LMFDB - Embedded newform 7872.2.a.ca.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7872,2,Mod(1,7872)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7872, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("7872.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7872 = 2^{6} \cdot 3 \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7872.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:	$62.8582364712$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.892.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 8x + 10 $ x^3 - x^2 - 8*x + 10
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 984)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.31955$ of defining polynomial
Character	$\chi$	$=$	7872.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -2.93923 q^{5} -2.93923 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -2.93923 q^{5} -2.93923 q^{7} +1.00000 q^{9} +1.68045 q^{11} +5.57834 q^{13} -2.93923 q^{15} +0.319551 q^{17} +4.93923 q^{19} -2.93923 q^{21} +0.939235 q^{23} +3.63910 q^{25} +1.00000 q^{27} +2.31955 q^{29} -7.89789 q^{31} +1.68045 q^{33} +8.63910 q^{35} -5.89789 q^{37} +5.57834 q^{39} -1.00000 q^{41} -9.77636 q^{43} -2.93923 q^{45} -4.31955 q^{47} +1.63910 q^{49} +0.319551 q^{51} +4.00000 q^{53} -4.93923 q^{55} +4.93923 q^{57} +2.12153 q^{59} +9.89789 q^{61} -2.93923 q^{63} -16.3960 q^{65} +4.63910 q^{67} +0.939235 q^{69} +2.19802 q^{71} -5.89789 q^{73} +3.63910 q^{75} -4.93923 q^{77} +3.27820 q^{79} +1.00000 q^{81} +2.42166 q^{83} -0.939235 q^{85} +2.31955 q^{87} -9.75694 q^{89} -16.3960 q^{91} -7.89789 q^{93} -14.5176 q^{95} +6.30013 q^{97} +1.68045 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{3} + 3 q^{9}+O(q^{10}) $ 3 * q + 3 * q^3 + 3 * q^9	$ 3 q + 3 q^{3} + 3 q^{9} + 8 q^{11} + 2 q^{13} - 2 q^{17} + 6 q^{19} - 6 q^{23} + 5 q^{25} + 3 q^{27} + 4 q^{29} - 6 q^{31} + 8 q^{33} + 20 q^{35} + 2 q^{39} - 3 q^{41} + 6 q^{43} - 10 q^{47} - q^{49} - 2 q^{51} + 12 q^{53} - 6 q^{55} + 6 q^{57} + 24 q^{59} + 12 q^{61} - 8 q^{65} + 8 q^{67} - 6 q^{69} - 14 q^{71} + 5 q^{75} - 6 q^{77} - 2 q^{79} + 3 q^{81} + 22 q^{83} + 6 q^{85} + 4 q^{87} + 6 q^{89} - 8 q^{91} - 6 q^{93} - 20 q^{95} + 16 q^{97} + 8 q^{99}+O(q^{100}) $ 3 * q + 3 * q^3 + 3 * q^9 + 8 * q^11 + 2 * q^13 - 2 * q^17 + 6 * q^19 - 6 * q^23 + 5 * q^25 + 3 * q^27 + 4 * q^29 - 6 * q^31 + 8 * q^33 + 20 * q^35 + 2 * q^39 - 3 * q^41 + 6 * q^43 - 10 * q^47 - q^49 - 2 * q^51 + 12 * q^53 - 6 * q^55 + 6 * q^57 + 24 * q^59 + 12 * q^61 - 8 * q^65 + 8 * q^67 - 6 * q^69 - 14 * q^71 + 5 * q^75 - 6 * q^77 - 2 * q^79 + 3 * q^81 + 22 * q^83 + 6 * q^85 + 4 * q^87 + 6 * q^89 - 8 * q^91 - 6 * q^93 - 20 * q^95 + 16 * q^97 + 8 * 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
$5$	−2.93923	−1.31447	−0.657233	−	0.753688i	$-0.728273\pi$
$5$	−2.93923	−1.31447	−0.657233	+	0.753688i	$0.728273\pi$
$6$	0	0
$7$	−2.93923	−1.11093	−0.555463	−	0.831541i	$-0.687459\pi$
$7$	−2.93923	−1.11093	−0.555463	+	0.831541i	$0.687459\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.68045	0.506674	0.253337	−	0.967378i	$-0.418472\pi$
$11$	1.68045	0.506674	0.253337	+	0.967378i	$0.418472\pi$
$12$	0	0
$13$	5.57834	1.54715	0.773576	−	0.633703i	$-0.218466\pi$
$13$	5.57834	1.54715	0.773576	+	0.633703i	$0.218466\pi$
$14$	0	0
$15$	−2.93923	−0.758907
$16$	0	0
$17$	0.319551	0.0775025	0.0387512	−	0.999249i	$-0.487662\pi$
$17$	0.319551	0.0775025	0.0387512	+	0.999249i	$0.487662\pi$
$18$	0	0
$19$	4.93923	1.13314	0.566569	−	0.824014i	$-0.308270\pi$
$19$	4.93923	1.13314	0.566569	+	0.824014i	$0.308270\pi$
$20$	0	0
$21$	−2.93923	−0.641394
$22$	0	0
$23$	0.939235	0.195844	0.0979220	−	0.995194i	$-0.468780\pi$
$23$	0.939235	0.195844	0.0979220	+	0.995194i	$0.468780\pi$
$24$	0	0
$25$	3.63910	0.727820
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	2.31955	0.430730	0.215365	−	0.976534i	$-0.430906\pi$
$29$	2.31955	0.430730	0.215365	+	0.976534i	$0.430906\pi$
$30$	0	0
$31$	−7.89789	−1.41850	−0.709251	−	0.704956i	$-0.750967\pi$
$31$	−7.89789	−1.41850	−0.709251	+	0.704956i	$0.750967\pi$
$32$	0	0
$33$	1.68045	0.292529
$34$	0	0
$35$	8.63910	1.46027
$36$	0	0
$37$	−5.89789	−0.969607	−0.484803	−	0.874623i	$-0.661109\pi$
$37$	−5.89789	−0.969607	−0.484803	+	0.874623i	$0.661109\pi$
$38$	0	0
$39$	5.57834	0.893249
$40$	0	0
$41$	−1.00000	−0.156174
$41$	−1.00000	−0.156174
$42$	0	0
$43$	−9.77636	−1.49088	−0.745440	−	0.666572i	$-0.767761\pi$
$43$	−9.77636	−1.49088	−0.745440	+	0.666572i	$0.767761\pi$
$44$	0	0
$45$	−2.93923	−0.438155
$46$	0	0
$47$	−4.31955	−0.630071	−0.315036	−	0.949080i	$-0.602016\pi$
$47$	−4.31955	−0.630071	−0.315036	+	0.949080i	$0.602016\pi$
$48$	0	0
$49$	1.63910	0.234157
$50$	0	0
$51$	0.319551	0.0447461
$52$	0	0
$53$	4.00000	0.549442	0.274721	−	0.961524i	$-0.411414\pi$
$53$	4.00000	0.549442	0.274721	+	0.961524i	$0.411414\pi$
$54$	0	0
$55$	−4.93923	−0.666006
$56$	0	0
$57$	4.93923	0.654218
$58$	0	0
$59$	2.12153	0.276200	0.138100	−	0.990418i	$-0.455901\pi$
$59$	2.12153	0.276200	0.138100	+	0.990418i	$0.455901\pi$
$60$	0	0
$61$	9.89789	1.26729	0.633647	−	0.773622i	$-0.281557\pi$
$61$	9.89789	1.26729	0.633647	+	0.773622i	$0.281557\pi$
$62$	0	0
$63$	−2.93923	−0.370309
$64$	0	0
$65$	−16.3960	−2.03368
$66$	0	0
$67$	4.63910	0.566756	0.283378	−	0.959008i	$-0.408545\pi$
$67$	4.63910	0.566756	0.283378	+	0.959008i	$0.408545\pi$
$68$	0	0
$69$	0.939235	0.113071
$70$	0	0
$71$	2.19802	0.260857	0.130429	−	0.991458i	$-0.458365\pi$
$71$	2.19802	0.260857	0.130429	+	0.991458i	$0.458365\pi$
$72$	0	0
$73$	−5.89789	−0.690295	−0.345148	−	0.938548i	$-0.612171\pi$
$73$	−5.89789	−0.690295	−0.345148	+	0.938548i	$0.612171\pi$
$74$	0	0
$75$	3.63910	0.420207
$76$	0	0
$77$	−4.93923	−0.562878
$78$	0	0
$79$	3.27820	0.368827	0.184413	−	0.982849i	$-0.440961\pi$
$79$	3.27820	0.368827	0.184413	+	0.982849i	$0.440961\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	2.42166	0.265812	0.132906	−	0.991129i	$-0.457569\pi$
$83$	2.42166	0.265812	0.132906	+	0.991129i	$0.457569\pi$
$84$	0	0
$85$	−0.939235	−0.101874
$86$	0	0
$87$	2.31955	0.248682
$88$	0	0
$89$	−9.75694	−1.03423	−0.517117	−	0.855915i	$-0.672995\pi$
$89$	−9.75694	−1.03423	−0.517117	+	0.855915i	$0.672995\pi$
$90$	0	0
$91$	−16.3960	−1.71877
$92$	0	0
$93$	−7.89789	−0.818973
$94$	0	0
$95$	−14.5176	−1.48947
$96$	0	0
$97$	6.30013	0.639682	0.319841	−	0.947471i	$-0.396371\pi$
$97$	6.30013	0.639682	0.319841	+	0.947471i	$0.396371\pi$
$98$	0	0
$99$	1.68045	0.168891
$100$	0	0
$101$	6.91982	0.688548	0.344274	−	0.938869i	$-0.388125\pi$
$101$	6.91982	0.688548	0.344274	+	0.938869i	$0.388125\pi$
$102$	0	0
$103$	8.49815	0.837348	0.418674	−	0.908137i	$-0.362495\pi$
$103$	8.49815	0.837348	0.418674	+	0.908137i	$0.362495\pi$
$104$	0	0
$105$	8.63910	0.843090
$106$	0	0
$107$	9.23937	0.893203	0.446602	−	0.894733i	$-0.352634\pi$
$107$	9.23937	0.893203	0.446602	+	0.894733i	$0.352634\pi$
$108$	0	0
$109$	4.30013	0.411878	0.205939	−	0.978565i	$-0.433975\pi$
$109$	4.30013	0.411878	0.205939	+	0.978565i	$0.433975\pi$
$110$	0	0
$111$	−5.89789	−0.559803
$112$	0	0
$113$	10.9392	1.02908	0.514538	−	0.857467i	$-0.327963\pi$
$113$	10.9392	1.02908	0.514538	+	0.857467i	$0.327963\pi$
$114$	0	0
$115$	−2.76063	−0.257430
$116$	0	0
$117$	5.57834	0.515717
$118$	0	0
$119$	−0.939235	−0.0860995
$120$	0	0
$121$	−8.17609	−0.743281
$122$	0	0
$123$	−1.00000	−0.0901670
$124$	0	0
$125$	4.00000	0.357771
$126$	0	0
$127$	−4.39604	−0.390086	−0.195043	−	0.980795i	$-0.562485\pi$
$127$	−4.39604	−0.390086	−0.195043	+	0.980795i	$0.562485\pi$
$128$	0	0
$129$	−9.77636	−0.860760
$130$	0	0
$131$	21.9744	1.91991	0.959955	−	0.280154i	$-0.0903855\pi$
$131$	21.9744	1.91991	0.959955	+	0.280154i	$0.0903855\pi$
$132$	0	0
$133$	−14.5176	−1.25883
$134$	0	0
$135$	−2.93923	−0.252969
$136$	0	0
$137$	14.2369	1.21634	0.608168	−	0.793808i	$-0.291905\pi$
$137$	14.2369	1.21634	0.608168	+	0.793808i	$0.291905\pi$
$138$	0	0
$139$	−19.1178	−1.62155	−0.810777	−	0.585355i	$-0.800955\pi$
$139$	−19.1178	−1.62155	−0.810777	+	0.585355i	$0.800955\pi$
$140$	0	0
$141$	−4.31955	−0.363772
$142$	0	0
$143$	9.37411	0.783903
$144$	0	0
$145$	−6.81770	−0.566180
$146$	0	0
$147$	1.63910	0.135191
$148$	0	0
$149$	−9.23937	−0.756919	−0.378459	−	0.925618i	$-0.623546\pi$
$149$	−9.23937	−0.756919	−0.378459	+	0.925618i	$0.623546\pi$
$150$	0	0
$151$	11.2394	0.914647	0.457323	−	0.889300i	$-0.348808\pi$
$151$	11.2394	0.914647	0.457323	+	0.889300i	$0.348808\pi$
$152$	0	0
$153$	0.319551	0.0258342
$154$	0	0
$155$	23.2137	1.86457
$156$	0	0
$157$	−23.5527	−1.87971	−0.939856	−	0.341572i	$-0.889041\pi$
$157$	−23.5527	−1.87971	−0.939856	+	0.341572i	$0.889041\pi$
$158$	0	0
$159$	4.00000	0.317221
$160$	0	0
$161$	−2.76063	−0.217568
$162$	0	0
$163$	−2.86274	−0.224227	−0.112114	−	0.993695i	$-0.535762\pi$
$163$	−2.86274	−0.224227	−0.112114	+	0.993695i	$0.535762\pi$
$164$	0	0
$165$	−4.93923	−0.384519
$166$	0	0
$167$	19.6354	1.51943	0.759717	−	0.650254i	$-0.225338\pi$
$167$	19.6354	1.51943	0.759717	+	0.650254i	$0.225338\pi$
$168$	0	0
$169$	18.1178	1.39368
$170$	0	0
$171$	4.93923	0.377713
$172$	0	0
$173$	−7.53950	−0.573218	−0.286609	−	0.958048i	$-0.592528\pi$
$173$	−7.53950	−0.573218	−0.286609	+	0.958048i	$0.592528\pi$
$174$	0	0
$175$	−10.6962	−0.808555
$176$	0	0
$177$	2.12153	0.159464
$178$	0	0
$179$	12.8371	0.959492	0.479746	−	0.877408i	$-0.340729\pi$
$179$	12.8371	0.959492	0.479746	+	0.877408i	$0.340729\pi$
$180$	0	0
$181$	16.0000	1.18927	0.594635	−	0.803996i	$-0.297296\pi$
$181$	16.0000	1.18927	0.594635	+	0.803996i	$0.297296\pi$
$182$	0	0
$183$	9.89789	0.731673
$184$	0	0
$185$	17.3353	1.27451
$186$	0	0
$187$	0.536989	0.0392685
$188$	0	0
$189$	−2.93923	−0.213798
$190$	0	0
$191$	−3.91730	−0.283446	−0.141723	−	0.989906i	$-0.545264\pi$
$191$	−3.91730	−0.283446	−0.141723	+	0.989906i	$0.545264\pi$
$192$	0	0
$193$	25.2526	1.81772	0.908860	−	0.417101i	$-0.136954\pi$
$193$	25.2526	1.81772	0.908860	+	0.417101i	$0.136954\pi$
$194$	0	0
$195$	−16.3960	−1.17414
$196$	0	0
$197$	13.4956	0.961525	0.480762	−	0.876851i	$-0.340360\pi$
$197$	13.4956	0.961525	0.480762	+	0.876851i	$0.340360\pi$
$198$	0	0
$199$	−11.5783	−0.820767	−0.410383	−	0.911913i	$-0.634605\pi$
$199$	−11.5783	−0.820767	−0.410383	+	0.911913i	$0.634605\pi$
$200$	0	0
$201$	4.63910	0.327217
$202$	0	0
$203$	−6.81770	−0.478509
$204$	0	0
$205$	2.93923	0.205285
$206$	0	0
$207$	0.939235	0.0652813
$208$	0	0
$209$	8.30013	0.574132
$210$	0	0
$211$	−28.4349	−1.95754	−0.978769	−	0.204967i	$-0.934291\pi$
$211$	−28.4349	−1.95754	−0.978769	+	0.204967i	$0.934291\pi$
$212$	0	0
$213$	2.19802	0.150606
$214$	0	0
$215$	28.7350	1.95971
$216$	0	0
$217$	23.2137	1.57585
$218$	0	0
$219$	−5.89789	−0.398542
$220$	0	0
$221$	1.78256	0.119908
$222$	0	0
$223$	19.7569	1.32302	0.661511	−	0.749935i	$-0.269915\pi$
$223$	19.7569	1.32302	0.661511	+	0.749935i	$0.269915\pi$
$224$	0	0
$225$	3.63910	0.242607
$226$	0	0
$227$	−6.35839	−0.422021	−0.211010	−	0.977484i	$-0.567675\pi$
$227$	−6.35839	−0.422021	−0.211010	+	0.977484i	$0.567675\pi$
$228$	0	0
$229$	28.4531	1.88023	0.940117	−	0.340851i	$-0.110715\pi$
$229$	28.4531	1.88023	0.940117	+	0.340851i	$0.110715\pi$
$230$	0	0
$231$	−4.93923	−0.324978
$232$	0	0
$233$	−22.7921	−1.49316	−0.746579	−	0.665296i	$-0.768305\pi$
$233$	−22.7921	−1.49316	−0.746579	+	0.665296i	$0.768305\pi$
$234$	0	0
$235$	12.6962	0.828207
$236$	0	0
$237$	3.27820	0.212942
$238$	0	0
$239$	14.3960	0.931202	0.465601	−	0.884995i	$-0.345838\pi$
$239$	14.3960	0.931202	0.465601	+	0.884995i	$0.345838\pi$
$240$	0	0
$241$	3.38032	0.217745	0.108873	−	0.994056i	$-0.465276\pi$
$241$	3.38032	0.217745	0.108873	+	0.994056i	$0.465276\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−4.81770	−0.307792
$246$	0	0
$247$	27.5527	1.75314
$248$	0	0
$249$	2.42166	0.153467
$250$	0	0
$251$	−12.3960	−0.782431	−0.391216	−	0.920299i	$-0.627945\pi$
$251$	−12.3960	−0.782431	−0.391216	+	0.920299i	$0.627945\pi$
$252$	0	0
$253$	1.57834	0.0992292
$254$	0	0
$255$	−0.939235	−0.0588172
$256$	0	0
$257$	22.7983	1.42212	0.711059	−	0.703132i	$-0.248216\pi$
$257$	22.7983	1.42212	0.711059	+	0.703132i	$0.248216\pi$
$258$	0	0
$259$	17.3353	1.07716
$260$	0	0
$261$	2.31955	0.143577
$262$	0	0
$263$	−28.5114	−1.75809	−0.879043	−	0.476742i	$-0.841817\pi$
$263$	−28.5114	−1.75809	−0.879043	+	0.476742i	$0.841817\pi$
$264$	0	0
$265$	−11.7569	−0.722223
$266$	0	0
$267$	−9.75694	−0.597115
$268$	0	0
$269$	−10.6391	−0.648677	−0.324339	−	0.945941i	$-0.605142\pi$
$269$	−10.6391	−0.648677	−0.324339	+	0.945941i	$0.605142\pi$
$270$	0	0
$271$	−23.0157	−1.39811	−0.699053	−	0.715070i	$-0.746395\pi$
$271$	−23.0157	−1.39811	−0.699053	+	0.715070i	$0.746395\pi$
$272$	0	0
$273$	−16.3960	−0.992334
$274$	0	0
$275$	6.11533	0.368768
$276$	0	0
$277$	24.8115	1.49078	0.745389	−	0.666629i	$-0.232264\pi$
$277$	24.8115	1.49078	0.745389	+	0.666629i	$0.232264\pi$
$278$	0	0
$279$	−7.89789	−0.472834
$280$	0	0
$281$	27.6292	1.64822	0.824110	−	0.566430i	$-0.191676\pi$
$281$	27.6292	1.64822	0.824110	+	0.566430i	$0.191676\pi$
$282$	0	0
$283$	2.05825	0.122350	0.0611752	−	0.998127i	$-0.480515\pi$
$283$	2.05825	0.122350	0.0611752	+	0.998127i	$0.480515\pi$
$284$	0	0
$285$	−14.5176	−0.859947
$286$	0	0
$287$	2.93923	0.173498
$288$	0	0
$289$	−16.8979	−0.993993
$290$	0	0
$291$	6.30013	0.369320
$292$	0	0
$293$	0.159185	0.00929971	0.00464986	−	0.999989i	$-0.498520\pi$
$293$	0.159185	0.00929971	0.00464986	+	0.999989i	$0.498520\pi$
$294$	0	0
$295$	−6.23568	−0.363055
$296$	0	0
$297$	1.68045	0.0975096
$298$	0	0
$299$	5.23937	0.303000
$300$	0	0
$301$	28.7350	1.65626
$302$	0	0
$303$	6.91982	0.397533
$304$	0	0
$305$	−29.0922	−1.66582
$306$	0	0
$307$	21.1373	1.20637	0.603183	−	0.797602i	$-0.293899\pi$
$307$	21.1373	1.20637	0.603183	+	0.797602i	$0.293899\pi$
$308$	0	0
$309$	8.49815	0.483443
$310$	0	0
$311$	−9.36090	−0.530808	−0.265404	−	0.964137i	$-0.585505\pi$
$311$	−9.36090	−0.530808	−0.265404	+	0.964137i	$0.585505\pi$
$312$	0	0
$313$	18.3390	1.03658	0.518290	−	0.855205i	$-0.326569\pi$
$313$	18.3390	1.03658	0.518290	+	0.855205i	$0.326569\pi$
$314$	0	0
$315$	8.63910	0.486758
$316$	0	0
$317$	31.3159	1.75887	0.879437	−	0.476015i	$-0.157919\pi$
$317$	31.3159	1.75887	0.879437	+	0.476015i	$0.157919\pi$
$318$	0	0
$319$	3.89789	0.218240
$320$	0	0
$321$	9.23937	0.515691
$322$	0	0
$323$	1.57834	0.0878210
$324$	0	0
$325$	20.3001	1.12605
$326$	0	0
$327$	4.30013	0.237798
$328$	0	0
$329$	12.6962	0.699963
$330$	0	0
$331$	−27.8528	−1.53093	−0.765465	−	0.643477i	$-0.777491\pi$
$331$	−27.8528	−1.53093	−0.765465	+	0.643477i	$0.777491\pi$
$332$	0	0
$333$	−5.89789	−0.323202
$334$	0	0
$335$	−13.6354	−0.744982
$336$	0	0
$337$	−3.98058	−0.216836	−0.108418	−	0.994105i	$-0.534579\pi$
$337$	−3.98058	−0.216836	−0.108418	+	0.994105i	$0.534579\pi$
$338$	0	0
$339$	10.9392	0.594138
$340$	0	0
$341$	−13.2720	−0.718719
$342$	0	0
$343$	15.7569	0.850795
$344$	0	0
$345$	−2.76063	−0.148627
$346$	0	0
$347$	30.6768	1.64681	0.823407	−	0.567451i	$-0.192070\pi$
$347$	30.6768	1.64681	0.823407	+	0.567451i	$0.192070\pi$
$348$	0	0
$349$	25.2588	1.35207	0.676036	−	0.736869i	$-0.263696\pi$
$349$	25.2588	1.35207	0.676036	+	0.736869i	$0.263696\pi$
$350$	0	0
$351$	5.57834	0.297750
$352$	0	0
$353$	12.4787	0.664176	0.332088	−	0.943248i	$-0.392247\pi$
$353$	12.4787	0.664176	0.332088	+	0.943248i	$0.392247\pi$
$354$	0	0
$355$	−6.46050	−0.342888
$356$	0	0
$357$	−0.939235	−0.0497096
$358$	0	0
$359$	32.6962	1.72564	0.862819	−	0.505513i	$-0.168697\pi$
$359$	32.6962	1.72564	0.862819	+	0.505513i	$0.168697\pi$
$360$	0	0
$361$	5.39604	0.284002
$362$	0	0
$363$	−8.17609	−0.429133
$364$	0	0
$365$	17.3353	0.907370
$366$	0	0
$367$	8.25509	0.430912	0.215456	−	0.976514i	$-0.430876\pi$
$367$	8.25509	0.430912	0.215456	+	0.976514i	$0.430876\pi$
$368$	0	0
$369$	−1.00000	−0.0520579
$370$	0	0
$371$	−11.7569	−0.610390
$372$	0	0
$373$	24.7727	1.28268	0.641340	−	0.767257i	$-0.278379\pi$
$373$	24.7727	1.28268	0.641340	+	0.767257i	$0.278379\pi$
$374$	0	0
$375$	4.00000	0.206559
$376$	0	0
$377$	12.9392	0.666404
$378$	0	0
$379$	5.67424	0.291466	0.145733	−	0.989324i	$-0.453446\pi$
$379$	5.67424	0.291466	0.145733	+	0.989324i	$0.453446\pi$
$380$	0	0
$381$	−4.39604	−0.225216
$382$	0	0
$383$	−3.08018	−0.157390	−0.0786950	−	0.996899i	$-0.525075\pi$
$383$	−3.08018	−0.157390	−0.0786950	+	0.996899i	$0.525075\pi$
$384$	0	0
$385$	14.5176	0.739884
$386$	0	0
$387$	−9.77636	−0.496960
$388$	0	0
$389$	20.8177	1.05550	0.527750	−	0.849400i	$-0.323036\pi$
$389$	20.8177	1.05550	0.527750	+	0.849400i	$0.323036\pi$
$390$	0	0
$391$	0.300133	0.0151784
$392$	0	0
$393$	21.9744	1.10846
$394$	0	0
$395$	−9.63541	−0.484810
$396$	0	0
$397$	16.3001	0.818080	0.409040	−	0.912516i	$-0.365864\pi$
$397$	16.3001	0.818080	0.409040	+	0.912516i	$0.365864\pi$
$398$	0	0
$399$	−14.5176	−0.726788
$400$	0	0
$401$	15.0740	0.752759	0.376379	−	0.926466i	$-0.377169\pi$
$401$	15.0740	0.752759	0.376379	+	0.926466i	$0.377169\pi$
$402$	0	0
$403$	−44.0571	−2.19464
$404$	0	0
$405$	−2.93923	−0.146052
$406$	0	0
$407$	−9.91110	−0.491275
$408$	0	0
$409$	10.2551	0.507082	0.253541	−	0.967325i	$-0.418405\pi$
$409$	10.2551	0.507082	0.253541	+	0.967325i	$0.418405\pi$
$410$	0	0
$411$	14.2369	0.702252
$412$	0	0
$413$	−6.23568	−0.306838
$414$	0	0
$415$	−7.11784	−0.349401
$416$	0	0
$417$	−19.1178	−0.936205
$418$	0	0
$419$	−31.8528	−1.55611	−0.778057	−	0.628194i	$-0.783794\pi$
$419$	−31.8528	−1.55611	−0.778057	+	0.628194i	$0.783794\pi$
$420$	0	0
$421$	−11.7569	−0.572998	−0.286499	−	0.958081i	$-0.592492\pi$
$421$	−11.7569	−0.572998	−0.286499	+	0.958081i	$0.592492\pi$
$422$	0	0
$423$	−4.31955	−0.210024
$424$	0	0
$425$	1.16288	0.0564079
$426$	0	0
$427$	−29.0922	−1.40787
$428$	0	0
$429$	9.37411	0.452586
$430$	0	0
$431$	−34.5746	−1.66540	−0.832701	−	0.553723i	$-0.813207\pi$
$431$	−34.5746	−1.66540	−0.832701	+	0.553723i	$0.813207\pi$
$432$	0	0
$433$	−5.89789	−0.283434	−0.141717	−	0.989907i	$-0.545262\pi$
$433$	−5.89789	−0.283434	−0.141717	+	0.989907i	$0.545262\pi$
$434$	0	0
$435$	−6.81770	−0.326884
$436$	0	0
$437$	4.63910	0.221918
$438$	0	0
$439$	3.03514	0.144859	0.0724297	−	0.997374i	$-0.476925\pi$
$439$	3.03514	0.144859	0.0724297	+	0.997374i	$0.476925\pi$
$440$	0	0
$441$	1.63910	0.0780525
$442$	0	0
$443$	5.20053	0.247085	0.123542	−	0.992339i	$-0.460575\pi$
$443$	5.20053	0.247085	0.123542	+	0.992339i	$0.460575\pi$
$444$	0	0
$445$	28.6779	1.35946
$446$	0	0
$447$	−9.23937	−0.437007
$448$	0	0
$449$	34.3960	1.62325	0.811625	−	0.584179i	$-0.198583\pi$
$449$	34.3960	1.62325	0.811625	+	0.584179i	$0.198583\pi$
$450$	0	0
$451$	−1.68045	−0.0791293
$452$	0	0
$453$	11.2394	0.528072
$454$	0	0
$455$	48.1918	2.25927
$456$	0	0
$457$	−5.55271	−0.259745	−0.129873	−	0.991531i	$-0.541457\pi$
$457$	−5.55271	−0.259745	−0.129873	+	0.991531i	$0.541457\pi$
$458$	0	0
$459$	0.319551	0.0149154
$460$	0	0
$461$	36.3133	1.69128	0.845641	−	0.533753i	$-0.179219\pi$
$461$	36.3133	1.69128	0.845641	+	0.533753i	$0.179219\pi$
$462$	0	0
$463$	−2.33897	−0.108701	−0.0543505	−	0.998522i	$-0.517309\pi$
$463$	−2.33897	−0.108701	−0.0543505	+	0.998522i	$0.517309\pi$
$464$	0	0
$465$	23.2137	1.07651
$466$	0	0
$467$	6.27451	0.290350	0.145175	−	0.989406i	$-0.453626\pi$
$467$	6.27451	0.290350	0.145175	+	0.989406i	$0.453626\pi$
$468$	0	0
$469$	−13.6354	−0.629625
$470$	0	0
$471$	−23.5527	−1.08525
$472$	0	0
$473$	−16.4287	−0.755391
$474$	0	0
$475$	17.9744	0.824721
$476$	0	0
$477$	4.00000	0.183147
$478$	0	0
$479$	2.23686	0.102205	0.0511023	−	0.998693i	$-0.483727\pi$
$479$	2.23686	0.102205	0.0511023	+	0.998693i	$0.483727\pi$
$480$	0	0
$481$	−32.9004	−1.50013
$482$	0	0
$483$	−2.76063	−0.125613
$484$	0	0
$485$	−18.5176	−0.840840
$486$	0	0
$487$	8.89419	0.403034	0.201517	−	0.979485i	$-0.435413\pi$
$487$	8.89419	0.403034	0.201517	+	0.979485i	$0.435413\pi$
$488$	0	0
$489$	−2.86274	−0.129458
$490$	0	0
$491$	24.0571	1.08568	0.542840	−	0.839836i	$-0.317349\pi$
$491$	24.0571	1.08568	0.542840	+	0.839836i	$0.317349\pi$
$492$	0	0
$493$	0.741214	0.0333826
$494$	0	0
$495$	−4.93923	−0.222002
$496$	0	0
$497$	−6.46050	−0.289793
$498$	0	0
$499$	−30.9136	−1.38388	−0.691942	−	0.721953i	$-0.743244\pi$
$499$	−30.9136	−1.38388	−0.691942	+	0.721953i	$0.743244\pi$
$500$	0	0
$501$	19.6354	0.877245
$502$	0	0
$503$	−9.99380	−0.445601	−0.222801	−	0.974864i	$-0.571520\pi$
$503$	−9.99380	−0.445601	−0.222801	+	0.974864i	$0.571520\pi$
$504$	0	0
$505$	−20.3390	−0.905072
$506$	0	0
$507$	18.1178	0.804641
$508$	0	0
$509$	−8.23686	−0.365092	−0.182546	−	0.983197i	$-0.558434\pi$
$509$	−8.23686	−0.365092	−0.182546	+	0.983197i	$0.558434\pi$
$510$	0	0
$511$	17.3353	0.766867
$512$	0	0
$513$	4.93923	0.218073
$514$	0	0
$515$	−24.9781	−1.10067
$516$	0	0
$517$	−7.25879	−0.319241
$518$	0	0
$519$	−7.53950	−0.330947
$520$	0	0
$521$	−27.8334	−1.21940	−0.609702	−	0.792630i	$-0.708711\pi$
$521$	−27.8334	−1.21940	−0.609702	+	0.792630i	$0.708711\pi$
$522$	0	0
$523$	−22.9524	−1.00364	−0.501820	−	0.864972i	$-0.667336\pi$
$523$	−22.9524	−1.00364	−0.501820	+	0.864972i	$0.667336\pi$
$524$	0	0
$525$	−10.6962	−0.466819
$526$	0	0
$527$	−2.52378	−0.109937
$528$	0	0
$529$	−22.1178	−0.961645
$530$	0	0
$531$	2.12153	0.0920666
$532$	0	0
$533$	−5.57834	−0.241625
$534$	0	0
$535$	−27.1567	−1.17409
$536$	0	0
$537$	12.8371	0.553963
$538$	0	0
$539$	2.75443	0.118642
$540$	0	0
$541$	24.7218	1.06287	0.531437	−	0.847098i	$-0.321652\pi$
$541$	24.7218	1.06287	0.531437	+	0.847098i	$0.321652\pi$
$542$	0	0
$543$	16.0000	0.686626
$544$	0	0
$545$	−12.6391	−0.541400
$546$	0	0
$547$	23.9488	1.02397	0.511987	−	0.858993i	$-0.328909\pi$
$547$	23.9488	1.02397	0.511987	+	0.858993i	$0.328909\pi$
$548$	0	0
$549$	9.89789	0.422432
$550$	0	0
$551$	11.4568	0.488076
$552$	0	0
$553$	−9.63541	−0.409739
$554$	0	0
$555$	17.3353	0.735841
$556$	0	0
$557$	34.0377	1.44222	0.721111	−	0.692820i	$-0.243632\pi$
$557$	34.0377	1.44222	0.721111	+	0.692820i	$0.243632\pi$
$558$	0	0
$559$	−54.5358	−2.30662
$560$	0	0
$561$	0.536989	0.0226717
$562$	0	0
$563$	8.23686	0.347142	0.173571	−	0.984821i	$-0.444469\pi$
$563$	8.23686	0.347142	0.173571	+	0.984821i	$0.444469\pi$
$564$	0	0
$565$	−32.1530	−1.35269
$566$	0	0
$567$	−2.93923	−0.123436
$568$	0	0
$569$	32.8565	1.37742	0.688709	−	0.725038i	$-0.258178\pi$
$569$	32.8565	1.37742	0.688709	+	0.725038i	$0.258178\pi$
$570$	0	0
$571$	18.4787	0.773311	0.386656	−	0.922224i	$-0.373630\pi$
$571$	18.4787	0.773311	0.386656	+	0.922224i	$0.373630\pi$
$572$	0	0
$573$	−3.91730	−0.163648
$574$	0	0
$575$	3.41797	0.142539
$576$	0	0
$577$	6.15298	0.256152	0.128076	−	0.991764i	$-0.459120\pi$
$577$	6.15298	0.256152	0.128076	+	0.991764i	$0.459120\pi$
$578$	0	0
$579$	25.2526	1.04946
$580$	0	0
$581$	−7.11784	−0.295298
$582$	0	0
$583$	6.72180	0.278388
$584$	0	0
$585$	−16.3960	−0.677893
$586$	0	0
$587$	7.20171	0.297247	0.148623	−	0.988894i	$-0.452516\pi$
$587$	7.20171	0.297247	0.148623	+	0.988894i	$0.452516\pi$
$588$	0	0
$589$	−39.0095	−1.60736
$590$	0	0
$591$	13.4956	0.555137
$592$	0	0
$593$	−18.9901	−0.779830	−0.389915	−	0.920851i	$-0.627496\pi$
$593$	−18.9901	−0.779830	−0.389915	+	0.920851i	$0.627496\pi$
$594$	0	0
$595$	2.76063	0.113175
$596$	0	0
$597$	−11.5783	−0.473870
$598$	0	0
$599$	−31.8528	−1.30147	−0.650736	−	0.759304i	$-0.725540\pi$
$599$	−31.8528	−1.30147	−0.650736	+	0.759304i	$0.725540\pi$
$600$	0	0
$601$	−18.3572	−0.748806	−0.374403	−	0.927266i	$-0.622152\pi$
$601$	−18.3572	−0.748806	−0.374403	+	0.927266i	$0.622152\pi$
$602$	0	0
$603$	4.63910	0.188919
$604$	0	0
$605$	24.0315	0.977017
$606$	0	0
$607$	−24.9963	−1.01457	−0.507284	−	0.861779i	$-0.669351\pi$
$607$	−24.9963	−1.01457	−0.507284	+	0.861779i	$0.669351\pi$
$608$	0	0
$609$	−6.81770	−0.276267
$610$	0	0
$611$	−24.0959	−0.974816
$612$	0	0
$613$	−21.1955	−0.856079	−0.428039	−	0.903760i	$-0.640796\pi$
$613$	−21.1955	−0.856079	−0.428039	+	0.903760i	$0.640796\pi$
$614$	0	0
$615$	2.93923	0.118521
$616$	0	0
$617$	−16.5746	−0.667270	−0.333635	−	0.942702i	$-0.608275\pi$
$617$	−16.5746	−0.667270	−0.333635	+	0.942702i	$0.608275\pi$
$618$	0	0
$619$	−16.1021	−0.647199	−0.323599	−	0.946194i	$-0.604893\pi$
$619$	−16.1021	−0.647199	−0.323599	+	0.946194i	$0.604893\pi$
$620$	0	0
$621$	0.939235	0.0376902
$622$	0	0
$623$	28.6779	1.14896
$624$	0	0
$625$	−29.9524	−1.19810
$626$	0	0
$627$	8.30013	0.331475
$628$	0	0
$629$	−1.88467	−0.0751469
$630$	0	0
$631$	15.2588	0.607443	0.303721	−	0.952761i	$-0.401771\pi$
$631$	15.2588	0.607443	0.303721	+	0.952761i	$0.401771\pi$
$632$	0	0
$633$	−28.4349	−1.13018
$634$	0	0
$635$	12.9210	0.512754
$636$	0	0
$637$	9.14346	0.362277
$638$	0	0
$639$	2.19802	0.0869523
$640$	0	0
$641$	0.0376552	0.00148729	0.000743645	−	1.00000i	$-0.499763\pi$
$641$	0.0376552	0.00148729	0.000743645		1.00000i	$0.499763\pi$
$642$	0	0
$643$	−21.6354	−0.853217	−0.426609	−	0.904436i	$-0.640292\pi$
$643$	−21.6354	−0.853217	−0.426609	+	0.904436i	$0.640292\pi$
$644$	0	0
$645$	28.7350	1.13144
$646$	0	0
$647$	−17.7313	−0.697090	−0.348545	−	0.937292i	$-0.613324\pi$
$647$	−17.7313	−0.697090	−0.348545	+	0.937292i	$0.613324\pi$
$648$	0	0
$649$	3.56512	0.139943
$650$	0	0
$651$	23.2137	0.909818
$652$	0	0
$653$	−45.6680	−1.78713	−0.893564	−	0.448935i	$-0.851803\pi$
$653$	−45.6680	−1.78713	−0.893564	+	0.448935i	$0.851803\pi$
$654$	0	0
$655$	−64.5879	−2.52366
$656$	0	0
$657$	−5.89789	−0.230098
$658$	0	0
$659$	−33.3923	−1.30078	−0.650391	−	0.759600i	$-0.725395\pi$
$659$	−33.3923	−1.30078	−0.650391	+	0.759600i	$0.725395\pi$
$660$	0	0
$661$	−6.99631	−0.272125	−0.136062	−	0.990700i	$-0.543445\pi$
$661$	−6.99631	−0.272125	−0.136062	+	0.990700i	$0.543445\pi$
$662$	0	0
$663$	1.78256	0.0692290
$664$	0	0
$665$	42.6706	1.65469
$666$	0	0
$667$	2.17860	0.0843558
$668$	0	0
$669$	19.7569	0.763847
$670$	0	0
$671$	16.6329	0.642106
$672$	0	0
$673$	−23.6354	−0.911078	−0.455539	−	0.890216i	$-0.650553\pi$
$673$	−23.6354	−0.911078	−0.455539	+	0.890216i	$0.650553\pi$
$674$	0	0
$675$	3.63910	0.140069
$676$	0	0
$677$	32.2174	1.23822	0.619108	−	0.785306i	$-0.287494\pi$
$677$	32.2174	1.23822	0.619108	+	0.785306i	$0.287494\pi$
$678$	0	0
$679$	−18.5176	−0.710639
$680$	0	0
$681$	−6.35839	−0.243654
$682$	0	0
$683$	−43.9099	−1.68017	−0.840083	−	0.542458i	$-0.817494\pi$
$683$	−43.9099	−1.68017	−0.840083	+	0.542458i	$0.817494\pi$
$684$	0	0
$685$	−41.8455	−1.59883
$686$	0	0
$687$	28.4531	1.08555
$688$	0	0
$689$	22.3133	0.850071
$690$	0	0
$691$	29.0922	1.10672	0.553360	−	0.832942i	$-0.313345\pi$
$691$	29.0922	1.10672	0.553360	+	0.832942i	$0.313345\pi$
$692$	0	0
$693$	−4.93923	−0.187626
$694$	0	0
$695$	56.1918	2.13148
$696$	0	0
$697$	−0.319551	−0.0121038
$698$	0	0
$699$	−22.7921	−0.862076
$700$	0	0
$701$	9.15667	0.345843	0.172921	−	0.984936i	$-0.444679\pi$
$701$	9.15667	0.345843	0.172921	+	0.984936i	$0.444679\pi$
$702$	0	0
$703$	−29.1311	−1.09870
$704$	0	0
$705$	12.6962	0.478166
$706$	0	0
$707$	−20.3390	−0.764926
$708$	0	0
$709$	−33.5966	−1.26175	−0.630873	−	0.775886i	$-0.717303\pi$
$709$	−33.5966	−1.26175	−0.630873	+	0.775886i	$0.717303\pi$
$710$	0	0
$711$	3.27820	0.122942
$712$	0	0
$713$	−7.41797	−0.277805
$714$	0	0
$715$	−27.5527	−1.03041
$716$	0	0
$717$	14.3960	0.537630
$718$	0	0
$719$	13.9161	0.518984	0.259492	−	0.965745i	$-0.416445\pi$
$719$	13.9161	0.518984	0.259492	+	0.965745i	$0.416445\pi$
$720$	0	0
$721$	−24.9781	−0.930232
$722$	0	0
$723$	3.38032	0.125715
$724$	0	0
$725$	8.44108	0.313494
$726$	0	0
$727$	23.0922	0.856443	0.428221	−	0.903674i	$-0.359140\pi$
$727$	23.0922	0.856443	0.428221	+	0.903674i	$0.359140\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−3.12404	−0.115547
$732$	0	0
$733$	4.77266	0.176282	0.0881412	−	0.996108i	$-0.471907\pi$
$733$	4.77266	0.176282	0.0881412	+	0.996108i	$0.471907\pi$
$734$	0	0
$735$	−4.81770	−0.177704
$736$	0	0
$737$	7.79577	0.287161
$738$	0	0
$739$	35.0546	1.28950	0.644751	−	0.764392i	$-0.276961\pi$
$739$	35.0546	1.28950	0.644751	+	0.764392i	$0.276961\pi$
$740$	0	0
$741$	27.5527	1.01217
$742$	0	0
$743$	22.7606	0.835007	0.417503	−	0.908675i	$-0.362905\pi$
$743$	22.7606	0.835007	0.417503	+	0.908675i	$0.362905\pi$
$744$	0	0
$745$	27.1567	0.994944
$746$	0	0
$747$	2.42166	0.0886040
$748$	0	0
$749$	−27.1567	−0.992283
$750$	0	0
$751$	−32.7218	−1.19404	−0.597018	−	0.802228i	$-0.703648\pi$
$751$	−32.7218	−1.19404	−0.597018	+	0.802228i	$0.703648\pi$
$752$	0	0
$753$	−12.3960	−0.451737
$754$	0	0
$755$	−33.0351	−1.20227
$756$	0	0
$757$	−15.4180	−0.560376	−0.280188	−	0.959945i	$-0.590397\pi$
$757$	−15.4180	−0.560376	−0.280188	+	0.959945i	$0.590397\pi$
$758$	0	0
$759$	1.57834	0.0572900
$760$	0	0
$761$	−2.39604	−0.0868564	−0.0434282	−	0.999057i	$-0.513828\pi$
$761$	−2.39604	−0.0868564	−0.0434282	+	0.999057i	$0.513828\pi$
$762$	0	0
$763$	−12.6391	−0.457566
$764$	0	0
$765$	−0.939235	−0.0339581
$766$	0	0
$767$	11.8346	0.427323
$768$	0	0
$769$	36.8892	1.33026	0.665129	−	0.746729i	$-0.268377\pi$
$769$	36.8892	1.33026	0.665129	+	0.746729i	$0.268377\pi$
$770$	0	0
$771$	22.7983	0.821060
$772$	0	0
$773$	−23.0728	−0.829871	−0.414935	−	0.909851i	$-0.636196\pi$
$773$	−23.0728	−0.829871	−0.414935	+	0.909851i	$0.636196\pi$
$774$	0	0
$775$	−28.7412	−1.03241
$776$	0	0
$777$	17.3353	0.621900
$778$	0	0
$779$	−4.93923	−0.176966
$780$	0	0
$781$	3.69366	0.132170
$782$	0	0
$783$	2.31955	0.0828940
$784$	0	0
$785$	69.2270	2.47082
$786$	0	0
$787$	0.293928	0.0104774	0.00523871	−	0.999986i	$-0.498332\pi$
$787$	0.293928	0.0104774	0.00523871	+	0.999986i	$0.498332\pi$
$788$	0	0
$789$	−28.5114	−1.01503
$790$	0	0
$791$	−32.1530	−1.14323
$792$	0	0
$793$	55.2137	1.96070
$794$	0	0
$795$	−11.7569	−0.416976
$796$	0	0
$797$	39.1493	1.38674	0.693369	−	0.720582i	$-0.256125\pi$
$797$	39.1493	1.38674	0.693369	+	0.720582i	$0.256125\pi$
$798$	0	0
$799$	−1.38032	−0.0488321
$800$	0	0
$801$	−9.75694	−0.344745
$802$	0	0
$803$	−9.91110	−0.349755
$804$	0	0
$805$	8.11415	0.285986
$806$	0	0
$807$	−10.6391	−0.374514
$808$	0	0
$809$	−40.9136	−1.43845	−0.719223	−	0.694779i	$-0.755502\pi$
$809$	−40.9136	−1.43845	−0.719223	+	0.694779i	$0.755502\pi$
$810$	0	0
$811$	−15.4118	−0.541180	−0.270590	−	0.962695i	$-0.587219\pi$
$811$	−15.4118	−0.541180	−0.270590	+	0.962695i	$0.587219\pi$
$812$	0	0
$813$	−23.0157	−0.807197
$814$	0	0
$815$	8.41428	0.294739
$816$	0	0
$817$	−48.2877	−1.68937
$818$	0	0
$819$	−16.3960	−0.572924
$820$	0	0
$821$	37.8917	1.32243	0.661214	−	0.750197i	$-0.270041\pi$
$821$	37.8917	1.32243	0.661214	+	0.750197i	$0.270041\pi$
$822$	0	0
$823$	−26.1141	−0.910282	−0.455141	−	0.890419i	$-0.650411\pi$
$823$	−26.1141	−0.910282	−0.455141	+	0.890419i	$0.650411\pi$
$824$	0	0
$825$	6.11533	0.212908
$826$	0	0
$827$	25.2332	0.877443	0.438722	−	0.898623i	$-0.355431\pi$
$827$	25.2332	0.877443	0.438722	+	0.898623i	$0.355431\pi$
$828$	0	0
$829$	23.1761	0.804939	0.402469	−	0.915433i	$-0.368152\pi$
$829$	23.1761	0.804939	0.402469	+	0.915433i	$0.368152\pi$
$830$	0	0
$831$	24.8115	0.860701
$832$	0	0
$833$	0.523776	0.0181478
$834$	0	0
$835$	−57.7131	−1.99724
$836$	0	0
$837$	−7.89789	−0.272991
$838$	0	0
$839$	−25.9876	−0.897191	−0.448596	−	0.893735i	$-0.648076\pi$
$839$	−25.9876	−0.897191	−0.448596	+	0.893735i	$0.648076\pi$
$840$	0	0
$841$	−23.6197	−0.814472
$842$	0	0
$843$	27.6292	0.951600
$844$	0	0
$845$	−53.2526	−1.83194
$846$	0	0
$847$	24.0315	0.825730
$848$	0	0
$849$	2.05825	0.0706390
$850$	0	0
$851$	−5.53950	−0.189892
$852$	0	0
$853$	−15.9173	−0.544998	−0.272499	−	0.962156i	$-0.587850\pi$
$853$	−15.9173	−0.544998	−0.272499	+	0.962156i	$0.587850\pi$
$854$	0	0
$855$	−14.5176	−0.496490
$856$	0	0
$857$	−48.4663	−1.65558	−0.827789	−	0.561039i	$-0.810402\pi$
$857$	−48.4663	−1.65558	−0.827789	+	0.561039i	$0.810402\pi$
$858$	0	0
$859$	22.2236	0.758261	0.379130	−	0.925343i	$-0.376223\pi$
$859$	22.2236	0.758261	0.379130	+	0.925343i	$0.376223\pi$
$860$	0	0
$861$	2.93923	0.100169
$862$	0	0
$863$	−27.0484	−0.920737	−0.460368	−	0.887728i	$-0.652283\pi$
$863$	−27.0484	−0.920737	−0.460368	+	0.887728i	$0.652283\pi$
$864$	0	0
$865$	22.1604	0.753475
$866$	0	0
$867$	−16.8979	−0.573882
$868$	0	0
$869$	5.50885	0.186875
$870$	0	0
$871$	25.8785	0.876858
$872$	0	0
$873$	6.30013	0.213227
$874$	0	0
$875$	−11.7569	−0.397457
$876$	0	0
$877$	−26.9843	−0.911194	−0.455597	−	0.890186i	$-0.650574\pi$
$877$	−26.9843	−0.911194	−0.455597	+	0.890186i	$0.650574\pi$
$878$	0	0
$879$	0.159185	0.00536919
$880$	0	0
$881$	−19.4824	−0.656380	−0.328190	−	0.944612i	$-0.606439\pi$
$881$	−19.4824	−0.656380	−0.328190	+	0.944612i	$0.606439\pi$
$882$	0	0
$883$	12.6962	0.427260	0.213630	−	0.976915i	$-0.431471\pi$
$883$	12.6962	0.427260	0.213630	+	0.976915i	$0.431471\pi$
$884$	0	0
$885$	−6.23568	−0.209610
$886$	0	0
$887$	−9.55892	−0.320957	−0.160479	−	0.987039i	$-0.551304\pi$
$887$	−9.55892	−0.320957	−0.160479	+	0.987039i	$0.551304\pi$
$888$	0	0
$889$	12.9210	0.433356
$890$	0	0
$891$	1.68045	0.0562972
$892$	0	0
$893$	−21.3353	−0.713958
$894$	0	0
$895$	−37.7313	−1.26122
$896$	0	0
$897$	5.23937	0.174937
$898$	0	0
$899$	−18.3196	−0.610991
$900$	0	0
$901$	1.27820	0.0425831
$902$	0	0
$903$	28.7350	0.956241
$904$	0	0
$905$	−47.0278	−1.56326
$906$	0	0
$907$	25.4312	0.844429	0.422214	−	0.906496i	$-0.361253\pi$
$907$	25.4312	0.844429	0.422214	+	0.906496i	$0.361253\pi$
$908$	0	0
$909$	6.91982	0.229516
$910$	0	0
$911$	−36.0447	−1.19421	−0.597106	−	0.802162i	$-0.703683\pi$
$911$	−36.0447	−1.19421	−0.597106	+	0.802162i	$0.703683\pi$
$912$	0	0
$913$	4.06948	0.134680
$914$	0	0
$915$	−29.0922	−0.961759
$916$	0	0
$917$	−64.5879	−2.13288
$918$	0	0
$919$	12.5746	0.414799	0.207400	−	0.978256i	$-0.433500\pi$
$919$	12.5746	0.414799	0.207400	+	0.978256i	$0.433500\pi$
$920$	0	0
$921$	21.1373	0.696496
$922$	0	0
$923$	12.2613	0.403586
$924$	0	0
$925$	−21.4630	−0.705699
$926$	0	0
$927$	8.49815	0.279116
$928$	0	0
$929$	59.3861	1.94840	0.974198	−	0.225695i	$-0.0724652\pi$
$929$	59.3861	1.94840	0.974198	+	0.225695i	$0.0724652\pi$
$930$	0	0
$931$	8.09591	0.265333
$932$	0	0
$933$	−9.36090	−0.306462
$934$	0	0
$935$	−1.57834	−0.0516171
$936$	0	0
$937$	19.0740	0.623120	0.311560	−	0.950227i	$-0.399149\pi$
$937$	19.0740	0.623120	0.311560	+	0.950227i	$0.399149\pi$
$938$	0	0
$939$	18.3390	0.598470
$940$	0	0
$941$	40.8624	1.33208	0.666038	−	0.745918i	$-0.267989\pi$
$941$	40.8624	1.33208	0.666038	+	0.745918i	$0.267989\pi$
$942$	0	0
$943$	−0.939235	−0.0305857
$944$	0	0
$945$	8.63910	0.281030
$946$	0	0
$947$	−27.7181	−0.900717	−0.450359	−	0.892848i	$-0.648704\pi$
$947$	−27.7181	−0.900717	−0.450359	+	0.892848i	$0.648704\pi$
$948$	0	0
$949$	−32.9004	−1.06799
$950$	0	0
$951$	31.3159	1.01549
$952$	0	0
$953$	48.2050	1.56151	0.780757	−	0.624835i	$-0.214834\pi$
$953$	48.2050	1.56151	0.780757	+	0.624835i	$0.214834\pi$
$954$	0	0
$955$	11.5139	0.372580
$956$	0	0
$957$	3.89789	0.126001
$958$	0	0
$959$	−41.8455	−1.35126
$960$	0	0
$961$	31.3766	1.01215
$962$	0	0
$963$	9.23937	0.297734
$964$	0	0
$965$	−74.2233	−2.38933
$966$	0	0
$967$	−18.0000	−0.578841	−0.289420	−	0.957202i	$-0.593463\pi$
$967$	−18.0000	−0.578841	−0.289420	+	0.957202i	$0.593463\pi$
$968$	0	0
$969$	1.57834	0.0507035
$970$	0	0
$971$	57.1943	1.83545	0.917727	−	0.397212i	$-0.130022\pi$
$971$	57.1943	1.83545	0.917727	+	0.397212i	$0.130022\pi$
$972$	0	0
$973$	56.1918	1.80143
$974$	0	0
$975$	20.3001	0.650125
$976$	0	0
$977$	24.3196	0.778051	0.389026	−	0.921227i	$-0.372812\pi$
$977$	24.3196	0.778051	0.389026	+	0.921227i	$0.372812\pi$
$978$	0	0
$979$	−16.3960	−0.524020
$980$	0	0
$981$	4.30013	0.137293
$982$	0	0
$983$	29.1699	0.930375	0.465187	−	0.885212i	$-0.345987\pi$
$983$	29.1699	0.930375	0.465187	+	0.885212i	$0.345987\pi$
$984$	0	0
$985$	−39.6669	−1.26389
$986$	0	0
$987$	12.6962	0.404124
$988$	0	0
$989$	−9.18230	−0.291980
$990$	0	0
$991$	32.4663	1.03133	0.515663	−	0.856791i	$-0.327546\pi$
$991$	32.4663	1.03133	0.515663	+	0.856791i	$0.327546\pi$
$992$	0	0
$993$	−27.8528	−0.883883
$994$	0	0
$995$	34.0315	1.07887
$996$	0	0
$997$	−19.8917	−0.629976	−0.314988	−	0.949096i	$-0.602000\pi$
$997$	−19.8917	−0.629976	−0.314988	+	0.949096i	$0.602000\pi$
$998$	0	0
$999$	−5.89789	−0.186601

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7872.2.a.ca.1.1		3
4.3	odd	2		7872.2.a.bv.1.1		3
8.3	odd	2		1968.2.a.u.1.3		3
8.5	even	2		984.2.a.f.1.3	✓	3
24.5	odd	2		2952.2.a.n.1.1		3
24.11	even	2		5904.2.a.bj.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
984.2.a.f.1.3	✓	3	8.5	even	2
1968.2.a.u.1.3		3	8.3	odd	2
2952.2.a.n.1.1		3	24.5	odd	2
5904.2.a.bj.1.1		3	24.11	even	2
7872.2.a.bv.1.1		3	4.3	odd	2
7872.2.a.ca.1.1		3	1.1	even	1	trivial

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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:	\( N \)	\(=\)	\( 7872 = 2^{6} \cdot 3 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7872.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -2.93923 q^{5} -2.93923 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -2.93923 q^{5} -2.93923 q^{7} +1.00000 q^{9} +1.68045 q^{11} +5.57834 q^{13} -2.93923 q^{15} +0.319551 q^{17} +4.93923 q^{19} -2.93923 q^{21} +0.939235 q^{23} +3.63910 q^{25} +1.00000 q^{27} +2.31955 q^{29} -7.89789 q^{31} +1.68045 q^{33} +8.63910 q^{35} -5.89789 q^{37} +5.57834 q^{39} -1.00000 q^{41} -9.77636 q^{43} -2.93923 q^{45} -4.31955 q^{47} +1.63910 q^{49} +0.319551 q^{51} +4.00000 q^{53} -4.93923 q^{55} +4.93923 q^{57} +2.12153 q^{59} +9.89789 q^{61} -2.93923 q^{63} -16.3960 q^{65} +4.63910 q^{67} +0.939235 q^{69} +2.19802 q^{71} -5.89789 q^{73} +3.63910 q^{75} -4.93923 q^{77} +3.27820 q^{79} +1.00000 q^{81} +2.42166 q^{83} -0.939235 q^{85} +2.31955 q^{87} -9.75694 q^{89} -16.3960 q^{91} -7.89789 q^{93} -14.5176 q^{95} +6.30013 q^{97} +1.68045 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^3 + 3 * q^9	\( 3 q + 3 q^{3} + 3 q^{9} + 8 q^{11} + 2 q^{13} - 2 q^{17} + 6 q^{19} - 6 q^{23} + 5 q^{25} + 3 q^{27} + 4 q^{29} - 6 q^{31} + 8 q^{33} + 20 q^{35} + 2 q^{39} - 3 q^{41} + 6 q^{43} - 10 q^{47} - q^{49} - 2 q^{51} + 12 q^{53} - 6 q^{55} + 6 q^{57} + 24 q^{59} + 12 q^{61} - 8 q^{65} + 8 q^{67} - 6 q^{69} - 14 q^{71} + 5 q^{75} - 6 q^{77} - 2 q^{79} + 3 q^{81} + 22 q^{83} + 6 q^{85} + 4 q^{87} + 6 q^{89} - 8 q^{91} - 6 q^{93} - 20 q^{95} + 16 q^{97} + 8 q^{99}+O(q^{100}) \) 3 * q + 3 * q^3 + 3 * q^9 + 8 * q^11 + 2 * q^13 - 2 * q^17 + 6 * q^19 - 6 * q^23 + 5 * q^25 + 3 * q^27 + 4 * q^29 - 6 * q^31 + 8 * q^33 + 20 * q^35 + 2 * q^39 - 3 * q^41 + 6 * q^43 - 10 * q^47 - q^49 - 2 * q^51 + 12 * q^53 - 6 * q^55 + 6 * q^57 + 24 * q^59 + 12 * q^61 - 8 * q^65 + 8 * q^67 - 6 * q^69 - 14 * q^71 + 5 * q^75 - 6 * q^77 - 2 * q^79 + 3 * q^81 + 22 * q^83 + 6 * q^85 + 4 * q^87 + 6 * q^89 - 8 * q^91 - 6 * q^93 - 20 * q^95 + 16 * q^97 + 8 * q^99

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