LMFDB - Embedded newform 2720.2.l.a.2481.15

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2720,2,Mod(2481,2720)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2720, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 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("2720.2481"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 2720 = 2^{5} \cdot 5 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2720.l (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [36,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	no
Analytic conductor:	$21.7193093498$
Analytic rank:	$0$
Dimension:	$36$
Twist minimal:	no (minimal twist has level 680)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2481.15
Character	$\chi$	$=$	2720.2481
Dual form			2720.2.l.a.2481.16

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-0.700054 q^{3} -1.00000 q^{5} +0.472369i q^{7} -2.50992 q^{9} +3.76587 q^{11} +2.08475i q^{13} +0.700054 q^{15} +(3.62268 + 1.96881i) q^{17} -4.24061i q^{19} -0.330684i q^{21} -3.04244i q^{23} +1.00000 q^{25} +3.85724 q^{27} -8.44930 q^{29} +1.89633i q^{31} -2.63631 q^{33} -0.472369i q^{35} -4.90602 q^{37} -1.45943i q^{39} +10.3105i q^{41} -8.18245i q^{43} +2.50992 q^{45} +5.56596 q^{47} +6.77687 q^{49} +(-2.53607 - 1.37827i) q^{51} -6.11891i q^{53} -3.76587 q^{55} +2.96866i q^{57} -5.25557i q^{59} -4.08954 q^{61} -1.18561i q^{63} -2.08475i q^{65} +14.0350i q^{67} +2.12987i q^{69} +14.4641i q^{71} +11.7807i q^{73} -0.700054 q^{75} +1.77888i q^{77} +14.2573i q^{79} +4.82950 q^{81} +1.62221i q^{83} +(-3.62268 - 1.96881i) q^{85} +5.91497 q^{87} +4.13274 q^{89} -0.984770 q^{91} -1.32754i q^{93} +4.24061i q^{95} -1.51194i q^{97} -9.45205 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 36 q - 4 q^{3} - 36 q^{5} + 36 q^{9} - 8 q^{11} + 4 q^{15} + 36 q^{25} - 16 q^{27} - 8 q^{33} - 36 q^{45} - 20 q^{47} - 36 q^{49} + 8 q^{55} - 4 q^{75} + 44 q^{81} - 24 q^{87} - 56 q^{91} - 40 q^{99}+O(q^{100}) $ 36 * q - 4 * q^3 - 36 * q^5 + 36 * q^9 - 8 * q^11 + 4 * q^15 + 36 * q^25 - 16 * q^27 - 8 * q^33 - 36 * q^45 - 20 * q^47 - 36 * q^49 + 8 * q^55 - 4 * q^75 + 44 * q^81 - 24 * q^87 - 56 * q^91 - 40 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2720\mathbb{Z}\right)^\times$.

$n$	$511$	$1601$	$1701$	$2177$
$\chi(n)$	$1$	$-1$	$-1$	$1$

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
$3$	−0.700054			−0.404176			−0.202088	−	0.979367i	$-0.564773\pi$
$3$	−0.700054			−0.404176			−0.202088	+	0.979367i	$0.564773\pi$
$4$		0			0
$5$	−1.00000			−0.447214
$5$	−1.00000			−0.447214
$6$		0			0
$7$			0.472369i			0.178539i	0.996008	+	0.0892694i	$0.0284532\pi$
$7$			0.472369i			0.178539i	−0.996008	+	0.0892694i	$0.971547\pi$
$8$		0			0
$9$	−2.50992			−0.836642
$10$		0			0
$11$	3.76587			1.13545			0.567726	−	0.823217i	$-0.307823\pi$
$11$	3.76587			1.13545			0.567726	+	0.823217i	$0.307823\pi$
$12$		0			0
$13$			2.08475i			0.578205i	0.957298	+	0.289102i	$0.0933568\pi$
$13$			2.08475i			0.578205i	−0.957298	+	0.289102i	$0.906643\pi$
$14$		0			0
$15$	0.700054			0.180753
$16$		0			0
$17$	3.62268	+	1.96881i	0.878628	+	0.477507i
$17$	3.62268	+	1.96881i	0.878628	+	0.477507i
$18$		0			0
$19$		−	4.24061i		−	0.972863i	−0.873719	−	0.486432i	$-0.838298\pi$
$19$		−	4.24061i		−	0.972863i	0.873719	−	0.486432i	$-0.161702\pi$
$20$		0			0
$21$		−	0.330684i		−	0.0721611i
$22$		0			0
$23$		−	3.04244i		−	0.634393i	−0.948360	−	0.317197i	$-0.897259\pi$
$23$		−	3.04244i		−	0.634393i	0.948360	−	0.317197i	$-0.102741\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$	3.85724			0.742327
$28$		0			0
$29$	−8.44930			−1.56900			−0.784498	−	0.620131i	$-0.787079\pi$
$29$	−8.44930			−1.56900			−0.784498	+	0.620131i	$0.787079\pi$
$30$		0			0
$31$			1.89633i			0.340592i	0.985393	+	0.170296i	$0.0544723\pi$
$31$			1.89633i			0.340592i	−0.985393	+	0.170296i	$0.945528\pi$
$32$		0			0
$33$	−2.63631			−0.458923
$34$		0			0
$35$		−	0.472369i		−	0.0798449i
$36$		0			0
$37$	−4.90602			−0.806544			−0.403272	−	0.915080i	$-0.632127\pi$
$37$	−4.90602			−0.806544			−0.403272	+	0.915080i	$0.632127\pi$
$38$		0			0
$39$		−	1.45943i		−	0.233697i
$40$		0			0
$41$			10.3105i			1.61022i	0.593123	+	0.805112i	$0.297895\pi$
$41$			10.3105i			1.61022i	−0.593123	+	0.805112i	$0.702105\pi$
$42$		0			0
$43$		−	8.18245i		−	1.24781i	−0.781500	−	0.623906i	$-0.785545\pi$
$43$		−	8.18245i		−	1.24781i	0.781500	−	0.623906i	$-0.214455\pi$
$44$		0			0
$45$	2.50992			0.374157
$46$		0			0
$47$	5.56596			0.811879			0.405940	−	0.913900i	$-0.366944\pi$
$47$	5.56596			0.811879			0.405940	+	0.913900i	$0.366944\pi$
$48$		0			0
$49$	6.77687			0.968124
$50$		0			0
$51$	−2.53607	−	1.37827i	−0.355121	−	0.192997i
$52$		0			0
$53$		−	6.11891i		−	0.840497i	−0.907409	−	0.420249i	$-0.861943\pi$
$53$		−	6.11891i		−	0.840497i	0.907409	−	0.420249i	$-0.138057\pi$
$54$		0			0
$55$	−3.76587			−0.507790
$56$		0			0
$57$			2.96866i			0.393208i
$58$		0			0
$59$		−	5.25557i		−	0.684217i	−0.939660	−	0.342109i	$-0.888859\pi$
$59$		−	5.25557i		−	0.684217i	0.939660	−	0.342109i	$-0.111141\pi$
$60$		0			0
$61$	−4.08954			−0.523612			−0.261806	−	0.965121i	$-0.584318\pi$
$61$	−4.08954			−0.523612			−0.261806	+	0.965121i	$0.584318\pi$
$62$		0			0
$63$		−	1.18561i		−	0.149373i
$64$		0			0
$65$		−	2.08475i		−	0.258581i
$66$		0			0
$67$			14.0350i			1.71465i	0.514775	+	0.857325i	$0.327875\pi$
$67$			14.0350i			1.71465i	−0.514775	+	0.857325i	$0.672125\pi$
$68$		0			0
$69$			2.12987i			0.256407i
$70$		0			0
$71$			14.4641i			1.71657i	0.513171	+	0.858287i	$0.328471\pi$
$71$			14.4641i			1.71657i	−0.513171	+	0.858287i	$0.671529\pi$
$72$		0			0
$73$			11.7807i			1.37883i	0.724368	+	0.689413i	$0.242132\pi$
$73$			11.7807i			1.37883i	−0.724368	+	0.689413i	$0.757868\pi$
$74$		0			0
$75$	−0.700054			−0.0808353
$76$		0			0
$77$			1.77888i			0.202722i
$78$		0			0
$79$			14.2573i			1.60408i	0.597273	+	0.802038i	$0.296251\pi$
$79$			14.2573i			1.60408i	−0.597273	+	0.802038i	$0.703749\pi$
$80$		0			0
$81$	4.82950			0.536611
$82$		0			0
$83$			1.62221i			0.178060i	0.996029	+	0.0890302i	$0.0283768\pi$
$83$			1.62221i			0.178060i	−0.996029	+	0.0890302i	$0.971623\pi$
$84$		0			0
$85$	−3.62268	−	1.96881i	−0.392934	−	0.213547i
$86$		0			0
$87$	5.91497			0.634151
$88$		0			0
$89$	4.13274			0.438070			0.219035	−	0.975717i	$-0.429709\pi$
$89$	4.13274			0.438070			0.219035	+	0.975717i	$0.429709\pi$
$90$		0			0
$91$	−0.984770			−0.103232
$92$		0			0
$93$		−	1.32754i		−	0.137659i
$94$		0			0
$95$			4.24061i			0.435078i
$96$		0			0
$97$		−	1.51194i		−	0.153514i	−0.997050	−	0.0767570i	$-0.975543\pi$
$97$		−	1.51194i		−	0.153514i	0.997050	−	0.0767570i	$-0.0244566\pi$
$98$		0			0
$99$	−9.45205			−0.949967
$100$		0			0
$101$			15.9111i			1.58321i	0.611032	+	0.791606i	$0.290755\pi$
$101$			15.9111i			1.58321i	−0.611032	+	0.791606i	$0.709245\pi$
$102$		0			0
$103$	14.9491			1.47298			0.736488	−	0.676451i	$-0.236483\pi$
$103$	14.9491			1.47298			0.736488	+	0.676451i	$0.236483\pi$
$104$		0			0
$105$			0.330684i			0.0322714i
$106$		0			0
$107$	15.6850			1.51633			0.758165	−	0.652063i	$-0.226096\pi$
$107$	15.6850			1.51633			0.758165	+	0.652063i	$0.226096\pi$
$108$		0			0
$109$	16.0643			1.53869			0.769343	−	0.638836i	$-0.220584\pi$
$109$	16.0643			1.53869			0.769343	+	0.638836i	$0.220584\pi$
$110$		0			0
$111$	3.43448			0.325986
$112$		0			0
$113$		−	7.80211i		−	0.733961i	−0.930229	−	0.366980i	$-0.880392\pi$
$113$		−	7.80211i		−	0.733961i	0.930229	−	0.366980i	$-0.119608\pi$
$114$		0			0
$115$			3.04244i			0.283709i
$116$		0			0
$117$		−	5.23256i		−	0.483750i
$118$		0			0
$119$	−0.930005	+	1.71124i	−0.0852534	+	0.156869i
$120$		0			0
$121$	3.18178			0.289252
$122$		0			0
$123$		−	7.21788i		−	0.650814i
$124$		0			0
$125$	−1.00000			−0.0894427
$126$		0			0
$127$	−10.8800			−0.965447			−0.482724	−	0.875773i	$-0.660352\pi$
$127$	−10.8800			−0.965447			−0.482724	+	0.875773i	$0.660352\pi$
$128$		0			0
$129$			5.72815i			0.504336i
$130$		0			0
$131$	13.9563			1.21937			0.609685	−	0.792644i	$-0.291296\pi$
$131$	13.9563			1.21937			0.609685	+	0.792644i	$0.291296\pi$
$132$		0			0
$133$	2.00313			0.173694
$134$		0			0
$135$	−3.85724			−0.331979
$136$		0			0
$137$	−16.2479			−1.38815			−0.694075	−	0.719903i	$-0.744186\pi$
$137$	−16.2479			−1.38815			−0.694075	+	0.719903i	$0.744186\pi$
$138$		0			0
$139$	3.86189			0.327561			0.163780	−	0.986497i	$-0.447631\pi$
$139$	3.86189			0.327561			0.163780	+	0.986497i	$0.447631\pi$
$140$		0			0
$141$	−3.89647			−0.328142
$142$		0			0
$143$			7.85089i			0.656524i
$144$		0			0
$145$	8.44930			0.701676
$146$		0			0
$147$	−4.74417			−0.391293
$148$		0			0
$149$			15.2325i			1.24789i	0.781467	+	0.623946i	$0.214472\pi$
$149$			15.2325i			1.24789i	−0.781467	+	0.623946i	$0.785528\pi$
$150$		0			0
$151$	1.45467			0.118380			0.0591899	−	0.998247i	$-0.481148\pi$
$151$	1.45467			0.118380			0.0591899	+	0.998247i	$0.481148\pi$
$152$		0			0
$153$	−9.09265	−	4.94157i	−0.735097	−	0.399502i
$154$		0			0
$155$		−	1.89633i		−	0.152317i
$156$		0			0
$157$			2.38317i			0.190198i	0.995468	+	0.0950989i	$0.0303167\pi$
$157$			2.38317i			0.190198i	−0.995468	+	0.0950989i	$0.969683\pi$
$158$		0			0
$159$			4.28357i			0.339709i
$160$		0			0
$161$	1.43716			0.113264
$162$		0			0
$163$	−10.0630			−0.788196			−0.394098	−	0.919068i	$-0.628943\pi$
$163$	−10.0630			−0.788196			−0.394098	+	0.919068i	$0.628943\pi$
$164$		0			0
$165$	2.63631			0.205237
$166$		0			0
$167$			4.09470i			0.316857i	0.987370	+	0.158429i	$0.0506428\pi$
$167$			4.09470i			0.316857i	−0.987370	+	0.158429i	$0.949357\pi$
$168$		0			0
$169$	8.65383			0.665679
$170$		0			0
$171$			10.6436i			0.813938i
$172$		0			0
$173$	−16.3781			−1.24520			−0.622601	−	0.782540i	$-0.713924\pi$
$173$	−16.3781			−1.24520			−0.622601	+	0.782540i	$0.713924\pi$
$174$		0			0
$175$			0.472369i			0.0357077i
$176$		0			0
$177$			3.67918i			0.276544i
$178$		0			0
$179$			8.16713i			0.610440i	0.952282	+	0.305220i	$0.0987300\pi$
$179$			8.16713i			0.610440i	−0.952282	+	0.305220i	$0.901270\pi$
$180$		0			0
$181$	12.6490			0.940190			0.470095	−	0.882616i	$-0.344220\pi$
$181$	12.6490			0.940190			0.470095	+	0.882616i	$0.344220\pi$
$182$		0			0
$183$	2.86290			0.211631
$184$		0			0
$185$	4.90602			0.360698
$186$		0			0
$187$	13.6425	+	7.41428i	0.997641	+	0.542186i
$188$		0			0
$189$			1.82204i			0.132534i
$190$		0			0
$191$	9.84032			0.712021			0.356010	−	0.934482i	$-0.384137\pi$
$191$	9.84032			0.712021			0.356010	+	0.934482i	$0.384137\pi$
$192$		0			0
$193$			19.3479i			1.39270i	0.717705	+	0.696348i	$0.245193\pi$
$193$			19.3479i			1.39270i	−0.717705	+	0.696348i	$0.754807\pi$
$194$		0			0
$195$			1.45943i			0.104512i
$196$		0			0
$197$	−2.53115			−0.180337			−0.0901685	−	0.995927i	$-0.528741\pi$
$197$	−2.53115			−0.180337			−0.0901685	+	0.995927i	$0.528741\pi$
$198$		0			0
$199$		−	7.75580i		−	0.549794i	−0.961474	−	0.274897i	$-0.911356\pi$
$199$		−	7.75580i		−	0.549794i	0.961474	−	0.274897i	$-0.0886437\pi$
$200$		0			0
$201$		−	9.82527i		−	0.693021i
$202$		0			0
$203$		−	3.99119i		−	0.280127i
$204$		0			0
$205$		−	10.3105i		−	0.720114i
$206$		0			0
$207$			7.63630i			0.530760i
$208$		0			0
$209$		−	15.9696i		−	1.10464i
$210$		0			0
$211$	−10.3268			−0.710923			−0.355462	−	0.934691i	$-0.615676\pi$
$211$	−10.3268			−0.710923			−0.355462	+	0.934691i	$0.615676\pi$
$212$		0			0
$213$		−	10.1257i		−	0.693798i
$214$		0			0
$215$			8.18245i			0.558038i
$216$		0			0
$217$	−0.895769			−0.0608088
$218$		0			0
$219$		−	8.24712i		−	0.557289i
$220$		0			0
$221$	−4.10447	+	7.55236i	−0.276097	+	0.508027i
$222$		0			0
$223$	−0.853240			−0.0571372			−0.0285686	−	0.999592i	$-0.509095\pi$
$223$	−0.853240			−0.0571372			−0.0285686	+	0.999592i	$0.509095\pi$
$224$		0			0
$225$	−2.50992			−0.167328
$226$		0			0
$227$	−21.4110			−1.42110			−0.710549	−	0.703648i	$-0.751553\pi$
$227$	−21.4110			−1.42110			−0.710549	+	0.703648i	$0.751553\pi$
$228$		0			0
$229$		−	0.280780i		−	0.0185545i	−0.999957	−	0.00927725i	$-0.997047\pi$
$229$		−	0.280780i		−	0.0185545i	0.999957	−	0.00927725i	$-0.00295308\pi$
$230$		0			0
$231$		−	1.24531i		−	0.0819355i
$232$		0			0
$233$		−	2.80188i		−	0.183557i	−0.995779	−	0.0917786i	$-0.970745\pi$
$233$		−	2.80188i		−	0.183557i	0.995779	−	0.0917786i	$-0.0292552\pi$
$234$		0			0
$235$	−5.56596			−0.363083
$236$		0			0
$237$		−	9.98091i		−	0.648329i
$238$		0			0
$239$	−7.25286			−0.469148			−0.234574	−	0.972098i	$-0.575370\pi$
$239$	−7.25286			−0.469148			−0.234574	+	0.972098i	$0.575370\pi$
$240$		0			0
$241$			21.7072i			1.39828i	0.714983	+	0.699142i	$0.246435\pi$
$241$			21.7072i			1.39828i	−0.714983	+	0.699142i	$0.753565\pi$
$242$		0			0
$243$	−14.9526			−0.959212
$244$		0			0
$245$	−6.77687			−0.432958
$246$		0			0
$247$	8.84060			0.562514
$248$		0			0
$249$		−	1.13563i		−	0.0719678i
$250$		0			0
$251$			7.87719i			0.497204i	0.968606	+	0.248602i	$0.0799711\pi$
$251$			7.87719i			0.497204i	−0.968606	+	0.248602i	$0.920029\pi$
$252$		0			0
$253$		−	11.4574i		−	0.720323i
$254$		0			0
$255$	2.53607	+	1.37827i	0.158815	+	0.0863108i
$256$		0			0
$257$	6.86958			0.428513			0.214256	−	0.976777i	$-0.431267\pi$
$257$	6.86958			0.428513			0.214256	+	0.976777i	$0.431267\pi$
$258$		0			0
$259$		−	2.31745i		−	0.143999i
$260$		0			0
$261$	21.2071			1.31269
$262$		0			0
$263$	26.2544			1.61891			0.809457	−	0.587179i	$-0.199761\pi$
$263$	26.2544			1.61891			0.809457	+	0.587179i	$0.199761\pi$
$264$		0			0
$265$			6.11891i			0.375882i
$266$		0			0
$267$	−2.89314			−0.177057
$268$		0			0
$269$	19.7359			1.20332			0.601661	−	0.798752i	$-0.294506\pi$
$269$	19.7359			1.20332			0.601661	+	0.798752i	$0.294506\pi$
$270$		0			0
$271$	9.82641			0.596912			0.298456	−	0.954423i	$-0.403528\pi$
$271$	9.82641			0.596912			0.298456	+	0.954423i	$0.403528\pi$
$272$		0			0
$273$	0.689392			0.0417239
$274$		0			0
$275$	3.76587			0.227091
$276$		0			0
$277$	7.12825			0.428295			0.214148	−	0.976801i	$-0.431303\pi$
$277$	7.12825			0.428295			0.214148	+	0.976801i	$0.431303\pi$
$278$		0			0
$279$		−	4.75966i		−	0.284953i
$280$		0			0
$281$	−19.5618			−1.16696			−0.583480	−	0.812127i	$-0.698309\pi$
$281$	−19.5618			−1.16696			−0.583480	+	0.812127i	$0.698309\pi$
$282$		0			0
$283$	−8.55218			−0.508374			−0.254187	−	0.967155i	$-0.581808\pi$
$283$	−8.55218			−0.508374			−0.254187	+	0.967155i	$0.581808\pi$
$284$		0			0
$285$		−	2.96866i		−	0.175848i
$286$		0			0
$287$	−4.87034			−0.287487
$288$		0			0
$289$	9.24757	+	14.2647i	0.543975	+	0.839102i
$290$		0			0
$291$			1.05844i			0.0620467i
$292$		0			0
$293$		−	22.8881i		−	1.33714i	−0.743649	−	0.668570i	$-0.766907\pi$
$293$		−	22.8881i		−	1.33714i	0.743649	−	0.668570i	$-0.233093\pi$
$294$		0			0
$295$			5.25557i			0.305991i
$296$		0			0
$297$	14.5259			0.842877
$298$		0			0
$299$	6.34272			0.366809
$300$		0			0
$301$	3.86513			0.222783
$302$		0			0
$303$		−	11.1386i		−	0.639897i
$304$		0			0
$305$	4.08954			0.234166
$306$		0			0
$307$			4.75160i			0.271188i	0.990764	+	0.135594i	$0.0432943\pi$
$307$			4.75160i			0.271188i	−0.990764	+	0.135594i	$0.956706\pi$
$308$		0			0
$309$	−10.4652			−0.595342
$310$		0			0
$311$		−	18.5567i		−	1.05225i	−0.850407	−	0.526126i	$-0.823644\pi$
$311$		−	18.5567i		−	1.05225i	0.850407	−	0.526126i	$-0.176356\pi$
$312$		0			0
$313$			27.7728i			1.56981i	0.619614	+	0.784907i	$0.287289\pi$
$313$			27.7728i			1.56981i	−0.619614	+	0.784907i	$0.712711\pi$
$314$		0			0
$315$			1.18561i			0.0668016i
$316$		0			0
$317$	6.90582			0.387869			0.193935	−	0.981014i	$-0.437875\pi$
$317$	6.90582			0.387869			0.193935	+	0.981014i	$0.437875\pi$
$318$		0			0
$319$	−31.8190			−1.78152
$320$		0			0
$321$	−10.9804			−0.612864
$322$		0			0
$323$	8.34896	−	15.3624i	0.464549	−	0.854785i
$324$		0			0
$325$			2.08475i			0.115641i
$326$		0			0
$327$	−11.2459			−0.621900
$328$		0			0
$329$			2.62919i			0.144952i
$330$		0			0
$331$			17.7274i			0.974385i	0.873295	+	0.487193i	$0.161979\pi$
$331$			17.7274i			0.974385i	−0.873295	+	0.487193i	$0.838021\pi$
$332$		0			0
$333$	12.3137			0.674789
$334$		0			0
$335$		−	14.0350i		−	0.766815i
$336$		0			0
$337$		−	6.64499i		−	0.361975i	−0.983485	−	0.180988i	$-0.942071\pi$
$337$		−	6.64499i		−	0.361975i	0.983485	−	0.180988i	$-0.0579294\pi$
$338$		0			0
$339$			5.46190i			0.296650i
$340$		0			0
$341$			7.14135i			0.386726i
$342$		0			0
$343$			6.50777i			0.351386i
$344$		0			0
$345$		−	2.12987i		−	0.114669i
$346$		0			0
$347$	−6.94078			−0.372600			−0.186300	−	0.982493i	$-0.559650\pi$
$347$	−6.94078			−0.372600			−0.186300	+	0.982493i	$0.559650\pi$
$348$		0			0
$349$			27.3230i			1.46257i	0.682072	+	0.731285i	$0.261079\pi$
$349$			27.3230i			1.46257i	−0.682072	+	0.731285i	$0.738921\pi$
$350$		0			0
$351$			8.04138i			0.429217i
$352$		0			0
$353$	5.69659			0.303199			0.151599	−	0.988442i	$-0.451558\pi$
$353$	5.69659			0.303199			0.151599	+	0.988442i	$0.451558\pi$
$354$		0			0
$355$		−	14.4641i		−	0.767675i
$356$		0			0
$357$	0.651054	−	1.19796i	0.0344574	−	0.0634028i
$358$		0			0
$359$	−28.5022			−1.50429			−0.752143	−	0.659000i	$-0.770980\pi$
$359$	−28.5022			−1.50429			−0.752143	+	0.659000i	$0.770980\pi$
$360$		0			0
$361$	1.01720			0.0535367
$362$		0			0
$363$	−2.22742			−0.116909
$364$		0			0
$365$		−	11.7807i		−	0.616630i
$366$		0			0
$367$		−	30.1453i		−	1.57357i	−0.617227	−	0.786786i	$-0.711744\pi$
$367$		−	30.1453i		−	1.57357i	0.617227	−	0.786786i	$-0.288256\pi$
$368$		0			0
$369$		−	25.8785i		−	1.34718i
$370$		0			0
$371$	2.89038			0.150061
$372$		0			0
$373$			8.99135i			0.465555i	0.972530	+	0.232777i	$0.0747813\pi$
$373$			8.99135i			0.465555i	−0.972530	+	0.232777i	$0.925219\pi$
$374$		0			0
$375$	0.700054			0.0361506
$376$		0			0
$377$		−	17.6147i		−	0.907201i
$378$		0			0
$379$	−11.8768			−0.610068			−0.305034	−	0.952341i	$-0.598668\pi$
$379$	−11.8768			−0.610068			−0.305034	+	0.952341i	$0.598668\pi$
$380$		0			0
$381$	7.61661			0.390211
$382$		0			0
$383$	−10.6980			−0.546644			−0.273322	−	0.961923i	$-0.588123\pi$
$383$	−10.6980			−0.546644			−0.273322	+	0.961923i	$0.588123\pi$
$384$		0			0
$385$		−	1.77888i		−	0.0906601i
$386$		0			0
$387$			20.5373i			1.04397i
$388$		0			0
$389$		−	14.5550i		−	0.737970i	−0.929435	−	0.368985i	$-0.879705\pi$
$389$		−	14.5550i		−	0.737970i	0.929435	−	0.368985i	$-0.120295\pi$
$390$		0			0
$391$	5.98999	−	11.0218i	0.302927	−	0.557396i
$392$		0			0
$393$	−9.77018			−0.492840
$394$		0			0
$395$		−	14.2573i		−	0.717365i
$396$		0			0
$397$	5.14070			0.258004			0.129002	−	0.991644i	$-0.458823\pi$
$397$	5.14070			0.258004			0.129002	+	0.991644i	$0.458823\pi$
$398$		0			0
$399$	−1.40230			−0.0702029
$400$		0			0
$401$			14.8456i			0.741354i	0.928762	+	0.370677i	$0.120874\pi$
$401$			14.8456i			0.741354i	−0.928762	+	0.370677i	$0.879126\pi$
$402$		0			0
$403$	−3.95338			−0.196932
$404$		0			0
$405$	−4.82950			−0.239980
$406$		0			0
$407$	−18.4754			−0.915793
$408$		0			0
$409$	28.8169			1.42490			0.712451	−	0.701722i	$-0.247585\pi$
$409$	28.8169			1.42490			0.712451	+	0.701722i	$0.247585\pi$
$410$		0			0
$411$	11.3744			0.561057
$412$		0			0
$413$	2.48257			0.122159
$414$		0			0
$415$		−	1.62221i		−	0.0796310i
$416$		0			0
$417$	−2.70353			−0.132392
$418$		0			0
$419$	−15.8132			−0.772525			−0.386263	−	0.922389i	$-0.626234\pi$
$419$	−15.8132			−0.772525			−0.386263	+	0.922389i	$0.626234\pi$
$420$		0			0
$421$			0.972923i			0.0474174i	0.999719	+	0.0237087i	$0.00754742\pi$
$421$			0.972923i			0.0474174i	−0.999719	+	0.0237087i	$0.992453\pi$
$422$		0			0
$423$	−13.9702			−0.679252
$424$		0			0
$425$	3.62268	+	1.96881i	0.175726	+	0.0955013i
$426$		0			0
$427$		−	1.93177i		−	0.0934849i
$428$		0			0
$429$		−	5.49604i		−	0.265351i
$430$		0			0
$431$		−	29.2081i		−	1.40690i	−0.710744	−	0.703451i	$-0.751641\pi$
$431$		−	29.2081i		−	1.40690i	0.710744	−	0.703451i	$-0.248359\pi$
$432$		0			0
$433$	−23.8697			−1.14710			−0.573552	−	0.819169i	$-0.694435\pi$
$433$	−23.8697			−1.14710			−0.573552	+	0.819169i	$0.694435\pi$
$434$		0			0
$435$	−5.91497			−0.283601
$436$		0			0
$437$	−12.9018			−0.617178
$438$		0			0
$439$			24.6752i			1.17768i	0.808249	+	0.588841i	$0.200415\pi$
$439$			24.6752i			1.17768i	−0.808249	+	0.588841i	$0.799585\pi$
$440$		0			0
$441$	−17.0094			−0.809973
$442$		0			0
$443$		−	6.58438i		−	0.312834i	−0.987691	−	0.156417i	$-0.950006\pi$
$443$		−	6.58438i		−	0.312834i	0.987691	−	0.156417i	$-0.0499943\pi$
$444$		0			0
$445$	−4.13274			−0.195911
$446$		0			0
$447$		−	10.6636i		−	0.504369i
$448$		0			0
$449$		−	17.6917i		−	0.834925i	−0.908694	−	0.417462i	$-0.862920\pi$
$449$		−	17.6917i		−	0.834925i	0.908694	−	0.417462i	$-0.137080\pi$
$450$		0			0
$451$			38.8279i			1.82833i
$452$		0			0
$453$	−1.01835			−0.0478463
$454$		0			0
$455$	0.984770			0.0461667
$456$		0			0
$457$	8.25279			0.386049			0.193025	−	0.981194i	$-0.438170\pi$
$457$	8.25279			0.386049			0.193025	+	0.981194i	$0.438170\pi$
$458$		0			0
$459$	13.9735	+	7.59418i	0.652229	+	0.354466i
$460$		0			0
$461$			8.12505i			0.378421i	0.981937	+	0.189211i	$0.0605929\pi$
$461$			8.12505i			0.378421i	−0.981937	+	0.189211i	$0.939407\pi$
$462$		0			0
$463$	25.7721			1.19773			0.598864	−	0.800850i	$-0.295619\pi$
$463$	25.7721			1.19773			0.598864	+	0.800850i	$0.295619\pi$
$464$		0			0
$465$			1.32754i			0.0615630i
$466$		0			0
$467$		−	31.6325i		−	1.46378i	−0.681425	−	0.731888i	$-0.738639\pi$
$467$		−	31.6325i		−	1.46378i	0.681425	−	0.731888i	$-0.261361\pi$
$468$		0			0
$469$	−6.62971			−0.306131
$470$		0			0
$471$		−	1.66835i		−	0.0768734i
$472$		0			0
$473$		−	30.8140i		−	1.41683i
$474$		0			0
$475$		−	4.24061i		−	0.194573i
$476$		0			0
$477$			15.3580i			0.703195i
$478$		0			0
$479$			10.9120i			0.498584i	0.968428	+	0.249292i	$0.0801978\pi$
$479$			10.9120i			0.498584i	−0.968428	+	0.249292i	$0.919802\pi$
$480$		0			0
$481$		−	10.2278i		−	0.466348i
$482$		0			0
$483$	−1.00609			−0.0457785
$484$		0			0
$485$			1.51194i			0.0686536i
$486$		0			0
$487$		−	8.42188i		−	0.381632i	−0.981626	−	0.190816i	$-0.938887\pi$
$487$		−	8.42188i		−	0.381632i	0.981626	−	0.190816i	$-0.0611134\pi$
$488$		0			0
$489$	7.04465			0.318570
$490$		0			0
$491$		−	1.68337i		−	0.0759696i	−0.999278	−	0.0379848i	$-0.987906\pi$
$491$		−	1.68337i		−	0.0759696i	0.999278	−	0.0379848i	$-0.0120938\pi$
$492$		0			0
$493$	−30.6091	−	16.6351i	−1.37856	−	0.749206i
$494$		0			0
$495$	9.45205			0.424838
$496$		0			0
$497$	−6.83239			−0.306475
$498$		0			0
$499$	−2.69995			−0.120866			−0.0604331	−	0.998172i	$-0.519248\pi$
$499$	−2.69995			−0.120866			−0.0604331	+	0.998172i	$0.519248\pi$
$500$		0			0
$501$		−	2.86651i		−	0.128066i
$502$		0			0
$503$		−	41.6996i		−	1.85929i	−0.368452	−	0.929647i	$-0.620112\pi$
$503$		−	41.6996i		−	1.85929i	0.368452	−	0.929647i	$-0.379888\pi$
$504$		0			0
$505$		−	15.9111i		−	0.708034i
$506$		0			0
$507$	−6.05815			−0.269052
$508$		0			0
$509$		−	21.3267i		−	0.945291i	−0.881253	−	0.472646i	$-0.843299\pi$
$509$		−	21.3267i		−	0.945291i	0.881253	−	0.472646i	$-0.156701\pi$
$510$		0			0
$511$	−5.56484			−0.246174
$512$		0			0
$513$		−	16.3571i		−	0.722183i
$514$		0			0
$515$	−14.9491			−0.658735
$516$		0			0
$517$	20.9607			0.921850
$518$		0			0
$519$	11.4655			0.503281
$520$		0			0
$521$		−	5.58565i		−	0.244712i	−0.992486	−	0.122356i	$-0.960955\pi$
$521$		−	5.58565i		−	0.244712i	0.992486	−	0.122356i	$-0.0390449\pi$
$522$		0			0
$523$		−	30.3049i		−	1.32514i	−0.748999	−	0.662571i	$-0.769465\pi$
$523$		−	30.3049i		−	1.32514i	0.748999	−	0.662571i	$-0.230535\pi$
$524$		0			0
$525$		−	0.330684i		−	0.0144322i
$526$		0			0
$527$	−3.73352	+	6.86981i	−0.162635	+	0.299253i
$528$		0			0
$529$	13.7435			0.597545
$530$		0			0
$531$			13.1911i			0.572445i
$532$		0			0
$533$	−21.4947			−0.931039
$534$		0			0
$535$	−15.6850			−0.678123
$536$		0			0
$537$		−	5.71743i		−	0.246725i
$538$		0			0
$539$	25.5208			1.09926
$540$		0			0
$541$	−5.91685			−0.254385			−0.127193	−	0.991878i	$-0.540597\pi$
$541$	−5.91685			−0.254385			−0.127193	+	0.991878i	$0.540597\pi$
$542$		0			0
$543$	−8.85496			−0.380003
$544$		0			0
$545$	−16.0643			−0.688121
$546$		0			0
$547$	39.6019			1.69325			0.846627	−	0.532187i	$-0.178629\pi$
$547$	39.6019			1.69325			0.846627	+	0.532187i	$0.178629\pi$
$548$		0			0
$549$	10.2644			0.438075
$550$		0			0
$551$			35.8302i			1.52642i
$552$		0			0
$553$	−6.73473			−0.286390
$554$		0			0
$555$	−3.43448			−0.145785
$556$		0			0
$557$			23.3121i			0.987763i	0.869529	+	0.493882i	$0.164422\pi$
$557$			23.3121i			0.987763i	−0.869529	+	0.493882i	$0.835578\pi$
$558$		0			0
$559$	17.0583			0.721490
$560$		0			0
$561$	−9.55051	−	5.19040i	−0.403223	−	0.219139i
$562$		0			0
$563$		−	8.22003i		−	0.346433i	−0.984884	−	0.173216i	$-0.944584\pi$
$563$		−	8.22003i		−	0.346433i	0.984884	−	0.173216i	$-0.0554160\pi$
$564$		0			0
$565$			7.80211i			0.328237i
$566$		0			0
$567$			2.28130i			0.0958058i
$568$		0			0
$569$	−42.0664			−1.76352			−0.881758	−	0.471703i	$-0.843639\pi$
$569$	−42.0664			−1.76352			−0.881758	+	0.471703i	$0.843639\pi$
$570$		0			0
$571$	45.1866			1.89100			0.945500	−	0.325623i	$-0.105574\pi$
$571$	45.1866			1.89100			0.945500	+	0.325623i	$0.105574\pi$
$572$		0			0
$573$	−6.88876			−0.287782
$574$		0			0
$575$		−	3.04244i		−	0.126879i
$576$		0			0
$577$	−9.00464			−0.374868			−0.187434	−	0.982277i	$-0.560017\pi$
$577$	−9.00464			−0.374868			−0.187434	+	0.982277i	$0.560017\pi$
$578$		0			0
$579$		−	13.5446i		−	0.562894i
$580$		0			0
$581$	−0.766280			−0.0317907
$582$		0			0
$583$		−	23.0430i		−	0.954345i
$584$		0			0
$585$			5.23256i			0.216340i
$586$		0			0
$587$		−	19.0434i		−	0.786005i	−0.919537	−	0.393002i	$-0.871436\pi$
$587$		−	19.0434i		−	0.786005i	0.919537	−	0.393002i	$-0.128564\pi$
$588$		0			0
$589$	8.04162			0.331349
$590$		0			0
$591$	1.77194			0.0728879
$592$		0			0
$593$	−15.4544			−0.634636			−0.317318	−	0.948319i	$-0.602782\pi$
$593$	−15.4544			−0.634636			−0.317318	+	0.948319i	$0.602782\pi$
$594$		0			0
$595$	0.930005	−	1.71124i	0.0381265	−	0.0701540i
$596$		0			0
$597$			5.42948i			0.222214i
$598$		0			0
$599$	−9.04450			−0.369548			−0.184774	−	0.982781i	$-0.559155\pi$
$599$	−9.04450			−0.369548			−0.184774	+	0.982781i	$0.559155\pi$
$600$		0			0
$601$			19.6183i			0.800246i	0.916461	+	0.400123i	$0.131033\pi$
$601$			19.6183i			0.800246i	−0.916461	+	0.400123i	$0.868967\pi$
$602$		0			0
$603$		−	35.2268i		−	1.43455i
$604$		0			0
$605$	−3.18178			−0.129358
$606$		0			0
$607$			25.1180i			1.01951i	0.860320	+	0.509754i	$0.170264\pi$
$607$			25.1180i			1.01951i	−0.860320	+	0.509754i	$0.829736\pi$
$608$		0			0
$609$			2.79405i			0.113220i
$610$		0			0
$611$			11.6036i			0.469432i
$612$		0			0
$613$		−	5.63031i		−	0.227406i	−0.993515	−	0.113703i	$-0.963729\pi$
$613$		−	5.63031i		−	0.227406i	0.993515	−	0.113703i	$-0.0362712\pi$
$614$		0			0
$615$			7.21788i			0.291053i
$616$		0			0
$617$		−	14.6134i		−	0.588314i	−0.955757	−	0.294157i	$-0.904961\pi$
$617$		−	14.6134i		−	0.588314i	0.955757	−	0.294157i	$-0.0950389\pi$
$618$		0			0
$619$	−10.0040			−0.402095			−0.201048	−	0.979581i	$-0.564435\pi$
$619$	−10.0040			−0.402095			−0.201048	+	0.979581i	$0.564435\pi$
$620$		0			0
$621$		−	11.7354i		−	0.470927i
$622$		0			0
$623$			1.95218i			0.0782124i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$			11.1796i			0.446469i
$628$		0			0
$629$	−17.7729	−	9.65902i	−0.708653	−	0.385130i
$630$		0			0
$631$	−13.4572			−0.535722			−0.267861	−	0.963458i	$-0.586317\pi$
$631$	−13.4572			−0.535722			−0.267861	+	0.963458i	$0.586317\pi$
$632$		0			0
$633$	7.22929			0.287338
$634$		0			0
$635$	10.8800			0.431761
$636$		0			0
$637$			14.1281i			0.559774i
$638$		0			0
$639$		−	36.3038i		−	1.43616i
$640$		0			0
$641$		−	42.8148i		−	1.69108i	−0.533909	−	0.845542i	$-0.679278\pi$
$641$		−	42.8148i		−	1.69108i	0.533909	−	0.845542i	$-0.320722\pi$
$642$		0			0
$643$	−22.4425			−0.885044			−0.442522	−	0.896758i	$-0.645916\pi$
$643$	−22.4425			−0.885044			−0.442522	+	0.896758i	$0.645916\pi$
$644$		0			0
$645$		−	5.72815i		−	0.225546i
$646$		0			0
$647$	31.4314			1.23570			0.617849	−	0.786297i	$-0.288004\pi$
$647$	31.4314			1.23570			0.617849	+	0.786297i	$0.288004\pi$
$648$		0			0
$649$		−	19.7918i		−	0.776896i
$650$		0			0
$651$	0.627087			0.0245775
$652$		0			0
$653$	−24.8465			−0.972318			−0.486159	−	0.873871i	$-0.661602\pi$
$653$	−24.8465			−0.972318			−0.486159	+	0.873871i	$0.661602\pi$
$654$		0			0
$655$	−13.9563			−0.545319
$656$		0			0
$657$		−	29.5687i		−	1.15358i
$658$		0			0
$659$			36.5507i			1.42381i	0.702273	+	0.711907i	$0.252168\pi$
$659$			36.5507i			1.42381i	−0.702273	+	0.711907i	$0.747832\pi$
$660$		0			0
$661$			17.4285i			0.677889i	0.940806	+	0.338944i	$0.110070\pi$
$661$			17.4285i			0.677889i	−0.940806	+	0.338944i	$0.889930\pi$
$662$		0			0
$663$	2.87335	−	5.28706i	0.111592	−	0.205332i
$664$		0			0
$665$	−2.00313			−0.0776782
$666$		0			0
$667$			25.7065i			0.995360i
$668$		0			0
$669$	0.597314			0.0230935
$670$		0			0
$671$	−15.4007			−0.594536
$672$		0			0
$673$		−	26.9124i		−	1.03740i	−0.854957	−	0.518699i	$-0.826417\pi$
$673$		−	26.9124i		−	1.03740i	0.854957	−	0.518699i	$-0.173583\pi$
$674$		0			0
$675$	3.85724			0.148465
$676$		0			0
$677$	15.5242			0.596642			0.298321	−	0.954466i	$-0.403573\pi$
$677$	15.5242			0.596642			0.298321	+	0.954466i	$0.403573\pi$
$678$		0			0
$679$	0.714193			0.0274082
$680$		0			0
$681$	14.9889			0.574374
$682$		0			0
$683$	32.9726			1.26166			0.630830	−	0.775921i	$-0.282715\pi$
$683$	32.9726			1.26166			0.630830	+	0.775921i	$0.282715\pi$
$684$		0			0
$685$	16.2479			0.620799
$686$		0			0
$687$			0.196561i			0.00749929i
$688$		0			0
$689$	12.7564			0.485979
$690$		0			0
$691$	−38.9502			−1.48174			−0.740869	−	0.671650i	$-0.765586\pi$
$691$	−38.9502			−1.48174			−0.740869	+	0.671650i	$0.765586\pi$
$692$		0			0
$693$		−	4.46486i		−	0.169606i
$694$		0			0
$695$	−3.86189			−0.146490
$696$		0			0
$697$	−20.2993	+	37.3515i	−0.768893	+	1.41479i
$698$		0			0
$699$			1.96147i			0.0741895i
$700$		0			0
$701$			39.7185i			1.50015i	0.661354	+	0.750074i	$0.269982\pi$
$701$			39.7185i			1.50015i	−0.661354	+	0.750074i	$0.730018\pi$
$702$		0			0
$703$			20.8045i			0.784658i
$704$		0			0
$705$	3.89647			0.146750
$706$		0			0
$707$	−7.51590			−0.282665
$708$		0			0
$709$	0.684050			0.0256901			0.0128450	−	0.999917i	$-0.495911\pi$
$709$	0.684050			0.0256901			0.0128450	+	0.999917i	$0.495911\pi$
$710$		0			0
$711$		−	35.7848i		−	1.34204i
$712$		0			0
$713$	5.76949			0.216069
$714$		0			0
$715$		−	7.85089i		−	0.293606i
$716$		0			0
$717$	5.07739			0.189619
$718$		0			0
$719$		−	38.6497i		−	1.44139i	−0.693252	−	0.720695i	$-0.743823\pi$
$719$		−	38.6497i		−	1.44139i	0.693252	−	0.720695i	$-0.256177\pi$
$720$		0			0
$721$			7.06148i			0.262983i
$722$		0			0
$723$		−	15.1962i		−	0.565153i
$724$		0			0
$725$	−8.44930			−0.313799
$726$		0			0
$727$	25.3357			0.939651			0.469825	−	0.882759i	$-0.344317\pi$
$727$	25.3357			0.939651			0.469825	+	0.882759i	$0.344317\pi$
$728$		0			0
$729$	−4.02083			−0.148920
$730$		0			0
$731$	16.1097	−	29.6424i	0.595838	−	1.09636i
$732$		0			0
$733$		−	53.4938i		−	1.97584i	−0.154968	−	0.987920i	$-0.549527\pi$
$733$		−	53.4938i		−	1.97584i	0.154968	−	0.987920i	$-0.450473\pi$
$734$		0			0
$735$	4.74417			0.174991
$736$		0			0
$737$			52.8540i			1.94690i
$738$		0			0
$739$			10.7854i			0.396746i	0.980127	+	0.198373i	$0.0635658\pi$
$739$			10.7854i			0.396746i	−0.980127	+	0.198373i	$0.936434\pi$
$740$		0			0
$741$	−6.18890			−0.227355
$742$		0			0
$743$		−	10.8676i		−	0.398694i	−0.979929	−	0.199347i	$-0.936118\pi$
$743$		−	10.8676i		−	0.398694i	0.979929	−	0.199347i	$-0.0638821\pi$
$744$		0			0
$745$		−	15.2325i		−	0.558075i
$746$		0			0
$747$		−	4.07162i		−	0.148973i
$748$		0			0
$749$			7.40912i			0.270724i
$750$		0			0
$751$			43.1284i			1.57378i	0.617095	+	0.786889i	$0.288309\pi$
$751$			43.1284i			1.57378i	−0.617095	+	0.786889i	$0.711691\pi$
$752$		0			0
$753$		−	5.51446i		−	0.200958i
$754$		0			0
$755$	−1.45467			−0.0529410
$756$		0			0
$757$			22.5791i			0.820652i	0.911939	+	0.410326i	$0.134585\pi$
$757$			22.5791i			0.820652i	−0.911939	+	0.410326i	$0.865415\pi$
$758$		0			0
$759$			8.02083i			0.291138i
$760$		0			0
$761$	−10.4373			−0.378353			−0.189177	−	0.981943i	$-0.560582\pi$
$761$	−10.4373			−0.378353			−0.189177	+	0.981943i	$0.560582\pi$
$762$		0			0
$763$			7.58830i			0.274715i
$764$		0			0
$765$	9.09265	+	4.94157i	0.328745	+	0.178663i
$766$		0			0
$767$	10.9565			0.395618
$768$		0			0
$769$	−30.5474			−1.10157			−0.550784	−	0.834648i	$-0.685671\pi$
$769$	−30.5474			−1.10157			−0.550784	+	0.834648i	$0.685671\pi$
$770$		0			0
$771$	−4.80908			−0.173195
$772$		0			0
$773$		−	40.8348i		−	1.46873i	−0.678756	−	0.734364i	$-0.737481\pi$
$773$		−	40.8348i		−	1.46873i	0.678756	−	0.734364i	$-0.262519\pi$
$774$		0			0
$775$			1.89633i			0.0681183i
$776$		0			0
$777$			1.62234i			0.0582011i
$778$		0			0
$779$	43.7227			1.56653
$780$		0			0
$781$			54.4699i			1.94909i
$782$		0			0
$783$	−32.5910			−1.16471
$784$		0			0
$785$		−	2.38317i		−	0.0850590i
$786$		0			0
$787$	−40.1369			−1.43073			−0.715363	−	0.698753i	$-0.753739\pi$
$787$	−40.1369			−1.43073			−0.715363	+	0.698753i	$0.753739\pi$
$788$		0			0
$789$	−18.3795			−0.654327
$790$		0			0
$791$	3.68547			0.131040
$792$		0			0
$793$		−	8.52565i		−	0.302755i
$794$		0			0
$795$		−	4.28357i		−	0.151923i
$796$		0			0
$797$			36.6352i			1.29768i	0.760923	+	0.648842i	$0.224746\pi$
$797$			36.6352i			1.29768i	−0.760923	+	0.648842i	$0.775254\pi$
$798$		0			0
$799$	20.1637	+	10.9583i	0.713340	+	0.387678i
$800$		0			0
$801$	−10.3729			−0.366507
$802$		0			0
$803$			44.3646i			1.56559i
$804$		0			0
$805$	−1.43716			−0.0506531
$806$		0			0
$807$	−13.8162			−0.486354
$808$		0			0
$809$		−	31.5268i		−	1.10842i	−0.832376	−	0.554212i	$-0.813020\pi$
$809$		−	31.5268i		−	1.10842i	0.832376	−	0.554212i	$-0.186980\pi$
$810$		0			0
$811$	−2.16993			−0.0761965			−0.0380982	−	0.999274i	$-0.512130\pi$
$811$	−2.16993			−0.0761965			−0.0380982	+	0.999274i	$0.512130\pi$
$812$		0			0
$813$	−6.87901			−0.241258
$814$		0			0
$815$	10.0630			0.352492
$816$		0			0
$817$	−34.6986			−1.21395
$818$		0			0
$819$	2.47170			0.0863681
$820$		0			0
$821$	43.9566			1.53410			0.767048	−	0.641590i	$-0.221725\pi$
$821$	43.9566			1.53410			0.767048	+	0.641590i	$0.221725\pi$
$822$		0			0
$823$			4.32095i			0.150619i	0.997160	+	0.0753094i	$0.0239945\pi$
$823$			4.32095i			0.150619i	−0.997160	+	0.0753094i	$0.976006\pi$
$824$		0			0
$825$	−2.63631			−0.0917846
$826$		0			0
$827$	13.4583			0.467990			0.233995	−	0.972238i	$-0.424820\pi$
$827$	13.4583			0.467990			0.233995	+	0.972238i	$0.424820\pi$
$828$		0			0
$829$		−	36.3519i		−	1.26255i	−0.775558	−	0.631277i	$-0.782531\pi$
$829$		−	36.3519i		−	1.26255i	0.775558	−	0.631277i	$-0.217469\pi$
$830$		0			0
$831$	−4.99016			−0.173107
$832$		0			0
$833$	24.5504	+	13.3424i	0.850621	+	0.462286i
$834$		0			0
$835$		−	4.09470i		−	0.141703i
$836$		0			0
$837$			7.31462i			0.252830i
$838$		0			0
$839$			10.1963i			0.352016i	0.984389	+	0.176008i	$0.0563185\pi$
$839$			10.1963i			0.352016i	−0.984389	+	0.176008i	$0.943682\pi$
$840$		0			0
$841$	42.3907			1.46175
$842$		0			0
$843$	13.6943			0.471658
$844$		0			0
$845$	−8.65383			−0.297701
$846$		0			0
$847$			1.50297i			0.0516428i
$848$		0			0
$849$	5.98698			0.205473
$850$		0			0
$851$			14.9263i			0.511666i
$852$		0			0
$853$	−19.0192			−0.651205			−0.325602	−	0.945507i	$-0.605567\pi$
$853$	−19.0192			−0.651205			−0.325602	+	0.945507i	$0.605567\pi$
$854$		0			0
$855$		−	10.6436i		−	0.364004i
$856$		0			0
$857$		−	15.6163i		−	0.533443i	−0.963774	−	0.266721i	$-0.914060\pi$
$857$		−	15.6163i		−	0.533443i	0.963774	−	0.266721i	$-0.0859404\pi$
$858$		0			0
$859$			8.33388i			0.284348i	0.989842	+	0.142174i	$0.0454093\pi$
$859$			8.33388i			0.284348i	−0.989842	+	0.142174i	$0.954591\pi$
$860$		0			0
$861$	3.40950			0.116196
$862$		0			0
$863$	−5.60477			−0.190789			−0.0953943	−	0.995440i	$-0.530411\pi$
$863$	−5.60477			−0.190789			−0.0953943	+	0.995440i	$0.530411\pi$
$864$		0			0
$865$	16.3781			0.556871
$866$		0			0
$867$	−6.47380	−	9.98608i	−0.219862	−	0.339145i
$868$		0			0
$869$			53.6913i			1.82135i
$870$		0			0
$871$	−29.2595			−0.991419
$872$		0			0
$873$			3.79485i			0.128436i
$874$		0			0
$875$		−	0.472369i		−	0.0159690i
$876$		0			0
$877$	−27.8925			−0.941864			−0.470932	−	0.882169i	$-0.656082\pi$
$877$	−27.8925			−0.941864			−0.470932	+	0.882169i	$0.656082\pi$
$878$		0			0
$879$			16.0229i			0.540440i
$880$		0			0
$881$		−	31.1567i		−	1.04969i	−0.851196	−	0.524847i	$-0.824122\pi$
$881$		−	31.1567i		−	1.04969i	0.851196	−	0.524847i	$-0.175878\pi$
$882$		0			0
$883$		−	26.5634i		−	0.893931i	−0.894551	−	0.446965i	$-0.852505\pi$
$883$		−	26.5634i		−	0.893931i	0.894551	−	0.446965i	$-0.147495\pi$
$884$		0			0
$885$		−	3.67918i		−	0.123674i
$886$		0			0
$887$		−	25.6641i		−	0.861717i	−0.902419	−	0.430859i	$-0.858211\pi$
$887$		−	25.6641i		−	0.861717i	0.902419	−	0.430859i	$-0.141789\pi$
$888$		0			0
$889$		−	5.13939i		−	0.172370i
$890$		0			0
$891$	18.1873			0.609296
$892$		0			0
$893$		−	23.6031i		−	0.789848i
$894$		0			0
$895$		−	8.16713i		−	0.272997i
$896$		0			0
$897$	−4.44025			−0.148256
$898$		0			0
$899$		−	16.0227i		−	0.534387i
$900$		0			0
$901$	12.0470	−	22.1668i	0.401343	−	0.738485i
$902$		0			0
$903$	−2.70580			−0.0900435
$904$		0			0
$905$	−12.6490			−0.420466
$906$		0			0
$907$	20.5456			0.682206			0.341103	−	0.940026i	$-0.389200\pi$
$907$	20.5456			0.682206			0.341103	+	0.940026i	$0.389200\pi$
$908$		0			0
$909$		−	39.9356i		−	1.32458i
$910$		0			0
$911$		−	34.5273i		−	1.14394i	−0.820274	−	0.571970i	$-0.806179\pi$
$911$		−	34.5273i		−	1.14394i	0.820274	−	0.571970i	$-0.193821\pi$
$912$		0			0
$913$			6.10902i			0.202179i
$914$		0			0
$915$	−2.86290			−0.0946444
$916$		0			0
$917$			6.59253i			0.217705i
$918$		0			0
$919$	−18.5422			−0.611650			−0.305825	−	0.952088i	$-0.598932\pi$
$919$	−18.5422			−0.611650			−0.305825	+	0.952088i	$0.598932\pi$
$920$		0			0
$921$		−	3.32638i		−	0.109608i
$922$		0			0
$923$	−30.1540			−0.992531
$924$		0			0
$925$	−4.90602			−0.161309
$926$		0			0
$927$	−37.5210			−1.23235
$928$		0			0
$929$			18.2002i			0.597130i	0.954389	+	0.298565i	$0.0965080\pi$
$929$			18.2002i			0.597130i	−0.954389	+	0.298565i	$0.903492\pi$
$930$		0			0
$931$		−	28.7381i		−	0.941852i
$932$		0			0
$933$			12.9907i			0.425295i
$934$		0			0
$935$	−13.6425	−	7.41428i	−0.446158	−	0.242473i
$936$		0			0
$937$	−35.1844			−1.14942			−0.574711	−	0.818356i	$-0.694886\pi$
$937$	−35.1844			−1.14942			−0.574711	+	0.818356i	$0.694886\pi$
$938$		0			0
$939$		−	19.4425i		−	0.634481i
$940$		0			0
$941$	6.97595			0.227409			0.113705	−	0.993515i	$-0.463728\pi$
$941$	6.97595			0.227409			0.113705	+	0.993515i	$0.463728\pi$
$942$		0			0
$943$	31.3690			1.02152
$944$		0			0
$945$		−	1.82204i		−	0.0592710i
$946$		0			0
$947$	51.4262			1.67113			0.835564	−	0.549394i	$-0.185141\pi$
$947$	51.4262			1.67113			0.835564	+	0.549394i	$0.185141\pi$
$948$		0			0
$949$	−24.5598			−0.797244
$950$		0			0
$951$	−4.83444			−0.156768
$952$		0			0
$953$	2.19024			0.0709487			0.0354743	−	0.999371i	$-0.488706\pi$
$953$	2.19024			0.0709487			0.0354743	+	0.999371i	$0.488706\pi$
$954$		0			0
$955$	−9.84032			−0.318425
$956$		0			0
$957$	22.2750			0.720048
$958$		0			0
$959$		−	7.67499i		−	0.247838i
$960$		0			0
$961$	27.4039			0.883997
$962$		0			0
$963$	−39.3683			−1.26862
$964$		0			0
$965$		−	19.3479i		−	0.622832i
$966$		0			0
$967$	23.0610			0.741592			0.370796	−	0.928714i	$-0.379085\pi$
$967$	23.0610			0.741592			0.370796	+	0.928714i	$0.379085\pi$
$968$		0			0
$969$	−5.84472	+	10.7545i	−0.187760	+	0.345484i
$970$		0			0
$971$			24.4595i			0.784944i	0.919764	+	0.392472i	$0.128380\pi$
$971$			24.4595i			0.784944i	−0.919764	+	0.392472i	$0.871620\pi$
$972$		0			0
$973$			1.82423i			0.0584823i
$974$		0			0
$975$		−	1.45943i		−	0.0467393i
$976$		0			0
$977$	59.5006			1.90359			0.951797	−	0.306730i	$-0.0992347\pi$
$977$	59.5006			1.90359			0.951797	+	0.306730i	$0.0992347\pi$
$978$		0			0
$979$	15.5634			0.497408
$980$		0			0
$981$	−40.3203			−1.28733
$982$		0			0
$983$			45.4634i			1.45006i	0.688718	+	0.725029i	$0.258174\pi$
$983$			45.4634i			1.45006i	−0.688718	+	0.725029i	$0.741826\pi$
$984$		0			0
$985$	2.53115			0.0806491
$986$		0			0
$987$		−	1.84057i		−	0.0585861i
$988$		0			0
$989$	−24.8946			−0.791603
$990$		0			0
$991$		−	3.88975i		−	0.123562i	−0.998090	−	0.0617809i	$-0.980322\pi$
$991$		−	3.88975i		−	0.123562i	0.998090	−	0.0617809i	$-0.0196780\pi$
$992$		0			0
$993$		−	12.4101i		−	0.393823i
$994$		0			0
$995$			7.75580i			0.245875i
$996$		0			0
$997$	32.4838			1.02877			0.514385	−	0.857559i	$-0.328020\pi$
$997$	32.4838			1.02877			0.514385	+	0.857559i	$0.328020\pi$
$998$		0			0
$999$	−18.9237			−0.598720

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2720.2.l.a.2481.15		36
4.3	odd	2		680.2.l.b.101.17	yes	36
8.3	odd	2		680.2.l.a.101.18	yes	36
8.5	even	2		2720.2.l.b.2481.21		36
17.16	even	2		2720.2.l.b.2481.22		36
68.67	odd	2		680.2.l.a.101.17	✓	36
136.67	odd	2		680.2.l.b.101.18	yes	36
136.101	even	2	inner	2720.2.l.a.2481.16		36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
680.2.l.a.101.17	✓	36	68.67	odd	2
680.2.l.a.101.18	yes	36	8.3	odd	2
680.2.l.b.101.17	yes	36	4.3	odd	2
680.2.l.b.101.18	yes	36	136.67	odd	2
2720.2.l.a.2481.15		36	1.1	even	1	trivial
2720.2.l.a.2481.16		36	136.101	even	2	inner
2720.2.l.b.2481.21		36	8.5	even	2
2720.2.l.b.2481.22		36	17.16	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2720 = 2^{5} \cdot 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2720.l (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-0.700054 q^{3} -1.00000 q^{5} +0.472369i q^{7} -2.50992 q^{9} +3.76587 q^{11} +2.08475i q^{13} +0.700054 q^{15} +(3.62268 + 1.96881i) q^{17} -4.24061i q^{19} -0.330684i q^{21} -3.04244i q^{23} +1.00000 q^{25} +3.85724 q^{27} -8.44930 q^{29} +1.89633i q^{31} -2.63631 q^{33} -0.472369i q^{35} -4.90602 q^{37} -1.45943i q^{39} +10.3105i q^{41} -8.18245i q^{43} +2.50992 q^{45} +5.56596 q^{47} +6.77687 q^{49} +(-2.53607 - 1.37827i) q^{51} -6.11891i q^{53} -3.76587 q^{55} +2.96866i q^{57} -5.25557i q^{59} -4.08954 q^{61} -1.18561i q^{63} -2.08475i q^{65} +14.0350i q^{67} +2.12987i q^{69} +14.4641i q^{71} +11.7807i q^{73} -0.700054 q^{75} +1.77888i q^{77} +14.2573i q^{79} +4.82950 q^{81} +1.62221i q^{83} +(-3.62268 - 1.96881i) q^{85} +5.91497 q^{87} +4.13274 q^{89} -0.984770 q^{91} -1.32754i q^{93} +4.24061i q^{95} -1.51194i q^{97} -9.45205 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q - 4 q^{3} - 36 q^{5} + 36 q^{9} - 8 q^{11} + 4 q^{15} + 36 q^{25} - 16 q^{27} - 8 q^{33} - 36 q^{45} - 20 q^{47} - 36 q^{49} + 8 q^{55} - 4 q^{75} + 44 q^{81} - 24 q^{87} - 56 q^{91} - 40 q^{99}+O(q^{100}) \) 36 * q - 4 * q^3 - 36 * q^5 + 36 * q^9 - 8 * q^11 + 4 * q^15 + 36 * q^25 - 16 * q^27 - 8 * q^33 - 36 * q^45 - 20 * q^47 - 36 * q^49 + 8 * q^55 - 4 * q^75 + 44 * q^81 - 24 * q^87 - 56 * q^91 - 40 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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