LMFDB - Embedded newform 959.1.r.a.734.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [959,1,Mod(202,959)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(959, base_ring=CyclotomicField(34))

chi = DirichletCharacter(H, H._module([17, 25]))

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

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

chi := DirichletCharacter("959.202");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 959 = 7 \cdot 137 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	959.r (of order $34$, degree $16$, 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:	$0.478603347115$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\Q(\zeta_{34})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - x^{15} + x^{14} - x^{13} + x^{12} - x^{11} + x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + \cdots + 1 $ x^16 - x^15 + x^14 - x^13 + x^12 - x^11 + x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{34}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{34} - \cdots)$

Embedding invariants

Embedding label			734.1
Root			$0.602635 + 0.798017i$ of defining polynomial
Character	$\chi$	$=$	959.734
Dual form			959.1.r.a.699.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.25664 - 0.778076i) q^{2} +(0.527993 + 1.06035i) q^{4} +(0.982973 - 0.183750i) q^{7} +(0.0251661 - 0.271585i) q^{8} +(-0.739009 - 0.673696i) q^{9} +O(q^{10})$	\(q+(-1.25664 - 0.778076i) q^{2} +(0.527993 + 1.06035i) q^{4} +(0.982973 - 0.183750i) q^{7} +(0.0251661 - 0.271585i) q^{8} +(-0.739009 - 0.673696i) q^{9} +(0.757949 + 1.52217i) q^{11} +(-1.37821 - 0.533922i) q^{14} +(0.470904 - 0.623578i) q^{16} +(0.404479 + 1.42160i) q^{18} +(0.231896 - 2.50255i) q^{22} +(1.27366 + 0.961826i) q^{23} +(-0.445738 + 0.895163i) q^{25} +(0.713843 + 0.945281i) q^{28} +(-0.840204 - 0.634493i) q^{29} +(-1.33128 + 0.515740i) q^{32} +(0.324164 - 1.13932i) q^{36} +0.891477 q^{37} +(1.53511 - 0.436776i) q^{43} +(-1.21384 + 1.60739i) q^{44} +(-0.852157 - 2.19967i) q^{46} +(0.932472 - 0.361242i) q^{49} +(1.25664 - 0.778076i) q^{50} +(0.694903 - 0.197717i) q^{53} +(-0.0251661 - 0.271585i) q^{56} +(0.562147 + 1.45107i) q^{58} +(-0.850217 - 0.526432i) q^{63} +(1.30611 + 0.244155i) q^{64} +(-0.247582 - 1.32445i) q^{67} +(-1.60263 - 0.798017i) q^{71} +(-0.201564 + 0.183750i) q^{72} +(-1.12026 - 0.693637i) q^{74} +(1.02474 + 1.35698i) q^{77} +(-0.694903 - 1.79375i) q^{79} +(0.0922684 + 0.995734i) q^{81} +(-2.26892 - 0.645562i) q^{86} +(0.432472 - 0.167541i) q^{88} +(-0.347390 + 1.85837i) q^{92} +(-1.45285 - 0.271585i) q^{98} +(0.465346 - 1.63552i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 2 q^{2} - 3 q^{4} + q^{7} - 4 q^{8} + q^{9}+O(q^{10}) $ 16 * q - 2 * q^2 - 3 * q^4 + q^7 - 4 * q^8 + q^9	$ 16 q - 2 q^{2} - 3 q^{4} + q^{7} - 4 q^{8} + q^{9} + 2 q^{11} + 2 q^{14} - 5 q^{16} + 2 q^{18} + 4 q^{22} + 17 q^{23} + q^{25} + 3 q^{28} - 6 q^{32} + 3 q^{36} - 2 q^{37} - 11 q^{44} - q^{49} + 2 q^{50} + 4 q^{56} - q^{63} + 10 q^{64} - 17 q^{71} - 13 q^{72} - 4 q^{74} - 2 q^{77} - q^{81} - 9 q^{88} - 2 q^{98} - 2 q^{99}+O(q^{100}) $ 16 * q - 2 * q^2 - 3 * q^4 + q^7 - 4 * q^8 + q^9 + 2 * q^11 + 2 * q^14 - 5 * q^16 + 2 * q^18 + 4 * q^22 + 17 * q^23 + q^25 + 3 * q^28 - 6 * q^32 + 3 * q^36 - 2 * q^37 - 11 * q^44 - q^49 + 2 * q^50 + 4 * q^56 - q^63 + 10 * q^64 - 17 * q^71 - 13 * q^72 - 4 * q^74 - 2 * q^77 - q^81 - 9 * q^88 - 2 * q^98 - 2 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/959\mathbb{Z}\right)^\times$.

$n$	$414$	$549$
$\chi(n)$	$e\left(\frac{21}{34}\right)$	$-1$

Coefficient data

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

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

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

$2$	−1.25664	−	0.778076i	−1.25664	−	0.778076i	−0.273663	−	0.961826i	$-0.588235\pi$
$2$	−1.25664	−	0.778076i	−1.25664	−	0.778076i	−0.982973	+	0.183750i	$0.941176\pi$
$3$		0			0		0.361242	−	0.932472i	$-0.382353\pi$
$3$		0			0		−0.361242	+	0.932472i	$0.617647\pi$
$4$	0.527993	+	1.06035i	0.527993	+	1.06035i
$5$		0			0		−0.526432	−	0.850217i	$-0.676471\pi$
$5$		0			0		0.526432	+	0.850217i	$0.323529\pi$
$6$		0			0
$7$	0.982973	−	0.183750i	0.982973	−	0.183750i
$7$	0.982973	−	0.183750i	0.982973	−	0.183750i
$8$	0.0251661	−	0.271585i	0.0251661	−	0.271585i
$9$	−0.739009	−	0.673696i	−0.739009	−	0.673696i
$10$		0			0
$11$	0.757949	+	1.52217i	0.757949	+	1.52217i	0.850217	+	0.526432i	$0.176471\pi$
$11$	0.757949	+	1.52217i	0.757949	+	1.52217i	−0.0922684	+	0.995734i	$0.529412\pi$
$12$		0			0
$13$		0			0		−0.183750	−	0.982973i	$-0.558824\pi$
$13$		0			0		0.183750	+	0.982973i	$0.441176\pi$
$14$	−1.37821	−	0.533922i	−1.37821	−	0.533922i
$15$		0			0
$16$	0.470904	−	0.623578i	0.470904	−	0.623578i
$17$		0			0		−0.0922684	−	0.995734i	$-0.529412\pi$
$17$		0			0		0.0922684	+	0.995734i	$0.470588\pi$
$18$	0.404479	+	1.42160i	0.404479	+	1.42160i
$19$		0			0		−0.273663	−	0.961826i	$-0.588235\pi$
$19$		0			0		0.273663	+	0.961826i	$0.411765\pi$
$20$		0			0
$21$		0			0
$22$	0.231896	−	2.50255i	0.231896	−	2.50255i
$23$	1.27366	+	0.961826i	1.27366	+	0.961826i	1.00000			$0$
$23$	1.27366	+	0.961826i	1.27366	+	0.961826i	0.273663	+	0.961826i	$0.411765\pi$
$24$		0			0
$25$	−0.445738	+	0.895163i	−0.445738	+	0.895163i
$26$		0			0
$27$		0			0
$28$	0.713843	+	0.945281i	0.713843	+	0.945281i
$29$	−0.840204	−	0.634493i	−0.840204	−	0.634493i	0.0922684	−	0.995734i	$-0.470588\pi$
$29$	−0.840204	−	0.634493i	−0.840204	−	0.634493i	−0.932472	+	0.361242i	$0.882353\pi$
$30$		0			0
$31$		0			0		−0.961826	−	0.273663i	$-0.911765\pi$
$31$		0			0		0.961826	+	0.273663i	$0.0882353\pi$
$32$	−1.33128	+	0.515740i	−1.33128	+	0.515740i
$33$		0			0
$34$		0			0
$35$		0			0
$36$	0.324164	−	1.13932i	0.324164	−	1.13932i
$37$	0.891477			0.891477			0.445738	−	0.895163i	$-0.352941\pi$
$37$	0.891477			0.891477			0.445738	+	0.895163i	$0.352941\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0			−	1.00000i	$-0.5\pi$
$41$		0			0				1.00000i	$0.5\pi$
$42$		0			0
$43$	1.53511	−	0.436776i	1.53511	−	0.436776i	0.602635	−	0.798017i	$-0.294118\pi$
$43$	1.53511	−	0.436776i	1.53511	−	0.436776i	0.932472	+	0.361242i	$0.117647\pi$
$44$	−1.21384	+	1.60739i	−1.21384	+	1.60739i
$45$		0			0
$46$	−0.852157	−	2.19967i	−0.852157	−	2.19967i
$47$		0			0		0.673696	−	0.739009i	$-0.264706\pi$
$47$		0			0		−0.673696	+	0.739009i	$0.735294\pi$
$48$		0			0
$49$	0.932472	−	0.361242i	0.932472	−	0.361242i
$50$	1.25664	−	0.778076i	1.25664	−	0.778076i
$51$		0			0
$52$		0			0
$53$	0.694903	−	0.197717i	0.694903	−	0.197717i	0.0922684	−	0.995734i	$-0.470588\pi$
$53$	0.694903	−	0.197717i	0.694903	−	0.197717i	0.602635	+	0.798017i	$0.294118\pi$
$54$		0			0
$55$		0			0
$56$	−0.0251661	−	0.271585i	−0.0251661	−	0.271585i
$57$		0			0
$58$	0.562147	+	1.45107i	0.562147	+	1.45107i
$59$		0			0		−0.739009	−	0.673696i	$-0.764706\pi$
$59$		0			0		0.739009	+	0.673696i	$0.235294\pi$
$60$		0			0
$61$		0			0		0.739009	−	0.673696i	$-0.235294\pi$
$61$		0			0		−0.739009	+	0.673696i	$0.764706\pi$
$62$		0			0
$63$	−0.850217	−	0.526432i	−0.850217	−	0.526432i
$64$	1.30611	+	0.244155i	1.30611	+	0.244155i
$65$		0			0
$66$		0			0
$67$	−0.247582	−	1.32445i	−0.247582	−	1.32445i	−0.850217	−	0.526432i	$-0.823529\pi$
$67$	−0.247582	−	1.32445i	−0.247582	−	1.32445i	0.602635	−	0.798017i	$-0.294118\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−1.60263	−	0.798017i	−1.60263	−	0.798017i	−0.602635	−	0.798017i	$-0.705882\pi$
$71$	−1.60263	−	0.798017i	−1.60263	−	0.798017i	−1.00000			$\pi$
$72$	−0.201564	+	0.183750i	−0.201564	+	0.183750i
$73$		0			0		−0.982973	−	0.183750i	$-0.941176\pi$
$73$		0			0		0.982973	+	0.183750i	$0.0588235\pi$
$74$	−1.12026	−	0.693637i	−1.12026	−	0.693637i
$75$		0			0
$76$		0			0
$77$	1.02474	+	1.35698i	1.02474	+	1.35698i
$78$		0			0
$79$	−0.694903	−	1.79375i	−0.694903	−	1.79375i	−0.602635	−	0.798017i	$-0.705882\pi$
$79$	−0.694903	−	1.79375i	−0.694903	−	1.79375i	−0.0922684	−	0.995734i	$-0.529412\pi$
$80$		0			0
$81$	0.0922684	+	0.995734i	0.0922684	+	0.995734i
$82$		0			0
$83$		0			0		−0.995734	−	0.0922684i	$-0.970588\pi$
$83$		0			0		0.995734	+	0.0922684i	$0.0294118\pi$
$84$		0			0
$85$		0			0
$86$	−2.26892	−	0.645562i	−2.26892	−	0.645562i
$87$		0			0
$88$	0.432472	−	0.167541i	0.432472	−	0.167541i
$89$		0			0		−0.526432	−	0.850217i	$-0.676471\pi$
$89$		0			0		0.526432	+	0.850217i	$0.323529\pi$
$90$		0			0
$91$		0			0
$92$	−0.347390	+	1.85837i	−0.347390	+	1.85837i
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		0			0		0.895163	−	0.445738i	$-0.147059\pi$
$97$		0			0		−0.895163	+	0.445738i	$0.852941\pi$
$98$	−1.45285	−	0.271585i	−1.45285	−	0.271585i
$99$	0.465346	−	1.63552i	0.465346	−	1.63552i
$100$	−1.18454			−1.18454
$101$		0			0		0.273663	−	0.961826i	$-0.411765\pi$
$101$		0			0		−0.273663	+	0.961826i	$0.588235\pi$
$102$		0			0
$103$		0			0		0.445738	−	0.895163i	$-0.352941\pi$
$103$		0			0		−0.445738	+	0.895163i	$0.647059\pi$
$104$		0			0
$105$		0			0
$106$	−1.02708	−	0.292229i	−1.02708	−	0.292229i
$107$	1.12388	+	0.435393i	1.12388	+	0.435393i	0.850217	−	0.526432i	$-0.176471\pi$
$107$	1.12388	+	0.435393i	1.12388	+	0.435393i	0.273663	+	0.961826i	$0.411765\pi$
$108$		0			0
$109$	0.890705	+	1.17948i	0.890705	+	1.17948i	0.982973	+	0.183750i	$0.0588235\pi$
$109$	0.890705	+	1.17948i	0.890705	+	1.17948i	−0.0922684	+	0.995734i	$0.529412\pi$
$110$		0			0
$111$		0			0
$112$	0.348304	−	0.699489i	0.348304	−	0.699489i
$113$	−0.328972	−	0.163808i	−0.328972	−	0.163808i	0.273663	−	0.961826i	$-0.411765\pi$
$113$	−0.328972	−	0.163808i	−0.328972	−	0.163808i	−0.602635	+	0.798017i	$0.705882\pi$
$114$		0			0
$115$		0			0
$116$	0.229164	−	1.22592i	0.229164	−	1.22592i
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−1.13987	+	1.50943i	−1.13987	+	1.50943i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$	0.658809	+	1.32307i	0.658809	+	1.32307i
$127$			1.34739i			1.34739i	0.739009	+	0.673696i	$0.235294\pi$
$127$			1.34739i			1.34739i	−0.739009	+	0.673696i	$0.764706\pi$
$128$	−0.396263	−	0.361242i	−0.396263	−	0.361242i
$129$		0			0
$130$		0			0
$131$		0			0		0.798017	−	0.602635i	$-0.205882\pi$
$131$		0			0		−0.798017	+	0.602635i	$0.794118\pi$
$132$		0			0
$133$		0			0
$134$	−0.719401	+	1.85699i	−0.719401	+	1.85699i
$135$		0			0
$136$		0			0
$137$	−1.00000			−1.00000
$137$	−1.00000			−1.00000
$138$		0			0
$139$		0			0		−0.850217	−	0.526432i	$-0.823529\pi$
$139$		0			0		0.850217	+	0.526432i	$0.176471\pi$
$140$		0			0
$141$		0			0
$142$	1.39301	+	2.24979i	1.39301	+	2.24979i
$143$		0			0
$144$	−0.768104	+	0.143584i	−0.768104	+	0.143584i
$145$		0			0
$146$		0			0
$147$		0			0
$148$	0.470694	+	0.945281i	0.470694	+	0.945281i
$149$	−0.719401	+	0.0666624i	−0.719401	+	0.0666624i	−0.445738	−	0.895163i	$-0.647059\pi$
$149$	−0.719401	+	0.0666624i	−0.719401	+	0.0666624i	−0.273663	+	0.961826i	$0.588235\pi$
$150$		0			0
$151$	−1.86494	−	0.722483i	−1.86494	−	0.722483i	−0.932472	−	0.361242i	$-0.882353\pi$
$151$	−1.86494	−	0.722483i	−1.86494	−	0.722483i	−0.932472	−	0.361242i	$-0.882353\pi$
$152$		0			0
$153$		0			0
$154$	−0.231896	−	2.50255i	−0.231896	−	2.50255i
$155$		0			0
$156$		0			0
$157$		0			0		0.526432	−	0.850217i	$-0.323529\pi$
$157$		0			0		−0.526432	+	0.850217i	$0.676471\pi$
$158$	−0.522435	+	2.79478i	−0.522435	+	2.79478i
$159$		0			0
$160$		0			0
$161$	1.42871	+	0.711414i	1.42871	+	0.711414i
$162$	0.658809	−	1.32307i	0.658809	−	1.32307i
$163$	−0.719401	+	1.85699i	−0.719401	+	1.85699i	−0.273663	+	0.961826i	$0.588235\pi$
$163$	−0.719401	+	1.85699i	−0.719401	+	1.85699i	−0.445738	+	0.895163i	$0.647059\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.932472	−	0.361242i	$-0.882353\pi$
$167$		0			0		0.932472	+	0.361242i	$0.117647\pi$
$168$		0			0
$169$	−0.932472	+	0.361242i	−0.932472	+	0.361242i
$170$		0			0
$171$		0			0
$172$	1.27366	+	1.39714i	1.27366	+	1.39714i
$173$		0			0		0.273663	−	0.961826i	$-0.411765\pi$
$173$		0			0		−0.273663	+	0.961826i	$0.588235\pi$
$174$		0			0
$175$	−0.273663	+	0.961826i	−0.273663	+	0.961826i
$176$	1.30611	+	0.244155i	1.30611	+	0.244155i
$177$		0			0
$178$		0			0
$179$		0			0		−0.739009	−	0.673696i	$-0.764706\pi$
$179$		0			0		0.739009	+	0.673696i	$0.235294\pi$
$180$		0			0
$181$		0			0		0.602635	−	0.798017i	$-0.294118\pi$
$181$		0			0		−0.602635	+	0.798017i	$0.705882\pi$
$182$		0			0
$183$		0			0
$184$	0.293271	−	0.321702i	0.293271	−	0.321702i
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	1.34164	+	0.124322i	1.34164	+	0.124322i	0.739009	−	0.673696i	$-0.235294\pi$
$191$	1.34164	+	0.124322i	1.34164	+	0.124322i	0.602635	+	0.798017i	$0.294118\pi$
$192$		0			0
$193$	0.136374	+	1.47171i	0.136374	+	1.47171i	0.739009	+	0.673696i	$0.235294\pi$
$193$	0.136374	+	1.47171i	0.136374	+	1.47171i	−0.602635	+	0.798017i	$0.705882\pi$
$194$		0			0
$195$		0			0
$196$	0.875383	+	0.798017i	0.875383	+	0.798017i
$197$	0.111208	+	0.147263i	0.111208	+	0.147263i	0.850217	−	0.526432i	$-0.176471\pi$
$197$	0.111208	+	0.147263i	0.111208	+	0.147263i	−0.739009	+	0.673696i	$0.764706\pi$
$198$	−1.85733	+	1.69318i	−1.85733	+	1.69318i
$199$		0			0		−0.673696	−	0.739009i	$-0.735294\pi$
$199$		0			0		0.673696	+	0.739009i	$0.264706\pi$
$200$	0.231896	+	0.143584i	0.231896	+	0.143584i
$201$		0			0
$202$		0			0
$203$	−0.942485	−	0.469302i	−0.942485	−	0.469302i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−0.293271	−	1.56886i	−0.293271	−	1.56886i
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−1.58561	−	0.981767i	−1.58561	−	0.981767i	−0.982973	−	0.183750i	$-0.941176\pi$
$211$	−1.58561	−	0.981767i	−1.58561	−	0.981767i	−0.602635	−	0.798017i	$-0.705882\pi$
$212$	0.576554	+	0.632450i	0.576554	+	0.632450i
$213$		0			0
$214$	−1.07354	−	1.42160i	−1.07354	−	1.42160i
$215$		0			0
$216$		0			0
$217$		0			0
$218$	−0.201564	−	2.17522i	−0.201564	−	2.17522i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		−0.961826	−	0.273663i	$-0.911765\pi$
$223$		0			0		0.961826	+	0.273663i	$0.0882353\pi$
$224$	−1.21384	+	0.751580i	−1.21384	+	0.751580i
$225$	0.932472	−	0.361242i	0.932472	−	0.361242i
$226$	0.285942	+	0.461813i	0.285942	+	0.461813i
$227$		0			0		0.673696	−	0.739009i	$-0.264706\pi$
$227$		0			0		−0.673696	+	0.739009i	$0.735294\pi$
$228$		0			0
$229$		0			0		0.183750	−	0.982973i	$-0.441176\pi$
$229$		0			0		−0.183750	+	0.982973i	$0.558824\pi$
$230$		0			0
$231$		0			0
$232$	−0.193463	+	0.212219i	−0.193463	+	0.212219i
$233$		−	1.92365i		−	1.92365i	−0.273663	−	0.961826i	$-0.588235\pi$
$233$		−	1.92365i		−	1.92365i	0.273663	−	0.961826i	$-0.411765\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−1.29596	−	1.42160i	−1.29596	−	1.42160i	−0.850217	−	0.526432i	$-0.823529\pi$
$239$	−1.29596	−	1.42160i	−1.29596	−	1.42160i	−0.445738	−	0.895163i	$-0.647059\pi$
$240$		0			0
$241$		0			0		0.995734	−	0.0922684i	$-0.0294118\pi$
$241$		0			0		−0.995734	+	0.0922684i	$0.970588\pi$
$242$	2.60685	−	1.00990i	2.60685	−	1.00990i
$243$		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		−0.798017	−	0.602635i	$-0.794118\pi$
$251$		0			0		0.798017	+	0.602635i	$0.205882\pi$
$252$	0.109295	−	1.17948i	0.109295	−	1.17948i
$253$	−0.498687	+	2.66774i	−0.498687	+	2.66774i
$254$	1.04837	−	1.69318i	1.04837	−	1.69318i
$255$		0			0
$256$	−0.146741	−	0.515740i	−0.146741	−	0.515740i
$257$		0			0		−0.0922684	−	0.995734i	$-0.529412\pi$
$257$		0			0		0.0922684	+	0.995734i	$0.470588\pi$
$258$		0			0
$259$	0.876298	−	0.163808i	0.876298	−	0.163808i
$260$		0			0
$261$	0.193463	+	1.03494i	0.193463	+	1.03494i
$262$		0			0
$263$	−0.243964	−	0.489946i	−0.243964	−	0.489946i	0.739009	−	0.673696i	$-0.235294\pi$
$263$	−0.243964	−	0.489946i	−0.243964	−	0.489946i	−0.982973	+	0.183750i	$0.941176\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	1.27366	−	0.961826i	1.27366	−	0.961826i
$269$		0			0		−0.526432	−	0.850217i	$-0.676471\pi$
$269$		0			0		0.526432	+	0.850217i	$0.323529\pi$
$270$		0			0
$271$		0			0		0.361242	−	0.932472i	$-0.382353\pi$
$271$		0			0		−0.361242	+	0.932472i	$0.617647\pi$
$272$		0			0
$273$		0			0
$274$	1.25664	+	0.778076i	1.25664	+	0.778076i
$275$	−1.70043			−1.70043
$276$		0			0
$277$	−0.380338	+	0.981767i	−0.380338	+	0.981767i	0.602635	+	0.798017i	$0.294118\pi$
$277$	−0.380338	+	0.981767i	−0.380338	+	0.981767i	−0.982973	+	0.183750i	$0.941176\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−1.83319	+	0.342683i	−1.83319	+	0.342683i	−0.982973	−	0.183750i	$-0.941176\pi$
$281$	−1.83319	+	0.342683i	−1.83319	+	0.342683i	−0.850217	+	0.526432i	$0.823529\pi$
$282$		0			0
$283$		0			0		−0.739009	−	0.673696i	$-0.764706\pi$
$283$		0			0		0.739009	+	0.673696i	$0.235294\pi$
$284$		−	2.12071i		−	2.12071i
$285$		0			0
$286$		0			0
$287$		0			0
$288$	1.33128	+	0.515740i	1.33128	+	0.515740i
$289$	−0.982973	+	0.183750i	−0.982973	+	0.183750i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0		−0.273663	−	0.961826i	$-0.588235\pi$
$293$		0			0		0.273663	+	0.961826i	$0.411765\pi$
$294$		0			0
$295$		0			0
$296$	0.0224350	−	0.242112i	0.0224350	−	0.242112i
$297$		0			0
$298$	0.955894	+	0.475979i	0.955894	+	0.475979i
$299$		0			0
$300$		0			0
$301$	1.42871	−	0.711414i	1.42871	−	0.711414i
$302$	1.78141	+	2.35897i	1.78141	+	2.35897i
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0		0.995734	−	0.0922684i	$-0.0294118\pi$
$307$		0			0		−0.995734	+	0.0922684i	$0.970588\pi$
$308$	−0.897818	+	1.80306i	−0.897818	+	1.80306i
$309$		0			0
$310$		0			0
$311$		0			0		1.00000			$0$
$311$		0			0		−1.00000			$\pi$
$312$		0			0
$313$		0			0		−0.982973	−	0.183750i	$-0.941176\pi$
$313$		0			0		0.982973	+	0.183750i	$0.0588235\pi$
$314$		0			0
$315$		0			0
$316$	1.53511	−	1.68393i	1.53511	−	1.68393i
$317$	1.72198	−	0.489946i	1.72198	−	0.489946i	0.739009	−	0.673696i	$-0.235294\pi$
$317$	1.72198	−	0.489946i	1.72198	−	0.489946i	0.982973	+	0.183750i	$0.0588235\pi$
$318$		0			0
$319$	0.328972	−	1.75984i	0.328972	−	1.75984i
$320$		0			0
$321$		0			0
$322$	−1.24184	−	2.00563i	−1.24184	−	2.00563i
$323$		0			0
$324$	−1.00711	+	0.623578i	−1.00711	+	0.623578i
$325$		0			0
$326$	2.34890	−	1.77381i	2.34890	−	1.77381i
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	1.78269	+	0.165190i	1.78269	+	0.165190i	0.932472	−	0.361242i	$-0.117647\pi$
$331$	1.78269	+	0.165190i	1.78269	+	0.165190i	0.850217	+	0.526432i	$0.176471\pi$
$332$		0			0
$333$	−0.658809	−	0.600584i	−0.658809	−	0.600584i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−1.44574	−	0.895163i	−1.44574	−	0.895163i	−0.445738	−	0.895163i	$-0.647059\pi$
$337$	−1.44574	−	0.895163i	−1.44574	−	0.895163i	−1.00000			$\pi$
$338$	1.45285	+	0.271585i	1.45285	+	0.271585i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	0.850217	−	0.526432i	0.850217	−	0.526432i
$344$	−0.0799891	−	0.427904i	−0.0799891	−	0.427904i
$345$		0			0
$346$		0			0
$347$	−1.93247	−	0.361242i	−1.93247	−	0.361242i	−0.932472	−	0.361242i	$-0.882353\pi$
$347$	−1.93247	−	0.361242i	−1.93247	−	0.361242i	−1.00000			$\pi$
$348$		0			0
$349$		0			0		−0.673696	−	0.739009i	$-0.735294\pi$
$349$		0			0		0.673696	+	0.739009i	$0.264706\pi$
$350$	1.09227	−	0.995734i	1.09227	−	0.995734i
$351$		0			0
$352$	−1.79408	−	1.63552i	−1.79408	−	1.63552i
$353$		0			0		−0.361242	−	0.932472i	$-0.617647\pi$
$353$		0			0		0.361242	+	0.932472i	$0.382353\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	0.840204	−	0.634493i	0.840204	−	0.634493i	−0.0922684	−	0.995734i	$-0.529412\pi$
$359$	0.840204	−	0.634493i	0.840204	−	0.634493i	0.932472	+	0.361242i	$0.117647\pi$
$360$		0			0
$361$	−0.850217	+	0.526432i	−0.850217	+	0.526432i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		0.602635	−	0.798017i	$-0.294118\pi$
$367$		0			0		−0.602635	+	0.798017i	$0.705882\pi$
$368$	1.19955	−	0.341301i	1.19955	−	0.341301i
$369$		0			0
$370$		0			0
$371$	0.646741	−	0.322039i	0.646741	−	0.322039i
$372$		0			0
$373$	0.538007	−	1.89090i	0.538007	−	1.89090i	0.0922684	−	0.995734i	$-0.470588\pi$
$373$	0.538007	−	1.89090i	0.538007	−	1.89090i	0.445738	−	0.895163i	$-0.352941\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	1.73901	−	0.673696i	1.73901	−	0.673696i	0.739009	−	0.673696i	$-0.235294\pi$
$379$	1.73901	−	0.673696i	1.73901	−	0.673696i	1.00000			$0$
$380$		0			0
$381$		0			0
$382$	−1.58923	−	1.20013i	−1.58923	−	1.20013i
$383$		0			0		−0.602635	−	0.798017i	$-0.705882\pi$
$383$		0			0		0.602635	+	0.798017i	$0.294118\pi$
$384$		0			0
$385$		0			0
$386$	0.973732	−	1.95552i	0.973732	−	1.95552i
$387$	−1.42871	−	0.711414i	−1.42871	−	0.711414i
$388$		0			0
$389$	0.181395	−	1.95756i	0.181395	−	1.95756i	−0.0922684	−	0.995734i	$-0.529412\pi$
$389$	0.181395	−	1.95756i	0.181395	−	1.95756i	0.273663	−	0.961826i	$-0.411765\pi$
$390$		0			0
$391$		0			0
$392$	−0.0746412	−	0.262337i	−0.0746412	−	0.262337i
$393$		0			0
$394$	−0.0251661	−	0.271585i	−0.0251661	−	0.271585i
$395$		0			0
$396$	1.97993	−	0.370113i	1.97993	−	0.370113i
$397$		0			0		−0.932472	−	0.361242i	$-0.882353\pi$
$397$		0			0		0.932472	+	0.361242i	$0.117647\pi$
$398$		0			0
$399$		0			0
$400$	0.348304	+	0.699489i	0.348304	+	0.699489i
$401$		−	0.367499i		−	0.367499i	−0.982973	−	0.183750i	$-0.941176\pi$
$401$		−	0.367499i		−	0.367499i	0.982973	−	0.183750i	$-0.0588235\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$	0.819209	+	1.32307i	0.819209	+	1.32307i
$407$	0.675694	+	1.35698i	0.675694	+	1.35698i
$408$		0			0
$409$		0			0		−0.850217	−	0.526432i	$-0.823529\pi$
$409$		0			0		0.850217	+	0.526432i	$0.176471\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$	−0.852157	+	2.19967i	−0.852157	+	2.19967i
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		0.0922684	−	0.995734i	$-0.470588\pi$
$419$		0			0		−0.0922684	+	0.995734i	$0.529412\pi$
$420$		0			0
$421$			1.79033i			1.79033i	0.445738	+	0.895163i	$0.352941\pi$
$421$			1.79033i			1.79033i	−0.445738	+	0.895163i	$0.647059\pi$
$422$	1.22864	+	2.46745i	1.22864	+	2.46745i
$423$		0			0
$424$	−0.0362090	−	0.193701i	−0.0362090	−	0.193701i
$425$		0			0
$426$		0			0
$427$		0			0
$428$	0.131730	+	1.42160i	0.131730	+	1.42160i
$429$		0			0
$430$		0			0
$431$	0.709310	−	1.14558i	0.709310	−	1.14558i	−0.273663	−	0.961826i	$-0.588235\pi$
$431$	0.709310	−	1.14558i	0.709310	−	1.14558i	0.982973	−	0.183750i	$-0.0588235\pi$
$432$		0			0
$433$		0			0		0.0922684	−	0.995734i	$-0.470588\pi$
$433$		0			0		−0.0922684	+	0.995734i	$0.529412\pi$
$434$		0			0
$435$		0			0
$436$	−0.780384	+	1.56722i	−0.780384	+	1.56722i
$437$		0			0
$438$		0			0
$439$		0			0		−0.602635	−	0.798017i	$-0.705882\pi$
$439$		0			0		0.602635	+	0.798017i	$0.294118\pi$
$440$		0			0
$441$	−0.932472	−	0.361242i	−0.932472	−	0.361242i
$442$		0			0
$443$	1.83319	−	0.710182i	1.83319	−	0.710182i	0.850217	−	0.526432i	$-0.176471\pi$
$443$	1.83319	−	0.710182i	1.83319	−	0.710182i	0.982973	−	0.183750i	$-0.0588235\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	1.32874			1.32874
$449$	−0.510366	+	1.79375i	−0.510366	+	1.79375i	0.0922684	+	0.995734i	$0.470588\pi$
$449$	−0.510366	+	1.79375i	−0.510366	+	1.79375i	−0.602635	+	0.798017i	$0.705882\pi$
$450$	−1.45285	−	0.271585i	−1.45285	−	0.271585i
$451$		0			0
$452$		−	0.435316i		−	0.435316i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$		0			0		0.932472	−	0.361242i	$-0.117647\pi$
$457$		0			0		−0.932472	+	0.361242i	$0.882353\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		0.850217	−	0.526432i	$-0.176471\pi$
$461$		0			0		−0.850217	+	0.526432i	$0.823529\pi$
$462$		0			0
$463$	−1.58923	+	1.20013i	−1.58923	+	1.20013i	−0.739009	+	0.673696i	$0.764706\pi$
$463$	−1.58923	+	1.20013i	−1.58923	+	1.20013i	−0.850217	+	0.526432i	$0.823529\pi$
$464$	−0.791311	+	0.225148i	−0.791311	+	0.225148i
$465$		0			0
$466$	−1.49675	+	2.41733i	−1.49675	+	2.41733i
$467$		0			0		−0.0922684	−	0.995734i	$-0.529412\pi$
$467$		0			0		0.0922684	+	0.995734i	$0.470588\pi$
$468$		0			0
$469$	−0.486734	−	1.25640i	−0.486734	−	1.25640i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	1.82838	+	2.00563i	1.82838	+	2.00563i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−0.646741	−	0.322039i	−0.646741	−	0.322039i
$478$	0.522435	+	2.79478i	0.522435	+	2.79478i
$479$		0			0		0.850217	−	0.526432i	$-0.176471\pi$
$479$		0			0		−0.850217	+	0.526432i	$0.823529\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	−2.20237	−	0.411695i	−2.20237	−	0.411695i
$485$		0			0
$486$		0			0
$487$	0.404479	−	0.368731i	0.404479	−	0.368731i	−0.445738	−	0.895163i	$-0.647059\pi$
$487$	0.404479	−	0.368731i	0.404479	−	0.368731i	0.850217	+	0.526432i	$0.176471\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−1.04837	−	0.0971461i	−1.04837	−	0.0971461i	−0.445738	−	0.895163i	$-0.647059\pi$
$491$	−1.04837	−	0.0971461i	−1.04837	−	0.0971461i	−0.602635	+	0.798017i	$0.705882\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−1.72198	−	0.489946i	−1.72198	−	0.489946i
$498$		0			0
$499$	−1.73901	+	0.673696i	−1.73901	+	0.673696i	−0.739009	+	0.673696i	$0.764706\pi$
$499$	−1.73901	+	0.673696i	−1.73901	+	0.673696i	−1.00000			$\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		0.183750	−	0.982973i	$-0.441176\pi$
$503$		0			0		−0.183750	+	0.982973i	$0.558824\pi$
$504$	−0.164368	+	0.217658i	−0.164368	+	0.217658i
$505$		0			0
$506$	2.70237	−	2.96436i	2.70237	−	2.96436i
$507$		0			0
$508$	−1.42871	+	0.711414i	−1.42871	+	0.711414i
$509$		0			0		−0.982973	−	0.183750i	$-0.941176\pi$
$509$		0			0		0.982973	+	0.183750i	$0.0588235\pi$
$510$		0			0
$511$		0			0
$512$	−0.363626	+	1.27801i	−0.363626	+	1.27801i
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$	−1.22864	−	0.475979i	−1.22864	−	0.475979i
$519$		0			0
$520$		0			0
$521$		0			0		0.895163	−	0.445738i	$-0.147059\pi$
$521$		0			0		−0.895163	+	0.445738i	$0.852941\pi$
$522$	0.562147	−	1.45107i	0.562147	−	1.45107i
$523$		0			0		0.445738	−	0.895163i	$-0.352941\pi$
$523$		0			0		−0.445738	+	0.895163i	$0.647059\pi$
$524$		0			0
$525$		0			0
$526$	−0.0746412	+	0.805507i	−0.0746412	+	0.805507i
$527$		0			0
$528$		0			0
$529$	0.423446	+	1.48826i	0.423446	+	1.48826i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$	−0.365931	+	0.0339085i	−0.365931	+	0.0339085i
$537$		0			0
$538$		0			0
$539$	1.25664	+	1.14558i	1.25664	+	1.14558i
$540$		0			0
$541$	−0.181395	+	0.0339085i	−0.181395	+	0.0339085i	−0.273663	−	0.961826i	$-0.588235\pi$
$541$	−0.181395	+	0.0339085i	−0.181395	+	0.0339085i	0.0922684	+	0.995734i	$0.470588\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−0.547326			−0.547326			−0.273663	−	0.961826i	$-0.588235\pi$
$547$	−0.547326			−0.547326			−0.273663	+	0.961826i	$0.588235\pi$
$548$	−0.527993	−	1.06035i	−0.527993	−	1.06035i
$549$		0			0
$550$	2.13683	+	1.32307i	2.13683	+	1.32307i
$551$		0			0
$552$		0			0
$553$	−1.01267	−	1.63552i	−1.01267	−	1.63552i
$554$	1.24184	−	0.937791i	1.24184	−	0.937791i
$555$		0			0
$556$		0			0
$557$	0.404479	+	0.368731i	0.404479	+	0.368731i	0.850217	−	0.526432i	$-0.176471\pi$
$557$	0.404479	+	0.368731i	0.404479	+	0.368731i	−0.445738	+	0.895163i	$0.647059\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	2.57029	+	0.995734i	2.57029	+	0.995734i
$563$		0			0		0.982973	−	0.183750i	$-0.0588235\pi$
$563$		0			0		−0.982973	+	0.183750i	$0.941176\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	0.273663	+	0.961826i	0.273663	+	0.961826i
$568$	−0.257062	+	0.415169i	−0.257062	+	0.415169i
$569$	0.0675278	−	0.361242i	0.0675278	−	0.361242i	−0.932472	−	0.361242i	$-0.882353\pi$
$569$	0.0675278	−	0.361242i	0.0675278	−	0.361242i	1.00000			$0$
$570$		0			0
$571$	−1.53511	−	1.15926i	−1.53511	−	1.15926i	−0.932472	−	0.361242i	$-0.882353\pi$
$571$	−1.53511	−	1.15926i	−1.53511	−	1.15926i	−0.602635	−	0.798017i	$-0.705882\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−1.42871	+	0.711414i	−1.42871	+	0.711414i
$576$	−0.800742	−	1.06035i	−0.800742	−	1.06035i
$577$		0			0		−0.798017	−	0.602635i	$-0.794118\pi$
$577$		0			0		0.798017	+	0.602635i	$0.205882\pi$
$578$	1.37821	+	0.533922i	1.37821	+	0.533922i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	0.827659	+	0.907899i	0.827659	+	0.907899i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		−0.982973	−	0.183750i	$-0.941176\pi$
$587$		0			0		0.982973	+	0.183750i	$0.0588235\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	0.419800	−	0.555905i	0.419800	−	0.555905i
$593$		0			0		0.183750	−	0.982973i	$-0.441176\pi$
$593$		0			0		−0.183750	+	0.982973i	$0.558824\pi$
$594$		0			0
$595$		0			0
$596$	−0.450525	−	0.727623i	−0.450525	−	0.727623i
$597$		0			0
$598$		0			0
$599$	0.694903	+	0.197717i	0.694903	+	0.197717i	0.602635	−	0.798017i	$-0.294118\pi$
$599$	0.694903	+	0.197717i	0.694903	+	0.197717i	0.0922684	+	0.995734i	$0.470588\pi$
$600$		0			0
$601$		0			0		0.961826	−	0.273663i	$-0.0882353\pi$
$601$		0			0		−0.961826	+	0.273663i	$0.911765\pi$
$602$	−2.34890	−	0.217658i	−2.34890	−	0.217658i
$603$	−0.709310	+	1.14558i	−0.709310	+	1.14558i
$604$	−0.218591	−	2.35897i	−0.218591	−	2.35897i
$605$		0			0
$606$		0			0
$607$		0			0		−0.739009	−	0.673696i	$-0.764706\pi$
$607$		0			0		0.739009	+	0.673696i	$0.235294\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	0.890705	−	0.811985i	0.890705	−	0.811985i	−0.0922684	−	0.995734i	$-0.529412\pi$
$613$	0.890705	−	0.811985i	0.890705	−	0.811985i	0.982973	+	0.183750i	$0.0588235\pi$
$614$		0			0
$615$		0			0
$616$	0.394323	−	0.244155i	0.394323	−	0.244155i
$617$	−0.465346	+	0.288130i	−0.465346	+	0.288130i	−0.739009	−	0.673696i	$-0.764706\pi$
$617$	−0.465346	+	0.288130i	−0.465346	+	0.288130i	0.273663	+	0.961826i	$0.411765\pi$
$618$		0			0
$619$		0			0		−0.895163	−	0.445738i	$-0.852941\pi$
$619$		0			0		0.895163	+	0.445738i	$0.147059\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−0.602635	−	0.798017i	−0.602635	−	0.798017i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	1.78269	+	0.165190i	1.78269	+	0.165190i	0.932472	−	0.361242i	$-0.117647\pi$
$631$	1.78269	+	0.165190i	1.78269	+	0.165190i	0.850217	+	0.526432i	$0.176471\pi$
$632$	−0.504644	+	0.143584i	−0.504644	+	0.143584i
$633$		0			0
$634$	−2.54512	−	0.724149i	−2.54512	−	0.724149i
$635$		0			0
$636$		0			0
$637$		0			0
$638$	−1.78269	+	1.95552i	−1.78269	+	1.95552i
$639$	0.646741	+	1.66943i	0.646741	+	1.66943i
$640$		0			0
$641$	−1.12388	+	1.48826i	−1.12388	+	1.48826i	−0.273663	+	0.961826i	$0.588235\pi$
$641$	−1.12388	+	1.48826i	−1.12388	+	1.48826i	−0.850217	+	0.526432i	$0.823529\pi$
$642$		0			0
$643$		0			0		0.673696	−	0.739009i	$-0.264706\pi$
$643$		0			0		−0.673696	+	0.739009i	$0.735294\pi$
$644$			1.89056i			1.89056i
$645$		0			0
$646$		0			0
$647$		0			0		0.273663	−	0.961826i	$-0.411765\pi$
$647$		0			0		−0.273663	+	0.961826i	$0.588235\pi$
$648$	0.272749			0.272749
$649$		0			0
$650$		0			0
$651$		0			0
$652$	−2.34890	+	0.217658i	−2.34890	+	0.217658i
$653$	−1.73901	+	0.673696i	−1.73901	+	0.673696i	−0.739009	+	0.673696i	$0.764706\pi$
$653$	−1.73901	+	0.673696i	−1.73901	+	0.673696i	−1.00000			$\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	0.719401	−	1.85699i	0.719401	−	1.85699i	0.273663	−	0.961826i	$-0.411765\pi$
$659$	0.719401	−	1.85699i	0.719401	−	1.85699i	0.445738	−	0.895163i	$-0.352941\pi$
$660$		0			0
$661$		0			0		−0.895163	−	0.445738i	$-0.852941\pi$
$661$		0			0		0.895163	+	0.445738i	$0.147059\pi$
$662$	−2.11166	−	1.59465i	−2.11166	−	1.59465i
$663$		0			0
$664$		0			0
$665$		0			0
$666$	0.360583	+	1.26732i	0.360583	+	1.26732i
$667$	−0.459865	−	1.61626i	−0.459865	−	1.61626i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−1.04837	+	0.0971461i	−1.04837	+	0.0971461i	−0.602635	−	0.798017i	$-0.705882\pi$
$673$	−1.04837	+	0.0971461i	−1.04837	+	0.0971461i	−0.445738	+	0.895163i	$0.647059\pi$
$674$	1.12026	+	2.24979i	1.12026	+	2.24979i
$675$		0			0
$676$	−0.875383	−	0.798017i	−0.875383	−	0.798017i
$677$		0			0		0.0922684	−	0.995734i	$-0.470588\pi$
$677$		0			0		−0.0922684	+	0.995734i	$0.529412\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	1.02474	+	0.634493i	1.02474	+	0.634493i	0.932472	−	0.361242i	$-0.117647\pi$
$683$	1.02474	+	0.634493i	1.02474	+	0.634493i	0.0922684	+	0.995734i	$0.470588\pi$
$684$		0			0
$685$		0			0
$686$	−1.47802			−1.47802
$687$		0			0
$688$	0.450525	−	1.16294i	0.450525	−	1.16294i
$689$		0			0
$690$		0			0
$691$		0			0		0.798017	−	0.602635i	$-0.205882\pi$
$691$		0			0		−0.798017	+	0.602635i	$0.794118\pi$
$692$		0			0
$693$	0.156896	−	1.69318i	0.156896	−	1.69318i
$694$	2.14734	+	1.95756i	2.14734	+	1.95756i
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−1.16437	+	0.217658i	−1.16437	+	0.217658i
$701$	0.537235	−	0.711414i	0.537235	−	0.711414i	−0.445738	−	0.895163i	$-0.647059\pi$
$701$	0.537235	−	0.711414i	0.537235	−	0.711414i	0.982973	+	0.183750i	$0.0588235\pi$
$702$		0			0
$703$		0			0
$704$	0.618322	+	2.17318i	0.618322	+	2.17318i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	0.942485	+	0.469302i	0.942485	+	0.469302i	0.850217	−	0.526432i	$-0.176471\pi$
$709$	0.942485	+	0.469302i	0.942485	+	0.469302i	0.0922684	+	0.995734i	$0.470588\pi$
$710$		0			0
$711$	−0.694903	+	1.79375i	−0.694903	+	1.79375i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$	−1.54951	+	0.143584i	−1.54951	+	0.143584i
$719$		0			0		0.445738	−	0.895163i	$-0.352941\pi$
$719$		0			0		−0.445738	+	0.895163i	$0.647059\pi$
$720$		0			0
$721$		0			0
$722$	1.47802			1.47802
$723$		0			0
$724$		0			0
$725$	0.942485	−	0.469302i	0.942485	−	0.469302i
$726$		0			0
$727$		0			0		0.673696	−	0.739009i	$-0.264706\pi$
$727$		0			0		−0.673696	+	0.739009i	$0.735294\pi$
$728$		0			0
$729$	0.602635	−	0.798017i	0.602635	−	0.798017i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		−0.526432	−	0.850217i	$-0.676471\pi$
$733$		0			0		0.526432	+	0.850217i	$0.323529\pi$
$734$		0			0
$735$		0			0
$736$	−2.19165	−	0.623578i	−2.19165	−	0.623578i
$737$	1.82838	−	1.38073i	1.82838	−	1.38073i
$738$		0			0
$739$	1.91545	+	0.177492i	1.91545	+	0.177492i	0.982973	−	0.183750i	$-0.0588235\pi$
$739$	1.91545	+	0.177492i	1.91545	+	0.177492i	0.932472	+	0.361242i	$0.117647\pi$
$740$		0			0
$741$		0			0
$742$	−1.06329	−	0.0985281i	−1.06329	−	0.0985281i
$743$	0.719401	+	1.85699i	0.719401	+	1.85699i	0.445738	+	0.895163i	$0.352941\pi$
$743$	0.719401	+	1.85699i	0.719401	+	1.85699i	0.273663	+	0.961826i	$0.411765\pi$
$744$		0			0
$745$		0			0
$746$	−2.14734	+	1.95756i	−2.14734	+	1.95756i
$747$		0			0
$748$		0			0
$749$	1.18475	+	0.221468i	1.18475	+	0.221468i
$750$		0			0
$751$	−0.328972	−	0.163808i	−0.328972	−	0.163808i	0.273663	−	0.961826i	$-0.411765\pi$
$751$	−0.328972	−	0.163808i	−0.328972	−	0.163808i	−0.602635	+	0.798017i	$0.705882\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	0.658809	−	0.600584i	0.658809	−	0.600584i	−0.273663	−	0.961826i	$-0.588235\pi$
$757$	0.658809	−	0.600584i	0.658809	−	0.600584i	0.932472	+	0.361242i	$0.117647\pi$
$758$	−2.70949	−	0.506491i	−2.70949	−	0.506491i
$759$		0			0
$760$		0			0
$761$		0			0		0.739009	−	0.673696i	$-0.235294\pi$
$761$		0			0		−0.739009	+	0.673696i	$0.764706\pi$
$762$		0			0
$763$	1.09227	+	0.995734i	1.09227	+	0.995734i
$764$	0.576554	+	1.48826i	0.576554	+	1.48826i
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$		0			0		0.961826	−	0.273663i	$-0.0882353\pi$
$769$		0			0		−0.961826	+	0.273663i	$0.911765\pi$
$770$		0			0
$771$		0			0
$772$	−1.48853	+	0.921660i	−1.48853	+	0.921660i
$773$		0			0		0.932472	−	0.361242i	$-0.117647\pi$
$773$		0			0		−0.932472	+	0.361242i	$0.882353\pi$
$774$	1.24184	+	2.00563i	1.24184	+	2.00563i
$775$		0			0
$776$		0			0
$777$		0			0
$778$	−1.75108	+	2.31880i	−1.75108	+	2.31880i
$779$		0			0
$780$		0			0
$781$		−	3.04433i		−	3.04433i
$782$		0			0
$783$		0			0
$784$	0.213843	−	0.751580i	0.213843	−	0.751580i
$785$		0			0
$786$		0			0
$787$		0			0		−0.673696	−	0.739009i	$-0.735294\pi$
$787$		0			0		0.673696	+	0.739009i	$0.264706\pi$
$788$	−0.0974342	+	0.195674i	−0.0974342	+	0.195674i
$789$		0			0
$790$		0			0
$791$	−0.353470	−	0.100571i	−0.353470	−	0.100571i
$792$	−0.432472	−	0.167541i	−0.432472	−	0.167541i
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		0.445738	−	0.895163i	$-0.352941\pi$
$797$		0			0		−0.445738	+	0.895163i	$0.647059\pi$
$798$		0			0
$799$		0			0
$800$	0.131730	−	1.42160i	0.131730	−	1.42160i
$801$		0			0
$802$	−0.285942	+	0.461813i	−0.285942	+	0.461813i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	0.365931	+	1.95756i	0.365931	+	1.95756i	0.273663	+	0.961826i	$0.411765\pi$
$809$	0.365931	+	1.95756i	0.365931	+	1.95756i	0.0922684	+	0.995734i	$0.470588\pi$
$810$		0			0
$811$		0			0		−0.445738	−	0.895163i	$-0.647059\pi$
$811$		0			0		0.445738	+	0.895163i	$0.352941\pi$
$812$		−	1.24716i		−	1.24716i
$813$		0			0
$814$	0.206729	−	2.23097i	0.206729	−	2.23097i
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−1.86494			−1.86494			−0.932472	−	0.361242i	$-0.882353\pi$
$821$	−1.86494			−1.86494			−0.932472	+	0.361242i	$0.882353\pi$
$822$		0			0
$823$	1.86494			1.86494			0.932472	−	0.361242i	$-0.117647\pi$
$823$	1.86494			1.86494			0.932472	+	0.361242i	$0.117647\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−0.840204	−	1.35698i	−0.840204	−	1.35698i	−0.932472	−	0.361242i	$-0.882353\pi$
$827$	−0.840204	−	1.35698i	−0.840204	−	1.35698i	0.0922684	−	0.995734i	$-0.470588\pi$
$828$	1.50870	−	1.13932i	1.50870	−	1.13932i
$829$		0			0		0.982973	−	0.183750i	$-0.0588235\pi$
$829$		0			0		−0.982973	+	0.183750i	$0.941176\pi$
$830$		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		−0.0922684	−	0.995734i	$-0.529412\pi$
$839$		0			0		0.0922684	+	0.995734i	$0.470588\pi$
$840$		0			0
$841$	0.0296988	+	0.104380i	0.0296988	+	0.104380i
$842$	1.39301	−	2.24979i	1.39301	−	2.24979i
$843$		0			0
$844$	0.203830	−	2.19967i	0.203830	−	2.19967i
$845$		0			0
$846$		0			0
$847$	−0.843104	+	1.69318i	−0.843104	+	1.69318i
$848$	0.203941	−	0.526432i	0.203941	−	0.526432i
$849$		0			0
$850$		0			0
$851$	1.13544	+	0.857445i	1.13544	+	0.857445i
$852$		0			0
$853$		0			0		−0.961826	−	0.273663i	$-0.911765\pi$
$853$		0			0		0.961826	+	0.273663i	$0.0882353\pi$
$854$		0			0
$855$		0			0
$856$	0.146530	−	0.294272i	0.146530	−	0.294272i
$857$		0			0		−0.673696	−	0.739009i	$-0.735294\pi$
$857$		0			0		0.673696	+	0.739009i	$0.264706\pi$
$858$		0			0
$859$		0			0		1.00000			$0$
$859$		0			0		−1.00000			$\pi$
$860$		0			0
$861$		0			0
$862$	−1.78269	+	0.887674i	−1.78269	+	0.887674i
$863$		−	1.92365i		−	1.92365i	−0.273663	−	0.961826i	$-0.588235\pi$
$863$		−	1.92365i		−	1.92365i	0.273663	−	0.961826i	$-0.411765\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	2.20369	−	2.41733i	2.20369	−	2.41733i
$870$		0			0
$871$		0			0
$872$	0.342746	−	0.212219i	0.342746	−	0.212219i
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		−0.850217	−	0.526432i	$-0.823529\pi$
$877$		0			0		0.850217	+	0.526432i	$0.176471\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		−0.739009	−	0.673696i	$-0.764706\pi$
$881$		0			0		0.739009	+	0.673696i	$0.235294\pi$
$882$	0.890705	+	1.17948i	0.890705	+	1.17948i
$883$	0.890705	−	0.811985i	0.890705	−	0.811985i	−0.0922684	−	0.995734i	$-0.529412\pi$
$883$	0.890705	−	0.811985i	0.890705	−	0.811985i	0.982973	+	0.183750i	$0.0588235\pi$
$884$		0			0
$885$		0			0
$886$	−2.85623	−	0.533922i	−2.85623	−	0.533922i
$887$		0			0		0.739009	−	0.673696i	$-0.235294\pi$
$887$		0			0		−0.739009	+	0.673696i	$0.764706\pi$
$888$		0			0
$889$	0.247582	+	1.32445i	0.247582	+	1.32445i
$890$		0			0
$891$	−1.44574	+	0.895163i	−1.44574	+	0.895163i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$	−0.455894	−	0.282278i	−0.455894	−	0.282278i
$897$		0			0
$898$	2.03702	−	1.85699i	2.03702	−	1.85699i
$899$		0			0
$900$	0.875383	+	0.798017i	0.875383	+	0.798017i
$901$		0			0
$902$		0			0
$903$		0			0
$904$	−0.0527668	+	0.0852214i	−0.0527668	+	0.0852214i
$905$		0			0
$906$		0			0
$907$	−0.293271	+	0.221468i	−0.293271	+	0.221468i	−0.739009	−	0.673696i	$-0.764706\pi$
$907$	−0.293271	+	0.221468i	−0.293271	+	0.221468i	0.445738	+	0.895163i	$0.352941\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−0.380338	−	0.614268i	−0.380338	−	0.614268i	0.602635	−	0.798017i	$-0.294118\pi$
$911$	−0.380338	−	0.614268i	−0.380338	−	0.614268i	−0.982973	+	0.183750i	$0.941176\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−1.78269	+	0.887674i	−1.78269	+	0.887674i	−0.850217	+	0.526432i	$0.823529\pi$
$919$	−1.78269	+	0.887674i	−1.78269	+	0.887674i	−0.932472	+	0.361242i	$0.882353\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−0.397365	+	0.798017i	−0.397365	+	0.798017i
$926$	2.93087	−	0.271585i	2.93087	−	0.271585i
$927$		0			0
$928$	1.44578	+	0.411359i	1.44578	+	0.411359i
$929$		0			0		−0.932472	−	0.361242i	$-0.882353\pi$
$929$		0			0		0.932472	+	0.361242i	$0.117647\pi$
$930$		0			0
$931$		0			0
$932$	2.03975	−	1.01568i	2.03975	−	1.01568i
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		0.0922684	−	0.995734i	$-0.470588\pi$
$937$		0			0		−0.0922684	+	0.995734i	$0.529412\pi$
$938$	−0.365931	+	1.95756i	−0.365931	+	1.95756i
$939$		0			0
$940$		0			0
$941$		0			0		−0.273663	−	0.961826i	$-0.588235\pi$
$941$		0			0		0.273663	+	0.961826i	$0.411765\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$	−0.737069	−	3.94297i	−0.737069	−	3.94297i
$947$	−1.58923	+	0.147263i	−1.58923	+	0.147263i	−0.850217	−	0.526432i	$-0.823529\pi$
$947$	−1.58923	+	0.147263i	−1.58923	+	0.147263i	−0.739009	+	0.673696i	$0.764706\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		−0.602635	−	0.798017i	$-0.705882\pi$
$953$		0			0		0.602635	+	0.798017i	$0.294118\pi$
$954$	0.562147	+	0.907899i	0.562147	+	0.907899i
$955$		0			0
$956$	0.823138	−	2.12476i	0.823138	−	2.12476i
$957$		0			0
$958$		0			0
$959$	−0.982973	+	0.183750i	−0.982973	+	0.183750i
$960$		0			0
$961$	0.850217	+	0.526432i	0.850217	+	0.526432i
$962$		0			0
$963$	−0.537235	−	1.07891i	−0.537235	−	1.07891i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	0.111208	−	1.20013i	0.111208	−	1.20013i	−0.739009	−	0.673696i	$-0.764706\pi$
$967$	0.111208	−	1.20013i	0.111208	−	1.20013i	0.850217	−	0.526432i	$-0.176471\pi$
$968$	0.381253	+	0.347558i	0.381253	+	0.347558i
$969$		0			0
$970$		0			0
$971$		0			0		0.995734	−	0.0922684i	$-0.0294118\pi$
$971$		0			0		−0.995734	+	0.0922684i	$0.970588\pi$
$972$		0			0
$973$		0			0
$974$	−0.795184	+	0.148646i	−0.795184	+	0.148646i
$975$		0			0
$976$		0			0
$977$	0.0505009	+	0.177492i	0.0505009	+	0.177492i	0.982973	−	0.183750i	$-0.0588235\pi$
$977$	0.0505009	+	0.177492i	0.0505009	+	0.177492i	−0.932472	+	0.361242i	$0.882353\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	0.136374	−	1.47171i	0.136374	−	1.47171i
$982$	1.24184	+	0.937791i	1.24184	+	0.937791i
$983$		0			0		−0.895163	−	0.445738i	$-0.852941\pi$
$983$		0			0		0.895163	+	0.445738i	$0.147059\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	2.37531	+	0.920200i	2.37531	+	0.920200i
$990$		0			0
$991$	−1.12388	+	0.435393i	−1.12388	+	0.435393i	−0.850217	−	0.526432i	$-0.823529\pi$
$991$	−1.12388	+	0.435393i	−1.12388	+	0.435393i	−0.273663	+	0.961826i	$0.588235\pi$
$992$		0			0
$993$		0			0
$994$	1.78269	+	1.95552i	1.78269	+	1.95552i
$995$		0			0
$996$		0			0
$997$		0			0		0.273663	−	0.961826i	$-0.411765\pi$
$997$		0			0		−0.273663	+	0.961826i	$0.588235\pi$
$998$	2.70949	+	0.506491i	2.70949	+	0.506491i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	959.1.r.a.734.1	yes	16
7.6	odd	2	CM	959.1.r.a.734.1	yes	16
137.14	even	34	inner	959.1.r.a.699.1	✓	16
959.699	odd	34	inner	959.1.r.a.699.1	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
959.1.r.a.699.1	✓	16	137.14	even	34	inner
959.1.r.a.699.1	✓	16	959.699	odd	34	inner
959.1.r.a.734.1	yes	16	1.1	even	1	trivial
959.1.r.a.734.1	yes	16	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:	\( N \)	\(=\)	\( 959 = 7 \cdot 137 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	959.r (of order \(34\), degree \(16\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.25664 - 0.778076i) q^{2} +(0.527993 + 1.06035i) q^{4} +(0.982973 - 0.183750i) q^{7} +(0.0251661 - 0.271585i) q^{8} +(-0.739009 - 0.673696i) q^{9} +O(q^{10})\)	\(q+(-1.25664 - 0.778076i) q^{2} +(0.527993 + 1.06035i) q^{4} +(0.982973 - 0.183750i) q^{7} +(0.0251661 - 0.271585i) q^{8} +(-0.739009 - 0.673696i) q^{9} +(0.757949 + 1.52217i) q^{11} +(-1.37821 - 0.533922i) q^{14} +(0.470904 - 0.623578i) q^{16} +(0.404479 + 1.42160i) q^{18} +(0.231896 - 2.50255i) q^{22} +(1.27366 + 0.961826i) q^{23} +(-0.445738 + 0.895163i) q^{25} +(0.713843 + 0.945281i) q^{28} +(-0.840204 - 0.634493i) q^{29} +(-1.33128 + 0.515740i) q^{32} +(0.324164 - 1.13932i) q^{36} +0.891477 q^{37} +(1.53511 - 0.436776i) q^{43} +(-1.21384 + 1.60739i) q^{44} +(-0.852157 - 2.19967i) q^{46} +(0.932472 - 0.361242i) q^{49} +(1.25664 - 0.778076i) q^{50} +(0.694903 - 0.197717i) q^{53} +(-0.0251661 - 0.271585i) q^{56} +(0.562147 + 1.45107i) q^{58} +(-0.850217 - 0.526432i) q^{63} +(1.30611 + 0.244155i) q^{64} +(-0.247582 - 1.32445i) q^{67} +(-1.60263 - 0.798017i) q^{71} +(-0.201564 + 0.183750i) q^{72} +(-1.12026 - 0.693637i) q^{74} +(1.02474 + 1.35698i) q^{77} +(-0.694903 - 1.79375i) q^{79} +(0.0922684 + 0.995734i) q^{81} +(-2.26892 - 0.645562i) q^{86} +(0.432472 - 0.167541i) q^{88} +(-0.347390 + 1.85837i) q^{92} +(-1.45285 - 0.271585i) q^{98} +(0.465346 - 1.63552i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 2 q^{2} - 3 q^{4} + q^{7} - 4 q^{8} + q^{9}+O(q^{10}) \) 16 * q - 2 * q^2 - 3 * q^4 + q^7 - 4 * q^8 + q^9	\( 16 q - 2 q^{2} - 3 q^{4} + q^{7} - 4 q^{8} + q^{9} + 2 q^{11} + 2 q^{14} - 5 q^{16} + 2 q^{18} + 4 q^{22} + 17 q^{23} + q^{25} + 3 q^{28} - 6 q^{32} + 3 q^{36} - 2 q^{37} - 11 q^{44} - q^{49} + 2 q^{50} + 4 q^{56} - q^{63} + 10 q^{64} - 17 q^{71} - 13 q^{72} - 4 q^{74} - 2 q^{77} - q^{81} - 9 q^{88} - 2 q^{98} - 2 q^{99}+O(q^{100}) \) 16 * q - 2 * q^2 - 3 * q^4 + q^7 - 4 * q^8 + q^9 + 2 * q^11 + 2 * q^14 - 5 * q^16 + 2 * q^18 + 4 * q^22 + 17 * q^23 + q^25 + 3 * q^28 - 6 * q^32 + 3 * q^36 - 2 * q^37 - 11 * q^44 - q^49 + 2 * q^50 + 4 * q^56 - q^63 + 10 * q^64 - 17 * q^71 - 13 * q^72 - 4 * q^74 - 2 * q^77 - q^81 - 9 * q^88 - 2 * q^98 - 2 * q^99

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