357$	−5.59454	−0.296094
$358$	0	0
$359$	−21.8620	−1.15383	−0.576916	−	0.816803i	$-0.695744\pi$
$359$	−21.8620	−1.15383	−0.576916	+	0.816803i	$0.695744\pi$
$360$	0	0
$361$	−1.50213	−0.0790595
$362$	0	0
$363$	5.13978	0.269768
$364$	0	0
$365$	33.8118	1.76979
$366$	0	0
$367$	12.7030	0.663090	0.331545	−	0.943439i	$-0.392430\pi$
$367$	12.7030	0.663090	0.331545	+	0.943439i	$0.392430\pi$
$368$	0	0
$369$	−2.22539	−0.115849
$370$	0	0
$371$	0.876797	0.0455211
$372$	0	0
$373$	20.4999	1.06144	0.530722	−	0.847546i	$-0.321921\pi$
$373$	20.4999	1.06144	0.530722	+	0.847546i	$0.321921\pi$
$374$	0	0
$375$	−9.50955	−0.491071
$376$	0	0
$377$	10.9330	0.563079
$378$	0	0
$379$	−9.86234	−0.506594	−0.253297	−	0.967389i	$-0.581515\pi$
$379$	−9.86234	−0.506594	−0.253297	+	0.967389i	$0.581515\pi$
$380$	0	0
$381$	−1.98049	−0.101464
$382$	0	0
$383$	18.0799	0.923840	0.461920	−	0.886922i	$-0.347161\pi$
$383$	18.0799	0.923840	0.461920	+	0.886922i	$0.347161\pi$
$384$	0	0
$385$	−10.5265	−0.536479
$386$	0	0
$387$	−0.812728	−0.0413133
$388$	0	0
$389$	−34.5027	−1.74936	−0.874679	−	0.484703i	$-0.838928\pi$
$389$	−34.5027	−1.74936	−0.874679	+	0.484703i	$0.838928\pi$
$390$	0	0
$391$	31.7140	1.60384
$392$	0	0
$393$	19.0312	0.959998
$394$	0	0
$395$	−10.4683	−0.526719
$396$	0	0
$397$	30.3398	1.52271	0.761356	−	0.648335i	$-0.224534\pi$
$397$	30.3398	1.52271	0.761356	+	0.648335i	$0.224534\pi$
$398$	0	0
$399$	4.41167	0.220860
$400$	0	0
$401$	−11.7104	−0.584790	−0.292395	−	0.956298i	$-0.594452\pi$
$401$	−11.7104	−0.584790	−0.292395	+	0.956298i	$0.594452\pi$
$402$	0	0
$403$	4.28428	0.213415
$404$	0	0
$405$	2.48442	0.123452
$406$	0	0
$407$	−0.478714	−0.0237290
$408$	0	0
$409$	15.5699	0.769880	0.384940	−	0.922942i	$-0.374222\pi$
$409$	15.5699	0.769880	0.384940	+	0.922942i	$0.374222\pi$
$410$	0	0
$411$	15.9477	0.786641
$412$	0	0
$413$	4.58711	0.225717
$414$	0	0
$415$	−24.3062	−1.19314
$416$	0	0
$417$	−12.5090	−0.612566
$418$	0	0
$419$	14.7342	0.719815	0.359907	−	0.932988i	$-0.382808\pi$
$419$	14.7342	0.719815	0.359907	+	0.932988i	$0.382808\pi$
$420$	0	0
$421$	2.76643	0.134828	0.0674139	−	0.997725i	$-0.478525\pi$
$421$	2.76643	0.134828	0.0674139	+	0.997725i	$0.478525\pi$
$422$	0	0
$423$	−4.81548	−0.234136
$424$	0	0
$425$	−6.21871	−0.301652
$426$	0	0
$427$	3.28787	0.159111
$428$	0	0
$429$	17.5596	0.847787
$430$	0	0
$431$	−2.30117	−0.110843	−0.0554216	−	0.998463i	$-0.517650\pi$
$431$	−2.30117	−0.110843	−0.0554216	+	0.998463i	$0.517650\pi$
$432$	0	0
$433$	−25.8309	−1.24135	−0.620676	−	0.784067i	$-0.713142\pi$
$433$	−25.8309	−1.24135	−0.620676	+	0.784067i	$0.713142\pi$
$434$	0	0
$435$	−6.21438	−0.297957
$436$	0	0
$437$	−25.0086	−1.19632
$438$	0	0
$439$	−13.8806	−0.662485	−0.331242	−	0.943546i	$-0.607468\pi$
$439$	−13.8806	−0.662485	−0.331242	+	0.943546i	$0.607468\pi$
$440$	0	0
$441$	−5.88770	−0.280367
$442$	0	0
$443$	−10.2269	−0.485897	−0.242948	−	0.970039i	$-0.578115\pi$
$443$	−10.2269	−0.485897	−0.242948	+	0.970039i	$0.578115\pi$
$444$	0	0
$445$	37.9689	1.79990
$446$	0	0
$447$	−21.4384	−1.01400
$448$	0	0
$449$	−35.2839	−1.66515	−0.832576	−	0.553911i	$-0.813135\pi$
$449$	−35.2839	−1.66515	−0.832576	+	0.553911i	$0.813135\pi$
$450$	0	0
$451$	8.94036	0.420985
$452$	0	0
$453$	−9.67697	−0.454664
$454$	0	0
$455$	−11.4525	−0.536902
$456$	0	0
$457$	−8.66677	−0.405414	−0.202707	−	0.979239i	$-0.564974\pi$
$457$	−8.66677	−0.405414	−0.202707	+	0.979239i	$0.564974\pi$
$458$	0	0
$459$	−5.30462	−0.247598
$460$	0	0
$461$	−17.1021	−0.796525	−0.398263	−	0.917271i	$-0.630387\pi$
$461$	−17.1021	−0.796525	−0.398263	+	0.917271i	$0.630387\pi$
$462$	0	0
$463$	−18.8628	−0.876629	−0.438315	−	0.898822i	$-0.644424\pi$
$463$	−18.8628	−0.876629	−0.438315	+	0.898822i	$0.644424\pi$
$464$	0	0
$465$	−2.43521	−0.112930
$466$	0	0
$467$	−23.1846	−1.07286	−0.536428	−	0.843946i	$-0.680227\pi$
$467$	−23.1846	−1.07286	−0.536428	+	0.843946i	$0.680227\pi$
$468$	0	0
$469$	−2.22398	−0.102694
$470$	0	0
$471$	−12.0099	−0.553389
$472$	0	0
$473$	3.26508	0.150129
$474$	0	0
$475$	4.90387	0.225005
$476$	0	0
$477$	0.831360	0.0380654
$478$	0	0
$479$	−29.2657	−1.33718	−0.668592	−	0.743629i	$-0.733103\pi$
$479$	−29.2657	−1.33718	−0.668592	+	0.743629i	$0.733103\pi$
$480$	0	0
$481$	−0.520828	−0.0237477
$482$	0	0
$483$	−6.30531	−0.286902
$484$	0	0
$485$	−28.6748	−1.30205
$486$	0	0
$487$	23.1260	1.04794	0.523969	−	0.851737i	$-0.324451\pi$
$487$	23.1260	1.04794	0.523969	+	0.851737i	$0.324451\pi$
$488$	0	0
$489$	−2.49487	−0.112822
$490$	0	0
$491$	−5.75991	−0.259941	−0.129970	−	0.991518i	$-0.541488\pi$
$491$	−5.75991	−0.259941	−0.129970	+	0.991518i	$0.541488\pi$
$492$	0	0
$493$	13.2687	0.597591
$494$	0	0
$495$	−9.98098	−0.448611
$496$	0	0
$497$	−12.5363	−0.562330
$498$	0	0
$499$	11.5592	0.517461	0.258731	−	0.965950i	$-0.416696\pi$
$499$	11.5592	0.517461	0.258731	+	0.965950i	$0.416696\pi$
$500$	0	0
$501$	12.9914	0.580414
$502$	0	0
$503$	−1.00000	−0.0445878
$503$	−1.00000	−0.0445878
$504$	0	0
$505$	−49.0319	−2.18189
$506$	0	0
$507$	6.10439	0.271105
$508$	0	0
$509$	−35.3469	−1.56672	−0.783361	−	0.621567i	$-0.786496\pi$
$509$	−35.3469	−1.56672	−0.783361	+	0.621567i	$0.786496\pi$
$510$	0	0
$511$	14.3534	0.634957
$512$	0	0
$513$	4.18305	0.184686
$514$	0	0
$515$	21.0781	0.928812
$516$	0	0
$517$	19.3459	0.850830
$518$	0	0
$519$	11.1085	0.487610
$520$	0	0
$521$	−9.01232	−0.394837	−0.197418	−	0.980319i	$-0.563256\pi$
$521$	−9.01232	−0.394837	−0.197418	+	0.980319i	$0.563256\pi$
$522$	0	0
$523$	10.5324	0.460550	0.230275	−	0.973126i	$-0.426037\pi$
$523$	10.5324	0.460550	0.230275	+	0.973126i	$0.426037\pi$
$524$	0	0
$525$	1.23639	0.0539606
$526$	0	0
$527$	5.19955	0.226496
$528$	0	0
$529$	12.7432	0.554051
$530$	0	0
$531$	4.34939	0.188748
$532$	0	0
$533$	9.72686	0.421317
$534$	0	0
$535$	−31.2920	−1.35287
$536$	0	0
$537$	0.128562	0.00554786
$538$	0	0
$539$	23.6535	1.01883
$540$	0	0
$541$	−20.2771	−0.871779	−0.435889	−	0.900000i	$-0.643566\pi$
$541$	−20.2771	−0.871779	−0.435889	+	0.900000i	$0.643566\pi$
$542$	0	0
$543$	−1.48913	−0.0639049
$544$	0	0
$545$	31.9040	1.36662
$546$	0	0
$547$	37.3092	1.59523	0.797613	−	0.603170i	$-0.206096\pi$
$547$	37.3092	1.59523	0.797613	+	0.603170i	$0.206096\pi$
$548$	0	0
$549$	3.11748	0.133051
$550$	0	0
$551$	−10.4632	−0.445749
$552$	0	0
$553$	−4.44389	−0.188973
$554$	0	0
$555$	0.296041	0.0125662
$556$	0	0
$557$	−29.4072	−1.24602	−0.623011	−	0.782213i	$-0.714091\pi$
$557$	−29.4072	−1.24602	−0.623011	+	0.782213i	$0.714091\pi$
$558$	0	0
$559$	3.55232	0.150247
$560$	0	0
$561$	21.3110	0.899749
$562$	0	0
$563$	25.0200	1.05447	0.527233	−	0.849721i	$-0.323229\pi$
$563$	25.0200	1.05447	0.527233	+	0.849721i	$0.323229\pi$
$564$	0	0
$565$	−26.9148	−1.13232
$566$	0	0
$567$	1.05465	0.0442913
$568$	0	0
$569$	−8.67473	−0.363664	−0.181832	−	0.983330i	$-0.558203\pi$
$569$	−8.67473	−0.363664	−0.181832	+	0.983330i	$0.558203\pi$
$570$	0	0
$571$	−31.2510	−1.30781	−0.653907	−	0.756575i	$-0.726871\pi$
$571$	−31.2510	−1.30781	−0.653907	+	0.756575i	$0.726871\pi$
$572$	0	0
$573$	24.1225	1.00773
$574$	0	0
$575$	−7.00879	−0.292287
$576$	0	0
$577$	−1.79266	−0.0746295	−0.0373148	−	0.999304i	$-0.511880\pi$
$577$	−1.79266	−0.0746295	−0.0373148	+	0.999304i	$0.511880\pi$
$578$	0	0
$579$	−3.41216	−0.141804
$580$	0	0
$581$	−10.3182	−0.428070
$582$	0	0
$583$	−3.33993	−0.138326
$584$	0	0
$585$	−10.8590	−0.448965
$586$	0	0
$587$	−2.03057	−0.0838105	−0.0419052	−	0.999122i	$-0.513343\pi$
$587$	−2.03057	−0.0838105	−0.0419052	+	0.999122i	$0.513343\pi$
$588$	0	0
$589$	−4.10019	−0.168945
$590$	0	0
$591$	4.66110	0.191732
$592$	0	0
$593$	32.6066	1.33899	0.669497	−	0.742815i	$-0.266510\pi$
$593$	32.6066	1.33899	0.669497	+	0.742815i	$0.266510\pi$
$594$	0	0
$595$	−13.8992	−0.569810
$596$	0	0
$597$	10.4261	0.426713
$598$	0	0
$599$	−34.2306	−1.39862	−0.699311	−	0.714817i	$-0.746510\pi$
$599$	−34.2306	−1.39862	−0.699311	+	0.714817i	$0.746510\pi$
$600$	0	0
$601$	3.20890	0.130894	0.0654468	−	0.997856i	$-0.479153\pi$
$601$	3.20890	0.130894	0.0654468	+	0.997856i	$0.479153\pi$
$602$	0	0
$603$	−2.10873	−0.0858741
$604$	0	0
$605$	12.7693	0.519148
$606$	0	0
$607$	−11.2556	−0.456851	−0.228425	−	0.973561i	$-0.573358\pi$
$607$	−11.2556	−0.456851	−0.228425	+	0.973561i	$0.573358\pi$
$608$	0	0
$609$	−2.63805	−0.106899
$610$	0	0
$611$	21.0478	0.851501
$612$	0	0
$613$	26.4653	1.06892	0.534461	−	0.845193i	$-0.320515\pi$
$613$	26.4653	1.06892	0.534461	+	0.845193i	$0.320515\pi$
$614$	0	0
$615$	−5.52880	−0.222943
$616$	0	0
$617$	−11.3482	−0.456863	−0.228431	−	0.973560i	$-0.573360\pi$
$617$	−11.3482	−0.456863	−0.228431	+	0.973560i	$0.573360\pi$
$618$	0	0
$619$	−39.5468	−1.58952	−0.794760	−	0.606924i	$-0.792403\pi$
$619$	−39.5468	−1.58952	−0.794760	+	0.606924i	$0.792403\pi$
$620$	0	0
$621$	−5.97856	−0.239911
$622$	0	0
$623$	16.1181	0.645759
$624$	0	0
$625$	−29.4873	−1.17949
$626$	0	0
$627$	−16.8051	−0.671131
$628$	0	0
$629$	−0.632094	−0.0252032
$630$	0	0
$631$	11.9826	0.477022	0.238511	−	0.971140i	$-0.423341\pi$
$631$	11.9826	0.477022	0.238511	+	0.971140i	$0.423341\pi$
$632$	0	0
$633$	−15.9315	−0.633220
$634$	0	0
$635$	−4.92037	−0.195259
$636$	0	0
$637$	25.7343	1.01963
$638$	0	0
$639$	−11.8866	−0.470228
$640$	0	0
$641$	16.5716	0.654540	0.327270	−	0.944931i	$-0.393871\pi$
$641$	16.5716	0.654540	0.327270	+	0.944931i	$0.393871\pi$
$642$	0	0
$643$	−2.28524	−0.0901211	−0.0450606	−	0.998984i	$-0.514348\pi$
$643$	−2.28524	−0.0901211	−0.0450606	+	0.998984i	$0.514348\pi$
$644$	0	0
$645$	−2.01915	−0.0795041
$646$	0	0
$647$	23.3807	0.919191	0.459596	−	0.888128i	$-0.347994\pi$
$647$	23.3807	0.919191	0.459596	+	0.888128i	$0.347994\pi$
$648$	0	0
$649$	−17.4734	−0.685891
$650$	0	0
$651$	−1.03376	−0.0405164
$652$	0	0
$653$	−8.17792	−0.320027	−0.160013	−	0.987115i	$-0.551154\pi$
$653$	−8.17792	−0.320027	−0.160013	+	0.987115i	$0.551154\pi$
$654$	0	0
$655$	47.2815	1.84744
$656$	0	0
$657$	13.6096	0.530960
$658$	0	0
$659$	8.67048	0.337754	0.168877	−	0.985637i	$-0.445986\pi$
$659$	8.67048	0.337754	0.168877	+	0.985637i	$0.445986\pi$
$660$	0	0
$661$	−8.16325	−0.317514	−0.158757	−	0.987318i	$-0.550749\pi$
$661$	−8.16325	−0.317514	−0.158757	+	0.987318i	$0.550749\pi$
$662$	0	0
$663$	23.1857	0.900459
$664$	0	0
$665$	10.9604	0.425027
$666$	0	0
$667$	14.9544	0.579038
$668$	0	0
$669$	13.5783	0.524968
$670$	0	0
$671$	−12.5243	−0.483495
$672$	0	0
$673$	32.8917	1.26788	0.633942	−	0.773381i	$-0.281436\pi$
$673$	32.8917	1.26788	0.633942	+	0.773381i	$0.281436\pi$
$674$	0	0
$675$	1.17232	0.0451226
$676$	0	0
$677$	−22.4717	−0.863658	−0.431829	−	0.901956i	$-0.642132\pi$
$677$	−22.4717	−0.863658	−0.431829	+	0.901956i	$0.642132\pi$
$678$	0	0
$679$	−12.1727	−0.467144
$680$	0	0
$681$	1.76364	0.0675827
$682$	0	0
$683$	−7.64770	−0.292631	−0.146316	−	0.989238i	$-0.546741\pi$
$683$	−7.64770	−0.292631	−0.146316	+	0.989238i	$0.546741\pi$
$684$	0	0
$685$	39.6206	1.51383
$686$	0	0
$687$	−3.21524	−0.122669
$688$	0	0
$689$	−3.63376	−0.138435
$690$	0	0
$691$	−30.6141	−1.16462	−0.582308	−	0.812968i	$-0.697850\pi$
$691$	−30.6141	−1.16462	−0.582308	+	0.812968i	$0.697850\pi$
$692$	0	0
$693$	−4.23700	−0.160950
$694$	0	0
$695$	−31.0774	−1.17883
$696$	0	0
$697$	11.8049	0.447141
$698$	0	0
$699$	25.8981	0.979557
$700$	0	0
$701$	8.36913	0.316098	0.158049	−	0.987431i	$-0.449480\pi$
$701$	8.36913	0.316098	0.158049	+	0.987431i	$0.449480\pi$
$702$	0	0
$703$	0.498448	0.0187993
$704$	0	0
$705$	−11.9636	−0.450577
$706$	0	0
$707$	−20.8144	−0.782808
$708$	0	0
$709$	27.3941	1.02881	0.514403	−	0.857549i	$-0.328014\pi$
$709$	27.3941	1.02881	0.514403	+	0.857549i	$0.328014\pi$
$710$	0	0
$711$	−4.21360	−0.158022
$712$	0	0
$713$	5.86014	0.219464
$714$	0	0
$715$	43.6254	1.63150
$716$	0	0
$717$	−2.26966	−0.0847621
$718$	0	0
$719$	22.5167	0.839732	0.419866	−	0.907586i	$-0.362077\pi$
$719$	22.5167	0.839732	0.419866	+	0.907586i	$0.362077\pi$
$720$	0	0
$721$	8.94782	0.333234
$722$	0	0
$723$	5.61835	0.208949
$724$	0	0
$725$	−2.93238	−0.108906
$726$	0	0
$727$	17.0213	0.631285	0.315643	−	0.948878i	$-0.397780\pi$
$727$	17.0213	0.631285	0.315643	+	0.948878i	$0.397780\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	4.31121	0.159456
$732$	0	0
$733$	−10.6392	−0.392968	−0.196484	−	0.980507i	$-0.562952\pi$
$733$	−10.6392	−0.392968	−0.196484	+	0.980507i	$0.562952\pi$
$734$	0	0
$735$	−14.6275	−0.539544
$736$	0	0
$737$	8.47168	0.312058
$738$	0	0
$739$	50.9689	1.87492	0.937460	−	0.348092i	$-0.113170\pi$
$739$	50.9689	1.87492	0.937460	+	0.348092i	$0.113170\pi$
$740$	0	0
$741$	−18.2835	−0.671661
$742$	0	0
$743$	−15.7645	−0.578344	−0.289172	−	0.957277i	$-0.593380\pi$
$743$	−15.7645	−0.578344	−0.289172	+	0.957277i	$0.593380\pi$
$744$	0	0
$745$	−53.2619	−1.95137
$746$	0	0
$747$	−9.78346	−0.357958
$748$	0	0
$749$	−13.2837	−0.485377
$750$	0	0
$751$	−13.1094	−0.478368	−0.239184	−	0.970974i	$-0.576880\pi$
$751$	−13.1094	−0.478368	−0.239184	+	0.970974i	$0.576880\pi$
$752$	0	0
$753$	14.3387	0.522530
$754$	0	0
$755$	−24.0416	−0.874964
$756$	0	0
$757$	−11.6733	−0.424274	−0.212137	−	0.977240i	$-0.568042\pi$
$757$	−11.6733	−0.424274	−0.212137	+	0.977240i	$0.568042\pi$
$758$	0	0
$759$	24.0185	0.871815
$760$	0	0
$761$	−13.4298	−0.486831	−0.243415	−	0.969922i	$-0.578268\pi$
$761$	−13.4298	−0.486831	−0.243415	+	0.969922i	$0.578268\pi$
$762$	0	0
$763$	13.5435	0.490308
$764$	0	0
$765$	−13.1789	−0.476483
$766$	0	0
$767$	−19.0106	−0.686432
$768$	0	0
$769$	2.33048	0.0840393	0.0420197	−	0.999117i	$-0.486621\pi$
$769$	2.33048	0.0840393	0.0420197	+	0.999117i	$0.486621\pi$
$770$	0	0
$771$	1.56506	0.0563642
$772$	0	0
$773$	25.0390	0.900591	0.450296	−	0.892879i	$-0.351319\pi$
$773$	25.0390	0.900591	0.450296	+	0.892879i	$0.351319\pi$
$774$	0	0
$775$	−1.14910	−0.0412769
$776$	0	0
$777$	0.125672	0.00450845
$778$	0	0
$779$	−9.30891	−0.333526
$780$	0	0
$781$	47.7538	1.70876
$782$	0	0
$783$	−2.50134	−0.0893907
$784$	0	0
$785$	−29.8377	−1.06495
$786$	0	0
$787$	29.5202	1.05228	0.526141	−	0.850398i	$-0.323639\pi$
$787$	29.5202	1.05228	0.526141	+	0.850398i	$0.323639\pi$
$788$	0	0
$789$	26.0080	0.925910
$790$	0	0
$791$	−11.4256	−0.406246
$792$	0	0
$793$	−13.6261	−0.483876
$794$	0	0
$795$	2.06544	0.0732537
$796$	0	0
$797$	44.1160	1.56267	0.781334	−	0.624113i	$-0.214540\pi$
$797$	44.1160	1.56267	0.781334	+	0.624113i	$0.214540\pi$
$798$	0	0
$799$	25.5443	0.903691
$800$	0	0
$801$	15.2828	0.539993
$802$	0	0
$803$	−54.6755	−1.92946
$804$	0	0
$805$	−15.6650	−0.552119
$806$	0	0
$807$	−25.3580	−0.892643
$808$	0	0
$809$	18.2833	0.642806	0.321403	−	0.946942i	$-0.395846\pi$
$809$	18.2833	0.642806	0.321403	+	0.946942i	$0.395846\pi$
$810$	0	0
$811$	−1.91833	−0.0673616	−0.0336808	−	0.999433i	$-0.510723\pi$
$811$	−1.91833	−0.0673616	−0.0336808	+	0.999433i	$0.510723\pi$
$812$	0	0
$813$	5.93004	0.207976
$814$	0	0
$815$	−6.19829	−0.217116
$816$	0	0
$817$	−3.39968	−0.118940
$818$	0	0
$819$	−4.60974	−0.161077
$820$	0	0
$821$	−34.8161	−1.21509	−0.607545	−	0.794286i	$-0.707845\pi$
$821$	−34.8161	−1.21509	−0.607545	+	0.794286i	$0.707845\pi$
$822$	0	0
$823$	30.0431	1.04724	0.523618	−	0.851953i	$-0.324582\pi$
$823$	30.0431	1.04724	0.523618	+	0.851953i	$0.324582\pi$
$824$	0	0
$825$	−4.70972	−0.163971
$826$	0	0
$827$	−20.4809	−0.712192	−0.356096	−	0.934449i	$-0.615892\pi$
$827$	−20.4809	−0.712192	−0.356096	+	0.934449i	$0.615892\pi$
$828$	0	0
$829$	8.60006	0.298692	0.149346	−	0.988785i	$-0.452283\pi$
$829$	8.60006	0.298692	0.149346	+	0.988785i	$0.452283\pi$
$830$	0	0
$831$	−3.14081	−0.108954
$832$	0	0
$833$	31.2320	1.08213
$834$	0	0
$835$	32.2761	1.11696
$836$	0	0
$837$	−0.980193	−0.0338804
$838$	0	0
$839$	11.0056	0.379957	0.189978	−	0.981788i	$-0.439158\pi$
$839$	11.0056	0.379957	0.189978	+	0.981788i	$0.439158\pi$
$840$	0	0
$841$	−22.7433	−0.784251
$842$	0	0
$843$	−22.5471	−0.776563
$844$	0	0
$845$	15.1658	0.521720
$846$	0	0
$847$	5.42069	0.186257
$848$	0	0
$849$	31.0880	1.06694
$850$	0	0
$851$	−0.712400	−0.0244208
$852$	0	0
$853$	24.6178	0.842899	0.421449	−	0.906852i	$-0.361522\pi$
$853$	24.6178	0.842899	0.421449	+	0.906852i	$0.361522\pi$
$854$	0	0
$855$	10.3924	0.355413
$856$	0	0
$857$	−6.92536	−0.236566	−0.118283	−	0.992980i	$-0.537739\pi$
$857$	−6.92536	−0.236566	−0.118283	+	0.992980i	$0.537739\pi$
$858$	0	0
$859$	−1.13144	−0.0386041	−0.0193021	−	0.999814i	$-0.506144\pi$
$859$	−1.13144	−0.0386041	−0.0193021	+	0.999814i	$0.506144\pi$
$860$	0	0
$861$	−2.34702	−0.0799862
$862$	0	0
$863$	−2.55593	−0.0870048	−0.0435024	−	0.999053i	$-0.513852\pi$
$863$	−2.55593	−0.0870048	−0.0435024	+	0.999053i	$0.513852\pi$
$864$	0	0
$865$	27.5982	0.938367
$866$	0	0
$867$	11.1390	0.378300
$868$	0	0
$869$	16.9278	0.574238
$870$	0	0
$871$	9.21695	0.312304
$872$	0	0
$873$	−11.5419	−0.390633
$874$	0	0
$875$	−10.0293	−0.339052
$876$	0	0
$877$	−32.6626	−1.10294	−0.551468	−	0.834196i	$-0.685932\pi$
$877$	−32.6626	−1.10294	−0.551468	+	0.834196i	$0.685932\pi$
$878$	0	0
$879$	12.6151	0.425498
$880$	0	0
$881$	1.75551	0.0591446	0.0295723	−	0.999563i	$-0.490585\pi$
$881$	1.75551	0.0591446	0.0295723	+	0.999563i	$0.490585\pi$
$882$	0	0
$883$	−27.9416	−0.940308	−0.470154	−	0.882584i	$-0.655802\pi$
$883$	−27.9416	−0.940308	−0.470154	+	0.882584i	$0.655802\pi$
$884$	0	0
$885$	10.8057	0.363230
$886$	0	0
$887$	−28.0057	−0.940338	−0.470169	−	0.882576i	$-0.655807\pi$
$887$	−28.0057	−0.940338	−0.470169	+	0.882576i	$0.655807\pi$
$888$	0	0
$889$	−2.08874	−0.0700539
$890$	0	0
$891$	−4.01743	−0.134589
$892$	0	0
$893$	−20.1434	−0.674072
$894$	0	0
$895$	0.319401	0.0106764
$896$	0	0
$897$	26.1314	0.872503
$898$	0	0
$899$	2.45180	0.0817721
$900$	0	0
$901$	−4.41005	−0.146920
$902$	0	0
$903$	−0.857147	−0.0285241
$904$	0	0
$905$	−3.69963	−0.122980
$906$	0	0
$907$	19.9592	0.662734	0.331367	−	0.943502i	$-0.392490\pi$
$907$	19.9592	0.662734	0.331367	+	0.943502i	$0.392490\pi$
$908$	0	0
$909$	−19.7358	−0.654595
$910$	0	0
$911$	−20.8739	−0.691582	−0.345791	−	0.938312i	$-0.612389\pi$
$911$	−20.8739	−0.691582	−0.345791	+	0.938312i	$0.612389\pi$
$912$	0	0
$913$	39.3044	1.30079
$914$	0	0
$915$	7.74512	0.256046
$916$	0	0
$917$	20.0714	0.662815
$918$	0	0
$919$	53.5443	1.76626	0.883132	−	0.469124i	$-0.155430\pi$
$919$	53.5443	1.76626	0.883132	+	0.469124i	$0.155430\pi$
$920$	0	0
$921$	12.4460	0.410110
$922$	0	0
$923$	51.9548	1.71011
$924$	0	0
$925$	0.139693	0.00459307
$926$	0	0
$927$	8.48413	0.278655
$928$	0	0
$929$	54.6576	1.79326	0.896629	−	0.442783i	$-0.146009\pi$
$929$	54.6576	1.79326	0.896629	+	0.442783i	$0.146009\pi$
$930$	0	0
$931$	−24.6285	−0.807168
$932$	0	0
$933$	17.8323	0.583803
$934$	0	0
$935$	52.9453	1.73150
$936$	0	0
$937$	−44.2994	−1.44720	−0.723599	−	0.690221i	$-0.757513\pi$
$937$	−44.2994	−1.44720	−0.723599	+	0.690221i	$0.757513\pi$
$938$	0	0
$939$	−26.8191	−0.875209
$940$	0	0
$941$	−53.7847	−1.75333	−0.876666	−	0.481100i	$-0.840237\pi$
$941$	−53.7847	−1.75333	−0.876666	+	0.481100i	$0.840237\pi$
$942$	0	0
$943$	13.3046	0.433258
$944$	0	0
$945$	2.62020	0.0852351
$946$	0	0
$947$	39.2958	1.27694	0.638471	−	0.769646i	$-0.279567\pi$
$947$	39.2958	1.27694	0.638471	+	0.769646i	$0.279567\pi$
$948$	0	0
$949$	−59.4855	−1.93098
$950$	0	0
$951$	−18.9847	−0.615622
$952$	0	0
$953$	27.4444	0.889013	0.444506	−	0.895776i	$-0.353379\pi$
$953$	27.4444	0.889013	0.444506	+	0.895776i	$0.353379\pi$
$954$	0	0
$955$	59.9302	1.93930
$956$	0	0
$957$	10.0490	0.324838
$958$	0	0
$959$	16.8193	0.543123
$960$	0	0
$961$	−30.0392	−0.969007
$962$	0	0
$963$	−12.5953	−0.405879
$964$	0	0
$965$	−8.47722	−0.272891
$966$	0	0
$967$	−42.0590	−1.35253	−0.676264	−	0.736660i	$-0.736402\pi$
$967$	−42.0590	−1.35253	−0.676264	+	0.736660i	$0.736402\pi$
$968$	0	0
$969$	−22.1895	−0.712828
$970$	0	0
$971$	−22.7144	−0.728940	−0.364470	−	0.931215i	$-0.618750\pi$
$971$	−22.7144	−0.728940	−0.364470	+	0.931215i	$0.618750\pi$
$972$	0	0
$973$	−13.1926	−0.422936
$974$	0	0
$975$	−5.12404	−0.164101
$976$	0	0
$977$	−22.1820	−0.709664	−0.354832	−	0.934930i	$-0.615462\pi$
$977$	−22.1820	−0.709664	−0.354832	+	0.934930i	$0.615462\pi$
$978$	0	0
$979$	−61.3978	−1.96228
$980$	0	0
$981$	12.8417	0.410003
$982$	0	0
$983$	−0.722876	−0.0230562	−0.0115281	−	0.999934i	$-0.503670\pi$
$983$	−0.722876	−0.0230562	−0.0115281	+	0.999934i	$0.503670\pi$
$984$	0	0
$985$	11.5801	0.368973
$986$	0	0
$987$	−5.07866	−0.161656
$988$	0	0
$989$	4.85894	0.154505
$990$	0	0
$991$	43.1987	1.37225	0.686126	−	0.727482i	$-0.259310\pi$
$991$	43.1987	1.37225	0.686126	+	0.727482i	$0.259310\pi$
$992$	0	0
$993$	14.3796	0.456321
$994$	0	0
$995$	25.9028	0.821175
$996$	0	0
$997$	49.6840	1.57351	0.786755	−	0.617266i	$-0.211760\pi$
$997$	49.6840	1.57351	0.786755	+	0.617266i	$0.211760\pi$
$998$	0	0
$999$	0.119159	0.00377003

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6036.2.a.g.1.13	✓	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6036.2.a.g.1.13	✓	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +2.48442 q^{5} +1.05465 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +2.48442 q^{5} +1.05465 q^{7} +1.00000 q^{9} -4.01743 q^{11} -4.37086 q^{13} +2.48442 q^{15} -5.30462 q^{17} +4.18305 q^{19} +1.05465 q^{21} -5.97856 q^{23} +1.17232 q^{25} +1.00000 q^{27} -2.50134 q^{29} -0.980193 q^{31} -4.01743 q^{33} +2.62020 q^{35} +0.119159 q^{37} -4.37086 q^{39} -2.22539 q^{41} -0.812728 q^{43} +2.48442 q^{45} -4.81548 q^{47} -5.88770 q^{49} -5.30462 q^{51} +0.831360 q^{53} -9.98098 q^{55} +4.18305 q^{57} +4.34939 q^{59} +3.11748 q^{61} +1.05465 q^{63} -10.8590 q^{65} -2.10873 q^{67} -5.97856 q^{69} -11.8866 q^{71} +13.6096 q^{73} +1.17232 q^{75} -4.23700 q^{77} -4.21360 q^{79} +1.00000 q^{81} -9.78346 q^{83} -13.1789 q^{85} -2.50134 q^{87} +15.2828 q^{89} -4.60974 q^{91} -0.980193 q^{93} +10.3924 q^{95} -11.5419 q^{97} -4.01743 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q + 15 q^{3} - 11 q^{5} - 4 q^{7} + 15 q^{9}+O(q^{10}) \) 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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