LMFDB - Embedded newform 3584.1.cd.a.125.1

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			125.1
Root			$0.671559 + 0.740951i$ of defining polynomial
Character	$\chi$	$=$	3584.125
Dual form			3584.1.cd.a.1749.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.514103 + 0.857729i) q^{2} +(-0.471397 - 0.881921i) q^{4} +(-0.671559 + 0.740951i) q^{7} +(0.998795 + 0.0490677i) q^{8} +(0.427555 - 0.903989i) q^{9} +O(q^{10})$	\(q+(-0.514103 + 0.857729i) q^{2} +(-0.471397 - 0.881921i) q^{4} +(-0.671559 + 0.740951i) q^{7} +(0.998795 + 0.0490677i) q^{8} +(0.427555 - 0.903989i) q^{9} +(0.244748 - 0.00600822i) q^{11} +(-0.290285 - 0.956940i) q^{14} +(-0.555570 + 0.831470i) q^{16} +(0.555570 + 0.831470i) q^{18} +(-0.120672 + 0.213016i) q^{22} +(-0.968101 - 1.61518i) q^{23} +(0.989177 + 0.146730i) q^{25} +(0.970031 + 0.242980i) q^{28} +(1.21881 - 0.470147i) q^{29} +(-0.427555 - 0.903989i) q^{32} +(-0.998795 + 0.0490677i) q^{36} +(-0.533265 + 1.92697i) q^{37} +(-0.400609 - 1.78399i) q^{43} +(-0.120672 - 0.213016i) q^{44} +1.88309 q^{46} +(-0.0980171 - 0.995185i) q^{49} +(-0.634393 + 0.773010i) q^{50} +(1.07445 + 0.414461i) q^{53} +(-0.707107 + 0.707107i) q^{56} +(-0.223335 + 1.28711i) q^{58} +(0.382683 + 0.923880i) q^{63} +(0.995185 + 0.0980171i) q^{64} +(1.12363 - 1.59544i) q^{67} +(1.19462 + 0.427441i) q^{71} +(0.471397 - 0.881921i) q^{72} +(-1.37867 - 1.44806i) q^{74} +(-0.159911 + 0.185381i) q^{77} +(-0.920964 - 0.755815i) q^{79} +(-0.634393 - 0.773010i) q^{81} +(1.73614 + 0.573542i) q^{86} +(0.244748 + 0.00600822i) q^{88} +(-0.968101 + 1.61518i) q^{92} +(0.903989 + 0.427555i) q^{98} +(0.0992117 - 0.223818i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/3584\mathbb{Z}\right)^\times$.

$n$	$1023$	$1025$	$1541$
$\chi(n)$	$1$	$-1$	$e\left(\frac{3}{128}\right)$

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 $

$2$	−0.514103	+	0.857729i	−0.514103	+	0.857729i
$2$	−0.514103	+	0.857729i	−0.514103	+	0.857729i
$3$		0			0		0.844854	−	0.534998i	$-0.179688\pi$
$3$		0			0		−0.844854	+	0.534998i	$0.820312\pi$
$4$	−0.471397	−	0.881921i	−0.471397	−	0.881921i
$5$		0			0		−0.997290	−	0.0735646i	$-0.976562\pi$
$5$		0			0		0.997290	+	0.0735646i	$0.0234375\pi$
$6$		0			0
$7$	−0.671559	+	0.740951i	−0.671559	+	0.740951i
$7$	−0.671559	+	0.740951i	−0.671559	+	0.740951i
$8$	0.998795	+	0.0490677i	0.998795	+	0.0490677i
$9$	0.427555	−	0.903989i	0.427555	−	0.903989i
$10$		0			0
$11$	0.244748	−	0.00600822i	0.244748	−	0.00600822i	0.0980171	−	0.995185i	$-0.468750\pi$
$11$	0.244748	−	0.00600822i	0.244748	−	0.00600822i	0.146730	+	0.989177i	$0.453125\pi$
$12$		0			0
$13$		0			0		0.313682	−	0.949528i	$-0.398438\pi$
$13$		0			0		−0.313682	+	0.949528i	$0.601562\pi$
$14$	−0.290285	−	0.956940i	−0.290285	−	0.956940i
$15$		0			0
$16$	−0.555570	+	0.831470i	−0.555570	+	0.831470i
$17$		0			0		0.471397	−	0.881921i	$-0.343750\pi$
$17$		0			0		−0.471397	+	0.881921i	$0.656250\pi$
$18$	0.555570	+	0.831470i	0.555570	+	0.831470i
$19$		0			0		0.122411	−	0.992480i	$-0.460938\pi$
$19$		0			0		−0.122411	+	0.992480i	$0.539062\pi$
$20$		0			0
$21$		0			0
$22$	−0.120672	+	0.213016i	−0.120672	+	0.213016i
$23$	−0.968101	−	1.61518i	−0.968101	−	1.61518i	−0.773010	−	0.634393i	$-0.781250\pi$
$23$	−0.968101	−	1.61518i	−0.968101	−	1.61518i	−0.195090	−	0.980785i	$-0.562500\pi$
$24$		0			0
$25$	0.989177	+	0.146730i	0.989177	+	0.146730i
$26$		0			0
$27$		0			0
$28$	0.970031	+	0.242980i	0.970031	+	0.242980i
$29$	1.21881	−	0.470147i	1.21881	−	0.470147i	0.336890	−	0.941544i	$-0.390625\pi$
$29$	1.21881	−	0.470147i	1.21881	−	0.470147i	0.881921	+	0.471397i	$0.156250\pi$
$30$		0			0
$31$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$31$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$32$	−0.427555	−	0.903989i	−0.427555	−	0.903989i
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−0.998795	+	0.0490677i	−0.998795	+	0.0490677i
$37$	−0.533265	+	1.92697i	−0.533265	+	1.92697i	−0.242980	+	0.970031i	$0.578125\pi$
$37$	−0.533265	+	1.92697i	−0.533265	+	1.92697i	−0.290285	+	0.956940i	$0.593750\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		0.989177	−	0.146730i	$-0.0468750\pi$
$41$		0			0		−0.989177	+	0.146730i	$0.953125\pi$
$42$		0			0
$43$	−0.400609	−	1.78399i	−0.400609	−	1.78399i	−0.595699	−	0.803208i	$-0.703125\pi$
$43$	−0.400609	−	1.78399i	−0.400609	−	1.78399i	0.195090	−	0.980785i	$-0.437500\pi$
$44$	−0.120672	−	0.213016i	−0.120672	−	0.213016i
$45$		0			0
$46$	1.88309			1.88309
$47$		0			0		−0.995185	−	0.0980171i	$-0.968750\pi$
$47$		0			0		0.995185	+	0.0980171i	$0.0312500\pi$
$48$		0			0
$49$	−0.0980171	−	0.995185i	−0.0980171	−	0.995185i
$50$	−0.634393	+	0.773010i	−0.634393	+	0.773010i
$51$		0			0
$52$		0			0
$53$	1.07445	+	0.414461i	1.07445	+	0.414461i	0.831470	−	0.555570i	$-0.187500\pi$
$53$	1.07445	+	0.414461i	1.07445	+	0.414461i	0.242980	+	0.970031i	$0.421875\pi$
$54$		0			0
$55$		0			0
$56$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$57$		0			0
$58$	−0.223335	+	1.28711i	−0.223335	+	1.28711i
$59$		0			0		0.949528	−	0.313682i	$-0.101562\pi$
$59$		0			0		−0.949528	+	0.313682i	$0.898438\pi$
$60$		0			0
$61$		0			0		0.985278	−	0.170962i	$-0.0546875\pi$
$61$		0			0		−0.985278	+	0.170962i	$0.945312\pi$
$62$		0			0
$63$	0.382683	+	0.923880i	0.382683	+	0.923880i
$64$	0.995185	+	0.0980171i	0.995185	+	0.0980171i
$65$		0			0
$66$		0			0
$67$	1.12363	−	1.59544i	1.12363	−	1.59544i	0.382683	−	0.923880i	$-0.375000\pi$
$67$	1.12363	−	1.59544i	1.12363	−	1.59544i	0.740951	−	0.671559i	$-0.234375\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	1.19462	+	0.427441i	1.19462	+	0.427441i	0.857729	−	0.514103i	$-0.171875\pi$
$71$	1.19462	+	0.427441i	1.19462	+	0.427441i	0.336890	+	0.941544i	$0.390625\pi$
$72$	0.471397	−	0.881921i	0.471397	−	0.881921i
$73$		0			0		−0.671559	−	0.740951i	$-0.734375\pi$
$73$		0			0		0.671559	+	0.740951i	$0.265625\pi$
$74$	−1.37867	−	1.44806i	−1.37867	−	1.44806i
$75$		0			0
$76$		0			0
$77$	−0.159911	+	0.185381i	−0.159911	+	0.185381i
$78$		0			0
$79$	−0.920964	−	0.755815i	−0.920964	−	0.755815i	0.0490677	−	0.998795i	$-0.484375\pi$
$79$	−0.920964	−	0.755815i	−0.920964	−	0.755815i	−0.970031	+	0.242980i	$0.921875\pi$
$80$		0			0
$81$	−0.634393	−	0.773010i	−0.634393	−	0.773010i
$82$		0			0
$83$		0			0		−0.266713	−	0.963776i	$-0.585938\pi$
$83$		0			0		0.266713	+	0.963776i	$0.414062\pi$
$84$		0			0
$85$		0			0
$86$	1.73614	+	0.573542i	1.73614	+	0.573542i
$87$		0			0
$88$	0.244748	+	0.00600822i	0.244748	+	0.00600822i
$89$		0			0		0.514103	−	0.857729i	$-0.328125\pi$
$89$		0			0		−0.514103	+	0.857729i	$0.671875\pi$
$90$		0			0
$91$		0			0
$92$	−0.968101	+	1.61518i	−0.968101	+	1.61518i
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$97$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$98$	0.903989	+	0.427555i	0.903989	+	0.427555i
$99$	0.0992117	−	0.223818i	0.0992117	−	0.223818i
$100$	−0.336890	−	0.941544i	−0.336890	−	0.941544i
$101$		0			0		0.615232	−	0.788346i	$-0.289062\pi$
$101$		0			0		−0.615232	+	0.788346i	$0.710938\pi$
$102$		0			0
$103$		0			0		0.803208	−	0.595699i	$-0.203125\pi$
$103$		0			0		−0.803208	+	0.595699i	$0.796875\pi$
$104$		0			0
$105$		0			0
$106$	−0.907873	+	0.708511i	−0.907873	+	0.708511i
$107$	1.13466	+	1.61110i	1.13466	+	1.61110i	0.707107	+	0.707107i	$0.250000\pi$
$107$	1.13466	+	1.61110i	1.13466	+	1.61110i	0.427555	+	0.903989i	$0.359375\pi$
$108$		0			0
$109$	1.51958	−	1.18589i	1.51958	−	1.18589i	0.595699	−	0.803208i	$-0.296875\pi$
$109$	1.51958	−	1.18589i	1.51958	−	1.18589i	0.923880	−	0.382683i	$-0.125000\pi$
$110$		0			0
$111$		0			0
$112$	−0.242980	−	0.970031i	−0.242980	−	0.970031i
$113$	1.41809	+	0.430174i	1.41809	+	0.430174i	0.903989	−	0.427555i	$-0.140625\pi$
$113$	1.41809	+	0.430174i	1.41809	+	0.430174i	0.514103	+	0.857729i	$0.328125\pi$
$114$		0			0
$115$		0			0
$116$	−0.989177	−	0.853270i	−0.989177	−	0.853270i
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−0.938930	+	0.0461267i	−0.938930	+	0.0461267i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$	−0.989177	−	0.146730i	−0.989177	−	0.146730i
$127$	−1.35332	+	1.35332i	−1.35332	+	1.35332i	−0.471397	+	0.881921i	$0.656250\pi$
$127$	−1.35332	+	1.35332i	−1.35332	+	1.35332i	−0.881921	+	0.471397i	$0.843750\pi$
$128$	−0.595699	+	0.803208i	−0.595699	+	0.803208i
$129$		0			0
$130$		0			0
$131$		0			0		0.219101	−	0.975702i	$-0.429688\pi$
$131$		0			0		−0.219101	+	0.975702i	$0.570312\pi$
$132$		0			0
$133$		0			0
$134$	0.790790	+	1.78399i	0.790790	+	1.78399i
$135$		0			0
$136$		0			0
$137$	1.86271	−	0.666487i	1.86271	−	0.666487i	0.881921	−	0.471397i	$-0.156250\pi$
$137$	1.86271	−	0.666487i	1.86271	−	0.666487i	0.980785	−	0.195090i	$-0.0625000\pi$
$138$		0			0
$139$		0			0		0.689541	−	0.724247i	$-0.257812\pi$
$139$		0			0		−0.689541	+	0.724247i	$0.742188\pi$
$140$		0			0
$141$		0			0
$142$	−0.980785	+	0.804910i	−0.980785	+	0.804910i
$143$		0			0
$144$	0.514103	+	0.857729i	0.514103	+	0.857729i
$145$		0			0
$146$		0			0
$147$		0			0
$148$	1.95082	−	0.438071i	1.95082	−	0.438071i
$149$	0.874812	−	0.616112i	0.874812	−	0.616112i	−0.0490677	−	0.998795i	$-0.515625\pi$
$149$	0.874812	−	0.616112i	0.874812	−	0.616112i	0.923880	+	0.382683i	$0.125000\pi$
$150$		0			0
$151$	0.997391	+	0.249834i	0.997391	+	0.249834i	0.707107	−	0.707107i	$-0.250000\pi$
$151$	0.997391	+	0.249834i	0.997391	+	0.249834i	0.290285	+	0.956940i	$0.406250\pi$
$152$		0			0
$153$		0			0
$154$	−0.0767960	−	0.232465i	−0.0767960	−	0.232465i
$155$		0			0
$156$		0			0
$157$		0			0		−0.914210	−	0.405241i	$-0.867188\pi$
$157$		0			0		0.914210	+	0.405241i	$0.132812\pi$
$158$	1.12175	−	0.401370i	1.12175	−	0.401370i
$159$		0			0
$160$		0			0
$161$	1.84691	+	0.367372i	1.84691	+	0.367372i
$162$	0.989177	−	0.146730i	0.989177	−	0.146730i
$163$	0.0153963	−	0.627175i	0.0153963	−	0.627175i	−0.941544	−	0.336890i	$-0.890625\pi$
$163$	0.0153963	−	0.627175i	0.0153963	−	0.627175i	0.956940	−	0.290285i	$-0.0937500\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.857729	−	0.514103i	$-0.828125\pi$
$167$		0			0		0.857729	+	0.514103i	$0.171875\pi$
$168$		0			0
$169$	−0.803208	−	0.595699i	−0.803208	−	0.595699i
$170$		0			0
$171$		0			0
$172$	−1.38450	+	1.19427i	−1.38450	+	1.19427i
$173$		0			0		0.963776	−	0.266713i	$-0.0859375\pi$
$173$		0			0		−0.963776	+	0.266713i	$0.914062\pi$
$174$		0			0
$175$	−0.773010	+	0.634393i	−0.773010	+	0.634393i
$176$	−0.130979	+	0.206838i	−0.130979	+	0.206838i
$177$		0			0
$178$		0			0
$179$	1.04425	+	0.900778i	1.04425	+	0.900778i	0.995185	−	0.0980171i	$-0.0312500\pi$
$179$	1.04425	+	0.900778i	1.04425	+	0.900778i	0.0490677	+	0.998795i	$0.484375\pi$
$180$		0			0
$181$		0			0		−0.405241	−	0.914210i	$-0.632812\pi$
$181$		0			0		0.405241	+	0.914210i	$0.367188\pi$
$182$		0			0
$183$		0			0
$184$	−0.887682	−	1.66074i	−0.887682	−	1.66074i
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−0.448969	−	0.185969i	−0.448969	−	0.185969i	0.146730	−	0.989177i	$-0.453125\pi$
$191$	−0.448969	−	0.185969i	−0.448969	−	0.185969i	−0.595699	+	0.803208i	$0.703125\pi$
$192$		0			0
$193$	−0.790019	+	0.327237i	−0.790019	+	0.327237i	−0.740951	−	0.671559i	$-0.765625\pi$
$193$	−0.790019	+	0.327237i	−0.790019	+	0.327237i	−0.0490677	+	0.998795i	$0.515625\pi$
$194$		0			0
$195$		0			0
$196$	−0.831470	+	0.555570i	−0.831470	+	0.555570i
$197$	−0.225785	−	0.683461i	−0.225785	−	0.683461i	−0.998795	−	0.0490677i	$-0.984375\pi$
$197$	−0.225785	−	0.683461i	−0.225785	−	0.683461i	0.773010	−	0.634393i	$-0.218750\pi$
$198$	0.140970	+	0.200162i	0.140970	+	0.200162i
$199$		0			0		0.903989	−	0.427555i	$-0.140625\pi$
$199$		0			0		−0.903989	+	0.427555i	$0.859375\pi$
$200$	0.980785	+	0.195090i	0.980785	+	0.195090i
$201$		0			0
$202$		0			0
$203$	−0.470147	+	1.21881i	−0.470147	+	1.21881i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−1.87402	+	0.184575i	−1.87402	+	0.184575i
$208$		0			0
$209$		0			0
$210$		0			0
$211$	1.73546	+	0.983125i	1.73546	+	0.983125i	0.903989	+	0.427555i	$0.140625\pi$
$211$	1.73546	+	0.983125i	1.73546	+	0.983125i	0.831470	+	0.555570i	$0.187500\pi$
$212$	−0.140970	−	1.14296i	−0.140970	−	1.14296i
$213$		0			0
$214$	−1.96522	+	0.144963i	−1.96522	+	0.144963i
$215$		0			0
$216$		0			0
$217$		0			0
$218$	0.235953	+	1.91306i	0.235953	+	1.91306i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$223$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$224$	0.956940	+	0.290285i	0.956940	+	0.290285i
$225$	0.555570	−	0.831470i	0.555570	−	0.831470i
$226$	−1.09802	+	0.995185i	−1.09802	+	0.995185i
$227$		0			0		−0.359895	−	0.932993i	$-0.617188\pi$
$227$		0			0		0.359895	+	0.932993i	$0.382812\pi$
$228$		0			0
$229$		0			0		0.992480	−	0.122411i	$-0.0390625\pi$
$229$		0			0		−0.992480	+	0.122411i	$0.960938\pi$
$230$		0			0
$231$		0			0
$232$	1.24041	−	0.409777i	1.24041	−	0.409777i
$233$	−1.37787	+	0.825862i	−1.37787	+	0.825862i	−0.995185	−	0.0980171i	$-0.968750\pi$
$233$	−1.37787	+	0.825862i	−1.37787	+	0.825862i	−0.382683	+	0.923880i	$0.625000\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−0.831470	−	0.444430i	−0.831470	−	0.444430i		−	1.00000i	$-0.5\pi$
$239$	−0.831470	−	0.444430i	−0.831470	−	0.444430i	−0.831470	+	0.555570i	$0.812500\pi$
$240$		0			0
$241$		0			0		−0.471397	−	0.881921i	$-0.656250\pi$
$241$		0			0		0.471397	+	0.881921i	$0.343750\pi$
$242$	0.443142	−	0.829061i	0.443142	−	0.829061i
$243$		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		0.0735646	−	0.997290i	$-0.476562\pi$
$251$		0			0		−0.0735646	+	0.997290i	$0.523438\pi$
$252$	0.634393	−	0.773010i	0.634393	−	0.773010i
$253$	−0.246645	−	0.389495i	−0.246645	−	0.389495i
$254$	−0.465035	−	1.85652i	−0.465035	−	1.85652i
$255$		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
$259$	−1.06967	−	1.68920i	−1.06967	−	1.68920i
$260$		0			0
$261$	0.0961008	−	1.30281i	0.0961008	−	1.30281i
$262$		0			0
$263$	−0.289105	−	0.262029i	−0.289105	−	0.262029i	0.514103	−	0.857729i	$-0.328125\pi$
$263$	−0.289105	−	0.262029i	−0.289105	−	0.262029i	−0.803208	+	0.595699i	$0.796875\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−1.93673	−	0.238873i	−1.93673	−	0.238873i
$269$		0			0		−0.949528	−	0.313682i	$-0.898438\pi$
$269$		0			0		0.949528	+	0.313682i	$0.101562\pi$
$270$		0			0
$271$		0			0		−0.471397	−	0.881921i	$-0.656250\pi$
$271$		0			0		0.471397	+	0.881921i	$0.343750\pi$
$272$		0			0
$273$		0			0
$274$	−0.385958	+	1.94034i	−0.385958	+	1.94034i
$275$	0.242980	+	0.0299687i	0.242980	+	0.0299687i
$276$		0			0
$277$	−0.971283	−	0.168534i	−0.971283	−	0.168534i	−0.336890	−	0.941544i	$-0.609375\pi$
$277$	−0.971283	−	0.168534i	−0.971283	−	0.168534i	−0.634393	+	0.773010i	$0.718750\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	0.251710	−	1.69689i	0.251710	−	1.69689i	−0.382683	−	0.923880i	$-0.625000\pi$
$281$	0.251710	−	1.69689i	0.251710	−	1.69689i	0.634393	−	0.773010i	$-0.281250\pi$
$282$		0			0
$283$		0			0		0.992480	−	0.122411i	$-0.0390625\pi$
$283$		0			0		−0.992480	+	0.122411i	$0.960938\pi$
$284$	−0.186170	−	1.25505i	−0.186170	−	1.25505i
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−1.00000			−1.00000
$289$	−0.555570	−	0.831470i	−0.555570	−	0.831470i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0		−0.963776	−	0.266713i	$-0.914062\pi$
$293$		0			0		0.963776	+	0.266713i	$0.0859375\pi$
$294$		0			0
$295$		0			0
$296$	−0.627175	+	1.89848i	−0.627175	+	1.89848i
$297$		0			0
$298$	0.0787137	+	1.06710i	0.0787137	+	1.06710i
$299$		0			0
$300$		0			0
$301$	1.59088	+	0.901225i	1.59088	+	0.901225i
$302$	−0.727051	+	0.727051i	−0.727051	+	0.727051i
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0		0.997290	−	0.0735646i	$-0.0234375\pi$
$307$		0			0		−0.997290	+	0.0735646i	$0.976562\pi$
$308$	0.238873	+	0.0536407i	0.238873	+	0.0536407i
$309$		0			0
$310$		0			0
$311$		0			0		0.0490677	−	0.998795i	$-0.484375\pi$
$311$		0			0		−0.0490677	+	0.998795i	$0.515625\pi$
$312$		0			0
$313$		0			0		0.903989	−	0.427555i	$-0.140625\pi$
$313$		0			0		−0.903989	+	0.427555i	$0.859375\pi$
$314$		0			0
$315$		0			0
$316$	−0.232430	+	1.16851i	−0.232430	+	1.16851i
$317$	0.324666	+	1.87110i	0.324666	+	1.87110i	0.471397	+	0.881921i	$0.343750\pi$
$317$	0.324666	+	1.87110i	0.324666	+	1.87110i	−0.146730	+	0.989177i	$0.546875\pi$
$318$		0			0
$319$	0.295476	−	0.122390i	0.295476	−	0.122390i
$320$		0			0
$321$		0			0
$322$	−1.26460	+	1.39528i	−1.26460	+	1.39528i
$323$		0			0
$324$	−0.382683	+	0.923880i	−0.382683	+	0.923880i
$325$		0			0
$326$	0.530030	+	0.335638i	0.530030	+	0.335638i
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−0.364402	−	0.822078i	−0.364402	−	0.822078i	−0.998795	−	0.0490677i	$-0.984375\pi$
$331$	−0.364402	−	0.822078i	−0.364402	−	0.822078i	0.634393	−	0.773010i	$-0.281250\pi$
$332$		0			0
$333$	1.51396	+	1.30595i	1.51396	+	1.30595i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−1.54416	+	1.26726i	−1.54416	+	1.26726i	−0.740951	+	0.671559i	$0.765625\pi$
$337$	−1.54416	+	1.26726i	−1.54416	+	1.26726i	−0.803208	+	0.595699i	$0.796875\pi$
$338$	0.923880	−	0.382683i	0.923880	−	0.382683i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	0.803208	+	0.595699i	0.803208	+	0.595699i
$344$	−0.312590	−	1.80150i	−0.312590	−	1.80150i
$345$		0			0
$346$		0			0
$347$	−1.55437	+	0.880537i	−1.55437	+	0.880537i	−0.555570	+	0.831470i	$0.687500\pi$
$347$	−1.55437	+	0.880537i	−1.55437	+	0.880537i	−0.998795	+	0.0490677i	$0.984375\pi$
$348$		0			0
$349$		0			0		0.0245412	−	0.999699i	$-0.492188\pi$
$349$		0			0		−0.0245412	+	0.999699i	$0.507812\pi$
$350$	−0.146730	−	0.989177i	−0.146730	−	0.989177i
$351$		0			0
$352$	−0.110074	−	0.218680i	−0.110074	−	0.218680i
$353$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$353$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$	−1.30948	+	0.432593i	−1.30948	+	0.432593i
$359$	1.18996	+	1.60448i	1.18996	+	1.60448i	0.634393	+	0.773010i	$0.281250\pi$
$359$	1.18996	+	1.60448i	1.18996	+	1.60448i	0.555570	+	0.831470i	$0.312500\pi$
$360$		0			0
$361$	−0.970031	−	0.242980i	−0.970031	−	0.242980i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		0.956940	−	0.290285i	$-0.0937500\pi$
$367$		0			0		−0.956940	+	0.290285i	$0.906250\pi$
$368$	1.88082	+	0.0923988i	1.88082	+	0.0923988i
$369$		0			0
$370$		0			0
$371$	−1.02865	+	0.517780i	−1.02865	+	0.517780i
$372$		0			0
$373$	−1.08719	+	1.14192i	−1.08719	+	1.14192i	−0.0980171	+	0.995185i	$0.531250\pi$
$373$	−1.08719	+	1.14192i	−1.08719	+	1.14192i	−0.989177	+	0.146730i	$0.953125\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	1.06805	+	1.23816i	1.06805	+	1.23816i	0.970031	+	0.242980i	$0.0781250\pi$
$379$	1.06805	+	1.23816i	1.06805	+	1.23816i	0.0980171	+	0.995185i	$0.468750\pi$
$380$		0			0
$381$		0			0
$382$	0.390327	−	0.289486i	0.390327	−	0.289486i
$383$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$383$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$384$		0			0
$385$		0			0
$386$	0.125471	−	0.845855i	0.125471	−	0.845855i
$387$	−1.78399	−	0.400609i	−1.78399	−	0.400609i
$388$		0			0
$389$	−0.613705	+	0.529385i	−0.613705	+	0.529385i	−0.903989	−	0.427555i	$-0.859375\pi$
$389$	−0.613705	+	0.529385i	−0.613705	+	0.529385i	0.290285	+	0.956940i	$0.406250\pi$
$390$		0			0
$391$		0			0
$392$	−0.0490677	−	0.998795i	−0.0490677	−	0.998795i
$393$		0			0
$394$	0.702301	+	0.157707i	0.702301	+	0.157707i
$395$		0			0
$396$	−0.244158	+	0.0180102i	−0.244158	+	0.0180102i
$397$		0			0		−0.449611	−	0.893224i	$-0.648438\pi$
$397$		0			0		0.449611	+	0.893224i	$0.351562\pi$
$398$		0			0
$399$		0			0
$400$	−0.671559	+	0.740951i	−0.671559	+	0.740951i
$401$	0.0284872	+	0.0939097i	0.0284872	+	0.0939097i	0.970031	−	0.242980i	$-0.0781250\pi$
$401$	0.0284872	+	0.0939097i	0.0284872	+	0.0939097i	−0.941544	+	0.336890i	$0.890625\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$	−0.803705	−	1.02985i	−0.803705	−	1.02985i
$407$	−0.118938	+	0.474826i	−0.118938	+	0.474826i
$408$		0			0
$409$		0			0		0.803208	−	0.595699i	$-0.203125\pi$
$409$		0			0		−0.803208	+	0.595699i	$0.796875\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$	0.805124	−	1.70229i	0.805124	−	1.70229i
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		−0.999699	−	0.0245412i	$-0.992188\pi$
$419$		0			0		0.999699	+	0.0245412i	$0.00781250\pi$
$420$		0			0
$421$	−0.309226	−	0.545861i	−0.309226	−	0.545861i	0.671559	−	0.740951i	$-0.265625\pi$
$421$	−0.309226	−	0.545861i	−0.309226	−	0.545861i	−0.980785	+	0.195090i	$0.937500\pi$
$422$	−1.73546	+	0.983125i	−1.73546	+	0.983125i
$423$		0			0
$424$	1.05282	+	0.466683i	1.05282	+	0.466683i
$425$		0			0
$426$		0			0
$427$		0			0
$428$	0.885984	−	1.76015i	0.885984	−	1.76015i
$429$		0			0
$430$		0			0
$431$	−0.704900	−	0.858923i	−0.704900	−	0.858923i	0.290285	−	0.956940i	$-0.406250\pi$
$431$	−0.704900	−	0.858923i	−0.704900	−	0.858923i	−0.995185	+	0.0980171i	$0.968750\pi$
$432$		0			0
$433$		0			0		−0.773010	−	0.634393i	$-0.781250\pi$
$433$		0			0		0.773010	+	0.634393i	$0.218750\pi$
$434$		0			0
$435$		0			0
$436$	−1.76219	−	0.781124i	−1.76219	−	0.781124i
$437$		0			0
$438$		0			0
$439$		0			0		−0.671559	−	0.740951i	$-0.734375\pi$
$439$		0			0		0.671559	+	0.740951i	$0.265625\pi$
$440$		0			0
$441$	−0.941544	−	0.336890i	−0.941544	−	0.336890i
$442$		0			0
$443$	0.0438416	+	0.0220680i	0.0438416	+	0.0220680i	0.471397	−	0.881921i	$-0.343750\pi$
$443$	0.0438416	+	0.0220680i	0.0438416	+	0.0220680i	−0.427555	+	0.903989i	$0.640625\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−0.740951	+	0.671559i	−0.740951	+	0.671559i
$449$	−0.761681	−	1.83886i	−0.761681	−	1.83886i	−0.471397	−	0.881921i	$-0.656250\pi$
$449$	−0.761681	−	1.83886i	−0.761681	−	1.83886i	−0.290285	−	0.956940i	$-0.593750\pi$
$450$	0.427555	+	0.903989i	0.427555	+	0.903989i
$451$		0			0
$452$	−0.289105	−	1.45343i	−0.289105	−	1.45343i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−0.941658	−	0.0462607i	−0.941658	−	0.0462607i	−0.427555	−	0.903989i	$-0.640625\pi$
$457$	−0.941658	−	0.0462607i	−0.941658	−	0.0462607i	−0.514103	+	0.857729i	$0.671875\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		−0.0735646	−	0.997290i	$-0.523438\pi$
$461$		0			0		0.0735646	+	0.997290i	$0.476562\pi$
$462$		0			0
$463$	0.00961895	+	0.0976628i	0.00961895	+	0.0976628i	0.998795	−	0.0490677i	$-0.0156250\pi$
$463$	0.00961895	+	0.0976628i	0.00961895	+	0.0976628i	−0.989177	+	0.146730i	$0.953125\pi$
$464$	−0.286222	+	1.27460i	−0.286222	+	1.27460i
$465$		0			0
$466$		−	1.60642i		−	1.60642i
$467$		0			0		0.492898	−	0.870087i	$-0.335938\pi$
$467$		0			0		−0.492898	+	0.870087i	$0.664062\pi$
$468$		0			0
$469$	0.427555	+	1.90399i	0.427555	+	1.90399i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−0.108767	−	0.434221i	−0.108767	−	0.434221i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	0.834055	−	0.794086i	0.834055	−	0.794086i
$478$	0.808661	−	0.484693i	0.808661	−	0.484693i
$479$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$479$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	0.483289	+	0.806319i	0.483289	+	0.806319i
$485$		0			0
$486$		0			0
$487$	−1.95694	−	0.290285i	−1.95694	−	0.290285i	−0.956940	−	0.290285i	$-0.906250\pi$
$487$	−1.95694	−	0.290285i	−1.95694	−	0.290285i	−1.00000			$\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	0.182928	−	1.05424i	0.182928	−	1.05424i	−0.740951	−	0.671559i	$-0.765625\pi$
$491$	0.182928	−	1.05424i	0.182928	−	1.05424i	0.923880	−	0.382683i	$-0.125000\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−1.11897	+	0.598102i	−1.11897	+	0.598102i
$498$		0			0
$499$	0.530030	−	1.60442i	0.530030	−	1.60442i	−0.242980	−	0.970031i	$-0.578125\pi$
$499$	0.530030	−	1.60442i	0.530030	−	1.60442i	0.773010	−	0.634393i	$-0.218750\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		0.427555	−	0.903989i	$-0.359375\pi$
$503$		0			0		−0.427555	+	0.903989i	$0.640625\pi$
$504$	0.336890	+	0.941544i	0.336890	+	0.941544i
$505$		0			0
$506$	0.460881	−	0.0113140i	0.460881	−	0.0113140i
$507$		0			0
$508$	1.83147	+	0.555570i	1.83147	+	0.555570i
$509$		0			0		0.844854	−	0.534998i	$-0.179688\pi$
$509$		0			0		−0.844854	+	0.534998i	$0.820312\pi$
$510$		0			0
$511$		0			0
$512$	0.989177	+	0.146730i	0.989177	+	0.146730i
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$	1.99880	−	0.0490677i	1.99880	−	0.0490677i
$519$		0			0
$520$		0			0
$521$		0			0		0.427555	−	0.903989i	$-0.359375\pi$
$521$		0			0		−0.427555	+	0.903989i	$0.640625\pi$
$522$	1.06805	+	0.752205i	1.06805	+	0.752205i
$523$		0			0		0.999699	−	0.0245412i	$-0.00781250\pi$
$523$		0			0		−0.999699	+	0.0245412i	$0.992188\pi$
$524$		0			0
$525$		0			0
$526$	0.373380	−	0.113263i	0.373380	−	0.113263i
$527$		0			0
$528$		0			0
$529$	−1.20019	+	2.24539i	−1.20019	+	2.24539i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$	1.20057	−	1.53838i	1.20057	−	1.53838i
$537$		0			0
$538$		0			0
$539$	−0.0299687	−	0.242980i	−0.0299687	−	0.242980i
$540$		0			0
$541$	−0.319012	+	0.123057i	−0.319012	+	0.123057i	−0.514103	−	0.857729i	$-0.671875\pi$
$541$	−0.319012	+	0.123057i	−0.319012	+	0.123057i	0.195090	+	0.980785i	$0.437500\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	0.106558	−	0.101451i	0.106558	−	0.101451i	−0.634393	−	0.773010i	$-0.718750\pi$
$547$	0.106558	−	0.101451i	0.106558	−	0.101451i	0.740951	+	0.671559i	$0.234375\pi$
$548$	−1.46586	−	1.32858i	−1.46586	−	1.32858i
$549$		0			0
$550$	−0.150622	+	0.193004i	−0.150622	+	0.193004i
$551$		0			0
$552$		0			0
$553$	1.17850	−	0.174814i	1.17850	−	0.174814i
$554$	0.643895	−	0.746454i	0.643895	−	0.746454i
$555$		0			0
$556$		0			0
$557$	0.857729	−	1.51410i	0.857729	−	1.51410i		−	1.00000i	$-0.5\pi$
$557$	0.857729	−	1.51410i	0.857729	−	1.51410i	0.857729	−	0.514103i	$-0.171875\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	1.32607	+	1.08827i	1.32607	+	1.08827i
$563$		0			0		−0.0735646	−	0.997290i	$-0.523438\pi$
$563$		0			0		0.0735646	+	0.997290i	$0.476562\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	0.998795	+	0.0490677i	0.998795	+	0.0490677i
$568$	1.17221	+	0.485544i	1.17221	+	0.485544i
$569$	−0.818289	−	1.73013i	−0.818289	−	1.73013i	−0.671559	−	0.740951i	$-0.734375\pi$
$569$	−0.818289	−	1.73013i	−0.818289	−	1.73013i	−0.146730	−	0.989177i	$-0.546875\pi$
$570$		0			0
$571$	−1.43798	+	0.475045i	−1.43798	+	0.475045i	−0.923880	−	0.382683i	$-0.875000\pi$
$571$	−1.43798	+	0.475045i	−1.43798	+	0.475045i	−0.514103	+	0.857729i	$0.671875\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−0.720627	−	1.73975i	−0.720627	−	1.73975i
$576$	0.514103	−	0.857729i	0.514103	−	0.857729i
$577$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$577$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$578$	0.998795	−	0.0490677i	0.998795	−	0.0490677i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	0.265459	+	0.0949828i	0.265459	+	0.0949828i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		0.914210	−	0.405241i	$-0.132812\pi$
$587$		0			0		−0.914210	+	0.405241i	$0.867188\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	−1.30595	−	1.51396i	−1.30595	−	1.51396i
$593$		0			0		−0.634393	−	0.773010i	$-0.718750\pi$
$593$		0			0		0.634393	+	0.773010i	$0.281250\pi$
$594$		0			0
$595$		0			0
$596$	−0.955746	−	0.481082i	−0.955746	−	0.481082i
$597$		0			0
$598$		0			0
$599$	−1.16851	+	1.57555i	−1.16851	+	1.57555i	−0.427555	+	0.903989i	$0.640625\pi$
$599$	−1.16851	+	1.57555i	−1.16851	+	1.57555i	−0.740951	+	0.671559i	$0.765625\pi$
$600$		0			0
$601$		0			0		0.514103	−	0.857729i	$-0.328125\pi$
$601$		0			0		−0.514103	+	0.857729i	$0.671875\pi$
$602$	−1.59088	+	0.901225i	−1.59088	+	0.901225i
$603$	−0.961844	−	1.69789i	−0.961844	−	1.69789i
$604$	−0.249834	−	0.997391i	−0.249834	−	0.997391i
$605$		0			0
$606$		0			0
$607$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$607$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−1.09908	+	1.40834i	−1.09908	+	1.40834i	−0.195090	+	0.980785i	$0.562500\pi$
$613$	−1.09908	+	1.40834i	−1.09908	+	1.40834i	−0.903989	+	0.427555i	$0.859375\pi$
$614$		0			0
$615$		0			0
$616$	−0.168814	+	0.177311i	−0.168814	+	0.177311i
$617$	0.428579	−	1.71098i	0.428579	−	1.71098i	−0.242980	−	0.970031i	$-0.578125\pi$
$617$	0.428579	−	1.71098i	0.428579	−	1.71098i	0.671559	−	0.740951i	$-0.265625\pi$
$618$		0			0
$619$		0			0		−0.575808	−	0.817585i	$-0.695312\pi$
$619$		0			0		0.575808	+	0.817585i	$0.304688\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	0.956940	+	0.290285i	0.956940	+	0.290285i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−0.670535	−	1.87402i	−0.670535	−	1.87402i	−0.427555	−	0.903989i	$-0.640625\pi$
$631$	−0.670535	−	1.87402i	−0.670535	−	1.87402i	−0.242980	−	0.970031i	$-0.578125\pi$
$632$	−0.882768	−	0.800094i	−0.882768	−	0.800094i
$633$		0			0
$634$	−1.77181	−	0.683461i	−1.77181	−	0.683461i
$635$		0			0
$636$		0			0
$637$		0			0
$638$	−0.0469275	+	0.316360i	−0.0469275	+	0.316360i
$639$	0.897168	−	0.897168i	0.897168	−	0.897168i
$640$		0			0
$641$	1.27843	+	1.27843i	1.27843	+	1.27843i	0.941544	+	0.336890i	$0.109375\pi$
$641$	1.27843	+	1.27843i	1.27843	+	1.27843i	0.336890	+	0.941544i	$0.390625\pi$
$642$		0			0
$643$		0			0		0.219101	−	0.975702i	$-0.429688\pi$
$643$		0			0		−0.219101	+	0.975702i	$0.570312\pi$
$644$	−0.546632	−	1.80200i	−0.546632	−	1.80200i
$645$		0			0
$646$		0			0
$647$		0			0		−0.0490677	−	0.998795i	$-0.515625\pi$
$647$		0			0		0.0490677	+	0.998795i	$0.484375\pi$
$648$	−0.595699	−	0.803208i	−0.595699	−	0.803208i
$649$		0			0
$650$		0			0
$651$		0			0
$652$	−0.560376	+	0.282070i	−0.560376	+	0.282070i
$653$	1.09908	−	0.553230i	1.09908	−	0.553230i	0.195090	−	0.980785i	$-0.437500\pi$
$653$	1.09908	−	0.553230i	1.09908	−	0.553230i	0.903989	+	0.427555i	$0.140625\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−0.328180	−	0.420524i	−0.328180	−	0.420524i	0.595699	−	0.803208i	$-0.296875\pi$
$659$	−0.328180	−	0.420524i	−0.328180	−	0.420524i	−0.923880	+	0.382683i	$0.875000\pi$
$660$		0			0
$661$		0			0		0.817585	−	0.575808i	$-0.195312\pi$
$661$		0			0		−0.817585	+	0.575808i	$0.804688\pi$
$662$	0.892460	+	0.110074i	0.892460	+	0.110074i
$663$		0			0
$664$		0			0
$665$		0			0
$666$	−1.89848	+	0.627175i	−1.89848	+	0.627175i
$667$	−1.93930	−	1.51345i	−1.93930	−	1.51345i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	0.192268	+	0.0382444i	0.192268	+	0.0382444i	0.290285	−	0.956940i	$-0.406250\pi$
$673$	0.192268	+	0.0382444i	0.192268	+	0.0382444i	−0.0980171	+	0.995185i	$0.531250\pi$
$674$	−0.293107	−	1.97597i	−0.293107	−	1.97597i
$675$		0			0
$676$	−0.146730	+	0.989177i	−0.146730	+	0.989177i
$677$		0			0		0.870087	−	0.492898i	$-0.164062\pi$
$677$		0			0		−0.870087	+	0.492898i	$0.835938\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	1.22377	+	0.774941i	1.22377	+	0.774941i	0.980785	−	0.195090i	$-0.0625000\pi$
$683$	1.22377	+	0.774941i	1.22377	+	0.774941i	0.242980	+	0.970031i	$0.421875\pi$
$684$		0			0
$685$		0			0
$686$	−0.923880	+	0.382683i	−0.923880	+	0.382683i
$687$		0			0
$688$	1.70590	+	0.658039i	1.70590	+	0.658039i
$689$		0			0
$690$		0			0
$691$		0			0		−0.757209	−	0.653173i	$-0.773438\pi$
$691$		0			0		0.757209	+	0.653173i	$0.226562\pi$
$692$		0			0
$693$	0.0992117	+	0.223818i	0.0992117	+	0.223818i
$694$	0.0438416	−	1.78591i	0.0438416	−	1.78591i
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	0.923880	+	0.382683i	0.923880	+	0.382683i
$701$	1.52560	+	1.07445i	1.52560	+	1.07445i	0.970031	+	0.242980i	$0.0781250\pi$
$701$	1.52560	+	1.07445i	1.52560	+	1.07445i	0.555570	+	0.831470i	$0.312500\pi$
$702$		0			0
$703$		0			0
$704$	0.244158	+	0.0180102i	0.244158	+	0.0180102i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	0.385974	+	1.16836i	0.385974	+	1.16836i	0.941544	+	0.336890i	$0.109375\pi$
$709$	0.385974	+	1.16836i	0.385974	+	1.16836i	−0.555570	+	0.831470i	$0.687500\pi$
$710$		0			0
$711$	−1.07701	+	0.509389i	−1.07701	+	0.509389i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$	0.302158	−	1.34557i	0.302158	−	1.34557i
$717$		0			0
$718$	−1.98797	+	0.195798i	−1.98797	+	0.195798i
$719$		0			0		0.995185	−	0.0980171i	$-0.0312500\pi$
$719$		0			0		−0.995185	+	0.0980171i	$0.968750\pi$
$720$		0			0
$721$		0			0
$722$	0.707107	−	0.707107i	0.707107	−	0.707107i
$723$		0			0
$724$		0			0
$725$	1.27460	−	0.286222i	1.27460	−	0.286222i
$726$		0			0
$727$		0			0		−0.146730	−	0.989177i	$-0.546875\pi$
$727$		0			0		0.146730	+	0.989177i	$0.453125\pi$
$728$		0			0
$729$	−0.970031	+	0.242980i	−0.970031	+	0.242980i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		−0.689541	−	0.724247i	$-0.742188\pi$
$733$		0			0		0.689541	+	0.724247i	$0.257812\pi$
$734$		0			0
$735$		0			0
$736$	−1.04619	+	1.56573i	−1.04619	+	1.56573i
$737$	0.265421	−	0.397231i	0.265421	−	0.397231i
$738$		0			0
$739$	0.354783	+	0.919741i	0.354783	+	0.919741i	0.989177	+	0.146730i	$0.0468750\pi$
$739$	0.354783	+	0.919741i	0.354783	+	0.919741i	−0.634393	+	0.773010i	$0.718750\pi$
$740$		0			0
$741$		0			0
$742$	0.0847182	−	1.14850i	0.0847182	−	1.14850i
$743$	−0.284666	+	1.91906i	−0.284666	+	1.91906i	0.0980171	+	0.995185i	$0.468750\pi$
$743$	−0.284666	+	1.91906i	−0.284666	+	1.91906i	−0.382683	+	0.923880i	$0.625000\pi$
$744$		0			0
$745$		0			0
$746$	−0.420524	−	1.51958i	−0.420524	−	1.51958i
$747$		0			0
$748$		0			0
$749$	−1.95574	−	0.241217i	−1.95574	−	0.241217i
$750$		0			0
$751$	−1.75535	−	0.938254i	−1.75535	−	0.938254i	−0.923880	−	0.382683i	$-0.875000\pi$
$751$	−1.75535	−	0.938254i	−1.75535	−	0.938254i	−0.831470	−	0.555570i	$-0.812500\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−0.0483598	−	1.96996i	−0.0483598	−	1.96996i	−0.195090	−	0.980785i	$-0.562500\pi$
$757$	−0.0483598	−	1.96996i	−0.0483598	−	1.96996i	0.146730	−	0.989177i	$-0.453125\pi$
$758$	−1.61110	+	0.279552i	−1.61110	+	0.279552i
$759$		0			0
$760$		0			0
$761$		0			0		−0.740951	−	0.671559i	$-0.765625\pi$
$761$		0			0		0.740951	+	0.671559i	$0.234375\pi$
$762$		0			0
$763$	−0.141800	+	1.92233i	−0.141800	+	1.92233i
$764$	0.0476324	+	0.483620i	0.0476324	+	0.483620i
$765$		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
$772$	0.661009	+	0.542476i	0.661009	+	0.542476i
$773$		0			0		0.0735646	−	0.997290i	$-0.476562\pi$
$773$		0			0		−0.0735646	+	0.997290i	$0.523438\pi$
$774$	1.26077	−	1.32423i	1.26077	−	1.32423i
$775$		0			0
$776$		0			0
$777$		0			0
$778$	−0.138562	−	0.798550i	−0.138562	−	0.798550i
$779$		0			0
$780$		0			0
$781$	0.294948	+	0.0974377i	0.294948	+	0.0974377i
$782$		0			0
$783$		0			0
$784$	0.881921	+	0.471397i	0.881921	+	0.471397i
$785$		0			0
$786$		0			0
$787$		0			0		−0.992480	−	0.122411i	$-0.960938\pi$
$787$		0			0		0.992480	+	0.122411i	$0.0390625\pi$
$788$	−0.496324	+	0.521306i	−0.496324	+	0.521306i
$789$		0			0
$790$		0			0
$791$	−1.27107	+	0.761850i	−1.27107	+	0.761850i
$792$	0.110074	−	0.218680i	0.110074	−	0.218680i
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		−0.359895	−	0.932993i	$-0.617188\pi$
$797$		0			0		0.359895	+	0.932993i	$0.382812\pi$
$798$		0			0
$799$		0			0
$800$	−0.290285	−	0.956940i	−0.290285	−	0.956940i
$801$		0			0
$802$	−0.0951944	−	0.0238449i	−0.0951944	−	0.0238449i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	0.258809	+	1.74475i	0.258809	+	1.74475i	0.595699	+	0.803208i	$0.296875\pi$
$809$	0.258809	+	1.74475i	0.258809	+	1.74475i	−0.336890	+	0.941544i	$0.609375\pi$
$810$		0			0
$811$		0			0		0.975702	−	0.219101i	$-0.0703125\pi$
$811$		0			0		−0.975702	+	0.219101i	$0.929688\pi$
$812$	1.29652	−	0.159911i	1.29652	−	0.159911i
$813$		0			0
$814$	−0.346125	−	0.346125i	−0.346125	−	0.346125i
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−0.642934	+	1.66674i	−0.642934	+	1.66674i	0.0980171	+	0.995185i	$0.468750\pi$
$821$	−0.642934	+	1.66674i	−0.642934	+	1.66674i	−0.740951	+	0.671559i	$0.765625\pi$
$822$		0			0
$823$	0.0962497	−	1.95921i	0.0962497	−	1.95921i	−0.146730	−	0.989177i	$-0.546875\pi$
$823$	0.0962497	−	1.95921i	0.0962497	−	1.95921i	0.242980	−	0.970031i	$-0.421875\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	0.585326	+	1.77181i	0.585326	+	1.77181i	0.634393	+	0.773010i	$0.281250\pi$
$827$	0.585326	+	1.77181i	0.585326	+	1.77181i	−0.0490677	+	0.998795i	$0.515625\pi$
$828$	1.04619	+	1.56573i	1.04619	+	1.56573i
$829$		0			0		−0.170962	−	0.985278i	$-0.554688\pi$
$829$		0			0		0.170962	+	0.985278i	$0.445312\pi$
$830$		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		0.336890	−	0.941544i	$-0.390625\pi$
$839$		0			0		−0.336890	+	0.941544i	$0.609375\pi$
$840$		0			0
$841$	0.523511	−	0.474483i	0.523511	−	0.474483i
$842$	0.627175	+	0.0153963i	0.627175	+	0.0153963i
$843$		0			0
$844$	0.0489495	−	1.99398i	0.0489495	−	1.99398i
$845$		0			0
$846$		0			0
$847$	0.596369	−	0.726678i	0.596369	−	0.726678i
$848$	−0.941544	+	0.663110i	−0.941544	+	0.663110i
$849$		0			0
$850$		0			0
$851$	3.62866	−	1.00418i	3.62866	−	1.00418i
$852$		0			0
$853$		0			0		−0.844854	−	0.534998i	$-0.820312\pi$
$853$		0			0		0.844854	+	0.534998i	$0.179688\pi$
$854$		0			0
$855$		0			0
$856$	1.05424	+	1.66483i	1.05424	+	1.66483i
$857$		0			0		−0.857729	−	0.514103i	$-0.828125\pi$
$857$		0			0		0.857729	+	0.514103i	$0.171875\pi$
$858$		0			0
$859$		0			0		0.870087	−	0.492898i	$-0.164062\pi$
$859$		0			0		−0.870087	+	0.492898i	$0.835938\pi$
$860$		0			0
$861$		0			0
$862$	1.09911	−	0.163038i	1.09911	−	0.163038i
$863$	1.68250	+	0.334669i	1.68250	+	0.334669i	0.941544	−	0.336890i	$-0.109375\pi$
$863$	1.68250	+	0.334669i	1.68250	+	0.334669i	0.740951	+	0.671559i	$0.234375\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−0.229945	−	0.179451i	−0.229945	−	0.179451i
$870$		0			0
$871$		0			0
$872$	1.57594	−	1.10990i	1.57594	−	1.10990i
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−1.14801	−	1.47104i	−1.14801	−	1.47104i	−0.857729	−	0.514103i	$-0.828125\pi$
$877$	−1.14801	−	1.47104i	−1.14801	−	1.47104i	−0.290285	−	0.956940i	$-0.593750\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		0.290285	−	0.956940i	$-0.406250\pi$
$881$		0			0		−0.290285	+	0.956940i	$0.593750\pi$
$882$	0.773010	−	0.634393i	0.773010	−	0.634393i
$883$	−1.69628	+	0.853837i	−1.69628	+	0.853837i	−0.707107	+	0.707107i	$0.750000\pi$
$883$	−1.69628	+	0.853837i	−1.69628	+	0.853837i	−0.989177	+	0.146730i	$0.953125\pi$
$884$		0			0
$885$		0			0
$886$	−0.0414675	+	0.0262590i	−0.0414675	+	0.0262590i
$887$		0			0		0.941544	−	0.336890i	$-0.109375\pi$
$887$		0			0		−0.941544	+	0.336890i	$0.890625\pi$
$888$		0			0
$889$	−0.0939097	−	1.91158i	−0.0939097	−	1.91158i
$890$		0			0
$891$	−0.159911	−	0.185381i	−0.159911	−	0.185381i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$	−0.195090	−	0.980785i	−0.195090	−	0.980785i
$897$		0			0
$898$	1.96883	+	0.292048i	1.96883	+	0.292048i
$899$		0			0
$900$	−0.995185	−	0.0980171i	−0.995185	−	0.0980171i
$901$		0			0
$902$		0			0
$903$		0			0
$904$	1.39528	+	0.499238i	1.39528	+	0.499238i
$905$		0			0
$906$		0			0
$907$	1.39602	+	1.32913i	1.39602	+	1.32913i	0.881921	+	0.471397i	$0.156250\pi$
$907$	1.39602	+	1.32913i	1.39602	+	1.32913i	0.514103	+	0.857729i	$0.328125\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−1.68789	−	0.512016i	−1.68789	−	0.512016i	−0.707107	−	0.707107i	$-0.750000\pi$
$911$	−1.68789	−	0.512016i	−1.68789	−	0.512016i	−0.980785	+	0.195090i	$0.937500\pi$
$912$		0			0
$913$		0			0
$914$	0.523788	−	0.783904i	0.523788	−	0.783904i
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−0.141067	+	0.563170i	−0.141067	+	0.563170i	0.857729	+	0.514103i	$0.171875\pi$
$919$	−0.141067	+	0.563170i	−0.141067	+	0.563170i	−0.998795	+	0.0490677i	$0.984375\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−0.810239	+	1.82787i	−0.810239	+	1.82787i
$926$	−0.0887133	−	0.0419583i	−0.0887133	−	0.0419583i
$927$		0			0
$928$	−0.946117	−	0.900778i	−0.946117	−	0.900778i
$929$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$929$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$930$		0			0
$931$		0			0
$932$	1.37787	+	0.825862i	1.37787	+	0.825862i
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		0.595699	−	0.803208i	$-0.296875\pi$
$937$		0			0		−0.595699	+	0.803208i	$0.703125\pi$
$938$	−1.85291	−	0.612120i	−1.85291	−	0.612120i
$939$		0			0
$940$		0			0
$941$		0			0		−0.266713	−	0.963776i	$-0.585938\pi$
$941$		0			0		0.266713	+	0.963776i	$0.414062\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$	0.428361	+	0.129942i	0.428361	+	0.129942i
$947$	0.643895	−	0.746454i	0.643895	−	0.746454i	−0.336890	−	0.941544i	$-0.609375\pi$
$947$	0.643895	−	0.746454i	0.643895	−	0.746454i	0.980785	+	0.195090i	$0.0625000\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−0.457553	−	0.163715i	−0.457553	−	0.163715i	0.0980171	−	0.995185i	$-0.468750\pi$
$953$	−0.457553	−	0.163715i	−0.457553	−	0.163715i	−0.555570	+	0.831470i	$0.687500\pi$
$954$	0.252321	+	1.12363i	0.252321	+	1.12363i
$955$		0			0
$956$			0.942793i			0.942793i
$957$		0			0
$958$		0			0
$959$	−0.757083	+	1.82776i	−0.757083	+	1.82776i
$960$		0			0
$961$	0.382683	+	0.923880i	0.382683	+	0.923880i
$962$		0			0
$963$	1.94154	−	0.336890i	1.94154	−	0.336890i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	0.733452	+	1.55075i	0.733452	+	1.55075i	0.831470	+	0.555570i	$0.187500\pi$
$967$	0.733452	+	1.55075i	0.733452	+	1.55075i	−0.0980171	+	0.995185i	$0.531250\pi$
$968$	−0.940063			−0.940063
$969$		0			0
$970$		0			0
$971$		0			0		−0.932993	−	0.359895i	$-0.882812\pi$
$971$		0			0		0.932993	+	0.359895i	$0.117188\pi$
$972$		0			0
$973$		0			0
$974$	1.25505	−	1.52929i	1.25505	−	1.52929i
$975$		0			0
$976$		0			0
$977$	−1.53858	−	0.151537i	−1.53858	−	0.151537i	−0.707107	−	0.707107i	$-0.750000\pi$
$977$	−1.53858	−	0.151537i	−1.53858	−	0.151537i	−0.831470	+	0.555570i	$0.812500\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−0.422329	−	1.88072i	−0.422329	−	1.88072i
$982$	0.810210	+	0.698892i	0.810210	+	0.698892i
$983$		0			0		0.989177	−	0.146730i	$-0.0468750\pi$
$983$		0			0		−0.989177	+	0.146730i	$0.953125\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−2.49364	+	2.37414i	−2.49364	+	2.37414i
$990$		0			0
$991$	0.0815966	−	0.0545211i	0.0815966	−	0.0545211i	−0.514103	−	0.857729i	$-0.671875\pi$
$991$	0.0815966	−	0.0545211i	0.0815966	−	0.0545211i	0.595699	+	0.803208i	$0.296875\pi$
$992$		0			0
$993$		0			0
$994$	0.0622564	−	1.26726i	0.0622564	−	1.26726i
$995$		0			0
$996$		0			0
$997$		0			0		−0.122411	−	0.992480i	$-0.539062\pi$
$997$		0			0		0.122411	+	0.992480i	$0.460938\pi$
$998$	1.10367	+	1.27946i	1.10367	+	1.27946i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3584.1.cd.a.125.1	✓	64
7.6	odd	2	CM	3584.1.cd.a.125.1	✓	64
512.213	even	128	inner	3584.1.cd.a.1749.1	yes	64
3584.1749	odd	128	inner	3584.1.cd.a.1749.1	yes	64

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3584.1.cd.a.125.1	✓	64	1.1	even	1	trivial
3584.1.cd.a.125.1	✓	64	7.6	odd	2	CM
3584.1.cd.a.1749.1	yes	64	512.213	even	128	inner
3584.1.cd.a.1749.1	yes	64	3584.1749	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)

\(f(q)\)	\(=\)	\(q+(-0.514103 + 0.857729i) q^{2} +(-0.471397 - 0.881921i) q^{4} +(-0.671559 + 0.740951i) q^{7} +(0.998795 + 0.0490677i) q^{8} +(0.427555 - 0.903989i) q^{9} +O(q^{10})\)	\(q+(-0.514103 + 0.857729i) q^{2} +(-0.471397 - 0.881921i) q^{4} +(-0.671559 + 0.740951i) q^{7} +(0.998795 + 0.0490677i) q^{8} +(0.427555 - 0.903989i) q^{9} +(0.244748 - 0.00600822i) q^{11} +(-0.290285 - 0.956940i) q^{14} +(-0.555570 + 0.831470i) q^{16} +(0.555570 + 0.831470i) q^{18} +(-0.120672 + 0.213016i) q^{22} +(-0.968101 - 1.61518i) q^{23} +(0.989177 + 0.146730i) q^{25} +(0.970031 + 0.242980i) q^{28} +(1.21881 - 0.470147i) q^{29} +(-0.427555 - 0.903989i) q^{32} +(-0.998795 + 0.0490677i) q^{36} +(-0.533265 + 1.92697i) q^{37} +(-0.400609 - 1.78399i) q^{43} +(-0.120672 - 0.213016i) q^{44} +1.88309 q^{46} +(-0.0980171 - 0.995185i) q^{49} +(-0.634393 + 0.773010i) q^{50} +(1.07445 + 0.414461i) q^{53} +(-0.707107 + 0.707107i) q^{56} +(-0.223335 + 1.28711i) q^{58} +(0.382683 + 0.923880i) q^{63} +(0.995185 + 0.0980171i) q^{64} +(1.12363 - 1.59544i) q^{67} +(1.19462 + 0.427441i) q^{71} +(0.471397 - 0.881921i) q^{72} +(-1.37867 - 1.44806i) q^{74} +(-0.159911 + 0.185381i) q^{77} +(-0.920964 - 0.755815i) q^{79} +(-0.634393 - 0.773010i) q^{81} +(1.73614 + 0.573542i) q^{86} +(0.244748 + 0.00600822i) q^{88} +(-0.968101 + 1.61518i) q^{92} +(0.903989 + 0.427555i) q^{98} +(0.0992117 - 0.223818i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q+O(q^{10}) \) 64 * q	\( 64 q+O(q^{100}) \) 64 * q

Introduction

L-functions

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