LMFDB - Embedded newform 4032.2.v.c.3599.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4032,2,Mod(1583,4032)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4032, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 3, 2, 0]))

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

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

chi := DirichletCharacter("4032.1583");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4032 = 2^{6} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4032.v (of order $4$, degree $2$, 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:	$12$
Relative dimension:	$6$ over $\Q(i)$
Coefficient field:	12.0.653473922154496.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 4x^{10} + 13x^{8} - 28x^{6} + 52x^{4} - 64x^{2} + 64 $ x^12 - 4x^10 + 13x^8 - 28x^6 + 52x^4 - 64*x^2 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 2^{12} $
Twist minimal:	no (minimal twist has level 1008)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			3599.1
Root			$1.35489 - 0.405301i$ of defining polynomial
Character	$\chi$	$=$	4032.3599
Dual form			4032.2.v.c.1583.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.41421 + 1.41421i) q^{5} +1.00000 q^{7} +O(q^{10})$	$q+(-1.41421 + 1.41421i) q^{5} +1.00000 q^{7} +(-1.89918 - 1.89918i) q^{11} +(0.853635 - 0.853635i) q^{13} +2.59114i q^{17} +0.206992i q^{23} +1.00000i q^{25} +(-3.10640 - 3.10640i) q^{29} +1.70727i q^{31} +(-1.41421 + 1.41421i) q^{35} +(-1.00000 - 1.00000i) q^{37} -11.0764 q^{41} +(7.12494 - 7.12494i) q^{43} +3.24241 q^{47} +1.00000 q^{49} +(-0.722255 + 0.722255i) q^{53} +5.37169 q^{55} +(-5.00558 - 5.00558i) q^{59} +(-3.14637 + 3.14637i) q^{61} +2.41444i q^{65} +(-4.14637 - 4.14637i) q^{67} +10.0762i q^{71} +5.66442i q^{73} +(-1.89918 - 1.89918i) q^{77} -3.41454i q^{79} +(7.27806 - 7.27806i) q^{83} +(-3.66442 - 3.66442i) q^{85} -17.1473 q^{89} +(0.853635 - 0.853635i) q^{91} -16.5426 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 12 q^{7}+O(q^{10}) $ 12 * q + 12 * q^7	$ 12 q + 12 q^{7} + 16 q^{13} - 12 q^{37} + 20 q^{43} + 12 q^{49} - 32 q^{55} - 32 q^{61} - 44 q^{67} + 64 q^{85} + 16 q^{91} - 56 q^{97}+O(q^{100}) $ 12 * q + 12 * q^7 + 16 * q^13 - 12 * q^37 + 20 * q^43 + 12 * q^49 - 32 * q^55 - 32 * q^61 - 44 * q^67 + 64 * q^85 + 16 * q^91 - 56 * q^97

Display 100 coefficients 10 coefficients

Character values

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

$n$	$127$	$577$	$1793$	$3781$
$\chi(n)$	$-1$	$1$	$-1$	$e\left(\frac{1}{4}\right)$

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
$5$	−1.41421	+	1.41421i	−0.632456	+	0.632456i	−0.948683	−	0.316228i	$-0.897584\pi$
$5$	−1.41421	+	1.41421i	−0.632456	+	0.632456i	0.316228	+	0.948683i	$0.397584\pi$
$6$		0			0
$7$	1.00000			0.377964
$7$	1.00000			0.377964
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−1.89918	−	1.89918i	−0.572624	−	0.572624i	0.360237	−	0.932861i	$-0.382696\pi$
$11$	−1.89918	−	1.89918i	−0.572624	−	0.572624i	−0.932861	+	0.360237i	$0.882696\pi$
$12$		0			0
$13$	0.853635	−	0.853635i	0.236756	−	0.236756i	−0.578750	−	0.815505i	$-0.696459\pi$
$13$	0.853635	−	0.853635i	0.236756	−	0.236756i	0.815505	+	0.578750i	$0.196459\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			2.59114i			0.628443i	0.949350	+	0.314222i	$0.101744\pi$
$17$			2.59114i			0.628443i	−0.949350	+	0.314222i	$0.898256\pi$
$18$		0			0
$19$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$19$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			0.206992i			0.0431608i	0.999767	+	0.0215804i	$0.00686979\pi$
$23$			0.206992i			0.0431608i	−0.999767	+	0.0215804i	$0.993130\pi$
$24$		0			0
$25$			1.00000i			0.200000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−3.10640	−	3.10640i	−0.576844	−	0.576844i	0.357188	−	0.934032i	$-0.383735\pi$
$29$	−3.10640	−	3.10640i	−0.576844	−	0.576844i	−0.934032	+	0.357188i	$0.883735\pi$
$30$		0			0
$31$			1.70727i			0.306635i	0.988177	+	0.153317i	$0.0489957\pi$
$31$			1.70727i			0.306635i	−0.988177	+	0.153317i	$0.951004\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−1.41421	+	1.41421i	−0.239046	+	0.239046i
$36$		0			0
$37$	−1.00000	−	1.00000i	−0.164399	−	0.164399i	0.620113	−	0.784512i	$-0.287087\pi$
$37$	−1.00000	−	1.00000i	−0.164399	−	0.164399i	−0.784512	+	0.620113i	$0.787087\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−11.0764			−1.72985			−0.864923	−	0.501904i	$-0.832633\pi$
$41$	−11.0764			−1.72985			−0.864923	+	0.501904i	$0.832633\pi$
$42$		0			0
$43$	7.12494	−	7.12494i	1.08654	−	1.08654i	0.0906618	−	0.995882i	$-0.471102\pi$
$43$	7.12494	−	7.12494i	1.08654	−	1.08654i	0.995882	−	0.0906618i	$-0.0288982\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	3.24241			0.472954			0.236477	−	0.971637i	$-0.424007\pi$
$47$	3.24241			0.472954			0.236477	+	0.971637i	$0.424007\pi$
$48$		0			0
$49$	1.00000			0.142857
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−0.722255	+	0.722255i	−0.0992094	+	0.0992094i	−0.754969	−	0.655760i	$-0.772348\pi$
$53$	−0.722255	+	0.722255i	−0.0992094	+	0.0992094i	0.655760	+	0.754969i	$0.272348\pi$
$54$		0			0
$55$	5.37169			0.724319
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−5.00558	−	5.00558i	−0.651671	−	0.651671i	0.301724	−	0.953395i	$-0.402438\pi$
$59$	−5.00558	−	5.00558i	−0.651671	−	0.651671i	−0.953395	+	0.301724i	$0.902438\pi$
$60$		0			0
$61$	−3.14637	+	3.14637i	−0.402851	+	0.402851i	−0.879236	−	0.476386i	$-0.841947\pi$
$61$	−3.14637	+	3.14637i	−0.402851	+	0.402851i	0.476386	+	0.879236i	$0.341947\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			2.41444i			0.299475i
$66$		0			0
$67$	−4.14637	−	4.14637i	−0.506559	−	0.506559i	0.406909	−	0.913469i	$-0.366606\pi$
$67$	−4.14637	−	4.14637i	−0.506559	−	0.506559i	−0.913469	+	0.406909i	$0.866606\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			10.0762i			1.19582i	0.801562	+	0.597912i	$0.204003\pi$
$71$			10.0762i			1.19582i	−0.801562	+	0.597912i	$0.795997\pi$
$72$		0			0
$73$			5.66442i			0.662971i	0.943460	+	0.331485i	$0.107550\pi$
$73$			5.66442i			0.662971i	−0.943460	+	0.331485i	$0.892450\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−1.89918	−	1.89918i	−0.216432	−	0.216432i
$78$		0			0
$79$		−	3.41454i		−	0.384166i	−0.981379	−	0.192083i	$-0.938476\pi$
$79$		−	3.41454i		−	0.384166i	0.981379	−	0.192083i	$-0.0615242\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	7.27806	−	7.27806i	0.798871	−	0.798871i	−0.184047	−	0.982918i	$-0.558920\pi$
$83$	7.27806	−	7.27806i	0.798871	−	0.798871i	0.982918	+	0.184047i	$0.0589198\pi$
$84$		0			0
$85$	−3.66442	−	3.66442i	−0.397463	−	0.397463i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−17.1473			−1.81761			−0.908803	−	0.417226i	$-0.863002\pi$
$89$	−17.1473			−1.81761			−0.908803	+	0.417226i	$0.863002\pi$
$90$		0			0
$91$	0.853635	−	0.853635i	0.0894852	−	0.0894852i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−16.5426			−1.67965			−0.839824	−	0.542859i	$-0.817342\pi$
$97$	−16.5426			−1.67965			−0.839824	+	0.542859i	$0.817342\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	0.206992	−	0.206992i	0.0205965	−	0.0205965i	−0.696734	−	0.717330i	$-0.745364\pi$
$101$	0.206992	−	0.206992i	0.0205965	−	0.0205965i	0.717330	+	0.696734i	$0.245364\pi$
$102$		0			0
$103$	5.95715			0.586976			0.293488	−	0.955963i	$-0.405184\pi$
$103$	5.95715			0.586976			0.293488	+	0.955963i	$0.405184\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−2.10617	−	2.10617i	−0.203611	−	0.203611i	0.597934	−	0.801545i	$-0.295988\pi$
$107$	−2.10617	−	2.10617i	−0.203611	−	0.203611i	−0.801545	+	0.597934i	$0.795988\pi$
$108$		0			0
$109$	1.00000	−	1.00000i	0.0957826	−	0.0957826i	−0.657592	−	0.753374i	$-0.728425\pi$
$109$	1.00000	−	1.00000i	0.0957826	−	0.0957826i	0.753374	+	0.657592i	$0.228425\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	4.79859i		−	0.451413i	−0.974195	−	0.225707i	$-0.927531\pi$
$113$		−	4.79859i		−	0.451413i	0.974195	−	0.225707i	$-0.0724691\pi$
$114$		0			0
$115$	−0.292731	−	0.292731i	−0.0272973	−	0.0272973i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			2.59114i			0.237529i
$120$		0			0
$121$		−	3.78623i		−	0.344203i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−8.48528	−	8.48528i	−0.758947	−	0.758947i
$126$		0			0
$127$		−	17.0790i		−	1.51551i	−0.652538	−	0.757756i	$-0.726296\pi$
$127$		−	17.0790i		−	1.51551i	0.652538	−	0.757756i	$-0.273704\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	1.62121	−	1.62121i	0.141645	−	0.141645i	−0.632728	−	0.774374i	$-0.718065\pi$
$131$	1.62121	−	1.62121i	0.141645	−	0.141645i	0.774374	+	0.632728i	$0.218065\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−15.6377			−1.33602			−0.668010	−	0.744152i	$-0.732854\pi$
$137$	−15.6377			−1.33602			−0.668010	+	0.744152i	$0.732854\pi$
$138$		0			0
$139$	−2.58546	+	2.58546i	−0.219296	+	0.219296i	−0.808202	−	0.588906i	$-0.799559\pi$
$139$	−2.58546	+	2.58546i	−0.219296	+	0.219296i	0.588906	+	0.808202i	$0.299559\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−3.24241			−0.271144
$144$		0			0
$145$	8.78623			0.729657
$146$		0			0
$147$		0			0
$148$		0			0
$149$	9.31920	−	9.31920i	0.763459	−	0.763459i	−0.213487	−	0.976946i	$-0.568482\pi$
$149$	9.31920	−	9.31920i	0.763459	−	0.763459i	0.976946	+	0.213487i	$0.0684821\pi$
$150$		0			0
$151$	15.0361			1.22362			0.611811	−	0.791004i	$-0.290441\pi$
$151$	15.0361			1.22362			0.611811	+	0.791004i	$0.290441\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−2.41444	−	2.41444i	−0.193933	−	0.193933i
$156$		0			0
$157$	8.81079	−	8.81079i	0.703177	−	0.703177i	−0.261914	−	0.965091i	$-0.584354\pi$
$157$	8.81079	−	8.81079i	0.703177	−	0.703177i	0.965091	+	0.261914i	$0.0843536\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			0.206992i			0.0163133i
$162$		0			0
$163$	−15.1825	−	15.1825i	−1.18918	−	1.18918i	−0.977293	−	0.211890i	$-0.932038\pi$
$163$	−15.1825	−	15.1825i	−1.18918	−	1.18918i	−0.211890	−	0.977293i	$-0.567962\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	2.00046i		−	0.154800i	−0.997000	−	0.0774001i	$-0.975338\pi$
$167$		−	2.00046i		−	0.154800i	0.997000	−	0.0774001i	$-0.0246619\pi$
$168$		0			0
$169$			11.5426i			0.887894i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	3.27271	+	3.27271i	0.248819	+	0.248819i	0.820486	−	0.571667i	$-0.193703\pi$
$173$	3.27271	+	3.27271i	0.248819	+	0.248819i	−0.571667	+	0.820486i	$0.693703\pi$
$174$		0			0
$175$			1.00000i			0.0755929i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−14.3898	+	14.3898i	−1.07555	+	1.07555i	−0.0786423	+	0.996903i	$0.525058\pi$
$179$	−14.3898	+	14.3898i	−1.07555	+	1.07555i	−0.996903	+	0.0786423i	$0.974942\pi$
$180$		0			0
$181$	−10.2253	−	10.2253i	−0.760043	−	0.760043i	0.216287	−	0.976330i	$-0.430605\pi$
$181$	−10.2253	−	10.2253i	−0.760043	−	0.760043i	−0.976330	+	0.216287i	$0.930605\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	2.82843			0.207950
$186$		0			0
$187$	4.92104	−	4.92104i	0.359862	−	0.359862i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−3.51000			−0.253974			−0.126987	−	0.991904i	$-0.540531\pi$
$191$	−3.51000			−0.253974			−0.126987	+	0.991904i	$0.540531\pi$
$192$		0			0
$193$	−2.00000			−0.143963			−0.0719816	−	0.997406i	$-0.522932\pi$
$193$	−2.00000			−0.143963			−0.0719816	+	0.997406i	$0.522932\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	10.5612	−	10.5612i	0.752451	−	0.752451i	−0.222485	−	0.974936i	$-0.571417\pi$
$197$	10.5612	−	10.5612i	0.752451	−	0.752451i	0.974936	+	0.222485i	$0.0714168\pi$
$198$		0			0
$199$	0.921039			0.0652907			0.0326453	−	0.999467i	$-0.489607\pi$
$199$	0.921039			0.0652907			0.0326453	+	0.999467i	$0.489607\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−3.10640	−	3.10640i	−0.218027	−	0.218027i
$204$		0			0
$205$	15.6644	−	15.6644i	1.09405	−	1.09405i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−14.8322	−	14.8322i	−1.02109	−	1.02109i	−0.999773	−	0.0213188i	$-0.993213\pi$
$211$	−14.8322	−	14.8322i	−1.02109	−	1.02109i	−0.0213188	−	0.999773i	$-0.506787\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			20.1524i			1.37438i
$216$		0			0
$217$			1.70727i			0.115897i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	2.21189	+	2.21189i	0.148788	+	0.148788i
$222$		0			0
$223$		−	26.4078i		−	1.76840i	−0.467111	−	0.884199i	$-0.654705\pi$
$223$		−	26.4078i		−	1.76840i	0.467111	−	0.884199i	$-0.345295\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	1.38392	−	1.38392i	0.0918538	−	0.0918538i	−0.659687	−	0.751541i	$-0.729311\pi$
$227$	1.38392	−	1.38392i	0.0918538	−	0.0918538i	0.751541	+	0.659687i	$0.229311\pi$
$228$		0			0
$229$	1.77467	+	1.77467i	0.117274	+	0.117274i	0.763308	−	0.646035i	$-0.223574\pi$
$229$	1.77467	+	1.77467i	0.117274	+	0.117274i	−0.646035	+	0.763308i	$0.723574\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	14.5864			0.955588			0.477794	−	0.878472i	$-0.341437\pi$
$233$	14.5864			0.955588			0.477794	+	0.878472i	$0.341437\pi$
$234$		0			0
$235$	−4.58546	+	4.58546i	−0.299123	+	0.299123i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−28.9866			−1.87499			−0.937494	−	0.348001i	$-0.886861\pi$
$239$	−28.9866			−1.87499			−0.937494	+	0.348001i	$0.886861\pi$
$240$		0			0
$241$	11.3717			0.732515			0.366258	−	0.930514i	$-0.380639\pi$
$241$	11.3717			0.732515			0.366258	+	0.930514i	$0.380639\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−1.41421	+	1.41421i	−0.0903508	+	0.0903508i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−11.8091	−	11.8091i	−0.745382	−	0.745382i	0.228227	−	0.973608i	$-0.426707\pi$
$251$	−11.8091	−	11.8091i	−0.745382	−	0.745382i	−0.973608	+	0.228227i	$0.926707\pi$
$252$		0			0
$253$	0.393115	−	0.393115i	0.0247149	−	0.0247149i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	19.0871i		−	1.19062i	−0.803495	−	0.595311i	$-0.797029\pi$
$257$		−	19.0871i		−	1.19062i	0.803495	−	0.595311i	$-0.202971\pi$
$258$		0			0
$259$	−1.00000	−	1.00000i	−0.0621370	−	0.0621370i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	17.2589i		−	1.06423i	−0.846672	−	0.532116i	$-0.821397\pi$
$263$		−	17.2589i		−	1.06423i	0.846672	−	0.532116i	$-0.178603\pi$
$264$		0			0
$265$		−	2.04285i		−	0.125491i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	9.66221	+	9.66221i	0.589115	+	0.589115i	0.937392	−	0.348277i	$-0.113233\pi$
$269$	9.66221	+	9.66221i	0.589115	+	0.589115i	−0.348277	+	0.937392i	$0.613233\pi$
$270$		0			0
$271$		−	13.9572i		−	0.847837i	−0.905700	−	0.423918i	$-0.860654\pi$
$271$		−	13.9572i		−	0.847837i	0.905700	−	0.423918i	$-0.139346\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	1.89918	−	1.89918i	0.114525	−	0.114525i
$276$		0			0
$277$	−14.2713	−	14.2713i	−0.857480	−	0.857480i	0.133561	−	0.991041i	$-0.457359\pi$
$277$	−14.2713	−	14.2713i	−0.857480	−	0.857480i	−0.991041	+	0.133561i	$0.957359\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−24.9510			−1.48845			−0.744225	−	0.667929i	$-0.767181\pi$
$281$	−24.9510			−1.48845			−0.744225	+	0.667929i	$0.767181\pi$
$282$		0			0
$283$	−23.0361	+	23.0361i	−1.36935	+	1.36935i	−0.507995	+	0.861360i	$0.669613\pi$
$283$	−23.0361	+	23.0361i	−1.36935	+	1.36935i	−0.861360	+	0.507995i	$0.830387\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−11.0764			−0.653820
$288$		0			0
$289$	10.2860			0.605059
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−19.5314	+	19.5314i	−1.14104	+	1.14104i	−0.152776	+	0.988261i	$0.548821\pi$
$293$	−19.5314	+	19.5314i	−1.14104	+	1.14104i	−0.988261	+	0.152776i	$0.951179\pi$
$294$		0			0
$295$	14.1579			0.824306
$296$		0			0
$297$		0			0
$298$		0			0
$299$	0.176696	+	0.176696i	0.0102186	+	0.0102186i
$300$		0			0
$301$	7.12494	−	7.12494i	0.410675	−	0.410675i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	8.89927i		−	0.509570i
$306$		0			0
$307$	14.7434	+	14.7434i	0.841449	+	0.841449i	0.989047	−	0.147598i	$-0.0471542\pi$
$307$	14.7434	+	14.7434i	0.841449	+	0.841449i	−0.147598	+	0.989047i	$0.547154\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			6.76875i			0.383821i	0.981412	+	0.191910i	$0.0614683\pi$
$311$			6.76875i			0.383821i	−0.981412	+	0.191910i	$0.938532\pi$
$312$		0			0
$313$			27.0361i			1.52817i	0.645115	+	0.764086i	$0.276810\pi$
$313$			27.0361i			1.52817i	−0.645115	+	0.764086i	$0.723190\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	4.93460	+	4.93460i	0.277155	+	0.277155i	0.831972	−	0.554817i	$-0.187212\pi$
$317$	4.93460	+	4.93460i	0.277155	+	0.277155i	−0.554817	+	0.831972i	$0.687212\pi$
$318$		0			0
$319$			11.7992i			0.660630i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	0.853635	+	0.853635i	0.0473511	+	0.0473511i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	3.24241			0.178760
$330$		0			0
$331$	−18.4538	+	18.4538i	−1.01431	+	1.01431i	−0.0144159	+	0.999896i	$0.504589\pi$
$331$	−18.4538	+	18.4538i	−1.01431	+	1.01431i	−0.999896	+	0.0144159i	$0.995411\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	11.7277			0.640752
$336$		0			0
$337$	8.15792			0.444390			0.222195	−	0.975002i	$-0.428678\pi$
$337$	8.15792			0.444390			0.222195	+	0.975002i	$0.428678\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	3.24241	−	3.24241i	0.175586	−	0.175586i
$342$		0			0
$343$	1.00000			0.0539949
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−21.0166	−	21.0166i	−1.12823	−	1.12823i	−0.990465	−	0.137765i	$-0.956008\pi$
$347$	−21.0166	−	21.0166i	−1.12823	−	1.12823i	−0.137765	−	0.990465i	$-0.543992\pi$
$348$		0			0
$349$	20.4752	−	20.4752i	1.09601	−	1.09601i	0.101141	−	0.994872i	$-0.467751\pi$
$349$	20.4752	−	20.4752i	1.09601	−	1.09601i	0.994872	−	0.101141i	$-0.0322493\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			18.6731i			0.993871i	0.867787	+	0.496935i	$0.165541\pi$
$353$			18.6731i			0.993871i	−0.867787	+	0.496935i	$0.834459\pi$
$354$		0			0
$355$	−14.2499	−	14.2499i	−0.756305	−	0.756305i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			1.31889i			0.0696083i	0.999394	+	0.0348042i	$0.0110807\pi$
$359$			1.31889i			0.0696083i	−0.999394	+	0.0348042i	$0.988919\pi$
$360$		0			0
$361$		−	19.0000i		−	1.00000i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−8.01070	−	8.01070i	−0.419299	−	0.419299i
$366$		0			0
$367$			27.6644i			1.44407i	0.691856	+	0.722036i	$0.256793\pi$
$367$			27.6644i			1.44407i	−0.691856	+	0.722036i	$0.743207\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−0.722255	+	0.722255i	−0.0374976	+	0.0374976i
$372$		0			0
$373$	−21.6430	−	21.6430i	−1.12063	−	1.12063i	−0.991646	−	0.128986i	$-0.958828\pi$
$373$	−21.6430	−	21.6430i	−1.12063	−	1.12063i	−0.128986	−	0.991646i	$-0.541172\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−5.30346			−0.273142
$378$		0			0
$379$	0.582326	−	0.582326i	0.0299121	−	0.0299121i	−0.691993	−	0.721905i	$-0.743267\pi$
$379$	0.582326	−	0.582326i	0.0299121	−	0.0299121i	0.721905	+	0.691993i	$0.243267\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−12.8396			−0.656072			−0.328036	−	0.944665i	$-0.606387\pi$
$383$	−12.8396			−0.656072			−0.328036	+	0.944665i	$0.606387\pi$
$384$		0			0
$385$	5.37169			0.273767
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−11.8334	+	11.8334i	−0.599977	+	0.599977i	−0.940306	−	0.340329i	$-0.889462\pi$
$389$	−11.8334	+	11.8334i	−0.599977	+	0.599977i	0.340329	+	0.940306i	$0.389462\pi$
$390$		0			0
$391$	−0.536345			−0.0271241
$392$		0			0
$393$		0			0
$394$		0			0
$395$	4.82889	+	4.82889i	0.242968	+	0.242968i
$396$		0			0
$397$	−19.8898	+	19.8898i	−0.998238	+	0.998238i	−0.999998	−	0.00176051i	$-0.999440\pi$
$397$	−19.8898	+	19.8898i	−0.998238	+	0.998238i	0.00176051	+	0.999998i	$0.499440\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	2.44474i		−	0.122084i	−0.998135	−	0.0610422i	$-0.980558\pi$
$401$		−	2.44474i		−	0.122084i	0.998135	−	0.0610422i	$-0.0194424\pi$
$402$		0			0
$403$	1.45738	+	1.45738i	0.0725975	+	0.0725975i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			3.79836i			0.188278i
$408$		0			0
$409$			22.1151i			1.09352i	0.837289	+	0.546760i	$0.184139\pi$
$409$			22.1151i			1.09352i	−0.837289	+	0.546760i	$0.815861\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−5.00558	−	5.00558i	−0.246309	−	0.246309i
$414$		0			0
$415$			20.5855i			1.01050i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	13.0163	−	13.0163i	0.635887	−	0.635887i	−0.313651	−	0.949538i	$-0.601552\pi$
$419$	13.0163	−	13.0163i	0.635887	−	0.635887i	0.949538	+	0.313651i	$0.101552\pi$
$420$		0			0
$421$	−7.64300	−	7.64300i	−0.372497	−	0.372497i	0.495889	−	0.868386i	$-0.334842\pi$
$421$	−7.64300	−	7.64300i	−0.372497	−	0.372497i	−0.868386	+	0.495889i	$0.834842\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−2.59114			−0.125689
$426$		0			0
$427$	−3.14637	+	3.14637i	−0.152263	+	0.152263i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	4.83332			0.232813			0.116406	−	0.993202i	$-0.462862\pi$
$431$	4.83332			0.232813			0.116406	+	0.993202i	$0.462862\pi$
$432$		0			0
$433$	−3.03612			−0.145906			−0.0729532	−	0.997335i	$-0.523242\pi$
$433$	−3.03612			−0.145906			−0.0729532	+	0.997335i	$0.523242\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	26.6577			1.27230			0.636151	−	0.771564i	$-0.280525\pi$
$439$	26.6577			1.27230			0.636151	+	0.771564i	$0.280525\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	2.45513	+	2.45513i	0.116647	+	0.116647i	0.763021	−	0.646374i	$-0.223715\pi$
$443$	2.45513	+	2.45513i	0.116647	+	0.116647i	−0.646374	+	0.763021i	$0.723715\pi$
$444$		0			0
$445$	24.2499	−	24.2499i	1.14955	−	1.14955i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			1.41421i			0.0667409i	0.999443	+	0.0333704i	$0.0106241\pi$
$449$			1.41421i			0.0667409i	−0.999443	+	0.0333704i	$0.989376\pi$
$450$		0			0
$451$	21.0361	+	21.0361i	0.990552	+	0.990552i
$452$		0			0
$453$		0			0
$454$		0			0
$455$			2.41444i			0.113191i
$456$		0			0
$457$			4.78623i			0.223890i	0.993714	+	0.111945i	$0.0357081\pi$
$457$			4.78623i			0.223890i	−0.993714	+	0.111945i	$0.964292\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	14.5258	+	14.5258i	0.676535	+	0.676535i	0.959214	−	0.282679i	$-0.0912233\pi$
$461$	14.5258	+	14.5258i	0.676535	+	0.676535i	−0.282679	+	0.959214i	$0.591223\pi$
$462$		0			0
$463$			1.17092i			0.0544174i	0.999630	+	0.0272087i	$0.00866187\pi$
$463$			1.17092i			0.0544174i	−0.999630	+	0.0272087i	$0.991338\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−1.38392	+	1.38392i	−0.0640400	+	0.0640400i	−0.738401	−	0.674361i	$-0.764419\pi$
$467$	−1.38392	+	1.38392i	−0.0640400	+	0.0640400i	0.674361	+	0.738401i	$0.264419\pi$
$468$		0			0
$469$	−4.14637	−	4.14637i	−0.191461	−	0.191461i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−27.0631			−1.24436
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$	20.4969			0.936527			0.468264	−	0.883589i	$-0.344880\pi$
$479$	20.4969			0.936527			0.468264	+	0.883589i	$0.344880\pi$
$480$		0			0
$481$	−1.70727			−0.0778448
$482$		0			0
$483$		0			0
$484$		0			0
$485$	23.3948	−	23.3948i	1.06230	−	1.06230i
$486$		0			0
$487$	23.7073			1.07428			0.537139	−	0.843493i	$-0.319505\pi$
$487$	23.7073			1.07428			0.537139	+	0.843493i	$0.319505\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−1.07121	−	1.07121i	−0.0483431	−	0.0483431i	0.682522	−	0.730865i	$-0.260883\pi$
$491$	−1.07121	−	1.07121i	−0.0483431	−	0.0483431i	−0.730865	+	0.682522i	$0.760883\pi$
$492$		0			0
$493$	8.04912	−	8.04912i	0.362514	−	0.362514i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			10.0762i			0.451979i
$498$		0			0
$499$	−6.53948	−	6.53948i	−0.292747	−	0.292747i	0.545417	−	0.838165i	$-0.316371\pi$
$499$	−6.53948	−	6.53948i	−0.292747	−	0.292747i	−0.838165	+	0.545417i	$0.816371\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			20.6270i			0.919711i	0.887994	+	0.459855i	$0.152099\pi$
$503$			20.6270i			0.919711i	−0.887994	+	0.459855i	$0.847901\pi$
$504$		0			0
$505$			0.585462i			0.0260527i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	3.11679	+	3.11679i	0.138149	+	0.138149i	0.772800	−	0.634650i	$-0.218856\pi$
$509$	3.11679	+	3.11679i	0.138149	+	0.138149i	−0.634650	+	0.772800i	$0.718856\pi$
$510$		0			0
$511$			5.66442i			0.250579i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−8.42469	+	8.42469i	−0.371236	+	0.371236i
$516$		0			0
$517$	−6.15792	−	6.15792i	−0.270825	−	0.270825i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	29.5729			1.29561			0.647805	−	0.761806i	$-0.275687\pi$
$521$	29.5729			1.29561			0.647805	+	0.761806i	$0.275687\pi$
$522$		0			0
$523$	3.46365	−	3.46365i	0.151455	−	0.151455i	−0.627313	−	0.778768i	$-0.715845\pi$
$523$	3.46365	−	3.46365i	0.151455	−	0.151455i	0.778768	+	0.627313i	$0.215845\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−4.42377			−0.192702
$528$		0			0
$529$	22.9572			0.998137
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−9.45521	+	9.45521i	−0.409551	+	0.409551i
$534$		0			0
$535$	5.95715			0.257550
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−1.89918	−	1.89918i	−0.0818035	−	0.0818035i
$540$		0			0
$541$	−21.8438	+	21.8438i	−0.939137	+	0.939137i	−0.998251	−	0.0591142i	$-0.981172\pi$
$541$	−21.8438	+	21.8438i	−0.939137	+	0.939137i	0.0591142	+	0.998251i	$0.481172\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			2.82843i			0.121157i
$546$		0			0
$547$	1.89648	+	1.89648i	0.0810876	+	0.0810876i	0.746487	−	0.665400i	$-0.231739\pi$
$547$	1.89648	+	1.89648i	0.0810876	+	0.0810876i	−0.665400	+	0.746487i	$0.731739\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		−	3.41454i		−	0.145201i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	31.2695	+	31.2695i	1.32493	+	1.32493i	0.909730	+	0.415201i	$0.136289\pi$
$557$	31.2695	+	31.2695i	1.32493	+	1.32493i	0.415201	+	0.909730i	$0.363711\pi$
$558$		0			0
$559$		−	12.1642i		−	0.514491i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	31.2288	−	31.2288i	1.31614	−	1.31614i	0.399330	−	0.916807i	$-0.369243\pi$
$563$	31.2288	−	31.2288i	1.31614	−	1.31614i	0.916807	−	0.399330i	$-0.130757\pi$
$564$		0			0
$565$	6.78623	+	6.78623i	0.285499	+	0.285499i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	4.94055			0.207119			0.103559	−	0.994623i	$-0.466977\pi$
$569$	4.94055			0.207119			0.103559	+	0.994623i	$0.466977\pi$
$570$		0			0
$571$	−25.1396	+	25.1396i	−1.05206	+	1.05206i	−0.0534928	+	0.998568i	$0.517035\pi$
$571$	−25.1396	+	25.1396i	−1.05206	+	1.05206i	−0.998568	+	0.0534928i	$0.982965\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−0.206992			−0.00863217
$576$		0			0
$577$	−36.1579			−1.50527			−0.752637	−	0.658436i	$-0.771218\pi$
$577$	−36.1579			−1.50527			−0.752637	+	0.658436i	$0.771218\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	7.27806	−	7.27806i	0.301945	−	0.301945i
$582$		0			0
$583$	2.74338			0.113619
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−11.8091	−	11.8091i	−0.487412	−	0.487412i	0.420076	−	0.907489i	$-0.362003\pi$
$587$	−11.8091	−	11.8091i	−0.487412	−	0.487412i	−0.907489	+	0.420076i	$0.862003\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$			11.3603i			0.466514i	0.972415	+	0.233257i	$0.0749383\pi$
$593$			11.3603i			0.466514i	−0.972415	+	0.233257i	$0.925062\pi$
$594$		0			0
$595$	−3.66442	−	3.66442i	−0.150227	−	0.150227i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	16.8449i		−	0.688266i	−0.938921	−	0.344133i	$-0.888173\pi$
$599$		−	16.8449i		−	0.688266i	0.938921	−	0.344133i	$-0.111827\pi$
$600$		0			0
$601$			16.0722i			0.655600i	0.944747	+	0.327800i	$0.106307\pi$
$601$			16.0722i			0.655600i	−0.944747	+	0.327800i	$0.893693\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	5.35454	+	5.35454i	0.217693	+	0.217693i
$606$		0			0
$607$			6.99327i			0.283848i	0.989878	+	0.141924i	$0.0453289\pi$
$607$			6.99327i			0.283848i	−0.989878	+	0.141924i	$0.954671\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	2.76783	−	2.76783i	0.111975	−	0.111975i
$612$		0			0
$613$	5.97858	+	5.97858i	0.241472	+	0.241472i	0.817459	−	0.575987i	$-0.195382\pi$
$613$	5.97858	+	5.97858i	0.241472	+	0.241472i	−0.575987	+	0.817459i	$0.695382\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	8.59695			0.346100			0.173050	−	0.984913i	$-0.444638\pi$
$617$	8.59695			0.346100			0.173050	+	0.984913i	$0.444638\pi$
$618$		0			0
$619$	−9.70727	+	9.70727i	−0.390168	+	0.390168i	−0.874747	−	0.484579i	$-0.838973\pi$
$619$	−9.70727	+	9.70727i	−0.390168	+	0.390168i	0.484579	+	0.874747i	$0.338973\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−17.1473			−0.686990
$624$		0			0
$625$	19.0000			0.760000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	2.59114	−	2.59114i	0.103315	−	0.103315i
$630$		0			0
$631$	−20.4935			−0.815833			−0.407917	−	0.913019i	$-0.633745\pi$
$631$	−20.4935			−0.815833			−0.407917	+	0.913019i	$0.633745\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	24.1533	+	24.1533i	0.958494	+	0.958494i
$636$		0			0
$637$	0.853635	−	0.853635i	0.0338222	−	0.0338222i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			6.04054i			0.238587i	0.992859	+	0.119294i	$0.0380630\pi$
$641$			6.04054i			0.238587i	−0.992859	+	0.119294i	$0.961937\pi$
$642$		0			0
$643$	−7.12181	−	7.12181i	−0.280857	−	0.280857i	0.552594	−	0.833451i	$-0.313638\pi$
$643$	−7.12181	−	7.12181i	−0.280857	−	0.280857i	−0.833451	+	0.552594i	$0.813638\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			13.6676i			0.537327i	0.963234	+	0.268664i	$0.0865820\pi$
$647$			13.6676i			0.537327i	−0.963234	+	0.268664i	$0.913418\pi$
$648$		0			0
$649$			19.0130i			0.746326i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	24.2893	+	24.2893i	0.950514	+	0.950514i	0.998832	−	0.0483179i	$-0.0153861\pi$
$653$	24.2893	+	24.2893i	0.950514	+	0.950514i	−0.0483179	+	0.998832i	$0.515386\pi$
$654$		0			0
$655$			4.58546i			0.179169i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	19.4301	−	19.4301i	0.756890	−	0.756890i	−0.218865	−	0.975755i	$-0.570235\pi$
$659$	19.4301	−	19.4301i	0.756890	−	0.756890i	0.975755	+	0.218865i	$0.0702354\pi$
$660$		0			0
$661$	−7.34713	−	7.34713i	−0.285770	−	0.285770i	0.549635	−	0.835405i	$-0.314767\pi$
$661$	−7.34713	−	7.34713i	−0.285770	−	0.285770i	−0.835405	+	0.549635i	$0.814767\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	0.643000	−	0.643000i	0.0248971	−	0.0248971i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	11.9510			0.461364
$672$		0			0
$673$	2.15792			0.0831818			0.0415909	−	0.999135i	$-0.486757\pi$
$673$	2.15792			0.0831818			0.0415909	+	0.999135i	$0.486757\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	29.3053	−	29.3053i	1.12629	−	1.12629i	0.135519	−	0.990775i	$-0.456730\pi$
$677$	29.3053	−	29.3053i	1.12629	−	1.12629i	0.990775	−	0.135519i	$-0.0432700\pi$
$678$		0			0
$679$	−16.5426			−0.634847
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−0.312705	−	0.312705i	−0.0119653	−	0.0119653i	0.701099	−	0.713064i	$-0.252693\pi$
$683$	−0.312705	−	0.312705i	−0.0119653	−	0.0119653i	−0.713064	+	0.701099i	$0.752693\pi$
$684$		0			0
$685$	22.1151	−	22.1151i	0.844974	−	0.844974i
$686$		0			0
$687$		0			0
$688$		0			0
$689$			1.23308i			0.0469767i
$690$		0			0
$691$	−11.9572	−	11.9572i	−0.454872	−	0.454872i	0.442096	−	0.896968i	$-0.354235\pi$
$691$	−11.9572	−	11.9572i	−0.454872	−	0.454872i	−0.896968	+	0.442096i	$0.854235\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	7.31279i		−	0.277390i
$696$		0			0
$697$		−	28.7005i		−	1.08711i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−31.8557	−	31.8557i	−1.20317	−	1.20317i	−0.973196	−	0.229979i	$-0.926134\pi$
$701$	−31.8557	−	31.8557i	−1.20317	−	1.20317i	−0.229979	−	0.973196i	$-0.573866\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$	0.206992	−	0.206992i	0.00778474	−	0.00778474i
$708$		0			0
$709$	16.3717	+	16.3717i	0.614852	+	0.614852i	0.944206	−	0.329355i	$-0.106831\pi$
$709$	16.3717	+	16.3717i	0.614852	+	0.614852i	−0.329355	+	0.944206i	$0.606831\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−0.353391			−0.0132346
$714$		0			0
$715$	4.58546	−	4.58546i	0.171487	−	0.171487i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	45.0036			1.67835			0.839175	−	0.543861i	$-0.183038\pi$
$719$	45.0036			1.67835			0.839175	+	0.543861i	$0.183038\pi$
$720$		0			0
$721$	5.95715			0.221856
$722$		0			0
$723$		0			0
$724$		0			0
$725$	3.10640	−	3.10640i	0.115369	−	0.115369i
$726$		0			0
$727$	30.6283			1.13594			0.567971	−	0.823049i	$-0.307729\pi$
$727$	30.6283			1.13594			0.567971	+	0.823049i	$0.307729\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	18.4617	+	18.4617i	0.682831	+	0.682831i
$732$		0			0
$733$	−23.3534	+	23.3534i	−0.862578	+	0.862578i	−0.991637	−	0.129059i	$-0.958804\pi$
$733$	−23.3534	+	23.3534i	−0.862578	+	0.862578i	0.129059	+	0.991637i	$0.458804\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			15.7494i			0.580136i
$738$		0			0
$739$	−8.93260	−	8.93260i	−0.328591	−	0.328591i	0.523460	−	0.852050i	$-0.324641\pi$
$739$	−8.93260	−	8.93260i	−0.328591	−	0.328591i	−0.852050	+	0.523460i	$0.824641\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			18.5823i			0.681717i	0.940115	+	0.340858i	$0.110718\pi$
$743$			18.5823i			0.681717i	−0.940115	+	0.340858i	$0.889282\pi$
$744$		0			0
$745$			26.3587i			0.965708i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−2.10617	−	2.10617i	−0.0769579	−	0.0769579i
$750$		0			0
$751$			45.9865i			1.67807i	0.544075	+	0.839036i	$0.316881\pi$
$751$			45.9865i			1.67807i	−0.544075	+	0.839036i	$0.683119\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−21.2643	+	21.2643i	−0.773886	+	0.773886i
$756$		0			0
$757$	−17.6283	−	17.6283i	−0.640712	−	0.640712i	0.310019	−	0.950730i	$-0.399665\pi$
$757$	−17.6283	−	17.6283i	−0.640712	−	0.640712i	−0.950730	+	0.310019i	$0.899665\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−20.3291			−0.736928			−0.368464	−	0.929642i	$-0.620116\pi$
$761$	−20.3291			−0.736928			−0.368464	+	0.929642i	$0.620116\pi$
$762$		0			0
$763$	1.00000	−	1.00000i	0.0362024	−	0.0362024i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−8.54587			−0.308574
$768$		0			0
$769$	−51.5359			−1.85843			−0.929216	−	0.369538i	$-0.879516\pi$
$769$	−51.5359			−1.85843			−0.929216	+	0.369538i	$0.879516\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	27.6027	−	27.6027i	0.992800	−	0.992800i	−0.00717408	−	0.999974i	$-0.502284\pi$
$773$	27.6027	−	27.6027i	0.992800	−	0.992800i	0.999974	+	0.00717408i	$0.00228360\pi$
$774$		0			0
$775$	−1.70727			−0.0613269
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$	19.1365	−	19.1365i	0.684758	−	0.684758i
$782$		0			0
$783$		0			0
$784$		0			0
$785$			24.9207i			0.889457i
$786$		0			0
$787$	15.1709	+	15.1709i	0.540785	+	0.540785i	0.923759	−	0.382974i	$-0.125100\pi$
$787$	15.1709	+	15.1709i	0.540785	+	0.540785i	−0.382974	+	0.923759i	$0.625100\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	4.79859i		−	0.170618i
$792$		0			0
$793$			5.37169i			0.190754i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−1.27225	−	1.27225i	−0.0450654	−	0.0450654i	0.684215	−	0.729280i	$-0.260145\pi$
$797$	−1.27225	−	1.27225i	−0.0450654	−	0.0450654i	−0.729280	+	0.684215i	$0.760145\pi$
$798$		0			0
$799$			8.40154i			0.297225i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	10.7578	−	10.7578i	0.379633	−	0.379633i
$804$		0			0
$805$	−0.292731	−	0.292731i	−0.0103174	−	0.0103174i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	7.95963			0.279846			0.139923	−	0.990162i	$-0.455315\pi$
$809$	7.95963			0.279846			0.139923	+	0.990162i	$0.455315\pi$
$810$		0			0
$811$	−14.5426	+	14.5426i	−0.510660	+	0.510660i	−0.914729	−	0.404068i	$-0.867596\pi$
$811$	−14.5426	+	14.5426i	−0.510660	+	0.510660i	0.404068	+	0.914729i	$0.367596\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	42.9425			1.50421
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	5.76257	−	5.76257i	0.201115	−	0.201115i	−0.599363	−	0.800478i	$-0.704579\pi$
$821$	5.76257	−	5.76257i	0.201115	−	0.201115i	0.800478	+	0.599363i	$0.204579\pi$
$822$		0			0
$823$	−45.6644			−1.59176			−0.795881	−	0.605453i	$-0.792992\pi$
$823$	−45.6644			−1.59176			−0.795881	+	0.605453i	$0.792992\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−11.2894	−	11.2894i	−0.392570	−	0.392570i	0.483033	−	0.875602i	$-0.339535\pi$
$827$	−11.2894	−	11.2894i	−0.392570	−	0.392570i	−0.875602	+	0.483033i	$0.839535\pi$
$828$		0			0
$829$	−21.2614	+	21.2614i	−0.738440	+	0.738440i	−0.972276	−	0.233836i	$-0.924872\pi$
$829$	−21.2614	+	21.2614i	−0.738440	+	0.738440i	0.233836	+	0.972276i	$0.424872\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			2.59114i			0.0897776i
$834$		0			0
$835$	2.82908	+	2.82908i	0.0979042	+	0.0979042i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	11.7277i		−	0.404885i	−0.979294	−	0.202442i	$-0.935112\pi$
$839$		−	11.7277i		−	0.404885i	0.979294	−	0.202442i	$-0.0648879\pi$
$840$		0			0
$841$		−	9.70054i		−	0.334501i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−16.3237	−	16.3237i	−0.561553	−	0.561553i
$846$		0			0
$847$		−	3.78623i		−	0.130096i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	0.206992	−	0.206992i	0.00709560	−	0.00709560i
$852$		0			0
$853$	38.0968	+	38.0968i	1.30441	+	1.30441i	0.925388	+	0.379021i	$0.123739\pi$
$853$	38.0968	+	38.0968i	1.30441	+	1.30441i	0.379021	+	0.925388i	$0.376261\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	41.9379			1.43257			0.716285	−	0.697808i	$-0.245841\pi$
$857$	41.9379			1.43257			0.716285	+	0.697808i	$0.245841\pi$
$858$		0			0
$859$	10.2070	−	10.2070i	0.348260	−	0.348260i	−0.511201	−	0.859461i	$-0.670799\pi$
$859$	10.2070	−	10.2070i	0.348260	−	0.348260i	0.859461	+	0.511201i	$0.170799\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	16.7636			0.570639			0.285319	−	0.958433i	$-0.407900\pi$
$863$	16.7636			0.570639			0.285319	+	0.958433i	$0.407900\pi$
$864$		0			0
$865$	−9.25662			−0.314734
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−6.48482	+	6.48482i	−0.219983	+	0.219983i
$870$		0			0
$871$	−7.07896			−0.239861
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−8.48528	−	8.48528i	−0.286855	−	0.286855i
$876$		0			0
$877$	−28.7648	+	28.7648i	−0.971319	+	0.971319i	−0.999600	−	0.0282815i	$-0.990997\pi$
$877$	−28.7648	+	28.7648i	−0.971319	+	0.971319i	0.0282815	+	0.999600i	$0.490997\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	38.0581i		−	1.28221i	−0.767453	−	0.641106i	$-0.778476\pi$
$881$		−	38.0581i		−	1.28221i	0.767453	−	0.641106i	$-0.221524\pi$
$882$		0			0
$883$	−23.3257	−	23.3257i	−0.784973	−	0.784973i	0.195693	−	0.980665i	$-0.437305\pi$
$883$	−23.3257	−	23.3257i	−0.784973	−	0.784973i	−0.980665	+	0.195693i	$0.937305\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	16.2726i		−	0.546382i	−0.961960	−	0.273191i	$-0.911921\pi$
$887$		−	16.2726i		−	0.546382i	0.961960	−	0.273191i	$-0.0880791\pi$
$888$		0			0
$889$		−	17.0790i		−	0.572810i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		−	40.7005i		−	1.36047i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	5.30346	−	5.30346i	0.176880	−	0.176880i
$900$		0			0
$901$	−1.87146	−	1.87146i	−0.0623475	−	0.0623475i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	28.9216			0.961386
$906$		0			0
$907$	5.16779	−	5.16779i	0.171594	−	0.171594i	−0.616086	−	0.787679i	$-0.711282\pi$
$907$	5.16779	−	5.16779i	0.171594	−	0.171594i	0.787679	+	0.616086i	$0.211282\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	17.8030			0.589839			0.294919	−	0.955522i	$-0.404707\pi$
$911$	17.8030			0.589839			0.294919	+	0.955522i	$0.404707\pi$
$912$		0			0
$913$	−27.6447			−0.914906
$914$		0			0
$915$		0			0
$916$		0			0
$917$	1.62121	−	1.62121i	0.0535369	−	0.0535369i
$918$		0			0
$919$	−14.6577			−0.483513			−0.241756	−	0.970337i	$-0.577723\pi$
$919$	−14.6577			−0.483513			−0.241756	+	0.970337i	$0.577723\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	8.60138	+	8.60138i	0.283118	+	0.283118i
$924$		0			0
$925$	1.00000	−	1.00000i	0.0328798	−	0.0328798i
$926$		0			0
$927$		0			0
$928$		0			0
$929$			13.4909i			0.442621i	0.975203	+	0.221310i	$0.0710334\pi$
$929$			13.4909i			0.442621i	−0.975203	+	0.221310i	$0.928967\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$			13.9188i			0.455193i
$936$		0			0
$937$			16.8291i			0.549782i	0.961475	+	0.274891i	$0.0886418\pi$
$937$			16.8291i			0.549782i	−0.961475	+	0.274891i	$0.911358\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	41.2355	+	41.2355i	1.34424	+	1.34424i	0.891790	+	0.452450i	$0.149450\pi$
$941$	41.2355	+	41.2355i	1.34424	+	1.34424i	0.452450	+	0.891790i	$0.350550\pi$
$942$		0			0
$943$		−	2.29273i		−	0.0746616i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−2.93414	+	2.93414i	−0.0953467	+	0.0953467i	−0.753171	−	0.657824i	$-0.771477\pi$
$947$	−2.93414	+	2.93414i	−0.0953467	+	0.0953467i	0.657824	+	0.753171i	$0.271477\pi$
$948$		0			0
$949$	4.83535	+	4.83535i	0.156962	+	0.156962i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−24.6582			−0.798757			−0.399378	−	0.916786i	$-0.630774\pi$
$953$	−24.6582			−0.798757			−0.399378	+	0.916786i	$0.630774\pi$
$954$		0			0
$955$	4.96388	−	4.96388i	0.160628	−	0.160628i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−15.6377			−0.504968
$960$		0			0
$961$	28.0852			0.905975
$962$		0			0
$963$		0			0
$964$		0			0
$965$	2.82843	−	2.82843i	0.0910503	−	0.0910503i
$966$		0			0
$967$	27.9143			0.897664			0.448832	−	0.893616i	$-0.351840\pi$
$967$	27.9143			0.897664			0.448832	+	0.893616i	$0.351840\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	24.9813	+	24.9813i	0.801687	+	0.801687i	0.983359	−	0.181672i	$-0.0581510\pi$
$971$	24.9813	+	24.9813i	0.801687	+	0.801687i	−0.181672	+	0.983359i	$0.558151\pi$
$972$		0			0
$973$	−2.58546	+	2.58546i	−0.0828861	+	0.0828861i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	39.5279i		−	1.26461i	−0.774720	−	0.632304i	$-0.782109\pi$
$977$		−	39.5279i		−	1.26461i	0.774720	−	0.632304i	$-0.217891\pi$
$978$		0			0
$979$	32.5657	+	32.5657i	1.04081	+	1.04081i
$980$		0			0
$981$		0			0
$982$		0			0
$983$			48.6600i			1.55201i	0.630725	+	0.776006i	$0.282758\pi$
$983$			48.6600i			1.55201i	−0.630725	+	0.776006i	$0.717242\pi$
$984$		0			0
$985$			29.8715i			0.951784i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	1.47481	+	1.47481i	0.0468961	+	0.0468961i
$990$		0			0
$991$		−	36.2070i		−	1.15015i	−0.818099	−	0.575077i	$-0.804972\pi$
$991$		−	36.2070i		−	1.15015i	0.818099	−	0.575077i	$-0.195028\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−1.30255	+	1.30255i	−0.0412935	+	0.0412935i
$996$		0			0
$997$	35.9326	+	35.9326i	1.13800	+	1.13800i	0.988809	+	0.149188i	$0.0476659\pi$
$997$	35.9326	+	35.9326i	1.13800	+	1.13800i	0.149188	+	0.988809i	$0.452334\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4032.2.v.c.3599.1		12
3.2	odd	2	inner	4032.2.v.c.3599.6		12
4.3	odd	2		1008.2.v.c.827.1	yes	12
12.11	even	2		1008.2.v.c.827.6	yes	12
16.3	odd	4	inner	4032.2.v.c.1583.6		12
16.13	even	4		1008.2.v.c.323.6	yes	12
48.29	odd	4		1008.2.v.c.323.1	✓	12
48.35	even	4	inner	4032.2.v.c.1583.1		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1008.2.v.c.323.1	✓	12	48.29	odd	4
1008.2.v.c.323.6	yes	12	16.13	even	4
1008.2.v.c.827.1	yes	12	4.3	odd	2
1008.2.v.c.827.6	yes	12	12.11	even	2
4032.2.v.c.1583.1		12	48.35	even	4	inner
4032.2.v.c.1583.6		12	16.3	odd	4	inner
4032.2.v.c.3599.1		12	1.1	even	1	trivial
4032.2.v.c.3599.6		12	3.2	odd	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.v (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.41421 + 1.41421i) q^{5} +1.00000 q^{7} +O(q^{10})\)	\(q+(-1.41421 + 1.41421i) q^{5} +1.00000 q^{7} +(-1.89918 - 1.89918i) q^{11} +(0.853635 - 0.853635i) q^{13} +2.59114i q^{17} +0.206992i q^{23} +1.00000i q^{25} +(-3.10640 - 3.10640i) q^{29} +1.70727i q^{31} +(-1.41421 + 1.41421i) q^{35} +(-1.00000 - 1.00000i) q^{37} -11.0764 q^{41} +(7.12494 - 7.12494i) q^{43} +3.24241 q^{47} +1.00000 q^{49} +(-0.722255 + 0.722255i) q^{53} +5.37169 q^{55} +(-5.00558 - 5.00558i) q^{59} +(-3.14637 + 3.14637i) q^{61} +2.41444i q^{65} +(-4.14637 - 4.14637i) q^{67} +10.0762i q^{71} +5.66442i q^{73} +(-1.89918 - 1.89918i) q^{77} -3.41454i q^{79} +(7.27806 - 7.27806i) q^{83} +(-3.66442 - 3.66442i) q^{85} -17.1473 q^{89} +(0.853635 - 0.853635i) q^{91} -16.5426 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 12 q^{7}+O(q^{10}) \) 12 * q + 12 * q^7	\( 12 q + 12 q^{7} + 16 q^{13} - 12 q^{37} + 20 q^{43} + 12 q^{49} - 32 q^{55} - 32 q^{61} - 44 q^{67} + 64 q^{85} + 16 q^{91} - 56 q^{97}+O(q^{100}) \) 12 * q + 12 * q^7 + 16 * q^13 - 12 * q^37 + 20 * q^43 + 12 * q^49 - 32 * q^55 - 32 * q^61 - 44 * q^67 + 64 * q^85 + 16 * q^91 - 56 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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