LMFDB - Embedded newform 7872.2.a.ca.1.3

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			1.3
Root			$2.59774$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} +3.34596 q^{5} +3.34596 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +3.34596 q^{5} +3.34596 q^{7} +1.00000 q^{9} +0.402265 q^{11} +1.84951 q^{13} +3.34596 q^{15} +1.59774 q^{17} -1.34596 q^{19} +3.34596 q^{21} -5.34596 q^{23} +6.19547 q^{25} +1.00000 q^{27} +3.59774 q^{29} -5.44724 q^{31} +0.402265 q^{33} +11.1955 q^{35} -3.44724 q^{37} +1.84951 q^{39} -1.00000 q^{41} +5.24468 q^{43} +3.34596 q^{45} -5.59774 q^{47} +4.19547 q^{49} +1.59774 q^{51} +4.00000 q^{53} +1.34596 q^{55} -1.34596 q^{57} +14.6919 q^{59} +7.44724 q^{61} +3.34596 q^{63} +6.18838 q^{65} +7.19547 q^{67} -5.34596 q^{69} -9.09419 q^{71} -3.44724 q^{73} +6.19547 q^{75} +1.34596 q^{77} +8.39094 q^{79} +1.00000 q^{81} +6.15049 q^{83} +5.34596 q^{85} +3.59774 q^{87} +15.3839 q^{89} +6.18838 q^{91} -5.44724 q^{93} -4.50354 q^{95} -2.54143 q^{97} +0.402265 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	3.34596	1.49636	0.748180	−	0.663496i	$-0.230928\pi$
$5$	3.34596	1.49636	0.748180	+	0.663496i	$0.230928\pi$
$6$	0	0
$7$	3.34596	1.26466	0.632328	−	0.774701i	$-0.282100\pi$
$7$	3.34596	1.26466	0.632328	+	0.774701i	$0.282100\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	0.402265	0.121287	0.0606437	−	0.998159i	$-0.480685\pi$
$11$	0.402265	0.121287	0.0606437	+	0.998159i	$0.480685\pi$
$12$	0	0
$13$	1.84951	0.512961	0.256480	−	0.966549i	$-0.417437\pi$
$13$	1.84951	0.512961	0.256480	+	0.966549i	$0.417437\pi$
$14$	0	0
$15$	3.34596	0.863924
$16$	0	0
$17$	1.59774	0.387508	0.193754	−	0.981050i	$-0.437934\pi$
$17$	1.59774	0.387508	0.193754	+	0.981050i	$0.437934\pi$
$18$	0	0
$19$	−1.34596	−0.308785	−0.154393	−	0.988010i	$-0.549342\pi$
$19$	−1.34596	−0.308785	−0.154393	+	0.988010i	$0.549342\pi$
$20$	0	0
$21$	3.34596	0.730149
$22$	0	0
$23$	−5.34596	−1.11471	−0.557355	−	0.830274i	$-0.688184\pi$
$23$	−5.34596	−1.11471	−0.557355	+	0.830274i	$0.688184\pi$
$24$	0	0
$25$	6.19547	1.23909
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	3.59774	0.668083	0.334041	−	0.942558i	$-0.391587\pi$
$29$	3.59774	0.668083	0.334041	+	0.942558i	$0.391587\pi$
$30$	0	0
$31$	−5.44724	−0.978354	−0.489177	−	0.872185i	$-0.662703\pi$
$31$	−5.44724	−0.978354	−0.489177	+	0.872185i	$0.662703\pi$
$32$	0	0
$33$	0.402265	0.0700253
$34$	0	0
$35$	11.1955	1.89238
$36$	0	0
$37$	−3.44724	−0.566723	−0.283362	−	0.959013i	$-0.591450\pi$
$37$	−3.44724	−0.566723	−0.283362	+	0.959013i	$0.591450\pi$
$38$	0	0
$39$	1.84951	0.296158
$40$	0	0
$41$	−1.00000	−0.156174
$41$	−1.00000	−0.156174
$42$	0	0
$43$	5.24468	0.799807	0.399903	−	0.916557i	$-0.369044\pi$
$43$	5.24468	0.799807	0.399903	+	0.916557i	$0.369044\pi$
$44$	0	0
$45$	3.34596	0.498787
$46$	0	0
$47$	−5.59774	−0.816514	−0.408257	−	0.912867i	$-0.633863\pi$
$47$	−5.59774	−0.816514	−0.408257	+	0.912867i	$0.633863\pi$
$48$	0	0
$49$	4.19547	0.599353
$50$	0	0
$51$	1.59774	0.223728
$52$	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
$55$	1.34596	0.181490
$56$	0	0
$57$	−1.34596	−0.178277
$58$	0	0
$59$	14.6919	1.91273	0.956363	−	0.292181i	$-0.0943811\pi$
$59$	14.6919	1.91273	0.956363	+	0.292181i	$0.0943811\pi$
$60$	0	0
$61$	7.44724	0.953522	0.476761	−	0.879033i	$-0.341811\pi$
$61$	7.44724	0.953522	0.476761	+	0.879033i	$0.341811\pi$
$62$	0	0
$63$	3.34596	0.421552
$64$	0	0
$65$	6.18838	0.767574
$66$	0	0
$67$	7.19547	0.879067	0.439533	−	0.898226i	$-0.355144\pi$
$67$	7.19547	0.879067	0.439533	+	0.898226i	$0.355144\pi$
$68$	0	0
$69$	−5.34596	−0.643578
$70$	0	0
$71$	−9.09419	−1.07928	−0.539641	−	0.841895i	$-0.681440\pi$
$71$	−9.09419	−1.07928	−0.539641	+	0.841895i	$0.681440\pi$
$72$	0	0
$73$	−3.44724	−0.403469	−0.201735	−	0.979440i	$-0.564658\pi$
$73$	−3.44724	−0.403469	−0.201735	+	0.979440i	$0.564658\pi$
$74$	0	0
$75$	6.19547	0.715391
$76$	0	0
$77$	1.34596	0.153387
$78$	0	0
$79$	8.39094	0.944055	0.472027	−	0.881584i	$-0.343522\pi$
$79$	8.39094	0.944055	0.472027	+	0.881584i	$0.343522\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	6.15049	0.675104	0.337552	−	0.941307i	$-0.390401\pi$
$83$	6.15049	0.675104	0.337552	+	0.941307i	$0.390401\pi$
$84$	0	0
$85$	5.34596	0.579851
$86$	0	0
$87$	3.59774	0.385718
$88$	0	0
$89$	15.3839	1.63069	0.815343	−	0.578979i	$-0.196549\pi$
$89$	15.3839	1.63069	0.815343	+	0.578979i	$0.196549\pi$
$90$	0	0
$91$	6.18838	0.648719
$92$	0	0
$93$	−5.44724	−0.564853
$94$	0	0
$95$	−4.50354	−0.462054
$96$	0	0
$97$	−2.54143	−0.258043	−0.129022	−	0.991642i	$-0.541184\pi$
$97$	−2.54143	−0.258043	−0.129022	+	0.991642i	$0.541184\pi$
$98$	0	0
$99$	0.402265	0.0404291
$100$	0	0
$101$	−9.48513	−0.943806	−0.471903	−	0.881650i	$-0.656433\pi$
$101$	−9.48513	−0.943806	−0.471903	+	0.881650i	$0.656433\pi$
$102$	0	0
$103$	−11.6356	−1.14649	−0.573246	−	0.819383i	$-0.694316\pi$
$103$	−11.6356	−1.14649	−0.573246	+	0.819383i	$0.694316\pi$
$104$	0	0
$105$	11.1955	1.09257
$106$	0	0
$107$	−5.88740	−0.569156	−0.284578	−	0.958653i	$-0.591853\pi$
$107$	−5.88740	−0.569156	−0.284578	+	0.958653i	$0.591853\pi$
$108$	0	0
$109$	−4.54143	−0.434990	−0.217495	−	0.976061i	$-0.569789\pi$
$109$	−4.54143	−0.434990	−0.217495	+	0.976061i	$0.569789\pi$
$110$	0	0
$111$	−3.44724	−0.327198
$112$	0	0
$113$	4.65404	0.437815	0.218907	−	0.975746i	$-0.429751\pi$
$113$	4.65404	0.437815	0.218907	+	0.975746i	$0.429751\pi$
$114$	0	0
$115$	−17.8874	−1.66801
$116$	0	0
$117$	1.84951	0.170987
$118$	0	0
$119$	5.34596	0.490064
$120$	0	0
$121$	−10.8382	−0.985289
$122$	0	0
$123$	−1.00000	−0.0901670
$124$	0	0
$125$	4.00000	0.357771
$126$	0	0
$127$	18.1884	1.61396	0.806979	−	0.590580i	$-0.201101\pi$
$127$	18.1884	1.61396	0.806979	+	0.590580i	$0.201101\pi$
$128$	0	0
$129$	5.24468	0.461769
$130$	0	0
$131$	−4.33888	−0.379089	−0.189545	−	0.981872i	$-0.560701\pi$
$131$	−4.33888	−0.379089	−0.189545	+	0.981872i	$0.560701\pi$
$132$	0	0
$133$	−4.50354	−0.390507
$134$	0	0
$135$	3.34596	0.287975
$136$	0	0
$137$	23.1841	1.98076	0.990378	−	0.138391i	$-0.0441931\pi$
$137$	23.1841	1.98076	0.990378	+	0.138391i	$0.0441931\pi$
$138$	0	0
$139$	8.57932	0.727689	0.363844	−	0.931460i	$-0.381464\pi$
$139$	8.57932	0.727689	0.363844	+	0.931460i	$0.381464\pi$
$140$	0	0
$141$	−5.59774	−0.471414
$142$	0	0
$143$	0.743992	0.0622157
$144$	0	0
$145$	12.0379	0.999692
$146$	0	0
$147$	4.19547	0.346037
$148$	0	0
$149$	5.88740	0.482314	0.241157	−	0.970486i	$-0.422473\pi$
$149$	5.88740	0.482314	0.241157	+	0.970486i	$0.422473\pi$
$150$	0	0
$151$	−3.88740	−0.316352	−0.158176	−	0.987411i	$-0.550561\pi$
$151$	−3.88740	−0.316352	−0.158176	+	0.987411i	$0.550561\pi$
$152$	0	0
$153$	1.59774	0.129169
$154$	0	0
$155$	−18.2263	−1.46397
$156$	0	0
$157$	6.48937	0.517908	0.258954	−	0.965890i	$-0.416622\pi$
$157$	6.48937	0.517908	0.258954	+	0.965890i	$0.416622\pi$
$158$	0	0
$159$	4.00000	0.317221
$160$	0	0
$161$	−17.8874	−1.40972
$162$	0	0
$163$	−20.4402	−1.60100	−0.800498	−	0.599335i	$-0.795432\pi$
$163$	−20.4402	−1.60100	−0.800498	+	0.599335i	$0.795432\pi$
$164$	0	0
$165$	1.34596	0.104783
$166$	0	0
$167$	−18.0758	−1.39875	−0.699373	−	0.714757i	$-0.746537\pi$
$167$	−18.0758	−1.39875	−0.699373	+	0.714757i	$0.746537\pi$
$168$	0	0
$169$	−9.57932	−0.736871
$170$	0	0
$171$	−1.34596	−0.102928
$172$	0	0
$173$	16.4288	1.24906	0.624530	−	0.781000i	$-0.285290\pi$
$173$	16.4288	1.24906	0.624530	+	0.781000i	$0.285290\pi$
$174$	0	0
$175$	20.7298	1.56703
$176$	0	0
$177$	14.6919	1.10431
$178$	0	0
$179$	4.10128	0.306544	0.153272	−	0.988184i	$-0.451019\pi$
$179$	4.10128	0.306544	0.153272	+	0.988184i	$0.451019\pi$
$180$	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
$183$	7.44724	0.550516
$184$	0	0
$185$	−11.5343	−0.848022
$186$	0	0
$187$	0.642713	0.0469998
$188$	0	0
$189$	3.34596	0.243383
$190$	0	0
$191$	−11.5864	−0.838363	−0.419182	−	0.907902i	$-0.637683\pi$
$191$	−11.5864	−0.838363	−0.419182	+	0.907902i	$0.637683\pi$
$192$	0	0
$193$	4.05207	0.291674	0.145837	−	0.989309i	$-0.453412\pi$
$193$	4.05207	0.291674	0.145837	+	0.989309i	$0.453412\pi$
$194$	0	0
$195$	6.18838	0.443159
$196$	0	0
$197$	17.4359	1.24226	0.621129	−	0.783708i	$-0.286674\pi$
$197$	17.4359	1.24226	0.621129	+	0.783708i	$0.286674\pi$
$198$	0	0
$199$	−7.84951	−0.556437	−0.278218	−	0.960518i	$-0.589744\pi$
$199$	−7.84951	−0.556437	−0.278218	+	0.960518i	$0.589744\pi$
$200$	0	0
$201$	7.19547	0.507529
$202$	0	0
$203$	12.0379	0.844894
$204$	0	0
$205$	−3.34596	−0.233692
$206$	0	0
$207$	−5.34596	−0.371570
$208$	0	0
$209$	−0.541434	−0.0374518
$210$	0	0
$211$	−26.0900	−1.79611	−0.898053	−	0.439887i	$-0.855019\pi$
$211$	−26.0900	−1.79611	−0.898053	+	0.439887i	$0.855019\pi$
$212$	0	0
$213$	−9.09419	−0.623124
$214$	0	0
$215$	17.5485	1.19680
$216$	0	0
$217$	−18.2263	−1.23728
$218$	0	0
$219$	−3.44724	−0.232943
$220$	0	0
$221$	2.95502	0.198776
$222$	0	0
$223$	−5.38385	−0.360529	−0.180265	−	0.983618i	$-0.557695\pi$
$223$	−5.38385	−0.360529	−0.180265	+	0.983618i	$0.557695\pi$
$224$	0	0
$225$	6.19547	0.413031
$226$	0	0
$227$	−27.8761	−1.85020	−0.925100	−	0.379724i	$-0.876019\pi$
$227$	−27.8761	−1.85020	−0.925100	+	0.379724i	$0.876019\pi$
$228$	0	0
$229$	−28.1137	−1.85780	−0.928902	−	0.370326i	$-0.879246\pi$
$229$	−28.1137	−1.85780	−0.928902	+	0.370326i	$0.879246\pi$
$230$	0	0
$231$	1.34596	0.0885579
$232$	0	0
$233$	22.3768	1.46595	0.732975	−	0.680255i	$-0.238131\pi$
$233$	22.3768	1.46595	0.732975	+	0.680255i	$0.238131\pi$
$234$	0	0
$235$	−18.7298	−1.22180
$236$	0	0
$237$	8.39094	0.545050
$238$	0	0
$239$	−8.18838	−0.529662	−0.264831	−	0.964295i	$-0.585316\pi$
$239$	−8.18838	−0.529662	−0.264831	+	0.964295i	$0.585316\pi$
$240$	0	0
$241$	10.9437	0.704946	0.352473	−	0.935822i	$-0.385341\pi$
$241$	10.9437	0.704946	0.352473	+	0.935822i	$0.385341\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	14.0379	0.896848
$246$	0	0
$247$	−2.48937	−0.158395
$248$	0	0
$249$	6.15049	0.389772
$250$	0	0
$251$	10.1884	0.643085	0.321543	−	0.946895i	$-0.395799\pi$
$251$	10.1884	0.643085	0.321543	+	0.946895i	$0.395799\pi$
$252$	0	0
$253$	−2.15049	−0.135200
$254$	0	0
$255$	5.34596	0.334777
$256$	0	0
$257$	−6.17706	−0.385314	−0.192657	−	0.981266i	$-0.561711\pi$
$257$	−6.17706	−0.385314	−0.192657	+	0.981266i	$0.561711\pi$
$258$	0	0
$259$	−11.5343	−0.716709
$260$	0	0
$261$	3.59774	0.222694
$262$	0	0
$263$	−2.30384	−0.142061	−0.0710303	−	0.997474i	$-0.522629\pi$
$263$	−2.30384	−0.142061	−0.0710303	+	0.997474i	$0.522629\pi$
$264$	0	0
$265$	13.3839	0.822164
$266$	0	0
$267$	15.3839	0.941477
$268$	0	0
$269$	−13.1955	−0.804542	−0.402271	−	0.915521i	$-0.631779\pi$
$269$	−13.1955	−0.804542	−0.402271	+	0.915521i	$0.631779\pi$
$270$	0	0
$271$	7.13208	0.433243	0.216622	−	0.976256i	$-0.430496\pi$
$271$	7.13208	0.433243	0.216622	+	0.976256i	$0.430496\pi$
$272$	0	0
$273$	6.18838	0.374538
$274$	0	0
$275$	2.49222	0.150287
$276$	0	0
$277$	−10.2376	−0.615118	−0.307559	−	0.951529i	$-0.599512\pi$
$277$	−10.2376	−0.615118	−0.307559	+	0.951529i	$0.599512\pi$
$278$	0	0
$279$	−5.44724	−0.326118
$280$	0	0
$281$	−26.2755	−1.56746	−0.783732	−	0.621099i	$-0.786686\pi$
$281$	−26.2755	−1.56746	−0.783732	+	0.621099i	$0.786686\pi$
$282$	0	0
$283$	32.4175	1.92702	0.963510	−	0.267671i	$-0.0862540\pi$
$283$	32.4175	1.92702	0.963510	+	0.267671i	$0.0862540\pi$
$284$	0	0
$285$	−4.50354	−0.266767
$286$	0	0
$287$	−3.34596	−0.197506
$288$	0	0
$289$	−14.4472	−0.849838
$290$	0	0
$291$	−2.54143	−0.148981
$292$	0	0
$293$	−31.3725	−1.83280	−0.916401	−	0.400261i	$-0.868920\pi$
$293$	−31.3725	−1.83280	−0.916401	+	0.400261i	$0.868920\pi$
$294$	0	0
$295$	49.1586	2.86213
$296$	0	0
$297$	0.402265	0.0233418
$298$	0	0
$299$	−9.88740	−0.571803
$300$	0	0
$301$	17.5485	1.01148
$302$	0	0
$303$	−9.48513	−0.544907
$304$	0	0
$305$	24.9182	1.42681
$306$	0	0
$307$	3.55985	0.203171	0.101586	−	0.994827i	$-0.467608\pi$
$307$	3.55985	0.203171	0.101586	+	0.994827i	$0.467608\pi$
$308$	0	0
$309$	−11.6356	−0.661928
$310$	0	0
$311$	−6.80453	−0.385849	−0.192925	−	0.981214i	$-0.561797\pi$
$311$	−6.80453	−0.385849	−0.192925	+	0.981214i	$0.561797\pi$
$312$	0	0
$313$	29.7369	1.68083	0.840415	−	0.541944i	$-0.182312\pi$
$313$	29.7369	1.68083	0.840415	+	0.541944i	$0.182312\pi$
$314$	0	0
$315$	11.1955	0.630793
$316$	0	0
$317$	−7.67351	−0.430988	−0.215494	−	0.976505i	$-0.569136\pi$
$317$	−7.67351	−0.430988	−0.215494	+	0.976505i	$0.569136\pi$
$318$	0	0
$319$	1.44724	0.0810300
$320$	0	0
$321$	−5.88740	−0.328602
$322$	0	0
$323$	−2.15049	−0.119657
$324$	0	0
$325$	11.4586	0.635607
$326$	0	0
$327$	−4.54143	−0.251142
$328$	0	0
$329$	−18.7298	−1.03261
$330$	0	0
$331$	11.0308	0.606308	0.303154	−	0.952942i	$-0.401960\pi$
$331$	11.0308	0.606308	0.303154	+	0.952942i	$0.401960\pi$
$332$	0	0
$333$	−3.44724	−0.188908
$334$	0	0
$335$	24.0758	1.31540
$336$	0	0
$337$	6.13917	0.334422	0.167211	−	0.985921i	$-0.446524\pi$
$337$	6.13917	0.334422	0.167211	+	0.985921i	$0.446524\pi$
$338$	0	0
$339$	4.65404	0.252773
$340$	0	0
$341$	−2.19123	−0.118662
$342$	0	0
$343$	−9.38385	−0.506680
$344$	0	0
$345$	−17.8874	−0.963025
$346$	0	0
$347$	−10.8690	−0.583478	−0.291739	−	0.956498i	$-0.594234\pi$
$347$	−10.8690	−0.583478	−0.291739	+	0.956498i	$0.594234\pi$
$348$	0	0
$349$	20.2518	1.08405	0.542026	−	0.840362i	$-0.317657\pi$
$349$	20.2518	1.08405	0.542026	+	0.840362i	$0.317657\pi$
$350$	0	0
$351$	1.84951	0.0987194
$352$	0	0
$353$	−17.7748	−0.946057	−0.473028	−	0.881047i	$-0.656839\pi$
$353$	−17.7748	−0.946057	−0.473028	+	0.881047i	$0.656839\pi$
$354$	0	0
$355$	−30.4288	−1.61499
$356$	0	0
$357$	5.34596	0.282938
$358$	0	0
$359$	1.27018	0.0670377	0.0335189	−	0.999438i	$-0.489329\pi$
$359$	1.27018	0.0670377	0.0335189	+	0.999438i	$0.489329\pi$
$360$	0	0
$361$	−17.1884	−0.904652
$362$	0	0
$363$	−10.8382	−0.568857
$364$	0	0
$365$	−11.5343	−0.603735
$366$	0	0
$367$	−37.0195	−1.93240	−0.966201	−	0.257792i	$-0.917005\pi$
$367$	−37.0195	−1.93240	−0.966201	+	0.257792i	$0.917005\pi$
$368$	0	0
$369$	−1.00000	−0.0520579
$370$	0	0
$371$	13.3839	0.694855
$372$	0	0
$373$	−30.5159	−1.58006	−0.790028	−	0.613071i	$-0.789934\pi$
$373$	−30.5159	−1.58006	−0.790028	+	0.613071i	$0.789934\pi$
$374$	0	0
$375$	4.00000	0.206559
$376$	0	0
$377$	6.65404	0.342700
$378$	0	0
$379$	−11.7974	−0.605994	−0.302997	−	0.952992i	$-0.597987\pi$
$379$	−11.7974	−0.605994	−0.302997	+	0.952992i	$0.597987\pi$
$380$	0	0
$381$	18.1884	0.931819
$382$	0	0
$383$	−19.4851	−0.995644	−0.497822	−	0.867279i	$-0.665867\pi$
$383$	−19.4851	−0.995644	−0.497822	+	0.867279i	$0.665867\pi$
$384$	0	0
$385$	4.50354	0.229522
$386$	0	0
$387$	5.24468	0.266602
$388$	0	0
$389$	1.96211	0.0994829	0.0497415	−	0.998762i	$-0.484160\pi$
$389$	1.96211	0.0994829	0.0497415	+	0.998762i	$0.484160\pi$
$390$	0	0
$391$	−8.54143	−0.431959
$392$	0	0
$393$	−4.33888	−0.218867
$394$	0	0
$395$	28.0758	1.41265
$396$	0	0
$397$	7.45857	0.374335	0.187167	−	0.982328i	$-0.440069\pi$
$397$	7.45857	0.374335	0.187167	+	0.982328i	$0.440069\pi$
$398$	0	0
$399$	−4.50354	−0.225459
$400$	0	0
$401$	15.2854	0.763318	0.381659	−	0.924303i	$-0.375353\pi$
$401$	15.2854	0.763318	0.381659	+	0.924303i	$0.375353\pi$
$402$	0	0
$403$	−10.0747	−0.501857
$404$	0	0
$405$	3.34596	0.166262
$406$	0	0
$407$	−1.38670	−0.0687364
$408$	0	0
$409$	−35.0195	−1.73160	−0.865801	−	0.500389i	$-0.833190\pi$
$409$	−35.0195	−1.73160	−0.865801	+	0.500389i	$0.833190\pi$
$410$	0	0
$411$	23.1841	1.14359
$412$	0	0
$413$	49.1586	2.41894
$414$	0	0
$415$	20.5793	1.01020
$416$	0	0
$417$	8.57932	0.420131
$418$	0	0
$419$	7.03080	0.343477	0.171739	−	0.985143i	$-0.445062\pi$
$419$	7.03080	0.343477	0.171739	+	0.985143i	$0.445062\pi$
$420$	0	0
$421$	13.3839	0.652289	0.326145	−	0.945320i	$-0.394250\pi$
$421$	13.3839	0.652289	0.326145	+	0.945320i	$0.394250\pi$
$422$	0	0
$423$	−5.59774	−0.272171
$424$	0	0
$425$	9.89872	0.480158
$426$	0	0
$427$	24.9182	1.20588
$428$	0	0
$429$	0.743992	0.0359203
$430$	0	0
$431$	9.42174	0.453829	0.226915	−	0.973915i	$-0.427136\pi$
$431$	9.42174	0.453829	0.226915	+	0.973915i	$0.427136\pi$
$432$	0	0
$433$	−3.44724	−0.165664	−0.0828319	−	0.996564i	$-0.526396\pi$
$433$	−3.44724	−0.165664	−0.0828319	+	0.996564i	$0.526396\pi$
$434$	0	0
$435$	12.0379	0.577173
$436$	0	0
$437$	7.19547	0.344206
$438$	0	0
$439$	−16.9929	−0.811027	−0.405514	−	0.914089i	$-0.632907\pi$
$439$	−16.9929	−0.811027	−0.405514	+	0.914089i	$0.632907\pi$
$440$	0	0
$441$	4.19547	0.199784
$442$	0	0
$443$	−30.1657	−1.43322	−0.716609	−	0.697475i	$-0.754307\pi$
$443$	−30.1657	−1.43322	−0.716609	+	0.697475i	$0.754307\pi$
$444$	0	0
$445$	51.4738	2.44009
$446$	0	0
$447$	5.88740	0.278464
$448$	0	0
$449$	11.8116	0.557425	0.278712	−	0.960375i	$-0.410092\pi$
$449$	11.8116	0.557425	0.278712	+	0.960375i	$0.410092\pi$
$450$	0	0
$451$	−0.402265	−0.0189419
$452$	0	0
$453$	−3.88740	−0.182646
$454$	0	0
$455$	20.7061	0.970717
$456$	0	0
$457$	24.4894	1.14556	0.572782	−	0.819708i	$-0.305864\pi$
$457$	24.4894	1.14556	0.572782	+	0.819708i	$0.305864\pi$
$458$	0	0
$459$	1.59774	0.0745759
$460$	0	0
$461$	21.3980	0.996606	0.498303	−	0.867003i	$-0.333957\pi$
$461$	21.3980	0.996606	0.498303	+	0.867003i	$0.333957\pi$
$462$	0	0
$463$	−13.7369	−0.638408	−0.319204	−	0.947686i	$-0.603416\pi$
$463$	−13.7369	−0.638408	−0.319204	+	0.947686i	$0.603416\pi$
$464$	0	0
$465$	−18.2263	−0.845223
$466$	0	0
$467$	−28.8803	−1.33642	−0.668211	−	0.743972i	$-0.732940\pi$
$467$	−28.8803	−1.33642	−0.668211	+	0.743972i	$0.732940\pi$
$468$	0	0
$469$	24.0758	1.11172
$470$	0	0
$471$	6.48937	0.299014
$472$	0	0
$473$	2.10975	0.0970065
$474$	0	0
$475$	−8.33888	−0.382614
$476$	0	0
$477$	4.00000	0.183147
$478$	0	0
$479$	11.1841	0.511017	0.255508	−	0.966807i	$-0.417757\pi$
$479$	11.1841	0.511017	0.255508	+	0.966807i	$0.417757\pi$
$480$	0	0
$481$	−6.37570	−0.290707
$482$	0	0
$483$	−17.8874	−0.813905
$484$	0	0
$485$	−8.50354	−0.386126
$486$	0	0
$487$	−33.8240	−1.53271	−0.766356	−	0.642416i	$-0.777932\pi$
$487$	−33.8240	−1.53271	−0.766356	+	0.642416i	$0.777932\pi$
$488$	0	0
$489$	−20.4402	−0.924336
$490$	0	0
$491$	−9.92529	−0.447922	−0.223961	−	0.974598i	$-0.571899\pi$
$491$	−9.92529	−0.447922	−0.223961	+	0.974598i	$0.571899\pi$
$492$	0	0
$493$	5.74823	0.258887
$494$	0	0
$495$	1.34596	0.0604966
$496$	0	0
$497$	−30.4288	−1.36492
$498$	0	0
$499$	1.68484	0.0754238	0.0377119	−	0.999289i	$-0.487993\pi$
$499$	1.68484	0.0754238	0.0377119	+	0.999289i	$0.487993\pi$
$500$	0	0
$501$	−18.0758	−0.807566
$502$	0	0
$503$	6.19971	0.276431	0.138216	−	0.990402i	$-0.455863\pi$
$503$	6.19971	0.276431	0.138216	+	0.990402i	$0.455863\pi$
$504$	0	0
$505$	−31.7369	−1.41227
$506$	0	0
$507$	−9.57932	−0.425433
$508$	0	0
$509$	−17.1841	−0.761674	−0.380837	−	0.924642i	$-0.624364\pi$
$509$	−17.1841	−0.761674	−0.380837	+	0.924642i	$0.624364\pi$
$510$	0	0
$511$	−11.5343	−0.510249
$512$	0	0
$513$	−1.34596	−0.0594257
$514$	0	0
$515$	−38.9324	−1.71557
$516$	0	0
$517$	−2.25177	−0.0990328
$518$	0	0
$519$	16.4288	0.721146
$520$	0	0
$521$	21.1700	0.927473	0.463737	−	0.885973i	$-0.346508\pi$
$521$	21.1700	0.927473	0.463737	+	0.885973i	$0.346508\pi$
$522$	0	0
$523$	−10.5935	−0.463221	−0.231611	−	0.972809i	$-0.574400\pi$
$523$	−10.5935	−0.463221	−0.231611	+	0.972809i	$0.574400\pi$
$524$	0	0
$525$	20.7298	0.904723
$526$	0	0
$527$	−8.70325	−0.379120
$528$	0	0
$529$	5.57932	0.242579
$530$	0	0
$531$	14.6919	0.637575
$532$	0	0
$533$	−1.84951	−0.0801110
$534$	0	0
$535$	−19.6990	−0.851663
$536$	0	0
$537$	4.10128	0.176983
$538$	0	0
$539$	1.68769	0.0726940
$540$	0	0
$541$	19.6091	0.843059	0.421530	−	0.906815i	$-0.361493\pi$
$541$	19.6091	0.843059	0.421530	+	0.906815i	$0.361493\pi$
$542$	0	0
$543$	16.0000	0.686626
$544$	0	0
$545$	−15.1955	−0.650902
$546$	0	0
$547$	−28.6778	−1.22617	−0.613086	−	0.790016i	$-0.710072\pi$
$547$	−28.6778	−1.22617	−0.613086	+	0.790016i	$0.710072\pi$
$548$	0	0
$549$	7.44724	0.317841
$550$	0	0
$551$	−4.84242	−0.206294
$552$	0	0
$553$	28.0758	1.19390
$554$	0	0
$555$	−11.5343	−0.489606
$556$	0	0
$557$	−10.0645	−0.426445	−0.213222	−	0.977004i	$-0.568396\pi$
$557$	−10.0645	−0.426445	−0.213222	+	0.977004i	$0.568396\pi$
$558$	0	0
$559$	9.70008	0.410270
$560$	0	0
$561$	0.642713	0.0271353
$562$	0	0
$563$	17.1841	0.724225	0.362113	−	0.932134i	$-0.382056\pi$
$563$	17.1841	0.724225	0.362113	+	0.932134i	$0.382056\pi$
$564$	0	0
$565$	15.5722	0.655129
$566$	0	0
$567$	3.34596	0.140517
$568$	0	0
$569$	34.2404	1.43543	0.717717	−	0.696335i	$-0.245187\pi$
$569$	34.2404	1.43543	0.717717	+	0.696335i	$0.245187\pi$
$570$	0	0
$571$	−11.7748	−0.492760	−0.246380	−	0.969173i	$-0.579241\pi$
$571$	−11.7748	−0.492760	−0.246380	+	0.969173i	$0.579241\pi$
$572$	0	0
$573$	−11.5864	−0.484029
$574$	0	0
$575$	−33.1208	−1.38123
$576$	0	0
$577$	−41.5722	−1.73067	−0.865337	−	0.501190i	$-0.832896\pi$
$577$	−41.5722	−1.73067	−0.865337	+	0.501190i	$0.832896\pi$
$578$	0	0
$579$	4.05207	0.168398
$580$	0	0
$581$	20.5793	0.853774
$582$	0	0
$583$	1.60906	0.0666404
$584$	0	0
$585$	6.18838	0.255858
$586$	0	0
$587$	36.1771	1.49319	0.746594	−	0.665280i	$-0.231688\pi$
$587$	36.1771	1.49319	0.746594	+	0.665280i	$0.231688\pi$
$588$	0	0
$589$	7.33179	0.302101
$590$	0	0
$591$	17.4359	0.717218
$592$	0	0
$593$	37.4710	1.53875	0.769374	−	0.638799i	$-0.220568\pi$
$593$	37.4710	1.53875	0.769374	+	0.638799i	$0.220568\pi$
$594$	0	0
$595$	17.8874	0.733312
$596$	0	0
$597$	−7.84951	−0.321259
$598$	0	0
$599$	7.03080	0.287271	0.143635	−	0.989631i	$-0.454121\pi$
$599$	7.03080	0.287271	0.143635	+	0.989631i	$0.454121\pi$
$600$	0	0
$601$	24.4667	0.998018	0.499009	−	0.866597i	$-0.333697\pi$
$601$	24.4667	0.998018	0.499009	+	0.866597i	$0.333697\pi$
$602$	0	0
$603$	7.19547	0.293022
$604$	0	0
$605$	−36.2642	−1.47435
$606$	0	0
$607$	15.2713	0.619841	0.309920	−	0.950762i	$-0.399698\pi$
$607$	15.2713	0.619841	0.309920	+	0.950762i	$0.399698\pi$
$608$	0	0
$609$	12.0379	0.487800
$610$	0	0
$611$	−10.3531	−0.418840
$612$	0	0
$613$	−33.9774	−1.37233	−0.686166	−	0.727445i	$-0.740708\pi$
$613$	−33.9774	−1.37233	−0.686166	+	0.727445i	$0.740708\pi$
$614$	0	0
$615$	−3.34596	−0.134922
$616$	0	0
$617$	27.4217	1.10396	0.551979	−	0.833858i	$-0.313873\pi$
$617$	27.4217	1.10396	0.551979	+	0.833858i	$0.313873\pi$
$618$	0	0
$619$	−18.5528	−0.745698	−0.372849	−	0.927892i	$-0.621619\pi$
$619$	−18.5528	−0.745698	−0.372849	+	0.927892i	$0.621619\pi$
$620$	0	0
$621$	−5.34596	−0.214526
$622$	0	0
$623$	51.4738	2.06225
$624$	0	0
$625$	−17.5935	−0.703740
$626$	0	0
$627$	−0.541434	−0.0216228
$628$	0	0
$629$	−5.50778	−0.219610
$630$	0	0
$631$	10.2518	0.408117	0.204058	−	0.978959i	$-0.434587\pi$
$631$	10.2518	0.408117	0.204058	+	0.978959i	$0.434587\pi$
$632$	0	0
$633$	−26.0900	−1.03698
$634$	0	0
$635$	60.8577	2.41506
$636$	0	0
$637$	7.75955	0.307445
$638$	0	0
$639$	−9.09419	−0.359761
$640$	0	0
$641$	−44.0645	−1.74044	−0.870221	−	0.492662i	$-0.836024\pi$
$641$	−44.0645	−1.74044	−0.870221	+	0.492662i	$0.836024\pi$
$642$	0	0
$643$	16.0758	0.633967	0.316983	−	0.948431i	$-0.397330\pi$
$643$	16.0758	0.633967	0.316983	+	0.948431i	$0.397330\pi$
$644$	0	0
$645$	17.5485	0.690972
$646$	0	0
$647$	33.7227	1.32578	0.662889	−	0.748718i	$-0.269330\pi$
$647$	33.7227	1.32578	0.662889	+	0.748718i	$0.269330\pi$
$648$	0	0
$649$	5.91005	0.231990
$650$	0	0
$651$	−18.2263	−0.714344
$652$	0	0
$653$	−12.0029	−0.469708	−0.234854	−	0.972031i	$-0.575461\pi$
$653$	−12.0029	−0.469708	−0.234854	+	0.972031i	$0.575461\pi$
$654$	0	0
$655$	−14.5177	−0.567254
$656$	0	0
$657$	−3.44724	−0.134490
$658$	0	0
$659$	29.4596	1.14758	0.573792	−	0.819001i	$-0.305472\pi$
$659$	29.4596	1.14758	0.573792	+	0.819001i	$0.305472\pi$
$660$	0	0
$661$	33.2713	1.29410	0.647051	−	0.762447i	$-0.276002\pi$
$661$	33.2713	1.29410	0.647051	+	0.762447i	$0.276002\pi$
$662$	0	0
$663$	2.95502	0.114764
$664$	0	0
$665$	−15.0687	−0.584339
$666$	0	0
$667$	−19.2334	−0.744719
$668$	0	0
$669$	−5.38385	−0.208152
$670$	0	0
$671$	2.99576	0.115650
$672$	0	0
$673$	14.0758	0.542581	0.271291	−	0.962497i	$-0.412550\pi$
$673$	14.0758	0.542581	0.271291	+	0.962497i	$0.412550\pi$
$674$	0	0
$675$	6.19547	0.238464
$676$	0	0
$677$	31.0450	1.19316	0.596578	−	0.802555i	$-0.296527\pi$
$677$	31.0450	1.19316	0.596578	+	0.802555i	$0.296527\pi$
$678$	0	0
$679$	−8.50354	−0.326336
$680$	0	0
$681$	−27.8761	−1.06821
$682$	0	0
$683$	28.9561	1.10797	0.553987	−	0.832525i	$-0.313106\pi$
$683$	28.9561	1.10797	0.553987	+	0.832525i	$0.313106\pi$
$684$	0	0
$685$	77.5733	2.96392
$686$	0	0
$687$	−28.1137	−1.07260
$688$	0	0
$689$	7.39803	0.281842
$690$	0	0
$691$	−24.9182	−0.947933	−0.473966	−	0.880543i	$-0.657178\pi$
$691$	−24.9182	−0.947933	−0.473966	+	0.880543i	$0.657178\pi$
$692$	0	0
$693$	1.34596	0.0511289
$694$	0	0
$695$	28.7061	1.08888
$696$	0	0
$697$	−1.59774	−0.0605185
$698$	0	0
$699$	22.3768	0.846367
$700$	0	0
$701$	1.69901	0.0641709	0.0320854	−	0.999485i	$-0.489785\pi$
$701$	1.69901	0.0641709	0.0320854	+	0.999485i	$0.489785\pi$
$702$	0	0
$703$	4.63986	0.174996
$704$	0	0
$705$	−18.7298	−0.705406
$706$	0	0
$707$	−31.7369	−1.19359
$708$	0	0
$709$	24.3541	0.914638	0.457319	−	0.889303i	$-0.348810\pi$
$709$	24.3541	0.914638	0.457319	+	0.889303i	$0.348810\pi$
$710$	0	0
$711$	8.39094	0.314685
$712$	0	0
$713$	29.1208	1.09058
$714$	0	0
$715$	2.48937	0.0930971
$716$	0	0
$717$	−8.18838	−0.305801
$718$	0	0
$719$	−42.7564	−1.59454	−0.797272	−	0.603620i	$-0.793724\pi$
$719$	−42.7564	−1.59454	−0.797272	+	0.603620i	$0.793724\pi$
$720$	0	0
$721$	−38.9324	−1.44992
$722$	0	0
$723$	10.9437	0.407001
$724$	0	0
$725$	22.2897	0.827817
$726$	0	0
$727$	−30.9182	−1.14669	−0.573346	−	0.819313i	$-0.694355\pi$
$727$	−30.9182	−1.14669	−0.573346	+	0.819313i	$0.694355\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	8.37962	0.309931
$732$	0	0
$733$	−50.5159	−1.86585	−0.932924	−	0.360073i	$-0.882752\pi$
$733$	−50.5159	−1.86585	−0.932924	+	0.360073i	$0.882752\pi$
$734$	0	0
$735$	14.0379	0.517795
$736$	0	0
$737$	2.89448	0.106620
$738$	0	0
$739$	25.1463	0.925020	0.462510	−	0.886614i	$-0.346949\pi$
$739$	25.1463	0.925020	0.462510	+	0.886614i	$0.346949\pi$
$740$	0	0
$741$	−2.48937	−0.0914492
$742$	0	0
$743$	37.8874	1.38995	0.694977	−	0.719032i	$-0.255415\pi$
$743$	37.8874	1.38995	0.694977	+	0.719032i	$0.255415\pi$
$744$	0	0
$745$	19.6990	0.721716
$746$	0	0
$747$	6.15049	0.225035
$748$	0	0
$749$	−19.6990	−0.719786
$750$	0	0
$751$	−27.6091	−1.00747	−0.503734	−	0.863859i	$-0.668041\pi$
$751$	−27.6091	−1.00747	−0.503734	+	0.863859i	$0.668041\pi$
$752$	0	0
$753$	10.1884	0.371285
$754$	0	0
$755$	−13.0071	−0.473376
$756$	0	0
$757$	21.1208	0.767647	0.383823	−	0.923406i	$-0.374607\pi$
$757$	21.1208	0.767647	0.383823	+	0.923406i	$0.374607\pi$
$758$	0	0
$759$	−2.15049	−0.0780579
$760$	0	0
$761$	20.1884	0.731828	0.365914	−	0.930649i	$-0.380756\pi$
$761$	20.1884	0.731828	0.365914	+	0.930649i	$0.380756\pi$
$762$	0	0
$763$	−15.1955	−0.550113
$764$	0	0
$765$	5.34596	0.193284
$766$	0	0
$767$	27.1728	0.981154
$768$	0	0
$769$	42.3191	1.52607	0.763033	−	0.646360i	$-0.223710\pi$
$769$	42.3191	1.52607	0.763033	+	0.646360i	$0.223710\pi$
$770$	0	0
$771$	−6.17706	−0.222461
$772$	0	0
$773$	41.0574	1.47673	0.738365	−	0.674401i	$-0.235598\pi$
$773$	41.0574	1.47673	0.738365	+	0.674401i	$0.235598\pi$
$774$	0	0
$775$	−33.7482	−1.21227
$776$	0	0
$777$	−11.5343	−0.413792
$778$	0	0
$779$	1.34596	0.0482241
$780$	0	0
$781$	−3.65827	−0.130903
$782$	0	0
$783$	3.59774	0.128573
$784$	0	0
$785$	21.7132	0.774977
$786$	0	0
$787$	−24.7411	−0.881926	−0.440963	−	0.897525i	$-0.645363\pi$
$787$	−24.7411	−0.881926	−0.440963	+	0.897525i	$0.645363\pi$
$788$	0	0
$789$	−2.30384	−0.0820188
$790$	0	0
$791$	15.5722	0.553685
$792$	0	0
$793$	13.7737	0.489119
$794$	0	0
$795$	13.3839	0.474676
$796$	0	0
$797$	−48.8435	−1.73013	−0.865063	−	0.501664i	$-0.832721\pi$
$797$	−48.8435	−1.73013	−0.865063	+	0.501664i	$0.832721\pi$
$798$	0	0
$799$	−8.94370	−0.316405
$800$	0	0
$801$	15.3839	0.543562
$802$	0	0
$803$	−1.38670	−0.0489357
$804$	0	0
$805$	−59.8506	−2.10946
$806$	0	0
$807$	−13.1955	−0.464503
$808$	0	0
$809$	−8.31516	−0.292345	−0.146173	−	0.989259i	$-0.546696\pi$
$809$	−8.31516	−0.292345	−0.146173	+	0.989259i	$0.546696\pi$
$810$	0	0
$811$	37.3205	1.31050	0.655249	−	0.755413i	$-0.272564\pi$
$811$	37.3205	1.31050	0.655249	+	0.755413i	$0.272564\pi$
$812$	0	0
$813$	7.13208	0.250133
$814$	0	0
$815$	−68.3920	−2.39567
$816$	0	0
$817$	−7.05915	−0.246968
$818$	0	0
$819$	6.18838	0.216240
$820$	0	0
$821$	19.2475	0.671744	0.335872	−	0.941908i	$-0.390969\pi$
$821$	19.2475	0.671744	0.335872	+	0.941908i	$0.390969\pi$
$822$	0	0
$823$	41.8506	1.45882	0.729410	−	0.684077i	$-0.239795\pi$
$823$	41.8506	1.45882	0.729410	+	0.684077i	$0.239795\pi$
$824$	0	0
$825$	2.49222	0.0867680
$826$	0	0
$827$	−6.08710	−0.211669	−0.105835	−	0.994384i	$-0.533751\pi$
$827$	−6.08710	−0.211669	−0.105835	+	0.994384i	$0.533751\pi$
$828$	0	0
$829$	25.8382	0.897397	0.448699	−	0.893683i	$-0.351888\pi$
$829$	25.8382	0.897397	0.448699	+	0.893683i	$0.351888\pi$
$830$	0	0
$831$	−10.2376	−0.355138
$832$	0	0
$833$	6.70325	0.232254
$834$	0	0
$835$	−60.4809	−2.09303
$836$	0	0
$837$	−5.44724	−0.188284
$838$	0	0
$839$	6.39941	0.220932	0.110466	−	0.993880i	$-0.464766\pi$
$839$	6.39941	0.220932	0.110466	+	0.993880i	$0.464766\pi$
$840$	0	0
$841$	−16.0563	−0.553666
$842$	0	0
$843$	−26.2755	−0.904976
$844$	0	0
$845$	−32.0521	−1.10262
$846$	0	0
$847$	−36.2642	−1.24605
$848$	0	0
$849$	32.4175	1.11257
$850$	0	0
$851$	18.4288	0.631732
$852$	0	0
$853$	−23.5864	−0.807583	−0.403792	−	0.914851i	$-0.632308\pi$
$853$	−23.5864	−0.807583	−0.403792	+	0.914851i	$0.632308\pi$
$854$	0	0
$855$	−4.50354	−0.154018
$856$	0	0
$857$	14.1742	0.484182	0.242091	−	0.970254i	$-0.422167\pi$
$857$	14.1742	0.484182	0.242091	+	0.970254i	$0.422167\pi$
$858$	0	0
$859$	37.2447	1.27077	0.635386	−	0.772195i	$-0.280841\pi$
$859$	37.2447	1.27077	0.635386	+	0.772195i	$0.280841\pi$
$860$	0	0
$861$	−3.34596	−0.114030
$862$	0	0
$863$	−0.946550	−0.0322209	−0.0161105	−	0.999870i	$-0.505128\pi$
$863$	−0.946550	−0.0322209	−0.0161105	+	0.999870i	$0.505128\pi$
$864$	0	0
$865$	54.9703	1.86905
$866$	0	0
$867$	−14.4472	−0.490654
$868$	0	0
$869$	3.37538	0.114502
$870$	0	0
$871$	13.3081	0.450927
$872$	0	0
$873$	−2.54143	−0.0860145
$874$	0	0
$875$	13.3839	0.452457
$876$	0	0
$877$	−57.1321	−1.92921	−0.964607	−	0.263693i	$-0.915059\pi$
$877$	−57.1321	−1.92921	−0.964607	+	0.263693i	$0.915059\pi$
$878$	0	0
$879$	−31.3725	−1.05817
$880$	0	0
$881$	−29.4965	−0.993761	−0.496880	−	0.867819i	$-0.665521\pi$
$881$	−29.4965	−0.993761	−0.496880	+	0.867819i	$0.665521\pi$
$882$	0	0
$883$	−18.7298	−0.630309	−0.315154	−	0.949040i	$-0.602056\pi$
$883$	−18.7298	−0.630309	−0.315154	+	0.949040i	$0.602056\pi$
$884$	0	0
$885$	49.1586	1.65245
$886$	0	0
$887$	4.28966	0.144033	0.0720164	−	0.997403i	$-0.477057\pi$
$887$	4.28966	0.144033	0.0720164	+	0.997403i	$0.477057\pi$
$888$	0	0
$889$	60.8577	2.04110
$890$	0	0
$891$	0.402265	0.0134764
$892$	0	0
$893$	7.53435	0.252127
$894$	0	0
$895$	13.7227	0.458700
$896$	0	0
$897$	−9.88740	−0.330131
$898$	0	0
$899$	−19.5977	−0.653621
$900$	0	0
$901$	6.39094	0.212913
$902$	0	0
$903$	17.5485	0.583978
$904$	0	0
$905$	53.5354	1.77958
$906$	0	0
$907$	−17.1813	−0.570496	−0.285248	−	0.958454i	$-0.592076\pi$
$907$	−17.1813	−0.570496	−0.285248	+	0.958454i	$0.592076\pi$
$908$	0	0
$909$	−9.48513	−0.314602
$910$	0	0
$911$	30.3247	1.00470	0.502351	−	0.864664i	$-0.332469\pi$
$911$	30.3247	1.00470	0.502351	+	0.864664i	$0.332469\pi$
$912$	0	0
$913$	2.47413	0.0818816
$914$	0	0
$915$	24.9182	0.823770
$916$	0	0
$917$	−14.5177	−0.479417
$918$	0	0
$919$	−31.4217	−1.03651	−0.518254	−	0.855227i	$-0.673418\pi$
$919$	−31.4217	−1.03651	−0.518254	+	0.855227i	$0.673418\pi$
$920$	0	0
$921$	3.55985	0.117301
$922$	0	0
$923$	−16.8198	−0.553630
$924$	0	0
$925$	−21.3573	−0.702223
$926$	0	0
$927$	−11.6356	−0.382164
$928$	0	0
$929$	−19.6593	−0.645002	−0.322501	−	0.946569i	$-0.604524\pi$
$929$	−19.6593	−0.645002	−0.322501	+	0.946569i	$0.604524\pi$
$930$	0	0
$931$	−5.64695	−0.185071
$932$	0	0
$933$	−6.80453	−0.222770
$934$	0	0
$935$	2.15049	0.0703286
$936$	0	0
$937$	19.2854	0.630027	0.315014	−	0.949087i	$-0.397991\pi$
$937$	19.2854	0.630027	0.315014	+	0.949087i	$0.397991\pi$
$938$	0	0
$939$	29.7369	0.970427
$940$	0	0
$941$	−44.3626	−1.44618	−0.723090	−	0.690754i	$-0.757279\pi$
$941$	−44.3626	−1.44618	−0.723090	+	0.690754i	$0.757279\pi$
$942$	0	0
$943$	5.34596	0.174089
$944$	0	0
$945$	11.1955	0.364189
$946$	0	0
$947$	17.6622	0.573944	0.286972	−	0.957939i	$-0.407351\pi$
$947$	17.6622	0.573944	0.286972	+	0.957939i	$0.407351\pi$
$948$	0	0
$949$	−6.37570	−0.206964
$950$	0	0
$951$	−7.67351	−0.248831
$952$	0	0
$953$	14.6456	0.474416	0.237208	−	0.971459i	$-0.423768\pi$
$953$	14.6456	0.474416	0.237208	+	0.971459i	$0.423768\pi$
$954$	0	0
$955$	−38.7677	−1.25449
$956$	0	0
$957$	1.44724	0.0467827
$958$	0	0
$959$	77.5733	2.50497
$960$	0	0
$961$	−1.32755	−0.0428242
$962$	0	0
$963$	−5.88740	−0.189719
$964$	0	0
$965$	13.5581	0.436449
$966$	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
$969$	−2.15049	−0.0690838
$970$	0	0
$971$	5.63456	0.180822	0.0904108	−	0.995905i	$-0.471182\pi$
$971$	5.63456	0.180822	0.0904108	+	0.995905i	$0.471182\pi$
$972$	0	0
$973$	28.7061	0.920275
$974$	0	0
$975$	11.4586	0.366968
$976$	0	0
$977$	25.5977	0.818944	0.409472	−	0.912323i	$-0.365713\pi$
$977$	25.5977	0.818944	0.409472	+	0.912323i	$0.365713\pi$
$978$	0	0
$979$	6.18838	0.197782
$980$	0	0
$981$	−4.54143	−0.144997
$982$	0	0
$983$	15.6385	0.498790	0.249395	−	0.968402i	$-0.419768\pi$
$983$	15.6385	0.498790	0.249395	+	0.968402i	$0.419768\pi$
$984$	0	0
$985$	58.3399	1.85887
$986$	0	0
$987$	−18.7298	−0.596177
$988$	0	0
$989$	−28.0379	−0.891553
$990$	0	0
$991$	−30.1742	−0.958515	−0.479258	−	0.877674i	$-0.659094\pi$
$991$	−30.1742	−0.958515	−0.479258	+	0.877674i	$0.659094\pi$
$992$	0	0
$993$	11.0308	0.350052
$994$	0	0
$995$	−26.2642	−0.832630
$996$	0	0
$997$	−1.24754	−0.0395098	−0.0197549	−	0.999805i	$-0.506289\pi$
$997$	−1.24754	−0.0395098	−0.0197549	+	0.999805i	$0.506289\pi$
$998$	0	0
$999$	−3.44724	−0.109066

Twists

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

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

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +3.34596 q^{5} +3.34596 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +3.34596 q^{5} +3.34596 q^{7} +1.00000 q^{9} +0.402265 q^{11} +1.84951 q^{13} +3.34596 q^{15} +1.59774 q^{17} -1.34596 q^{19} +3.34596 q^{21} -5.34596 q^{23} +6.19547 q^{25} +1.00000 q^{27} +3.59774 q^{29} -5.44724 q^{31} +0.402265 q^{33} +11.1955 q^{35} -3.44724 q^{37} +1.84951 q^{39} -1.00000 q^{41} +5.24468 q^{43} +3.34596 q^{45} -5.59774 q^{47} +4.19547 q^{49} +1.59774 q^{51} +4.00000 q^{53} +1.34596 q^{55} -1.34596 q^{57} +14.6919 q^{59} +7.44724 q^{61} +3.34596 q^{63} +6.18838 q^{65} +7.19547 q^{67} -5.34596 q^{69} -9.09419 q^{71} -3.44724 q^{73} +6.19547 q^{75} +1.34596 q^{77} +8.39094 q^{79} +1.00000 q^{81} +6.15049 q^{83} +5.34596 q^{85} +3.59774 q^{87} +15.3839 q^{89} +6.18838 q^{91} -5.44724 q^{93} -4.50354 q^{95} -2.54143 q^{97} +0.402265 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} + 3 q^{9}+O(q^{10}) \) 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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