LMFDB - Embedded newform 3584.1.cd.a.1077.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3584,1,Mod(13,3584)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3584, base_ring=CyclotomicField(128))

chi = DirichletCharacter(H, H._module([0, 111, 64]))

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("3584.13");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 3584 = 2^{9} \cdot 7 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3584.cd (of order $128$, degree $64$, 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:	$1.78864900528$
Analytic rank:	$0$
Dimension:	$64$
Coefficient field:	$\Q(\zeta_{128})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{64} + 1 $ x^64 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{128}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{128} - \cdots)$

Embedding invariants

Embedding label			1077.1
Root			$-0.941544 - 0.336890i$ of defining polynomial
Character	$\chi$	$=$	3584.1077
Dual form			3584.1.cd.a.2589.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.146730 + 0.989177i) q^{2} +(-0.956940 + 0.290285i) q^{4} +(0.941544 - 0.336890i) q^{7} +(-0.427555 - 0.903989i) q^{8} +(-0.671559 + 0.740951i) q^{9} +O(q^{10})$	\(q+(0.146730 + 0.989177i) q^{2} +(-0.956940 + 0.290285i) q^{4} +(0.941544 - 0.336890i) q^{7} +(-0.427555 - 0.903989i) q^{8} +(-0.671559 + 0.740951i) q^{9} +(-1.01599 - 1.60442i) q^{11} +(0.471397 + 0.881921i) q^{14} +(0.831470 - 0.555570i) q^{16} +(-0.831470 - 0.555570i) q^{18} +(1.43798 - 1.24041i) q^{22} +(0.0143994 - 0.0970732i) q^{23} +(0.970031 + 0.242980i) q^{25} +(-0.803208 + 0.595699i) q^{28} +(1.28908 - 0.907873i) q^{29} +(0.671559 + 0.740951i) q^{32} +(0.427555 - 0.903989i) q^{36} +(1.06710 - 0.0787137i) q^{37} +(0.123057 + 0.319012i) q^{43} +(1.43798 + 1.24041i) q^{44} +0.0981353 q^{46} +(0.773010 - 0.634393i) q^{49} +(-0.0980171 + 0.995185i) q^{50} +(-0.0401291 - 0.0282621i) q^{53} +(-0.707107 - 0.707107i) q^{56} +(1.08719 + 1.14192i) q^{58} +(-0.382683 + 0.923880i) q^{63} +(-0.634393 + 0.773010i) q^{64} +(-0.0457936 - 1.86542i) q^{67} +(0.00961895 - 0.195798i) q^{71} +(0.956940 + 0.290285i) q^{72} +(0.234437 + 1.04400i) q^{74} +(-1.49711 - 1.16836i) q^{77} +(1.70720 + 0.168144i) q^{79} +(-0.0980171 - 0.995185i) q^{81} +(-0.297503 + 0.168534i) q^{86} +(-1.01599 + 1.60442i) q^{88} +(0.0143994 + 0.0970732i) q^{92} +(0.740951 + 0.671559i) q^{98} +(1.87110 + 0.324666i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 64 q+O(q^{10}) $ 64 * q	$ 64 q+O(q^{100}) $ 64 * q

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3584\mathbb{Z}\right)^\times$.

$n$	$1023$	$1025$	$1541$
$\chi(n)$	$1$	$-1$	$e\left(\frac{5}{128}\right)$

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 $

$2$	0.146730	+	0.989177i	0.146730	+	0.989177i
$2$	0.146730	+	0.989177i	0.146730	+	0.989177i
$3$		0			0		−0.405241	−	0.914210i	$-0.632812\pi$
$3$		0			0		0.405241	+	0.914210i	$0.367188\pi$
$4$	−0.956940	+	0.290285i	−0.956940	+	0.290285i
$5$		0			0		−0.992480	−	0.122411i	$-0.960938\pi$
$5$		0			0		0.992480	+	0.122411i	$0.0390625\pi$
$6$		0			0
$7$	0.941544	−	0.336890i	0.941544	−	0.336890i
$7$	0.941544	−	0.336890i	0.941544	−	0.336890i
$8$	−0.427555	−	0.903989i	−0.427555	−	0.903989i
$9$	−0.671559	+	0.740951i	−0.671559	+	0.740951i
$10$		0			0
$11$	−1.01599	−	1.60442i	−1.01599	−	1.60442i	−0.773010	−	0.634393i	$-0.781250\pi$
$11$	−1.01599	−	1.60442i	−1.01599	−	1.60442i	−0.242980	−	0.970031i	$-0.578125\pi$
$12$		0			0
$13$		0			0		−0.492898	−	0.870087i	$-0.664062\pi$
$13$		0			0		0.492898	+	0.870087i	$0.335938\pi$
$14$	0.471397	+	0.881921i	0.471397	+	0.881921i
$15$		0			0
$16$	0.831470	−	0.555570i	0.831470	−	0.555570i
$17$		0			0		−0.956940	−	0.290285i	$-0.906250\pi$
$17$		0			0		0.956940	+	0.290285i	$0.0937500\pi$
$18$	−0.831470	−	0.555570i	−0.831470	−	0.555570i
$19$		0			0		0.949528	−	0.313682i	$-0.101562\pi$
$19$		0			0		−0.949528	+	0.313682i	$0.898438\pi$
$20$		0			0
$21$		0			0
$22$	1.43798	−	1.24041i	1.43798	−	1.24041i
$23$	0.0143994	−	0.0970732i	0.0143994	−	0.0970732i	−0.980785	−	0.195090i	$-0.937500\pi$
$23$	0.0143994	−	0.0970732i	0.0143994	−	0.0970732i	0.995185	+	0.0980171i	$0.0312500\pi$
$24$		0			0
$25$	0.970031	+	0.242980i	0.970031	+	0.242980i
$26$		0			0
$27$		0			0
$28$	−0.803208	+	0.595699i	−0.803208	+	0.595699i
$29$	1.28908	−	0.907873i	1.28908	−	0.907873i	0.290285	−	0.956940i	$-0.406250\pi$
$29$	1.28908	−	0.907873i	1.28908	−	0.907873i	0.998795	+	0.0490677i	$0.0156250\pi$
$30$		0			0
$31$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$31$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$32$	0.671559	+	0.740951i	0.671559	+	0.740951i
$33$		0			0
$34$		0			0
$35$		0			0
$36$	0.427555	−	0.903989i	0.427555	−	0.903989i
$37$	1.06710	−	0.0787137i	1.06710	−	0.0787137i	0.471397	−	0.881921i	$-0.343750\pi$
$37$	1.06710	−	0.0787137i	1.06710	−	0.0787137i	0.595699	+	0.803208i	$0.296875\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		0.970031	−	0.242980i	$-0.0781250\pi$
$41$		0			0		−0.970031	+	0.242980i	$0.921875\pi$
$42$		0			0
$43$	0.123057	+	0.319012i	0.123057	+	0.319012i	0.980785	−	0.195090i	$-0.0625000\pi$
$43$	0.123057	+	0.319012i	0.123057	+	0.319012i	−0.857729	+	0.514103i	$0.828125\pi$
$44$	1.43798	+	1.24041i	1.43798	+	1.24041i
$45$		0			0
$46$	0.0981353			0.0981353
$47$		0			0		0.634393	−	0.773010i	$-0.281250\pi$
$47$		0			0		−0.634393	+	0.773010i	$0.718750\pi$
$48$		0			0
$49$	0.773010	−	0.634393i	0.773010	−	0.634393i
$50$	−0.0980171	+	0.995185i	−0.0980171	+	0.995185i
$51$		0			0
$52$		0			0
$53$	−0.0401291	−	0.0282621i	−0.0401291	−	0.0282621i	0.555570	−	0.831470i	$-0.312500\pi$
$53$	−0.0401291	−	0.0282621i	−0.0401291	−	0.0282621i	−0.595699	+	0.803208i	$0.703125\pi$
$54$		0			0
$55$		0			0
$56$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$57$		0			0
$58$	1.08719	+	1.14192i	1.08719	+	1.14192i
$59$		0			0		−0.870087	−	0.492898i	$-0.835938\pi$
$59$		0			0		0.870087	+	0.492898i	$0.164062\pi$
$60$		0			0
$61$		0			0		−0.724247	−	0.689541i	$-0.757812\pi$
$61$		0			0		0.724247	+	0.689541i	$0.242188\pi$
$62$		0			0
$63$	−0.382683	+	0.923880i	−0.382683	+	0.923880i
$64$	−0.634393	+	0.773010i	−0.634393	+	0.773010i
$65$		0			0
$66$		0			0
$67$	−0.0457936	−	1.86542i	−0.0457936	−	1.86542i	−0.382683	−	0.923880i	$-0.625000\pi$
$67$	−0.0457936	−	1.86542i	−0.0457936	−	1.86542i	0.336890	−	0.941544i	$-0.390625\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	0.00961895	−	0.195798i	0.00961895	−	0.195798i	−0.989177	−	0.146730i	$-0.953125\pi$
$71$	0.00961895	−	0.195798i	0.00961895	−	0.195798i	0.998795	−	0.0490677i	$-0.0156250\pi$
$72$	0.956940	+	0.290285i	0.956940	+	0.290285i
$73$		0			0		−0.941544	−	0.336890i	$-0.890625\pi$
$73$		0			0		0.941544	+	0.336890i	$0.109375\pi$
$74$	0.234437	+	1.04400i	0.234437	+	1.04400i
$75$		0			0
$76$		0			0
$77$	−1.49711	−	1.16836i	−1.49711	−	1.16836i
$78$		0			0
$79$	1.70720	+	0.168144i	1.70720	+	0.168144i	0.903989	−	0.427555i	$-0.140625\pi$
$79$	1.70720	+	0.168144i	1.70720	+	0.168144i	0.803208	+	0.595699i	$0.203125\pi$
$80$		0			0
$81$	−0.0980171	−	0.995185i	−0.0980171	−	0.995185i
$82$		0			0
$83$		0			0		−0.997290	−	0.0735646i	$-0.976562\pi$
$83$		0			0		0.997290	+	0.0735646i	$0.0234375\pi$
$84$		0			0
$85$		0			0
$86$	−0.297503	+	0.168534i	−0.297503	+	0.168534i
$87$		0			0
$88$	−1.01599	+	1.60442i	−1.01599	+	1.60442i
$89$		0			0		−0.146730	−	0.989177i	$-0.546875\pi$
$89$		0			0		0.146730	+	0.989177i	$0.453125\pi$
$90$		0			0
$91$		0			0
$92$	0.0143994	+	0.0970732i	0.0143994	+	0.0970732i
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$97$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$98$	0.740951	+	0.671559i	0.740951	+	0.671559i
$99$	1.87110	+	0.324666i	1.87110	+	0.324666i
$100$	−0.998795	+	0.0490677i	−0.998795	+	0.0490677i
$101$		0			0		0.893224	−	0.449611i	$-0.148438\pi$
$101$		0			0		−0.893224	+	0.449611i	$0.851562\pi$
$102$		0			0
$103$		0			0		−0.514103	−	0.857729i	$-0.671875\pi$
$103$		0			0		0.514103	+	0.857729i	$0.328125\pi$
$104$		0			0
$105$		0			0
$106$	0.0220680	−	0.0438416i	0.0220680	−	0.0438416i
$107$	0.0355478	−	1.44806i	0.0355478	−	1.44806i	−0.671559	−	0.740951i	$-0.734375\pi$
$107$	0.0355478	−	1.44806i	0.0355478	−	1.44806i	0.707107	−	0.707107i	$-0.250000\pi$
$108$		0			0
$109$	−0.0661509	+	0.131419i	−0.0661509	+	0.131419i	−0.923880	−	0.382683i	$-0.875000\pi$
$109$	−0.0661509	+	0.131419i	−0.0661509	+	0.131419i	0.857729	+	0.514103i	$0.171875\pi$
$110$		0			0
$111$		0			0
$112$	0.595699	−	0.803208i	0.595699	−	0.803208i
$113$	0.594221	+	0.317618i	0.594221	+	0.317618i	0.740951	−	0.671559i	$-0.234375\pi$
$113$	0.594221	+	0.317618i	0.594221	+	0.317618i	−0.146730	+	0.989177i	$0.546875\pi$
$114$		0			0
$115$		0			0
$116$	−0.970031	+	1.24298i	−0.970031	+	1.24298i
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−1.11439	+	2.35617i	−1.11439	+	2.35617i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$	−0.970031	−	0.242980i	−0.970031	−	0.242980i
$127$	−1.24723	−	1.24723i	−1.24723	−	1.24723i	−0.956940	−	0.290285i	$-0.906250\pi$
$127$	−1.24723	−	1.24723i	−1.24723	−	1.24723i	−0.290285	−	0.956940i	$-0.593750\pi$
$128$	−0.857729	−	0.514103i	−0.857729	−	0.514103i
$129$		0			0
$130$		0			0
$131$		0			0		0.359895	−	0.932993i	$-0.382812\pi$
$131$		0			0		−0.359895	+	0.932993i	$0.617188\pi$
$132$		0			0
$133$		0			0
$134$	1.83851	−	0.319012i	1.83851	−	0.319012i
$135$		0			0
$136$		0			0
$137$	0.0951944	+	1.93773i	0.0951944	+	1.93773i	0.290285	+	0.956940i	$0.406250\pi$
$137$	0.0951944	+	1.93773i	0.0951944	+	1.93773i	−0.195090	+	0.980785i	$0.562500\pi$
$138$		0			0
$139$		0			0		0.219101	−	0.975702i	$-0.429688\pi$
$139$		0			0		−0.219101	+	0.975702i	$0.570312\pi$
$140$		0			0
$141$		0			0
$142$	0.195090	−	0.0192147i	0.195090	−	0.0192147i
$143$		0			0
$144$	−0.146730	+	0.989177i	−0.146730	+	0.989177i
$145$		0			0
$146$		0			0
$147$		0			0
$148$	−0.998298	+	0.385086i	−0.998298	+	0.385086i
$149$	−1.82787	−	0.0448717i	−1.82787	−	0.0448717i	−0.903989	−	0.427555i	$-0.859375\pi$
$149$	−1.82787	−	0.0448717i	−1.82787	−	0.0448717i	−0.923880	+	0.382683i	$0.875000\pi$
$150$		0			0
$151$	0.235710	−	0.174814i	0.235710	−	0.174814i	−0.471397	−	0.881921i	$-0.656250\pi$
$151$	0.235710	−	0.174814i	0.235710	−	0.174814i	0.707107	+	0.707107i	$0.250000\pi$
$152$		0			0
$153$		0			0
$154$	0.936041	−	1.65234i	0.936041	−	1.65234i
$155$		0			0
$156$		0			0
$157$		0			0		0.170962	−	0.985278i	$-0.445312\pi$
$157$		0			0		−0.170962	+	0.985278i	$0.554688\pi$
$158$	0.0841735	+	1.71339i	0.0841735	+	1.71339i
$159$		0			0
$160$		0			0
$161$	−0.0191453	−	0.0962497i	−0.0191453	−	0.0962497i
$162$	0.970031	−	0.242980i	0.970031	−	0.242980i
$163$	0.832854	+	0.527399i	0.832854	+	0.527399i	0.881921	−	0.471397i	$-0.156250\pi$
$163$	0.832854	+	0.527399i	0.832854	+	0.527399i	−0.0490677	+	0.998795i	$0.515625\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.989177	−	0.146730i	$-0.0468750\pi$
$167$		0			0		−0.989177	+	0.146730i	$0.953125\pi$
$168$		0			0
$169$	−0.514103	+	0.857729i	−0.514103	+	0.857729i
$170$		0			0
$171$		0			0
$172$	−0.210362	−	0.269554i	−0.210362	−	0.269554i
$173$		0			0		0.0735646	−	0.997290i	$-0.476562\pi$
$173$		0			0		−0.0735646	+	0.997290i	$0.523438\pi$
$174$		0			0
$175$	0.995185	−	0.0980171i	0.995185	−	0.0980171i
$176$	−1.73614	−	0.769576i	−1.73614	−	0.769576i
$177$		0			0
$178$		0			0
$179$	0.269596	−	0.345455i	0.269596	−	0.345455i	−0.634393	−	0.773010i	$-0.718750\pi$
$179$	0.269596	−	0.345455i	0.269596	−	0.345455i	0.903989	+	0.427555i	$0.140625\pi$
$180$		0			0
$181$		0			0		0.985278	−	0.170962i	$-0.0546875\pi$
$181$		0			0		−0.985278	+	0.170962i	$0.945312\pi$
$182$		0			0
$183$		0			0
$184$	−0.0939097	+	0.0284872i	−0.0939097	+	0.0284872i
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−1.10071	+	0.455929i	−1.10071	+	0.455929i	−0.857729	−	0.514103i	$-0.828125\pi$
$191$	−1.10071	+	0.455929i	−1.10071	+	0.455929i	−0.242980	+	0.970031i	$0.578125\pi$
$192$		0			0
$193$	−1.24088	−	0.513989i	−1.24088	−	0.513989i	−0.336890	−	0.941544i	$-0.609375\pi$
$193$	−1.24088	−	0.513989i	−1.24088	−	0.513989i	−0.903989	+	0.427555i	$0.859375\pi$
$194$		0			0
$195$		0			0
$196$	−0.555570	+	0.831470i	−0.555570	+	0.831470i
$197$	−0.567630	+	1.00201i	−0.567630	+	1.00201i	0.427555	+	0.903989i	$0.359375\pi$
$197$	−0.567630	+	1.00201i	−0.567630	+	1.00201i	−0.995185	+	0.0980171i	$0.968750\pi$
$198$	−0.0466052	+	1.89848i	−0.0466052	+	1.89848i
$199$		0			0		0.740951	−	0.671559i	$-0.234375\pi$
$199$		0			0		−0.740951	+	0.671559i	$0.765625\pi$
$200$	−0.195090	−	0.980785i	−0.195090	−	0.980785i
$201$		0			0
$202$		0			0
$203$	0.907873	−	1.28908i	0.907873	−	1.28908i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	0.0622564	+	0.0758597i	0.0622564	+	0.0758597i
$208$		0			0
$209$		0			0
$210$		0			0
$211$	1.29652	+	1.50303i	1.29652	+	1.50303i	0.740951	+	0.671559i	$0.234375\pi$
$211$	1.29652	+	1.50303i	1.29652	+	1.50303i	0.555570	+	0.831470i	$0.312500\pi$
$212$	0.0466052	+	0.0153963i	0.0466052	+	0.0153963i
$213$		0			0
$214$	1.43760	−	0.177311i	1.43760	−	0.177311i
$215$		0			0
$216$		0			0
$217$		0			0
$218$	−0.139703	−	0.0461517i	−0.139703	−	0.0461517i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$223$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$224$	0.881921	+	0.471397i	0.881921	+	0.471397i
$225$	−0.831470	+	0.555570i	−0.831470	+	0.555570i
$226$	−0.226990	+	0.634393i	−0.226990	+	0.634393i
$227$		0			0		−0.575808	−	0.817585i	$-0.695312\pi$
$227$		0			0		0.575808	+	0.817585i	$0.304688\pi$
$228$		0			0
$229$		0			0		0.313682	−	0.949528i	$-0.398438\pi$
$229$		0			0		−0.313682	+	0.949528i	$0.601562\pi$
$230$		0			0
$231$		0			0
$232$	−1.37186	−	0.777149i	−1.37186	−	0.777149i
$233$	1.01708	+	0.150869i	1.01708	+	0.150869i	0.634393	−	0.773010i	$-0.281250\pi$
$233$	1.01708	+	0.150869i	1.01708	+	0.150869i	0.382683	+	0.923880i	$0.375000\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−0.555570	+	1.83147i	−0.555570	+	1.83147i			1.00000i	$0.5\pi$
$239$	−0.555570	+	1.83147i	−0.555570	+	1.83147i	−0.555570	+	0.831470i	$0.687500\pi$
$240$		0			0
$241$		0			0		0.956940	−	0.290285i	$-0.0937500\pi$
$241$		0			0		−0.956940	+	0.290285i	$0.906250\pi$
$242$	−2.49418	−	0.756602i	−2.49418	−	0.756602i
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		0.122411	−	0.992480i	$-0.460938\pi$
$251$		0			0		−0.122411	+	0.992480i	$0.539062\pi$
$252$	0.0980171	−	0.995185i	0.0980171	−	0.995185i
$253$	−0.170376	+	0.0755226i	−0.170376	+	0.0755226i
$254$	1.05072	−	1.41673i	1.05072	−	1.41673i
$255$		0			0
$256$	0.382683	−	0.923880i	0.382683	−	0.923880i
$257$		0			0			−	1.00000i	$-0.5\pi$
$257$		0			0				1.00000i	$0.5\pi$
$258$		0			0
$259$	0.978200	−	0.433606i	0.978200	−	0.433606i
$260$		0			0
$261$	−0.193004	+	1.56484i	−0.193004	+	1.56484i
$262$		0			0
$263$	−0.660833	−	1.84691i	−0.660833	−	1.84691i	−0.514103	−	0.857729i	$-0.671875\pi$
$263$	−0.660833	−	1.84691i	−0.660833	−	1.84691i	−0.146730	−	0.989177i	$-0.546875\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	0.585326	+	1.77181i	0.585326	+	1.77181i
$269$		0			0		0.870087	−	0.492898i	$-0.164062\pi$
$269$		0			0		−0.870087	+	0.492898i	$0.835938\pi$
$270$		0			0
$271$		0			0		0.956940	−	0.290285i	$-0.0937500\pi$
$271$		0			0		−0.956940	+	0.290285i	$0.906250\pi$
$272$		0			0
$273$		0			0
$274$	−1.90278	+	0.378487i	−1.90278	+	0.378487i
$275$	−0.595699	−	1.80321i	−0.595699	−	1.80321i
$276$		0			0
$277$	−1.09681	+	1.04425i	−1.09681	+	1.04425i	−0.0980171	+	0.995185i	$0.531250\pi$
$277$	−1.09681	+	1.04425i	−1.09681	+	1.04425i	−0.998795	+	0.0490677i	$0.984375\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	0.480701	−	1.91906i	0.480701	−	1.91906i	0.0980171	−	0.995185i	$-0.468750\pi$
$281$	0.480701	−	1.91906i	0.480701	−	1.91906i	0.382683	−	0.923880i	$-0.375000\pi$
$282$		0			0
$283$		0			0		0.313682	−	0.949528i	$-0.398438\pi$
$283$		0			0		−0.313682	+	0.949528i	$0.601562\pi$
$284$	0.0476324	+	0.190159i	0.0476324	+	0.190159i
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−1.00000			−1.00000
$289$	0.831470	+	0.555570i	0.831470	+	0.555570i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0		−0.0735646	−	0.997290i	$-0.523438\pi$
$293$		0			0		0.0735646	+	0.997290i	$0.476562\pi$
$294$		0			0
$295$		0			0
$296$	−0.527399	−	0.930989i	−0.527399	−	0.930989i
$297$		0			0
$298$	−0.223818	−	1.81467i	−0.223818	−	1.81467i
$299$		0			0
$300$		0			0
$301$	0.223335	+	0.258908i	0.223335	+	0.258908i
$302$	0.207508	+	0.207508i	0.207508	+	0.207508i
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0		0.992480	−	0.122411i	$-0.0390625\pi$
$307$		0			0		−0.992480	+	0.122411i	$0.960938\pi$
$308$	1.77181	+	0.683461i	1.77181	+	0.683461i
$309$		0			0
$310$		0			0
$311$		0			0		0.903989	−	0.427555i	$-0.140625\pi$
$311$		0			0		−0.903989	+	0.427555i	$0.859375\pi$
$312$		0			0
$313$		0			0		0.740951	−	0.671559i	$-0.234375\pi$
$313$		0			0		−0.740951	+	0.671559i	$0.765625\pi$
$314$		0			0
$315$		0			0
$316$	−1.68250	+	0.334669i	−1.68250	+	0.334669i
$317$	1.19992	−	1.26032i	1.19992	−	1.26032i	0.242980	−	0.970031i	$-0.421875\pi$
$317$	1.19992	−	1.26032i	1.19992	−	1.26032i	0.956940	−	0.290285i	$-0.0937500\pi$
$318$		0			0
$319$	−2.76631	−	1.14584i	−2.76631	−	1.14584i
$320$		0			0
$321$		0			0
$322$	0.0923988	−	0.0330608i	0.0923988	−	0.0330608i
$323$		0			0
$324$	0.382683	+	0.923880i	0.382683	+	0.923880i
$325$		0			0
$326$	−0.399485	+	0.901225i	−0.399485	+	0.901225i
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	0.525572	−	0.0911954i	0.525572	−	0.0911954i	0.0980171	−	0.995185i	$-0.468750\pi$
$331$	0.525572	−	0.0911954i	0.525572	−	0.0911954i	0.427555	+	0.903989i	$0.359375\pi$
$332$		0			0
$333$	−0.658295	+	0.843527i	−0.658295	+	0.843527i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−0.850993	+	0.0838155i	−0.850993	+	0.0838155i	−0.514103	−	0.857729i	$-0.671875\pi$
$337$	−0.850993	+	0.0838155i	−0.850993	+	0.0838155i	−0.336890	+	0.941544i	$0.609375\pi$
$338$	−0.923880	−	0.382683i	−0.923880	−	0.382683i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	0.514103	−	0.857729i	0.514103	−	0.857729i
$344$	0.235770	−	0.247637i	0.235770	−	0.247637i
$345$		0			0
$346$		0			0
$347$	1.25902	−	1.45956i	1.25902	−	1.45956i	0.427555	−	0.903989i	$-0.359375\pi$
$347$	1.25902	−	1.45956i	1.25902	−	1.45956i	0.831470	−	0.555570i	$-0.187500\pi$
$348$		0			0
$349$		0			0		−0.844854	−	0.534998i	$-0.820312\pi$
$349$		0			0		0.844854	+	0.534998i	$0.179688\pi$
$350$	0.242980	+	0.970031i	0.242980	+	0.970031i
$351$		0			0
$352$	0.506503	−	1.83027i	0.506503	−	1.83027i
$353$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$353$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$	0.381274	+	0.215989i	0.381274	+	0.215989i
$359$	−0.733452	+	0.439614i	−0.733452	+	0.439614i	−0.831470	−	0.555570i	$-0.812500\pi$
$359$	−0.733452	+	0.439614i	−0.733452	+	0.439614i	0.0980171	+	0.995185i	$0.468750\pi$
$360$		0			0
$361$	0.803208	−	0.595699i	0.803208	−	0.595699i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		0.881921	−	0.471397i	$-0.156250\pi$
$367$		0			0		−0.881921	+	0.471397i	$0.843750\pi$
$368$	−0.0419583	−	0.0887133i	−0.0419583	−	0.0887133i
$369$		0			0
$370$		0			0
$371$	−0.0473045	−	0.0130909i	−0.0473045	−	0.0130909i
$372$		0			0
$373$	−0.197021	+	0.877373i	−0.197021	+	0.877373i	0.773010	+	0.634393i	$0.218750\pi$
$373$	−0.197021	+	0.877373i	−0.197021	+	0.877373i	−0.970031	+	0.242980i	$0.921875\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−1.57622	+	1.23009i	−1.57622	+	1.23009i	−0.773010	+	0.634393i	$0.781250\pi$
$379$	−1.57622	+	1.23009i	−1.57622	+	1.23009i	−0.803208	+	0.595699i	$0.796875\pi$
$380$		0			0
$381$		0			0
$382$	−0.612501	−	1.02190i	−0.612501	−	1.02190i
$383$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$383$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$384$		0			0
$385$		0			0
$386$	0.326351	−	1.30287i	0.326351	−	1.30287i
$387$	−0.319012	−	0.123057i	−0.319012	−	0.123057i
$388$		0			0
$389$	−1.21235	−	1.55348i	−1.21235	−	1.55348i	−0.740951	−	0.671559i	$-0.765625\pi$
$389$	−1.21235	−	1.55348i	−1.21235	−	1.55348i	−0.471397	−	0.881921i	$-0.656250\pi$
$390$		0			0
$391$		0			0
$392$	−0.903989	−	0.427555i	−0.903989	−	0.427555i
$393$		0			0
$394$	−1.07445	−	0.414461i	−1.07445	−	0.414461i
$395$		0			0
$396$	−1.88477	+	0.232465i	−1.88477	+	0.232465i
$397$		0			0		0.266713	−	0.963776i	$-0.414062\pi$
$397$		0			0		−0.266713	+	0.963776i	$0.585938\pi$
$398$		0			0
$399$		0			0
$400$	0.941544	−	0.336890i	0.941544	−	0.336890i
$401$	−0.852275	−	1.59449i	−0.852275	−	1.59449i	−0.803208	−	0.595699i	$-0.796875\pi$
$401$	−0.852275	−	1.59449i	−0.852275	−	1.59449i	−0.0490677	−	0.998795i	$-0.515625\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$	1.40834	+	0.708899i	1.40834	+	0.708899i
$407$	−1.21045	−	1.63210i	−1.21045	−	1.63210i
$408$		0			0
$409$		0			0		−0.514103	−	0.857729i	$-0.671875\pi$
$409$		0			0		0.514103	+	0.857729i	$0.328125\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$	−0.0659037	+	0.0727135i	−0.0659037	+	0.0727135i
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		0.534998	−	0.844854i	$-0.320312\pi$
$419$		0			0		−0.534998	+	0.844854i	$0.679688\pi$
$420$		0			0
$421$	−0.746454	−	0.643895i	−0.746454	−	0.643895i	0.195090	−	0.980785i	$-0.437500\pi$
$421$	−0.746454	−	0.643895i	−0.746454	−	0.643895i	−0.941544	+	0.336890i	$0.890625\pi$
$422$	−1.29652	+	1.50303i	−1.29652	+	1.50303i
$423$		0			0
$424$	−0.00839123	+	0.0483598i	−0.00839123	+	0.0483598i
$425$		0			0
$426$		0			0
$427$		0			0
$428$	0.386332	+	1.39602i	0.386332	+	1.39602i
$429$		0			0
$430$		0			0
$431$	0.162997	+	1.65493i	0.162997	+	1.65493i	0.634393	+	0.773010i	$0.281250\pi$
$431$	0.162997	+	1.65493i	0.162997	+	1.65493i	−0.471397	+	0.881921i	$0.656250\pi$
$432$		0			0
$433$		0			0		−0.995185	−	0.0980171i	$-0.968750\pi$
$433$		0			0		0.995185	+	0.0980171i	$0.0312500\pi$
$434$		0			0
$435$		0			0
$436$	0.0251535	−	0.144963i	0.0251535	−	0.144963i
$437$		0			0
$438$		0			0
$439$		0			0		−0.941544	−	0.336890i	$-0.890625\pi$
$439$		0			0		0.941544	+	0.336890i	$0.109375\pi$
$440$		0			0
$441$	−0.0490677	+	0.998795i	−0.0490677	+	0.998795i
$442$		0			0
$443$	1.62850	−	0.450666i	1.62850	−	0.450666i	0.671559	−	0.740951i	$-0.265625\pi$
$443$	1.62850	−	0.450666i	1.62850	−	0.450666i	0.956940	+	0.290285i	$0.0937500\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−0.336890	+	0.941544i	−0.336890	+	0.941544i
$449$	−0.485544	+	1.17221i	−0.485544	+	1.17221i	0.471397	+	0.881921i	$0.343750\pi$
$449$	−0.485544	+	1.17221i	−0.485544	+	1.17221i	−0.956940	+	0.290285i	$0.906250\pi$
$450$	−0.671559	−	0.740951i	−0.671559	−	0.740951i
$451$		0			0
$452$	−0.660833	−	0.131448i	−0.660833	−	0.131448i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	0.818289	+	1.73013i	0.818289	+	1.73013i	0.671559	+	0.740951i	$0.265625\pi$
$457$	0.818289	+	1.73013i	0.818289	+	1.73013i	0.146730	+	0.989177i	$0.453125\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		−0.122411	−	0.992480i	$-0.539062\pi$
$461$		0			0		0.122411	+	0.992480i	$0.460938\pi$
$462$		0			0
$463$	−1.39759	+	1.14697i	−1.39759	+	1.14697i	−0.427555	+	0.903989i	$0.640625\pi$
$463$	−1.39759	+	1.14697i	−1.39759	+	1.14697i	−0.970031	+	0.242980i	$0.921875\pi$
$464$	0.567444	−	1.47104i	0.567444	−	1.47104i
$465$		0			0
$466$			1.02821i			1.02821i
$467$		0			0		0.757209	−	0.653173i	$-0.226562\pi$
$467$		0			0		−0.757209	+	0.653173i	$0.773438\pi$
$468$		0			0
$469$	−0.671559	−	1.74095i	−0.671559	−	1.74095i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	0.386807	−	0.521549i	0.386807	−	0.521549i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	0.0478899	−	0.0107540i	0.0478899	−	0.0107540i
$478$	−1.89317	−	0.280825i	−1.89317	−	0.280825i
$479$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$479$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	0.382441	−	2.57820i	0.382441	−	2.57820i
$485$		0			0
$486$		0			0
$487$	−1.88192	−	0.471397i	−1.88192	−	0.471397i	−0.881921	−	0.471397i	$-0.843750\pi$
$487$	−1.88192	−	0.471397i	−1.88192	−	0.471397i	−1.00000			$\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−1.26077	−	1.32423i	−1.26077	−	1.32423i	−0.923880	−	0.382683i	$-0.875000\pi$
$491$	−1.26077	−	1.32423i	−1.26077	−	1.32423i	−0.336890	−	0.941544i	$-0.609375\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−0.0569057	−	0.187593i	−0.0569057	−	0.187593i
$498$		0			0
$499$	−0.399485	−	0.705190i	−0.399485	−	0.705190i	0.595699	−	0.803208i	$-0.296875\pi$
$499$	−0.399485	−	0.705190i	−0.399485	−	0.705190i	−0.995185	+	0.0980171i	$0.968750\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		0.671559	−	0.740951i	$-0.265625\pi$
$503$		0			0		−0.671559	+	0.740951i	$0.734375\pi$
$504$	0.998795	−	0.0490677i	0.998795	−	0.0490677i
$505$		0			0
$506$	−0.0997046	−	0.157451i	−0.0997046	−	0.157451i
$507$		0			0
$508$	1.55557	+	0.831470i	1.55557	+	0.831470i
$509$		0			0		−0.405241	−	0.914210i	$-0.632812\pi$
$509$		0			0		0.405241	+	0.914210i	$0.367188\pi$
$510$		0			0
$511$		0			0
$512$	0.970031	+	0.242980i	0.970031	+	0.242980i
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$	0.572445	+	0.903989i	0.572445	+	0.903989i
$519$		0			0
$520$		0			0
$521$		0			0		0.671559	−	0.740951i	$-0.265625\pi$
$521$		0			0		−0.671559	+	0.740951i	$0.734375\pi$
$522$	−1.57622	+	0.0386940i	−1.57622	+	0.0386940i
$523$		0			0		−0.534998	−	0.844854i	$-0.679688\pi$
$523$		0			0		0.534998	+	0.844854i	$0.320312\pi$
$524$		0			0
$525$		0			0
$526$	1.72995	−	0.924678i	1.72995	−	0.924678i
$527$		0			0
$528$		0			0
$529$	0.947724	+	0.287489i	0.947724	+	0.287489i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$	−1.66674	+	0.838968i	−1.66674	+	0.838968i
$537$		0			0
$538$		0			0
$539$	−1.80321	−	0.595699i	−1.80321	−	0.595699i
$540$		0			0
$541$	1.12752	−	0.794086i	1.12752	−	0.794086i	0.146730	−	0.989177i	$-0.453125\pi$
$541$	1.12752	−	0.794086i	1.12752	−	0.794086i	0.980785	+	0.195090i	$0.0625000\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	0.238873	−	0.0536407i	0.238873	−	0.0536407i	−0.0980171	−	0.995185i	$-0.531250\pi$
$547$	0.238873	−	0.0536407i	0.238873	−	0.0536407i	0.336890	+	0.941544i	$0.390625\pi$
$548$	−0.653587	−	1.82665i	−0.653587	−	1.82665i
$549$		0			0
$550$	1.69628	−	0.853837i	1.69628	−	0.853837i
$551$		0			0
$552$		0			0
$553$	1.66405	−	0.416822i	1.66405	−	0.416822i
$554$	−1.19389	−	0.931718i	−1.19389	−	0.931718i
$555$		0			0
$556$		0			0
$557$	−0.989177	+	0.853270i	−0.989177	+	0.853270i	−0.989177	−	0.146730i	$-0.953125\pi$
$557$	−0.989177	+	0.853270i	−0.989177	+	0.853270i			1.00000i	$0.5\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	1.96883	+	0.193913i	1.96883	+	0.193913i
$563$		0			0		−0.122411	−	0.992480i	$-0.539062\pi$
$563$		0			0		0.122411	+	0.992480i	$0.460938\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−0.427555	−	0.903989i	−0.427555	−	0.903989i
$568$	−0.181112	+	0.0750191i	−0.181112	+	0.0750191i
$569$	1.18452	+	1.30692i	1.18452	+	1.30692i	0.941544	+	0.336890i	$0.109375\pi$
$569$	1.18452	+	1.30692i	1.18452	+	1.30692i	0.242980	+	0.970031i	$0.421875\pi$
$570$		0			0
$571$	1.07061	+	0.606493i	1.07061	+	0.606493i	0.923880	−	0.382683i	$-0.125000\pi$
$571$	1.07061	+	0.606493i	1.07061	+	0.606493i	0.146730	+	0.989177i	$0.453125\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	0.0375548	−	0.0906652i	0.0375548	−	0.0906652i
$576$	−0.146730	−	0.989177i	−0.146730	−	0.989177i
$577$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$577$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$578$	−0.427555	+	0.903989i	−0.427555	+	0.903989i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−0.00457362	+	0.0930981i	−0.00457362	+	0.0930981i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		−0.170962	−	0.985278i	$-0.554688\pi$
$587$		0			0		0.170962	+	0.985278i	$0.445312\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	0.843527	−	0.658295i	0.843527	−	0.658295i
$593$		0			0		−0.0980171	−	0.995185i	$-0.531250\pi$
$593$		0			0		0.0980171	+	0.995185i	$0.468750\pi$
$594$		0			0
$595$		0			0
$596$	1.76219	−	0.487663i	1.76219	−	0.487663i
$597$		0			0
$598$		0			0
$599$	0.334669	+	0.200593i	0.334669	+	0.200593i	0.671559	−	0.740951i	$-0.265625\pi$
$599$	0.334669	+	0.200593i	0.334669	+	0.200593i	−0.336890	+	0.941544i	$0.609375\pi$
$600$		0			0
$601$		0			0		−0.146730	−	0.989177i	$-0.546875\pi$
$601$		0			0		0.146730	+	0.989177i	$0.453125\pi$
$602$	−0.223335	+	0.258908i	−0.223335	+	0.258908i
$603$	1.41294	+	1.21881i	1.41294	+	1.21881i
$604$	−0.174814	+	0.235710i	−0.174814	+	0.235710i
$605$		0			0
$606$		0			0
$607$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$607$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−1.72174	+	0.866649i	−1.72174	+	0.866649i	−0.740951	+	0.671559i	$0.765625\pi$
$613$	−1.72174	+	0.866649i	−1.72174	+	0.866649i	−0.980785	+	0.195090i	$0.937500\pi$
$614$		0			0
$615$		0			0
$616$	−0.416086	+	1.85291i	−0.416086	+	1.85291i
$617$	−0.345845	−	0.466318i	−0.345845	−	0.466318i	0.595699	−	0.803208i	$-0.296875\pi$
$617$	−0.345845	−	0.466318i	−0.345845	−	0.466318i	−0.941544	+	0.336890i	$0.890625\pi$
$618$		0			0
$619$		0			0		0.0245412	−	0.999699i	$-0.492188\pi$
$619$		0			0		−0.0245412	+	0.999699i	$0.507812\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	0.881921	+	0.471397i	0.881921	+	0.471397i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	1.26726	−	0.0622564i	1.26726	−	0.0622564i	0.595699	−	0.803208i	$-0.296875\pi$
$631$	1.26726	−	0.0622564i	1.26726	−	0.0622564i	0.671559	+	0.740951i	$0.265625\pi$
$632$	−0.577920	−	1.61518i	−0.577920	−	1.61518i
$633$		0			0
$634$	1.42274	+	1.00201i	1.42274	+	1.00201i
$635$		0			0
$636$		0			0
$637$		0			0
$638$	0.727538	−	2.90450i	0.727538	−	2.90450i
$639$	0.138617	+	0.138617i	0.138617	+	0.138617i
$640$		0			0
$641$	1.04786	−	1.04786i	1.04786	−	1.04786i	0.0490677	−	0.998795i	$-0.484375\pi$
$641$	1.04786	−	1.04786i	1.04786	−	1.04786i	0.998795	−	0.0490677i	$-0.0156250\pi$
$642$		0			0
$643$		0			0		0.359895	−	0.932993i	$-0.382812\pi$
$643$		0			0		−0.359895	+	0.932993i	$0.617188\pi$
$644$	0.0462607	+	0.0865477i	0.0462607	+	0.0865477i
$645$		0			0
$646$		0			0
$647$		0			0		−0.903989	−	0.427555i	$-0.859375\pi$
$647$		0			0		0.903989	+	0.427555i	$0.140625\pi$
$648$	−0.857729	+	0.514103i	−0.857729	+	0.514103i
$649$		0			0
$650$		0			0
$651$		0			0
$652$	−0.950087	−	0.262924i	−0.950087	−	0.262924i
$653$	1.72174	+	0.476469i	1.72174	+	0.476469i	0.980785	−	0.195090i	$-0.0625000\pi$
$653$	1.72174	+	0.476469i	1.72174	+	0.476469i	0.740951	+	0.671559i	$0.234375\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	1.78161	+	0.896786i	1.78161	+	0.896786i	0.923880	+	0.382683i	$0.125000\pi$
$659$	1.78161	+	0.896786i	1.78161	+	0.896786i	0.857729	+	0.514103i	$0.171875\pi$
$660$		0			0
$661$		0			0		−0.999699	−	0.0245412i	$-0.992188\pi$
$661$		0			0		0.999699	+	0.0245412i	$0.00781250\pi$
$662$	0.167326	+	0.506503i	0.167326	+	0.506503i
$663$		0			0
$664$		0			0
$665$		0			0
$666$	−0.930989	−	0.527399i	−0.930989	−	0.527399i
$667$	−0.0695680	−	0.138208i	−0.0695680	−	0.138208i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	0.301614	+	1.51631i	0.301614	+	1.51631i	0.773010	+	0.634393i	$0.218750\pi$
$673$	0.301614	+	1.51631i	0.301614	+	1.51631i	−0.471397	+	0.881921i	$0.656250\pi$
$674$	−0.207775	−	0.829484i	−0.207775	−	0.829484i
$675$		0			0
$676$	0.242980	−	0.970031i	0.242980	−	0.970031i
$677$		0			0		0.653173	−	0.757209i	$-0.273438\pi$
$677$		0			0		−0.653173	+	0.757209i	$0.726562\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−0.790790	+	1.78399i	−0.790790	+	1.78399i	−0.195090	+	0.980785i	$0.562500\pi$
$683$	−0.790790	+	1.78399i	−0.790790	+	1.78399i	−0.595699	+	0.803208i	$0.703125\pi$
$684$		0			0
$685$		0			0
$686$	0.923880	+	0.382683i	0.923880	+	0.382683i
$687$		0			0
$688$	0.279552	+	0.196883i	0.279552	+	0.196883i
$689$		0			0
$690$		0			0
$691$		0			0		0.615232	−	0.788346i	$-0.289062\pi$
$691$		0			0		−0.615232	+	0.788346i	$0.710938\pi$
$692$		0			0
$693$	1.87110	−	0.324666i	1.87110	−	0.324666i
$694$	1.62850	+	1.03124i	1.62850	+	1.03124i
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−0.923880	+	0.382683i	−0.923880	+	0.382683i
$701$	−1.63468	+	0.0401291i	−1.63468	+	0.0401291i	−0.831470	−	0.555570i	$-0.812500\pi$
$701$	−1.63468	+	0.0401291i	−1.63468	+	0.0401291i	−0.803208	+	0.595699i	$0.796875\pi$
$702$		0			0
$703$		0			0
$704$	1.88477	+	0.232465i	1.88477	+	0.232465i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	0.880537	−	1.55437i	0.880537	−	1.55437i	0.0490677	−	0.998795i	$-0.484375\pi$
$709$	0.880537	−	1.55437i	0.880537	−	1.55437i	0.831470	−	0.555570i	$-0.187500\pi$
$710$		0			0
$711$	−1.27107	+	1.15203i	−1.27107	+	1.15203i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$	−0.157707	+	0.408840i	−0.157707	+	0.408840i
$717$		0			0
$718$	−0.542476	−	0.661009i	−0.542476	−	0.661009i
$719$		0			0		−0.634393	−	0.773010i	$-0.718750\pi$
$719$		0			0		0.634393	+	0.773010i	$0.281250\pi$
$720$		0			0
$721$		0			0
$722$	0.707107	+	0.707107i	0.707107	+	0.707107i
$723$		0			0
$724$		0			0
$725$	1.47104	−	0.567444i	1.47104	−	0.567444i
$726$		0			0
$727$		0			0		−0.242980	−	0.970031i	$-0.578125\pi$
$727$		0			0		0.242980	+	0.970031i	$0.421875\pi$
$728$		0			0
$729$	0.803208	+	0.595699i	0.803208	+	0.595699i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		−0.219101	−	0.975702i	$-0.570312\pi$
$733$		0			0		0.219101	+	0.975702i	$0.429688\pi$
$734$		0			0
$735$		0			0
$736$	0.0815966	−	0.0545211i	0.0815966	−	0.0545211i
$737$	−2.94641	+	1.96873i	−2.94641	+	1.96873i
$738$		0			0
$739$	0.872014	+	1.23816i	0.872014	+	1.23816i	0.970031	+	0.242980i	$0.0781250\pi$
$739$	0.872014	+	1.23816i	0.872014	+	1.23816i	−0.0980171	+	0.995185i	$0.531250\pi$
$740$		0			0
$741$		0			0
$742$	0.00600822	−	0.0487133i	0.00600822	−	0.0487133i
$743$	−0.390327	+	1.55827i	−0.390327	+	1.55827i	0.382683	+	0.923880i	$0.375000\pi$
$743$	−0.390327	+	1.55827i	−0.390327	+	1.55827i	−0.773010	+	0.634393i	$0.781250\pi$
$744$		0			0
$745$		0			0
$746$	−0.896786	−	0.0661509i	−0.896786	−	0.0661509i
$747$		0			0
$748$		0			0
$749$	−0.454366	−	1.37539i	−0.454366	−	1.37539i
$750$		0			0
$751$	0.368309	−	1.21415i	0.368309	−	1.21415i	−0.555570	−	0.831470i	$-0.687500\pi$
$751$	0.368309	−	1.21415i	0.368309	−	1.21415i	0.923880	−	0.382683i	$-0.125000\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−1.22377	+	0.774941i	−1.22377	+	0.774941i	−0.980785	−	0.195090i	$-0.937500\pi$
$757$	−1.22377	+	0.774941i	−1.22377	+	0.774941i	−0.242980	+	0.970031i	$0.578125\pi$
$758$	−1.44806	−	1.37867i	−1.44806	−	1.37867i
$759$		0			0
$760$		0			0
$761$		0			0		−0.336890	−	0.941544i	$-0.609375\pi$
$761$		0			0		0.336890	+	0.941544i	$0.390625\pi$
$762$		0			0
$763$	−0.0180102	+	0.146023i	−0.0180102	+	0.146023i
$764$	0.920964	−	0.755815i	0.920964	−	0.755815i
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$		0			0			−	1.00000i	$-0.5\pi$
$769$		0			0				1.00000i	$0.5\pi$
$770$		0			0
$771$		0			0
$772$	1.33665	+	0.131649i	1.33665	+	0.131649i
$773$		0			0		0.122411	−	0.992480i	$-0.460938\pi$
$773$		0			0		−0.122411	+	0.992480i	$0.539062\pi$
$774$	0.0749159	−	0.333616i	0.0749159	−	0.333616i
$775$		0			0
$776$		0			0
$777$		0			0
$778$	1.35878	−	1.42717i	1.35878	−	1.42717i
$779$		0			0
$780$		0			0
$781$	−0.323916	+	0.183496i	−0.323916	+	0.183496i
$782$		0			0
$783$		0			0
$784$	0.290285	−	0.956940i	0.290285	−	0.956940i
$785$		0			0
$786$		0			0
$787$		0			0		−0.313682	−	0.949528i	$-0.601562\pi$
$787$		0			0		0.313682	+	0.949528i	$0.398438\pi$
$788$	0.252321	−	1.12363i	0.252321	−	1.12363i
$789$		0			0
$790$		0			0
$791$	0.666487	+	0.0988640i	0.666487	+	0.0988640i
$792$	−0.506503	−	1.83027i	−0.506503	−	1.83027i
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		−0.575808	−	0.817585i	$-0.695312\pi$
$797$		0			0		0.575808	+	0.817585i	$0.304688\pi$
$798$		0			0
$799$		0			0
$800$	0.471397	+	0.881921i	0.471397	+	0.881921i
$801$		0			0
$802$	1.45218	−	1.07701i	1.45218	−	1.07701i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−0.141067	−	0.563170i	−0.141067	−	0.563170i	−0.998795	−	0.0490677i	$-0.984375\pi$
$809$	−0.141067	−	0.563170i	−0.141067	−	0.563170i	0.857729	−	0.514103i	$-0.171875\pi$
$810$		0			0
$811$		0			0		0.932993	−	0.359895i	$-0.117188\pi$
$811$		0			0		−0.932993	+	0.359895i	$0.882812\pi$
$812$	−0.494580	+	1.49711i	−0.494580	+	1.49711i
$813$		0			0
$814$	1.43683	−	1.43683i	1.43683	−	1.43683i
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−1.10990	+	1.57594i	−1.10990	+	1.57594i	−0.336890	+	0.941544i	$0.609375\pi$
$821$	−1.10990	+	1.57594i	−1.10990	+	1.57594i	−0.773010	+	0.634393i	$0.781250\pi$
$822$		0			0
$823$	−0.352719	+	0.166824i	−0.352719	+	0.166824i	−0.595699	−	0.803208i	$-0.703125\pi$
$823$	−0.352719	+	0.166824i	−0.352719	+	0.166824i	0.242980	+	0.970031i	$0.421875\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−0.805972	+	1.42274i	−0.805972	+	1.42274i	0.0980171	+	0.995185i	$0.468750\pi$
$827$	−0.805972	+	1.42274i	−0.805972	+	1.42274i	−0.903989	+	0.427555i	$0.859375\pi$
$828$	−0.0815966	−	0.0545211i	−0.0815966	−	0.0545211i
$829$		0			0		0.689541	−	0.724247i	$-0.257812\pi$
$829$		0			0		−0.689541	+	0.724247i	$0.742188\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		−0.998795	−	0.0490677i	$-0.984375\pi$
$839$		0			0		0.998795	+	0.0490677i	$0.0156250\pi$
$840$		0			0
$841$	0.500605	−	1.39910i	0.500605	−	1.39910i
$842$	0.527399	−	0.832854i	0.527399	−	0.832854i
$843$		0			0
$844$	−1.67700	−	1.06195i	−1.67700	−	1.06195i
$845$		0			0
$846$		0			0
$847$	−0.255473	+	2.59386i	−0.255473	+	2.59386i
$848$	−0.0490677	−	0.00120454i	−0.0490677	−	0.00120454i
$849$		0			0
$850$		0			0
$851$	0.00772460	−	0.104720i	0.00772460	−	0.104720i
$852$		0			0
$853$		0			0		0.405241	−	0.914210i	$-0.367188\pi$
$853$		0			0		−0.405241	+	0.914210i	$0.632812\pi$
$854$		0			0
$855$		0			0
$856$	−1.32423	+	0.586990i	−1.32423	+	0.586990i
$857$		0			0		0.989177	−	0.146730i	$-0.0468750\pi$
$857$		0			0		−0.989177	+	0.146730i	$0.953125\pi$
$858$		0			0
$859$		0			0		0.653173	−	0.757209i	$-0.273438\pi$
$859$		0			0		−0.653173	+	0.757209i	$0.726562\pi$
$860$		0			0
$861$		0			0
$862$	−1.61310	+	0.404061i	−1.61310	+	0.404061i
$863$	0.385958	+	1.94034i	0.385958	+	1.94034i	0.336890	+	0.941544i	$0.390625\pi$
$863$	0.385958	+	1.94034i	0.385958	+	1.94034i	0.0490677	+	0.998795i	$0.484375\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−1.46472	−	2.90990i	−1.46472	−	2.90990i
$870$		0			0
$871$		0			0
$872$	0.147085	+	0.00361073i	0.147085	+	0.00361073i
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	1.46057	+	0.735191i	1.46057	+	0.735191i	0.989177	−	0.146730i	$-0.0468750\pi$
$877$	1.46057	+	0.735191i	1.46057	+	0.735191i	0.471397	+	0.881921i	$0.343750\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		0.471397	−	0.881921i	$-0.343750\pi$
$881$		0			0		−0.471397	+	0.881921i	$0.656250\pi$
$882$	−0.995185	+	0.0980171i	−0.995185	+	0.0980171i
$883$	−1.67714	−	0.464127i	−1.67714	−	0.464127i	−0.707107	−	0.707107i	$-0.750000\pi$
$883$	−1.67714	−	0.464127i	−1.67714	−	0.464127i	−0.970031	+	0.242980i	$0.921875\pi$
$884$		0			0
$885$		0			0
$886$	0.684739	+	1.54475i	0.684739	+	1.54475i
$887$		0			0		−0.0490677	−	0.998795i	$-0.515625\pi$
$887$		0			0		0.0490677	+	0.998795i	$0.484375\pi$
$888$		0			0
$889$	−1.59449	−	0.754140i	−1.59449	−	0.754140i
$890$		0			0
$891$	−1.49711	+	1.16836i	−1.49711	+	1.16836i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$	−0.980785	−	0.195090i	−0.980785	−	0.195090i
$897$		0			0
$898$	−1.23076	−	0.308290i	−1.23076	−	0.308290i
$899$		0			0
$900$	0.634393	−	0.773010i	0.634393	−	0.773010i
$901$		0			0
$902$		0			0
$903$		0			0
$904$	0.0330608	−	0.672968i	0.0330608	−	0.672968i
$905$		0			0
$906$		0			0
$907$	0.143554	+	0.0322362i	0.143554	+	0.0322362i	0.290285	−	0.956940i	$-0.406250\pi$
$907$	0.143554	+	0.0322362i	0.143554	+	0.0322362i	−0.146730	+	0.989177i	$0.546875\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−0.512016	−	0.273678i	−0.512016	−	0.273678i	0.195090	−	0.980785i	$-0.437500\pi$
$911$	−0.512016	−	0.273678i	−0.512016	−	0.273678i	−0.707107	+	0.707107i	$0.750000\pi$
$912$		0			0
$913$		0			0
$914$	−1.59133	+	1.06330i	−1.59133	+	1.06330i
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−0.561621	−	0.757259i	−0.561621	−	0.757259i	0.427555	−	0.903989i	$-0.359375\pi$
$919$	−0.561621	−	0.757259i	−0.561621	−	0.757259i	−0.989177	+	0.146730i	$0.953125\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	1.05424	+	0.182928i	1.05424	+	0.182928i
$926$	−1.33962	−	1.21416i	−1.33962	−	1.21416i
$927$		0			0
$928$	1.53838	+	0.345455i	1.53838	+	0.345455i
$929$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$929$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$930$		0			0
$931$		0			0
$932$	−1.01708	+	0.150869i	−1.01708	+	0.150869i
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		−0.857729	−	0.514103i	$-0.828125\pi$
$937$		0			0		0.857729	+	0.514103i	$0.171875\pi$
$938$	1.62357	−	0.919741i	1.62357	−	0.919741i
$939$		0			0
$940$		0			0
$941$		0			0		−0.997290	−	0.0735646i	$-0.976562\pi$
$941$		0			0		0.997290	+	0.0735646i	$0.0234375\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$	0.572660	+	0.306093i	0.572660	+	0.306093i
$947$	−1.19389	−	0.931718i	−1.19389	−	0.931718i	−0.195090	−	0.980785i	$-0.562500\pi$
$947$	−1.19389	−	0.931718i	−1.19389	−	0.931718i	−0.998795	+	0.0490677i	$0.984375\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	0.0584592	−	1.18996i	0.0584592	−	1.18996i	−0.773010	−	0.634393i	$-0.781250\pi$
$953$	0.0584592	−	1.18996i	0.0584592	−	1.18996i	0.831470	−	0.555570i	$-0.187500\pi$
$954$	0.0176645	+	0.0457936i	0.0176645	+	0.0457936i
$955$		0			0
$956$		−	1.91388i		−	1.91388i
$957$		0			0
$958$		0			0
$959$	0.742430	+	1.79238i	0.742430	+	1.79238i
$960$		0			0
$961$	−0.382683	+	0.923880i	−0.382683	+	0.923880i
$962$		0			0
$963$	1.04907	+	0.998795i	1.04907	+	0.998795i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	1.32858	+	1.46586i	1.32858	+	1.46586i	0.773010	+	0.634393i	$0.218750\pi$
$967$	1.32858	+	1.46586i	1.32858	+	1.46586i	0.555570	+	0.831470i	$0.312500\pi$
$968$	2.60642			2.60642
$969$		0			0
$970$		0			0
$971$		0			0		−0.817585	−	0.575808i	$-0.804688\pi$
$971$		0			0		0.817585	+	0.575808i	$0.195312\pi$
$972$		0			0
$973$		0			0
$974$	0.190159	−	1.93072i	0.190159	−	1.93072i
$975$		0			0
$976$		0			0
$977$	−1.26268	+	1.53858i	−1.26268	+	1.53858i	−0.555570	+	0.831470i	$0.687500\pi$
$977$	−1.26268	+	1.53858i	−1.26268	+	1.53858i	−0.707107	+	0.707107i	$0.750000\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−0.0529510	−	0.137270i	−0.0529510	−	0.137270i
$982$	1.12490	−	1.44143i	1.12490	−	1.44143i
$983$		0			0		0.970031	−	0.242980i	$-0.0781250\pi$
$983$		0			0		−0.970031	+	0.242980i	$0.921875\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	0.0327395	−	0.00735190i	0.0327395	−	0.00735190i
$990$		0			0
$991$	1.00446	−	1.50328i	1.00446	−	1.50328i	0.146730	−	0.989177i	$-0.453125\pi$
$991$	1.00446	−	1.50328i	1.00446	−	1.50328i	0.857729	−	0.514103i	$-0.171875\pi$
$992$		0			0
$993$		0			0
$994$	0.177213	−	0.0838155i	0.177213	−	0.0838155i
$995$		0			0
$996$		0			0
$997$		0			0		−0.949528	−	0.313682i	$-0.898438\pi$
$997$		0			0		0.949528	+	0.313682i	$0.101562\pi$
$998$	0.638941	−	0.498635i	0.638941	−	0.498635i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3584.1.cd.a.1077.1	✓	64
7.6	odd	2	CM	3584.1.cd.a.1077.1	✓	64
512.29	even	128	inner	3584.1.cd.a.2589.1	yes	64
3584.2589	odd	128	inner	3584.1.cd.a.2589.1	yes	64

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3584.1.cd.a.1077.1	✓	64	1.1	even	1	trivial
3584.1.cd.a.1077.1	✓	64	7.6	odd	2	CM
3584.1.cd.a.2589.1	yes	64	512.29	even	128	inner
3584.1.cd.a.2589.1	yes	64	3584.2589	odd	128	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:	\( N \)	\(=\)	\( 3584 = 2^{9} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3584.cd (of order \(128\), degree \(64\), minimal)

\(f(q)\)	\(=\)	\(q+(0.146730 + 0.989177i) q^{2} +(-0.956940 + 0.290285i) q^{4} +(0.941544 - 0.336890i) q^{7} +(-0.427555 - 0.903989i) q^{8} +(-0.671559 + 0.740951i) q^{9} +O(q^{10})\)	\(q+(0.146730 + 0.989177i) q^{2} +(-0.956940 + 0.290285i) q^{4} +(0.941544 - 0.336890i) q^{7} +(-0.427555 - 0.903989i) q^{8} +(-0.671559 + 0.740951i) q^{9} +(-1.01599 - 1.60442i) q^{11} +(0.471397 + 0.881921i) q^{14} +(0.831470 - 0.555570i) q^{16} +(-0.831470 - 0.555570i) q^{18} +(1.43798 - 1.24041i) q^{22} +(0.0143994 - 0.0970732i) q^{23} +(0.970031 + 0.242980i) q^{25} +(-0.803208 + 0.595699i) q^{28} +(1.28908 - 0.907873i) q^{29} +(0.671559 + 0.740951i) q^{32} +(0.427555 - 0.903989i) q^{36} +(1.06710 - 0.0787137i) q^{37} +(0.123057 + 0.319012i) q^{43} +(1.43798 + 1.24041i) q^{44} +0.0981353 q^{46} +(0.773010 - 0.634393i) q^{49} +(-0.0980171 + 0.995185i) q^{50} +(-0.0401291 - 0.0282621i) q^{53} +(-0.707107 - 0.707107i) q^{56} +(1.08719 + 1.14192i) q^{58} +(-0.382683 + 0.923880i) q^{63} +(-0.634393 + 0.773010i) q^{64} +(-0.0457936 - 1.86542i) q^{67} +(0.00961895 - 0.195798i) q^{71} +(0.956940 + 0.290285i) q^{72} +(0.234437 + 1.04400i) q^{74} +(-1.49711 - 1.16836i) q^{77} +(1.70720 + 0.168144i) q^{79} +(-0.0980171 - 0.995185i) q^{81} +(-0.297503 + 0.168534i) q^{86} +(-1.01599 + 1.60442i) q^{88} +(0.0143994 + 0.0970732i) q^{92} +(0.740951 + 0.671559i) q^{98} +(1.87110 + 0.324666i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q+O(q^{10}) \) 64 * q	\( 64 q+O(q^{100}) \) 64 * q

Introduction

L-functions

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