LMFDB - Embedded newform 1274.2.d.i.883.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1274,2,Mod(883,1274)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1274.883"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1274, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1274 = 2 \cdot 7^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1274.d (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,-4,0,0,0,0,20,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(10)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$10.1729412175$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			883.4
Root			$0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1274.883
Dual form			1274.2.d.i.883.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000i q^{2} +2.82843 q^{3} -1.00000 q^{4} -1.41421i q^{5} +2.82843i q^{6} -1.00000i q^{8} +5.00000 q^{9} +1.41421 q^{10} -2.82843 q^{12} +(-0.707107 - 3.53553i) q^{13} -4.00000i q^{15} +1.00000 q^{16} +1.41421 q^{17} +5.00000i q^{18} -2.82843i q^{19} +1.41421i q^{20} +8.00000 q^{23} -2.82843i q^{24} +3.00000 q^{25} +(3.53553 - 0.707107i) q^{26} +5.65685 q^{27} +4.00000 q^{30} +1.00000i q^{32} +1.41421i q^{34} -5.00000 q^{36} +10.0000i q^{37} +2.82843 q^{38} +(-2.00000 - 10.0000i) q^{39} -1.41421 q^{40} -9.89949i q^{41} -8.00000 q^{43} -7.07107i q^{45} +8.00000i q^{46} +11.3137i q^{47} +2.82843 q^{48} +3.00000i q^{50} +4.00000 q^{51} +(0.707107 + 3.53553i) q^{52} -6.00000 q^{53} +5.65685i q^{54} -8.00000i q^{57} -2.82843i q^{59} +4.00000i q^{60} +12.7279 q^{61} -1.00000 q^{64} +(-5.00000 + 1.00000i) q^{65} -12.0000i q^{67} -1.41421 q^{68} +22.6274 q^{69} +8.00000i q^{71} -5.00000i q^{72} +12.7279i q^{73} -10.0000 q^{74} +8.48528 q^{75} +2.82843i q^{76} +(10.0000 - 2.00000i) q^{78} -4.00000 q^{79} -1.41421i q^{80} +1.00000 q^{81} +9.89949 q^{82} +2.82843i q^{83} -2.00000i q^{85} -8.00000i q^{86} -12.7279i q^{89} +7.07107 q^{90} -8.00000 q^{92} -11.3137 q^{94} -4.00000 q^{95} +2.82843i q^{96} +4.24264i q^{97} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4} + 20 q^{9} + 4 q^{16} + 32 q^{23} + 12 q^{25} + 16 q^{30} - 20 q^{36} - 8 q^{39} - 32 q^{43} + 16 q^{51} - 24 q^{53} - 4 q^{64} - 20 q^{65} - 40 q^{74} + 40 q^{78} - 16 q^{79} + 4 q^{81} - 32 q^{92}+ \cdots - 16 q^{95}+O(q^{100}) $ 4 * q - 4 * q^4 + 20 * q^9 + 4 * q^16 + 32 * q^23 + 12 * q^25 + 16 * q^30 - 20 * q^36 - 8 * q^39 - 32 * q^43 + 16 * q^51 - 24 * q^53 - 4 * q^64 - 20 * q^65 - 40 * q^74 + 40 * q^78 - 16 * q^79 + 4 * q^81 - 32 * q^92 - 16 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1274\mathbb{Z}\right)^\times$.

$n$	$197$	$885$
$\chi(n)$	$-1$	$1$

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 $

$2$			1.00000i			0.707107i
$2$			1.00000i			0.707107i
$3$	2.82843			1.63299			0.816497	−	0.577350i	$-0.195913\pi$
$3$	2.82843			1.63299			0.816497	+	0.577350i	$0.195913\pi$
$4$	−1.00000			−0.500000
$5$		−	1.41421i		−	0.632456i	−0.948683	−	0.316228i	$-0.897584\pi$
$5$		−	1.41421i		−	0.632456i	0.948683	−	0.316228i	$-0.102416\pi$
$6$			2.82843i			1.15470i
$7$		0			0
$7$		0			0
$8$		−	1.00000i		−	0.353553i
$9$	5.00000			1.66667
$10$	1.41421			0.447214
$11$		0			0		1.00000			$0$
$11$		0			0		−1.00000			$\pi$
$12$	−2.82843			−0.816497
$13$	−0.707107	−	3.53553i	−0.196116	−	0.980581i
$13$	−0.707107	−	3.53553i	−0.196116	−	0.980581i
$14$		0			0
$15$		−	4.00000i		−	1.03280i
$16$	1.00000			0.250000
$17$	1.41421			0.342997			0.171499	−	0.985184i	$-0.445139\pi$
$17$	1.41421			0.342997			0.171499	+	0.985184i	$0.445139\pi$
$18$			5.00000i			1.17851i
$19$		−	2.82843i		−	0.648886i	−0.945905	−	0.324443i	$-0.894823\pi$
$19$		−	2.82843i		−	0.648886i	0.945905	−	0.324443i	$-0.105177\pi$
$20$			1.41421i			0.316228i
$21$		0			0
$22$		0			0
$23$	8.00000			1.66812			0.834058	−	0.551677i	$-0.186012\pi$
$23$	8.00000			1.66812			0.834058	+	0.551677i	$0.186012\pi$
$24$		−	2.82843i		−	0.577350i
$25$	3.00000			0.600000
$26$	3.53553	−	0.707107i	0.693375	−	0.138675i
$27$	5.65685			1.08866
$28$		0			0
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$	4.00000			0.730297
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$			1.00000i			0.176777i
$33$		0			0
$34$			1.41421i			0.242536i
$35$		0			0
$36$	−5.00000			−0.833333
$37$			10.0000i			1.64399i	0.569495	+	0.821995i	$0.307139\pi$
$37$			10.0000i			1.64399i	−0.569495	+	0.821995i	$0.692861\pi$
$38$	2.82843			0.458831
$39$	−2.00000	−	10.0000i	−0.320256	−	1.60128i
$40$	−1.41421			−0.223607
$41$		−	9.89949i		−	1.54604i	−0.634381	−	0.773021i	$-0.718745\pi$
$41$		−	9.89949i		−	1.54604i	0.634381	−	0.773021i	$-0.281255\pi$
$42$		0			0
$43$	−8.00000			−1.21999			−0.609994	−	0.792406i	$-0.708828\pi$
$43$	−8.00000			−1.21999			−0.609994	+	0.792406i	$0.708828\pi$
$44$		0			0
$45$		−	7.07107i		−	1.05409i
$46$			8.00000i			1.17954i
$47$			11.3137i			1.65027i	0.564933	+	0.825137i	$0.308902\pi$
$47$			11.3137i			1.65027i	−0.564933	+	0.825137i	$0.691098\pi$
$48$	2.82843			0.408248
$49$		0			0
$50$			3.00000i			0.424264i
$51$	4.00000			0.560112
$52$	0.707107	+	3.53553i	0.0980581	+	0.490290i
$53$	−6.00000			−0.824163			−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000			−0.824163			−0.412082	+	0.911147i	$0.635198\pi$
$54$			5.65685i			0.769800i
$55$		0			0
$56$		0			0
$57$		−	8.00000i		−	1.05963i
$58$		0			0
$59$		−	2.82843i		−	0.368230i	−0.982905	−	0.184115i	$-0.941058\pi$
$59$		−	2.82843i		−	0.368230i	0.982905	−	0.184115i	$-0.0589419\pi$
$60$			4.00000i			0.516398i
$61$	12.7279			1.62964			0.814822	−	0.579712i	$-0.196835\pi$
$61$	12.7279			1.62964			0.814822	+	0.579712i	$0.196835\pi$
$62$		0			0
$63$		0			0
$64$	−1.00000			−0.125000
$65$	−5.00000	+	1.00000i	−0.620174	+	0.124035i
$66$		0			0
$67$		−	12.0000i		−	1.46603i	−0.680211	−	0.733017i	$-0.738112\pi$
$67$		−	12.0000i		−	1.46603i	0.680211	−	0.733017i	$-0.261888\pi$
$68$	−1.41421			−0.171499
$69$	22.6274			2.72402
$70$		0			0
$71$			8.00000i			0.949425i	0.880141	+	0.474713i	$0.157448\pi$
$71$			8.00000i			0.949425i	−0.880141	+	0.474713i	$0.842552\pi$
$72$		−	5.00000i		−	0.589256i
$73$			12.7279i			1.48969i	0.667237	+	0.744845i	$0.267477\pi$
$73$			12.7279i			1.48969i	−0.667237	+	0.744845i	$0.732523\pi$
$74$	−10.0000			−1.16248
$75$	8.48528			0.979796
$76$			2.82843i			0.324443i
$77$		0			0
$78$	10.0000	−	2.00000i	1.13228	−	0.226455i
$79$	−4.00000			−0.450035			−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000			−0.450035			−0.225018	+	0.974355i	$0.572244\pi$
$80$		−	1.41421i		−	0.158114i
$81$	1.00000			0.111111
$82$	9.89949			1.09322
$83$			2.82843i			0.310460i	0.987878	+	0.155230i	$0.0496119\pi$
$83$			2.82843i			0.310460i	−0.987878	+	0.155230i	$0.950388\pi$
$84$		0			0
$85$		−	2.00000i		−	0.216930i
$86$		−	8.00000i		−	0.862662i
$87$		0			0
$88$		0			0
$89$		−	12.7279i		−	1.34916i	−0.738203	−	0.674579i	$-0.764325\pi$
$89$		−	12.7279i		−	1.34916i	0.738203	−	0.674579i	$-0.235675\pi$
$90$	7.07107			0.745356
$91$		0			0
$92$	−8.00000			−0.834058
$93$		0			0
$94$	−11.3137			−1.16692
$95$	−4.00000			−0.410391
$96$			2.82843i			0.288675i
$97$			4.24264i			0.430775i	0.976529	+	0.215387i	$0.0691014\pi$
$97$			4.24264i			0.430775i	−0.976529	+	0.215387i	$0.930899\pi$
$98$		0			0
$99$		0			0
$100$	−3.00000			−0.300000
$101$	−4.24264			−0.422159			−0.211079	−	0.977469i	$-0.567698\pi$
$101$	−4.24264			−0.422159			−0.211079	+	0.977469i	$0.567698\pi$
$102$			4.00000i			0.396059i
$103$		0			0			−	1.00000i	$-0.5\pi$
$103$		0			0				1.00000i	$0.5\pi$
$104$	−3.53553	+	0.707107i	−0.346688	+	0.0693375i
$105$		0			0
$106$		−	6.00000i		−	0.582772i
$107$	−12.0000			−1.16008			−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000			−1.16008			−0.580042	+	0.814587i	$0.696964\pi$
$108$	−5.65685			−0.544331
$109$			2.00000i			0.191565i	0.995402	+	0.0957826i	$0.0305354\pi$
$109$			2.00000i			0.191565i	−0.995402	+	0.0957826i	$0.969465\pi$
$110$		0			0
$111$			28.2843i			2.68462i
$112$		0			0
$113$	−6.00000			−0.564433			−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000			−0.564433			−0.282216	+	0.959351i	$0.591070\pi$
$114$	8.00000			0.749269
$115$		−	11.3137i		−	1.05501i
$116$		0			0
$117$	−3.53553	−	17.6777i	−0.326860	−	1.63430i
$118$	2.82843			0.260378
$119$		0			0
$120$	−4.00000			−0.365148
$121$	11.0000			1.00000
$122$			12.7279i			1.15233i
$123$		−	28.0000i		−	2.52467i
$124$		0			0
$125$		−	11.3137i		−	1.01193i
$126$		0			0
$127$	−16.0000			−1.41977			−0.709885	−	0.704317i	$-0.751253\pi$
$127$	−16.0000			−1.41977			−0.709885	+	0.704317i	$0.751253\pi$
$128$		−	1.00000i		−	0.0883883i
$129$	−22.6274			−1.99223
$130$	−1.00000	−	5.00000i	−0.0877058	−	0.438529i
$131$	8.48528			0.741362			0.370681	−	0.928760i	$-0.379124\pi$
$131$	8.48528			0.741362			0.370681	+	0.928760i	$0.379124\pi$
$132$		0			0
$133$		0			0
$134$	12.0000			1.03664
$135$		−	8.00000i		−	0.688530i
$136$		−	1.41421i		−	0.121268i
$137$			18.0000i			1.53784i	0.639343	+	0.768922i	$0.279207\pi$
$137$			18.0000i			1.53784i	−0.639343	+	0.768922i	$0.720793\pi$
$138$			22.6274i			1.92617i
$139$	−14.1421			−1.19952			−0.599760	−	0.800180i	$-0.704737\pi$
$139$	−14.1421			−1.19952			−0.599760	+	0.800180i	$0.704737\pi$
$140$		0			0
$141$			32.0000i			2.69489i
$142$	−8.00000			−0.671345
$143$		0			0
$144$	5.00000			0.416667
$145$		0			0
$146$	−12.7279			−1.05337
$147$		0			0
$148$		−	10.0000i		−	0.821995i
$149$			8.00000i			0.655386i	0.944784	+	0.327693i	$0.106271\pi$
$149$			8.00000i			0.655386i	−0.944784	+	0.327693i	$0.893729\pi$
$150$			8.48528i			0.692820i
$151$			20.0000i			1.62758i	0.581161	+	0.813788i	$0.302599\pi$
$151$			20.0000i			1.62758i	−0.581161	+	0.813788i	$0.697401\pi$
$152$	−2.82843			−0.229416
$153$	7.07107			0.571662
$154$		0			0
$155$		0			0
$156$	2.00000	+	10.0000i	0.160128	+	0.800641i
$157$	12.7279			1.01580			0.507899	−	0.861416i	$-0.330422\pi$
$157$	12.7279			1.01580			0.507899	+	0.861416i	$0.330422\pi$
$158$		−	4.00000i		−	0.318223i
$159$	−16.9706			−1.34585
$160$	1.41421			0.111803
$161$		0			0
$162$			1.00000i			0.0785674i
$163$		−	8.00000i		−	0.626608i	−0.949653	−	0.313304i	$-0.898564\pi$
$163$		−	8.00000i		−	0.626608i	0.949653	−	0.313304i	$-0.101436\pi$
$164$			9.89949i			0.773021i
$165$		0			0
$166$	−2.82843			−0.219529
$167$		−	5.65685i		−	0.437741i	−0.975754	−	0.218870i	$-0.929763\pi$
$167$		−	5.65685i		−	0.437741i	0.975754	−	0.218870i	$-0.0702371\pi$
$168$		0			0
$169$	−12.0000	+	5.00000i	−0.923077	+	0.384615i
$170$	2.00000			0.153393
$171$		−	14.1421i		−	1.08148i
$172$	8.00000			0.609994
$173$	−12.7279			−0.967686			−0.483843	−	0.875155i	$-0.660759\pi$
$173$	−12.7279			−0.967686			−0.483843	+	0.875155i	$0.660759\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		−	8.00000i		−	0.601317i
$178$	12.7279			0.953998
$179$	−20.0000			−1.49487			−0.747435	−	0.664335i	$-0.768715\pi$
$179$	−20.0000			−1.49487			−0.747435	+	0.664335i	$0.768715\pi$
$180$			7.07107i			0.527046i
$181$	−15.5563			−1.15629			−0.578147	−	0.815933i	$-0.696224\pi$
$181$	−15.5563			−1.15629			−0.578147	+	0.815933i	$0.696224\pi$
$182$		0			0
$183$	36.0000			2.66120
$184$		−	8.00000i		−	0.589768i
$185$	14.1421			1.03975
$186$		0			0
$187$		0			0
$188$		−	11.3137i		−	0.825137i
$189$		0			0
$190$		−	4.00000i		−	0.290191i
$191$	12.0000			0.868290			0.434145	−	0.900843i	$-0.357051\pi$
$191$	12.0000			0.868290			0.434145	+	0.900843i	$0.357051\pi$
$192$	−2.82843			−0.204124
$193$		−	8.00000i		−	0.575853i	−0.957653	−	0.287926i	$-0.907034\pi$
$193$		−	8.00000i		−	0.575853i	0.957653	−	0.287926i	$-0.0929658\pi$
$194$	−4.24264			−0.304604
$195$	−14.1421	+	2.82843i	−1.01274	+	0.202548i
$196$		0			0
$197$			24.0000i			1.70993i	0.518686	+	0.854965i	$0.326421\pi$
$197$			24.0000i			1.70993i	−0.518686	+	0.854965i	$0.673579\pi$
$198$		0			0
$199$	16.9706			1.20301			0.601506	−	0.798869i	$-0.294568\pi$
$199$	16.9706			1.20301			0.601506	+	0.798869i	$0.294568\pi$
$200$		−	3.00000i		−	0.212132i
$201$		−	33.9411i		−	2.39402i
$202$		−	4.24264i		−	0.298511i
$203$		0			0
$204$	−4.00000			−0.280056
$205$	−14.0000			−0.977802
$206$		0			0
$207$	40.0000			2.78019
$208$	−0.707107	−	3.53553i	−0.0490290	−	0.245145i
$209$		0			0
$210$		0			0
$211$	12.0000			0.826114			0.413057	−	0.910705i	$-0.364461\pi$
$211$	12.0000			0.826114			0.413057	+	0.910705i	$0.364461\pi$
$212$	6.00000			0.412082
$213$			22.6274i			1.55041i
$214$		−	12.0000i		−	0.820303i
$215$			11.3137i			0.771589i
$216$		−	5.65685i		−	0.384900i
$217$		0			0
$218$	−2.00000			−0.135457
$219$			36.0000i			2.43265i
$220$		0			0
$221$	−1.00000	−	5.00000i	−0.0672673	−	0.336336i
$222$	−28.2843			−1.89832
$223$			5.65685i			0.378811i	0.981899	+	0.189405i	$0.0606561\pi$
$223$			5.65685i			0.378811i	−0.981899	+	0.189405i	$0.939344\pi$
$224$		0			0
$225$	15.0000			1.00000
$226$		−	6.00000i		−	0.399114i
$227$		−	25.4558i		−	1.68956i	−0.535111	−	0.844782i	$-0.679730\pi$
$227$		−	25.4558i		−	1.68956i	0.535111	−	0.844782i	$-0.320270\pi$
$228$			8.00000i			0.529813i
$229$			24.0416i			1.58872i	0.607450	+	0.794358i	$0.292192\pi$
$229$			24.0416i			1.58872i	−0.607450	+	0.794358i	$0.707808\pi$
$230$	11.3137			0.746004
$231$		0			0
$232$		0			0
$233$		0			0			−	1.00000i	$-0.5\pi$
$233$		0			0				1.00000i	$0.5\pi$
$234$	17.6777	−	3.53553i	1.15563	−	0.231125i
$235$	16.0000			1.04372
$236$			2.82843i			0.184115i
$237$	−11.3137			−0.734904
$238$		0			0
$239$		−	12.0000i		−	0.776215i	−0.921614	−	0.388108i	$-0.873129\pi$
$239$		−	12.0000i		−	0.776215i	0.921614	−	0.388108i	$-0.126871\pi$
$240$		−	4.00000i		−	0.258199i
$241$			15.5563i			1.00207i	0.865426	+	0.501036i	$0.167048\pi$
$241$			15.5563i			1.00207i	−0.865426	+	0.501036i	$0.832952\pi$
$242$			11.0000i			0.707107i
$243$	−14.1421			−0.907218
$244$	−12.7279			−0.814822
$245$		0			0
$246$	28.0000			1.78521
$247$	−10.0000	+	2.00000i	−0.636285	+	0.127257i
$248$		0			0
$249$			8.00000i			0.506979i
$250$	11.3137			0.715542
$251$	−14.1421			−0.892644			−0.446322	−	0.894873i	$-0.647266\pi$
$251$	−14.1421			−0.892644			−0.446322	+	0.894873i	$0.647266\pi$
$252$		0			0
$253$		0			0
$254$		−	16.0000i		−	1.00393i
$255$		−	5.65685i		−	0.354246i
$256$	1.00000			0.0625000
$257$	1.41421			0.0882162			0.0441081	−	0.999027i	$-0.485955\pi$
$257$	1.41421			0.0882162			0.0441081	+	0.999027i	$0.485955\pi$
$258$		−	22.6274i		−	1.40872i
$259$		0			0
$260$	5.00000	−	1.00000i	0.310087	−	0.0620174i
$261$		0			0
$262$			8.48528i			0.524222i
$263$	−4.00000			−0.246651			−0.123325	−	0.992366i	$-0.539356\pi$
$263$	−4.00000			−0.246651			−0.123325	+	0.992366i	$0.539356\pi$
$264$		0			0
$265$			8.48528i			0.521247i
$266$		0			0
$267$		−	36.0000i		−	2.20316i
$268$			12.0000i			0.733017i
$269$	9.89949			0.603583			0.301791	−	0.953374i	$-0.402415\pi$
$269$	9.89949			0.603583			0.301791	+	0.953374i	$0.402415\pi$
$270$	8.00000			0.486864
$271$			16.9706i			1.03089i	0.856923	+	0.515444i	$0.172373\pi$
$271$			16.9706i			1.03089i	−0.856923	+	0.515444i	$0.827627\pi$
$272$	1.41421			0.0857493
$273$		0			0
$274$	−18.0000			−1.08742
$275$		0			0
$276$	−22.6274			−1.36201
$277$	22.0000			1.32185			0.660926	−	0.750451i	$-0.270164\pi$
$277$	22.0000			1.32185			0.660926	+	0.750451i	$0.270164\pi$
$278$		−	14.1421i		−	0.848189i
$279$		0			0
$280$		0			0
$281$		−	2.00000i		−	0.119310i	−0.998219	−	0.0596550i	$-0.981000\pi$
$281$		−	2.00000i		−	0.119310i	0.998219	−	0.0596550i	$-0.0190001\pi$
$282$	−32.0000			−1.90557
$283$	8.48528			0.504398			0.252199	−	0.967675i	$-0.418846\pi$
$283$	8.48528			0.504398			0.252199	+	0.967675i	$0.418846\pi$
$284$		−	8.00000i		−	0.474713i
$285$	−11.3137			−0.670166
$286$		0			0
$287$		0			0
$288$			5.00000i			0.294628i
$289$	−15.0000			−0.882353
$290$		0			0
$291$			12.0000i			0.703452i
$292$		−	12.7279i		−	0.744845i
$293$		−	24.0416i		−	1.40453i	−0.711917	−	0.702264i	$-0.752173\pi$
$293$		−	24.0416i		−	1.40453i	0.711917	−	0.702264i	$-0.247827\pi$
$294$		0			0
$295$	−4.00000			−0.232889
$296$	10.0000			0.581238
$297$		0			0
$298$	−8.00000			−0.463428
$299$	−5.65685	−	28.2843i	−0.327144	−	1.63572i
$300$	−8.48528			−0.489898
$301$		0			0
$302$	−20.0000			−1.15087
$303$	−12.0000			−0.689382
$304$		−	2.82843i		−	0.162221i
$305$		−	18.0000i		−	1.03068i
$306$			7.07107i			0.404226i
$307$			8.48528i			0.484281i	0.970241	+	0.242140i	$0.0778494\pi$
$307$			8.48528i			0.484281i	−0.970241	+	0.242140i	$0.922151\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−28.2843			−1.60385			−0.801927	−	0.597422i	$-0.796192\pi$
$311$	−28.2843			−1.60385			−0.801927	+	0.597422i	$0.796192\pi$
$312$	−10.0000	+	2.00000i	−0.566139	+	0.113228i
$313$	15.5563			0.879297			0.439648	−	0.898170i	$-0.355103\pi$
$313$	15.5563			0.879297			0.439648	+	0.898170i	$0.355103\pi$
$314$			12.7279i			0.718278i
$315$		0			0
$316$	4.00000			0.225018
$317$		−	24.0000i		−	1.34797i	−0.738743	−	0.673987i	$-0.764580\pi$
$317$		−	24.0000i		−	1.34797i	0.738743	−	0.673987i	$-0.235420\pi$
$318$		−	16.9706i		−	0.951662i
$319$		0			0
$320$			1.41421i			0.0790569i
$321$	−33.9411			−1.89441
$322$		0			0
$323$		−	4.00000i		−	0.222566i
$324$	−1.00000			−0.0555556
$325$	−2.12132	−	10.6066i	−0.117670	−	0.588348i
$326$	8.00000			0.443079
$327$			5.65685i			0.312825i
$328$	−9.89949			−0.546608
$329$		0			0
$330$		0			0
$331$			16.0000i			0.879440i	0.898135	+	0.439720i	$0.144922\pi$
$331$			16.0000i			0.879440i	−0.898135	+	0.439720i	$0.855078\pi$
$332$		−	2.82843i		−	0.155230i
$333$			50.0000i			2.73998i
$334$	5.65685			0.309529
$335$	−16.9706			−0.927201
$336$		0			0
$337$	−8.00000			−0.435788			−0.217894	−	0.975972i	$-0.569919\pi$
$337$	−8.00000			−0.435788			−0.217894	+	0.975972i	$0.569919\pi$
$338$	−5.00000	−	12.0000i	−0.271964	−	0.652714i
$339$	−16.9706			−0.921714
$340$			2.00000i			0.108465i
$341$		0			0
$342$	14.1421			0.764719
$343$		0			0
$344$			8.00000i			0.431331i
$345$		−	32.0000i		−	1.72282i
$346$		−	12.7279i		−	0.684257i
$347$	−16.0000			−0.858925			−0.429463	−	0.903085i	$-0.641297\pi$
$347$	−16.0000			−0.858925			−0.429463	+	0.903085i	$0.641297\pi$
$348$		0			0
$349$		−	29.6985i		−	1.58972i	−0.606791	−	0.794862i	$-0.707543\pi$
$349$		−	29.6985i		−	1.58972i	0.606791	−	0.794862i	$-0.292457\pi$
$350$		0			0
$351$	−4.00000	−	20.0000i	−0.213504	−	1.06752i
$352$		0			0
$353$		−	7.07107i		−	0.376355i	−0.982135	−	0.188177i	$-0.939742\pi$
$353$		−	7.07107i		−	0.376355i	0.982135	−	0.188177i	$-0.0602580\pi$
$354$	8.00000			0.425195
$355$	11.3137			0.600469
$356$			12.7279i			0.674579i
$357$		0			0
$358$		−	20.0000i		−	1.05703i
$359$		−	4.00000i		−	0.211112i	−0.994413	−	0.105556i	$-0.966338\pi$
$359$		−	4.00000i		−	0.211112i	0.994413	−	0.105556i	$-0.0336622\pi$
$360$	−7.07107			−0.372678
$361$	11.0000			0.578947
$362$		−	15.5563i		−	0.817624i
$363$	31.1127			1.63299
$364$		0			0
$365$	18.0000			0.942163
$366$			36.0000i			1.88175i
$367$	16.9706			0.885856			0.442928	−	0.896557i	$-0.353940\pi$
$367$	16.9706			0.885856			0.442928	+	0.896557i	$0.353940\pi$
$368$	8.00000			0.417029
$369$		−	49.4975i		−	2.57674i
$370$			14.1421i			0.735215i
$371$		0			0
$372$		0			0
$373$	−10.0000			−0.517780			−0.258890	−	0.965907i	$-0.583357\pi$
$373$	−10.0000			−0.517780			−0.258890	+	0.965907i	$0.583357\pi$
$374$		0			0
$375$		−	32.0000i		−	1.65247i
$376$	11.3137			0.583460
$377$		0			0
$378$		0			0
$379$			8.00000i			0.410932i	0.978664	+	0.205466i	$0.0658711\pi$
$379$			8.00000i			0.410932i	−0.978664	+	0.205466i	$0.934129\pi$
$380$	4.00000			0.205196
$381$	−45.2548			−2.31848
$382$			12.0000i			0.613973i
$383$			5.65685i			0.289052i	0.989501	+	0.144526i	$0.0461657\pi$
$383$			5.65685i			0.289052i	−0.989501	+	0.144526i	$0.953834\pi$
$384$		−	2.82843i		−	0.144338i
$385$		0			0
$386$	8.00000			0.407189
$387$	−40.0000			−2.03331
$388$		−	4.24264i		−	0.215387i
$389$	−16.0000			−0.811232			−0.405616	−	0.914044i	$-0.632943\pi$
$389$	−16.0000			−0.811232			−0.405616	+	0.914044i	$0.632943\pi$
$390$	−2.82843	−	14.1421i	−0.143223	−	0.716115i
$391$	11.3137			0.572159
$392$		0			0
$393$	24.0000			1.21064
$394$	−24.0000			−1.20910
$395$			5.65685i			0.284627i
$396$		0			0
$397$			18.3848i			0.922705i	0.887217	+	0.461353i	$0.152636\pi$
$397$			18.3848i			0.922705i	−0.887217	+	0.461353i	$0.847364\pi$
$398$			16.9706i			0.850657i
$399$		0			0
$400$	3.00000			0.150000
$401$			2.00000i			0.0998752i	0.998752	+	0.0499376i	$0.0159023\pi$
$401$			2.00000i			0.0998752i	−0.998752	+	0.0499376i	$0.984098\pi$
$402$	33.9411			1.69283
$403$		0			0
$404$	4.24264			0.211079
$405$		−	1.41421i		−	0.0702728i
$406$		0			0
$407$		0			0
$408$		−	4.00000i		−	0.198030i
$409$		−	12.7279i		−	0.629355i	−0.949199	−	0.314678i	$-0.898104\pi$
$409$		−	12.7279i		−	0.629355i	0.949199	−	0.314678i	$-0.101896\pi$
$410$		−	14.0000i		−	0.691411i
$411$			50.9117i			2.51129i
$412$		0			0
$413$		0			0
$414$			40.0000i			1.96589i
$415$	4.00000			0.196352
$416$	3.53553	−	0.707107i	0.173344	−	0.0346688i
$417$	−40.0000			−1.95881
$418$		0			0
$419$	36.7696			1.79631			0.898155	−	0.439679i	$-0.144908\pi$
$419$	36.7696			1.79631			0.898155	+	0.439679i	$0.144908\pi$
$420$		0			0
$421$		−	8.00000i		−	0.389896i	−0.980814	−	0.194948i	$-0.937546\pi$
$421$		−	8.00000i		−	0.389896i	0.980814	−	0.194948i	$-0.0624538\pi$
$422$			12.0000i			0.584151i
$423$			56.5685i			2.75046i
$424$			6.00000i			0.291386i
$425$	4.24264			0.205798
$426$	−22.6274			−1.09630
$427$		0			0
$428$	12.0000			0.580042
$429$		0			0
$430$	−11.3137			−0.545595
$431$			20.0000i			0.963366i	0.876346	+	0.481683i	$0.159974\pi$
$431$			20.0000i			0.963366i	−0.876346	+	0.481683i	$0.840026\pi$
$432$	5.65685			0.272166
$433$	21.2132			1.01944			0.509721	−	0.860340i	$-0.329749\pi$
$433$	21.2132			1.01944			0.509721	+	0.860340i	$0.329749\pi$
$434$		0			0
$435$		0			0
$436$		−	2.00000i		−	0.0957826i
$437$		−	22.6274i		−	1.08242i
$438$	−36.0000			−1.72015
$439$	−11.3137			−0.539974			−0.269987	−	0.962864i	$-0.587019\pi$
$439$	−11.3137			−0.539974			−0.269987	+	0.962864i	$0.587019\pi$
$440$		0			0
$441$		0			0
$442$	5.00000	−	1.00000i	0.237826	−	0.0475651i
$443$	−28.0000			−1.33032			−0.665160	−	0.746701i	$-0.731637\pi$
$443$	−28.0000			−1.33032			−0.665160	+	0.746701i	$0.731637\pi$
$444$		−	28.2843i		−	1.34231i
$445$	−18.0000			−0.853282
$446$	−5.65685			−0.267860
$447$			22.6274i			1.07024i
$448$		0			0
$449$		−	32.0000i		−	1.51017i	−0.655625	−	0.755087i	$-0.727595\pi$
$449$		−	32.0000i		−	1.51017i	0.655625	−	0.755087i	$-0.272405\pi$
$450$			15.0000i			0.707107i
$451$		0			0
$452$	6.00000			0.282216
$453$			56.5685i			2.65782i
$454$	25.4558			1.19470
$455$		0			0
$456$	−8.00000			−0.374634
$457$		−	24.0000i		−	1.12267i	−0.827588	−	0.561336i	$-0.810287\pi$
$457$		−	24.0000i		−	1.12267i	0.827588	−	0.561336i	$-0.189713\pi$
$458$	−24.0416			−1.12339
$459$	8.00000			0.373408
$460$			11.3137i			0.527504i
$461$		−	26.8701i		−	1.25146i	−0.780038	−	0.625732i	$-0.784800\pi$
$461$		−	26.8701i		−	1.25146i	0.780038	−	0.625732i	$-0.215200\pi$
$462$		0			0
$463$			32.0000i			1.48717i	0.668644	+	0.743583i	$0.266875\pi$
$463$			32.0000i			1.48717i	−0.668644	+	0.743583i	$0.733125\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−25.4558			−1.17796			−0.588978	−	0.808149i	$-0.700470\pi$
$467$	−25.4558			−1.17796			−0.588978	+	0.808149i	$0.700470\pi$
$468$	3.53553	+	17.6777i	0.163430	+	0.817151i
$469$		0			0
$470$			16.0000i			0.738025i
$471$	36.0000			1.65879
$472$	−2.82843			−0.130189
$473$		0			0
$474$		−	11.3137i		−	0.519656i
$475$		−	8.48528i		−	0.389331i
$476$		0			0
$477$	−30.0000			−1.37361
$478$	12.0000			0.548867
$479$		−	11.3137i		−	0.516937i	−0.966020	−	0.258468i	$-0.916782\pi$
$479$		−	11.3137i		−	0.516937i	0.966020	−	0.258468i	$-0.0832177\pi$
$480$	4.00000			0.182574
$481$	35.3553	−	7.07107i	1.61206	−	0.322413i
$482$	−15.5563			−0.708572
$483$		0			0
$484$	−11.0000			−0.500000
$485$	6.00000			0.272446
$486$		−	14.1421i		−	0.641500i
$487$		−	12.0000i		−	0.543772i	−0.962329	−	0.271886i	$-0.912353\pi$
$487$		−	12.0000i		−	0.543772i	0.962329	−	0.271886i	$-0.0876473\pi$
$488$		−	12.7279i		−	0.576166i
$489$		−	22.6274i		−	1.02325i
$490$		0			0
$491$	20.0000			0.902587			0.451294	−	0.892375i	$-0.350963\pi$
$491$	20.0000			0.902587			0.451294	+	0.892375i	$0.350963\pi$
$492$			28.0000i			1.26234i
$493$		0			0
$494$	−2.00000	−	10.0000i	−0.0899843	−	0.449921i
$495$		0			0
$496$		0			0
$497$		0			0
$498$	−8.00000			−0.358489
$499$			20.0000i			0.895323i	0.894203	+	0.447661i	$0.147743\pi$
$499$			20.0000i			0.895323i	−0.894203	+	0.447661i	$0.852257\pi$
$500$			11.3137i			0.505964i
$501$		−	16.0000i		−	0.714827i
$502$		−	14.1421i		−	0.631194i
$503$	−22.6274			−1.00891			−0.504453	−	0.863439i	$-0.668306\pi$
$503$	−22.6274			−1.00891			−0.504453	+	0.863439i	$0.668306\pi$
$504$		0			0
$505$			6.00000i			0.266996i
$506$		0			0
$507$	−33.9411	+	14.1421i	−1.50738	+	0.628074i
$508$	16.0000			0.709885
$509$			7.07107i			0.313420i	0.987645	+	0.156710i	$0.0500887\pi$
$509$			7.07107i			0.313420i	−0.987645	+	0.156710i	$0.949911\pi$
$510$	5.65685			0.250490
$511$		0			0
$512$			1.00000i			0.0441942i
$513$		−	16.0000i		−	0.706417i
$514$			1.41421i			0.0623783i
$515$		0			0
$516$	22.6274			0.996116
$517$		0			0
$518$		0			0
$519$	−36.0000			−1.58022
$520$	1.00000	+	5.00000i	0.0438529	+	0.219265i
$521$	−7.07107			−0.309789			−0.154895	−	0.987931i	$-0.549504\pi$
$521$	−7.07107			−0.309789			−0.154895	+	0.987931i	$0.549504\pi$
$522$		0			0
$523$	25.4558			1.11311			0.556553	−	0.830812i	$-0.312124\pi$
$523$	25.4558			1.11311			0.556553	+	0.830812i	$0.312124\pi$
$524$	−8.48528			−0.370681
$525$		0			0
$526$		−	4.00000i		−	0.174408i
$527$		0			0
$528$		0			0
$529$	41.0000			1.78261
$530$	−8.48528			−0.368577
$531$		−	14.1421i		−	0.613716i
$532$		0			0
$533$	−35.0000	+	7.00000i	−1.51602	+	0.303204i
$534$	36.0000			1.55787
$535$			16.9706i			0.733701i
$536$	−12.0000			−0.518321
$537$	−56.5685			−2.44111
$538$			9.89949i			0.426798i
$539$		0			0
$540$			8.00000i			0.344265i
$541$		−	8.00000i		−	0.343947i	−0.985102	−	0.171973i	$-0.944986\pi$
$541$		−	8.00000i		−	0.343947i	0.985102	−	0.171973i	$-0.0550143\pi$
$542$	−16.9706			−0.728948
$543$	−44.0000			−1.88822
$544$			1.41421i			0.0606339i
$545$	2.82843			0.121157
$546$		0			0
$547$	−16.0000			−0.684111			−0.342055	−	0.939680i	$-0.611123\pi$
$547$	−16.0000			−0.684111			−0.342055	+	0.939680i	$0.611123\pi$
$548$		−	18.0000i		−	0.768922i
$549$	63.6396			2.71607
$550$		0			0
$551$		0			0
$552$		−	22.6274i		−	0.963087i
$553$		0			0
$554$			22.0000i			0.934690i
$555$	40.0000			1.69791
$556$	14.1421			0.599760
$557$			8.00000i			0.338971i	0.985533	+	0.169485i	$0.0542106\pi$
$557$			8.00000i			0.338971i	−0.985533	+	0.169485i	$0.945789\pi$
$558$		0			0
$559$	5.65685	+	28.2843i	0.239259	+	1.19630i
$560$		0			0
$561$		0			0
$562$	2.00000			0.0843649
$563$	8.48528			0.357612			0.178806	−	0.983884i	$-0.442777\pi$
$563$	8.48528			0.357612			0.178806	+	0.983884i	$0.442777\pi$
$564$		−	32.0000i		−	1.34744i
$565$			8.48528i			0.356978i
$566$			8.48528i			0.356663i
$567$		0			0
$568$	8.00000			0.335673
$569$	24.0000			1.00613			0.503066	−	0.864248i	$-0.332205\pi$
$569$	24.0000			1.00613			0.503066	+	0.864248i	$0.332205\pi$
$570$		−	11.3137i		−	0.473879i
$571$	40.0000			1.67395			0.836974	−	0.547243i	$-0.184323\pi$
$571$	40.0000			1.67395			0.836974	+	0.547243i	$0.184323\pi$
$572$		0			0
$573$	33.9411			1.41791
$574$		0			0
$575$	24.0000			1.00087
$576$	−5.00000			−0.208333
$577$			29.6985i			1.23636i	0.786035	+	0.618182i	$0.212131\pi$
$577$			29.6985i			1.23636i	−0.786035	+	0.618182i	$0.787869\pi$
$578$		−	15.0000i		−	0.623918i
$579$		−	22.6274i		−	0.940363i
$580$		0			0
$581$		0			0
$582$	−12.0000			−0.497416
$583$		0			0
$584$	12.7279			0.526685
$585$	−25.0000	+	5.00000i	−1.03362	+	0.206725i
$586$	24.0416			0.993151
$587$			8.48528i			0.350225i	0.984548	+	0.175113i	$0.0560289\pi$
$587$			8.48528i			0.350225i	−0.984548	+	0.175113i	$0.943971\pi$
$588$		0			0
$589$		0			0
$590$		−	4.00000i		−	0.164677i
$591$			67.8823i			2.79230i
$592$			10.0000i			0.410997i
$593$			7.07107i			0.290374i	0.989404	+	0.145187i	$0.0463784\pi$
$593$			7.07107i			0.290374i	−0.989404	+	0.145187i	$0.953622\pi$
$594$		0			0
$595$		0			0
$596$		−	8.00000i		−	0.327693i
$597$	48.0000			1.96451
$598$	28.2843	−	5.65685i	1.15663	−	0.231326i
$599$	12.0000			0.490307			0.245153	−	0.969484i	$-0.421162\pi$
$599$	12.0000			0.490307			0.245153	+	0.969484i	$0.421162\pi$
$600$		−	8.48528i		−	0.346410i
$601$	4.24264			0.173061			0.0865305	−	0.996249i	$-0.472422\pi$
$601$	4.24264			0.173061			0.0865305	+	0.996249i	$0.472422\pi$
$602$		0			0
$603$		−	60.0000i		−	2.44339i
$604$		−	20.0000i		−	0.813788i
$605$		−	15.5563i		−	0.632456i
$606$		−	12.0000i		−	0.487467i
$607$	−16.9706			−0.688814			−0.344407	−	0.938820i	$-0.611920\pi$
$607$	−16.9706			−0.688814			−0.344407	+	0.938820i	$0.611920\pi$
$608$	2.82843			0.114708
$609$		0			0
$610$	18.0000			0.728799
$611$	40.0000	−	8.00000i	1.61823	−	0.323645i
$612$	−7.07107			−0.285831
$613$			6.00000i			0.242338i	0.992632	+	0.121169i	$0.0386643\pi$
$613$			6.00000i			0.242338i	−0.992632	+	0.121169i	$0.961336\pi$
$614$	−8.48528			−0.342438
$615$	−39.5980			−1.59674
$616$		0			0
$617$		−	26.0000i		−	1.04672i	−0.852111	−	0.523360i	$-0.824678\pi$
$617$		−	26.0000i		−	1.04672i	0.852111	−	0.523360i	$-0.175322\pi$
$618$		0			0
$619$			31.1127i			1.25052i	0.780415	+	0.625262i	$0.215008\pi$
$619$			31.1127i			1.25052i	−0.780415	+	0.625262i	$0.784992\pi$
$620$		0			0
$621$	45.2548			1.81601
$622$		−	28.2843i		−	1.13410i
$623$		0			0
$624$	−2.00000	−	10.0000i	−0.0800641	−	0.400320i
$625$	−1.00000			−0.0400000
$626$			15.5563i			0.621757i
$627$		0			0
$628$	−12.7279			−0.507899
$629$			14.1421i			0.563884i
$630$		0			0
$631$		−	8.00000i		−	0.318475i	−0.987240	−	0.159237i	$-0.949096\pi$
$631$		−	8.00000i		−	0.318475i	0.987240	−	0.159237i	$-0.0509036\pi$
$632$			4.00000i			0.159111i
$633$	33.9411			1.34904
$634$	24.0000			0.953162
$635$			22.6274i			0.897942i
$636$	16.9706			0.672927
$637$		0			0
$638$		0			0
$639$			40.0000i			1.58238i
$640$	−1.41421			−0.0559017
$641$	24.0000			0.947943			0.473972	−	0.880540i	$-0.342820\pi$
$641$	24.0000			0.947943			0.473972	+	0.880540i	$0.342820\pi$
$642$		−	33.9411i		−	1.33955i
$643$			36.7696i			1.45005i	0.688723	+	0.725025i	$0.258172\pi$
$643$			36.7696i			1.45005i	−0.688723	+	0.725025i	$0.741828\pi$
$644$		0			0
$645$			32.0000i			1.26000i
$646$	4.00000			0.157378
$647$		0			0			−	1.00000i	$-0.5\pi$
$647$		0			0				1.00000i	$0.5\pi$
$648$		−	1.00000i		−	0.0392837i
$649$		0			0
$650$	10.6066	−	2.12132i	0.416025	−	0.0832050i
$651$		0			0
$652$			8.00000i			0.313304i
$653$	48.0000			1.87839			0.939193	−	0.343391i	$-0.111576\pi$
$653$	48.0000			1.87839			0.939193	+	0.343391i	$0.111576\pi$
$654$	−5.65685			−0.221201
$655$		−	12.0000i		−	0.468879i
$656$		−	9.89949i		−	0.386510i
$657$			63.6396i			2.48282i
$658$		0			0
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$		−	35.3553i		−	1.37516i	−0.726107	−	0.687582i	$-0.758672\pi$
$661$		−	35.3553i		−	1.37516i	0.726107	−	0.687582i	$-0.241328\pi$
$662$	−16.0000			−0.621858
$663$	−2.82843	−	14.1421i	−0.109847	−	0.549235i
$664$	2.82843			0.109764
$665$		0			0
$666$	−50.0000			−1.93746
$667$		0			0
$668$			5.65685i			0.218870i
$669$			16.0000i			0.618596i
$670$		−	16.9706i		−	0.655630i
$671$		0			0
$672$		0			0
$673$	48.0000			1.85026			0.925132	−	0.379646i	$-0.123954\pi$
$673$	48.0000			1.85026			0.925132	+	0.379646i	$0.123954\pi$
$674$		−	8.00000i		−	0.308148i
$675$	16.9706			0.653197
$676$	12.0000	−	5.00000i	0.461538	−	0.192308i
$677$	−41.0122			−1.57623			−0.788113	−	0.615530i	$-0.788942\pi$
$677$	−41.0122			−1.57623			−0.788113	+	0.615530i	$0.788942\pi$
$678$		−	16.9706i		−	0.651751i
$679$		0			0
$680$	−2.00000			−0.0766965
$681$		−	72.0000i		−	2.75905i
$682$		0			0
$683$			20.0000i			0.765279i	0.923898	+	0.382639i	$0.124985\pi$
$683$			20.0000i			0.765279i	−0.923898	+	0.382639i	$0.875015\pi$
$684$			14.1421i			0.540738i
$685$	25.4558			0.972618
$686$		0			0
$687$			68.0000i			2.59436i
$688$	−8.00000			−0.304997
$689$	4.24264	+	21.2132i	0.161632	+	0.808159i
$690$	32.0000			1.21822
$691$			19.7990i			0.753189i	0.926378	+	0.376595i	$0.122905\pi$
$691$			19.7990i			0.753189i	−0.926378	+	0.376595i	$0.877095\pi$
$692$	12.7279			0.483843
$693$		0			0
$694$		−	16.0000i		−	0.607352i
$695$			20.0000i			0.758643i
$696$		0			0
$697$		−	14.0000i		−	0.530288i
$698$	29.6985			1.12410
$699$		0			0
$700$		0			0
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$	20.0000	−	4.00000i	0.754851	−	0.150970i
$703$	28.2843			1.06676
$704$		0			0
$705$	45.2548			1.70440
$706$	7.07107			0.266123
$707$		0			0
$708$			8.00000i			0.300658i
$709$			22.0000i			0.826227i	0.910679	+	0.413114i	$0.135559\pi$
$709$			22.0000i			0.826227i	−0.910679	+	0.413114i	$0.864441\pi$
$710$			11.3137i			0.424596i
$711$	−20.0000			−0.750059
$712$	−12.7279			−0.476999
$713$		0			0
$714$		0			0
$715$		0			0
$716$	20.0000			0.747435
$717$		−	33.9411i		−	1.26755i
$718$	4.00000			0.149279
$719$	39.5980			1.47676			0.738378	−	0.674387i	$-0.235592\pi$
$719$	39.5980			1.47676			0.738378	+	0.674387i	$0.235592\pi$
$720$		−	7.07107i		−	0.263523i
$721$		0			0
$722$			11.0000i			0.409378i
$723$			44.0000i			1.63638i
$724$	15.5563			0.578147
$725$		0			0
$726$			31.1127i			1.15470i
$727$	39.5980			1.46861			0.734304	−	0.678821i	$-0.237509\pi$
$727$	39.5980			1.46861			0.734304	+	0.678821i	$0.237509\pi$
$728$		0			0
$729$	−43.0000			−1.59259
$730$			18.0000i			0.666210i
$731$	−11.3137			−0.418453
$732$	−36.0000			−1.33060
$733$		−	4.24264i		−	0.156706i	−0.996926	−	0.0783528i	$-0.975034\pi$
$733$		−	4.24264i		−	0.156706i	0.996926	−	0.0783528i	$-0.0249660\pi$
$734$			16.9706i			0.626395i
$735$		0			0
$736$			8.00000i			0.294884i
$737$		0			0
$738$	49.4975			1.82203
$739$			48.0000i			1.76571i	0.469647	+	0.882854i	$0.344381\pi$
$739$			48.0000i			1.76571i	−0.469647	+	0.882854i	$0.655619\pi$
$740$	−14.1421			−0.519875
$741$	−28.2843	+	5.65685i	−1.03905	+	0.207810i
$742$		0			0
$743$		−	4.00000i		−	0.146746i	−0.997305	−	0.0733729i	$-0.976624\pi$
$743$		−	4.00000i		−	0.146746i	0.997305	−	0.0733729i	$-0.0233763\pi$
$744$		0			0
$745$	11.3137			0.414502
$746$		−	10.0000i		−	0.366126i
$747$			14.1421i			0.517434i
$748$		0			0
$749$		0			0
$750$	32.0000			1.16847
$751$	−24.0000			−0.875772			−0.437886	−	0.899030i	$-0.644273\pi$
$751$	−24.0000			−0.875772			−0.437886	+	0.899030i	$0.644273\pi$
$752$			11.3137i			0.412568i
$753$	−40.0000			−1.45768
$754$		0			0
$755$	28.2843			1.02937
$756$		0			0
$757$	−16.0000			−0.581530			−0.290765	−	0.956795i	$-0.593910\pi$
$757$	−16.0000			−0.581530			−0.290765	+	0.956795i	$0.593910\pi$
$758$	−8.00000			−0.290573
$759$		0			0
$760$			4.00000i			0.145095i
$761$		−	43.8406i		−	1.58922i	−0.607119	−	0.794611i	$-0.707675\pi$
$761$		−	43.8406i		−	1.58922i	0.607119	−	0.794611i	$-0.292325\pi$
$762$		−	45.2548i		−	1.63941i
$763$		0			0
$764$	−12.0000			−0.434145
$765$		−	10.0000i		−	0.361551i
$766$	−5.65685			−0.204390
$767$	−10.0000	+	2.00000i	−0.361079	+	0.0722158i
$768$	2.82843			0.102062
$769$		−	18.3848i		−	0.662972i	−0.943460	−	0.331486i	$-0.892450\pi$
$769$		−	18.3848i		−	0.662972i	0.943460	−	0.331486i	$-0.107550\pi$
$770$		0			0
$771$	4.00000			0.144056
$772$			8.00000i			0.287926i
$773$			9.89949i			0.356060i	0.984025	+	0.178030i	$0.0569724\pi$
$773$			9.89949i			0.356060i	−0.984025	+	0.178030i	$0.943028\pi$
$774$		−	40.0000i		−	1.43777i
$775$		0			0
$776$	4.24264			0.152302
$777$		0			0
$778$		−	16.0000i		−	0.573628i
$779$	−28.0000			−1.00320
$780$	14.1421	−	2.82843i	0.506370	−	0.101274i
$781$		0			0
$782$			11.3137i			0.404577i
$783$		0			0
$784$		0			0
$785$		−	18.0000i		−	0.642448i
$786$			24.0000i			0.856052i
$787$		−	25.4558i		−	0.907403i	−0.891154	−	0.453701i	$-0.850103\pi$
$787$		−	25.4558i		−	0.907403i	0.891154	−	0.453701i	$-0.149897\pi$
$788$		−	24.0000i		−	0.854965i
$789$	−11.3137			−0.402779
$790$	−5.65685			−0.201262
$791$		0			0
$792$		0			0
$793$	−9.00000	−	45.0000i	−0.319599	−	1.59800i
$794$	−18.3848			−0.652451
$795$			24.0000i			0.851192i
$796$	−16.9706			−0.601506
$797$	−1.41421			−0.0500940			−0.0250470	−	0.999686i	$-0.507974\pi$
$797$	−1.41421			−0.0500940			−0.0250470	+	0.999686i	$0.507974\pi$
$798$		0			0
$799$			16.0000i			0.566039i
$800$			3.00000i			0.106066i
$801$		−	63.6396i		−	2.24860i
$802$	−2.00000			−0.0706225
$803$		0			0
$804$			33.9411i			1.19701i
$805$		0			0
$806$		0			0
$807$	28.0000			0.985647
$808$			4.24264i			0.149256i
$809$	−34.0000			−1.19538			−0.597688	−	0.801729i	$-0.703914\pi$
$809$	−34.0000			−1.19538			−0.597688	+	0.801729i	$0.703914\pi$
$810$	1.41421			0.0496904
$811$		−	19.7990i		−	0.695237i	−0.937636	−	0.347618i	$-0.886991\pi$
$811$		−	19.7990i		−	0.695237i	0.937636	−	0.347618i	$-0.113009\pi$
$812$		0			0
$813$			48.0000i			1.68343i
$814$		0			0
$815$	−11.3137			−0.396302
$816$	4.00000			0.140028
$817$			22.6274i			0.791633i
$818$	12.7279			0.445021
$819$		0			0
$820$	14.0000			0.488901
$821$			24.0000i			0.837606i	0.908077	+	0.418803i	$0.137550\pi$
$821$			24.0000i			0.837606i	−0.908077	+	0.418803i	$0.862450\pi$
$822$	−50.9117			−1.77575
$823$	28.0000			0.976019			0.488009	−	0.872838i	$-0.337723\pi$
$823$	28.0000			0.976019			0.488009	+	0.872838i	$0.337723\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			4.00000i			0.139094i	0.997579	+	0.0695468i	$0.0221553\pi$
$827$			4.00000i			0.139094i	−0.997579	+	0.0695468i	$0.977845\pi$
$828$	−40.0000			−1.39010
$829$	−32.5269			−1.12971			−0.564853	−	0.825191i	$-0.691067\pi$
$829$	−32.5269			−1.12971			−0.564853	+	0.825191i	$0.691067\pi$
$830$			4.00000i			0.138842i
$831$	62.2254			2.15858
$832$	0.707107	+	3.53553i	0.0245145	+	0.122573i
$833$		0			0
$834$		−	40.0000i		−	1.38509i
$835$	−8.00000			−0.276851
$836$		0			0
$837$		0			0
$838$			36.7696i			1.27018i
$839$			11.3137i			0.390593i	0.980744	+	0.195296i	$0.0625668\pi$
$839$			11.3137i			0.390593i	−0.980744	+	0.195296i	$0.937433\pi$
$840$		0			0
$841$	−29.0000			−1.00000
$842$	8.00000			0.275698
$843$		−	5.65685i		−	0.194832i
$844$	−12.0000			−0.413057
$845$	7.07107	+	16.9706i	0.243252	+	0.583805i
$846$	−56.5685			−1.94487
$847$		0			0
$848$	−6.00000			−0.206041
$849$	24.0000			0.823678
$850$			4.24264i			0.145521i
$851$			80.0000i			2.74236i
$852$		−	22.6274i		−	0.775203i
$853$			1.41421i			0.0484218i	0.999707	+	0.0242109i	$0.00770731\pi$
$853$			1.41421i			0.0484218i	−0.999707	+	0.0242109i	$0.992293\pi$
$854$		0			0
$855$	−20.0000			−0.683986
$856$			12.0000i			0.410152i
$857$	−41.0122			−1.40095			−0.700475	−	0.713677i	$-0.747028\pi$
$857$	−41.0122			−1.40095			−0.700475	+	0.713677i	$0.747028\pi$
$858$		0			0
$859$	25.4558			0.868542			0.434271	−	0.900782i	$-0.357006\pi$
$859$	25.4558			0.868542			0.434271	+	0.900782i	$0.357006\pi$
$860$		−	11.3137i		−	0.385794i
$861$		0			0
$862$	−20.0000			−0.681203
$863$			24.0000i			0.816970i	0.912765	+	0.408485i	$0.133943\pi$
$863$			24.0000i			0.816970i	−0.912765	+	0.408485i	$0.866057\pi$
$864$			5.65685i			0.192450i
$865$			18.0000i			0.612018i
$866$			21.2132i			0.720854i
$867$	−42.4264			−1.44088
$868$		0			0
$869$		0			0
$870$		0			0
$871$	−42.4264	+	8.48528i	−1.43756	+	0.287513i
$872$	2.00000			0.0677285
$873$			21.2132i			0.717958i
$874$	22.6274			0.765384
$875$		0			0
$876$		−	36.0000i		−	1.21633i
$877$		−	46.0000i		−	1.55331i	−0.629926	−	0.776655i	$-0.716915\pi$
$877$		−	46.0000i		−	1.55331i	0.629926	−	0.776655i	$-0.283085\pi$
$878$		−	11.3137i		−	0.381819i
$879$		−	68.0000i		−	2.29358i
$880$		0			0
$881$	43.8406			1.47703			0.738514	−	0.674238i	$-0.235528\pi$
$881$	43.8406			1.47703			0.738514	+	0.674238i	$0.235528\pi$
$882$		0			0
$883$	−4.00000			−0.134611			−0.0673054	−	0.997732i	$-0.521440\pi$
$883$	−4.00000			−0.134611			−0.0673054	+	0.997732i	$0.521440\pi$
$884$	1.00000	+	5.00000i	0.0336336	+	0.168168i
$885$	−11.3137			−0.380306
$886$		−	28.0000i		−	0.940678i
$887$	−45.2548			−1.51951			−0.759754	−	0.650210i	$-0.774681\pi$
$887$	−45.2548			−1.51951			−0.759754	+	0.650210i	$0.774681\pi$
$888$	28.2843			0.949158
$889$		0			0
$890$		−	18.0000i		−	0.603361i
$891$		0			0
$892$		−	5.65685i		−	0.189405i
$893$	32.0000			1.07084
$894$	−22.6274			−0.756774
$895$			28.2843i			0.945439i
$896$		0			0
$897$	−16.0000	−	80.0000i	−0.534224	−	2.67112i
$898$	32.0000			1.06785
$899$		0			0
$900$	−15.0000			−0.500000
$901$	−8.48528			−0.282686
$902$		0			0
$903$		0			0
$904$			6.00000i			0.199557i
$905$			22.0000i			0.731305i
$906$	−56.5685			−1.87936
$907$	−52.0000			−1.72663			−0.863316	−	0.504664i	$-0.831616\pi$
$907$	−52.0000			−1.72663			−0.863316	+	0.504664i	$0.831616\pi$
$908$			25.4558i			0.844782i
$909$	−21.2132			−0.703598
$910$		0			0
$911$	−16.0000			−0.530104			−0.265052	−	0.964234i	$-0.585389\pi$
$911$	−16.0000			−0.530104			−0.265052	+	0.964234i	$0.585389\pi$
$912$		−	8.00000i		−	0.264906i
$913$		0			0
$914$	24.0000			0.793849
$915$		−	50.9117i		−	1.68309i
$916$		−	24.0416i		−	0.794358i
$917$		0			0
$918$			8.00000i			0.264039i
$919$	−20.0000			−0.659739			−0.329870	−	0.944027i	$-0.607005\pi$
$919$	−20.0000			−0.659739			−0.329870	+	0.944027i	$0.607005\pi$
$920$	−11.3137			−0.373002
$921$			24.0000i			0.790827i
$922$	26.8701			0.884918
$923$	28.2843	−	5.65685i	0.930988	−	0.186198i
$924$		0			0
$925$			30.0000i			0.986394i
$926$	−32.0000			−1.05159
$927$		0			0
$928$		0			0
$929$		−	7.07107i		−	0.231994i	−0.993250	−	0.115997i	$-0.962994\pi$
$929$		−	7.07107i		−	0.231994i	0.993250	−	0.115997i	$-0.0370063\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	−80.0000			−2.61908
$934$		−	25.4558i		−	0.832941i
$935$		0			0
$936$	−17.6777	+	3.53553i	−0.577813	+	0.115563i
$937$	−4.24264			−0.138601			−0.0693005	−	0.997596i	$-0.522077\pi$
$937$	−4.24264			−0.138601			−0.0693005	+	0.997596i	$0.522077\pi$
$938$		0			0
$939$	44.0000			1.43589
$940$	−16.0000			−0.521862
$941$		−	41.0122i		−	1.33696i	−0.743730	−	0.668480i	$-0.766945\pi$
$941$		−	41.0122i		−	1.33696i	0.743730	−	0.668480i	$-0.233055\pi$
$942$			36.0000i			1.17294i
$943$		−	79.1960i		−	2.57898i
$944$		−	2.82843i		−	0.0920575i
$945$		0			0
$946$		0			0
$947$			48.0000i			1.55979i	0.625910	+	0.779895i	$0.284728\pi$
$947$			48.0000i			1.55979i	−0.625910	+	0.779895i	$0.715272\pi$
$948$	11.3137			0.367452
$949$	45.0000	−	9.00000i	1.46076	−	0.292152i
$950$	8.48528			0.275299
$951$		−	67.8823i		−	2.20123i
$952$		0			0
$953$	58.0000			1.87880			0.939402	−	0.342817i	$-0.111381\pi$
$953$	58.0000			1.87880			0.939402	+	0.342817i	$0.111381\pi$
$954$		−	30.0000i		−	0.971286i
$955$		−	16.9706i		−	0.549155i
$956$			12.0000i			0.388108i
$957$		0			0
$958$	11.3137			0.365529
$959$		0			0
$960$			4.00000i			0.129099i
$961$	31.0000			1.00000
$962$	7.07107	+	35.3553i	0.227980	+	1.13990i
$963$	−60.0000			−1.93347
$964$		−	15.5563i		−	0.501036i
$965$	−11.3137			−0.364201
$966$		0			0
$967$		−	4.00000i		−	0.128631i	−0.997930	−	0.0643157i	$-0.979514\pi$
$967$		−	4.00000i		−	0.128631i	0.997930	−	0.0643157i	$-0.0204865\pi$
$968$		−	11.0000i		−	0.353553i
$969$		−	11.3137i		−	0.363449i
$970$			6.00000i			0.192648i
$971$	−36.7696			−1.17999			−0.589996	−	0.807406i	$-0.700871\pi$
$971$	−36.7696			−1.17999			−0.589996	+	0.807406i	$0.700871\pi$
$972$	14.1421			0.453609
$973$		0			0
$974$	12.0000			0.384505
$975$	−6.00000	−	30.0000i	−0.192154	−	0.960769i
$976$	12.7279			0.407411
$977$		−	54.0000i		−	1.72761i	−0.503824	−	0.863807i	$-0.668074\pi$
$977$		−	54.0000i		−	1.72761i	0.503824	−	0.863807i	$-0.331926\pi$
$978$	22.6274			0.723545
$979$		0			0
$980$		0			0
$981$			10.0000i			0.319275i
$982$			20.0000i			0.638226i
$983$			5.65685i			0.180426i	0.995923	+	0.0902128i	$0.0287547\pi$
$983$			5.65685i			0.180426i	−0.995923	+	0.0902128i	$0.971245\pi$
$984$	−28.0000			−0.892607
$985$	33.9411			1.08145
$986$		0			0
$987$		0			0
$988$	10.0000	−	2.00000i	0.318142	−	0.0636285i
$989$	−64.0000			−2.03508
$990$		0			0
$991$	44.0000			1.39771			0.698853	−	0.715265i	$-0.253694\pi$
$991$	44.0000			1.39771			0.698853	+	0.715265i	$0.253694\pi$
$992$		0			0
$993$			45.2548i			1.43612i
$994$		0			0
$995$		−	24.0000i		−	0.760851i
$996$		−	8.00000i		−	0.253490i
$997$	26.8701			0.850983			0.425492	−	0.904962i	$-0.360101\pi$
$997$	26.8701			0.850983			0.425492	+	0.904962i	$0.360101\pi$
$998$	−20.0000			−0.633089
$999$			56.5685i			1.78975i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1274.2.d.i.883.4	yes	4
7.2	even	3		1274.2.n.i.753.3		8
7.3	odd	6		1274.2.n.i.961.2		8
7.4	even	3		1274.2.n.i.961.1		8
7.5	odd	6		1274.2.n.i.753.4		8
7.6	odd	2	inner	1274.2.d.i.883.3	yes	4
13.12	even	2	inner	1274.2.d.i.883.2	yes	4
91.12	odd	6		1274.2.n.i.753.2		8
91.25	even	6		1274.2.n.i.961.3		8
91.38	odd	6		1274.2.n.i.961.4		8
91.51	even	6		1274.2.n.i.753.1		8
91.90	odd	2	inner	1274.2.d.i.883.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1274.2.d.i.883.1	✓	4	91.90	odd	2	inner
1274.2.d.i.883.2	yes	4	13.12	even	2	inner
1274.2.d.i.883.3	yes	4	7.6	odd	2	inner
1274.2.d.i.883.4	yes	4	1.1	even	1	trivial
1274.2.n.i.753.1		8	91.51	even	6
1274.2.n.i.753.2		8	91.12	odd	6
1274.2.n.i.753.3		8	7.2	even	3
1274.2.n.i.753.4		8	7.5	odd	6
1274.2.n.i.961.1		8	7.4	even	3
1274.2.n.i.961.2		8	7.3	odd	6
1274.2.n.i.961.3		8	91.25	even	6
1274.2.n.i.961.4		8	91.38	odd	6

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1274 = 2 \cdot 7^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1274.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} +2.82843 q^{3} -1.00000 q^{4} -1.41421i q^{5} +2.82843i q^{6} -1.00000i q^{8} +5.00000 q^{9} +1.41421 q^{10} -2.82843 q^{12} +(-0.707107 - 3.53553i) q^{13} -4.00000i q^{15} +1.00000 q^{16} +1.41421 q^{17} +5.00000i q^{18} -2.82843i q^{19} +1.41421i q^{20} +8.00000 q^{23} -2.82843i q^{24} +3.00000 q^{25} +(3.53553 - 0.707107i) q^{26} +5.65685 q^{27} +4.00000 q^{30} +1.00000i q^{32} +1.41421i q^{34} -5.00000 q^{36} +10.0000i q^{37} +2.82843 q^{38} +(-2.00000 - 10.0000i) q^{39} -1.41421 q^{40} -9.89949i q^{41} -8.00000 q^{43} -7.07107i q^{45} +8.00000i q^{46} +11.3137i q^{47} +2.82843 q^{48} +3.00000i q^{50} +4.00000 q^{51} +(0.707107 + 3.53553i) q^{52} -6.00000 q^{53} +5.65685i q^{54} -8.00000i q^{57} -2.82843i q^{59} +4.00000i q^{60} +12.7279 q^{61} -1.00000 q^{64} +(-5.00000 + 1.00000i) q^{65} -12.0000i q^{67} -1.41421 q^{68} +22.6274 q^{69} +8.00000i q^{71} -5.00000i q^{72} +12.7279i q^{73} -10.0000 q^{74} +8.48528 q^{75} +2.82843i q^{76} +(10.0000 - 2.00000i) q^{78} -4.00000 q^{79} -1.41421i q^{80} +1.00000 q^{81} +9.89949 q^{82} +2.82843i q^{83} -2.00000i q^{85} -8.00000i q^{86} -12.7279i q^{89} +7.07107 q^{90} -8.00000 q^{92} -11.3137 q^{94} -4.00000 q^{95} +2.82843i q^{96} +4.24264i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4} + 20 q^{9} + 4 q^{16} + 32 q^{23} + 12 q^{25} + 16 q^{30} - 20 q^{36} - 8 q^{39} - 32 q^{43} + 16 q^{51} - 24 q^{53} - 4 q^{64} - 20 q^{65} - 40 q^{74} + 40 q^{78} - 16 q^{79} + 4 q^{81} - 32 q^{92}+ \cdots - 16 q^{95}+O(q^{100}) \) 4 * q - 4 * q^4 + 20 * q^9 + 4 * q^16 + 32 * q^23 + 12 * q^25 + 16 * q^30 - 20 * q^36 - 8 * q^39 - 32 * q^43 + 16 * q^51 - 24 * q^53 - 4 * q^64 - 20 * q^65 - 40 * q^74 + 40 * q^78 - 16 * q^79 + 4 * q^81 - 32 * q^92 - 16 * q^95

Introduction

L-functions

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