LMFDB - Embedded newform 3520.1.db.a.1869.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3520,1,Mod(109,3520)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3520.109"); S:= CuspForms(chi, 1); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3520, base_ring=CyclotomicField(16)) chi = DirichletCharacter(H, H._module([0, 7, 8, 8])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]

Level:	$ N $	$=$	$ 3520 = 2^{6} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3520.db (of order $16$, degree $8$, minimal)

Newform invariants

comment:select newform

sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$1.75670884447$
Analytic rank:	$0$
Dimension:	$32$
Relative dimension:	$4$ over $\Q(\zeta_{16})$
Coefficient field:	$\Q(\zeta_{64})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{32} + 1 $ x^32 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{32}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{32} - \cdots)$

Embedding invariants

Embedding label			1869.3
Root			$0.471397 + 0.881921i$ of defining polynomial
Character	$\chi$	$=$	3520.1869
Dual form			3520.1.db.a.2309.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(0.634393 + 0.773010i) q^{2} +(-0.195090 + 0.980785i) q^{4} +(-0.980785 + 0.195090i) q^{5} +(0.674993 + 1.62958i) q^{7} +(-0.881921 + 0.471397i) q^{8} +(0.382683 - 0.923880i) q^{9} +(-0.773010 - 0.634393i) q^{10} +(0.555570 - 0.831470i) q^{11} +(1.95213 + 0.388302i) q^{13} +(-0.831470 + 1.55557i) q^{14} +(-0.923880 - 0.382683i) q^{16} +(0.410525 + 0.410525i) q^{17} +(0.956940 - 0.290285i) q^{18} -1.00000i q^{20} +(0.995185 - 0.0980171i) q^{22} +(0.923880 - 0.382683i) q^{25} +(0.938254 + 1.75535i) q^{26} +(-1.72995 + 0.344109i) q^{28} +1.11114i q^{31} +(-0.290285 - 0.956940i) q^{32} +(-0.0569057 + 0.577774i) q^{34} +(-0.979938 - 1.46658i) q^{35} +(0.831470 + 0.555570i) q^{36} +(0.773010 - 0.634393i) q^{40} +(-1.28547 - 0.858923i) q^{43} +(0.707107 + 0.707107i) q^{44} +(-0.195090 + 0.980785i) q^{45} +(-1.49280 + 1.49280i) q^{49} +(0.881921 + 0.471397i) q^{50} +(-0.761681 + 1.83886i) q^{52} +(-0.382683 + 0.923880i) q^{55} +(-1.36347 - 1.11897i) q^{56} +(-1.92388 + 0.382683i) q^{59} +(-0.858923 + 0.704900i) q^{62} +1.76384 q^{63} +(0.555570 - 0.831470i) q^{64} -1.99037 q^{65} +(-0.482726 + 0.322547i) q^{68} +(0.512016 - 1.68789i) q^{70} +(-0.292893 - 0.707107i) q^{71} +(0.0980171 + 0.995185i) q^{72} +(-0.485544 + 1.17221i) q^{73} +(1.72995 + 0.344109i) q^{77} +(0.980785 + 0.195090i) q^{80} +(-0.707107 - 0.707107i) q^{81} +(0.388302 - 1.95213i) q^{83} +(-0.482726 - 0.322547i) q^{85} +(-0.151537 - 1.53858i) q^{86} +(-0.0980171 + 0.995185i) q^{88} +(0.360480 - 0.149316i) q^{89} +(-0.881921 + 0.471397i) q^{90} +(0.684903 + 3.44324i) q^{91} +(-2.10097 - 0.206928i) q^{98} +(-0.555570 - 0.831470i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q - 32 q^{59} - 32 q^{71}+O(q^{100}) $ 32 * q - 32 * q^59 - 32 * q^71

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3520\mathbb{Z}\right)^\times$.

$n$	$321$	$1541$	$2751$	$2817$
$\chi(n)$	$-1$	$e\left(\frac{15}{16}\right)$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	0.634393	+	0.773010i	0.634393	+	0.773010i
$2$	0.634393	+	0.773010i	0.634393	+	0.773010i
$3$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$3$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$4$	−0.195090	+	0.980785i	−0.195090	+	0.980785i
$5$	−0.980785	+	0.195090i	−0.980785	+	0.195090i
$5$	−0.980785	+	0.195090i	−0.980785	+	0.195090i
$6$		0			0
$7$	0.674993	+	1.62958i	0.674993	+	1.62958i	0.773010	+	0.634393i	$0.218750\pi$
$7$	0.674993	+	1.62958i	0.674993	+	1.62958i	−0.0980171	+	0.995185i	$0.531250\pi$
$8$	−0.881921	+	0.471397i	−0.881921	+	0.471397i
$9$	0.382683	−	0.923880i	0.382683	−	0.923880i
$10$	−0.773010	−	0.634393i	−0.773010	−	0.634393i
$11$	0.555570	−	0.831470i	0.555570	−	0.831470i
$11$	0.555570	−	0.831470i	0.555570	−	0.831470i
$12$		0			0
$13$	1.95213	+	0.388302i	1.95213	+	0.388302i	0.995185	+	0.0980171i	$0.0312500\pi$
$13$	1.95213	+	0.388302i	1.95213	+	0.388302i	0.956940	+	0.290285i	$0.0937500\pi$
$14$	−0.831470	+	1.55557i	−0.831470	+	1.55557i
$15$		0			0
$16$	−0.923880	−	0.382683i	−0.923880	−	0.382683i
$17$	0.410525	+	0.410525i	0.410525	+	0.410525i	0.881921	−	0.471397i	$-0.156250\pi$
$17$	0.410525	+	0.410525i	0.410525	+	0.410525i	−0.471397	+	0.881921i	$0.656250\pi$
$18$	0.956940	−	0.290285i	0.956940	−	0.290285i
$19$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$19$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$20$		−	1.00000i		−	1.00000i
$21$		0			0
$22$	0.995185	−	0.0980171i	0.995185	−	0.0980171i
$23$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$23$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$24$		0			0
$25$	0.923880	−	0.382683i	0.923880	−	0.382683i
$26$	0.938254	+	1.75535i	0.938254	+	1.75535i
$27$		0			0
$28$	−1.72995	+	0.344109i	−1.72995	+	0.344109i
$29$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$29$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$30$		0			0
$31$			1.11114i			1.11114i	0.831470	+	0.555570i	$0.187500\pi$
$31$			1.11114i			1.11114i	−0.831470	+	0.555570i	$0.812500\pi$
$32$	−0.290285	−	0.956940i	−0.290285	−	0.956940i
$33$		0			0
$34$	−0.0569057	+	0.577774i	−0.0569057	+	0.577774i
$35$	−0.979938	−	1.46658i	−0.979938	−	1.46658i
$36$	0.831470	+	0.555570i	0.831470	+	0.555570i
$37$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$37$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$38$		0			0
$39$		0			0
$40$	0.773010	−	0.634393i	0.773010	−	0.634393i
$41$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$41$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$42$		0			0
$43$	−1.28547	−	0.858923i	−1.28547	−	0.858923i	−0.290285	−	0.956940i	$-0.593750\pi$
$43$	−1.28547	−	0.858923i	−1.28547	−	0.858923i	−0.995185	+	0.0980171i	$0.968750\pi$
$44$	0.707107	+	0.707107i	0.707107	+	0.707107i
$45$	−0.195090	+	0.980785i	−0.195090	+	0.980785i
$46$		0			0
$47$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$47$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$48$		0			0
$49$	−1.49280	+	1.49280i	−1.49280	+	1.49280i
$50$	0.881921	+	0.471397i	0.881921	+	0.471397i
$51$		0			0
$52$	−0.761681	+	1.83886i	−0.761681	+	1.83886i
$53$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$53$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$54$		0			0
$55$	−0.382683	+	0.923880i	−0.382683	+	0.923880i
$56$	−1.36347	−	1.11897i	−1.36347	−	1.11897i
$57$		0			0
$58$		0			0
$59$	−1.92388	+	0.382683i	−1.92388	+	0.382683i	−0.923880	+	0.382683i	$0.875000\pi$
$59$	−1.92388	+	0.382683i	−1.92388	+	0.382683i	−1.00000			$\pi$
$60$		0			0
$61$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$61$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$62$	−0.858923	+	0.704900i	−0.858923	+	0.704900i
$63$	1.76384			1.76384
$64$	0.555570	−	0.831470i	0.555570	−	0.831470i
$65$	−1.99037			−1.99037
$66$		0			0
$67$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$67$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$68$	−0.482726	+	0.322547i	−0.482726	+	0.322547i
$69$		0			0
$70$	0.512016	−	1.68789i	0.512016	−	1.68789i
$71$	−0.292893	−	0.707107i	−0.292893	−	0.707107i	0.707107	−	0.707107i	$-0.250000\pi$
$71$	−0.292893	−	0.707107i	−0.292893	−	0.707107i	−1.00000			$\pi$
$72$	0.0980171	+	0.995185i	0.0980171	+	0.995185i
$73$	−0.485544	+	1.17221i	−0.485544	+	1.17221i	0.471397	+	0.881921i	$0.343750\pi$
$73$	−0.485544	+	1.17221i	−0.485544	+	1.17221i	−0.956940	+	0.290285i	$0.906250\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	1.72995	+	0.344109i	1.72995	+	0.344109i
$78$		0			0
$79$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$79$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$80$	0.980785	+	0.195090i	0.980785	+	0.195090i
$81$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$82$		0			0
$83$	0.388302	−	1.95213i	0.388302	−	1.95213i	0.0980171	−	0.995185i	$-0.468750\pi$
$83$	0.388302	−	1.95213i	0.388302	−	1.95213i	0.290285	−	0.956940i	$-0.406250\pi$
$84$		0			0
$85$	−0.482726	−	0.322547i	−0.482726	−	0.322547i
$86$	−0.151537	−	1.53858i	−0.151537	−	1.53858i
$87$		0			0
$88$	−0.0980171	+	0.995185i	−0.0980171	+	0.995185i
$89$	0.360480	−	0.149316i	0.360480	−	0.149316i	−0.195090	−	0.980785i	$-0.562500\pi$
$89$	0.360480	−	0.149316i	0.360480	−	0.149316i	0.555570	+	0.831470i	$0.312500\pi$
$90$	−0.881921	+	0.471397i	−0.881921	+	0.471397i
$91$	0.684903	+	3.44324i	0.684903	+	3.44324i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		0			0			−	1.00000i	$-0.5\pi$
$97$		0			0				1.00000i	$0.5\pi$
$98$	−2.10097	−	0.206928i	−2.10097	−	0.206928i
$99$	−0.555570	−	0.831470i	−0.555570	−	0.831470i
$100$	0.195090	+	0.980785i	0.195090	+	0.980785i
$101$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$101$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$102$		0			0
$103$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$103$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$104$	−1.90466	+	0.577774i	−1.90466	+	0.577774i
$105$		0			0
$106$		0			0
$107$	1.59133	+	1.06330i	1.59133	+	1.06330i	0.956940	+	0.290285i	$0.0937500\pi$
$107$	1.59133	+	1.06330i	1.59133	+	1.06330i	0.634393	+	0.773010i	$0.281250\pi$
$108$		0			0
$109$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$109$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$110$	−0.956940	+	0.290285i	−0.956940	+	0.290285i
$111$		0			0
$112$		−	1.76384i		−	1.76384i
$113$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$113$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	1.10579	−	1.65493i	1.10579	−	1.65493i
$118$	−1.51631	−	1.24441i	−1.51631	−	1.24441i
$119$	−0.391880	+	0.946083i	−0.391880	+	0.946083i
$120$		0			0
$121$	−0.382683	−	0.923880i	−0.382683	−	0.923880i
$122$		0			0
$123$		0			0
$124$	−1.08979	−	0.216773i	−1.08979	−	0.216773i
$125$	−0.831470	+	0.555570i	−0.831470	+	0.555570i
$126$	1.11897	+	1.36347i	1.11897	+	1.36347i
$127$	−1.26879			−1.26879			−0.634393	−	0.773010i	$-0.718750\pi$
$127$	−1.26879			−1.26879			−0.634393	+	0.773010i	$0.718750\pi$
$128$	0.995185	−	0.0980171i	0.995185	−	0.0980171i
$129$		0			0
$130$	−1.26268	−	1.53858i	−1.26268	−	1.53858i
$131$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$131$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$	−0.555570	−	0.168530i	−0.555570	−	0.168530i
$137$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$137$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$138$		0			0
$139$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$139$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$140$	1.62958	−	0.674993i	1.62958	−	0.674993i
$141$		0			0
$142$	0.360791	−	0.674993i	0.360791	−	0.674993i
$143$	1.40740	−	1.40740i	1.40740	−	1.40740i
$144$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$145$		0			0
$146$	−1.21415	+	0.368309i	−1.21415	+	0.368309i
$147$		0			0
$148$		0			0
$149$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$149$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$150$		0			0
$151$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$151$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$152$		0			0
$153$	0.536376	−	0.222174i	0.536376	−	0.222174i
$154$	0.831470	+	1.55557i	0.831470	+	1.55557i
$155$	−0.216773	−	1.08979i	−0.216773	−	1.08979i
$156$		0			0
$157$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$157$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$158$		0			0
$159$		0			0
$160$	0.471397	+	0.881921i	0.471397	+	0.881921i
$161$		0			0
$162$	0.0980171	−	0.995185i	0.0980171	−	0.995185i
$163$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$163$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$164$		0			0
$165$		0			0
$166$	1.75535	−	0.938254i	1.75535	−	0.938254i
$167$	−0.181112	+	0.0750191i	−0.181112	+	0.0750191i	−0.471397	−	0.881921i	$-0.656250\pi$
$167$	−0.181112	+	0.0750191i	−0.181112	+	0.0750191i	0.290285	+	0.956940i	$0.406250\pi$
$168$		0			0
$169$	2.73613	+	1.13334i	2.73613	+	1.13334i
$170$	−0.0569057	−	0.577774i	−0.0569057	−	0.577774i
$171$		0			0
$172$	1.09320	−	1.09320i	1.09320	−	1.09320i
$173$	0.373380	−	1.87711i	0.373380	−	1.87711i	−0.0980171	−	0.995185i	$-0.531250\pi$
$173$	0.373380	−	1.87711i	0.373380	−	1.87711i	0.471397	−	0.881921i	$-0.343750\pi$
$174$		0			0
$175$	1.24723	+	1.24723i	1.24723	+	1.24723i
$176$	−0.831470	+	0.555570i	−0.831470	+	0.555570i
$177$		0			0
$178$	0.344109	+	0.183930i	0.344109	+	0.183930i
$179$	−0.750661	−	0.149316i	−0.750661	−	0.149316i	−0.195090	−	0.980785i	$-0.562500\pi$
$179$	−0.750661	−	0.149316i	−0.750661	−	0.149316i	−0.555570	+	0.831470i	$0.687500\pi$
$180$	−0.923880	−	0.382683i	−0.923880	−	0.382683i
$181$	1.02656	−	1.53636i	1.02656	−	1.53636i	0.195090	−	0.980785i	$-0.437500\pi$
$181$	1.02656	−	1.53636i	1.02656	−	1.53636i	0.831470	−	0.555570i	$-0.187500\pi$
$182$	−2.22716	+	2.71381i	−2.22716	+	2.71381i
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	0.569414	−	0.113263i	0.569414	−	0.113263i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−1.96157			−1.96157			−0.980785	−	0.195090i	$-0.937500\pi$
$191$	−1.96157			−1.96157			−0.980785	+	0.195090i	$0.937500\pi$
$192$		0			0
$193$	1.54602			1.54602			0.773010	−	0.634393i	$-0.218750\pi$
$193$	1.54602			1.54602			0.773010	+	0.634393i	$0.218750\pi$
$194$		0			0
$195$		0			0
$196$	−1.17289	−	1.75535i	−1.17289	−	1.75535i
$197$	−0.924678	+	0.183930i	−0.924678	+	0.183930i	−0.634393	−	0.773010i	$-0.718750\pi$
$197$	−0.924678	+	0.183930i	−0.924678	+	0.183930i	−0.290285	+	0.956940i	$0.593750\pi$
$198$	0.290285	−	0.956940i	0.290285	−	0.956940i
$199$	0.707107	+	1.70711i	0.707107	+	1.70711i	0.707107	+	0.707107i	$0.250000\pi$
$199$	0.707107	+	1.70711i	0.707107	+	1.70711i			1.00000i	$0.5\pi$
$200$	−0.634393	+	0.773010i	−0.634393	+	0.773010i
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	−1.65493	−	1.10579i	−1.65493	−	1.10579i
$209$		0			0
$210$		0			0
$211$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$211$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$212$		0			0
$213$		0			0
$214$	0.187593	+	1.90466i	0.187593	+	1.90466i
$215$	1.42834	+	0.591637i	1.42834	+	0.591637i
$216$		0			0
$217$	−1.81069	+	0.750012i	−1.81069	+	0.750012i
$218$		0			0
$219$		0			0
$220$	−0.831470	−	0.555570i	−0.831470	−	0.555570i
$221$	0.641988	+	0.960803i	0.641988	+	0.960803i
$222$		0			0
$223$		0			0			−	1.00000i	$-0.5\pi$
$223$		0			0				1.00000i	$0.5\pi$
$224$	1.36347	−	1.11897i	1.36347	−	1.11897i
$225$		−	1.00000i		−	1.00000i
$226$		0			0
$227$	0.108911	+	0.162997i	0.108911	+	0.162997i	0.881921	−	0.471397i	$-0.156250\pi$
$227$	0.108911	+	0.162997i	0.108911	+	0.162997i	−0.773010	+	0.634393i	$0.781250\pi$
$228$		0			0
$229$	−0.324423	−	1.63099i	−0.324423	−	1.63099i	−0.707107	−	0.707107i	$-0.750000\pi$
$229$	−0.324423	−	1.63099i	−0.324423	−	1.63099i	0.382683	−	0.923880i	$-0.375000\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	0.871028	+	0.360791i	0.871028	+	0.360791i	0.773010	−	0.634393i	$-0.218750\pi$
$233$	0.871028	+	0.360791i	0.871028	+	0.360791i	0.0980171	+	0.995185i	$0.468750\pi$
$234$	1.98079	−	0.195090i	1.98079	−	0.195090i
$235$		0			0
$236$		−	1.96157i		−	1.96157i
$237$		0			0
$238$	−0.979938	+	0.297261i	−0.979938	+	0.297261i
$239$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$239$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$240$		0			0
$241$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$241$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$242$	0.471397	−	0.881921i	0.471397	−	0.881921i
$243$		0			0
$244$		0			0
$245$	1.17289	−	1.75535i	1.17289	−	1.75535i
$246$		0			0
$247$		0			0
$248$	−0.523788	−	0.979938i	−0.523788	−	0.979938i
$249$		0			0
$250$	−0.956940	−	0.290285i	−0.956940	−	0.290285i
$251$	0.382683	−	0.0761205i	0.382683	−	0.0761205i		−	1.00000i	$-0.5\pi$
$251$	0.382683	−	0.0761205i	0.382683	−	0.0761205i	0.382683	+	0.923880i	$0.375000\pi$
$252$	−0.344109	+	1.72995i	−0.344109	+	1.72995i
$253$		0			0
$254$	−0.804910	−	0.980785i	−0.804910	−	0.980785i
$255$		0			0
$256$	0.707107	+	0.707107i	0.707107	+	0.707107i
$257$		0			0		1.00000			$0$
$257$		0			0		−1.00000			$\pi$
$258$		0			0
$259$		0			0
$260$	0.388302	−	1.95213i	0.388302	−	1.95213i
$261$		0			0
$262$		0			0
$263$	0.591637	+	1.42834i	0.591637	+	1.42834i	0.881921	+	0.471397i	$0.156250\pi$
$263$	0.591637	+	1.42834i	0.591637	+	1.42834i	−0.290285	+	0.956940i	$0.593750\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$	1.92388	+	0.382683i	1.92388	+	0.382683i	1.00000			$0$
$269$	1.92388	+	0.382683i	1.92388	+	0.382683i	0.923880	+	0.382683i	$0.125000\pi$
$270$		0			0
$271$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$271$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$272$	−0.222174	−	0.536376i	−0.222174	−	0.536376i
$273$		0			0
$274$		0			0
$275$	0.195090	−	0.980785i	0.195090	−	0.980785i
$276$		0			0
$277$	−1.28547	−	0.858923i	−1.28547	−	0.858923i	−0.290285	−	0.956940i	$-0.593750\pi$
$277$	−1.28547	−	0.858923i	−1.28547	−	0.858923i	−0.995185	+	0.0980171i	$0.968750\pi$
$278$		0			0
$279$	1.02656	+	0.425215i	1.02656	+	0.425215i
$280$	1.55557	+	0.831470i	1.55557	+	0.831470i
$281$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$281$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$282$		0			0
$283$	0.0382444	+	0.192268i	0.0382444	+	0.192268i	0.995185	−	0.0980171i	$-0.0312500\pi$
$283$	0.0382444	+	0.192268i	0.0382444	+	0.192268i	−0.956940	+	0.290285i	$0.906250\pi$
$284$	0.750661	−	0.149316i	0.750661	−	0.149316i
$285$		0			0
$286$	1.98079	+	0.195090i	1.98079	+	0.195090i
$287$		0			0
$288$	−0.995185	−	0.0980171i	−0.995185	−	0.0980171i
$289$		−	0.662939i		−	0.662939i
$290$		0			0
$291$		0			0
$292$	−1.05496	−	0.704900i	−1.05496	−	0.704900i
$293$	0.247528	+	1.24441i	0.247528	+	1.24441i	0.881921	+	0.471397i	$0.156250\pi$
$293$	0.247528	+	1.24441i	0.247528	+	1.24441i	−0.634393	+	0.773010i	$0.718750\pi$
$294$		0			0
$295$	1.81225	−	0.750661i	1.81225	−	0.750661i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	0.531999	−	2.67454i	0.531999	−	2.67454i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$	0.512016	+	0.273678i	0.512016	+	0.273678i
$307$	0.569414	+	0.113263i	0.569414	+	0.113263i	0.471397	−	0.881921i	$-0.343750\pi$
$307$	0.569414	+	0.113263i	0.569414	+	0.113263i	0.0980171	+	0.995185i	$0.468750\pi$
$308$	−0.674993	+	1.62958i	−0.674993	+	1.62958i
$309$		0			0
$310$	0.704900	−	0.858923i	0.704900	−	0.858923i
$311$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$311$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$312$		0			0
$313$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$313$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$314$		0			0
$315$	−1.72995	+	0.344109i	−1.72995	+	0.344109i
$316$		0			0
$317$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$317$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$318$		0			0
$319$		0			0
$320$	−0.382683	+	0.923880i	−0.382683	+	0.923880i
$321$		0			0
$322$		0			0
$323$		0			0
$324$	0.831470	−	0.555570i	0.831470	−	0.555570i
$325$	1.95213	−	0.388302i	1.95213	−	0.388302i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−0.785695	+	1.17588i	−0.785695	+	1.17588i	0.195090	+	0.980785i	$0.437500\pi$
$331$	−0.785695	+	1.17588i	−0.785695	+	1.17588i	−0.980785	+	0.195090i	$0.937500\pi$
$332$	1.83886	+	0.761681i	1.83886	+	0.761681i
$333$		0			0
$334$	−0.172887	−	0.0924099i	−0.172887	−	0.0924099i
$335$		0			0
$336$		0			0
$337$	−0.666656	−	0.666656i	−0.666656	−	0.666656i	0.290285	−	0.956940i	$-0.406250\pi$
$337$	−0.666656	−	0.666656i	−0.666656	−	0.666656i	−0.956940	+	0.290285i	$0.906250\pi$
$338$	0.859699	+	2.83405i	0.859699	+	2.83405i
$339$		0			0
$340$	0.410525	−	0.410525i	0.410525	−	0.410525i
$341$	0.923880	+	0.617317i	0.923880	+	0.617317i
$342$		0			0
$343$	−1.81069	−	0.750012i	−1.81069	−	0.750012i
$344$	1.53858	+	0.151537i	1.53858	+	0.151537i
$345$		0			0
$346$	1.68789	−	0.902197i	1.68789	−	0.902197i
$347$	−0.247528	−	1.24441i	−0.247528	−	1.24441i	−0.881921	−	0.471397i	$-0.843750\pi$
$347$	−0.247528	−	1.24441i	−0.247528	−	1.24441i	0.634393	−	0.773010i	$-0.281250\pi$
$348$		0			0
$349$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$349$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$350$	−0.172887	+	1.75535i	−0.172887	+	1.75535i
$351$		0			0
$352$	−0.956940	−	0.290285i	−0.956940	−	0.290285i
$353$		0			0			−	1.00000i	$-0.5\pi$
$353$		0			0				1.00000i	$0.5\pi$
$354$		0			0
$355$	0.425215	+	0.636379i	0.425215	+	0.636379i
$356$	0.0761205	+	0.382683i	0.0761205	+	0.382683i
$357$		0			0
$358$	−0.360791	−	0.674993i	−0.360791	−	0.674993i
$359$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$359$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$360$	−0.290285	−	0.956940i	−0.290285	−	0.956940i
$361$	−0.923880	−	0.382683i	−0.923880	−	0.382683i
$362$	1.83886	−	0.181112i	1.83886	−	0.181112i
$363$		0			0
$364$	−3.51070			−3.51070
$365$	0.247528	−	1.24441i	0.247528	−	1.24441i
$366$		0			0
$367$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$367$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	0.523788	−	0.783904i	0.523788	−	0.783904i	−0.471397	−	0.881921i	$-0.656250\pi$
$373$	0.523788	−	0.783904i	0.523788	−	0.783904i	0.995185	+	0.0980171i	$0.0312500\pi$
$374$	0.448786	+	0.368309i	0.448786	+	0.368309i
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−0.382683	+	0.0761205i	−0.382683	+	0.0761205i	−0.382683	−	0.923880i	$-0.625000\pi$
$379$	−0.382683	+	0.0761205i	−0.382683	+	0.0761205i			1.00000i	$0.5\pi$
$380$		0			0
$381$		0			0
$382$	−1.24441	−	1.51631i	−1.24441	−	1.51631i
$383$		0			0		1.00000			$0$
$383$		0			0		−1.00000			$\pi$
$384$		0			0
$385$	−1.76384			−1.76384
$386$	0.980785	+	1.19509i	0.980785	+	1.19509i
$387$	−1.28547	+	0.858923i	−1.28547	+	0.858923i
$388$		0			0
$389$	−1.63099	+	0.324423i	−1.63099	+	0.324423i	−0.923880	−	0.382683i	$-0.875000\pi$
$389$	−1.63099	+	0.324423i	−1.63099	+	0.324423i	−0.707107	+	0.707107i	$0.750000\pi$
$390$		0			0
$391$		0			0
$392$	0.612832	−	2.02024i	0.612832	−	2.02024i
$393$		0			0
$394$	−0.728789	−	0.598102i	−0.728789	−	0.598102i
$395$		0			0
$396$	0.923880	−	0.382683i	0.923880	−	0.382683i
$397$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$397$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$398$	−0.871028	+	1.62958i	−0.871028	+	1.62958i
$399$		0			0
$400$	−1.00000			−1.00000
$401$	−1.38704	−	1.38704i	−1.38704	−	1.38704i	−0.831470	−	0.555570i	$-0.812500\pi$
$401$	−1.38704	−	1.38704i	−1.38704	−	1.38704i	−0.555570	−	0.831470i	$-0.687500\pi$
$402$		0			0
$403$	−0.431458	+	2.16909i	−0.431458	+	2.16909i
$404$		0			0
$405$	0.831470	+	0.555570i	0.831470	+	0.555570i
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$409$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−1.92222	−	2.87680i	−1.92222	−	2.87680i
$414$		0			0
$415$			1.99037i			1.99037i
$416$	−0.195090	−	1.98079i	−0.195090	−	1.98079i
$417$		0			0
$418$		0			0
$419$	−0.785695	−	1.17588i	−0.785695	−	1.17588i	−0.980785	−	0.195090i	$-0.937500\pi$
$419$	−0.785695	−	1.17588i	−0.785695	−	1.17588i	0.195090	−	0.980785i	$-0.437500\pi$
$420$		0			0
$421$	−0.360480	−	1.81225i	−0.360480	−	1.81225i	−0.555570	−	0.831470i	$-0.687500\pi$
$421$	−0.360480	−	1.81225i	−0.360480	−	1.81225i	0.195090	−	0.980785i	$-0.437500\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	0.536376	+	0.222174i	0.536376	+	0.222174i
$426$		0			0
$427$		0			0
$428$	−1.35332	+	1.35332i	−1.35332	+	1.35332i
$429$		0			0
$430$	0.448786	+	1.47945i	0.448786	+	1.47945i
$431$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$431$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$432$		0			0
$433$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$433$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$434$	−1.72846	−	0.923880i	−1.72846	−	0.923880i
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$439$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$440$	−0.0980171	−	0.995185i	−0.0980171	−	0.995185i
$441$	0.807898	+	1.95044i	0.807898	+	1.95044i
$442$	−0.335438	+	1.10579i	−0.335438	+	1.10579i
$443$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$443$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$444$		0			0
$445$	−0.324423	+	0.216773i	−0.324423	+	0.216773i
$446$		0			0
$447$		0			0
$448$	1.72995	+	0.344109i	1.72995	+	0.344109i
$449$	−1.66294			−1.66294			−0.831470	−	0.555570i	$-0.812500\pi$
$449$	−1.66294			−1.66294			−0.831470	+	0.555570i	$0.812500\pi$
$450$	0.773010	−	0.634393i	0.773010	−	0.634393i
$451$		0			0
$452$		0			0
$453$		0			0
$454$	−0.0569057	+	0.187593i	−0.0569057	+	0.187593i
$455$	−1.34349	−	3.24346i	−1.34349	−	3.24346i
$456$		0			0
$457$	0.0750191	−	0.181112i	0.0750191	−	0.181112i	−0.881921	−	0.471397i	$-0.843750\pi$
$457$	0.0750191	−	0.181112i	0.0750191	−	0.181112i	0.956940	+	0.290285i	$0.0937500\pi$
$458$	1.05496	−	1.28547i	1.05496	−	1.28547i
$459$		0			0
$460$		0			0
$461$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$461$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$462$		0			0
$463$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$463$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$464$		0			0
$465$		0			0
$466$	0.273678	+	0.902197i	0.273678	+	0.902197i
$467$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$467$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$468$	1.40740	+	1.40740i	1.40740	+	1.40740i
$469$		0			0
$470$		0			0
$471$		0			0
$472$	1.51631	−	1.24441i	1.51631	−	1.24441i
$473$	−1.42834	+	0.591637i	−1.42834	+	0.591637i
$474$		0			0
$475$		0			0
$476$	−0.851452	−	0.568922i	−0.851452	−	0.568922i
$477$		0			0
$478$		0			0
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	0.980785	−	0.195090i	0.980785	−	0.195090i
$485$		0			0
$486$		0			0
$487$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$487$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$488$		0			0
$489$		0			0
$490$	2.10097	−	0.206928i	2.10097	−	0.206928i
$491$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$491$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$	0.707107	+	0.707107i	0.707107	+	0.707107i
$496$	0.425215	−	1.02656i	0.425215	−	1.02656i
$497$	0.954585	−	0.954585i	0.954585	−	0.954585i
$498$		0			0
$499$	1.08979	+	0.216773i	1.08979	+	0.216773i	0.707107	−	0.707107i	$-0.250000\pi$
$499$	1.08979	+	0.216773i	1.08979	+	0.216773i	0.382683	+	0.923880i	$0.375000\pi$
$500$	−0.382683	−	0.923880i	−0.382683	−	0.923880i
$501$		0			0
$502$	0.301614	+	0.247528i	0.301614	+	0.247528i
$503$	0.222174	−	0.536376i	0.222174	−	0.536376i	−0.773010	−	0.634393i	$-0.781250\pi$
$503$	0.222174	−	0.536376i	0.222174	−	0.536376i	0.995185	+	0.0980171i	$0.0312500\pi$
$504$	−1.55557	+	0.831470i	−1.55557	+	0.831470i
$505$		0			0
$506$		0			0
$507$		0			0
$508$	0.247528	−	1.24441i	0.247528	−	1.24441i
$509$	0.324423	−	0.216773i	0.324423	−	0.216773i	−0.382683	−	0.923880i	$-0.625000\pi$
$509$	0.324423	−	0.216773i	0.324423	−	0.216773i	0.707107	+	0.707107i	$0.250000\pi$
$510$		0			0
$511$	−2.23794			−2.23794
$512$	−0.0980171	+	0.995185i	−0.0980171	+	0.995185i
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$	1.75535	−	0.938254i	1.75535	−	0.938254i
$521$	0.636379	−	1.53636i	0.636379	−	1.53636i	−0.195090	−	0.980785i	$-0.562500\pi$
$521$	0.636379	−	1.53636i	0.636379	−	1.53636i	0.831470	−	0.555570i	$-0.187500\pi$
$522$		0			0
$523$	0.523788	−	0.783904i	0.523788	−	0.783904i	−0.471397	−	0.881921i	$-0.656250\pi$
$523$	0.523788	−	0.783904i	0.523788	−	0.783904i	0.995185	+	0.0980171i	$0.0312500\pi$
$524$		0			0
$525$		0			0
$526$	−0.728789	+	1.36347i	−0.728789	+	1.36347i
$527$	−0.456150	+	0.456150i	−0.456150	+	0.456150i
$528$		0			0
$529$	0.707107	+	0.707107i	0.707107	+	0.707107i
$530$		0			0
$531$	−0.382683	+	1.92388i	−0.382683	+	1.92388i
$532$		0			0
$533$		0			0
$534$		0			0
$535$	−1.76820	−	0.732410i	−1.76820	−	0.732410i
$536$		0			0
$537$		0			0
$538$	0.924678	+	1.72995i	0.924678	+	1.72995i
$539$	0.411863	+	2.07058i	0.411863	+	2.07058i
$540$		0			0
$541$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$541$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$542$		0			0
$543$		0			0
$544$	0.273678	−	0.512016i	0.273678	−	0.512016i
$545$		0			0
$546$		0			0
$547$	−1.06330	−	1.59133i	−1.06330	−	1.59133i	−0.773010	−	0.634393i	$-0.781250\pi$
$547$	−1.06330	−	1.59133i	−1.06330	−	1.59133i	−0.290285	−	0.956940i	$-0.593750\pi$
$548$		0			0
$549$		0			0
$550$	0.881921	−	0.471397i	0.881921	−	0.471397i
$551$		0			0
$552$		0			0
$553$		0			0
$554$	−0.151537	−	1.53858i	−0.151537	−	1.53858i
$555$		0			0
$556$		0			0
$557$	−0.301614	+	1.51631i	−0.301614	+	1.51631i	0.471397	+	0.881921i	$0.343750\pi$
$557$	−0.301614	+	1.51631i	−0.301614	+	1.51631i	−0.773010	+	0.634393i	$0.781250\pi$
$558$	0.322547	+	1.06330i	0.322547	+	1.06330i
$559$	−2.17588	−	2.17588i	−2.17588	−	2.17588i
$560$	0.344109	+	1.72995i	0.344109	+	1.72995i
$561$		0			0
$562$		0			0
$563$	−0.924678	−	0.183930i	−0.924678	−	0.183930i	−0.290285	−	0.956940i	$-0.593750\pi$
$563$	−0.924678	−	0.183930i	−0.924678	−	0.183930i	−0.634393	+	0.773010i	$0.718750\pi$
$564$		0			0
$565$		0			0
$566$	−0.124363	+	0.151537i	−0.124363	+	0.151537i
$567$	0.674993	−	1.62958i	0.674993	−	1.62958i
$568$	0.591637	+	0.485544i	0.591637	+	0.485544i
$569$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$569$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$570$		0			0
$571$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$571$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$572$	1.10579	+	1.65493i	1.10579	+	1.65493i
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−0.555570	−	0.831470i	−0.555570	−	0.831470i
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$	0.512459	−	0.420564i	0.512459	−	0.420564i
$579$		0			0
$580$		0			0
$581$	3.44324	−	0.684903i	3.44324	−	0.684903i
$582$		0			0
$583$		0			0
$584$	−0.124363	−	1.26268i	−0.124363	−	1.26268i
$585$	−0.761681	+	1.83886i	−0.761681	+	1.83886i
$586$	−0.804910	+	0.980785i	−0.804910	+	0.980785i
$587$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$587$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$588$		0			0
$589$		0			0
$590$	1.72995	+	0.924678i	1.72995	+	0.924678i
$591$		0			0
$592$		0			0
$593$	1.24723	+	1.24723i	1.24723	+	1.24723i	0.956940	+	0.290285i	$0.0937500\pi$
$593$	1.24723	+	1.24723i	1.24723	+	1.24723i	0.290285	+	0.956940i	$0.406250\pi$
$594$		0			0
$595$	0.199779	−	1.00436i	0.199779	−	1.00436i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	1.53636	+	0.636379i	1.53636	+	0.636379i	0.980785	−	0.195090i	$-0.0625000\pi$
$599$	1.53636	+	0.636379i	1.53636	+	0.636379i	0.555570	+	0.831470i	$0.312500\pi$
$600$		0			0
$601$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$601$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$602$	2.40494	−	1.28547i	2.40494	−	1.28547i
$603$		0			0
$604$		0			0
$605$	0.555570	+	0.831470i	0.555570	+	0.831470i
$606$		0			0
$607$		−	0.580569i		−	0.580569i	−0.956940	−	0.290285i	$-0.906250\pi$
$607$		−	0.580569i		−	0.580569i	0.956940	−	0.290285i	$-0.0937500\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$	0.113263	+	0.569414i	0.113263	+	0.569414i
$613$	0.373380	+	1.87711i	0.373380	+	1.87711i	0.471397	+	0.881921i	$0.343750\pi$
$613$	0.373380	+	1.87711i	0.373380	+	1.87711i	−0.0980171	+	0.995185i	$0.531250\pi$
$614$	0.273678	+	0.512016i	0.273678	+	0.512016i
$615$		0			0
$616$	−1.68789	+	0.512016i	−1.68789	+	0.512016i
$617$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$617$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$618$		0			0
$619$	1.53636	+	1.02656i	1.53636	+	1.02656i	0.980785	+	0.195090i	$0.0625000\pi$
$619$	1.53636	+	1.02656i	1.53636	+	1.02656i	0.555570	+	0.831470i	$0.312500\pi$
$620$	1.11114			1.11114
$621$		0			0
$622$		0			0
$623$	0.486643	+	0.486643i	0.486643	+	0.486643i
$624$		0			0
$625$	0.707107	−	0.707107i	0.707107	−	0.707107i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$	−1.36347	−	1.11897i	−1.36347	−	1.11897i
$631$	0.541196	−	1.30656i	0.541196	−	1.30656i	−0.382683	−	0.923880i	$-0.625000\pi$
$631$	0.541196	−	1.30656i	0.541196	−	1.30656i	0.923880	−	0.382683i	$-0.125000\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	1.24441	−	0.247528i	1.24441	−	0.247528i
$636$		0			0
$637$	−3.49379	+	2.33448i	−3.49379	+	2.33448i
$638$		0			0
$639$	−0.765367			−0.765367
$640$	−0.956940	+	0.290285i	−0.956940	+	0.290285i
$641$	0.390181			0.390181			0.195090	−	0.980785i	$-0.437500\pi$
$641$	0.390181			0.390181			0.195090	+	0.980785i	$0.437500\pi$
$642$		0			0
$643$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$643$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$647$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$648$	0.956940	+	0.290285i	0.956940	+	0.290285i
$649$	−0.750661	+	1.81225i	−0.750661	+	1.81225i
$650$	1.53858	+	1.26268i	1.53858	+	1.26268i
$651$		0			0
$652$		0			0
$653$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$653$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	0.897168	+	0.897168i	0.897168	+	0.897168i
$658$		0			0
$659$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$659$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$660$		0			0
$661$	−0.923880	−	0.617317i	−0.923880	−	0.617317i		−	1.00000i	$-0.5\pi$
$661$	−0.923880	−	0.617317i	−0.923880	−	0.617317i	−0.923880	+	0.382683i	$0.875000\pi$
$662$	−1.40740	+	0.138617i	−1.40740	+	0.138617i
$663$		0			0
$664$	0.577774	+	1.90466i	0.577774	+	1.90466i
$665$		0			0
$666$		0			0
$667$		0			0
$668$	−0.0382444	−	0.192268i	−0.0382444	−	0.192268i
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$			0.942793i			0.942793i	0.881921	+	0.471397i	$0.156250\pi$
$673$			0.942793i			0.942793i	−0.881921	+	0.471397i	$0.843750\pi$
$674$	0.0924099	−	0.938254i	0.0924099	−	0.938254i
$675$		0			0
$676$	−1.64536	+	2.46246i	−1.64536	+	2.46246i
$677$	0.113263	+	0.569414i	0.113263	+	0.569414i	0.995185	+	0.0980171i	$0.0312500\pi$
$677$	0.113263	+	0.569414i	0.113263	+	0.569414i	−0.881921	+	0.471397i	$0.843750\pi$
$678$		0			0
$679$		0			0
$680$	0.577774	+	0.0569057i	0.577774	+	0.0569057i
$681$		0			0
$682$	0.108911	+	1.10579i	0.108911	+	1.10579i
$683$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$683$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$684$		0			0
$685$		0			0
$686$	−0.568922	−	1.87549i	−0.568922	−	1.87549i
$687$		0			0
$688$	0.858923	+	1.28547i	0.858923	+	1.28547i
$689$		0			0
$690$		0			0
$691$	−1.63099	−	0.324423i	−1.63099	−	0.324423i	−0.707107	−	0.707107i	$-0.750000\pi$
$691$	−1.63099	−	0.324423i	−1.63099	−	0.324423i	−0.923880	+	0.382683i	$0.875000\pi$
$692$	1.76820	+	0.732410i	1.76820	+	0.732410i
$693$	0.979938	−	1.46658i	0.979938	−	1.46658i
$694$	0.804910	−	0.980785i	0.804910	−	0.980785i
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−1.46658	+	0.979938i	−1.46658	+	0.979938i
$701$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$701$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$702$		0			0
$703$		0			0
$704$	−0.382683	−	0.923880i	−0.382683	−	0.923880i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	1.38704	−	0.275899i	1.38704	−	0.275899i	0.555570	−	0.831470i	$-0.312500\pi$
$709$	1.38704	−	0.275899i	1.38704	−	0.275899i	0.831470	+	0.555570i	$0.187500\pi$
$710$	−0.222174	+	0.732410i	−0.222174	+	0.732410i
$711$		0			0
$712$	−0.247528	+	0.301614i	−0.247528	+	0.301614i
$713$		0			0
$714$		0			0
$715$	−1.10579	+	1.65493i	−1.10579	+	1.65493i
$716$	0.292893	−	0.707107i	0.292893	−	0.707107i
$717$		0			0
$718$		0			0
$719$	0.275899	−	0.275899i	0.275899	−	0.275899i	−0.555570	−	0.831470i	$-0.687500\pi$
$719$	0.275899	−	0.275899i	0.275899	−	0.275899i	0.831470	+	0.555570i	$0.187500\pi$
$720$	0.555570	−	0.831470i	0.555570	−	0.831470i
$721$		0			0
$722$	−0.290285	−	0.956940i	−0.290285	−	0.956940i
$723$		0			0
$724$	1.30656	+	1.30656i	1.30656	+	1.30656i
$725$		0			0
$726$		0			0
$727$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$727$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$728$	−2.22716	−	2.71381i	−2.22716	−	2.71381i
$729$	−0.923880	+	0.382683i	−0.923880	+	0.382683i
$730$	1.11897	−	0.598102i	1.11897	−	0.598102i
$731$	−0.175108	−	0.880326i	−0.175108	−	0.880326i
$732$		0			0
$733$	0.979938	+	1.46658i	0.979938	+	1.46658i	0.881921	+	0.471397i	$0.156250\pi$
$733$	0.979938	+	1.46658i	0.979938	+	1.46658i	0.0980171	+	0.995185i	$0.468750\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$739$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	1.76820	−	0.732410i	1.76820	−	0.732410i	0.773010	−	0.634393i	$-0.218750\pi$
$743$	1.76820	−	0.732410i	1.76820	−	0.732410i	0.995185	−	0.0980171i	$-0.0312500\pi$
$744$		0			0
$745$		0			0
$746$	0.938254	−	0.0924099i	0.938254	−	0.0924099i
$747$	−1.65493	−	1.10579i	−1.65493	−	1.10579i
$748$			0.580569i			0.580569i
$749$	−0.658583	+	3.31092i	−0.658583	+	3.31092i
$750$		0			0
$751$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$751$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$757$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$758$	−0.301614	−	0.247528i	−0.301614	−	0.247528i
$759$		0			0
$760$		0			0
$761$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$761$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$762$		0			0
$763$		0			0
$764$	0.382683	−	1.92388i	0.382683	−	1.92388i
$765$	−0.482726	+	0.322547i	−0.482726	+	0.322547i
$766$		0			0
$767$	−3.90425			−3.90425
$768$		0			0
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$	−1.11897	−	1.36347i	−1.11897	−	1.36347i
$771$		0			0
$772$	−0.301614	+	1.51631i	−0.301614	+	1.51631i
$773$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$773$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$774$	−1.47945	−	0.448786i	−1.47945	−	0.448786i
$775$	0.425215	+	1.02656i	0.425215	+	1.02656i
$776$		0			0
$777$		0			0
$778$	−1.28547	−	1.05496i	−1.28547	−	1.05496i
$779$		0			0
$780$		0			0
$781$	−0.750661	−	0.149316i	−0.750661	−	0.149316i
$782$		0			0
$783$		0			0
$784$	1.95044	−	0.807898i	1.95044	−	0.807898i
$785$		0			0
$786$		0			0
$787$	−0.373380	+	1.87711i	−0.373380	+	1.87711i	0.0980171	+	0.995185i	$0.468750\pi$
$787$	−0.373380	+	1.87711i	−0.373380	+	1.87711i	−0.471397	+	0.881921i	$0.656250\pi$
$788$		−	0.942793i		−	0.942793i
$789$		0			0
$790$		0			0
$791$		0			0
$792$	0.881921	+	0.471397i	0.881921	+	0.471397i
$793$		0			0
$794$		0			0
$795$		0			0
$796$	−1.81225	+	0.360480i	−1.81225	+	0.360480i
$797$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$797$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$798$		0			0
$799$		0			0
$800$	−0.634393	−	0.773010i	−0.634393	−	0.773010i
$801$		−	0.390181i		−	0.390181i
$802$	0.192268	−	1.95213i	0.192268	−	1.95213i
$803$	0.704900	+	1.05496i	0.704900	+	1.05496i
$804$		0			0
$805$		0			0
$806$	−1.95044	+	1.04253i	−1.95044	+	1.04253i
$807$		0			0
$808$		0			0
$809$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$809$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$810$	0.0980171	+	0.995185i	0.0980171	+	0.995185i
$811$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$811$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$	3.44324	+	0.684903i	3.44324	+	0.684903i
$820$		0			0
$821$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$821$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$822$		0			0
$823$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$823$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$824$		0			0
$825$		0			0
$826$	1.00436	−	3.31092i	1.00436	−	3.31092i
$827$	1.95213	−	0.388302i	1.95213	−	0.388302i	0.956940	−	0.290285i	$-0.0937500\pi$
$827$	1.95213	−	0.388302i	1.95213	−	0.388302i	0.995185	−	0.0980171i	$-0.0312500\pi$
$828$		0			0
$829$	−1.17588	+	0.785695i	−1.17588	+	0.785695i	−0.980785	−	0.195090i	$-0.937500\pi$
$829$	−1.17588	+	0.785695i	−1.17588	+	0.785695i	−0.195090	+	0.980785i	$0.562500\pi$
$830$	−1.53858	+	1.26268i	−1.53858	+	1.26268i
$831$		0			0
$832$	1.40740	−	1.40740i	1.40740	−	1.40740i
$833$	−1.22566			−1.22566
$834$		0			0
$835$	0.162997	−	0.108911i	0.162997	−	0.108911i
$836$		0			0
$837$		0			0
$838$	0.410525	−	1.35332i	0.410525	−	1.35332i
$839$	0.292893	+	0.707107i	0.292893	+	0.707107i	1.00000			$0$
$839$	0.292893	+	0.707107i	0.292893	+	0.707107i	−0.707107	+	0.707107i	$0.750000\pi$
$840$		0			0
$841$	−0.382683	+	0.923880i	−0.382683	+	0.923880i
$842$	1.17221	−	1.42834i	1.17221	−	1.42834i
$843$		0			0
$844$		0			0
$845$	−2.90466	−	0.577774i	−2.90466	−	0.577774i
$846$		0			0
$847$	1.24723	−	1.24723i	1.24723	−	1.24723i
$848$		0			0
$849$		0			0
$850$	0.168530	+	0.555570i	0.168530	+	0.555570i
$851$		0			0
$852$		0			0
$853$	1.05496	+	0.704900i	1.05496	+	0.704900i	0.956940	−	0.290285i	$-0.0937500\pi$
$853$	1.05496	+	0.704900i	1.05496	+	0.704900i	0.0980171	+	0.995185i	$0.468750\pi$
$854$		0			0
$855$		0			0
$856$	−1.90466	−	0.187593i	−1.90466	−	0.187593i
$857$	1.17221	−	0.485544i	1.17221	−	0.485544i	0.290285	−	0.956940i	$-0.406250\pi$
$857$	1.17221	−	0.485544i	1.17221	−	0.485544i	0.881921	+	0.471397i	$0.156250\pi$
$858$		0			0
$859$	−0.275899	−	1.38704i	−0.275899	−	1.38704i	−0.831470	−	0.555570i	$-0.812500\pi$
$859$	−0.275899	−	1.38704i	−0.275899	−	1.38704i	0.555570	−	0.831470i	$-0.312500\pi$
$860$	−0.858923	+	1.28547i	−0.858923	+	1.28547i
$861$		0			0
$862$		0			0
$863$		0			0			−	1.00000i	$-0.5\pi$
$863$		0			0				1.00000i	$0.5\pi$
$864$		0			0
$865$			1.91388i			1.91388i
$866$		0			0
$867$		0			0
$868$	−0.382353	−	1.92222i	−0.382353	−	1.92222i
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−1.46658	−	0.979938i	−1.46658	−	0.979938i
$876$		0			0
$877$	−0.113263	+	0.569414i	−0.113263	+	0.569414i	0.881921	+	0.471397i	$0.156250\pi$
$877$	−0.113263	+	0.569414i	−0.113263	+	0.569414i	−0.995185	+	0.0980171i	$0.968750\pi$
$878$		0			0
$879$		0			0
$880$	0.707107	−	0.707107i	0.707107	−	0.707107i
$881$	0.541196	−	0.541196i	0.541196	−	0.541196i	−0.382683	−	0.923880i	$-0.625000\pi$
$881$	0.541196	−	0.541196i	0.541196	−	0.541196i	0.923880	+	0.382683i	$0.125000\pi$
$882$	−0.995185	+	1.86186i	−0.995185	+	1.86186i
$883$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$883$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$884$	−1.06759	+	0.442209i	−1.06759	+	0.442209i
$885$		0			0
$886$		0			0
$887$	−0.0750191	+	0.181112i	−0.0750191	+	0.181112i	−0.956940	−	0.290285i	$-0.906250\pi$
$887$	−0.0750191	+	0.181112i	−0.0750191	+	0.181112i	0.881921	+	0.471397i	$0.156250\pi$
$888$		0			0
$889$	−0.856422	−	2.06759i	−0.856422	−	2.06759i
$890$	−0.373380	−	0.113263i	−0.373380	−	0.113263i
$891$	−0.980785	+	0.195090i	−0.980785	+	0.195090i
$892$		0			0
$893$		0			0
$894$		0			0
$895$	0.765367			0.765367
$896$	0.831470	+	1.55557i	0.831470	+	1.55557i
$897$		0			0
$898$	−1.05496	−	1.28547i	−1.05496	−	1.28547i
$899$		0			0
$900$	0.980785	+	0.195090i	0.980785	+	0.195090i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−0.707107	+	1.70711i	−0.707107	+	1.70711i
$906$		0			0
$907$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$907$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$908$	−0.181112	+	0.0750191i	−0.181112	+	0.0750191i
$909$		0			0
$910$	1.65493	−	3.09616i	1.65493	−	3.09616i
$911$	−1.17588	+	1.17588i	−1.17588	+	1.17588i	−0.195090	+	0.980785i	$0.562500\pi$
$911$	−1.17588	+	1.17588i	−1.17588	+	1.17588i	−0.980785	+	0.195090i	$0.937500\pi$
$912$		0			0
$913$	−1.40740	−	1.40740i	−1.40740	−	1.40740i
$914$	0.187593	−	0.0569057i	0.187593	−	0.0569057i
$915$		0			0
$916$	1.66294			1.66294
$917$		0			0
$918$		0			0
$919$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$919$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−0.297193	−	1.49409i	−0.297193	−	1.49409i
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	0.765367i		−	0.765367i	−0.923880	−	0.382683i	$-0.875000\pi$
$929$		−	0.765367i		−	0.765367i	0.923880	−	0.382683i	$-0.125000\pi$
$930$		0			0
$931$		0			0
$932$	−0.523788	+	0.783904i	−0.523788	+	0.783904i
$933$		0			0
$934$		0			0
$935$	−0.536376	+	0.222174i	−0.536376	+	0.222174i
$936$	−0.195090	+	1.98079i	−0.195090	+	1.98079i
$937$	−1.76820	−	0.732410i	−1.76820	−	0.732410i	−0.995185	−	0.0980171i	$-0.968750\pi$
$937$	−1.76820	−	0.732410i	−1.76820	−	0.732410i	−0.773010	−	0.634393i	$-0.781250\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$941$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$942$		0			0
$943$		0			0
$944$	1.92388	+	0.382683i	1.92388	+	0.382683i
$945$		0			0
$946$	−1.36347	−	0.728789i	−1.36347	−	0.728789i
$947$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$947$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$948$		0			0
$949$	−1.40301	+	2.09976i	−1.40301	+	2.09976i
$950$		0			0
$951$		0			0
$952$	−0.100373	−	1.01910i	−0.100373	−	1.01910i
$953$	−0.761681	−	1.83886i	−0.761681	−	1.83886i	−0.471397	−	0.881921i	$-0.656250\pi$
$953$	−0.761681	−	1.83886i	−0.761681	−	1.83886i	−0.290285	−	0.956940i	$-0.593750\pi$
$954$		0			0
$955$	1.92388	−	0.382683i	1.92388	−	0.382683i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−0.234633			−0.234633
$962$		0			0
$963$	1.59133	−	1.06330i	1.59133	−	1.06330i
$964$		0			0
$965$	−1.51631	+	0.301614i	−1.51631	+	0.301614i
$966$		0			0
$967$	0.222174	+	0.536376i	0.222174	+	0.536376i	0.995185	−	0.0980171i	$-0.0312500\pi$
$967$	0.222174	+	0.536376i	0.222174	+	0.536376i	−0.773010	+	0.634393i	$0.781250\pi$
$968$	0.773010	+	0.634393i	0.773010	+	0.634393i
$969$		0			0
$970$		0			0
$971$	−1.02656	+	1.53636i	−1.02656	+	1.53636i	−0.195090	+	0.980785i	$0.562500\pi$
$971$	−1.02656	+	1.53636i	−1.02656	+	1.53636i	−0.831470	+	0.555570i	$0.812500\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$977$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$978$		0			0
$979$	0.0761205	−	0.382683i	0.0761205	−	0.382683i
$980$	1.49280	+	1.49280i	1.49280	+	1.49280i
$981$		0			0
$982$		0			0
$983$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$983$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$984$		0			0
$985$	0.871028	−	0.360791i	0.871028	−	0.360791i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$	−0.0980171	+	0.995185i	−0.0980171	+	0.995185i
$991$		−	1.66294i		−	1.66294i	−0.555570	−	0.831470i	$-0.687500\pi$
$991$		−	1.66294i		−	1.66294i	0.555570	−	0.831470i	$-0.312500\pi$
$992$	1.06330	−	0.322547i	1.06330	−	0.322547i
$993$		0			0
$994$	1.34349	+	0.132322i	1.34349	+	0.132322i
$995$	−1.02656	−	1.53636i	−1.02656	−	1.53636i
$996$		0			0
$997$	0.344109	+	1.72995i	0.344109	+	1.72995i	0.634393	+	0.773010i	$0.281250\pi$
$997$	0.344109	+	1.72995i	0.344109	+	1.72995i	−0.290285	+	0.956940i	$0.593750\pi$
$998$	0.523788	+	0.979938i	0.523788	+	0.979938i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3520.1.db.a.1869.3	yes	32
5.4	even	2	inner	3520.1.db.a.1869.2	✓	32
11.10	odd	2	inner	3520.1.db.a.1869.2	✓	32
55.54	odd	2	CM	3520.1.db.a.1869.3	yes	32
64.5	even	16	inner	3520.1.db.a.2309.3	yes	32
320.69	even	16	inner	3520.1.db.a.2309.2	yes	32
704.197	odd	16	inner	3520.1.db.a.2309.2	yes	32
3520.2309	odd	16	inner	3520.1.db.a.2309.3	yes	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3520.1.db.a.1869.2	✓	32	5.4	even	2	inner
3520.1.db.a.1869.2	✓	32	11.10	odd	2	inner
3520.1.db.a.1869.3	yes	32	1.1	even	1	trivial
3520.1.db.a.1869.3	yes	32	55.54	odd	2	CM
3520.1.db.a.2309.2	yes	32	320.69	even	16	inner
3520.1.db.a.2309.2	yes	32	704.197	odd	16	inner
3520.1.db.a.2309.3	yes	32	64.5	even	16	inner
3520.1.db.a.2309.3	yes	32	3520.2309	odd	16	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 3520 = 2^{6} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3520.db (of order \(16\), degree \(8\), minimal)

\(f(q)\)	\(=\)	\(q+(0.634393 + 0.773010i) q^{2} +(-0.195090 + 0.980785i) q^{4} +(-0.980785 + 0.195090i) q^{5} +(0.674993 + 1.62958i) q^{7} +(-0.881921 + 0.471397i) q^{8} +(0.382683 - 0.923880i) q^{9} +(-0.773010 - 0.634393i) q^{10} +(0.555570 - 0.831470i) q^{11} +(1.95213 + 0.388302i) q^{13} +(-0.831470 + 1.55557i) q^{14} +(-0.923880 - 0.382683i) q^{16} +(0.410525 + 0.410525i) q^{17} +(0.956940 - 0.290285i) q^{18} -1.00000i q^{20} +(0.995185 - 0.0980171i) q^{22} +(0.923880 - 0.382683i) q^{25} +(0.938254 + 1.75535i) q^{26} +(-1.72995 + 0.344109i) q^{28} +1.11114i q^{31} +(-0.290285 - 0.956940i) q^{32} +(-0.0569057 + 0.577774i) q^{34} +(-0.979938 - 1.46658i) q^{35} +(0.831470 + 0.555570i) q^{36} +(0.773010 - 0.634393i) q^{40} +(-1.28547 - 0.858923i) q^{43} +(0.707107 + 0.707107i) q^{44} +(-0.195090 + 0.980785i) q^{45} +(-1.49280 + 1.49280i) q^{49} +(0.881921 + 0.471397i) q^{50} +(-0.761681 + 1.83886i) q^{52} +(-0.382683 + 0.923880i) q^{55} +(-1.36347 - 1.11897i) q^{56} +(-1.92388 + 0.382683i) q^{59} +(-0.858923 + 0.704900i) q^{62} +1.76384 q^{63} +(0.555570 - 0.831470i) q^{64} -1.99037 q^{65} +(-0.482726 + 0.322547i) q^{68} +(0.512016 - 1.68789i) q^{70} +(-0.292893 - 0.707107i) q^{71} +(0.0980171 + 0.995185i) q^{72} +(-0.485544 + 1.17221i) q^{73} +(1.72995 + 0.344109i) q^{77} +(0.980785 + 0.195090i) q^{80} +(-0.707107 - 0.707107i) q^{81} +(0.388302 - 1.95213i) q^{83} +(-0.482726 - 0.322547i) q^{85} +(-0.151537 - 1.53858i) q^{86} +(-0.0980171 + 0.995185i) q^{88} +(0.360480 - 0.149316i) q^{89} +(-0.881921 + 0.471397i) q^{90} +(0.684903 + 3.44324i) q^{91} +(-2.10097 - 0.206928i) q^{98} +(-0.555570 - 0.831470i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 32 q^{59} - 32 q^{71}+O(q^{100}) \) 32 * q - 32 * q^59 - 32 * q^71

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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Database

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