LMFDB - Embedded newform 3584.1.cd.a.181.1

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			181.1
Root			$0.336890 - 0.941544i$ of defining polynomial
Character	$\chi$	$=$	3584.181
Dual form			3584.1.cd.a.3485.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.989177 - 0.146730i) q^{2} +(0.956940 - 0.290285i) q^{4} +(-0.336890 - 0.941544i) q^{7} +(0.903989 - 0.427555i) q^{8} +(-0.740951 - 0.671559i) q^{9} +O(q^{10})$	\(q+(0.989177 - 0.146730i) q^{2} +(0.956940 - 0.290285i) q^{4} +(-0.336890 - 0.941544i) q^{7} +(0.903989 - 0.427555i) q^{8} +(-0.740951 - 0.671559i) q^{9} +(1.74304 + 0.391413i) q^{11} +(-0.471397 - 0.881921i) q^{14} +(0.831470 - 0.555570i) q^{16} +(-0.831470 - 0.555570i) q^{18} +(1.78161 + 0.131419i) q^{22} +(-1.97597 - 0.293107i) q^{23} +(0.242980 - 0.970031i) q^{25} +(-0.595699 - 0.803208i) q^{28} +(-0.241217 - 0.0418551i) q^{29} +(0.740951 - 0.671559i) q^{32} +(-0.903989 - 0.427555i) q^{36} +(-1.27460 + 1.47762i) q^{37} +(1.49489 + 0.662638i) q^{43} +(1.78161 - 0.131419i) q^{44} -1.99759 q^{46} +(-0.773010 + 0.634393i) q^{49} +(0.0980171 - 0.995185i) q^{50} +(1.35878 - 0.235770i) q^{53} +(-0.707107 - 0.707107i) q^{56} +(-0.244748 - 0.00600822i) q^{58} +(-0.382683 + 0.923880i) q^{63} +(0.634393 - 0.773010i) q^{64} +(0.558861 - 0.586990i) q^{67} +(0.195798 + 0.00961895i) q^{71} +(-0.956940 - 0.290285i) q^{72} +(-1.04400 + 1.64865i) q^{74} +(-0.218680 - 1.77301i) q^{77} +(1.02325 + 0.100782i) q^{79} +(0.0980171 + 0.995185i) q^{81} +(1.57594 + 0.436121i) q^{86} +(1.74304 - 0.391413i) q^{88} +(-1.97597 + 0.293107i) q^{92} +(-0.671559 + 0.740951i) q^{98} +(-1.02865 - 1.46057i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/3584\mathbb{Z}\right)^\times$.

$n$	$1023$	$1025$	$1541$
$\chi(n)$	$1$	$-1$	$e\left(\frac{101}{128}\right)$

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 $

$2$	0.989177	−	0.146730i	0.989177	−	0.146730i
$2$	0.989177	−	0.146730i	0.989177	−	0.146730i
$3$		0			0		0.359895	−	0.932993i	$-0.382812\pi$
$3$		0			0		−0.359895	+	0.932993i	$0.617188\pi$
$4$	0.956940	−	0.290285i	0.956940	−	0.290285i
$5$		0			0		0.788346	−	0.615232i	$-0.210938\pi$
$5$		0			0		−0.788346	+	0.615232i	$0.789062\pi$
$6$		0			0
$7$	−0.336890	−	0.941544i	−0.336890	−	0.941544i
$7$	−0.336890	−	0.941544i	−0.336890	−	0.941544i
$8$	0.903989	−	0.427555i	0.903989	−	0.427555i
$9$	−0.740951	−	0.671559i	−0.740951	−	0.671559i
$10$		0			0
$11$	1.74304	+	0.391413i	1.74304	+	0.391413i	0.970031	−	0.242980i	$-0.0781250\pi$
$11$	1.74304	+	0.391413i	1.74304	+	0.391413i	0.773010	+	0.634393i	$0.218750\pi$
$12$		0			0
$13$		0			0		0.266713	−	0.963776i	$-0.414062\pi$
$13$		0			0		−0.266713	+	0.963776i	$0.585938\pi$
$14$	−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		−0.893224	−	0.449611i	$-0.851562\pi$
$19$		0			0		0.893224	+	0.449611i	$0.148438\pi$
$20$		0			0
$21$		0			0
$22$	1.78161	+	0.131419i	1.78161	+	0.131419i
$23$	−1.97597	−	0.293107i	−1.97597	−	0.293107i	−0.995185	−	0.0980171i	$-0.968750\pi$
$23$	−1.97597	−	0.293107i	−1.97597	−	0.293107i	−0.980785	−	0.195090i	$-0.937500\pi$
$24$		0			0
$25$	0.242980	−	0.970031i	0.242980	−	0.970031i
$26$		0			0
$27$		0			0
$28$	−0.595699	−	0.803208i	−0.595699	−	0.803208i
$29$	−0.241217	−	0.0418551i	−0.241217	−	0.0418551i	0.0490677	−	0.998795i	$-0.484375\pi$
$29$	−0.241217	−	0.0418551i	−0.241217	−	0.0418551i	−0.290285	+	0.956940i	$0.593750\pi$
$30$		0			0
$31$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$31$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$32$	0.740951	−	0.671559i	0.740951	−	0.671559i
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−0.903989	−	0.427555i	−0.903989	−	0.427555i
$37$	−1.27460	+	1.47762i	−1.27460	+	1.47762i	−0.471397	+	0.881921i	$0.656250\pi$
$37$	−1.27460	+	1.47762i	−1.27460	+	1.47762i	−0.803208	+	0.595699i	$0.796875\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		−0.242980	−	0.970031i	$-0.578125\pi$
$41$		0			0		0.242980	+	0.970031i	$0.421875\pi$
$42$		0			0
$43$	1.49489	+	0.662638i	1.49489	+	0.662638i	0.980785	−	0.195090i	$-0.0625000\pi$
$43$	1.49489	+	0.662638i	1.49489	+	0.662638i	0.514103	+	0.857729i	$0.328125\pi$
$44$	1.78161	−	0.131419i	1.78161	−	0.131419i
$45$		0			0
$46$	−1.99759			−1.99759
$47$		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
$53$	1.35878	−	0.235770i	1.35878	−	0.235770i	0.555570	−	0.831470i	$-0.312500\pi$
$53$	1.35878	−	0.235770i	1.35878	−	0.235770i	0.803208	+	0.595699i	$0.203125\pi$
$54$		0			0
$55$		0			0
$56$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$57$		0			0
$58$	−0.244748	−	0.00600822i	−0.244748	−	0.00600822i
$59$		0			0		0.963776	−	0.266713i	$-0.0859375\pi$
$59$		0			0		−0.963776	+	0.266713i	$0.914062\pi$
$60$		0			0
$61$		0			0		−0.0245412	−	0.999699i	$-0.507812\pi$
$61$		0			0		0.0245412	+	0.999699i	$0.492188\pi$
$62$		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
$67$	0.558861	−	0.586990i	0.558861	−	0.586990i	−0.382683	−	0.923880i	$-0.625000\pi$
$67$	0.558861	−	0.586990i	0.558861	−	0.586990i	0.941544	+	0.336890i	$0.109375\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	0.195798	+	0.00961895i	0.195798	+	0.00961895i	0.146730	−	0.989177i	$-0.453125\pi$
$71$	0.195798	+	0.00961895i	0.195798	+	0.00961895i	0.0490677	+	0.998795i	$0.484375\pi$
$72$	−0.956940	−	0.290285i	−0.956940	−	0.290285i
$73$		0			0		0.336890	−	0.941544i	$-0.390625\pi$
$73$		0			0		−0.336890	+	0.941544i	$0.609375\pi$
$74$	−1.04400	+	1.64865i	−1.04400	+	1.64865i
$75$		0			0
$76$		0			0
$77$	−0.218680	−	1.77301i	−0.218680	−	1.77301i
$78$		0			0
$79$	1.02325	+	0.100782i	1.02325	+	0.100782i	0.595699	−	0.803208i	$-0.296875\pi$
$79$	1.02325	+	0.100782i	1.02325	+	0.100782i	0.427555	+	0.903989i	$0.359375\pi$
$80$		0			0
$81$	0.0980171	+	0.995185i	0.0980171	+	0.995185i
$82$		0			0
$83$		0			0		−0.653173	−	0.757209i	$-0.726562\pi$
$83$		0			0		0.653173	+	0.757209i	$0.273438\pi$
$84$		0			0
$85$		0			0
$86$	1.57594	+	0.436121i	1.57594	+	0.436121i
$87$		0			0
$88$	1.74304	−	0.391413i	1.74304	−	0.391413i
$89$		0			0		0.989177	−	0.146730i	$-0.0468750\pi$
$89$		0			0		−0.989177	+	0.146730i	$0.953125\pi$
$90$		0			0
$91$		0			0
$92$	−1.97597	+	0.293107i	−1.97597	+	0.293107i
$93$		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	$0.0625000\pi$
$98$	−0.671559	+	0.740951i	−0.671559	+	0.740951i
$99$	−1.02865	−	1.46057i	−1.02865	−	1.46057i
$100$	−0.0490677	−	0.998795i	−0.0490677	−	0.998795i
$101$		0			0		0.313682	−	0.949528i	$-0.398438\pi$
$101$		0			0		−0.313682	+	0.949528i	$0.601562\pi$
$102$		0			0
$103$		0			0		0.857729	−	0.514103i	$-0.171875\pi$
$103$		0			0		−0.857729	+	0.514103i	$0.828125\pi$
$104$		0			0
$105$		0			0
$106$	1.30948	−	0.432593i	1.30948	−	0.432593i
$107$	−0.0338443	−	0.0355478i	−0.0338443	−	0.0355478i	0.707107	−	0.707107i	$-0.250000\pi$
$107$	−0.0338443	−	0.0355478i	−0.0338443	−	0.0355478i	−0.740951	+	0.671559i	$0.765625\pi$
$108$		0			0
$109$	−1.43798	+	0.475045i	−1.43798	+	0.475045i	−0.923880	−	0.382683i	$-0.875000\pi$
$109$	−1.43798	+	0.475045i	−1.43798	+	0.475045i	−0.514103	+	0.857729i	$0.671875\pi$
$110$		0			0
$111$		0			0
$112$	−0.803208	−	0.595699i	−0.803208	−	0.595699i
$113$	−1.66074	−	0.887682i	−1.66074	−	0.887682i	−0.989177	−	0.146730i	$-0.953125\pi$
$113$	−1.66074	−	0.887682i	−1.66074	−	0.887682i	−0.671559	−	0.740951i	$-0.734375\pi$
$114$		0			0
$115$		0			0
$116$	−0.242980	+	0.0299687i	−0.242980	+	0.0299687i
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	1.98100	+	0.936944i	1.98100	+	0.936944i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$	−0.242980	+	0.970031i	−0.242980	+	0.970031i
$127$	1.24723	+	1.24723i	1.24723	+	1.24723i	0.956940	+	0.290285i	$0.0937500\pi$
$127$	1.24723	+	1.24723i	1.24723	+	1.24723i	0.290285	+	0.956940i	$0.406250\pi$
$128$	0.514103	−	0.857729i	0.514103	−	0.857729i
$129$		0			0
$130$		0			0
$131$		0			0		0.914210	−	0.405241i	$-0.132812\pi$
$131$		0			0		−0.914210	+	0.405241i	$0.867188\pi$
$132$		0			0
$133$		0			0
$134$	0.466683	−	0.662638i	0.466683	−	0.662638i
$135$		0			0
$136$		0			0
$137$	−0.485375	+	0.0238449i	−0.485375	+	0.0238449i	−0.290285	−	0.956940i	$-0.593750\pi$
$137$	−0.485375	+	0.0238449i	−0.485375	+	0.0238449i	−0.195090	+	0.980785i	$0.562500\pi$
$138$		0			0
$139$		0			0		−0.534998	−	0.844854i	$-0.679688\pi$
$139$		0			0		0.534998	+	0.844854i	$0.320312\pi$
$140$		0			0
$141$		0			0
$142$	0.195090	−	0.0192147i	0.195090	−	0.0192147i
$143$		0			0
$144$	−0.989177	−	0.146730i	−0.989177	−	0.146730i
$145$		0			0
$146$		0			0
$147$		0			0
$148$	−0.790790	+	1.78399i	−0.790790	+	1.78399i
$149$	−1.35143	+	1.28667i	−1.35143	+	1.28667i	−0.427555	+	0.903989i	$0.640625\pi$
$149$	−1.35143	+	1.28667i	−1.35143	+	1.28667i	−0.923880	+	0.382683i	$0.875000\pi$
$150$		0			0
$151$	1.17850	+	1.58903i	1.17850	+	1.58903i	0.707107	+	0.707107i	$0.250000\pi$
$151$	1.17850	+	1.58903i	1.17850	+	1.58903i	0.471397	+	0.881921i	$0.343750\pi$
$152$		0			0
$153$		0			0
$154$	−0.476469	−	1.72174i	−0.476469	−	1.72174i
$155$		0			0
$156$		0			0
$157$		0			0		0.817585	−	0.575808i	$-0.195312\pi$
$157$		0			0		−0.817585	+	0.575808i	$0.804688\pi$
$158$	1.02697	−	0.0504517i	1.02697	−	0.0504517i
$159$		0			0
$160$		0			0
$161$	0.389711	+	1.95921i	0.389711	+	1.95921i
$162$	0.242980	+	0.970031i	0.242980	+	0.970031i
$163$	0.116874	+	0.520464i	0.116874	+	0.520464i	0.998795	+	0.0490677i	$0.0156250\pi$
$163$	0.116874	+	0.520464i	0.116874	+	0.520464i	−0.881921	+	0.471397i	$0.843750\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.146730	−	0.989177i	$-0.546875\pi$
$167$		0			0		0.146730	+	0.989177i	$0.453125\pi$
$168$		0			0
$169$	−0.857729	−	0.514103i	−0.857729	−	0.514103i
$170$		0			0
$171$		0			0
$172$	1.62287	+	0.200162i	1.62287	+	0.200162i
$173$		0			0		0.757209	−	0.653173i	$-0.226562\pi$
$173$		0			0		−0.757209	+	0.653173i	$0.773438\pi$
$174$		0			0
$175$	−0.995185	+	0.0980171i	−0.995185	+	0.0980171i
$176$	1.66674	−	0.642934i	1.66674	−	0.642934i
$177$		0			0
$178$		0			0
$179$	1.06195	−	0.130979i	1.06195	−	0.130979i	0.427555	−	0.903989i	$-0.359375\pi$
$179$	1.06195	−	0.130979i	1.06195	−	0.130979i	0.634393	+	0.773010i	$0.281250\pi$
$180$		0			0
$181$		0			0		0.575808	−	0.817585i	$-0.304688\pi$
$181$		0			0		−0.575808	+	0.817585i	$0.695312\pi$
$182$		0			0
$183$		0			0
$184$	−1.91158	+	0.579870i	−1.91158	+	0.579870i
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	1.48413	−	0.614748i	1.48413	−	0.614748i	0.514103	−	0.857729i	$-0.328125\pi$
$191$	1.48413	−	0.614748i	1.48413	−	0.614748i	0.970031	+	0.242980i	$0.0781250\pi$
$192$		0			0
$193$	−1.36910	−	0.567099i	−1.36910	−	0.567099i	−0.427555	−	0.903989i	$-0.640625\pi$
$193$	−1.36910	−	0.567099i	−1.36910	−	0.567099i	−0.941544	+	0.336890i	$0.890625\pi$
$194$		0			0
$195$		0			0
$196$	−0.555570	+	0.831470i	−0.555570	+	0.831470i
$197$	0.0911954	+	0.329538i	0.0911954	+	0.329538i	0.995185	−	0.0980171i	$-0.0312500\pi$
$197$	0.0911954	+	0.329538i	0.0911954	+	0.329538i	−0.903989	+	0.427555i	$0.859375\pi$
$198$	−1.23183	−	1.29383i	−1.23183	−	1.29383i
$199$		0			0		−0.671559	−	0.740951i	$-0.734375\pi$
$199$		0			0		0.671559	+	0.740951i	$0.265625\pi$
$200$	−0.195090	−	0.980785i	−0.195090	−	0.980785i
$201$		0			0
$202$		0			0
$203$	0.0418551	+	0.241217i	0.0418551	+	0.241217i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	1.26726	+	1.54416i	1.26726	+	1.54416i
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−0.115989	+	1.57242i	−0.115989	+	1.57242i	0.555570	+	0.831470i	$0.312500\pi$
$211$	−0.115989	+	1.57242i	−0.115989	+	1.57242i	−0.671559	+	0.740951i	$0.734375\pi$
$212$	1.23183	−	0.620050i	1.23183	−	0.620050i
$213$		0			0
$214$	−0.0386940	−	0.0301971i	−0.0386940	−	0.0301971i
$215$		0			0
$216$		0			0
$217$		0			0
$218$	−1.35271	+	0.680899i	−1.35271	+	0.680899i
$219$		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
$226$	−1.77301	−	0.634393i	−1.77301	−	0.634393i
$227$		0			0		0.170962	−	0.985278i	$-0.445312\pi$
$227$		0			0		−0.170962	+	0.985278i	$0.554688\pi$
$228$		0			0
$229$		0			0		−0.449611	−	0.893224i	$-0.648438\pi$
$229$		0			0		0.449611	+	0.893224i	$0.351562\pi$
$230$		0			0
$231$		0			0
$232$	−0.235953	+	0.0652970i	−0.235953	+	0.0652970i
$233$	−0.251710	+	1.69689i	−0.251710	+	1.69689i	0.382683	+	0.923880i	$0.375000\pi$
$233$	−0.251710	+	1.69689i	−0.251710	+	1.69689i	−0.634393	+	0.773010i	$0.718750\pi$
$234$		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	$0.906250\pi$
$242$	2.09704	+	0.636129i	2.09704	+	0.636129i
$243$		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		−0.615232	−	0.788346i	$-0.710938\pi$
$251$		0			0		0.615232	+	0.788346i	$0.289062\pi$
$252$	−0.0980171	+	0.995185i	−0.0980171	+	0.995185i
$253$	−3.32947	−	1.28432i	−3.32947	−	1.28432i
$254$	1.41673	+	1.05072i	1.41673	+	1.05072i
$255$		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
$259$	1.82065	+	0.702301i	1.82065	+	0.702301i
$260$		0			0
$261$	0.150622	+	0.193004i	0.150622	+	0.193004i
$262$		0			0
$263$	−1.84691	+	0.660833i	−1.84691	+	0.660833i	−0.857729	+	0.514103i	$0.828125\pi$
$263$	−1.84691	+	0.660833i	−1.84691	+	0.660833i	−0.989177	+	0.146730i	$0.953125\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	0.364402	−	0.723943i	0.364402	−	0.723943i
$269$		0			0		−0.963776	−	0.266713i	$-0.914062\pi$
$269$		0			0		0.963776	+	0.266713i	$0.0859375\pi$
$270$		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
$274$	−0.476623	+	0.0948062i	−0.476623	+	0.0948062i
$275$	0.803208	−	1.59570i	0.803208	−	1.59570i
$276$		0			0
$277$	0.0489495	−	1.99398i	0.0489495	−	1.99398i	−0.0490677	−	0.998795i	$-0.515625\pi$
$277$	0.0489495	−	1.99398i	0.0489495	−	1.99398i	0.0980171	−	0.995185i	$-0.468750\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	0.284666	+	0.0713052i	0.284666	+	0.0713052i	0.382683	−	0.923880i	$-0.375000\pi$
$281$	0.284666	+	0.0713052i	0.284666	+	0.0713052i	−0.0980171	+	0.995185i	$0.531250\pi$
$282$		0			0
$283$		0			0		−0.449611	−	0.893224i	$-0.648438\pi$
$283$		0			0		0.449611	+	0.893224i	$0.351562\pi$
$284$	0.190159	−	0.0476324i	0.190159	−	0.0476324i
$285$		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		−0.757209	−	0.653173i	$-0.773438\pi$
$293$		0			0		0.757209	+	0.653173i	$0.226562\pi$
$294$		0			0
$295$		0			0
$296$	−0.520464	+	1.88072i	−0.520464	+	1.88072i
$297$		0			0
$298$	−1.14801	+	1.47104i	−1.14801	+	1.47104i
$299$		0			0
$300$		0			0
$301$	0.120291	−	1.63074i	0.120291	−	1.63074i
$302$	1.39891	+	1.39891i	1.39891	+	1.39891i
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0		−0.788346	−	0.615232i	$-0.789062\pi$
$307$		0			0		0.788346	+	0.615232i	$0.210938\pi$
$308$	−0.723943	−	1.63319i	−0.723943	−	1.63319i
$309$		0			0
$310$		0			0
$311$		0			0		−0.427555	−	0.903989i	$-0.640625\pi$
$311$		0			0		0.427555	+	0.903989i	$0.359375\pi$
$312$		0			0
$313$		0			0		−0.671559	−	0.740951i	$-0.734375\pi$
$313$		0			0		0.671559	+	0.740951i	$0.265625\pi$
$314$		0			0
$315$		0			0
$316$	1.00845	−	0.200593i	1.00845	−	0.200593i
$317$	−1.92697	+	0.0473045i	−1.92697	+	0.0473045i	−0.970031	−	0.242980i	$-0.921875\pi$
$317$	−1.92697	+	0.0473045i	−1.92697	+	0.0473045i	−0.956940	+	0.290285i	$0.906250\pi$
$318$		0			0
$319$	−0.404069	−	0.167371i	−0.404069	−	0.167371i
$320$		0			0
$321$		0			0
$322$	0.672968	+	1.88082i	0.672968	+	1.88082i
$323$		0			0
$324$	0.382683	+	0.923880i	0.382683	+	0.923880i
$325$		0			0
$326$	0.191977	+	0.497682i	0.191977	+	0.497682i
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−1.00201	+	1.42274i	−1.00201	+	1.42274i	−0.0980171	+	0.995185i	$0.531250\pi$
$331$	−1.00201	+	1.42274i	−1.00201	+	1.42274i	−0.903989	+	0.427555i	$0.859375\pi$
$332$		0			0
$333$	1.93673	−	0.238873i	1.93673	−	0.238873i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−1.79927	+	0.177213i	−1.79927	+	0.177213i	−0.941544	−	0.336890i	$-0.890625\pi$
$337$	−1.79927	+	0.177213i	−1.79927	+	0.177213i	−0.857729	+	0.514103i	$0.828125\pi$
$338$	−0.923880	−	0.382683i	−0.923880	−	0.382683i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	0.857729	+	0.514103i	0.857729	+	0.514103i
$344$	1.63468	−	0.0401291i	1.63468	−	0.0401291i
$345$		0			0
$346$		0			0
$347$	−0.0725197	−	0.983125i	−0.0725197	−	0.983125i	−0.903989	−	0.427555i	$-0.859375\pi$
$347$	−0.0725197	−	0.983125i	−0.0725197	−	0.983125i	0.831470	−	0.555570i	$-0.187500\pi$
$348$		0			0
$349$		0			0		−0.219101	−	0.975702i	$-0.570312\pi$
$349$		0			0		0.219101	+	0.975702i	$0.429688\pi$
$350$	−0.970031	+	0.242980i	−0.970031	+	0.242980i
$351$		0			0
$352$	1.55437	−	0.880537i	1.55437	−	0.880537i
$353$		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
$358$	1.03124	−	0.285381i	1.03124	−	0.285381i
$359$	−0.929487	−	1.55075i	−0.929487	−	1.55075i	−0.831470	−	0.555570i	$-0.812500\pi$
$359$	−0.929487	−	1.55075i	−0.929487	−	1.55075i	−0.0980171	−	0.995185i	$-0.531250\pi$
$360$		0			0
$361$	0.595699	+	0.803208i	0.595699	+	0.803208i
$362$		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	$0.843750\pi$
$368$	−1.80580	+	0.854080i	−1.80580	+	0.854080i
$369$		0			0
$370$		0			0
$371$	−0.679747	−	1.19992i	−0.679747	−	1.19992i
$372$		0			0
$373$	−1.01599	−	1.60442i	−1.01599	−	1.60442i	−0.773010	−	0.634393i	$-0.781250\pi$
$373$	−1.01599	−	1.60442i	−1.01599	−	1.60442i	−0.242980	−	0.970031i	$-0.578125\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	0.177311	−	1.43760i	0.177311	−	1.43760i	−0.595699	−	0.803208i	$-0.703125\pi$
$379$	0.177311	−	1.43760i	0.177311	−	1.43760i	0.773010	−	0.634393i	$-0.218750\pi$
$380$		0			0
$381$		0			0
$382$	1.37787	−	0.825862i	1.37787	−	0.825862i
$383$		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
$386$	−1.43749	−	0.360073i	−1.43749	−	0.360073i
$387$	−0.662638	−	1.49489i	−0.662638	−	1.49489i
$388$		0			0
$389$	1.14296	+	0.140970i	1.14296	+	0.140970i	0.671559	−	0.740951i	$-0.265625\pi$
$389$	1.14296	+	0.140970i	1.14296	+	0.140970i	0.471397	+	0.881921i	$0.343750\pi$
$390$		0			0
$391$		0			0
$392$	−0.427555	+	0.903989i	−0.427555	+	0.903989i
$393$		0			0
$394$	0.138562	+	0.312590i	0.138562	+	0.312590i
$395$		0			0
$396$	−1.40834	−	1.09908i	−1.40834	−	1.09908i
$397$		0			0		0.870087	−	0.492898i	$-0.164062\pi$
$397$		0			0		−0.870087	+	0.492898i	$0.835938\pi$
$398$		0			0
$399$		0			0
$400$	−0.336890	−	0.941544i	−0.336890	−	0.941544i
$401$	0.403096	+	0.754140i	0.403096	+	0.754140i	0.998795	−	0.0490677i	$-0.0156250\pi$
$401$	0.403096	+	0.754140i	0.403096	+	0.754140i	−0.595699	+	0.803208i	$0.703125\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$	0.0767960	+	0.232465i	0.0767960	+	0.232465i
$407$	−2.80005	+	2.07666i	−2.80005	+	2.07666i
$408$		0			0
$409$		0			0		0.857729	−	0.514103i	$-0.171875\pi$
$409$		0			0		−0.857729	+	0.514103i	$0.828125\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$	1.48012	+	1.34150i	1.48012	+	1.34150i
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		0.975702	−	0.219101i	$-0.0703125\pi$
$419$		0			0		−0.975702	+	0.219101i	$0.929688\pi$
$420$		0			0
$421$	0.531980	−	0.0392412i	0.531980	−	0.0392412i	0.195090	−	0.980785i	$-0.437500\pi$
$421$	0.531980	−	0.0392412i	0.531980	−	0.0392412i	0.336890	+	0.941544i	$0.390625\pi$
$422$	0.115989	+	1.57242i	0.115989	+	1.57242i
$423$		0			0
$424$	1.12752	−	0.794086i	1.12752	−	0.794086i
$425$		0			0
$426$		0			0
$427$		0			0
$428$	−0.0427060	−	0.0241927i	−0.0427060	−	0.0241927i
$429$		0			0
$430$		0			0
$431$	−0.162997	−	1.65493i	−0.162997	−	1.65493i	−0.634393	−	0.773010i	$-0.718750\pi$
$431$	−0.162997	−	1.65493i	−0.162997	−	1.65493i	0.471397	−	0.881921i	$-0.343750\pi$
$432$		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
$436$	−1.23816	+	0.872014i	−1.23816	+	0.872014i
$437$		0			0
$438$		0			0
$439$		0			0		0.336890	−	0.941544i	$-0.390625\pi$
$439$		0			0		−0.336890	+	0.941544i	$0.609375\pi$
$440$		0			0
$441$	0.998795	+	0.0490677i	0.998795	+	0.0490677i
$442$		0			0
$443$	−0.215989	+	0.381274i	−0.215989	+	0.381274i	−0.956940	−	0.290285i	$-0.906250\pi$
$443$	−0.215989	+	0.381274i	−0.215989	+	0.381274i	0.740951	+	0.671559i	$0.234375\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−0.941544	−	0.336890i	−0.941544	−	0.336890i
$449$	0.485544	−	1.17221i	0.485544	−	1.17221i	−0.471397	−	0.881921i	$-0.656250\pi$
$449$	0.485544	−	1.17221i	0.485544	−	1.17221i	0.956940	−	0.290285i	$-0.0937500\pi$
$450$	−0.740951	+	0.671559i	−0.740951	+	0.671559i
$451$		0			0
$452$	−1.84691	−	0.367372i	−1.84691	−	0.367372i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	1.73013	−	0.818289i	1.73013	−	0.818289i	0.740951	−	0.671559i	$-0.234375\pi$
$457$	1.73013	−	0.818289i	1.73013	−	0.818289i	0.989177	−	0.146730i	$-0.0468750\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		0.615232	−	0.788346i	$-0.289062\pi$
$461$		0			0		−0.615232	+	0.788346i	$0.710938\pi$
$462$		0			0
$463$	0.661009	−	0.542476i	0.661009	−	0.542476i	−0.242980	−	0.970031i	$-0.578125\pi$
$463$	0.661009	−	0.542476i	0.661009	−	0.542476i	0.903989	+	0.427555i	$0.140625\pi$
$464$	−0.223818	+	0.0992117i	−0.223818	+	0.0992117i
$465$		0			0
$466$			1.71546i			1.71546i
$467$		0			0		−0.997290	−	0.0735646i	$-0.976562\pi$
$467$		0			0		0.997290	+	0.0735646i	$0.0234375\pi$
$468$		0			0
$469$	−0.740951	−	0.328441i	−0.740951	−	0.328441i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	2.34629	+	1.74012i	2.34629	+	1.74012i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−1.16512	−	0.737805i	−1.16512	−	0.737805i
$478$	−0.280825	+	1.89317i	−0.280825	+	1.89317i
$479$		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
$484$	2.16768	+	0.321545i	2.16768	+	0.321545i
$485$		0			0
$486$		0			0
$487$	−0.118079	+	0.471397i	−0.118079	+	0.471397i	0.881921	+	0.471397i	$0.156250\pi$
$487$	−0.118079	+	0.471397i	−0.118079	+	0.471397i	−1.00000			$\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−1.86542	−	0.0457936i	−1.86542	−	0.0457936i	−0.923880	−	0.382683i	$-0.875000\pi$
$491$	−1.86542	−	0.0457936i	−1.86542	−	0.0457936i	−0.941544	+	0.336890i	$0.890625\pi$
$492$		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
$499$	0.191977	−	0.693716i	0.191977	−	0.693716i	−0.803208	−	0.595699i	$-0.796875\pi$
$499$	0.191977	−	0.693716i	0.191977	−	0.693716i	0.995185	−	0.0980171i	$-0.0312500\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		−0.740951	−	0.671559i	$-0.765625\pi$
$503$		0			0		0.740951	+	0.671559i	$0.234375\pi$
$504$	0.0490677	+	0.998795i	0.0490677	+	0.998795i
$505$		0			0
$506$	−3.48188	−	0.781883i	−3.48188	−	0.781883i
$507$		0			0
$508$	1.55557	+	0.831470i	1.55557	+	0.831470i
$509$		0			0		0.359895	−	0.932993i	$-0.382812\pi$
$509$		0			0		−0.359895	+	0.932993i	$0.617188\pi$
$510$		0			0
$511$		0			0
$512$	0.242980	−	0.970031i	0.242980	−	0.970031i
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$	1.90399	+	0.427555i	1.90399	+	0.427555i
$519$		0			0
$520$		0			0
$521$		0			0		−0.740951	−	0.671559i	$-0.765625\pi$
$521$		0			0		0.740951	+	0.671559i	$0.234375\pi$
$522$	0.177311	+	0.168814i	0.177311	+	0.168814i
$523$		0			0		−0.975702	−	0.219101i	$-0.929688\pi$
$523$		0			0		0.975702	+	0.219101i	$0.0703125\pi$
$524$		0			0
$525$		0			0
$526$	−1.72995	+	0.924678i	−1.72995	+	0.924678i
$527$		0			0
$528$		0			0
$529$	2.86161	+	0.868058i	2.86161	+	0.868058i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$	0.254234	−	0.769576i	0.254234	−	0.769576i
$537$		0			0
$538$		0			0
$539$	−1.59570	+	0.803208i	−1.59570	+	0.803208i
$540$		0			0
$541$	1.96996	+	0.341821i	1.96996	+	0.341821i	0.989177	+	0.146730i	$0.0468750\pi$
$541$	1.96996	+	0.341821i	1.96996	+	0.341821i	0.980785	+	0.195090i	$0.0625000\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	1.03956	+	0.658295i	1.03956	+	0.658295i	0.941544	−	0.336890i	$-0.109375\pi$
$547$	1.03956	+	0.658295i	1.03956	+	0.658295i	0.0980171	+	0.995185i	$0.468750\pi$
$548$	−0.457553	+	0.163715i	−0.457553	+	0.163715i
$549$		0			0
$550$	0.560376	−	1.69628i	0.560376	−	1.69628i
$551$		0			0
$552$		0			0
$553$	−0.249834	−	0.997391i	−0.249834	−	0.997391i
$554$	−0.244158	−	1.97958i	−0.244158	−	1.97958i
$555$		0			0
$556$		0			0
$557$	0.146730	+	0.0108235i	0.146730	+	0.0108235i	0.146730	−	0.989177i	$-0.453125\pi$
$557$	0.146730	+	0.0108235i	0.146730	+	0.0108235i			1.00000i	$0.5\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	0.292048	+	0.0287642i	0.292048	+	0.0287642i
$563$		0			0		0.615232	−	0.788346i	$-0.289062\pi$
$563$		0			0		−0.615232	+	0.788346i	$0.710938\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	0.903989	−	0.427555i	0.903989	−	0.427555i
$568$	0.181112	−	0.0750191i	0.181112	−	0.0750191i
$569$	−1.30692	+	1.18452i	−1.30692	+	1.18452i	−0.336890	+	0.941544i	$0.609375\pi$
$569$	−1.30692	+	1.18452i	−1.30692	+	1.18452i	−0.970031	+	0.242980i	$0.921875\pi$
$570$		0			0
$571$	1.91306	−	0.529414i	1.91306	−	0.529414i	0.923880	−	0.382683i	$-0.125000\pi$
$571$	1.91306	−	0.529414i	1.91306	−	0.529414i	0.989177	−	0.146730i	$-0.0468750\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−0.764445	+	1.84553i	−0.764445	+	1.84553i
$576$	−0.989177	+	0.146730i	−0.989177	+	0.146730i
$577$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$577$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$578$	0.903989	+	0.427555i	0.903989	+	0.427555i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	2.46069	+	0.120886i	2.46069	+	0.120886i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		−0.817585	−	0.575808i	$-0.804688\pi$
$587$		0			0		0.817585	+	0.575808i	$0.195312\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	−0.238873	+	1.93673i	−0.238873	+	1.93673i
$593$		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
$596$	−0.919741	+	1.62357i	−0.919741	+	1.62357i
$597$		0			0
$598$		0			0
$599$	−0.200593	+	0.334669i	−0.200593	+	0.334669i	−0.941544	−	0.336890i	$-0.890625\pi$
$599$	−0.200593	+	0.334669i	−0.200593	+	0.334669i	0.740951	+	0.671559i	$0.234375\pi$
$600$		0			0
$601$		0			0		0.989177	−	0.146730i	$-0.0468750\pi$
$601$		0			0		−0.989177	+	0.146730i	$0.953125\pi$
$602$	−0.120291	−	1.63074i	−0.120291	−	1.63074i
$603$	−0.808287	+	0.0596228i	−0.808287	+	0.0596228i
$604$	1.58903	+	1.17850i	1.58903	+	1.17850i
$605$		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
$613$	−0.309226	+	0.936041i	−0.309226	+	0.936041i	0.671559	+	0.740951i	$0.265625\pi$
$613$	−0.309226	+	0.936041i	−0.309226	+	0.936041i	−0.980785	+	0.195090i	$0.937500\pi$
$614$		0			0
$615$		0			0
$616$	−0.955746	−	1.50929i	−0.955746	−	1.50929i
$617$	−0.466318	+	0.345845i	−0.466318	+	0.345845i	−0.803208	−	0.595699i	$-0.796875\pi$
$617$	−0.466318	+	0.345845i	−0.466318	+	0.345845i	0.336890	+	0.941544i	$0.390625\pi$
$618$		0			0
$619$		0			0		−0.689541	−	0.724247i	$-0.742188\pi$
$619$		0			0		0.689541	+	0.724247i	$0.257812\pi$
$620$		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
$631$	−0.0622564	−	1.26726i	−0.0622564	−	1.26726i	−0.803208	−	0.595699i	$-0.796875\pi$
$631$	−0.0622564	−	1.26726i	−0.0622564	−	1.26726i	0.740951	−	0.671559i	$-0.234375\pi$
$632$	0.968101	−	0.346392i	0.968101	−	0.346392i
$633$		0			0
$634$	−1.89917	+	0.329538i	−1.89917	+	0.329538i
$635$		0			0
$636$		0			0
$637$		0			0
$638$	−0.424254	−	0.106270i	−0.424254	−	0.106270i
$639$	−0.138617	−	0.138617i	−0.138617	−	0.138617i
$640$		0			0
$641$	−0.949728	+	0.949728i	−0.949728	+	0.949728i	−0.998795	−	0.0490677i	$-0.984375\pi$
$641$	−0.949728	+	0.949728i	−0.949728	+	0.949728i	0.0490677	+	0.998795i	$0.484375\pi$
$642$		0			0
$643$		0			0		0.914210	−	0.405241i	$-0.132812\pi$
$643$		0			0		−0.914210	+	0.405241i	$0.867188\pi$
$644$	0.941658	+	1.76172i	0.941658	+	1.76172i
$645$		0			0
$646$		0			0
$647$		0			0		0.427555	−	0.903989i	$-0.359375\pi$
$647$		0			0		−0.427555	+	0.903989i	$0.640625\pi$
$648$	0.514103	+	0.857729i	0.514103	+	0.857729i
$649$		0			0
$650$		0			0
$651$		0			0
$652$	0.262924	+	0.464127i	0.262924	+	0.464127i
$653$	0.309226	+	0.545861i	0.309226	+	0.545861i	0.980785	−	0.195090i	$-0.0625000\pi$
$653$	0.309226	+	0.545861i	0.309226	+	0.545861i	−0.671559	+	0.740951i	$0.734375\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	0.409777	+	1.24041i	0.409777	+	1.24041i	0.923880	+	0.382683i	$0.125000\pi$
$659$	0.409777	+	1.24041i	0.409777	+	1.24041i	−0.514103	+	0.857729i	$0.671875\pi$
$660$		0			0
$661$		0			0		0.724247	−	0.689541i	$-0.242188\pi$
$661$		0			0		−0.724247	+	0.689541i	$0.757812\pi$
$662$	−0.782402	+	1.55437i	−0.782402	+	1.55437i
$663$		0			0
$664$		0			0
$665$		0			0
$666$	1.88072	−	0.520464i	1.88072	−	0.520464i
$667$	0.464370	+	0.153407i	0.464370	+	0.153407i
$668$		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	$-0.781250\pi$
$673$	−0.301614	−	1.51631i	−0.301614	−	1.51631i	0.471397	−	0.881921i	$-0.343750\pi$
$674$	−1.75380	+	0.439303i	−1.75380	+	0.439303i
$675$		0			0
$676$	−0.970031	−	0.242980i	−0.970031	−	0.242980i
$677$		0			0		−0.0735646	−	0.997290i	$-0.523438\pi$
$677$		0			0		0.0735646	+	0.997290i	$0.476562\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	0.608117	+	1.57648i	0.608117	+	1.57648i	0.803208	+	0.595699i	$0.203125\pi$
$683$	0.608117	+	1.57648i	0.608117	+	1.57648i	−0.195090	+	0.980785i	$0.562500\pi$
$684$		0			0
$685$		0			0
$686$	0.923880	+	0.382683i	0.923880	+	0.382683i
$687$		0			0
$688$	1.61110	−	0.279552i	1.61110	−	0.279552i
$689$		0			0
$690$		0			0
$691$		0			0		0.992480	−	0.122411i	$-0.0390625\pi$
$691$		0			0		−0.992480	+	0.122411i	$0.960938\pi$
$692$		0			0
$693$	−1.02865	+	1.46057i	−1.02865	+	1.46057i
$694$	−0.215989	−	0.961844i	−0.215989	−	0.961844i
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−0.923880	+	0.382683i	−0.923880	+	0.382683i
$701$	−1.42717	−	1.35878i	−1.42717	−	1.35878i	−0.831470	−	0.555570i	$-0.812500\pi$
$701$	−1.42717	−	1.35878i	−1.42717	−	1.35878i	−0.595699	−	0.803208i	$-0.703125\pi$
$702$		0			0
$703$		0			0
$704$	1.40834	−	1.09908i	1.40834	−	1.09908i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−0.167326	−	0.604638i	−0.167326	−	0.604638i	−0.998795	−	0.0490677i	$-0.984375\pi$
$709$	−0.167326	−	0.604638i	−0.167326	−	0.604638i	0.831470	−	0.555570i	$-0.187500\pi$
$710$		0			0
$711$	−0.690501	−	0.761850i	−0.690501	−	0.761850i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$	0.978200	−	0.433606i	0.978200	−	0.433606i
$717$		0			0
$718$	−1.14697	−	1.39759i	−1.14697	−	1.39759i
$719$		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
$725$	−0.0992117	+	0.223818i	−0.0992117	+	0.223818i
$726$		0			0
$727$		0			0		0.970031	−	0.242980i	$-0.0781250\pi$
$727$		0			0		−0.970031	+	0.242980i	$0.921875\pi$
$728$		0			0
$729$	0.595699	−	0.803208i	0.595699	−	0.803208i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		0.534998	−	0.844854i	$-0.320312\pi$
$733$		0			0		−0.534998	+	0.844854i	$0.679688\pi$
$734$		0			0
$735$		0			0
$736$	−1.66094	+	1.10980i	−1.66094	+	1.10980i
$737$	1.20387	−	0.804402i	1.20387	−	0.804402i
$738$		0			0
$739$	0.340997	−	1.96522i	0.340997	−	1.96522i	0.0980171	−	0.995185i	$-0.468750\pi$
$739$	0.340997	−	1.96522i	0.340997	−	1.96522i	0.242980	−	0.970031i	$-0.421875\pi$
$740$		0			0
$741$		0			0
$742$	−0.848454	−	1.08719i	−0.848454	−	1.08719i
$743$	1.15569	+	0.289486i	1.15569	+	0.289486i	0.773010	−	0.634393i	$-0.218750\pi$
$743$	1.15569	+	0.289486i	1.15569	+	0.289486i	0.382683	+	0.923880i	$0.375000\pi$
$744$		0			0
$745$		0			0
$746$	−1.24041	−	1.43798i	−1.24041	−	1.43798i
$747$		0			0
$748$		0			0
$749$	−0.0220680	+	0.0438416i	−0.0220680	+	0.0438416i
$750$		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
$757$	−0.0107540	+	0.0478899i	−0.0107540	+	0.0478899i	−0.980785	−	0.195090i	$-0.937500\pi$
$757$	−0.0107540	+	0.0478899i	−0.0107540	+	0.0478899i	0.970031	+	0.242980i	$0.0781250\pi$
$758$	−0.0355478	−	1.44806i	−0.0355478	−	1.44806i
$759$		0			0
$760$		0			0
$761$		0			0		0.941544	−	0.336890i	$-0.109375\pi$
$761$		0			0		−0.941544	+	0.336890i	$0.890625\pi$
$762$		0			0
$763$	0.931718	+	1.19389i	0.931718	+	1.19389i
$764$	1.24178	−	1.01910i	1.24178	−	1.01910i
$765$		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
$772$	−1.47477	−	0.145252i	−1.47477	−	0.145252i
$773$		0			0		−0.615232	−	0.788346i	$-0.710938\pi$
$773$		0			0		0.615232	+	0.788346i	$0.289062\pi$
$774$	−0.874812	−	1.38148i	−0.874812	−	1.38148i
$775$		0			0
$776$		0			0
$777$		0			0
$778$	1.15127	−	0.0282621i	1.15127	−	0.0282621i
$779$		0			0
$780$		0			0
$781$	0.337519	+	0.0934042i	0.337519	+	0.0934042i
$782$		0			0
$783$		0			0
$784$	−0.290285	+	0.956940i	−0.290285	+	0.956940i
$785$		0			0
$786$		0			0
$787$		0			0		0.449611	−	0.893224i	$-0.351562\pi$
$787$		0			0		−0.449611	+	0.893224i	$0.648438\pi$
$788$	0.182928	+	0.288876i	0.182928	+	0.288876i
$789$		0			0
$790$		0			0
$791$	−0.276306	+	1.86271i	−0.276306	+	1.86271i
$792$	−1.55437	−	0.880537i	−1.55437	−	0.880537i
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		0.170962	−	0.985278i	$-0.445312\pi$
$797$		0			0		−0.170962	+	0.985278i	$0.554688\pi$
$798$		0			0
$799$		0			0
$800$	−0.471397	−	0.881921i	−0.471397	−	0.881921i
$801$		0			0
$802$	0.509389	+	0.686831i	0.509389	+	0.686831i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−0.563170	+	0.141067i	−0.563170	+	0.141067i	−0.514103	−	0.857729i	$-0.671875\pi$
$809$	−0.563170	+	0.141067i	−0.563170	+	0.141067i	−0.0490677	+	0.998795i	$0.515625\pi$
$810$		0			0
$811$		0			0		0.405241	−	0.914210i	$-0.367188\pi$
$811$		0			0		−0.405241	+	0.914210i	$0.632812\pi$
$812$	0.110074	+	0.218680i	0.110074	+	0.218680i
$813$		0			0
$814$	−2.46503	+	2.46503i	−2.46503	+	2.46503i
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−0.168534	−	0.971283i	−0.168534	−	0.971283i	−0.941544	−	0.336890i	$-0.890625\pi$
$821$	−0.168534	−	0.971283i	−0.168534	−	0.971283i	0.773010	−	0.634393i	$-0.218750\pi$
$822$		0			0
$823$	−0.166824	−	0.352719i	−0.166824	−	0.352719i	0.803208	−	0.595699i	$-0.203125\pi$
$823$	−0.166824	−	0.352719i	−0.166824	−	0.352719i	−0.970031	+	0.242980i	$0.921875\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−0.525572	−	1.89917i	−0.525572	−	1.89917i	−0.427555	−	0.903989i	$-0.640625\pi$
$827$	−0.525572	−	1.89917i	−0.525572	−	1.89917i	−0.0980171	−	0.995185i	$-0.531250\pi$
$828$	1.66094	+	1.10980i	1.66094	+	1.10980i
$829$		0			0		0.999699	−	0.0245412i	$-0.00781250\pi$
$829$		0			0		−0.999699	+	0.0245412i	$0.992188\pi$
$830$		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		0.0490677	−	0.998795i	$-0.484375\pi$
$839$		0			0		−0.0490677	+	0.998795i	$0.515625\pi$
$840$		0			0
$841$	−0.885110	−	0.316698i	−0.885110	−	0.316698i
$842$	0.520464	−	0.116874i	0.520464	−	0.116874i
$843$		0			0
$844$	0.345455	+	1.53838i	0.345455	+	1.53838i
$845$		0			0
$846$		0			0
$847$	0.214795	−	2.18085i	0.214795	−	2.18085i
$848$	0.998795	−	0.950932i	0.998795	−	0.950932i
$849$		0			0
$850$		0			0
$851$	2.95168	−	2.54614i	2.95168	−	2.54614i
$852$		0			0
$853$		0			0		−0.359895	−	0.932993i	$-0.617188\pi$
$853$		0			0		0.359895	+	0.932993i	$0.382812\pi$
$854$		0			0
$855$		0			0
$856$	−0.0457936	−	0.0176645i	−0.0457936	−	0.0176645i
$857$		0			0		−0.146730	−	0.989177i	$-0.546875\pi$
$857$		0			0		0.146730	+	0.989177i	$0.453125\pi$
$858$		0			0
$859$		0			0		−0.0735646	−	0.997290i	$-0.523438\pi$
$859$		0			0		0.0735646	+	0.997290i	$0.476562\pi$
$860$		0			0
$861$		0			0
$862$	−0.404061	−	1.61310i	−0.404061	−	1.61310i
$863$	−0.0572514	−	0.287822i	−0.0572514	−	0.287822i	0.941544	−	0.336890i	$-0.109375\pi$
$863$	−0.0572514	−	0.287822i	−0.0572514	−	0.287822i	−0.998795	+	0.0490677i	$0.984375\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	1.74413	+	0.576182i	1.74413	+	0.576182i
$870$		0			0
$871$		0			0
$872$	−1.09681	+	1.04425i	−1.09681	+	1.04425i
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−0.618127	−	1.87110i	−0.618127	−	1.87110i	−0.471397	−	0.881921i	$-0.656250\pi$
$877$	−0.618127	−	1.87110i	−0.618127	−	1.87110i	−0.146730	−	0.989177i	$-0.546875\pi$
$878$		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
$883$	−0.950087	−	1.67714i	−0.950087	−	1.67714i	−0.707107	−	0.707107i	$-0.750000\pi$
$883$	−0.950087	−	1.67714i	−0.950087	−	1.67714i	−0.242980	−	0.970031i	$-0.578125\pi$
$884$		0			0
$885$		0			0
$886$	−0.157707	+	0.408840i	−0.157707	+	0.408840i
$887$		0			0		0.998795	−	0.0490677i	$-0.0156250\pi$
$887$		0			0		−0.998795	+	0.0490677i	$0.984375\pi$
$888$		0			0
$889$	0.754140	−	1.59449i	0.754140	−	1.59449i
$890$		0			0
$891$	−0.218680	+	1.77301i	−0.218680	+	1.77301i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$	−0.980785	−	0.195090i	−0.980785	−	0.195090i
$897$		0			0
$898$	0.308290	−	1.23076i	0.308290	−	1.23076i
$899$		0			0
$900$	−0.634393	+	0.773010i	−0.634393	+	0.773010i
$901$		0			0
$902$		0			0
$903$		0			0
$904$	−1.88082	−	0.0923988i	−1.88082	−	0.0923988i
$905$		0			0
$906$		0			0
$907$	−1.27946	+	0.810210i	−1.27946	+	0.810210i	−0.989177	−	0.146730i	$-0.953125\pi$
$907$	−1.27946	+	0.810210i	−1.27946	+	0.810210i	−0.290285	+	0.956940i	$0.593750\pi$
$908$		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
$919$	−0.757259	+	0.561621i	−0.757259	+	0.561621i	−0.903989	−	0.427555i	$-0.859375\pi$
$919$	−0.757259	+	0.561621i	−0.757259	+	0.561621i	0.146730	+	0.989177i	$0.453125\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	1.12363	+	1.59544i	1.12363	+	1.59544i
$926$	0.574257	−	0.633595i	0.574257	−	0.633595i
$927$		0			0
$928$	−0.206838	+	0.130979i	−0.206838	+	0.130979i
$929$		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
$932$	0.251710	+	1.69689i	0.251710	+	1.69689i
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		0.514103	−	0.857729i	$-0.328125\pi$
$937$		0			0		−0.514103	+	0.857729i	$0.671875\pi$
$938$	−0.781124	−	0.216166i	−0.781124	−	0.216166i
$939$		0			0
$940$		0			0
$941$		0			0		−0.653173	−	0.757209i	$-0.726562\pi$
$941$		0			0		0.653173	+	0.757209i	$0.273438\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$	2.57622	+	1.37702i	2.57622	+	1.37702i
$947$	−0.244158	−	1.97958i	−0.244158	−	1.97958i	−0.195090	−	0.980785i	$-0.562500\pi$
$947$	−0.244158	−	1.97958i	−0.244158	−	1.97958i	−0.0490677	−	0.998795i	$-0.515625\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	1.60448	+	0.0788231i	1.60448	+	0.0788231i	0.831470	−	0.555570i	$-0.187500\pi$
$953$	1.60448	+	0.0788231i	1.60448	+	0.0788231i	0.773010	+	0.634393i	$0.218750\pi$
$954$	−1.26077	−	0.558861i	−1.26077	−	0.558861i
$955$		0			0
$956$			1.91388i			1.91388i
$957$		0			0
$958$		0			0
$959$	0.185969	+	0.448969i	0.185969	+	0.448969i
$960$		0			0
$961$	−0.382683	+	0.923880i	−0.382683	+	0.923880i
$962$		0			0
$963$	0.00120454	+	0.0490677i	0.00120454	+	0.0490677i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−0.217440	+	0.197076i	−0.217440	+	0.197076i	−0.773010	−	0.634393i	$-0.781250\pi$
$967$	−0.217440	+	0.197076i	−0.217440	+	0.197076i	0.555570	+	0.831470i	$0.312500\pi$
$968$	2.19140			2.19140
$969$		0			0
$970$		0			0
$971$		0			0		0.985278	−	0.170962i	$-0.0546875\pi$
$971$		0			0		−0.985278	+	0.170962i	$0.945312\pi$
$972$		0			0
$973$		0			0
$974$	−0.0476324	+	0.483620i	−0.0476324	+	0.483620i
$975$		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
$981$	1.38450	+	0.613705i	1.38450	+	0.613705i
$982$	−1.85195	+	0.228417i	−1.85195	+	0.228417i
$983$		0			0		−0.242980	−	0.970031i	$-0.578125\pi$
$983$		0			0		0.242980	+	0.970031i	$0.421875\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−2.75963	−	1.74752i	−2.75963	−	1.74752i
$990$		0			0
$991$	0.475074	−	0.710998i	0.475074	−	0.710998i	−0.514103	−	0.857729i	$-0.671875\pi$
$991$	0.475074	−	0.710998i	0.475074	−	0.710998i	0.989177	+	0.146730i	$0.0468750\pi$
$992$		0			0
$993$		0			0
$994$	−0.0838155	−	0.177213i	−0.0838155	−	0.177213i
$995$		0			0
$996$		0			0
$997$		0			0		0.893224	−	0.449611i	$-0.148438\pi$
$997$		0			0		−0.893224	+	0.449611i	$0.851562\pi$
$998$	0.0881100	−	0.714377i	0.0881100	−	0.714377i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3584.1.cd.a.181.1	✓	64
7.6	odd	2	CM	3584.1.cd.a.181.1	✓	64
512.413	even	128	inner	3584.1.cd.a.3485.1	yes	64
3584.3485	odd	128	inner	3584.1.cd.a.3485.1	yes	64

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3584.1.cd.a.181.1	✓	64	1.1	even	1	trivial
3584.1.cd.a.181.1	✓	64	7.6	odd	2	CM
3584.1.cd.a.3485.1	yes	64	512.413	even	128	inner
3584.1.cd.a.3485.1	yes	64	3584.3485	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)

\(f(q)\)	\(=\)	\(q+(0.989177 - 0.146730i) q^{2} +(0.956940 - 0.290285i) q^{4} +(-0.336890 - 0.941544i) q^{7} +(0.903989 - 0.427555i) q^{8} +(-0.740951 - 0.671559i) q^{9} +O(q^{10})\)	\(q+(0.989177 - 0.146730i) q^{2} +(0.956940 - 0.290285i) q^{4} +(-0.336890 - 0.941544i) q^{7} +(0.903989 - 0.427555i) q^{8} +(-0.740951 - 0.671559i) q^{9} +(1.74304 + 0.391413i) q^{11} +(-0.471397 - 0.881921i) q^{14} +(0.831470 - 0.555570i) q^{16} +(-0.831470 - 0.555570i) q^{18} +(1.78161 + 0.131419i) q^{22} +(-1.97597 - 0.293107i) q^{23} +(0.242980 - 0.970031i) q^{25} +(-0.595699 - 0.803208i) q^{28} +(-0.241217 - 0.0418551i) q^{29} +(0.740951 - 0.671559i) q^{32} +(-0.903989 - 0.427555i) q^{36} +(-1.27460 + 1.47762i) q^{37} +(1.49489 + 0.662638i) q^{43} +(1.78161 - 0.131419i) q^{44} -1.99759 q^{46} +(-0.773010 + 0.634393i) q^{49} +(0.0980171 - 0.995185i) q^{50} +(1.35878 - 0.235770i) q^{53} +(-0.707107 - 0.707107i) q^{56} +(-0.244748 - 0.00600822i) q^{58} +(-0.382683 + 0.923880i) q^{63} +(0.634393 - 0.773010i) q^{64} +(0.558861 - 0.586990i) q^{67} +(0.195798 + 0.00961895i) q^{71} +(-0.956940 - 0.290285i) q^{72} +(-1.04400 + 1.64865i) q^{74} +(-0.218680 - 1.77301i) q^{77} +(1.02325 + 0.100782i) q^{79} +(0.0980171 + 0.995185i) q^{81} +(1.57594 + 0.436121i) q^{86} +(1.74304 - 0.391413i) q^{88} +(-1.97597 + 0.293107i) q^{92} +(-0.671559 + 0.740951i) q^{98} +(-1.02865 - 1.46057i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q+O(q^{10}) \) 64 * q	\( 64 q+O(q^{100}) \) 64 * q

Introduction

L-functions

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