LMFDB - Embedded newform 4608.2.k.bh.1153.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4608,2,Mod(1153,4608)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4608, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 3, 0]))

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

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

chi := DirichletCharacter("4608.1153");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4608 = 2^{9} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4608.k (of order $4$, degree $2$, 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:	$36.7950652514$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(i)$
Coefficient field:	$\Q(\zeta_{16})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 1 $ x^8 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 1536)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			1153.1
Root			$-0.382683 + 0.923880i$ of defining polynomial
Character	$\chi$	$=$	4608.1153
Dual form			4608.2.k.bh.3457.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.847759 + 0.847759i) q^{5} +0.613126i q^{7} +O(q^{10})$	$q+(-0.847759 + 0.847759i) q^{5} +0.613126i q^{7} +(1.08239 - 1.08239i) q^{11} +(0.883480 + 0.883480i) q^{13} -4.35916 q^{17} +(-0.917608 - 0.917608i) q^{19} -4.00000i q^{23} +3.56261i q^{25} +(0.682975 + 0.682975i) q^{29} +6.77791 q^{31} +(-0.519783 - 0.519783i) q^{35} +(7.64047 - 7.64047i) q^{37} -8.68873i q^{41} +(-6.14386 + 6.14386i) q^{43} +9.65685 q^{47} +6.62408 q^{49} +(-8.07401 + 8.07401i) q^{53} +1.83522i q^{55} +(-6.39782 + 6.39782i) q^{59} +(-1.07786 - 1.07786i) q^{61} -1.49796 q^{65} +(8.56261 + 8.56261i) q^{67} +3.56940i q^{71} +5.15800i q^{73} +(0.663643 + 0.663643i) q^{77} +7.49623 q^{79} +(5.08239 + 5.08239i) q^{83} +(3.69552 - 3.69552i) q^{85} -0.672715i q^{89} +(-0.541684 + 0.541684i) q^{91} +1.55582 q^{95} +4.71832 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{5}+O(q^{10}) $ 8 * q + 8 * q^5	$ 8 q + 8 q^{5} + 8 q^{13} - 16 q^{19} + 8 q^{29} + 16 q^{31} - 32 q^{35} + 8 q^{37} - 16 q^{43} + 32 q^{47} - 8 q^{49} - 8 q^{53} - 32 q^{59} + 8 q^{61} + 16 q^{65} + 32 q^{67} - 48 q^{79} + 32 q^{83} + 48 q^{91} - 64 q^{95} - 32 q^{97}+O(q^{100}) $ 8 * q + 8 * q^5 + 8 * q^13 - 16 * q^19 + 8 * q^29 + 16 * q^31 - 32 * q^35 + 8 * q^37 - 16 * q^43 + 32 * q^47 - 8 * q^49 - 8 * q^53 - 32 * q^59 + 8 * q^61 + 16 * q^65 + 32 * q^67 - 48 * q^79 + 32 * q^83 + 48 * q^91 - 64 * q^95 - 32 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4608\mathbb{Z}\right)^\times$.

$n$	$2053$	$3583$	$4097$
$\chi(n)$	$e\left(\frac{3}{4}\right)$	$1$	$1$

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
$5$	−0.847759	+	0.847759i	−0.379129	+	0.379129i	−0.870788	−	0.491659i	$-0.836391\pi$
$5$	−0.847759	+	0.847759i	−0.379129	+	0.379129i	0.491659	+	0.870788i	$0.336391\pi$
$6$		0			0
$7$			0.613126i			0.231740i	0.993264	+	0.115870i	$0.0369656\pi$
$7$			0.613126i			0.231740i	−0.993264	+	0.115870i	$0.963034\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	1.08239	−	1.08239i	0.326354	−	0.326354i	−0.524845	−	0.851198i	$-0.675877\pi$
$11$	1.08239	−	1.08239i	0.326354	−	0.326354i	0.851198	+	0.524845i	$0.175877\pi$
$12$		0			0
$13$	0.883480	+	0.883480i	0.245033	+	0.245033i	0.818929	−	0.573895i	$-0.194568\pi$
$13$	0.883480	+	0.883480i	0.245033	+	0.245033i	−0.573895	+	0.818929i	$0.694568\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−4.35916			−1.05725			−0.528626	−	0.848855i	$-0.677293\pi$
$17$	−4.35916			−1.05725			−0.528626	+	0.848855i	$0.677293\pi$
$18$		0			0
$19$	−0.917608	−	0.917608i	−0.210514	−	0.210514i	0.593972	−	0.804486i	$-0.297559\pi$
$19$	−0.917608	−	0.917608i	−0.210514	−	0.210514i	−0.804486	+	0.593972i	$0.797559\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	4.00000i		−	0.834058i	−0.908893	−	0.417029i	$-0.863071\pi$
$23$		−	4.00000i		−	0.834058i	0.908893	−	0.417029i	$-0.136929\pi$
$24$		0			0
$25$			3.56261i			0.712522i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	0.682975	+	0.682975i	0.126825	+	0.126825i	0.767670	−	0.640845i	$-0.221416\pi$
$29$	0.682975	+	0.682975i	0.126825	+	0.126825i	−0.640845	+	0.767670i	$0.721416\pi$
$30$		0			0
$31$	6.77791			1.21735			0.608674	−	0.793420i	$-0.291702\pi$
$31$	6.77791			1.21735			0.608674	+	0.793420i	$0.291702\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−0.519783	−	0.519783i	−0.0878594	−	0.0878594i
$36$		0			0
$37$	7.64047	−	7.64047i	1.25608	−	1.25608i	0.303138	−	0.952947i	$-0.401966\pi$
$37$	7.64047	−	7.64047i	1.25608	−	1.25608i	0.952947	−	0.303138i	$-0.0980344\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	8.68873i		−	1.35695i	−0.734623	−	0.678476i	$-0.762641\pi$
$41$		−	8.68873i		−	1.35695i	0.734623	−	0.678476i	$-0.237359\pi$
$42$		0			0
$43$	−6.14386	+	6.14386i	−0.936930	+	0.936930i	−0.998126	−	0.0611960i	$-0.980509\pi$
$43$	−6.14386	+	6.14386i	−0.936930	+	0.936930i	0.0611960	+	0.998126i	$0.480509\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	9.65685			1.40860			0.704298	−	0.709904i	$-0.251262\pi$
$47$	9.65685			1.40860			0.704298	+	0.709904i	$0.251262\pi$
$48$		0			0
$49$	6.62408			0.946297
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−8.07401	+	8.07401i	−1.10905	+	1.10905i	−0.115775	+	0.993275i	$0.536935\pi$
$53$	−8.07401	+	8.07401i	−1.10905	+	1.10905i	−0.993275	+	0.115775i	$0.963065\pi$
$54$		0			0
$55$			1.83522i			0.247460i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−6.39782	+	6.39782i	−0.832926	+	0.832926i	−0.987916	−	0.154990i	$-0.950465\pi$
$59$	−6.39782	+	6.39782i	−0.832926	+	0.832926i	0.154990	+	0.987916i	$0.450465\pi$
$60$		0			0
$61$	−1.07786	−	1.07786i	−0.138005	−	0.138005i	0.634729	−	0.772735i	$-0.281112\pi$
$61$	−1.07786	−	1.07786i	−0.138005	−	0.138005i	−0.772735	+	0.634729i	$0.781112\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−1.49796			−0.185799
$66$		0			0
$67$	8.56261	+	8.56261i	1.04609	+	1.04609i	0.998885	+	0.0472039i	$0.0150311\pi$
$67$	8.56261	+	8.56261i	1.04609	+	1.04609i	0.0472039	+	0.998885i	$0.484969\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			3.56940i			0.423610i	0.977312	+	0.211805i	$0.0679341\pi$
$71$			3.56940i			0.423610i	−0.977312	+	0.211805i	$0.932066\pi$
$72$		0			0
$73$			5.15800i			0.603698i	0.953356	+	0.301849i	$0.0976038\pi$
$73$			5.15800i			0.603698i	−0.953356	+	0.301849i	$0.902396\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	0.663643	+	0.663643i	0.0756291	+	0.0756291i
$78$		0			0
$79$	7.49623			0.843392			0.421696	−	0.906737i	$-0.361435\pi$
$79$	7.49623			0.843392			0.421696	+	0.906737i	$0.361435\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	5.08239	+	5.08239i	0.557865	+	0.557865i	0.928699	−	0.370834i	$-0.120928\pi$
$83$	5.08239	+	5.08239i	0.557865	+	0.557865i	−0.370834	+	0.928699i	$0.620928\pi$
$84$		0			0
$85$	3.69552	−	3.69552i	0.400835	−	0.400835i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	0.672715i		−	0.0713076i	−0.999364	−	0.0356538i	$-0.988649\pi$
$89$		−	0.672715i		−	0.0713076i	0.999364	−	0.0356538i	$-0.0113514\pi$
$90$		0			0
$91$	−0.541684	+	0.541684i	−0.0567840	+	0.0567840i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	1.55582			0.159624
$96$		0			0
$97$	4.71832			0.479073			0.239536	−	0.970887i	$-0.423005\pi$
$97$	4.71832			0.479073			0.239536	+	0.970887i	$0.423005\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−4.01933	+	4.01933i	−0.399938	+	0.399938i	−0.878211	−	0.478273i	$-0.841263\pi$
$101$	−4.01933	+	4.01933i	−0.399938	+	0.399938i	0.478273	+	0.878211i	$0.341263\pi$
$102$		0			0
$103$		−	15.5381i		−	1.53101i	−0.643428	−	0.765506i	$-0.722489\pi$
$103$		−	15.5381i		−	1.53101i	0.643428	−	0.765506i	$-0.277511\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−0.0546790	+	0.0546790i	−0.00528602	+	0.00528602i	−0.709745	−	0.704459i	$-0.751190\pi$
$107$	−0.0546790	+	0.0546790i	−0.00528602	+	0.00528602i	0.704459	+	0.709745i	$0.251190\pi$
$108$		0			0
$109$	−2.28809	−	2.28809i	−0.219160	−	0.219160i	0.588985	−	0.808144i	$-0.299528\pi$
$109$	−2.28809	−	2.28809i	−0.219160	−	0.219160i	−0.808144	+	0.588985i	$0.799528\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−7.65685			−0.720296			−0.360148	−	0.932895i	$-0.617274\pi$
$113$	−7.65685			−0.720296			−0.360148	+	0.932895i	$0.617274\pi$
$114$		0			0
$115$	3.39104	+	3.39104i	0.316216	+	0.316216i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	2.67271i		−	0.245007i
$120$		0			0
$121$			8.65685i			0.786987i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−7.25903	−	7.25903i	−0.649267	−	0.649267i
$126$		0			0
$127$	20.9008			1.85465			0.927325	−	0.374257i	$-0.122102\pi$
$127$	20.9008			1.85465			0.927325	+	0.374257i	$0.122102\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−5.45929	−	5.45929i	−0.476981	−	0.476981i	0.427184	−	0.904165i	$-0.359506\pi$
$131$	−5.45929	−	5.45929i	−0.476981	−	0.476981i	−0.904165	+	0.427184i	$0.859506\pi$
$132$		0			0
$133$	0.562609	−	0.562609i	0.0487844	−	0.0487844i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	7.42063i		−	0.633987i	−0.948428	−	0.316994i	$-0.897327\pi$
$137$		−	7.42063i		−	0.633987i	0.948428	−	0.316994i	$-0.102673\pi$
$138$		0			0
$139$	−8.66364	+	8.66364i	−0.734841	+	0.734841i	−0.971575	−	0.236734i	$-0.923923\pi$
$139$	−8.66364	+	8.66364i	−0.734841	+	0.734841i	0.236734	+	0.971575i	$0.423923\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	1.91254			0.159935
$144$		0			0
$145$	−1.15800			−0.0961663
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−2.50461	+	2.50461i	−0.205186	+	0.205186i	−0.802218	−	0.597032i	$-0.796347\pi$
$149$	−2.50461	+	2.50461i	−0.205186	+	0.205186i	0.597032	+	0.802218i	$0.296347\pi$
$150$		0			0
$151$			13.0019i			1.05808i	0.848598	+	0.529039i	$0.177447\pi$
$151$			13.0019i			1.05808i	−0.848598	+	0.529039i	$0.822553\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−5.74603	+	5.74603i	−0.461533	+	0.461533i
$156$		0			0
$157$	0.812038	+	0.812038i	0.0648077	+	0.0648077i	0.738768	−	0.673960i	$-0.235408\pi$
$157$	0.812038	+	0.812038i	0.0648077	+	0.0648077i	−0.673960	+	0.738768i	$0.735408\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	2.45250			0.193284
$162$		0			0
$163$	13.0688	+	13.0688i	1.02363	+	1.02363i	0.999714	+	0.0239146i	$0.00761298\pi$
$163$	13.0688	+	13.0688i	1.02363	+	1.02363i	0.0239146	+	0.999714i	$0.492387\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			20.2104i			1.56393i	0.623324	+	0.781964i	$0.285782\pi$
$167$			20.2104i			1.56393i	−0.623324	+	0.781964i	$0.714218\pi$
$168$		0			0
$169$		−	11.4389i		−	0.879917i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	7.16826	+	7.16826i	0.544992	+	0.544992i	0.924988	−	0.379996i	$-0.124075\pi$
$173$	7.16826	+	7.16826i	0.544992	+	0.544992i	−0.379996	+	0.924988i	$0.624075\pi$
$174$		0			0
$175$	−2.18433			−0.165120
$176$		0			0
$177$		0			0
$178$		0			0
$179$	14.2195	+	14.2195i	1.06281	+	1.06281i	0.997890	+	0.0649223i	$0.0206799\pi$
$179$	14.2195	+	14.2195i	1.06281	+	1.06281i	0.0649223	+	0.997890i	$0.479320\pi$
$180$		0			0
$181$	−6.77337	+	6.77337i	−0.503461	+	0.503461i	−0.912512	−	0.409051i	$-0.865860\pi$
$181$	−6.77337	+	6.77337i	−0.503461	+	0.503461i	0.409051	+	0.912512i	$0.365860\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			12.9545i			0.952437i
$186$		0			0
$187$	−4.71832	+	4.71832i	−0.345038	+	0.345038i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−2.30767			−0.166977			−0.0834885	−	0.996509i	$-0.526606\pi$
$191$	−2.30767			−0.166977			−0.0834885	+	0.996509i	$0.526606\pi$
$192$		0			0
$193$	2.03278			0.146323			0.0731613	−	0.997320i	$-0.476691\pi$
$193$	2.03278			0.146323			0.0731613	+	0.997320i	$0.476691\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	18.6694	−	18.6694i	1.33014	−	1.33014i	0.424899	−	0.905241i	$-0.360310\pi$
$197$	18.6694	−	18.6694i	1.33014	−	1.33014i	0.905241	−	0.424899i	$-0.139690\pi$
$198$		0			0
$199$			10.1927i			0.722538i	0.932462	+	0.361269i	$0.117656\pi$
$199$			10.1927i			0.722538i	−0.932462	+	0.361269i	$0.882344\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−0.418749	+	0.418749i	−0.0293905	+	0.0293905i
$204$		0			0
$205$	7.36595	+	7.36595i	0.514460	+	0.514460i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−1.98642			−0.137404
$210$		0			0
$211$	4.99321	+	4.99321i	0.343747	+	0.343747i	0.857774	−	0.514027i	$-0.171847\pi$
$211$	4.99321	+	4.99321i	0.343747	+	0.343747i	−0.514027	+	0.857774i	$0.671847\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	10.4170i		−	0.710435i
$216$		0			0
$217$			4.15571i			0.282108i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−3.85123	−	3.85123i	−0.259062	−	0.259062i
$222$		0			0
$223$	18.9646			1.26996			0.634982	−	0.772527i	$-0.281008\pi$
$223$	18.9646			1.26996			0.634982	+	0.772527i	$0.281008\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	11.0270	+	11.0270i	0.731885	+	0.731885i	0.970993	−	0.239108i	$-0.0768549\pi$
$227$	11.0270	+	11.0270i	0.731885	+	0.731885i	−0.239108	+	0.970993i	$0.576855\pi$
$228$		0			0
$229$	−4.50756	+	4.50756i	−0.297868	+	0.297868i	−0.840178	−	0.542310i	$-0.817550\pi$
$229$	−4.50756	+	4.50756i	−0.297868	+	0.297868i	0.542310	+	0.840178i	$0.317550\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			1.13880i			0.0746050i	0.999304	+	0.0373025i	$0.0118765\pi$
$233$			1.13880i			0.0746050i	−0.999304	+	0.0373025i	$0.988123\pi$
$234$		0			0
$235$	−8.18669	+	8.18669i	−0.534040	+	0.534040i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	25.6652			1.66014			0.830071	−	0.557657i	$-0.188300\pi$
$239$	25.6652			1.66014			0.830071	+	0.557657i	$0.188300\pi$
$240$		0			0
$241$	−19.2172			−1.23789			−0.618944	−	0.785435i	$-0.712439\pi$
$241$	−19.2172			−1.23789			−0.618944	+	0.785435i	$0.712439\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−5.61562	+	5.61562i	−0.358769	+	0.358769i
$246$		0			0
$247$		−	1.62138i		−	0.103166i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−4.57446	+	4.57446i	−0.288737	+	0.288737i	−0.836581	−	0.547843i	$-0.815449\pi$
$251$	−4.57446	+	4.57446i	−0.288737	+	0.288737i	0.547843	+	0.836581i	$0.315449\pi$
$252$		0			0
$253$	−4.32957	−	4.32957i	−0.272198	−	0.272198i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	22.1094			1.37914			0.689572	−	0.724217i	$-0.257799\pi$
$257$	22.1094			1.37914			0.689572	+	0.724217i	$0.257799\pi$
$258$		0			0
$259$	4.68457	+	4.68457i	0.291085	+	0.291085i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	22.8914i		−	1.41155i	−0.708438	−	0.705773i	$-0.750600\pi$
$263$		−	22.8914i		−	1.41155i	0.708438	−	0.705773i	$-0.249400\pi$
$264$		0			0
$265$		−	13.6896i		−	0.840947i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	16.6011	+	16.6011i	1.01219	+	1.01219i	0.999925	+	0.0122647i	$0.00390407\pi$
$269$	16.6011	+	16.6011i	1.01219	+	1.01219i	0.0122647	+	0.999925i	$0.496096\pi$
$270$		0			0
$271$	25.5837			1.55410			0.777049	−	0.629440i	$-0.216716\pi$
$271$	25.5837			1.55410			0.777049	+	0.629440i	$0.216716\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	3.85614	+	3.85614i	0.232534	+	0.232534i
$276$		0			0
$277$	−9.66555	+	9.66555i	−0.580747	+	0.580747i	−0.935108	−	0.354362i	$-0.884698\pi$
$277$	−9.66555	+	9.66555i	−0.580747	+	0.580747i	0.354362	+	0.935108i	$0.384698\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			21.1116i			1.25941i	0.776832	+	0.629707i	$0.216825\pi$
$281$			21.1116i			1.25941i	−0.776832	+	0.629707i	$0.783175\pi$
$282$		0			0
$283$	−12.0547	+	12.0547i	−0.716576	+	0.716576i	−0.967902	−	0.251326i	$-0.919133\pi$
$283$	−12.0547	+	12.0547i	−0.716576	+	0.716576i	0.251326	+	0.967902i	$0.419133\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	5.32729			0.314460
$288$		0			0
$289$	2.00228			0.117781
$290$		0			0
$291$		0			0
$292$		0			0
$293$	11.3877	−	11.3877i	0.665278	−	0.665278i	−0.291341	−	0.956619i	$-0.594102\pi$
$293$	11.3877	−	11.3877i	0.665278	−	0.665278i	0.956619	+	0.291341i	$0.0941016\pi$
$294$		0			0
$295$		−	10.8476i		−	0.631573i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	3.53392	−	3.53392i	0.204372	−	0.204372i
$300$		0			0
$301$	−3.76696	−	3.76696i	−0.217124	−	0.217124i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	1.82752			0.104644
$306$		0			0
$307$	4.89218	+	4.89218i	0.279211	+	0.279211i	0.832794	−	0.553583i	$-0.186740\pi$
$307$	4.89218	+	4.89218i	0.279211	+	0.279211i	−0.553583	+	0.832794i	$0.686740\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			20.4389i			1.15899i	0.814977	+	0.579493i	$0.196749\pi$
$311$			20.4389i			1.15899i	−0.814977	+	0.579493i	$0.803251\pi$
$312$		0			0
$313$		−	11.6241i		−	0.657032i	−0.944498	−	0.328516i	$-0.893452\pi$
$313$		−	11.6241i		−	0.657032i	0.944498	−	0.328516i	$-0.106548\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−15.2288	−	15.2288i	−0.855336	−	0.855336i	0.135449	−	0.990784i	$-0.456752\pi$
$317$	−15.2288	−	15.2288i	−0.855336	−	0.855336i	−0.990784	+	0.135449i	$0.956752\pi$
$318$		0			0
$319$	1.47849			0.0827797
$320$		0			0
$321$		0			0
$322$		0			0
$323$	4.00000	+	4.00000i	0.222566	+	0.222566i
$324$		0			0
$325$	−3.14749	+	3.14749i	−0.174592	+	0.174592i
$326$		0			0
$327$		0			0
$328$		0			0
$329$			5.92087i			0.326428i
$330$		0			0
$331$	−9.78886	+	9.78886i	−0.538044	+	0.538044i	−0.922954	−	0.384910i	$-0.874233\pi$
$331$	−9.78886	+	9.78886i	−0.538044	+	0.538044i	0.384910	+	0.922954i	$0.374233\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−14.5181			−0.793206
$336$		0			0
$337$	−33.8428			−1.84353			−0.921767	−	0.387744i	$-0.873255\pi$
$337$	−33.8428			−1.84353			−0.921767	+	0.387744i	$0.873255\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	7.33636	−	7.33636i	0.397286	−	0.397286i
$342$		0			0
$343$			8.35327i			0.451034i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−7.53490	+	7.53490i	−0.404494	+	0.404494i	−0.879813	−	0.475319i	$-0.842333\pi$
$347$	−7.53490	+	7.53490i	−0.404494	+	0.404494i	0.475319	+	0.879813i	$0.342333\pi$
$348$		0			0
$349$	−11.0315	−	11.0315i	−0.590503	−	0.590503i	0.347264	−	0.937767i	$-0.387111\pi$
$349$	−11.0315	−	11.0315i	−0.590503	−	0.590503i	−0.937767	+	0.347264i	$0.887111\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	28.2978			1.50614			0.753071	−	0.657939i	$-0.228572\pi$
$353$	28.2978			1.50614			0.753071	+	0.657939i	$0.228572\pi$
$354$		0			0
$355$	−3.02599	−	3.02599i	−0.160603	−	0.160603i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			3.41474i			0.180223i	0.995932	+	0.0901116i	$0.0287224\pi$
$359$			3.41474i			0.180223i	−0.995932	+	0.0901116i	$0.971278\pi$
$360$		0			0
$361$		−	17.3160i		−	0.911368i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−4.37274	−	4.37274i	−0.228880	−	0.228880i
$366$		0			0
$367$	−10.0916			−0.526778			−0.263389	−	0.964690i	$-0.584840\pi$
$367$	−10.0916			−0.526778			−0.263389	+	0.964690i	$0.584840\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−4.95039	−	4.95039i	−0.257011	−	0.257011i
$372$		0			0
$373$	−10.5790	+	10.5790i	−0.547760	+	0.547760i	−0.925792	−	0.378033i	$-0.876601\pi$
$373$	−10.5790	+	10.5790i	−0.547760	+	0.547760i	0.378033	+	0.925792i	$0.376601\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			1.20679i			0.0621528i
$378$		0			0
$379$	9.16985	−	9.16985i	0.471023	−	0.471023i	−0.431222	−	0.902246i	$-0.641918\pi$
$379$	9.16985	−	9.16985i	0.471023	−	0.471023i	0.902246	+	0.431222i	$0.141918\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−3.81527			−0.194951			−0.0974755	−	0.995238i	$-0.531077\pi$
$383$	−3.81527			−0.194951			−0.0974755	+	0.995238i	$0.531077\pi$
$384$		0			0
$385$	−1.12522			−0.0573464
$386$		0			0
$387$		0			0
$388$		0			0
$389$	19.8866	−	19.8866i	1.00829	−	1.00829i	0.00832416	−	0.999965i	$-0.497350\pi$
$389$	19.8866	−	19.8866i	1.00829	−	1.00829i	0.999965	−	0.00832416i	$-0.00264969\pi$
$390$		0			0
$391$			17.4366i			0.881809i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−6.35500	+	6.35500i	−0.319755	+	0.319755i
$396$		0			0
$397$	22.6884	+	22.6884i	1.13870	+	1.13870i	0.988684	+	0.150012i	$0.0479311\pi$
$397$	22.6884	+	22.6884i	1.13870	+	1.13870i	0.150012	+	0.988684i	$0.452069\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−14.9545			−0.746794			−0.373397	−	0.927672i	$-0.621807\pi$
$401$	−14.9545			−0.746794			−0.373397	+	0.927672i	$0.621807\pi$
$402$		0			0
$403$	5.98815	+	5.98815i	0.298291	+	0.298291i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	16.5400i		−	0.819855i
$408$		0			0
$409$		−	10.5298i		−	0.520667i	−0.965519	−	0.260333i	$-0.916168\pi$
$409$		−	10.5298i		−	0.520667i	0.965519	−	0.260333i	$-0.0838325\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−3.92267	−	3.92267i	−0.193022	−	0.193022i
$414$		0			0
$415$	−8.61729			−0.423006
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−14.6974	−	14.6974i	−0.718015	−	0.718015i	0.250184	−	0.968198i	$-0.419509\pi$
$419$	−14.6974	−	14.6974i	−0.718015	−	0.718015i	−0.968198	+	0.250184i	$0.919509\pi$
$420$		0			0
$421$	−9.91217	+	9.91217i	−0.483090	+	0.483090i	−0.906117	−	0.423027i	$-0.860967\pi$
$421$	−9.91217	+	9.91217i	−0.483090	+	0.483090i	0.423027	+	0.906117i	$0.360967\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		−	15.5300i		−	0.753315i
$426$		0			0
$427$	0.660862	−	0.660862i	0.0319813	−	0.0319813i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	13.8318			0.666253			0.333126	−	0.942882i	$-0.391896\pi$
$431$	13.8318			0.666253			0.333126	+	0.942882i	$0.391896\pi$
$432$		0			0
$433$	14.8476			0.713531			0.356766	−	0.934194i	$-0.383879\pi$
$433$	14.8476			0.713531			0.356766	+	0.934194i	$0.383879\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−3.67043	+	3.67043i	−0.175581	+	0.175581i
$438$		0			0
$439$		−	35.7840i		−	1.70787i	−0.520376	−	0.853937i	$-0.674208\pi$
$439$		−	35.7840i		−	1.70787i	0.520376	−	0.853937i	$-0.325792\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	23.3565	−	23.3565i	1.10970	−	1.10970i	0.116513	−	0.993189i	$-0.462828\pi$
$443$	23.3565	−	23.3565i	1.10970	−	1.10970i	0.993189	−	0.116513i	$-0.0371718\pi$
$444$		0			0
$445$	0.570300	+	0.570300i	0.0270348	+	0.0270348i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	7.27775			0.343458			0.171729	−	0.985144i	$-0.445065\pi$
$449$	7.27775			0.343458			0.171729	+	0.985144i	$0.445065\pi$
$450$		0			0
$451$	−9.40461	−	9.40461i	−0.442846	−	0.442846i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		−	0.918436i		−	0.0430569i
$456$		0			0
$457$			14.3096i			0.669376i	0.942329	+	0.334688i	$0.108631\pi$
$457$			14.3096i			0.669376i	−0.942329	+	0.334688i	$0.891369\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	16.6115	+	16.6115i	0.773676	+	0.773676i	0.978747	−	0.205071i	$-0.0657425\pi$
$461$	16.6115	+	16.6115i	0.773676	+	0.773676i	−0.205071	+	0.978747i	$0.565742\pi$
$462$		0			0
$463$	29.7704			1.38355			0.691773	−	0.722115i	$-0.256830\pi$
$463$	29.7704			1.38355			0.691773	+	0.722115i	$0.256830\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−16.5745	−	16.5745i	−0.766975	−	0.766975i	0.210598	−	0.977573i	$-0.432459\pi$
$467$	−16.5745	−	16.5745i	−0.766975	−	0.766975i	−0.977573	+	0.210598i	$0.932459\pi$
$468$		0			0
$469$	−5.24996	+	5.24996i	−0.242421	+	0.242421i
$470$		0			0
$471$		0			0
$472$		0			0
$473$			13.3001i			0.611541i
$474$		0			0
$475$	3.26908	−	3.26908i	0.149996	−	0.149996i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	24.2104			1.10620			0.553101	−	0.833115i	$-0.313445\pi$
$479$	24.2104			1.10620			0.553101	+	0.833115i	$0.313445\pi$
$480$		0			0
$481$	13.5004			0.615565
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−4.00000	+	4.00000i	−0.181631	+	0.181631i
$486$		0			0
$487$		−	10.0498i		−	0.455399i	−0.973732	−	0.227699i	$-0.926880\pi$
$487$		−	10.0498i		−	0.455399i	0.973732	−	0.227699i	$-0.0731203\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−25.9401	+	25.9401i	−1.17066	+	1.17066i	−0.188606	+	0.982053i	$0.560397\pi$
$491$	−25.9401	+	25.9401i	−1.17066	+	1.17066i	−0.982053	+	0.188606i	$0.939603\pi$
$492$		0			0
$493$	−2.97720	−	2.97720i	−0.134086	−	0.134086i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−2.18849			−0.0981672
$498$		0			0
$499$	−1.40011	−	1.40011i	−0.0626774	−	0.0626774i	0.675073	−	0.737751i	$-0.264112\pi$
$499$	−1.40011	−	1.40011i	−0.0626774	−	0.0626774i	−0.737751	+	0.675073i	$0.764112\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			12.0739i			0.538348i	0.963092	+	0.269174i	$0.0867506\pi$
$503$			12.0739i			0.538348i	−0.963092	+	0.269174i	$0.913249\pi$
$504$		0			0
$505$		−	6.81485i		−	0.303257i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	3.17414	+	3.17414i	0.140691	+	0.140691i	0.773945	−	0.633253i	$-0.218281\pi$
$509$	3.17414	+	3.17414i	0.140691	+	0.140691i	−0.633253	+	0.773945i	$0.718281\pi$
$510$		0			0
$511$	−3.16250			−0.139901
$512$		0			0
$513$		0			0
$514$		0			0
$515$	13.1725	+	13.1725i	0.580452	+	0.580452i
$516$		0			0
$517$	10.4525	−	10.4525i	0.459701	−	0.459701i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	4.75428i		−	0.208289i	−0.994562	−	0.104145i	$-0.966790\pi$
$521$		−	4.75428i		−	0.208289i	0.994562	−	0.104145i	$-0.0332104\pi$
$522$		0			0
$523$	24.5609	−	24.5609i	1.07397	−	1.07397i	0.0769365	−	0.997036i	$-0.475486\pi$
$523$	24.5609	−	24.5609i	1.07397	−	1.07397i	0.997036	−	0.0769365i	$-0.0245139\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−29.5460			−1.28704
$528$		0			0
$529$	7.00000			0.304348
$530$		0			0
$531$		0			0
$532$		0			0
$533$	7.67632	−	7.67632i	0.332498	−	0.332498i
$534$		0			0
$535$		−	0.0927092i		−	0.00400817i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	7.16985	−	7.16985i	0.308827	−	0.308827i
$540$		0			0
$541$	−11.5245	−	11.5245i	−0.495476	−	0.495476i	0.414550	−	0.910026i	$-0.363939\pi$
$541$	−11.5245	−	11.5245i	−0.495476	−	0.495476i	−0.910026	+	0.414550i	$0.863939\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	3.87950			0.166180
$546$		0			0
$547$	6.25322	+	6.25322i	0.267368	+	0.267368i	0.828039	−	0.560671i	$-0.189457\pi$
$547$	6.25322	+	6.25322i	0.267368	+	0.267368i	−0.560671	+	0.828039i	$0.689457\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	1.25341i		−	0.0533969i
$552$		0			0
$553$			4.59613i			0.195448i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−24.2775	−	24.2775i	−1.02867	−	1.02867i	−0.999577	−	0.0290922i	$-0.990738\pi$
$557$	−24.2775	−	24.2775i	−1.02867	−	1.02867i	−0.0290922	−	0.999577i	$-0.509262\pi$
$558$		0			0
$559$	−10.8560			−0.459158
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−19.7133	−	19.7133i	−0.830815	−	0.830815i	0.156813	−	0.987628i	$-0.449878\pi$
$563$	−19.7133	−	19.7133i	−0.830815	−	0.830815i	−0.987628	+	0.156813i	$0.949878\pi$
$564$		0			0
$565$	6.49117	−	6.49117i	0.273085	−	0.273085i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			22.1525i			0.928682i	0.885656	+	0.464341i	$0.153709\pi$
$569$			22.1525i			0.928682i	−0.885656	+	0.464341i	$0.846291\pi$
$570$		0			0
$571$	−24.3842	+	24.3842i	−1.02045	+	1.02045i	−0.0206625	+	0.999787i	$0.506578\pi$
$571$	−24.3842	+	24.3842i	−1.02045	+	1.02045i	−0.999787	+	0.0206625i	$0.993422\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	14.2504			0.594284
$576$		0			0
$577$	−22.3488			−0.930391			−0.465196	−	0.885208i	$-0.654016\pi$
$577$	−22.3488			−0.930391			−0.465196	+	0.885208i	$0.654016\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−3.11615	+	3.11615i	−0.129280	+	0.129280i
$582$		0			0
$583$			17.4785i			0.723885i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	10.1184	−	10.1184i	0.417632	−	0.417632i	−0.466755	−	0.884387i	$-0.654577\pi$
$587$	10.1184	−	10.1184i	0.417632	−	0.417632i	0.884387	+	0.466755i	$0.154577\pi$
$588$		0			0
$589$	−6.21946	−	6.21946i	−0.256269	−	0.256269i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−27.3439			−1.12288			−0.561440	−	0.827517i	$-0.689753\pi$
$593$	−27.3439			−1.12288			−0.561440	+	0.827517i	$0.689753\pi$
$594$		0			0
$595$	2.26582	+	2.26582i	0.0928895	+	0.0928895i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			23.4684i			0.958891i	0.877572	+	0.479446i	$0.159162\pi$
$599$			23.4684i			0.958891i	−0.877572	+	0.479446i	$0.840838\pi$
$600$		0			0
$601$			20.4717i			0.835058i	0.908664	+	0.417529i	$0.137104\pi$
$601$			20.4717i			0.835058i	−0.908664	+	0.417529i	$0.862896\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−7.33893	−	7.33893i	−0.298370	−	0.298370i
$606$		0			0
$607$	−32.6633			−1.32576			−0.662881	−	0.748725i	$-0.730667\pi$
$607$	−32.6633			−1.32576			−0.662881	+	0.748725i	$0.730667\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	8.53164	+	8.53164i	0.345153	+	0.345153i
$612$		0			0
$613$	5.03150	−	5.03150i	0.203220	−	0.203220i	−0.598158	−	0.801378i	$-0.704100\pi$
$613$	5.03150	−	5.03150i	0.203220	−	0.203220i	0.801378	+	0.598158i	$0.204100\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	20.5619i		−	0.827789i	−0.910325	−	0.413895i	$-0.864168\pi$
$617$		−	20.5619i		−	0.827789i	0.910325	−	0.413895i	$-0.135832\pi$
$618$		0			0
$619$	25.0607	−	25.0607i	1.00728	−	1.00728i	0.00730204	−	0.999973i	$-0.497676\pi$
$619$	25.0607	−	25.0607i	1.00728	−	1.00728i	0.999973	−	0.00730204i	$-0.00232433\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	0.412459			0.0165248
$624$		0			0
$625$	−5.50523			−0.220209
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−33.3060	+	33.3060i	−1.32800	+	1.32800i
$630$		0			0
$631$		−	3.05731i		−	0.121709i	−0.998147	−	0.0608547i	$-0.980617\pi$
$631$		−	3.05731i		−	0.121709i	0.998147	−	0.0608547i	$-0.0193826\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−17.7189	+	17.7189i	−0.703152	+	0.703152i
$636$		0			0
$637$	5.85224	+	5.85224i	0.231874	+	0.231874i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−10.8890			−0.430089			−0.215045	−	0.976604i	$-0.568990\pi$
$641$	−10.8890			−0.430089			−0.215045	+	0.976604i	$0.568990\pi$
$642$		0			0
$643$	−16.7392	−	16.7392i	−0.660131	−	0.660131i	0.295279	−	0.955411i	$-0.404587\pi$
$643$	−16.7392	−	16.7392i	−0.660131	−	0.660131i	−0.955411	+	0.295279i	$0.904587\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	42.3417i		−	1.66462i	−0.554309	−	0.832311i	$-0.687017\pi$
$647$		−	42.3417i		−	1.66462i	0.554309	−	0.832311i	$-0.312983\pi$
$648$		0			0
$649$			13.8499i			0.543657i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	34.1911	+	34.1911i	1.33800	+	1.33800i	0.897996	+	0.440003i	$0.145023\pi$
$653$	34.1911	+	34.1911i	1.33800	+	1.33800i	0.440003	+	0.897996i	$0.354977\pi$
$654$		0			0
$655$	9.25633			0.361675
$656$		0			0
$657$		0			0
$658$		0			0
$659$	16.2651	+	16.2651i	0.633597	+	0.633597i	0.948968	−	0.315371i	$-0.102129\pi$
$659$	16.2651	+	16.2651i	0.633597	+	0.633597i	−0.315371	+	0.948968i	$0.602129\pi$
$660$		0			0
$661$	15.1880	−	15.1880i	0.590743	−	0.590743i	−0.347089	−	0.937832i	$-0.612830\pi$
$661$	15.1880	−	15.1880i	0.590743	−	0.590743i	0.937832	+	0.347089i	$0.112830\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$			0.953914i			0.0369912i
$666$		0			0
$667$	2.73190	−	2.73190i	0.105780	−	0.105780i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−2.33333			−0.0900771
$672$		0			0
$673$	17.6569			0.680622			0.340311	−	0.940313i	$-0.389468\pi$
$673$	17.6569			0.680622			0.340311	+	0.940313i	$0.389468\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−22.7891	+	22.7891i	−0.875858	+	0.875858i	−0.993103	−	0.117245i	$-0.962594\pi$
$677$	−22.7891	+	22.7891i	−0.875858	+	0.875858i	0.117245	+	0.993103i	$0.462594\pi$
$678$		0			0
$679$			2.89293i			0.111020i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	13.0134	−	13.0134i	0.497943	−	0.497943i	−0.412854	−	0.910797i	$-0.635468\pi$
$683$	13.0134	−	13.0134i	0.497943	−	0.497943i	0.910797	+	0.412854i	$0.135468\pi$
$684$		0			0
$685$	6.29090	+	6.29090i	0.240363	+	0.240363i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−14.2665			−0.543509
$690$		0			0
$691$	−8.84860	−	8.84860i	−0.336617	−	0.336617i	0.518476	−	0.855092i	$-0.326500\pi$
$691$	−8.84860	−	8.84860i	−0.336617	−	0.336617i	−0.855092	+	0.518476i	$0.826500\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	14.6894i		−	0.557199i
$696$		0			0
$697$			37.8756i			1.43464i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	3.11946	+	3.11946i	0.117821	+	0.117821i	0.763559	−	0.645738i	$-0.223450\pi$
$701$	3.11946	+	3.11946i	0.117821	+	0.117821i	−0.645738	+	0.763559i	$0.723450\pi$
$702$		0			0
$703$	−14.0219			−0.528846
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−2.46436	−	2.46436i	−0.0926817	−	0.0926817i
$708$		0			0
$709$	28.8371	−	28.8371i	1.08300	−	1.08300i	0.0867728	−	0.996228i	$-0.472345\pi$
$709$	28.8371	−	28.8371i	1.08300	−	1.08300i	0.996228	−	0.0867728i	$-0.0276554\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	27.1116i		−	1.01534i
$714$		0			0
$715$	−1.62138	+	1.62138i	−0.0606360	+	0.0606360i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−38.4170			−1.43271			−0.716357	−	0.697734i	$-0.754192\pi$
$719$	−38.4170			−1.43271			−0.716357	+	0.697734i	$0.754192\pi$
$720$		0			0
$721$	9.52680			0.354797
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−2.43317	+	2.43317i	−0.0903657	+	0.0903657i
$726$		0			0
$727$		−	1.48610i		−	0.0551165i	−0.999620	−	0.0275583i	$-0.991227\pi$
$727$		−	1.48610i		−	0.0551165i	0.999620	−	0.0275583i	$-0.00877318\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	26.7821	−	26.7821i	0.990571	−	0.990571i
$732$		0			0
$733$	5.47887	+	5.47887i	0.202367	+	0.202367i	0.801013	−	0.598647i	$-0.204295\pi$
$733$	5.47887	+	5.47887i	0.202367	+	0.202367i	−0.598647	+	0.801013i	$0.704295\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	18.5362			0.682790
$738$		0			0
$739$	−30.1505	−	30.1505i	−1.10910	−	1.10910i	−0.993269	−	0.115834i	$-0.963046\pi$
$739$	−30.1505	−	30.1505i	−1.10910	−	1.10910i	−0.115834	−	0.993269i	$-0.536954\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			43.6516i			1.60142i	0.599051	+	0.800711i	$0.295545\pi$
$743$			43.6516i			1.60142i	−0.599051	+	0.800711i	$0.704455\pi$
$744$		0			0
$745$		−	4.24662i		−	0.155584i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−0.0335251	−	0.0335251i	−0.00122498	−	0.00122498i
$750$		0			0
$751$	−43.3854			−1.58315			−0.791577	−	0.611069i	$-0.790740\pi$
$751$	−43.3854			−1.58315			−0.791577	+	0.611069i	$0.790740\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−11.0225	−	11.0225i	−0.401148	−	0.401148i
$756$		0			0
$757$	−16.0415	+	16.0415i	−0.583037	+	0.583037i	−0.935737	−	0.352699i	$-0.885264\pi$
$757$	−16.0415	+	16.0415i	−0.583037	+	0.583037i	0.352699	+	0.935737i	$0.385264\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	31.0047i		−	1.12392i	−0.827164	−	0.561960i	$-0.810047\pi$
$761$		−	31.0047i		−	1.12392i	0.827164	−	0.561960i	$-0.189953\pi$
$762$		0			0
$763$	1.40289	−	1.40289i	0.0507880	−	0.0507880i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−11.3047			−0.408189
$768$		0			0
$769$	43.3767			1.56420			0.782102	−	0.623150i	$-0.214148\pi$
$769$	43.3767			1.56420			0.782102	+	0.623150i	$0.214148\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	3.83778	−	3.83778i	0.138036	−	0.138036i	−0.634713	−	0.772748i	$-0.718882\pi$
$773$	3.83778	−	3.83778i	0.138036	−	0.138036i	0.772748	+	0.634713i	$0.218882\pi$
$774$		0			0
$775$			24.1470i			0.867387i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−7.97285	+	7.97285i	−0.285657	+	0.285657i
$780$		0			0
$781$	3.86349	+	3.86349i	0.138246	+	0.138246i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−1.37683			−0.0491410
$786$		0			0
$787$	−8.96778	−	8.96778i	−0.319667	−	0.319667i	0.528972	−	0.848639i	$-0.322578\pi$
$787$	−8.96778	−	8.96778i	−0.319667	−	0.319667i	−0.848639	+	0.528972i	$0.822578\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	4.69462i		−	0.166921i
$792$		0			0
$793$		−	1.90453i		−	0.0676318i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−17.4682	−	17.4682i	−0.618757	−	0.618757i	0.326456	−	0.945212i	$-0.394146\pi$
$797$	−17.4682	−	17.4682i	−0.618757	−	0.618757i	−0.945212	+	0.326456i	$0.894146\pi$
$798$		0			0
$799$	−42.0958			−1.48924
$800$		0			0
$801$		0			0
$802$		0			0
$803$	5.58297	+	5.58297i	0.197019	+	0.197019i
$804$		0			0
$805$	−2.07913	+	2.07913i	−0.0732798	+	0.0732798i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			6.81166i			0.239485i	0.992805	+	0.119743i	$0.0382070\pi$
$809$			6.81166i			0.239485i	−0.992805	+	0.119743i	$0.961793\pi$
$810$		0			0
$811$	1.99265	−	1.99265i	0.0699715	−	0.0699715i	−0.671255	−	0.741226i	$-0.734244\pi$
$811$	1.99265	−	1.99265i	0.0699715	−	0.0699715i	0.741226	+	0.671255i	$0.234244\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−22.1584			−0.776175
$816$		0			0
$817$	11.2753			0.394473
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−25.8629	+	25.8629i	−0.902620	+	0.902620i	−0.995662	−	0.0930418i	$-0.970341\pi$
$821$	−25.8629	+	25.8629i	−0.902620	+	0.902620i	0.0930418	+	0.995662i	$0.470341\pi$
$822$		0			0
$823$		−	5.15309i		−	0.179625i	−0.995959	−	0.0898126i	$-0.971373\pi$
$823$		−	5.15309i		−	0.179625i	0.995959	−	0.0898126i	$-0.0286268\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	25.2036	−	25.2036i	0.876415	−	0.876415i	−0.116747	−	0.993162i	$-0.537247\pi$
$827$	25.2036	−	25.2036i	0.876415	−	0.876415i	0.993162	+	0.116747i	$0.0372465\pi$
$828$		0			0
$829$	−29.9306	−	29.9306i	−1.03953	−	1.03953i	−0.999186	−	0.0403478i	$-0.987153\pi$
$829$	−29.9306	−	29.9306i	−1.03953	−	1.03953i	−0.0403478	−	0.999186i	$-0.512847\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−28.8754			−1.00047
$834$		0			0
$835$	−17.1335	−	17.1335i	−0.592931	−	0.592931i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	48.7768i		−	1.68396i	−0.539507	−	0.841981i	$-0.681389\pi$
$839$		−	48.7768i		−	1.68396i	0.539507	−	0.841981i	$-0.318611\pi$
$840$		0			0
$841$		−	28.0671i		−	0.967831i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	9.69745	+	9.69745i	0.333603	+	0.333603i
$846$		0			0
$847$	−5.30774			−0.182376
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−30.5619	−	30.5619i	−1.04765	−	1.04765i
$852$		0			0
$853$	28.5647	−	28.5647i	0.978036	−	0.978036i	−0.0217281	−	0.999764i	$-0.506917\pi$
$853$	28.5647	−	28.5647i	0.978036	−	0.978036i	0.999764	+	0.0217281i	$0.00691682\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	25.2977i		−	0.864153i	−0.901837	−	0.432076i	$-0.857781\pi$
$857$		−	25.2977i		−	0.864153i	0.901837	−	0.432076i	$-0.142219\pi$
$858$		0			0
$859$	24.8486	−	24.8486i	0.847823	−	0.847823i	−0.142038	−	0.989861i	$-0.545365\pi$
$859$	24.8486	−	24.8486i	0.847823	−	0.847823i	0.989861	+	0.142038i	$0.0453655\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−29.4193			−1.00144			−0.500722	−	0.865608i	$-0.666932\pi$
$863$	−29.4193			−1.00144			−0.500722	+	0.865608i	$0.666932\pi$
$864$		0			0
$865$	−12.1539			−0.413245
$866$		0			0
$867$		0			0
$868$		0			0
$869$	8.11386	−	8.11386i	0.275244	−	0.275244i
$870$		0			0
$871$			15.1298i			0.512653i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	4.45070	−	4.45070i	0.150461	−	0.150461i
$876$		0			0
$877$	2.75285	+	2.75285i	0.0929573	+	0.0929573i	0.752056	−	0.659099i	$-0.229062\pi$
$877$	2.75285	+	2.75285i	0.0929573	+	0.0929573i	−0.659099	+	0.752056i	$0.729062\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−21.6049			−0.727887			−0.363943	−	0.931421i	$-0.618570\pi$
$881$	−21.6049			−0.727887			−0.363943	+	0.931421i	$0.618570\pi$
$882$		0			0
$883$	16.3323	+	16.3323i	0.549627	+	0.549627i	0.926333	−	0.376706i	$-0.122943\pi$
$883$	16.3323	+	16.3323i	0.549627	+	0.549627i	−0.376706	+	0.926333i	$0.622943\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			56.1260i			1.88453i	0.334873	+	0.942263i	$0.391307\pi$
$887$			56.1260i			1.88453i	−0.334873	+	0.942263i	$0.608693\pi$
$888$		0			0
$889$			12.8149i			0.429796i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−8.86120	−	8.86120i	−0.296529	−	0.296529i
$894$		0			0
$895$	−24.1094			−0.805887
$896$		0			0
$897$		0			0
$898$		0			0
$899$	4.62914	+	4.62914i	0.154390	+	0.154390i
$900$		0			0
$901$	35.1959	−	35.1959i	1.17255	−	1.17255i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	11.4844i		−	0.381754i
$906$		0			0
$907$	18.5745	−	18.5745i	0.616755	−	0.616755i	−0.327943	−	0.944698i	$-0.606355\pi$
$907$	18.5745	−	18.5745i	0.616755	−	0.616755i	0.944698	+	0.327943i	$0.106355\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−22.8560			−0.757251			−0.378626	−	0.925550i	$-0.623603\pi$
$911$	−22.8560			−0.757251			−0.378626	+	0.925550i	$0.623603\pi$
$912$		0			0
$913$	11.0023			0.364122
$914$		0			0
$915$		0			0
$916$		0			0
$917$	3.34723	−	3.34723i	0.110535	−	0.110535i
$918$		0			0
$919$		−	0.319035i		−	0.0105240i	−0.999986	−	0.00526200i	$-0.998325\pi$
$919$		−	0.319035i		−	0.0105240i	0.999986	−	0.00526200i	$-0.00167496\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−3.15349	+	3.15349i	−0.103798	+	0.103798i
$924$		0			0
$925$	27.2200	+	27.2200i	0.894988	+	0.894988i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−57.3735			−1.88236			−0.941182	−	0.337900i	$-0.890284\pi$
$929$	−57.3735			−1.88236			−0.941182	+	0.337900i	$0.890284\pi$
$930$		0			0
$931$	−6.07830	−	6.07830i	−0.199208	−	0.199208i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	8.00000i		−	0.261628i
$936$		0			0
$937$			39.8100i			1.30054i	0.759705	+	0.650268i	$0.225343\pi$
$937$			39.8100i			1.30054i	−0.759705	+	0.650268i	$0.774657\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−31.9639	−	31.9639i	−1.04199	−	1.04199i	−0.999079	−	0.0429148i	$-0.986336\pi$
$941$	−31.9639	−	31.9639i	−1.04199	−	1.04199i	−0.0429148	−	0.999079i	$-0.513664\pi$
$942$		0			0
$943$	−34.7549			−1.13178
$944$		0			0
$945$		0			0
$946$		0			0
$947$	30.5309	+	30.5309i	0.992121	+	0.992121i	0.999969	−	0.00784863i	$-0.00249832\pi$
$947$	30.5309	+	30.5309i	0.992121	+	0.992121i	−0.00784863	+	0.999969i	$0.502498\pi$
$948$		0			0
$949$	−4.55699	+	4.55699i	−0.147926	+	0.147926i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			15.1594i			0.491060i	0.969389	+	0.245530i	$0.0789620\pi$
$953$			15.1594i			0.491060i	−0.969389	+	0.245530i	$0.921038\pi$
$954$		0			0
$955$	1.95635	−	1.95635i	0.0633059	−	0.0633059i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	4.54978			0.146920
$960$		0			0
$961$	14.9401			0.481938
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−1.72331	+	1.72331i	−0.0554752	+	0.0554752i
$966$		0			0
$967$		−	16.1170i		−	0.518287i	−0.965839	−	0.259143i	$-0.916560\pi$
$967$		−	16.1170i		−	0.518287i	0.965839	−	0.259143i	$-0.0834402\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	2.94131	−	2.94131i	0.0943912	−	0.0943912i	−0.658334	−	0.752726i	$-0.728739\pi$
$971$	2.94131	−	2.94131i	0.0943912	−	0.0943912i	0.752726	+	0.658334i	$0.228739\pi$
$972$		0			0
$973$	−5.31190	−	5.31190i	−0.170292	−	0.170292i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	41.2552			1.31987			0.659935	−	0.751323i	$-0.270584\pi$
$977$	41.2552			1.31987			0.659935	+	0.751323i	$0.270584\pi$
$978$		0			0
$979$	−0.728141	−	0.728141i	−0.0232715	−	0.0232715i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	32.0257i		−	1.02146i	−0.859741	−	0.510730i	$-0.829375\pi$
$983$		−	32.0257i		−	1.02146i	0.859741	−	0.510730i	$-0.170625\pi$
$984$		0			0
$985$			31.6543i			1.00859i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	24.5754	+	24.5754i	0.781453	+	0.781453i
$990$		0			0
$991$	−9.94041			−0.315768			−0.157884	−	0.987458i	$-0.550467\pi$
$991$	−9.94041			−0.315768			−0.157884	+	0.987458i	$0.550467\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−8.64091	−	8.64091i	−0.273935	−	0.273935i
$996$		0			0
$997$	−12.7393	+	12.7393i	−0.403457	+	0.403457i	−0.879449	−	0.475992i	$-0.842089\pi$
$997$	−12.7393	+	12.7393i	−0.403457	+	0.403457i	0.475992	+	0.879449i	$0.342089\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4608.2.k.bh.1153.1		8
3.2	odd	2		1536.2.j.e.1153.2	yes	8
4.3	odd	2		4608.2.k.bj.1153.1		8
8.3	odd	2		4608.2.k.bc.1153.4		8
8.5	even	2		4608.2.k.be.1153.4		8
12.11	even	2		1536.2.j.f.1153.4	yes	8
16.3	odd	4		4608.2.k.bc.3457.4		8
16.5	even	4	inner	4608.2.k.bh.3457.1		8
16.11	odd	4		4608.2.k.bj.3457.1		8
16.13	even	4		4608.2.k.be.3457.4		8
24.5	odd	2		1536.2.j.j.1153.3	yes	8
24.11	even	2		1536.2.j.i.1153.1	yes	8
32.5	even	8		9216.2.a.bm.1.2		4
32.11	odd	8		9216.2.a.z.1.3		4
32.21	even	8		9216.2.a.bl.1.3		4
32.27	odd	8		9216.2.a.ba.1.2		4
48.5	odd	4		1536.2.j.e.385.2	✓	8
48.11	even	4		1536.2.j.f.385.4	yes	8
48.29	odd	4		1536.2.j.j.385.3	yes	8
48.35	even	4		1536.2.j.i.385.1	yes	8
96.5	odd	8		3072.2.a.s.1.3		4
96.11	even	8		3072.2.a.p.1.2		4
96.29	odd	8		3072.2.d.e.1537.3		8
96.35	even	8		3072.2.d.j.1537.7		8
96.53	odd	8		3072.2.a.m.1.2		4
96.59	even	8		3072.2.a.j.1.3		4
96.77	odd	8		3072.2.d.e.1537.6		8
96.83	even	8		3072.2.d.j.1537.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1536.2.j.e.385.2	✓	8	48.5	odd	4
1536.2.j.e.1153.2	yes	8	3.2	odd	2
1536.2.j.f.385.4	yes	8	48.11	even	4
1536.2.j.f.1153.4	yes	8	12.11	even	2
1536.2.j.i.385.1	yes	8	48.35	even	4
1536.2.j.i.1153.1	yes	8	24.11	even	2
1536.2.j.j.385.3	yes	8	48.29	odd	4
1536.2.j.j.1153.3	yes	8	24.5	odd	2
3072.2.a.j.1.3		4	96.59	even	8
3072.2.a.m.1.2		4	96.53	odd	8
3072.2.a.p.1.2		4	96.11	even	8
3072.2.a.s.1.3		4	96.5	odd	8
3072.2.d.e.1537.3		8	96.29	odd	8
3072.2.d.e.1537.6		8	96.77	odd	8
3072.2.d.j.1537.2		8	96.83	even	8
3072.2.d.j.1537.7		8	96.35	even	8
4608.2.k.bc.1153.4		8	8.3	odd	2
4608.2.k.bc.3457.4		8	16.3	odd	4
4608.2.k.be.1153.4		8	8.5	even	2
4608.2.k.be.3457.4		8	16.13	even	4
4608.2.k.bh.1153.1		8	1.1	even	1	trivial
4608.2.k.bh.3457.1		8	16.5	even	4	inner
4608.2.k.bj.1153.1		8	4.3	odd	2
4608.2.k.bj.3457.1		8	16.11	odd	4
9216.2.a.z.1.3		4	32.11	odd	8
9216.2.a.ba.1.2		4	32.27	odd	8
9216.2.a.bl.1.3		4	32.21	even	8
9216.2.a.bm.1.2		4	32.5	even	8

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:	\( N \)	\(=\)	\( 4608 = 2^{9} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4608.k (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.847759 + 0.847759i) q^{5} +0.613126i q^{7} +O(q^{10})\)	\(q+(-0.847759 + 0.847759i) q^{5} +0.613126i q^{7} +(1.08239 - 1.08239i) q^{11} +(0.883480 + 0.883480i) q^{13} -4.35916 q^{17} +(-0.917608 - 0.917608i) q^{19} -4.00000i q^{23} +3.56261i q^{25} +(0.682975 + 0.682975i) q^{29} +6.77791 q^{31} +(-0.519783 - 0.519783i) q^{35} +(7.64047 - 7.64047i) q^{37} -8.68873i q^{41} +(-6.14386 + 6.14386i) q^{43} +9.65685 q^{47} +6.62408 q^{49} +(-8.07401 + 8.07401i) q^{53} +1.83522i q^{55} +(-6.39782 + 6.39782i) q^{59} +(-1.07786 - 1.07786i) q^{61} -1.49796 q^{65} +(8.56261 + 8.56261i) q^{67} +3.56940i q^{71} +5.15800i q^{73} +(0.663643 + 0.663643i) q^{77} +7.49623 q^{79} +(5.08239 + 5.08239i) q^{83} +(3.69552 - 3.69552i) q^{85} -0.672715i q^{89} +(-0.541684 + 0.541684i) q^{91} +1.55582 q^{95} +4.71832 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{5}+O(q^{10}) \) 8 * q + 8 * q^5	\( 8 q + 8 q^{5} + 8 q^{13} - 16 q^{19} + 8 q^{29} + 16 q^{31} - 32 q^{35} + 8 q^{37} - 16 q^{43} + 32 q^{47} - 8 q^{49} - 8 q^{53} - 32 q^{59} + 8 q^{61} + 16 q^{65} + 32 q^{67} - 48 q^{79} + 32 q^{83} + 48 q^{91} - 64 q^{95} - 32 q^{97}+O(q^{100}) \) 8 * q + 8 * q^5 + 8 * q^13 - 16 * q^19 + 8 * q^29 + 16 * q^31 - 32 * q^35 + 8 * q^37 - 16 * q^43 + 32 * q^47 - 8 * q^49 - 8 * q^53 - 32 * q^59 + 8 * q^61 + 16 * q^65 + 32 * q^67 - 48 * q^79 + 32 * q^83 + 48 * q^91 - 64 * q^95 - 32 * q^97

Introduction

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