LMFDB - Embedded newform 6036.2.a.g.1.5

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			1.5
Root			$-1.01946$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} -2.01946 q^{5} -2.53250 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -2.01946 q^{5} -2.53250 q^{7} +1.00000 q^{9} +4.24449 q^{11} -2.95131 q^{13} -2.01946 q^{15} +7.86103 q^{17} -7.02817 q^{19} -2.53250 q^{21} -2.38097 q^{23} -0.921780 q^{25} +1.00000 q^{27} -0.629927 q^{29} +4.91752 q^{31} +4.24449 q^{33} +5.11428 q^{35} +1.38091 q^{37} -2.95131 q^{39} +4.74319 q^{41} -1.92065 q^{43} -2.01946 q^{45} +4.57547 q^{47} -0.586439 q^{49} +7.86103 q^{51} -5.90347 q^{53} -8.57157 q^{55} -7.02817 q^{57} +7.09876 q^{59} -11.3443 q^{61} -2.53250 q^{63} +5.96004 q^{65} +5.20369 q^{67} -2.38097 q^{69} -15.1860 q^{71} +5.38038 q^{73} -0.921780 q^{75} -10.7492 q^{77} +8.24659 q^{79} +1.00000 q^{81} -3.55142 q^{83} -15.8750 q^{85} -0.629927 q^{87} -6.50278 q^{89} +7.47418 q^{91} +4.91752 q^{93} +14.1931 q^{95} -17.5190 q^{97} +4.24449 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−2.01946	−0.903130	−0.451565	−	0.892238i	$-0.649134\pi$
$5$	−2.01946	−0.903130	−0.451565	+	0.892238i	$0.649134\pi$
$6$	0	0
$7$	−2.53250	−0.957195	−0.478598	−	0.878034i	$-0.658855\pi$
$7$	−2.53250	−0.957195	−0.478598	+	0.878034i	$0.658855\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	4.24449	1.27976	0.639881	−	0.768474i	$-0.278984\pi$
$11$	4.24449	1.27976	0.639881	+	0.768474i	$0.278984\pi$
$12$	0	0
$13$	−2.95131	−0.818545	−0.409272	−	0.912412i	$-0.634217\pi$
$13$	−2.95131	−0.818545	−0.409272	+	0.912412i	$0.634217\pi$
$14$	0	0
$15$	−2.01946	−0.521422
$16$	0	0
$17$	7.86103	1.90658	0.953290	−	0.302057i	$-0.0976734\pi$
$17$	7.86103	1.90658	0.953290	+	0.302057i	$0.0976734\pi$
$18$	0	0
$19$	−7.02817	−1.61237	−0.806186	−	0.591662i	$-0.798472\pi$
$19$	−7.02817	−1.61237	−0.806186	+	0.591662i	$0.798472\pi$
$20$	0	0
$21$	−2.53250	−0.552637
$22$	0	0
$23$	−2.38097	−0.496466	−0.248233	−	0.968700i	$-0.579850\pi$
$23$	−2.38097	−0.496466	−0.248233	+	0.968700i	$0.579850\pi$
$24$	0	0
$25$	−0.921780	−0.184356
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−0.629927	−0.116974	−0.0584872	−	0.998288i	$-0.518628\pi$
$29$	−0.629927	−0.116974	−0.0584872	+	0.998288i	$0.518628\pi$
$30$	0	0
$31$	4.91752	0.883213	0.441606	−	0.897209i	$-0.354409\pi$
$31$	4.91752	0.883213	0.441606	+	0.897209i	$0.354409\pi$
$32$	0	0
$33$	4.24449	0.738870
$34$	0	0
$35$	5.11428	0.864472
$36$	0	0
$37$	1.38091	0.227020	0.113510	−	0.993537i	$-0.463791\pi$
$37$	1.38091	0.227020	0.113510	+	0.993537i	$0.463791\pi$
$38$	0	0
$39$	−2.95131	−0.472587
$40$	0	0
$41$	4.74319	0.740762	0.370381	−	0.928880i	$-0.379227\pi$
$41$	4.74319	0.740762	0.370381	+	0.928880i	$0.379227\pi$
$42$	0	0
$43$	−1.92065	−0.292897	−0.146448	−	0.989218i	$-0.546784\pi$
$43$	−1.92065	−0.292897	−0.146448	+	0.989218i	$0.546784\pi$
$44$	0	0
$45$	−2.01946	−0.301043
$46$	0	0
$47$	4.57547	0.667401	0.333701	−	0.942679i	$-0.391703\pi$
$47$	4.57547	0.667401	0.333701	+	0.942679i	$0.391703\pi$
$48$	0	0
$49$	−0.586439	−0.0837771
$50$	0	0
$51$	7.86103	1.10076
$52$	0	0
$53$	−5.90347	−0.810904	−0.405452	−	0.914116i	$-0.632886\pi$
$53$	−5.90347	−0.810904	−0.405452	+	0.914116i	$0.632886\pi$
$54$	0	0
$55$	−8.57157	−1.15579
$56$	0	0
$57$	−7.02817	−0.930904
$58$	0	0
$59$	7.09876	0.924179	0.462090	−	0.886833i	$-0.347100\pi$
$59$	7.09876	0.924179	0.462090	+	0.886833i	$0.347100\pi$
$60$	0	0
$61$	−11.3443	−1.45249	−0.726243	−	0.687438i	$-0.758735\pi$
$61$	−11.3443	−1.45249	−0.726243	+	0.687438i	$0.758735\pi$
$62$	0	0
$63$	−2.53250	−0.319065
$64$	0	0
$65$	5.96004	0.739253
$66$	0	0
$67$	5.20369	0.635731	0.317866	−	0.948136i	$-0.397034\pi$
$67$	5.20369	0.635731	0.317866	+	0.948136i	$0.397034\pi$
$68$	0	0
$69$	−2.38097	−0.286635
$70$	0	0
$71$	−15.1860	−1.80225	−0.901125	−	0.433559i	$-0.857257\pi$
$71$	−15.1860	−1.80225	−0.901125	+	0.433559i	$0.857257\pi$
$72$	0	0
$73$	5.38038	0.629726	0.314863	−	0.949137i	$-0.398041\pi$
$73$	5.38038	0.629726	0.314863	+	0.949137i	$0.398041\pi$
$74$	0	0
$75$	−0.921780	−0.106438
$76$	0	0
$77$	−10.7492	−1.22498
$78$	0	0
$79$	8.24659	0.927814	0.463907	−	0.885884i	$-0.346447\pi$
$79$	8.24659	0.927814	0.463907	+	0.885884i	$0.346447\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−3.55142	−0.389819	−0.194909	−	0.980821i	$-0.562441\pi$
$83$	−3.55142	−0.389819	−0.194909	+	0.980821i	$0.562441\pi$
$84$	0	0
$85$	−15.8750	−1.72189
$86$	0	0
$87$	−0.629927	−0.0675352
$88$	0	0
$89$	−6.50278	−0.689293	−0.344647	−	0.938732i	$-0.612001\pi$
$89$	−6.50278	−0.689293	−0.344647	+	0.938732i	$0.612001\pi$
$90$	0	0
$91$	7.47418	0.783507
$92$	0	0
$93$	4.91752	0.509923
$94$	0	0
$95$	14.1931	1.45618
$96$	0	0
$97$	−17.5190	−1.77879	−0.889394	−	0.457141i	$-0.848874\pi$
$97$	−17.5190	−1.77879	−0.889394	+	0.457141i	$0.848874\pi$
$98$	0	0
$99$	4.24449	0.426587
$100$	0	0
$101$	2.11356	0.210307	0.105153	−	0.994456i	$-0.466467\pi$
$101$	2.11356	0.210307	0.105153	+	0.994456i	$0.466467\pi$
$102$	0	0
$103$	−16.2304	−1.59923	−0.799613	−	0.600515i	$-0.794962\pi$
$103$	−16.2304	−1.59923	−0.799613	+	0.600515i	$0.794962\pi$
$104$	0	0
$105$	5.11428	0.499103
$106$	0	0
$107$	8.13701	0.786634	0.393317	−	0.919403i	$-0.371327\pi$
$107$	8.13701	0.786634	0.393317	+	0.919403i	$0.371327\pi$
$108$	0	0
$109$	−9.54930	−0.914657	−0.457329	−	0.889298i	$-0.651194\pi$
$109$	−9.54930	−0.914657	−0.457329	+	0.889298i	$0.651194\pi$
$110$	0	0
$111$	1.38091	0.131070
$112$	0	0
$113$	−13.0671	−1.22925	−0.614627	−	0.788818i	$-0.710693\pi$
$113$	−13.0671	−1.22925	−0.614627	+	0.788818i	$0.710693\pi$
$114$	0	0
$115$	4.80827	0.448374
$116$	0	0
$117$	−2.95131	−0.272848
$118$	0	0
$119$	−19.9081	−1.82497
$120$	0	0
$121$	7.01567	0.637789
$122$	0	0
$123$	4.74319	0.427679
$124$	0	0
$125$	11.9588	1.06963
$126$	0	0
$127$	−5.03653	−0.446920	−0.223460	−	0.974713i	$-0.571735\pi$
$127$	−5.03653	−0.446920	−0.223460	+	0.974713i	$0.571735\pi$
$128$	0	0
$129$	−1.92065	−0.169104
$130$	0	0
$131$	−5.79467	−0.506283	−0.253141	−	0.967429i	$-0.581464\pi$
$131$	−5.79467	−0.506283	−0.253141	+	0.967429i	$0.581464\pi$
$132$	0	0
$133$	17.7988	1.54336
$134$	0	0
$135$	−2.01946	−0.173807
$136$	0	0
$137$	6.64234	0.567494	0.283747	−	0.958899i	$-0.408422\pi$
$137$	6.64234	0.567494	0.283747	+	0.958899i	$0.408422\pi$
$138$	0	0
$139$	11.6369	0.987026	0.493513	−	0.869738i	$-0.335713\pi$
$139$	11.6369	0.987026	0.493513	+	0.869738i	$0.335713\pi$
$140$	0	0
$141$	4.57547	0.385324
$142$	0	0
$143$	−12.5268	−1.04754
$144$	0	0
$145$	1.27211	0.105643
$146$	0	0
$147$	−0.586439	−0.0483687
$148$	0	0
$149$	1.73821	0.142400	0.0711999	−	0.997462i	$-0.477317\pi$
$149$	1.73821	0.142400	0.0711999	+	0.997462i	$0.477317\pi$
$150$	0	0
$151$	−15.4779	−1.25957	−0.629786	−	0.776769i	$-0.716858\pi$
$151$	−15.4779	−1.25957	−0.629786	+	0.776769i	$0.716858\pi$
$152$	0	0
$153$	7.86103	0.635526
$154$	0	0
$155$	−9.93074	−0.797656
$156$	0	0
$157$	12.9643	1.03467	0.517334	−	0.855784i	$-0.326925\pi$
$157$	12.9643	1.03467	0.517334	+	0.855784i	$0.326925\pi$
$158$	0	0
$159$	−5.90347	−0.468175
$160$	0	0
$161$	6.02980	0.475215
$162$	0	0
$163$	2.07407	0.162454	0.0812268	−	0.996696i	$-0.474116\pi$
$163$	2.07407	0.162454	0.0812268	+	0.996696i	$0.474116\pi$
$164$	0	0
$165$	−8.57157	−0.667296
$166$	0	0
$167$	−17.0029	−1.31572	−0.657861	−	0.753140i	$-0.728538\pi$
$167$	−17.0029	−1.31572	−0.657861	+	0.753140i	$0.728538\pi$
$168$	0	0
$169$	−4.28980	−0.329984
$170$	0	0
$171$	−7.02817	−0.537457
$172$	0	0
$173$	4.09680	0.311474	0.155737	−	0.987799i	$-0.450225\pi$
$173$	4.09680	0.311474	0.155737	+	0.987799i	$0.450225\pi$
$174$	0	0
$175$	2.33441	0.176465
$176$	0	0
$177$	7.09876	0.533575
$178$	0	0
$179$	−16.1857	−1.20978	−0.604888	−	0.796311i	$-0.706782\pi$
$179$	−16.1857	−1.20978	−0.604888	+	0.796311i	$0.706782\pi$
$180$	0	0
$181$	8.02017	0.596134	0.298067	−	0.954545i	$-0.403658\pi$
$181$	8.02017	0.596134	0.298067	+	0.954545i	$0.403658\pi$
$182$	0	0
$183$	−11.3443	−0.838593
$184$	0	0
$185$	−2.78869	−0.205029
$186$	0	0
$187$	33.3660	2.43997
$188$	0	0
$189$	−2.53250	−0.184212
$190$	0	0
$191$	−0.505177	−0.0365534	−0.0182767	−	0.999833i	$-0.505818\pi$
$191$	−0.505177	−0.0365534	−0.0182767	+	0.999833i	$0.505818\pi$
$192$	0	0
$193$	−5.10206	−0.367254	−0.183627	−	0.982996i	$-0.558784\pi$
$193$	−5.10206	−0.367254	−0.183627	+	0.982996i	$0.558784\pi$
$194$	0	0
$195$	5.96004	0.426808
$196$	0	0
$197$	11.6387	0.829224	0.414612	−	0.909998i	$-0.363917\pi$
$197$	11.6387	0.829224	0.414612	+	0.909998i	$0.363917\pi$
$198$	0	0
$199$	−25.0047	−1.77254	−0.886269	−	0.463171i	$-0.846712\pi$
$199$	−25.0047	−1.77254	−0.886269	+	0.463171i	$0.846712\pi$
$200$	0	0
$201$	5.20369	0.367040
$202$	0	0
$203$	1.59529	0.111967
$204$	0	0
$205$	−9.57868	−0.669004
$206$	0	0
$207$	−2.38097	−0.165489
$208$	0	0
$209$	−29.8310	−2.06345
$210$	0	0
$211$	−0.649648	−0.0447236	−0.0223618	−	0.999750i	$-0.507119\pi$
$211$	−0.649648	−0.0447236	−0.0223618	+	0.999750i	$0.507119\pi$
$212$	0	0
$213$	−15.1860	−1.04053
$214$	0	0
$215$	3.87868	0.264524
$216$	0	0
$217$	−12.4536	−0.845407
$218$	0	0
$219$	5.38038	0.363573
$220$	0	0
$221$	−23.2003	−1.56062
$222$	0	0
$223$	7.21727	0.483304	0.241652	−	0.970363i	$-0.422311\pi$
$223$	7.21727	0.483304	0.241652	+	0.970363i	$0.422311\pi$
$224$	0	0
$225$	−0.921780	−0.0614520
$226$	0	0
$227$	−19.6905	−1.30690	−0.653451	−	0.756969i	$-0.726680\pi$
$227$	−19.6905	−1.30690	−0.653451	+	0.756969i	$0.726680\pi$
$228$	0	0
$229$	−7.73068	−0.510858	−0.255429	−	0.966828i	$-0.582217\pi$
$229$	−7.73068	−0.510858	−0.255429	+	0.966828i	$0.582217\pi$
$230$	0	0
$231$	−10.7492	−0.707243
$232$	0	0
$233$	−13.4122	−0.878659	−0.439330	−	0.898326i	$-0.644784\pi$
$233$	−13.4122	−0.878659	−0.439330	+	0.898326i	$0.644784\pi$
$234$	0	0
$235$	−9.23999	−0.602750
$236$	0	0
$237$	8.24659	0.535674
$238$	0	0
$239$	−9.18032	−0.593826	−0.296913	−	0.954905i	$-0.595957\pi$
$239$	−9.18032	−0.593826	−0.296913	+	0.954905i	$0.595957\pi$
$240$	0	0
$241$	−4.16316	−0.268173	−0.134086	−	0.990970i	$-0.542810\pi$
$241$	−4.16316	−0.268173	−0.134086	+	0.990970i	$0.542810\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	1.18429	0.0756616
$246$	0	0
$247$	20.7423	1.31980
$248$	0	0
$249$	−3.55142	−0.225062
$250$	0	0
$251$	2.70130	0.170505	0.0852524	−	0.996359i	$-0.472830\pi$
$251$	2.70130	0.170505	0.0852524	+	0.996359i	$0.472830\pi$
$252$	0	0
$253$	−10.1060	−0.635358
$254$	0	0
$255$	−15.8750	−0.994133
$256$	0	0
$257$	−21.0351	−1.31213	−0.656067	−	0.754703i	$-0.727781\pi$
$257$	−21.0351	−1.31213	−0.656067	+	0.754703i	$0.727781\pi$
$258$	0	0
$259$	−3.49716	−0.217303
$260$	0	0
$261$	−0.629927	−0.0389915
$262$	0	0
$263$	8.02727	0.494983	0.247491	−	0.968890i	$-0.420394\pi$
$263$	8.02727	0.494983	0.247491	+	0.968890i	$0.420394\pi$
$264$	0	0
$265$	11.9218	0.732352
$266$	0	0
$267$	−6.50278	−0.397964
$268$	0	0
$269$	−11.5170	−0.702201	−0.351101	−	0.936338i	$-0.614193\pi$
$269$	−11.5170	−0.702201	−0.351101	+	0.936338i	$0.614193\pi$
$270$	0	0
$271$	1.05095	0.0638410	0.0319205	−	0.999490i	$-0.489838\pi$
$271$	1.05095	0.0638410	0.0319205	+	0.999490i	$0.489838\pi$
$272$	0	0
$273$	7.47418	0.452358
$274$	0	0
$275$	−3.91248	−0.235932
$276$	0	0
$277$	−26.6531	−1.60143	−0.800716	−	0.599044i	$-0.795547\pi$
$277$	−26.6531	−1.60143	−0.800716	+	0.599044i	$0.795547\pi$
$278$	0	0
$279$	4.91752	0.294404
$280$	0	0
$281$	−15.9460	−0.951257	−0.475629	−	0.879646i	$-0.657779\pi$
$281$	−15.9460	−0.951257	−0.475629	+	0.879646i	$0.657779\pi$
$282$	0	0
$283$	4.19913	0.249612	0.124806	−	0.992181i	$-0.460169\pi$
$283$	4.19913	0.249612	0.124806	+	0.992181i	$0.460169\pi$
$284$	0	0
$285$	14.1931	0.840727
$286$	0	0
$287$	−12.0121	−0.709054
$288$	0	0
$289$	44.7958	2.63505
$290$	0	0
$291$	−17.5190	−1.02698
$292$	0	0
$293$	−22.2635	−1.30065	−0.650323	−	0.759657i	$-0.725367\pi$
$293$	−22.2635	−1.30065	−0.650323	+	0.759657i	$0.725367\pi$
$294$	0	0
$295$	−14.3357	−0.834654
$296$	0	0
$297$	4.24449	0.246290
$298$	0	0
$299$	7.02696	0.406380
$300$	0	0
$301$	4.86405	0.280359
$302$	0	0
$303$	2.11356	0.121421
$304$	0	0
$305$	22.9093	1.31178
$306$	0	0
$307$	−12.5229	−0.714718	−0.357359	−	0.933967i	$-0.616323\pi$
$307$	−12.5229	−0.714718	−0.357359	+	0.933967i	$0.616323\pi$
$308$	0	0
$309$	−16.2304	−0.923314
$310$	0	0
$311$	−8.00220	−0.453763	−0.226882	−	0.973922i	$-0.572853\pi$
$311$	−8.00220	−0.453763	−0.226882	+	0.973922i	$0.572853\pi$
$312$	0	0
$313$	−15.7311	−0.889173	−0.444587	−	0.895736i	$-0.646649\pi$
$313$	−15.7311	−0.889173	−0.444587	+	0.895736i	$0.646649\pi$
$314$	0	0
$315$	5.11428	0.288157
$316$	0	0
$317$	17.7878	0.999063	0.499532	−	0.866296i	$-0.333505\pi$
$317$	17.7878	0.999063	0.499532	+	0.866296i	$0.333505\pi$
$318$	0	0
$319$	−2.67372	−0.149699
$320$	0	0
$321$	8.13701	0.454164
$322$	0	0
$323$	−55.2486	−3.07412
$324$	0	0
$325$	2.72045	0.150904
$326$	0	0
$327$	−9.54930	−0.528078
$328$	0	0
$329$	−11.5874	−0.638834
$330$	0	0
$331$	18.4270	1.01284	0.506419	−	0.862288i	$-0.330969\pi$
$331$	18.4270	1.01284	0.506419	+	0.862288i	$0.330969\pi$
$332$	0	0
$333$	1.38091	0.0756734
$334$	0	0
$335$	−10.5086	−0.574148
$336$	0	0
$337$	14.5829	0.794381	0.397191	−	0.917736i	$-0.369985\pi$
$337$	14.5829	0.794381	0.397191	+	0.917736i	$0.369985\pi$
$338$	0	0
$339$	−13.0671	−0.709710
$340$	0	0
$341$	20.8724	1.13030
$342$	0	0
$343$	19.2127	1.03739
$344$	0	0
$345$	4.80827	0.258869
$346$	0	0
$347$	−28.1874	−1.51318	−0.756589	−	0.653890i	$-0.773136\pi$
$347$	−28.1874	−1.51318	−0.756589	+	0.653890i	$0.773136\pi$
$348$	0	0
$349$	−20.1091	−1.07642	−0.538208	−	0.842812i	$-0.680899\pi$
$349$	−20.1091	−1.07642	−0.538208	+	0.842812i	$0.680899\pi$
$350$	0	0
$351$	−2.95131	−0.157529
$352$	0	0
$353$	−20.8412	−1.10927	−0.554634	−	0.832095i	$-0.687142\pi$
$353$	−20.8412	−1.10927	−0.554634	+	0.832095i	$0.687142\pi$
$354$	0	0
$355$	30.6676	1.62767
$356$	0	0
$357$	−19.9081	−1.05365
$358$	0	0
$359$	−20.7585	−1.09559	−0.547796	−	0.836612i	$-0.684533\pi$
$359$	−20.7585	−1.09559	−0.547796	+	0.836612i	$0.684533\pi$
$360$	0	0
$361$	30.3951	1.59974
$362$	0	0
$363$	7.01567	0.368227
$364$	0	0
$365$	−10.8655	−0.568725
$366$	0	0
$367$	11.2571	0.587616	0.293808	−	0.955864i	$-0.405077\pi$
$367$	11.2571	0.587616	0.293808	+	0.955864i	$0.405077\pi$
$368$	0	0
$369$	4.74319	0.246921
$370$	0	0
$371$	14.9505	0.776193
$372$	0	0
$373$	32.8855	1.70275	0.851373	−	0.524561i	$-0.175770\pi$
$373$	32.8855	1.70275	0.851373	+	0.524561i	$0.175770\pi$
$374$	0	0
$375$	11.9588	0.617550
$376$	0	0
$377$	1.85911	0.0957488
$378$	0	0
$379$	32.4684	1.66779	0.833895	−	0.551923i	$-0.186106\pi$
$379$	32.4684	1.66779	0.833895	+	0.551923i	$0.186106\pi$
$380$	0	0
$381$	−5.03653	−0.258029
$382$	0	0
$383$	10.0388	0.512958	0.256479	−	0.966550i	$-0.417438\pi$
$383$	10.0388	0.512958	0.256479	+	0.966550i	$0.417438\pi$
$384$	0	0
$385$	21.7075	1.10632
$386$	0	0
$387$	−1.92065	−0.0976323
$388$	0	0
$389$	−0.0747660	−0.00379079	−0.00189539	−	0.999998i	$-0.500603\pi$
$389$	−0.0747660	−0.00379079	−0.00189539	+	0.999998i	$0.500603\pi$
$390$	0	0
$391$	−18.7169	−0.946552
$392$	0	0
$393$	−5.79467	−0.292302
$394$	0	0
$395$	−16.6537	−0.837937
$396$	0	0
$397$	−8.22255	−0.412678	−0.206339	−	0.978481i	$-0.566155\pi$
$397$	−8.22255	−0.412678	−0.206339	+	0.978481i	$0.566155\pi$
$398$	0	0
$399$	17.7988	0.891057
$400$	0	0
$401$	30.3843	1.51732	0.758661	−	0.651486i	$-0.225854\pi$
$401$	30.3843	1.51732	0.758661	+	0.651486i	$0.225854\pi$
$402$	0	0
$403$	−14.5131	−0.722949
$404$	0	0
$405$	−2.01946	−0.100348
$406$	0	0
$407$	5.86126	0.290532
$408$	0	0
$409$	11.7817	0.582568	0.291284	−	0.956637i	$-0.405917\pi$
$409$	11.7817	0.582568	0.291284	+	0.956637i	$0.405917\pi$
$410$	0	0
$411$	6.64234	0.327643
$412$	0	0
$413$	−17.9776	−0.884620
$414$	0	0
$415$	7.17194	0.352057
$416$	0	0
$417$	11.6369	0.569860
$418$	0	0
$419$	11.3422	0.554102	0.277051	−	0.960855i	$-0.410643\pi$
$419$	11.3422	0.554102	0.277051	+	0.960855i	$0.410643\pi$
$420$	0	0
$421$	9.49711	0.462861	0.231430	−	0.972851i	$-0.425659\pi$
$421$	9.49711	0.462861	0.231430	+	0.972851i	$0.425659\pi$
$422$	0	0
$423$	4.57547	0.222467
$424$	0	0
$425$	−7.24614	−0.351489
$426$	0	0
$427$	28.7294	1.39031
$428$	0	0
$429$	−12.5268	−0.604799
$430$	0	0
$431$	4.48878	0.216217	0.108108	−	0.994139i	$-0.465521\pi$
$431$	4.48878	0.216217	0.108108	+	0.994139i	$0.465521\pi$
$432$	0	0
$433$	−24.5817	−1.18132	−0.590661	−	0.806920i	$-0.701133\pi$
$433$	−24.5817	−1.18132	−0.590661	+	0.806920i	$0.701133\pi$
$434$	0	0
$435$	1.27211	0.0609931
$436$	0	0
$437$	16.7338	0.800488
$438$	0	0
$439$	−30.0677	−1.43505	−0.717526	−	0.696532i	$-0.754725\pi$
$439$	−30.0677	−1.43505	−0.717526	+	0.696532i	$0.754725\pi$
$440$	0	0
$441$	−0.586439	−0.0279257
$442$	0	0
$443$	−6.71285	−0.318937	−0.159468	−	0.987203i	$-0.550978\pi$
$443$	−6.71285	−0.318937	−0.159468	+	0.987203i	$0.550978\pi$
$444$	0	0
$445$	13.1321	0.622522
$446$	0	0
$447$	1.73821	0.0822146
$448$	0	0
$449$	−23.0759	−1.08902	−0.544509	−	0.838755i	$-0.683284\pi$
$449$	−23.0759	−1.08902	−0.544509	+	0.838755i	$0.683284\pi$
$450$	0	0
$451$	20.1324	0.947998
$452$	0	0
$453$	−15.4779	−0.727214
$454$	0	0
$455$	−15.0938	−0.707609
$456$	0	0
$457$	−37.4921	−1.75381	−0.876904	−	0.480665i	$-0.840395\pi$
$457$	−37.4921	−1.75381	−0.876904	+	0.480665i	$0.840395\pi$
$458$	0	0
$459$	7.86103	0.366921
$460$	0	0
$461$	29.0705	1.35395	0.676973	−	0.736008i	$-0.263291\pi$
$461$	29.0705	1.35395	0.676973	+	0.736008i	$0.263291\pi$
$462$	0	0
$463$	−8.40645	−0.390681	−0.195340	−	0.980736i	$-0.562581\pi$
$463$	−8.40645	−0.390681	−0.195340	+	0.980736i	$0.562581\pi$
$464$	0	0
$465$	−9.93074	−0.460527
$466$	0	0
$467$	13.4203	0.621018	0.310509	−	0.950570i	$-0.399501\pi$
$467$	13.4203	0.621018	0.310509	+	0.950570i	$0.399501\pi$
$468$	0	0
$469$	−13.1783	−0.608519
$470$	0	0
$471$	12.9643	0.597366
$472$	0	0
$473$	−8.15219	−0.374838
$474$	0	0
$475$	6.47843	0.297251
$476$	0	0
$477$	−5.90347	−0.270301
$478$	0	0
$479$	−5.25266	−0.240000	−0.120000	−	0.992774i	$-0.538289\pi$
$479$	−5.25266	−0.240000	−0.120000	+	0.992774i	$0.538289\pi$
$480$	0	0
$481$	−4.07549	−0.185826
$482$	0	0
$483$	6.02980	0.274366
$484$	0	0
$485$	35.3790	1.60648
$486$	0	0
$487$	20.3929	0.924092	0.462046	−	0.886856i	$-0.347116\pi$
$487$	20.3929	0.924092	0.462046	+	0.886856i	$0.347116\pi$
$488$	0	0
$489$	2.07407	0.0937926
$490$	0	0
$491$	9.72205	0.438750	0.219375	−	0.975641i	$-0.429598\pi$
$491$	9.72205	0.438750	0.219375	+	0.975641i	$0.429598\pi$
$492$	0	0
$493$	−4.95187	−0.223021
$494$	0	0
$495$	−8.57157	−0.385264
$496$	0	0
$497$	38.4586	1.72511
$498$	0	0
$499$	15.7348	0.704387	0.352194	−	0.935927i	$-0.385436\pi$
$499$	15.7348	0.704387	0.352194	+	0.935927i	$0.385436\pi$
$500$	0	0
$501$	−17.0029	−0.759632
$502$	0	0
$503$	−1.00000	−0.0445878
$503$	−1.00000	−0.0445878
$504$	0	0
$505$	−4.26825	−0.189935
$506$	0	0
$507$	−4.28980	−0.190517
$508$	0	0
$509$	−12.4472	−0.551714	−0.275857	−	0.961199i	$-0.588962\pi$
$509$	−12.4472	−0.551714	−0.275857	+	0.961199i	$0.588962\pi$
$510$	0	0
$511$	−13.6258	−0.602771
$512$	0	0
$513$	−7.02817	−0.310301
$514$	0	0
$515$	32.7766	1.44431
$516$	0	0
$517$	19.4205	0.854114
$518$	0	0
$519$	4.09680	0.179830
$520$	0	0
$521$	27.3762	1.19937	0.599687	−	0.800235i	$-0.295292\pi$
$521$	27.3762	1.19937	0.599687	+	0.800235i	$0.295292\pi$
$522$	0	0
$523$	10.0862	0.441037	0.220519	−	0.975383i	$-0.429225\pi$
$523$	10.0862	0.441037	0.220519	+	0.975383i	$0.429225\pi$
$524$	0	0
$525$	2.33441	0.101882
$526$	0	0
$527$	38.6568	1.68392
$528$	0	0
$529$	−17.3310	−0.753521
$530$	0	0
$531$	7.09876	0.308060
$532$	0	0
$533$	−13.9986	−0.606347
$534$	0	0
$535$	−16.4324	−0.710433
$536$	0	0
$537$	−16.1857	−0.698464
$538$	0	0
$539$	−2.48913	−0.107215
$540$	0	0
$541$	−16.5271	−0.710555	−0.355277	−	0.934761i	$-0.615614\pi$
$541$	−16.5271	−0.710555	−0.355277	+	0.934761i	$0.615614\pi$
$542$	0	0
$543$	8.02017	0.344178
$544$	0	0
$545$	19.2844	0.826054
$546$	0	0
$547$	4.75516	0.203316	0.101658	−	0.994819i	$-0.467585\pi$
$547$	4.75516	0.203316	0.101658	+	0.994819i	$0.467585\pi$
$548$	0	0
$549$	−11.3443	−0.484162
$550$	0	0
$551$	4.42723	0.188606
$552$	0	0
$553$	−20.8845	−0.888099
$554$	0	0
$555$	−2.78869	−0.118373
$556$	0	0
$557$	−38.7255	−1.64085	−0.820426	−	0.571753i	$-0.806264\pi$
$557$	−38.7255	−1.64085	−0.820426	+	0.571753i	$0.806264\pi$
$558$	0	0
$559$	5.66843	0.239749
$560$	0	0
$561$	33.3660	1.40872
$562$	0	0
$563$	−13.1048	−0.552301	−0.276151	−	0.961114i	$-0.589059\pi$
$563$	−13.1048	−0.552301	−0.276151	+	0.961114i	$0.589059\pi$
$564$	0	0
$565$	26.3886	1.11018
$566$	0	0
$567$	−2.53250	−0.106355
$568$	0	0
$569$	−5.96578	−0.250098	−0.125049	−	0.992151i	$-0.539909\pi$
$569$	−5.96578	−0.250098	−0.125049	+	0.992151i	$0.539909\pi$
$570$	0	0
$571$	9.60891	0.402120	0.201060	−	0.979579i	$-0.435561\pi$
$571$	9.60891	0.402120	0.201060	+	0.979579i	$0.435561\pi$
$572$	0	0
$573$	−0.505177	−0.0211041
$574$	0	0
$575$	2.19473	0.0915265
$576$	0	0
$577$	−2.67626	−0.111414	−0.0557071	−	0.998447i	$-0.517741\pi$
$577$	−2.67626	−0.111414	−0.0557071	+	0.998447i	$0.517741\pi$
$578$	0	0
$579$	−5.10206	−0.212034
$580$	0	0
$581$	8.99396	0.373132
$582$	0	0
$583$	−25.0572	−1.03776
$584$	0	0
$585$	5.96004	0.246418
$586$	0	0
$587$	17.3459	0.715942	0.357971	−	0.933733i	$-0.383469\pi$
$587$	17.3459	0.715942	0.357971	+	0.933733i	$0.383469\pi$
$588$	0	0
$589$	−34.5612	−1.42407
$590$	0	0
$591$	11.6387	0.478753
$592$	0	0
$593$	43.9187	1.80352	0.901762	−	0.432234i	$-0.142275\pi$
$593$	43.9187	1.80352	0.901762	+	0.432234i	$0.142275\pi$
$594$	0	0
$595$	40.2035	1.64818
$596$	0	0
$597$	−25.0047	−1.02338
$598$	0	0
$599$	19.2550	0.786737	0.393368	−	0.919381i	$-0.371310\pi$
$599$	19.2550	0.786737	0.393368	+	0.919381i	$0.371310\pi$
$600$	0	0
$601$	−22.9669	−0.936838	−0.468419	−	0.883506i	$-0.655176\pi$
$601$	−22.9669	−0.936838	−0.468419	+	0.883506i	$0.655176\pi$
$602$	0	0
$603$	5.20369	0.211910
$604$	0	0
$605$	−14.1679	−0.576006
$606$	0	0
$607$	7.69084	0.312161	0.156081	−	0.987744i	$-0.450114\pi$
$607$	7.69084	0.312161	0.156081	+	0.987744i	$0.450114\pi$
$608$	0	0
$609$	1.59529	0.0646444
$610$	0	0
$611$	−13.5036	−0.546298
$612$	0	0
$613$	1.95217	0.0788473	0.0394237	−	0.999223i	$-0.487448\pi$
$613$	1.95217	0.0788473	0.0394237	+	0.999223i	$0.487448\pi$
$614$	0	0
$615$	−9.57868	−0.386250
$616$	0	0
$617$	21.1141	0.850023	0.425011	−	0.905188i	$-0.360270\pi$
$617$	21.1141	0.850023	0.425011	+	0.905188i	$0.360270\pi$
$618$	0	0
$619$	−26.4155	−1.06173	−0.530865	−	0.847456i	$-0.678133\pi$
$619$	−26.4155	−1.06173	−0.530865	+	0.847456i	$0.678133\pi$
$620$	0	0
$621$	−2.38097	−0.0955450
$622$	0	0
$623$	16.4683	0.659788
$624$	0	0
$625$	−19.5414	−0.781657
$626$	0	0
$627$	−29.8310	−1.19133
$628$	0	0
$629$	10.8554	0.432832
$630$	0	0
$631$	−27.0893	−1.07841	−0.539204	−	0.842175i	$-0.681275\pi$
$631$	−27.0893	−1.07841	−0.539204	+	0.842175i	$0.681275\pi$
$632$	0	0
$633$	−0.649648	−0.0258212
$634$	0	0
$635$	10.1711	0.403626
$636$	0	0
$637$	1.73076	0.0685753
$638$	0	0
$639$	−15.1860	−0.600750
$640$	0	0
$641$	−44.8094	−1.76987	−0.884933	−	0.465719i	$-0.845796\pi$
$641$	−44.8094	−1.76987	−0.884933	+	0.465719i	$0.845796\pi$
$642$	0	0
$643$	40.7483	1.60695	0.803477	−	0.595335i	$-0.202981\pi$
$643$	40.7483	1.60695	0.803477	+	0.595335i	$0.202981\pi$
$644$	0	0
$645$	3.87868	0.152723
$646$	0	0
$647$	39.4355	1.55037	0.775184	−	0.631736i	$-0.217657\pi$
$647$	39.4355	1.55037	0.775184	+	0.631736i	$0.217657\pi$
$648$	0	0
$649$	30.1306	1.18273
$650$	0	0
$651$	−12.4536	−0.488096
$652$	0	0
$653$	−17.2143	−0.673647	−0.336824	−	0.941568i	$-0.609353\pi$
$653$	−17.2143	−0.673647	−0.336824	+	0.941568i	$0.609353\pi$
$654$	0	0
$655$	11.7021	0.457239
$656$	0	0
$657$	5.38038	0.209909
$658$	0	0
$659$	13.6754	0.532717	0.266359	−	0.963874i	$-0.414179\pi$
$659$	13.6754	0.532717	0.266359	+	0.963874i	$0.414179\pi$
$660$	0	0
$661$	0.396413	0.0154187	0.00770934	−	0.999970i	$-0.497546\pi$
$661$	0.396413	0.0154187	0.00770934	+	0.999970i	$0.497546\pi$
$662$	0	0
$663$	−23.2003	−0.901025
$664$	0	0
$665$	−35.9441	−1.39385
$666$	0	0
$667$	1.49984	0.0580738
$668$	0	0
$669$	7.21727	0.279036
$670$	0	0
$671$	−48.1507	−1.85884
$672$	0	0
$673$	14.7027	0.566747	0.283373	−	0.959010i	$-0.408546\pi$
$673$	14.7027	0.566747	0.283373	+	0.959010i	$0.408546\pi$
$674$	0	0
$675$	−0.921780	−0.0354793
$676$	0	0
$677$	42.9339	1.65009	0.825043	−	0.565071i	$-0.191151\pi$
$677$	42.9339	1.65009	0.825043	+	0.565071i	$0.191151\pi$
$678$	0	0
$679$	44.3670	1.70265
$680$	0	0
$681$	−19.6905	−0.754541
$682$	0	0
$683$	15.9889	0.611797	0.305899	−	0.952064i	$-0.401043\pi$
$683$	15.9889	0.611797	0.305899	+	0.952064i	$0.401043\pi$
$684$	0	0
$685$	−13.4139	−0.512521
$686$	0	0
$687$	−7.73068	−0.294944
$688$	0	0
$689$	17.4229	0.663761
$690$	0	0
$691$	10.3477	0.393645	0.196823	−	0.980439i	$-0.436938\pi$
$691$	10.3477	0.393645	0.196823	+	0.980439i	$0.436938\pi$
$692$	0	0
$693$	−10.7492	−0.408327
$694$	0	0
$695$	−23.5002	−0.891413
$696$	0	0
$697$	37.2863	1.41232
$698$	0	0
$699$	−13.4122	−0.507294
$700$	0	0
$701$	0.407304	0.0153837	0.00769183	−	0.999970i	$-0.497552\pi$
$701$	0.407304	0.0153837	0.00769183	+	0.999970i	$0.497552\pi$
$702$	0	0
$703$	−9.70527	−0.366041
$704$	0	0
$705$	−9.23999	−0.347998
$706$	0	0
$707$	−5.35259	−0.201305
$708$	0	0
$709$	−15.7842	−0.592789	−0.296394	−	0.955066i	$-0.595784\pi$
$709$	−15.7842	−0.592789	−0.296394	+	0.955066i	$0.595784\pi$
$710$	0	0
$711$	8.24659	0.309271
$712$	0	0
$713$	−11.7085	−0.438485
$714$	0	0
$715$	25.2973	0.946067
$716$	0	0
$717$	−9.18032	−0.342845
$718$	0	0
$719$	−32.6429	−1.21737	−0.608687	−	0.793410i	$-0.708303\pi$
$719$	−32.6429	−1.21737	−0.608687	+	0.793410i	$0.708303\pi$
$720$	0	0
$721$	41.1035	1.53077
$722$	0	0
$723$	−4.16316	−0.154830
$724$	0	0
$725$	0.580654	0.0215649
$726$	0	0
$727$	17.9242	0.664770	0.332385	−	0.943144i	$-0.392147\pi$
$727$	17.9242	0.664770	0.332385	+	0.943144i	$0.392147\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−15.0983	−0.558431
$732$	0	0
$733$	5.28982	0.195384	0.0976920	−	0.995217i	$-0.468854\pi$
$733$	5.28982	0.195384	0.0976920	+	0.995217i	$0.468854\pi$
$734$	0	0
$735$	1.18429	0.0436832
$736$	0	0
$737$	22.0870	0.813584
$738$	0	0
$739$	43.9774	1.61773	0.808867	−	0.587992i	$-0.200081\pi$
$739$	43.9774	1.61773	0.808867	+	0.587992i	$0.200081\pi$
$740$	0	0
$741$	20.7423	0.761986
$742$	0	0
$743$	35.0566	1.28610	0.643050	−	0.765824i	$-0.277669\pi$
$743$	35.0566	1.28610	0.643050	+	0.765824i	$0.277669\pi$
$744$	0	0
$745$	−3.51025	−0.128606
$746$	0	0
$747$	−3.55142	−0.129940
$748$	0	0
$749$	−20.6070	−0.752963
$750$	0	0
$751$	39.6506	1.44687	0.723436	−	0.690392i	$-0.242562\pi$
$751$	39.6506	1.44687	0.723436	+	0.690392i	$0.242562\pi$
$752$	0	0
$753$	2.70130	0.0984410
$754$	0	0
$755$	31.2570	1.13756
$756$	0	0
$757$	−15.5005	−0.563374	−0.281687	−	0.959506i	$-0.590894\pi$
$757$	−15.5005	−0.563374	−0.281687	+	0.959506i	$0.590894\pi$
$758$	0	0
$759$	−10.1060	−0.366824
$760$	0	0
$761$	−3.67337	−0.133159	−0.0665797	−	0.997781i	$-0.521209\pi$
$761$	−3.67337	−0.133159	−0.0665797	+	0.997781i	$0.521209\pi$
$762$	0	0
$763$	24.1836	0.875506
$764$	0	0
$765$	−15.8750	−0.573963
$766$	0	0
$767$	−20.9506	−0.756482
$768$	0	0
$769$	28.4920	1.02745	0.513723	−	0.857956i	$-0.328266\pi$
$769$	28.4920	1.02745	0.513723	+	0.857956i	$0.328266\pi$
$770$	0	0
$771$	−21.0351	−0.757561
$772$	0	0
$773$	15.6900	0.564331	0.282165	−	0.959366i	$-0.408947\pi$
$773$	15.6900	0.564331	0.282165	+	0.959366i	$0.408947\pi$
$774$	0	0
$775$	−4.53287	−0.162826
$776$	0	0
$777$	−3.49716	−0.125460
$778$	0	0
$779$	−33.3359	−1.19438
$780$	0	0
$781$	−64.4569	−2.30645
$782$	0	0
$783$	−0.629927	−0.0225117
$784$	0	0
$785$	−26.1810	−0.934439
$786$	0	0
$787$	21.7844	0.776531	0.388265	−	0.921548i	$-0.373074\pi$
$787$	21.7844	0.776531	0.388265	+	0.921548i	$0.373074\pi$
$788$	0	0
$789$	8.02727	0.285779
$790$	0	0
$791$	33.0926	1.17664
$792$	0	0
$793$	33.4804	1.18893
$794$	0	0
$795$	11.9218	0.422823
$796$	0	0
$797$	48.2495	1.70908	0.854542	−	0.519382i	$-0.173838\pi$
$797$	48.2495	1.70908	0.854542	+	0.519382i	$0.173838\pi$
$798$	0	0
$799$	35.9679	1.27245
$800$	0	0
$801$	−6.50278	−0.229764
$802$	0	0
$803$	22.8370	0.805899
$804$	0	0
$805$	−12.1769	−0.429181
$806$	0	0
$807$	−11.5170	−0.405416
$808$	0	0
$809$	38.6407	1.35854	0.679268	−	0.733891i	$-0.262298\pi$
$809$	38.6407	1.35854	0.679268	+	0.733891i	$0.262298\pi$
$810$	0	0
$811$	−36.2377	−1.27248	−0.636239	−	0.771492i	$-0.719511\pi$
$811$	−36.2377	−1.27248	−0.636239	+	0.771492i	$0.719511\pi$
$812$	0	0
$813$	1.05095	0.0368586
$814$	0	0
$815$	−4.18850	−0.146717
$816$	0	0
$817$	13.4987	0.472259
$818$	0	0
$819$	7.47418	0.261169
$820$	0	0
$821$	−29.8347	−1.04124	−0.520618	−	0.853789i	$-0.674299\pi$
$821$	−29.8347	−1.04124	−0.520618	+	0.853789i	$0.674299\pi$
$822$	0	0
$823$	−12.5527	−0.437560	−0.218780	−	0.975774i	$-0.570208\pi$
$823$	−12.5527	−0.437560	−0.218780	+	0.975774i	$0.570208\pi$
$824$	0	0
$825$	−3.91248	−0.136215
$826$	0	0
$827$	16.4591	0.572340	0.286170	−	0.958179i	$-0.407618\pi$
$827$	16.4591	0.572340	0.286170	+	0.958179i	$0.407618\pi$
$828$	0	0
$829$	−47.9337	−1.66481	−0.832404	−	0.554169i	$-0.813036\pi$
$829$	−47.9337	−1.66481	−0.832404	+	0.554169i	$0.813036\pi$
$830$	0	0
$831$	−26.6531	−0.924587
$832$	0	0
$833$	−4.61002	−0.159728
$834$	0	0
$835$	34.3366	1.18827
$836$	0	0
$837$	4.91752	0.169974
$838$	0	0
$839$	−25.9577	−0.896159	−0.448079	−	0.893994i	$-0.647892\pi$
$839$	−25.9577	−0.896159	−0.448079	+	0.893994i	$0.647892\pi$
$840$	0	0
$841$	−28.6032	−0.986317
$842$	0	0
$843$	−15.9460	−0.549209
$844$	0	0
$845$	8.66307	0.298019
$846$	0	0
$847$	−17.7672	−0.610488
$848$	0	0
$849$	4.19913	0.144114
$850$	0	0
$851$	−3.28790	−0.112708
$852$	0	0
$853$	−39.3407	−1.34700	−0.673500	−	0.739187i	$-0.735210\pi$
$853$	−39.3407	−1.34700	−0.673500	+	0.739187i	$0.735210\pi$
$854$	0	0
$855$	14.1931	0.485394
$856$	0	0
$857$	−24.9662	−0.852830	−0.426415	−	0.904528i	$-0.640224\pi$
$857$	−24.9662	−0.852830	−0.426415	+	0.904528i	$0.640224\pi$
$858$	0	0
$859$	10.6182	0.362287	0.181144	−	0.983457i	$-0.442020\pi$
$859$	10.6182	0.362287	0.181144	+	0.983457i	$0.442020\pi$
$860$	0	0
$861$	−12.0121	−0.409372
$862$	0	0
$863$	−19.3732	−0.659470	−0.329735	−	0.944073i	$-0.606959\pi$
$863$	−19.3732	−0.659470	−0.329735	+	0.944073i	$0.606959\pi$
$864$	0	0
$865$	−8.27333	−0.281302
$866$	0	0
$867$	44.7958	1.52134
$868$	0	0
$869$	35.0025	1.18738
$870$	0	0
$871$	−15.3577	−0.520375
$872$	0	0
$873$	−17.5190	−0.592930
$874$	0	0
$875$	−30.2857	−1.02384
$876$	0	0
$877$	−11.9970	−0.405110	−0.202555	−	0.979271i	$-0.564924\pi$
$877$	−11.9970	−0.405110	−0.202555	+	0.979271i	$0.564924\pi$
$878$	0	0
$879$	−22.2635	−0.750929
$880$	0	0
$881$	8.86347	0.298618	0.149309	−	0.988791i	$-0.452295\pi$
$881$	8.86347	0.298618	0.149309	+	0.988791i	$0.452295\pi$
$882$	0	0
$883$	40.5950	1.36613	0.683065	−	0.730358i	$-0.260647\pi$
$883$	40.5950	1.36613	0.683065	+	0.730358i	$0.260647\pi$
$884$	0	0
$885$	−14.3357	−0.481888
$886$	0	0
$887$	1.46229	0.0490988	0.0245494	−	0.999699i	$-0.492185\pi$
$887$	1.46229	0.0490988	0.0245494	+	0.999699i	$0.492185\pi$
$888$	0	0
$889$	12.7550	0.427789
$890$	0	0
$891$	4.24449	0.142196
$892$	0	0
$893$	−32.1572	−1.07610
$894$	0	0
$895$	32.6864	1.09258
$896$	0	0
$897$	7.02696	0.234624
$898$	0	0
$899$	−3.09768	−0.103313
$900$	0	0
$901$	−46.4073	−1.54605
$902$	0	0
$903$	4.86405	0.161866
$904$	0	0
$905$	−16.1964	−0.538387
$906$	0	0
$907$	−23.9893	−0.796551	−0.398276	−	0.917266i	$-0.630391\pi$
$907$	−23.9893	−0.796551	−0.398276	+	0.917266i	$0.630391\pi$
$908$	0	0
$909$	2.11356	0.0701023
$910$	0	0
$911$	9.33223	0.309191	0.154595	−	0.987978i	$-0.450593\pi$
$911$	9.33223	0.309191	0.154595	+	0.987978i	$0.450593\pi$
$912$	0	0
$913$	−15.0739	−0.498875
$914$	0	0
$915$	22.9093	0.757359
$916$	0	0
$917$	14.6750	0.484611
$918$	0	0
$919$	−16.3333	−0.538787	−0.269394	−	0.963030i	$-0.586823\pi$
$919$	−16.3333	−0.538787	−0.269394	+	0.963030i	$0.586823\pi$
$920$	0	0
$921$	−12.5229	−0.412642
$922$	0	0
$923$	44.8186	1.47522
$924$	0	0
$925$	−1.27290	−0.0418526
$926$	0	0
$927$	−16.2304	−0.533076
$928$	0	0
$929$	48.1109	1.57847	0.789233	−	0.614093i	$-0.210478\pi$
$929$	48.1109	1.57847	0.789233	+	0.614093i	$0.210478\pi$
$930$	0	0
$931$	4.12159	0.135080
$932$	0	0
$933$	−8.00220	−0.261980
$934$	0	0
$935$	−67.3814	−2.20361
$936$	0	0
$937$	10.7224	0.350285	0.175142	−	0.984543i	$-0.443961\pi$
$937$	10.7224	0.350285	0.175142	+	0.984543i	$0.443961\pi$
$938$	0	0
$939$	−15.7311	−0.513364
$940$	0	0
$941$	22.6505	0.738386	0.369193	−	0.929353i	$-0.379634\pi$
$941$	22.6505	0.738386	0.369193	+	0.929353i	$0.379634\pi$
$942$	0	0
$943$	−11.2934	−0.367763
$944$	0	0
$945$	5.11428	0.166368
$946$	0	0
$947$	−55.7265	−1.81087	−0.905434	−	0.424486i	$-0.860455\pi$
$947$	−55.7265	−1.81087	−0.905434	+	0.424486i	$0.860455\pi$
$948$	0	0
$949$	−15.8792	−0.515459
$950$	0	0
$951$	17.7878	0.576809
$952$	0	0
$953$	4.80440	0.155630	0.0778150	−	0.996968i	$-0.475206\pi$
$953$	4.80440	0.155630	0.0778150	+	0.996968i	$0.475206\pi$
$954$	0	0
$955$	1.02019	0.0330124
$956$	0	0
$957$	−2.67372	−0.0864289
$958$	0	0
$959$	−16.8217	−0.543202
$960$	0	0
$961$	−6.81800	−0.219935
$962$	0	0
$963$	8.13701	0.262211
$964$	0	0
$965$	10.3034	0.331678
$966$	0	0
$967$	8.55364	0.275067	0.137533	−	0.990497i	$-0.456083\pi$
$967$	8.55364	0.275067	0.137533	+	0.990497i	$0.456083\pi$
$968$	0	0
$969$	−55.2486	−1.77484
$970$	0	0
$971$	−24.0553	−0.771973	−0.385986	−	0.922504i	$-0.626139\pi$
$971$	−24.0553	−0.771973	−0.385986	+	0.922504i	$0.626139\pi$
$972$	0	0
$973$	−29.4704	−0.944777
$974$	0	0
$975$	2.72045	0.0871243
$976$	0	0
$977$	−18.4717	−0.590961	−0.295480	−	0.955349i	$-0.595480\pi$
$977$	−18.4717	−0.590961	−0.295480	+	0.955349i	$0.595480\pi$
$978$	0	0
$979$	−27.6010	−0.882131
$980$	0	0
$981$	−9.54930	−0.304886
$982$	0	0
$983$	9.68346	0.308855	0.154427	−	0.988004i	$-0.450647\pi$
$983$	9.68346	0.308855	0.154427	+	0.988004i	$0.450647\pi$
$984$	0	0
$985$	−23.5039	−0.748897
$986$	0	0
$987$	−11.5874	−0.368831
$988$	0	0
$989$	4.57301	0.145413
$990$	0	0
$991$	21.0292	0.668015	0.334007	−	0.942570i	$-0.391599\pi$
$991$	21.0292	0.668015	0.334007	+	0.942570i	$0.391599\pi$
$992$	0	0
$993$	18.4270	0.584762
$994$	0	0
$995$	50.4961	1.60083
$996$	0	0
$997$	16.5281	0.523452	0.261726	−	0.965142i	$-0.415708\pi$
$997$	16.5281	0.523452	0.261726	+	0.965142i	$0.415708\pi$
$998$	0	0
$999$	1.38091	0.0436901

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6036.2.a.g.1.5	✓	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6036.2.a.g.1.5	✓	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -2.01946 q^{5} -2.53250 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -2.01946 q^{5} -2.53250 q^{7} +1.00000 q^{9} +4.24449 q^{11} -2.95131 q^{13} -2.01946 q^{15} +7.86103 q^{17} -7.02817 q^{19} -2.53250 q^{21} -2.38097 q^{23} -0.921780 q^{25} +1.00000 q^{27} -0.629927 q^{29} +4.91752 q^{31} +4.24449 q^{33} +5.11428 q^{35} +1.38091 q^{37} -2.95131 q^{39} +4.74319 q^{41} -1.92065 q^{43} -2.01946 q^{45} +4.57547 q^{47} -0.586439 q^{49} +7.86103 q^{51} -5.90347 q^{53} -8.57157 q^{55} -7.02817 q^{57} +7.09876 q^{59} -11.3443 q^{61} -2.53250 q^{63} +5.96004 q^{65} +5.20369 q^{67} -2.38097 q^{69} -15.1860 q^{71} +5.38038 q^{73} -0.921780 q^{75} -10.7492 q^{77} +8.24659 q^{79} +1.00000 q^{81} -3.55142 q^{83} -15.8750 q^{85} -0.629927 q^{87} -6.50278 q^{89} +7.47418 q^{91} +4.91752 q^{93} +14.1931 q^{95} -17.5190 q^{97} +4.24449 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q + 15 q^{3} - 11 q^{5} - 4 q^{7} + 15 q^{9}+O(q^{10}) \) 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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