LMFDB - Embedded newform 1632.2.l.b.1393.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1632,2,Mod(1393,1632)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1632, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 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("1632.1393");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1632 = 2^{5} \cdot 3 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1632.l (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:	$13.0315856099$
Analytic rank:	$0$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - x^{16} - 2x^{14} + 2x^{12} - 4x^{11} + 4x^{10} + 8x^{8} - 16x^{7} + 16x^{6} - 64x^{4} - 128x^{2} + 512 $ x^18 - x^16 - 2x^14 + 2x^12 - 4x^11 + 4x^10 + 8x^8 - 16x^7 + 16x^6 - 64x^4 - 128*x^2 + 512
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{17} $
Twist minimal:	no (minimal twist has level 408)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1393.4
Root			$-0.943929 + 1.05309i$ of defining polynomial
Character	$\chi$	$=$	1632.1393
Dual form			1632.2.l.b.1393.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -1.57273 q^{5} +1.68207i q^{7} +1.00000 q^{9} +3.33403 q^{11} -0.177520i q^{13} -1.57273 q^{15} +(-3.53017 + 2.13024i) q^{17} +2.58556i q^{19} +1.68207i q^{21} +3.40373i q^{23} -2.52653 q^{25} +1.00000 q^{27} +2.46715 q^{29} +4.84803i q^{31} +3.33403 q^{33} -2.64545i q^{35} +5.33914 q^{37} -0.177520i q^{39} -4.79275i q^{41} +5.48185i q^{43} -1.57273 q^{45} -5.72051 q^{47} +4.17063 q^{49} +(-3.53017 + 2.13024i) q^{51} +4.52038i q^{53} -5.24353 q^{55} +2.58556i q^{57} -6.32370i q^{59} +6.28670 q^{61} +1.68207i q^{63} +0.279190i q^{65} +15.8300i q^{67} +3.40373i q^{69} +13.3194i q^{71} +4.38096i q^{73} -2.52653 q^{75} +5.60809i q^{77} +7.47987i q^{79} +1.00000 q^{81} -6.58394i q^{83} +(5.55199 - 3.35029i) q^{85} +2.46715 q^{87} -8.64940 q^{89} +0.298601 q^{91} +4.84803i q^{93} -4.06638i q^{95} +13.1686i q^{97} +3.33403 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q + 18 q^{3} + 4 q^{5} + 18 q^{9} + 4 q^{15} - 2 q^{17} + 22 q^{25} + 18 q^{27} - 12 q^{29} + 16 q^{37} + 4 q^{45} - 18 q^{49} - 2 q^{51} + 16 q^{55} + 16 q^{61} + 22 q^{75} + 18 q^{81} - 16 q^{85} - 12 q^{87}+ \cdots + 20 q^{89}+O(q^{100}) $ 18 * q + 18 * q^3 + 4 * q^5 + 18 * q^9 + 4 * q^15 - 2 * q^17 + 22 * q^25 + 18 * q^27 - 12 * q^29 + 16 * q^37 + 4 * q^45 - 18 * q^49 - 2 * q^51 + 16 * q^55 + 16 * q^61 + 22 * q^75 + 18 * q^81 - 16 * q^85 - 12 * q^87 + 20 * q^89

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1632\mathbb{Z}\right)^\times$.

$n$	$511$	$545$	$613$	$1057$
$\chi(n)$	$1$	$1$	$-1$	$-1$

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
$3$	1.00000			0.577350
$3$	1.00000			0.577350
$4$		0			0
$5$	−1.57273			−0.703346			−0.351673	−	0.936123i	$-0.614387\pi$
$5$	−1.57273			−0.703346			−0.351673	+	0.936123i	$0.614387\pi$
$6$		0			0
$7$			1.68207i			0.635764i	0.948130	+	0.317882i	$0.102972\pi$
$7$			1.68207i			0.635764i	−0.948130	+	0.317882i	$0.897028\pi$
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$	3.33403			1.00525			0.502624	−	0.864505i	$-0.332368\pi$
$11$	3.33403			1.00525			0.502624	+	0.864505i	$0.332368\pi$
$12$		0			0
$13$		−	0.177520i		−	0.0492351i	−0.999697	−	0.0246175i	$-0.992163\pi$
$13$		−	0.177520i		−	0.0492351i	0.999697	−	0.0246175i	$-0.00783680\pi$
$14$		0			0
$15$	−1.57273			−0.406077
$16$		0			0
$17$	−3.53017	+	2.13024i	−0.856191	+	0.516659i
$17$	−3.53017	+	2.13024i	−0.856191	+	0.516659i
$18$		0			0
$19$			2.58556i			0.593168i	0.955007	+	0.296584i	$0.0958475\pi$
$19$			2.58556i			0.593168i	−0.955007	+	0.296584i	$0.904152\pi$
$20$		0			0
$21$			1.68207i			0.367059i
$22$		0			0
$23$			3.40373i			0.709726i	0.934918	+	0.354863i	$0.115473\pi$
$23$			3.40373i			0.709726i	−0.934918	+	0.354863i	$0.884527\pi$
$24$		0			0
$25$	−2.52653			−0.505305
$26$		0			0
$27$	1.00000			0.192450
$28$		0			0
$29$	2.46715			0.458138			0.229069	−	0.973410i	$-0.426432\pi$
$29$	2.46715			0.458138			0.229069	+	0.973410i	$0.426432\pi$
$30$		0			0
$31$			4.84803i			0.870731i	0.900254	+	0.435366i	$0.143381\pi$
$31$			4.84803i			0.870731i	−0.900254	+	0.435366i	$0.856619\pi$
$32$		0			0
$33$	3.33403			0.580380
$34$		0			0
$35$		−	2.64545i		−	0.447162i
$36$		0			0
$37$	5.33914			0.877750			0.438875	−	0.898548i	$-0.355377\pi$
$37$	5.33914			0.877750			0.438875	+	0.898548i	$0.355377\pi$
$38$		0			0
$39$		−	0.177520i		−	0.0284259i
$40$		0			0
$41$		−	4.79275i		−	0.748502i	−0.927327	−	0.374251i	$-0.877900\pi$
$41$		−	4.79275i		−	0.748502i	0.927327	−	0.374251i	$-0.122100\pi$
$42$		0			0
$43$			5.48185i			0.835975i	0.908453	+	0.417987i	$0.137264\pi$
$43$			5.48185i			0.835975i	−0.908453	+	0.417987i	$0.862736\pi$
$44$		0			0
$45$	−1.57273			−0.234449
$46$		0			0
$47$	−5.72051			−0.834422			−0.417211	−	0.908810i	$-0.636992\pi$
$47$	−5.72051			−0.834422			−0.417211	+	0.908810i	$0.636992\pi$
$48$		0			0
$49$	4.17063			0.595804
$50$		0			0
$51$	−3.53017	+	2.13024i	−0.494322	+	0.298293i
$52$		0			0
$53$			4.52038i			0.620922i	0.950586	+	0.310461i	$0.100483\pi$
$53$			4.52038i			0.620922i	−0.950586	+	0.310461i	$0.899517\pi$
$54$		0			0
$55$	−5.24353			−0.707037
$56$		0			0
$57$			2.58556i			0.342466i
$58$		0			0
$59$		−	6.32370i		−	0.823276i	−0.911348	−	0.411638i	$-0.864957\pi$
$59$		−	6.32370i		−	0.823276i	0.911348	−	0.411638i	$-0.135043\pi$
$60$		0			0
$61$	6.28670			0.804929			0.402464	−	0.915436i	$-0.368154\pi$
$61$	6.28670			0.804929			0.402464	+	0.915436i	$0.368154\pi$
$62$		0			0
$63$			1.68207i			0.211921i
$64$		0			0
$65$			0.279190i			0.0346293i
$66$		0			0
$67$			15.8300i			1.93395i	0.254875	+	0.966974i	$0.417966\pi$
$67$			15.8300i			1.93395i	−0.254875	+	0.966974i	$0.582034\pi$
$68$		0			0
$69$			3.40373i			0.409761i
$70$		0			0
$71$			13.3194i			1.58073i	0.612638	+	0.790364i	$0.290108\pi$
$71$			13.3194i			1.58073i	−0.612638	+	0.790364i	$0.709892\pi$
$72$		0			0
$73$			4.38096i			0.512752i	0.966577	+	0.256376i	$0.0825286\pi$
$73$			4.38096i			0.512752i	−0.966577	+	0.256376i	$0.917471\pi$
$74$		0			0
$75$	−2.52653			−0.291738
$76$		0			0
$77$			5.60809i			0.639101i
$78$		0			0
$79$			7.47987i			0.841551i	0.907165	+	0.420776i	$0.138242\pi$
$79$			7.47987i			0.841551i	−0.907165	+	0.420776i	$0.861758\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$		−	6.58394i		−	0.722681i	−0.932434	−	0.361341i	$-0.882319\pi$
$83$		−	6.58394i		−	0.722681i	0.932434	−	0.361341i	$-0.117681\pi$
$84$		0			0
$85$	5.55199	−	3.35029i	0.602198	−	0.363390i
$86$		0			0
$87$	2.46715			0.264506
$88$		0			0
$89$	−8.64940			−0.916835			−0.458417	−	0.888737i	$-0.651584\pi$
$89$	−8.64940			−0.916835			−0.458417	+	0.888737i	$0.651584\pi$
$90$		0			0
$91$	0.298601			0.0313019
$92$		0			0
$93$			4.84803i			0.502717i
$94$		0			0
$95$		−	4.06638i		−	0.417202i
$96$		0			0
$97$			13.1686i			1.33707i	0.743680	+	0.668536i	$0.233079\pi$
$97$			13.1686i			1.33707i	−0.743680	+	0.668536i	$0.766921\pi$
$98$		0			0
$99$	3.33403			0.335083
$100$		0			0
$101$		−	1.10135i		−	0.109588i	−0.998498	−	0.0547940i	$-0.982550\pi$
$101$		−	1.10135i		−	0.109588i	0.998498	−	0.0547940i	$-0.0174502\pi$
$102$		0			0
$103$	−9.03960			−0.890698			−0.445349	−	0.895357i	$-0.646920\pi$
$103$	−9.03960			−0.890698			−0.445349	+	0.895357i	$0.646920\pi$
$104$		0			0
$105$		−	2.64545i		−	0.258169i
$106$		0			0
$107$	18.3379			1.77280			0.886398	−	0.462923i	$-0.153200\pi$
$107$	18.3379			1.77280			0.886398	+	0.462923i	$0.153200\pi$
$108$		0			0
$109$	1.67742			0.160668			0.0803340	−	0.996768i	$-0.474401\pi$
$109$	1.67742			0.160668			0.0803340	+	0.996768i	$0.474401\pi$
$110$		0			0
$111$	5.33914			0.506769
$112$		0			0
$113$		−	6.87985i		−	0.647202i	−0.946194	−	0.323601i	$-0.895106\pi$
$113$		−	6.87985i		−	0.647202i	0.946194	−	0.323601i	$-0.104894\pi$
$114$		0			0
$115$		−	5.35314i		−	0.499183i
$116$		0			0
$117$		−	0.177520i		−	0.0164117i
$118$		0			0
$119$	−3.58322	−	5.93800i	−0.328473	−	0.544336i
$120$		0			0
$121$	0.115770			0.0105245
$122$		0			0
$123$		−	4.79275i		−	0.432148i
$124$		0			0
$125$	11.8372			1.05875
$126$		0			0
$127$	4.19770			0.372486			0.186243	−	0.982504i	$-0.440369\pi$
$127$	4.19770			0.372486			0.186243	+	0.982504i	$0.440369\pi$
$128$		0			0
$129$			5.48185i			0.482650i
$130$		0			0
$131$	3.59013			0.313671			0.156835	−	0.987625i	$-0.449871\pi$
$131$	3.59013			0.313671			0.156835	+	0.987625i	$0.449871\pi$
$132$		0			0
$133$	−4.34910			−0.377115
$134$		0			0
$135$	−1.57273			−0.135359
$136$		0			0
$137$	−11.9023			−1.01688			−0.508441	−	0.861097i	$-0.669778\pi$
$137$	−11.9023			−1.01688			−0.508441	+	0.861097i	$0.669778\pi$
$138$		0			0
$139$	−5.96225			−0.505711			−0.252856	−	0.967504i	$-0.581370\pi$
$139$	−5.96225			−0.505711			−0.252856	+	0.967504i	$0.581370\pi$
$140$		0			0
$141$	−5.72051			−0.481754
$142$		0			0
$143$		−	0.591856i		−	0.0494935i
$144$		0			0
$145$	−3.88016			−0.322230
$146$		0			0
$147$	4.17063			0.343987
$148$		0			0
$149$		−	22.1583i		−	1.81528i	−0.419751	−	0.907639i	$-0.637883\pi$
$149$		−	22.1583i		−	1.81528i	0.419751	−	0.907639i	$-0.362117\pi$
$150$		0			0
$151$	4.16246			0.338736			0.169368	−	0.985553i	$-0.445827\pi$
$151$	4.16246			0.338736			0.169368	+	0.985553i	$0.445827\pi$
$152$		0			0
$153$	−3.53017	+	2.13024i	−0.285397	+	0.172220i
$154$		0			0
$155$		−	7.62463i		−	0.612425i
$156$		0			0
$157$		−	1.66640i		−	0.132993i	−0.997787	−	0.0664964i	$-0.978818\pi$
$157$		−	1.66640i		−	0.132993i	0.997787	−	0.0664964i	$-0.0211821\pi$
$158$		0			0
$159$			4.52038i			0.358489i
$160$		0			0
$161$	−5.72532			−0.451219
$162$		0			0
$163$	5.18018			0.405743			0.202871	−	0.979205i	$-0.434973\pi$
$163$	5.18018			0.405743			0.202871	+	0.979205i	$0.434973\pi$
$164$		0			0
$165$	−5.24353			−0.408208
$166$		0			0
$167$		−	20.8328i		−	1.61209i	−0.591854	−	0.806046i	$-0.701604\pi$
$167$		−	20.8328i		−	1.61209i	0.591854	−	0.806046i	$-0.298396\pi$
$168$		0			0
$169$	12.9685			0.997576
$170$		0			0
$171$			2.58556i			0.197723i
$172$		0			0
$173$	21.8618			1.66212			0.831062	−	0.556179i	$-0.187733\pi$
$173$	21.8618			1.66212			0.831062	+	0.556179i	$0.187733\pi$
$174$		0			0
$175$		−	4.24980i		−	0.321255i
$176$		0			0
$177$		−	6.32370i		−	0.475318i
$178$		0			0
$179$		−	4.38920i		−	0.328064i	−0.986455	−	0.164032i	$-0.947550\pi$
$179$		−	4.38920i		−	0.328064i	0.986455	−	0.164032i	$-0.0524500\pi$
$180$		0			0
$181$	−11.1107			−0.825852			−0.412926	−	0.910765i	$-0.635493\pi$
$181$	−11.1107			−0.825852			−0.412926	+	0.910765i	$0.635493\pi$
$182$		0			0
$183$	6.28670			0.464726
$184$		0			0
$185$	−8.39702			−0.617361
$186$		0			0
$187$	−11.7697	+	7.10229i	−0.860685	+	0.519371i
$188$		0			0
$189$			1.68207i			0.122353i
$190$		0			0
$191$	−11.5334			−0.834530			−0.417265	−	0.908785i	$-0.637011\pi$
$191$	−11.5334			−0.834530			−0.417265	+	0.908785i	$0.637011\pi$
$192$		0			0
$193$			13.5081i			0.972334i	0.873866	+	0.486167i	$0.161605\pi$
$193$			13.5081i			0.972334i	−0.873866	+	0.486167i	$0.838395\pi$
$194$		0			0
$195$			0.279190i			0.0199932i
$196$		0			0
$197$	−20.9839			−1.49504			−0.747519	−	0.664240i	$-0.768755\pi$
$197$	−20.9839			−1.49504			−0.747519	+	0.664240i	$0.768755\pi$
$198$		0			0
$199$		−	27.9971i		−	1.98466i	−0.123605	−	0.992332i	$-0.539445\pi$
$199$		−	27.9971i		−	1.98466i	0.123605	−	0.992332i	$-0.460555\pi$
$200$		0			0
$201$			15.8300i			1.11657i
$202$		0			0
$203$			4.14993i			0.291268i
$204$		0			0
$205$			7.53770i			0.526456i
$206$		0			0
$207$			3.40373i			0.236575i
$208$		0			0
$209$			8.62034i			0.596282i
$210$		0			0
$211$	−14.4756			−0.996540			−0.498270	−	0.867022i	$-0.666031\pi$
$211$	−14.4756			−0.996540			−0.498270	+	0.867022i	$0.666031\pi$
$212$		0			0
$213$			13.3194i			0.912633i
$214$		0			0
$215$		−	8.62147i		−	0.587979i
$216$		0			0
$217$	−8.15474			−0.553580
$218$		0			0
$219$			4.38096i			0.296038i
$220$		0			0
$221$	0.378159	+	0.626674i	0.0254377	+	0.0421546i
$222$		0			0
$223$	27.2591			1.82540			0.912702	−	0.408626i	$-0.133992\pi$
$223$	27.2591			1.82540			0.912702	+	0.408626i	$0.133992\pi$
$224$		0			0
$225$	−2.52653			−0.168435
$226$		0			0
$227$	9.26818			0.615151			0.307575	−	0.951524i	$-0.400482\pi$
$227$	9.26818			0.615151			0.307575	+	0.951524i	$0.400482\pi$
$228$		0			0
$229$		−	19.8306i		−	1.31044i	−0.755437	−	0.655222i	$-0.772575\pi$
$229$		−	19.8306i		−	1.31044i	0.755437	−	0.655222i	$-0.227425\pi$
$230$		0			0
$231$			5.60809i			0.368985i
$232$		0			0
$233$			9.37047i			0.613880i	0.951729	+	0.306940i	$0.0993051\pi$
$233$			9.37047i			0.613880i	−0.951729	+	0.306940i	$0.900695\pi$
$234$		0			0
$235$	8.99681			0.586887
$236$		0			0
$237$			7.47987i			0.485870i
$238$		0			0
$239$	7.48309			0.484041			0.242021	−	0.970271i	$-0.422190\pi$
$239$	7.48309			0.484041			0.242021	+	0.970271i	$0.422190\pi$
$240$		0			0
$241$		−	12.5710i		−	0.809768i	−0.914368	−	0.404884i	$-0.867312\pi$
$241$		−	12.5710i		−	0.809768i	0.914368	−	0.404884i	$-0.132688\pi$
$242$		0			0
$243$	1.00000			0.0641500
$244$		0			0
$245$	−6.55926			−0.419056
$246$		0			0
$247$	0.458988			0.0292047
$248$		0			0
$249$		−	6.58394i		−	0.417240i
$250$		0			0
$251$			1.95634i			0.123483i	0.998092	+	0.0617416i	$0.0196655\pi$
$251$			1.95634i			0.123483i	−0.998092	+	0.0617416i	$0.980335\pi$
$252$		0			0
$253$			11.3481i			0.713451i
$254$		0			0
$255$	5.55199	−	3.35029i	0.347679	−	0.209803i
$256$		0			0
$257$	13.3326			0.831664			0.415832	−	0.909441i	$-0.363490\pi$
$257$	13.3326			0.831664			0.415832	+	0.909441i	$0.363490\pi$
$258$		0			0
$259$			8.98083i			0.558042i
$260$		0			0
$261$	2.46715			0.152713
$262$		0			0
$263$	−19.5100			−1.20304			−0.601518	−	0.798859i	$-0.705437\pi$
$263$	−19.5100			−1.20304			−0.601518	+	0.798859i	$0.705437\pi$
$264$		0			0
$265$		−	7.10933i		−	0.436723i
$266$		0			0
$267$	−8.64940			−0.529335
$268$		0			0
$269$	6.00877			0.366361			0.183180	−	0.983079i	$-0.441361\pi$
$269$	6.00877			0.366361			0.183180	+	0.983079i	$0.441361\pi$
$270$		0			0
$271$	13.9230			0.845760			0.422880	−	0.906186i	$-0.361019\pi$
$271$	13.9230			0.845760			0.422880	+	0.906186i	$0.361019\pi$
$272$		0			0
$273$	0.298601			0.0180722
$274$		0			0
$275$	−8.42352			−0.507957
$276$		0			0
$277$	−27.7674			−1.66838			−0.834192	−	0.551474i	$-0.814066\pi$
$277$	−27.7674			−1.66838			−0.834192	+	0.551474i	$0.814066\pi$
$278$		0			0
$279$			4.84803i			0.290244i
$280$		0			0
$281$	16.6843			0.995302			0.497651	−	0.867377i	$-0.334196\pi$
$281$	16.6843			0.995302			0.497651	+	0.867377i	$0.334196\pi$
$282$		0			0
$283$	−9.70263			−0.576761			−0.288381	−	0.957516i	$-0.593117\pi$
$283$	−9.70263			−0.576761			−0.288381	+	0.957516i	$0.593117\pi$
$284$		0			0
$285$		−	4.06638i		−	0.240872i
$286$		0			0
$287$	8.06176			0.475871
$288$		0			0
$289$	7.92416	−	15.0402i	0.466127	−	0.884718i
$290$		0			0
$291$			13.1686i			0.771959i
$292$		0			0
$293$		−	31.3940i		−	1.83406i	−0.398824	−	0.917028i	$-0.630582\pi$
$293$		−	31.3940i		−	1.83406i	0.398824	−	0.917028i	$-0.369418\pi$
$294$		0			0
$295$			9.94546i			0.579047i
$296$		0			0
$297$	3.33403			0.193460
$298$		0			0
$299$	0.604228			0.0349434
$300$		0			0
$301$	−9.22089			−0.531483
$302$		0			0
$303$		−	1.10135i		−	0.0632707i
$304$		0			0
$305$	−9.88727			−0.566143
$306$		0			0
$307$			1.53330i			0.0875099i	0.999042	+	0.0437549i	$0.0139321\pi$
$307$			1.53330i			0.0875099i	−0.999042	+	0.0437549i	$0.986068\pi$
$308$		0			0
$309$	−9.03960			−0.514245
$310$		0			0
$311$			1.74672i			0.0990472i	0.998773	+	0.0495236i	$0.0157703\pi$
$311$			1.74672i			0.0990472i	−0.998773	+	0.0495236i	$0.984230\pi$
$312$		0			0
$313$		−	31.9640i		−	1.80671i	−0.428894	−	0.903355i	$-0.641097\pi$
$313$		−	31.9640i		−	1.80671i	0.428894	−	0.903355i	$-0.358903\pi$
$314$		0			0
$315$		−	2.64545i		−	0.149054i
$316$		0			0
$317$	−3.80550			−0.213738			−0.106869	−	0.994273i	$-0.534083\pi$
$317$	−3.80550			−0.213738			−0.106869	+	0.994273i	$0.534083\pi$
$318$		0			0
$319$	8.22556			0.460543
$320$		0			0
$321$	18.3379			1.02352
$322$		0			0
$323$	−5.50786	−	9.12746i	−0.306466	−	0.507865i
$324$		0			0
$325$			0.448508i			0.0248787i
$326$		0			0
$327$	1.67742			0.0927617
$328$		0			0
$329$		−	9.62232i		−	0.530496i
$330$		0			0
$331$		−	10.4319i		−	0.573388i	−0.958022	−	0.286694i	$-0.907444\pi$
$331$		−	10.4319i		−	0.573388i	0.958022	−	0.286694i	$-0.0925563\pi$
$332$		0			0
$333$	5.33914			0.292583
$334$		0			0
$335$		−	24.8964i		−	1.36023i
$336$		0			0
$337$			21.4237i			1.16702i	0.812105	+	0.583511i	$0.198321\pi$
$337$			21.4237i			1.16702i	−0.812105	+	0.583511i	$0.801679\pi$
$338$		0			0
$339$		−	6.87985i		−	0.373662i
$340$		0			0
$341$			16.1635i			0.875301i
$342$		0			0
$343$			18.7898i			1.01456i
$344$		0			0
$345$		−	5.35314i		−	0.288203i
$346$		0			0
$347$	−16.4230			−0.881632			−0.440816	−	0.897598i	$-0.645311\pi$
$347$	−16.4230			−0.881632			−0.440816	+	0.897598i	$0.645311\pi$
$348$		0			0
$349$			7.84717i			0.420050i	0.977696	+	0.210025i	$0.0673545\pi$
$349$			7.84717i			0.420050i	−0.977696	+	0.210025i	$0.932646\pi$
$350$		0			0
$351$		−	0.177520i		−	0.00947529i
$352$		0			0
$353$	15.0952			0.803438			0.401719	−	0.915763i	$-0.368413\pi$
$353$	15.0952			0.803438			0.401719	+	0.915763i	$0.368413\pi$
$354$		0			0
$355$		−	20.9479i		−	1.11180i
$356$		0			0
$357$	−3.58322	−	5.93800i	−0.189644	−	0.314272i
$358$		0			0
$359$	−17.8645			−0.942850			−0.471425	−	0.881906i	$-0.656260\pi$
$359$	−17.8645			−0.942850			−0.471425	+	0.881906i	$0.656260\pi$
$360$		0			0
$361$	12.3149			0.648151
$362$		0			0
$363$	0.115770			0.00607635
$364$		0			0
$365$		−	6.89006i		−	0.360642i
$366$		0			0
$367$			17.8393i			0.931205i	0.884994	+	0.465602i	$0.154162\pi$
$367$			17.8393i			0.931205i	−0.884994	+	0.465602i	$0.845838\pi$
$368$		0			0
$369$		−	4.79275i		−	0.249501i
$370$		0			0
$371$	−7.60361			−0.394760
$372$		0			0
$373$			4.99138i			0.258444i	0.991616	+	0.129222i	$0.0412480\pi$
$373$			4.99138i			0.258444i	−0.991616	+	0.129222i	$0.958752\pi$
$374$		0			0
$375$	11.8372			0.611269
$376$		0			0
$377$		−	0.437968i		−	0.0225565i
$378$		0			0
$379$	1.52394			0.0782793			0.0391396	−	0.999234i	$-0.487538\pi$
$379$	1.52394			0.0782793			0.0391396	+	0.999234i	$0.487538\pi$
$380$		0			0
$381$	4.19770			0.215055
$382$		0			0
$383$	−17.0544			−0.871438			−0.435719	−	0.900083i	$-0.643506\pi$
$383$	−17.0544			−0.871438			−0.435719	+	0.900083i	$0.643506\pi$
$384$		0			0
$385$		−	8.82000i		−	0.449509i
$386$		0			0
$387$			5.48185i			0.278658i
$388$		0			0
$389$		−	18.6459i		−	0.945383i	−0.881228	−	0.472691i	$-0.843283\pi$
$389$		−	18.6459i		−	0.945383i	0.881228	−	0.472691i	$-0.156717\pi$
$390$		0			0
$391$	−7.25075	−	12.0157i	−0.366686	−	0.607661i
$392$		0			0
$393$	3.59013			0.181098
$394$		0			0
$395$		−	11.7638i		−	0.591901i
$396$		0			0
$397$	22.9158			1.15011			0.575057	−	0.818114i	$-0.304980\pi$
$397$	22.9158			1.15011			0.575057	+	0.818114i	$0.304980\pi$
$398$		0			0
$399$	−4.34910			−0.217728
$400$		0			0
$401$		−	31.7440i		−	1.58522i	−0.609728	−	0.792610i	$-0.708721\pi$
$401$		−	31.7440i		−	1.58522i	0.609728	−	0.792610i	$-0.291279\pi$
$402$		0			0
$403$	0.860619			0.0428705
$404$		0			0
$405$	−1.57273			−0.0781495
$406$		0			0
$407$	17.8009			0.882357
$408$		0			0
$409$	27.8277			1.37599			0.687996	−	0.725715i	$-0.258491\pi$
$409$	27.8277			1.37599			0.687996	+	0.725715i	$0.258491\pi$
$410$		0			0
$411$	−11.9023			−0.587097
$412$		0			0
$413$	10.6369			0.523409
$414$		0			0
$415$			10.3547i			0.508295i
$416$		0			0
$417$	−5.96225			−0.291972
$418$		0			0
$419$	−24.9157			−1.21721			−0.608605	−	0.793473i	$-0.708271\pi$
$419$	−24.9157			−1.21721			−0.608605	+	0.793473i	$0.708271\pi$
$420$		0			0
$421$			33.7977i			1.64720i	0.567171	+	0.823600i	$0.308038\pi$
$421$			33.7977i			1.64720i	−0.567171	+	0.823600i	$0.691962\pi$
$422$		0			0
$423$	−5.72051			−0.278141
$424$		0			0
$425$	8.91906	−	5.38210i	0.432638	−	0.261070i
$426$		0			0
$427$			10.5747i			0.511745i
$428$		0			0
$429$		−	0.591856i		−	0.0285751i
$430$		0			0
$431$			6.65096i			0.320366i	0.987087	+	0.160183i	$0.0512084\pi$
$431$			6.65096i			0.320366i	−0.987087	+	0.160183i	$0.948792\pi$
$432$		0			0
$433$	−19.5349			−0.938786			−0.469393	−	0.882989i	$-0.655527\pi$
$433$	−19.5349			−0.938786			−0.469393	+	0.882989i	$0.655527\pi$
$434$		0			0
$435$	−3.88016			−0.186039
$436$		0			0
$437$	−8.80054			−0.420987
$438$		0			0
$439$			0.431972i			0.0206169i	0.999947	+	0.0103085i	$0.00328134\pi$
$439$			0.431972i			0.0206169i	−0.999947	+	0.0103085i	$0.996719\pi$
$440$		0			0
$441$	4.17063			0.198601
$442$		0			0
$443$		−	0.940818i		−	0.0446996i	−0.999750	−	0.0223498i	$-0.992885\pi$
$443$		−	0.940818i		−	0.0446996i	0.999750	−	0.0223498i	$-0.00711476\pi$
$444$		0			0
$445$	13.6032			0.644852
$446$		0			0
$447$		−	22.1583i		−	1.04805i
$448$		0			0
$449$		−	40.6689i		−	1.91928i	−0.281227	−	0.959641i	$-0.590741\pi$
$449$		−	40.6689i		−	1.91928i	0.281227	−	0.959641i	$-0.409259\pi$
$450$		0			0
$451$		−	15.9792i		−	0.752431i
$452$		0			0
$453$	4.16246			0.195569
$454$		0			0
$455$	−0.469618			−0.0220161
$456$		0			0
$457$	−10.6020			−0.495940			−0.247970	−	0.968768i	$-0.579763\pi$
$457$	−10.6020			−0.495940			−0.247970	+	0.968768i	$0.579763\pi$
$458$		0			0
$459$	−3.53017	+	2.13024i	−0.164774	+	0.0994311i
$460$		0			0
$461$			6.90435i			0.321568i	0.986990	+	0.160784i	$0.0514022\pi$
$461$			6.90435i			0.321568i	−0.986990	+	0.160784i	$0.948598\pi$
$462$		0			0
$463$	−5.01760			−0.233188			−0.116594	−	0.993180i	$-0.537198\pi$
$463$	−5.01760			−0.233188			−0.116594	+	0.993180i	$0.537198\pi$
$464$		0			0
$465$		−	7.62463i		−	0.353584i
$466$		0			0
$467$		−	35.3420i		−	1.63543i	−0.575621	−	0.817717i	$-0.695240\pi$
$467$		−	35.3420i		−	1.63543i	0.575621	−	0.817717i	$-0.304760\pi$
$468$		0			0
$469$	−26.6273			−1.22953
$470$		0			0
$471$		−	1.66640i		−	0.0767835i
$472$		0			0
$473$			18.2767i			0.840363i
$474$		0			0
$475$		−	6.53248i		−	0.299731i
$476$		0			0
$477$			4.52038i			0.206974i
$478$		0			0
$479$			40.1158i			1.83294i	0.400106	+	0.916469i	$0.368973\pi$
$479$			40.1158i			1.83294i	−0.400106	+	0.916469i	$0.631027\pi$
$480$		0			0
$481$		−	0.947802i		−	0.0432161i
$482$		0			0
$483$	−5.72532			−0.260511
$484$		0			0
$485$		−	20.7107i		−	0.940424i
$486$		0			0
$487$		−	22.2766i		−	1.00945i	−0.863281	−	0.504724i	$-0.831594\pi$
$487$		−	22.2766i		−	1.00945i	0.863281	−	0.504724i	$-0.168406\pi$
$488$		0			0
$489$	5.18018			0.234256
$490$		0			0
$491$			23.2548i			1.04947i	0.851265	+	0.524736i	$0.175836\pi$
$491$			23.2548i			1.04947i	−0.851265	+	0.524736i	$0.824164\pi$
$492$		0			0
$493$	−8.70946	+	5.25562i	−0.392254	+	0.236701i
$494$		0			0
$495$	−5.24353			−0.235679
$496$		0			0
$497$	−22.4043			−1.00497
$498$		0			0
$499$	−4.16978			−0.186665			−0.0933324	−	0.995635i	$-0.529752\pi$
$499$	−4.16978			−0.186665			−0.0933324	+	0.995635i	$0.529752\pi$
$500$		0			0
$501$		−	20.8328i		−	0.930741i
$502$		0			0
$503$			2.70786i			0.120737i	0.998176	+	0.0603687i	$0.0192276\pi$
$503$			2.70786i			0.120737i	−0.998176	+	0.0603687i	$0.980772\pi$
$504$		0			0
$505$			1.73212i			0.0770782i
$506$		0			0
$507$	12.9685			0.575951
$508$		0			0
$509$		−	15.0471i		−	0.666952i	−0.942759	−	0.333476i	$-0.891778\pi$
$509$		−	15.0471i		−	0.666952i	0.942759	−	0.333476i	$-0.108222\pi$
$510$		0			0
$511$	−7.36910			−0.325990
$512$		0			0
$513$			2.58556i			0.114155i
$514$		0			0
$515$	14.2168			0.626469
$516$		0			0
$517$	−19.0724			−0.838802
$518$		0			0
$519$	21.8618			0.959628
$520$		0			0
$521$		−	29.0171i		−	1.27126i	−0.771993	−	0.635631i	$-0.780740\pi$
$521$		−	29.0171i		−	1.27126i	0.771993	−	0.635631i	$-0.219260\pi$
$522$		0			0
$523$		−	29.2097i		−	1.27725i	−0.769518	−	0.638625i	$-0.779503\pi$
$523$		−	29.2097i		−	1.27725i	0.769518	−	0.638625i	$-0.220497\pi$
$524$		0			0
$525$		−	4.24980i		−	0.185477i
$526$		0			0
$527$	−10.3275	−	17.1143i	−0.449871	−	0.745512i
$528$		0			0
$529$	11.4146			0.496289
$530$		0			0
$531$		−	6.32370i		−	0.274425i
$532$		0			0
$533$	−0.850807			−0.0368526
$534$		0			0
$535$	−28.8406			−1.24689
$536$		0			0
$537$		−	4.38920i		−	0.189408i
$538$		0			0
$539$	13.9050			0.598931
$540$		0			0
$541$	30.5463			1.31329			0.656643	−	0.754202i	$-0.271976\pi$
$541$	30.5463			1.31329			0.656643	+	0.754202i	$0.271976\pi$
$542$		0			0
$543$	−11.1107			−0.476806
$544$		0			0
$545$	−2.63813			−0.113005
$546$		0			0
$547$	36.8589			1.57597			0.787986	−	0.615693i	$-0.211124\pi$
$547$	36.8589			1.57597			0.787986	+	0.615693i	$0.211124\pi$
$548$		0			0
$549$	6.28670			0.268310
$550$		0			0
$551$			6.37897i			0.271753i
$552$		0			0
$553$	−12.5817			−0.535028
$554$		0			0
$555$	−8.39702			−0.356434
$556$		0			0
$557$			11.1786i			0.473652i	0.971552	+	0.236826i	$0.0761071\pi$
$557$			11.1786i			0.473652i	−0.971552	+	0.236826i	$0.923893\pi$
$558$		0			0
$559$	0.973137			0.0411593
$560$		0			0
$561$	−11.7697	+	7.10229i	−0.496917	+	0.299859i
$562$		0			0
$563$			1.85190i			0.0780483i	0.999238	+	0.0390242i	$0.0124249\pi$
$563$			1.85190i			0.0780483i	−0.999238	+	0.0390242i	$0.987575\pi$
$564$		0			0
$565$			10.8201i			0.455207i
$566$		0			0
$567$			1.68207i			0.0706405i
$568$		0			0
$569$	−3.17782			−0.133221			−0.0666106	−	0.997779i	$-0.521219\pi$
$569$	−3.17782			−0.133221			−0.0666106	+	0.997779i	$0.521219\pi$
$570$		0			0
$571$	20.2381			0.846937			0.423468	−	0.905911i	$-0.360812\pi$
$571$	20.2381			0.846937			0.423468	+	0.905911i	$0.360812\pi$
$572$		0			0
$573$	−11.5334			−0.481816
$574$		0			0
$575$		−	8.59960i		−	0.358628i
$576$		0			0
$577$	−5.53369			−0.230371			−0.115185	−	0.993344i	$-0.536746\pi$
$577$	−5.53369			−0.230371			−0.115185	+	0.993344i	$0.536746\pi$
$578$		0			0
$579$			13.5081i			0.561377i
$580$		0			0
$581$	11.0747			0.459455
$582$		0			0
$583$			15.0711i			0.624181i
$584$		0			0
$585$			0.279190i			0.0115431i
$586$		0			0
$587$			32.8487i			1.35581i	0.735149	+	0.677905i	$0.237112\pi$
$587$			32.8487i			1.35581i	−0.735149	+	0.677905i	$0.762888\pi$
$588$		0			0
$589$	−12.5349			−0.516490
$590$		0			0
$591$	−20.9839			−0.863161
$592$		0			0
$593$	−4.38985			−0.180269			−0.0901347	−	0.995930i	$-0.528730\pi$
$593$	−4.38985			−0.180269			−0.0901347	+	0.995930i	$0.528730\pi$
$594$		0			0
$595$	5.63543	+	9.33887i	0.231030	+	0.382856i
$596$		0			0
$597$		−	27.9971i		−	1.14585i
$598$		0			0
$599$	24.8193			1.01409			0.507044	−	0.861920i	$-0.330738\pi$
$599$	24.8193			1.01409			0.507044	+	0.861920i	$0.330738\pi$
$600$		0			0
$601$			43.9610i			1.79321i	0.442834	+	0.896603i	$0.353973\pi$
$601$			43.9610i			1.79321i	−0.442834	+	0.896603i	$0.646027\pi$
$602$		0			0
$603$			15.8300i			0.644649i
$604$		0			0
$605$	−0.182075			−0.00740240
$606$		0			0
$607$		−	44.7140i		−	1.81488i	−0.420177	−	0.907442i	$-0.638032\pi$
$607$		−	44.7140i		−	1.81488i	0.420177	−	0.907442i	$-0.361968\pi$
$608$		0			0
$609$			4.14993i			0.168164i
$610$		0			0
$611$			1.01550i			0.0410828i
$612$		0			0
$613$		−	39.2562i		−	1.58554i	−0.609518	−	0.792772i	$-0.708637\pi$
$613$		−	39.2562i		−	1.58554i	0.609518	−	0.792772i	$-0.291363\pi$
$614$		0			0
$615$			7.53770i			0.303949i
$616$		0			0
$617$			33.5704i			1.35149i	0.737133	+	0.675747i	$0.236179\pi$
$617$			33.5704i			1.35149i	−0.737133	+	0.675747i	$0.763821\pi$
$618$		0			0
$619$	−29.2701			−1.17647			−0.588233	−	0.808692i	$-0.700176\pi$
$619$	−29.2701			−1.17647			−0.588233	+	0.808692i	$0.700176\pi$
$620$		0			0
$621$			3.40373i			0.136587i
$622$		0			0
$623$		−	14.5489i		−	0.582891i
$624$		0			0
$625$	−5.98405			−0.239362
$626$		0			0
$627$			8.62034i			0.344263i
$628$		0			0
$629$	−18.8481	+	11.3737i	−0.751522	+	0.453497i
$630$		0			0
$631$	−11.2429			−0.447574			−0.223787	−	0.974638i	$-0.571842\pi$
$631$	−11.2429			−0.447574			−0.223787	+	0.974638i	$0.571842\pi$
$632$		0			0
$633$	−14.4756			−0.575353
$634$		0			0
$635$	−6.60185			−0.261986
$636$		0			0
$637$		−	0.740368i		−	0.0293344i
$638$		0			0
$639$			13.3194i			0.526909i
$640$		0			0
$641$			33.9330i			1.34027i	0.742237	+	0.670137i	$0.233765\pi$
$641$			33.9330i			1.34027i	−0.742237	+	0.670137i	$0.766235\pi$
$642$		0			0
$643$	35.2361			1.38958			0.694788	−	0.719214i	$-0.255498\pi$
$643$	35.2361			1.38958			0.694788	+	0.719214i	$0.255498\pi$
$644$		0			0
$645$		−	8.62147i		−	0.339470i
$646$		0			0
$647$	41.8365			1.64476			0.822381	−	0.568937i	$-0.192645\pi$
$647$	41.8365			1.64476			0.822381	+	0.568937i	$0.192645\pi$
$648$		0			0
$649$		−	21.0834i		−	0.827596i
$650$		0			0
$651$	−8.15474			−0.319609
$652$		0			0
$653$	22.6804			0.887554			0.443777	−	0.896137i	$-0.353638\pi$
$653$	22.6804			0.887554			0.443777	+	0.896137i	$0.353638\pi$
$654$		0			0
$655$	−5.64629			−0.220619
$656$		0			0
$657$			4.38096i			0.170917i
$658$		0			0
$659$			34.4905i			1.34356i	0.740751	+	0.671779i	$0.234470\pi$
$659$			34.4905i			1.34356i	−0.740751	+	0.671779i	$0.765530\pi$
$660$		0			0
$661$		−	0.923741i		−	0.0359293i	−0.999839	−	0.0179647i	$-0.994281\pi$
$661$		−	0.923741i		−	0.0359293i	0.999839	−	0.0179647i	$-0.00571864\pi$
$662$		0			0
$663$	0.378159	+	0.626674i	0.0146865	+	0.0243380i
$664$		0			0
$665$	6.83996			0.265242
$666$		0			0
$667$			8.39751i			0.325153i
$668$		0			0
$669$	27.2591			1.05390
$670$		0			0
$671$	20.9600			0.809154
$672$		0			0
$673$			3.91400i			0.150874i	0.997151	+	0.0754368i	$0.0240351\pi$
$673$			3.91400i			0.150874i	−0.997151	+	0.0754368i	$0.975965\pi$
$674$		0			0
$675$	−2.52653			−0.0972460
$676$		0			0
$677$	−22.1006			−0.849394			−0.424697	−	0.905336i	$-0.639619\pi$
$677$	−22.1006			−0.849394			−0.424697	+	0.905336i	$0.639619\pi$
$678$		0			0
$679$	−22.1506			−0.850063
$680$		0			0
$681$	9.26818			0.355157
$682$		0			0
$683$	42.9736			1.64434			0.822169	−	0.569244i	$-0.192764\pi$
$683$	42.9736			1.64434			0.822169	+	0.569244i	$0.192764\pi$
$684$		0			0
$685$	18.7191			0.715219
$686$		0			0
$687$		−	19.8306i		−	0.756585i
$688$		0			0
$689$	0.802456			0.0305711
$690$		0			0
$691$	−17.7933			−0.676890			−0.338445	−	0.940986i	$-0.609901\pi$
$691$	−17.7933			−0.676890			−0.338445	+	0.940986i	$0.609901\pi$
$692$		0			0
$693$			5.60809i			0.213034i
$694$		0			0
$695$	9.37699			0.355690
$696$		0			0
$697$	10.2097	+	16.9192i	0.386720	+	0.640861i
$698$		0			0
$699$			9.37047i			0.354424i
$700$		0			0
$701$			18.7151i			0.706859i	0.935461	+	0.353430i	$0.114985\pi$
$701$			18.7151i			0.706859i	−0.935461	+	0.353430i	$0.885015\pi$
$702$		0			0
$703$			13.8047i			0.520653i
$704$		0			0
$705$	8.99681			0.338839
$706$		0			0
$707$	1.85255			0.0696721
$708$		0			0
$709$	−30.0583			−1.12886			−0.564431	−	0.825480i	$-0.690905\pi$
$709$	−30.0583			−1.12886			−0.564431	+	0.825480i	$0.690905\pi$
$710$		0			0
$711$			7.47987i			0.280517i
$712$		0			0
$713$	−16.5014			−0.617981
$714$		0			0
$715$			0.930829i			0.0348110i
$716$		0			0
$717$	7.48309			0.279461
$718$		0			0
$719$			5.50072i			0.205142i	0.994726	+	0.102571i	$0.0327070\pi$
$719$			5.50072i			0.205142i	−0.994726	+	0.102571i	$0.967293\pi$
$720$		0			0
$721$		−	15.2053i		−	0.566274i
$722$		0			0
$723$		−	12.5710i		−	0.467520i
$724$		0			0
$725$	−6.23332			−0.231500
$726$		0			0
$727$	48.9745			1.81636			0.908182	−	0.418576i	$-0.137471\pi$
$727$	48.9745			1.81636			0.908182	+	0.418576i	$0.137471\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$	−11.6777	−	19.3519i	−0.431914	−	0.715754i
$732$		0			0
$733$			20.3595i			0.751996i	0.926621	+	0.375998i	$0.122700\pi$
$733$			20.3595i			0.751996i	−0.926621	+	0.375998i	$0.877300\pi$
$734$		0			0
$735$	−6.55926			−0.241942
$736$		0			0
$737$			52.7779i			1.94410i
$738$		0			0
$739$		−	13.0474i		−	0.479955i	−0.970778	−	0.239977i	$-0.922860\pi$
$739$		−	13.0474i		−	0.479955i	0.970778	−	0.239977i	$-0.0771400\pi$
$740$		0			0
$741$	0.458988			0.0168613
$742$		0			0
$743$			15.7329i			0.577186i	0.957452	+	0.288593i	$0.0931874\pi$
$743$			15.7329i			0.577186i	−0.957452	+	0.288593i	$0.906813\pi$
$744$		0			0
$745$			34.8490i			1.27677i
$746$		0			0
$747$		−	6.58394i		−	0.240894i
$748$		0			0
$749$			30.8458i			1.12708i
$750$		0			0
$751$			45.6534i			1.66591i	0.553337	+	0.832957i	$0.313354\pi$
$751$			45.6534i			1.66591i	−0.553337	+	0.832957i	$0.686646\pi$
$752$		0			0
$753$			1.95634i			0.0712931i
$754$		0			0
$755$	−6.54642			−0.238249
$756$		0			0
$757$		−	33.5775i		−	1.22039i	−0.792250	−	0.610197i	$-0.791090\pi$
$757$		−	33.5775i		−	1.22039i	0.792250	−	0.610197i	$-0.208910\pi$
$758$		0			0
$759$			11.3481i			0.411911i
$760$		0			0
$761$	−1.39682			−0.0506348			−0.0253174	−	0.999679i	$-0.508060\pi$
$761$	−1.39682			−0.0506348			−0.0253174	+	0.999679i	$0.508060\pi$
$762$		0			0
$763$			2.82155i			0.102147i
$764$		0			0
$765$	5.55199	−	3.35029i	0.200733	−	0.121130i
$766$		0			0
$767$	−1.12258			−0.0405340
$768$		0			0
$769$	20.8164			0.750658			0.375329	−	0.926892i	$-0.377530\pi$
$769$	20.8164			0.750658			0.375329	+	0.926892i	$0.377530\pi$
$770$		0			0
$771$	13.3326			0.480161
$772$		0			0
$773$			45.0123i			1.61898i	0.587133	+	0.809490i	$0.300256\pi$
$773$			45.0123i			1.61898i	−0.587133	+	0.809490i	$0.699744\pi$
$774$		0			0
$775$		−	12.2487i		−	0.439985i
$776$		0			0
$777$			8.98083i			0.322186i
$778$		0			0
$779$	12.3919			0.443988
$780$		0			0
$781$			44.4075i			1.58902i
$782$		0			0
$783$	2.46715			0.0881688
$784$		0			0
$785$			2.62079i			0.0935400i
$786$		0			0
$787$	−16.8464			−0.600509			−0.300255	−	0.953859i	$-0.597072\pi$
$787$	−16.8464			−0.600509			−0.300255	+	0.953859i	$0.597072\pi$
$788$		0			0
$789$	−19.5100			−0.694573
$790$		0			0
$791$	11.5724			0.411468
$792$		0			0
$793$		−	1.11601i		−	0.0396307i
$794$		0			0
$795$		−	7.10933i		−	0.252142i
$796$		0			0
$797$			6.68748i			0.236883i	0.992961	+	0.118441i	$0.0377897\pi$
$797$			6.68748i			0.236883i	−0.992961	+	0.118441i	$0.962210\pi$
$798$		0			0
$799$	20.1944	−	12.1861i	0.714425	−	0.431112i
$800$		0			0
$801$	−8.64940			−0.305612
$802$		0			0
$803$			14.6063i			0.515444i
$804$		0			0
$805$	9.00438			0.317363
$806$		0			0
$807$	6.00877			0.211519
$808$		0			0
$809$		−	16.7530i		−	0.589004i	−0.955651	−	0.294502i	$-0.904846\pi$
$809$		−	16.7530i		−	0.589004i	0.955651	−	0.294502i	$-0.0951537\pi$
$810$		0			0
$811$	10.8001			0.379243			0.189622	−	0.981857i	$-0.439274\pi$
$811$	10.8001			0.379243			0.189622	+	0.981857i	$0.439274\pi$
$812$		0			0
$813$	13.9230			0.488300
$814$		0			0
$815$	−8.14701			−0.285377
$816$		0			0
$817$	−14.1737			−0.495874
$818$		0			0
$819$	0.298601			0.0104340
$820$		0			0
$821$	38.5393			1.34503			0.672516	−	0.740083i	$-0.265214\pi$
$821$	38.5393			1.34503			0.672516	+	0.740083i	$0.265214\pi$
$822$		0			0
$823$		−	18.0592i		−	0.629505i	−0.949174	−	0.314752i	$-0.898079\pi$
$823$		−	18.0592i		−	0.629505i	0.949174	−	0.314752i	$-0.101921\pi$
$824$		0			0
$825$	−8.42352			−0.293269
$826$		0			0
$827$	−27.1647			−0.944611			−0.472305	−	0.881435i	$-0.656578\pi$
$827$	−27.1647			−0.944611			−0.472305	+	0.881435i	$0.656578\pi$
$828$		0			0
$829$			24.5873i			0.853952i	0.904263	+	0.426976i	$0.140421\pi$
$829$			24.5873i			0.853952i	−0.904263	+	0.426976i	$0.859579\pi$
$830$		0			0
$831$	−27.7674			−0.963242
$832$		0			0
$833$	−14.7230	+	8.88443i	−0.510122	+	0.307827i
$834$		0			0
$835$			32.7644i			1.13386i
$836$		0			0
$837$			4.84803i			0.167572i
$838$		0			0
$839$			5.06106i			0.174727i	0.996176	+	0.0873637i	$0.0278442\pi$
$839$			5.06106i			0.174727i	−0.996176	+	0.0873637i	$0.972156\pi$
$840$		0			0
$841$	−22.9132			−0.790109
$842$		0			0
$843$	16.6843			0.574638
$844$		0			0
$845$	−20.3959			−0.701641
$846$		0			0
$847$			0.194734i			0.00669113i
$848$		0			0
$849$	−9.70263			−0.332993
$850$		0			0
$851$			18.1730i			0.622962i
$852$		0			0
$853$	19.5592			0.669695			0.334847	−	0.942272i	$-0.391315\pi$
$853$	19.5592			0.669695			0.334847	+	0.942272i	$0.391315\pi$
$854$		0			0
$855$		−	4.06638i		−	0.139067i
$856$		0			0
$857$			15.7887i			0.539333i	0.962954	+	0.269666i	$0.0869134\pi$
$857$			15.7887i			0.539333i	−0.962954	+	0.269666i	$0.913087\pi$
$858$		0			0
$859$		−	32.5990i		−	1.11226i	−0.831094	−	0.556132i	$-0.812285\pi$
$859$		−	32.5990i		−	1.11226i	0.831094	−	0.556132i	$-0.187715\pi$
$860$		0			0
$861$	8.06176			0.274744
$862$		0			0
$863$	−43.1734			−1.46964			−0.734819	−	0.678263i	$-0.762733\pi$
$863$	−43.1734			−1.46964			−0.734819	+	0.678263i	$0.762733\pi$
$864$		0			0
$865$	−34.3827			−1.16905
$866$		0			0
$867$	7.92416	−	15.0402i	0.269119	−	0.510792i
$868$		0			0
$869$			24.9381i			0.845968i
$870$		0			0
$871$	2.81014			0.0952181
$872$		0			0
$873$			13.1686i			0.445691i
$874$		0			0
$875$			19.9110i			0.673115i
$876$		0			0
$877$	49.3608			1.66680			0.833399	−	0.552672i	$-0.186392\pi$
$877$	49.3608			1.66680			0.833399	+	0.552672i	$0.186392\pi$
$878$		0			0
$879$		−	31.3940i		−	1.05889i
$880$		0			0
$881$			44.3363i			1.49373i	0.664977	+	0.746864i	$0.268441\pi$
$881$			44.3363i			1.49373i	−0.664977	+	0.746864i	$0.731559\pi$
$882$		0			0
$883$		−	12.1954i		−	0.410407i	−0.978719	−	0.205203i	$-0.934214\pi$
$883$		−	12.1954i		−	0.410407i	0.978719	−	0.205203i	$-0.0657856\pi$
$884$		0			0
$885$			9.94546i			0.334313i
$886$		0			0
$887$			39.8250i			1.33719i	0.743626	+	0.668596i	$0.233104\pi$
$887$			39.8250i			1.33719i	−0.743626	+	0.668596i	$0.766896\pi$
$888$		0			0
$889$			7.06085i			0.236813i
$890$		0			0
$891$	3.33403			0.111694
$892$		0			0
$893$		−	14.7907i		−	0.494953i
$894$		0			0
$895$			6.90301i			0.230742i
$896$		0			0
$897$	0.604228			0.0201746
$898$		0			0
$899$			11.9608i			0.398915i
$900$		0			0
$901$	−9.62949	−	15.9577i	−0.320805	−	0.531628i
$902$		0			0
$903$	−9.22089			−0.306852
$904$		0			0
$905$	17.4741			0.580860
$906$		0			0
$907$	31.3002			1.03931			0.519653	−	0.854377i	$-0.326061\pi$
$907$	31.3002			1.03931			0.519653	+	0.854377i	$0.326061\pi$
$908$		0			0
$909$		−	1.10135i		−	0.0365293i
$910$		0			0
$911$			19.0074i			0.629745i	0.949134	+	0.314872i	$0.101962\pi$
$911$			19.0074i			0.629745i	−0.949134	+	0.314872i	$0.898038\pi$
$912$		0			0
$913$		−	21.9511i		−	0.726474i
$914$		0			0
$915$	−9.88727			−0.326863
$916$		0			0
$917$			6.03886i			0.199421i
$918$		0			0
$919$	−60.5097			−1.99603			−0.998016	−	0.0629608i	$-0.979946\pi$
$919$	−60.5097			−1.99603			−0.998016	+	0.0629608i	$0.979946\pi$
$920$		0			0
$921$			1.53330i			0.0505238i
$922$		0			0
$923$	2.36446			0.0778272
$924$		0			0
$925$	−13.4895			−0.443531
$926$		0			0
$927$	−9.03960			−0.296899
$928$		0			0
$929$		−	35.6403i		−	1.16932i	−0.811279	−	0.584660i	$-0.801228\pi$
$929$		−	35.6403i		−	1.16932i	0.811279	−	0.584660i	$-0.198772\pi$
$930$		0			0
$931$			10.7834i			0.353412i
$932$		0			0
$933$			1.74672i			0.0571849i
$934$		0			0
$935$	18.5105	−	11.1700i	0.605359	−	0.365297i
$936$		0			0
$937$	−44.8079			−1.46381			−0.731906	−	0.681406i	$-0.761369\pi$
$937$	−44.8079			−1.46381			−0.731906	+	0.681406i	$0.761369\pi$
$938$		0			0
$939$		−	31.9640i		−	1.04310i
$940$		0			0
$941$	51.2637			1.67115			0.835575	−	0.549377i	$-0.185135\pi$
$941$	51.2637			1.67115			0.835575	+	0.549377i	$0.185135\pi$
$942$		0			0
$943$	16.3132			0.531231
$944$		0			0
$945$		−	2.64545i		−	0.0860564i
$946$		0			0
$947$	−32.5760			−1.05858			−0.529289	−	0.848442i	$-0.677541\pi$
$947$	−32.5760			−1.05858			−0.529289	+	0.848442i	$0.677541\pi$
$948$		0			0
$949$	0.777706			0.0252454
$950$		0			0
$951$	−3.80550			−0.123402
$952$		0			0
$953$	36.4757			1.18156			0.590782	−	0.806831i	$-0.298819\pi$
$953$	36.4757			1.18156			0.590782	+	0.806831i	$0.298819\pi$
$954$		0			0
$955$	18.1390			0.586963
$956$		0			0
$957$	8.22556			0.265895
$958$		0			0
$959$		−	20.0205i		−	0.646497i
$960$		0			0
$961$	7.49665			0.241827
$962$		0			0
$963$	18.3379			0.590932
$964$		0			0
$965$		−	21.2446i		−	0.683887i
$966$		0			0
$967$	−22.8837			−0.735891			−0.367946	−	0.929847i	$-0.619939\pi$
$967$	−22.8837			−0.735891			−0.367946	+	0.929847i	$0.619939\pi$
$968$		0			0
$969$	−5.50786	−	9.12746i	−0.176938	−	0.293216i
$970$		0			0
$971$		−	34.4721i		−	1.10626i	−0.833094	−	0.553131i	$-0.813433\pi$
$971$		−	34.4721i		−	1.10626i	0.833094	−	0.553131i	$-0.186567\pi$
$972$		0			0
$973$		−	10.0289i		−	0.321513i
$974$		0			0
$975$			0.448508i			0.0143637i
$976$		0			0
$977$	16.3524			0.523160			0.261580	−	0.965182i	$-0.415756\pi$
$977$	16.3524			0.523160			0.261580	+	0.965182i	$0.415756\pi$
$978$		0			0
$979$	−28.8374			−0.921647
$980$		0			0
$981$	1.67742			0.0535560
$982$		0			0
$983$			39.1096i			1.24740i	0.781662	+	0.623702i	$0.214372\pi$
$983$			39.1096i			1.24740i	−0.781662	+	0.623702i	$0.785628\pi$
$984$		0			0
$985$	33.0019			1.05153
$986$		0			0
$987$		−	9.62232i		−	0.306282i
$988$		0			0
$989$	−18.6587			−0.593313
$990$		0			0
$991$			28.4092i			0.902449i	0.892411	+	0.451224i	$0.149013\pi$
$991$			28.4092i			0.902449i	−0.892411	+	0.451224i	$0.850987\pi$
$992$		0			0
$993$		−	10.4319i		−	0.331046i
$994$		0			0
$995$			44.0319i			1.39590i
$996$		0			0
$997$	9.78760			0.309976			0.154988	−	0.987916i	$-0.450466\pi$
$997$	9.78760			0.309976			0.154988	+	0.987916i	$0.450466\pi$
$998$		0			0
$999$	5.33914			0.168923

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1632.2.l.b.1393.4		18
3.2	odd	2		4896.2.l.c.3025.16		18
4.3	odd	2		408.2.l.a.373.13	✓	18
8.3	odd	2		408.2.l.b.373.14	yes	18
8.5	even	2		1632.2.l.a.1393.16		18
12.11	even	2		1224.2.l.c.1189.6		18
17.16	even	2		1632.2.l.a.1393.15		18
24.5	odd	2		4896.2.l.d.3025.4		18
24.11	even	2		1224.2.l.d.1189.5		18
51.50	odd	2		4896.2.l.d.3025.3		18
68.67	odd	2		408.2.l.b.373.13	yes	18
136.67	odd	2		408.2.l.a.373.14	yes	18
136.101	even	2	inner	1632.2.l.b.1393.3		18
204.203	even	2		1224.2.l.d.1189.6		18
408.101	odd	2		4896.2.l.c.3025.15		18
408.203	even	2		1224.2.l.c.1189.5		18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
408.2.l.a.373.13	✓	18	4.3	odd	2
408.2.l.a.373.14	yes	18	136.67	odd	2
408.2.l.b.373.13	yes	18	68.67	odd	2
408.2.l.b.373.14	yes	18	8.3	odd	2
1224.2.l.c.1189.5		18	408.203	even	2
1224.2.l.c.1189.6		18	12.11	even	2
1224.2.l.d.1189.5		18	24.11	even	2
1224.2.l.d.1189.6		18	204.203	even	2
1632.2.l.a.1393.15		18	17.16	even	2
1632.2.l.a.1393.16		18	8.5	even	2
1632.2.l.b.1393.3		18	136.101	even	2	inner
1632.2.l.b.1393.4		18	1.1	even	1	trivial
4896.2.l.c.3025.15		18	408.101	odd	2
4896.2.l.c.3025.16		18	3.2	odd	2
4896.2.l.d.3025.3		18	51.50	odd	2
4896.2.l.d.3025.4		18	24.5	odd	2

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:	\( N \)	\(=\)	\( 1632 = 2^{5} \cdot 3 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1632.l (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.57273 q^{5} +1.68207i q^{7} +1.00000 q^{9} +3.33403 q^{11} -0.177520i q^{13} -1.57273 q^{15} +(-3.53017 + 2.13024i) q^{17} +2.58556i q^{19} +1.68207i q^{21} +3.40373i q^{23} -2.52653 q^{25} +1.00000 q^{27} +2.46715 q^{29} +4.84803i q^{31} +3.33403 q^{33} -2.64545i q^{35} +5.33914 q^{37} -0.177520i q^{39} -4.79275i q^{41} +5.48185i q^{43} -1.57273 q^{45} -5.72051 q^{47} +4.17063 q^{49} +(-3.53017 + 2.13024i) q^{51} +4.52038i q^{53} -5.24353 q^{55} +2.58556i q^{57} -6.32370i q^{59} +6.28670 q^{61} +1.68207i q^{63} +0.279190i q^{65} +15.8300i q^{67} +3.40373i q^{69} +13.3194i q^{71} +4.38096i q^{73} -2.52653 q^{75} +5.60809i q^{77} +7.47987i q^{79} +1.00000 q^{81} -6.58394i q^{83} +(5.55199 - 3.35029i) q^{85} +2.46715 q^{87} -8.64940 q^{89} +0.298601 q^{91} +4.84803i q^{93} -4.06638i q^{95} +13.1686i q^{97} +3.33403 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 18 q^{3} + 4 q^{5} + 18 q^{9} + 4 q^{15} - 2 q^{17} + 22 q^{25} + 18 q^{27} - 12 q^{29} + 16 q^{37} + 4 q^{45} - 18 q^{49} - 2 q^{51} + 16 q^{55} + 16 q^{61} + 22 q^{75} + 18 q^{81} - 16 q^{85} - 12 q^{87}+ \cdots + 20 q^{89}+O(q^{100}) \) 18 * q + 18 * q^3 + 4 * q^5 + 18 * q^9 + 4 * q^15 - 2 * q^17 + 22 * q^25 + 18 * q^27 - 12 * q^29 + 16 * q^37 + 4 * q^45 - 18 * q^49 - 2 * q^51 + 16 * q^55 + 16 * q^61 + 22 * q^75 + 18 * q^81 - 16 * q^85 - 12 * q^87 + 20 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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