LMFDB - Embedded newform 6724.2.a.k.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6724,2,Mod(1,6724)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6724, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("6724.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6724 = 2^{2} \cdot 41^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6724.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:	$53.6914103191$
Analytic rank:	$1$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - 5 x^{17} - 19 x^{16} + 125 x^{15} + 97 x^{14} - 1213 x^{13} + 139 x^{12} + 6021 x^{11} + \cdots - 419 $ x^18 - 5x^17 - 19x^16 + 125x^15 + 97x^14 - 1213x^13 + 139x^12 + 6021x^11 - 2731x^10 - 16810x^9 + 8920x^8 + 27346x^7 - 11767x^6 - 25826x^5 + 5697x^4 + 12793x^3 + 90x^2 - 2396*x - 419
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.56056$ of defining polynomial
Character	$\chi$	$=$	6724.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.56056 q^{3} +1.07147 q^{5} -1.61718 q^{7} +3.55649 q^{9} +O(q^{10})$	$q-2.56056 q^{3} +1.07147 q^{5} -1.61718 q^{7} +3.55649 q^{9} -0.256374 q^{11} -4.03130 q^{13} -2.74358 q^{15} +4.59606 q^{17} +3.87182 q^{19} +4.14089 q^{21} -2.28409 q^{23} -3.85195 q^{25} -1.42494 q^{27} -9.89677 q^{29} +7.10848 q^{31} +0.656463 q^{33} -1.73276 q^{35} +8.97686 q^{37} +10.3224 q^{39} -3.51961 q^{43} +3.81069 q^{45} +1.49652 q^{47} -4.38474 q^{49} -11.7685 q^{51} -12.7005 q^{53} -0.274698 q^{55} -9.91405 q^{57} +14.7582 q^{59} -0.543990 q^{61} -5.75148 q^{63} -4.31943 q^{65} +4.41597 q^{67} +5.84855 q^{69} +8.18937 q^{71} +15.0112 q^{73} +9.86316 q^{75} +0.414603 q^{77} -4.68332 q^{79} -7.02084 q^{81} +0.168230 q^{83} +4.92456 q^{85} +25.3413 q^{87} -4.75445 q^{89} +6.51933 q^{91} -18.2017 q^{93} +4.14855 q^{95} -10.2932 q^{97} -0.911794 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q - 5 q^{3} + 2 q^{5} - 7 q^{7} + 9 q^{9}+O(q^{10}) $ 18 * q - 5 * q^3 + 2 * q^5 - 7 * q^7 + 9 * q^9	$ 18 q - 5 q^{3} + 2 q^{5} - 7 q^{7} + 9 q^{9} - 12 q^{11} - 12 q^{13} + 14 q^{15} - 21 q^{17} - 15 q^{19} + 5 q^{21} + 3 q^{23} + 8 q^{25} - 5 q^{27} - 38 q^{29} + 12 q^{31} + 25 q^{33} - 9 q^{35} + 12 q^{37} - 3 q^{39} + 21 q^{43} - 13 q^{45} + 29 q^{47} - 9 q^{49} + 9 q^{51} - 27 q^{53} - 17 q^{55} - 13 q^{57} + 25 q^{59} - 9 q^{61} - 68 q^{63} - 45 q^{65} + 5 q^{67} - 46 q^{69} - 12 q^{71} + 19 q^{73} - 24 q^{75} - 3 q^{77} + 27 q^{79} - 46 q^{81} - 46 q^{83} - 54 q^{85} - 11 q^{87} - 55 q^{89} - 31 q^{91} - 59 q^{93} - 27 q^{95} - 31 q^{97} - 15 q^{99}+O(q^{100}) $ 18 * q - 5 * q^3 + 2 * q^5 - 7 * q^7 + 9 * q^9 - 12 * q^11 - 12 * q^13 + 14 * q^15 - 21 * q^17 - 15 * q^19 + 5 * q^21 + 3 * q^23 + 8 * q^25 - 5 * q^27 - 38 * q^29 + 12 * q^31 + 25 * q^33 - 9 * q^35 + 12 * q^37 - 3 * q^39 + 21 * q^43 - 13 * q^45 + 29 * q^47 - 9 * q^49 + 9 * q^51 - 27 * q^53 - 17 * q^55 - 13 * q^57 + 25 * q^59 - 9 * q^61 - 68 * q^63 - 45 * q^65 + 5 * q^67 - 46 * q^69 - 12 * q^71 + 19 * q^73 - 24 * q^75 - 3 * q^77 + 27 * q^79 - 46 * q^81 - 46 * q^83 - 54 * q^85 - 11 * q^87 - 55 * q^89 - 31 * q^91 - 59 * q^93 - 27 * q^95 - 31 * q^97 - 15 * 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
$3$	−2.56056	−1.47834	−0.739171	−	0.673517i	$-0.764783\pi$
$3$	−2.56056	−1.47834	−0.739171	+	0.673517i	$0.764783\pi$
$4$	0	0
$5$	1.07147	0.479177	0.239589	−	0.970875i	$-0.422987\pi$
$5$	1.07147	0.479177	0.239589	+	0.970875i	$0.422987\pi$
$6$	0	0
$7$	−1.61718	−0.611235	−0.305618	−	0.952154i	$-0.598863\pi$
$7$	−1.61718	−0.611235	−0.305618	+	0.952154i	$0.598863\pi$
$8$	0	0
$9$	3.55649	1.18550
$10$	0	0
$11$	−0.256374	−0.0772998	−0.0386499	−	0.999253i	$-0.512306\pi$
$11$	−0.256374	−0.0772998	−0.0386499	+	0.999253i	$0.512306\pi$
$12$	0	0
$13$	−4.03130	−1.11808	−0.559041	−	0.829140i	$-0.688831\pi$
$13$	−4.03130	−1.11808	−0.559041	+	0.829140i	$0.688831\pi$
$14$	0	0
$15$	−2.74358	−0.708388
$16$	0	0
$17$	4.59606	1.11471	0.557355	−	0.830275i	$-0.311816\pi$
$17$	4.59606	1.11471	0.557355	+	0.830275i	$0.311816\pi$
$18$	0	0
$19$	3.87182	0.888257	0.444128	−	0.895963i	$-0.353513\pi$
$19$	3.87182	0.888257	0.444128	+	0.895963i	$0.353513\pi$
$20$	0	0
$21$	4.14089	0.903615
$22$	0	0
$23$	−2.28409	−0.476265	−0.238133	−	0.971233i	$-0.576535\pi$
$23$	−2.28409	−0.476265	−0.238133	+	0.971233i	$0.576535\pi$
$24$	0	0
$25$	−3.85195	−0.770389
$26$	0	0
$27$	−1.42494	−0.274229
$28$	0	0
$29$	−9.89677	−1.83778	−0.918892	−	0.394509i	$-0.870915\pi$
$29$	−9.89677	−1.83778	−0.918892	+	0.394509i	$0.870915\pi$
$30$	0	0
$31$	7.10848	1.27672	0.638360	−	0.769738i	$-0.279613\pi$
$31$	7.10848	1.27672	0.638360	+	0.769738i	$0.279613\pi$
$32$	0	0
$33$	0.656463	0.114276
$34$	0	0
$35$	−1.73276	−0.292890
$36$	0	0
$37$	8.97686	1.47579	0.737893	−	0.674918i	$-0.235821\pi$
$37$	8.97686	1.47579	0.737893	+	0.674918i	$0.235821\pi$
$38$	0	0
$39$	10.3224	1.65291
$40$	0	0
$41$	0	0
$41$	0	0
$42$	0	0
$43$	−3.51961	−0.536736	−0.268368	−	0.963316i	$-0.586484\pi$
$43$	−3.51961	−0.536736	−0.268368	+	0.963316i	$0.586484\pi$
$44$	0	0
$45$	3.81069	0.568063
$46$	0	0
$47$	1.49652	0.218290	0.109145	−	0.994026i	$-0.465189\pi$
$47$	1.49652	0.218290	0.109145	+	0.994026i	$0.465189\pi$
$48$	0	0
$49$	−4.38474	−0.626391
$50$	0	0
$51$	−11.7685	−1.64792
$52$	0	0
$53$	−12.7005	−1.74455	−0.872275	−	0.489016i	$-0.837356\pi$
$53$	−12.7005	−1.74455	−0.872275	+	0.489016i	$0.837356\pi$
$54$	0	0
$55$	−0.274698	−0.0370403
$56$	0	0
$57$	−9.91405	−1.31315
$58$	0	0
$59$	14.7582	1.92135	0.960676	−	0.277671i	$-0.0895624\pi$
$59$	14.7582	1.92135	0.960676	+	0.277671i	$0.0895624\pi$
$60$	0	0
$61$	−0.543990	−0.0696508	−0.0348254	−	0.999393i	$-0.511088\pi$
$61$	−0.543990	−0.0696508	−0.0348254	+	0.999393i	$0.511088\pi$
$62$	0	0
$63$	−5.75148	−0.724618
$64$	0	0
$65$	−4.31943	−0.535760
$66$	0	0
$67$	4.41597	0.539496	0.269748	−	0.962931i	$-0.413060\pi$
$67$	4.41597	0.539496	0.269748	+	0.962931i	$0.413060\pi$
$68$	0	0
$69$	5.84855	0.704083
$70$	0	0
$71$	8.18937	0.971900	0.485950	−	0.873987i	$-0.338474\pi$
$71$	8.18937	0.971900	0.485950	+	0.873987i	$0.338474\pi$
$72$	0	0
$73$	15.0112	1.75692	0.878462	−	0.477813i	$-0.158570\pi$
$73$	15.0112	1.75692	0.878462	+	0.477813i	$0.158570\pi$
$74$	0	0
$75$	9.86316	1.13890
$76$	0	0
$77$	0.414603	0.0472484
$78$	0	0
$79$	−4.68332	−0.526914	−0.263457	−	0.964671i	$-0.584863\pi$
$79$	−4.68332	−0.526914	−0.263457	+	0.964671i	$0.584863\pi$
$80$	0	0
$81$	−7.02084	−0.780093
$82$	0	0
$83$	0.168230	0.0184657	0.00923284	−	0.999957i	$-0.497061\pi$
$83$	0.168230	0.0184657	0.00923284	+	0.999957i	$0.497061\pi$
$84$	0	0
$85$	4.92456	0.534143
$86$	0	0
$87$	25.3413	2.71687
$88$	0	0
$89$	−4.75445	−0.503971	−0.251985	−	0.967731i	$-0.581083\pi$
$89$	−4.75445	−0.503971	−0.251985	+	0.967731i	$0.581083\pi$
$90$	0	0
$91$	6.51933	0.683411
$92$	0	0
$93$	−18.2017	−1.88743
$94$	0	0
$95$	4.14855	0.425632
$96$	0	0
$97$	−10.2932	−1.04512	−0.522560	−	0.852603i	$-0.675023\pi$
$97$	−10.2932	−1.04512	−0.522560	+	0.852603i	$0.675023\pi$
$98$	0	0
$99$	−0.911794	−0.0916387
$100$	0	0
$101$	−6.20858	−0.617777	−0.308889	−	0.951098i	$-0.599957\pi$
$101$	−6.20858	−0.617777	−0.308889	+	0.951098i	$0.599957\pi$
$102$	0	0
$103$	16.5946	1.63512	0.817558	−	0.575846i	$-0.195327\pi$
$103$	16.5946	1.63512	0.817558	+	0.575846i	$0.195327\pi$
$104$	0	0
$105$	4.43685	0.432992
$106$	0	0
$107$	10.0529	0.971854	0.485927	−	0.873999i	$-0.338482\pi$
$107$	10.0529	0.971854	0.485927	+	0.873999i	$0.338482\pi$
$108$	0	0
$109$	−10.4427	−1.00023	−0.500115	−	0.865959i	$-0.666709\pi$
$109$	−10.4427	−1.00023	−0.500115	+	0.865959i	$0.666709\pi$
$110$	0	0
$111$	−22.9858	−2.18172
$112$	0	0
$113$	5.79934	0.545556	0.272778	−	0.962077i	$-0.412058\pi$
$113$	5.79934	0.545556	0.272778	+	0.962077i	$0.412058\pi$
$114$	0	0
$115$	−2.44734	−0.228215
$116$	0	0
$117$	−14.3373	−1.32548
$118$	0	0
$119$	−7.43265	−0.681350
$120$	0	0
$121$	−10.9343	−0.994025
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−9.48462	−0.848330
$126$	0	0
$127$	1.68645	0.149649	0.0748243	−	0.997197i	$-0.476160\pi$
$127$	1.68645	0.149649	0.0748243	+	0.997197i	$0.476160\pi$
$128$	0	0
$129$	9.01220	0.793480
$130$	0	0
$131$	−19.6900	−1.72032	−0.860160	−	0.510025i	$-0.829636\pi$
$131$	−19.6900	−1.72032	−0.860160	+	0.510025i	$0.829636\pi$
$132$	0	0
$133$	−6.26142	−0.542934
$134$	0	0
$135$	−1.52678	−0.131404
$136$	0	0
$137$	12.2395	1.04569	0.522846	−	0.852427i	$-0.324870\pi$
$137$	12.2395	1.04569	0.522846	+	0.852427i	$0.324870\pi$
$138$	0	0
$139$	−19.0898	−1.61917	−0.809587	−	0.587000i	$-0.800309\pi$
$139$	−19.0898	−1.61917	−0.809587	+	0.587000i	$0.800309\pi$
$140$	0	0
$141$	−3.83194	−0.322707
$142$	0	0
$143$	1.03352	0.0864276
$144$	0	0
$145$	−10.6041	−0.880624
$146$	0	0
$147$	11.2274	0.926021
$148$	0	0
$149$	0.415495	0.0340386	0.0170193	−	0.999855i	$-0.494582\pi$
$149$	0.415495	0.0340386	0.0170193	+	0.999855i	$0.494582\pi$
$150$	0	0
$151$	6.17661	0.502645	0.251323	−	0.967903i	$-0.419134\pi$
$151$	6.17661	0.502645	0.251323	+	0.967903i	$0.419134\pi$
$152$	0	0
$153$	16.3459	1.32149
$154$	0	0
$155$	7.61654	0.611775
$156$	0	0
$157$	23.4041	1.86785	0.933926	−	0.357466i	$-0.116359\pi$
$157$	23.4041	1.86785	0.933926	+	0.357466i	$0.116359\pi$
$158$	0	0
$159$	32.5205	2.57904
$160$	0	0
$161$	3.69377	0.291110
$162$	0	0
$163$	−12.7586	−0.999327	−0.499664	−	0.866219i	$-0.666543\pi$
$163$	−12.7586	−0.999327	−0.499664	+	0.866219i	$0.666543\pi$
$164$	0	0
$165$	0.703383	0.0547583
$166$	0	0
$167$	0.953041	0.0737485	0.0368742	−	0.999320i	$-0.488260\pi$
$167$	0.953041	0.0737485	0.0368742	+	0.999320i	$0.488260\pi$
$168$	0	0
$169$	3.25141	0.250108
$170$	0	0
$171$	13.7701	1.05303
$172$	0	0
$173$	−3.13438	−0.238303	−0.119151	−	0.992876i	$-0.538017\pi$
$173$	−3.13438	−0.238303	−0.119151	+	0.992876i	$0.538017\pi$
$174$	0	0
$175$	6.22928	0.470889
$176$	0	0
$177$	−37.7893	−2.84042
$178$	0	0
$179$	12.2277	0.913938	0.456969	−	0.889483i	$-0.348935\pi$
$179$	12.2277	0.913938	0.456969	+	0.889483i	$0.348935\pi$
$180$	0	0
$181$	−1.74937	−0.130030	−0.0650148	−	0.997884i	$-0.520709\pi$
$181$	−1.74937	−0.130030	−0.0650148	+	0.997884i	$0.520709\pi$
$182$	0	0
$183$	1.39292	0.102968
$184$	0	0
$185$	9.61846	0.707163
$186$	0	0
$187$	−1.17831	−0.0861668
$188$	0	0
$189$	2.30437	0.167618
$190$	0	0
$191$	−19.8593	−1.43697	−0.718483	−	0.695545i	$-0.755163\pi$
$191$	−19.8593	−1.43697	−0.718483	+	0.695545i	$0.755163\pi$
$192$	0	0
$193$	1.97164	0.141922	0.0709610	−	0.997479i	$-0.477393\pi$
$193$	1.97164	0.141922	0.0709610	+	0.997479i	$0.477393\pi$
$194$	0	0
$195$	11.0602	0.792036
$196$	0	0
$197$	8.59858	0.612623	0.306312	−	0.951931i	$-0.400905\pi$
$197$	8.59858	0.612623	0.306312	+	0.951931i	$0.400905\pi$
$198$	0	0
$199$	13.5726	0.962137	0.481069	−	0.876683i	$-0.340249\pi$
$199$	13.5726	0.962137	0.481069	+	0.876683i	$0.340249\pi$
$200$	0	0
$201$	−11.3074	−0.797560
$202$	0	0
$203$	16.0048	1.12332
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−8.12334	−0.564611
$208$	0	0
$209$	−0.992636	−0.0686621
$210$	0	0
$211$	6.63854	0.457016	0.228508	−	0.973542i	$-0.426615\pi$
$211$	6.63854	0.457016	0.228508	+	0.973542i	$0.426615\pi$
$212$	0	0
$213$	−20.9694	−1.43680
$214$	0	0
$215$	−3.77117	−0.257192
$216$	0	0
$217$	−11.4957	−0.780377
$218$	0	0
$219$	−38.4371	−2.59734
$220$	0	0
$221$	−18.5281	−1.24634
$222$	0	0
$223$	3.24417	0.217246	0.108623	−	0.994083i	$-0.465356\pi$
$223$	3.24417	0.217246	0.108623	+	0.994083i	$0.465356\pi$
$224$	0	0
$225$	−13.6994	−0.913295
$226$	0	0
$227$	12.0396	0.799098	0.399549	−	0.916712i	$-0.369167\pi$
$227$	12.0396	0.799098	0.399549	+	0.916712i	$0.369167\pi$
$228$	0	0
$229$	6.62030	0.437482	0.218741	−	0.975783i	$-0.429805\pi$
$229$	6.62030	0.437482	0.218741	+	0.975783i	$0.429805\pi$
$230$	0	0
$231$	−1.06162	−0.0698493
$232$	0	0
$233$	3.41052	0.223431	0.111715	−	0.993740i	$-0.464366\pi$
$233$	3.41052	0.223431	0.111715	+	0.993740i	$0.464366\pi$
$234$	0	0
$235$	1.60348	0.104600
$236$	0	0
$237$	11.9919	0.778960
$238$	0	0
$239$	−21.5127	−1.39154	−0.695769	−	0.718265i	$-0.744936\pi$
$239$	−21.5127	−1.39154	−0.695769	+	0.718265i	$0.744936\pi$
$240$	0	0
$241$	−11.2374	−0.723861	−0.361931	−	0.932205i	$-0.617882\pi$
$241$	−11.2374	−0.723861	−0.361931	+	0.932205i	$0.617882\pi$
$242$	0	0
$243$	22.2521	1.42747
$244$	0	0
$245$	−4.69813	−0.300153
$246$	0	0
$247$	−15.6085	−0.993144
$248$	0	0
$249$	−0.430765	−0.0272986
$250$	0	0
$251$	5.99145	0.378177	0.189088	−	0.981960i	$-0.439447\pi$
$251$	5.99145	0.378177	0.189088	+	0.981960i	$0.439447\pi$
$252$	0	0
$253$	0.585582	0.0368152
$254$	0	0
$255$	−12.6096	−0.789647
$256$	0	0
$257$	18.8431	1.17540	0.587699	−	0.809079i	$-0.300034\pi$
$257$	18.8431	1.17540	0.587699	+	0.809079i	$0.300034\pi$
$258$	0	0
$259$	−14.5172	−0.902052
$260$	0	0
$261$	−35.1978	−2.17869
$262$	0	0
$263$	8.27865	0.510483	0.255242	−	0.966877i	$-0.417845\pi$
$263$	8.27865	0.510483	0.255242	+	0.966877i	$0.417845\pi$
$264$	0	0
$265$	−13.6082	−0.835948
$266$	0	0
$267$	12.1741	0.745041
$268$	0	0
$269$	12.3562	0.753372	0.376686	−	0.926341i	$-0.377064\pi$
$269$	12.3562	0.753372	0.376686	+	0.926341i	$0.377064\pi$
$270$	0	0
$271$	−14.0082	−0.850937	−0.425468	−	0.904973i	$-0.639891\pi$
$271$	−14.0082	−0.850937	−0.425468	+	0.904973i	$0.639891\pi$
$272$	0	0
$273$	−16.6932	−1.01032
$274$	0	0
$275$	0.987541	0.0595509
$276$	0	0
$277$	−10.6151	−0.637797	−0.318898	−	0.947789i	$-0.603313\pi$
$277$	−10.6151	−0.637797	−0.318898	+	0.947789i	$0.603313\pi$
$278$	0	0
$279$	25.2813	1.51355
$280$	0	0
$281$	−30.4630	−1.81727	−0.908635	−	0.417591i	$-0.862874\pi$
$281$	−30.4630	−1.81727	−0.908635	+	0.417591i	$0.862874\pi$
$282$	0	0
$283$	−30.4513	−1.81014	−0.905072	−	0.425258i	$-0.860183\pi$
$283$	−30.4513	−1.81014	−0.905072	+	0.425258i	$0.860183\pi$
$284$	0	0
$285$	−10.6226	−0.629230
$286$	0	0
$287$	0	0
$288$	0	0
$289$	4.12380	0.242577
$290$	0	0
$291$	26.3565	1.54504
$292$	0	0
$293$	−16.0435	−0.937270	−0.468635	−	0.883392i	$-0.655254\pi$
$293$	−16.0435	−0.937270	−0.468635	+	0.883392i	$0.655254\pi$
$294$	0	0
$295$	15.8130	0.920668
$296$	0	0
$297$	0.365317	0.0211978
$298$	0	0
$299$	9.20785	0.532504
$300$	0	0
$301$	5.69184	0.328072
$302$	0	0
$303$	15.8975	0.913286
$304$	0	0
$305$	−0.582871	−0.0333751
$306$	0	0
$307$	−22.6972	−1.29540	−0.647700	−	0.761896i	$-0.724269\pi$
$307$	−22.6972	−1.29540	−0.647700	+	0.761896i	$0.724269\pi$
$308$	0	0
$309$	−42.4916	−2.41726
$310$	0	0
$311$	−2.14567	−0.121670	−0.0608349	−	0.998148i	$-0.519376\pi$
$311$	−2.14567	−0.121670	−0.0608349	+	0.998148i	$0.519376\pi$
$312$	0	0
$313$	−20.8239	−1.17703	−0.588517	−	0.808485i	$-0.700288\pi$
$313$	−20.8239	−1.17703	−0.588517	+	0.808485i	$0.700288\pi$
$314$	0	0
$315$	−6.16255	−0.347220
$316$	0	0
$317$	24.0917	1.35312	0.676562	−	0.736386i	$-0.263469\pi$
$317$	24.0917	1.35312	0.676562	+	0.736386i	$0.263469\pi$
$318$	0	0
$319$	2.53728	0.142060
$320$	0	0
$321$	−25.7412	−1.43673
$322$	0	0
$323$	17.7951	0.990148
$324$	0	0
$325$	15.5284	0.861359
$326$	0	0
$327$	26.7392	1.47868
$328$	0	0
$329$	−2.42014	−0.133427
$330$	0	0
$331$	−23.0209	−1.26534	−0.632671	−	0.774420i	$-0.718042\pi$
$331$	−23.0209	−1.26534	−0.632671	+	0.774420i	$0.718042\pi$
$332$	0	0
$333$	31.9261	1.74954
$334$	0	0
$335$	4.73159	0.258514
$336$	0	0
$337$	−19.7649	−1.07666	−0.538331	−	0.842734i	$-0.680945\pi$
$337$	−19.7649	−1.07666	−0.538331	+	0.842734i	$0.680945\pi$
$338$	0	0
$339$	−14.8496	−0.806518
$340$	0	0
$341$	−1.82243	−0.0986902
$342$	0	0
$343$	18.4111	0.994108
$344$	0	0
$345$	6.26657	0.337381
$346$	0	0
$347$	−35.9927	−1.93219	−0.966095	−	0.258188i	$-0.916875\pi$
$347$	−35.9927	−1.93219	−0.966095	+	0.258188i	$0.916875\pi$
$348$	0	0
$349$	−8.95138	−0.479156	−0.239578	−	0.970877i	$-0.577009\pi$
$349$	−8.95138	−0.479156	−0.239578	+	0.970877i	$0.577009\pi$
$350$	0	0
$351$	5.74435	0.306611
$352$	0	0
$353$	−25.3090	−1.34706	−0.673530	−	0.739160i	$-0.735223\pi$
$353$	−25.3090	−1.34706	−0.673530	+	0.739160i	$0.735223\pi$
$354$	0	0
$355$	8.77469	0.465712
$356$	0	0
$357$	19.0318	1.00727
$358$	0	0
$359$	35.1789	1.85667	0.928335	−	0.371745i	$-0.121240\pi$
$359$	35.1789	1.85667	0.928335	+	0.371745i	$0.121240\pi$
$360$	0	0
$361$	−4.00900	−0.211000
$362$	0	0
$363$	27.9979	1.46951
$364$	0	0
$365$	16.0840	0.841878
$366$	0	0
$367$	2.34228	0.122266	0.0611330	−	0.998130i	$-0.480529\pi$
$367$	2.34228	0.122266	0.0611330	+	0.998130i	$0.480529\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	20.5390	1.06633
$372$	0	0
$373$	−14.6246	−0.757234	−0.378617	−	0.925554i	$-0.623600\pi$
$373$	−14.6246	−0.757234	−0.378617	+	0.925554i	$0.623600\pi$
$374$	0	0
$375$	24.2860	1.25412
$376$	0	0
$377$	39.8969	2.05479
$378$	0	0
$379$	−12.0341	−0.618151	−0.309076	−	0.951037i	$-0.600020\pi$
$379$	−12.0341	−0.618151	−0.309076	+	0.951037i	$0.600020\pi$
$380$	0	0
$381$	−4.31828	−0.221232
$382$	0	0
$383$	−1.92158	−0.0981884	−0.0490942	−	0.998794i	$-0.515633\pi$
$383$	−1.92158	−0.0981884	−0.0490942	+	0.998794i	$0.515633\pi$
$384$	0	0
$385$	0.444235	0.0226403
$386$	0	0
$387$	−12.5175	−0.636299
$388$	0	0
$389$	−32.2098	−1.63310	−0.816551	−	0.577273i	$-0.804117\pi$
$389$	−32.2098	−1.63310	−0.816551	+	0.577273i	$0.804117\pi$
$390$	0	0
$391$	−10.4978	−0.530897
$392$	0	0
$393$	50.4174	2.54322
$394$	0	0
$395$	−5.01805	−0.252485
$396$	0	0
$397$	18.0271	0.904756	0.452378	−	0.891826i	$-0.350576\pi$
$397$	18.0271	0.904756	0.452378	+	0.891826i	$0.350576\pi$
$398$	0	0
$399$	16.0328	0.802642
$400$	0	0
$401$	−6.61896	−0.330535	−0.165268	−	0.986249i	$-0.552849\pi$
$401$	−6.61896	−0.330535	−0.165268	+	0.986249i	$0.552849\pi$
$402$	0	0
$403$	−28.6564	−1.42748
$404$	0	0
$405$	−7.52264	−0.373803
$406$	0	0
$407$	−2.30144	−0.114078
$408$	0	0
$409$	18.3299	0.906355	0.453177	−	0.891420i	$-0.350290\pi$
$409$	18.3299	0.906355	0.453177	+	0.891420i	$0.350290\pi$
$410$	0	0
$411$	−31.3400	−1.54589
$412$	0	0
$413$	−23.8666	−1.17440
$414$	0	0
$415$	0.180254	0.00884833
$416$	0	0
$417$	48.8806	2.39369
$418$	0	0
$419$	−36.6036	−1.78820	−0.894102	−	0.447864i	$-0.852185\pi$
$419$	−36.6036	−1.78820	−0.894102	+	0.447864i	$0.852185\pi$
$420$	0	0
$421$	−2.46895	−0.120329	−0.0601646	−	0.998188i	$-0.519163\pi$
$421$	−2.46895	−0.120329	−0.0601646	+	0.998188i	$0.519163\pi$
$422$	0	0
$423$	5.32236	0.258782
$424$	0	0
$425$	−17.7038	−0.858760
$426$	0	0
$427$	0.879728	0.0425730
$428$	0	0
$429$	−2.64640	−0.127770
$430$	0	0
$431$	−28.3374	−1.36497	−0.682483	−	0.730902i	$-0.739100\pi$
$431$	−28.3374	−1.36497	−0.682483	+	0.730902i	$0.739100\pi$
$432$	0	0
$433$	0.0805642	0.00387167	0.00193583	−	0.999998i	$-0.499384\pi$
$433$	0.0805642	0.00387167	0.00193583	+	0.999998i	$0.499384\pi$
$434$	0	0
$435$	27.1525	1.30186
$436$	0	0
$437$	−8.84358	−0.423046
$438$	0	0
$439$	−17.5435	−0.837304	−0.418652	−	0.908147i	$-0.637497\pi$
$439$	−17.5435	−0.837304	−0.418652	+	0.908147i	$0.637497\pi$
$440$	0	0
$441$	−15.5943	−0.742586
$442$	0	0
$443$	−13.2439	−0.629234	−0.314617	−	0.949219i	$-0.601876\pi$
$443$	−13.2439	−0.629234	−0.314617	+	0.949219i	$0.601876\pi$
$444$	0	0
$445$	−5.09426	−0.241491
$446$	0	0
$447$	−1.06390	−0.0503208
$448$	0	0
$449$	−20.0759	−0.947439	−0.473719	−	0.880676i	$-0.657089\pi$
$449$	−20.0759	−0.947439	−0.473719	+	0.880676i	$0.657089\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−15.8156	−0.743082
$454$	0	0
$455$	6.98528	0.327475
$456$	0	0
$457$	−12.3694	−0.578616	−0.289308	−	0.957236i	$-0.593425\pi$
$457$	−12.3694	−0.578616	−0.289308	+	0.957236i	$0.593425\pi$
$458$	0	0
$459$	−6.54910	−0.305686
$460$	0	0
$461$	36.4639	1.69829	0.849147	−	0.528157i	$-0.177117\pi$
$461$	36.4639	1.69829	0.849147	+	0.528157i	$0.177117\pi$
$462$	0	0
$463$	20.9668	0.974408	0.487204	−	0.873288i	$-0.338017\pi$
$463$	20.9668	0.974408	0.487204	+	0.873288i	$0.338017\pi$
$464$	0	0
$465$	−19.5026	−0.904414
$466$	0	0
$467$	−35.9259	−1.66245	−0.831225	−	0.555936i	$-0.812360\pi$
$467$	−35.9259	−1.66245	−0.831225	+	0.555936i	$0.812360\pi$
$468$	0	0
$469$	−7.14140	−0.329759
$470$	0	0
$471$	−59.9278	−2.76133
$472$	0	0
$473$	0.902339	0.0414896
$474$	0	0
$475$	−14.9140	−0.684303
$476$	0	0
$477$	−45.1693	−2.06816
$478$	0	0
$479$	−20.3660	−0.930548	−0.465274	−	0.885167i	$-0.654044\pi$
$479$	−20.3660	−0.930548	−0.465274	+	0.885167i	$0.654044\pi$
$480$	0	0
$481$	−36.1884	−1.65005
$482$	0	0
$483$	−9.45814	−0.430360
$484$	0	0
$485$	−11.0289	−0.500797
$486$	0	0
$487$	−5.84041	−0.264654	−0.132327	−	0.991206i	$-0.542245\pi$
$487$	−5.84041	−0.264654	−0.132327	+	0.991206i	$0.542245\pi$
$488$	0	0
$489$	32.6691	1.47735
$490$	0	0
$491$	−20.7260	−0.935353	−0.467677	−	0.883900i	$-0.654909\pi$
$491$	−20.7260	−0.935353	−0.467677	+	0.883900i	$0.654909\pi$
$492$	0	0
$493$	−45.4862	−2.04859
$494$	0	0
$495$	−0.976962	−0.0439112
$496$	0	0
$497$	−13.2437	−0.594059
$498$	0	0
$499$	−27.6046	−1.23575	−0.617877	−	0.786275i	$-0.712007\pi$
$499$	−27.6046	−1.23575	−0.617877	+	0.786275i	$0.712007\pi$
$500$	0	0
$501$	−2.44032	−0.109026
$502$	0	0
$503$	7.46797	0.332980	0.166490	−	0.986043i	$-0.446757\pi$
$503$	7.46797	0.332980	0.166490	+	0.986043i	$0.446757\pi$
$504$	0	0
$505$	−6.65233	−0.296025
$506$	0	0
$507$	−8.32544	−0.369746
$508$	0	0
$509$	−25.3211	−1.12234	−0.561169	−	0.827701i	$-0.689648\pi$
$509$	−25.3211	−1.12234	−0.561169	+	0.827701i	$0.689648\pi$
$510$	0	0
$511$	−24.2757	−1.07389
$512$	0	0
$513$	−5.51710	−0.243586
$514$	0	0
$515$	17.7807	0.783510
$516$	0	0
$517$	−0.383670	−0.0168738
$518$	0	0
$519$	8.02579	0.352293
$520$	0	0
$521$	24.4474	1.07106	0.535529	−	0.844517i	$-0.320112\pi$
$521$	24.4474	1.07106	0.535529	+	0.844517i	$0.320112\pi$
$522$	0	0
$523$	17.1410	0.749523	0.374761	−	0.927121i	$-0.377725\pi$
$523$	17.1410	0.749523	0.374761	+	0.927121i	$0.377725\pi$
$524$	0	0
$525$	−15.9505	−0.696135
$526$	0	0
$527$	32.6710	1.42317
$528$	0	0
$529$	−17.7829	−0.773172
$530$	0	0
$531$	52.4874	2.27776
$532$	0	0
$533$	0	0
$534$	0	0
$535$	10.7714	0.465690
$536$	0	0
$537$	−31.3097	−1.35111
$538$	0	0
$539$	1.12414	0.0484199
$540$	0	0
$541$	−22.7495	−0.978075	−0.489038	−	0.872263i	$-0.662652\pi$
$541$	−22.7495	−0.978075	−0.489038	+	0.872263i	$0.662652\pi$
$542$	0	0
$543$	4.47938	0.192228
$544$	0	0
$545$	−11.1891	−0.479287
$546$	0	0
$547$	−37.6593	−1.61019	−0.805097	−	0.593144i	$-0.797887\pi$
$547$	−37.6593	−1.61019	−0.805097	+	0.593144i	$0.797887\pi$
$548$	0	0
$549$	−1.93470	−0.0825709
$550$	0	0
$551$	−38.3185	−1.63242
$552$	0	0
$553$	7.57375	0.322069
$554$	0	0
$555$	−24.6287	−1.04543
$556$	0	0
$557$	28.9783	1.22785	0.613925	−	0.789365i	$-0.289590\pi$
$557$	28.9783	1.22785	0.613925	+	0.789365i	$0.289590\pi$
$558$	0	0
$559$	14.1886	0.600115
$560$	0	0
$561$	3.01715	0.127384
$562$	0	0
$563$	−22.9289	−0.966336	−0.483168	−	0.875528i	$-0.660514\pi$
$563$	−22.9289	−0.966336	−0.483168	+	0.875528i	$0.660514\pi$
$564$	0	0
$565$	6.21383	0.261418
$566$	0	0
$567$	11.3539	0.476820
$568$	0	0
$569$	−19.8066	−0.830334	−0.415167	−	0.909745i	$-0.636277\pi$
$569$	−19.8066	−0.830334	−0.415167	+	0.909745i	$0.636277\pi$
$570$	0	0
$571$	32.4895	1.35964	0.679821	−	0.733378i	$-0.262057\pi$
$571$	32.4895	1.35964	0.679821	+	0.733378i	$0.262057\pi$
$572$	0	0
$573$	50.8509	2.12433
$574$	0	0
$575$	8.79818	0.366909
$576$	0	0
$577$	−30.3524	−1.26359	−0.631794	−	0.775137i	$-0.717681\pi$
$577$	−30.3524	−1.26359	−0.631794	+	0.775137i	$0.717681\pi$
$578$	0	0
$579$	−5.04852	−0.209809
$580$	0	0
$581$	−0.272058	−0.0112869
$582$	0	0
$583$	3.25609	0.134853
$584$	0	0
$585$	−15.3620	−0.635142
$586$	0	0
$587$	5.31138	0.219224	0.109612	−	0.993974i	$-0.465039\pi$
$587$	5.31138	0.219224	0.109612	+	0.993974i	$0.465039\pi$
$588$	0	0
$589$	27.5228	1.13406
$590$	0	0
$591$	−22.0172	−0.905667
$592$	0	0
$593$	−23.1836	−0.952036	−0.476018	−	0.879436i	$-0.657920\pi$
$593$	−23.1836	−0.952036	−0.476018	+	0.879436i	$0.657920\pi$
$594$	0	0
$595$	−7.96388	−0.326487
$596$	0	0
$597$	−34.7536	−1.42237
$598$	0	0
$599$	1.25809	0.0514042	0.0257021	−	0.999670i	$-0.491818\pi$
$599$	1.25809	0.0514042	0.0257021	+	0.999670i	$0.491818\pi$
$600$	0	0
$601$	−34.0211	−1.38775	−0.693874	−	0.720097i	$-0.744097\pi$
$601$	−34.0211	−1.38775	−0.693874	+	0.720097i	$0.744097\pi$
$602$	0	0
$603$	15.7054	0.639572
$604$	0	0
$605$	−11.7158	−0.476314
$606$	0	0
$607$	32.5116	1.31960	0.659802	−	0.751439i	$-0.270640\pi$
$607$	32.5116	1.31960	0.659802	+	0.751439i	$0.270640\pi$
$608$	0	0
$609$	−40.9814	−1.66065
$610$	0	0
$611$	−6.03293	−0.244066
$612$	0	0
$613$	−34.6045	−1.39766	−0.698831	−	0.715287i	$-0.746296\pi$
$613$	−34.6045	−1.39766	−0.698831	+	0.715287i	$0.746296\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	10.6644	0.429334	0.214667	−	0.976687i	$-0.431133\pi$
$617$	10.6644	0.429334	0.214667	+	0.976687i	$0.431133\pi$
$618$	0	0
$619$	−33.4548	−1.34466	−0.672330	−	0.740251i	$-0.734707\pi$
$619$	−33.4548	−1.34466	−0.672330	+	0.740251i	$0.734707\pi$
$620$	0	0
$621$	3.25468	0.130606
$622$	0	0
$623$	7.68878	0.308045
$624$	0	0
$625$	9.09722	0.363889
$626$	0	0
$627$	2.54171	0.101506
$628$	0	0
$629$	41.2582	1.64507
$630$	0	0
$631$	36.2612	1.44353	0.721767	−	0.692136i	$-0.243330\pi$
$631$	36.2612	1.44353	0.721767	+	0.692136i	$0.243330\pi$
$632$	0	0
$633$	−16.9984	−0.675626
$634$	0	0
$635$	1.80699	0.0717082
$636$	0	0
$637$	17.6762	0.700357
$638$	0	0
$639$	29.1254	1.15218
$640$	0	0
$641$	13.9262	0.550054	0.275027	−	0.961437i	$-0.411313\pi$
$641$	13.9262	0.550054	0.275027	+	0.961437i	$0.411313\pi$
$642$	0	0
$643$	−24.6740	−0.973050	−0.486525	−	0.873667i	$-0.661736\pi$
$643$	−24.6740	−0.973050	−0.486525	+	0.873667i	$0.661736\pi$
$644$	0	0
$645$	9.65632	0.380217
$646$	0	0
$647$	26.0605	1.02454	0.512272	−	0.858823i	$-0.328804\pi$
$647$	26.0605	1.02454	0.512272	+	0.858823i	$0.328804\pi$
$648$	0	0
$649$	−3.78362	−0.148520
$650$	0	0
$651$	29.4354	1.15366
$652$	0	0
$653$	−11.2207	−0.439098	−0.219549	−	0.975601i	$-0.570459\pi$
$653$	−11.2207	−0.439098	−0.219549	+	0.975601i	$0.570459\pi$
$654$	0	0
$655$	−21.0972	−0.824338
$656$	0	0
$657$	53.3871	2.08283
$658$	0	0
$659$	5.39575	0.210189	0.105094	−	0.994462i	$-0.466486\pi$
$659$	5.39575	0.210189	0.105094	+	0.994462i	$0.466486\pi$
$660$	0	0
$661$	15.6088	0.607110	0.303555	−	0.952814i	$-0.401826\pi$
$661$	15.6088	0.607110	0.303555	+	0.952814i	$0.401826\pi$
$662$	0	0
$663$	47.4425	1.84251
$664$	0	0
$665$	−6.70894	−0.260161
$666$	0	0
$667$	22.6051	0.875272
$668$	0	0
$669$	−8.30691	−0.321164
$670$	0	0
$671$	0.139465	0.00538399
$672$	0	0
$673$	19.0154	0.732990	0.366495	−	0.930420i	$-0.380558\pi$
$673$	19.0154	0.732990	0.366495	+	0.930420i	$0.380558\pi$
$674$	0	0
$675$	5.48878	0.211263
$676$	0	0
$677$	−2.28006	−0.0876298	−0.0438149	−	0.999040i	$-0.513951\pi$
$677$	−2.28006	−0.0876298	−0.0438149	+	0.999040i	$0.513951\pi$
$678$	0	0
$679$	16.6460	0.638814
$680$	0	0
$681$	−30.8282	−1.18134
$682$	0	0
$683$	−22.7257	−0.869574	−0.434787	−	0.900533i	$-0.643176\pi$
$683$	−22.7257	−0.869574	−0.434787	+	0.900533i	$0.643176\pi$
$684$	0	0
$685$	13.1143	0.501071
$686$	0	0
$687$	−16.9517	−0.646748
$688$	0	0
$689$	51.1996	1.95055
$690$	0	0
$691$	3.01885	0.114843	0.0574213	−	0.998350i	$-0.481712\pi$
$691$	3.01885	0.114843	0.0574213	+	0.998350i	$0.481712\pi$
$692$	0	0
$693$	1.47453	0.0560128
$694$	0	0
$695$	−20.4542	−0.775871
$696$	0	0
$697$	0	0
$698$	0	0
$699$	−8.73286	−0.330307
$700$	0	0
$701$	17.0569	0.644228	0.322114	−	0.946701i	$-0.395606\pi$
$701$	17.0569	0.644228	0.322114	+	0.946701i	$0.395606\pi$
$702$	0	0
$703$	34.7568	1.31088
$704$	0	0
$705$	−4.10582	−0.154634
$706$	0	0
$707$	10.0404	0.377607
$708$	0	0
$709$	28.8465	1.08335	0.541676	−	0.840588i	$-0.317790\pi$
$709$	28.8465	1.08335	0.541676	+	0.840588i	$0.317790\pi$
$710$	0	0
$711$	−16.6562	−0.624656
$712$	0	0
$713$	−16.2364	−0.608057
$714$	0	0
$715$	1.10739	0.0414141
$716$	0	0
$717$	55.0846	2.05717
$718$	0	0
$719$	31.2227	1.16441	0.582205	−	0.813042i	$-0.302190\pi$
$719$	31.2227	1.16441	0.582205	+	0.813042i	$0.302190\pi$
$720$	0	0
$721$	−26.8364	−0.999440
$722$	0	0
$723$	28.7740	1.07012
$724$	0	0
$725$	38.1218	1.41581
$726$	0	0
$727$	4.38131	0.162494	0.0812470	−	0.996694i	$-0.474110\pi$
$727$	4.38131	0.162494	0.0812470	+	0.996694i	$0.474110\pi$
$728$	0	0
$729$	−35.9155	−1.33020
$730$	0	0
$731$	−16.1764	−0.598304
$732$	0	0
$733$	−7.91901	−0.292495	−0.146248	−	0.989248i	$-0.546720\pi$
$733$	−7.91901	−0.292495	−0.146248	+	0.989248i	$0.546720\pi$
$734$	0	0
$735$	12.0299	0.443728
$736$	0	0
$737$	−1.13214	−0.0417030
$738$	0	0
$739$	11.4222	0.420174	0.210087	−	0.977683i	$-0.432625\pi$
$739$	11.4222	0.420174	0.210087	+	0.977683i	$0.432625\pi$
$740$	0	0
$741$	39.9665	1.46821
$742$	0	0
$743$	−33.4726	−1.22799	−0.613995	−	0.789310i	$-0.710438\pi$
$743$	−33.4726	−1.22799	−0.613995	+	0.789310i	$0.710438\pi$
$744$	0	0
$745$	0.445191	0.0163105
$746$	0	0
$747$	0.598310	0.0218910
$748$	0	0
$749$	−16.2574	−0.594031
$750$	0	0
$751$	34.5343	1.26017	0.630087	−	0.776525i	$-0.283019\pi$
$751$	34.5343	1.26017	0.630087	+	0.776525i	$0.283019\pi$
$752$	0	0
$753$	−15.3415	−0.559075
$754$	0	0
$755$	6.61807	0.240856
$756$	0	0
$757$	−39.5830	−1.43867	−0.719335	−	0.694664i	$-0.755553\pi$
$757$	−39.5830	−1.43867	−0.719335	+	0.694664i	$0.755553\pi$
$758$	0	0
$759$	−1.49942	−0.0544255
$760$	0	0
$761$	0.0387723	0.00140549	0.000702747	−	1.00000i	$-0.499776\pi$
$761$	0.0387723	0.00140549	0.000702747		1.00000i	$0.499776\pi$
$762$	0	0
$763$	16.8877	0.611376
$764$	0	0
$765$	17.5142	0.633225
$766$	0	0
$767$	−59.4947	−2.14823
$768$	0	0
$769$	25.5399	0.920991	0.460496	−	0.887662i	$-0.347672\pi$
$769$	25.5399	0.920991	0.460496	+	0.887662i	$0.347672\pi$
$770$	0	0
$771$	−48.2489	−1.73764
$772$	0	0
$773$	−17.6372	−0.634364	−0.317182	−	0.948365i	$-0.602737\pi$
$773$	−17.6372	−0.634364	−0.317182	+	0.948365i	$0.602737\pi$
$774$	0	0
$775$	−27.3815	−0.983572
$776$	0	0
$777$	37.1721	1.33354
$778$	0	0
$779$	0	0
$780$	0	0
$781$	−2.09955	−0.0751277
$782$	0	0
$783$	14.1023	0.503974
$784$	0	0
$785$	25.0769	0.895032
$786$	0	0
$787$	29.5029	1.05166	0.525832	−	0.850588i	$-0.323754\pi$
$787$	29.5029	1.05166	0.525832	+	0.850588i	$0.323754\pi$
$788$	0	0
$789$	−21.1980	−0.754669
$790$	0	0
$791$	−9.37855	−0.333463
$792$	0	0
$793$	2.19299	0.0778754
$794$	0	0
$795$	34.8448	1.23582
$796$	0	0
$797$	21.4202	0.758743	0.379372	−	0.925244i	$-0.376140\pi$
$797$	21.4202	0.758743	0.379372	+	0.925244i	$0.376140\pi$
$798$	0	0
$799$	6.87810	0.243330
$800$	0	0
$801$	−16.9092	−0.597456
$802$	0	0
$803$	−3.84848	−0.135810
$804$	0	0
$805$	3.95778	0.139493
$806$	0	0
$807$	−31.6389	−1.11374
$808$	0	0
$809$	−37.0407	−1.30228	−0.651140	−	0.758957i	$-0.725709\pi$
$809$	−37.0407	−1.30228	−0.651140	+	0.758957i	$0.725709\pi$
$810$	0	0
$811$	1.44861	0.0508675	0.0254338	−	0.999677i	$-0.491903\pi$
$811$	1.44861	0.0508675	0.0254338	+	0.999677i	$0.491903\pi$
$812$	0	0
$813$	35.8689	1.25798
$814$	0	0
$815$	−13.6704	−0.478855
$816$	0	0
$817$	−13.6273	−0.476759
$818$	0	0
$819$	23.1859	0.810183
$820$	0	0
$821$	27.3319	0.953889	0.476944	−	0.878933i	$-0.341744\pi$
$821$	27.3319	0.953889	0.476944	+	0.878933i	$0.341744\pi$
$822$	0	0
$823$	22.9395	0.799619	0.399810	−	0.916598i	$-0.369076\pi$
$823$	22.9395	0.799619	0.399810	+	0.916598i	$0.369076\pi$
$824$	0	0
$825$	−2.52866	−0.0880367
$826$	0	0
$827$	−36.0059	−1.25205	−0.626024	−	0.779804i	$-0.715319\pi$
$827$	−36.0059	−1.25205	−0.626024	+	0.779804i	$0.715319\pi$
$828$	0	0
$829$	29.8022	1.03507	0.517537	−	0.855661i	$-0.326849\pi$
$829$	29.8022	1.03507	0.517537	+	0.855661i	$0.326849\pi$
$830$	0	0
$831$	27.1805	0.942883
$832$	0	0
$833$	−20.1525	−0.698244
$834$	0	0
$835$	1.02116	0.0353386
$836$	0	0
$837$	−10.1291	−0.350114
$838$	0	0
$839$	29.3207	1.01226	0.506132	−	0.862456i	$-0.331075\pi$
$839$	29.3207	1.01226	0.506132	+	0.862456i	$0.331075\pi$
$840$	0	0
$841$	68.9461	2.37745
$842$	0	0
$843$	78.0025	2.68655
$844$	0	0
$845$	3.48380	0.119846
$846$	0	0
$847$	17.6826	0.607583
$848$	0	0
$849$	77.9726	2.67601
$850$	0	0
$851$	−20.5039	−0.702865
$852$	0	0
$853$	−36.4560	−1.24823	−0.624115	−	0.781333i	$-0.714540\pi$
$853$	−36.4560	−1.24823	−0.624115	+	0.781333i	$0.714540\pi$
$854$	0	0
$855$	14.7543	0.504586
$856$	0	0
$857$	−14.9266	−0.509884	−0.254942	−	0.966956i	$-0.582056\pi$
$857$	−14.9266	−0.509884	−0.254942	+	0.966956i	$0.582056\pi$
$858$	0	0
$859$	11.2212	0.382863	0.191431	−	0.981506i	$-0.438687\pi$
$859$	11.2212	0.382863	0.191431	+	0.981506i	$0.438687\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−24.6529	−0.839196	−0.419598	−	0.907710i	$-0.637829\pi$
$863$	−24.6529	−0.839196	−0.419598	+	0.907710i	$0.637829\pi$
$864$	0	0
$865$	−3.35840	−0.114189
$866$	0	0
$867$	−10.5593	−0.358611
$868$	0	0
$869$	1.20068	0.0407304
$870$	0	0
$871$	−17.8021	−0.603201
$872$	0	0
$873$	−36.6078	−1.23899
$874$	0	0
$875$	15.3383	0.518529
$876$	0	0
$877$	−26.5619	−0.896933	−0.448466	−	0.893800i	$-0.648030\pi$
$877$	−26.5619	−0.896933	−0.448466	+	0.893800i	$0.648030\pi$
$878$	0	0
$879$	41.0804	1.38561
$880$	0	0
$881$	12.7722	0.430307	0.215153	−	0.976580i	$-0.430975\pi$
$881$	12.7722	0.430307	0.215153	+	0.976580i	$0.430975\pi$
$882$	0	0
$883$	24.8650	0.836774	0.418387	−	0.908269i	$-0.362596\pi$
$883$	24.8650	0.836774	0.418387	+	0.908269i	$0.362596\pi$
$884$	0	0
$885$	−40.4902	−1.36106
$886$	0	0
$887$	5.10092	0.171272	0.0856361	−	0.996326i	$-0.472708\pi$
$887$	5.10092	0.171272	0.0856361	+	0.996326i	$0.472708\pi$
$888$	0	0
$889$	−2.72729	−0.0914705
$890$	0	0
$891$	1.79996	0.0603010
$892$	0	0
$893$	5.79426	0.193898
$894$	0	0
$895$	13.1016	0.437938
$896$	0	0
$897$	−23.5773	−0.787223
$898$	0	0
$899$	−70.3510	−2.34634
$900$	0	0
$901$	−58.3724	−1.94466
$902$	0	0
$903$	−14.5743	−0.485003
$904$	0	0
$905$	−1.87440	−0.0623072
$906$	0	0
$907$	−5.82884	−0.193543	−0.0967717	−	0.995307i	$-0.530852\pi$
$907$	−5.82884	−0.193543	−0.0967717	+	0.995307i	$0.530852\pi$
$908$	0	0
$909$	−22.0808	−0.732373
$910$	0	0
$911$	−27.1621	−0.899919	−0.449960	−	0.893049i	$-0.648562\pi$
$911$	−27.1621	−0.899919	−0.449960	+	0.893049i	$0.648562\pi$
$912$	0	0
$913$	−0.0431300	−0.00142739
$914$	0	0
$915$	1.49248	0.0493398
$916$	0	0
$917$	31.8421	1.05152
$918$	0	0
$919$	4.13524	0.136409	0.0682045	−	0.997671i	$-0.478273\pi$
$919$	4.13524	0.136409	0.0682045	+	0.997671i	$0.478273\pi$
$920$	0	0
$921$	58.1177	1.91504
$922$	0	0
$923$	−33.0139	−1.08666
$924$	0	0
$925$	−34.5784	−1.13693
$926$	0	0
$927$	59.0186	1.93843
$928$	0	0
$929$	6.03088	0.197867	0.0989334	−	0.995094i	$-0.468457\pi$
$929$	6.03088	0.197867	0.0989334	+	0.995094i	$0.468457\pi$
$930$	0	0
$931$	−16.9769	−0.556396
$932$	0	0
$933$	5.49413	0.179870
$934$	0	0
$935$	−1.26253	−0.0412892
$936$	0	0
$937$	48.2987	1.57785	0.788924	−	0.614490i	$-0.210638\pi$
$937$	48.2987	1.57785	0.788924	+	0.614490i	$0.210638\pi$
$938$	0	0
$939$	53.3209	1.74006
$940$	0	0
$941$	−45.7341	−1.49089	−0.745444	−	0.666569i	$-0.767762\pi$
$941$	−45.7341	−1.49089	−0.745444	+	0.666569i	$0.767762\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	2.46907	0.0803189
$946$	0	0
$947$	44.2304	1.43729	0.718647	−	0.695375i	$-0.244762\pi$
$947$	44.2304	1.43729	0.718647	+	0.695375i	$0.244762\pi$
$948$	0	0
$949$	−60.5145	−1.96439
$950$	0	0
$951$	−61.6883	−2.00038
$952$	0	0
$953$	−25.2297	−0.817269	−0.408634	−	0.912698i	$-0.633995\pi$
$953$	−25.2297	−0.817269	−0.408634	+	0.912698i	$0.633995\pi$
$954$	0	0
$955$	−21.2787	−0.688561
$956$	0	0
$957$	−6.49687	−0.210014
$958$	0	0
$959$	−19.7934	−0.639163
$960$	0	0
$961$	19.5305	0.630015
$962$	0	0
$963$	35.7532	1.15213
$964$	0	0
$965$	2.11256	0.0680058
$966$	0	0
$967$	−10.6881	−0.343705	−0.171852	−	0.985123i	$-0.554975\pi$
$967$	−10.6881	−0.343705	−0.171852	+	0.985123i	$0.554975\pi$
$968$	0	0
$969$	−45.5656	−1.46378
$970$	0	0
$971$	17.4427	0.559763	0.279882	−	0.960035i	$-0.409705\pi$
$971$	17.4427	0.559763	0.279882	+	0.960035i	$0.409705\pi$
$972$	0	0
$973$	30.8715	0.989696
$974$	0	0
$975$	−39.7614	−1.27338
$976$	0	0
$977$	−5.11471	−0.163634	−0.0818170	−	0.996647i	$-0.526072\pi$
$977$	−5.11471	−0.163634	−0.0818170	+	0.996647i	$0.526072\pi$
$978$	0	0
$979$	1.21892	0.0389568
$980$	0	0
$981$	−37.1394	−1.18577
$982$	0	0
$983$	−32.9278	−1.05023	−0.525117	−	0.851030i	$-0.675979\pi$
$983$	−32.9278	−1.05023	−0.525117	+	0.851030i	$0.675979\pi$
$984$	0	0
$985$	9.21314	0.293555
$986$	0	0
$987$	6.19692	0.197250
$988$	0	0
$989$	8.03910	0.255629
$990$	0	0
$991$	16.2440	0.516007	0.258004	−	0.966144i	$-0.416935\pi$
$991$	16.2440	0.516007	0.258004	+	0.966144i	$0.416935\pi$
$992$	0	0
$993$	58.9465	1.87061
$994$	0	0
$995$	14.5427	0.461034
$996$	0	0
$997$	39.1578	1.24014	0.620069	−	0.784547i	$-0.287104\pi$
$997$	39.1578	1.24014	0.620069	+	0.784547i	$0.287104\pi$
$998$	0	0
$999$	−12.7914	−0.404703

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6724.2.a.k.1.2	✓	18
41.40	even	2		6724.2.a.l.1.17	yes	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6724.2.a.k.1.2	✓	18	1.1	even	1	trivial
6724.2.a.l.1.17	yes	18	41.40	even	2

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:	\( N \)	\(=\)	\( 6724 = 2^{2} \cdot 41^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6724.a (trivial)

\(f(q)\)	\(=\)	\(q-2.56056 q^{3} +1.07147 q^{5} -1.61718 q^{7} +3.55649 q^{9} +O(q^{10})\)	\(q-2.56056 q^{3} +1.07147 q^{5} -1.61718 q^{7} +3.55649 q^{9} -0.256374 q^{11} -4.03130 q^{13} -2.74358 q^{15} +4.59606 q^{17} +3.87182 q^{19} +4.14089 q^{21} -2.28409 q^{23} -3.85195 q^{25} -1.42494 q^{27} -9.89677 q^{29} +7.10848 q^{31} +0.656463 q^{33} -1.73276 q^{35} +8.97686 q^{37} +10.3224 q^{39} -3.51961 q^{43} +3.81069 q^{45} +1.49652 q^{47} -4.38474 q^{49} -11.7685 q^{51} -12.7005 q^{53} -0.274698 q^{55} -9.91405 q^{57} +14.7582 q^{59} -0.543990 q^{61} -5.75148 q^{63} -4.31943 q^{65} +4.41597 q^{67} +5.84855 q^{69} +8.18937 q^{71} +15.0112 q^{73} +9.86316 q^{75} +0.414603 q^{77} -4.68332 q^{79} -7.02084 q^{81} +0.168230 q^{83} +4.92456 q^{85} +25.3413 q^{87} -4.75445 q^{89} +6.51933 q^{91} -18.2017 q^{93} +4.14855 q^{95} -10.2932 q^{97} -0.911794 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q - 5 q^{3} + 2 q^{5} - 7 q^{7} + 9 q^{9}+O(q^{10}) \) 18 * q - 5 * q^3 + 2 * q^5 - 7 * q^7 + 9 * q^9	\( 18 q - 5 q^{3} + 2 q^{5} - 7 q^{7} + 9 q^{9} - 12 q^{11} - 12 q^{13} + 14 q^{15} - 21 q^{17} - 15 q^{19} + 5 q^{21} + 3 q^{23} + 8 q^{25} - 5 q^{27} - 38 q^{29} + 12 q^{31} + 25 q^{33} - 9 q^{35} + 12 q^{37} - 3 q^{39} + 21 q^{43} - 13 q^{45} + 29 q^{47} - 9 q^{49} + 9 q^{51} - 27 q^{53} - 17 q^{55} - 13 q^{57} + 25 q^{59} - 9 q^{61} - 68 q^{63} - 45 q^{65} + 5 q^{67} - 46 q^{69} - 12 q^{71} + 19 q^{73} - 24 q^{75} - 3 q^{77} + 27 q^{79} - 46 q^{81} - 46 q^{83} - 54 q^{85} - 11 q^{87} - 55 q^{89} - 31 q^{91} - 59 q^{93} - 27 q^{95} - 31 q^{97} - 15 q^{99}+O(q^{100}) \) 18 * q - 5 * q^3 + 2 * q^5 - 7 * q^7 + 9 * q^9 - 12 * q^11 - 12 * q^13 + 14 * q^15 - 21 * q^17 - 15 * q^19 + 5 * q^21 + 3 * q^23 + 8 * q^25 - 5 * q^27 - 38 * q^29 + 12 * q^31 + 25 * q^33 - 9 * q^35 + 12 * q^37 - 3 * q^39 + 21 * q^43 - 13 * q^45 + 29 * q^47 - 9 * q^49 + 9 * q^51 - 27 * q^53 - 17 * q^55 - 13 * q^57 + 25 * q^59 - 9 * q^61 - 68 * q^63 - 45 * q^65 + 5 * q^67 - 46 * q^69 - 12 * q^71 + 19 * q^73 - 24 * q^75 - 3 * q^77 + 27 * q^79 - 46 * q^81 - 46 * q^83 - 54 * q^85 - 11 * q^87 - 55 * q^89 - 31 * q^91 - 59 * q^93 - 27 * q^95 - 31 * q^97 - 15 * q^99

Introduction

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