LMFDB - Embedded newform 2496.2.c.r.961.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2496,2,Mod(961,2496)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2496, 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("2496.961"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 2496 = 2^{6} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2496.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,8,0,0,0,0,0,8,0,0,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	no
Analytic conductor:	$19.9306603445$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.134560000.4
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 7x^{6} + 13x^{4} + 7x^{2} + 1 $ x^8 + 7x^6 + 13x^4 + 7*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{11} $
Twist minimal:	no (minimal twist has level 1248)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			961.5
Root			$0.477260i$ of defining polynomial
Character	$\chi$	$=$	2496.961
Dual form			2496.2.c.r.961.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +0.954520i q^{5} -4.78051i q^{7} +1.00000 q^{9} -2.64614i q^{11} +(1.58993 + 3.23607i) q^{13} +0.954520i q^{15} +4.47214 q^{17} +2.30838i q^{19} -4.78051i q^{21} -8.38118 q^{23} +4.08889 q^{25} +1.00000 q^{27} +3.90904 q^{29} -2.87147i q^{31} -2.64614i q^{33} +4.56310 q^{35} -8.38118i q^{37} +(1.58993 + 3.23607i) q^{39} +4.95452i q^{41} +7.65199 q^{43} +0.954520i q^{45} -10.2981i q^{47} -15.8533 q^{49} +4.47214 q^{51} -10.3812 q^{53} +2.52580 q^{55} +2.30838i q^{57} -4.55518i q^{59} +14.0332 q^{61} -4.78051i q^{63} +(-3.08889 + 1.51762i) q^{65} +0.399338i q^{67} -8.38118 q^{69} -3.44482i q^{71} -12.2687i q^{73} +4.08889 q^{75} -12.6499 q^{77} -12.7624 q^{79} +1.00000 q^{81} -0.737102i q^{83} +4.26874i q^{85} +3.90904 q^{87} -11.8078i q^{89} +(15.4701 - 7.60066i) q^{91} -2.87147i q^{93} -2.20339 q^{95} -6.09096i q^{97} -2.64614i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{3} + 8 q^{9} - 8 q^{23} - 16 q^{25} + 8 q^{27} + 8 q^{29} + 24 q^{35} - 32 q^{49} - 24 q^{53} - 16 q^{55} - 8 q^{61} + 24 q^{65} - 8 q^{69} - 16 q^{75} + 32 q^{77} + 16 q^{79} + 8 q^{81} + 8 q^{87}+ \cdots - 56 q^{95}+O(q^{100}) $ 8 * q + 8 * q^3 + 8 * q^9 - 8 * q^23 - 16 * q^25 + 8 * q^27 + 8 * q^29 + 24 * q^35 - 32 * q^49 - 24 * q^53 - 16 * q^55 - 8 * q^61 + 24 * q^65 - 8 * q^69 - 16 * q^75 + 32 * q^77 + 16 * q^79 + 8 * q^81 + 8 * q^87 + 16 * q^91 - 56 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2496\mathbb{Z}\right)^\times$.

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

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
$5$			0.954520i			0.426874i	0.976957	+	0.213437i	$0.0684659\pi$
$5$			0.954520i			0.426874i	−0.976957	+	0.213437i	$0.931534\pi$
$6$		0			0
$7$		−	4.78051i		−	1.80686i	−0.428731	−	0.903432i	$-0.641039\pi$
$7$		−	4.78051i		−	1.80686i	0.428731	−	0.903432i	$-0.358961\pi$
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$		−	2.64614i		−	0.797842i	−0.916985	−	0.398921i	$-0.869385\pi$
$11$		−	2.64614i		−	0.797842i	0.916985	−	0.398921i	$-0.130615\pi$
$12$		0			0
$13$	1.58993	+	3.23607i	0.440966	+	0.897524i
$13$	1.58993	+	3.23607i	0.440966	+	0.897524i
$14$		0			0
$15$			0.954520i			0.246456i
$16$		0			0
$17$	4.47214			1.08465			0.542326	−	0.840168i	$-0.317544\pi$
$17$	4.47214			1.08465			0.542326	+	0.840168i	$0.317544\pi$
$18$		0			0
$19$			2.30838i			0.529578i	0.964306	+	0.264789i	$0.0853023\pi$
$19$			2.30838i			0.529578i	−0.964306	+	0.264789i	$0.914698\pi$
$20$		0			0
$21$		−	4.78051i		−	1.04319i
$22$		0			0
$23$	−8.38118			−1.74760			−0.873798	−	0.486289i	$-0.838350\pi$
$23$	−8.38118			−1.74760			−0.873798	+	0.486289i	$0.838350\pi$
$24$		0			0
$25$	4.08889			0.817778
$26$		0			0
$27$	1.00000			0.192450
$28$		0			0
$29$	3.90904			0.725891			0.362945	−	0.931810i	$-0.381771\pi$
$29$	3.90904			0.725891			0.362945	+	0.931810i	$0.381771\pi$
$30$		0			0
$31$		−	2.87147i		−	0.515732i	−0.966181	−	0.257866i	$-0.916981\pi$
$31$		−	2.87147i		−	0.515732i	0.966181	−	0.257866i	$-0.0830193\pi$
$32$		0			0
$33$		−	2.64614i		−	0.460634i
$34$		0			0
$35$	4.56310			0.771304
$36$		0			0
$37$		−	8.38118i		−	1.37786i	−0.724829	−	0.688928i	$-0.758081\pi$
$37$		−	8.38118i		−	1.37786i	0.724829	−	0.688928i	$-0.241919\pi$
$38$		0			0
$39$	1.58993	+	3.23607i	0.254592	+	0.518186i
$40$		0			0
$41$			4.95452i			0.773766i	0.922129	+	0.386883i	$0.126448\pi$
$41$			4.95452i			0.773766i	−0.922129	+	0.386883i	$0.873552\pi$
$42$		0			0
$43$	7.65199			1.16692			0.583459	−	0.812143i	$-0.301699\pi$
$43$	7.65199			1.16692			0.583459	+	0.812143i	$0.301699\pi$
$44$		0			0
$45$			0.954520i			0.142291i
$46$		0			0
$47$		−	10.2981i		−	1.50214i	−0.660225	−	0.751068i	$-0.729539\pi$
$47$		−	10.2981i		−	1.50214i	0.660225	−	0.751068i	$-0.270461\pi$
$48$		0			0
$49$	−15.8533			−2.26476
$50$		0			0
$51$	4.47214			0.626224
$52$		0			0
$53$	−10.3812			−1.42596			−0.712982	−	0.701182i	$-0.752656\pi$
$53$	−10.3812			−1.42596			−0.712982	+	0.701182i	$0.752656\pi$
$54$		0			0
$55$	2.52580			0.340578
$56$		0			0
$57$			2.30838i			0.305752i
$58$		0			0
$59$		−	4.55518i		−	0.593034i	−0.955028	−	0.296517i	$-0.904175\pi$
$59$		−	4.55518i		−	0.593034i	0.955028	−	0.296517i	$-0.0958252\pi$
$60$		0			0
$61$	14.0332			1.79676			0.898381	−	0.439217i	$-0.144744\pi$
$61$	14.0332			1.79676			0.898381	+	0.439217i	$0.144744\pi$
$62$		0			0
$63$		−	4.78051i		−	0.602288i
$64$		0			0
$65$	−3.08889	+	1.51762i	−0.383130	+	0.188237i
$66$		0			0
$67$			0.399338i			0.0487869i	0.999702	+	0.0243934i	$0.00776544\pi$
$67$			0.399338i			0.0487869i	−0.999702	+	0.0243934i	$0.992235\pi$
$68$		0			0
$69$	−8.38118			−1.00898
$70$		0			0
$71$		−	3.44482i		−	0.408825i	−0.978885	−	0.204412i	$-0.934472\pi$
$71$		−	3.44482i		−	0.408825i	0.978885	−	0.204412i	$-0.0655283\pi$
$72$		0			0
$73$		−	12.2687i		−	1.43595i	−0.696070	−	0.717974i	$-0.745070\pi$
$73$		−	12.2687i		−	1.43595i	0.696070	−	0.717974i	$-0.254930\pi$
$74$		0			0
$75$	4.08889			0.472145
$76$		0			0
$77$	−12.6499			−1.44159
$78$		0			0
$79$	−12.7624			−1.43588			−0.717938	−	0.696107i	$-0.754914\pi$
$79$	−12.7624			−1.43588			−0.717938	+	0.696107i	$0.754914\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$		−	0.737102i		−	0.0809075i	−0.999181	−	0.0404537i	$-0.987120\pi$
$83$		−	0.737102i		−	0.0809075i	0.999181	−	0.0404537i	$-0.0128803\pi$
$84$		0			0
$85$			4.26874i			0.463010i
$86$		0			0
$87$	3.90904			0.419093
$88$		0			0
$89$		−	11.8078i		−	1.25163i	−0.779972	−	0.625814i	$-0.784767\pi$
$89$		−	11.8078i		−	1.25163i	0.779972	−	0.625814i	$-0.215233\pi$
$90$		0			0
$91$	15.4701	−	7.60066i	1.62170	−	0.796766i
$92$		0			0
$93$		−	2.87147i		−	0.297758i
$94$		0			0
$95$	−2.20339			−0.226063
$96$		0			0
$97$		−	6.09096i		−	0.618443i	−0.950990	−	0.309222i	$-0.899932\pi$
$97$		−	6.09096i		−	0.618443i	0.950990	−	0.309222i	$-0.100068\pi$
$98$		0			0
$99$		−	2.64614i		−	0.265947i
$100$		0			0
$101$	6.38118			0.634951			0.317475	−	0.948267i	$-0.397165\pi$
$101$	6.38118			0.634951			0.317475	+	0.948267i	$0.397165\pi$
$102$		0			0
$103$	−0.348012			−0.0342907			−0.0171453	−	0.999853i	$-0.505458\pi$
$103$	−0.348012			−0.0342907			−0.0171453	+	0.999853i	$0.505458\pi$
$104$		0			0
$105$	4.56310			0.445313
$106$		0			0
$107$	−2.02147			−0.195423			−0.0977116	−	0.995215i	$-0.531152\pi$
$107$	−2.02147			−0.195423			−0.0977116	+	0.995215i	$0.531152\pi$
$108$		0			0
$109$			8.83184i			0.845937i	0.906144	+	0.422968i	$0.139012\pi$
$109$			8.83184i			0.845937i	−0.906144	+	0.422968i	$0.860988\pi$
$110$		0			0
$111$		−	8.38118i		−	0.795506i
$112$		0			0
$113$	−10.8318			−1.01897			−0.509487	−	0.860478i	$-0.670165\pi$
$113$	−10.8318			−1.01897			−0.509487	+	0.860478i	$0.670165\pi$
$114$		0			0
$115$		−	8.00000i		−	0.746004i
$116$		0			0
$117$	1.58993	+	3.23607i	0.146989	+	0.299175i
$118$		0			0
$119$		−	21.3791i		−	1.95982i
$120$		0			0
$121$	3.99793			0.363448
$122$		0			0
$123$			4.95452i			0.446734i
$124$		0			0
$125$			8.67553i			0.775963i
$126$		0			0
$127$	19.3040			1.71295			0.856475	−	0.516188i	$-0.172649\pi$
$127$	19.3040			1.71295			0.856475	+	0.516188i	$0.172649\pi$
$128$		0			0
$129$	7.65199			0.673720
$130$		0			0
$131$	−0.563096			−0.0491979			−0.0245990	−	0.999697i	$-0.507831\pi$
$131$	−0.563096			−0.0491979			−0.0245990	+	0.999697i	$0.507831\pi$
$132$		0			0
$133$	11.0352			0.956876
$134$		0			0
$135$			0.954520i			0.0821520i
$136$		0			0
$137$			14.1675i			1.21041i	0.796068	+	0.605207i	$0.206910\pi$
$137$			14.1675i			1.21041i	−0.796068	+	0.605207i	$0.793090\pi$
$138$		0			0
$139$	−1.11036			−0.0941799			−0.0470899	−	0.998891i	$-0.514995\pi$
$139$	−1.11036			−0.0941799			−0.0470899	+	0.998891i	$0.514995\pi$
$140$		0			0
$141$		−	10.2981i		−	0.867259i
$142$		0			0
$143$	8.56310	−	4.20717i	0.716082	−	0.351821i
$144$		0			0
$145$			3.73126i			0.309864i
$146$		0			0
$147$	−15.8533			−1.30756
$148$		0			0
$149$		−	0.954520i		−	0.0781973i	−0.999235	−	0.0390987i	$-0.987551\pi$
$149$		−	0.954520i		−	0.0781973i	0.999235	−	0.0390987i	$-0.0124487\pi$
$150$		0			0
$151$			10.0728i			0.819713i	0.912150	+	0.409856i	$0.134421\pi$
$151$			10.0728i			0.819713i	−0.912150	+	0.409856i	$0.865579\pi$
$152$		0			0
$153$	4.47214			0.361551
$154$		0			0
$155$	2.74088			0.220153
$156$		0			0
$157$	7.90904			0.631210			0.315605	−	0.948891i	$-0.397793\pi$
$157$	7.90904			0.631210			0.315605	+	0.948891i	$0.397793\pi$
$158$		0			0
$159$	−10.3812			−0.823281
$160$		0			0
$161$			40.0663i			3.15767i
$162$		0			0
$163$		−	3.86941i		−	0.303075i	−0.988451	−	0.151538i	$-0.951578\pi$
$163$		−	3.86941i		−	0.303075i	0.988451	−	0.151538i	$-0.0484225\pi$
$164$		0			0
$165$	2.52580			0.196633
$166$		0			0
$167$			24.8239i			1.92093i	0.278396	+	0.960467i	$0.410197\pi$
$167$			24.8239i			1.92093i	−0.278396	+	0.960467i	$0.589803\pi$
$168$		0			0
$169$	−7.94427	+	10.2902i	−0.611098	+	0.791555i
$170$		0			0
$171$			2.30838i			0.176526i
$172$		0			0
$173$	−16.9228			−1.28662			−0.643308	−	0.765607i	$-0.722439\pi$
$173$	−16.9228			−1.28662			−0.643308	+	0.765607i	$0.722439\pi$
$174$		0			0
$175$		−	19.5470i		−	1.47761i
$176$		0			0
$177$		−	4.55518i		−	0.342388i
$178$		0			0
$179$	16.1993			1.21079			0.605395	−	0.795925i	$-0.293015\pi$
$179$	16.1993			1.21079			0.605395	+	0.795925i	$0.293015\pi$
$180$		0			0
$181$	7.90904			0.587874			0.293937	−	0.955825i	$-0.405034\pi$
$181$	7.90904			0.587874			0.293937	+	0.955825i	$0.405034\pi$
$182$		0			0
$183$	14.0332			1.03736
$184$		0			0
$185$	8.00000			0.588172
$186$		0			0
$187$		−	11.8339i		−	0.865381i
$188$		0			0
$189$		−	4.78051i		−	0.347731i
$190$		0			0
$191$	22.5590			1.63231			0.816154	−	0.577834i	$-0.196102\pi$
$191$	22.5590			1.63231			0.816154	+	0.577834i	$0.196102\pi$
$192$		0			0
$193$		−	19.1221i		−	1.37644i	−0.725504	−	0.688218i	$-0.758393\pi$
$193$		−	19.1221i		−	1.37644i	0.725504	−	0.688218i	$-0.241607\pi$
$194$		0			0
$195$	−3.08889	+	1.51762i	−0.221200	+	0.108679i
$196$		0			0
$197$			3.53909i			0.252150i	0.992021	+	0.126075i	$0.0402379\pi$
$197$			3.53909i			0.252150i	−0.992021	+	0.126075i	$0.959762\pi$
$198$		0			0
$199$	6.58457			0.466768			0.233384	−	0.972385i	$-0.425020\pi$
$199$	6.58457			0.466768			0.233384	+	0.972385i	$0.425020\pi$
$200$		0			0
$201$			0.399338i			0.0281671i
$202$		0			0
$203$		−	18.6872i		−	1.31159i
$204$		0			0
$205$	−4.72919			−0.330301
$206$		0			0
$207$	−8.38118			−0.582532
$208$		0			0
$209$	6.10830			0.422520
$210$		0			0
$211$	−10.1778			−0.700667			−0.350334	−	0.936625i	$-0.613932\pi$
$211$	−10.1778			−0.700667			−0.350334	+	0.936625i	$0.613932\pi$
$212$		0			0
$213$		−	3.44482i		−	0.236035i
$214$		0			0
$215$			7.30398i			0.498127i
$216$		0			0
$217$	−13.7271			−0.931858
$218$		0			0
$219$		−	12.2687i		−	0.829045i
$220$		0			0
$221$	7.11036	+	14.4721i	0.478295	+	0.973501i
$222$		0			0
$223$		−	8.51177i		−	0.569990i	−0.958529	−	0.284995i	$-0.908008\pi$
$223$		−	8.51177i		−	0.569990i	0.958529	−	0.284995i	$-0.0919920\pi$
$224$		0			0
$225$	4.08889			0.272593
$226$		0			0
$227$			19.7565i			1.31129i	0.755071	+	0.655643i	$0.227602\pi$
$227$			19.7565i			1.31129i	−0.755071	+	0.655643i	$0.772398\pi$
$228$		0			0
$229$		−	3.41641i		−	0.225763i	−0.993608	−	0.112881i	$-0.963992\pi$
$229$		−	3.41641i		−	0.225763i	0.993608	−	0.112881i	$-0.0360080\pi$
$230$		0			0
$231$	−12.6499			−0.832304
$232$		0			0
$233$	12.5846			0.824443			0.412221	−	0.911084i	$-0.364753\pi$
$233$	12.5846			0.824443			0.412221	+	0.911084i	$0.364753\pi$
$234$		0			0
$235$	9.82977			0.641224
$236$		0			0
$237$	−12.7624			−0.829004
$238$		0			0
$239$			13.7887i			0.891916i	0.895054	+	0.445958i	$0.147137\pi$
$239$			13.7887i			0.891916i	−0.895054	+	0.445958i	$0.852863\pi$
$240$		0			0
$241$		−	3.03523i		−	0.195516i	−0.995210	−	0.0977582i	$-0.968833\pi$
$241$		−	3.03523i		−	0.195516i	0.995210	−	0.0977582i	$-0.0311672\pi$
$242$		0			0
$243$	1.00000			0.0641500
$244$		0			0
$245$		−	15.1323i		−	0.966767i
$246$		0			0
$247$	−7.47007	+	3.67015i	−0.475309	+	0.233526i
$248$		0			0
$249$		−	0.737102i		−	0.0467120i
$250$		0			0
$251$	−10.1778			−0.642416			−0.321208	−	0.947009i	$-0.604089\pi$
$251$	−10.1778			−0.642416			−0.321208	+	0.947009i	$0.604089\pi$
$252$		0			0
$253$			22.1778i			1.39431i
$254$		0			0
$255$			4.26874i			0.267319i
$256$		0			0
$257$	4.58457			0.285978			0.142989	−	0.989724i	$-0.454329\pi$
$257$	4.58457			0.285978			0.142989	+	0.989724i	$0.454329\pi$
$258$		0			0
$259$	−40.0663			−2.48960
$260$		0			0
$261$	3.90904			0.241964
$262$		0			0
$263$	10.5846			0.652672			0.326336	−	0.945254i	$-0.394186\pi$
$263$	10.5846			0.652672			0.326336	+	0.945254i	$0.394186\pi$
$264$		0			0
$265$		−	9.90904i		−	0.608707i
$266$		0			0
$267$		−	11.8078i		−	0.722628i
$268$		0			0
$269$	−27.0311			−1.64811			−0.824057	−	0.566506i	$-0.808295\pi$
$269$	−27.0311			−1.64811			−0.824057	+	0.566506i	$0.808295\pi$
$270$		0			0
$271$		−	5.54287i		−	0.336705i	−0.985727	−	0.168353i	$-0.946155\pi$
$271$		−	5.54287i		−	0.336705i	0.985727	−	0.168353i	$-0.0538447\pi$
$272$		0			0
$273$	15.4701	−	7.60066i	0.936291	−	0.460013i
$274$		0			0
$275$		−	10.8198i		−	0.652458i
$276$		0			0
$277$	3.12619			0.187835			0.0939173	−	0.995580i	$-0.470061\pi$
$277$	3.12619			0.187835			0.0939173	+	0.995580i	$0.470061\pi$
$278$		0			0
$279$		−	2.87147i		−	0.171911i
$280$		0			0
$281$		−	11.1323i		−	0.664098i	−0.943262	−	0.332049i	$-0.892260\pi$
$281$		−	11.1323i		−	0.664098i	0.943262	−	0.332049i	$-0.107740\pi$
$282$		0			0
$283$	−28.0663			−1.66837			−0.834185	−	0.551485i	$-0.814061\pi$
$283$	−28.0663			−1.66837			−0.834185	+	0.551485i	$0.814061\pi$
$284$		0			0
$285$	−2.20339			−0.130518
$286$		0			0
$287$	23.6852			1.39809
$288$		0			0
$289$	3.00000			0.176471
$290$		0			0
$291$		−	6.09096i		−	0.357058i
$292$		0			0
$293$			28.0766i			1.64025i	0.572184	+	0.820126i	$0.306096\pi$
$293$			28.0766i			1.64025i	−0.572184	+	0.820126i	$0.693904\pi$
$294$		0			0
$295$	4.34801			0.253151
$296$		0			0
$297$		−	2.64614i		−	0.153545i
$298$		0			0
$299$	−13.3254	−	27.1221i	−0.770631	−	1.56851i
$300$		0			0
$301$		−	36.5804i		−	2.10846i
$302$		0			0
$303$	6.38118			0.366589
$304$		0			0
$305$			13.3949i			0.766992i
$306$		0			0
$307$			19.7463i			1.12698i	0.826123	+	0.563489i	$0.190541\pi$
$307$			19.7463i			1.12698i	−0.826123	+	0.563489i	$0.809459\pi$
$308$		0			0
$309$	−0.348012			−0.0197977
$310$		0			0
$311$	−13.7966			−0.782334			−0.391167	−	0.920320i	$-0.627928\pi$
$311$	−13.7966			−0.782334			−0.391167	+	0.920320i	$0.627928\pi$
$312$		0			0
$313$	−18.5268			−1.04720			−0.523598	−	0.851966i	$-0.675410\pi$
$313$	−18.5268			−1.04720			−0.523598	+	0.851966i	$0.675410\pi$
$314$		0			0
$315$	4.56310			0.257101
$316$		0			0
$317$			17.0208i			0.955986i	0.878364	+	0.477993i	$0.158636\pi$
$317$			17.0208i			0.955986i	−0.878364	+	0.477993i	$0.841364\pi$
$318$		0			0
$319$		−	10.3439i		−	0.579146i
$320$		0			0
$321$	−2.02147			−0.112828
$322$		0			0
$323$			10.3234i			0.574408i
$324$		0			0
$325$	6.50103	+	13.2319i	0.360612	+	0.733975i
$326$		0			0
$327$			8.83184i			0.488402i
$328$		0			0
$329$	−49.2304			−2.71416
$330$		0			0
$331$			7.27315i			0.399768i	0.979820	+	0.199884i	$0.0640566\pi$
$331$			7.27315i			0.399768i	−0.979820	+	0.199884i	$0.935943\pi$
$332$		0			0
$333$		−	8.38118i		−	0.459286i
$334$		0			0
$335$	−0.381176			−0.0208259
$336$		0			0
$337$	−7.94220			−0.432639			−0.216320	−	0.976323i	$-0.569405\pi$
$337$	−7.94220			−0.432639			−0.216320	+	0.976323i	$0.569405\pi$
$338$		0			0
$339$	−10.8318			−0.588305
$340$		0			0
$341$	−7.59833			−0.411473
$342$		0			0
$343$			42.3234i			2.28525i
$344$		0			0
$345$		−	8.00000i		−	0.430706i
$346$		0			0
$347$	10.8738			0.583737			0.291868	−	0.956459i	$-0.405723\pi$
$347$	10.8738			0.583737			0.291868	+	0.956459i	$0.405723\pi$
$348$		0			0
$349$		−	21.3254i		−	1.14153i	−0.821115	−	0.570763i	$-0.806648\pi$
$349$		−	21.3254i		−	1.14153i	0.821115	−	0.570763i	$-0.193352\pi$
$350$		0			0
$351$	1.58993	+	3.23607i	0.0848640	+	0.172729i
$352$		0			0
$353$			3.72101i			0.198049i	0.995085	+	0.0990247i	$0.0315723\pi$
$353$			3.72101i			0.198049i	−0.995085	+	0.0990247i	$0.968428\pi$
$354$		0			0
$355$	3.28815			0.174517
$356$		0			0
$357$		−	21.3791i		−	1.13150i
$358$		0			0
$359$			13.0058i			0.686422i	0.939258	+	0.343211i	$0.111515\pi$
$359$			13.0058i			0.686422i	−0.939258	+	0.343211i	$0.888485\pi$
$360$		0			0
$361$	13.6714			0.719547
$362$		0			0
$363$	3.99793			0.209837
$364$		0			0
$365$	11.7108			0.612969
$366$		0			0
$367$	25.5405			1.33320			0.666602	−	0.745413i	$-0.267748\pi$
$367$	25.5405			1.33320			0.666602	+	0.745413i	$0.267748\pi$
$368$		0			0
$369$			4.95452i			0.257922i
$370$		0			0
$371$			49.6274i			2.57652i
$372$		0			0
$373$	−10.9979			−0.569451			−0.284726	−	0.958609i	$-0.591903\pi$
$373$	−10.9979			−0.569451			−0.284726	+	0.958609i	$0.591903\pi$
$374$		0			0
$375$			8.67553i			0.448002i
$376$		0			0
$377$	6.21508	+	12.6499i	0.320093	+	0.651504i
$378$		0			0
$379$			35.0707i			1.80146i	0.434377	+	0.900731i	$0.356969\pi$
$379$			35.0707i			1.80146i	−0.434377	+	0.900731i	$0.643031\pi$
$380$		0			0
$381$	19.3040			0.988973
$382$		0			0
$383$		−	8.40959i		−	0.429710i	−0.976646	−	0.214855i	$-0.931072\pi$
$383$		−	8.40959i		−	0.429710i	0.976646	−	0.214855i	$-0.0689279\pi$
$384$		0			0
$385$		−	12.0746i		−	0.615379i
$386$		0			0
$387$	7.65199			0.388972
$388$		0			0
$389$	27.2130			1.37975			0.689877	−	0.723926i	$-0.257664\pi$
$389$	27.2130			1.37975			0.689877	+	0.723926i	$0.257664\pi$
$390$		0			0
$391$	−37.4818			−1.89553
$392$		0			0
$393$	−0.563096			−0.0284044
$394$		0			0
$395$		−	12.1819i		−	0.612939i
$396$		0			0
$397$		−	26.7409i		−	1.34209i	−0.741418	−	0.671043i	$-0.765847\pi$
$397$		−	26.7409i		−	1.34209i	0.741418	−	0.671043i	$-0.234153\pi$
$398$		0			0
$399$	11.0352			0.552453
$400$		0			0
$401$			24.3014i			1.21356i	0.794871	+	0.606778i	$0.207538\pi$
$401$			24.3014i			1.21356i	−0.794871	+	0.606778i	$0.792462\pi$
$402$		0			0
$403$	9.29228	−	4.56543i	0.462882	−	0.227420i
$404$		0			0
$405$			0.954520i			0.0474305i
$406$		0			0
$407$	−22.1778			−1.09931
$408$		0			0
$409$		−	21.3949i		−	1.05791i	−0.848649	−	0.528956i	$-0.822584\pi$
$409$		−	21.3949i		−	1.05791i	0.848649	−	0.528956i	$-0.177416\pi$
$410$		0			0
$411$			14.1675i			0.698833i
$412$		0			0
$413$	−21.7761			−1.07153
$414$		0			0
$415$	0.703579			0.0345373
$416$		0			0
$417$	−1.11036			−0.0543748
$418$		0			0
$419$	−5.97853			−0.292070			−0.146035	−	0.989279i	$-0.546651\pi$
$419$	−5.97853			−0.292070			−0.146035	+	0.989279i	$0.546651\pi$
$420$		0			0
$421$			3.35008i			0.163273i	0.996662	+	0.0816365i	$0.0260146\pi$
$421$			3.35008i			0.163273i	−0.996662	+	0.0816365i	$0.973985\pi$
$422$		0			0
$423$		−	10.2981i		−	0.500712i
$424$		0			0
$425$	18.2861			0.887005
$426$		0			0
$427$		−	67.0857i		−	3.24651i
$428$		0			0
$429$	8.56310	−	4.20717i	0.413430	−	0.203124i
$430$		0			0
$431$		−	23.7136i		−	1.14224i	−0.820866	−	0.571121i	$-0.806509\pi$
$431$		−	23.7136i		−	1.14224i	0.820866	−	0.571121i	$-0.193491\pi$
$432$		0			0
$433$	−13.0889			−0.629012			−0.314506	−	0.949255i	$-0.601839\pi$
$433$	−13.0889			−0.629012			−0.314506	+	0.949255i	$0.601839\pi$
$434$		0			0
$435$			3.73126i			0.178900i
$436$		0			0
$437$		−	19.3469i		−	0.925489i
$438$		0			0
$439$	25.8727			1.23484			0.617419	−	0.786635i	$-0.288178\pi$
$439$	25.8727			1.23484			0.617419	+	0.786635i	$0.288178\pi$
$440$		0			0
$441$	−15.8533			−0.754920
$442$		0			0
$443$	−21.6147			−1.02694			−0.513472	−	0.858106i	$-0.671641\pi$
$443$	−21.6147			−1.02694			−0.513472	+	0.858106i	$0.671641\pi$
$444$		0			0
$445$	11.2708			0.534288
$446$		0			0
$447$		−	0.954520i		−	0.0451472i
$448$		0			0
$449$		−	13.0413i		−	0.615459i	−0.951474	−	0.307730i	$-0.900431\pi$
$449$		−	13.0413i		−	0.615459i	0.951474	−	0.307730i	$-0.0995692\pi$
$450$		0			0
$451$	13.1104			0.617343
$452$		0			0
$453$			10.0728i			0.473261i
$454$		0			0
$455$	7.25498	+	14.7665i	0.340119	+	0.692264i
$456$		0			0
$457$		−	12.1298i		−	0.567407i	−0.958912	−	0.283703i	$-0.908437\pi$
$457$		−	12.1298i		−	0.567407i	0.958912	−	0.283703i	$-0.0915631\pi$
$458$		0			0
$459$	4.47214			0.208741
$460$		0			0
$461$			26.8431i			1.25021i	0.780542	+	0.625103i	$0.214943\pi$
$461$			26.8431i			1.25021i	−0.780542	+	0.625103i	$0.785057\pi$
$462$		0			0
$463$			23.7999i			1.10608i	0.833156	+	0.553038i	$0.186532\pi$
$463$			23.7999i			1.10608i	−0.833156	+	0.553038i	$0.813468\pi$
$464$		0			0
$465$	2.74088			0.127105
$466$		0			0
$467$	−27.3040			−1.26348			−0.631739	−	0.775181i	$-0.717658\pi$
$467$	−27.3040			−1.26348			−0.631739	+	0.775181i	$0.717658\pi$
$468$		0			0
$469$	1.90904			0.0881513
$470$		0			0
$471$	7.90904			0.364429
$472$		0			0
$473$		−	20.2482i		−	0.931015i
$474$		0			0
$475$			9.43871i			0.433078i
$476$		0			0
$477$	−10.3812			−0.475321
$478$		0			0
$479$			32.0411i			1.46399i	0.681308	+	0.731997i	$0.261412\pi$
$479$			32.0411i			1.46399i	−0.681308	+	0.731997i	$0.738588\pi$
$480$		0			0
$481$	27.1221	−	13.3254i	1.23666	−	0.607588i
$482$		0			0
$483$			40.0663i			1.82308i
$484$		0			0
$485$	5.81394			0.263998
$486$		0			0
$487$		−	13.9272i		−	0.631102i	−0.948909	−	0.315551i	$-0.897811\pi$
$487$		−	13.9272i		−	0.631102i	0.948909	−	0.315551i	$-0.102189\pi$
$488$		0			0
$489$		−	3.86941i		−	0.174981i
$490$		0			0
$491$	28.4302			1.28304			0.641518	−	0.767108i	$-0.278305\pi$
$491$	28.4302			1.28304			0.641518	+	0.767108i	$0.278305\pi$
$492$		0			0
$493$	17.4818			0.787339
$494$		0			0
$495$	2.52580			0.113526
$496$		0			0
$497$	−16.4680			−0.738691
$498$		0			0
$499$			31.9945i			1.43227i	0.697961	+	0.716135i	$0.254091\pi$
$499$			31.9945i			1.43227i	−0.697961	+	0.716135i	$0.745909\pi$
$500$		0			0
$501$			24.8239i			1.10905i
$502$		0			0
$503$	−22.2441			−0.991816			−0.495908	−	0.868375i	$-0.665165\pi$
$503$	−22.2441			−0.991816			−0.495908	+	0.868375i	$0.665165\pi$
$504$		0			0
$505$			6.09096i			0.271044i
$506$		0			0
$507$	−7.94427	+	10.2902i	−0.352818	+	0.457005i
$508$		0			0
$509$		−	1.63005i		−	0.0722506i	−0.999347	−	0.0361253i	$-0.988498\pi$
$509$		−	1.63005i		−	0.0722506i	0.999347	−	0.0361253i	$-0.0115015\pi$
$510$		0			0
$511$	−58.6509			−2.59456
$512$		0			0
$513$			2.30838i			0.101917i
$514$		0			0
$515$		−	0.332185i		−	0.0146378i
$516$		0			0
$517$	−27.2503			−1.19847
$518$		0			0
$519$	−16.9228			−0.742828
$520$		0			0
$521$	5.93051			0.259820			0.129910	−	0.991526i	$-0.458531\pi$
$521$	5.93051			0.259820			0.129910	+	0.991526i	$0.458531\pi$
$522$		0			0
$523$	31.2881			1.36814			0.684068	−	0.729419i	$-0.260209\pi$
$523$	31.2881			1.36814			0.684068	+	0.729419i	$0.260209\pi$
$524$		0			0
$525$		−	19.5470i		−	0.853101i
$526$		0			0
$527$		−	12.8416i		−	0.559390i
$528$		0			0
$529$	47.2441			2.05409
$530$		0			0
$531$		−	4.55518i		−	0.197678i
$532$		0			0
$533$	−16.0332	+	7.87732i	−0.694473	+	0.341205i
$534$		0			0
$535$		−	1.92954i		−	0.0834211i
$536$		0			0
$537$	16.1993			0.699049
$538$		0			0
$539$			41.9501i			1.80692i
$540$		0			0
$541$			28.5109i			1.22578i	0.790168	+	0.612891i	$0.209993\pi$
$541$			28.5109i			1.22578i	−0.790168	+	0.612891i	$0.790007\pi$
$542$		0			0
$543$	7.90904			0.339409
$544$		0			0
$545$	−8.43017			−0.361109
$546$		0			0
$547$	18.2366			0.779739			0.389869	−	0.920870i	$-0.372520\pi$
$547$	18.2366			0.779739			0.389869	+	0.920870i	$0.372520\pi$
$548$		0			0
$549$	14.0332			0.598921
$550$		0			0
$551$			9.02354i			0.384416i
$552$		0			0
$553$			61.0106i			2.59443i
$554$		0			0
$555$	8.00000			0.339581
$556$		0			0
$557$			29.5350i			1.25144i	0.780049	+	0.625718i	$0.215194\pi$
$557$			29.5350i			1.25144i	−0.780049	+	0.625718i	$0.784806\pi$
$558$		0			0
$559$	12.1661	+	24.7624i	0.514571	+	1.04734i
$560$		0			0
$561$		−	11.8339i		−	0.499628i
$562$		0			0
$563$	−38.7409			−1.63273			−0.816367	−	0.577534i	$-0.804015\pi$
$563$	−38.7409			−1.63273			−0.816367	+	0.577534i	$0.804015\pi$
$564$		0			0
$565$		−	10.3392i		−	0.434974i
$566$		0			0
$567$		−	4.78051i		−	0.200763i
$568$		0			0
$569$	−0.584569			−0.0245064			−0.0122532	−	0.999925i	$-0.503900\pi$
$569$	−0.584569			−0.0245064			−0.0122532	+	0.999925i	$0.503900\pi$
$570$		0			0
$571$	−8.28924			−0.346894			−0.173447	−	0.984843i	$-0.555491\pi$
$571$	−8.28924			−0.346894			−0.173447	+	0.984843i	$0.555491\pi$
$572$		0			0
$573$	22.5590			0.942414
$574$		0			0
$575$	−34.2697			−1.42915
$576$		0			0
$577$		−	0.696025i		−	0.0289759i	−0.999895	−	0.0144879i	$-0.995388\pi$
$577$		−	0.696025i		−	0.0289759i	0.999895	−	0.0144879i	$-0.00461182\pi$
$578$		0			0
$579$		−	19.1221i		−	0.794685i
$580$		0			0
$581$	−3.52373			−0.146189
$582$		0			0
$583$			27.4701i			1.13769i
$584$		0			0
$585$	−3.08889	+	1.51762i	−0.127710	+	0.0627457i
$586$		0			0
$587$		−	2.62565i		−	0.108372i	−0.998531	−	0.0541860i	$-0.982744\pi$
$587$		−	2.62565i		−	0.108372i	0.998531	−	0.0541860i	$-0.0172564\pi$
$588$		0			0
$589$	6.62845			0.273120
$590$		0			0
$591$			3.53909i			0.145579i
$592$		0			0
$593$			18.8001i			0.772028i	0.922493	+	0.386014i	$0.126148\pi$
$593$			18.8001i			0.772028i	−0.922493	+	0.386014i	$0.873852\pi$
$594$		0			0
$595$	20.4068			0.836597
$596$		0			0
$597$	6.58457			0.269488
$598$		0			0
$599$	−22.5416			−0.921026			−0.460513	−	0.887653i	$-0.652334\pi$
$599$	−22.5416			−0.921026			−0.460513	+	0.887653i	$0.652334\pi$
$600$		0			0
$601$	40.8865			1.66779			0.833897	−	0.551920i	$-0.186105\pi$
$601$	40.8865			1.66779			0.833897	+	0.551920i	$0.186105\pi$
$602$		0			0
$603$			0.399338i			0.0162623i
$604$		0			0
$605$			3.81611i			0.155147i
$606$		0			0
$607$	6.88964			0.279642			0.139821	−	0.990177i	$-0.455347\pi$
$607$	6.88964			0.279642			0.139821	+	0.990177i	$0.455347\pi$
$608$		0			0
$609$		−	18.6872i		−	0.757244i
$610$		0			0
$611$	33.3254	−	16.3733i	1.34820	−	0.662391i
$612$		0			0
$613$		−	2.31071i		−	0.0933288i	−0.998911	−	0.0466644i	$-0.985141\pi$
$613$		−	2.31071i		−	0.0933288i	0.998911	−	0.0466644i	$-0.0148591\pi$
$614$		0			0
$615$	−4.72919			−0.190699
$616$		0			0
$617$			4.41703i			0.177823i	0.996040	+	0.0889115i	$0.0283388\pi$
$617$			4.41703i			0.177823i	−0.996040	+	0.0889115i	$0.971661\pi$
$618$		0			0
$619$			12.9798i			0.521701i	0.965379	+	0.260850i	$0.0840030\pi$
$619$			12.9798i			0.521701i	−0.965379	+	0.260850i	$0.915997\pi$
$620$		0			0
$621$	−8.38118			−0.336325
$622$		0			0
$623$	−56.4475			−2.26152
$624$		0			0
$625$	12.1635			0.486540
$626$		0			0
$627$	6.10830			0.243942
$628$		0			0
$629$		−	37.4818i		−	1.49450i
$630$		0			0
$631$			1.17147i			0.0466356i	0.999728	+	0.0233178i	$0.00742295\pi$
$631$			1.17147i			0.0466356i	−0.999728	+	0.0233178i	$0.992577\pi$
$632$		0			0
$633$	−10.1778			−0.404531
$634$		0			0
$635$			18.4260i			0.731215i
$636$		0			0
$637$	−25.2056	−	51.3024i	−0.998682	−	2.03267i
$638$		0			0
$639$		−	3.44482i		−	0.136275i
$640$		0			0
$641$	−18.8318			−0.743813			−0.371906	−	0.928270i	$-0.621296\pi$
$641$	−18.8318			−0.743813			−0.371906	+	0.928270i	$0.621296\pi$
$642$		0			0
$643$			26.7590i			1.05527i	0.849470	+	0.527637i	$0.176922\pi$
$643$			26.7590i			1.05527i	−0.849470	+	0.527637i	$0.823078\pi$
$644$		0			0
$645$			7.30398i			0.287594i
$646$		0			0
$647$	9.45838			0.371847			0.185924	−	0.982564i	$-0.440472\pi$
$647$	9.45838			0.371847			0.185924	+	0.982564i	$0.440472\pi$
$648$		0			0
$649$	−12.0537			−0.473148
$650$		0			0
$651$	−13.7271			−0.538008
$652$		0			0
$653$	33.0782			1.29445			0.647225	−	0.762299i	$-0.275930\pi$
$653$	33.0782			1.29445			0.647225	+	0.762299i	$0.275930\pi$
$654$		0			0
$655$		−	0.537486i		−	0.0210013i
$656$		0			0
$657$		−	12.2687i		−	0.478649i
$658$		0			0
$659$	−0.719406			−0.0280241			−0.0140120	−	0.999902i	$-0.504460\pi$
$659$	−0.719406			−0.0280241			−0.0140120	+	0.999902i	$0.504460\pi$
$660$		0			0
$661$			28.8543i			1.12230i	0.827714	+	0.561151i	$0.189641\pi$
$661$			28.8543i			1.12230i	−0.827714	+	0.561151i	$0.810359\pi$
$662$		0			0
$663$	7.11036	+	14.4721i	0.276144	+	0.562051i
$664$		0			0
$665$			10.5333i			0.408466i
$666$		0			0
$667$	−32.7624			−1.26856
$668$		0			0
$669$		−	8.51177i		−	0.329084i
$670$		0			0
$671$		−	37.1337i		−	1.43353i
$672$		0			0
$673$	−20.1339			−0.776105			−0.388053	−	0.921637i	$-0.626852\pi$
$673$	−20.1339			−0.776105			−0.388053	+	0.921637i	$0.626852\pi$
$674$		0			0
$675$	4.08889			0.157382
$676$		0			0
$677$	−27.8396			−1.06996			−0.534980	−	0.844864i	$-0.679681\pi$
$677$	−27.8396			−1.06996			−0.534980	+	0.844864i	$0.679681\pi$
$678$		0			0
$679$	−29.1179			−1.11744
$680$		0			0
$681$			19.7565i			0.757071i
$682$		0			0
$683$			20.6578i			0.790450i	0.918584	+	0.395225i	$0.129333\pi$
$683$			20.6578i			0.790450i	−0.918584	+	0.395225i	$0.870667\pi$
$684$		0			0
$685$	−13.5232			−0.516695
$686$		0			0
$687$		−	3.41641i		−	0.130344i
$688$		0			0
$689$	−16.5053	−	33.5942i	−0.628802	−	1.27984i
$690$		0			0
$691$			43.1892i			1.64299i	0.570212	+	0.821497i	$0.306861\pi$
$691$			43.1892i			1.64299i	−0.570212	+	0.821497i	$0.693139\pi$
$692$		0			0
$693$	−12.6499			−0.480531
$694$		0			0
$695$		−	1.05986i		−	0.0402030i
$696$		0			0
$697$			22.1573i			0.839267i
$698$		0			0
$699$	12.5846			0.475992
$700$		0			0
$701$	22.4936			0.849572			0.424786	−	0.905294i	$-0.360349\pi$
$701$	22.4936			0.849572			0.424786	+	0.905294i	$0.360349\pi$
$702$		0			0
$703$	19.3469			0.729683
$704$		0			0
$705$	9.82977			0.370211
$706$		0			0
$707$		−	30.5053i		−	1.14727i
$708$		0			0
$709$		−	14.9433i		−	0.561207i	−0.959824	−	0.280604i	$-0.909465\pi$
$709$		−	14.9433i		−	0.561207i	0.959824	−	0.280604i	$-0.0905346\pi$
$710$		0			0
$711$	−12.7624			−0.478626
$712$		0			0
$713$			24.0663i			0.901291i
$714$		0			0
$715$	4.01583	+	8.17365i	0.150183	+	0.305677i
$716$		0			0
$717$			13.7887i			0.514948i
$718$		0			0
$719$	10.2697			0.382996			0.191498	−	0.981493i	$-0.438665\pi$
$719$	10.2697			0.382996			0.191498	+	0.981493i	$0.438665\pi$
$720$		0			0
$721$			1.66368i			0.0619586i
$722$		0			0
$723$		−	3.03523i		−	0.112881i
$724$		0			0
$725$	15.9836			0.593618
$726$		0			0
$727$	−15.5932			−0.578320			−0.289160	−	0.957281i	$-0.593376\pi$
$727$	−15.5932			−0.578320			−0.289160	+	0.957281i	$0.593376\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$	34.2207			1.26570
$732$		0			0
$733$			8.76551i			0.323762i	0.986810	+	0.161881i	$0.0517560\pi$
$733$			8.76551i			0.323762i	−0.986810	+	0.161881i	$0.948244\pi$
$734$		0			0
$735$		−	15.1323i		−	0.558163i
$736$		0			0
$737$	1.05670			0.0389242
$738$		0			0
$739$			21.7421i			0.799797i	0.916559	+	0.399898i	$0.130955\pi$
$739$			21.7421i			0.799797i	−0.916559	+	0.399898i	$0.869045\pi$
$740$		0			0
$741$	−7.47007	+	3.67015i	−0.274420	+	0.134826i
$742$		0			0
$743$		−	45.5070i		−	1.66949i	−0.550637	−	0.834745i	$-0.685615\pi$
$743$		−	45.5070i		−	1.66949i	0.550637	−	0.834745i	$-0.314385\pi$
$744$		0			0
$745$	0.911108			0.0333804
$746$		0			0
$747$		−	0.737102i		−	0.0269692i
$748$		0			0
$749$			9.66368i			0.353103i
$750$		0			0
$751$	−3.36275			−0.122708			−0.0613542	−	0.998116i	$-0.519542\pi$
$751$	−3.36275			−0.122708			−0.0613542	+	0.998116i	$0.519542\pi$
$752$		0			0
$753$	−10.1778			−0.370899
$754$		0			0
$755$	−9.61469			−0.349914
$756$		0			0
$757$	−35.1594			−1.27789			−0.638944	−	0.769253i	$-0.720629\pi$
$757$	−35.1594			−1.27789			−0.638944	+	0.769253i	$0.720629\pi$
$758$		0			0
$759$			22.1778i			0.805003i
$760$		0			0
$761$		−	26.8001i		−	0.971504i	−0.874097	−	0.485752i	$-0.838546\pi$
$761$		−	26.8001i		−	0.971504i	0.874097	−	0.485752i	$-0.161454\pi$
$762$		0			0
$763$	42.2207			1.52849
$764$		0			0
$765$			4.26874i			0.154337i
$766$		0			0
$767$	14.7409	−	7.24240i	0.532262	−	0.261508i
$768$		0			0
$769$		−	2.22073i		−	0.0800815i	−0.999198	−	0.0400408i	$-0.987251\pi$
$769$		−	2.22073i		−	0.0800815i	0.999198	−	0.0400408i	$-0.0127488\pi$
$770$		0			0
$771$	4.58457			0.165109
$772$		0			0
$773$		−	47.8947i		−	1.72265i	−0.508054	−	0.861326i	$-0.669635\pi$
$773$		−	47.8947i		−	1.72265i	0.508054	−	0.861326i	$-0.330365\pi$
$774$		0			0
$775$		−	11.7411i		−	0.421754i
$776$		0			0
$777$	−40.0663			−1.43737
$778$		0			0
$779$	−11.4369			−0.409770
$780$		0			0
$781$	−9.11548			−0.326177
$782$		0			0
$783$	3.90904			0.139698
$784$		0			0
$785$			7.54934i			0.269447i
$786$		0			0
$787$			9.49387i			0.338420i	0.985580	+	0.169210i	$0.0541216\pi$
$787$			9.49387i			0.338420i	−0.985580	+	0.169210i	$0.945878\pi$
$788$		0			0
$789$	10.5846			0.376821
$790$		0			0
$791$			51.7818i			1.84115i
$792$		0			0
$793$	22.3117	+	45.4123i	0.792311	+	1.61264i
$794$		0			0
$795$		−	9.90904i		−	0.351437i
$796$		0			0
$797$	−8.38020			−0.296842			−0.148421	−	0.988924i	$-0.547419\pi$
$797$	−8.38020			−0.296842			−0.148421	+	0.988924i	$0.547419\pi$
$798$		0			0
$799$		−	46.0546i		−	1.62930i
$800$		0			0
$801$		−	11.8078i		−	0.417209i
$802$		0			0
$803$	−32.4648			−1.14566
$804$		0			0
$805$	−38.2441			−1.34793
$806$		0			0
$807$	−27.0311			−0.951540
$808$		0			0
$809$	11.7098			0.411694			0.205847	−	0.978584i	$-0.434005\pi$
$809$	11.7098			0.411694			0.205847	+	0.978584i	$0.434005\pi$
$810$		0			0
$811$		−	41.8009i		−	1.46783i	−0.679242	−	0.733914i	$-0.737692\pi$
$811$		−	41.8009i		−	1.46783i	0.679242	−	0.733914i	$-0.262308\pi$
$812$		0			0
$813$		−	5.54287i		−	0.194397i
$814$		0			0
$815$	3.69342			0.129375
$816$		0			0
$817$			17.6637i			0.617974i
$818$		0			0
$819$	15.4701	−	7.60066i	0.540568	−	0.265589i
$820$		0			0
$821$			49.5788i			1.73031i	0.501502	+	0.865157i	$0.332781\pi$
$821$			49.5788i			1.73031i	−0.501502	+	0.865157i	$0.667219\pi$
$822$		0			0
$823$	−17.4742			−0.609112			−0.304556	−	0.952494i	$-0.598508\pi$
$823$	−17.4742			−0.609112			−0.304556	+	0.952494i	$0.598508\pi$
$824$		0			0
$825$		−	10.8198i		−	0.376697i
$826$		0			0
$827$		−	29.0722i		−	1.01094i	−0.862845	−	0.505469i	$-0.831319\pi$
$827$		−	29.0722i		−	1.01094i	0.862845	−	0.505469i	$-0.168681\pi$
$828$		0			0
$829$	0.981572			0.0340914			0.0170457	−	0.999855i	$-0.494574\pi$
$829$	0.981572			0.0340914			0.0170457	+	0.999855i	$0.494574\pi$
$830$		0			0
$831$	3.12619			0.108446
$832$		0			0
$833$	−70.8982			−2.45648
$834$		0			0
$835$	−23.6949			−0.819997
$836$		0			0
$837$		−	2.87147i		−	0.0992527i
$838$		0			0
$839$			49.2019i			1.69864i	0.527879	+	0.849320i	$0.322988\pi$
$839$			49.2019i			1.69864i	−0.527879	+	0.849320i	$0.677012\pi$
$840$		0			0
$841$	−13.7194			−0.473083
$842$		0			0
$843$		−	11.1323i		−	0.383417i
$844$		0			0
$845$	−9.82222	−	7.58297i	−0.337895	−	0.260862i
$846$		0			0
$847$		−	19.1122i		−	0.656702i
$848$		0			0
$849$	−28.0663			−0.963234
$850$		0			0
$851$			70.2441i			2.40794i
$852$		0			0
$853$			7.68515i			0.263135i	0.991307	+	0.131567i	$0.0420009\pi$
$853$			7.68515i			0.263135i	−0.991307	+	0.131567i	$0.957999\pi$
$854$		0			0
$855$	−2.20339			−0.0753544
$856$		0			0
$857$	−3.16914			−0.108256			−0.0541278	−	0.998534i	$-0.517238\pi$
$857$	−3.16914			−0.108256			−0.0541278	+	0.998534i	$0.517238\pi$
$858$		0			0
$859$	−35.3040			−1.20456			−0.602278	−	0.798286i	$-0.705740\pi$
$859$	−35.3040			−1.20456			−0.602278	+	0.798286i	$0.705740\pi$
$860$		0			0
$861$	23.6852			0.807188
$862$		0			0
$863$			13.0058i			0.442724i	0.975192	+	0.221362i	$0.0710503\pi$
$863$			13.0058i			0.442724i	−0.975192	+	0.221362i	$0.928950\pi$
$864$		0			0
$865$		−	16.1532i		−	0.549224i
$866$		0			0
$867$	3.00000			0.101885
$868$		0			0
$869$			33.7710i			1.14560i
$870$		0			0
$871$	−1.29228	+	0.634918i	−0.0437874	+	0.0215134i
$872$		0			0
$873$		−	6.09096i		−	0.206148i
$874$		0			0
$875$	41.4735			1.40206
$876$		0			0
$877$			42.4946i			1.43494i	0.696589	+	0.717470i	$0.254700\pi$
$877$			42.4946i			1.43494i	−0.696589	+	0.717470i	$0.745300\pi$
$878$		0			0
$879$			28.0766i			0.946999i
$880$		0			0
$881$	3.82222			0.128774			0.0643869	−	0.997925i	$-0.479491\pi$
$881$	3.82222			0.128774			0.0643869	+	0.997925i	$0.479491\pi$
$882$		0			0
$883$	−3.23765			−0.108956			−0.0544778	−	0.998515i	$-0.517349\pi$
$883$	−3.23765			−0.108956			−0.0544778	+	0.998515i	$0.517349\pi$
$884$		0			0
$885$	4.34801			0.146157
$886$		0			0
$887$	20.3382			0.682891			0.341445	−	0.939902i	$-0.389084\pi$
$887$	20.3382			0.682891			0.341445	+	0.939902i	$0.389084\pi$
$888$		0			0
$889$		−	92.2829i		−	3.09507i
$890$		0			0
$891$		−	2.64614i		−	0.0886491i
$892$		0			0
$893$	23.7720			0.795499
$894$		0			0
$895$			15.4625i			0.516855i
$896$		0			0
$897$	−13.3254	−	27.1221i	−0.444924	−	0.905579i
$898$		0			0
$899$		−	11.2247i		−	0.374365i
$900$		0			0
$901$	−46.4260			−1.54668
$902$		0			0
$903$		−	36.5804i		−	1.21732i
$904$		0			0
$905$			7.54934i			0.250948i
$906$		0			0
$907$	6.13484			0.203704			0.101852	−	0.994800i	$-0.467523\pi$
$907$	6.13484			0.203704			0.101852	+	0.994800i	$0.467523\pi$
$908$		0			0
$909$	6.38118			0.211650
$910$		0			0
$911$	−44.1583			−1.46303			−0.731514	−	0.681826i	$-0.761186\pi$
$911$	−44.1583			−1.46303			−0.731514	+	0.681826i	$0.761186\pi$
$912$		0			0
$913$	−1.95048			−0.0645514
$914$		0			0
$915$			13.3949i			0.442823i
$916$		0			0
$917$			2.69189i			0.0888940i
$918$		0			0
$919$	53.8885			1.77762			0.888810	−	0.458277i	$-0.151533\pi$
$919$	53.8885			1.77762			0.888810	+	0.458277i	$0.151533\pi$
$920$		0			0
$921$			19.7463i			0.650661i
$922$		0			0
$923$	11.1477	−	5.47700i	0.366930	−	0.180278i
$924$		0			0
$925$		−	34.2697i		−	1.12678i
$926$		0			0
$927$	−0.348012			−0.0114302
$928$		0			0
$929$			33.5350i			1.10025i	0.835084	+	0.550123i	$0.185419\pi$
$929$			33.5350i			1.10025i	−0.835084	+	0.550123i	$0.814581\pi$
$930$		0			0
$931$		−	36.5954i		−	1.19937i
$932$		0			0
$933$	−13.7966			−0.451681
$934$		0			0
$935$	11.2957			0.369409
$936$		0			0
$937$	52.0720			1.70112			0.850559	−	0.525880i	$-0.176264\pi$
$937$	52.0720			1.70112			0.850559	+	0.525880i	$0.176264\pi$
$938$		0			0
$939$	−18.5268			−0.604598
$940$		0			0
$941$			23.5830i			0.768783i	0.923170	+	0.384391i	$0.125589\pi$
$941$			23.5830i			0.768783i	−0.923170	+	0.384391i	$0.874411\pi$
$942$		0			0
$943$		−	41.5247i		−	1.35223i
$944$		0			0
$945$	4.56310			0.148438
$946$		0			0
$947$		−	49.2746i		−	1.60121i	−0.599193	−	0.800604i	$-0.704512\pi$
$947$		−	49.2746i		−	1.60121i	0.599193	−	0.800604i	$-0.295488\pi$
$948$		0			0
$949$	39.7025	−	19.5064i	1.28880	−	0.633204i
$950$		0			0
$951$			17.0208i			0.551939i
$952$		0			0
$953$	50.1327			1.62396			0.811978	−	0.583688i	$-0.198391\pi$
$953$	50.1327			1.62396			0.811978	+	0.583688i	$0.198391\pi$
$954$		0			0
$955$			21.5330i			0.696791i
$956$		0			0
$957$		−	10.3439i		−	0.334370i
$958$		0			0
$959$	67.7281			2.18705
$960$		0			0
$961$	22.7546			0.734021
$962$		0			0
$963$	−2.02147			−0.0651410
$964$		0			0
$965$	18.2524			0.587565
$966$		0			0
$967$		−	4.55472i		−	0.146470i	−0.997315	−	0.0732349i	$-0.976668\pi$
$967$		−	4.55472i		−	0.146470i	0.997315	−	0.0732349i	$-0.0233323\pi$
$968$		0			0
$969$			10.3234i			0.331635i
$970$		0			0
$971$	48.0234			1.54114			0.770572	−	0.637353i	$-0.219971\pi$
$971$	48.0234			1.54114			0.770572	+	0.637353i	$0.219971\pi$
$972$		0			0
$973$			5.30811i			0.170170i
$974$		0			0
$975$	6.50103	+	13.2319i	0.208200	+	0.423761i
$976$		0			0
$977$		−	22.9820i		−	0.735261i	−0.929972	−	0.367630i	$-0.880169\pi$
$977$		−	22.9820i		−	0.735261i	0.929972	−	0.367630i	$-0.119831\pi$
$978$		0			0
$979$	−31.2452			−0.998601
$980$		0			0
$981$			8.83184i			0.281979i
$982$		0			0
$983$			9.63843i			0.307418i	0.988116	+	0.153709i	$0.0491219\pi$
$983$			9.63843i			0.307418i	−0.988116	+	0.153709i	$0.950878\pi$
$984$		0			0
$985$	−3.37813			−0.107636
$986$		0			0
$987$	−49.2304			−1.56702
$988$		0			0
$989$	−64.1327			−2.03930
$990$		0			0
$991$	−10.2366			−0.325175			−0.162587	−	0.986694i	$-0.551984\pi$
$991$	−10.2366			−0.325175			−0.162587	+	0.986694i	$0.551984\pi$
$992$		0			0
$993$			7.27315i			0.230806i
$994$		0			0
$995$			6.28510i			0.199251i
$996$		0			0
$997$	30.3556			0.961370			0.480685	−	0.876893i	$-0.340388\pi$
$997$	30.3556			0.961370			0.480685	+	0.876893i	$0.340388\pi$
$998$		0			0
$999$		−	8.38118i		−	0.265169i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2496.2.c.r.961.5		8
4.3	odd	2		2496.2.c.q.961.5		8
8.3	odd	2		1248.2.c.d.961.4	yes	8
8.5	even	2		1248.2.c.c.961.4	✓	8
13.12	even	2	inner	2496.2.c.r.961.4		8
24.5	odd	2		3744.2.c.p.3457.5		8
24.11	even	2		3744.2.c.o.3457.5		8
52.51	odd	2		2496.2.c.q.961.4		8
104.51	odd	2		1248.2.c.d.961.5	yes	8
104.77	even	2		1248.2.c.c.961.5	yes	8
312.77	odd	2		3744.2.c.p.3457.4		8
312.155	even	2		3744.2.c.o.3457.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1248.2.c.c.961.4	✓	8	8.5	even	2
1248.2.c.c.961.5	yes	8	104.77	even	2
1248.2.c.d.961.4	yes	8	8.3	odd	2
1248.2.c.d.961.5	yes	8	104.51	odd	2
2496.2.c.q.961.4		8	52.51	odd	2
2496.2.c.q.961.5		8	4.3	odd	2
2496.2.c.r.961.4		8	13.12	even	2	inner
2496.2.c.r.961.5		8	1.1	even	1	trivial
3744.2.c.o.3457.4		8	312.155	even	2
3744.2.c.o.3457.5		8	24.11	even	2
3744.2.c.p.3457.4		8	312.77	odd	2
3744.2.c.p.3457.5		8	24.5	odd	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:	\( N \)	\(=\)	\( 2496 = 2^{6} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2496.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +0.954520i q^{5} -4.78051i q^{7} +1.00000 q^{9} -2.64614i q^{11} +(1.58993 + 3.23607i) q^{13} +0.954520i q^{15} +4.47214 q^{17} +2.30838i q^{19} -4.78051i q^{21} -8.38118 q^{23} +4.08889 q^{25} +1.00000 q^{27} +3.90904 q^{29} -2.87147i q^{31} -2.64614i q^{33} +4.56310 q^{35} -8.38118i q^{37} +(1.58993 + 3.23607i) q^{39} +4.95452i q^{41} +7.65199 q^{43} +0.954520i q^{45} -10.2981i q^{47} -15.8533 q^{49} +4.47214 q^{51} -10.3812 q^{53} +2.52580 q^{55} +2.30838i q^{57} -4.55518i q^{59} +14.0332 q^{61} -4.78051i q^{63} +(-3.08889 + 1.51762i) q^{65} +0.399338i q^{67} -8.38118 q^{69} -3.44482i q^{71} -12.2687i q^{73} +4.08889 q^{75} -12.6499 q^{77} -12.7624 q^{79} +1.00000 q^{81} -0.737102i q^{83} +4.26874i q^{85} +3.90904 q^{87} -11.8078i q^{89} +(15.4701 - 7.60066i) q^{91} -2.87147i q^{93} -2.20339 q^{95} -6.09096i q^{97} -2.64614i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{3} + 8 q^{9} - 8 q^{23} - 16 q^{25} + 8 q^{27} + 8 q^{29} + 24 q^{35} - 32 q^{49} - 24 q^{53} - 16 q^{55} - 8 q^{61} + 24 q^{65} - 8 q^{69} - 16 q^{75} + 32 q^{77} + 16 q^{79} + 8 q^{81} + 8 q^{87}+ \cdots - 56 q^{95}+O(q^{100}) \) 8 * q + 8 * q^3 + 8 * q^9 - 8 * q^23 - 16 * q^25 + 8 * q^27 + 8 * q^29 + 24 * q^35 - 32 * q^49 - 24 * q^53 - 16 * q^55 - 8 * q^61 + 24 * q^65 - 8 * q^69 - 16 * q^75 + 32 * q^77 + 16 * q^79 + 8 * q^81 + 8 * q^87 + 16 * q^91 - 56 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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