LMFDB - Embedded newform 2640.2.d.h.529.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2640,2,Mod(529,2640)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2640, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 1, 0]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("2640.529");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2640 = 2^{4} \cdot 3 \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2640.d (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$21.0805061336$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.350464.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 $ x^6 - 2x^5 + 2x^4 + 2x^3 + 4x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 165)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			529.5
Root			$1.45161 - 1.45161i$ of defining polynomial
Character	$\chi$	$=$	2640.529
Dual form			2640.2.d.h.529.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} +(0.311108 + 2.21432i) q^{5} +0.903212i q^{7} -1.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{3} +(0.311108 + 2.21432i) q^{5} +0.903212i q^{7} -1.00000 q^{9} -1.00000 q^{11} -2.90321i q^{13} +(-2.21432 + 0.311108i) q^{15} +2.28100i q^{17} +2.42864 q^{19} -0.903212 q^{21} +4.00000i q^{23} +(-4.80642 + 1.37778i) q^{25} -1.00000i q^{27} -7.05086 q^{29} +2.62222 q^{31} -1.00000i q^{33} +(-2.00000 + 0.280996i) q^{35} +5.80642i q^{37} +2.90321 q^{39} -10.6637 q^{41} +10.7096i q^{43} +(-0.311108 - 2.21432i) q^{45} -0.949145i q^{47} +6.18421 q^{49} -2.28100 q^{51} -0.815792i q^{53} +(-0.311108 - 2.21432i) q^{55} +2.42864i q^{57} -1.67307 q^{59} -7.24443 q^{61} -0.903212i q^{63} +(6.42864 - 0.903212i) q^{65} -12.8573i q^{67} -4.00000 q^{69} -9.28592 q^{71} +5.65878i q^{73} +(-1.37778 - 4.80642i) q^{75} -0.903212i q^{77} -16.5303 q^{79} +1.00000 q^{81} +7.76049i q^{83} +(-5.05086 + 0.709636i) q^{85} -7.05086i q^{87} -6.13335 q^{89} +2.62222 q^{91} +2.62222i q^{93} +(0.755569 + 5.37778i) q^{95} -12.4701i q^{97} +1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 2 q^{5} - 6 q^{9}+O(q^{10}) $ 6 * q + 2 * q^5 - 6 * q^9	$ 6 q + 2 q^{5} - 6 q^{9} - 6 q^{11} - 12 q^{19} + 8 q^{21} - 2 q^{25} - 16 q^{29} + 16 q^{31} - 12 q^{35} + 4 q^{39} + 16 q^{41} - 2 q^{45} + 10 q^{49} - 2 q^{55} + 16 q^{59} - 44 q^{61} + 12 q^{65} - 24 q^{69} + 24 q^{71} - 8 q^{75} - 20 q^{79} + 6 q^{81} - 4 q^{85} - 36 q^{89} + 16 q^{91} + 4 q^{95} + 6 q^{99}+O(q^{100}) $ 6 * q + 2 * q^5 - 6 * q^9 - 6 * q^11 - 12 * q^19 + 8 * q^21 - 2 * q^25 - 16 * q^29 + 16 * q^31 - 12 * q^35 + 4 * q^39 + 16 * q^41 - 2 * q^45 + 10 * q^49 - 2 * q^55 + 16 * q^59 - 44 * q^61 + 12 * q^65 - 24 * q^69 + 24 * q^71 - 8 * q^75 - 20 * q^79 + 6 * q^81 - 4 * q^85 - 36 * q^89 + 16 * q^91 + 4 * q^95 + 6 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2640\mathbb{Z}\right)^\times$.

$n$	$661$	$881$	$991$	$1057$	$1201$
$\chi(n)$	$1$	$1$	$1$	$-1$	$1$

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
$3$			1.00000i			0.577350i
$3$			1.00000i			0.577350i
$4$		0			0
$5$	0.311108	+	2.21432i	0.139132	+	0.990274i
$5$	0.311108	+	2.21432i	0.139132	+	0.990274i
$6$		0			0
$7$			0.903212i			0.341382i	0.985325	+	0.170691i	$0.0546000\pi$
$7$			0.903212i			0.341382i	−0.985325	+	0.170691i	$0.945400\pi$
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	−1.00000			−0.301511
$11$	−1.00000			−0.301511
$12$		0			0
$13$		−	2.90321i		−	0.805206i	−0.915375	−	0.402603i	$-0.868106\pi$
$13$		−	2.90321i		−	0.805206i	0.915375	−	0.402603i	$-0.131894\pi$
$14$		0			0
$15$	−2.21432	+	0.311108i	−0.571735	+	0.0803277i
$16$		0			0
$17$			2.28100i			0.553223i	0.960982	+	0.276611i	$0.0892115\pi$
$17$			2.28100i			0.553223i	−0.960982	+	0.276611i	$0.910789\pi$
$18$		0			0
$19$	2.42864			0.557168			0.278584	−	0.960412i	$-0.410135\pi$
$19$	2.42864			0.557168			0.278584	+	0.960412i	$0.410135\pi$
$20$		0			0
$21$	−0.903212			−0.197097
$22$		0			0
$23$			4.00000i			0.834058i	0.908893	+	0.417029i	$0.136929\pi$
$23$			4.00000i			0.834058i	−0.908893	+	0.417029i	$0.863071\pi$
$24$		0			0
$25$	−4.80642	+	1.37778i	−0.961285	+	0.275557i
$26$		0			0
$27$		−	1.00000i		−	0.192450i
$28$		0			0
$29$	−7.05086			−1.30931			−0.654655	−	0.755927i	$-0.727186\pi$
$29$	−7.05086			−1.30931			−0.654655	+	0.755927i	$0.727186\pi$
$30$		0			0
$31$	2.62222			0.470964			0.235482	−	0.971879i	$-0.424333\pi$
$31$	2.62222			0.470964			0.235482	+	0.971879i	$0.424333\pi$
$32$		0			0
$33$		−	1.00000i		−	0.174078i
$34$		0			0
$35$	−2.00000	+	0.280996i	−0.338062	+	0.0474970i
$36$		0			0
$37$			5.80642i			0.954570i	0.878749	+	0.477285i	$0.158379\pi$
$37$			5.80642i			0.954570i	−0.878749	+	0.477285i	$0.841621\pi$
$38$		0			0
$39$	2.90321			0.464886
$40$		0			0
$41$	−10.6637			−1.66539			−0.832695	−	0.553731i	$-0.813203\pi$
$41$	−10.6637			−1.66539			−0.832695	+	0.553731i	$0.813203\pi$
$42$		0			0
$43$			10.7096i			1.63320i	0.577201	+	0.816602i	$0.304145\pi$
$43$			10.7096i			1.63320i	−0.577201	+	0.816602i	$0.695855\pi$
$44$		0			0
$45$	−0.311108	−	2.21432i	−0.0463772	−	0.330091i
$46$		0			0
$47$		−	0.949145i		−	0.138447i	−0.997601	−	0.0692235i	$-0.977948\pi$
$47$		−	0.949145i		−	0.138447i	0.997601	−	0.0692235i	$-0.0220522\pi$
$48$		0			0
$49$	6.18421			0.883458
$50$		0			0
$51$	−2.28100			−0.319403
$52$		0			0
$53$		−	0.815792i		−	0.112058i	−0.998429	−	0.0560288i	$-0.982156\pi$
$53$		−	0.815792i		−	0.112058i	0.998429	−	0.0560288i	$-0.0178439\pi$
$54$		0			0
$55$	−0.311108	−	2.21432i	−0.0419498	−	0.298579i
$56$		0			0
$57$			2.42864i			0.321681i
$58$		0			0
$59$	−1.67307			−0.217815			−0.108908	−	0.994052i	$-0.534735\pi$
$59$	−1.67307			−0.217815			−0.108908	+	0.994052i	$0.534735\pi$
$60$		0			0
$61$	−7.24443			−0.927554			−0.463777	−	0.885952i	$-0.653506\pi$
$61$	−7.24443			−0.927554			−0.463777	+	0.885952i	$0.653506\pi$
$62$		0			0
$63$		−	0.903212i		−	0.113794i
$64$		0			0
$65$	6.42864	−	0.903212i	0.797375	−	0.112030i
$66$		0			0
$67$		−	12.8573i		−	1.57077i	−0.619010	−	0.785383i	$-0.712466\pi$
$67$		−	12.8573i		−	1.57077i	0.619010	−	0.785383i	$-0.287534\pi$
$68$		0			0
$69$	−4.00000			−0.481543
$70$		0			0
$71$	−9.28592			−1.10204			−0.551018	−	0.834493i	$-0.685760\pi$
$71$	−9.28592			−1.10204			−0.551018	+	0.834493i	$0.685760\pi$
$72$		0			0
$73$			5.65878i			0.662310i	0.943576	+	0.331155i	$0.107438\pi$
$73$			5.65878i			0.662310i	−0.943576	+	0.331155i	$0.892562\pi$
$74$		0			0
$75$	−1.37778	−	4.80642i	−0.159093	−	0.554998i
$76$		0			0
$77$		−	0.903212i		−	0.102931i
$78$		0			0
$79$	−16.5303			−1.85981			−0.929905	−	0.367800i	$-0.880111\pi$
$79$	−16.5303			−1.85981			−0.929905	+	0.367800i	$0.880111\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$			7.76049i			0.851825i	0.904764	+	0.425912i	$0.140047\pi$
$83$			7.76049i			0.851825i	−0.904764	+	0.425912i	$0.859953\pi$
$84$		0			0
$85$	−5.05086	+	0.709636i	−0.547842	+	0.0769708i
$86$		0			0
$87$		−	7.05086i		−	0.755931i
$88$		0			0
$89$	−6.13335			−0.650134			−0.325067	−	0.945691i	$-0.605387\pi$
$89$	−6.13335			−0.650134			−0.325067	+	0.945691i	$0.605387\pi$
$90$		0			0
$91$	2.62222			0.274883
$92$		0			0
$93$			2.62222i			0.271911i
$94$		0			0
$95$	0.755569	+	5.37778i	0.0775197	+	0.551749i
$96$		0			0
$97$		−	12.4701i		−	1.26615i	−0.774091	−	0.633075i	$-0.781793\pi$
$97$		−	12.4701i		−	1.26615i	0.774091	−	0.633075i	$-0.218207\pi$
$98$		0			0
$99$	1.00000			0.100504
$100$		0			0
$101$	16.1748			1.60946			0.804728	−	0.593643i	$-0.202311\pi$
$101$	16.1748			1.60946			0.804728	+	0.593643i	$0.202311\pi$
$102$		0			0
$103$		−	17.1526i		−	1.69009i	−0.534693	−	0.845046i	$-0.679573\pi$
$103$		−	17.1526i		−	1.69009i	0.534693	−	0.845046i	$-0.320427\pi$
$104$		0			0
$105$	−0.280996	−	2.00000i	−0.0274224	−	0.195180i
$106$		0			0
$107$			13.5669i			1.31156i	0.754951	+	0.655782i	$0.227661\pi$
$107$			13.5669i			1.31156i	−0.754951	+	0.655782i	$0.772339\pi$
$108$		0			0
$109$	10.0000			0.957826			0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000			0.957826			0.478913	+	0.877862i	$0.341031\pi$
$110$		0			0
$111$	−5.80642			−0.551121
$112$		0			0
$113$		−	14.2351i		−	1.33912i	−0.742757	−	0.669561i	$-0.766482\pi$
$113$		−	14.2351i		−	1.33912i	0.742757	−	0.669561i	$-0.233518\pi$
$114$		0			0
$115$	−8.85728	+	1.24443i	−0.825946	+	0.116044i
$116$		0			0
$117$			2.90321i			0.268402i
$118$		0			0
$119$	−2.06022			−0.188860
$120$		0			0
$121$	1.00000			0.0909091
$122$		0			0
$123$		−	10.6637i		−	0.961514i
$124$		0			0
$125$	−4.54617	−	10.2143i	−0.406622	−	0.913597i
$126$		0			0
$127$		−	11.0049i		−	0.976529i	−0.872696	−	0.488264i	$-0.837630\pi$
$127$		−	11.0049i		−	0.976529i	0.872696	−	0.488264i	$-0.162370\pi$
$128$		0			0
$129$	−10.7096			−0.942931
$130$		0			0
$131$	−1.24443			−0.108726			−0.0543632	−	0.998521i	$-0.517313\pi$
$131$	−1.24443			−0.108726			−0.0543632	+	0.998521i	$0.517313\pi$
$132$		0			0
$133$			2.19358i			0.190207i
$134$		0			0
$135$	2.21432	−	0.311108i	0.190578	−	0.0267759i
$136$		0			0
$137$		−	4.42864i		−	0.378364i	−0.981942	−	0.189182i	$-0.939416\pi$
$137$		−	4.42864i		−	0.378364i	0.981942	−	0.189182i	$-0.0605836\pi$
$138$		0			0
$139$	0.917502			0.0778215			0.0389108	−	0.999243i	$-0.487611\pi$
$139$	0.917502			0.0778215			0.0389108	+	0.999243i	$0.487611\pi$
$140$		0			0
$141$	0.949145			0.0799324
$142$		0			0
$143$			2.90321i			0.242779i
$144$		0			0
$145$	−2.19358	−	15.6128i	−0.182167	−	1.29658i
$146$		0			0
$147$			6.18421i			0.510065i
$148$		0			0
$149$	−2.19358			−0.179705			−0.0898524	−	0.995955i	$-0.528640\pi$
$149$	−2.19358			−0.179705			−0.0898524	+	0.995955i	$0.528640\pi$
$150$		0			0
$151$	10.4286			0.848671			0.424335	−	0.905505i	$-0.360508\pi$
$151$	10.4286			0.848671			0.424335	+	0.905505i	$0.360508\pi$
$152$		0			0
$153$		−	2.28100i		−	0.184408i
$154$		0			0
$155$	0.815792	+	5.80642i	0.0655260	+	0.466383i
$156$		0			0
$157$			5.80642i			0.463403i	0.972787	+	0.231702i	$0.0744293\pi$
$157$			5.80642i			0.463403i	−0.972787	+	0.231702i	$0.925571\pi$
$158$		0			0
$159$	0.815792			0.0646965
$160$		0			0
$161$	−3.61285			−0.284732
$162$		0			0
$163$			11.0509i			0.865570i	0.901497	+	0.432785i	$0.142469\pi$
$163$			11.0509i			0.865570i	−0.901497	+	0.432785i	$0.857531\pi$
$164$		0			0
$165$	2.21432	−	0.311108i	0.172385	−	0.0242197i
$166$		0			0
$167$		−	13.9541i		−	1.07980i	−0.841730	−	0.539899i	$-0.818462\pi$
$167$		−	13.9541i		−	1.07980i	0.841730	−	0.539899i	$-0.181538\pi$
$168$		0			0
$169$	4.57136			0.351643
$170$		0			0
$171$	−2.42864			−0.185723
$172$		0			0
$173$			18.8430i			1.43261i	0.697789	+	0.716303i	$0.254167\pi$
$173$			18.8430i			1.43261i	−0.697789	+	0.716303i	$0.745833\pi$
$174$		0			0
$175$	−1.24443	−	4.34122i	−0.0940702	−	0.328165i
$176$		0			0
$177$		−	1.67307i		−	0.125756i
$178$		0			0
$179$	4.85728			0.363050			0.181525	−	0.983386i	$-0.441897\pi$
$179$	4.85728			0.363050			0.181525	+	0.983386i	$0.441897\pi$
$180$		0			0
$181$	−16.7971			−1.24852			−0.624258	−	0.781219i	$-0.714598\pi$
$181$	−16.7971			−1.24852			−0.624258	+	0.781219i	$0.714598\pi$
$182$		0			0
$183$		−	7.24443i		−	0.535524i
$184$		0			0
$185$	−12.8573	+	1.80642i	−0.945286	+	0.132811i
$186$		0			0
$187$		−	2.28100i		−	0.166803i
$188$		0			0
$189$	0.903212			0.0656990
$190$		0			0
$191$	−16.8573			−1.21975			−0.609875	−	0.792498i	$-0.708780\pi$
$191$	−16.8573			−1.21975			−0.609875	+	0.792498i	$0.708780\pi$
$192$		0			0
$193$			24.6178i			1.77203i	0.463661	+	0.886013i	$0.346536\pi$
$193$			24.6178i			1.77203i	−0.463661	+	0.886013i	$0.653464\pi$
$194$		0			0
$195$	0.903212	+	6.42864i	0.0646803	+	0.460364i
$196$		0			0
$197$			14.8716i			1.05956i	0.848137	+	0.529778i	$0.177725\pi$
$197$			14.8716i			1.05956i	−0.848137	+	0.529778i	$0.822275\pi$
$198$		0			0
$199$	−11.2257			−0.795768			−0.397884	−	0.917436i	$-0.630255\pi$
$199$	−11.2257			−0.795768			−0.397884	+	0.917436i	$0.630255\pi$
$200$		0			0
$201$	12.8573			0.906883
$202$		0			0
$203$		−	6.36842i		−	0.446975i
$204$		0			0
$205$	−3.31756	−	23.6128i	−0.231709	−	1.64919i
$206$		0			0
$207$		−	4.00000i		−	0.278019i
$208$		0			0
$209$	−2.42864			−0.167993
$210$		0			0
$211$	11.9398			0.821968			0.410984	−	0.911643i	$-0.365185\pi$
$211$	11.9398			0.821968			0.410984	+	0.911643i	$0.365185\pi$
$212$		0			0
$213$		−	9.28592i		−	0.636261i
$214$		0			0
$215$	−23.7146	+	3.33185i	−1.61732	+	0.227230i
$216$		0			0
$217$			2.36842i			0.160779i
$218$		0			0
$219$	−5.65878			−0.382385
$220$		0			0
$221$	6.62222			0.445458
$222$		0			0
$223$			21.8064i			1.46027i	0.683305	+	0.730133i	$0.260542\pi$
$223$			21.8064i			1.46027i	−0.683305	+	0.730133i	$0.739458\pi$
$224$		0			0
$225$	4.80642	−	1.37778i	0.320428	−	0.0918523i
$226$		0			0
$227$		−	3.19850i		−	0.212292i	−0.994351	−	0.106146i	$-0.966149\pi$
$227$		−	3.19850i		−	0.212292i	0.994351	−	0.106146i	$-0.0338511\pi$
$228$		0			0
$229$	−7.12399			−0.470766			−0.235383	−	0.971903i	$-0.575634\pi$
$229$	−7.12399			−0.470766			−0.235383	+	0.971903i	$0.575634\pi$
$230$		0			0
$231$	0.903212			0.0594270
$232$		0			0
$233$			19.5254i			1.27915i	0.768727	+	0.639577i	$0.220890\pi$
$233$			19.5254i			1.27915i	−0.768727	+	0.639577i	$0.779110\pi$
$234$		0			0
$235$	2.10171	−	0.295286i	0.137100	−	0.0192624i
$236$		0			0
$237$		−	16.5303i		−	1.07376i
$238$		0			0
$239$	−21.9813			−1.42185			−0.710925	−	0.703268i	$-0.751723\pi$
$239$	−21.9813			−1.42185			−0.710925	+	0.703268i	$0.751723\pi$
$240$		0			0
$241$	5.34614			0.344375			0.172188	−	0.985064i	$-0.444916\pi$
$241$	5.34614			0.344375			0.172188	+	0.985064i	$0.444916\pi$
$242$		0			0
$243$			1.00000i			0.0641500i
$244$		0			0
$245$	1.92396	+	13.6938i	0.122917	+	0.874866i
$246$		0			0
$247$		−	7.05086i		−	0.448635i
$248$		0			0
$249$	−7.76049			−0.491801
$250$		0			0
$251$	23.7748			1.50065			0.750325	−	0.661069i	$-0.229897\pi$
$251$	23.7748			1.50065			0.750325	+	0.661069i	$0.229897\pi$
$252$		0			0
$253$		−	4.00000i		−	0.251478i
$254$		0			0
$255$	−0.709636	−	5.05086i	−0.0444391	−	0.316297i
$256$		0			0
$257$		−	8.13335i		−	0.507345i	−0.967290	−	0.253672i	$-0.918362\pi$
$257$		−	8.13335i		−	0.507345i	0.967290	−	0.253672i	$-0.0816385\pi$
$258$		0			0
$259$	−5.24443			−0.325873
$260$		0			0
$261$	7.05086			0.436437
$262$		0			0
$263$			22.9032i			1.41227i	0.708076	+	0.706136i	$0.249563\pi$
$263$			22.9032i			1.41227i	−0.708076	+	0.706136i	$0.750437\pi$
$264$		0			0
$265$	1.80642	−	0.253799i	0.110968	−	0.0155908i
$266$		0			0
$267$		−	6.13335i		−	0.375355i
$268$		0			0
$269$	−11.8350			−0.721593			−0.360796	−	0.932645i	$-0.617495\pi$
$269$	−11.8350			−0.721593			−0.360796	+	0.932645i	$0.617495\pi$
$270$		0			0
$271$	−14.8988			−0.905036			−0.452518	−	0.891755i	$-0.649474\pi$
$271$	−14.8988			−0.905036			−0.452518	+	0.891755i	$0.649474\pi$
$272$		0			0
$273$			2.62222i			0.158704i
$274$		0			0
$275$	4.80642	−	1.37778i	0.289838	−	0.0830835i
$276$		0			0
$277$		−	27.6686i		−	1.66245i	−0.555939	−	0.831223i	$-0.687641\pi$
$277$		−	27.6686i		−	1.66245i	0.555939	−	0.831223i	$-0.312359\pi$
$278$		0			0
$279$	−2.62222			−0.156988
$280$		0			0
$281$	9.80642			0.585002			0.292501	−	0.956265i	$-0.405512\pi$
$281$	9.80642			0.585002			0.292501	+	0.956265i	$0.405512\pi$
$282$		0			0
$283$		−	19.0049i		−	1.12973i	−0.825185	−	0.564863i	$-0.808929\pi$
$283$		−	19.0049i		−	1.12973i	0.825185	−	0.564863i	$-0.191071\pi$
$284$		0			0
$285$	−5.37778	+	0.755569i	−0.318552	+	0.0447560i
$286$		0			0
$287$		−	9.63158i		−	0.568534i
$288$		0			0
$289$	11.7971			0.693944
$290$		0			0
$291$	12.4701			0.731012
$292$		0			0
$293$			30.7511i			1.79650i	0.439485	+	0.898250i	$0.355161\pi$
$293$			30.7511i			1.79650i	−0.439485	+	0.898250i	$0.644839\pi$
$294$		0			0
$295$	−0.520505	−	3.70471i	−0.0303050	−	0.215697i
$296$		0			0
$297$			1.00000i			0.0580259i
$298$		0			0
$299$	11.6128			0.671588
$300$		0			0
$301$	−9.67307			−0.557547
$302$		0			0
$303$			16.1748i			0.929220i
$304$		0			0
$305$	−2.25380	−	16.0415i	−0.129052	−	0.918533i
$306$		0			0
$307$			13.4938i			0.770131i	0.922889	+	0.385065i	$0.125821\pi$
$307$			13.4938i			0.770131i	−0.922889	+	0.385065i	$0.874179\pi$
$308$		0			0
$309$	17.1526			0.975775
$310$		0			0
$311$	17.5526			0.995318			0.497659	−	0.867373i	$-0.334193\pi$
$311$	17.5526			0.995318			0.497659	+	0.867373i	$0.334193\pi$
$312$		0			0
$313$		−	14.3970i		−	0.813766i	−0.913480	−	0.406883i	$-0.866616\pi$
$313$		−	14.3970i		−	0.813766i	0.913480	−	0.406883i	$-0.133384\pi$
$314$		0			0
$315$	2.00000	−	0.280996i	0.112687	−	0.0158323i
$316$		0			0
$317$			29.4608i			1.65468i	0.561701	+	0.827341i	$0.310147\pi$
$317$			29.4608i			1.65468i	−0.561701	+	0.827341i	$0.689853\pi$
$318$		0			0
$319$	7.05086			0.394772
$320$		0			0
$321$	−13.5669			−0.757231
$322$		0			0
$323$			5.53972i			0.308238i
$324$		0			0
$325$	4.00000	+	13.9541i	0.221880	+	0.774032i
$326$		0			0
$327$			10.0000i			0.553001i
$328$		0			0
$329$	0.857279			0.0472633
$330$		0			0
$331$	2.62222			0.144130			0.0720650	−	0.997400i	$-0.477041\pi$
$331$	2.62222			0.144130			0.0720650	+	0.997400i	$0.477041\pi$
$332$		0			0
$333$		−	5.80642i		−	0.318190i
$334$		0			0
$335$	28.4701	−	4.00000i	1.55549	−	0.218543i
$336$		0			0
$337$		−	5.00492i		−	0.272635i	−0.990665	−	0.136318i	$-0.956473\pi$
$337$		−	5.00492i		−	0.272635i	0.990665	−	0.136318i	$-0.0435268\pi$
$338$		0			0
$339$	14.2351			0.773143
$340$		0			0
$341$	−2.62222			−0.142001
$342$		0			0
$343$			11.9081i			0.642979i
$344$		0			0
$345$	−1.24443	−	8.85728i	−0.0669979	−	0.476860i
$346$		0			0
$347$			22.8113i			1.22458i	0.790634	+	0.612289i	$0.209751\pi$
$347$			22.8113i			1.22458i	−0.790634	+	0.612289i	$0.790249\pi$
$348$		0			0
$349$	−21.2257			−1.13619			−0.568093	−	0.822965i	$-0.692319\pi$
$349$	−21.2257			−1.13619			−0.568093	+	0.822965i	$0.692319\pi$
$350$		0			0
$351$	−2.90321			−0.154962
$352$		0			0
$353$		−	7.18421i		−	0.382377i	−0.981553	−	0.191188i	$-0.938766\pi$
$353$		−	7.18421i		−	0.382377i	0.981553	−	0.191188i	$-0.0612341\pi$
$354$		0			0
$355$	−2.88892	−	20.5620i	−0.153328	−	1.09132i
$356$		0			0
$357$		−	2.06022i		−	0.109039i
$358$		0			0
$359$	−14.1017			−0.744260			−0.372130	−	0.928181i	$-0.621372\pi$
$359$	−14.1017			−0.744260			−0.372130	+	0.928181i	$0.621372\pi$
$360$		0			0
$361$	−13.1017			−0.689564
$362$		0			0
$363$			1.00000i			0.0524864i
$364$		0			0
$365$	−12.5303	+	1.76049i	−0.655868	+	0.0921483i
$366$		0			0
$367$			3.90813i			0.204003i	0.994784	+	0.102001i	$0.0325246\pi$
$367$			3.90813i			0.204003i	−0.994784	+	0.102001i	$0.967475\pi$
$368$		0			0
$369$	10.6637			0.555130
$370$		0			0
$371$	0.736833			0.0382545
$372$		0			0
$373$		−	12.9763i		−	0.671890i	−0.941882	−	0.335945i	$-0.890944\pi$
$373$		−	12.9763i		−	0.671890i	0.941882	−	0.335945i	$-0.109056\pi$
$374$		0			0
$375$	10.2143	−	4.54617i	0.527465	−	0.234763i
$376$		0			0
$377$			20.4701i			1.05427i
$378$		0			0
$379$	36.0830			1.85346			0.926729	−	0.375731i	$-0.122608\pi$
$379$	36.0830			1.85346			0.926729	+	0.375731i	$0.122608\pi$
$380$		0			0
$381$	11.0049			0.563799
$382$		0			0
$383$			20.2953i			1.03704i	0.855065	+	0.518520i	$0.173517\pi$
$383$			20.2953i			1.03704i	−0.855065	+	0.518520i	$0.826483\pi$
$384$		0			0
$385$	2.00000	−	0.280996i	0.101929	−	0.0143209i
$386$		0			0
$387$		−	10.7096i		−	0.544401i
$388$		0			0
$389$	−30.4701			−1.54490			−0.772448	−	0.635078i	$-0.780968\pi$
$389$	−30.4701			−1.54490			−0.772448	+	0.635078i	$0.780968\pi$
$390$		0			0
$391$	−9.12399			−0.461420
$392$		0			0
$393$		−	1.24443i		−	0.0627733i
$394$		0			0
$395$	−5.14272	−	36.6035i	−0.258758	−	1.84172i
$396$		0			0
$397$			4.97773i			0.249825i	0.992168	+	0.124912i	$0.0398650\pi$
$397$			4.97773i			0.249825i	−0.992168	+	0.124912i	$0.960135\pi$
$398$		0			0
$399$	−2.19358			−0.109816
$400$		0			0
$401$	−1.86665			−0.0932159			−0.0466079	−	0.998913i	$-0.514841\pi$
$401$	−1.86665			−0.0932159			−0.0466079	+	0.998913i	$0.514841\pi$
$402$		0			0
$403$		−	7.61285i		−	0.379223i
$404$		0			0
$405$	0.311108	+	2.21432i	0.0154591	+	0.110030i
$406$		0			0
$407$		−	5.80642i		−	0.287814i
$408$		0			0
$409$	3.63158			0.179570			0.0897851	−	0.995961i	$-0.471382\pi$
$409$	3.63158			0.179570			0.0897851	+	0.995961i	$0.471382\pi$
$410$		0			0
$411$	4.42864			0.218449
$412$		0			0
$413$		−	1.51114i		−	0.0743582i
$414$		0			0
$415$	−17.1842	+	2.41435i	−0.843540	+	0.118516i
$416$		0			0
$417$			0.917502i			0.0449303i
$418$		0			0
$419$	4.85728			0.237294			0.118647	−	0.992937i	$-0.462144\pi$
$419$	4.85728			0.237294			0.118647	+	0.992937i	$0.462144\pi$
$420$		0			0
$421$	22.6321			1.10302			0.551510	−	0.834169i	$-0.314052\pi$
$421$	22.6321			1.10302			0.551510	+	0.834169i	$0.314052\pi$
$422$		0			0
$423$			0.949145i			0.0461490i
$424$		0			0
$425$	−3.14272	−	10.9634i	−0.152444	−	0.531805i
$426$		0			0
$427$		−	6.54326i		−	0.316650i
$428$		0			0
$429$	−2.90321			−0.140168
$430$		0			0
$431$	−1.24443			−0.0599421			−0.0299711	−	0.999551i	$-0.509542\pi$
$431$	−1.24443			−0.0599421			−0.0299711	+	0.999551i	$0.509542\pi$
$432$		0			0
$433$			16.0000i			0.768911i	0.923144	+	0.384455i	$0.125611\pi$
$433$			16.0000i			0.768911i	−0.923144	+	0.384455i	$0.874389\pi$
$434$		0			0
$435$	15.6128	−	2.19358i	0.748579	−	0.105174i
$436$		0			0
$437$			9.71456i			0.464710i
$438$		0			0
$439$	−2.42864			−0.115913			−0.0579563	−	0.998319i	$-0.518458\pi$
$439$	−2.42864			−0.115913			−0.0579563	+	0.998319i	$0.518458\pi$
$440$		0			0
$441$	−6.18421			−0.294486
$442$		0			0
$443$			31.0509i			1.47527i	0.675199	+	0.737635i	$0.264058\pi$
$443$			31.0509i			1.47527i	−0.675199	+	0.737635i	$0.735942\pi$
$444$		0			0
$445$	−1.90813	−	13.5812i	−0.0904542	−	0.643811i
$446$		0			0
$447$		−	2.19358i		−	0.103753i
$448$		0			0
$449$	37.3590			1.76308			0.881541	−	0.472107i	$-0.156506\pi$
$449$	37.3590			1.76308			0.881541	+	0.472107i	$0.156506\pi$
$450$		0			0
$451$	10.6637			0.502134
$452$		0			0
$453$			10.4286i			0.489980i
$454$		0			0
$455$	0.815792	+	5.80642i	0.0382449	+	0.272209i
$456$		0			0
$457$			8.73822i			0.408757i	0.978892	+	0.204378i	$0.0655172\pi$
$457$			8.73822i			0.408757i	−0.978892	+	0.204378i	$0.934483\pi$
$458$		0			0
$459$	2.28100			0.106468
$460$		0			0
$461$	31.7877			1.48050			0.740250	−	0.672332i	$-0.234707\pi$
$461$	31.7877			1.48050			0.740250	+	0.672332i	$0.234707\pi$
$462$		0			0
$463$			12.0919i			0.561957i	0.959714	+	0.280978i	$0.0906589\pi$
$463$			12.0919i			0.561957i	−0.959714	+	0.280978i	$0.909341\pi$
$464$		0			0
$465$	−5.80642	+	0.815792i	−0.269266	+	0.0378314i
$466$		0			0
$467$			15.3461i			0.710135i	0.934841	+	0.355067i	$0.115542\pi$
$467$			15.3461i			0.710135i	−0.934841	+	0.355067i	$0.884458\pi$
$468$		0			0
$469$	11.6128			0.536231
$470$		0			0
$471$	−5.80642			−0.267546
$472$		0			0
$473$		−	10.7096i		−	0.492430i
$474$		0			0
$475$	−11.6731	+	3.34614i	−0.535597	+	0.153532i
$476$		0			0
$477$			0.815792i			0.0373525i
$478$		0			0
$479$	5.89829			0.269500			0.134750	−	0.990880i	$-0.456977\pi$
$479$	5.89829			0.269500			0.134750	+	0.990880i	$0.456977\pi$
$480$		0			0
$481$	16.8573			0.768626
$482$		0			0
$483$		−	3.61285i		−	0.164390i
$484$		0			0
$485$	27.6128	−	3.87955i	1.25383	−	0.176161i
$486$		0			0
$487$		−	31.3461i		−	1.42043i	−0.703985	−	0.710215i	$-0.748598\pi$
$487$		−	31.3461i		−	1.42043i	0.703985	−	0.710215i	$-0.251402\pi$
$488$		0			0
$489$	−11.0509			−0.499737
$490$		0			0
$491$	8.00000			0.361035			0.180517	−	0.983572i	$-0.442223\pi$
$491$	8.00000			0.361035			0.180517	+	0.983572i	$0.442223\pi$
$492$		0			0
$493$		−	16.0830i		−	0.724341i
$494$		0			0
$495$	0.311108	+	2.21432i	0.0139833	+	0.0995263i
$496$		0			0
$497$		−	8.38715i		−	0.376215i
$498$		0			0
$499$	15.1427			0.677881			0.338941	−	0.940808i	$-0.389931\pi$
$499$	15.1427			0.677881			0.338941	+	0.940808i	$0.389931\pi$
$500$		0			0
$501$	13.9541			0.623422
$502$		0			0
$503$			26.0370i			1.16093i	0.814284	+	0.580467i	$0.197130\pi$
$503$			26.0370i			1.16093i	−0.814284	+	0.580467i	$0.802870\pi$
$504$		0			0
$505$	5.03212	+	35.8163i	0.223926	+	1.59380i
$506$		0			0
$507$			4.57136i			0.203021i
$508$		0			0
$509$	24.5718			1.08913			0.544564	−	0.838719i	$-0.316695\pi$
$509$	24.5718			1.08913			0.544564	+	0.838719i	$0.316695\pi$
$510$		0			0
$511$	−5.11108			−0.226101
$512$		0			0
$513$		−	2.42864i		−	0.107227i
$514$		0			0
$515$	37.9813	−	5.33630i	1.67365	−	0.235145i
$516$		0			0
$517$			0.949145i			0.0417433i
$518$		0			0
$519$	−18.8430			−0.827115
$520$		0			0
$521$	−4.88892			−0.214188			−0.107094	−	0.994249i	$-0.534155\pi$
$521$	−4.88892			−0.214188			−0.107094	+	0.994249i	$0.534155\pi$
$522$		0			0
$523$		−	4.22077i		−	0.184562i	−0.995733	−	0.0922808i	$-0.970584\pi$
$523$		−	4.22077i		−	0.184562i	0.995733	−	0.0922808i	$-0.0294157\pi$
$524$		0			0
$525$	4.34122	−	1.24443i	0.189466	−	0.0543114i
$526$		0			0
$527$			5.98126i			0.260548i
$528$		0			0
$529$	7.00000			0.304348
$530$		0			0
$531$	1.67307			0.0726051
$532$		0			0
$533$			30.9590i			1.34098i
$534$		0			0
$535$	−30.0415	+	4.22077i	−1.29881	+	0.182480i
$536$		0			0
$537$			4.85728i			0.209607i
$538$		0			0
$539$	−6.18421			−0.266373
$540$		0			0
$541$	−13.6128			−0.585262			−0.292631	−	0.956225i	$-0.594531\pi$
$541$	−13.6128			−0.585262			−0.292631	+	0.956225i	$0.594531\pi$
$542$		0			0
$543$		−	16.7971i		−	0.720831i
$544$		0			0
$545$	3.11108	+	22.1432i	0.133264	+	0.948510i
$546$		0			0
$547$			7.48394i			0.319990i	0.987118	+	0.159995i	$0.0511478\pi$
$547$			7.48394i			0.319990i	−0.987118	+	0.159995i	$0.948852\pi$
$548$		0			0
$549$	7.24443			0.309185
$550$		0			0
$551$	−17.1240			−0.729506
$552$		0			0
$553$		−	14.9304i		−	0.634906i
$554$		0			0
$555$	−1.80642	−	12.8573i	−0.0766784	−	0.545761i
$556$		0			0
$557$		−	32.2908i		−	1.36821i	−0.729385	−	0.684103i	$-0.760194\pi$
$557$		−	32.2908i		−	1.36821i	0.729385	−	0.684103i	$-0.239806\pi$
$558$		0			0
$559$	31.0923			1.31507
$560$		0			0
$561$	2.28100			0.0963037
$562$		0			0
$563$		−	7.49378i		−	0.315825i	−0.987453	−	0.157913i	$-0.949524\pi$
$563$		−	7.49378i		−	0.315825i	0.987453	−	0.157913i	$-0.0504765\pi$
$564$		0			0
$565$	31.5210	−	4.42864i	1.32610	−	0.186314i
$566$		0			0
$567$			0.903212i			0.0379313i
$568$		0			0
$569$	12.9491			0.542856			0.271428	−	0.962459i	$-0.412504\pi$
$569$	12.9491			0.542856			0.271428	+	0.962459i	$0.412504\pi$
$570$		0			0
$571$	15.2859			0.639696			0.319848	−	0.947469i	$-0.396368\pi$
$571$	15.2859			0.639696			0.319848	+	0.947469i	$0.396368\pi$
$572$		0			0
$573$		−	16.8573i		−	0.704223i
$574$		0			0
$575$	−5.51114	−	19.2257i	−0.229830	−	0.801767i
$576$		0			0
$577$		−	28.4415i		−	1.18404i	−0.805924	−	0.592019i	$-0.798331\pi$
$577$		−	28.4415i		−	1.18404i	0.805924	−	0.592019i	$-0.201669\pi$
$578$		0			0
$579$	−24.6178			−1.02308
$580$		0			0
$581$	−7.00937			−0.290798
$582$		0			0
$583$			0.815792i			0.0337866i
$584$		0			0
$585$	−6.42864	+	0.903212i	−0.265792	+	0.0373432i
$586$		0			0
$587$		−	8.47013i		−	0.349600i	−0.984604	−	0.174800i	$-0.944072\pi$
$587$		−	8.47013i		−	0.349600i	0.984604	−	0.174800i	$-0.0559278\pi$
$588$		0			0
$589$	6.36842			0.262406
$590$		0			0
$591$	−14.8716			−0.611735
$592$		0			0
$593$			26.5763i			1.09136i	0.837995	+	0.545679i	$0.183728\pi$
$593$			26.5763i			1.09136i	−0.837995	+	0.545679i	$0.816272\pi$
$594$		0			0
$595$	−0.640951	−	4.56199i	−0.0262764	−	0.187023i
$596$		0			0
$597$		−	11.2257i		−	0.459437i
$598$		0			0
$599$	−8.77430			−0.358508			−0.179254	−	0.983803i	$-0.557368\pi$
$599$	−8.77430			−0.358508			−0.179254	+	0.983803i	$0.557368\pi$
$600$		0			0
$601$	−41.8163			−1.70572			−0.852861	−	0.522139i	$-0.825134\pi$
$601$	−41.8163			−1.70572			−0.852861	+	0.522139i	$0.825134\pi$
$602$		0			0
$603$			12.8573i			0.523589i
$604$		0			0
$605$	0.311108	+	2.21432i	0.0126483	+	0.0900249i
$606$		0			0
$607$			29.9353i			1.21504i	0.794305	+	0.607519i	$0.207835\pi$
$607$			29.9353i			1.21504i	−0.794305	+	0.607519i	$0.792165\pi$
$608$		0			0
$609$	6.36842			0.258061
$610$		0			0
$611$	−2.75557			−0.111478
$612$		0			0
$613$			26.1289i			1.05534i	0.849450	+	0.527668i	$0.176934\pi$
$613$			26.1289i			1.05534i	−0.849450	+	0.527668i	$0.823066\pi$
$614$		0			0
$615$	23.6128	−	3.31756i	0.952162	−	0.133777i
$616$		0			0
$617$			3.66323i			0.147476i	0.997278	+	0.0737380i	$0.0234928\pi$
$617$			3.66323i			0.147476i	−0.997278	+	0.0737380i	$0.976507\pi$
$618$		0			0
$619$	43.2958			1.74020			0.870102	−	0.492872i	$-0.164053\pi$
$619$	43.2958			1.74020			0.870102	+	0.492872i	$0.164053\pi$
$620$		0			0
$621$	4.00000			0.160514
$622$		0			0
$623$		−	5.53972i		−	0.221944i
$624$		0			0
$625$	21.2034	−	13.2444i	0.848137	−	0.529777i
$626$		0			0
$627$		−	2.42864i		−	0.0969905i
$628$		0			0
$629$	−13.2444			−0.528090
$630$		0			0
$631$	−8.97773			−0.357398			−0.178699	−	0.983904i	$-0.557189\pi$
$631$	−8.97773			−0.357398			−0.178699	+	0.983904i	$0.557189\pi$
$632$		0			0
$633$			11.9398i			0.474564i
$634$		0			0
$635$	24.3684	−	3.42372i	0.967031	−	0.135866i
$636$		0			0
$637$		−	17.9541i		−	0.711366i
$638$		0			0
$639$	9.28592			0.367345
$640$		0			0
$641$	9.21279			0.363883			0.181942	−	0.983309i	$-0.441762\pi$
$641$	9.21279			0.363883			0.181942	+	0.983309i	$0.441762\pi$
$642$		0			0
$643$			16.3783i			0.645896i	0.946417	+	0.322948i	$0.104674\pi$
$643$			16.3783i			0.645896i	−0.946417	+	0.322948i	$0.895326\pi$
$644$		0			0
$645$	−3.33185	−	23.7146i	−0.131192	−	0.933760i
$646$		0			0
$647$			9.80642i			0.385530i	0.981245	+	0.192765i	$0.0617456\pi$
$647$			9.80642i			0.385530i	−0.981245	+	0.192765i	$0.938254\pi$
$648$		0			0
$649$	1.67307			0.0656738
$650$		0			0
$651$	−2.36842			−0.0928256
$652$		0			0
$653$			33.0736i			1.29427i	0.762375	+	0.647135i	$0.224033\pi$
$653$			33.0736i			1.29427i	−0.762375	+	0.647135i	$0.775967\pi$
$654$		0			0
$655$	−0.387152	−	2.75557i	−0.0151273	−	0.107669i
$656$		0			0
$657$		−	5.65878i		−	0.220770i
$658$		0			0
$659$	34.1017			1.32841			0.664207	−	0.747549i	$-0.268769\pi$
$659$	34.1017			1.32841			0.664207	+	0.747549i	$0.268769\pi$
$660$		0			0
$661$	−5.40943			−0.210402			−0.105201	−	0.994451i	$-0.533549\pi$
$661$	−5.40943			−0.210402			−0.105201	+	0.994451i	$0.533549\pi$
$662$		0			0
$663$			6.62222i			0.257186i
$664$		0			0
$665$	−4.85728	+	0.682439i	−0.188357	+	0.0264638i
$666$		0			0
$667$		−	28.2034i		−	1.09204i
$668$		0			0
$669$	−21.8064			−0.843085
$670$		0			0
$671$	7.24443			0.279668
$672$		0			0
$673$			24.1476i			0.930823i	0.885094	+	0.465412i	$0.154094\pi$
$673$			24.1476i			0.930823i	−0.885094	+	0.465412i	$0.845906\pi$
$674$		0			0
$675$	1.37778	+	4.80642i	0.0530309	+	0.184999i
$676$		0			0
$677$		−	26.2810i		−	1.01006i	−0.863102	−	0.505030i	$-0.831481\pi$
$677$		−	26.2810i		−	1.01006i	0.863102	−	0.505030i	$-0.168519\pi$
$678$		0			0
$679$	11.2632			0.432241
$680$		0			0
$681$	3.19850			0.122567
$682$		0			0
$683$		−	15.3176i		−	0.586110i	−0.956096	−	0.293055i	$-0.905328\pi$
$683$		−	15.3176i		−	0.586110i	0.956096	−	0.293055i	$-0.0946719\pi$
$684$		0			0
$685$	9.80642	−	1.37778i	0.374684	−	0.0526424i
$686$		0			0
$687$		−	7.12399i		−	0.271797i
$688$		0			0
$689$	−2.36842			−0.0902295
$690$		0			0
$691$	−15.0223			−0.571474			−0.285737	−	0.958308i	$-0.592238\pi$
$691$	−15.0223			−0.571474			−0.285737	+	0.958308i	$0.592238\pi$
$692$		0			0
$693$			0.903212i			0.0343102i
$694$		0			0
$695$	0.285442	+	2.03164i	0.0108274	+	0.0770646i
$696$		0			0
$697$		−	24.3239i		−	0.921332i
$698$		0			0
$699$	−19.5254			−0.738519
$700$		0			0
$701$	−19.9081			−0.751920			−0.375960	−	0.926636i	$-0.622687\pi$
$701$	−19.9081			−0.751920			−0.375960	+	0.926636i	$0.622687\pi$
$702$		0			0
$703$			14.1017i			0.531856i
$704$		0			0
$705$	0.295286	+	2.10171i	0.0111211	+	0.0791550i
$706$		0			0
$707$			14.6093i			0.549440i
$708$		0			0
$709$	13.5081			0.507306			0.253653	−	0.967295i	$-0.418368\pi$
$709$	13.5081			0.507306			0.253653	+	0.967295i	$0.418368\pi$
$710$		0			0
$711$	16.5303			0.619937
$712$		0			0
$713$			10.4889i			0.392811i
$714$		0			0
$715$	−6.42864	+	0.903212i	−0.240417	+	0.0337782i
$716$		0			0
$717$		−	21.9813i		−	0.820905i
$718$		0			0
$719$	−16.0830			−0.599794			−0.299897	−	0.953972i	$-0.596952\pi$
$719$	−16.0830			−0.599794			−0.299897	+	0.953972i	$0.596952\pi$
$720$		0			0
$721$	15.4924			0.576967
$722$		0			0
$723$			5.34614i			0.198825i
$724$		0			0
$725$	33.8894	−	9.71456i	1.25862	−	0.360790i
$726$		0			0
$727$		−	23.6128i		−	0.875752i	−0.899035	−	0.437876i	$-0.855731\pi$
$727$		−	23.6128i		−	0.875752i	0.899035	−	0.437876i	$-0.144269\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$	−24.4286			−0.903526
$732$		0			0
$733$			30.0459i			1.10977i	0.831926	+	0.554886i	$0.187238\pi$
$733$			30.0459i			1.10977i	−0.831926	+	0.554886i	$0.812762\pi$
$734$		0			0
$735$	−13.6938	+	1.92396i	−0.505104	+	0.0709662i
$736$		0			0
$737$			12.8573i			0.473604i
$738$		0			0
$739$	−24.4099			−0.897933			−0.448966	−	0.893549i	$-0.648208\pi$
$739$	−24.4099			−0.897933			−0.448966	+	0.893549i	$0.648208\pi$
$740$		0			0
$741$	7.05086			0.259020
$742$		0			0
$743$		−	33.1798i		−	1.21725i	−0.793459	−	0.608624i	$-0.791722\pi$
$743$		−	33.1798i		−	1.21725i	0.793459	−	0.608624i	$-0.208278\pi$
$744$		0			0
$745$	−0.682439	−	4.85728i	−0.0250026	−	0.177957i
$746$		0			0
$747$		−	7.76049i		−	0.283942i
$748$		0			0
$749$	−12.2538			−0.447744
$750$		0			0
$751$	22.5718			0.823658			0.411829	−	0.911261i	$-0.364890\pi$
$751$	22.5718			0.823658			0.411829	+	0.911261i	$0.364890\pi$
$752$		0			0
$753$			23.7748i			0.866401i
$754$		0			0
$755$	3.24443	+	23.0923i	0.118077	+	0.840416i
$756$		0			0
$757$		−	4.94914i		−	0.179880i	−0.995947	−	0.0899399i	$-0.971333\pi$
$757$		−	4.94914i		−	0.179880i	0.995947	−	0.0899399i	$-0.0286675\pi$
$758$		0			0
$759$	4.00000			0.145191
$760$		0			0
$761$	14.6637			0.531559			0.265779	−	0.964034i	$-0.414371\pi$
$761$	14.6637			0.531559			0.265779	+	0.964034i	$0.414371\pi$
$762$		0			0
$763$			9.03212i			0.326985i
$764$		0			0
$765$	5.05086	−	0.709636i	0.182614	−	0.0256569i
$766$		0			0
$767$			4.85728i			0.175386i
$768$		0			0
$769$	44.5718			1.60730			0.803651	−	0.595101i	$-0.202888\pi$
$769$	44.5718			1.60730			0.803651	+	0.595101i	$0.202888\pi$
$770$		0			0
$771$	8.13335			0.292916
$772$		0			0
$773$			17.3145i			0.622759i	0.950286	+	0.311380i	$0.100791\pi$
$773$			17.3145i			0.622759i	−0.950286	+	0.311380i	$0.899209\pi$
$774$		0			0
$775$	−12.6035	+	3.61285i	−0.452730	+	0.129777i
$776$		0			0
$777$		−	5.24443i		−	0.188143i
$778$		0			0
$779$	−25.8983			−0.927903
$780$		0			0
$781$	9.28592			0.332276
$782$		0			0
$783$			7.05086i			0.251977i
$784$		0			0
$785$	−12.8573	+	1.80642i	−0.458896	+	0.0644740i
$786$		0			0
$787$			36.5161i			1.30166i	0.759225	+	0.650828i	$0.225578\pi$
$787$			36.5161i			1.30166i	−0.759225	+	0.650828i	$0.774422\pi$
$788$		0			0
$789$	−22.9032			−0.815376
$790$		0			0
$791$	12.8573			0.457152
$792$		0			0
$793$			21.0321i			0.746872i
$794$		0			0
$795$	0.253799	+	1.80642i	0.00900133	+	0.0640673i
$796$		0			0
$797$			14.3180i			0.507171i	0.967313	+	0.253585i	$0.0816099\pi$
$797$			14.3180i			0.507171i	−0.967313	+	0.253585i	$0.918390\pi$
$798$		0			0
$799$	2.16500			0.0765921
$800$		0			0
$801$	6.13335			0.216711
$802$		0			0
$803$		−	5.65878i		−	0.199694i
$804$		0			0
$805$	−1.12399	−	8.00000i	−0.0396153	−	0.281963i
$806$		0			0
$807$		−	11.8350i		−	0.416612i
$808$		0			0
$809$	−32.0544			−1.12697			−0.563486	−	0.826125i	$-0.690540\pi$
$809$	−32.0544			−1.12697			−0.563486	+	0.826125i	$0.690540\pi$
$810$		0			0
$811$	8.44738			0.296627			0.148314	−	0.988940i	$-0.452615\pi$
$811$	8.44738			0.296627			0.148314	+	0.988940i	$0.452615\pi$
$812$		0			0
$813$		−	14.8988i		−	0.522523i
$814$		0			0
$815$	−24.4701	+	3.43801i	−0.857151	+	0.120428i
$816$		0			0
$817$			26.0098i			0.909969i
$818$		0			0
$819$	−2.62222			−0.0916276
$820$		0			0
$821$	17.2159			0.600837			0.300419	−	0.953807i	$-0.402874\pi$
$821$	17.2159			0.600837			0.300419	+	0.953807i	$0.402874\pi$
$822$		0			0
$823$		−	12.7654i		−	0.444974i	−0.974936	−	0.222487i	$-0.928582\pi$
$823$		−	12.7654i		−	0.444974i	0.974936	−	0.222487i	$-0.0714175\pi$
$824$		0			0
$825$	1.37778	+	4.80642i	0.0479683	+	0.167338i
$826$		0			0
$827$			8.70964i			0.302864i	0.988468	+	0.151432i	$0.0483884\pi$
$827$			8.70964i			0.302864i	−0.988468	+	0.151432i	$0.951612\pi$
$828$		0			0
$829$	8.32693			0.289206			0.144603	−	0.989490i	$-0.453809\pi$
$829$	8.32693			0.289206			0.144603	+	0.989490i	$0.453809\pi$
$830$		0			0
$831$	27.6686			0.959814
$832$		0			0
$833$			14.1062i			0.488749i
$834$		0			0
$835$	30.8988	−	4.34122i	1.06930	−	0.150234i
$836$		0			0
$837$		−	2.62222i		−	0.0906370i
$838$		0			0
$839$	12.8988			0.445315			0.222657	−	0.974897i	$-0.428527\pi$
$839$	12.8988			0.445315			0.222657	+	0.974897i	$0.428527\pi$
$840$		0			0
$841$	20.7146			0.714295
$842$		0			0
$843$			9.80642i			0.337751i
$844$		0			0
$845$	1.42219	+	10.1225i	0.0489247	+	0.348223i
$846$		0			0
$847$			0.903212i			0.0310347i
$848$		0			0
$849$	19.0049			0.652247
$850$		0			0
$851$	−23.2257			−0.796167
$852$		0			0
$853$		−	19.6686i		−	0.673441i	−0.941605	−	0.336720i	$-0.890682\pi$
$853$		−	19.6686i		−	0.673441i	0.941605	−	0.336720i	$-0.109318\pi$
$854$		0			0
$855$	−0.755569	−	5.37778i	−0.0258399	−	0.183916i
$856$		0			0
$857$		−	31.8207i		−	1.08697i	−0.839417	−	0.543487i	$-0.817104\pi$
$857$		−	31.8207i		−	1.08697i	0.839417	−	0.543487i	$-0.182896\pi$
$858$		0			0
$859$	27.8292			0.949519			0.474760	−	0.880116i	$-0.342535\pi$
$859$	27.8292			0.949519			0.474760	+	0.880116i	$0.342535\pi$
$860$		0			0
$861$	9.63158			0.328243
$862$		0			0
$863$			4.82870i			0.164371i	0.996617	+	0.0821854i	$0.0261900\pi$
$863$			4.82870i			0.164371i	−0.996617	+	0.0821854i	$0.973810\pi$
$864$		0			0
$865$	−41.7244	+	5.86220i	−1.41867	+	0.199321i
$866$		0			0
$867$			11.7971i			0.400649i
$868$		0			0
$869$	16.5303			0.560754
$870$		0			0
$871$	−37.3274			−1.26479
$872$		0			0
$873$			12.4701i			0.422050i
$874$		0			0
$875$	9.22570	−	4.10616i	0.311885	−	0.138813i
$876$		0			0
$877$		−	21.9826i		−	0.742301i	−0.928573	−	0.371151i	$-0.878963\pi$
$877$		−	21.9826i		−	0.742301i	0.928573	−	0.371151i	$-0.121037\pi$
$878$		0			0
$879$	−30.7511			−1.03721
$880$		0			0
$881$	−12.1017			−0.407717			−0.203858	−	0.979000i	$-0.565348\pi$
$881$	−12.1017			−0.407717			−0.203858	+	0.979000i	$0.565348\pi$
$882$		0			0
$883$		−	8.73683i		−	0.294018i	−0.989135	−	0.147009i	$-0.953035\pi$
$883$		−	8.73683i		−	0.294018i	0.989135	−	0.147009i	$-0.0469646\pi$
$884$		0			0
$885$	3.70471	−	0.520505i	0.124533	−	0.0174966i
$886$		0			0
$887$		−	19.8524i		−	0.666577i	−0.942825	−	0.333288i	$-0.891842\pi$
$887$		−	19.8524i		−	0.666577i	0.942825	−	0.333288i	$-0.108158\pi$
$888$		0			0
$889$	9.93978			0.333369
$890$		0			0
$891$	−1.00000			−0.0335013
$892$		0			0
$893$		−	2.30513i		−	0.0771383i
$894$		0			0
$895$	1.51114	+	10.7556i	0.0505118	+	0.359519i
$896$		0			0
$897$			11.6128i			0.387742i
$898$		0			0
$899$	−18.4889			−0.616638
$900$		0			0
$901$	1.86082			0.0619928
$902$		0			0
$903$		−	9.67307i		−	0.321900i
$904$		0			0
$905$	−5.22570	−	37.1941i	−0.173708	−	1.23637i
$906$		0			0
$907$			32.8287i			1.09006i	0.838417	+	0.545030i	$0.183482\pi$
$907$			32.8287i			1.09006i	−0.838417	+	0.545030i	$0.816518\pi$
$908$		0			0
$909$	−16.1748			−0.536486
$910$		0			0
$911$	16.3497			0.541689			0.270845	−	0.962623i	$-0.412697\pi$
$911$	16.3497			0.541689			0.270845	+	0.962623i	$0.412697\pi$
$912$		0			0
$913$		−	7.76049i		−	0.256835i
$914$		0			0
$915$	16.0415	−	2.25380i	0.530315	−	0.0745083i
$916$		0			0
$917$		−	1.12399i		−	0.0371173i
$918$		0			0
$919$	20.0228			0.660490			0.330245	−	0.943895i	$-0.392869\pi$
$919$	20.0228			0.660490			0.330245	+	0.943895i	$0.392869\pi$
$920$		0			0
$921$	−13.4938			−0.444635
$922$		0			0
$923$			26.9590i			0.887366i
$924$		0			0
$925$	−8.00000	−	27.9081i	−0.263038	−	0.917614i
$926$		0			0
$927$			17.1526i			0.563364i
$928$		0			0
$929$	43.5308			1.42820			0.714100	−	0.700044i	$-0.246836\pi$
$929$	43.5308			1.42820			0.714100	+	0.700044i	$0.246836\pi$
$930$		0			0
$931$	15.0192			0.492235
$932$		0			0
$933$			17.5526i			0.574647i
$934$		0			0
$935$	5.05086	−	0.709636i	0.165181	−	0.0232076i
$936$		0			0
$937$			43.4563i			1.41966i	0.704375	+	0.709828i	$0.251227\pi$
$937$			43.4563i			1.41966i	−0.704375	+	0.709828i	$0.748773\pi$
$938$		0			0
$939$	14.3970			0.469828
$940$		0			0
$941$	−23.7244			−0.773393			−0.386697	−	0.922207i	$-0.626384\pi$
$941$	−23.7244			−0.773393			−0.386697	+	0.922207i	$0.626384\pi$
$942$		0			0
$943$		−	42.6548i		−	1.38903i
$944$		0			0
$945$	0.280996	+	2.00000i	0.00914081	+	0.0650600i
$946$		0			0
$947$		−	11.7047i		−	0.380352i	−0.981750	−	0.190176i	$-0.939094\pi$
$947$		−	11.7047i		−	0.380352i	0.981750	−	0.190176i	$-0.0609059\pi$
$948$		0			0
$949$	16.4286			0.533296
$950$		0			0
$951$	−29.4608			−0.955331
$952$		0			0
$953$		−	46.1258i		−	1.49416i	−0.664733	−	0.747081i	$-0.731455\pi$
$953$		−	46.1258i		−	1.49416i	0.664733	−	0.747081i	$-0.268545\pi$
$954$		0			0
$955$	−5.24443	−	37.3274i	−0.169706	−	1.20789i
$956$		0			0
$957$			7.05086i			0.227922i
$958$		0			0
$959$	4.00000			0.129167
$960$		0			0
$961$	−24.1240			−0.778193
$962$		0			0
$963$		−	13.5669i		−	0.437188i
$964$		0			0
$965$	−54.5116	+	7.65878i	−1.75479	+	0.246545i
$966$		0			0
$967$		−	17.0495i		−	0.548274i	−0.961691	−	0.274137i	$-0.911608\pi$
$967$		−	17.0495i		−	0.548274i	0.961691	−	0.274137i	$-0.0883922\pi$
$968$		0			0
$969$	−5.53972			−0.177961
$970$		0			0
$971$	58.1847			1.86724			0.933618	−	0.358271i	$-0.116634\pi$
$971$	58.1847			1.86724			0.933618	+	0.358271i	$0.116634\pi$
$972$		0			0
$973$			0.828699i			0.0265669i
$974$		0			0
$975$	−13.9541	+	4.00000i	−0.446888	+	0.128103i
$976$		0			0
$977$		−	51.7373i		−	1.65522i	−0.561301	−	0.827612i	$-0.689699\pi$
$977$		−	51.7373i		−	1.65522i	0.561301	−	0.827612i	$-0.310301\pi$
$978$		0			0
$979$	6.13335			0.196023
$980$		0			0
$981$	−10.0000			−0.319275
$982$		0			0
$983$		−	26.3970i		−	0.841933i	−0.907076	−	0.420967i	$-0.861691\pi$
$983$		−	26.3970i		−	0.841933i	0.907076	−	0.420967i	$-0.138309\pi$
$984$		0			0
$985$	−32.9304	+	4.62666i	−1.04925	+	0.147418i
$986$		0			0
$987$			0.857279i			0.0272875i
$988$		0			0
$989$	−42.8385			−1.36219
$990$		0			0
$991$	23.0923			0.733552			0.366776	−	0.930309i	$-0.380461\pi$
$991$	23.0923			0.733552			0.366776	+	0.930309i	$0.380461\pi$
$992$		0			0
$993$			2.62222i			0.0832135i
$994$		0			0
$995$	−3.49240	−	24.8573i	−0.110717	−	0.788029i
$996$		0			0
$997$			12.9131i			0.408961i	0.978871	+	0.204480i	$0.0655504\pi$
$997$			12.9131i			0.408961i	−0.978871	+	0.204480i	$0.934450\pi$
$998$		0			0
$999$	5.80642			0.183707

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2640.2.d.h.529.5		6
4.3	odd	2		165.2.c.b.34.4	yes	6
5.4	even	2	inner	2640.2.d.h.529.2		6
12.11	even	2		495.2.c.e.199.3		6
20.3	even	4		825.2.a.l.1.2		3
20.7	even	4		825.2.a.j.1.2		3
20.19	odd	2		165.2.c.b.34.3	✓	6
44.43	even	2		1815.2.c.e.364.3		6
60.23	odd	4		2475.2.a.ba.1.2		3
60.47	odd	4		2475.2.a.bc.1.2		3
60.59	even	2		495.2.c.e.199.4		6
220.43	odd	4		9075.2.a.cg.1.2		3
220.87	odd	4		9075.2.a.ch.1.2		3
220.219	even	2		1815.2.c.e.364.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.2.c.b.34.3	✓	6	20.19	odd	2
165.2.c.b.34.4	yes	6	4.3	odd	2
495.2.c.e.199.3		6	12.11	even	2
495.2.c.e.199.4		6	60.59	even	2
825.2.a.j.1.2		3	20.7	even	4
825.2.a.l.1.2		3	20.3	even	4
1815.2.c.e.364.3		6	44.43	even	2
1815.2.c.e.364.4		6	220.219	even	2
2475.2.a.ba.1.2		3	60.23	odd	4
2475.2.a.bc.1.2		3	60.47	odd	4
2640.2.d.h.529.2		6	5.4	even	2	inner
2640.2.d.h.529.5		6	1.1	even	1	trivial
9075.2.a.cg.1.2		3	220.43	odd	4
9075.2.a.ch.1.2		3	220.87	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2640 = 2^{4} \cdot 3 \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2640.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} +(0.311108 + 2.21432i) q^{5} +0.903212i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{3} +(0.311108 + 2.21432i) q^{5} +0.903212i q^{7} -1.00000 q^{9} -1.00000 q^{11} -2.90321i q^{13} +(-2.21432 + 0.311108i) q^{15} +2.28100i q^{17} +2.42864 q^{19} -0.903212 q^{21} +4.00000i q^{23} +(-4.80642 + 1.37778i) q^{25} -1.00000i q^{27} -7.05086 q^{29} +2.62222 q^{31} -1.00000i q^{33} +(-2.00000 + 0.280996i) q^{35} +5.80642i q^{37} +2.90321 q^{39} -10.6637 q^{41} +10.7096i q^{43} +(-0.311108 - 2.21432i) q^{45} -0.949145i q^{47} +6.18421 q^{49} -2.28100 q^{51} -0.815792i q^{53} +(-0.311108 - 2.21432i) q^{55} +2.42864i q^{57} -1.67307 q^{59} -7.24443 q^{61} -0.903212i q^{63} +(6.42864 - 0.903212i) q^{65} -12.8573i q^{67} -4.00000 q^{69} -9.28592 q^{71} +5.65878i q^{73} +(-1.37778 - 4.80642i) q^{75} -0.903212i q^{77} -16.5303 q^{79} +1.00000 q^{81} +7.76049i q^{83} +(-5.05086 + 0.709636i) q^{85} -7.05086i q^{87} -6.13335 q^{89} +2.62222 q^{91} +2.62222i q^{93} +(0.755569 + 5.37778i) q^{95} -12.4701i q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{5} - 6 q^{9}+O(q^{10}) \) 6 * q + 2 * q^5 - 6 * q^9	\( 6 q + 2 q^{5} - 6 q^{9} - 6 q^{11} - 12 q^{19} + 8 q^{21} - 2 q^{25} - 16 q^{29} + 16 q^{31} - 12 q^{35} + 4 q^{39} + 16 q^{41} - 2 q^{45} + 10 q^{49} - 2 q^{55} + 16 q^{59} - 44 q^{61} + 12 q^{65} - 24 q^{69} + 24 q^{71} - 8 q^{75} - 20 q^{79} + 6 q^{81} - 4 q^{85} - 36 q^{89} + 16 q^{91} + 4 q^{95} + 6 q^{99}+O(q^{100}) \) 6 * q + 2 * q^5 - 6 * q^9 - 6 * q^11 - 12 * q^19 + 8 * q^21 - 2 * q^25 - 16 * q^29 + 16 * q^31 - 12 * q^35 + 4 * q^39 + 16 * q^41 - 2 * q^45 + 10 * q^49 - 2 * q^55 + 16 * q^59 - 44 * q^61 + 12 * q^65 - 24 * q^69 + 24 * q^71 - 8 * q^75 - 20 * q^79 + 6 * q^81 - 4 * q^85 - 36 * q^89 + 16 * q^91 + 4 * q^95 + 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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