LMFDB - Embedded newform 2112.2.o.e.703.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2112,2,Mod(703,2112)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2112, 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("2112.703");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2112 = 2^{6} \cdot 3 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2112.o (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:	$16.8644049069$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.454201344.7
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 4x^{6} + 24x^{5} - 25x^{4} - 12x^{3} + 128x^{2} - 182x + 169 $ x^8 - 2x^7 - 4x^6 + 24x^5 - 25x^4 - 12x^3 + 128x^2 - 182*x + 169
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 528)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			703.4
Root			$-2.39244 - 0.0909984i$ of defining polynomial
Character	$\chi$	$=$	2112.703
Dual form			2112.2.o.e.703.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} +0.732051 q^{5} +2.36806 q^{7} -1.00000 q^{9} +(-3.23483 + 0.732051i) q^{11} +2.36806i q^{13} -0.732051i q^{15} +6.46965i q^{17} -6.46965 q^{19} -2.36806i q^{21} +4.73205i q^{23} -4.46410 q^{25} +1.00000i q^{27} +2.00000i q^{31} +(0.732051 + 3.23483i) q^{33} +1.73354 q^{35} -7.46410 q^{37} +2.36806 q^{39} +4.73611i q^{41} -6.46965 q^{43} -0.732051 q^{45} -6.19615i q^{47} -1.39230 q^{49} +6.46965 q^{51} +7.26795 q^{53} +(-2.36806 + 0.535898i) q^{55} +6.46965i q^{57} -2.53590i q^{59} +15.3074i q^{61} -2.36806 q^{63} +1.73354i q^{65} -10.3923i q^{67} +4.73205 q^{69} -7.26795i q^{71} -4.73611i q^{73} +4.46410i q^{75} +(-7.66025 + 1.73354i) q^{77} -2.36806 q^{79} +1.00000 q^{81} +4.73611 q^{83} +4.73611i q^{85} +10.3923 q^{89} +5.60770i q^{91} +2.00000 q^{93} -4.73611 q^{95} -1.46410 q^{97} +(3.23483 - 0.732051i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{5} - 8 q^{9} - 8 q^{25} - 8 q^{33} - 32 q^{37} + 8 q^{45} + 72 q^{49} + 72 q^{53} + 24 q^{69} + 8 q^{77} + 8 q^{81} + 16 q^{93} + 16 q^{97}+O(q^{100}) $ 8 * q - 8 * q^5 - 8 * q^9 - 8 * q^25 - 8 * q^33 - 32 * q^37 + 8 * q^45 + 72 * q^49 + 72 * q^53 + 24 * q^69 + 8 * q^77 + 8 * q^81 + 16 * q^93 + 16 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2112\mathbb{Z}\right)^\times$.

$n$	$133$	$1409$	$1729$	$2047$
$\chi(n)$	$1$	$1$	$-1$	$-1$

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
$3$		−	1.00000i		−	0.577350i
$3$		−	1.00000i		−	0.577350i
$4$		0			0
$5$	0.732051			0.327383			0.163692	−	0.986512i	$-0.447660\pi$
$5$	0.732051			0.327383			0.163692	+	0.986512i	$0.447660\pi$
$6$		0			0
$7$	2.36806			0.895042			0.447521	−	0.894274i	$-0.352307\pi$
$7$	2.36806			0.895042			0.447521	+	0.894274i	$0.352307\pi$
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	−3.23483	+	0.732051i	−0.975337	+	0.220722i
$11$	−3.23483	+	0.732051i	−0.975337	+	0.220722i
$12$		0			0
$13$			2.36806i			0.656781i	0.944542	+	0.328390i	$0.106506\pi$
$13$			2.36806i			0.656781i	−0.944542	+	0.328390i	$0.893494\pi$
$14$		0			0
$15$		−	0.732051i		−	0.189015i
$16$		0			0
$17$			6.46965i			1.56912i	0.620052	+	0.784561i	$0.287111\pi$
$17$			6.46965i			1.56912i	−0.620052	+	0.784561i	$0.712889\pi$
$18$		0			0
$19$	−6.46965			−1.48424			−0.742120	−	0.670267i	$-0.766180\pi$
$19$	−6.46965			−1.48424			−0.742120	+	0.670267i	$0.766180\pi$
$20$		0			0
$21$		−	2.36806i		−	0.516752i
$22$		0			0
$23$			4.73205i			0.986701i	0.869831	+	0.493350i	$0.164228\pi$
$23$			4.73205i			0.986701i	−0.869831	+	0.493350i	$0.835772\pi$
$24$		0			0
$25$	−4.46410			−0.892820
$26$		0			0
$27$			1.00000i			0.192450i
$28$		0			0
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$			2.00000i			0.359211i	0.983739	+	0.179605i	$0.0574821\pi$
$31$			2.00000i			0.359211i	−0.983739	+	0.179605i	$0.942518\pi$
$32$		0			0
$33$	0.732051	+	3.23483i	0.127434	+	0.563111i
$34$		0			0
$35$	1.73354			0.293021
$36$		0			0
$37$	−7.46410			−1.22709			−0.613545	−	0.789659i	$-0.710257\pi$
$37$	−7.46410			−1.22709			−0.613545	+	0.789659i	$0.710257\pi$
$38$		0			0
$39$	2.36806			0.379193
$40$		0			0
$41$			4.73611i			0.739657i	0.929100	+	0.369828i	$0.120584\pi$
$41$			4.73611i			0.739657i	−0.929100	+	0.369828i	$0.879416\pi$
$42$		0			0
$43$	−6.46965			−0.986613			−0.493306	−	0.869856i	$-0.664212\pi$
$43$	−6.46965			−0.986613			−0.493306	+	0.869856i	$0.664212\pi$
$44$		0			0
$45$	−0.732051			−0.109128
$46$		0			0
$47$		−	6.19615i		−	0.903802i	−0.892068	−	0.451901i	$-0.850746\pi$
$47$		−	6.19615i		−	0.903802i	0.892068	−	0.451901i	$-0.149254\pi$
$48$		0			0
$49$	−1.39230			−0.198901
$50$		0			0
$51$	6.46965			0.905933
$52$		0			0
$53$	7.26795			0.998330			0.499165	−	0.866507i	$-0.333640\pi$
$53$	7.26795			0.998330			0.499165	+	0.866507i	$0.333640\pi$
$54$		0			0
$55$	−2.36806	+	0.535898i	−0.319309	+	0.0722605i
$56$		0			0
$57$			6.46965i			0.856926i
$58$		0			0
$59$		−	2.53590i		−	0.330146i	−0.986281	−	0.165073i	$-0.947214\pi$
$59$		−	2.53590i		−	0.330146i	0.986281	−	0.165073i	$-0.0527859\pi$
$60$		0			0
$61$			15.3074i			1.95991i	0.199226	+	0.979953i	$0.436157\pi$
$61$			15.3074i			1.95991i	−0.199226	+	0.979953i	$0.563843\pi$
$62$		0			0
$63$	−2.36806			−0.298347
$64$		0			0
$65$			1.73354i			0.215019i
$66$		0			0
$67$		−	10.3923i		−	1.26962i	−0.772667	−	0.634811i	$-0.781078\pi$
$67$		−	10.3923i		−	1.26962i	0.772667	−	0.634811i	$-0.218922\pi$
$68$		0			0
$69$	4.73205			0.569672
$70$		0			0
$71$		−	7.26795i		−	0.862547i	−0.902221	−	0.431273i	$-0.858064\pi$
$71$		−	7.26795i		−	0.862547i	0.902221	−	0.431273i	$-0.141936\pi$
$72$		0			0
$73$		−	4.73611i		−	0.554320i	−0.960824	−	0.277160i	$-0.910607\pi$
$73$		−	4.73611i		−	0.554320i	0.960824	−	0.277160i	$-0.0893933\pi$
$74$		0			0
$75$			4.46410i			0.515470i
$76$		0			0
$77$	−7.66025	+	1.73354i	−0.872967	+	0.197555i
$78$		0			0
$79$	−2.36806			−0.266427			−0.133214	−	0.991087i	$-0.542530\pi$
$79$	−2.36806			−0.266427			−0.133214	+	0.991087i	$0.542530\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$	4.73611			0.519856			0.259928	−	0.965628i	$-0.416301\pi$
$83$	4.73611			0.519856			0.259928	+	0.965628i	$0.416301\pi$
$84$		0			0
$85$			4.73611i			0.513704i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	10.3923			1.10158			0.550791	−	0.834643i	$-0.314326\pi$
$89$	10.3923			1.10158			0.550791	+	0.834643i	$0.314326\pi$
$90$		0			0
$91$			5.60770i			0.587846i
$92$		0			0
$93$	2.00000			0.207390
$94$		0			0
$95$	−4.73611			−0.485915
$96$		0			0
$97$	−1.46410			−0.148657			−0.0743285	−	0.997234i	$-0.523681\pi$
$97$	−1.46410			−0.148657			−0.0743285	+	0.997234i	$0.523681\pi$
$98$		0			0
$99$	3.23483	−	0.732051i	0.325112	−	0.0735739i
$100$		0			0
$101$			8.20319i			0.816248i	0.912927	+	0.408124i	$0.133817\pi$
$101$			8.20319i			0.816248i	−0.912927	+	0.408124i	$0.866183\pi$
$102$		0			0
$103$		−	10.0000i		−	0.985329i	−0.870219	−	0.492665i	$-0.836023\pi$
$103$		−	10.0000i		−	0.985329i	0.870219	−	0.492665i	$-0.163977\pi$
$104$		0			0
$105$		−	1.73354i		−	0.169176i
$106$		0			0
$107$	6.46965			0.625445			0.312722	−	0.949845i	$-0.398759\pi$
$107$	6.46965			0.625445			0.312722	+	0.949845i	$0.398759\pi$
$108$		0			0
$109$			10.5712i			1.01254i	0.862374	+	0.506271i	$0.168976\pi$
$109$			10.5712i			1.01254i	−0.862374	+	0.506271i	$0.831024\pi$
$110$		0			0
$111$			7.46410i			0.708461i
$112$		0			0
$113$	−17.3205			−1.62938			−0.814688	−	0.579899i	$-0.803092\pi$
$113$	−17.3205			−1.62938			−0.814688	+	0.579899i	$0.803092\pi$
$114$		0			0
$115$			3.46410i			0.323029i
$116$		0			0
$117$		−	2.36806i		−	0.218927i
$118$		0			0
$119$			15.3205i			1.40443i
$120$		0			0
$121$	9.92820	−	4.73611i	0.902564	−	0.430556i
$122$		0			0
$123$	4.73611			0.427041
$124$		0			0
$125$	−6.92820			−0.619677
$126$		0			0
$127$	2.36806			0.210131			0.105066	−	0.994465i	$-0.466495\pi$
$127$	2.36806			0.210131			0.105066	+	0.994465i	$0.466495\pi$
$128$		0			0
$129$			6.46965i			0.569621i
$130$		0			0
$131$	17.6754			1.54431			0.772154	−	0.635435i	$-0.219179\pi$
$131$	17.6754			1.54431			0.772154	+	0.635435i	$0.219179\pi$
$132$		0			0
$133$	−15.3205			−1.32846
$134$		0			0
$135$			0.732051i			0.0630049i
$136$		0			0
$137$	−8.92820			−0.762788			−0.381394	−	0.924413i	$-0.624556\pi$
$137$	−8.92820			−0.762788			−0.381394	+	0.924413i	$0.624556\pi$
$138$		0			0
$139$	−1.73354			−0.147037			−0.0735184	−	0.997294i	$-0.523423\pi$
$139$	−1.73354			−0.147037			−0.0735184	+	0.997294i	$0.523423\pi$
$140$		0			0
$141$	−6.19615			−0.521810
$142$		0			0
$143$	−1.73354	−	7.66025i	−0.144966	−	0.640583i
$144$		0			0
$145$		0			0
$146$		0			0
$147$			1.39230i			0.114835i
$148$		0			0
$149$		−	8.20319i		−	0.672032i	−0.941856	−	0.336016i	$-0.890920\pi$
$149$		−	8.20319i		−	0.672032i	0.941856	−	0.336016i	$-0.109080\pi$
$150$		0			0
$151$	−15.3074			−1.24570			−0.622848	−	0.782343i	$-0.714024\pi$
$151$	−15.3074			−1.24570			−0.622848	+	0.782343i	$0.714024\pi$
$152$		0			0
$153$		−	6.46965i		−	0.523040i
$154$		0			0
$155$			1.46410i			0.117599i
$156$		0			0
$157$	15.8564			1.26548			0.632740	−	0.774365i	$-0.281930\pi$
$157$	15.8564			1.26548			0.632740	+	0.774365i	$0.281930\pi$
$158$		0			0
$159$		−	7.26795i		−	0.576386i
$160$		0			0
$161$			11.2058i			0.883138i
$162$		0			0
$163$			10.9282i			0.855963i	0.903788	+	0.427981i	$0.140775\pi$
$163$			10.9282i			0.855963i	−0.903788	+	0.427981i	$0.859225\pi$
$164$		0			0
$165$	0.535898	+	2.36806i	0.0417196	+	0.184353i
$166$		0			0
$167$	8.20319			0.634782			0.317391	−	0.948295i	$-0.397193\pi$
$167$	8.20319			0.634782			0.317391	+	0.948295i	$0.397193\pi$
$168$		0			0
$169$	7.39230			0.568639
$170$		0			0
$171$	6.46965			0.494747
$172$		0			0
$173$			22.4115i			1.70392i	0.523609	+	0.851959i	$0.324585\pi$
$173$			22.4115i			1.70392i	−0.523609	+	0.851959i	$0.675415\pi$
$174$		0			0
$175$	−10.5712			−0.799111
$176$		0			0
$177$	−2.53590			−0.190610
$178$		0			0
$179$			20.3923i			1.52419i	0.647464	+	0.762096i	$0.275830\pi$
$179$			20.3923i			1.52419i	−0.647464	+	0.762096i	$0.724170\pi$
$180$		0			0
$181$	18.3923			1.36709			0.683545	−	0.729909i	$-0.260437\pi$
$181$	18.3923			1.36709			0.683545	+	0.729909i	$0.260437\pi$
$182$		0			0
$183$	15.3074			1.13155
$184$		0			0
$185$	−5.46410			−0.401729
$186$		0			0
$187$	−4.73611	−	20.9282i	−0.346339	−	1.53042i
$188$		0			0
$189$			2.36806i			0.172251i
$190$		0			0
$191$			13.1244i			0.949645i	0.880082	+	0.474823i	$0.157488\pi$
$191$			13.1244i			0.949645i	−0.880082	+	0.474823i	$0.842512\pi$
$192$		0			0
$193$			22.4115i			1.61322i	0.591086	+	0.806609i	$0.298700\pi$
$193$			22.4115i			1.61322i	−0.591086	+	0.806609i	$0.701300\pi$
$194$		0			0
$195$	1.73354			0.124141
$196$		0			0
$197$		−	17.6754i		−	1.25932i	−0.776870	−	0.629661i	$-0.783194\pi$
$197$		−	17.6754i		−	1.25932i	0.776870	−	0.629661i	$-0.216806\pi$
$198$		0			0
$199$			16.5359i			1.17220i	0.810239	+	0.586099i	$0.199337\pi$
$199$			16.5359i			1.17220i	−0.810239	+	0.586099i	$0.800663\pi$
$200$		0			0
$201$	−10.3923			−0.733017
$202$		0			0
$203$		0			0
$204$		0			0
$205$			3.46708i			0.242151i
$206$		0			0
$207$		−	4.73205i		−	0.328900i
$208$		0			0
$209$	20.9282	−	4.73611i	1.44763	−	0.327604i
$210$		0			0
$211$	−14.6728			−1.01012			−0.505060	−	0.863084i	$-0.668530\pi$
$211$	−14.6728			−1.01012			−0.505060	+	0.863084i	$0.668530\pi$
$212$		0			0
$213$	−7.26795			−0.497992
$214$		0			0
$215$	−4.73611			−0.323000
$216$		0			0
$217$			4.73611i			0.321508i
$218$		0			0
$219$	−4.73611			−0.320037
$220$		0			0
$221$	−15.3205			−1.03057
$222$		0			0
$223$		−	4.53590i		−	0.303746i	−0.988400	−	0.151873i	$-0.951469\pi$
$223$		−	4.53590i		−	0.303746i	0.988400	−	0.151873i	$-0.0485305\pi$
$224$		0			0
$225$	4.46410			0.297607
$226$		0			0
$227$	−15.9419			−1.05810			−0.529050	−	0.848591i	$-0.677452\pi$
$227$	−15.9419			−1.05810			−0.529050	+	0.848591i	$0.677452\pi$
$228$		0			0
$229$	3.85641			0.254839			0.127419	−	0.991849i	$-0.459331\pi$
$229$	3.85641			0.254839			0.127419	+	0.991849i	$0.459331\pi$
$230$		0			0
$231$	1.73354	+	7.66025i	0.114058	+	0.504008i
$232$		0			0
$233$			4.73611i			0.310273i	0.987893	+	0.155137i	$0.0495818\pi$
$233$			4.73611i			0.310273i	−0.987893	+	0.155137i	$0.950418\pi$
$234$		0			0
$235$		−	4.53590i		−	0.295889i
$236$		0			0
$237$			2.36806i			0.153822i
$238$		0			0
$239$	27.1476			1.75604			0.878018	−	0.478628i	$-0.158866\pi$
$239$	27.1476			1.75604			0.878018	+	0.478628i	$0.158866\pi$
$240$		0			0
$241$			3.46708i			0.223334i	0.993746	+	0.111667i	$0.0356190\pi$
$241$			3.46708i			0.223334i	−0.993746	+	0.111667i	$0.964381\pi$
$242$		0			0
$243$		−	1.00000i		−	0.0641500i
$244$		0			0
$245$	−1.01924			−0.0651167
$246$		0			0
$247$		−	15.3205i		−	0.974821i
$248$		0			0
$249$		−	4.73611i		−	0.300139i
$250$		0			0
$251$		−	5.85641i		−	0.369653i	−0.982771	−	0.184827i	$-0.940828\pi$
$251$		−	5.85641i		−	0.369653i	0.982771	−	0.184827i	$-0.0591723\pi$
$252$		0			0
$253$	−3.46410	−	15.3074i	−0.217786	−	0.962366i
$254$		0			0
$255$	4.73611			0.296587
$256$		0			0
$257$	−8.53590			−0.532455			−0.266227	−	0.963910i	$-0.585777\pi$
$257$	−8.53590			−0.532455			−0.266227	+	0.963910i	$0.585777\pi$
$258$		0			0
$259$	−17.6754			−1.09830
$260$		0			0
$261$		0			0
$262$		0			0
$263$	30.6147			1.88778			0.943892	−	0.330253i	$-0.107134\pi$
$263$	30.6147			1.88778			0.943892	+	0.330253i	$0.107134\pi$
$264$		0			0
$265$	5.32051			0.326836
$266$		0			0
$267$		−	10.3923i		−	0.635999i
$268$		0			0
$269$	25.1244			1.53186			0.765930	−	0.642925i	$-0.222279\pi$
$269$	25.1244			1.53186			0.765930	+	0.642925i	$0.222279\pi$
$270$		0			0
$271$	−15.3074			−0.929856			−0.464928	−	0.885348i	$-0.653920\pi$
$271$	−15.3074			−0.929856			−0.464928	+	0.885348i	$0.653920\pi$
$272$		0			0
$273$	5.60770			0.339393
$274$		0			0
$275$	14.4406	−	3.26795i	0.870801	−	0.197065i
$276$		0			0
$277$		−	28.2467i		−	1.69718i	−0.529053	−	0.848589i	$-0.677453\pi$
$277$		−	28.2467i		−	1.69718i	0.529053	−	0.848589i	$-0.322547\pi$
$278$		0			0
$279$		−	2.00000i		−	0.119737i
$280$		0			0
$281$		−	28.8812i		−	1.72291i	−0.507836	−	0.861454i	$-0.669555\pi$
$281$		−	28.8812i		−	1.72291i	0.507836	−	0.861454i	$-0.330445\pi$
$282$		0			0
$283$	1.73354			0.103048			0.0515241	−	0.998672i	$-0.483592\pi$
$283$	1.73354			0.103048			0.0515241	+	0.998672i	$0.483592\pi$
$284$		0			0
$285$			4.73611i			0.280543i
$286$		0			0
$287$			11.2154i			0.662024i
$288$		0			0
$289$	−24.8564			−1.46214
$290$		0			0
$291$			1.46410i			0.0858272i
$292$		0			0
$293$			9.47223i			0.553374i	0.960960	+	0.276687i	$0.0892364\pi$
$293$			9.47223i			0.553374i	−0.960960	+	0.276687i	$0.910764\pi$
$294$		0			0
$295$		−	1.85641i		−	0.108084i
$296$		0			0
$297$	−0.732051	−	3.23483i	−0.0424779	−	0.187704i
$298$		0			0
$299$	−11.2058			−0.648046
$300$		0			0
$301$	−15.3205			−0.883059
$302$		0			0
$303$	8.20319			0.471261
$304$		0			0
$305$			11.2058i			0.641640i
$306$		0			0
$307$	−11.2058			−0.639547			−0.319773	−	0.947494i	$-0.603607\pi$
$307$	−11.2058			−0.639547			−0.319773	+	0.947494i	$0.603607\pi$
$308$		0			0
$309$	−10.0000			−0.568880
$310$		0			0
$311$		−	30.9808i		−	1.75676i	−0.477965	−	0.878379i	$-0.658625\pi$
$311$		−	30.9808i		−	1.75676i	0.477965	−	0.878379i	$-0.341375\pi$
$312$		0			0
$313$	−15.8564			−0.896257			−0.448129	−	0.893969i	$-0.647909\pi$
$313$	−15.8564			−0.896257			−0.448129	+	0.893969i	$0.647909\pi$
$314$		0			0
$315$	−1.73354			−0.0976738
$316$		0			0
$317$	7.66025			0.430243			0.215121	−	0.976587i	$-0.430985\pi$
$317$	7.66025			0.430243			0.215121	+	0.976587i	$0.430985\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		−	6.46965i		−	0.361101i
$322$		0			0
$323$		−	41.8564i		−	2.32895i
$324$		0			0
$325$		−	10.5712i		−	0.586387i
$326$		0			0
$327$	10.5712			0.584591
$328$		0			0
$329$		−	14.6728i		−	0.808940i
$330$		0			0
$331$			14.9282i			0.820528i	0.911967	+	0.410264i	$0.134563\pi$
$331$			14.9282i			0.820528i	−0.911967	+	0.410264i	$0.865437\pi$
$332$		0			0
$333$	7.46410			0.409030
$334$		0			0
$335$		−	7.60770i		−	0.415653i
$336$		0			0
$337$			17.6754i			0.962841i	0.876490	+	0.481421i	$0.159879\pi$
$337$			17.6754i			0.962841i	−0.876490	+	0.481421i	$0.840121\pi$
$338$		0			0
$339$			17.3205i			0.940721i
$340$		0			0
$341$	−1.46410	−	6.46965i	−0.0792855	−	0.350351i
$342$		0			0
$343$	−19.8735			−1.07307
$344$		0			0
$345$	3.46410			0.186501
$346$		0			0
$347$	−14.2083			−0.762744			−0.381372	−	0.924422i	$-0.624548\pi$
$347$	−14.2083			−0.762744			−0.381372	+	0.924422i	$0.624548\pi$
$348$		0			0
$349$		−	20.0435i		−	1.07290i	−0.843931	−	0.536451i	$-0.819764\pi$
$349$		−	20.0435i		−	1.07290i	0.843931	−	0.536451i	$-0.180236\pi$
$350$		0			0
$351$	−2.36806			−0.126398
$352$		0			0
$353$	35.8564			1.90844			0.954222	−	0.299099i	$-0.0966862\pi$
$353$	35.8564			1.90844			0.954222	+	0.299099i	$0.0966862\pi$
$354$		0			0
$355$		−	5.32051i		−	0.282383i
$356$		0			0
$357$	15.3205			0.810847
$358$		0			0
$359$	−12.9393			−0.682910			−0.341455	−	0.939898i	$-0.610920\pi$
$359$	−12.9393			−0.682910			−0.341455	+	0.939898i	$0.610920\pi$
$360$		0			0
$361$	22.8564			1.20297
$362$		0			0
$363$	−4.73611	−	9.92820i	−0.248582	−	0.521096i
$364$		0			0
$365$		−	3.46708i		−	0.181475i
$366$		0			0
$367$		−	10.3923i		−	0.542474i	−0.962513	−	0.271237i	$-0.912567\pi$
$367$		−	10.3923i		−	0.542474i	0.962513	−	0.271237i	$-0.0874327\pi$
$368$		0			0
$369$		−	4.73611i		−	0.246552i
$370$		0			0
$371$	17.2109			0.893546
$372$		0			0
$373$		−	24.7796i		−	1.28304i	−0.767107	−	0.641519i	$-0.778304\pi$
$373$		−	24.7796i		−	1.28304i	0.767107	−	0.641519i	$-0.221696\pi$
$374$		0			0
$375$			6.92820i			0.357771i
$376$		0			0
$377$		0			0
$378$		0			0
$379$		−	10.9282i		−	0.561344i	−0.959804	−	0.280672i	$-0.909443\pi$
$379$		−	10.9282i		−	0.561344i	0.959804	−	0.280672i	$-0.0905573\pi$
$380$		0			0
$381$		−	2.36806i		−	0.121319i
$382$		0			0
$383$			30.9808i			1.58304i	0.611141	+	0.791521i	$0.290711\pi$
$383$			30.9808i			1.58304i	−0.611141	+	0.791521i	$0.709289\pi$
$384$		0			0
$385$	−5.60770	+	1.26904i	−0.285795	+	0.0646762i
$386$		0			0
$387$	6.46965			0.328871
$388$		0			0
$389$	4.73205			0.239924			0.119962	−	0.992778i	$-0.461723\pi$
$389$	4.73205			0.239924			0.119962	+	0.992778i	$0.461723\pi$
$390$		0			0
$391$	−30.6147			−1.54825
$392$		0			0
$393$		−	17.6754i		−	0.891607i
$394$		0			0
$395$	−1.73354			−0.0872238
$396$		0			0
$397$	−13.3205			−0.668537			−0.334269	−	0.942478i	$-0.608489\pi$
$397$	−13.3205			−0.668537			−0.334269	+	0.942478i	$0.608489\pi$
$398$		0			0
$399$			15.3205i			0.766985i
$400$		0			0
$401$	−3.46410			−0.172989			−0.0864945	−	0.996252i	$-0.527566\pi$
$401$	−3.46410			−0.172989			−0.0864945	+	0.996252i	$0.527566\pi$
$402$		0			0
$403$	−4.73611			−0.235923
$404$		0			0
$405$	0.732051			0.0363759
$406$		0			0
$407$	24.1451	−	5.46410i	1.19683	−	0.270845i
$408$		0			0
$409$		−	21.1425i		−	1.04543i	−0.852508	−	0.522715i	$-0.824919\pi$
$409$		−	21.1425i		−	1.04543i	0.852508	−	0.522715i	$-0.175081\pi$
$410$		0			0
$411$			8.92820i			0.440396i
$412$		0			0
$413$		−	6.00515i		−	0.295494i
$414$		0			0
$415$	3.46708			0.170192
$416$		0			0
$417$			1.73354i			0.0848917i
$418$		0			0
$419$			33.8564i			1.65399i	0.562207	+	0.826997i	$0.309953\pi$
$419$			33.8564i			1.65399i	−0.562207	+	0.826997i	$0.690047\pi$
$420$		0			0
$421$	25.3205			1.23405			0.617023	−	0.786945i	$-0.288339\pi$
$421$	25.3205			1.23405			0.617023	+	0.786945i	$0.288339\pi$
$422$		0			0
$423$			6.19615i			0.301267i
$424$		0			0
$425$		−	28.8812i		−	1.40094i
$426$		0			0
$427$			36.2487i			1.75420i
$428$		0			0
$429$	−7.66025	+	1.73354i	−0.369841	+	0.0836960i
$430$		0			0
$431$	−27.1476			−1.30766			−0.653828	−	0.756643i	$-0.726838\pi$
$431$	−27.1476			−1.30766			−0.653828	+	0.756643i	$0.726838\pi$
$432$		0			0
$433$	−3.07180			−0.147621			−0.0738106	−	0.997272i	$-0.523516\pi$
$433$	−3.07180			−0.147621			−0.0738106	+	0.997272i	$0.523516\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	30.6147i		−	1.46450i
$438$		0			0
$439$	−29.5157			−1.40871			−0.704354	−	0.709849i	$-0.748763\pi$
$439$	−29.5157			−1.40871			−0.704354	+	0.709849i	$0.748763\pi$
$440$		0			0
$441$	1.39230			0.0663002
$442$		0			0
$443$			29.8564i			1.41852i	0.704947	+	0.709260i	$0.250971\pi$
$443$			29.8564i			1.41852i	−0.704947	+	0.709260i	$0.749029\pi$
$444$		0			0
$445$	7.60770			0.360639
$446$		0			0
$447$	−8.20319			−0.387998
$448$		0			0
$449$	−28.9282			−1.36521			−0.682603	−	0.730789i	$-0.739152\pi$
$449$	−28.9282			−1.36521			−0.682603	+	0.730789i	$0.739152\pi$
$450$		0			0
$451$	−3.46708	−	15.3205i	−0.163258	−	0.721415i
$452$		0			0
$453$			15.3074i			0.719203i
$454$		0			0
$455$			4.10512i			0.192451i
$456$		0			0
$457$		−	17.6754i		−	0.826821i	−0.910545	−	0.413411i	$-0.864337\pi$
$457$		−	17.6754i		−	0.826821i	0.910545	−	0.413411i	$-0.135663\pi$
$458$		0			0
$459$	−6.46965			−0.301978
$460$		0			0
$461$			35.3508i			1.64645i	0.567713	+	0.823226i	$0.307828\pi$
$461$			35.3508i			1.64645i	−0.567713	+	0.823226i	$0.692172\pi$
$462$		0			0
$463$			19.0718i			0.886342i	0.896437	+	0.443171i	$0.146146\pi$
$463$			19.0718i			0.886342i	−0.896437	+	0.443171i	$0.853854\pi$
$464$		0			0
$465$	1.46410			0.0678961
$466$		0			0
$467$			2.53590i			0.117347i	0.998277	+	0.0586737i	$0.0186871\pi$
$467$			2.53590i			0.117347i	−0.998277	+	0.0586737i	$0.981313\pi$
$468$		0			0
$469$		−	24.6096i		−	1.13636i
$470$		0			0
$471$		−	15.8564i		−	0.730625i
$472$		0			0
$473$	20.9282	−	4.73611i	0.962280	−	0.217767i
$474$		0			0
$475$	28.8812			1.32516
$476$		0			0
$477$	−7.26795			−0.332777
$478$		0			0
$479$	4.73611			0.216399			0.108199	−	0.994129i	$-0.465492\pi$
$479$	4.73611			0.216399			0.108199	+	0.994129i	$0.465492\pi$
$480$		0			0
$481$		−	17.6754i		−	0.805930i
$482$		0			0
$483$	11.2058			0.509880
$484$		0			0
$485$	−1.07180			−0.0486678
$486$		0			0
$487$		−	2.39230i		−	0.108406i	−0.998530	−	0.0542028i	$-0.982738\pi$
$487$		−	2.39230i		−	0.108406i	0.998530	−	0.0542028i	$-0.0172618\pi$
$488$		0			0
$489$	10.9282			0.494190
$490$		0			0
$491$	19.4090			0.875914			0.437957	−	0.898996i	$-0.355702\pi$
$491$	19.4090			0.875914			0.437957	+	0.898996i	$0.355702\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$	2.36806	−	0.535898i	0.106436	−	0.0240868i
$496$		0			0
$497$		−	17.2109i		−	0.772015i
$498$		0			0
$499$		−	30.9282i		−	1.38454i	−0.721640	−	0.692268i	$-0.756611\pi$
$499$		−	30.9282i		−	1.38454i	0.721640	−	0.692268i	$-0.243389\pi$
$500$		0			0
$501$		−	8.20319i		−	0.366492i
$502$		0			0
$503$	−17.6754			−0.788108			−0.394054	−	0.919087i	$-0.628928\pi$
$503$	−17.6754			−0.788108			−0.394054	+	0.919087i	$0.628928\pi$
$504$		0			0
$505$			6.00515i			0.267226i
$506$		0			0
$507$		−	7.39230i		−	0.328304i
$508$		0			0
$509$	−23.2679			−1.03133			−0.515667	−	0.856789i	$-0.672456\pi$
$509$	−23.2679			−1.03133			−0.515667	+	0.856789i	$0.672456\pi$
$510$		0			0
$511$		−	11.2154i		−	0.496140i
$512$		0			0
$513$		−	6.46965i		−	0.285642i
$514$		0			0
$515$		−	7.32051i		−	0.322580i
$516$		0			0
$517$	4.53590	+	20.0435i	0.199489	+	0.881511i
$518$		0			0
$519$	22.4115			0.983757
$520$		0			0
$521$	−6.67949			−0.292634			−0.146317	−	0.989238i	$-0.546742\pi$
$521$	−6.67949			−0.292634			−0.146317	+	0.989238i	$0.546742\pi$
$522$		0			0
$523$	25.4141			1.11128			0.555641	−	0.831422i	$-0.312473\pi$
$523$	25.4141			1.11128			0.555641	+	0.831422i	$0.312473\pi$
$524$		0			0
$525$			10.5712i			0.461367i
$526$		0			0
$527$	−12.9393			−0.563645
$528$		0			0
$529$	0.607695			0.0264215
$530$		0			0
$531$			2.53590i			0.110049i
$532$		0			0
$533$	−11.2154			−0.485792
$534$		0			0
$535$	4.73611			0.204760
$536$		0			0
$537$	20.3923			0.879993
$538$		0			0
$539$	4.50386	−	1.01924i	0.193995	−	0.0439017i
$540$		0			0
$541$			36.4499i			1.56710i	0.621328	+	0.783551i	$0.286594\pi$
$541$			36.4499i			1.56710i	−0.621328	+	0.783551i	$0.713406\pi$
$542$		0			0
$543$		−	18.3923i		−	0.789289i
$544$		0			0
$545$			7.73869i			0.331489i
$546$		0			0
$547$	−28.8812			−1.23487			−0.617435	−	0.786622i	$-0.711828\pi$
$547$	−28.8812			−1.23487			−0.617435	+	0.786622i	$0.711828\pi$
$548$		0			0
$549$		−	15.3074i		−	0.653302i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−5.60770			−0.238463
$554$		0			0
$555$			5.46410i			0.231938i
$556$		0			0
$557$		−	9.47223i		−	0.401351i	−0.979658	−	0.200676i	$-0.935686\pi$
$557$		−	9.47223i		−	0.401351i	0.979658	−	0.200676i	$-0.0643137\pi$
$558$		0			0
$559$		−	15.3205i		−	0.647988i
$560$		0			0
$561$	−20.9282	+	4.73611i	−0.883589	+	0.199959i
$562$		0			0
$563$	9.93673			0.418783			0.209392	−	0.977832i	$-0.432852\pi$
$563$	9.93673			0.418783			0.209392	+	0.977832i	$0.432852\pi$
$564$		0			0
$565$	−12.6795			−0.533430
$566$		0			0
$567$	2.36806			0.0994491
$568$		0			0
$569$		−	28.8812i		−	1.21076i	−0.795936	−	0.605381i	$-0.793021\pi$
$569$		−	28.8812i		−	1.21076i	0.795936	−	0.605381i	$-0.206979\pi$
$570$		0			0
$571$	1.73354			0.0725463			0.0362732	−	0.999342i	$-0.488451\pi$
$571$	1.73354			0.0725463			0.0362732	+	0.999342i	$0.488451\pi$
$572$		0			0
$573$	13.1244			0.548278
$574$		0			0
$575$		−	21.1244i		−	0.880947i
$576$		0			0
$577$	19.6077			0.816279			0.408140	−	0.912920i	$-0.366178\pi$
$577$	19.6077			0.816279			0.408140	+	0.912920i	$0.366178\pi$
$578$		0			0
$579$	22.4115			0.931392
$580$		0			0
$581$	11.2154			0.465293
$582$		0			0
$583$	−23.5106	+	5.32051i	−0.973708	+	0.220353i
$584$		0			0
$585$		−	1.73354i		−	0.0716730i
$586$		0			0
$587$		−	29.1769i		−	1.20426i	−0.798398	−	0.602130i	$-0.794319\pi$
$587$		−	29.1769i		−	1.20426i	0.798398	−	0.602130i	$-0.205681\pi$
$588$		0			0
$589$		−	12.9393i		−	0.533155i
$590$		0			0
$591$	−17.6754			−0.727070
$592$		0			0
$593$			3.00258i			0.123301i	0.998098	+	0.0616505i	$0.0196364\pi$
$593$			3.00258i			0.123301i	−0.998098	+	0.0616505i	$0.980364\pi$
$594$		0			0
$595$			11.2154i			0.459786i
$596$		0			0
$597$	16.5359			0.676769
$598$		0			0
$599$			26.1962i			1.07035i	0.844743	+	0.535173i	$0.179754\pi$
$599$			26.1962i			1.07035i	−0.844743	+	0.535173i	$0.820246\pi$
$600$		0			0
$601$		−	34.0818i		−	1.39023i	−0.718901	−	0.695113i	$-0.755354\pi$
$601$		−	34.0818i		−	1.39023i	0.718901	−	0.695113i	$-0.244646\pi$
$602$		0			0
$603$			10.3923i			0.423207i
$604$		0			0
$605$	7.26795	−	3.46708i	0.295484	−	0.140957i
$606$		0			0
$607$	36.4499			1.47945			0.739727	−	0.672907i	$-0.234955\pi$
$607$	36.4499			1.47945			0.739727	+	0.672907i	$0.234955\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	14.6728			0.593600
$612$		0			0
$613$			1.09902i			0.0443890i	0.999754	+	0.0221945i	$0.00706530\pi$
$613$			1.09902i			0.0443890i	−0.999754	+	0.0221945i	$0.992935\pi$
$614$		0			0
$615$	3.46708			0.139806
$616$		0			0
$617$	15.4641			0.622561			0.311281	−	0.950318i	$-0.399242\pi$
$617$	15.4641			0.622561			0.311281	+	0.950318i	$0.399242\pi$
$618$		0			0
$619$		−	15.4641i		−	0.621555i	−0.950483	−	0.310777i	$-0.899411\pi$
$619$		−	15.4641i		−	0.621555i	0.950483	−	0.310777i	$-0.100589\pi$
$620$		0			0
$621$	−4.73205			−0.189891
$622$		0			0
$623$	24.6096			0.985962
$624$		0			0
$625$	17.2487			0.689948
$626$		0			0
$627$	−4.73611	−	20.9282i	−0.189142	−	0.835792i
$628$		0			0
$629$		−	48.2901i		−	1.92545i
$630$		0			0
$631$		−	34.7846i		−	1.38475i	−0.721536	−	0.692377i	$-0.756564\pi$
$631$		−	34.7846i		−	1.38475i	0.721536	−	0.692377i	$-0.243436\pi$
$632$		0			0
$633$			14.6728i			0.583193i
$634$		0			0
$635$	1.73354			0.0687934
$636$		0			0
$637$		−	3.29706i		−	0.130634i
$638$		0			0
$639$			7.26795i			0.287516i
$640$		0			0
$641$	−11.4641			−0.452805			−0.226402	−	0.974034i	$-0.572696\pi$
$641$	−11.4641			−0.452805			−0.226402	+	0.974034i	$0.572696\pi$
$642$		0			0
$643$			8.53590i			0.336623i	0.985734	+	0.168311i	$0.0538314\pi$
$643$			8.53590i			0.336623i	−0.985734	+	0.168311i	$0.946169\pi$
$644$		0			0
$645$			4.73611i			0.186484i
$646$		0			0
$647$			37.1244i			1.45951i	0.683709	+	0.729755i	$0.260366\pi$
$647$			37.1244i			1.45951i	−0.683709	+	0.729755i	$0.739634\pi$
$648$		0			0
$649$	1.85641	+	8.20319i	0.0728703	+	0.322003i
$650$		0			0
$651$	4.73611			0.185623
$652$		0			0
$653$	24.7321			0.967840			0.483920	−	0.875112i	$-0.339213\pi$
$653$	24.7321			0.967840			0.483920	+	0.875112i	$0.339213\pi$
$654$		0			0
$655$	12.9393			0.505581
$656$		0			0
$657$			4.73611i			0.184773i
$658$		0			0
$659$	−25.4141			−0.989993			−0.494997	−	0.868895i	$-0.664831\pi$
$659$	−25.4141			−0.989993			−0.494997	+	0.868895i	$0.664831\pi$
$660$		0			0
$661$	−4.14359			−0.161167			−0.0805836	−	0.996748i	$-0.525678\pi$
$661$	−4.14359			−0.161167			−0.0805836	+	0.996748i	$0.525678\pi$
$662$		0			0
$663$			15.3205i			0.594999i
$664$		0			0
$665$	−11.2154			−0.434914
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−4.53590			−0.175368
$670$		0			0
$671$	−11.2058	−	49.5167i	−0.432594	−	1.91157i
$672$		0			0
$673$			34.0818i			1.31376i	0.753996	+	0.656878i	$0.228124\pi$
$673$			34.0818i			1.31376i	−0.753996	+	0.656878i	$0.771876\pi$
$674$		0			0
$675$		−	4.46410i		−	0.171823i
$676$		0			0
$677$			30.6147i			1.17662i	0.808636	+	0.588310i	$0.200206\pi$
$677$			30.6147i			1.17662i	−0.808636	+	0.588310i	$0.799794\pi$
$678$		0			0
$679$	−3.46708			−0.133054
$680$		0			0
$681$			15.9419i			0.610894i
$682$		0			0
$683$			35.7128i			1.36651i	0.730179	+	0.683256i	$0.239437\pi$
$683$			35.7128i			1.36651i	−0.730179	+	0.683256i	$0.760563\pi$
$684$		0			0
$685$	−6.53590			−0.249724
$686$		0			0
$687$		−	3.85641i		−	0.147131i
$688$		0			0
$689$			17.2109i			0.655684i
$690$		0			0
$691$			9.60770i			0.365494i	0.983160	+	0.182747i	$0.0584989\pi$
$691$			9.60770i			0.365494i	−0.983160	+	0.182747i	$0.941501\pi$
$692$		0			0
$693$	7.66025	−	1.73354i	0.290989	−	0.0658517i
$694$		0			0
$695$	−1.26904			−0.0481374
$696$		0			0
$697$	−30.6410			−1.16061
$698$		0			0
$699$	4.73611			0.179136
$700$		0			0
$701$		−	22.4115i		−	0.846472i	−0.906019	−	0.423236i	$-0.860894\pi$
$701$		−	22.4115i		−	0.846472i	0.906019	−	0.423236i	$-0.139106\pi$
$702$		0			0
$703$	48.2901			1.82130
$704$		0			0
$705$	−4.53590			−0.170832
$706$		0			0
$707$			19.4256i			0.730576i
$708$		0			0
$709$	−18.7846			−0.705471			−0.352735	−	0.935723i	$-0.614748\pi$
$709$	−18.7846			−0.705471			−0.352735	+	0.935723i	$0.614748\pi$
$710$		0			0
$711$	2.36806			0.0888091
$712$		0			0
$713$	−9.46410			−0.354433
$714$		0			0
$715$	−1.26904	−	5.60770i	−0.0474593	−	0.209716i
$716$		0			0
$717$		−	27.1476i		−	1.01385i
$718$		0			0
$719$		−	11.6603i		−	0.434854i	−0.976077	−	0.217427i	$-0.930234\pi$
$719$		−	11.6603i		−	0.434854i	0.976077	−	0.217427i	$-0.0697664\pi$
$720$		0			0
$721$		−	23.6806i		−	0.881911i
$722$		0			0
$723$	3.46708			0.128942
$724$		0			0
$725$		0			0
$726$		0			0
$727$			48.2487i			1.78945i	0.446622	+	0.894723i	$0.352627\pi$
$727$			48.2487i			1.78945i	−0.446622	+	0.894723i	$0.647373\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$		−	41.8564i		−	1.54812i
$732$		0			0
$733$			10.5712i			0.390458i	0.980758	+	0.195229i	$0.0625450\pi$
$733$			10.5712i			0.390458i	−0.980758	+	0.195229i	$0.937455\pi$
$734$		0			0
$735$			1.01924i			0.0375952i
$736$		0			0
$737$	7.60770	+	33.6173i	0.280233	+	1.23831i
$738$		0			0
$739$	−24.1451			−0.888191			−0.444095	−	0.895979i	$-0.646475\pi$
$739$	−24.1451			−0.888191			−0.444095	+	0.895979i	$0.646475\pi$
$740$		0			0
$741$	−15.3205			−0.562813
$742$		0			0
$743$	38.8179			1.42409			0.712046	−	0.702133i	$-0.247769\pi$
$743$	38.8179			1.42409			0.712046	+	0.702133i	$0.247769\pi$
$744$		0			0
$745$		−	6.00515i		−	0.220012i
$746$		0			0
$747$	−4.73611			−0.173285
$748$		0			0
$749$	15.3205			0.559799
$750$		0			0
$751$			51.5692i			1.88179i	0.338702	+	0.940894i	$0.390012\pi$
$751$			51.5692i			1.88179i	−0.338702	+	0.940894i	$0.609988\pi$
$752$		0			0
$753$	−5.85641			−0.213419
$754$		0			0
$755$	−11.2058			−0.407820
$756$		0			0
$757$	19.1769			0.696997			0.348498	−	0.937309i	$-0.386692\pi$
$757$	19.1769			0.696997			0.348498	+	0.937309i	$0.386692\pi$
$758$		0			0
$759$	−15.3074	+	3.46410i	−0.555622	+	0.125739i
$760$		0			0
$761$			25.4141i			0.921261i	0.887592	+	0.460630i	$0.152377\pi$
$761$			25.4141i			0.921261i	−0.887592	+	0.460630i	$0.847623\pi$
$762$		0			0
$763$			25.0333i			0.906267i
$764$		0			0
$765$		−	4.73611i		−	0.171235i
$766$		0			0
$767$	6.00515			0.216833
$768$		0			0
$769$		−	35.3508i		−	1.27478i	−0.770540	−	0.637392i	$-0.780013\pi$
$769$		−	35.3508i		−	1.27478i	0.770540	−	0.637392i	$-0.219987\pi$
$770$		0			0
$771$			8.53590i			0.307413i
$772$		0			0
$773$	−44.8372			−1.61268			−0.806340	−	0.591452i	$-0.798555\pi$
$773$	−44.8372			−1.61268			−0.806340	+	0.591452i	$0.798555\pi$
$774$		0			0
$775$		−	8.92820i		−	0.320711i
$776$		0			0
$777$			17.6754i			0.634102i
$778$		0			0
$779$		−	30.6410i		−	1.09783i
$780$		0			0
$781$	5.32051	+	23.5106i	0.190383	+	0.841274i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	11.6077			0.414296
$786$		0			0
$787$	−28.8812			−1.02950			−0.514752	−	0.857339i	$-0.672116\pi$
$787$	−28.8812			−1.02950			−0.514752	+	0.857339i	$0.672116\pi$
$788$		0			0
$789$		−	30.6147i		−	1.08991i
$790$		0			0
$791$	−41.0160			−1.45836
$792$		0			0
$793$	−36.2487			−1.28723
$794$		0			0
$795$		−	5.32051i		−	0.188699i
$796$		0			0
$797$	−0.339746			−0.0120344			−0.00601721	−	0.999982i	$-0.501915\pi$
$797$	−0.339746			−0.0120344			−0.00601721	+	0.999982i	$0.501915\pi$
$798$		0			0
$799$	40.0870			1.41817
$800$		0			0
$801$	−10.3923			−0.367194
$802$		0			0
$803$	3.46708	+	15.3205i	0.122350	+	0.540649i
$804$		0			0
$805$			8.20319i			0.289124i
$806$		0			0
$807$		−	25.1244i		−	0.884419i
$808$		0			0
$809$			21.1425i			0.743331i	0.928367	+	0.371665i	$0.121213\pi$
$809$			21.1425i			0.743331i	−0.928367	+	0.371665i	$0.878787\pi$
$810$		0			0
$811$	6.46965			0.227180			0.113590	−	0.993528i	$-0.463765\pi$
$811$	6.46965			0.227180			0.113590	+	0.993528i	$0.463765\pi$
$812$		0			0
$813$			15.3074i			0.536853i
$814$		0			0
$815$			8.00000i			0.280228i
$816$		0			0
$817$	41.8564			1.46437
$818$		0			0
$819$		−	5.60770i		−	0.195949i
$820$		0			0
$821$		−	8.20319i		−	0.286293i	−0.989701	−	0.143147i	$-0.954278\pi$
$821$		−	8.20319i		−	0.286293i	0.989701	−	0.143147i	$-0.0457221\pi$
$822$		0			0
$823$		−	4.24871i		−	0.148101i	−0.997255	−	0.0740504i	$-0.976407\pi$
$823$		−	4.24871i		−	0.148101i	0.997255	−	0.0740504i	$-0.0235926\pi$
$824$		0			0
$825$	−3.26795	−	14.4406i	−0.113775	−	0.502757i
$826$		0			0
$827$	−30.6147			−1.06458			−0.532289	−	0.846563i	$-0.678668\pi$
$827$	−30.6147			−1.06458			−0.532289	+	0.846563i	$0.678668\pi$
$828$		0			0
$829$	−12.1436			−0.421764			−0.210882	−	0.977511i	$-0.567634\pi$
$829$	−12.1436			−0.421764			−0.210882	+	0.977511i	$0.567634\pi$
$830$		0			0
$831$	−28.2467			−0.979866
$832$		0			0
$833$		−	9.00773i		−	0.312099i
$834$		0			0
$835$	6.00515			0.207817
$836$		0			0
$837$	−2.00000			−0.0691301
$838$		0			0
$839$		−	1.12436i		−	0.0388171i	−0.999812	−	0.0194085i	$-0.993822\pi$
$839$		−	1.12436i		−	0.0388171i	0.999812	−	0.0194085i	$-0.00617832\pi$
$840$		0			0
$841$	29.0000			1.00000
$842$		0			0
$843$	−28.8812			−0.994721
$844$		0			0
$845$	5.41154			0.186163
$846$		0			0
$847$	23.5106	−	11.2154i	0.807832	−	0.385365i
$848$		0			0
$849$		−	1.73354i		−	0.0594949i
$850$		0			0
$851$		−	35.3205i		−	1.21077i
$852$		0			0
$853$		−	16.5764i		−	0.567565i	−0.958889	−	0.283783i	$-0.908411\pi$
$853$		−	16.5764i		−	0.567565i	0.958889	−	0.283783i	$-0.0915894\pi$
$854$		0			0
$855$	4.73611			0.161972
$856$		0			0
$857$			30.6147i			1.04578i	0.852400	+	0.522890i	$0.175146\pi$
$857$			30.6147i			1.04578i	−0.852400	+	0.522890i	$0.824854\pi$
$858$		0			0
$859$			34.9282i			1.19173i	0.803083	+	0.595867i	$0.203192\pi$
$859$			34.9282i			1.19173i	−0.803083	+	0.595867i	$0.796808\pi$
$860$		0			0
$861$	11.2154			0.382219
$862$		0			0
$863$		−	29.1244i		−	0.991405i	−0.868493	−	0.495702i	$-0.834911\pi$
$863$		−	29.1244i		−	0.991405i	0.868493	−	0.495702i	$-0.165089\pi$
$864$		0			0
$865$			16.4064i			0.557834i
$866$		0			0
$867$			24.8564i			0.844168i
$868$		0			0
$869$	7.66025	−	1.73354i	0.259856	−	0.0588063i
$870$		0			0
$871$	24.6096			0.833864
$872$		0			0
$873$	1.46410			0.0495523
$874$		0			0
$875$	−16.4064			−0.554637
$876$		0			0
$877$			24.7796i			0.836747i	0.908275	+	0.418374i	$0.137400\pi$
$877$			24.7796i			0.836747i	−0.908275	+	0.418374i	$0.862600\pi$
$878$		0			0
$879$	9.47223			0.319490
$880$		0			0
$881$	−22.7846			−0.767633			−0.383817	−	0.923409i	$-0.625390\pi$
$881$	−22.7846			−0.767633			−0.383817	+	0.923409i	$0.625390\pi$
$882$		0			0
$883$		−	36.5359i		−	1.22953i	−0.788710	−	0.614765i	$-0.789251\pi$
$883$		−	36.5359i		−	1.22953i	0.788710	−	0.614765i	$-0.210749\pi$
$884$		0			0
$885$	−1.85641			−0.0624024
$886$		0			0
$887$	12.9393			0.434459			0.217230	−	0.976121i	$-0.430298\pi$
$887$	12.9393			0.434459			0.217230	+	0.976121i	$0.430298\pi$
$888$		0			0
$889$	5.60770			0.188076
$890$		0			0
$891$	−3.23483	+	0.732051i	−0.108371	+	0.0245246i
$892$		0			0
$893$			40.0870i			1.34146i
$894$		0			0
$895$			14.9282i			0.498995i
$896$		0			0
$897$			11.2058i			0.374150i
$898$		0			0
$899$		0			0
$900$		0			0
$901$			47.0211i			1.56650i
$902$		0			0
$903$			15.3205i			0.509835i
$904$		0			0
$905$	13.4641			0.447562
$906$		0			0
$907$		−	38.3923i		−	1.27480i	−0.770535	−	0.637398i	$-0.780011\pi$
$907$		−	38.3923i		−	1.27480i	0.770535	−	0.637398i	$-0.219989\pi$
$908$		0			0
$909$		−	8.20319i		−	0.272083i
$910$		0			0
$911$			50.9808i			1.68907i	0.535502	+	0.844534i	$0.320122\pi$
$911$			50.9808i			1.68907i	−0.535502	+	0.844534i	$0.679878\pi$
$912$		0			0
$913$	−15.3205	+	3.46708i	−0.507035	+	0.114743i
$914$		0			0
$915$	11.2058			0.370451
$916$		0			0
$917$	41.8564			1.38222
$918$		0			0
$919$	29.5157			0.973633			0.486817	−	0.873504i	$-0.338158\pi$
$919$	29.5157			0.973633			0.486817	+	0.873504i	$0.338158\pi$
$920$		0			0
$921$			11.2058i			0.369243i
$922$		0			0
$923$	17.2109			0.566504
$924$		0			0
$925$	33.3205			1.09557
$926$		0			0
$927$			10.0000i			0.328443i
$928$		0			0
$929$	11.8564			0.388996			0.194498	−	0.980903i	$-0.437692\pi$
$929$	11.8564			0.388996			0.194498	+	0.980903i	$0.437692\pi$
$930$		0			0
$931$	9.00773			0.295216
$932$		0			0
$933$	−30.9808			−1.01426
$934$		0			0
$935$	−3.46708	−	15.3205i	−0.113386	−	0.501034i
$936$		0			0
$937$		−	12.9393i		−	0.422709i	−0.977410	−	0.211354i	$-0.932213\pi$
$937$		−	12.9393i		−	0.422709i	0.977410	−	0.211354i	$-0.0677874\pi$
$938$		0			0
$939$			15.8564i			0.517454i
$940$		0			0
$941$		−	4.73611i		−	0.154393i	−0.997016	−	0.0771965i	$-0.975403\pi$
$941$		−	4.73611i		−	0.154393i	0.997016	−	0.0771965i	$-0.0245969\pi$
$942$		0			0
$943$	−22.4115			−0.729820
$944$		0			0
$945$			1.73354i			0.0563920i
$946$		0			0
$947$			22.1436i			0.719570i	0.933035	+	0.359785i	$0.117150\pi$
$947$			22.1436i			0.719570i	−0.933035	+	0.359785i	$0.882850\pi$
$948$		0			0
$949$	11.2154			0.364067
$950$		0			0
$951$		−	7.66025i		−	0.248401i
$952$		0			0
$953$			47.0211i			1.52316i	0.648069	+	0.761582i	$0.275577\pi$
$953$			47.0211i			1.52316i	−0.648069	+	0.761582i	$0.724423\pi$
$954$		0			0
$955$			9.60770i			0.310898i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−21.1425			−0.682727
$960$		0			0
$961$	27.0000			0.870968
$962$		0			0
$963$	−6.46965			−0.208482
$964$		0			0
$965$			16.4064i			0.528140i
$966$		0			0
$967$	7.10417			0.228455			0.114227	−	0.993455i	$-0.463561\pi$
$967$	7.10417			0.228455			0.114227	+	0.993455i	$0.463561\pi$
$968$		0			0
$969$	−41.8564			−1.34462
$970$		0			0
$971$			40.0000i			1.28366i	0.766846	+	0.641831i	$0.221825\pi$
$971$			40.0000i			1.28366i	−0.766846	+	0.641831i	$0.778175\pi$
$972$		0			0
$973$	−4.10512			−0.131604
$974$		0			0
$975$	−10.5712			−0.338551
$976$		0			0
$977$	−40.2487			−1.28767			−0.643835	−	0.765164i	$-0.722658\pi$
$977$	−40.2487			−1.28767			−0.643835	+	0.765164i	$0.722658\pi$
$978$		0			0
$979$	−33.6173	+	7.60770i	−1.07441	+	0.243143i
$980$		0			0
$981$		−	10.5712i		−	0.337514i
$982$		0			0
$983$		−	32.3397i		−	1.03148i	−0.856746	−	0.515739i	$-0.827517\pi$
$983$		−	32.3397i		−	1.03148i	0.856746	−	0.515739i	$-0.172483\pi$
$984$		0			0
$985$		−	12.9393i		−	0.412281i
$986$		0			0
$987$	−14.6728			−0.467042
$988$		0			0
$989$		−	30.6147i		−	0.973492i
$990$		0			0
$991$			18.7846i			0.596713i	0.954455	+	0.298356i	$0.0964384\pi$
$991$			18.7846i			0.596713i	−0.954455	+	0.298356i	$0.903562\pi$
$992$		0			0
$993$	14.9282			0.473732
$994$		0			0
$995$			12.1051i			0.383758i
$996$		0			0
$997$			54.1253i			1.71416i	0.515179	+	0.857082i	$0.327725\pi$
$997$			54.1253i			1.71416i	−0.515179	+	0.857082i	$0.672275\pi$
$998$		0			0
$999$		−	7.46410i		−	0.236154i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2112.2.o.e.703.4		8
4.3	odd	2	inner	2112.2.o.e.703.7		8
8.3	odd	2		528.2.o.b.175.1	✓	8
8.5	even	2		528.2.o.b.175.6	yes	8
11.10	odd	2	inner	2112.2.o.e.703.3		8
24.5	odd	2		1584.2.o.g.703.8		8
24.11	even	2		1584.2.o.g.703.5		8
44.43	even	2	inner	2112.2.o.e.703.8		8
88.21	odd	2		528.2.o.b.175.5	yes	8
88.43	even	2		528.2.o.b.175.2	yes	8
264.131	odd	2		1584.2.o.g.703.7		8
264.197	even	2		1584.2.o.g.703.6		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
528.2.o.b.175.1	✓	8	8.3	odd	2
528.2.o.b.175.2	yes	8	88.43	even	2
528.2.o.b.175.5	yes	8	88.21	odd	2
528.2.o.b.175.6	yes	8	8.5	even	2
1584.2.o.g.703.5		8	24.11	even	2
1584.2.o.g.703.6		8	264.197	even	2
1584.2.o.g.703.7		8	264.131	odd	2
1584.2.o.g.703.8		8	24.5	odd	2
2112.2.o.e.703.3		8	11.10	odd	2	inner
2112.2.o.e.703.4		8	1.1	even	1	trivial
2112.2.o.e.703.7		8	4.3	odd	2	inner
2112.2.o.e.703.8		8	44.43	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:	\( N \)	\(=\)	\( 2112 = 2^{6} \cdot 3 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2112.o (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} +0.732051 q^{5} +2.36806 q^{7} -1.00000 q^{9} +(-3.23483 + 0.732051i) q^{11} +2.36806i q^{13} -0.732051i q^{15} +6.46965i q^{17} -6.46965 q^{19} -2.36806i q^{21} +4.73205i q^{23} -4.46410 q^{25} +1.00000i q^{27} +2.00000i q^{31} +(0.732051 + 3.23483i) q^{33} +1.73354 q^{35} -7.46410 q^{37} +2.36806 q^{39} +4.73611i q^{41} -6.46965 q^{43} -0.732051 q^{45} -6.19615i q^{47} -1.39230 q^{49} +6.46965 q^{51} +7.26795 q^{53} +(-2.36806 + 0.535898i) q^{55} +6.46965i q^{57} -2.53590i q^{59} +15.3074i q^{61} -2.36806 q^{63} +1.73354i q^{65} -10.3923i q^{67} +4.73205 q^{69} -7.26795i q^{71} -4.73611i q^{73} +4.46410i q^{75} +(-7.66025 + 1.73354i) q^{77} -2.36806 q^{79} +1.00000 q^{81} +4.73611 q^{83} +4.73611i q^{85} +10.3923 q^{89} +5.60770i q^{91} +2.00000 q^{93} -4.73611 q^{95} -1.46410 q^{97} +(3.23483 - 0.732051i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{5} - 8 q^{9} - 8 q^{25} - 8 q^{33} - 32 q^{37} + 8 q^{45} + 72 q^{49} + 72 q^{53} + 24 q^{69} + 8 q^{77} + 8 q^{81} + 16 q^{93} + 16 q^{97}+O(q^{100}) \) 8 * q - 8 * q^5 - 8 * q^9 - 8 * q^25 - 8 * q^33 - 32 * q^37 + 8 * q^45 + 72 * q^49 + 72 * q^53 + 24 * q^69 + 8 * q^77 + 8 * q^81 + 16 * q^93 + 16 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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