LMFDB - Embedded newform 3584.1.cd.a.1301.1

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			1301.1
Root			$0.998795 - 0.0490677i$ of defining polynomial
Character	$\chi$	$=$	3584.1301
Dual form			3584.1.cd.a.573.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.970031 + 0.242980i) q^{2} +(0.881921 + 0.471397i) q^{4} +(-0.998795 - 0.0490677i) q^{7} +(0.740951 + 0.671559i) q^{8} +(-0.336890 + 0.941544i) q^{9} +O(q^{10})$	\(q+(0.970031 + 0.242980i) q^{2} +(0.881921 + 0.471397i) q^{4} +(-0.998795 - 0.0490677i) q^{7} +(0.740951 + 0.671559i) q^{8} +(-0.336890 + 0.941544i) q^{9} +(-1.79839 + 0.693716i) q^{11} +(-0.956940 - 0.290285i) q^{14} +(0.555570 + 0.831470i) q^{16} +(-0.555570 + 0.831470i) q^{18} +(-1.91306 + 0.235953i) q^{22} +(0.829484 - 0.207775i) q^{23} +(-0.595699 + 0.803208i) q^{25} +(-0.857729 - 0.514103i) q^{28} +(1.37539 + 1.30948i) q^{29} +(0.336890 + 0.941544i) q^{32} +(-0.740951 + 0.671559i) q^{36} +(-1.47104 + 1.14801i) q^{37} +(-0.341821 - 1.96996i) q^{43} +(-1.91306 - 0.235953i) q^{44} +0.855110 q^{46} +(0.995185 + 0.0980171i) q^{49} +(-0.773010 + 0.634393i) q^{50} +(-0.317367 + 0.302158i) q^{53} +(-0.707107 - 0.707107i) q^{56} +(1.01599 + 1.60442i) q^{58} +(0.382683 - 0.923880i) q^{63} +(0.0980171 + 0.995185i) q^{64} +(0.431751 + 1.92267i) q^{67} +(0.661009 - 1.39759i) q^{71} +(-0.881921 + 0.471397i) q^{72} +(-1.70590 + 0.756174i) q^{74} +(1.83027 - 0.604638i) q^{77} +(0.186170 + 0.226848i) q^{79} +(-0.773010 - 0.634393i) q^{81} +(0.147085 - 1.99398i) q^{86} +(-1.79839 - 0.693716i) q^{88} +(0.829484 + 0.207775i) q^{92} +(0.941544 + 0.336890i) q^{98} +(-0.0473045 - 1.92697i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/3584\mathbb{Z}\right)^\times$.

$n$	$1023$	$1025$	$1541$
$\chi(n)$	$1$	$-1$	$e\left(\frac{45}{128}\right)$

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 $

$2$	0.970031	+	0.242980i	0.970031	+	0.242980i
$2$	0.970031	+	0.242980i	0.970031	+	0.242980i
$3$		0			0		−0.575808	−	0.817585i	$-0.695312\pi$
$3$		0			0		0.575808	+	0.817585i	$0.304688\pi$
$4$	0.881921	+	0.471397i	0.881921	+	0.471397i
$5$		0			0		−0.449611	−	0.893224i	$-0.648438\pi$
$5$		0			0		0.449611	+	0.893224i	$0.351562\pi$
$6$		0			0
$7$	−0.998795	−	0.0490677i	−0.998795	−	0.0490677i
$7$	−0.998795	−	0.0490677i	−0.998795	−	0.0490677i
$8$	0.740951	+	0.671559i	0.740951	+	0.671559i
$9$	−0.336890	+	0.941544i	−0.336890	+	0.941544i
$10$		0			0
$11$	−1.79839	+	0.693716i	−1.79839	+	0.693716i	−0.803208	+	0.595699i	$0.796875\pi$
$11$	−1.79839	+	0.693716i	−1.79839	+	0.693716i	−0.995185	+	0.0980171i	$0.968750\pi$
$12$		0			0
$13$		0			0		−0.997290	−	0.0735646i	$-0.976562\pi$
$13$		0			0		0.997290	+	0.0735646i	$0.0234375\pi$
$14$	−0.956940	−	0.290285i	−0.956940	−	0.290285i
$15$		0			0
$16$	0.555570	+	0.831470i	0.555570	+	0.831470i
$17$		0			0		0.881921	−	0.471397i	$-0.156250\pi$
$17$		0			0		−0.881921	+	0.471397i	$0.843750\pi$
$18$	−0.555570	+	0.831470i	−0.555570	+	0.831470i
$19$		0			0		−0.963776	−	0.266713i	$-0.914062\pi$
$19$		0			0		0.963776	+	0.266713i	$0.0859375\pi$
$20$		0			0
$21$		0			0
$22$	−1.91306	+	0.235953i	−1.91306	+	0.235953i
$23$	0.829484	−	0.207775i	0.829484	−	0.207775i	0.195090	−	0.980785i	$-0.437500\pi$
$23$	0.829484	−	0.207775i	0.829484	−	0.207775i	0.634393	+	0.773010i	$0.281250\pi$
$24$		0			0
$25$	−0.595699	+	0.803208i	−0.595699	+	0.803208i
$26$		0			0
$27$		0			0
$28$	−0.857729	−	0.514103i	−0.857729	−	0.514103i
$29$	1.37539	+	1.30948i	1.37539	+	1.30948i	0.903989	+	0.427555i	$0.140625\pi$
$29$	1.37539	+	1.30948i	1.37539	+	1.30948i	0.471397	+	0.881921i	$0.343750\pi$
$30$		0			0
$31$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$31$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$32$	0.336890	+	0.941544i	0.336890	+	0.941544i
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−0.740951	+	0.671559i	−0.740951	+	0.671559i
$37$	−1.47104	+	1.14801i	−1.47104	+	1.14801i	−0.514103	+	0.857729i	$0.671875\pi$
$37$	−1.47104	+	1.14801i	−1.47104	+	1.14801i	−0.956940	+	0.290285i	$0.906250\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		−0.595699	−	0.803208i	$-0.703125\pi$
$41$		0			0		0.595699	+	0.803208i	$0.296875\pi$
$42$		0			0
$43$	−0.341821	−	1.96996i	−0.341821	−	1.96996i	−0.195090	−	0.980785i	$-0.562500\pi$
$43$	−0.341821	−	1.96996i	−0.341821	−	1.96996i	−0.146730	−	0.989177i	$-0.546875\pi$
$44$	−1.91306	−	0.235953i	−1.91306	−	0.235953i
$45$		0			0
$46$	0.855110			0.855110
$47$		0			0		−0.0980171	−	0.995185i	$-0.531250\pi$
$47$		0			0		0.0980171	+	0.995185i	$0.468750\pi$
$48$		0			0
$49$	0.995185	+	0.0980171i	0.995185	+	0.0980171i
$50$	−0.773010	+	0.634393i	−0.773010	+	0.634393i
$51$		0			0
$52$		0			0
$53$	−0.317367	+	0.302158i	−0.317367	+	0.302158i	−0.831470	−	0.555570i	$-0.812500\pi$
$53$	−0.317367	+	0.302158i	−0.317367	+	0.302158i	0.514103	+	0.857729i	$0.328125\pi$
$54$		0			0
$55$		0			0
$56$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$57$		0			0
$58$	1.01599	+	1.60442i	1.01599	+	1.60442i
$59$		0			0		−0.0735646	−	0.997290i	$-0.523438\pi$
$59$		0			0		0.0735646	+	0.997290i	$0.476562\pi$
$60$		0			0
$61$		0			0		−0.844854	−	0.534998i	$-0.820312\pi$
$61$		0			0		0.844854	+	0.534998i	$0.179688\pi$
$62$		0			0
$63$	0.382683	−	0.923880i	0.382683	−	0.923880i
$64$	0.0980171	+	0.995185i	0.0980171	+	0.995185i
$65$		0			0
$66$		0			0
$67$	0.431751	+	1.92267i	0.431751	+	1.92267i	0.382683	+	0.923880i	$0.375000\pi$
$67$	0.431751	+	1.92267i	0.431751	+	1.92267i	0.0490677	+	0.998795i	$0.484375\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	0.661009	−	1.39759i	0.661009	−	1.39759i	−0.242980	−	0.970031i	$-0.578125\pi$
$71$	0.661009	−	1.39759i	0.661009	−	1.39759i	0.903989	−	0.427555i	$-0.140625\pi$
$72$	−0.881921	+	0.471397i	−0.881921	+	0.471397i
$73$		0			0		0.998795	−	0.0490677i	$-0.0156250\pi$
$73$		0			0		−0.998795	+	0.0490677i	$0.984375\pi$
$74$	−1.70590	+	0.756174i	−1.70590	+	0.756174i
$75$		0			0
$76$		0			0
$77$	1.83027	−	0.604638i	1.83027	−	0.604638i
$78$		0			0
$79$	0.186170	+	0.226848i	0.186170	+	0.226848i	0.857729	−	0.514103i	$-0.171875\pi$
$79$	0.186170	+	0.226848i	0.186170	+	0.226848i	−0.671559	+	0.740951i	$0.734375\pi$
$80$		0			0
$81$	−0.773010	−	0.634393i	−0.773010	−	0.634393i
$82$		0			0
$83$		0			0		−0.788346	−	0.615232i	$-0.789062\pi$
$83$		0			0		0.788346	+	0.615232i	$0.210938\pi$
$84$		0			0
$85$		0			0
$86$	0.147085	−	1.99398i	0.147085	−	1.99398i
$87$		0			0
$88$	−1.79839	−	0.693716i	−1.79839	−	0.693716i
$89$		0			0		−0.970031	−	0.242980i	$-0.921875\pi$
$89$		0			0		0.970031	+	0.242980i	$0.0781250\pi$
$90$		0			0
$91$		0			0
$92$	0.829484	+	0.207775i	0.829484	+	0.207775i
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$97$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$98$	0.941544	+	0.336890i	0.941544	+	0.336890i
$99$	−0.0473045	−	1.92697i	−0.0473045	−	1.92697i
$100$	−0.903989	+	0.427555i	−0.903989	+	0.427555i
$101$		0			0		0.492898	−	0.870087i	$-0.335938\pi$
$101$		0			0		−0.492898	+	0.870087i	$0.664062\pi$
$102$		0			0
$103$		0			0		0.989177	−	0.146730i	$-0.0468750\pi$
$103$		0			0		−0.989177	+	0.146730i	$0.953125\pi$
$104$		0			0
$105$		0			0
$106$	−0.381274	+	0.215989i	−0.381274	+	0.215989i
$107$	0.370217	−	1.64865i	0.370217	−	1.64865i	−0.336890	−	0.941544i	$-0.609375\pi$
$107$	0.370217	−	1.64865i	0.370217	−	1.64865i	0.707107	−	0.707107i	$-0.250000\pi$
$108$		0			0
$109$	1.07061	−	0.606493i	1.07061	−	0.606493i	0.146730	−	0.989177i	$-0.453125\pi$
$109$	1.07061	−	0.606493i	1.07061	−	0.606493i	0.923880	+	0.382683i	$0.125000\pi$
$110$		0			0
$111$		0			0
$112$	−0.514103	−	0.857729i	−0.514103	−	0.857729i
$113$	−0.0284872	−	0.0939097i	−0.0284872	−	0.0939097i	0.941544	−	0.336890i	$-0.109375\pi$
$113$	−0.0284872	−	0.0939097i	−0.0284872	−	0.0939097i	−0.970031	+	0.242980i	$0.921875\pi$
$114$		0			0
$115$		0			0
$116$	0.595699	+	1.80321i	0.595699	+	1.80321i
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	2.01202	−	1.82359i	2.01202	−	1.82359i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$	0.595699	−	0.803208i	0.595699	−	0.803208i
$127$	0.410525	+	0.410525i	0.410525	+	0.410525i	0.881921	−	0.471397i	$-0.156250\pi$
$127$	0.410525	+	0.410525i	0.410525	+	0.410525i	−0.471397	+	0.881921i	$0.656250\pi$
$128$	−0.146730	+	0.989177i	−0.146730	+	0.989177i
$129$		0			0
$130$		0			0
$131$		0			0		0.170962	−	0.985278i	$-0.445312\pi$
$131$		0			0		−0.170962	+	0.985278i	$0.554688\pi$
$132$		0			0
$133$		0			0
$134$	−0.0483598	+	1.96996i	−0.0483598	+	1.96996i
$135$		0			0
$136$		0			0
$137$	−0.509389	−	1.07701i	−0.509389	−	1.07701i	−0.980785	−	0.195090i	$-0.937500\pi$
$137$	−0.509389	−	1.07701i	−0.509389	−	1.07701i	0.471397	−	0.881921i	$-0.343750\pi$
$138$		0			0
$139$		0			0		−0.914210	−	0.405241i	$-0.867188\pi$
$139$		0			0		0.914210	+	0.405241i	$0.132812\pi$
$140$		0			0
$141$		0			0
$142$	0.980785	−	1.19509i	0.980785	−	1.19509i
$143$		0			0
$144$	−0.970031	+	0.242980i	−0.970031	+	0.242980i
$145$		0			0
$146$		0			0
$147$		0			0
$148$	−1.83851	+	0.319012i	−1.83851	+	0.319012i
$149$	1.59544	+	0.358268i	1.59544	+	0.358268i	0.923880	−	0.382683i	$-0.125000\pi$
$149$	1.59544	+	0.358268i	1.59544	+	0.358268i	0.671559	+	0.740951i	$0.265625\pi$
$150$		0			0
$151$	1.66405	+	0.997391i	1.66405	+	0.997391i	0.956940	+	0.290285i	$0.0937500\pi$
$151$	1.66405	+	0.997391i	1.66405	+	0.997391i	0.707107	+	0.707107i	$0.250000\pi$
$152$		0			0
$153$		0			0
$154$	1.92233	−	0.141800i	1.92233	−	0.141800i
$155$		0			0
$156$		0			0
$157$		0			0		0.999699	−	0.0245412i	$-0.00781250\pi$
$157$		0			0		−0.999699	+	0.0245412i	$0.992188\pi$
$158$	0.125471	+	0.265286i	0.125471	+	0.265286i
$159$		0			0
$160$		0			0
$161$	−0.838679	+	0.166824i	−0.838679	+	0.166824i
$162$	−0.595699	−	0.803208i	−0.595699	−	0.803208i
$163$	−0.717840	+	1.86093i	−0.717840	+	1.86093i	−0.290285	+	0.956940i	$0.593750\pi$
$163$	−0.717840	+	1.86093i	−0.717840	+	1.86093i	−0.427555	+	0.903989i	$0.640625\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.242980	−	0.970031i	$-0.421875\pi$
$167$		0			0		−0.242980	+	0.970031i	$0.578125\pi$
$168$		0			0
$169$	0.989177	+	0.146730i	0.989177	+	0.146730i
$170$		0			0
$171$		0			0
$172$	0.627175	−	1.89848i	0.627175	−	1.89848i
$173$		0			0		0.615232	−	0.788346i	$-0.289062\pi$
$173$		0			0		−0.615232	+	0.788346i	$0.710938\pi$
$174$		0			0
$175$	0.634393	−	0.773010i	0.634393	−	0.773010i
$176$	−1.57594	−	1.10990i	−1.57594	−	1.10990i
$177$		0			0
$178$		0			0
$179$	−0.573542	−	1.73614i	−0.573542	−	1.73614i	−0.671559	−	0.740951i	$-0.734375\pi$
$179$	−0.573542	−	1.73614i	−0.573542	−	1.73614i	0.0980171	−	0.995185i	$-0.468750\pi$
$180$		0			0
$181$		0			0		0.0245412	−	0.999699i	$-0.492188\pi$
$181$		0			0		−0.0245412	+	0.999699i	$0.507812\pi$
$182$		0			0
$183$		0			0
$184$	0.754140	+	0.403096i	0.754140	+	0.403096i
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−0.949938	+	0.393477i	−0.949938	+	0.393477i	−0.803208	−	0.595699i	$-0.796875\pi$
$191$	−0.949938	+	0.393477i	−0.949938	+	0.393477i	−0.146730	+	0.989177i	$0.546875\pi$
$192$		0			0
$193$	0.622491	+	0.257844i	0.622491	+	0.257844i	0.671559	−	0.740951i	$-0.265625\pi$
$193$	0.622491	+	0.257844i	0.622491	+	0.257844i	−0.0490677	+	0.998795i	$0.515625\pi$
$194$		0			0
$195$		0			0
$196$	0.831470	+	0.555570i	0.831470	+	0.555570i
$197$	−1.37534	+	0.101451i	−1.37534	+	0.101451i	−0.740951	−	0.671559i	$-0.765625\pi$
$197$	−1.37534	+	0.101451i	−1.37534	+	0.101451i	−0.634393	+	0.773010i	$0.718750\pi$
$198$	0.422329	−	1.88072i	0.422329	−	1.88072i
$199$		0			0		0.941544	−	0.336890i	$-0.109375\pi$
$199$		0			0		−0.941544	+	0.336890i	$0.890625\pi$
$200$	−0.980785	+	0.195090i	−0.980785	+	0.195090i
$201$		0			0
$202$		0			0
$203$	−1.30948	−	1.37539i	−1.30948	−	1.37539i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−0.0838155	+	0.850993i	−0.0838155	+	0.850993i
$208$		0			0
$209$		0			0
$210$		0			0
$211$	0.110074	+	0.892460i	0.110074	+	0.892460i	0.941544	+	0.336890i	$0.109375\pi$
$211$	0.110074	+	0.892460i	0.110074	+	0.892460i	−0.831470	+	0.555570i	$0.812500\pi$
$212$	−0.422329	+	0.116874i	−0.422329	+	0.116874i
$213$		0			0
$214$	0.759711	−	1.50929i	0.759711	−	1.50929i
$215$		0			0
$216$		0			0
$217$		0			0
$218$	1.18589	−	0.328180i	1.18589	−	0.328180i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$223$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$224$	−0.290285	−	0.956940i	−0.290285	−	0.956940i
$225$	−0.555570	−	0.831470i	−0.555570	−	0.831470i
$226$	−0.00481527	−	0.0980171i	−0.00481527	−	0.0980171i
$227$		0			0		0.689541	−	0.724247i	$-0.257812\pi$
$227$		0			0		−0.689541	+	0.724247i	$0.742188\pi$
$228$		0			0
$229$		0			0		−0.266713	−	0.963776i	$-0.585938\pi$
$229$		0			0		0.266713	+	0.963776i	$0.414062\pi$
$230$		0			0
$231$		0			0
$232$	0.139703	+	1.89391i	0.139703	+	1.89391i
$233$	−0.480701	−	1.91906i	−0.480701	−	1.91906i	−0.382683	−	0.923880i	$-0.625000\pi$
$233$	−0.480701	−	1.91906i	−0.480701	−	1.91906i	−0.0980171	−	0.995185i	$-0.531250\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	0.831470	+	1.55557i	0.831470	+	1.55557i	0.831470	+	0.555570i	$0.187500\pi$
$239$	0.831470	+	1.55557i	0.831470	+	1.55557i			1.00000i	$0.5\pi$
$240$		0			0
$241$		0			0		−0.881921	−	0.471397i	$-0.843750\pi$
$241$		0			0		0.881921	+	0.471397i	$0.156250\pi$
$242$	2.39482	−	1.28006i	2.39482	−	1.28006i
$243$		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		0.893224	−	0.449611i	$-0.148438\pi$
$251$		0			0		−0.893224	+	0.449611i	$0.851562\pi$
$252$	0.773010	−	0.634393i	0.773010	−	0.634393i
$253$	−1.34760	+	0.949087i	−1.34760	+	0.949087i
$254$	0.298472	+	0.497971i	0.298472	+	0.497971i
$255$		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
$259$	1.52560	−	1.07445i	1.52560	−	1.07445i
$260$		0			0
$261$	−1.69628	+	0.853837i	−1.69628	+	0.853837i
$262$		0			0
$263$	0.0191453	−	0.389711i	0.0191453	−	0.389711i	−0.970031	−	0.242980i	$-0.921875\pi$
$263$	0.0191453	−	0.389711i	0.0191453	−	0.389711i	0.989177	−	0.146730i	$-0.0468750\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−0.525572	+	1.89917i	−0.525572	+	1.89917i
$269$		0			0		0.0735646	−	0.997290i	$-0.476562\pi$
$269$		0			0		−0.0735646	+	0.997290i	$0.523438\pi$
$270$		0			0
$271$		0			0		−0.881921	−	0.471397i	$-0.843750\pi$
$271$		0			0		0.881921	+	0.471397i	$0.156250\pi$
$272$		0			0
$273$		0			0
$274$	−0.232430	−	1.16851i	−0.232430	−	1.16851i
$275$	0.514103	−	1.85773i	0.514103	−	1.85773i
$276$		0			0
$277$	−1.67700	+	1.06195i	−1.67700	+	1.06195i	−0.773010	+	0.634393i	$0.781250\pi$
$277$	−1.67700	+	1.06195i	−1.67700	+	1.06195i	−0.903989	+	0.427555i	$0.859375\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	0.390327	+	0.289486i	0.390327	+	0.289486i	0.773010	−	0.634393i	$-0.218750\pi$
$281$	0.390327	+	0.289486i	0.390327	+	0.289486i	−0.382683	+	0.923880i	$0.625000\pi$
$282$		0			0
$283$		0			0		−0.266713	−	0.963776i	$-0.585938\pi$
$283$		0			0		0.266713	+	0.963776i	$0.414062\pi$
$284$	1.24178	−	0.920964i	1.24178	−	0.920964i
$285$		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		−0.615232	−	0.788346i	$-0.710938\pi$
$293$		0			0		0.615232	+	0.788346i	$0.289062\pi$
$294$		0			0
$295$		0			0
$296$	−1.86093	−	0.137270i	−1.86093	−	0.137270i
$297$		0			0
$298$	1.46057	+	0.735191i	1.46057	+	0.735191i
$299$		0			0
$300$		0			0
$301$	0.244748	+	1.98436i	0.244748	+	1.98436i
$302$	1.37183	+	1.37183i	1.37183	+	1.37183i
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0		0.449611	−	0.893224i	$-0.351562\pi$
$307$		0			0		−0.449611	+	0.893224i	$0.648438\pi$
$308$	1.89917	+	0.329538i	1.89917	+	0.329538i
$309$		0			0
$310$		0			0
$311$		0			0		0.671559	−	0.740951i	$-0.265625\pi$
$311$		0			0		−0.671559	+	0.740951i	$0.734375\pi$
$312$		0			0
$313$		0			0		0.941544	−	0.336890i	$-0.109375\pi$
$313$		0			0		−0.941544	+	0.336890i	$0.890625\pi$
$314$		0			0
$315$		0			0
$316$	0.0572514	+	0.287822i	0.0572514	+	0.287822i
$317$	−0.0787137	+	0.124303i	−0.0787137	+	0.124303i	−0.881921	−	0.471397i	$-0.843750\pi$
$317$	−0.0787137	+	0.124303i	−0.0787137	+	0.124303i	0.803208	+	0.595699i	$0.203125\pi$
$318$		0			0
$319$	−3.38189	−	1.40082i	−3.38189	−	1.40082i
$320$		0			0
$321$		0			0
$322$	−0.854080	−	0.0419583i	−0.854080	−	0.0419583i
$323$		0			0
$324$	−0.382683	−	0.923880i	−0.382683	−	0.923880i
$325$		0			0
$326$	−1.14850	+	1.63074i	−1.14850	+	1.63074i
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	0.0320593	−	1.30595i	0.0320593	−	1.30595i	−0.740951	−	0.671559i	$-0.765625\pi$
$331$	0.0320593	−	1.30595i	0.0320593	−	1.30595i	0.773010	−	0.634393i	$-0.218750\pi$
$332$		0			0
$333$	−0.585326	−	1.77181i	−0.585326	−	1.77181i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	0.940109	−	1.14553i	0.940109	−	1.14553i	−0.0490677	−	0.998795i	$-0.515625\pi$
$337$	0.940109	−	1.14553i	0.940109	−	1.14553i	0.989177	−	0.146730i	$-0.0468750\pi$
$338$	0.923880	+	0.382683i	0.923880	+	0.382683i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−0.989177	−	0.146730i	−0.989177	−	0.146730i
$344$	1.06967	−	1.68920i	1.06967	−	1.68920i
$345$		0			0
$346$		0			0
$347$	−0.185381	+	1.50303i	−0.185381	+	1.50303i	0.555570	+	0.831470i	$0.312500\pi$
$347$	−0.185381	+	1.50303i	−0.185381	+	1.50303i	−0.740951	+	0.671559i	$0.765625\pi$
$348$		0			0
$349$		0			0		0.359895	−	0.932993i	$-0.382812\pi$
$349$		0			0		−0.359895	+	0.932993i	$0.617188\pi$
$350$	0.803208	−	0.595699i	0.803208	−	0.595699i
$351$		0			0
$352$	−1.25902	−	1.45956i	−1.25902	−	1.45956i
$353$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$353$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$	−0.134507	−	1.82347i	−0.134507	−	1.82347i
$359$	0.217440	+	1.46586i	0.217440	+	1.46586i	0.773010	+	0.634393i	$0.218750\pi$
$359$	0.217440	+	1.46586i	0.217440	+	1.46586i	−0.555570	+	0.831470i	$0.687500\pi$
$360$		0			0
$361$	0.857729	+	0.514103i	0.857729	+	0.514103i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		0.290285	−	0.956940i	$-0.406250\pi$
$367$		0			0		−0.290285	+	0.956940i	$0.593750\pi$
$368$	0.633595	+	0.574257i	0.633595	+	0.574257i
$369$		0			0
$370$		0			0
$371$	0.331811	−	0.286222i	0.331811	−	0.286222i
$372$		0			0
$373$	1.59088	+	0.705190i	1.59088	+	0.705190i	0.995185	−	0.0980171i	$-0.0312500\pi$
$373$	1.59088	+	0.705190i	1.59088	+	0.705190i	0.595699	+	0.803208i	$0.296875\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−1.85291	−	0.612120i	−1.85291	−	0.612120i	−0.995185	−	0.0980171i	$-0.968750\pi$
$379$	−1.85291	−	0.612120i	−1.85291	−	0.612120i	−0.857729	−	0.514103i	$-0.828125\pi$
$380$		0			0
$381$		0			0
$382$	−1.01708	+	0.150869i	−1.01708	+	0.150869i
$383$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$383$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$384$		0			0
$385$		0			0
$386$	0.541185	+	0.401370i	0.541185	+	0.401370i
$387$	1.96996	+	0.341821i	1.96996	+	0.341821i
$388$		0			0
$389$	0.0153963	−	0.0466052i	0.0153963	−	0.0466052i	−0.941544	−	0.336890i	$-0.890625\pi$
$389$	0.0153963	−	0.0466052i	0.0153963	−	0.0466052i	0.956940	+	0.290285i	$0.0937500\pi$
$390$		0			0
$391$		0			0
$392$	0.671559	+	0.740951i	0.671559	+	0.740951i
$393$		0			0
$394$	−1.35878	−	0.235770i	−1.35878	−	0.235770i
$395$		0			0
$396$	0.866649	−	1.72174i	0.866649	−	1.72174i
$397$		0			0		−0.653173	−	0.757209i	$-0.726562\pi$
$397$		0			0		0.653173	+	0.757209i	$0.273438\pi$
$398$		0			0
$399$		0			0
$400$	−0.998795	−	0.0490677i	−0.998795	−	0.0490677i
$401$	−1.28528	−	0.389887i	−1.28528	−	0.389887i	−0.427555	−	0.903989i	$-0.640625\pi$
$401$	−1.28528	−	0.389887i	−1.28528	−	0.389887i	−0.857729	+	0.514103i	$0.828125\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$	−0.936041	−	1.65234i	−0.936041	−	1.65234i
$407$	1.84912	−	3.08506i	1.84912	−	3.08506i
$408$		0			0
$409$		0			0		0.989177	−	0.146730i	$-0.0468750\pi$
$409$		0			0		−0.989177	+	0.146730i	$0.953125\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$	−0.288078	+	0.805124i	−0.288078	+	0.805124i
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		−0.932993	−	0.359895i	$-0.882812\pi$
$419$		0			0		0.932993	+	0.359895i	$0.117188\pi$
$420$		0			0
$421$	1.97958	+	0.244158i	1.97958	+	0.244158i	0.998795	+	0.0490677i	$0.0156250\pi$
$421$	1.97958	+	0.244158i	1.97958	+	0.244158i	0.980785	+	0.195090i	$0.0625000\pi$
$422$	−0.110074	+	0.892460i	−0.110074	+	0.892460i
$423$		0			0
$424$	−0.438071	+	0.0107540i	−0.438071	+	0.0107540i
$425$		0			0
$426$		0			0
$427$		0			0
$428$	1.10367	−	1.27946i	1.10367	−	1.27946i
$429$		0			0
$430$		0			0
$431$	0.858923	+	0.704900i	0.858923	+	0.704900i	0.956940	−	0.290285i	$-0.0937500\pi$
$431$	0.858923	+	0.704900i	0.858923	+	0.704900i	−0.0980171	+	0.995185i	$0.531250\pi$
$432$		0			0
$433$		0			0		−0.634393	−	0.773010i	$-0.718750\pi$
$433$		0			0		0.634393	+	0.773010i	$0.281250\pi$
$434$		0			0
$435$		0			0
$436$	1.23009	−	0.0301971i	1.23009	−	0.0301971i
$437$		0			0
$438$		0			0
$439$		0			0		0.998795	−	0.0490677i	$-0.0156250\pi$
$439$		0			0		−0.998795	+	0.0490677i	$0.984375\pi$
$440$		0			0
$441$	−0.427555	+	0.903989i	−0.427555	+	0.903989i
$442$		0			0
$443$	−0.545031	−	0.470147i	−0.545031	−	0.470147i	0.336890	−	0.941544i	$-0.390625\pi$
$443$	−0.545031	−	0.470147i	−0.545031	−	0.470147i	−0.881921	+	0.471397i	$0.843750\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−0.0490677	−	0.998795i	−0.0490677	−	0.998795i
$449$	−0.0750191	+	0.181112i	−0.0750191	+	0.181112i	−0.956940	−	0.290285i	$-0.906250\pi$
$449$	−0.0750191	+	0.181112i	−0.0750191	+	0.181112i	0.881921	+	0.471397i	$0.156250\pi$
$450$	−0.336890	−	0.941544i	−0.336890	−	0.941544i
$451$		0			0
$452$	0.0191453	−	0.0962497i	0.0191453	−	0.0962497i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	1.30692	+	1.18452i	1.30692	+	1.18452i	0.970031	+	0.242980i	$0.0781250\pi$
$457$	1.30692	+	1.18452i	1.30692	+	1.18452i	0.336890	+	0.941544i	$0.390625\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		−0.893224	−	0.449611i	$-0.851562\pi$
$461$		0			0		0.893224	+	0.449611i	$0.148438\pi$
$462$		0			0
$463$	1.33665	+	0.131649i	1.33665	+	0.131649i	0.740951	−	0.671559i	$-0.234375\pi$
$463$	1.33665	+	0.131649i	1.33665	+	0.131649i	0.595699	+	0.803208i	$0.296875\pi$
$464$	−0.324666	+	1.87110i	−0.324666	+	1.87110i
$465$		0			0
$466$		−	1.97835i		−	1.97835i
$467$		0			0		0.992480	−	0.122411i	$-0.0390625\pi$
$467$		0			0		−0.992480	+	0.122411i	$0.960938\pi$
$468$		0			0
$469$	−0.336890	−	1.94154i	−0.336890	−	1.94154i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	1.98132	+	3.30564i	1.98132	+	3.30564i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−0.177578	−	0.400609i	−0.177578	−	0.400609i
$478$	0.428579	+	1.71098i	0.428579	+	1.71098i
$479$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$479$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	2.63408	−	0.659802i	2.63408	−	0.659802i
$485$		0			0
$486$		0			0
$487$	−0.709715	+	0.956940i	−0.709715	+	0.956940i	0.290285	+	0.956940i	$0.406250\pi$
$487$	−0.709715	+	0.956940i	−0.709715	+	0.956940i	−1.00000			$\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	0.874812	+	1.38148i	0.874812	+	1.38148i	0.923880	+	0.382683i	$0.125000\pi$
$491$	0.874812	+	1.38148i	0.874812	+	1.38148i	−0.0490677	+	0.998795i	$0.515625\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−0.728789	+	1.36347i	−0.728789	+	1.36347i
$498$		0			0
$499$	−1.14850	−	0.0847182i	−1.14850	−	0.0847182i	−0.514103	−	0.857729i	$-0.671875\pi$
$499$	−1.14850	−	0.0847182i	−1.14850	−	0.0847182i	−0.634393	+	0.773010i	$0.718750\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		0.336890	−	0.941544i	$-0.390625\pi$
$503$		0			0		−0.336890	+	0.941544i	$0.609375\pi$
$504$	0.903989	−	0.427555i	0.903989	−	0.427555i
$505$		0			0
$506$	−1.53782	+	0.593204i	−1.53782	+	0.593204i
$507$		0			0
$508$	0.168530	+	0.555570i	0.168530	+	0.555570i
$509$		0			0		−0.575808	−	0.817585i	$-0.695312\pi$
$509$		0			0		0.575808	+	0.817585i	$0.304688\pi$
$510$		0			0
$511$		0			0
$512$	−0.595699	+	0.803208i	−0.595699	+	0.803208i
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$	1.74095	−	0.671559i	1.74095	−	0.671559i
$519$		0			0
$520$		0			0
$521$		0			0		0.336890	−	0.941544i	$-0.390625\pi$
$521$		0			0		−0.336890	+	0.941544i	$0.609375\pi$
$522$	−1.85291	+	0.416086i	−1.85291	+	0.416086i
$523$		0			0		0.932993	−	0.359895i	$-0.117188\pi$
$523$		0			0		−0.932993	+	0.359895i	$0.882812\pi$
$524$		0			0
$525$		0			0
$526$	0.113263	−	0.373380i	0.113263	−	0.373380i
$527$		0			0
$528$		0			0
$529$	−0.237049	+	0.126705i	−0.237049	+	0.126705i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$	−0.971283	+	1.71455i	−0.971283	+	1.71455i
$537$		0			0
$538$		0			0
$539$	−1.85773	+	0.514103i	−1.85773	+	0.514103i
$540$		0			0
$541$	0.774941	+	0.737805i	0.774941	+	0.737805i	0.970031	−	0.242980i	$-0.0781250\pi$
$541$	0.774941	+	0.737805i	0.774941	+	0.737805i	−0.195090	+	0.980785i	$0.562500\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−0.723943	−	1.63319i	−0.723943	−	1.63319i	−0.773010	−	0.634393i	$-0.781250\pi$
$547$	−0.723943	−	1.63319i	−0.723943	−	1.63319i	0.0490677	−	0.998795i	$-0.484375\pi$
$548$	0.0584592	−	1.18996i	0.0584592	−	1.18996i
$549$		0			0
$550$	0.950087	−	1.67714i	0.950087	−	1.67714i
$551$		0			0
$552$		0			0
$553$	−0.174814	−	0.235710i	−0.174814	−	0.235710i
$554$	−1.88477	+	0.622645i	−1.88477	+	0.622645i
$555$		0			0
$556$		0			0
$557$	−0.242980	+	0.0299687i	−0.242980	+	0.0299687i	−0.242980	−	0.970031i	$-0.578125\pi$
$557$	−0.242980	+	0.0299687i	−0.242980	+	0.0299687i			1.00000i	$0.5\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	0.308290	+	0.375652i	0.308290	+	0.375652i
$563$		0			0		−0.893224	−	0.449611i	$-0.851562\pi$
$563$		0			0		0.893224	+	0.449611i	$0.148438\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	0.740951	+	0.671559i	0.740951	+	0.671559i
$568$	1.42834	−	0.591637i	1.42834	−	0.591637i
$569$	−0.195588	−	0.546632i	−0.195588	−	0.546632i	0.803208	−	0.595699i	$-0.203125\pi$
$569$	−0.195588	−	0.546632i	−0.195588	−	0.546632i	−0.998795	+	0.0490677i	$0.984375\pi$
$570$		0			0
$571$	0.0461517	+	0.625664i	0.0461517	+	0.625664i	0.970031	+	0.242980i	$0.0781250\pi$
$571$	0.0461517	+	0.625664i	0.0461517	+	0.625664i	−0.923880	+	0.382683i	$0.875000\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−0.327237	+	0.790019i	−0.327237	+	0.790019i
$576$	−0.970031	−	0.242980i	−0.970031	−	0.242980i
$577$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$577$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$578$	0.740951	−	0.671559i	0.740951	−	0.671559i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	0.361138	−	0.763562i	0.361138	−	0.763562i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		−0.999699	−	0.0245412i	$-0.992188\pi$
$587$		0			0		0.999699	+	0.0245412i	$0.00781250\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	−1.77181	−	0.585326i	−1.77181	−	0.585326i
$593$		0			0		−0.773010	−	0.634393i	$-0.781250\pi$
$593$		0			0		0.773010	+	0.634393i	$0.218750\pi$
$594$		0			0
$595$		0			0
$596$	1.23816	+	1.06805i	1.23816	+	1.06805i
$597$		0			0
$598$		0			0
$599$	0.287822	−	1.94034i	0.287822	−	1.94034i	−0.0490677	−	0.998795i	$-0.515625\pi$
$599$	0.287822	−	1.94034i	0.287822	−	1.94034i	0.336890	−	0.941544i	$-0.390625\pi$
$600$		0			0
$601$		0			0		−0.970031	−	0.242980i	$-0.921875\pi$
$601$		0			0		0.970031	+	0.242980i	$0.0781250\pi$
$602$	−0.244748	+	1.98436i	−0.244748	+	1.98436i
$603$	−1.95574	−	0.241217i	−1.95574	−	0.241217i
$604$	0.997391	+	1.66405i	0.997391	+	1.66405i
$605$		0			0
$606$		0			0
$607$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$607$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−0.746454	+	1.31768i	−0.746454	+	1.31768i	0.195090	+	0.980785i	$0.437500\pi$
$613$	−0.746454	+	1.31768i	−0.746454	+	1.31768i	−0.941544	+	0.336890i	$0.890625\pi$
$614$		0			0
$615$		0			0
$616$	1.76219	+	0.781124i	1.76219	+	0.781124i
$617$	0.484693	−	0.808661i	0.484693	−	0.808661i	−0.514103	−	0.857729i	$-0.671875\pi$
$617$	0.484693	−	0.808661i	0.484693	−	0.808661i	0.998795	+	0.0490677i	$0.0156250\pi$
$618$		0			0
$619$		0			0		0.219101	−	0.975702i	$-0.429688\pi$
$619$		0			0		−0.219101	+	0.975702i	$0.570312\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−0.290285	−	0.956940i	−0.290285	−	0.956940i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−0.177213	+	0.0838155i	−0.177213	+	0.0838155i	−0.514103	−	0.857729i	$-0.671875\pi$
$631$	−0.177213	+	0.0838155i	−0.177213	+	0.0838155i	0.336890	+	0.941544i	$0.390625\pi$
$632$	−0.0143994	+	0.293107i	−0.0143994	+	0.293107i
$633$		0			0
$634$	−0.106558	+	0.101451i	−0.106558	+	0.101451i
$635$		0			0
$636$		0			0
$637$		0			0
$638$	−2.94017	−	2.18058i	−2.94017	−	2.18058i
$639$	1.09320	+	1.09320i	1.09320	+	1.09320i
$640$		0			0
$641$	1.33154	−	1.33154i	1.33154	−	1.33154i	0.427555	−	0.903989i	$-0.359375\pi$
$641$	1.33154	−	1.33154i	1.33154	−	1.33154i	0.903989	−	0.427555i	$-0.140625\pi$
$642$		0			0
$643$		0			0		0.170962	−	0.985278i	$-0.445312\pi$
$643$		0			0		−0.170962	+	0.985278i	$0.554688\pi$
$644$	−0.818289	−	0.248225i	−0.818289	−	0.248225i
$645$		0			0
$646$		0			0
$647$		0			0		−0.671559	−	0.740951i	$-0.734375\pi$
$647$		0			0		0.671559	+	0.740951i	$0.265625\pi$
$648$	−0.146730	−	0.989177i	−0.146730	−	0.989177i
$649$		0			0
$650$		0			0
$651$		0			0
$652$	−1.51031	+	1.30281i	−1.51031	+	1.30281i
$653$	0.746454	−	0.643895i	0.746454	−	0.643895i	−0.195090	−	0.980785i	$-0.562500\pi$
$653$	0.746454	−	0.643895i	0.746454	−	0.643895i	0.941544	+	0.336890i	$0.109375\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−0.777149	−	1.37186i	−0.777149	−	1.37186i	−0.923880	−	0.382683i	$-0.875000\pi$
$659$	−0.777149	−	1.37186i	−0.777149	−	1.37186i	0.146730	−	0.989177i	$-0.453125\pi$
$660$		0			0
$661$		0			0		−0.975702	−	0.219101i	$-0.929688\pi$
$661$		0			0		0.975702	+	0.219101i	$0.0703125\pi$
$662$	0.348419	−	1.25902i	0.348419	−	1.25902i
$663$		0			0
$664$		0			0
$665$		0			0
$666$	−0.137270	−	1.86093i	−0.137270	−	1.86093i
$667$	1.41294	+	0.800419i	1.41294	+	0.800419i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	1.95213	−	0.388302i	1.95213	−	0.388302i	0.956940	−	0.290285i	$-0.0937500\pi$
$673$	1.95213	−	0.388302i	1.95213	−	0.388302i	0.995185	−	0.0980171i	$-0.0312500\pi$
$674$	1.19028	−	0.882768i	1.19028	−	0.882768i
$675$		0			0
$676$	0.803208	+	0.595699i	0.803208	+	0.595699i
$677$		0			0		0.122411	−	0.992480i	$-0.460938\pi$
$677$		0			0		−0.122411	+	0.992480i	$0.539062\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−0.466683	+	0.662638i	−0.466683	+	0.662638i	−0.980785	−	0.195090i	$-0.937500\pi$
$683$	−0.466683	+	0.662638i	−0.466683	+	0.662638i	0.514103	+	0.857729i	$0.328125\pi$
$684$		0			0
$685$		0			0
$686$	−0.923880	−	0.382683i	−0.923880	−	0.382683i
$687$		0			0
$688$	1.44806	−	1.37867i	1.44806	−	1.37867i
$689$		0			0
$690$		0			0
$691$		0			0		−0.313682	−	0.949528i	$-0.601562\pi$
$691$		0			0		0.313682	+	0.949528i	$0.398438\pi$
$692$		0			0
$693$	−0.0473045	+	1.92697i	−0.0473045	+	1.92697i
$694$	−0.545031	+	1.41294i	−0.545031	+	1.41294i
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	0.923880	−	0.382683i	0.923880	−	0.382683i
$701$	−1.41330	+	0.317367i	−1.41330	+	0.317367i	−0.857729	−	0.514103i	$-0.828125\pi$
$701$	−1.41330	+	0.317367i	−1.41330	+	0.317367i	−0.555570	+	0.831470i	$0.687500\pi$
$702$		0			0
$703$		0			0
$704$	−0.866649	−	1.72174i	−0.866649	−	1.72174i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	0.983125	−	0.0725197i	0.983125	−	0.0725197i	0.427555	−	0.903989i	$-0.359375\pi$
$709$	0.983125	−	0.0725197i	0.983125	−	0.0725197i	0.555570	+	0.831470i	$0.312500\pi$
$710$		0			0
$711$	−0.276306	+	0.0988640i	−0.276306	+	0.0988640i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$	0.312590	−	1.80150i	0.312590	−	1.80150i
$717$		0			0
$718$	−0.145252	+	1.47477i	−0.145252	+	1.47477i
$719$		0			0		0.0980171	−	0.995185i	$-0.468750\pi$
$719$		0			0		−0.0980171	+	0.995185i	$0.531250\pi$
$720$		0			0
$721$		0			0
$722$	0.707107	+	0.707107i	0.707107	+	0.707107i
$723$		0			0
$724$		0			0
$725$	−1.87110	+	0.324666i	−1.87110	+	0.324666i
$726$		0			0
$727$		0			0		0.803208	−	0.595699i	$-0.203125\pi$
$727$		0			0		−0.803208	+	0.595699i	$0.796875\pi$
$728$		0			0
$729$	0.857729	−	0.514103i	0.857729	−	0.514103i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		0.914210	−	0.405241i	$-0.132812\pi$
$733$		0			0		−0.914210	+	0.405241i	$0.867188\pi$
$734$		0			0
$735$		0			0
$736$	0.475074	+	0.710998i	0.475074	+	0.710998i
$737$	−2.11025	−	3.15821i	−2.11025	−	3.15821i
$738$		0			0
$739$	−1.36871	+	1.43760i	−1.36871	+	1.43760i	−0.595699	+	0.803208i	$0.703125\pi$
$739$	−1.36871	+	1.43760i	−1.36871	+	1.43760i	−0.773010	+	0.634393i	$0.781250\pi$
$740$		0			0
$741$		0			0
$742$	0.391413	−	0.197021i	0.391413	−	0.197021i
$743$	−1.37787	−	1.02190i	−1.37787	−	1.02190i	−0.995185	−	0.0980171i	$-0.968750\pi$
$743$	−1.37787	−	1.02190i	−1.37787	−	1.02190i	−0.382683	−	0.923880i	$-0.625000\pi$
$744$		0			0
$745$		0			0
$746$	1.37186	+	1.07061i	1.37186	+	1.07061i
$747$		0			0
$748$		0			0
$749$	−0.450666	+	1.62850i	−0.450666	+	1.62850i
$750$		0			0
$751$	−0.0924099	−	0.172887i	−0.0924099	−	0.172887i	0.831470	−	0.555570i	$-0.187500\pi$
$751$	−0.0924099	−	0.172887i	−0.0924099	−	0.172887i	−0.923880	+	0.382683i	$0.875000\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−0.608117	−	1.57648i	−0.608117	−	1.57648i	−0.803208	−	0.595699i	$-0.796875\pi$
$757$	−0.608117	−	1.57648i	−0.608117	−	1.57648i	0.195090	−	0.980785i	$-0.437500\pi$
$758$	−1.64865	−	1.04400i	−1.64865	−	1.04400i
$759$		0			0
$760$		0			0
$761$		0			0		0.0490677	−	0.998795i	$-0.484375\pi$
$761$		0			0		−0.0490677	+	0.998795i	$0.515625\pi$
$762$		0			0
$763$	−1.09908	+	0.553230i	−1.09908	+	0.553230i
$764$	−1.02325	−	0.100782i	−1.02325	−	0.100782i
$765$		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
$772$	0.427441	+	0.520839i	0.427441	+	0.520839i
$773$		0			0		0.893224	−	0.449611i	$-0.148438\pi$
$773$		0			0		−0.893224	+	0.449611i	$0.851562\pi$
$774$	1.82787	+	0.810239i	1.82787	+	0.810239i
$775$		0			0
$776$		0			0
$777$		0			0
$778$	0.0262590	−	0.0414675i	0.0262590	−	0.0414675i
$779$		0			0
$780$		0			0
$781$	−0.219225	+	2.97196i	−0.219225	+	2.97196i
$782$		0			0
$783$		0			0
$784$	0.471397	+	0.881921i	0.471397	+	0.881921i
$785$		0			0
$786$		0			0
$787$		0			0		0.266713	−	0.963776i	$-0.414062\pi$
$787$		0			0		−0.266713	+	0.963776i	$0.585938\pi$
$788$	−1.26077	−	0.558861i	−1.26077	−	0.558861i
$789$		0			0
$790$		0			0
$791$	0.0238449	+	0.0951944i	0.0238449	+	0.0951944i
$792$	1.25902	−	1.45956i	1.25902	−	1.45956i
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		0.689541	−	0.724247i	$-0.257812\pi$
$797$		0			0		−0.689541	+	0.724247i	$0.742188\pi$
$798$		0			0
$799$		0			0
$800$	−0.956940	−	0.290285i	−0.956940	−	0.290285i
$801$		0			0
$802$	−1.15203	−	0.690501i	−1.15203	−	0.690501i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−0.757259	+	0.561621i	−0.757259	+	0.561621i	−0.903989	−	0.427555i	$-0.859375\pi$
$809$	−0.757259	+	0.561621i	−0.757259	+	0.561621i	0.146730	+	0.989177i	$0.453125\pi$
$810$		0			0
$811$		0			0		0.985278	−	0.170962i	$-0.0546875\pi$
$811$		0			0		−0.985278	+	0.170962i	$0.945312\pi$
$812$	−0.506503	−	1.83027i	−0.506503	−	1.83027i
$813$		0			0
$814$	2.54331	−	2.54331i	2.54331	−	2.54331i
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−1.04425	−	1.09681i	−1.04425	−	1.09681i	−0.995185	−	0.0980171i	$-0.968750\pi$
$821$	−1.04425	−	1.09681i	−1.04425	−	1.09681i	−0.0490677	−	0.998795i	$-0.515625\pi$
$822$		0			0
$823$	1.31731	−	1.45343i	1.31731	−	1.45343i	0.514103	−	0.857729i	$-0.328125\pi$
$823$	1.31731	−	1.45343i	1.31731	−	1.45343i	0.803208	−	0.595699i	$-0.203125\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	1.44457	−	0.106558i	1.44457	−	0.106558i	0.671559	−	0.740951i	$-0.265625\pi$
$827$	1.44457	−	0.106558i	1.44457	−	0.106558i	0.773010	+	0.634393i	$0.218750\pi$
$828$	−0.475074	+	0.710998i	−0.475074	+	0.710998i
$829$		0			0		0.534998	−	0.844854i	$-0.320312\pi$
$829$		0			0		−0.534998	+	0.844854i	$0.679688\pi$
$830$		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		−0.903989	−	0.427555i	$-0.859375\pi$
$839$		0			0		0.903989	+	0.427555i	$0.140625\pi$
$840$		0			0
$841$	0.127891	+	2.60328i	0.127891	+	2.60328i
$842$	1.86093	+	0.717840i	1.86093	+	0.717840i
$843$		0			0
$844$	−0.323626	+	0.838968i	−0.323626	+	0.838968i
$845$		0			0
$846$		0			0
$847$	−2.09908	+	1.72267i	−2.09908	+	1.72267i
$848$	−0.427555	−	0.0960107i	−0.427555	−	0.0960107i
$849$		0			0
$850$		0			0
$851$	−0.981678	+	1.25790i	−0.981678	+	1.25790i
$852$		0			0
$853$		0			0		0.575808	−	0.817585i	$-0.304688\pi$
$853$		0			0		−0.575808	+	0.817585i	$0.695312\pi$
$854$		0			0
$855$		0			0
$856$	1.38148	−	0.972947i	1.38148	−	0.972947i
$857$		0			0		0.242980	−	0.970031i	$-0.421875\pi$
$857$		0			0		−0.242980	+	0.970031i	$0.578125\pi$
$858$		0			0
$859$		0			0		0.122411	−	0.992480i	$-0.460938\pi$
$859$		0			0		−0.122411	+	0.992480i	$0.539062\pi$
$860$		0			0
$861$		0			0
$862$	0.661906	+	0.892476i	0.661906	+	0.892476i
$863$	0.476623	−	0.0948062i	0.476623	−	0.0948062i	0.0490677	−	0.998795i	$-0.484375\pi$
$863$	0.476623	−	0.0948062i	0.476623	−	0.0948062i	0.427555	+	0.903989i	$0.359375\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−0.492175	−	0.278813i	−0.492175	−	0.278813i
$870$		0			0
$871$		0			0
$872$	1.20057	+	0.269596i	1.20057	+	0.269596i
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−0.713960	−	1.26032i	−0.713960	−	1.26032i	−0.956940	−	0.290285i	$-0.906250\pi$
$877$	−0.713960	−	1.26032i	−0.713960	−	1.26032i	0.242980	−	0.970031i	$-0.421875\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		0.956940	−	0.290285i	$-0.0937500\pi$
$881$		0			0		−0.956940	+	0.290285i	$0.906250\pi$
$882$	−0.634393	+	0.773010i	−0.634393	+	0.773010i
$883$	−0.111407	+	0.0961008i	−0.111407	+	0.0961008i	−0.707107	−	0.707107i	$-0.750000\pi$
$883$	−0.111407	+	0.0961008i	−0.111407	+	0.0961008i	0.595699	+	0.803208i	$0.296875\pi$
$884$		0			0
$885$		0			0
$886$	−0.414461	−	0.588489i	−0.414461	−	0.588489i
$887$		0			0		−0.427555	−	0.903989i	$-0.640625\pi$
$887$		0			0		0.427555	+	0.903989i	$0.359375\pi$
$888$		0			0
$889$	−0.389887	−	0.430174i	−0.389887	−	0.430174i
$890$		0			0
$891$	1.83027	+	0.604638i	1.83027	+	0.604638i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$	0.195090	−	0.980785i	0.195090	−	0.980785i
$897$		0			0
$898$	−0.116777	+	0.157456i	−0.116777	+	0.157456i
$899$		0			0
$900$	−0.0980171	−	0.995185i	−0.0980171	−	0.995185i
$901$		0			0
$902$		0			0
$903$		0			0
$904$	0.0419583	−	0.0887133i	0.0419583	−	0.0887133i
$905$		0			0
$906$		0			0
$907$	−0.498635	+	1.12490i	−0.498635	+	1.12490i	0.471397	+	0.881921i	$0.343750\pi$
$907$	−0.498635	+	1.12490i	−0.498635	+	1.12490i	−0.970031	+	0.242980i	$0.921875\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	0.273678	+	0.902197i	0.273678	+	0.902197i	0.980785	+	0.195090i	$0.0625000\pi$
$911$	0.273678	+	0.902197i	0.273678	+	0.902197i	−0.707107	+	0.707107i	$0.750000\pi$
$912$		0			0
$913$		0			0
$914$	0.979938	+	1.46658i	0.979938	+	1.46658i
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−0.983931	+	1.64159i	−0.983931	+	1.64159i	−0.242980	+	0.970031i	$0.578125\pi$
$919$	−0.983931	+	1.64159i	−0.983931	+	1.64159i	−0.740951	+	0.671559i	$0.765625\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−0.0457936	−	1.86542i	−0.0457936	−	1.86542i
$926$	1.26460	+	0.452483i	1.26460	+	0.452483i
$927$		0			0
$928$	−0.769576	+	1.73614i	−0.769576	+	1.73614i
$929$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$929$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$930$		0			0
$931$		0			0
$932$	0.480701	−	1.91906i	0.480701	−	1.91906i
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		0.146730	−	0.989177i	$-0.453125\pi$
$937$		0			0		−0.146730	+	0.989177i	$0.546875\pi$
$938$	0.144963	−	1.96522i	0.144963	−	1.96522i
$939$		0			0
$940$		0			0
$941$		0			0		−0.788346	−	0.615232i	$-0.789062\pi$
$941$		0			0		0.788346	+	0.615232i	$0.210938\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$	1.11874	+	3.68799i	1.11874	+	3.68799i
$947$	−1.88477	+	0.622645i	−1.88477	+	0.622645i	−0.903989	+	0.427555i	$0.859375\pi$
$947$	−1.88477	+	0.622645i	−1.88477	+	0.622645i	−0.980785	+	0.195090i	$0.937500\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−0.439614	+	0.929487i	−0.439614	+	0.929487i	0.555570	+	0.831470i	$0.312500\pi$
$953$	−0.439614	+	0.929487i	−0.439614	+	0.929487i	−0.995185	+	0.0980171i	$0.968750\pi$
$954$	−0.0749159	−	0.431751i	−0.0749159	−	0.431751i
$955$		0			0
$956$			1.76384i			1.76384i
$957$		0			0
$958$		0			0
$959$	0.455929	+	1.10071i	0.455929	+	1.10071i
$960$		0			0
$961$	0.382683	−	0.923880i	0.382683	−	0.923880i
$962$		0			0
$963$	1.42756	+	0.903989i	1.42756	+	0.903989i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	0.163715	+	0.457553i	0.163715	+	0.457553i	0.995185	−	0.0980171i	$-0.0312500\pi$
$967$	0.163715	+	0.457553i	0.163715	+	0.457553i	−0.831470	+	0.555570i	$0.812500\pi$
$968$	2.71546			2.71546
$969$		0			0
$970$		0			0
$971$		0			0		0.724247	−	0.689541i	$-0.242188\pi$
$971$		0			0		−0.724247	+	0.689541i	$0.757812\pi$
$972$		0			0
$973$		0			0
$974$	−0.920964	+	0.755815i	−0.920964	+	0.755815i
$975$		0			0
$976$		0			0
$977$	0.124363	+	1.26268i	0.124363	+	1.26268i	0.831470	+	0.555570i	$0.187500\pi$
$977$	0.124363	+	1.26268i	0.124363	+	1.26268i	−0.707107	+	0.707107i	$0.750000\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	0.210362	+	1.21235i	0.210362	+	1.21235i
$982$	0.512923	+	1.55264i	0.512923	+	1.55264i
$983$		0			0		−0.595699	−	0.803208i	$-0.703125\pi$
$983$		0			0		0.595699	+	0.803208i	$0.296875\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−0.692843	−	1.56303i	−0.692843	−	1.56303i
$990$		0			0
$991$	1.11676	+	0.746196i	1.11676	+	0.746196i	0.970031	−	0.242980i	$-0.0781250\pi$
$991$	1.11676	+	0.746196i	1.11676	+	0.746196i	0.146730	+	0.989177i	$0.453125\pi$
$992$		0			0
$993$		0			0
$994$	−1.03824	+	1.14553i	−1.03824	+	1.14553i
$995$		0			0
$996$		0			0
$997$		0			0		0.963776	−	0.266713i	$-0.0859375\pi$
$997$		0			0		−0.963776	+	0.266713i	$0.914062\pi$
$998$	−1.09349	−	0.361241i	−1.09349	−	0.361241i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3584.1.cd.a.1301.1	yes	64
7.6	odd	2	CM	3584.1.cd.a.1301.1	yes	64
512.61	even	128	inner	3584.1.cd.a.573.1	✓	64
3584.573	odd	128	inner	3584.1.cd.a.573.1	✓	64

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3584.1.cd.a.573.1	✓	64	512.61	even	128	inner
3584.1.cd.a.573.1	✓	64	3584.573	odd	128	inner
3584.1.cd.a.1301.1	yes	64	1.1	even	1	trivial
3584.1.cd.a.1301.1	yes	64	7.6	odd	2	CM

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)

\(f(q)\)	\(=\)	\(q+(0.970031 + 0.242980i) q^{2} +(0.881921 + 0.471397i) q^{4} +(-0.998795 - 0.0490677i) q^{7} +(0.740951 + 0.671559i) q^{8} +(-0.336890 + 0.941544i) q^{9} +O(q^{10})\)	\(q+(0.970031 + 0.242980i) q^{2} +(0.881921 + 0.471397i) q^{4} +(-0.998795 - 0.0490677i) q^{7} +(0.740951 + 0.671559i) q^{8} +(-0.336890 + 0.941544i) q^{9} +(-1.79839 + 0.693716i) q^{11} +(-0.956940 - 0.290285i) q^{14} +(0.555570 + 0.831470i) q^{16} +(-0.555570 + 0.831470i) q^{18} +(-1.91306 + 0.235953i) q^{22} +(0.829484 - 0.207775i) q^{23} +(-0.595699 + 0.803208i) q^{25} +(-0.857729 - 0.514103i) q^{28} +(1.37539 + 1.30948i) q^{29} +(0.336890 + 0.941544i) q^{32} +(-0.740951 + 0.671559i) q^{36} +(-1.47104 + 1.14801i) q^{37} +(-0.341821 - 1.96996i) q^{43} +(-1.91306 - 0.235953i) q^{44} +0.855110 q^{46} +(0.995185 + 0.0980171i) q^{49} +(-0.773010 + 0.634393i) q^{50} +(-0.317367 + 0.302158i) q^{53} +(-0.707107 - 0.707107i) q^{56} +(1.01599 + 1.60442i) q^{58} +(0.382683 - 0.923880i) q^{63} +(0.0980171 + 0.995185i) q^{64} +(0.431751 + 1.92267i) q^{67} +(0.661009 - 1.39759i) q^{71} +(-0.881921 + 0.471397i) q^{72} +(-1.70590 + 0.756174i) q^{74} +(1.83027 - 0.604638i) q^{77} +(0.186170 + 0.226848i) q^{79} +(-0.773010 - 0.634393i) q^{81} +(0.147085 - 1.99398i) q^{86} +(-1.79839 - 0.693716i) q^{88} +(0.829484 + 0.207775i) q^{92} +(0.941544 + 0.336890i) q^{98} +(-0.0473045 - 1.92697i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q+O(q^{10}) \) 64 * q	\( 64 q+O(q^{100}) \) 64 * q

Introduction

L-functions

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