LMFDB - Embedded newform 2640.2.t.c.1231.8

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2640,2,Mod(1231,2640)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2640.1231"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2640, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 0, 0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2640 = 2^{4} \cdot 3 \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2640.t (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,8,0,0,0,-8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0, 0,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(33)] == traces)

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

Self dual:	no
Analytic conductor:	$21.0805061336$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.1169858560000.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 73x^{4} + 16 $ x^8 + 73*x^4 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1231.8
Root			$2.06532 - 2.06532i$ of defining polynomial
Character	$\chi$	$=$	2640.1231
Dual form			2640.2.t.c.1231.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} +1.00000 q^{5} +4.13065 q^{7} -1.00000 q^{9} +(3.16228 + 1.00000i) q^{11} -0.968371i q^{13} +1.00000i q^{15} +4.13065i q^{17} -5.09902 q^{19} +4.13065i q^{21} +9.06226i q^{23} +1.00000 q^{25} -1.00000i q^{27} +3.16228i q^{29} +(-1.00000 + 3.16228i) q^{33} +4.13065 q^{35} -7.06226 q^{37} +0.968371 q^{39} +1.22554i q^{41} +4.13065 q^{43} -1.00000 q^{45} -8.00000i q^{47} +10.0623 q^{49} -4.13065 q^{51} +13.0623 q^{53} +(3.16228 + 1.00000i) q^{55} -5.09902i q^{57} -13.0623i q^{59} +14.5859i q^{61} -4.13065 q^{63} -0.968371i q^{65} -11.0623i q^{67} -9.06226 q^{69} -7.06226i q^{71} +15.5542i q^{73} +1.00000i q^{75} +(13.0623 + 4.13065i) q^{77} -3.16228 q^{79} +1.00000 q^{81} +7.29293 q^{83} +4.13065i q^{85} -3.16228 q^{87} -2.00000 q^{89} -4.00000i q^{91} -5.09902 q^{95} -7.06226 q^{97} +(-3.16228 - 1.00000i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{5} - 8 q^{9} + 8 q^{25} - 8 q^{33} + 8 q^{37} - 8 q^{45} + 16 q^{49} + 40 q^{53} - 8 q^{69} + 40 q^{77} + 8 q^{81} - 16 q^{89} + 8 q^{97}+O(q^{100}) $ 8 * q + 8 * q^5 - 8 * q^9 + 8 * q^25 - 8 * q^33 + 8 * q^37 - 8 * q^45 + 16 * q^49 + 40 * q^53 - 8 * q^69 + 40 * q^77 + 8 * q^81 - 16 * q^89 + 8 * q^97

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
$5$	1.00000			0.447214
$5$	1.00000			0.447214
$6$		0			0
$7$	4.13065			1.56124			0.780619	−	0.625007i	$-0.214904\pi$
$7$	4.13065			1.56124			0.780619	+	0.625007i	$0.214904\pi$
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	3.16228	+	1.00000i	0.953463	+	0.301511i
$11$	3.16228	+	1.00000i	0.953463	+	0.301511i
$12$		0			0
$13$		−	0.968371i		−	0.268578i	−0.990942	−	0.134289i	$-0.957125\pi$
$13$		−	0.968371i		−	0.268578i	0.990942	−	0.134289i	$-0.0428750\pi$
$14$		0			0
$15$			1.00000i			0.258199i
$16$		0			0
$17$			4.13065i			1.00183i	0.865497	+	0.500915i	$0.167003\pi$
$17$			4.13065i			1.00183i	−0.865497	+	0.500915i	$0.832997\pi$
$18$		0			0
$19$	−5.09902			−1.16980			−0.584898	−	0.811107i	$-0.698865\pi$
$19$	−5.09902			−1.16980			−0.584898	+	0.811107i	$0.698865\pi$
$20$		0			0
$21$			4.13065i			0.901381i
$22$		0			0
$23$			9.06226i			1.88961i	0.327631	+	0.944806i	$0.393750\pi$
$23$			9.06226i			1.88961i	−0.327631	+	0.944806i	$0.606250\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$		−	1.00000i		−	0.192450i
$28$		0			0
$29$			3.16228i			0.587220i	0.955925	+	0.293610i	$0.0948567\pi$
$29$			3.16228i			0.587220i	−0.955925	+	0.293610i	$0.905143\pi$
$30$		0			0
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$		0			0
$33$	−1.00000	+	3.16228i	−0.174078	+	0.550482i
$34$		0			0
$35$	4.13065			0.698207
$36$		0			0
$37$	−7.06226			−1.16103			−0.580514	−	0.814250i	$-0.697148\pi$
$37$	−7.06226			−1.16103			−0.580514	+	0.814250i	$0.697148\pi$
$38$		0			0
$39$	0.968371			0.155063
$40$		0			0
$41$			1.22554i			0.191397i	0.995410	+	0.0956983i	$0.0305084\pi$
$41$			1.22554i			0.191397i	−0.995410	+	0.0956983i	$0.969492\pi$
$42$		0			0
$43$	4.13065			0.629918			0.314959	−	0.949105i	$-0.398009\pi$
$43$	4.13065			0.629918			0.314959	+	0.949105i	$0.398009\pi$
$44$		0			0
$45$	−1.00000			−0.149071
$46$		0			0
$47$		−	8.00000i		−	1.16692i	−0.812142	−	0.583460i	$-0.801699\pi$
$47$		−	8.00000i		−	1.16692i	0.812142	−	0.583460i	$-0.198301\pi$
$48$		0			0
$49$	10.0623			1.43747
$50$		0			0
$51$	−4.13065			−0.578406
$52$		0			0
$53$	13.0623			1.79424			0.897120	−	0.441788i	$-0.145656\pi$
$53$	13.0623			1.79424			0.897120	+	0.441788i	$0.145656\pi$
$54$		0			0
$55$	3.16228	+	1.00000i	0.426401	+	0.134840i
$56$		0			0
$57$		−	5.09902i		−	0.675382i
$58$		0			0
$59$		−	13.0623i		−	1.70056i	−0.526330	−	0.850281i	$-0.676432\pi$
$59$		−	13.0623i		−	1.70056i	0.526330	−	0.850281i	$-0.323568\pi$
$60$		0			0
$61$			14.5859i			1.86753i	0.357891	+	0.933764i	$0.383496\pi$
$61$			14.5859i			1.86753i	−0.357891	+	0.933764i	$0.616504\pi$
$62$		0			0
$63$	−4.13065			−0.520413
$64$		0			0
$65$		−	0.968371i		−	0.120112i
$66$		0			0
$67$		−	11.0623i		−	1.35147i	−0.737145	−	0.675735i	$-0.763826\pi$
$67$		−	11.0623i		−	1.35147i	0.737145	−	0.675735i	$-0.236174\pi$
$68$		0			0
$69$	−9.06226			−1.09097
$70$		0			0
$71$		−	7.06226i		−	0.838136i	−0.907955	−	0.419068i	$-0.862357\pi$
$71$		−	7.06226i		−	0.838136i	0.907955	−	0.419068i	$-0.137643\pi$
$72$		0			0
$73$			15.5542i			1.82048i	0.414077	+	0.910242i	$0.364105\pi$
$73$			15.5542i			1.82048i	−0.414077	+	0.910242i	$0.635895\pi$
$74$		0			0
$75$			1.00000i			0.115470i
$76$		0			0
$77$	13.0623	+	4.13065i	1.48858	+	0.470731i
$78$		0			0
$79$	−3.16228			−0.355784			−0.177892	−	0.984050i	$-0.556928\pi$
$79$	−3.16228			−0.355784			−0.177892	+	0.984050i	$0.556928\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$	7.29293			0.800503			0.400251	−	0.916405i	$-0.368923\pi$
$83$	7.29293			0.800503			0.400251	+	0.916405i	$0.368923\pi$
$84$		0			0
$85$			4.13065i			0.448032i
$86$		0			0
$87$	−3.16228			−0.339032
$88$		0			0
$89$	−2.00000			−0.212000			−0.106000	−	0.994366i	$-0.533804\pi$
$89$	−2.00000			−0.212000			−0.106000	+	0.994366i	$0.533804\pi$
$90$		0			0
$91$		−	4.00000i		−	0.419314i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−5.09902			−0.523148
$96$		0			0
$97$	−7.06226			−0.717064			−0.358532	−	0.933518i	$-0.616723\pi$
$97$	−7.06226			−0.717064			−0.358532	+	0.933518i	$0.616723\pi$
$98$		0			0
$99$	−3.16228	−	1.00000i	−0.317821	−	0.100504i
$100$		0			0
$101$		−	7.03576i		−	0.700084i	−0.936734	−	0.350042i	$-0.886167\pi$
$101$		−	7.03576i		−	0.700084i	0.936734	−	0.350042i	$-0.113833\pi$
$102$		0			0
$103$			11.0623i			1.09000i	0.838437	+	0.544998i	$0.183470\pi$
$103$			11.0623i			1.09000i	−0.838437	+	0.544998i	$0.816530\pi$
$104$		0			0
$105$			4.13065i			0.403110i
$106$		0			0
$107$	3.41944			0.330570			0.165285	−	0.986246i	$-0.447146\pi$
$107$	3.41944			0.330570			0.165285	+	0.986246i	$0.447146\pi$
$108$		0			0
$109$		−	10.1980i		−	0.976795i	−0.872621	−	0.488397i	$-0.837582\pi$
$109$		−	10.1980i		−	0.976795i	0.872621	−	0.488397i	$-0.162418\pi$
$110$		0			0
$111$		−	7.06226i		−	0.670320i
$112$		0			0
$113$	2.93774			0.276360			0.138180	−	0.990407i	$-0.455875\pi$
$113$	2.93774			0.276360			0.138180	+	0.990407i	$0.455875\pi$
$114$		0			0
$115$			9.06226i			0.845060i
$116$		0			0
$117$			0.968371i			0.0895259i
$118$		0			0
$119$			17.0623i			1.56409i
$120$		0			0
$121$	9.00000	+	6.32456i	0.818182	+	0.574960i
$122$		0			0
$123$	−1.22554			−0.110503
$124$		0			0
$125$	1.00000			0.0894427
$126$		0			0
$127$	−14.3287			−1.27147			−0.635733	−	0.771909i	$-0.719302\pi$
$127$	−14.3287			−1.27147			−0.635733	+	0.771909i	$0.719302\pi$
$128$		0			0
$129$			4.13065i			0.363683i
$130$		0			0
$131$	4.38781			0.383365			0.191683	−	0.981457i	$-0.438606\pi$
$131$	4.38781			0.383365			0.191683	+	0.981457i	$0.438606\pi$
$132$		0			0
$133$	−21.0623			−1.82633
$134$		0			0
$135$		−	1.00000i		−	0.0860663i
$136$		0			0
$137$	−10.0000			−0.854358			−0.427179	−	0.904167i	$-0.640493\pi$
$137$	−10.0000			−0.854358			−0.427179	+	0.904167i	$0.640493\pi$
$138$		0			0
$139$	8.97250			0.761038			0.380519	−	0.924773i	$-0.375745\pi$
$139$	8.97250			0.761038			0.380519	+	0.924773i	$0.375745\pi$
$140$		0			0
$141$	8.00000			0.673722
$142$		0			0
$143$	0.968371	−	3.06226i	0.0809792	−	0.256079i
$144$		0			0
$145$			3.16228i			0.262613i
$146$		0			0
$147$			10.0623i			0.829921i
$148$		0			0
$149$			15.2971i			1.25318i	0.779348	+	0.626592i	$0.215551\pi$
$149$			15.2971i			1.25318i	−0.779348	+	0.626592i	$0.784449\pi$
$150$		0			0
$151$	−7.55009			−0.614418			−0.307209	−	0.951642i	$-0.599395\pi$
$151$	−7.55009			−0.614418			−0.307209	+	0.951642i	$0.599395\pi$
$152$		0			0
$153$		−	4.13065i		−	0.333943i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	12.1245			0.967642			0.483821	−	0.875167i	$-0.339249\pi$
$157$	12.1245			0.967642			0.483821	+	0.875167i	$0.339249\pi$
$158$		0			0
$159$			13.0623i			1.03590i
$160$		0			0
$161$			37.4330i			2.95013i
$162$		0			0
$163$			0.937742i			0.0734496i	0.999325	+	0.0367248i	$0.0116925\pi$
$163$			0.937742i			0.0734496i	−0.999325	+	0.0367248i	$0.988307\pi$
$164$		0			0
$165$	−1.00000	+	3.16228i	−0.0778499	+	0.246183i
$166$		0			0
$167$	17.4910			1.35349			0.676746	−	0.736217i	$-0.263390\pi$
$167$	17.4910			1.35349			0.676746	+	0.736217i	$0.263390\pi$
$168$		0			0
$169$	12.0623			0.927866
$170$		0			0
$171$	5.09902			0.389932
$172$		0			0
$173$		−	6.06739i		−	0.461295i	−0.973037	−	0.230648i	$-0.925916\pi$
$173$		−	6.06739i		−	0.461295i	0.973037	−	0.230648i	$-0.0740844\pi$
$174$		0			0
$175$	4.13065			0.312248
$176$		0			0
$177$	13.0623			0.981819
$178$		0			0
$179$		−	12.0000i		−	0.896922i	−0.893802	−	0.448461i	$-0.851972\pi$
$179$		−	12.0000i		−	0.896922i	0.893802	−	0.448461i	$-0.148028\pi$
$180$		0			0
$181$	−13.0623			−0.970910			−0.485455	−	0.874262i	$-0.661346\pi$
$181$	−13.0623			−0.970910			−0.485455	+	0.874262i	$0.661346\pi$
$182$		0			0
$183$	−14.5859			−1.07822
$184$		0			0
$185$	−7.06226			−0.519228
$186$		0			0
$187$	−4.13065	+	13.0623i	−0.302063	+	0.955207i
$188$		0			0
$189$		−	4.13065i		−	0.300460i
$190$		0			0
$191$		−	8.00000i		−	0.578860i	−0.957199	−	0.289430i	$-0.906534\pi$
$191$		−	8.00000i		−	0.578860i	0.957199	−	0.289430i	$-0.0934657\pi$
$192$		0			0
$193$		−	13.1032i		−	0.943186i	−0.881816	−	0.471593i	$-0.843679\pi$
$193$		−	13.1032i		−	0.943186i	0.881816	−	0.471593i	$-0.156321\pi$
$194$		0			0
$195$	0.968371			0.0693465
$196$		0			0
$197$		−	18.7165i		−	1.33350i	−0.745283	−	0.666748i	$-0.767686\pi$
$197$		−	18.7165i		−	1.33350i	0.745283	−	0.666748i	$-0.232314\pi$
$198$		0			0
$199$			12.0000i			0.850657i	0.905039	+	0.425329i	$0.139842\pi$
$199$			12.0000i			0.850657i	−0.905039	+	0.425329i	$0.860158\pi$
$200$		0			0
$201$	11.0623			0.780272
$202$		0			0
$203$			13.0623i			0.916791i
$204$		0			0
$205$			1.22554i			0.0855951i
$206$		0			0
$207$		−	9.06226i		−	0.629870i
$208$		0			0
$209$	−16.1245	−	5.09902i	−1.11536	−	0.352707i
$210$		0			0
$211$	−26.0094			−1.79056			−0.895281	−	0.445501i	$-0.853026\pi$
$211$	−26.0094			−1.79056			−0.895281	+	0.445501i	$0.853026\pi$
$212$		0			0
$213$	7.06226			0.483898
$214$		0			0
$215$	4.13065			0.281708
$216$		0			0
$217$		0			0
$218$		0			0
$219$	−15.5542			−1.05106
$220$		0			0
$221$	4.00000			0.269069
$222$		0			0
$223$		−	16.0000i		−	1.07144i	−0.844396	−	0.535720i	$-0.820040\pi$
$223$		−	16.0000i		−	1.07144i	0.844396	−	0.535720i	$-0.179960\pi$
$224$		0			0
$225$	−1.00000			−0.0666667
$226$		0			0
$227$	19.9420			1.32360			0.661800	−	0.749681i	$-0.269793\pi$
$227$	19.9420			1.32360			0.661800	+	0.749681i	$0.269793\pi$
$228$		0			0
$229$	26.1245			1.72636			0.863178	−	0.504899i	$-0.168470\pi$
$229$	26.1245			1.72636			0.863178	+	0.504899i	$0.168470\pi$
$230$		0			0
$231$	−4.13065	+	13.0623i	−0.271777	+	0.859433i
$232$		0			0
$233$			20.6532i			1.35304i	0.736425	+	0.676519i	$0.236513\pi$
$233$			20.6532i			1.35304i	−0.736425	+	0.676519i	$0.763487\pi$
$234$		0			0
$235$		−	8.00000i		−	0.521862i
$236$		0			0
$237$		−	3.16228i		−	0.205412i
$238$		0			0
$239$	8.26130			0.534379			0.267189	−	0.963644i	$-0.413905\pi$
$239$	8.26130			0.534379			0.267189	+	0.963644i	$0.413905\pi$
$240$		0			0
$241$			17.0369i			1.09744i	0.836005	+	0.548722i	$0.184886\pi$
$241$			17.0369i			1.09744i	−0.836005	+	0.548722i	$0.815114\pi$
$242$		0			0
$243$			1.00000i			0.0641500i
$244$		0			0
$245$	10.0623			0.642854
$246$		0			0
$247$			4.93774i			0.314181i
$248$		0			0
$249$			7.29293i			0.462170i
$250$		0			0
$251$			3.06226i			0.193288i	0.995319	+	0.0966440i	$0.0308108\pi$
$251$			3.06226i			0.193288i	−0.995319	+	0.0966440i	$0.969189\pi$
$252$		0			0
$253$	−9.06226	+	28.6574i	−0.569739	+	1.80167i
$254$		0			0
$255$	−4.13065			−0.258671
$256$		0			0
$257$	−6.00000			−0.374270			−0.187135	−	0.982334i	$-0.559920\pi$
$257$	−6.00000			−0.374270			−0.187135	+	0.982334i	$0.559920\pi$
$258$		0			0
$259$	−29.1717			−1.81264
$260$		0			0
$261$		−	3.16228i		−	0.195740i
$262$		0			0
$263$	−2.90511			−0.179137			−0.0895685	−	0.995981i	$-0.528549\pi$
$263$	−2.90511			−0.179137			−0.0895685	+	0.995981i	$0.528549\pi$
$264$		0			0
$265$	13.0623			0.802408
$266$		0			0
$267$		−	2.00000i		−	0.122398i
$268$		0			0
$269$	−24.1245			−1.47090			−0.735449	−	0.677580i	$-0.763029\pi$
$269$	−24.1245			−1.47090			−0.735449	+	0.677580i	$0.763029\pi$
$270$		0			0
$271$	7.03576			0.427392			0.213696	−	0.976900i	$-0.431450\pi$
$271$	7.03576			0.427392			0.213696	+	0.976900i	$0.431450\pi$
$272$		0			0
$273$	4.00000			0.242091
$274$		0			0
$275$	3.16228	+	1.00000i	0.190693	+	0.0603023i
$276$		0			0
$277$		−	26.2666i		−	1.57821i	−0.614261	−	0.789103i	$-0.710546\pi$
$277$		−	26.2666i		−	1.57821i	0.614261	−	0.789103i	$-0.289454\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			22.1359i			1.32052i	0.751037	+	0.660260i	$0.229554\pi$
$281$			22.1359i			1.32052i	−0.751037	+	0.660260i	$0.770446\pi$
$282$		0			0
$283$	−20.6532			−1.22771			−0.613854	−	0.789420i	$-0.710382\pi$
$283$	−20.6532			−1.22771			−0.613854	+	0.789420i	$0.710382\pi$
$284$		0			0
$285$		−	5.09902i		−	0.302040i
$286$		0			0
$287$			5.06226i			0.298816i
$288$		0			0
$289$	−0.0622577			−0.00366222
$290$		0			0
$291$		−	7.06226i		−	0.413997i
$292$		0			0
$293$		−	14.8430i		−	0.867138i	−0.901120	−	0.433569i	$-0.857254\pi$
$293$		−	14.8430i		−	0.867138i	0.901120	−	0.433569i	$-0.142746\pi$
$294$		0			0
$295$		−	13.0623i		−	0.760514i
$296$		0			0
$297$	1.00000	−	3.16228i	0.0580259	−	0.183494i
$298$		0			0
$299$	8.77563			0.507508
$300$		0			0
$301$	17.0623			0.983452
$302$		0			0
$303$	7.03576			0.404194
$304$		0			0
$305$			14.5859i			0.835183i
$306$		0			0
$307$	18.2022			1.03885			0.519426	−	0.854515i	$-0.326146\pi$
$307$	18.2022			1.03885			0.519426	+	0.854515i	$0.326146\pi$
$308$		0			0
$309$	−11.0623			−0.629310
$310$		0			0
$311$			1.06226i			0.0602351i	0.999546	+	0.0301176i	$0.00958817\pi$
$311$			1.06226i			0.0602351i	−0.999546	+	0.0301176i	$0.990412\pi$
$312$		0			0
$313$	−11.0623			−0.625276			−0.312638	−	0.949872i	$-0.601213\pi$
$313$	−11.0623			−0.625276			−0.312638	+	0.949872i	$0.601213\pi$
$314$		0			0
$315$	−4.13065			−0.232736
$316$		0			0
$317$	−31.1868			−1.75162			−0.875812	−	0.482653i	$-0.839673\pi$
$317$	−31.1868			−1.75162			−0.875812	+	0.482653i	$0.839673\pi$
$318$		0			0
$319$	−3.16228	+	10.0000i	−0.177054	+	0.559893i
$320$		0			0
$321$			3.41944i			0.190855i
$322$		0			0
$323$		−	21.0623i		−	1.17194i
$324$		0			0
$325$		−	0.968371i		−	0.0537156i
$326$		0			0
$327$	10.1980			0.563953
$328$		0			0
$329$		−	33.0452i		−	1.82184i
$330$		0			0
$331$			22.1245i			1.21607i	0.793909	+	0.608037i	$0.208043\pi$
$331$			22.1245i			1.21607i	−0.793909	+	0.608037i	$0.791957\pi$
$332$		0			0
$333$	7.06226			0.387009
$334$		0			0
$335$		−	11.0623i		−	0.604396i
$336$		0			0
$337$			2.90511i			0.158252i	0.996865	+	0.0791258i	$0.0252129\pi$
$337$			2.90511i			0.158252i	−0.996865	+	0.0791258i	$0.974787\pi$
$338$		0			0
$339$			2.93774i			0.159556i
$340$		0			0
$341$		0			0
$342$		0			0
$343$	12.6491			0.682988
$344$		0			0
$345$	−9.06226			−0.487896
$346$		0			0
$347$	34.5279			1.85355			0.926777	−	0.375612i	$-0.122567\pi$
$347$	34.5279			1.85355			0.926777	+	0.375612i	$0.122567\pi$
$348$		0			0
$349$			5.81023i			0.311014i	0.987835	+	0.155507i	$0.0497012\pi$
$349$			5.81023i			0.311014i	−0.987835	+	0.155507i	$0.950299\pi$
$350$		0			0
$351$	−0.968371			−0.0516878
$352$		0			0
$353$	9.06226			0.482335			0.241168	−	0.970483i	$-0.422470\pi$
$353$	9.06226			0.482335			0.241168	+	0.970483i	$0.422470\pi$
$354$		0			0
$355$		−	7.06226i		−	0.374826i
$356$		0			0
$357$	−17.0623			−0.903030
$358$		0			0
$359$	−18.4593			−0.974247			−0.487123	−	0.873333i	$-0.661954\pi$
$359$	−18.4593			−0.974247			−0.487123	+	0.873333i	$0.661954\pi$
$360$		0			0
$361$	7.00000			0.368421
$362$		0			0
$363$	−6.32456	+	9.00000i	−0.331953	+	0.472377i
$364$		0			0
$365$			15.5542i			0.814145i
$366$		0			0
$367$		−	26.1245i		−	1.36369i	−0.731497	−	0.681844i	$-0.761178\pi$
$367$		−	26.1245i		−	1.36369i	0.731497	−	0.681844i	$-0.238822\pi$
$368$		0			0
$369$		−	1.22554i		−	0.0637988i
$370$		0			0
$371$	53.9556			2.80123
$372$		0			0
$373$		−	34.5279i		−	1.78779i	−0.448280	−	0.893893i	$-0.647963\pi$
$373$		−	34.5279i		−	1.78779i	0.448280	−	0.893893i	$-0.352037\pi$
$374$		0			0
$375$			1.00000i			0.0516398i
$376$		0			0
$377$	3.06226			0.157714
$378$		0			0
$379$		−	14.1245i		−	0.725528i	−0.931881	−	0.362764i	$-0.881833\pi$
$379$		−	14.1245i		−	0.725528i	0.931881	−	0.362764i	$-0.118167\pi$
$380$		0			0
$381$		−	14.3287i		−	0.734081i
$382$		0			0
$383$		−	17.0623i		−	0.871841i	−0.899985	−	0.435920i	$-0.856423\pi$
$383$		−	17.0623i		−	0.871841i	0.899985	−	0.435920i	$-0.143577\pi$
$384$		0			0
$385$	13.0623	+	4.13065i	0.665714	+	0.210517i
$386$		0			0
$387$	−4.13065			−0.209973
$388$		0			0
$389$	−3.87548			−0.196495			−0.0982474	−	0.995162i	$-0.531324\pi$
$389$	−3.87548			−0.196495			−0.0982474	+	0.995162i	$0.531324\pi$
$390$		0			0
$391$	−37.4330			−1.89307
$392$		0			0
$393$			4.38781i			0.221336i
$394$		0			0
$395$	−3.16228			−0.159111
$396$		0			0
$397$	−8.93774			−0.448572			−0.224286	−	0.974523i	$-0.572005\pi$
$397$	−8.93774			−0.448572			−0.224286	+	0.974523i	$0.572005\pi$
$398$		0			0
$399$		−	21.0623i		−	1.05443i
$400$		0			0
$401$	20.1245			1.00497			0.502485	−	0.864586i	$-0.332419\pi$
$401$	20.1245			1.00497			0.502485	+	0.864586i	$0.332419\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	1.00000			0.0496904
$406$		0			0
$407$	−22.3328	−	7.06226i	−1.10700	−	0.350063i
$408$		0			0
$409$		−	2.45107i		−	0.121198i	−0.998162	−	0.0605988i	$-0.980699\pi$
$409$		−	2.45107i		−	0.121198i	0.998162	−	0.0605988i	$-0.0193010\pi$
$410$		0			0
$411$		−	10.0000i		−	0.493264i
$412$		0			0
$413$		−	53.9556i		−	2.65498i
$414$		0			0
$415$	7.29293			0.357996
$416$		0			0
$417$			8.97250i			0.439385i
$418$		0			0
$419$		−	4.00000i		−	0.195413i	−0.995215	−	0.0977064i	$-0.968849\pi$
$419$		−	4.00000i		−	0.195413i	0.995215	−	0.0977064i	$-0.0311506\pi$
$420$		0			0
$421$	−27.1868			−1.32500			−0.662501	−	0.749061i	$-0.730505\pi$
$421$	−27.1868			−1.32500			−0.662501	+	0.749061i	$0.730505\pi$
$422$		0			0
$423$			8.00000i			0.388973i
$424$		0			0
$425$			4.13065i			0.200366i
$426$		0			0
$427$			60.2490i			2.91565i
$428$		0			0
$429$	3.06226	+	0.968371i	0.147847	+	0.0467534i
$430$		0			0
$431$	2.45107			0.118064			0.0590320	−	0.998256i	$-0.481199\pi$
$431$	2.45107			0.118064			0.0590320	+	0.998256i	$0.481199\pi$
$432$		0			0
$433$	11.0623			0.531618			0.265809	−	0.964026i	$-0.414361\pi$
$433$	11.0623			0.531618			0.265809	+	0.964026i	$0.414361\pi$
$434$		0			0
$435$	−3.16228			−0.151620
$436$		0			0
$437$		−	46.2086i		−	2.21046i
$438$		0			0
$439$	−5.09902			−0.243363			−0.121681	−	0.992569i	$-0.538829\pi$
$439$	−5.09902			−0.243363			−0.121681	+	0.992569i	$0.538829\pi$
$440$		0			0
$441$	−10.0623			−0.479155
$442$		0			0
$443$		−	21.0623i		−	1.00070i	−0.865824	−	0.500349i	$-0.833205\pi$
$443$		−	21.0623i		−	1.00070i	0.865824	−	0.500349i	$-0.166795\pi$
$444$		0			0
$445$	−2.00000			−0.0948091
$446$		0			0
$447$	−15.2971			−0.723526
$448$		0			0
$449$	20.1245			0.949735			0.474867	−	0.880057i	$-0.342496\pi$
$449$	20.1245			0.949735			0.474867	+	0.880057i	$0.342496\pi$
$450$		0			0
$451$	−1.22554	+	3.87548i	−0.0577082	+	0.182489i
$452$		0			0
$453$		−	7.55009i		−	0.354734i
$454$		0			0
$455$		−	4.00000i		−	0.187523i
$456$		0			0
$457$		−	26.2666i		−	1.22870i	−0.789034	−	0.614350i	$-0.789418\pi$
$457$		−	26.2666i		−	1.22870i	0.789034	−	0.614350i	$-0.210582\pi$
$458$		0			0
$459$	4.13065			0.192802
$460$		0			0
$461$			33.7564i			1.57219i	0.618104	+	0.786096i	$0.287901\pi$
$461$			33.7564i			1.57219i	−0.618104	+	0.786096i	$0.712099\pi$
$462$		0			0
$463$		−	34.1245i		−	1.58590i	−0.609286	−	0.792950i	$-0.708544\pi$
$463$		−	34.1245i		−	1.58590i	0.609286	−	0.792950i	$-0.291456\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			13.0623i			0.604449i	0.953237	+	0.302225i	$0.0977293\pi$
$467$			13.0623i			0.604449i	−0.953237	+	0.302225i	$0.902271\pi$
$468$		0			0
$469$		−	45.6943i		−	2.10997i
$470$		0			0
$471$			12.1245i			0.558668i
$472$		0			0
$473$	13.0623	+	4.13065i	0.600603	+	0.189927i
$474$		0			0
$475$	−5.09902			−0.233959
$476$		0			0
$477$	−13.0623			−0.598080
$478$		0			0
$479$	41.3065			1.88734			0.943671	−	0.330886i	$-0.107348\pi$
$479$	41.3065			1.88734			0.943671	+	0.330886i	$0.107348\pi$
$480$		0			0
$481$			6.83889i			0.311826i
$482$		0			0
$483$	−37.4330			−1.70326
$484$		0			0
$485$	−7.06226			−0.320681
$486$		0			0
$487$		−	24.9377i		−	1.13004i	−0.825078	−	0.565018i	$-0.808869\pi$
$487$		−	24.9377i		−	1.13004i	0.825078	−	0.565018i	$-0.191131\pi$
$488$		0			0
$489$	−0.937742			−0.0424062
$490$		0			0
$491$	25.2982			1.14169			0.570846	−	0.821057i	$-0.306615\pi$
$491$	25.2982			1.14169			0.570846	+	0.821057i	$0.306615\pi$
$492$		0			0
$493$	−13.0623			−0.588295
$494$		0			0
$495$	−3.16228	−	1.00000i	−0.142134	−	0.0449467i
$496$		0			0
$497$		−	29.1717i		−	1.30853i
$498$		0			0
$499$		−	9.87548i		−	0.442087i	−0.975264	−	0.221044i	$-0.929054\pi$
$499$		−	9.87548i		−	0.442087i	0.975264	−	0.221044i	$-0.0709463\pi$
$500$		0			0
$501$			17.4910i			0.781439i
$502$		0			0
$503$	17.4910			0.779884			0.389942	−	0.920840i	$-0.372495\pi$
$503$	17.4910			0.779884			0.389942	+	0.920840i	$0.372495\pi$
$504$		0			0
$505$		−	7.03576i		−	0.313087i
$506$		0			0
$507$			12.0623i			0.535704i
$508$		0			0
$509$	10.0000			0.443242			0.221621	−	0.975133i	$-0.428865\pi$
$509$	10.0000			0.443242			0.221621	+	0.975133i	$0.428865\pi$
$510$		0			0
$511$			64.2490i			2.84221i
$512$		0			0
$513$			5.09902i			0.225127i
$514$		0			0
$515$			11.0623i			0.487461i
$516$		0			0
$517$	8.00000	−	25.2982i	0.351840	−	1.11261i
$518$		0			0
$519$	6.06739			0.266329
$520$		0			0
$521$	−24.1245			−1.05691			−0.528457	−	0.848960i	$-0.677229\pi$
$521$	−24.1245			−1.05691			−0.528457	+	0.848960i	$0.677229\pi$
$522$		0			0
$523$	23.1043			1.01028			0.505141	−	0.863037i	$-0.331441\pi$
$523$	23.1043			1.01028			0.505141	+	0.863037i	$0.331441\pi$
$524$		0			0
$525$			4.13065i			0.180276i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−59.1245			−2.57063
$530$		0			0
$531$			13.0623i			0.566854i
$532$		0			0
$533$	1.18677			0.0514049
$534$		0			0
$535$	3.41944			0.147835
$536$		0			0
$537$	12.0000			0.517838
$538$		0			0
$539$	31.8197	+	10.0623i	1.37057	+	0.433412i
$540$		0			0
$541$		−	6.32456i		−	0.271914i	−0.990715	−	0.135957i	$-0.956589\pi$
$541$		−	6.32456i		−	0.271914i	0.990715	−	0.135957i	$-0.0434109\pi$
$542$		0			0
$543$		−	13.0623i		−	0.560555i
$544$		0			0
$545$		−	10.1980i		−	0.436836i
$546$		0			0
$547$	−35.7534			−1.52871			−0.764353	−	0.644798i	$-0.776941\pi$
$547$	−35.7534			−1.52871			−0.764353	+	0.644798i	$0.776941\pi$
$548$		0			0
$549$		−	14.5859i		−	0.622509i
$550$		0			0
$551$		−	16.1245i		−	0.686927i
$552$		0			0
$553$	−13.0623			−0.555464
$554$		0			0
$555$		−	7.06226i		−	0.299776i
$556$		0			0
$557$		−	29.4289i		−	1.24694i	−0.781847	−	0.623471i	$-0.785722\pi$
$557$		−	29.4289i		−	1.24694i	0.781847	−	0.623471i	$-0.214278\pi$
$558$		0			0
$559$		−	4.00000i		−	0.169182i
$560$		0			0
$561$	−13.0623	−	4.13065i	−0.551489	−	0.174396i
$562$		0			0
$563$	−19.4277			−0.818780			−0.409390	−	0.912359i	$-0.634258\pi$
$563$	−19.4277			−0.818780			−0.409390	+	0.912359i	$0.634258\pi$
$564$		0			0
$565$	2.93774			0.123592
$566$		0			0
$567$	4.13065			0.173471
$568$		0			0
$569$		−	1.73987i		−	0.0729390i	−0.999335	−	0.0364695i	$-0.988389\pi$
$569$		−	1.73987i		−	0.0729390i	0.999335	−	0.0364695i	$-0.0116112\pi$
$570$		0			0
$571$	12.8460			0.537588			0.268794	−	0.963198i	$-0.413375\pi$
$571$	12.8460			0.537588			0.268794	+	0.963198i	$0.413375\pi$
$572$		0			0
$573$	8.00000			0.334205
$574$		0			0
$575$			9.06226i			0.377922i
$576$		0			0
$577$	−18.0000			−0.749350			−0.374675	−	0.927156i	$-0.622246\pi$
$577$	−18.0000			−0.749350			−0.374675	+	0.927156i	$0.622246\pi$
$578$		0			0
$579$	13.1032			0.544548
$580$		0			0
$581$	30.1245			1.24978
$582$		0			0
$583$	41.3065	+	13.0623i	1.71074	+	0.540983i
$584$		0			0
$585$			0.968371i			0.0400372i
$586$		0			0
$587$		−	10.9377i		−	0.451449i	−0.974191	−	0.225724i	$-0.927525\pi$
$587$		−	10.9377i		−	0.451449i	0.974191	−	0.225724i	$-0.0724749\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$	18.7165			0.769894
$592$		0			0
$593$			10.4552i			0.429344i	0.976686	+	0.214672i	$0.0688682\pi$
$593$			10.4552i			0.429344i	−0.976686	+	0.214672i	$0.931132\pi$
$594$		0			0
$595$			17.0623i			0.699484i
$596$		0			0
$597$	−12.0000			−0.491127
$598$		0			0
$599$			18.0000i			0.735460i	0.929933	+	0.367730i	$0.119865\pi$
$599$			18.0000i			0.735460i	−0.929933	+	0.367730i	$0.880135\pi$
$600$		0			0
$601$			31.1084i			1.26894i	0.772947	+	0.634470i	$0.218782\pi$
$601$			31.1084i			1.26894i	−0.772947	+	0.634470i	$0.781218\pi$
$602$		0			0
$603$			11.0623i			0.450490i
$604$		0			0
$605$	9.00000	+	6.32456i	0.365902	+	0.257130i
$606$		0			0
$607$	−14.8430			−0.602459			−0.301230	−	0.953552i	$-0.597397\pi$
$607$	−14.8430			−0.602459			−0.301230	+	0.953552i	$0.597397\pi$
$608$		0			0
$609$	−13.0623			−0.529309
$610$		0			0
$611$	−7.74697			−0.313409
$612$		0			0
$613$		−	14.1318i		−	0.570778i	−0.958412	−	0.285389i	$-0.907877\pi$
$613$		−	14.1318i		−	0.570778i	0.958412	−	0.285389i	$-0.0921229\pi$
$614$		0			0
$615$	−1.22554			−0.0494184
$616$		0			0
$617$	6.93774			0.279303			0.139651	−	0.990201i	$-0.455402\pi$
$617$	6.93774			0.279303			0.139651	+	0.990201i	$0.455402\pi$
$618$		0			0
$619$			22.1245i			0.889259i	0.895714	+	0.444630i	$0.146665\pi$
$619$			22.1245i			0.889259i	−0.895714	+	0.444630i	$0.853335\pi$
$620$		0			0
$621$	9.06226			0.363656
$622$		0			0
$623$	−8.26130			−0.330982
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	5.09902	−	16.1245i	0.203635	−	0.643951i
$628$		0			0
$629$		−	29.1717i		−	1.16315i
$630$		0			0
$631$		−	20.0000i		−	0.796187i	−0.917345	−	0.398094i	$-0.869672\pi$
$631$		−	20.0000i		−	0.796187i	0.917345	−	0.398094i	$-0.130328\pi$
$632$		0			0
$633$		−	26.0094i		−	1.03378i
$634$		0			0
$635$	−14.3287			−0.568617
$636$		0			0
$637$		−	9.74400i		−	0.386071i
$638$		0			0
$639$			7.06226i			0.279379i
$640$		0			0
$641$	38.0000			1.50091			0.750455	−	0.660922i	$-0.229834\pi$
$641$	38.0000			1.50091			0.750455	+	0.660922i	$0.229834\pi$
$642$		0			0
$643$			8.93774i			0.352470i	0.984348	+	0.176235i	$0.0563919\pi$
$643$			8.93774i			0.352470i	−0.984348	+	0.176235i	$0.943608\pi$
$644$		0			0
$645$			4.13065i			0.162644i
$646$		0			0
$647$		−	7.18677i		−	0.282541i	−0.989971	−	0.141271i	$-0.954881\pi$
$647$		−	7.18677i		−	0.282541i	0.989971	−	0.141271i	$-0.0451188\pi$
$648$		0			0
$649$	13.0623	−	41.3065i	0.512738	−	1.62142i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−50.2490			−1.96640			−0.983198	−	0.182541i	$-0.941568\pi$
$653$	−50.2490			−1.96640			−0.983198	+	0.182541i	$0.941568\pi$
$654$		0			0
$655$	4.38781			0.171446
$656$		0			0
$657$		−	15.5542i		−	0.606828i
$658$		0			0
$659$	−1.42241			−0.0554093			−0.0277047	−	0.999616i	$-0.508820\pi$
$659$	−1.42241			−0.0554093			−0.0277047	+	0.999616i	$0.508820\pi$
$660$		0			0
$661$	−7.87548			−0.306321			−0.153160	−	0.988201i	$-0.548945\pi$
$661$	−7.87548			−0.306321			−0.153160	+	0.988201i	$0.548945\pi$
$662$		0			0
$663$			4.00000i			0.155347i
$664$		0			0
$665$	−21.0623			−0.816759
$666$		0			0
$667$	−28.6574			−1.10962
$668$		0			0
$669$	16.0000			0.618596
$670$		0			0
$671$	−14.5859	+	46.1245i	−0.563081	+	1.78062i
$672$		0			0
$673$			36.4646i			1.40561i	0.711383	+	0.702804i	$0.248069\pi$
$673$			36.4646i			1.40561i	−0.711383	+	0.702804i	$0.751931\pi$
$674$		0			0
$675$		−	1.00000i		−	0.0384900i
$676$		0			0
$677$			15.7511i			0.605364i	0.953092	+	0.302682i	$0.0978820\pi$
$677$			15.7511i			0.605364i	−0.953092	+	0.302682i	$0.902118\pi$
$678$		0			0
$679$	−29.1717			−1.11951
$680$		0			0
$681$			19.9420i			0.764181i
$682$		0			0
$683$		−	26.1245i		−	0.999627i	−0.866133	−	0.499813i	$-0.833402\pi$
$683$		−	26.1245i		−	0.999627i	0.866133	−	0.499813i	$-0.166598\pi$
$684$		0			0
$685$	−10.0000			−0.382080
$686$		0			0
$687$			26.1245i			0.996712i
$688$		0			0
$689$		−	12.6491i		−	0.481893i
$690$		0			0
$691$		−	46.3735i		−	1.76413i	−0.471125	−	0.882066i	$-0.656152\pi$
$691$		−	46.3735i		−	1.76413i	0.471125	−	0.882066i	$-0.343848\pi$
$692$		0			0
$693$	−13.0623	−	4.13065i	−0.496194	−	0.156910i
$694$		0			0
$695$	8.97250			0.340346
$696$		0			0
$697$	−5.06226			−0.191747
$698$		0			0
$699$	−20.6532			−0.781177
$700$		0			0
$701$		−	12.8460i		−	0.485186i	−0.970128	−	0.242593i	$-0.922002\pi$
$701$		−	12.8460i		−	0.485186i	0.970128	−	0.242593i	$-0.0779980\pi$
$702$		0			0
$703$	36.0106			1.35817
$704$		0			0
$705$	8.00000			0.301297
$706$		0			0
$707$		−	29.0623i		−	1.09300i
$708$		0			0
$709$	11.0623			0.415452			0.207726	−	0.978187i	$-0.433394\pi$
$709$	11.0623			0.415452			0.207726	+	0.978187i	$0.433394\pi$
$710$		0			0
$711$	3.16228			0.118595
$712$		0			0
$713$		0			0
$714$		0			0
$715$	0.968371	−	3.06226i	0.0362150	−	0.114522i
$716$		0			0
$717$			8.26130i			0.308524i
$718$		0			0
$719$			12.2490i			0.456812i	0.973566	+	0.228406i	$0.0733513\pi$
$719$			12.2490i			0.456812i	−0.973566	+	0.228406i	$0.926649\pi$
$720$		0			0
$721$			45.6943i			1.70174i
$722$		0			0
$723$	−17.0369			−0.633610
$724$		0			0
$725$			3.16228i			0.117444i
$726$		0			0
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$			17.0623i			0.631070i
$732$		0			0
$733$		−	34.0136i		−	1.25632i	−0.778084	−	0.628160i	$-0.783808\pi$
$733$		−	34.0136i		−	1.25632i	0.778084	−	0.628160i	$-0.216192\pi$
$734$		0			0
$735$			10.0623i			0.371152i
$736$		0			0
$737$	11.0623	−	34.9819i	0.407484	−	1.28858i
$738$		0			0
$739$	9.48683			0.348979			0.174489	−	0.984659i	$-0.444173\pi$
$739$	9.48683			0.348979			0.174489	+	0.984659i	$0.444173\pi$
$740$		0			0
$741$	−4.93774			−0.181392
$742$		0			0
$743$	−18.0053			−0.660550			−0.330275	−	0.943885i	$-0.607142\pi$
$743$	−18.0053			−0.660550			−0.330275	+	0.943885i	$0.607142\pi$
$744$		0			0
$745$			15.2971i			0.560441i
$746$		0			0
$747$	−7.29293			−0.266834
$748$		0			0
$749$	14.1245			0.516099
$750$		0			0
$751$		−	20.2490i		−	0.738898i	−0.929251	−	0.369449i	$-0.879547\pi$
$751$		−	20.2490i		−	0.738898i	0.929251	−	0.369449i	$-0.120453\pi$
$752$		0			0
$753$	−3.06226			−0.111595
$754$		0			0
$755$	−7.55009			−0.274776
$756$		0			0
$757$	4.12452			0.149908			0.0749540	−	0.997187i	$-0.476119\pi$
$757$	4.12452			0.149908			0.0749540	+	0.997187i	$0.476119\pi$
$758$		0			0
$759$	−28.6574	−	9.06226i	−1.04020	−	0.328939i
$760$		0			0
$761$		−	12.8460i		−	0.465667i	−0.972517	−	0.232833i	$-0.925200\pi$
$761$		−	12.8460i		−	0.465667i	0.972517	−	0.232833i	$-0.0747997\pi$
$762$		0			0
$763$		−	42.1245i		−	1.52501i
$764$		0			0
$765$		−	4.13065i		−	0.149344i
$766$		0			0
$767$	−12.6491			−0.456733
$768$		0			0
$769$			42.7289i			1.54084i	0.637535	+	0.770422i	$0.279954\pi$
$769$			42.7289i			1.54084i	−0.637535	+	0.770422i	$0.720046\pi$
$770$		0			0
$771$		−	6.00000i		−	0.216085i
$772$		0			0
$773$	−6.00000			−0.215805			−0.107903	−	0.994161i	$-0.534413\pi$
$773$	−6.00000			−0.215805			−0.107903	+	0.994161i	$0.534413\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		−	29.1717i		−	1.04653i
$778$		0			0
$779$		−	6.24903i		−	0.223895i
$780$		0			0
$781$	7.06226	−	22.3328i	0.252707	−	0.799131i
$782$		0			0
$783$	3.16228			0.113011
$784$		0			0
$785$	12.1245			0.432743
$786$		0			0
$787$	−33.8167			−1.20543			−0.602717	−	0.797955i	$-0.705915\pi$
$787$	−33.8167			−1.20543			−0.602717	+	0.797955i	$0.705915\pi$
$788$		0			0
$789$		−	2.90511i		−	0.103425i
$790$		0			0
$791$	12.1348			0.431463
$792$		0			0
$793$	14.1245			0.501576
$794$		0			0
$795$			13.0623i			0.463271i
$796$		0			0
$797$	−6.93774			−0.245747			−0.122874	−	0.992422i	$-0.539211\pi$
$797$	−6.93774			−0.245747			−0.122874	+	0.992422i	$0.539211\pi$
$798$		0			0
$799$	33.0452			1.16905
$800$		0			0
$801$	2.00000			0.0706665
$802$		0			0
$803$	−15.5542	+	49.1868i	−0.548897	+	1.73576i
$804$		0			0
$805$			37.4330i			1.31934i
$806$		0			0
$807$		−	24.1245i		−	0.849223i
$808$		0			0
$809$		−	28.4605i		−	1.00062i	−0.865847	−	0.500309i	$-0.833220\pi$
$809$		−	28.4605i		−	1.00062i	0.865847	−	0.500309i	$-0.166780\pi$
$810$		0			0
$811$	−52.2157			−1.83354			−0.916771	−	0.399413i	$-0.869214\pi$
$811$	−52.2157			−1.83354			−0.916771	+	0.399413i	$0.869214\pi$
$812$		0			0
$813$			7.03576i			0.246755i
$814$		0			0
$815$			0.937742i			0.0328477i
$816$		0			0
$817$	−21.0623			−0.736875
$818$		0			0
$819$			4.00000i			0.139771i
$820$		0			0
$821$		−	31.8197i		−	1.11051i	−0.831679	−	0.555257i	$-0.812620\pi$
$821$		−	31.8197i		−	1.11051i	0.831679	−	0.555257i	$-0.187380\pi$
$822$		0			0
$823$			39.3113i			1.37031i	0.728399	+	0.685153i	$0.240265\pi$
$823$			39.3113i			1.37031i	−0.728399	+	0.685153i	$0.759735\pi$
$824$		0			0
$825$	−1.00000	+	3.16228i	−0.0348155	+	0.110096i
$826$		0			0
$827$	29.6257			1.03019			0.515094	−	0.857134i	$-0.327757\pi$
$827$	29.6257			1.03019			0.515094	+	0.857134i	$0.327757\pi$
$828$		0			0
$829$	−9.18677			−0.319070			−0.159535	−	0.987192i	$-0.550999\pi$
$829$	−9.18677			−0.319070			−0.159535	+	0.987192i	$0.550999\pi$
$830$		0			0
$831$	26.2666			0.911178
$832$		0			0
$833$			41.5637i			1.44010i
$834$		0			0
$835$	17.4910			0.605300
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	41.0623i		−	1.41763i	−0.705396	−	0.708813i	$-0.749231\pi$
$839$		−	41.0623i		−	1.41763i	0.705396	−	0.708813i	$-0.250769\pi$
$840$		0			0
$841$	19.0000			0.655172
$842$		0			0
$843$	−22.1359			−0.762402
$844$		0			0
$845$	12.0623			0.414954
$846$		0			0
$847$	37.1758	+	26.1245i	1.27738	+	0.897649i
$848$		0			0
$849$		−	20.6532i		−	0.708817i
$850$		0			0
$851$		−	64.0000i		−	2.19389i
$852$		0			0
$853$		−	34.5279i		−	1.18221i	−0.806594	−	0.591106i	$-0.798691\pi$
$853$		−	34.5279i		−	1.18221i	0.806594	−	0.591106i	$-0.201309\pi$
$854$		0			0
$855$	5.09902			0.174383
$856$		0			0
$857$			32.7880i			1.12002i	0.828487	+	0.560009i	$0.189202\pi$
$857$			32.7880i			1.12002i	−0.828487	+	0.560009i	$0.810798\pi$
$858$		0			0
$859$		−	34.1245i		−	1.16431i	−0.813077	−	0.582157i	$-0.802209\pi$
$859$		−	34.1245i		−	1.16431i	0.813077	−	0.582157i	$-0.197791\pi$
$860$		0			0
$861$	−5.06226			−0.172521
$862$		0			0
$863$			38.9377i			1.32546i	0.748860	+	0.662728i	$0.230601\pi$
$863$			38.9377i			1.32546i	−0.748860	+	0.662728i	$0.769399\pi$
$864$		0			0
$865$		−	6.06739i		−	0.206297i
$866$		0			0
$867$		−	0.0622577i		−	0.00211438i
$868$		0			0
$869$	−10.0000	−	3.16228i	−0.339227	−	0.107273i
$870$		0			0
$871$	−10.7124			−0.362975
$872$		0			0
$873$	7.06226			0.239021
$874$		0			0
$875$	4.13065			0.139641
$876$		0			0
$877$		−	46.1483i		−	1.55832i	−0.626826	−	0.779159i	$-0.715646\pi$
$877$		−	46.1483i		−	1.55832i	0.626826	−	0.779159i	$-0.284354\pi$
$878$		0			0
$879$	14.8430			0.500643
$880$		0			0
$881$	−4.12452			−0.138958			−0.0694792	−	0.997583i	$-0.522134\pi$
$881$	−4.12452			−0.138958			−0.0694792	+	0.997583i	$0.522134\pi$
$882$		0			0
$883$		−	9.87548i		−	0.332337i	−0.986097	−	0.166168i	$-0.946861\pi$
$883$		−	9.87548i		−	0.332337i	0.986097	−	0.166168i	$-0.0531395\pi$
$884$		0			0
$885$	13.0623			0.439083
$886$		0			0
$887$	15.0399			0.504990			0.252495	−	0.967598i	$-0.418749\pi$
$887$	15.0399			0.504990			0.252495	+	0.967598i	$0.418749\pi$
$888$		0			0
$889$	−59.1868			−1.98506
$890$		0			0
$891$	3.16228	+	1.00000i	0.105940	+	0.0335013i
$892$		0			0
$893$			40.7922i			1.36506i
$894$		0			0
$895$		−	12.0000i		−	0.401116i
$896$		0			0
$897$			8.77563i			0.293010i
$898$		0			0
$899$		0			0
$900$		0			0
$901$			53.9556i			1.79752i
$902$		0			0
$903$			17.0623i			0.567796i
$904$		0			0
$905$	−13.0623			−0.434204
$906$		0			0
$907$		−	16.2490i		−	0.539540i	−0.962925	−	0.269770i	$-0.913052\pi$
$907$		−	16.2490i		−	0.539540i	0.962925	−	0.269770i	$-0.0869477\pi$
$908$		0			0
$909$			7.03576i			0.233361i
$910$		0			0
$911$		−	32.0000i		−	1.06021i	−0.847933	−	0.530104i	$-0.822153\pi$
$911$		−	32.0000i		−	1.06021i	0.847933	−	0.530104i	$-0.177847\pi$
$912$		0			0
$913$	23.0623	+	7.29293i	0.763249	+	0.241361i
$914$		0			0
$915$	−14.5859			−0.482193
$916$		0			0
$917$	18.1245			0.598524
$918$		0			0
$919$	9.48683			0.312942			0.156471	−	0.987683i	$-0.449988\pi$
$919$	9.48683			0.312942			0.156471	+	0.987683i	$0.449988\pi$
$920$		0			0
$921$			18.2022i			0.599782i
$922$		0			0
$923$	−6.83889			−0.225105
$924$		0			0
$925$	−7.06226			−0.232206
$926$		0			0
$927$		−	11.0623i		−	0.363332i
$928$		0			0
$929$	−4.12452			−0.135321			−0.0676605	−	0.997708i	$-0.521553\pi$
$929$	−4.12452			−0.135321			−0.0676605	+	0.997708i	$0.521553\pi$
$930$		0			0
$931$	−51.3076			−1.68154
$932$		0			0
$933$	−1.06226			−0.0347768
$934$		0			0
$935$	−4.13065	+	13.0623i	−0.135087	+	0.427182i
$936$		0			0
$937$		−	19.9420i		−	0.651478i	−0.945460	−	0.325739i	$-0.894387\pi$
$937$		−	19.9420i		−	0.651478i	0.945460	−	0.325739i	$-0.105613\pi$
$938$		0			0
$939$		−	11.0623i		−	0.361003i
$940$		0			0
$941$			32.3340i			1.05406i	0.849847	+	0.527029i	$0.176694\pi$
$941$			32.3340i			1.05406i	−0.849847	+	0.527029i	$0.823306\pi$
$942$		0			0
$943$	−11.1061			−0.361665
$944$		0			0
$945$		−	4.13065i		−	0.134370i
$946$		0			0
$947$		−	20.0000i		−	0.649913i	−0.945729	−	0.324956i	$-0.894650\pi$
$947$		−	20.0000i		−	0.649913i	0.945729	−	0.324956i	$-0.105350\pi$
$948$		0			0
$949$	15.0623			0.488942
$950$		0			0
$951$		−	31.1868i		−	1.01130i
$952$		0			0
$953$		−	13.8144i		−	0.447491i	−0.974648	−	0.223745i	$-0.928172\pi$
$953$		−	13.8144i		−	0.447491i	0.974648	−	0.223745i	$-0.0718284\pi$
$954$		0			0
$955$		−	8.00000i		−	0.258874i
$956$		0			0
$957$	−10.0000	−	3.16228i	−0.323254	−	0.102222i
$958$		0			0
$959$	−41.3065			−1.33386
$960$		0			0
$961$	31.0000			1.00000
$962$		0			0
$963$	−3.41944			−0.110190
$964$		0			0
$965$		−	13.1032i		−	0.421805i
$966$		0			0
$967$	−23.1043			−0.742985			−0.371492	−	0.928436i	$-0.621154\pi$
$967$	−23.1043			−0.742985			−0.371492	+	0.928436i	$0.621154\pi$
$968$		0			0
$969$	21.0623			0.676617
$970$		0			0
$971$		−	8.12452i		−	0.260728i	−0.991466	−	0.130364i	$-0.958385\pi$
$971$		−	8.12452i		−	0.260728i	0.991466	−	0.130364i	$-0.0416146\pi$
$972$		0			0
$973$	37.0623			1.18816
$974$		0			0
$975$	0.968371			0.0310127
$976$		0			0
$977$	22.2490			0.711810			0.355905	−	0.934522i	$-0.384173\pi$
$977$	22.2490			0.711810			0.355905	+	0.934522i	$0.384173\pi$
$978$		0			0
$979$	−6.32456	−	2.00000i	−0.202134	−	0.0639203i
$980$		0			0
$981$			10.1980i			0.325598i
$982$		0			0
$983$			15.1868i			0.484383i	0.970228	+	0.242191i	$0.0778662\pi$
$983$			15.1868i			0.484383i	−0.970228	+	0.242191i	$0.922134\pi$
$984$		0			0
$985$		−	18.7165i		−	0.596357i
$986$		0			0
$987$	33.0452			1.05184
$988$		0			0
$989$			37.4330i			1.19030i
$990$		0			0
$991$			40.2490i			1.27855i	0.768977	+	0.639276i	$0.220766\pi$
$991$			40.2490i			1.27855i	−0.768977	+	0.639276i	$0.779234\pi$
$992$		0			0
$993$	−22.1245			−0.702100
$994$		0			0
$995$			12.0000i			0.380426i
$996$		0			0
$997$		−	3.41944i		−	0.108295i	−0.998533	−	0.0541474i	$-0.982756\pi$
$997$		−	3.41944i		−	0.108295i	0.998533	−	0.0541474i	$-0.0172441\pi$
$998$		0			0
$999$			7.06226i			0.223440i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2640.2.t.c.1231.8	yes	8
4.3	odd	2	inner	2640.2.t.c.1231.1	✓	8
11.10	odd	2	inner	2640.2.t.c.1231.5	yes	8
44.43	even	2	inner	2640.2.t.c.1231.4	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2640.2.t.c.1231.1	✓	8	4.3	odd	2	inner
2640.2.t.c.1231.4	yes	8	44.43	even	2	inner
2640.2.t.c.1231.5	yes	8	11.10	odd	2	inner
2640.2.t.c.1231.8	yes	8	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 2640 = 2^{4} \cdot 3 \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2640.t (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} +1.00000 q^{5} +4.13065 q^{7} -1.00000 q^{9} +(3.16228 + 1.00000i) q^{11} -0.968371i q^{13} +1.00000i q^{15} +4.13065i q^{17} -5.09902 q^{19} +4.13065i q^{21} +9.06226i q^{23} +1.00000 q^{25} -1.00000i q^{27} +3.16228i q^{29} +(-1.00000 + 3.16228i) q^{33} +4.13065 q^{35} -7.06226 q^{37} +0.968371 q^{39} +1.22554i q^{41} +4.13065 q^{43} -1.00000 q^{45} -8.00000i q^{47} +10.0623 q^{49} -4.13065 q^{51} +13.0623 q^{53} +(3.16228 + 1.00000i) q^{55} -5.09902i q^{57} -13.0623i q^{59} +14.5859i q^{61} -4.13065 q^{63} -0.968371i q^{65} -11.0623i q^{67} -9.06226 q^{69} -7.06226i q^{71} +15.5542i q^{73} +1.00000i q^{75} +(13.0623 + 4.13065i) q^{77} -3.16228 q^{79} +1.00000 q^{81} +7.29293 q^{83} +4.13065i q^{85} -3.16228 q^{87} -2.00000 q^{89} -4.00000i q^{91} -5.09902 q^{95} -7.06226 q^{97} +(-3.16228 - 1.00000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{5} - 8 q^{9} + 8 q^{25} - 8 q^{33} + 8 q^{37} - 8 q^{45} + 16 q^{49} + 40 q^{53} - 8 q^{69} + 40 q^{77} + 8 q^{81} - 16 q^{89} + 8 q^{97}+O(q^{100}) \) 8 * q + 8 * q^5 - 8 * q^9 + 8 * q^25 - 8 * q^33 + 8 * q^37 - 8 * q^45 + 16 * q^49 + 40 * q^53 - 8 * q^69 + 40 * q^77 + 8 * q^81 - 16 * q^89 + 8 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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