LMFDB - Embedded newform 1248.2.c.a.961.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1248,2,Mod(961,1248)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1248, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1248.961"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 1248 = 2^{5} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1248.c (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [6,0,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	no
Analytic conductor:	$9.96533017226$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.153664.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 5x^{4} + 6x^{2} + 1 $ x^6 + 5x^4 + 6x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{7} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			961.3
Root			$-0.445042i$ of defining polynomial
Character	$\chi$	$=$	1248.961
Dual form			1248.2.c.a.961.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -0.890084i q^{5} -1.60388i q^{7} +1.00000 q^{9} +2.49396i q^{11} +(0.109916 + 3.60388i) q^{13} +0.890084i q^{15} +2.00000 q^{17} +1.60388i q^{19} +1.60388i q^{21} -4.98792 q^{23} +4.20775 q^{25} -1.00000 q^{27} +6.98792 q^{29} -3.38404i q^{31} -2.49396i q^{33} -1.42758 q^{35} -0.987918i q^{37} +(-0.109916 - 3.60388i) q^{39} -2.67025i q^{41} +8.19567 q^{43} -0.890084i q^{45} -1.50604i q^{47} +4.42758 q^{49} -2.00000 q^{51} +3.42758 q^{53} +2.21983 q^{55} -1.60388i q^{57} +0.713792i q^{59} +8.76809 q^{61} -1.60388i q^{63} +(3.20775 - 0.0978347i) q^{65} +7.82371i q^{67} +4.98792 q^{69} -9.70171i q^{71} -1.78017i q^{73} -4.20775 q^{75} +4.00000 q^{77} +10.4155 q^{79} +1.00000 q^{81} +12.2741i q^{83} -1.78017i q^{85} -6.98792 q^{87} +15.3056i q^{89} +(5.78017 - 0.176292i) q^{91} +3.38404i q^{93} +1.42758 q^{95} -9.78017i q^{97} +2.49396i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{3} + 6 q^{9} + 2 q^{13} + 12 q^{17} + 8 q^{23} - 10 q^{25} - 6 q^{27} + 4 q^{29} + 24 q^{35} - 2 q^{39} - 24 q^{43} - 6 q^{49} - 12 q^{51} - 12 q^{53} + 16 q^{55} + 12 q^{61} - 16 q^{65} - 8 q^{69}+ \cdots - 24 q^{95}+O(q^{100}) $ 6 * q - 6 * q^3 + 6 * q^9 + 2 * q^13 + 12 * q^17 + 8 * q^23 - 10 * q^25 - 6 * q^27 + 4 * q^29 + 24 * q^35 - 2 * q^39 - 24 * q^43 - 6 * q^49 - 12 * q^51 - 12 * q^53 + 16 * q^55 + 12 * q^61 - 16 * q^65 - 8 * q^69 + 10 * q^75 + 24 * q^77 - 8 * q^79 + 6 * q^81 - 4 * q^87 + 32 * q^91 - 24 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1248\mathbb{Z}\right)^\times$.

$n$	$703$	$769$	$833$	$1093$
$\chi(n)$	$1$	$-1$	$1$	$1$

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 $

$2$		0			0
$2$		0			0
$3$	−1.00000			−0.577350
$3$	−1.00000			−0.577350
$4$		0			0
$5$		−	0.890084i		−	0.398058i	−0.979994	−	0.199029i	$-0.936221\pi$
$5$		−	0.890084i		−	0.398058i	0.979994	−	0.199029i	$-0.0637787\pi$
$6$		0			0
$7$		−	1.60388i		−	0.606208i	−0.952957	−	0.303104i	$-0.901977\pi$
$7$		−	1.60388i		−	0.606208i	0.952957	−	0.303104i	$-0.0980229\pi$
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$			2.49396i			0.751957i	0.926628	+	0.375978i	$0.122693\pi$
$11$			2.49396i			0.751957i	−0.926628	+	0.375978i	$0.877307\pi$
$12$		0			0
$13$	0.109916	+	3.60388i	0.0304853	+	0.999535i
$13$	0.109916	+	3.60388i	0.0304853	+	0.999535i
$14$		0			0
$15$			0.890084i			0.229819i
$16$		0			0
$17$	2.00000			0.485071			0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000			0.485071			0.242536	+	0.970143i	$0.422021\pi$
$18$		0			0
$19$			1.60388i			0.367954i	0.982931	+	0.183977i	$0.0588972\pi$
$19$			1.60388i			0.367954i	−0.982931	+	0.183977i	$0.941103\pi$
$20$		0			0
$21$			1.60388i			0.349994i
$22$		0			0
$23$	−4.98792			−1.04005			−0.520026	−	0.854150i	$-0.674078\pi$
$23$	−4.98792			−1.04005			−0.520026	+	0.854150i	$0.674078\pi$
$24$		0			0
$25$	4.20775			0.841550
$26$		0			0
$27$	−1.00000			−0.192450
$28$		0			0
$29$	6.98792			1.29762			0.648812	−	0.760949i	$-0.275266\pi$
$29$	6.98792			1.29762			0.648812	+	0.760949i	$0.275266\pi$
$30$		0			0
$31$		−	3.38404i		−	0.607792i	−0.952705	−	0.303896i	$-0.901712\pi$
$31$		−	3.38404i		−	0.607792i	0.952705	−	0.303896i	$-0.0982875\pi$
$32$		0			0
$33$		−	2.49396i		−	0.434143i
$34$		0			0
$35$	−1.42758			−0.241306
$36$		0			0
$37$		−	0.987918i		−	0.162413i	−0.996697	−	0.0812064i	$-0.974123\pi$
$37$		−	0.987918i		−	0.162413i	0.996697	−	0.0812064i	$-0.0258773\pi$
$38$		0			0
$39$	−0.109916	−	3.60388i	−0.0176007	−	0.577082i
$40$		0			0
$41$		−	2.67025i		−	0.417023i	−0.978020	−	0.208512i	$-0.933138\pi$
$41$		−	2.67025i		−	0.417023i	0.978020	−	0.208512i	$-0.0668619\pi$
$42$		0			0
$43$	8.19567			1.24983			0.624914	−	0.780694i	$-0.285134\pi$
$43$	8.19567			1.24983			0.624914	+	0.780694i	$0.285134\pi$
$44$		0			0
$45$		−	0.890084i		−	0.132686i
$46$		0			0
$47$		−	1.50604i		−	0.219679i	−0.993949	−	0.109839i	$-0.964966\pi$
$47$		−	1.50604i		−	0.219679i	0.993949	−	0.109839i	$-0.0350336\pi$
$48$		0			0
$49$	4.42758			0.632512
$50$		0			0
$51$	−2.00000			−0.280056
$52$		0			0
$53$	3.42758			0.470815			0.235407	−	0.971897i	$-0.424358\pi$
$53$	3.42758			0.470815			0.235407	+	0.971897i	$0.424358\pi$
$54$		0			0
$55$	2.21983			0.299322
$56$		0			0
$57$		−	1.60388i		−	0.212438i
$58$		0			0
$59$			0.713792i			0.0929278i	0.998920	+	0.0464639i	$0.0147952\pi$
$59$			0.713792i			0.0929278i	−0.998920	+	0.0464639i	$0.985205\pi$
$60$		0			0
$61$	8.76809			1.12264			0.561319	−	0.827599i	$-0.310294\pi$
$61$	8.76809			1.12264			0.561319	+	0.827599i	$0.310294\pi$
$62$		0			0
$63$		−	1.60388i		−	0.202069i
$64$		0			0
$65$	3.20775	−	0.0978347i	0.397873	−	0.0121349i
$66$		0			0
$67$			7.82371i			0.955818i	0.878409	+	0.477909i	$0.158605\pi$
$67$			7.82371i			0.955818i	−0.878409	+	0.477909i	$0.841395\pi$
$68$		0			0
$69$	4.98792			0.600475
$70$		0			0
$71$		−	9.70171i		−	1.15138i	−0.817668	−	0.575691i	$-0.804733\pi$
$71$		−	9.70171i		−	1.15138i	0.817668	−	0.575691i	$-0.195267\pi$
$72$		0			0
$73$		−	1.78017i		−	0.208353i	−0.994559	−	0.104176i	$-0.966779\pi$
$73$		−	1.78017i		−	0.208353i	0.994559	−	0.104176i	$-0.0332207\pi$
$74$		0			0
$75$	−4.20775			−0.485869
$76$		0			0
$77$	4.00000			0.455842
$78$		0			0
$79$	10.4155			1.17184			0.585918	−	0.810371i	$-0.300734\pi$
$79$	10.4155			1.17184			0.585918	+	0.810371i	$0.300734\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$			12.2741i			1.34726i	0.739069	+	0.673630i	$0.235266\pi$
$83$			12.2741i			1.34726i	−0.739069	+	0.673630i	$0.764734\pi$
$84$		0			0
$85$		−	1.78017i		−	0.193086i
$86$		0			0
$87$	−6.98792			−0.749183
$88$		0			0
$89$			15.3056i			1.62239i	0.584777	+	0.811194i	$0.301182\pi$
$89$			15.3056i			1.62239i	−0.584777	+	0.811194i	$0.698818\pi$
$90$		0			0
$91$	5.78017	−	0.176292i	0.605926	−	0.0184804i
$92$		0			0
$93$			3.38404i			0.350909i
$94$		0			0
$95$	1.42758			0.146467
$96$		0			0
$97$		−	9.78017i		−	0.993026i	−0.868029	−	0.496513i	$-0.834614\pi$
$97$		−	9.78017i		−	0.993026i	0.868029	−	0.496513i	$-0.165386\pi$
$98$		0			0
$99$			2.49396i			0.250652i
$100$		0			0
$101$	−6.54825			−0.651576			−0.325788	−	0.945443i	$-0.605629\pi$
$101$	−6.54825			−0.651576			−0.325788	+	0.945443i	$0.605629\pi$
$102$		0			0
$103$	12.6353			1.24500			0.622498	−	0.782621i	$-0.286118\pi$
$103$	12.6353			1.24500			0.622498	+	0.782621i	$0.286118\pi$
$104$		0			0
$105$	1.42758			0.139318
$106$		0			0
$107$	−8.98792			−0.868895			−0.434447	−	0.900697i	$-0.643056\pi$
$107$	−8.98792			−0.868895			−0.434447	+	0.900697i	$0.643056\pi$
$108$		0			0
$109$			7.20775i			0.690377i	0.938533	+	0.345189i	$0.112185\pi$
$109$			7.20775i			0.690377i	−0.938533	+	0.345189i	$0.887815\pi$
$110$		0			0
$111$			0.987918i			0.0937691i
$112$		0			0
$113$	5.56033			0.523072			0.261536	−	0.965194i	$-0.415771\pi$
$113$	5.56033			0.523072			0.261536	+	0.965194i	$0.415771\pi$
$114$		0			0
$115$			4.43967i			0.414001i
$116$		0			0
$117$	0.109916	+	3.60388i	0.0101618	+	0.333178i
$118$		0			0
$119$		−	3.20775i		−	0.294054i
$120$		0			0
$121$	4.78017			0.434561
$122$		0			0
$123$			2.67025i			0.240768i
$124$		0			0
$125$		−	8.19567i		−	0.733043i
$126$		0			0
$127$	4.00000			0.354943			0.177471	−	0.984126i	$-0.443208\pi$
$127$	4.00000			0.354943			0.177471	+	0.984126i	$0.443208\pi$
$128$		0			0
$129$	−8.19567			−0.721588
$130$		0			0
$131$	4.54825			0.397383			0.198691	−	0.980062i	$-0.436331\pi$
$131$	4.54825			0.397383			0.198691	+	0.980062i	$0.436331\pi$
$132$		0			0
$133$	2.57242			0.223057
$134$		0			0
$135$			0.890084i			0.0766062i
$136$		0			0
$137$			9.08575i			0.776248i	0.921607	+	0.388124i	$0.126877\pi$
$137$			9.08575i			0.776248i	−0.921607	+	0.388124i	$0.873123\pi$
$138$		0			0
$139$	6.21983			0.527559			0.263780	−	0.964583i	$-0.415031\pi$
$139$	6.21983			0.527559			0.263780	+	0.964583i	$0.415031\pi$
$140$		0			0
$141$			1.50604i			0.126832i
$142$		0			0
$143$	−8.98792	+	0.274127i	−0.751607	+	0.0229236i
$144$		0			0
$145$		−	6.21983i		−	0.516529i
$146$		0			0
$147$	−4.42758			−0.365181
$148$		0			0
$149$		−	15.1099i		−	1.23785i	−0.785449	−	0.618926i	$-0.787568\pi$
$149$		−	15.1099i		−	1.23785i	0.785449	−	0.618926i	$-0.212432\pi$
$150$		0			0
$151$		−	21.3599i		−	1.73824i	−0.494599	−	0.869121i	$-0.664685\pi$
$151$		−	21.3599i		−	1.73824i	0.494599	−	0.869121i	$-0.335315\pi$
$152$		0			0
$153$	2.00000			0.161690
$154$		0			0
$155$	−3.01208			−0.241936
$156$		0			0
$157$	−13.4034			−1.06971			−0.534855	−	0.844944i	$-0.679634\pi$
$157$	−13.4034			−1.06971			−0.534855	+	0.844944i	$0.679634\pi$
$158$		0			0
$159$	−3.42758			−0.271825
$160$		0			0
$161$			8.00000i			0.630488i
$162$		0			0
$163$			6.04354i			0.473367i	0.971587	+	0.236683i	$0.0760604\pi$
$163$			6.04354i			0.473367i	−0.971587	+	0.236683i	$0.923940\pi$
$164$		0			0
$165$	−2.21983			−0.172814
$166$		0			0
$167$		−	7.92154i		−	0.612987i	−0.951873	−	0.306494i	$-0.900844\pi$
$167$		−	7.92154i		−	0.612987i	0.951873	−	0.306494i	$-0.0991558\pi$
$168$		0			0
$169$	−12.9758	+	0.792249i	−0.998141	+	0.0609422i
$170$		0			0
$171$			1.60388i			0.122651i
$172$		0			0
$173$	−10.9879			−0.835396			−0.417698	−	0.908586i	$-0.637163\pi$
$173$	−10.9879			−0.835396			−0.417698	+	0.908586i	$0.637163\pi$
$174$		0			0
$175$		−	6.74871i		−	0.510154i
$176$		0			0
$177$		−	0.713792i		−	0.0536519i
$178$		0			0
$179$	−5.42758			−0.405677			−0.202838	−	0.979212i	$-0.565017\pi$
$179$	−5.42758			−0.405677			−0.202838	+	0.979212i	$0.565017\pi$
$180$		0			0
$181$	−0.572417			−0.0425474			−0.0212737	−	0.999774i	$-0.506772\pi$
$181$	−0.572417			−0.0425474			−0.0212737	+	0.999774i	$0.506772\pi$
$182$		0			0
$183$	−8.76809			−0.648156
$184$		0			0
$185$	−0.879330			−0.0646496
$186$		0			0
$187$			4.98792i			0.364753i
$188$		0			0
$189$			1.60388i			0.116665i
$190$		0			0
$191$	−9.42758			−0.682156			−0.341078	−	0.940035i	$-0.610792\pi$
$191$	−9.42758			−0.682156			−0.341078	+	0.940035i	$0.610792\pi$
$192$		0			0
$193$			17.9758i			1.29393i	0.762520	+	0.646965i	$0.223962\pi$
$193$			17.9758i			1.29393i	−0.762520	+	0.646965i	$0.776038\pi$
$194$		0			0
$195$	−3.20775	+	0.0978347i	−0.229712	+	0.00700609i
$196$		0			0
$197$		−	19.7453i		−	1.40679i	−0.710799	−	0.703395i	$-0.751666\pi$
$197$		−	19.7453i		−	1.40679i	0.710799	−	0.703395i	$-0.248334\pi$
$198$		0			0
$199$	12.0000			0.850657			0.425329	−	0.905039i	$-0.360158\pi$
$199$	12.0000			0.850657			0.425329	+	0.905039i	$0.360158\pi$
$200$		0			0
$201$		−	7.82371i		−	0.551842i
$202$		0			0
$203$		−	11.2078i		−	0.786630i
$204$		0			0
$205$	−2.37675			−0.165999
$206$		0			0
$207$	−4.98792			−0.346684
$208$		0			0
$209$	−4.00000			−0.276686
$210$		0			0
$211$	17.5362			1.20724			0.603620	−	0.797272i	$-0.293724\pi$
$211$	17.5362			1.20724			0.603620	+	0.797272i	$0.293724\pi$
$212$		0			0
$213$			9.70171i			0.664750i
$214$		0			0
$215$		−	7.29483i		−	0.497503i
$216$		0			0
$217$	−5.42758			−0.368448
$218$		0			0
$219$			1.78017i			0.120293i
$220$		0			0
$221$	0.219833	+	7.20775i	0.0147875	+	0.484846i
$222$		0			0
$223$			16.1763i			1.08324i	0.840622	+	0.541622i	$0.182190\pi$
$223$			16.1763i			1.08324i	−0.840622	+	0.541622i	$0.817810\pi$
$224$		0			0
$225$	4.20775			0.280517
$226$		0			0
$227$			12.4698i			0.827649i	0.910357	+	0.413825i	$0.135807\pi$
$227$			12.4698i			0.827649i	−0.910357	+	0.413825i	$0.864193\pi$
$228$		0			0
$229$			14.3284i			0.946849i	0.880834	+	0.473424i	$0.156982\pi$
$229$			14.3284i			0.946849i	−0.880834	+	0.473424i	$0.843018\pi$
$230$		0			0
$231$	−4.00000			−0.263181
$232$		0			0
$233$	−10.4397			−0.683925			−0.341963	−	0.939714i	$-0.611092\pi$
$233$	−10.4397			−0.683925			−0.341963	+	0.939714i	$0.611092\pi$
$234$		0			0
$235$	−1.34050			−0.0874447
$236$		0			0
$237$	−10.4155			−0.676560
$238$		0			0
$239$			9.15346i			0.592088i	0.955174	+	0.296044i	$0.0956675\pi$
$239$			9.15346i			0.592088i	−0.955174	+	0.296044i	$0.904333\pi$
$240$		0			0
$241$		−	12.6353i		−	0.813913i	−0.913447	−	0.406957i	$-0.866590\pi$
$241$		−	12.6353i		−	0.813913i	0.913447	−	0.406957i	$-0.133410\pi$
$242$		0			0
$243$	−1.00000			−0.0641500
$244$		0			0
$245$		−	3.94092i		−	0.251776i
$246$		0			0
$247$	−5.78017	+	0.176292i	−0.367783	+	0.0112172i
$248$		0			0
$249$		−	12.2741i		−	0.777841i
$250$		0			0
$251$	−25.5362			−1.61183			−0.805914	−	0.592032i	$-0.798326\pi$
$251$	−25.5362			−1.61183			−0.805914	+	0.592032i	$0.798326\pi$
$252$		0			0
$253$		−	12.4397i		−	0.782075i
$254$		0			0
$255$			1.78017i			0.111478i
$256$		0			0
$257$	−22.3913			−1.39673			−0.698367	−	0.715740i	$-0.746089\pi$
$257$	−22.3913			−1.39673			−0.698367	+	0.715740i	$0.746089\pi$
$258$		0			0
$259$	−1.58450			−0.0984559
$260$		0			0
$261$	6.98792			0.432541
$262$		0			0
$263$	−3.56033			−0.219540			−0.109770	−	0.993957i	$-0.535011\pi$
$263$	−3.56033			−0.219540			−0.109770	+	0.993957i	$0.535011\pi$
$264$		0			0
$265$		−	3.05084i		−	0.187411i
$266$		0			0
$267$		−	15.3056i		−	0.936687i
$268$		0			0
$269$	−17.4034			−1.06111			−0.530553	−	0.847652i	$-0.678016\pi$
$269$	−17.4034			−1.06111			−0.530553	+	0.847652i	$0.678016\pi$
$270$		0			0
$271$			12.8116i			0.778251i	0.921185	+	0.389125i	$0.127223\pi$
$271$			12.8116i			0.778251i	−0.921185	+	0.389125i	$0.872777\pi$
$272$		0			0
$273$	−5.78017	+	0.176292i	−0.349832	+	0.0106697i
$274$		0			0
$275$			10.4940i			0.632810i
$276$		0			0
$277$	−12.8552			−0.772392			−0.386196	−	0.922417i	$-0.626211\pi$
$277$	−12.8552			−0.772392			−0.386196	+	0.922417i	$0.626211\pi$
$278$		0			0
$279$		−	3.38404i		−	0.202597i
$280$		0			0
$281$			23.5013i			1.40197i	0.713177	+	0.700984i	$0.247255\pi$
$281$			23.5013i			1.40197i	−0.713177	+	0.700984i	$0.752745\pi$
$282$		0			0
$283$	18.4155			1.09469			0.547344	−	0.836908i	$-0.315639\pi$
$283$	18.4155			1.09469			0.547344	+	0.836908i	$0.315639\pi$
$284$		0			0
$285$	−1.42758			−0.0845627
$286$		0			0
$287$	−4.28275			−0.252803
$288$		0			0
$289$	−13.0000			−0.764706
$290$		0			0
$291$			9.78017i			0.573324i
$292$		0			0
$293$		−	23.3056i		−	1.36153i	−0.732503	−	0.680763i	$-0.761648\pi$
$293$		−	23.3056i		−	1.36153i	0.732503	−	0.680763i	$-0.238352\pi$
$294$		0			0
$295$	0.635334			0.0369906
$296$		0			0
$297$		−	2.49396i		−	0.144714i
$298$		0			0
$299$	−0.548253	−	17.9758i	−0.0317063	−	1.03957i
$300$		0			0
$301$		−	13.1448i		−	0.757656i
$302$		0			0
$303$	6.54825			0.376187
$304$		0			0
$305$		−	7.80433i		−	0.446875i
$306$		0			0
$307$			11.7754i			0.672057i	0.941852	+	0.336028i	$0.109084\pi$
$307$			11.7754i			0.672057i	−0.941852	+	0.336028i	$0.890916\pi$
$308$		0			0
$309$	−12.6353			−0.718799
$310$		0			0
$311$	14.5724			0.826326			0.413163	−	0.910657i	$-0.364424\pi$
$311$	14.5724			0.826326			0.413163	+	0.910657i	$0.364424\pi$
$312$		0			0
$313$	−20.1715			−1.14016			−0.570080	−	0.821589i	$-0.693088\pi$
$313$	−20.1715			−1.14016			−0.570080	+	0.821589i	$0.693088\pi$
$314$		0			0
$315$	−1.42758			−0.0804352
$316$		0			0
$317$			14.2306i			0.799269i	0.916674	+	0.399635i	$0.130863\pi$
$317$			14.2306i			0.799269i	−0.916674	+	0.399635i	$0.869137\pi$
$318$		0			0
$319$			17.4276i			0.975757i
$320$		0			0
$321$	8.98792			0.501657
$322$		0			0
$323$			3.20775i			0.178484i
$324$		0			0
$325$	0.462500	+	15.1642i	0.0256549	+	0.841159i
$326$		0			0
$327$		−	7.20775i		−	0.398590i
$328$		0			0
$329$	−2.41550			−0.133171
$330$		0			0
$331$		−	35.4228i		−	1.94701i	−0.228659	−	0.973507i	$-0.573434\pi$
$331$		−	35.4228i		−	1.94701i	0.228659	−	0.973507i	$-0.426566\pi$
$332$		0			0
$333$		−	0.987918i		−	0.0541376i
$334$		0			0
$335$	6.96376			0.380471
$336$		0			0
$337$	−3.78017			−0.205919			−0.102959	−	0.994686i	$-0.532831\pi$
$337$	−3.78017			−0.205919			−0.102959	+	0.994686i	$0.532831\pi$
$338$		0			0
$339$	−5.56033			−0.301996
$340$		0			0
$341$	8.43967			0.457033
$342$		0			0
$343$		−	18.3284i		−	0.989642i
$344$		0			0
$345$		−	4.43967i		−	0.239024i
$346$		0			0
$347$	21.9758			1.17972			0.589862	−	0.807504i	$-0.299182\pi$
$347$	21.9758			1.17972			0.589862	+	0.807504i	$0.299182\pi$
$348$		0			0
$349$			24.9879i			1.33757i	0.743455	+	0.668786i	$0.233186\pi$
$349$			24.9879i			1.33757i	−0.743455	+	0.668786i	$0.766814\pi$
$350$		0			0
$351$	−0.109916	−	3.60388i	−0.00586690	−	0.192361i
$352$		0			0
$353$		−	5.13408i		−	0.273259i	−0.990622	−	0.136630i	$-0.956373\pi$
$353$		−	5.13408i		−	0.273259i	0.990622	−	0.136630i	$-0.0436271\pi$
$354$		0			0
$355$	−8.63533			−0.458316
$356$		0			0
$357$			3.20775i			0.169772i
$358$		0			0
$359$			4.90946i			0.259111i	0.991572	+	0.129556i	$0.0413551\pi$
$359$			4.90946i			0.259111i	−0.991572	+	0.129556i	$0.958645\pi$
$360$		0			0
$361$	16.4276			0.864610
$362$		0			0
$363$	−4.78017			−0.250894
$364$		0			0
$365$	−1.58450			−0.0829364
$366$		0			0
$367$	3.75600			0.196062			0.0980309	−	0.995183i	$-0.468746\pi$
$367$	3.75600			0.196062			0.0980309	+	0.995183i	$0.468746\pi$
$368$		0			0
$369$		−	2.67025i		−	0.139008i
$370$		0			0
$371$		−	5.49742i		−	0.285412i
$372$		0			0
$373$	−14.1957			−0.735024			−0.367512	−	0.930019i	$-0.619790\pi$
$373$	−14.1957			−0.735024			−0.367512	+	0.930019i	$0.619790\pi$
$374$		0			0
$375$			8.19567i			0.423223i
$376$		0			0
$377$	0.768086	+	25.1836i	0.0395584	+	1.29702i
$378$		0			0
$379$		−	3.93230i		−	0.201988i	−0.994887	−	0.100994i	$-0.967798\pi$
$379$		−	3.93230i		−	0.201988i	0.994887	−	0.100994i	$-0.0322024\pi$
$380$		0			0
$381$	−4.00000			−0.204926
$382$		0			0
$383$			10.9336i			0.558682i	0.960192	+	0.279341i	$0.0901160\pi$
$383$			10.9336i			0.558682i	−0.960192	+	0.279341i	$0.909884\pi$
$384$		0			0
$385$		−	3.56033i		−	0.181451i
$386$		0			0
$387$	8.19567			0.416609
$388$		0			0
$389$	−1.40342			−0.0711562			−0.0355781	−	0.999367i	$-0.511327\pi$
$389$	−1.40342			−0.0711562			−0.0355781	+	0.999367i	$0.511327\pi$
$390$		0			0
$391$	−9.97584			−0.504500
$392$		0			0
$393$	−4.54825			−0.229429
$394$		0			0
$395$		−	9.27067i		−	0.466458i
$396$		0			0
$397$			2.96376i			0.148747i	0.997230	+	0.0743733i	$0.0236956\pi$
$397$			2.96376i			0.148747i	−0.997230	+	0.0743733i	$0.976304\pi$
$398$		0			0
$399$	−2.57242			−0.128782
$400$		0			0
$401$			21.3297i			1.06516i	0.846381	+	0.532578i	$0.178777\pi$
$401$			21.3297i			1.06516i	−0.846381	+	0.532578i	$0.821223\pi$
$402$		0			0
$403$	12.1957	−	0.371961i	0.607510	−	0.0185287i
$404$		0			0
$405$		−	0.890084i		−	0.0442286i
$406$		0			0
$407$	2.46383			0.122127
$408$		0			0
$409$		−	28.6353i		−	1.41593i	−0.706249	−	0.707963i	$-0.749614\pi$
$409$		−	28.6353i		−	1.41593i	0.706249	−	0.707963i	$-0.250386\pi$
$410$		0			0
$411$		−	9.08575i		−	0.448167i
$412$		0			0
$413$	1.14483			0.0563336
$414$		0			0
$415$	10.9250			0.536287
$416$		0			0
$417$	−6.21983			−0.304587
$418$		0			0
$419$	9.86725			0.482047			0.241023	−	0.970519i	$-0.422517\pi$
$419$	9.86725			0.482047			0.241023	+	0.970519i	$0.422517\pi$
$420$		0			0
$421$			22.5026i			1.09671i	0.836246	+	0.548354i	$0.184746\pi$
$421$			22.5026i			1.09671i	−0.836246	+	0.548354i	$0.815254\pi$
$422$		0			0
$423$		−	1.50604i		−	0.0732262i
$424$		0			0
$425$	8.41550			0.408212
$426$		0			0
$427$		−	14.0629i		−	0.680552i
$428$		0			0
$429$	8.98792	−	0.274127i	0.433941	−	0.0132350i
$430$		0			0
$431$			32.8611i			1.58287i	0.611256	+	0.791433i	$0.290664\pi$
$431$			32.8611i			1.58287i	−0.611256	+	0.791433i	$0.709336\pi$
$432$		0			0
$433$	2.79225			0.134187			0.0670935	−	0.997747i	$-0.478627\pi$
$433$	2.79225			0.134187			0.0670935	+	0.997747i	$0.478627\pi$
$434$		0			0
$435$			6.21983i			0.298218i
$436$		0			0
$437$		−	8.00000i		−	0.382692i
$438$		0			0
$439$	−7.80433			−0.372480			−0.186240	−	0.982504i	$-0.559630\pi$
$439$	−7.80433			−0.372480			−0.186240	+	0.982504i	$0.559630\pi$
$440$		0			0
$441$	4.42758			0.210837
$442$		0			0
$443$	−3.84309			−0.182590			−0.0912952	−	0.995824i	$-0.529101\pi$
$443$	−3.84309			−0.182590			−0.0912952	+	0.995824i	$0.529101\pi$
$444$		0			0
$445$	13.6233			0.645804
$446$		0			0
$447$			15.1099i			0.714675i
$448$		0			0
$449$		−	27.0616i		−	1.27712i	−0.769574	−	0.638558i	$-0.779531\pi$
$449$		−	27.0616i		−	1.27712i	0.769574	−	0.638558i	$-0.220469\pi$
$450$		0			0
$451$	6.65950			0.313583
$452$		0			0
$453$			21.3599i			1.00357i
$454$		0			0
$455$	−0.156915	−	5.14483i	−0.00735627	−	0.241193i
$456$		0			0
$457$		−	25.0750i		−	1.17296i	−0.809964	−	0.586480i	$-0.800513\pi$
$457$		−	25.0750i		−	1.17296i	0.809964	−	0.586480i	$-0.199487\pi$
$458$		0			0
$459$	−2.00000			−0.0933520
$460$		0			0
$461$		−	36.8418i		−	1.71589i	−0.513740	−	0.857946i	$-0.671740\pi$
$461$		−	36.8418i		−	1.71589i	0.513740	−	0.857946i	$-0.328260\pi$
$462$		0			0
$463$			4.81163i			0.223615i	0.993730	+	0.111808i	$0.0356640\pi$
$463$			4.81163i			0.223615i	−0.993730	+	0.111808i	$0.964336\pi$
$464$		0			0
$465$	3.01208			0.139682
$466$		0			0
$467$	−29.2707			−1.35449			−0.677243	−	0.735760i	$-0.736825\pi$
$467$	−29.2707			−1.35449			−0.677243	+	0.735760i	$0.736825\pi$
$468$		0			0
$469$	12.5483			0.579425
$470$		0			0
$471$	13.4034			0.617597
$472$		0			0
$473$			20.4397i			0.939817i
$474$		0			0
$475$			6.74871i			0.309652i
$476$		0			0
$477$	3.42758			0.156938
$478$		0			0
$479$			9.15346i			0.418232i	0.977891	+	0.209116i	$0.0670586\pi$
$479$			9.15346i			0.418232i	−0.977891	+	0.209116i	$0.932941\pi$
$480$		0			0
$481$	3.56033	−	0.108588i	0.162337	−	0.00495120i
$482$		0			0
$483$		−	8.00000i		−	0.364013i
$484$		0			0
$485$	−8.70517			−0.395281
$486$		0			0
$487$		−	5.84787i		−	0.264992i	−0.991184	−	0.132496i	$-0.957701\pi$
$487$		−	5.84787i		−	0.264992i	0.991184	−	0.132496i	$-0.0422992\pi$
$488$		0			0
$489$		−	6.04354i		−	0.273298i
$490$		0			0
$491$	27.2948			1.23180			0.615899	−	0.787825i	$-0.288793\pi$
$491$	27.2948			1.23180			0.615899	+	0.787825i	$0.288793\pi$
$492$		0			0
$493$	13.9758			0.629440
$494$		0			0
$495$	2.21983			0.0997741
$496$		0			0
$497$	−15.5603			−0.697976
$498$		0			0
$499$		−	9.05562i		−	0.405385i	−0.979242	−	0.202693i	$-0.935031\pi$
$499$		−	9.05562i		−	0.405385i	0.979242	−	0.202693i	$-0.0649692\pi$
$500$		0			0
$501$			7.92154i			0.353908i
$502$		0			0
$503$	−8.87933			−0.395910			−0.197955	−	0.980211i	$-0.563430\pi$
$503$	−8.87933			−0.395910			−0.197955	+	0.980211i	$0.563430\pi$
$504$		0			0
$505$			5.82849i			0.259365i
$506$		0			0
$507$	12.9758	−	0.792249i	0.576277	−	0.0351850i
$508$		0			0
$509$		−	6.91425i		−	0.306469i	−0.988190	−	0.153234i	$-0.951031\pi$
$509$		−	6.91425i		−	0.306469i	0.988190	−	0.153234i	$-0.0489689\pi$
$510$		0			0
$511$	−2.85517			−0.126305
$512$		0			0
$513$		−	1.60388i		−	0.0708128i
$514$		0			0
$515$		−	11.2465i		−	0.495580i
$516$		0			0
$517$	3.75600			0.165189
$518$		0			0
$519$	10.9879			0.482316
$520$		0			0
$521$	31.5362			1.38162			0.690812	−	0.723034i	$-0.257253\pi$
$521$	31.5362			1.38162			0.690812	+	0.723034i	$0.257253\pi$
$522$		0			0
$523$	−18.8767			−0.825419			−0.412710	−	0.910863i	$-0.635418\pi$
$523$	−18.8767			−0.825419			−0.412710	+	0.910863i	$0.635418\pi$
$524$		0			0
$525$			6.74871i			0.294538i
$526$		0			0
$527$		−	6.76809i		−	0.294822i
$528$		0			0
$529$	1.87933			0.0817100
$530$		0			0
$531$			0.713792i			0.0309759i
$532$		0			0
$533$	9.62325	−	0.293504i	0.416829	−	0.0127131i
$534$		0			0
$535$			8.00000i			0.345870i
$536$		0			0
$537$	5.42758			0.234218
$538$		0			0
$539$			11.0422i			0.475622i
$540$		0			0
$541$		−	19.1594i		−	0.823728i	−0.911245	−	0.411864i	$-0.864878\pi$
$541$		−	19.1594i		−	0.823728i	0.911245	−	0.411864i	$-0.135122\pi$
$542$		0			0
$543$	0.572417			0.0245647
$544$		0			0
$545$	6.41550			0.274810
$546$		0			0
$547$	−24.1957			−1.03453			−0.517266	−	0.855825i	$-0.673050\pi$
$547$	−24.1957			−1.03453			−0.517266	+	0.855825i	$0.673050\pi$
$548$		0			0
$549$	8.76809			0.374213
$550$		0			0
$551$			11.2078i			0.477466i
$552$		0			0
$553$		−	16.7052i		−	0.710376i
$554$		0			0
$555$	0.879330			0.0373255
$556$		0			0
$557$			29.5254i			1.25103i	0.780211	+	0.625516i	$0.215112\pi$
$557$			29.5254i			1.25103i	−0.780211	+	0.625516i	$0.784888\pi$
$558$		0			0
$559$	0.900837	+	29.5362i	0.0381014	+	1.24925i
$560$		0			0
$561$		−	4.98792i		−	0.210590i
$562$		0			0
$563$	3.84309			0.161967			0.0809834	−	0.996715i	$-0.474194\pi$
$563$	3.84309			0.161967			0.0809834	+	0.996715i	$0.474194\pi$
$564$		0			0
$565$		−	4.94916i		−	0.208213i
$566$		0			0
$567$		−	1.60388i		−	0.0673564i
$568$		0			0
$569$	17.5120			0.734142			0.367071	−	0.930193i	$-0.380361\pi$
$569$	17.5120			0.734142			0.367071	+	0.930193i	$0.380361\pi$
$570$		0			0
$571$	−46.3672			−1.94041			−0.970203	−	0.242294i	$-0.922100\pi$
$571$	−46.3672			−1.94041			−0.970203	+	0.242294i	$0.922100\pi$
$572$		0			0
$573$	9.42758			0.393843
$574$		0			0
$575$	−20.9879			−0.875257
$576$		0			0
$577$			45.9275i			1.91199i	0.293387	+	0.955994i	$0.405217\pi$
$577$			45.9275i			1.91199i	−0.293387	+	0.955994i	$0.594783\pi$
$578$		0			0
$579$		−	17.9758i		−	0.747050i
$580$		0			0
$581$	19.6862			0.816720
$582$		0			0
$583$			8.54825i			0.354032i
$584$		0			0
$585$	3.20775	−	0.0978347i	0.132624	−	0.00404497i
$586$		0			0
$587$		−	48.0689i		−	1.98402i	−0.126178	−	0.992008i	$-0.540271\pi$
$587$		−	48.0689i		−	1.98402i	0.126178	−	0.992008i	$-0.459729\pi$
$588$		0			0
$589$	5.42758			0.223640
$590$		0			0
$591$			19.7453i			0.812211i
$592$		0			0
$593$			2.47458i			0.101619i	0.998708	+	0.0508094i	$0.0161801\pi$
$593$			2.47458i			0.101619i	−0.998708	+	0.0508094i	$0.983820\pi$
$594$		0			0
$595$	−2.85517			−0.117050
$596$		0			0
$597$	−12.0000			−0.491127
$598$		0			0
$599$	34.3672			1.40420			0.702102	−	0.712076i	$-0.252245\pi$
$599$	34.3672			1.40420			0.702102	+	0.712076i	$0.252245\pi$
$600$		0			0
$601$	21.4905			0.876616			0.438308	−	0.898825i	$-0.355578\pi$
$601$	21.4905			0.876616			0.438308	+	0.898825i	$0.355578\pi$
$602$		0			0
$603$			7.82371i			0.318606i
$604$		0			0
$605$		−	4.25475i		−	0.172980i
$606$		0			0
$607$	−30.2198			−1.22659			−0.613293	−	0.789856i	$-0.710155\pi$
$607$	−30.2198			−1.22659			−0.613293	+	0.789856i	$0.710155\pi$
$608$		0			0
$609$			11.2078i			0.454161i
$610$		0			0
$611$	5.42758	−	0.165538i	0.219577	−	0.00669697i
$612$		0			0
$613$		−	40.4999i		−	1.63578i	−0.575377	−	0.817888i	$-0.695145\pi$
$613$		−	40.4999i		−	1.63578i	0.575377	−	0.817888i	$-0.304855\pi$
$614$		0			0
$615$	2.37675			0.0958397
$616$		0			0
$617$		−	15.1099i		−	0.608302i	−0.952624	−	0.304151i	$-0.901627\pi$
$617$		−	15.1099i		−	0.608302i	0.952624	−	0.304151i	$-0.0983728\pi$
$618$		0			0
$619$		−	30.5918i		−	1.22959i	−0.788688	−	0.614794i	$-0.789239\pi$
$619$		−	30.5918i		−	1.22959i	0.788688	−	0.614794i	$-0.210761\pi$
$620$		0			0
$621$	4.98792			0.200158
$622$		0			0
$623$	24.5483			0.983505
$624$		0			0
$625$	13.7439			0.549757
$626$		0			0
$627$	4.00000			0.159745
$628$		0			0
$629$		−	1.97584i		−	0.0787818i
$630$		0			0
$631$			29.7125i			1.18283i	0.806366	+	0.591417i	$0.201431\pi$
$631$			29.7125i			1.18283i	−0.806366	+	0.591417i	$0.798569\pi$
$632$		0			0
$633$	−17.5362			−0.697000
$634$		0			0
$635$		−	3.56033i		−	0.141288i
$636$		0			0
$637$	0.486663	+	15.9565i	0.0192823	+	0.632218i
$638$		0			0
$639$		−	9.70171i		−	0.383794i
$640$		0			0
$641$	42.3913			1.67436			0.837179	−	0.546930i	$-0.184203\pi$
$641$	42.3913			1.67436			0.837179	+	0.546930i	$0.184203\pi$
$642$		0			0
$643$		−	20.6160i		−	0.813014i	−0.913648	−	0.406507i	$-0.866747\pi$
$643$		−	20.6160i		−	0.813014i	0.913648	−	0.406507i	$-0.133253\pi$
$644$		0			0
$645$			7.29483i			0.287234i
$646$		0			0
$647$	−37.5362			−1.47570			−0.737850	−	0.674965i	$-0.764159\pi$
$647$	−37.5362			−1.47570			−0.737850	+	0.674965i	$0.764159\pi$
$648$		0			0
$649$	−1.78017			−0.0698777
$650$		0			0
$651$	5.42758			0.212724
$652$		0			0
$653$	−4.96376			−0.194247			−0.0971234	−	0.995272i	$-0.530964\pi$
$653$	−4.96376			−0.194247			−0.0971234	+	0.995272i	$0.530964\pi$
$654$		0			0
$655$		−	4.04833i		−	0.158181i
$656$		0			0
$657$		−	1.78017i		−	0.0694509i
$658$		0			0
$659$	47.9517			1.86793			0.933966	−	0.357362i	$-0.116324\pi$
$659$	47.9517			1.86793			0.933966	+	0.357362i	$0.116324\pi$
$660$		0			0
$661$			35.8431i			1.39413i	0.717006	+	0.697067i	$0.245512\pi$
$661$			35.8431i			1.39413i	−0.717006	+	0.697067i	$0.754488\pi$
$662$		0			0
$663$	−0.219833	−	7.20775i	−0.00853759	−	0.279926i
$664$		0			0
$665$		−	2.28967i		−	0.0887894i
$666$		0			0
$667$	−34.8552			−1.34960
$668$		0			0
$669$		−	16.1763i		−	0.625412i
$670$		0			0
$671$			21.8672i			0.844176i
$672$		0			0
$673$	38.4999			1.48406			0.742032	−	0.670365i	$-0.233862\pi$
$673$	38.4999			1.48406			0.742032	+	0.670365i	$0.233862\pi$
$674$		0			0
$675$	−4.20775			−0.161956
$676$		0			0
$677$	−15.8189			−0.607971			−0.303985	−	0.952677i	$-0.598317\pi$
$677$	−15.8189			−0.607971			−0.303985	+	0.952677i	$0.598317\pi$
$678$		0			0
$679$	−15.6862			−0.601980
$680$		0			0
$681$		−	12.4698i		−	0.477844i
$682$		0			0
$683$		−	20.8009i		−	0.795923i	−0.917402	−	0.397962i	$-0.869718\pi$
$683$		−	20.8009i		−	0.795923i	0.917402	−	0.397962i	$-0.130282\pi$
$684$		0			0
$685$	8.08708			0.308991
$686$		0			0
$687$		−	14.3284i		−	0.546663i
$688$		0			0
$689$	0.376747	+	12.3526i	0.0143529	+	0.470596i
$690$		0			0
$691$			38.4564i			1.46295i	0.681868	+	0.731475i	$0.261168\pi$
$691$			38.4564i			1.46295i	−0.681868	+	0.731475i	$0.738832\pi$
$692$		0			0
$693$	4.00000			0.151947
$694$		0			0
$695$		−	5.53617i		−	0.209999i
$696$		0			0
$697$		−	5.34050i		−	0.202286i
$698$		0			0
$699$	10.4397			0.394864
$700$		0			0
$701$	40.2586			1.52055			0.760273	−	0.649604i	$-0.225065\pi$
$701$	40.2586			1.52055			0.760273	+	0.649604i	$0.225065\pi$
$702$		0			0
$703$	1.58450			0.0597605
$704$		0			0
$705$	1.34050			0.0504862
$706$		0			0
$707$			10.5026i			0.394990i
$708$		0			0
$709$			4.35258i			0.163465i	0.996654	+	0.0817324i	$0.0260453\pi$
$709$			4.35258i			0.163465i	−0.996654	+	0.0817324i	$0.973955\pi$
$710$		0			0
$711$	10.4155			0.390612
$712$		0			0
$713$			16.8793i			0.632136i
$714$		0			0
$715$	0.243996	+	8.00000i	0.00912492	+	0.299183i
$716$		0			0
$717$		−	9.15346i		−	0.341842i
$718$		0			0
$719$	0.156915			0.00585193			0.00292596	−	0.999996i	$-0.499069\pi$
$719$	0.156915			0.00585193			0.00292596	+	0.999996i	$0.499069\pi$
$720$		0			0
$721$		−	20.2655i		−	0.754727i
$722$		0			0
$723$			12.6353i			0.469913i
$724$		0			0
$725$	29.4034			1.09202
$726$		0			0
$727$	−14.3672			−0.532849			−0.266425	−	0.963856i	$-0.585842\pi$
$727$	−14.3672			−0.532849			−0.266425	+	0.963856i	$0.585842\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$	16.3913			0.606256
$732$		0			0
$733$		−	17.5749i		−	0.649145i	−0.945861	−	0.324572i	$-0.894780\pi$
$733$		−	17.5749i		−	0.649145i	0.945861	−	0.324572i	$-0.105220\pi$
$734$		0			0
$735$			3.94092i			0.145363i
$736$		0			0
$737$	−19.5120			−0.718734
$738$		0			0
$739$		−	24.5676i		−	0.903735i	−0.892085	−	0.451867i	$-0.850758\pi$
$739$		−	24.5676i		−	0.903735i	0.892085	−	0.451867i	$-0.149242\pi$
$740$		0			0
$741$	5.78017	−	0.176292i	0.212340	−	0.00647625i
$742$		0			0
$743$		−	20.3827i		−	0.747769i	−0.927475	−	0.373885i	$-0.878026\pi$
$743$		−	20.3827i		−	0.747769i	0.927475	−	0.373885i	$-0.121974\pi$
$744$		0			0
$745$	−13.4491			−0.492737
$746$		0			0
$747$			12.2741i			0.449087i
$748$		0			0
$749$			14.4155i			0.526731i
$750$		0			0
$751$	−42.1715			−1.53886			−0.769430	−	0.638731i	$-0.779460\pi$
$751$	−42.1715			−1.53886			−0.769430	+	0.638731i	$0.779460\pi$
$752$		0			0
$753$	25.5362			0.930590
$754$		0			0
$755$	−19.0121			−0.691920
$756$		0			0
$757$	6.08708			0.221239			0.110619	−	0.993863i	$-0.464717\pi$
$757$	6.08708			0.221239			0.110619	+	0.993863i	$0.464717\pi$
$758$		0			0
$759$			12.4397i			0.451531i
$760$		0			0
$761$			13.5254i			0.490296i	0.969486	+	0.245148i	$0.0788366\pi$
$761$			13.5254i			0.490296i	−0.969486	+	0.245148i	$0.921163\pi$
$762$		0			0
$763$	11.5603			0.418512
$764$		0			0
$765$		−	1.78017i		−	0.0643621i
$766$		0			0
$767$	−2.57242	+	0.0784573i	−0.0928846	+	0.00283293i
$768$		0			0
$769$			5.53617i			0.199639i	0.995006	+	0.0998197i	$0.0318266\pi$
$769$			5.53617i			0.199639i	−0.995006	+	0.0998197i	$0.968173\pi$
$770$		0			0
$771$	22.3913			0.806404
$772$		0			0
$773$		−	21.1341i		−	0.760140i	−0.924958	−	0.380070i	$-0.875900\pi$
$773$		−	21.1341i		−	0.760140i	0.924958	−	0.380070i	$-0.124100\pi$
$774$		0			0
$775$		−	14.2392i		−	0.511488i
$776$		0			0
$777$	1.58450			0.0568436
$778$		0			0
$779$	4.28275			0.153445
$780$		0			0
$781$	24.1957			0.865789
$782$		0			0
$783$	−6.98792			−0.249728
$784$		0			0
$785$			11.9302i			0.425806i
$786$		0			0
$787$			35.3840i			1.26130i	0.776065	+	0.630652i	$0.217213\pi$
$787$			35.3840i			1.26130i	−0.776065	+	0.630652i	$0.782787\pi$
$788$		0			0
$789$	3.56033			0.126751
$790$		0			0
$791$		−	8.91808i		−	0.317091i
$792$		0			0
$793$	0.963755	+	31.5991i	0.0342240	+	1.12212i
$794$		0			0
$795$			3.05084i			0.108202i
$796$		0			0
$797$	4.52409			0.160251			0.0801257	−	0.996785i	$-0.474468\pi$
$797$	4.52409			0.160251			0.0801257	+	0.996785i	$0.474468\pi$
$798$		0			0
$799$		−	3.01208i		−	0.106560i
$800$		0			0
$801$			15.3056i			0.540796i
$802$		0			0
$803$	4.43967			0.156672
$804$		0			0
$805$	7.12067			0.250971
$806$		0			0
$807$	17.4034			0.612629
$808$		0			0
$809$	−23.0965			−0.812030			−0.406015	−	0.913866i	$-0.633082\pi$
$809$	−23.0965			−0.812030			−0.406015	+	0.913866i	$0.633082\pi$
$810$		0			0
$811$		−	49.2513i		−	1.72945i	−0.502248	−	0.864723i	$-0.667494\pi$
$811$		−	49.2513i		−	1.72945i	0.502248	−	0.864723i	$-0.332506\pi$
$812$		0			0
$813$		−	12.8116i		−	0.449323i
$814$		0			0
$815$	5.37926			0.188427
$816$		0			0
$817$			13.1448i			0.459879i
$818$		0			0
$819$	5.78017	−	0.176292i	0.201975	−	0.00616014i
$820$		0			0
$821$		−	43.0616i		−	1.50286i	−0.659813	−	0.751430i	$-0.729364\pi$
$821$		−	43.0616i		−	1.50286i	0.659813	−	0.751430i	$-0.270636\pi$
$822$		0			0
$823$	37.7318			1.31525			0.657625	−	0.753346i	$-0.271561\pi$
$823$	37.7318			1.31525			0.657625	+	0.753346i	$0.271561\pi$
$824$		0			0
$825$		−	10.4940i		−	0.365353i
$826$		0			0
$827$			31.3250i			1.08928i	0.838671	+	0.544638i	$0.183333\pi$
$827$			31.3250i			1.08928i	−0.838671	+	0.544638i	$0.816667\pi$
$828$		0			0
$829$	−43.5749			−1.51342			−0.756710	−	0.653751i	$-0.773194\pi$
$829$	−43.5749			−1.51342			−0.756710	+	0.653751i	$0.773194\pi$
$830$		0			0
$831$	12.8552			0.445941
$832$		0			0
$833$	8.85517			0.306813
$834$		0			0
$835$	−7.05084			−0.244004
$836$		0			0
$837$			3.38404i			0.116970i
$838$		0			0
$839$		−	11.2862i		−	0.389643i	−0.980839	−	0.194822i	$-0.937587\pi$
$839$		−	11.2862i		−	0.389643i	0.980839	−	0.194822i	$-0.0624128\pi$
$840$		0			0
$841$	19.8310			0.683828
$842$		0			0
$843$		−	23.5013i		−	0.809426i
$844$		0			0
$845$	0.705168	+	11.5496i	0.0242585	+	0.397318i
$846$		0			0
$847$		−	7.66679i		−	0.263434i
$848$		0			0
$849$	−18.4155			−0.632018
$850$		0			0
$851$			4.92766i			0.168918i
$852$		0			0
$853$			46.9154i			1.60635i	0.595741	+	0.803177i	$0.296859\pi$
$853$			46.9154i			1.60635i	−0.595741	+	0.803177i	$0.703141\pi$
$854$		0			0
$855$	1.42758			0.0488223
$856$		0			0
$857$	−55.6620			−1.90138			−0.950689	−	0.310146i	$-0.899622\pi$
$857$	−55.6620			−1.90138			−0.950689	+	0.310146i	$0.899622\pi$
$858$		0			0
$859$	4.39134			0.149831			0.0749153	−	0.997190i	$-0.476131\pi$
$859$	4.39134			0.149831			0.0749153	+	0.997190i	$0.476131\pi$
$860$		0			0
$861$	4.28275			0.145956
$862$		0			0
$863$			20.0301i			0.681834i	0.940093	+	0.340917i	$0.110737\pi$
$863$			20.0301i			0.681834i	−0.940093	+	0.340917i	$0.889263\pi$
$864$		0			0
$865$			9.78017i			0.332536i
$866$		0			0
$867$	13.0000			0.441503
$868$		0			0
$869$			25.9758i			0.881170i
$870$		0			0
$871$	−28.1957	+	0.859953i	−0.955374	+	0.0291384i
$872$		0			0
$873$		−	9.78017i		−	0.331009i
$874$		0			0
$875$	−13.1448			−0.444376
$876$		0			0
$877$		−	35.1379i		−	1.18652i	−0.805010	−	0.593262i	$-0.797840\pi$
$877$		−	35.1379i		−	1.18652i	0.805010	−	0.593262i	$-0.202160\pi$
$878$		0			0
$879$			23.3056i			0.786078i
$880$		0			0
$881$	7.53617			0.253900			0.126950	−	0.991909i	$-0.459481\pi$
$881$	7.53617			0.253900			0.126950	+	0.991909i	$0.459481\pi$
$882$		0			0
$883$	−18.0242			−0.606561			−0.303281	−	0.952901i	$-0.598082\pi$
$883$	−18.0242			−0.606561			−0.303281	+	0.952901i	$0.598082\pi$
$884$		0			0
$885$	−0.635334			−0.0213565
$886$		0			0
$887$	31.8431			1.06919			0.534593	−	0.845110i	$-0.320465\pi$
$887$	31.8431			1.06919			0.534593	+	0.845110i	$0.320465\pi$
$888$		0			0
$889$		−	6.41550i		−	0.215169i
$890$		0			0
$891$			2.49396i			0.0835508i
$892$		0			0
$893$	2.41550			0.0808317
$894$		0			0
$895$			4.83100i			0.161483i
$896$		0			0
$897$	0.548253	+	17.9758i	0.0183056	+	0.600196i
$898$		0			0
$899$		−	23.6474i		−	0.788685i
$900$		0			0
$901$	6.85517			0.228379
$902$		0			0
$903$			13.1448i			0.437433i
$904$		0			0
$905$			0.509499i			0.0169363i
$906$		0			0
$907$	3.60866			0.119824			0.0599118	−	0.998204i	$-0.480918\pi$
$907$	3.60866			0.119824			0.0599118	+	0.998204i	$0.480918\pi$
$908$		0			0
$909$	−6.54825			−0.217192
$910$		0			0
$911$	46.9638			1.55598			0.777989	−	0.628278i	$-0.216240\pi$
$911$	46.9638			1.55598			0.777989	+	0.628278i	$0.216240\pi$
$912$		0			0
$913$	−30.6112			−1.01308
$914$		0			0
$915$			7.80433i			0.258003i
$916$		0			0
$917$		−	7.29483i		−	0.240897i
$918$		0			0
$919$	−43.5120			−1.43533			−0.717664	−	0.696389i	$-0.754789\pi$
$919$	−43.5120			−1.43533			−0.717664	+	0.696389i	$0.754789\pi$
$920$		0			0
$921$		−	11.7754i		−	0.388012i
$922$		0			0
$923$	34.9638	−	1.06638i	1.15085	−	0.0351002i
$924$		0			0
$925$		−	4.15691i		−	0.136679i
$926$		0			0
$927$	12.6353			0.414999
$928$		0			0
$929$		−	41.9651i		−	1.37683i	−0.725317	−	0.688415i	$-0.758307\pi$
$929$		−	41.9651i		−	1.37683i	0.725317	−	0.688415i	$-0.241693\pi$
$930$		0			0
$931$			7.10129i			0.232735i
$932$		0			0
$933$	−14.5724			−0.477079
$934$		0			0
$935$	4.43967			0.145193
$936$		0			0
$937$	−1.25608			−0.0410343			−0.0205171	−	0.999790i	$-0.506531\pi$
$937$	−1.25608			−0.0410343			−0.0205171	+	0.999790i	$0.506531\pi$
$938$		0			0
$939$	20.1715			0.658272
$940$		0			0
$941$			20.4504i			0.666665i	0.942809	+	0.333332i	$0.108173\pi$
$941$			20.4504i			0.666665i	−0.942809	+	0.333332i	$0.891827\pi$
$942$		0			0
$943$			13.3190i			0.433726i
$944$		0			0
$945$	1.42758			0.0464393
$946$		0			0
$947$			37.9361i			1.23276i	0.787449	+	0.616379i	$0.211401\pi$
$947$			37.9361i			1.23276i	−0.787449	+	0.616379i	$0.788599\pi$
$948$		0			0
$949$	6.41550	−	0.195669i	0.208256	−	0.00635170i
$950$		0			0
$951$		−	14.2306i		−	0.461458i
$952$		0			0
$953$	21.9517			0.711084			0.355542	−	0.934660i	$-0.384296\pi$
$953$	21.9517			0.711084			0.355542	+	0.934660i	$0.384296\pi$
$954$		0			0
$955$			8.39134i			0.271537i
$956$		0			0
$957$		−	17.4276i		−	0.563354i
$958$		0			0
$959$	14.5724			0.470568
$960$		0			0
$961$	19.5483			0.630589
$962$		0			0
$963$	−8.98792			−0.289632
$964$		0			0
$965$	16.0000			0.515058
$966$		0			0
$967$		−	59.0702i		−	1.89957i	−0.312903	−	0.949785i	$-0.601302\pi$
$967$		−	59.0702i		−	1.89957i	0.312903	−	0.949785i	$-0.398698\pi$
$968$		0			0
$969$		−	3.20775i		−	0.103048i
$970$		0			0
$971$	−32.3430			−1.03794			−0.518968	−	0.854793i	$-0.673684\pi$
$971$	−32.3430			−1.03794			−0.518968	+	0.854793i	$0.673684\pi$
$972$		0			0
$973$		−	9.97584i		−	0.319811i
$974$		0			0
$975$	−0.462500	−	15.1642i	−0.0148119	−	0.485643i
$976$		0			0
$977$		−	47.6969i		−	1.52596i	−0.646422	−	0.762980i	$-0.723736\pi$
$977$		−	47.6969i		−	1.52596i	0.646422	−	0.762980i	$-0.276264\pi$
$978$		0			0
$979$	−38.1715			−1.21997
$980$		0			0
$981$			7.20775i			0.230126i
$982$		0			0
$983$			19.1293i			0.610130i	0.952332	+	0.305065i	$0.0986781\pi$
$983$			19.1293i			0.610130i	−0.952332	+	0.305065i	$0.901322\pi$
$984$		0			0
$985$	−17.5749			−0.559984
$986$		0			0
$987$	2.41550			0.0768863
$988$		0			0
$989$	−40.8793			−1.29989
$990$		0			0
$991$	−19.3647			−0.615139			−0.307569	−	0.951526i	$-0.599516\pi$
$991$	−19.3647			−0.615139			−0.307569	+	0.951526i	$0.599516\pi$
$992$		0			0
$993$			35.4228i			1.12411i
$994$		0			0
$995$		−	10.6810i		−	0.338611i
$996$		0			0
$997$	−46.8310			−1.48315			−0.741576	−	0.670868i	$-0.765922\pi$
$997$	−46.8310			−1.48315			−0.741576	+	0.670868i	$0.765922\pi$
$998$		0			0
$999$			0.987918i			0.0312564i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1248.2.c.a.961.3	✓	6
3.2	odd	2		3744.2.c.l.3457.4		6
4.3	odd	2		1248.2.c.b.961.3	yes	6
8.3	odd	2		2496.2.c.o.961.4		6
8.5	even	2		2496.2.c.p.961.4		6
12.11	even	2		3744.2.c.m.3457.4		6
13.12	even	2	inner	1248.2.c.a.961.4	yes	6
39.38	odd	2		3744.2.c.l.3457.3		6
52.51	odd	2		1248.2.c.b.961.4	yes	6
104.51	odd	2		2496.2.c.o.961.3		6
104.77	even	2		2496.2.c.p.961.3		6
156.155	even	2		3744.2.c.m.3457.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1248.2.c.a.961.3	✓	6	1.1	even	1	trivial
1248.2.c.a.961.4	yes	6	13.12	even	2	inner
1248.2.c.b.961.3	yes	6	4.3	odd	2
1248.2.c.b.961.4	yes	6	52.51	odd	2
2496.2.c.o.961.3		6	104.51	odd	2
2496.2.c.o.961.4		6	8.3	odd	2
2496.2.c.p.961.3		6	104.77	even	2
2496.2.c.p.961.4		6	8.5	even	2
3744.2.c.l.3457.3		6	39.38	odd	2
3744.2.c.l.3457.4		6	3.2	odd	2
3744.2.c.m.3457.3		6	156.155	even	2
3744.2.c.m.3457.4		6	12.11	even	2

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:	\( N \)	\(=\)	\( 1248 = 2^{5} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1248.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -0.890084i q^{5} -1.60388i q^{7} +1.00000 q^{9} +2.49396i q^{11} +(0.109916 + 3.60388i) q^{13} +0.890084i q^{15} +2.00000 q^{17} +1.60388i q^{19} +1.60388i q^{21} -4.98792 q^{23} +4.20775 q^{25} -1.00000 q^{27} +6.98792 q^{29} -3.38404i q^{31} -2.49396i q^{33} -1.42758 q^{35} -0.987918i q^{37} +(-0.109916 - 3.60388i) q^{39} -2.67025i q^{41} +8.19567 q^{43} -0.890084i q^{45} -1.50604i q^{47} +4.42758 q^{49} -2.00000 q^{51} +3.42758 q^{53} +2.21983 q^{55} -1.60388i q^{57} +0.713792i q^{59} +8.76809 q^{61} -1.60388i q^{63} +(3.20775 - 0.0978347i) q^{65} +7.82371i q^{67} +4.98792 q^{69} -9.70171i q^{71} -1.78017i q^{73} -4.20775 q^{75} +4.00000 q^{77} +10.4155 q^{79} +1.00000 q^{81} +12.2741i q^{83} -1.78017i q^{85} -6.98792 q^{87} +15.3056i q^{89} +(5.78017 - 0.176292i) q^{91} +3.38404i q^{93} +1.42758 q^{95} -9.78017i q^{97} +2.49396i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{3} + 6 q^{9} + 2 q^{13} + 12 q^{17} + 8 q^{23} - 10 q^{25} - 6 q^{27} + 4 q^{29} + 24 q^{35} - 2 q^{39} - 24 q^{43} - 6 q^{49} - 12 q^{51} - 12 q^{53} + 16 q^{55} + 12 q^{61} - 16 q^{65} - 8 q^{69}+ \cdots - 24 q^{95}+O(q^{100}) \) 6 * q - 6 * q^3 + 6 * q^9 + 2 * q^13 + 12 * q^17 + 8 * q^23 - 10 * q^25 - 6 * q^27 + 4 * q^29 + 24 * q^35 - 2 * q^39 - 24 * q^43 - 6 * q^49 - 12 * q^51 - 12 * q^53 + 16 * q^55 + 12 * q^61 - 16 * q^65 - 8 * q^69 + 10 * q^75 + 24 * q^77 - 8 * q^79 + 6 * q^81 - 4 * q^87 + 32 * q^91 - 24 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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