LMFDB - Embedded newform 5796.2.k.b.5473.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5796,2,Mod(5473,5796)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5796, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("5796.5473");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5796 = 2^{2} \cdot 3^{2} \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5796.k (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$46.2812930115$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 2x^{14} + 68x^{12} - 46x^{10} + 4950x^{8} - 2254x^{6} + 163268x^{4} - 235298x^{2} + 5764801 $ x^16 - 2x^14 + 68x^12 - 46x^10 + 4950x^8 - 2254x^6 + 163268x^4 - 235298*x^2 + 5764801
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 644)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			5473.9
Root			$2.33151 - 1.25062i$ of defining polynomial
Character	$\chi$	$=$	5796.5473
Dual form			5796.2.k.b.5473.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.529536 q^{5} +(-2.33151 - 1.25062i) q^{7} +O(q^{10})$	$q+0.529536 q^{5} +(-2.33151 - 1.25062i) q^{7} +0.302618i q^{11} -2.16040i q^{13} +6.13811 q^{17} -1.42940 q^{19} +(-4.10651 + 2.47721i) q^{23} -4.71959 q^{25} +7.35686 q^{29} +5.16943i q^{31} +(-1.23462 - 0.662249i) q^{35} +8.15959i q^{37} +5.58926i q^{41} -2.01310i q^{43} +10.2955i q^{47} +(3.87189 + 5.83168i) q^{49} -7.71951i q^{53} +0.160247i q^{55} -5.38571i q^{59} +5.56289 q^{61} -1.14401i q^{65} -15.3126i q^{67} -1.63727 q^{71} -3.85766i q^{73} +(0.378461 - 0.705557i) q^{77} -14.2673i q^{79} -4.75439 q^{83} +3.25035 q^{85} +12.6009 q^{89} +(-2.70184 + 5.03700i) q^{91} -0.756921 q^{95} +0.620903 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q + 16 q^{23} + 8 q^{25} + 4 q^{35} + 4 q^{49} + 8 q^{71} + 28 q^{77} + 16 q^{85} - 56 q^{95}+O(q^{100}) $ 16 * q + 16 * q^23 + 8 * q^25 + 4 * q^35 + 4 * q^49 + 8 * q^71 + 28 * q^77 + 16 * q^85 - 56 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5796\mathbb{Z}\right)^\times$.

$n$	$829$	$1289$	$2899$	$4789$
$\chi(n)$	$-1$	$1$	$1$	$-1$

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
$5$	0.529536			0.236816			0.118408	−	0.992965i	$-0.462221\pi$
$5$	0.529536			0.236816			0.118408	+	0.992965i	$0.462221\pi$
$6$		0			0
$7$	−2.33151	−	1.25062i	−0.881228	−	0.472691i
$7$	−2.33151	−	1.25062i	−0.881228	−	0.472691i
$8$		0			0
$9$		0			0
$10$		0			0
$11$			0.302618i			0.0912428i	0.998959	+	0.0456214i	$0.0145268\pi$
$11$			0.302618i			0.0912428i	−0.998959	+	0.0456214i	$0.985473\pi$
$12$		0			0
$13$		−	2.16040i		−	0.599187i	−0.954067	−	0.299594i	$-0.903149\pi$
$13$		−	2.16040i		−	0.599187i	0.954067	−	0.299594i	$-0.0968511\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	6.13811			1.48871			0.744355	−	0.667784i	$-0.232757\pi$
$17$	6.13811			1.48871			0.744355	+	0.667784i	$0.232757\pi$
$18$		0			0
$19$	−1.42940			−0.327928			−0.163964	−	0.986466i	$-0.552428\pi$
$19$	−1.42940			−0.327928			−0.163964	+	0.986466i	$0.552428\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−4.10651	+	2.47721i	−0.856267	+	0.516534i
$23$	−4.10651	+	2.47721i	−0.856267	+	0.516534i
$24$		0			0
$25$	−4.71959			−0.943918
$26$		0			0
$27$		0			0
$28$		0			0
$29$	7.35686			1.36614			0.683068	−	0.730355i	$-0.260645\pi$
$29$	7.35686			1.36614			0.683068	+	0.730355i	$0.260645\pi$
$30$		0			0
$31$			5.16943i			0.928457i	0.885716	+	0.464228i	$0.153668\pi$
$31$			5.16943i			0.928457i	−0.885716	+	0.464228i	$0.846332\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−1.23462	−	0.662249i	−0.208689	−	0.111941i
$36$		0			0
$37$			8.15959i			1.34143i	0.741716	+	0.670714i	$0.234012\pi$
$37$			8.15959i			1.34143i	−0.741716	+	0.670714i	$0.765988\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			5.58926i			0.872896i	0.899729	+	0.436448i	$0.143764\pi$
$41$			5.58926i			0.872896i	−0.899729	+	0.436448i	$0.856236\pi$
$42$		0			0
$43$		−	2.01310i		−	0.306994i	−0.988149	−	0.153497i	$-0.950946\pi$
$43$		−	2.01310i		−	0.306994i	0.988149	−	0.153497i	$-0.0490536\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			10.2955i			1.50176i	0.660439	+	0.750880i	$0.270370\pi$
$47$			10.2955i			1.50176i	−0.660439	+	0.750880i	$0.729630\pi$
$48$		0			0
$49$	3.87189	+	5.83168i	0.553127	+	0.833097i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	7.71951i		−	1.06036i	−0.847886	−	0.530178i	$-0.822125\pi$
$53$		−	7.71951i		−	1.06036i	0.847886	−	0.530178i	$-0.177875\pi$
$54$		0			0
$55$			0.160247i			0.0216077i
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	5.38571i		−	0.701160i	−0.936533	−	0.350580i	$-0.885985\pi$
$59$		−	5.38571i		−	0.701160i	0.936533	−	0.350580i	$-0.114015\pi$
$60$		0			0
$61$	5.56289			0.712255			0.356128	−	0.934437i	$-0.384097\pi$
$61$	5.56289			0.712255			0.356128	+	0.934437i	$0.384097\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	1.14401i		−	0.141897i
$66$		0			0
$67$		−	15.3126i		−	1.87074i	−0.353675	−	0.935368i	$-0.615068\pi$
$67$		−	15.3126i		−	1.87074i	0.353675	−	0.935368i	$-0.384932\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−1.63727			−0.194308			−0.0971542	−	0.995269i	$-0.530974\pi$
$71$	−1.63727			−0.194308			−0.0971542	+	0.995269i	$0.530974\pi$
$72$		0			0
$73$		−	3.85766i		−	0.451504i	−0.974185	−	0.225752i	$-0.927516\pi$
$73$		−	3.85766i		−	0.451504i	0.974185	−	0.225752i	$-0.0724840\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	0.378461	−	0.705557i	0.0431296	−	0.0804057i
$78$		0			0
$79$		−	14.2673i		−	1.60520i	−0.596518	−	0.802600i	$-0.703450\pi$
$79$		−	14.2673i		−	1.60520i	0.596518	−	0.802600i	$-0.296550\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−4.75439			−0.521862			−0.260931	−	0.965357i	$-0.584030\pi$
$83$	−4.75439			−0.521862			−0.260931	+	0.965357i	$0.584030\pi$
$84$		0			0
$85$	3.25035			0.352550
$86$		0			0
$87$		0			0
$88$		0			0
$89$	12.6009			1.33569			0.667845	−	0.744301i	$-0.267217\pi$
$89$	12.6009			1.33569			0.667845	+	0.744301i	$0.267217\pi$
$90$		0			0
$91$	−2.70184	+	5.03700i	−0.283230	+	0.528021i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−0.756921			−0.0776585
$96$		0			0
$97$	0.620903			0.0630431			0.0315216	−	0.999503i	$-0.489965\pi$
$97$	0.620903			0.0630431			0.0315216	+	0.999503i	$0.489965\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			3.42886i			0.341185i	0.985342	+	0.170592i	$0.0545681\pi$
$101$			3.42886i			0.341185i	−0.985342	+	0.170592i	$0.945432\pi$
$102$		0			0
$103$	16.6603			1.64159			0.820793	−	0.571226i	$-0.193532\pi$
$103$	16.6603			1.64159			0.820793	+	0.571226i	$0.193532\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			6.31165i			0.610170i	0.952325	+	0.305085i	$0.0986850\pi$
$107$			6.31165i			0.610170i	−0.952325	+	0.305085i	$0.901315\pi$
$108$		0			0
$109$		−	13.3298i		−	1.27676i	−0.769720	−	0.638381i	$-0.779604\pi$
$109$		−	13.3298i		−	1.27676i	0.769720	−	0.638381i	$-0.220396\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			9.46876i			0.890746i	0.895345	+	0.445373i	$0.146929\pi$
$113$			9.46876i			0.890746i	−0.895345	+	0.445373i	$0.853071\pi$
$114$		0			0
$115$	−2.17455	+	1.31177i	−0.202777	+	0.122323i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−14.3111	−	7.67645i	−1.31189	−	0.703699i
$120$		0			0
$121$	10.9084			0.991675
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−5.14688			−0.460351
$126$		0			0
$127$	9.07645			0.805405			0.402703	−	0.915331i	$-0.368071\pi$
$127$	9.07645			0.805405			0.402703	+	0.915331i	$0.368071\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			8.97105i			0.783804i	0.920007	+	0.391902i	$0.128183\pi$
$131$			8.97105i			0.783804i	−0.920007	+	0.391902i	$0.871817\pi$
$132$		0			0
$133$	3.33267	+	1.78764i	0.288979	+	0.155008i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			15.6153i			1.33410i	0.745012	+	0.667051i	$0.232444\pi$
$137$			15.6153i			1.33410i	−0.745012	+	0.667051i	$0.767556\pi$
$138$		0			0
$139$			8.69873i			0.737817i	0.929466	+	0.368908i	$0.120268\pi$
$139$			8.69873i			0.737817i	−0.929466	+	0.368908i	$0.879732\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	0.653776			0.0546715
$144$		0			0
$145$	3.89573			0.323522
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	12.0206i		−	0.984768i	−0.870378	−	0.492384i	$-0.836126\pi$
$149$		−	12.0206i		−	0.984768i	0.870378	−	0.492384i	$-0.163874\pi$
$150$		0			0
$151$	−7.34959			−0.598101			−0.299051	−	0.954237i	$-0.596670\pi$
$151$	−7.34959			−0.598101			−0.299051	+	0.954237i	$0.596670\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			2.73740i			0.219873i
$156$		0			0
$157$	10.0605			0.802913			0.401456	−	0.915878i	$-0.368504\pi$
$157$	10.0605			0.802913			0.401456	+	0.915878i	$0.368504\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	12.6724	−	0.639951i	0.998727	−	0.0504352i
$162$		0			0
$163$	12.3026			0.963615			0.481808	−	0.876277i	$-0.339980\pi$
$163$	12.3026			0.963615			0.481808	+	0.876277i	$0.339980\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	3.15545i		−	0.244176i	−0.992519	−	0.122088i	$-0.961041\pi$
$167$		−	3.15545i		−	0.244176i	0.992519	−	0.122088i	$-0.0389590\pi$
$168$		0			0
$169$	8.33267			0.640975
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	17.9421i		−	1.36411i	−0.731300	−	0.682056i	$-0.761086\pi$
$173$		−	17.9421i		−	1.36411i	0.731300	−	0.682056i	$-0.238914\pi$
$174$		0			0
$175$	11.0038	+	5.90242i	0.831808	+	0.446181i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	18.7719			1.40307			0.701537	−	0.712633i	$-0.252497\pi$
$179$	18.7719			1.40307			0.701537	+	0.712633i	$0.252497\pi$
$180$		0			0
$181$	7.93785			0.590015			0.295008	−	0.955495i	$-0.404678\pi$
$181$	7.93785			0.590015			0.295008	+	0.955495i	$0.404678\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			4.32080i			0.317672i
$186$		0			0
$187$			1.85750i			0.135834i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	13.7673i		−	0.996168i	−0.867129	−	0.498084i	$-0.834037\pi$
$191$		−	13.7673i		−	0.996168i	0.867129	−	0.498084i	$-0.165963\pi$
$192$		0			0
$193$	−4.12343			−0.296811			−0.148405	−	0.988927i	$-0.547414\pi$
$193$	−4.12343			−0.296811			−0.148405	+	0.988927i	$0.547414\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	9.54569			0.680103			0.340051	−	0.940407i	$-0.389556\pi$
$197$	9.54569			0.680103			0.340051	+	0.940407i	$0.389556\pi$
$198$		0			0
$199$	15.1395			1.07321			0.536606	−	0.843833i	$-0.319706\pi$
$199$	15.1395			1.07321			0.536606	+	0.843833i	$0.319706\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−17.1526	−	9.20065i	−1.20388	−	0.645759i
$204$		0			0
$205$			2.95972i			0.206716i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	0.432563i		−	0.0299210i
$210$		0			0
$211$	−10.5007			−0.722898			−0.361449	−	0.932392i	$-0.617718\pi$
$211$	−10.5007			−0.722898			−0.361449	+	0.932392i	$0.617718\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	1.06601i		−	0.0727011i
$216$		0			0
$217$	6.46500	−	12.0526i	0.438873	−	0.818182i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	13.2608i		−	0.892016i
$222$		0			0
$223$			23.4742i			1.57195i	0.618258	+	0.785975i	$0.287839\pi$
$223$			23.4742i			1.57195i	−0.618258	+	0.785975i	$0.712161\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	13.3398			0.885392			0.442696	−	0.896672i	$-0.354022\pi$
$227$	13.3398			0.885392			0.442696	+	0.896672i	$0.354022\pi$
$228$		0			0
$229$	2.07246			0.136952			0.0684761	−	0.997653i	$-0.478186\pi$
$229$	2.07246			0.136952			0.0684761	+	0.997653i	$0.478186\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−3.37378			−0.221024			−0.110512	−	0.993875i	$-0.535249\pi$
$233$	−3.37378			−0.221024			−0.110512	+	0.993875i	$0.535249\pi$
$234$		0			0
$235$			5.45187i			0.355640i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−1.58889			−0.102777			−0.0513884	−	0.998679i	$-0.516365\pi$
$239$	−1.58889			−0.102777			−0.0513884	+	0.998679i	$0.516365\pi$
$240$		0			0
$241$	25.6222			1.65047			0.825236	−	0.564787i	$-0.191042\pi$
$241$	25.6222			1.65047			0.825236	+	0.564787i	$0.191042\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	2.05031	+	3.08808i	0.130989	+	0.197290i
$246$		0			0
$247$			3.08808i			0.196490i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	19.6105			1.23780			0.618900	−	0.785469i	$-0.287578\pi$
$251$	19.6105			1.23780			0.618900	+	0.785469i	$0.287578\pi$
$252$		0			0
$253$	−0.749648	−	1.24270i	−0.0471300	−	0.0781281i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			23.0122i			1.43546i	0.696321	+	0.717730i	$0.254819\pi$
$257$			23.0122i			1.43546i	−0.696321	+	0.717730i	$0.745181\pi$
$258$		0			0
$259$	10.2046	−	19.0242i	0.634081	−	1.18211i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			31.8957i			1.96677i	0.181528	+	0.983386i	$0.441896\pi$
$263$			31.8957i			1.96677i	−0.181528	+	0.983386i	$0.558104\pi$
$264$		0			0
$265$		−	4.08776i		−	0.251109i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			20.2489i			1.23460i	0.786728	+	0.617299i	$0.211773\pi$
$269$			20.2489i			1.23460i	−0.786728	+	0.617299i	$0.788227\pi$
$270$		0			0
$271$			3.91479i			0.237806i	0.992906	+	0.118903i	$0.0379378\pi$
$271$			3.91479i			0.237806i	−0.992906	+	0.118903i	$0.962062\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		−	1.42823i		−	0.0861257i
$276$		0			0
$277$	−22.7960			−1.36968			−0.684841	−	0.728693i	$-0.740128\pi$
$277$	−22.7960			−1.36968			−0.684841	+	0.728693i	$0.740128\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	20.9803i		−	1.25158i	−0.779992	−	0.625789i	$-0.784777\pi$
$281$		−	20.9803i		−	1.25158i	0.779992	−	0.625789i	$-0.215223\pi$
$282$		0			0
$283$	−16.5295			−0.982575			−0.491287	−	0.870998i	$-0.663474\pi$
$283$	−16.5295			−0.982575			−0.491287	+	0.870998i	$0.663474\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	6.99005	−	13.0314i	0.412610	−	0.769221i
$288$		0			0
$289$	20.6764			1.21626
$290$		0			0
$291$		0			0
$292$		0			0
$293$	11.7512			0.686510			0.343255	−	0.939242i	$-0.388470\pi$
$293$	11.7512			0.686510			0.343255	+	0.939242i	$0.388470\pi$
$294$		0			0
$295$		−	2.85193i		−	0.166046i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	5.35176	+	8.87171i	0.309500	+	0.513064i
$300$		0			0
$301$	−2.51762	+	4.69356i	−0.145113	+	0.270532i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	2.94575			0.168673
$306$		0			0
$307$		−	11.3921i		−	0.650184i	−0.945682	−	0.325092i	$-0.894605\pi$
$307$		−	11.3921i		−	0.650184i	0.945682	−	0.325092i	$-0.105395\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			6.74079i			0.382235i	0.981567	+	0.191118i	$0.0612112\pi$
$311$			6.74079i			0.382235i	−0.981567	+	0.191118i	$0.938789\pi$
$312$		0			0
$313$	−6.10315			−0.344970			−0.172485	−	0.985012i	$-0.555180\pi$
$313$	−6.10315			−0.344970			−0.172485	+	0.985012i	$0.555180\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−8.94575			−0.502444			−0.251222	−	0.967930i	$-0.580832\pi$
$317$	−8.94575			−0.502444			−0.251222	+	0.967930i	$0.580832\pi$
$318$		0			0
$319$			2.22632i			0.124650i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−8.77384			−0.488189
$324$		0			0
$325$			10.1962i			0.565584i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	12.8758	−	24.0042i	0.709868	−	1.32339i
$330$		0			0
$331$	−5.78499			−0.317972			−0.158986	−	0.987281i	$-0.550822\pi$
$331$	−5.78499			−0.317972			−0.158986	+	0.987281i	$0.550822\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	8.10860i		−	0.443020i
$336$		0			0
$337$		−	18.1561i		−	0.989023i	−0.869171	−	0.494512i	$-0.835347\pi$
$337$		−	18.1561i		−	0.989023i	0.869171	−	0.494512i	$-0.164653\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−1.56436			−0.0847149
$342$		0			0
$343$	−1.73414	−	18.4389i	−0.0936346	−	0.995607i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	14.8614			0.797804			0.398902	−	0.916993i	$-0.369391\pi$
$347$	14.8614			0.797804			0.398902	+	0.916993i	$0.369391\pi$
$348$		0			0
$349$			31.5672i			1.68975i	0.534962	+	0.844876i	$0.320326\pi$
$349$			31.5672i			1.68975i	−0.534962	+	0.844876i	$0.679674\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			7.91777i			0.421420i	0.977549	+	0.210710i	$0.0675776\pi$
$353$			7.91777i			0.421420i	−0.977549	+	0.210710i	$0.932422\pi$
$354$		0			0
$355$	−0.866995			−0.0460153
$356$		0			0
$357$		0			0
$358$		0			0
$359$			4.43602i			0.234124i	0.993125	+	0.117062i	$0.0373476\pi$
$359$			4.43602i			0.234124i	−0.993125	+	0.117062i	$0.962652\pi$
$360$		0			0
$361$	−16.9568			−0.892463
$362$		0			0
$363$		0			0
$364$		0			0
$365$		−	2.04277i		−	0.106923i
$366$		0			0
$367$	−8.89931			−0.464540			−0.232270	−	0.972651i	$-0.574615\pi$
$367$	−8.89931			−0.464540			−0.232270	+	0.972651i	$0.574615\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−9.65419	+	17.9981i	−0.501221	+	0.934416i
$372$		0			0
$373$			6.04780i			0.313143i	0.987667	+	0.156572i	$0.0500442\pi$
$373$			6.04780i			0.313143i	−0.987667	+	0.156572i	$0.949956\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	15.8938i		−	0.818571i
$378$		0			0
$379$			13.3686i			0.686697i	0.939208	+	0.343348i	$0.111561\pi$
$379$			13.3686i			0.686697i	−0.939208	+	0.343348i	$0.888439\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−29.4989			−1.50732			−0.753662	−	0.657262i	$-0.771714\pi$
$383$	−29.4989			−1.50732			−0.753662	+	0.657262i	$0.771714\pi$
$384$		0			0
$385$	0.200409	−	0.373618i	0.0102138	−	0.0190413i
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	12.8982i		−	0.653966i	−0.945030	−	0.326983i	$-0.893968\pi$
$389$		−	12.8982i		−	0.653966i	0.945030	−	0.326983i	$-0.106032\pi$
$390$		0			0
$391$	−25.2062	+	15.2054i	−1.27473	+	0.768969i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	7.55506i		−	0.380137i
$396$		0			0
$397$			12.9340i			0.649140i	0.945862	+	0.324570i	$0.105220\pi$
$397$			12.9340i			0.649140i	−0.945862	+	0.324570i	$0.894780\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	26.0221i		−	1.29948i	−0.760155	−	0.649741i	$-0.774877\pi$
$401$		−	26.0221i		−	1.29948i	0.760155	−	0.649741i	$-0.225123\pi$
$402$		0			0
$403$	11.1680			0.556319
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−2.46924			−0.122396
$408$		0			0
$409$		−	22.8113i		−	1.12795i	−0.825793	−	0.563973i	$-0.809272\pi$
$409$		−	22.8113i		−	1.12795i	0.825793	−	0.563973i	$-0.190728\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−6.73548	+	12.5568i	−0.331432	+	0.617882i
$414$		0			0
$415$	−2.51762			−0.123585
$416$		0			0
$417$		0			0
$418$		0			0
$419$	25.2107			1.23162			0.615812	−	0.787893i	$-0.288828\pi$
$419$	25.2107			1.23162			0.615812	+	0.787893i	$0.288828\pi$
$420$		0			0
$421$			23.2657i			1.13390i	0.823751	+	0.566952i	$0.191877\pi$
$421$			23.2657i			1.13390i	−0.823751	+	0.566952i	$0.808123\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−28.9694			−1.40522
$426$		0			0
$427$	−12.9699	−	6.95707i	−0.627660	−	0.336676i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	13.4766i		−	0.649143i	−0.945861	−	0.324571i	$-0.894780\pi$
$431$		−	13.4766i		−	0.649143i	0.945861	−	0.324571i	$-0.105220\pi$
$432$		0			0
$433$	35.0054			1.68225			0.841126	−	0.540839i	$-0.181893\pi$
$433$	35.0054			1.68225			0.841126	+	0.540839i	$0.181893\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	5.86986	−	3.54093i	0.280794	−	0.169386i
$438$		0			0
$439$		−	14.3695i		−	0.685818i	−0.939369	−	0.342909i	$-0.888588\pi$
$439$		−	14.3695i		−	0.685818i	0.939369	−	0.342909i	$-0.111412\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	5.81883			0.276461			0.138230	−	0.990400i	$-0.455859\pi$
$443$	5.81883			0.276461			0.138230	+	0.990400i	$0.455859\pi$
$444$		0			0
$445$	6.67262			0.316312
$446$		0			0
$447$		0			0
$448$		0			0
$449$	23.4634			1.10731			0.553653	−	0.832748i	$-0.313234\pi$
$449$	23.4634			1.10731			0.553653	+	0.832748i	$0.313234\pi$
$450$		0			0
$451$	−1.69141			−0.0796455
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−1.43072	+	2.66727i	−0.0670734	+	0.125044i
$456$		0			0
$457$			3.05841i			0.143066i	0.997438	+	0.0715332i	$0.0227892\pi$
$457$			3.05841i			0.143066i	−0.997438	+	0.0715332i	$0.977211\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	17.1062i		−	0.796715i	−0.917230	−	0.398357i	$-0.869580\pi$
$461$		−	17.1062i		−	0.796715i	0.917230	−	0.398357i	$-0.130420\pi$
$462$		0			0
$463$	9.39599			0.436669			0.218334	−	0.975874i	$-0.429938\pi$
$463$	9.39599			0.436669			0.218334	+	0.975874i	$0.429938\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	11.7626			0.544308			0.272154	−	0.962254i	$-0.412264\pi$
$467$	11.7626			0.544308			0.272154	+	0.962254i	$0.412264\pi$
$468$		0			0
$469$	−19.1503	+	35.7016i	−0.884279	+	1.64855i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	0.609199			0.0280110
$474$		0			0
$475$	6.74620			0.309537
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−30.0399			−1.37256			−0.686279	−	0.727339i	$-0.740757\pi$
$479$	−30.0399			−1.37256			−0.686279	+	0.727339i	$0.740757\pi$
$480$		0			0
$481$	17.6280			0.803767
$482$		0			0
$483$		0			0
$484$		0			0
$485$	0.328790			0.0149296
$486$		0			0
$487$	−9.05954			−0.410527			−0.205263	−	0.978707i	$-0.565805\pi$
$487$	−9.05954			−0.410527			−0.205263	+	0.978707i	$0.565805\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−2.19809			−0.0991984			−0.0495992	−	0.998769i	$-0.515794\pi$
$491$	−2.19809			−0.0991984			−0.0495992	+	0.998769i	$0.515794\pi$
$492$		0			0
$493$	45.1572			2.03378
$494$		0			0
$495$		0			0
$496$		0			0
$497$	3.81732	+	2.04761i	0.171230	+	0.0918477i
$498$		0			0
$499$	−25.3379			−1.13428			−0.567139	−	0.823622i	$-0.691950\pi$
$499$	−25.3379			−1.13428			−0.567139	+	0.823622i	$0.691950\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	17.8720			0.796872			0.398436	−	0.917196i	$-0.369553\pi$
$503$	17.8720			0.796872			0.398436	+	0.917196i	$0.369553\pi$
$504$		0			0
$505$			1.81571i			0.0807979i
$506$		0			0
$507$		0			0
$508$		0			0
$509$			5.24196i			0.232346i	0.993229	+	0.116173i	$0.0370627\pi$
$509$			5.24196i			0.232346i	−0.993229	+	0.116173i	$0.962937\pi$
$510$		0			0
$511$	−4.82447	+	8.99417i	−0.213422	+	0.397879i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	8.82222			0.388754
$516$		0			0
$517$	−3.11562			−0.137025
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−0.721614			−0.0316145			−0.0158072	−	0.999875i	$-0.505032\pi$
$521$	−0.721614			−0.0316145			−0.0158072	+	0.999875i	$0.505032\pi$
$522$		0			0
$523$	−4.20132			−0.183711			−0.0918556	−	0.995772i	$-0.529280\pi$
$523$	−4.20132			−0.183711			−0.0918556	+	0.995772i	$0.529280\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			31.7305i			1.38220i
$528$		0			0
$529$	10.7269	−	20.3454i	0.466385	−	0.884582i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	12.0750			0.523028
$534$		0			0
$535$			3.34225i			0.144498i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−1.76477	+	1.17170i	−0.0760140	+	0.0504689i
$540$		0			0
$541$	26.2352			1.12794			0.563970	−	0.825795i	$-0.309273\pi$
$541$	26.2352			1.12794			0.563970	+	0.825795i	$0.309273\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	7.05861i		−	0.302358i
$546$		0			0
$547$	37.3064			1.59511			0.797553	−	0.603249i	$-0.206127\pi$
$547$	37.3064			1.59511			0.797553	+	0.603249i	$0.206127\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−10.5159			−0.447994
$552$		0			0
$553$	−17.8430	+	33.2644i	−0.758762	+	1.41455i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	16.8257i		−	0.712929i	−0.934309	−	0.356464i	$-0.883982\pi$
$557$		−	16.8257i		−	0.712929i	0.934309	−	0.356464i	$-0.116018\pi$
$558$		0			0
$559$	−4.34909			−0.183947
$560$		0			0
$561$		0			0
$562$		0			0
$563$	4.28821			0.180727			0.0903633	−	0.995909i	$-0.471197\pi$
$563$	4.28821			0.180727			0.0903633	+	0.995909i	$0.471197\pi$
$564$		0			0
$565$			5.01405i			0.210943i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			6.38067i			0.267492i	0.991016	+	0.133746i	$0.0427006\pi$
$569$			6.38067i			0.267492i	−0.991016	+	0.133746i	$0.957299\pi$
$570$		0			0
$571$		−	12.8872i		−	0.539311i	−0.962957	−	0.269655i	$-0.913090\pi$
$571$		−	12.8872i		−	0.539311i	0.962957	−	0.269655i	$-0.0869098\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	19.3811	−	11.6914i	0.808246	−	0.487566i
$576$		0			0
$577$			42.1391i			1.75427i	0.480242	+	0.877136i	$0.340549\pi$
$577$			42.1391i			1.75427i	−0.480242	+	0.877136i	$0.659451\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	11.0849	+	5.94594i	0.459880	+	0.246679i
$582$		0			0
$583$	2.33606			0.0967499
$584$		0			0
$585$		0			0
$586$		0			0
$587$			8.17955i			0.337606i	0.985650	+	0.168803i	$0.0539902\pi$
$587$			8.17955i			0.337606i	−0.985650	+	0.168803i	$0.946010\pi$
$588$		0			0
$589$		−	7.38920i		−	0.304467i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	29.2947i		−	1.20299i	−0.798877	−	0.601495i	$-0.794572\pi$
$593$		−	29.2947i		−	1.20299i	0.798877	−	0.601495i	$-0.205428\pi$
$594$		0			0
$595$	−7.57823	−	4.06496i	−0.310677	−	0.166647i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−10.9267			−0.446455			−0.223227	−	0.974766i	$-0.571659\pi$
$599$	−10.9267			−0.446455			−0.223227	+	0.974766i	$0.571659\pi$
$600$		0			0
$601$		−	13.5653i		−	0.553339i	−0.960965	−	0.276669i	$-0.910769\pi$
$601$		−	13.5653i		−	0.553339i	0.960965	−	0.276669i	$-0.0892307\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	5.77640			0.234844
$606$		0			0
$607$		−	43.6809i		−	1.77295i	−0.462772	−	0.886477i	$-0.653145\pi$
$607$		−	43.6809i		−	1.77295i	0.462772	−	0.886477i	$-0.346855\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	22.2425			0.899835
$612$		0			0
$613$		−	32.6680i		−	1.31945i	−0.751507	−	0.659725i	$-0.770673\pi$
$613$		−	32.6680i		−	1.31945i	0.751507	−	0.659725i	$-0.229327\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			28.8070i			1.15973i	0.814714	+	0.579863i	$0.196894\pi$
$617$			28.8070i			1.15973i	−0.814714	+	0.579863i	$0.803106\pi$
$618$		0			0
$619$	17.5536			0.705538			0.352769	−	0.935710i	$-0.385240\pi$
$619$	17.5536			0.705538			0.352769	+	0.935710i	$0.385240\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−29.3791	−	15.7589i	−1.17705	−	0.631368i
$624$		0			0
$625$	20.8725			0.834900
$626$		0			0
$627$		0			0
$628$		0			0
$629$			50.0845i			1.99700i
$630$		0			0
$631$			8.46477i			0.336977i	0.985704	+	0.168489i	$0.0538886\pi$
$631$			8.46477i			0.336977i	−0.985704	+	0.168489i	$0.946111\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	4.80631			0.190733
$636$		0			0
$637$	12.5988	−	8.36483i	0.499181	−	0.331427i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	7.89574i		−	0.311863i	−0.987768	−	0.155932i	$-0.950162\pi$
$641$		−	7.89574i		−	0.311863i	0.987768	−	0.155932i	$-0.0498379\pi$
$642$		0			0
$643$	18.2330			0.719038			0.359519	−	0.933138i	$-0.382941\pi$
$643$	18.2330			0.719038			0.359519	+	0.933138i	$0.382941\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	3.41237i		−	0.134154i	−0.997748	−	0.0670770i	$-0.978633\pi$
$647$		−	3.41237i		−	0.134154i	0.997748	−	0.0670770i	$-0.0213673\pi$
$648$		0			0
$649$	1.62981			0.0639757
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−3.80191			−0.148780			−0.0743901	−	0.997229i	$-0.523701\pi$
$653$	−3.80191			−0.148780			−0.0743901	+	0.997229i	$0.523701\pi$
$654$		0			0
$655$			4.75050i			0.185617i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			11.9997i			0.467441i	0.972304	+	0.233720i	$0.0750901\pi$
$659$			11.9997i			0.467441i	−0.972304	+	0.233720i	$0.924910\pi$
$660$		0			0
$661$	−31.0526			−1.20780			−0.603902	−	0.797058i	$-0.706388\pi$
$661$	−31.0526			−1.20780			−0.603902	+	0.797058i	$0.706388\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	1.76477	+	0.946622i	0.0684349	+	0.0367084i
$666$		0			0
$667$	−30.2110	+	18.2245i	−1.16978	+	0.705655i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			1.68343i			0.0649881i
$672$		0			0
$673$	36.9659			1.42493			0.712465	−	0.701708i	$-0.247579\pi$
$673$	36.9659			1.42493			0.712465	+	0.701708i	$0.247579\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−31.0082			−1.19174			−0.595872	−	0.803080i	$-0.703193\pi$
$677$	−31.0082			−1.19174			−0.595872	+	0.803080i	$0.703193\pi$
$678$		0			0
$679$	−1.44764	−	0.776514i	−0.0555554	−	0.0297999i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−41.6986			−1.59555			−0.797776	−	0.602954i	$-0.793990\pi$
$683$	−41.6986			−1.59555			−0.797776	+	0.602954i	$0.793990\pi$
$684$		0			0
$685$			8.26884i			0.315936i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−16.6772			−0.635352
$690$		0			0
$691$		−	32.1573i		−	1.22332i	−0.791120	−	0.611661i	$-0.790502\pi$
$691$		−	32.1573i		−	1.22332i	0.791120	−	0.611661i	$-0.209498\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			4.60629i			0.174727i
$696$		0			0
$697$			34.3075i			1.29949i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			47.0109i			1.77558i	0.460251	+	0.887789i	$0.347760\pi$
$701$			47.0109i			1.77558i	−0.460251	+	0.887789i	$0.652240\pi$
$702$		0			0
$703$		−	11.6634i		−	0.439892i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	4.28821	−	7.99444i	0.161275	−	0.300662i
$708$		0			0
$709$		−	24.3025i		−	0.912701i	−0.889800	−	0.456351i	$-0.849156\pi$
$709$		−	24.3025i		−	0.912701i	0.889800	−	0.456351i	$-0.150844\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−12.8058	−	21.2283i	−0.479579	−	0.795007i
$714$		0			0
$715$	0.346198			0.0129471
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	27.8021i		−	1.03685i	−0.855125	−	0.518423i	$-0.826519\pi$
$719$		−	27.8021i		−	1.03685i	0.855125	−	0.518423i	$-0.173481\pi$
$720$		0			0
$721$	−38.8436	−	20.8357i	−1.44661	−	0.775962i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−34.7214			−1.28952
$726$		0			0
$727$	−15.6926			−0.582005			−0.291003	−	0.956722i	$-0.593989\pi$
$727$	−15.6926			−0.582005			−0.291003	+	0.956722i	$0.593989\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	12.3566i		−	0.457026i
$732$		0			0
$733$	24.0623			0.888763			0.444382	−	0.895838i	$-0.353423\pi$
$733$	24.0623			0.888763			0.444382	+	0.895838i	$0.353423\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	4.63388			0.170691
$738$		0			0
$739$	−7.59408			−0.279353			−0.139676	−	0.990197i	$-0.544606\pi$
$739$	−7.59408			−0.279353			−0.139676	+	0.990197i	$0.544606\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			11.6193i			0.426271i	0.977023	+	0.213136i	$0.0683676\pi$
$743$			11.6193i			0.426271i	−0.977023	+	0.213136i	$0.931632\pi$
$744$		0			0
$745$		−	6.36536i		−	0.233209i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	7.89349	−	14.7157i	0.288422	−	0.537700i
$750$		0			0
$751$			6.37414i			0.232596i	0.993214	+	0.116298i	$0.0371027\pi$
$751$			6.37414i			0.232596i	−0.993214	+	0.116298i	$0.962897\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−3.89187			−0.141640
$756$		0			0
$757$			24.5775i			0.893283i	0.894713	+	0.446642i	$0.147380\pi$
$757$			24.5775i			0.893283i	−0.894713	+	0.446642i	$0.852620\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			19.2449i			0.697628i	0.937192	+	0.348814i	$0.113415\pi$
$761$			19.2449i			0.697628i	−0.937192	+	0.348814i	$0.886585\pi$
$762$		0			0
$763$	−16.6705	+	31.0786i	−0.603514	+	1.12512i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−11.6353			−0.420126
$768$		0			0
$769$	28.2110			1.01732			0.508658	−	0.860969i	$-0.330142\pi$
$769$	28.2110			1.01732			0.508658	+	0.860969i	$0.330142\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	30.6431			1.10216			0.551078	−	0.834454i	$-0.314217\pi$
$773$	30.6431			1.10216			0.551078	+	0.834454i	$0.314217\pi$
$774$		0			0
$775$		−	24.3976i		−	0.876387i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	7.98932i		−	0.286247i
$780$		0			0
$781$		−	0.495468i		−	0.0177292i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	5.32738			0.190142
$786$		0			0
$787$	−51.6542			−1.84127			−0.920637	−	0.390419i	$-0.872330\pi$
$787$	−51.6542			−1.84127			−0.920637	+	0.390419i	$0.872330\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	11.8418	−	22.0765i	0.421047	−	0.784951i
$792$		0			0
$793$		−	12.0181i		−	0.426774i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−34.7091			−1.22946			−0.614730	−	0.788737i	$-0.710735\pi$
$797$	−34.7091			−1.22946			−0.614730	+	0.788737i	$0.710735\pi$
$798$		0			0
$799$			63.1952i			2.23569i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	1.16740			0.0411965
$804$		0			0
$805$	6.71051	−	0.338877i	0.236514	−	0.0119439i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−22.5345			−0.792272			−0.396136	−	0.918192i	$-0.629649\pi$
$809$	−22.5345			−0.792272			−0.396136	+	0.918192i	$0.629649\pi$
$810$		0			0
$811$		−	42.2138i		−	1.48233i	−0.671325	−	0.741163i	$-0.734274\pi$
$811$		−	42.2138i		−	1.48233i	0.671325	−	0.741163i	$-0.265726\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	6.51468			0.228199
$816$		0			0
$817$			2.87753i			0.100672i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−14.5418			−0.507513			−0.253756	−	0.967268i	$-0.581666\pi$
$821$	−14.5418			−0.507513			−0.253756	+	0.967268i	$0.581666\pi$
$822$		0			0
$823$	−27.3641			−0.953854			−0.476927	−	0.878943i	$-0.658249\pi$
$823$	−27.3641			−0.953854			−0.476927	+	0.878943i	$0.658249\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	43.2242i		−	1.50305i	−0.659702	−	0.751527i	$-0.729318\pi$
$827$		−	43.2242i		−	1.50305i	0.659702	−	0.751527i	$-0.270682\pi$
$828$		0			0
$829$			10.7116i			0.372030i	0.982547	+	0.186015i	$0.0595572\pi$
$829$			10.7116i			0.372030i	−0.982547	+	0.186015i	$0.940443\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	23.7661	+	35.7955i	0.823446	+	1.24024i
$834$		0			0
$835$		−	1.67093i		−	0.0578248i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−39.6525			−1.36896			−0.684478	−	0.729033i	$-0.739970\pi$
$839$	−39.6525			−1.36896			−0.684478	+	0.729033i	$0.739970\pi$
$840$		0			0
$841$	25.1234			0.866325
$842$		0			0
$843$		0			0
$844$		0			0
$845$	4.41245			0.151793
$846$		0			0
$847$	−25.4331	−	13.6423i	−0.873892	−	0.468755i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−20.2130	−	33.5075i	−0.692893	−	1.14862i
$852$		0			0
$853$		−	37.3283i		−	1.27810i	−0.769166	−	0.639048i	$-0.779328\pi$
$853$		−	37.3283i		−	1.27810i	0.769166	−	0.639048i	$-0.220672\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	31.0203i		−	1.05963i	−0.848112	−	0.529817i	$-0.822260\pi$
$857$		−	31.0203i		−	1.05963i	0.848112	−	0.529817i	$-0.177740\pi$
$858$		0			0
$859$			42.8245i			1.46115i	0.682831	+	0.730577i	$0.260749\pi$
$859$			42.8245i			1.46115i	−0.682831	+	0.730577i	$0.739251\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	50.4724			1.71810			0.859051	−	0.511890i	$-0.171055\pi$
$863$	50.4724			1.71810			0.859051	+	0.511890i	$0.171055\pi$
$864$		0			0
$865$		−	9.50099i		−	0.323043i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	4.31755			0.146463
$870$		0			0
$871$	−33.0814			−1.12092
$872$		0			0
$873$		0			0
$874$		0			0
$875$	12.0000	+	6.43679i	0.405674	+	0.217603i
$876$		0			0
$877$	−25.1419			−0.848980			−0.424490	−	0.905433i	$-0.639547\pi$
$877$	−25.1419			−0.848980			−0.424490	+	0.905433i	$0.639547\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	33.6545			1.13385			0.566925	−	0.823769i	$-0.308133\pi$
$881$	33.6545			1.13385			0.566925	+	0.823769i	$0.308133\pi$
$882$		0			0
$883$	36.6705			1.23406			0.617031	−	0.786939i	$-0.288335\pi$
$883$	36.6705			1.23406			0.617031	+	0.786939i	$0.288335\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			19.7286i			0.662423i	0.943557	+	0.331211i	$0.107457\pi$
$887$			19.7286i			0.662423i	−0.943557	+	0.331211i	$0.892543\pi$
$888$		0			0
$889$	−21.1619	−	11.3512i	−0.709746	−	0.380707i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	14.7165i		−	0.492469i
$894$		0			0
$895$	9.94038			0.332270
$896$		0			0
$897$		0			0
$898$		0			0
$899$			38.0308i			1.26840i
$900$		0			0
$901$		−	47.3832i		−	1.57856i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	4.20338			0.139725
$906$		0			0
$907$			25.0274i			0.831022i	0.909588	+	0.415511i	$0.136397\pi$
$907$			25.0274i			0.831022i	−0.909588	+	0.415511i	$0.863603\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			46.9636i			1.55597i	0.628280	+	0.777987i	$0.283759\pi$
$911$			46.9636i			1.55597i	−0.628280	+	0.777987i	$0.716241\pi$
$912$		0			0
$913$		−	1.43876i		−	0.0476161i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	11.2194	−	20.9161i	0.370497	−	0.690711i
$918$		0			0
$919$			27.2532i			0.898999i	0.893281	+	0.449500i	$0.148398\pi$
$919$			27.2532i			0.898999i	−0.893281	+	0.449500i	$0.851602\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			3.53716i			0.116427i
$924$		0			0
$925$		−	38.5099i		−	1.26620i
$926$		0			0
$927$		0			0
$928$		0			0
$929$			28.4513i			0.933455i	0.884401	+	0.466728i	$0.154567\pi$
$929$			28.4513i			0.933455i	−0.884401	+	0.466728i	$0.845433\pi$
$930$		0			0
$931$	−5.53450	−	8.33582i	−0.181386	−	0.273196i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			0.983615i			0.0321676i
$936$		0			0
$937$	−52.0582			−1.70067			−0.850333	−	0.526245i	$-0.823599\pi$
$937$	−52.0582			−1.70067			−0.850333	+	0.526245i	$0.823599\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	29.7152			0.968689			0.484344	−	0.874877i	$-0.339058\pi$
$941$	29.7152			0.968689			0.484344	+	0.874877i	$0.339058\pi$
$942$		0			0
$943$	−13.8458	−	22.9524i	−0.450881	−	0.747432i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	5.11029			0.166062			0.0830311	−	0.996547i	$-0.473540\pi$
$947$	5.11029			0.166062			0.0830311	+	0.996547i	$0.473540\pi$
$948$		0			0
$949$	−8.33408			−0.270536
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	48.7234i		−	1.57831i	−0.614197	−	0.789153i	$-0.710520\pi$
$953$		−	48.7234i		−	1.57831i	0.614197	−	0.789153i	$-0.289480\pi$
$954$		0			0
$955$		−	7.29029i		−	0.235908i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	19.5288	−	36.4071i	0.630617	−	1.17565i
$960$		0			0
$961$	4.27702			0.137968
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−2.18351			−0.0702895
$966$		0			0
$967$	51.0671			1.64221			0.821104	−	0.570779i	$-0.193359\pi$
$967$	51.0671			1.64221			0.821104	+	0.570779i	$0.193359\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	50.7703			1.62930			0.814648	−	0.579956i	$-0.196930\pi$
$971$	50.7703			1.62930			0.814648	+	0.579956i	$0.196930\pi$
$972$		0			0
$973$	10.8788	−	20.2812i	0.348759	−	0.650185i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			9.42941i			0.301674i	0.988559	+	0.150837i	$0.0481968\pi$
$977$			9.42941i			0.301674i	−0.988559	+	0.150837i	$0.951803\pi$
$978$		0			0
$979$			3.81325i			0.121872i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−53.2664			−1.69893			−0.849467	−	0.527642i	$-0.823076\pi$
$983$	−53.2664			−1.69893			−0.849467	+	0.527642i	$0.823076\pi$
$984$		0			0
$985$	5.05479			0.161059
$986$		0			0
$987$		0			0
$988$		0			0
$989$	4.98686	+	8.26680i	0.158573	+	0.262869i
$990$		0			0
$991$	19.2431			0.611277			0.305638	−	0.952148i	$-0.401130\pi$
$991$	19.2431			0.611277			0.305638	+	0.952148i	$0.401130\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	8.01692			0.254153
$996$		0			0
$997$			33.4325i			1.05882i	0.848367	+	0.529409i	$0.177586\pi$
$997$			33.4325i			1.05882i	−0.848367	+	0.529409i	$0.822414\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5796.2.k.b.5473.9		16
3.2	odd	2		644.2.d.a.321.3	✓	16
7.6	odd	2	inner	5796.2.k.b.5473.7		16
12.11	even	2		2576.2.f.g.321.13		16
21.20	even	2		644.2.d.a.321.14	yes	16
23.22	odd	2	inner	5796.2.k.b.5473.8		16
69.68	even	2		644.2.d.a.321.4	yes	16
84.83	odd	2		2576.2.f.g.321.4		16
161.160	even	2	inner	5796.2.k.b.5473.10		16
276.275	odd	2		2576.2.f.g.321.14		16
483.482	odd	2		644.2.d.a.321.13	yes	16
1932.1931	even	2		2576.2.f.g.321.3		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
644.2.d.a.321.3	✓	16	3.2	odd	2
644.2.d.a.321.4	yes	16	69.68	even	2
644.2.d.a.321.13	yes	16	483.482	odd	2
644.2.d.a.321.14	yes	16	21.20	even	2
2576.2.f.g.321.3		16	1932.1931	even	2
2576.2.f.g.321.4		16	84.83	odd	2
2576.2.f.g.321.13		16	12.11	even	2
2576.2.f.g.321.14		16	276.275	odd	2
5796.2.k.b.5473.7		16	7.6	odd	2	inner
5796.2.k.b.5473.8		16	23.22	odd	2	inner
5796.2.k.b.5473.9		16	1.1	even	1	trivial
5796.2.k.b.5473.10		16	161.160	even	2	inner

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:	\( N \)	\(=\)	\( 5796 = 2^{2} \cdot 3^{2} \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5796.k (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+0.529536 q^{5} +(-2.33151 - 1.25062i) q^{7} +O(q^{10})\)	\(q+0.529536 q^{5} +(-2.33151 - 1.25062i) q^{7} +0.302618i q^{11} -2.16040i q^{13} +6.13811 q^{17} -1.42940 q^{19} +(-4.10651 + 2.47721i) q^{23} -4.71959 q^{25} +7.35686 q^{29} +5.16943i q^{31} +(-1.23462 - 0.662249i) q^{35} +8.15959i q^{37} +5.58926i q^{41} -2.01310i q^{43} +10.2955i q^{47} +(3.87189 + 5.83168i) q^{49} -7.71951i q^{53} +0.160247i q^{55} -5.38571i q^{59} +5.56289 q^{61} -1.14401i q^{65} -15.3126i q^{67} -1.63727 q^{71} -3.85766i q^{73} +(0.378461 - 0.705557i) q^{77} -14.2673i q^{79} -4.75439 q^{83} +3.25035 q^{85} +12.6009 q^{89} +(-2.70184 + 5.03700i) q^{91} -0.756921 q^{95} +0.620903 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q + 16 q^{23} + 8 q^{25} + 4 q^{35} + 4 q^{49} + 8 q^{71} + 28 q^{77} + 16 q^{85} - 56 q^{95}+O(q^{100}) \) 16 * q + 16 * q^23 + 8 * q^25 + 4 * q^35 + 4 * q^49 + 8 * q^71 + 28 * q^77 + 16 * q^85 - 56 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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