LMFDB - Embedded newform 6480.2.a.cb.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6480,2,Mod(1,6480)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6480, 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("6480.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6480 = 2^{4} \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6480.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:	$51.7430605098$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.29268.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 9x^{2} + 5x + 16 $ x^4 - x^3 - 9x^2 + 5x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 360)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.85121$ of defining polynomial
Character	$\chi$	$=$	6480.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{5} +0.331895 q^{7} -4.40761 q^{11} +6.98062 q^{13} -3.07139 q^{17} +7.55364 q^{19} -1.66811 q^{23} +1.00000 q^{25} +3.57301 q^{29} -5.64441 q^{31} +0.331895 q^{35} +7.64441 q^{37} +5.49838 q^{41} +7.07139 q^{43} -7.97630 q^{47} -6.88985 q^{49} -5.47900 q^{53} -4.40761 q^{55} +7.38823 q^{59} +0.592395 q^{61} +6.98062 q^{65} -5.22174 q^{67} -8.98062 q^{71} -8.05202 q^{73} -1.46286 q^{77} +10.9968 q^{79} +6.29314 q^{83} -3.07139 q^{85} +11.9060 q^{89} +2.31683 q^{91} +7.55364 q^{95} -1.24544 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{5} + q^{7} - q^{11} + 4 q^{13} + 5 q^{17} - q^{19} - 7 q^{23} + 4 q^{25} + 7 q^{29} + 2 q^{31} + q^{35} + 6 q^{37} + 12 q^{41} + 11 q^{43} - 7 q^{47} + 3 q^{49} + 12 q^{53} - q^{55} - 11 q^{59}+ \cdots + q^{97}+O(q^{100}) $ 4 * q + 4 * q^5 + q^7 - q^11 + 4 * q^13 + 5 * q^17 - q^19 - 7 * q^23 + 4 * q^25 + 7 * q^29 + 2 * q^31 + q^35 + 6 * q^37 + 12 * q^41 + 11 * q^43 - 7 * q^47 + 3 * q^49 + 12 * q^53 - q^55 - 11 * q^59 + 19 * q^61 + 4 * q^65 + 10 * q^67 - 12 * q^71 + 9 * q^73 + 32 * q^77 + 24 * q^79 - 23 * q^83 + 5 * q^85 + 21 * q^89 - 14 * q^91 - q^95 + q^97

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	0.331895	0.125444	0.0627222	−	0.998031i	$-0.480022\pi$
$7$	0.331895	0.125444	0.0627222	+	0.998031i	$0.480022\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−4.40761	−1.32894	−0.664472	−	0.747314i	$-0.731343\pi$
$11$	−4.40761	−1.32894	−0.664472	+	0.747314i	$0.731343\pi$
$12$	0	0
$13$	6.98062	1.93608	0.968038	−	0.250804i	$-0.0806950\pi$
$13$	6.98062	1.93608	0.968038	+	0.250804i	$0.0806950\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−3.07139	−0.744923	−0.372461	−	0.928048i	$-0.621486\pi$
$17$	−3.07139	−0.744923	−0.372461	+	0.928048i	$0.621486\pi$
$18$	0	0
$19$	7.55364	1.73292	0.866461	−	0.499244i	$-0.166389\pi$
$19$	7.55364	1.73292	0.866461	+	0.499244i	$0.166389\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−1.66811	−0.347824	−0.173912	−	0.984761i	$-0.555641\pi$
$23$	−1.66811	−0.347824	−0.173912	+	0.984761i	$0.555641\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	3.57301	0.663492	0.331746	−	0.943369i	$-0.392362\pi$
$29$	3.57301	0.663492	0.331746	+	0.943369i	$0.392362\pi$
$30$	0	0
$31$	−5.64441	−1.01377	−0.506883	−	0.862015i	$-0.669202\pi$
$31$	−5.64441	−1.01377	−0.506883	+	0.862015i	$0.669202\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0.331895	0.0561004
$36$	0	0
$37$	7.64441	1.25673	0.628367	−	0.777917i	$-0.283724\pi$
$37$	7.64441	1.25673	0.628367	+	0.777917i	$0.283724\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	5.49838	0.858703	0.429351	−	0.903138i	$-0.358742\pi$
$41$	5.49838	0.858703	0.429351	+	0.903138i	$0.358742\pi$
$42$	0	0
$43$	7.07139	1.07838	0.539189	−	0.842185i	$-0.318731\pi$
$43$	7.07139	1.07838	0.539189	+	0.842185i	$0.318731\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−7.97630	−1.16346	−0.581732	−	0.813381i	$-0.697625\pi$
$47$	−7.97630	−1.16346	−0.581732	+	0.813381i	$0.697625\pi$
$48$	0	0
$49$	−6.88985	−0.984264
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−5.47900	−0.752599	−0.376299	−	0.926498i	$-0.622804\pi$
$53$	−5.47900	−0.752599	−0.376299	+	0.926498i	$0.622804\pi$
$54$	0	0
$55$	−4.40761	−0.594321
$56$	0	0
$57$	0	0
$58$	0	0
$59$	7.38823	0.961865	0.480933	−	0.876758i	$-0.340298\pi$
$59$	7.38823	0.961865	0.480933	+	0.876758i	$0.340298\pi$
$60$	0	0
$61$	0.592395	0.0758484	0.0379242	−	0.999281i	$-0.487925\pi$
$61$	0.592395	0.0758484	0.0379242	+	0.999281i	$0.487925\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	6.98062	0.865839
$66$	0	0
$67$	−5.22174	−0.637937	−0.318969	−	0.947765i	$-0.603336\pi$
$67$	−5.22174	−0.637937	−0.318969	+	0.947765i	$0.603336\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−8.98062	−1.06580	−0.532902	−	0.846177i	$-0.678898\pi$
$71$	−8.98062	−1.06580	−0.532902	+	0.846177i	$0.678898\pi$
$72$	0	0
$73$	−8.05202	−0.942417	−0.471209	−	0.882022i	$-0.656182\pi$
$73$	−8.05202	−0.942417	−0.471209	+	0.882022i	$0.656182\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−1.46286	−0.166708
$78$	0	0
$79$	10.9968	1.23723	0.618616	−	0.785693i	$-0.287694\pi$
$79$	10.9968	1.23723	0.618616	+	0.785693i	$0.287694\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	6.29314	0.690761	0.345381	−	0.938463i	$-0.387750\pi$
$83$	6.29314	0.690761	0.345381	+	0.938463i	$0.387750\pi$
$84$	0	0
$85$	−3.07139	−0.333140
$86$	0	0
$87$	0	0
$88$	0	0
$89$	11.9060	1.26203	0.631016	−	0.775770i	$-0.282638\pi$
$89$	11.9060	1.26203	0.631016	+	0.775770i	$0.282638\pi$
$90$	0	0
$91$	2.31683	0.242870
$92$	0	0
$93$	0	0
$94$	0	0
$95$	7.55364	0.774987
$96$	0	0
$97$	−1.24544	−0.126455	−0.0632274	−	0.997999i	$-0.520139\pi$
$97$	−1.24544	−0.126455	−0.0632274	+	0.997999i	$0.520139\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	4.66379	0.464064	0.232032	−	0.972708i	$-0.425463\pi$
$101$	4.66379	0.464064	0.232032	+	0.972708i	$0.425463\pi$
$102$	0	0
$103$	4.16541	0.410430	0.205215	−	0.978717i	$-0.434211\pi$
$103$	4.16541	0.410430	0.205215	+	0.978717i	$0.434211\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	6.57409	0.635541	0.317771	−	0.948168i	$-0.397066\pi$
$107$	6.57409	0.635541	0.317771	+	0.948168i	$0.397066\pi$
$108$	0	0
$109$	0.743816	0.0712446	0.0356223	−	0.999365i	$-0.488659\pi$
$109$	0.743816	0.0712446	0.0356223	+	0.999365i	$0.488659\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	16.6250	1.56395	0.781976	−	0.623309i	$-0.214212\pi$
$113$	16.6250	1.56395	0.781976	+	0.623309i	$0.214212\pi$
$114$	0	0
$115$	−1.66811	−0.155552
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−1.01938	−0.0934464
$120$	0	0
$121$	8.42699	0.766090
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−4.79152	−0.425178	−0.212589	−	0.977142i	$-0.568190\pi$
$127$	−4.79152	−0.425178	−0.212589	+	0.977142i	$0.568190\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−19.7797	−1.72816	−0.864080	−	0.503355i	$-0.832099\pi$
$131$	−19.7797	−1.72816	−0.864080	+	0.503355i	$0.832099\pi$
$132$	0	0
$133$	2.50701	0.217385
$134$	0	0
$135$	0	0
$136$	0	0
$137$	18.5343	1.58349	0.791744	−	0.610853i	$-0.209173\pi$
$137$	18.5343	1.58349	0.791744	+	0.610853i	$0.209173\pi$
$138$	0	0
$139$	−5.37959	−0.456291	−0.228146	−	0.973627i	$-0.573266\pi$
$139$	−5.37959	−0.456291	−0.228146	+	0.973627i	$0.573266\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−30.7678	−2.57293
$144$	0	0
$145$	3.57301	0.296723
$146$	0	0
$147$	0	0
$148$	0	0
$149$	11.0359	0.904094	0.452047	−	0.891994i	$-0.350694\pi$
$149$	11.0359	0.904094	0.452047	+	0.891994i	$0.350694\pi$
$150$	0	0
$151$	1.50162	0.122200	0.0611001	−	0.998132i	$-0.480539\pi$
$151$	1.50162	0.122200	0.0611001	+	0.998132i	$0.480539\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−5.64441	−0.453370
$156$	0	0
$157$	−23.6218	−1.88522	−0.942612	−	0.333890i	$-0.891639\pi$
$157$	−23.6218	−1.88522	−0.942612	+	0.333890i	$0.891639\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−0.553635	−0.0436326
$162$	0	0
$163$	9.16217	0.717636	0.358818	−	0.933407i	$-0.383180\pi$
$163$	9.16217	0.717636	0.358818	+	0.933407i	$0.383180\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	0.648726	0.0501999	0.0250999	−	0.999685i	$-0.492010\pi$
$167$	0.648726	0.0501999	0.0250999	+	0.999685i	$0.492010\pi$
$168$	0	0
$169$	35.7291	2.74839
$170$	0	0
$171$	0	0
$172$	0	0
$173$	8.81521	0.670208	0.335104	−	0.942181i	$-0.391229\pi$
$173$	8.81521	0.670208	0.335104	+	0.942181i	$0.391229\pi$
$174$	0	0
$175$	0.331895	0.0250889
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−9.16217	−0.684813	−0.342406	−	0.939552i	$-0.611242\pi$
$179$	−9.16217	−0.684813	−0.342406	+	0.939552i	$0.611242\pi$
$180$	0	0
$181$	13.5730	1.00887	0.504437	−	0.863448i	$-0.331700\pi$
$181$	13.5730	1.00887	0.504437	+	0.863448i	$0.331700\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	7.64441	0.562028
$186$	0	0
$187$	13.5375	0.989960
$188$	0	0
$189$	0	0
$190$	0	0
$191$	10.2694	0.743071	0.371535	−	0.928419i	$-0.378831\pi$
$191$	10.2694	0.743071	0.371535	+	0.928419i	$0.378831\pi$
$192$	0	0
$193$	9.07139	0.652973	0.326487	−	0.945202i	$-0.394135\pi$
$193$	9.07139	0.652973	0.326487	+	0.945202i	$0.394135\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−4.80658	−0.342455	−0.171227	−	0.985232i	$-0.554773\pi$
$197$	−4.80658	−0.342455	−0.171227	+	0.985232i	$0.554773\pi$
$198$	0	0
$199$	18.4596	1.30857	0.654284	−	0.756249i	$-0.272970\pi$
$199$	18.4596	1.30857	0.654284	+	0.756249i	$0.272970\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	1.18586	0.0832314
$204$	0	0
$205$	5.49838	0.384024
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−33.2934	−2.30296
$210$	0	0
$211$	−0.355590	−0.0244798	−0.0122399	−	0.999925i	$-0.503896\pi$
$211$	−0.355590	−0.0244798	−0.0122399	+	0.999925i	$0.503896\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	7.07139	0.482265
$216$	0	0
$217$	−1.87335	−0.127171
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−21.4402	−1.44223
$222$	0	0
$223$	−3.30388	−0.221244	−0.110622	−	0.993863i	$-0.535284\pi$
$223$	−3.30388	−0.221244	−0.110622	+	0.993863i	$0.535284\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	4.73842	0.314500	0.157250	−	0.987559i	$-0.449737\pi$
$227$	4.73842	0.314500	0.157250	+	0.987559i	$0.449737\pi$
$228$	0	0
$229$	19.0606	1.25956	0.629781	−	0.776772i	$-0.283144\pi$
$229$	19.0606	1.25956	0.629781	+	0.776772i	$0.283144\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−13.3721	−0.876035	−0.438017	−	0.898967i	$-0.644319\pi$
$233$	−13.3721	−0.876035	−0.438017	+	0.898967i	$0.644319\pi$
$234$	0	0
$235$	−7.97630	−0.520317
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−19.2975	−1.24825	−0.624124	−	0.781325i	$-0.714544\pi$
$239$	−19.2975	−1.24825	−0.624124	+	0.781325i	$0.714544\pi$
$240$	0	0
$241$	19.9806	1.28706	0.643532	−	0.765419i	$-0.277468\pi$
$241$	19.9806	1.28706	0.643532	+	0.765419i	$0.277468\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−6.88985	−0.440176
$246$	0	0
$247$	52.7291	3.35507
$248$	0	0
$249$	0	0
$250$	0	0
$251$	0.889846	0.0561666	0.0280833	−	0.999606i	$-0.491060\pi$
$251$	0.889846	0.0561666	0.0280833	+	0.999606i	$0.491060\pi$
$252$	0	0
$253$	7.35235	0.462238
$254$	0	0
$255$	0	0
$256$	0	0
$257$	16.7244	1.04324	0.521621	−	0.853177i	$-0.325328\pi$
$257$	16.7244	1.04324	0.521621	+	0.853177i	$0.325328\pi$
$258$	0	0
$259$	2.53714	0.157650
$260$	0	0
$261$	0	0
$262$	0	0
$263$	7.12341	0.439248	0.219624	−	0.975585i	$-0.429517\pi$
$263$	7.12341	0.439248	0.219624	+	0.975585i	$0.429517\pi$
$264$	0	0
$265$	−5.47900	−0.336572
$266$	0	0
$267$	0	0
$268$	0	0
$269$	23.5030	1.43300	0.716502	−	0.697585i	$-0.245742\pi$
$269$	23.5030	1.43300	0.716502	+	0.697585i	$0.245742\pi$
$270$	0	0
$271$	8.49838	0.516240	0.258120	−	0.966113i	$-0.416897\pi$
$271$	8.49838	0.516240	0.258120	+	0.966113i	$0.416897\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−4.40761	−0.265789
$276$	0	0
$277$	−22.2307	−1.33571	−0.667856	−	0.744290i	$-0.732788\pi$
$277$	−22.2307	−1.33571	−0.667856	+	0.744290i	$0.732788\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−11.7406	−0.700384	−0.350192	−	0.936678i	$-0.613884\pi$
$281$	−11.7406	−0.700384	−0.350192	+	0.936678i	$0.613884\pi$
$282$	0	0
$283$	23.7334	1.41080	0.705401	−	0.708808i	$-0.250767\pi$
$283$	23.7334	1.41080	0.705401	+	0.708808i	$0.250767\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1.82488	0.107719
$288$	0	0
$289$	−7.56653	−0.445090
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−20.2694	−1.18415	−0.592077	−	0.805882i	$-0.701692\pi$
$293$	−20.2694	−1.18415	−0.592077	+	0.805882i	$0.701692\pi$
$294$	0	0
$295$	7.38823	0.430159
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−11.6444	−0.673414
$300$	0	0
$301$	2.34696	0.135276
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0.592395	0.0339204
$306$	0	0
$307$	−2.05094	−0.117053	−0.0585267	−	0.998286i	$-0.518640\pi$
$307$	−2.05094	−0.117053	−0.0585267	+	0.998286i	$0.518640\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	19.7958	1.12252	0.561259	−	0.827640i	$-0.310317\pi$
$311$	19.7958	1.12252	0.561259	+	0.827640i	$0.310317\pi$
$312$	0	0
$313$	−0.115545	−0.00653102	−0.00326551	−	0.999995i	$-0.501039\pi$
$313$	−0.115545	−0.00653102	−0.00326551	+	0.999995i	$0.501039\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	23.7872	1.33602	0.668011	−	0.744151i	$-0.267146\pi$
$317$	23.7872	1.33602	0.668011	+	0.744151i	$0.267146\pi$
$318$	0	0
$319$	−15.7484	−0.881743
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−23.2002	−1.29089
$324$	0	0
$325$	6.98062	0.387215
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−2.64729	−0.145950
$330$	0	0
$331$	−34.6250	−1.90316	−0.951582	−	0.307395i	$-0.900543\pi$
$331$	−34.6250	−1.90316	−0.951582	+	0.307395i	$0.900543\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−5.22174	−0.285294
$336$	0	0
$337$	−10.5343	−0.573837	−0.286919	−	0.957955i	$-0.592631\pi$
$337$	−10.5343	−0.573837	−0.286919	+	0.957955i	$0.592631\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	24.8783	1.34724
$342$	0	0
$343$	−4.60997	−0.248915
$344$	0	0
$345$	0	0
$346$	0	0
$347$	5.89309	0.316357	0.158179	−	0.987410i	$-0.449438\pi$
$347$	5.89309	0.316357	0.158179	+	0.987410i	$0.449438\pi$
$348$	0	0
$349$	−23.0133	−1.23187	−0.615936	−	0.787796i	$-0.711222\pi$
$349$	−23.0133	−1.23187	−0.615936	+	0.787796i	$0.711222\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−14.7244	−0.783703	−0.391851	−	0.920029i	$-0.628165\pi$
$353$	−14.7244	−0.783703	−0.391851	+	0.920029i	$0.628165\pi$
$354$	0	0
$355$	−8.98062	−0.476642
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−4.02262	−0.212306	−0.106153	−	0.994350i	$-0.533853\pi$
$359$	−4.02262	−0.212306	−0.106153	+	0.994350i	$0.533853\pi$
$360$	0	0
$361$	38.0574	2.00302
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−8.05202	−0.421462
$366$	0	0
$367$	8.87659	0.463354	0.231677	−	0.972793i	$-0.425579\pi$
$367$	8.87659	0.463354	0.231677	+	0.972793i	$0.425579\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−1.81845	−0.0944093
$372$	0	0
$373$	30.7291	1.59109	0.795545	−	0.605894i	$-0.207185\pi$
$373$	30.7291	1.59109	0.795545	+	0.605894i	$0.207185\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	24.9419	1.28457
$378$	0	0
$379$	4.24004	0.217796	0.108898	−	0.994053i	$-0.465268\pi$
$379$	4.24004	0.217796	0.108898	+	0.994053i	$0.465268\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−32.0879	−1.63961	−0.819807	−	0.572639i	$-0.805920\pi$
$383$	−32.0879	−1.63961	−0.819807	+	0.572639i	$0.805920\pi$
$384$	0	0
$385$	−1.46286	−0.0745543
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−28.3333	−1.43656	−0.718278	−	0.695756i	$-0.755070\pi$
$389$	−28.3333	−1.43656	−0.718278	+	0.695756i	$0.755070\pi$
$390$	0	0
$391$	5.12341	0.259102
$392$	0	0
$393$	0	0
$394$	0	0
$395$	10.9968	0.553307
$396$	0	0
$397$	13.1395	0.659455	0.329728	−	0.944076i	$-0.393043\pi$
$397$	13.1395	0.659455	0.329728	+	0.944076i	$0.393043\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	15.1894	0.758523	0.379262	−	0.925290i	$-0.376178\pi$
$401$	15.1894	0.758523	0.379262	+	0.925290i	$0.376178\pi$
$402$	0	0
$403$	−39.4015	−1.96273
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−33.6935	−1.67013
$408$	0	0
$409$	−10.2013	−0.504421	−0.252211	−	0.967672i	$-0.581158\pi$
$409$	−10.2013	−0.504421	−0.252211	+	0.967672i	$0.581158\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	2.45211	0.120661
$414$	0	0
$415$	6.29314	0.308918
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−12.6412	−0.617562	−0.308781	−	0.951133i	$-0.599921\pi$
$419$	−12.6412	−0.617562	−0.308781	+	0.951133i	$0.599921\pi$
$420$	0	0
$421$	1.00324	0.0488949	0.0244475	−	0.999701i	$-0.492217\pi$
$421$	1.00324	0.0488949	0.0244475	+	0.999701i	$0.492217\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−3.07139	−0.148985
$426$	0	0
$427$	0.196613	0.00951475
$428$	0	0
$429$	0	0
$430$	0	0
$431$	39.0685	1.88186	0.940932	−	0.338596i	$-0.109952\pi$
$431$	39.0685	1.88186	0.940932	+	0.338596i	$0.109952\pi$
$432$	0	0
$433$	−29.6663	−1.42567	−0.712836	−	0.701331i	$-0.752589\pi$
$433$	−29.6663	−1.42567	−0.712836	+	0.701331i	$0.752589\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−12.6003	−0.602752
$438$	0	0
$439$	32.2447	1.53895	0.769477	−	0.638674i	$-0.220517\pi$
$439$	32.2447	1.53895	0.769477	+	0.638674i	$0.220517\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	4.44744	0.211304	0.105652	−	0.994403i	$-0.466307\pi$
$443$	4.44744	0.211304	0.105652	+	0.994403i	$0.466307\pi$
$444$	0	0
$445$	11.9060	0.564398
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−13.0639	−0.616523	−0.308261	−	0.951302i	$-0.599747\pi$
$449$	−13.0639	−0.616523	−0.308261	+	0.951302i	$0.599747\pi$
$450$	0	0
$451$	−24.2347	−1.14117
$452$	0	0
$453$	0	0
$454$	0	0
$455$	2.31683	0.108615
$456$	0	0
$457$	16.3301	0.763889	0.381945	−	0.924185i	$-0.375255\pi$
$457$	16.3301	0.763889	0.381945	+	0.924185i	$0.375255\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	21.5278	1.00265	0.501324	−	0.865260i	$-0.332846\pi$
$461$	21.5278	1.00265	0.501324	+	0.865260i	$0.332846\pi$
$462$	0	0
$463$	22.3168	1.03715	0.518576	−	0.855032i	$-0.326462\pi$
$463$	22.3168	1.03715	0.518576	+	0.855032i	$0.326462\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	36.5203	1.68996	0.844978	−	0.534801i	$-0.179613\pi$
$467$	36.5203	1.68996	0.844978	+	0.534801i	$0.179613\pi$
$468$	0	0
$469$	−1.73307	−0.0800256
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−31.1679	−1.43310
$474$	0	0
$475$	7.55364	0.346585
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−8.95800	−0.409301	−0.204651	−	0.978835i	$-0.565606\pi$
$479$	−8.95800	−0.409301	−0.204651	+	0.978835i	$0.565606\pi$
$480$	0	0
$481$	53.3627	2.43313
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−1.24544	−0.0565523
$486$	0	0
$487$	29.5016	1.33685	0.668423	−	0.743781i	$-0.266970\pi$
$487$	29.5016	1.33685	0.668423	+	0.743781i	$0.266970\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	10.3527	0.467211	0.233606	−	0.972331i	$-0.424948\pi$
$491$	10.3527	0.467211	0.233606	+	0.972331i	$0.424948\pi$
$492$	0	0
$493$	−10.9741	−0.494250
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−2.98062	−0.133699
$498$	0	0
$499$	−22.3850	−1.00209	−0.501045	−	0.865421i	$-0.667051\pi$
$499$	−22.3850	−1.00209	−0.501045	+	0.865421i	$0.667051\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	42.8654	1.91127	0.955637	−	0.294546i	$-0.0951685\pi$
$503$	42.8654	1.91127	0.955637	+	0.294546i	$0.0951685\pi$
$504$	0	0
$505$	4.66379	0.207536
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−7.09077	−0.314293	−0.157147	−	0.987575i	$-0.550229\pi$
$509$	−7.09077	−0.314293	−0.157147	+	0.987575i	$0.550229\pi$
$510$	0	0
$511$	−2.67242	−0.118221
$512$	0	0
$513$	0	0
$514$	0	0
$515$	4.16541	0.183550
$516$	0	0
$517$	35.1564	1.54618
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−39.5863	−1.73431	−0.867153	−	0.498042i	$-0.834053\pi$
$521$	−39.5863	−1.73431	−0.867153	+	0.498042i	$0.834053\pi$
$522$	0	0
$523$	−25.9375	−1.13417	−0.567085	−	0.823659i	$-0.691929\pi$
$523$	−25.9375	−1.13417	−0.567085	+	0.823659i	$0.691929\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	17.3362	0.755177
$528$	0	0
$529$	−20.2174	−0.879018
$530$	0	0
$531$	0	0
$532$	0	0
$533$	38.3821	1.66251
$534$	0	0
$535$	6.57409	0.284223
$536$	0	0
$537$	0	0
$538$	0	0
$539$	30.3677	1.30803
$540$	0	0
$541$	−38.7929	−1.66784	−0.833920	−	0.551886i	$-0.813908\pi$
$541$	−38.7929	−1.66784	−0.833920	+	0.551886i	$0.813908\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0.743816	0.0318616
$546$	0	0
$547$	10.0531	0.429839	0.214920	−	0.976632i	$-0.431051\pi$
$547$	10.0531	0.429839	0.214920	+	0.976632i	$0.431051\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	26.9893	1.14978
$552$	0	0
$553$	3.64977	0.155204
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−23.4316	−0.992829	−0.496415	−	0.868086i	$-0.665350\pi$
$557$	−23.4316	−0.992829	−0.496415	+	0.868086i	$0.665350\pi$
$558$	0	0
$559$	49.3627	2.08782
$560$	0	0
$561$	0	0
$562$	0	0
$563$	41.5160	1.74969	0.874844	−	0.484404i	$-0.160964\pi$
$563$	41.5160	1.74969	0.874844	+	0.484404i	$0.160964\pi$
$564$	0	0
$565$	16.6250	0.699420
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−35.1980	−1.47558	−0.737789	−	0.675031i	$-0.764130\pi$
$569$	−35.1980	−1.47558	−0.737789	+	0.675031i	$0.764130\pi$
$570$	0	0
$571$	1.37959	0.0577342	0.0288671	−	0.999583i	$-0.490810\pi$
$571$	1.37959	0.0577342	0.0288671	+	0.999583i	$0.490810\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−1.66811	−0.0695648
$576$	0	0
$577$	−17.3322	−0.721549	−0.360775	−	0.932653i	$-0.617488\pi$
$577$	−17.3322	−0.721549	−0.360775	+	0.932653i	$0.617488\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	2.08866	0.0866521
$582$	0	0
$583$	24.1493	1.00016
$584$	0	0
$585$	0	0
$586$	0	0
$587$	45.6900	1.88583	0.942914	−	0.333037i	$-0.108073\pi$
$587$	45.6900	1.88583	0.942914	+	0.333037i	$0.108073\pi$
$588$	0	0
$589$	−42.6358	−1.75678
$590$	0	0
$591$	0	0
$592$	0	0
$593$	10.4661	0.429791	0.214896	−	0.976637i	$-0.431059\pi$
$593$	10.4661	0.429791	0.214896	+	0.976637i	$0.431059\pi$
$594$	0	0
$595$	−1.01938	−0.0417905
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−21.1395	−0.863739	−0.431869	−	0.901936i	$-0.642146\pi$
$599$	−21.1395	−0.863739	−0.431869	+	0.901936i	$0.642146\pi$
$600$	0	0
$601$	−10.6684	−0.435174	−0.217587	−	0.976041i	$-0.569819\pi$
$601$	−10.6684	−0.435174	−0.217587	+	0.976041i	$0.569819\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	8.42699	0.342606
$606$	0	0
$607$	−1.30388	−0.0529230	−0.0264615	−	0.999650i	$-0.508424\pi$
$607$	−1.30388	−0.0529230	−0.0264615	+	0.999650i	$0.508424\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−55.6796	−2.25255
$612$	0	0
$613$	30.9741	1.25103	0.625517	−	0.780211i	$-0.284888\pi$
$613$	30.9741	1.25103	0.625517	+	0.780211i	$0.284888\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−2.58380	−0.104020	−0.0520099	−	0.998647i	$-0.516563\pi$
$617$	−2.58380	−0.104020	−0.0520099	+	0.998647i	$0.516563\pi$
$618$	0	0
$619$	0.930756	0.0374102	0.0187051	−	0.999825i	$-0.494046\pi$
$619$	0.930756	0.0374102	0.0187051	+	0.999825i	$0.494046\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	3.95153	0.158315
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−23.4790	−0.936169
$630$	0	0
$631$	−20.6250	−0.821069	−0.410535	−	0.911845i	$-0.634658\pi$
$631$	−20.6250	−0.821069	−0.410535	+	0.911845i	$0.634658\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−4.79152	−0.190145
$636$	0	0
$637$	−48.0954	−1.90561
$638$	0	0
$639$	0	0
$640$	0	0
$641$	23.9031	0.944116	0.472058	−	0.881568i	$-0.343511\pi$
$641$	23.9031	0.944116	0.472058	+	0.881568i	$0.343511\pi$
$642$	0	0
$643$	−2.25726	−0.0890176	−0.0445088	−	0.999009i	$-0.514172\pi$
$643$	−2.25726	−0.0890176	−0.0445088	+	0.999009i	$0.514172\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	12.7262	0.500320	0.250160	−	0.968204i	$-0.419517\pi$
$647$	12.7262	0.500320	0.250160	+	0.968204i	$0.419517\pi$
$648$	0	0
$649$	−32.5644	−1.27826
$650$	0	0
$651$	0	0
$652$	0	0
$653$	1.17080	0.0458170	0.0229085	−	0.999738i	$-0.492707\pi$
$653$	1.17080	0.0458170	0.0229085	+	0.999738i	$0.492707\pi$
$654$	0	0
$655$	−19.7797	−0.772857
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−34.7355	−1.35310	−0.676552	−	0.736395i	$-0.736527\pi$
$659$	−34.7355	−1.35310	−0.676552	+	0.736395i	$0.736527\pi$
$660$	0	0
$661$	35.1460	1.36702	0.683511	−	0.729940i	$-0.260452\pi$
$661$	35.1460	1.36702	0.683511	+	0.729940i	$0.260452\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	2.50701	0.0972177
$666$	0	0
$667$	−5.96017	−0.230779
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−2.61104	−0.100798
$672$	0	0
$673$	−27.2748	−1.05137	−0.525684	−	0.850680i	$-0.676190\pi$
$673$	−27.2748	−1.05137	−0.525684	+	0.850680i	$0.676190\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	9.13204	0.350973	0.175486	−	0.984482i	$-0.443850\pi$
$677$	9.13204	0.350973	0.175486	+	0.984482i	$0.443850\pi$
$678$	0	0
$679$	−0.413354	−0.0158631
$680$	0	0
$681$	0	0
$682$	0	0
$683$	29.0326	1.11090	0.555451	−	0.831549i	$-0.312545\pi$
$683$	29.0326	1.11090	0.555451	+	0.831549i	$0.312545\pi$
$684$	0	0
$685$	18.5343	0.708158
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−38.2468	−1.45709
$690$	0	0
$691$	16.7765	0.638206	0.319103	−	0.947720i	$-0.396618\pi$
$691$	16.7765	0.638206	0.319103	+	0.947720i	$0.396618\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−5.37959	−0.204060
$696$	0	0
$697$	−16.8877	−0.639667
$698$	0	0
$699$	0	0
$700$	0	0
$701$	2.93400	0.110816	0.0554078	−	0.998464i	$-0.482354\pi$
$701$	2.93400	0.110816	0.0554078	+	0.998464i	$0.482354\pi$
$702$	0	0
$703$	57.7431	2.17782
$704$	0	0
$705$	0	0
$706$	0	0
$707$	1.54789	0.0582143
$708$	0	0
$709$	27.8586	1.04625	0.523126	−	0.852256i	$-0.324766\pi$
$709$	27.8586	1.04625	0.523126	+	0.852256i	$0.324766\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	9.41547	0.352612
$714$	0	0
$715$	−30.7678	−1.15065
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−6.95585	−0.259409	−0.129705	−	0.991553i	$-0.541403\pi$
$719$	−6.95585	−0.259409	−0.129705	+	0.991553i	$0.541403\pi$
$720$	0	0
$721$	1.38248	0.0514861
$722$	0	0
$723$	0	0
$724$	0	0
$725$	3.57301	0.132698
$726$	0	0
$727$	5.02046	0.186198	0.0930992	−	0.995657i	$-0.470323\pi$
$727$	5.02046	0.186198	0.0930992	+	0.995657i	$0.470323\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−21.7190	−0.803308
$732$	0	0
$733$	−20.4736	−0.756210	−0.378105	−	0.925763i	$-0.623424\pi$
$733$	−20.4736	−0.756210	−0.378105	+	0.925763i	$0.623424\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	23.0154	0.847782
$738$	0	0
$739$	−51.1614	−1.88200	−0.941002	−	0.338401i	$-0.890114\pi$
$739$	−51.1614	−1.88200	−0.941002	+	0.338401i	$0.890114\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	6.14171	0.225318	0.112659	−	0.993634i	$-0.464063\pi$
$743$	6.14171	0.225318	0.112659	+	0.993634i	$0.464063\pi$
$744$	0	0
$745$	11.0359	0.404323
$746$	0	0
$747$	0	0
$748$	0	0
$749$	2.18191	0.0797251
$750$	0	0
$751$	15.0911	0.550683	0.275342	−	0.961346i	$-0.411209\pi$
$751$	15.0911	0.550683	0.275342	+	0.961346i	$0.411209\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	1.50162	0.0546496
$756$	0	0
$757$	−2.87659	−0.104551	−0.0522757	−	0.998633i	$-0.516647\pi$
$757$	−2.87659	−0.104551	−0.0522757	+	0.998633i	$0.516647\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	0.238955	0.00866212	0.00433106	−	0.999991i	$-0.498621\pi$
$761$	0.238955	0.00866212	0.00433106	+	0.999991i	$0.498621\pi$
$762$	0	0
$763$	0.246869	0.00893724
$764$	0	0
$765$	0	0
$766$	0	0
$767$	51.5744	1.86224
$768$	0	0
$769$	−3.85859	−0.139144	−0.0695722	−	0.997577i	$-0.522163\pi$
$769$	−3.85859	−0.139144	−0.0695722	+	0.997577i	$0.522163\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−5.22355	−0.187878	−0.0939390	−	0.995578i	$-0.529946\pi$
$773$	−5.22355	−0.187878	−0.0939390	+	0.995578i	$0.529946\pi$
$774$	0	0
$775$	−5.64441	−0.202753
$776$	0	0
$777$	0	0
$778$	0	0
$779$	41.5328	1.48807
$780$	0	0
$781$	39.5830	1.41639
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−23.6218	−0.843098
$786$	0	0
$787$	21.7636	0.775787	0.387893	−	0.921704i	$-0.373203\pi$
$787$	21.7636	0.775787	0.387893	+	0.921704i	$0.373203\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	5.51776	0.196189
$792$	0	0
$793$	4.13528	0.146848
$794$	0	0
$795$	0	0
$796$	0	0
$797$	25.4456	0.901329	0.450665	−	0.892693i	$-0.351187\pi$
$797$	25.4456	0.901329	0.450665	+	0.892693i	$0.351187\pi$
$798$	0	0
$799$	24.4984	0.866690
$800$	0	0
$801$	0	0
$802$	0	0
$803$	35.4901	1.25242
$804$	0	0
$805$	−0.553635	−0.0195131
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−11.0854	−0.389741	−0.194871	−	0.980829i	$-0.562429\pi$
$809$	−11.0854	−0.389741	−0.194871	+	0.980829i	$0.562429\pi$
$810$	0	0
$811$	−42.4869	−1.49192	−0.745958	−	0.665993i	$-0.768008\pi$
$811$	−42.4869	−1.49192	−0.745958	+	0.665993i	$0.768008\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	9.16217	0.320937
$816$	0	0
$817$	53.4147	1.86875
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−30.1345	−1.05170	−0.525851	−	0.850577i	$-0.676253\pi$
$821$	−30.1345	−1.05170	−0.525851	+	0.850577i	$0.676253\pi$
$822$	0	0
$823$	−21.6218	−0.753690	−0.376845	−	0.926276i	$-0.622991\pi$
$823$	−21.6218	−0.753690	−0.376845	+	0.926276i	$0.622991\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1.82488	0.0634574	0.0317287	−	0.999497i	$-0.489899\pi$
$827$	1.82488	0.0634574	0.0317287	+	0.999497i	$0.489899\pi$
$828$	0	0
$829$	30.4374	1.05713	0.528567	−	0.848892i	$-0.322730\pi$
$829$	30.4374	1.05713	0.528567	+	0.848892i	$0.322730\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	21.1614	0.733200
$834$	0	0
$835$	0.648726	0.0224501
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−33.5669	−1.15886	−0.579429	−	0.815023i	$-0.696724\pi$
$839$	−33.5669	−1.15886	−0.579429	+	0.815023i	$0.696724\pi$
$840$	0	0
$841$	−16.2336	−0.559778
$842$	0	0
$843$	0	0
$844$	0	0
$845$	35.7291	1.22912
$846$	0	0
$847$	2.79687	0.0961016
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−12.7517	−0.437122
$852$	0	0
$853$	−23.7549	−0.813353	−0.406676	−	0.913572i	$-0.633312\pi$
$853$	−23.7549	−0.813353	−0.406676	+	0.913572i	$0.633312\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−26.0104	−0.888497	−0.444249	−	0.895904i	$-0.646529\pi$
$857$	−26.0104	−0.888497	−0.444249	+	0.895904i	$0.646529\pi$
$858$	0	0
$859$	50.7671	1.73215	0.866075	−	0.499914i	$-0.166635\pi$
$859$	50.7671	1.73215	0.866075	+	0.499914i	$0.166635\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−32.9343	−1.12110	−0.560548	−	0.828122i	$-0.689410\pi$
$863$	−32.9343	−1.12110	−0.560548	+	0.828122i	$0.689410\pi$
$864$	0	0
$865$	8.81521	0.299726
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−48.4694	−1.64421
$870$	0	0
$871$	−36.4510	−1.23509
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0.331895	0.0112201
$876$	0	0
$877$	17.1234	0.578216	0.289108	−	0.957296i	$-0.406641\pi$
$877$	17.1234	0.578216	0.289108	+	0.957296i	$0.406641\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−46.7617	−1.57544	−0.787721	−	0.616032i	$-0.788739\pi$
$881$	−46.7617	−1.57544	−0.787721	+	0.616032i	$0.788739\pi$
$882$	0	0
$883$	21.1356	0.711269	0.355635	−	0.934625i	$-0.384265\pi$
$883$	21.1356	0.711269	0.355635	+	0.934625i	$0.384265\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	45.9774	1.54377	0.771885	−	0.635763i	$-0.219314\pi$
$887$	45.9774	1.54377	0.771885	+	0.635763i	$0.219314\pi$
$888$	0	0
$889$	−1.59028	−0.0533362
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−60.2501	−2.01619
$894$	0	0
$895$	−9.16217	−0.306258
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−20.1676	−0.672626
$900$	0	0
$901$	16.8282	0.560628
$902$	0	0
$903$	0	0
$904$	0	0
$905$	13.5730	0.451182
$906$	0	0
$907$	6.58808	0.218754	0.109377	−	0.994000i	$-0.465114\pi$
$907$	6.58808	0.218754	0.109377	+	0.994000i	$0.465114\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	23.7711	0.787570	0.393785	−	0.919202i	$-0.371165\pi$
$911$	23.7711	0.787570	0.393785	+	0.919202i	$0.371165\pi$
$912$	0	0
$913$	−27.7377	−0.917982
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−6.56477	−0.216788
$918$	0	0
$919$	−46.1305	−1.52171	−0.760853	−	0.648924i	$-0.775219\pi$
$919$	−46.1305	−1.52171	−0.760853	+	0.648924i	$0.775219\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−62.6903	−2.06348
$924$	0	0
$925$	7.64441	0.251347
$926$	0	0
$927$	0	0
$928$	0	0
$929$	22.6164	0.742020	0.371010	−	0.928629i	$-0.379012\pi$
$929$	22.6164	0.742020	0.371010	+	0.928629i	$0.379012\pi$
$930$	0	0
$931$	−52.0434	−1.70565
$932$	0	0
$933$	0	0
$934$	0	0
$935$	13.5375	0.442723
$936$	0	0
$937$	−47.5830	−1.55447	−0.777235	−	0.629211i	$-0.783378\pi$
$937$	−47.5830	−1.55447	−0.777235	+	0.629211i	$0.783378\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−42.9981	−1.40170	−0.700850	−	0.713309i	$-0.747196\pi$
$941$	−42.9981	−1.40170	−0.700850	+	0.713309i	$0.747196\pi$
$942$	0	0
$943$	−9.17188	−0.298677
$944$	0	0
$945$	0	0
$946$	0	0
$947$	48.0548	1.56157	0.780786	−	0.624798i	$-0.214819\pi$
$947$	48.0548	1.56157	0.780786	+	0.624798i	$0.214819\pi$
$948$	0	0
$949$	−56.2081	−1.82459
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−39.0413	−1.26467	−0.632335	−	0.774695i	$-0.717904\pi$
$953$	−39.0413	−1.26467	−0.632335	+	0.774695i	$0.717904\pi$
$954$	0	0
$955$	10.2694	0.332311
$956$	0	0
$957$	0	0
$958$	0	0
$959$	6.15142	0.198640
$960$	0	0
$961$	0.859361	0.0277213
$962$	0	0
$963$	0	0
$964$	0	0
$965$	9.07139	0.292019
$966$	0	0
$967$	2.04847	0.0658742	0.0329371	−	0.999457i	$-0.489514\pi$
$967$	2.04847	0.0658742	0.0329371	+	0.999457i	$0.489514\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−27.4068	−0.879527	−0.439764	−	0.898114i	$-0.644938\pi$
$971$	−27.4068	−0.879527	−0.439764	+	0.898114i	$0.644938\pi$
$972$	0	0
$973$	−1.78546	−0.0572392
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−4.09828	−0.131116	−0.0655578	−	0.997849i	$-0.520883\pi$
$977$	−4.09828	−0.131116	−0.0655578	+	0.997849i	$0.520883\pi$
$978$	0	0
$979$	−52.4769	−1.67717
$980$	0	0
$981$	0	0
$982$	0	0
$983$	6.11807	0.195136	0.0975680	−	0.995229i	$-0.468894\pi$
$983$	6.11807	0.195136	0.0975680	+	0.995229i	$0.468894\pi$
$984$	0	0
$985$	−4.80658	−0.153150
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−11.7958	−0.375086
$990$	0	0
$991$	−14.9881	−0.476114	−0.238057	−	0.971251i	$-0.576510\pi$
$991$	−14.9881	−0.476114	−0.238057	+	0.971251i	$0.576510\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	18.4596	0.585209
$996$	0	0
$997$	−13.4963	−0.427431	−0.213715	−	0.976896i	$-0.568557\pi$
$997$	−13.4963	−0.427431	−0.213715	+	0.976896i	$0.568557\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6480.2.a.cb.1.3		4
3.2	odd	2		6480.2.a.bz.1.3		4
4.3	odd	2		3240.2.a.u.1.2		4
9.2	odd	6		2160.2.q.l.1441.2		8
9.4	even	3		720.2.q.l.241.4		8
9.5	odd	6		2160.2.q.l.721.2		8
9.7	even	3		720.2.q.l.481.4		8
12.11	even	2		3240.2.a.s.1.2		4
36.7	odd	6		360.2.q.e.121.1	✓	8
36.11	even	6		1080.2.q.e.361.3		8
36.23	even	6		1080.2.q.e.721.3		8
36.31	odd	6		360.2.q.e.241.1	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
360.2.q.e.121.1	✓	8	36.7	odd	6
360.2.q.e.241.1	yes	8	36.31	odd	6
720.2.q.l.241.4		8	9.4	even	3
720.2.q.l.481.4		8	9.7	even	3
1080.2.q.e.361.3		8	36.11	even	6
1080.2.q.e.721.3		8	36.23	even	6
2160.2.q.l.721.2		8	9.5	odd	6
2160.2.q.l.1441.2		8	9.2	odd	6
3240.2.a.s.1.2		4	12.11	even	2
3240.2.a.u.1.2		4	4.3	odd	2
6480.2.a.bz.1.3		4	3.2	odd	2
6480.2.a.cb.1.3		4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 6480 = 2^{4} \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6480.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{5} +0.331895 q^{7} -4.40761 q^{11} +6.98062 q^{13} -3.07139 q^{17} +7.55364 q^{19} -1.66811 q^{23} +1.00000 q^{25} +3.57301 q^{29} -5.64441 q^{31} +0.331895 q^{35} +7.64441 q^{37} +5.49838 q^{41} +7.07139 q^{43} -7.97630 q^{47} -6.88985 q^{49} -5.47900 q^{53} -4.40761 q^{55} +7.38823 q^{59} +0.592395 q^{61} +6.98062 q^{65} -5.22174 q^{67} -8.98062 q^{71} -8.05202 q^{73} -1.46286 q^{77} +10.9968 q^{79} +6.29314 q^{83} -3.07139 q^{85} +11.9060 q^{89} +2.31683 q^{91} +7.55364 q^{95} -1.24544 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{5} + q^{7} - q^{11} + 4 q^{13} + 5 q^{17} - q^{19} - 7 q^{23} + 4 q^{25} + 7 q^{29} + 2 q^{31} + q^{35} + 6 q^{37} + 12 q^{41} + 11 q^{43} - 7 q^{47} + 3 q^{49} + 12 q^{53} - q^{55} - 11 q^{59}+ \cdots + q^{97}+O(q^{100}) \) 4 * q + 4 * q^5 + q^7 - q^11 + 4 * q^13 + 5 * q^17 - q^19 - 7 * q^23 + 4 * q^25 + 7 * q^29 + 2 * q^31 + q^35 + 6 * q^37 + 12 * q^41 + 11 * q^43 - 7 * q^47 + 3 * q^49 + 12 * q^53 - q^55 - 11 * q^59 + 19 * q^61 + 4 * q^65 + 10 * q^67 - 12 * q^71 + 9 * q^73 + 32 * q^77 + 24 * q^79 - 23 * q^83 + 5 * q^85 + 21 * q^89 - 14 * q^91 - q^95 + q^97

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