LMFDB - Embedded newform 4032.2.b.r.3583.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4032,2,Mod(3583,4032)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4032, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4032.3583");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4032 = 2^{6} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4032.b (of order $2$, degree $1$, not 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:	$32.1956820950$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	16.0.29960650073923649536.7
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 7x^{12} + 40x^{8} - 112x^{4} + 256 $ x^16 - 7x^12 + 40x^8 - 112*x^4 + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{30} $
Twist minimal:	no (minimal twist has level 2016)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3583.10
Root			$-1.38588 + 0.281691i$ of defining polynomial
Character	$\chi$	$=$	4032.3583
Dual form			4032.2.b.r.3583.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.69614i q^{5} +(-2.13578 + 1.56155i) q^{7} +O(q^{10})$	$q+1.69614i q^{5} +(-2.13578 + 1.56155i) q^{7} -0.794156i q^{11} -1.87285i q^{13} -4.34475i q^{17} -2.39871 q^{19} +3.62258i q^{23} +2.12311 q^{25} -4.41674 q^{29} -1.87285 q^{31} +(-2.64861 - 3.62258i) q^{35} +3.12311 q^{37} -4.34475i q^{41} -0.876894i q^{43} +12.0818 q^{47} +(2.12311 - 6.67026i) q^{49} -10.0736 q^{53} +1.34700 q^{55} -6.78456 q^{59} -11.4677i q^{61} +3.17662 q^{65} +0.876894i q^{67} +5.21089i q^{71} -3.74571i q^{73} +(1.24012 + 1.69614i) q^{77} -3.12311i q^{79} -12.0818 q^{83} +7.36932 q^{85} -7.73704i q^{89} +(2.92456 + 4.00000i) q^{91} -4.06854i q^{95} -13.3405i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q - 32 q^{25} - 16 q^{37} - 32 q^{49} - 80 q^{85}+O(q^{100}) $ 16 * q - 32 * q^25 - 16 * q^37 - 32 * q^49 - 80 * q^85

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4032\mathbb{Z}\right)^\times$.

$n$	$127$	$577$	$1793$	$3781$
$\chi(n)$	$-1$	$-1$	$1$	$1$

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
$5$			1.69614i			0.758537i	0.925287	+	0.379269i	$0.123824\pi$
$5$			1.69614i			0.758537i	−0.925287	+	0.379269i	$0.876176\pi$
$6$		0			0
$7$	−2.13578	+	1.56155i	−0.807249	+	0.590211i
$7$	−2.13578	+	1.56155i	−0.807249	+	0.590211i
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	0.794156i		−	0.239447i	−0.992807	−	0.119723i	$-0.961799\pi$
$11$		−	0.794156i		−	0.239447i	0.992807	−	0.119723i	$-0.0382008\pi$
$12$		0			0
$13$		−	1.87285i		−	0.519436i	−0.965685	−	0.259718i	$-0.916370\pi$
$13$		−	1.87285i		−	0.519436i	0.965685	−	0.259718i	$-0.0836296\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	4.34475i		−	1.05376i	−0.849940	−	0.526879i	$-0.823362\pi$
$17$		−	4.34475i		−	1.05376i	0.849940	−	0.526879i	$-0.176638\pi$
$18$		0			0
$19$	−2.39871			−0.550301			−0.275150	−	0.961401i	$-0.588728\pi$
$19$	−2.39871			−0.550301			−0.275150	+	0.961401i	$0.588728\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			3.62258i			0.755361i	0.925936	+	0.377680i	$0.123278\pi$
$23$			3.62258i			0.755361i	−0.925936	+	0.377680i	$0.876722\pi$
$24$		0			0
$25$	2.12311			0.424621
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−4.41674			−0.820168			−0.410084	−	0.912048i	$-0.634501\pi$
$29$	−4.41674			−0.820168			−0.410084	+	0.912048i	$0.634501\pi$
$30$		0			0
$31$	−1.87285			−0.336374			−0.168187	−	0.985755i	$-0.553791\pi$
$31$	−1.87285			−0.336374			−0.168187	+	0.985755i	$0.553791\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−2.64861	−	3.62258i	−0.447697	−	0.612328i
$36$		0			0
$37$	3.12311			0.513435			0.256718	−	0.966486i	$-0.417359\pi$
$37$	3.12311			0.513435			0.256718	+	0.966486i	$0.417359\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	4.34475i		−	0.678537i	−0.940690	−	0.339268i	$-0.889821\pi$
$41$		−	4.34475i		−	0.678537i	0.940690	−	0.339268i	$-0.110179\pi$
$42$		0			0
$43$		−	0.876894i		−	0.133725i	−0.997762	−	0.0668626i	$-0.978701\pi$
$43$		−	0.876894i		−	0.133725i	0.997762	−	0.0668626i	$-0.0212989\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	12.0818			1.76231			0.881155	−	0.472827i	$-0.156766\pi$
$47$	12.0818			1.76231			0.881155	+	0.472827i	$0.156766\pi$
$48$		0			0
$49$	2.12311	−	6.67026i	0.303301	−	0.952895i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−10.0736			−1.38371			−0.691857	−	0.722034i	$-0.743207\pi$
$53$	−10.0736			−1.38371			−0.691857	+	0.722034i	$0.743207\pi$
$54$		0			0
$55$	1.34700			0.181629
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−6.78456			−0.883275			−0.441638	−	0.897194i	$-0.645602\pi$
$59$	−6.78456			−0.883275			−0.441638	+	0.897194i	$0.645602\pi$
$60$		0			0
$61$		−	11.4677i		−	1.46829i	−0.678995	−	0.734143i	$-0.737584\pi$
$61$		−	11.4677i		−	1.46829i	0.678995	−	0.734143i	$-0.262416\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	3.17662			0.394012
$66$		0			0
$67$			0.876894i			0.107130i	0.998564	+	0.0535648i	$0.0170584\pi$
$67$			0.876894i			0.107130i	−0.998564	+	0.0535648i	$0.982942\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			5.21089i			0.618419i	0.950994	+	0.309210i	$0.100064\pi$
$71$			5.21089i			0.618419i	−0.950994	+	0.309210i	$0.899936\pi$
$72$		0			0
$73$		−	3.74571i		−	0.438402i	−0.975680	−	0.219201i	$-0.929655\pi$
$73$		−	3.74571i		−	0.438402i	0.975680	−	0.219201i	$-0.0703450\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	1.24012	+	1.69614i	0.141324	+	0.193293i
$78$		0			0
$79$		−	3.12311i		−	0.351377i	−0.984446	−	0.175688i	$-0.943785\pi$
$79$		−	3.12311i		−	0.351377i	0.984446	−	0.175688i	$-0.0562151\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−12.0818			−1.32615			−0.663074	−	0.748553i	$-0.730749\pi$
$83$	−12.0818			−1.32615			−0.663074	+	0.748553i	$0.730749\pi$
$84$		0			0
$85$	7.36932			0.799315
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	7.73704i		−	0.820124i	−0.912058	−	0.410062i	$-0.865507\pi$
$89$		−	7.73704i		−	0.820124i	0.912058	−	0.410062i	$-0.134493\pi$
$90$		0			0
$91$	2.92456	+	4.00000i	0.306577	+	0.419314i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		−	4.06854i		−	0.417424i
$96$		0			0
$97$		−	13.3405i		−	1.35453i	−0.735741	−	0.677263i	$-0.763166\pi$
$97$		−	13.3405i		−	1.35453i	0.735741	−	0.677263i	$-0.236834\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			10.3857i			1.03341i	0.856163	+	0.516705i	$0.172842\pi$
$101$			10.3857i			1.03341i	−0.856163	+	0.516705i	$0.827158\pi$
$102$		0			0
$103$	11.4677			1.12994			0.564972	−	0.825110i	$-0.308887\pi$
$103$	11.4677			1.12994			0.564972	+	0.825110i	$0.308887\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	13.6962i		−	1.32406i	−0.749478	−	0.662030i	$-0.769695\pi$
$107$		−	13.6962i		−	1.32406i	0.749478	−	0.662030i	$-0.230305\pi$
$108$		0			0
$109$	8.24621			0.789844			0.394922	−	0.918715i	$-0.370772\pi$
$109$	8.24621			0.789844			0.394922	+	0.918715i	$0.370772\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	5.65685			0.532152			0.266076	−	0.963952i	$-0.414273\pi$
$113$	5.65685			0.532152			0.266076	+	0.963952i	$0.414273\pi$
$114$		0			0
$115$	−6.14441			−0.572969
$116$		0			0
$117$		0			0
$118$		0			0
$119$	6.78456	+	9.27944i	0.621940	+	0.850645i
$120$		0			0
$121$	10.3693			0.942665
$122$		0			0
$123$		0			0
$124$		0			0
$125$			12.0818i			1.08063i
$126$		0			0
$127$		−	11.1231i		−	0.987016i	−0.869741	−	0.493508i	$-0.835714\pi$
$127$		−	11.1231i		−	0.987016i	0.869741	−	0.493508i	$-0.164286\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	6.78456			0.592770			0.296385	−	0.955069i	$-0.404219\pi$
$131$	6.78456			0.592770			0.296385	+	0.955069i	$0.404219\pi$
$132$		0			0
$133$	5.12311	−	3.74571i	0.444230	−	0.324794i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	20.1472			1.72129			0.860645	−	0.509205i	$-0.170061\pi$
$137$	20.1472			1.72129			0.860645	+	0.509205i	$0.170061\pi$
$138$		0			0
$139$	17.0862			1.44924			0.724618	−	0.689151i	$-0.242016\pi$
$139$	17.0862			1.44924			0.724618	+	0.689151i	$0.242016\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−1.48734			−0.124377
$144$		0			0
$145$		−	7.49141i		−	0.622128i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	10.0736			0.825261			0.412630	−	0.910899i	$-0.364610\pi$
$149$	10.0736			0.825261			0.412630	+	0.910899i	$0.364610\pi$
$150$		0			0
$151$			11.1231i			0.905185i	0.891717	+	0.452593i	$0.149501\pi$
$151$			11.1231i			0.905185i	−0.891717	+	0.452593i	$0.850499\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		−	3.17662i		−	0.255152i
$156$		0			0
$157$		−	18.9591i		−	1.51310i	−0.653936	−	0.756550i	$-0.726883\pi$
$157$		−	18.9591i		−	1.51310i	0.653936	−	0.756550i	$-0.273117\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−5.65685	−	7.73704i	−0.445823	−	0.609764i
$162$		0			0
$163$		−	7.12311i		−	0.557925i	−0.960302	−	0.278962i	$-0.910010\pi$
$163$		−	7.12311i		−	0.557925i	0.960302	−	0.278962i	$-0.0899905\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0			−	1.00000i	$-0.5\pi$
$167$		0			0				1.00000i	$0.5\pi$
$168$		0			0
$169$	9.49242			0.730186
$170$		0			0
$171$		0			0
$172$		0			0
$173$			3.60109i			0.273786i	0.990586	+	0.136893i	$0.0437116\pi$
$173$			3.60109i			0.273786i	−0.990586	+	0.136893i	$0.956288\pi$
$174$		0			0
$175$	−4.53448	+	3.31534i	−0.342775	+	0.250616i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			16.1764i			1.20908i	0.796574	+	0.604541i	$0.206643\pi$
$179$			16.1764i			1.20908i	−0.796574	+	0.604541i	$0.793357\pi$
$180$		0			0
$181$		−	1.87285i		−	0.139208i	−0.997575	−	0.0696040i	$-0.977826\pi$
$181$		−	1.87285i		−	0.139208i	0.997575	−	0.0696040i	$-0.0221736\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			5.29723i			0.389460i
$186$		0			0
$187$	−3.45041			−0.252319
$188$		0			0
$189$		0			0
$190$		0			0
$191$			14.9363i			1.08075i	0.841424	+	0.540376i	$0.181718\pi$
$191$			14.9363i			1.08075i	−0.841424	+	0.540376i	$0.818282\pi$
$192$		0			0
$193$	13.3693			0.962344			0.481172	−	0.876626i	$-0.340211\pi$
$193$	13.3693			0.962344			0.481172	+	0.876626i	$0.340211\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−4.41674			−0.314680			−0.157340	−	0.987545i	$-0.550292\pi$
$197$	−4.41674			−0.314680			−0.157340	+	0.987545i	$0.550292\pi$
$198$		0			0
$199$	9.06897			0.642882			0.321441	−	0.946930i	$-0.395833\pi$
$199$	9.06897			0.642882			0.321441	+	0.946930i	$0.395833\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	9.43318	−	6.89697i	0.662079	−	0.484072i
$204$		0			0
$205$	7.36932			0.514695
$206$		0			0
$207$		0			0
$208$		0			0
$209$			1.90495i			0.131768i
$210$		0			0
$211$		−	21.3693i		−	1.47112i	−0.677457	−	0.735562i	$-0.736918\pi$
$211$		−	21.3693i		−	1.47112i	0.677457	−	0.735562i	$-0.263082\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	1.48734			0.101436
$216$		0			0
$217$	4.00000	−	2.92456i	0.271538	−	0.198532i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−8.13709			−0.547360
$222$		0			0
$223$	−12.8147			−0.858134			−0.429067	−	0.903273i	$-0.641158\pi$
$223$	−12.8147			−0.858134			−0.429067	+	0.903273i	$0.641158\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−18.8664			−1.25220			−0.626102	−	0.779741i	$-0.715351\pi$
$227$	−18.8664			−1.25220			−0.626102	+	0.779741i	$0.715351\pi$
$228$		0			0
$229$		−	18.9591i		−	1.25285i	−0.779481	−	0.626426i	$-0.784517\pi$
$229$		−	18.9591i		−	1.25285i	0.779481	−	0.626426i	$-0.215483\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−16.9706			−1.11178			−0.555889	−	0.831256i	$-0.687622\pi$
$233$	−16.9706			−1.11178			−0.555889	+	0.831256i	$0.687622\pi$
$234$		0			0
$235$			20.4924i			1.33678i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	19.7012i		−	1.27437i	−0.770712	−	0.637183i	$-0.780099\pi$
$239$		−	19.7012i		−	1.27437i	0.770712	−	0.637183i	$-0.219901\pi$
$240$		0			0
$241$		−	9.59482i		−	0.618057i	−0.951053	−	0.309028i	$-0.899996\pi$
$241$		−	9.59482i		−	0.618057i	0.951053	−	0.309028i	$-0.100004\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	11.3137	+	3.60109i	0.722806	+	0.230065i
$246$		0			0
$247$			4.49242i			0.285846i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	5.29723			0.334358			0.167179	−	0.985927i	$-0.446534\pi$
$251$	5.29723			0.334358			0.167179	+	0.985927i	$0.446534\pi$
$252$		0			0
$253$	2.87689			0.180869
$254$		0			0
$255$		0			0
$256$		0			0
$257$			14.5216i			0.905833i	0.891553	+	0.452916i	$0.149616\pi$
$257$			14.5216i			0.905833i	−0.891553	+	0.452916i	$0.850384\pi$
$258$		0			0
$259$	−6.67026	+	4.87689i	−0.414470	+	0.303035i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	20.5931i		−	1.26983i	−0.772583	−	0.634914i	$-0.781036\pi$
$263$		−	20.5931i		−	1.26983i	0.772583	−	0.634914i	$-0.218964\pi$
$264$		0			0
$265$		−	17.0862i		−	1.04960i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	19.0752i		−	1.16303i	−0.813535	−	0.581517i	$-0.802460\pi$
$269$		−	19.0752i		−	1.16303i	0.813535	−	0.581517i	$-0.197540\pi$
$270$		0			0
$271$	−15.2134			−0.924147			−0.462074	−	0.886842i	$-0.652894\pi$
$271$	−15.2134			−0.924147			−0.462074	+	0.886842i	$0.652894\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		−	1.68608i		−	0.101674i
$276$		0			0
$277$	1.36932			0.0822743			0.0411371	−	0.999154i	$-0.486902\pi$
$277$	1.36932			0.0822743			0.0411371	+	0.999154i	$0.486902\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	5.65685			0.337460			0.168730	−	0.985662i	$-0.446033\pi$
$281$	5.65685			0.337460			0.168730	+	0.985662i	$0.446033\pi$
$282$		0			0
$283$	6.14441			0.365247			0.182624	−	0.983183i	$-0.441541\pi$
$283$	6.14441			0.365247			0.182624	+	0.983183i	$0.441541\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	6.78456	+	9.27944i	0.400480	+	0.547748i
$288$		0			0
$289$	−1.87689			−0.110406
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	25.8597i		−	1.51074i	−0.655298	−	0.755371i	$-0.727457\pi$
$293$		−	25.8597i		−	1.51074i	0.655298	−	0.755371i	$-0.272543\pi$
$294$		0			0
$295$		−	11.5076i		−	0.669997i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	6.78456			0.392361
$300$		0			0
$301$	1.36932	+	1.87285i	0.0789261	+	0.107949i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	19.4508			1.11375
$306$		0			0
$307$	−23.2306			−1.32584			−0.662921	−	0.748689i	$-0.730684\pi$
$307$	−23.2306			−1.32584			−0.662921	+	0.748689i	$0.730684\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−25.6509			−1.45453			−0.727265	−	0.686357i	$-0.759209\pi$
$311$	−25.6509			−1.45453			−0.727265	+	0.686357i	$0.759209\pi$
$312$		0			0
$313$		0			0		1.00000			$0$
$313$		0			0		−1.00000			$\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	12.5538			0.705093			0.352547	−	0.935794i	$-0.385316\pi$
$317$	12.5538			0.705093			0.352547	+	0.935794i	$0.385316\pi$
$318$		0			0
$319$			3.50758i			0.196387i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			10.4218i			0.579884i
$324$		0			0
$325$		−	3.97626i		−	0.220563i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−25.8040	+	18.8664i	−1.42262	+	1.04014i
$330$		0			0
$331$			11.6155i			0.638447i	0.947679	+	0.319224i	$0.103422\pi$
$331$			11.6155i			0.638447i	−0.947679	+	0.319224i	$0.896578\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−1.48734			−0.0812619
$336$		0			0
$337$	8.24621			0.449200			0.224600	−	0.974451i	$-0.427892\pi$
$337$	8.24621			0.449200			0.224600	+	0.974451i	$0.427892\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			1.48734i			0.0805438i
$342$		0			0
$343$	5.88148	+	17.5616i	0.317570	+	0.948235i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			33.8434i			1.81681i	0.418095	+	0.908403i	$0.362698\pi$
$347$			33.8434i			1.81681i	−0.418095	+	0.908403i	$0.637302\pi$
$348$		0			0
$349$		−	15.2134i		−	0.814354i	−0.913349	−	0.407177i	$-0.866513\pi$
$349$		−	15.2134i		−	0.814354i	0.913349	−	0.407177i	$-0.133487\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	0.952473i		−	0.0506950i	−0.999679	−	0.0253475i	$-0.991931\pi$
$353$		−	0.952473i		−	0.0506950i	0.999679	−	0.0253475i	$-0.00806923\pi$
$354$		0			0
$355$	−8.83841			−0.469094
$356$		0			0
$357$		0			0
$358$		0			0
$359$			4.51450i			0.238266i	0.992878	+	0.119133i	$0.0380115\pi$
$359$			4.51450i			0.238266i	−0.992878	+	0.119133i	$0.961988\pi$
$360$		0			0
$361$	−13.2462			−0.697169
$362$		0			0
$363$		0			0
$364$		0			0
$365$	6.35324			0.332544
$366$		0			0
$367$	−18.6638			−0.974242			−0.487121	−	0.873334i	$-0.661953\pi$
$367$	−18.6638			−0.974242			−0.487121	+	0.873334i	$0.661953\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	21.5150	−	15.7304i	1.11700	−	0.816684i
$372$		0			0
$373$	10.0000			0.517780			0.258890	−	0.965907i	$-0.416643\pi$
$373$	10.0000			0.517780			0.258890	+	0.965907i	$0.416643\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			8.27190i			0.426025i
$378$		0			0
$379$		−	16.8769i		−	0.866908i	−0.901176	−	0.433454i	$-0.857295\pi$
$379$		−	16.8769i		−	0.866908i	0.901176	−	0.433454i	$-0.142705\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−22.6762			−1.15870			−0.579351	−	0.815078i	$-0.696694\pi$
$383$	−22.6762			−1.15870			−0.579351	+	0.815078i	$0.696694\pi$
$384$		0			0
$385$	−2.87689	+	2.10341i	−0.146620	+	0.107200i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−18.2107			−0.923318			−0.461659	−	0.887057i	$-0.652746\pi$
$389$	−18.2107			−0.923318			−0.461659	+	0.887057i	$0.652746\pi$
$390$		0			0
$391$	15.7392			0.795967
$392$		0			0
$393$		0			0
$394$		0			0
$395$	5.29723			0.266532
$396$		0			0
$397$			28.5539i			1.43308i	0.697546	+	0.716540i	$0.254275\pi$
$397$			28.5539i			1.43308i	−0.697546	+	0.716540i	$0.745725\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−37.8141			−1.88835			−0.944174	−	0.329447i	$-0.893138\pi$
$401$	−37.8141			−1.88835			−0.944174	+	0.329447i	$0.893138\pi$
$402$		0			0
$403$			3.50758i			0.174725i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	2.48023i		−	0.122941i
$408$		0			0
$409$		−	20.8319i		−	1.03007i	−0.857168	−	0.515037i	$-0.827778\pi$
$409$		−	20.8319i		−	1.03007i	0.857168	−	0.515037i	$-0.172222\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	14.4903	−	10.5945i	0.713023	−	0.521319i
$414$		0			0
$415$		−	20.4924i		−	1.00593i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	36.2454			1.77070			0.885351	−	0.464923i	$-0.153918\pi$
$419$	36.2454			1.77070			0.885351	+	0.464923i	$0.153918\pi$
$420$		0			0
$421$	−3.12311			−0.152211			−0.0761054	−	0.997100i	$-0.524249\pi$
$421$	−3.12311			−0.152211			−0.0761054	+	0.997100i	$0.524249\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		−	9.22437i		−	0.447448i
$426$		0			0
$427$	17.9074	+	24.4924i	0.866599	+	1.18527i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	39.1520i		−	1.88589i	−0.332954	−	0.942943i	$-0.608045\pi$
$431$		−	39.1520i		−	1.88589i	0.332954	−	0.942943i	$-0.391955\pi$
$432$		0			0
$433$			30.4268i			1.46222i	0.682261	+	0.731108i	$0.260997\pi$
$433$			30.4268i			1.46222i	−0.682261	+	0.731108i	$0.739003\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	8.68951i		−	0.415676i
$438$		0			0
$439$	16.5604			0.790384			0.395192	−	0.918599i	$-0.370678\pi$
$439$	16.5604			0.790384			0.395192	+	0.918599i	$0.370678\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	16.1764i		−	0.768564i	−0.923216	−	0.384282i	$-0.874449\pi$
$443$		−	16.1764i		−	0.768564i	0.923216	−	0.384282i	$-0.125551\pi$
$444$		0			0
$445$	13.1231			0.622095
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−14.4903			−0.683841			−0.341920	−	0.939729i	$-0.611077\pi$
$449$	−14.4903			−0.683841			−0.341920	+	0.939729i	$0.611077\pi$
$450$		0			0
$451$	−3.45041			−0.162474
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−6.78456	+	4.96046i	−0.318065	+	0.232550i
$456$		0			0
$457$	3.75379			0.175595			0.0877974	−	0.996138i	$-0.472017\pi$
$457$	3.75379			0.175595			0.0877974	+	0.996138i	$0.472017\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			8.48071i			0.394986i	0.980304	+	0.197493i	$0.0632799\pi$
$461$			8.48071i			0.394986i	−0.980304	+	0.197493i	$0.936720\pi$
$462$		0			0
$463$		−	39.6155i		−	1.84109i	−0.390637	−	0.920545i	$-0.627745\pi$
$463$		−	39.6155i		−	1.84109i	0.390637	−	0.920545i	$-0.372255\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−5.29723			−0.245126			−0.122563	−	0.992461i	$-0.539111\pi$
$467$	−5.29723			−0.245126			−0.122563	+	0.992461i	$0.539111\pi$
$468$		0			0
$469$	−1.36932	−	1.87285i	−0.0632292	−	0.0864803i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−0.696391			−0.0320201
$474$		0			0
$475$	−5.09271			−0.233669
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−22.6762			−1.03610			−0.518052	−	0.855349i	$-0.673343\pi$
$479$	−22.6762			−1.03610			−0.518052	+	0.855349i	$0.673343\pi$
$480$		0			0
$481$		−	5.84912i		−	0.266697i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	22.6274			1.02746
$486$		0			0
$487$			31.6155i			1.43264i	0.697774	+	0.716318i	$0.254174\pi$
$487$			31.6155i			1.43264i	−0.697774	+	0.716318i	$0.745826\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			32.2550i			1.45565i	0.685763	+	0.727825i	$0.259469\pi$
$491$			32.2550i			1.45565i	−0.685763	+	0.727825i	$0.740531\pi$
$492$		0			0
$493$			19.1896i			0.864258i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−8.13709	−	11.1293i	−0.364998	−	0.499218i
$498$		0			0
$499$		−	25.8617i		−	1.15773i	−0.815423	−	0.578865i	$-0.803496\pi$
$499$		−	25.8617i		−	1.15773i	0.815423	−	0.578865i	$-0.196504\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−12.0818			−0.538700			−0.269350	−	0.963042i	$-0.586809\pi$
$503$	−12.0818			−0.538700			−0.269350	+	0.963042i	$0.586809\pi$
$504$		0			0
$505$	−17.6155			−0.783881
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	44.7261i		−	1.98245i	−0.132191	−	0.991224i	$-0.542201\pi$
$509$		−	44.7261i		−	1.98245i	0.132191	−	0.991224i	$-0.457799\pi$
$510$		0			0
$511$	5.84912	+	8.00000i	0.258750	+	0.353899i
$512$		0			0
$513$		0			0
$514$		0			0
$515$			19.4508i			0.857104i
$516$		0			0
$517$		−	9.59482i		−	0.421980i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			11.1293i			0.487584i	0.969828	+	0.243792i	$0.0783915\pi$
$521$			11.1293i			0.487584i	−0.969828	+	0.243792i	$0.921609\pi$
$522$		0			0
$523$	40.0216			1.75002			0.875011	−	0.484103i	$-0.160854\pi$
$523$	40.0216			1.75002			0.875011	+	0.484103i	$0.160854\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			8.13709i			0.354457i
$528$		0			0
$529$	9.87689			0.429430
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−8.13709			−0.352456
$534$		0			0
$535$	23.2306			1.00435
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−5.29723	−	1.68608i	−0.228168	−	0.0726244i
$540$		0			0
$541$	−31.6155			−1.35926			−0.679629	−	0.733556i	$-0.737859\pi$
$541$	−31.6155			−1.35926			−0.679629	+	0.733556i	$0.737859\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			13.9867i			0.599126i
$546$		0			0
$547$		−	39.1231i		−	1.67278i	−0.548132	−	0.836392i	$-0.684661\pi$
$547$		−	39.1231i		−	1.67278i	0.548132	−	0.836392i	$-0.315339\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	10.5945			0.451339
$552$		0			0
$553$	4.87689	+	6.67026i	0.207387	+	0.283648i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−41.5345			−1.75987			−0.879936	−	0.475092i	$-0.842415\pi$
$557$	−41.5345			−1.75987			−0.879936	+	0.475092i	$0.842415\pi$
$558$		0			0
$559$	−1.64229			−0.0694616
$560$		0			0
$561$		0			0
$562$		0			0
$563$	22.6762			0.955690			0.477845	−	0.878444i	$-0.341418\pi$
$563$	22.6762			0.955690			0.477845	+	0.878444i	$0.341418\pi$
$564$		0			0
$565$			9.59482i			0.403657i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−14.4903			−0.607466			−0.303733	−	0.952757i	$-0.598233\pi$
$569$	−14.4903			−0.607466			−0.303733	+	0.952757i	$0.598233\pi$
$570$		0			0
$571$			24.8769i			1.04107i	0.853842	+	0.520533i	$0.174267\pi$
$571$			24.8769i			1.04107i	−0.853842	+	0.520533i	$0.825733\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			7.69113i			0.320742i
$576$		0			0
$577$			37.9182i			1.57855i	0.614037	+	0.789277i	$0.289544\pi$
$577$			37.9182i			1.57855i	−0.614037	+	0.789277i	$0.710456\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	25.8040	−	18.8664i	1.07053	−	0.782708i
$582$		0			0
$583$			8.00000i			0.331326i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−24.1636			−0.997338			−0.498669	−	0.866793i	$-0.666178\pi$
$587$	−24.1636			−0.997338			−0.498669	+	0.866793i	$0.666178\pi$
$588$		0			0
$589$	4.49242			0.185107
$590$		0			0
$591$		0			0
$592$		0			0
$593$			21.3062i			0.874939i	0.899233	+	0.437470i	$0.144125\pi$
$593$			21.3062i			0.874939i	−0.899233	+	0.437470i	$0.855875\pi$
$594$		0			0
$595$	−15.7392	+	11.5076i	−0.645246	+	0.471765i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	41.4367i		−	1.69306i	−0.532342	−	0.846529i	$-0.678688\pi$
$599$		−	41.4367i		−	1.69306i	0.532342	−	0.846529i	$-0.321312\pi$
$600$		0			0
$601$			13.3405i			0.544171i	0.962273	+	0.272086i	$0.0877134\pi$
$601$			13.3405i			0.544171i	−0.962273	+	0.272086i	$0.912287\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$			17.5878i			0.715047i
$606$		0			0
$607$	−39.4957			−1.60308			−0.801541	−	0.597940i	$-0.795986\pi$
$607$	−39.4957			−1.60308			−0.801541	+	0.597940i	$0.795986\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	22.6274i		−	0.915407i
$612$		0			0
$613$	−41.2311			−1.66531			−0.832653	−	0.553795i	$-0.813179\pi$
$613$	−41.2311			−1.66531			−0.832653	+	0.553795i	$0.813179\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	11.3137			0.455473			0.227736	−	0.973723i	$-0.426868\pi$
$617$	11.3137			0.455473			0.227736	+	0.973723i	$0.426868\pi$
$618$		0			0
$619$	43.7673			1.75916			0.879578	−	0.475755i	$-0.157825\pi$
$619$	43.7673			1.75916			0.879578	+	0.475755i	$0.157825\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	12.0818	+	16.5246i	0.484047	+	0.662044i
$624$		0			0
$625$	−9.87689			−0.395076
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	13.5691i		−	0.541037i
$630$		0			0
$631$		−	14.6307i		−	0.582438i	−0.956656	−	0.291219i	$-0.905939\pi$
$631$		−	14.6307i		−	0.582438i	0.956656	−	0.291219i	$-0.0940609\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	18.8664			0.748688
$636$		0			0
$637$	−12.4924	−	3.97626i	−0.494968	−	0.157545i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	31.4609			1.24263			0.621315	−	0.783561i	$-0.286599\pi$
$641$	31.4609			1.24263			0.621315	+	0.783561i	$0.286599\pi$
$642$		0			0
$643$	−16.7909			−0.662170			−0.331085	−	0.943601i	$-0.607415\pi$
$643$	−16.7909			−0.662170			−0.331085	+	0.943601i	$0.607415\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	24.1636			0.949968			0.474984	−	0.879994i	$-0.342454\pi$
$647$	24.1636			0.949968			0.474984	+	0.879994i	$0.342454\pi$
$648$		0			0
$649$			5.38800i			0.211497i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	32.7010			1.27969			0.639845	−	0.768504i	$-0.278999\pi$
$653$	32.7010			1.27969			0.639845	+	0.768504i	$0.278999\pi$
$654$		0			0
$655$			11.5076i			0.449638i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			0.794156i			0.0309359i	0.999880	+	0.0154680i	$0.00492380\pi$
$659$			0.794156i			0.0309359i	−0.999880	+	0.0154680i	$0.995076\pi$
$660$		0			0
$661$			15.2134i			0.591732i	0.955229	+	0.295866i	$0.0956082\pi$
$661$			15.2134i			0.591732i	−0.955229	+	0.295866i	$0.904392\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	6.35324	+	8.68951i	0.246368	+	0.336965i
$666$		0			0
$667$		−	16.0000i		−	0.619522i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−9.10712			−0.351576
$672$		0			0
$673$	−11.6155			−0.447746			−0.223873	−	0.974618i	$-0.571870\pi$
$673$	−11.6155			−0.447746			−0.223873	+	0.974618i	$0.571870\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			27.3471i			1.05103i	0.850783	+	0.525516i	$0.176128\pi$
$677$			27.3471i			1.05103i	−0.850783	+	0.525516i	$0.823872\pi$
$678$		0			0
$679$	20.8319	+	28.4924i	0.799456	+	1.09344i
$680$		0			0
$681$		0			0
$682$		0			0
$683$			29.0784i			1.11266i	0.830963	+	0.556328i	$0.187790\pi$
$683$			29.0784i			1.11266i	−0.830963	+	0.556328i	$0.812210\pi$
$684$		0			0
$685$			34.1725i			1.30566i
$686$		0			0
$687$		0			0
$688$		0			0
$689$			18.8664i			0.718751i
$690$		0			0
$691$	−13.3405			−0.507498			−0.253749	−	0.967270i	$-0.581664\pi$
$691$	−13.3405			−0.507498			−0.253749	+	0.967270i	$0.581664\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			28.9807i			1.09930i
$696$		0			0
$697$	−18.8769			−0.715013
$698$		0			0
$699$		0			0
$700$		0			0
$701$	1.24012			0.0468385			0.0234193	−	0.999726i	$-0.492545\pi$
$701$	1.24012			0.0468385			0.0234193	+	0.999726i	$0.492545\pi$
$702$		0			0
$703$	−7.49141			−0.282544
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−16.2177	−	22.1815i	−0.609931	−	0.834220i
$708$		0			0
$709$	6.00000			0.225335			0.112667	−	0.993633i	$-0.464061\pi$
$709$	6.00000			0.225335			0.112667	+	0.993633i	$0.464061\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	6.78456i		−	0.254084i
$714$		0			0
$715$		−	2.52273i		−	0.0943448i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−1.48734			−0.0554683			−0.0277341	−	0.999615i	$-0.508829\pi$
$719$	−1.48734			−0.0554683			−0.0277341	+	0.999615i	$0.508829\pi$
$720$		0			0
$721$	−24.4924	+	17.9074i	−0.912145	+	0.666906i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−9.37720			−0.348261
$726$		0			0
$727$	7.72197			0.286392			0.143196	−	0.989694i	$-0.454262\pi$
$727$	7.72197			0.286392			0.143196	+	0.989694i	$0.454262\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−3.80989			−0.140914
$732$		0			0
$733$			15.2134i			0.561919i	0.959720	+	0.280960i	$0.0906527\pi$
$733$			15.2134i			0.561919i	−0.959720	+	0.280960i	$0.909347\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	0.696391			0.0256519
$738$		0			0
$739$			16.8769i			0.620827i	0.950602	+	0.310413i	$0.100467\pi$
$739$			16.8769i			0.620827i	−0.950602	+	0.310413i	$0.899533\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	33.4952i		−	1.22882i	−0.788988	−	0.614409i	$-0.789394\pi$
$743$		−	33.4952i		−	1.22882i	0.788988	−	0.614409i	$-0.210606\pi$
$744$		0			0
$745$			17.0862i			0.625991i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	21.3873	+	29.2520i	0.781475	+	1.06885i
$750$		0			0
$751$		−	25.3693i		−	0.925740i	−0.886426	−	0.462870i	$-0.846820\pi$
$751$		−	25.3693i		−	0.925740i	0.886426	−	0.462870i	$-0.153180\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−18.8664			−0.686617
$756$		0			0
$757$	−34.9848			−1.27155			−0.635773	−	0.771876i	$-0.719319\pi$
$757$	−34.9848			−1.27155			−0.635773	+	0.771876i	$0.719319\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	0.534864i		−	0.0193888i	−0.999953	−	0.00969440i	$-0.996914\pi$
$761$		−	0.534864i		−	0.0193888i	0.999953	−	0.00969440i	$-0.00308587\pi$
$762$		0			0
$763$	−17.6121	+	12.8769i	−0.637600	+	0.466175i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			12.7065i			0.458805i
$768$		0			0
$769$			2.10341i			0.0758510i	0.999281	+	0.0379255i	$0.0120750\pi$
$769$			2.10341i			0.0758510i	−0.999281	+	0.0379255i	$0.987925\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			13.7779i			0.495558i	0.968817	+	0.247779i	$0.0797007\pi$
$773$			13.7779i			0.495558i	−0.968817	+	0.247779i	$0.920299\pi$
$774$		0			0
$775$	−3.97626			−0.142832
$776$		0			0
$777$		0			0
$778$		0			0
$779$			10.4218i			0.373399i
$780$		0			0
$781$	4.13826			0.148079
$782$		0			0
$783$		0			0
$784$		0			0
$785$	32.1573			1.14774
$786$		0			0
$787$	−45.4096			−1.61868			−0.809338	−	0.587343i	$-0.800174\pi$
$787$	−45.4096			−1.61868			−0.809338	+	0.587343i	$0.800174\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−12.0818	+	8.83348i	−0.429579	+	0.314082i
$792$		0			0
$793$	−21.4773			−0.762680
$794$		0			0
$795$		0			0
$796$		0			0
$797$			20.5625i			0.728361i	0.931328	+	0.364181i	$0.118651\pi$
$797$			20.5625i			0.728361i	−0.931328	+	0.364181i	$0.881349\pi$
$798$		0			0
$799$		−	52.4924i		−	1.85705i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−2.97467			−0.104974
$804$		0			0
$805$	13.1231	−	9.59482i	0.462529	−	0.338173i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	30.7645			1.08162			0.540811	−	0.841144i	$-0.318117\pi$
$809$	30.7645			1.08162			0.540811	+	0.841144i	$0.318117\pi$
$810$		0			0
$811$	28.3234			0.994567			0.497284	−	0.867588i	$-0.334331\pi$
$811$	28.3234			0.994567			0.497284	+	0.867588i	$0.334331\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	12.0818			0.423207
$816$		0			0
$817$			2.10341i			0.0735891i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−5.11313			−0.178449			−0.0892247	−	0.996012i	$-0.528439\pi$
$821$	−5.11313			−0.178449			−0.0892247	+	0.996012i	$0.528439\pi$
$822$		0			0
$823$			37.8617i			1.31978i	0.751363	+	0.659889i	$0.229397\pi$
$823$			37.8617i			1.31978i	−0.751363	+	0.659889i	$0.770603\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	38.6083i		−	1.34254i	−0.741212	−	0.671271i	$-0.765749\pi$
$827$		−	38.6083i		−	1.34254i	0.741212	−	0.671271i	$-0.234251\pi$
$828$		0			0
$829$			18.9591i			0.658476i	0.944247	+	0.329238i	$0.106792\pi$
$829$			18.9591i			0.658476i	−0.944247	+	0.329238i	$0.893208\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−28.9807	−	9.22437i	−1.00412	−	0.319606i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$	10.5945			0.365761			0.182881	−	0.983135i	$-0.441458\pi$
$839$	10.5945			0.365761			0.182881	+	0.983135i	$0.441458\pi$
$840$		0			0
$841$	−9.49242			−0.327325
$842$		0			0
$843$		0			0
$844$		0			0
$845$			16.1005i			0.553874i
$846$		0			0
$847$	−22.1466	+	16.1922i	−0.760965	+	0.556372i
$848$		0			0
$849$		0			0
$850$		0			0
$851$			11.3137i			0.387829i
$852$		0			0
$853$			26.4505i			0.905648i	0.891600	+	0.452824i	$0.149583\pi$
$853$			26.4505i			0.905648i	−0.891600	+	0.452824i	$0.850417\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	48.8620i		−	1.66910i	−0.550936	−	0.834548i	$-0.685729\pi$
$857$		−	48.8620i		−	1.66910i	0.550936	−	0.834548i	$-0.314271\pi$
$858$		0			0
$859$	−0.295294			−0.0100753			−0.00503765	−	0.999987i	$-0.501604\pi$
$859$	−0.295294			−0.0100753			−0.00503765	+	0.999987i	$0.501604\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			26.2500i			0.893560i	0.894644	+	0.446780i	$0.147429\pi$
$863$			26.2500i			0.893560i	−0.894644	+	0.446780i	$0.852571\pi$
$864$		0			0
$865$	−6.10795			−0.207677
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−2.48023			−0.0841361
$870$		0			0
$871$	1.64229			0.0556470
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−18.8664	−	25.8040i	−0.637799	−	0.872336i
$876$		0			0
$877$	−8.24621			−0.278455			−0.139227	−	0.990260i	$-0.544462\pi$
$877$	−8.24621			−0.278455			−0.139227	+	0.990260i	$0.544462\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	45.4697i		−	1.53191i	−0.642891	−	0.765957i	$-0.722265\pi$
$881$		−	45.4697i		−	1.53191i	0.642891	−	0.765957i	$-0.277735\pi$
$882$		0			0
$883$		−	49.8617i		−	1.67798i	−0.544146	−	0.838991i	$-0.683146\pi$
$883$		−	49.8617i		−	1.67798i	0.544146	−	0.838991i	$-0.316854\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	10.5945			0.355727			0.177863	−	0.984055i	$-0.443081\pi$
$887$	10.5945			0.355727			0.177863	+	0.984055i	$0.443081\pi$
$888$		0			0
$889$	17.3693	+	23.7565i	0.582548	+	0.796767i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−28.9807			−0.969801
$894$		0			0
$895$	−27.4375			−0.917133
$896$		0			0
$897$		0			0
$898$		0			0
$899$	8.27190			0.275883
$900$		0			0
$901$			43.7673i			1.45810i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	3.17662			0.105595
$906$		0			0
$907$			56.1080i			1.86303i	0.363698	+	0.931517i	$0.381514\pi$
$907$			56.1080i			1.86303i	−0.363698	+	0.931517i	$0.618486\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			5.21089i			0.172645i	0.996267	+	0.0863223i	$0.0275115\pi$
$911$			5.21089i			0.172645i	−0.996267	+	0.0863223i	$0.972489\pi$
$912$		0			0
$913$			9.59482i			0.317542i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−14.4903	+	10.5945i	−0.478513	+	0.349860i
$918$		0			0
$919$		−	8.38447i		−	0.276578i	−0.990392	−	0.138289i	$-0.955840\pi$
$919$		−	8.38447i		−	0.276578i	0.990392	−	0.138289i	$-0.0441603\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	9.75924			0.321229
$924$		0			0
$925$	6.63068			0.218016
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	8.15465i		−	0.267545i	−0.991012	−	0.133773i	$-0.957291\pi$
$929$		−	8.15465i		−	0.267545i	0.991012	−	0.133773i	$-0.0427092\pi$
$930$		0			0
$931$	−5.09271	+	16.0000i	−0.166907	+	0.524379i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	5.85238i		−	0.191393i
$936$		0			0
$937$			43.7673i			1.42982i	0.699219	+	0.714908i	$0.253531\pi$
$937$			43.7673i			1.42982i	−0.699219	+	0.714908i	$0.746469\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	27.7647i		−	0.905102i	−0.891738	−	0.452551i	$-0.850514\pi$
$941$		−	27.7647i		−	0.905102i	0.891738	−	0.452551i	$-0.149486\pi$
$942$		0			0
$943$	15.7392			0.512540
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	53.0986i		−	1.72547i	−0.505653	−	0.862737i	$-0.668749\pi$
$947$		−	53.0986i		−	1.72547i	0.505653	−	0.862737i	$-0.331251\pi$
$948$		0			0
$949$	−7.01515			−0.227722
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−2.48023			−0.0803426			−0.0401713	−	0.999193i	$-0.512790\pi$
$953$	−2.48023			−0.0803426			−0.0401713	+	0.999193i	$0.512790\pi$
$954$		0			0
$955$	−25.3341			−0.819791
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−43.0299	+	31.4609i	−1.38951	+	1.01593i
$960$		0			0
$961$	−27.4924			−0.886852
$962$		0			0
$963$		0			0
$964$		0			0
$965$			22.6762i			0.729974i
$966$		0			0
$967$		−	22.6307i		−	0.727754i	−0.931447	−	0.363877i	$-0.881453\pi$
$967$		−	22.6307i		−	0.727754i	0.931447	−	0.363877i	$-0.118547\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−32.4355			−1.04090			−0.520452	−	0.853891i	$-0.674237\pi$
$971$	−32.4355			−1.04090			−0.520452	+	0.853891i	$0.674237\pi$
$972$		0			0
$973$	−36.4924	+	26.6811i	−1.16989	+	0.855355i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	20.8436			0.666845			0.333423	−	0.942777i	$-0.391796\pi$
$977$	20.8436			0.666845			0.333423	+	0.942777i	$0.391796\pi$
$978$		0			0
$979$	−6.14441			−0.196376
$980$		0			0
$981$		0			0
$982$		0			0
$983$	51.3018			1.63627			0.818137	−	0.575023i	$-0.195007\pi$
$983$	51.3018			1.63627			0.818137	+	0.575023i	$0.195007\pi$
$984$		0			0
$985$		−	7.49141i		−	0.238696i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	3.17662			0.101011
$990$		0			0
$991$			14.6307i			0.464759i	0.972625	+	0.232379i	$0.0746511\pi$
$991$			14.6307i			0.464759i	−0.972625	+	0.232379i	$0.925349\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			15.3823i			0.487650i
$996$		0			0
$997$			30.6573i			0.970927i	0.874257	+	0.485464i	$0.161349\pi$
$997$			30.6573i			0.970927i	−0.874257	+	0.485464i	$0.838651\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4032.2.b.r.3583.10		16
3.2	odd	2	inner	4032.2.b.r.3583.6		16
4.3	odd	2	inner	4032.2.b.r.3583.11		16
7.6	odd	2	inner	4032.2.b.r.3583.8		16
8.3	odd	2		2016.2.b.d.1567.7	yes	16
8.5	even	2		2016.2.b.d.1567.6	yes	16
12.11	even	2	inner	4032.2.b.r.3583.7		16
21.20	even	2	inner	4032.2.b.r.3583.12		16
24.5	odd	2		2016.2.b.d.1567.10	yes	16
24.11	even	2		2016.2.b.d.1567.11	yes	16
28.27	even	2	inner	4032.2.b.r.3583.5		16
56.13	odd	2		2016.2.b.d.1567.12	yes	16
56.27	even	2		2016.2.b.d.1567.9	yes	16
84.83	odd	2	inner	4032.2.b.r.3583.9		16
168.83	odd	2		2016.2.b.d.1567.5	✓	16
168.125	even	2		2016.2.b.d.1567.8	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2016.2.b.d.1567.5	✓	16	168.83	odd	2
2016.2.b.d.1567.6	yes	16	8.5	even	2
2016.2.b.d.1567.7	yes	16	8.3	odd	2
2016.2.b.d.1567.8	yes	16	168.125	even	2
2016.2.b.d.1567.9	yes	16	56.27	even	2
2016.2.b.d.1567.10	yes	16	24.5	odd	2
2016.2.b.d.1567.11	yes	16	24.11	even	2
2016.2.b.d.1567.12	yes	16	56.13	odd	2
4032.2.b.r.3583.5		16	28.27	even	2	inner
4032.2.b.r.3583.6		16	3.2	odd	2	inner
4032.2.b.r.3583.7		16	12.11	even	2	inner
4032.2.b.r.3583.8		16	7.6	odd	2	inner
4032.2.b.r.3583.9		16	84.83	odd	2	inner
4032.2.b.r.3583.10		16	1.1	even	1	trivial
4032.2.b.r.3583.11		16	4.3	odd	2	inner
4032.2.b.r.3583.12		16	21.20	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:	\( N \)	\(=\)	\( 4032 = 2^{6} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4032.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.69614i q^{5} +(-2.13578 + 1.56155i) q^{7} +O(q^{10})\)	\(q+1.69614i q^{5} +(-2.13578 + 1.56155i) q^{7} -0.794156i q^{11} -1.87285i q^{13} -4.34475i q^{17} -2.39871 q^{19} +3.62258i q^{23} +2.12311 q^{25} -4.41674 q^{29} -1.87285 q^{31} +(-2.64861 - 3.62258i) q^{35} +3.12311 q^{37} -4.34475i q^{41} -0.876894i q^{43} +12.0818 q^{47} +(2.12311 - 6.67026i) q^{49} -10.0736 q^{53} +1.34700 q^{55} -6.78456 q^{59} -11.4677i q^{61} +3.17662 q^{65} +0.876894i q^{67} +5.21089i q^{71} -3.74571i q^{73} +(1.24012 + 1.69614i) q^{77} -3.12311i q^{79} -12.0818 q^{83} +7.36932 q^{85} -7.73704i q^{89} +(2.92456 + 4.00000i) q^{91} -4.06854i q^{95} -13.3405i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q - 32 q^{25} - 16 q^{37} - 32 q^{49} - 80 q^{85}+O(q^{100}) \) 16 * q - 32 * q^25 - 16 * q^37 - 32 * q^49 - 80 * q^85

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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