LMFDB - Embedded newform 672.2.p.a.559.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [672,2,Mod(559,672)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(672, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("672.559");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 672 = 2^{5} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	672.p (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:	$5.36594701583$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + x^{14} - 4x^{12} - 4x^{10} + 16x^{8} - 16x^{6} - 64x^{4} + 64x^{2} + 256 $ x^16 + x^14 - 4x^12 - 4x^10 + 16x^8 - 16x^6 - 64x^4 + 64x^2 + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{17} $
Twist minimal:	no (minimal twist has level 168)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			559.6
Root			$-0.474920 - 1.33209i$ of defining polynomial
Character	$\chi$	$=$	672.559
Dual form			672.2.p.a.559.14

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} +1.58069 q^{5} +(2.37995 - 1.15578i) q^{7} -1.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} +1.58069 q^{5} +(2.37995 - 1.15578i) q^{7} -1.00000 q^{9} +2.26057 q^{11} +0.548664 q^{13} -1.58069i q^{15} +0.433635i q^{17} +6.02255i q^{19} +(-1.15578 - 2.37995i) q^{21} -8.24028i q^{23} -2.50141 q^{25} +1.00000i q^{27} -0.548664i q^{29} +7.50941 q^{31} -2.26057i q^{33} +(3.76198 - 1.82694i) q^{35} -4.21124i q^{37} -0.548664i q^{39} +7.09032i q^{41} +1.82694 q^{43} -1.58069 q^{45} -11.5839 q^{47} +(4.32834 - 5.50141i) q^{49} +0.433635 q^{51} -3.71005i q^{53} +3.57327 q^{55} +6.02255 q^{57} -11.5240i q^{59} +12.1325 q^{61} +(-2.37995 + 1.15578i) q^{63} +0.867270 q^{65} -9.35089 q^{67} -8.24028 q^{69} -1.27953i q^{71} +0.867270i q^{73} +2.50141i q^{75} +(5.38005 - 2.61272i) q^{77} +10.3696i q^{79} +1.00000 q^{81} +6.13554i q^{83} +0.685444i q^{85} -0.548664 q^{87} -7.95759i q^{89} +(1.30579 - 0.634135i) q^{91} -7.50941i q^{93} +9.51981i q^{95} +19.1778i q^{97} -2.26057 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 16 q^{9}+O(q^{10}) $ 16 * q - 16 * q^9	$ 16 q - 16 q^{9} + 8 q^{11} + 16 q^{25} - 24 q^{35} + 8 q^{43} - 8 q^{49} - 16 q^{57} + 40 q^{67} + 16 q^{81} + 56 q^{91} - 8 q^{99}+O(q^{100}) $ 16 * q - 16 * q^9 + 8 * q^11 + 16 * q^25 - 24 * q^35 + 8 * q^43 - 8 * q^49 - 16 * q^57 + 40 * q^67 + 16 * q^81 + 56 * q^91 - 8 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/672\mathbb{Z}\right)^\times$.

$n$	$127$	$421$	$449$	$577$
$\chi(n)$	$-1$	$-1$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		−	1.00000i		−	0.577350i
$3$		−	1.00000i		−	0.577350i
$4$		0			0
$5$	1.58069			0.706908			0.353454	−	0.935452i	$-0.385007\pi$
$5$	1.58069			0.706908			0.353454	+	0.935452i	$0.385007\pi$
$6$		0			0
$7$	2.37995	−	1.15578i	0.899537	−	0.436844i
$7$	2.37995	−	1.15578i	0.899537	−	0.436844i
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	2.26057			0.681588			0.340794	−	0.940138i	$-0.389304\pi$
$11$	2.26057			0.681588			0.340794	+	0.940138i	$0.389304\pi$
$12$		0			0
$13$	0.548664			0.152172			0.0760860	−	0.997101i	$-0.475758\pi$
$13$	0.548664			0.152172			0.0760860	+	0.997101i	$0.475758\pi$
$14$		0			0
$15$		−	1.58069i		−	0.408134i
$16$		0			0
$17$			0.433635i			0.105172i	0.998616	+	0.0525860i	$0.0167464\pi$
$17$			0.433635i			0.105172i	−0.998616	+	0.0525860i	$0.983254\pi$
$18$		0			0
$19$			6.02255i			1.38167i	0.723014	+	0.690834i	$0.242756\pi$
$19$			6.02255i			1.38167i	−0.723014	+	0.690834i	$0.757244\pi$
$20$		0			0
$21$	−1.15578	−	2.37995i	−0.252212	−	0.519348i
$22$		0			0
$23$		−	8.24028i		−	1.71822i	−0.511794	−	0.859108i	$-0.671019\pi$
$23$		−	8.24028i		−	1.71822i	0.511794	−	0.859108i	$-0.328981\pi$
$24$		0			0
$25$	−2.50141			−0.500281
$26$		0			0
$27$			1.00000i			0.192450i
$28$		0			0
$29$		−	0.548664i		−	0.101884i	−0.998702	−	0.0509422i	$-0.983778\pi$
$29$		−	0.548664i		−	0.101884i	0.998702	−	0.0509422i	$-0.0162224\pi$
$30$		0			0
$31$	7.50941			1.34873			0.674365	−	0.738398i	$-0.264418\pi$
$31$	7.50941			1.34873			0.674365	+	0.738398i	$0.264418\pi$
$32$		0			0
$33$		−	2.26057i		−	0.393515i
$34$		0			0
$35$	3.76198	−	1.82694i	0.635890	−	0.308809i
$36$		0			0
$37$		−	4.21124i		−	0.692324i	−0.938175	−	0.346162i	$-0.887485\pi$
$37$		−	4.21124i		−	0.692324i	0.938175	−	0.346162i	$-0.112515\pi$
$38$		0			0
$39$		−	0.548664i		−	0.0878565i
$40$		0			0
$41$			7.09032i			1.10732i	0.832742	+	0.553661i	$0.186770\pi$
$41$			7.09032i			1.10732i	−0.832742	+	0.553661i	$0.813230\pi$
$42$		0			0
$43$	1.82694			0.278605			0.139303	−	0.990250i	$-0.455514\pi$
$43$	1.82694			0.278605			0.139303	+	0.990250i	$0.455514\pi$
$44$		0			0
$45$	−1.58069			−0.235636
$46$		0			0
$47$	−11.5839			−1.68968			−0.844840	−	0.535018i	$-0.820305\pi$
$47$	−11.5839			−1.68968			−0.844840	+	0.535018i	$0.820305\pi$
$48$		0			0
$49$	4.32834	−	5.50141i	0.618334	−	0.785915i
$50$		0			0
$51$	0.433635			0.0607210
$52$		0			0
$53$		−	3.71005i		−	0.509615i	−0.966992	−	0.254807i	$-0.917988\pi$
$53$		−	3.71005i		−	0.509615i	0.966992	−	0.254807i	$-0.0820121\pi$
$54$		0			0
$55$	3.57327			0.481820
$56$		0			0
$57$	6.02255			0.797706
$58$		0			0
$59$		−	11.5240i		−	1.50029i	−0.661273	−	0.750145i	$-0.729983\pi$
$59$		−	11.5240i		−	1.50029i	0.661273	−	0.750145i	$-0.270017\pi$
$60$		0			0
$61$	12.1325			1.55341			0.776706	−	0.629864i	$-0.216889\pi$
$61$	12.1325			1.55341			0.776706	+	0.629864i	$0.216889\pi$
$62$		0			0
$63$	−2.37995	+	1.15578i	−0.299846	+	0.145615i
$64$		0			0
$65$	0.867270			0.107572
$66$		0			0
$67$	−9.35089			−1.14239			−0.571196	−	0.820813i	$-0.693521\pi$
$67$	−9.35089			−1.14239			−0.571196	+	0.820813i	$0.693521\pi$
$68$		0			0
$69$	−8.24028			−0.992013
$70$		0			0
$71$		−	1.27953i		−	0.151852i	−0.997113	−	0.0759262i	$-0.975809\pi$
$71$		−	1.27953i		−	0.151852i	0.997113	−	0.0759262i	$-0.0241913\pi$
$72$		0			0
$73$			0.867270i			0.101506i	0.998711	+	0.0507531i	$0.0161622\pi$
$73$			0.867270i			0.101506i	−0.998711	+	0.0507531i	$0.983838\pi$
$74$		0			0
$75$			2.50141i			0.288837i
$76$		0			0
$77$	5.38005	−	2.61272i	0.613114	−	0.297748i
$78$		0			0
$79$			10.3696i			1.16668i	0.812230	+	0.583338i	$0.198253\pi$
$79$			10.3696i			1.16668i	−0.812230	+	0.583338i	$0.801747\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$			6.13554i			0.673463i	0.941601	+	0.336732i	$0.109321\pi$
$83$			6.13554i			0.673463i	−0.941601	+	0.336732i	$0.890679\pi$
$84$		0			0
$85$			0.685444i			0.0743469i
$86$		0			0
$87$	−0.548664			−0.0588229
$88$		0			0
$89$		−	7.95759i		−	0.843503i	−0.906712	−	0.421751i	$-0.861416\pi$
$89$		−	7.95759i		−	0.843503i	0.906712	−	0.421751i	$-0.138584\pi$
$90$		0			0
$91$	1.30579	−	0.634135i	0.136884	−	0.0664754i
$92$		0			0
$93$		−	7.50941i		−	0.778689i
$94$		0			0
$95$			9.51981i			0.976712i
$96$		0			0
$97$			19.1778i			1.94721i	0.228233	+	0.973607i	$0.426705\pi$
$97$			19.1778i			1.94721i	−0.228233	+	0.973607i	$0.573295\pi$
$98$		0			0
$99$	−2.26057			−0.227196
$100$		0			0
$101$	−4.74208			−0.471855			−0.235927	−	0.971771i	$-0.575813\pi$
$101$	−4.74208			−0.471855			−0.235927	+	0.971771i	$0.575813\pi$
$102$		0			0
$103$	2.01040			0.198090			0.0990452	−	0.995083i	$-0.468421\pi$
$103$	2.01040			0.198090			0.0990452	+	0.995083i	$0.468421\pi$
$104$		0			0
$105$	−1.82694	−	3.76198i	−0.178291	−	0.367131i
$106$		0			0
$107$	−17.7845			−1.71929			−0.859647	−	0.510888i	$-0.829317\pi$
$107$	−17.7845			−1.71929			−0.859647	+	0.510888i	$0.829317\pi$
$108$		0			0
$109$			12.0432i			1.15353i	0.816909	+	0.576766i	$0.195686\pi$
$109$			12.0432i			1.15353i	−0.816909	+	0.576766i	$0.804314\pi$
$110$		0			0
$111$	−4.21124			−0.399713
$112$		0			0
$113$	−2.86727			−0.269730			−0.134865	−	0.990864i	$-0.543060\pi$
$113$	−2.86727			−0.269730			−0.134865	+	0.990864i	$0.543060\pi$
$114$		0			0
$115$		−	13.0254i		−	1.21462i
$116$		0			0
$117$	−0.548664			−0.0507240
$118$		0			0
$119$	0.501187	+	1.03203i	0.0459437	+	0.0946061i
$120$		0			0
$121$	−5.88982			−0.535438
$122$		0			0
$123$	7.09032			0.639312
$124$		0			0
$125$	−11.8574			−1.06056
$126$		0			0
$127$			14.9928i			1.33039i	0.746669	+	0.665196i	$0.231652\pi$
$127$			14.9928i			1.33039i	−0.746669	+	0.665196i	$0.768348\pi$
$128$		0			0
$129$		−	1.82694i		−	0.160853i
$130$		0			0
$131$			8.52114i			0.744496i	0.928133	+	0.372248i	$0.121413\pi$
$131$			8.52114i			0.744496i	−0.928133	+	0.372248i	$0.878587\pi$
$132$		0			0
$133$	6.96075	+	14.3334i	0.603573	+	1.24286i
$134$		0			0
$135$			1.58069i			0.136045i
$136$		0			0
$137$	5.00281			0.427419			0.213709	−	0.976897i	$-0.431445\pi$
$137$	5.00281			0.427419			0.213709	+	0.976897i	$0.431445\pi$
$138$		0			0
$139$			14.3106i			1.21381i	0.794776	+	0.606903i	$0.207588\pi$
$139$			14.3106i			1.21381i	−0.794776	+	0.606903i	$0.792412\pi$
$140$		0			0
$141$			11.5839i			0.975538i
$142$		0			0
$143$	1.24029			0.103719
$144$		0			0
$145$		−	0.867270i		−	0.0720229i
$146$		0			0
$147$	−5.50141	−	4.32834i	−0.453748	−	0.356996i
$148$		0			0
$149$			18.4553i			1.51192i	0.654619	+	0.755959i	$0.272829\pi$
$149$			18.4553i			1.51192i	−0.654619	+	0.755959i	$0.727171\pi$
$150$		0			0
$151$		−	14.5334i		−	1.18271i	−0.806411	−	0.591356i	$-0.798593\pi$
$151$		−	14.5334i		−	1.18271i	0.806411	−	0.591356i	$-0.201407\pi$
$152$		0			0
$153$		−	0.433635i		−	0.0350573i
$154$		0			0
$155$	11.8701			0.953428
$156$		0			0
$157$	5.80975			0.463669			0.231834	−	0.972755i	$-0.425527\pi$
$157$	5.80975			0.463669			0.231834	+	0.972755i	$0.425527\pi$
$158$		0			0
$159$	−3.71005			−0.294226
$160$		0			0
$161$	−9.52395	−	19.6115i	−0.750593	−	1.54560i
$162$		0			0
$163$	17.3509			1.35903			0.679513	−	0.733663i	$-0.262191\pi$
$163$	17.3509			1.35903			0.679513	+	0.733663i	$0.262191\pi$
$164$		0			0
$165$		−	3.57327i		−	0.278179i
$166$		0			0
$167$	0.823767			0.0637450			0.0318725	−	0.999492i	$-0.489853\pi$
$167$	0.823767			0.0637450			0.0318725	+	0.999492i	$0.489853\pi$
$168$		0			0
$169$	−12.6990			−0.976844
$170$		0			0
$171$		−	6.02255i		−	0.460556i
$172$		0			0
$173$	−19.5230			−1.48430			−0.742152	−	0.670231i	$-0.766195\pi$
$173$	−19.5230			−1.48430			−0.742152	+	0.670231i	$0.766195\pi$
$174$		0			0
$175$	−5.95322	+	2.89108i	−0.450021	+	0.218545i
$176$		0			0
$177$	−11.5240			−0.866193
$178$		0			0
$179$	4.39611			0.328581			0.164290	−	0.986412i	$-0.447467\pi$
$179$	4.39611			0.328581			0.164290	+	0.986412i	$0.447467\pi$
$180$		0			0
$181$	−8.14738			−0.605590			−0.302795	−	0.953056i	$-0.597920\pi$
$181$	−8.14738			−0.605590			−0.302795	+	0.953056i	$0.597920\pi$
$182$		0			0
$183$		−	12.1325i		−	0.896863i
$184$		0			0
$185$		−	6.65668i		−	0.489409i
$186$		0			0
$187$			0.980263i			0.0716839i
$188$		0			0
$189$	1.15578	+	2.37995i	0.0840707	+	0.173116i
$190$		0			0
$191$		−	0.420123i		−	0.0303990i	−0.999884	−	0.0151995i	$-0.995162\pi$
$191$		−	0.420123i		−	0.0303990i	0.999884	−	0.0151995i	$-0.00483834\pi$
$192$		0			0
$193$	−11.1553			−0.802974			−0.401487	−	0.915865i	$-0.631507\pi$
$193$	−11.1553			−0.802974			−0.401487	+	0.915865i	$0.631507\pi$
$194$		0			0
$195$		−	0.867270i		−	0.0621065i
$196$		0			0
$197$		−	4.95035i		−	0.352698i	−0.984328	−	0.176349i	$-0.943571\pi$
$197$		−	4.95035i		−	0.352698i	0.984328	−	0.176349i	$-0.0564287\pi$
$198$		0			0
$199$	12.0851			0.856687			0.428343	−	0.903616i	$-0.359097\pi$
$199$	12.0851			0.856687			0.428343	+	0.903616i	$0.359097\pi$
$200$		0			0
$201$			9.35089i			0.659561i
$202$		0			0
$203$	−0.634135	−	1.30579i	−0.0445076	−	0.0916488i
$204$		0			0
$205$			11.2076i			0.782775i
$206$		0			0
$207$			8.24028i			0.572739i
$208$		0			0
$209$			13.6144i			0.941728i
$210$		0			0
$211$	−6.17306			−0.424971			−0.212486	−	0.977164i	$-0.568156\pi$
$211$	−6.17306			−0.424971			−0.212486	+	0.977164i	$0.568156\pi$
$212$		0			0
$213$	−1.27953			−0.0876720
$214$		0			0
$215$	2.88783			0.196948
$216$		0			0
$217$	17.8720	−	8.67923i	1.21323	−	0.589185i
$218$		0			0
$219$	0.867270			0.0586047
$220$		0			0
$221$			0.237920i			0.0160042i
$222$		0			0
$223$	−18.4078			−1.23268			−0.616340	−	0.787480i	$-0.711385\pi$
$223$	−18.4078			−1.23268			−0.616340	+	0.787480i	$0.711385\pi$
$224$		0			0
$225$	2.50141			0.166760
$226$		0			0
$227$		−	10.6567i		−	0.707309i	−0.935376	−	0.353654i	$-0.884939\pi$
$227$		−	10.6567i		−	0.707309i	0.935376	−	0.353654i	$-0.115061\pi$
$228$		0			0
$229$	−8.97114			−0.592830			−0.296415	−	0.955059i	$-0.595791\pi$
$229$	−8.97114			−0.592830			−0.296415	+	0.955059i	$0.595791\pi$
$230$		0			0
$231$	−2.61272	−	5.38005i	−0.171905	−	0.353981i
$232$		0			0
$233$	14.9124			0.976942			0.488471	−	0.872580i	$-0.337555\pi$
$233$	14.9124			0.976942			0.488471	+	0.872580i	$0.337555\pi$
$234$		0			0
$235$	−18.3106			−1.19445
$236$		0			0
$237$	10.3696			0.673580
$238$		0			0
$239$			13.1370i			0.849759i	0.905250	+	0.424880i	$0.139684\pi$
$239$			13.1370i			0.849759i	−0.905250	+	0.424880i	$0.860316\pi$
$240$		0			0
$241$		−	10.1355i		−	0.652888i	−0.945217	−	0.326444i	$-0.894150\pi$
$241$		−	10.1355i		−	0.652888i	0.945217	−	0.326444i	$-0.105850\pi$
$242$		0			0
$243$		−	1.00000i		−	0.0641500i
$244$		0			0
$245$	6.84179	−	8.69604i	0.437106	−	0.555570i
$246$		0			0
$247$			3.30435i			0.210251i
$248$		0			0
$249$	6.13554			0.388824
$250$		0			0
$251$			22.7018i			1.43292i	0.697626	+	0.716462i	$0.254240\pi$
$251$			22.7018i			1.43292i	−0.697626	+	0.716462i	$0.745760\pi$
$252$		0			0
$253$		−	18.6277i		−	1.17112i
$254$		0			0
$255$	0.685444			0.0429242
$256$		0			0
$257$			16.1326i			1.00632i	0.864192	+	0.503162i	$0.167830\pi$
$257$			16.1326i			1.00632i	−0.864192	+	0.503162i	$0.832170\pi$
$258$		0			0
$259$	−4.86727	−	10.0225i	−0.302437	−	0.622771i
$260$		0			0
$261$			0.548664i			0.0339614i
$262$		0			0
$263$		−	15.0581i		−	0.928519i	−0.885699	−	0.464260i	$-0.846320\pi$
$263$		−	15.0581i		−	0.928519i	0.885699	−	0.464260i	$-0.153680\pi$
$264$		0			0
$265$		−	5.86446i		−	0.360251i
$266$		0			0
$267$	−7.95759			−0.486996
$268$		0			0
$269$	−8.76288			−0.534282			−0.267141	−	0.963657i	$-0.586079\pi$
$269$	−8.76288			−0.534282			−0.267141	+	0.963657i	$0.586079\pi$
$270$		0			0
$271$	4.31238			0.261958			0.130979	−	0.991385i	$-0.458188\pi$
$271$	4.31238			0.261958			0.130979	+	0.991385i	$0.458188\pi$
$272$		0			0
$273$	−0.634135	−	1.30579i	−0.0383796	−	0.0790302i
$274$		0			0
$275$	−5.65460			−0.340985
$276$		0			0
$277$		−	25.5528i		−	1.53532i	−0.640857	−	0.767661i	$-0.721421\pi$
$277$		−	25.5528i		−	1.53532i	0.640857	−	0.767661i	$-0.278579\pi$
$278$		0			0
$279$	−7.50941			−0.449577
$280$		0			0
$281$	2.17501			0.129751			0.0648753	−	0.997893i	$-0.479335\pi$
$281$	2.17501			0.129751			0.0648753	+	0.997893i	$0.479335\pi$
$282$		0			0
$283$		−	12.2880i		−	0.730446i	−0.930920	−	0.365223i	$-0.880993\pi$
$283$		−	12.2880i		−	0.730446i	0.930920	−	0.365223i	$-0.119007\pi$
$284$		0			0
$285$	9.51981			0.563905
$286$		0			0
$287$	8.19485	+	16.8746i	0.483727	+	0.996077i
$288$		0			0
$289$	16.8120			0.988939
$290$		0			0
$291$	19.1778			1.12422
$292$		0			0
$293$	24.8790			1.45345			0.726724	−	0.686929i	$-0.241042\pi$
$293$	24.8790			1.45345			0.726724	+	0.686929i	$0.241042\pi$
$294$		0			0
$295$		−	18.2158i		−	1.06057i
$296$		0			0
$297$			2.26057i			0.131172i
$298$		0			0
$299$		−	4.52114i		−	0.261464i
$300$		0			0
$301$	4.34802	−	2.11154i	0.250616	−	0.121707i
$302$		0			0
$303$			4.74208i			0.272426i
$304$		0			0
$305$	19.1778			1.09812
$306$		0			0
$307$			17.0254i			0.971689i	0.874045	+	0.485844i	$0.161488\pi$
$307$			17.0254i			0.971689i	−0.874045	+	0.485844i	$0.838512\pi$
$308$		0			0
$309$		−	2.01040i		−	0.114368i
$310$		0			0
$311$	23.2983			1.32113			0.660564	−	0.750770i	$-0.270317\pi$
$311$	23.2983			1.32113			0.660564	+	0.750770i	$0.270317\pi$
$312$		0			0
$313$			15.0479i			0.850558i	0.905062	+	0.425279i	$0.139824\pi$
$313$			15.0479i			0.850558i	−0.905062	+	0.425279i	$0.860176\pi$
$314$		0			0
$315$	−3.76198	+	1.82694i	−0.211963	+	0.102936i
$316$		0			0
$317$		−	22.4761i		−	1.26238i	−0.775627	−	0.631192i	$-0.782566\pi$
$317$		−	22.4761i		−	1.26238i	0.775627	−	0.631192i	$-0.217434\pi$
$318$		0			0
$319$		−	1.24029i		−	0.0694431i
$320$		0			0
$321$			17.7845i			0.992635i
$322$		0			0
$323$	−2.61159			−0.145313
$324$		0			0
$325$	−1.37243			−0.0761288
$326$		0			0
$327$	12.0432			0.665992
$328$		0			0
$329$	−27.5690	+	13.3884i	−1.51993	+	0.738127i
$330$		0			0
$331$	23.5315			1.29341			0.646705	−	0.762740i	$-0.276147\pi$
$331$	23.5315			1.29341			0.646705	+	0.762740i	$0.276147\pi$
$332$		0			0
$333$			4.21124i			0.230775i
$334$		0			0
$335$	−14.7809			−0.807567
$336$		0			0
$337$	8.77682			0.478104			0.239052	−	0.971007i	$-0.423163\pi$
$337$	8.77682			0.478104			0.239052	+	0.971007i	$0.423163\pi$
$338$		0			0
$339$			2.86727i			0.155729i
$340$		0			0
$341$	16.9756			0.919278
$342$		0			0
$343$	3.94283	−	18.0957i	0.212893	−	0.977076i
$344$		0			0
$345$	−13.0254			−0.701262
$346$		0			0
$347$	4.00489			0.214994			0.107497	−	0.994205i	$-0.465716\pi$
$347$	4.00489			0.214994			0.107497	+	0.994205i	$0.465716\pi$
$348$		0			0
$349$	3.43649			0.183951			0.0919756	−	0.995761i	$-0.470682\pi$
$349$	3.43649			0.183951			0.0919756	+	0.995761i	$0.470682\pi$
$350$		0			0
$351$			0.548664i			0.0292855i
$352$		0			0
$353$		−	23.0903i		−	1.22897i	−0.788927	−	0.614487i	$-0.789363\pi$
$353$		−	23.0903i		−	1.22897i	0.788927	−	0.614487i	$-0.210637\pi$
$354$		0			0
$355$		−	2.02255i		−	0.107346i
$356$		0			0
$357$	1.03203	−	0.501187i	0.0546208	−	0.0265256i
$358$		0			0
$359$		−	21.8882i		−	1.15522i	−0.816315	−	0.577608i	$-0.803986\pi$
$359$		−	21.8882i		−	1.15522i	0.816315	−	0.577608i	$-0.196014\pi$
$360$		0			0
$361$	−17.2711			−0.909004
$362$		0			0
$363$			5.88982i			0.309135i
$364$		0			0
$365$			1.37089i			0.0717556i
$366$		0			0
$367$	−27.9276			−1.45781			−0.728905	−	0.684614i	$-0.759971\pi$
$367$	−27.9276			−1.45781			−0.728905	+	0.684614i	$0.759971\pi$
$368$		0			0
$369$		−	7.09032i		−	0.369107i
$370$		0			0
$371$	−4.28801	−	8.82975i	−0.222622	−	0.458418i
$372$		0			0
$373$		−	15.2403i		−	0.789111i	−0.918872	−	0.394555i	$-0.870899\pi$
$373$		−	15.2403i		−	0.789111i	0.918872	−	0.394555i	$-0.129101\pi$
$374$		0			0
$375$			11.8574i			0.612315i
$376$		0			0
$377$		−	0.301032i		−	0.0155039i
$378$		0			0
$379$	−11.4864			−0.590018			−0.295009	−	0.955494i	$-0.595323\pi$
$379$	−11.4864			−0.590018			−0.295009	+	0.955494i	$0.595323\pi$
$380$		0			0
$381$	14.9928			0.768102
$382$		0			0
$383$	22.4746			1.14840			0.574198	−	0.818716i	$-0.305314\pi$
$383$	22.4746			1.14840			0.574198	+	0.818716i	$0.305314\pi$
$384$		0			0
$385$	8.50422	−	4.12992i	0.433415	−	0.210480i
$386$		0			0
$387$	−1.82694			−0.0928684
$388$		0			0
$389$			20.5907i			1.04399i	0.852949	+	0.521994i	$0.174812\pi$
$389$			20.5907i			1.04399i	−0.852949	+	0.521994i	$0.825188\pi$
$390$		0			0
$391$	3.57327			0.180708
$392$		0			0
$393$	8.52114			0.429835
$394$		0			0
$395$			16.3912i			0.824732i
$396$		0			0
$397$	−26.8778			−1.34896			−0.674479	−	0.738294i	$-0.735632\pi$
$397$	−26.8778			−1.34896			−0.674479	+	0.738294i	$0.735632\pi$
$398$		0			0
$399$	14.3334	−	6.96075i	0.717566	−	0.348473i
$400$		0			0
$401$	9.00281			0.449579			0.224789	−	0.974407i	$-0.427831\pi$
$401$	9.00281			0.449579			0.224789	+	0.974407i	$0.427831\pi$
$402$		0			0
$403$	4.12014			0.205239
$404$		0			0
$405$	1.58069			0.0785453
$406$		0			0
$407$		−	9.51981i		−	0.471879i
$408$		0			0
$409$		−	4.91237i		−	0.242901i	−0.992597	−	0.121450i	$-0.961245\pi$
$409$		−	4.91237i		−	0.242901i	0.992597	−	0.121450i	$-0.0387545\pi$
$410$		0			0
$411$		−	5.00281i		−	0.246770i
$412$		0			0
$413$	−13.3192	−	27.4265i	−0.655393	−	1.34957i
$414$		0			0
$415$			9.69841i			0.476076i
$416$		0			0
$417$	14.3106			0.700791
$418$		0			0
$419$			4.17501i			0.203963i	0.994786	+	0.101981i	$0.0325182\pi$
$419$			4.17501i			0.203963i	−0.994786	+	0.101981i	$0.967482\pi$
$420$		0			0
$421$			3.11391i			0.151763i	0.997117	+	0.0758814i	$0.0241770\pi$
$421$			3.11391i			0.151763i	−0.997117	+	0.0758814i	$0.975823\pi$
$422$		0			0
$423$	11.5839			0.563227
$424$		0			0
$425$		−	1.08470i		−	0.0526155i
$426$		0			0
$427$	28.8748	−	14.0225i	1.39735	−	0.678599i
$428$		0			0
$429$		−	1.24029i		−	0.0598820i
$430$		0			0
$431$		−	0.641564i		−	0.0309030i	−0.999881	−	0.0154515i	$-0.995081\pi$
$431$		−	0.641564i		−	0.0309030i	0.999881	−	0.0154515i	$-0.00491857\pi$
$432$		0			0
$433$			6.95772i			0.334366i	0.985926	+	0.167183i	$0.0534671\pi$
$433$			6.95772i			0.334366i	−0.985926	+	0.167183i	$0.946533\pi$
$434$		0			0
$435$	−0.867270			−0.0415824
$436$		0			0
$437$	49.6275			2.37400
$438$		0			0
$439$	−17.4055			−0.830717			−0.415359	−	0.909658i	$-0.636344\pi$
$439$	−17.4055			−0.830717			−0.415359	+	0.909658i	$0.636344\pi$
$440$		0			0
$441$	−4.32834	+	5.50141i	−0.206111	+	0.261972i
$442$		0			0
$443$	−18.0856			−0.859271			−0.429635	−	0.903002i	$-0.641358\pi$
$443$	−18.0856			−0.859271			−0.429635	+	0.903002i	$0.641358\pi$
$444$		0			0
$445$		−	12.5785i		−	0.596279i
$446$		0			0
$447$	18.4553			0.872906
$448$		0			0
$449$	−21.2229			−1.00157			−0.500786	−	0.865571i	$-0.666956\pi$
$449$	−21.2229			−1.00157			−0.500786	+	0.865571i	$0.666956\pi$
$450$		0			0
$451$			16.0282i			0.754737i
$452$		0			0
$453$	−14.5334			−0.682839
$454$		0			0
$455$	2.06406	−	1.00237i	0.0967647	−	0.0469920i
$456$		0			0
$457$	27.7046			1.29597			0.647983	−	0.761655i	$-0.275613\pi$
$457$	27.7046			1.29597			0.647983	+	0.761655i	$0.275613\pi$
$458$		0			0
$459$	−0.433635			−0.0202403
$460$		0			0
$461$	−23.5438			−1.09654			−0.548272	−	0.836300i	$-0.684714\pi$
$461$	−23.5438			−1.09654			−0.548272	+	0.836300i	$0.684714\pi$
$462$		0			0
$463$		−	26.8145i		−	1.24618i	−0.782151	−	0.623089i	$-0.785878\pi$
$463$		−	26.8145i		−	1.24618i	0.782151	−	0.623089i	$-0.214122\pi$
$464$		0			0
$465$		−	11.8701i		−	0.550462i
$466$		0			0
$467$		−	17.3884i		−	0.804640i	−0.915499	−	0.402320i	$-0.868204\pi$
$467$		−	17.3884i		−	0.804640i	0.915499	−	0.402320i	$-0.131796\pi$
$468$		0			0
$469$	−22.2547	+	10.8076i	−1.02762	+	0.499048i
$470$		0			0
$471$		−	5.80975i		−	0.267699i
$472$		0			0
$473$	4.12992			0.189894
$474$		0			0
$475$		−	15.0648i		−	0.691222i
$476$		0			0
$477$			3.71005i			0.169872i
$478$		0			0
$479$	−25.6003			−1.16971			−0.584854	−	0.811139i	$-0.698848\pi$
$479$	−25.6003			−1.16971			−0.584854	+	0.811139i	$0.698848\pi$
$480$		0			0
$481$		−	2.31056i		−	0.105352i
$482$		0			0
$483$	−19.6115	+	9.52395i	−0.892352	+	0.433355i
$484$		0			0
$485$			30.3143i			1.37650i
$486$		0			0
$487$			0.940673i			0.0426260i	0.999773	+	0.0213130i	$0.00678464\pi$
$487$			0.940673i			0.0426260i	−0.999773	+	0.0213130i	$0.993215\pi$
$488$		0			0
$489$		−	17.3509i		−	0.784634i
$490$		0			0
$491$	24.0500			1.08536			0.542680	−	0.839939i	$-0.317410\pi$
$491$	24.0500			1.08536			0.542680	+	0.839939i	$0.317410\pi$
$492$		0			0
$493$	0.237920			0.0107154
$494$		0			0
$495$	−3.57327			−0.160607
$496$		0			0
$497$	−1.47886	−	3.04522i	−0.0663358	−	0.136597i
$498$		0			0
$499$	26.8354			1.20132			0.600658	−	0.799506i	$-0.294905\pi$
$499$	26.8354			1.20132			0.600658	+	0.799506i	$0.294905\pi$
$500$		0			0
$501$		−	0.823767i		−	0.0368032i
$502$		0			0
$503$	1.13297			0.0505166			0.0252583	−	0.999681i	$-0.491959\pi$
$503$	1.13297			0.0505166			0.0252583	+	0.999681i	$0.491959\pi$
$504$		0			0
$505$	−7.49578			−0.333558
$506$		0			0
$507$			12.6990i			0.563981i
$508$		0			0
$509$	11.7937			0.522745			0.261373	−	0.965238i	$-0.415825\pi$
$509$	11.7937			0.522745			0.261373	+	0.965238i	$0.415825\pi$
$510$		0			0
$511$	1.00237	+	2.06406i	0.0443424	+	0.0913087i
$512$		0			0
$513$	−6.02255			−0.265902
$514$		0			0
$515$	3.17783			0.140032
$516$		0			0
$517$	−26.1862			−1.15167
$518$		0			0
$519$			19.5230i			0.856964i
$520$		0			0
$521$			14.6991i			0.643979i	0.946743	+	0.321990i	$0.104352\pi$
$521$			14.6991i			0.643979i	−0.946743	+	0.321990i	$0.895648\pi$
$522$		0			0
$523$		−	7.69507i		−	0.336482i	−0.985746	−	0.168241i	$-0.946191\pi$
$523$		−	7.69507i		−	0.336482i	0.985746	−	0.168241i	$-0.0538086\pi$
$524$		0			0
$525$	2.89108	+	5.95322i	0.126177	+	0.259820i
$526$		0			0
$527$			3.25634i			0.141849i
$528$		0			0
$529$	−44.9022			−1.95227
$530$		0			0
$531$			11.5240i			0.500097i
$532$		0			0
$533$			3.89020i			0.168503i
$534$		0			0
$535$	−28.1119			−1.21538
$536$		0			0
$537$		−	4.39611i		−	0.189706i
$538$		0			0
$539$	9.78452	−	12.4363i	0.421449	−	0.535670i
$540$		0			0
$541$		−	11.6313i		−	0.500071i	−0.968237	−	0.250035i	$-0.919558\pi$
$541$		−	11.6313i		−	0.500071i	0.968237	−	0.250035i	$-0.0804422\pi$
$542$		0			0
$543$			8.14738i			0.349637i
$544$		0			0
$545$			19.0367i			0.815441i
$546$		0			0
$547$	−21.0048			−0.898099			−0.449049	−	0.893507i	$-0.648237\pi$
$547$	−21.0048			−0.898099			−0.449049	+	0.893507i	$0.648237\pi$
$548$		0			0
$549$	−12.1325			−0.517804
$550$		0			0
$551$	3.30435			0.140770
$552$		0			0
$553$	11.9850	+	24.6792i	0.509655	+	1.04947i
$554$		0			0
$555$	−6.65668			−0.282560
$556$		0			0
$557$			34.7501i			1.47241i	0.676760	+	0.736204i	$0.263383\pi$
$557$			34.7501i			1.47241i	−0.676760	+	0.736204i	$0.736617\pi$
$558$		0			0
$559$	1.00237			0.0423959
$560$		0			0
$561$	0.980263			0.0413867
$562$		0			0
$563$			16.9124i			0.712771i	0.934339	+	0.356386i	$0.115991\pi$
$563$			16.9124i			0.712771i	−0.934339	+	0.356386i	$0.884009\pi$
$564$		0			0
$565$	−4.53228			−0.190674
$566$		0			0
$567$	2.37995	−	1.15578i	0.0999486	−	0.0485382i
$568$		0			0
$569$	−20.4856			−0.858800			−0.429400	−	0.903114i	$-0.641275\pi$
$569$	−20.4856			−0.858800			−0.429400	+	0.903114i	$0.641275\pi$
$570$		0			0
$571$	−46.0132			−1.92559			−0.962796	−	0.270229i	$-0.912901\pi$
$571$	−46.0132			−1.92559			−0.962796	+	0.270229i	$0.912901\pi$
$572$		0			0
$573$	−0.420123			−0.0175509
$574$		0			0
$575$			20.6123i			0.859591i
$576$		0			0
$577$		−	35.8191i		−	1.49117i	−0.666411	−	0.745585i	$-0.732170\pi$
$577$		−	35.8191i		−	1.49117i	0.666411	−	0.745585i	$-0.267830\pi$
$578$		0			0
$579$			11.1553i			0.463598i
$580$		0			0
$581$	7.09134	+	14.6023i	0.294198	+	0.605805i
$582$		0			0
$583$		−	8.38684i		−	0.347347i
$584$		0			0
$585$	−0.867270			−0.0358572
$586$		0			0
$587$		−	30.4007i		−	1.25477i	−0.778708	−	0.627387i	$-0.784125\pi$
$587$		−	30.4007i		−	1.25477i	0.778708	−	0.627387i	$-0.215875\pi$
$588$		0			0
$589$			45.2258i			1.86350i
$590$		0			0
$591$	−4.95035			−0.203630
$592$		0			0
$593$			23.3065i			0.957084i	0.878065	+	0.478542i	$0.158835\pi$
$593$			23.3065i			0.957084i	−0.878065	+	0.478542i	$0.841165\pi$
$594$		0			0
$595$	0.792224	+	1.63132i	0.0324780	+	0.0668778i
$596$		0			0
$597$		−	12.0851i		−	0.494608i
$598$		0			0
$599$			19.9547i			0.815329i	0.913132	+	0.407664i	$0.133657\pi$
$599$			19.9547i			0.815329i	−0.913132	+	0.407664i	$0.866343\pi$
$600$		0			0
$601$		−	27.0028i		−	1.10147i	−0.834681	−	0.550734i	$-0.814348\pi$
$601$		−	27.0028i		−	1.10147i	0.834681	−	0.550734i	$-0.185652\pi$
$602$		0			0
$603$	9.35089			0.380798
$604$		0			0
$605$	−9.31000			−0.378505
$606$		0			0
$607$	−1.63416			−0.0663283			−0.0331642	−	0.999450i	$-0.510558\pi$
$607$	−1.63416			−0.0663283			−0.0331642	+	0.999450i	$0.510558\pi$
$608$		0			0
$609$	−1.30579	+	0.634135i	−0.0529134	+	0.0256965i
$610$		0			0
$611$	−6.35565			−0.257122
$612$		0			0
$613$		−	16.3500i		−	0.660369i	−0.943916	−	0.330184i	$-0.892889\pi$
$613$		−	16.3500i		−	0.660369i	0.943916	−	0.330184i	$-0.107111\pi$
$614$		0			0
$615$	11.2076			0.451935
$616$		0			0
$617$	0.220365			0.00887154			0.00443577	−	0.999990i	$-0.498588\pi$
$617$	0.220365			0.00887154			0.00443577	+	0.999990i	$0.498588\pi$
$618$		0			0
$619$			16.9972i			0.683175i	0.939850	+	0.341587i	$0.110965\pi$
$619$			16.9972i			0.683175i	−0.939850	+	0.341587i	$0.889035\pi$
$620$		0			0
$621$	8.24028			0.330671
$622$		0			0
$623$	−9.19723	−	18.9387i	−0.368479	−	0.758762i
$624$		0			0
$625$	−6.23595			−0.249438
$626$		0			0
$627$	13.6144			0.543707
$628$		0			0
$629$	1.82614			0.0728130
$630$		0			0
$631$			13.8402i			0.550971i	0.961305	+	0.275485i	$0.0888386\pi$
$631$			13.8402i			0.550971i	−0.961305	+	0.275485i	$0.911161\pi$
$632$		0			0
$633$			6.17306i			0.245357i
$634$		0			0
$635$			23.6990i			0.940465i
$636$		0			0
$637$	2.37480	−	3.01842i	0.0940932	−	0.119594i
$638$		0			0
$639$			1.27953i			0.0506175i
$640$		0			0
$641$	13.0479			0.515361			0.257681	−	0.966230i	$-0.417042\pi$
$641$	13.0479			0.515361			0.257681	+	0.966230i	$0.417042\pi$
$642$		0			0
$643$		−	12.2880i		−	0.484592i	−0.970202	−	0.242296i	$-0.922100\pi$
$643$		−	12.2880i		−	0.484592i	0.970202	−	0.242296i	$-0.0779005\pi$
$644$		0			0
$645$		−	2.88783i		−	0.113708i
$646$		0			0
$647$	−31.8638			−1.25269			−0.626347	−	0.779544i	$-0.715451\pi$
$647$	−31.8638			−1.25269			−0.626347	+	0.779544i	$0.715451\pi$
$648$		0			0
$649$		−	26.0507i		−	1.02258i
$650$		0			0
$651$	−8.67923	−	17.8720i	−0.340166	−	0.700460i
$652$		0			0
$653$			14.6007i			0.571371i	0.958323	+	0.285686i	$0.0922213\pi$
$653$			14.6007i			0.571371i	−0.958323	+	0.285686i	$0.907779\pi$
$654$		0			0
$655$			13.4693i			0.526290i
$656$		0			0
$657$		−	0.867270i		−	0.0338354i
$658$		0			0
$659$	−7.25776			−0.282722			−0.141361	−	0.989958i	$-0.545148\pi$
$659$	−7.25776			−0.282722			−0.141361	+	0.989958i	$0.545148\pi$
$660$		0			0
$661$	−15.6031			−0.606891			−0.303446	−	0.952849i	$-0.598137\pi$
$661$	−15.6031			−0.606891			−0.303446	+	0.952849i	$0.598137\pi$
$662$		0			0
$663$	0.237920			0.00924004
$664$		0			0
$665$	11.0028	+	22.6567i	0.426671	+	0.878588i
$666$		0			0
$667$	−4.52114			−0.175059
$668$		0			0
$669$			18.4078i			0.711688i
$670$		0			0
$671$	27.4265			1.05879
$672$		0			0
$673$	9.19475			0.354432			0.177216	−	0.984172i	$-0.443291\pi$
$673$	9.19475			0.354432			0.177216	+	0.984172i	$0.443291\pi$
$674$		0			0
$675$		−	2.50141i		−	0.0962791i
$676$		0			0
$677$	14.1669			0.544480			0.272240	−	0.962229i	$-0.412236\pi$
$677$	14.1669			0.544480			0.272240	+	0.962229i	$0.412236\pi$
$678$		0			0
$679$	22.1654	+	45.6423i	0.850629	+	1.75159i
$680$		0			0
$681$	−10.6567			−0.408365
$682$		0			0
$683$	35.9201			1.37444			0.687222	−	0.726448i	$-0.258830\pi$
$683$	35.9201			1.37444			0.687222	+	0.726448i	$0.258830\pi$
$684$		0			0
$685$	7.90791			0.302146
$686$		0			0
$687$			8.97114i			0.342270i
$688$		0			0
$689$		−	2.03557i		−	0.0775491i
$690$		0			0
$691$		−	24.0451i		−	0.914719i	−0.889282	−	0.457359i	$-0.848795\pi$
$691$		−	24.0451i		−	0.914719i	0.889282	−	0.457359i	$-0.151205\pi$
$692$		0			0
$693$	−5.38005	+	2.61272i	−0.204371	+	0.0992492i
$694$		0			0
$695$			22.6206i			0.858049i
$696$		0			0
$697$	−3.07461			−0.116459
$698$		0			0
$699$		−	14.9124i		−	0.564037i
$700$		0			0
$701$		−	39.3211i		−	1.48514i	−0.669771	−	0.742568i	$-0.733608\pi$
$701$		−	39.3211i		−	1.48514i	0.669771	−	0.742568i	$-0.266392\pi$
$702$		0			0
$703$	25.3624			0.956561
$704$		0			0
$705$			18.3106i			0.689615i
$706$		0			0
$707$	−11.2859	+	5.48081i	−0.424451	+	0.206127i
$708$		0			0
$709$			37.8578i			1.42178i	0.703304	+	0.710890i	$0.251707\pi$
$709$			37.8578i			1.42178i	−0.703304	+	0.710890i	$0.748293\pi$
$710$		0			0
$711$		−	10.3696i		−	0.388892i
$712$		0			0
$713$		−	61.8796i		−	2.31741i
$714$		0			0
$715$	1.96053			0.0733195
$716$		0			0
$717$	13.1370			0.490609
$718$		0			0
$719$	−25.6003			−0.954731			−0.477365	−	0.878705i	$-0.658408\pi$
$719$	−25.6003			−0.954731			−0.477365	+	0.878705i	$0.658408\pi$
$720$		0			0
$721$	4.78465	−	2.32358i	0.178190	−	0.0865346i
$722$		0			0
$723$	−10.1355			−0.376945
$724$		0			0
$725$			1.37243i			0.0509708i
$726$		0			0
$727$	2.11772			0.0785420			0.0392710	−	0.999229i	$-0.487496\pi$
$727$	2.11772			0.0785420			0.0392710	+	0.999229i	$0.487496\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$			0.792224i			0.0293014i
$732$		0			0
$733$	20.4120			0.753936			0.376968	−	0.926226i	$-0.376967\pi$
$733$	20.4120			0.753936			0.376968	+	0.926226i	$0.376967\pi$
$734$		0			0
$735$	−8.69604	−	6.84179i	−0.320758	−	0.252363i
$736$		0			0
$737$	−21.1384			−0.778641
$738$		0			0
$739$	−18.3932			−0.676604			−0.338302	−	0.941038i	$-0.609853\pi$
$739$	−18.3932			−0.676604			−0.338302	+	0.941038i	$0.609853\pi$
$740$		0			0
$741$	3.30435			0.121389
$742$		0			0
$743$			33.3812i			1.22464i	0.790611	+	0.612319i	$0.209763\pi$
$743$			33.3812i			1.22464i	−0.790611	+	0.612319i	$0.790237\pi$
$744$		0			0
$745$			29.1722i			1.06879i
$746$		0			0
$747$		−	6.13554i		−	0.224488i
$748$		0			0
$749$	−42.3263	+	20.5550i	−1.54657	+	0.751064i
$750$		0			0
$751$			9.83899i			0.359030i	0.983755	+	0.179515i	$0.0574528\pi$
$751$			9.83899i			0.359030i	−0.983755	+	0.179515i	$0.942547\pi$
$752$		0			0
$753$	22.7018			0.827299
$754$		0			0
$755$		−	22.9729i		−	0.836068i
$756$		0			0
$757$		−	23.9363i		−	0.869980i	−0.900435	−	0.434990i	$-0.856752\pi$
$757$		−	23.9363i		−	0.869980i	0.900435	−	0.434990i	$-0.143248\pi$
$758$		0			0
$759$	−18.6277			−0.676144
$760$		0			0
$761$		−	14.6991i		−	0.532842i	−0.963857	−	0.266421i	$-0.914159\pi$
$761$		−	14.6991i		−	0.532842i	0.963857	−	0.266421i	$-0.0858411\pi$
$762$		0			0
$763$	13.9193	+	28.6623i	0.503914	+	1.03765i
$764$		0			0
$765$		−	0.685444i		−	0.0247823i
$766$		0			0
$767$		−	6.32278i		−	0.228302i
$768$		0			0
$769$			22.7317i			0.819727i	0.912147	+	0.409864i	$0.134424\pi$
$769$			22.7317i			0.819727i	−0.912147	+	0.409864i	$0.865576\pi$
$770$		0			0
$771$	16.1326			0.581002
$772$		0			0
$773$	8.21267			0.295389			0.147695	−	0.989033i	$-0.452815\pi$
$773$	8.21267			0.295389			0.147695	+	0.989033i	$0.452815\pi$
$774$		0			0
$775$	−18.7841			−0.674744
$776$		0			0
$777$	−10.0225	+	4.86727i	−0.359557	+	0.174612i
$778$		0			0
$779$	−42.7018			−1.52995
$780$		0			0
$781$		−	2.89247i		−	0.103501i
$782$		0			0
$783$	0.548664			0.0196076
$784$		0			0
$785$	9.18345			0.327771
$786$		0			0
$787$			25.3134i			0.902324i	0.892442	+	0.451162i	$0.148990\pi$
$787$			25.3134i			0.902324i	−0.892442	+	0.451162i	$0.851010\pi$
$788$		0			0
$789$	−15.0581			−0.536081
$790$		0			0
$791$	−6.82396	+	3.31394i	−0.242632	+	0.117830i
$792$		0			0
$793$	6.65668			0.236386
$794$		0			0
$795$	−5.86446			−0.207991
$796$		0			0
$797$	10.2767			0.364021			0.182010	−	0.983297i	$-0.441740\pi$
$797$	10.2767			0.364021			0.182010	+	0.983297i	$0.441740\pi$
$798$		0			0
$799$		−	5.02317i		−	0.177707i
$800$		0			0
$801$			7.95759i			0.281168i
$802$		0			0
$803$			1.96053i			0.0691854i
$804$		0			0
$805$	−15.0545	−	30.9997i	−0.530600	−	1.09260i
$806$		0			0
$807$			8.76288i			0.308468i
$808$		0			0
$809$	28.4968			1.00189			0.500947	−	0.865478i	$-0.332985\pi$
$809$	28.4968			1.00189			0.500947	+	0.865478i	$0.332985\pi$
$810$		0			0
$811$		−	13.2683i		−	0.465912i	−0.972487	−	0.232956i	$-0.925160\pi$
$811$		−	13.2683i		−	0.465912i	0.972487	−	0.232956i	$-0.0748398\pi$
$812$		0			0
$813$		−	4.31238i		−	0.151242i
$814$		0			0
$815$	27.4265			0.960707
$816$		0			0
$817$			11.0028i			0.384940i
$818$		0			0
$819$	−1.30579	+	0.634135i	−0.0456281	+	0.0221585i
$820$		0			0
$821$		−	50.5926i		−	1.76570i	−0.469659	−	0.882848i	$-0.655623\pi$
$821$		−	50.5926i		−	1.76570i	0.469659	−	0.882848i	$-0.344377\pi$
$822$		0			0
$823$		−	19.5730i		−	0.682274i	−0.940014	−	0.341137i	$-0.889188\pi$
$823$		−	19.5730i		−	0.682274i	0.940014	−	0.341137i	$-0.110812\pi$
$824$		0			0
$825$			5.65460i			0.196868i
$826$		0			0
$827$	4.64617			0.161563			0.0807816	−	0.996732i	$-0.474258\pi$
$827$	4.64617			0.161563			0.0807816	+	0.996732i	$0.474258\pi$
$828$		0			0
$829$	22.2382			0.772364			0.386182	−	0.922423i	$-0.373794\pi$
$829$	22.2382			0.772364			0.386182	+	0.922423i	$0.373794\pi$
$830$		0			0
$831$	−25.5528			−0.886418
$832$		0			0
$833$	2.38560	+	1.87692i	0.0826562	+	0.0650314i
$834$		0			0
$835$	1.30212			0.0450619
$836$		0			0
$837$			7.50941i			0.259563i
$838$		0			0
$839$	−36.5297			−1.26115			−0.630573	−	0.776130i	$-0.717180\pi$
$839$	−36.5297			−1.26115			−0.630573	+	0.776130i	$0.717180\pi$
$840$		0			0
$841$	28.6990			0.989620
$842$		0			0
$843$		−	2.17501i		−	0.0749115i
$844$		0			0
$845$	−20.0732			−0.690539
$846$		0			0
$847$	−14.0175	+	6.80734i	−0.481646	+	0.233903i
$848$		0			0
$849$	−12.2880			−0.421723
$850$		0			0
$851$	−34.7018			−1.18956
$852$		0			0
$853$	42.1702			1.44388			0.721940	−	0.691956i	$-0.243251\pi$
$853$	42.1702			1.44388			0.721940	+	0.691956i	$0.243251\pi$
$854$		0			0
$855$		−	9.51981i		−	0.325571i
$856$		0			0
$857$			37.5720i			1.28343i	0.766941	+	0.641717i	$0.221778\pi$
$857$			37.5720i			1.28343i	−0.766941	+	0.641717i	$0.778222\pi$
$858$		0			0
$859$			12.7543i			0.435170i	0.976041	+	0.217585i	$0.0698180\pi$
$859$			12.7543i			0.435170i	−0.976041	+	0.217585i	$0.930182\pi$
$860$		0			0
$861$	16.8746	−	8.19485i	0.575085	−	0.279280i
$862$		0			0
$863$		−	51.1449i		−	1.74099i	−0.492175	−	0.870496i	$-0.663798\pi$
$863$		−	51.1449i		−	1.74099i	0.492175	−	0.870496i	$-0.336202\pi$
$864$		0			0
$865$	−30.8599			−1.04927
$866$		0			0
$867$		−	16.8120i		−	0.570964i
$868$		0			0
$869$			23.4413i			0.795192i
$870$		0			0
$871$	−5.13050			−0.173840
$872$		0			0
$873$		−	19.1778i		−	0.649071i
$874$		0			0
$875$	−28.2201	+	13.7046i	−0.954014	+	0.463300i
$876$		0			0
$877$			13.8694i			0.468335i	0.972196	+	0.234168i	$0.0752365\pi$
$877$			13.8694i			0.468335i	−0.972196	+	0.234168i	$0.924764\pi$
$878$		0			0
$879$		−	24.8790i		−	0.839149i
$880$		0			0
$881$		−	5.92202i		−	0.199518i	−0.995012	−	0.0997589i	$-0.968193\pi$
$881$		−	5.92202i		−	0.199518i	0.995012	−	0.0997589i	$-0.0318071\pi$
$882$		0			0
$883$	3.17025			0.106688			0.0533438	−	0.998576i	$-0.483012\pi$
$883$	3.17025			0.106688			0.0533438	+	0.998576i	$0.483012\pi$
$884$		0			0
$885$	−18.2158			−0.612319
$886$		0			0
$887$	49.9010			1.67551			0.837756	−	0.546045i	$-0.183867\pi$
$887$	49.9010			1.67551			0.837756	+	0.546045i	$0.183867\pi$
$888$		0			0
$889$	17.3283	+	35.6820i	0.581174	+	1.19674i
$890$		0			0
$891$	2.26057			0.0757320
$892$		0			0
$893$		−	69.7644i		−	2.33458i
$894$		0			0
$895$	6.94891			0.232276
$896$		0			0
$897$	−4.52114			−0.150957
$898$		0			0
$899$		−	4.12014i		−	0.137414i
$900$		0			0
$901$	1.60881			0.0535972
$902$		0			0
$903$	−2.11154	−	4.34802i	−0.0702676	−	0.144693i
$904$		0			0
$905$	−12.8785			−0.428096
$906$		0			0
$907$	2.60938			0.0866431			0.0433216	−	0.999061i	$-0.486206\pi$
$907$	2.60938			0.0866431			0.0433216	+	0.999061i	$0.486206\pi$
$908$		0			0
$909$	4.74208			0.157285
$910$		0			0
$911$		−	42.9082i		−	1.42161i	−0.703388	−	0.710807i	$-0.748330\pi$
$911$		−	42.9082i		−	1.42161i	0.703388	−	0.710807i	$-0.251670\pi$
$912$		0			0
$913$			13.8698i			0.459024i
$914$		0			0
$915$		−	19.1778i		−	0.633999i
$916$		0			0
$917$	9.84857	+	20.2799i	0.325229	+	0.669702i
$918$		0			0
$919$			24.4217i			0.805598i	0.915288	+	0.402799i	$0.131963\pi$
$919$			24.4217i			0.805598i	−0.915288	+	0.402799i	$0.868037\pi$
$920$		0			0
$921$	17.0254			0.561005
$922$		0			0
$923$		−	0.702033i		−	0.0231077i
$924$		0			0
$925$			10.5340i			0.346356i
$926$		0			0
$927$	−2.01040			−0.0660301
$928$		0			0
$929$		−	32.9150i		−	1.07991i	−0.841695	−	0.539954i	$-0.818442\pi$
$929$		−	32.9150i		−	1.07991i	0.841695	−	0.539954i	$-0.181558\pi$
$930$		0			0
$931$	33.1325	+	26.0676i	1.08587	+	0.854332i
$932$		0			0
$933$		−	23.2983i		−	0.762753i
$934$		0			0
$935$			1.54950i			0.0506739i
$936$		0			0
$937$		−	41.2683i		−	1.34818i	−0.738651	−	0.674088i	$-0.764537\pi$
$937$		−	41.2683i		−	1.34818i	0.738651	−	0.674088i	$-0.235463\pi$
$938$		0			0
$939$	15.0479			0.491070
$940$		0			0
$941$	−53.4508			−1.74245			−0.871223	−	0.490887i	$-0.836673\pi$
$941$	−53.4508			−1.74245			−0.871223	+	0.490887i	$0.836673\pi$
$942$		0			0
$943$	58.4262			1.90262
$944$		0			0
$945$	1.82694	+	3.76198i	0.0594302	+	0.122377i
$946$		0			0
$947$	55.2631			1.79581			0.897905	−	0.440189i	$-0.145089\pi$
$947$	55.2631			1.79581			0.897905	+	0.440189i	$0.145089\pi$
$948$		0			0
$949$			0.475840i			0.0154464i
$950$		0			0
$951$	−22.4761			−0.728838
$952$		0			0
$953$	−22.5760			−0.731309			−0.365654	−	0.930751i	$-0.619155\pi$
$953$	−22.5760			−0.731309			−0.365654	+	0.930751i	$0.619155\pi$
$954$		0			0
$955$		−	0.664086i		−	0.0214893i
$956$		0			0
$957$	−1.24029			−0.0400930
$958$		0			0
$959$	11.9064	−	5.78215i	0.384479	−	0.186715i
$960$		0			0
$961$	25.3912			0.819072
$962$		0			0
$963$	17.7845			0.573098
$964$		0			0
$965$	−17.6331			−0.567629
$966$		0			0
$967$		−	24.2911i		−	0.781150i	−0.920571	−	0.390575i	$-0.872276\pi$
$967$		−	24.2911i		−	0.781150i	0.920571	−	0.390575i	$-0.127724\pi$
$968$		0			0
$969$			2.61159i			0.0838963i
$970$		0			0
$971$		−	24.9221i		−	0.799790i	−0.916561	−	0.399895i	$-0.869047\pi$
$971$		−	24.9221i		−	0.799790i	0.916561	−	0.399895i	$-0.130953\pi$
$972$		0			0
$973$	16.5399	+	34.0584i	0.530244	+	1.09186i
$974$		0			0
$975$			1.37243i			0.0439530i
$976$		0			0
$977$	1.13273			0.0362392			0.0181196	−	0.999836i	$-0.494232\pi$
$977$	1.13273			0.0362392			0.0181196	+	0.999836i	$0.494232\pi$
$978$		0			0
$979$		−	17.9887i		−	0.574921i
$980$		0			0
$981$		−	12.0432i		−	0.384511i
$982$		0			0
$983$	11.8218			0.377056			0.188528	−	0.982068i	$-0.439628\pi$
$983$	11.8218			0.377056			0.188528	+	0.982068i	$0.439628\pi$
$984$		0			0
$985$		−	7.82499i		−	0.249325i
$986$		0			0
$987$	13.3884	+	27.5690i	0.426158	+	0.877532i
$988$		0			0
$989$		−	15.0545i		−	0.478704i
$990$		0			0
$991$		−	52.1176i		−	1.65557i	−0.561045	−	0.827785i	$-0.689600\pi$
$991$		−	52.1176i		−	1.65557i	0.561045	−	0.827785i	$-0.310400\pi$
$992$		0			0
$993$		−	23.5315i		−	0.746750i
$994$		0			0
$995$	19.1028			0.605599
$996$		0			0
$997$	−26.2326			−0.830796			−0.415398	−	0.909640i	$-0.636358\pi$
$997$	−26.2326			−0.830796			−0.415398	+	0.909640i	$0.636358\pi$
$998$		0			0
$999$	4.21124			0.133238

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	672.2.p.a.559.6		16
3.2	odd	2		2016.2.p.g.559.5		16
4.3	odd	2		168.2.p.a.139.14	yes	16
7.6	odd	2	inner	672.2.p.a.559.11		16
8.3	odd	2	inner	672.2.p.a.559.3		16
8.5	even	2		168.2.p.a.139.16	yes	16
12.11	even	2		504.2.p.g.307.3		16
21.20	even	2		2016.2.p.g.559.11		16
24.5	odd	2		504.2.p.g.307.2		16
24.11	even	2		2016.2.p.g.559.12		16
28.27	even	2		168.2.p.a.139.13	✓	16
56.13	odd	2		168.2.p.a.139.15	yes	16
56.27	even	2	inner	672.2.p.a.559.14		16
84.83	odd	2		504.2.p.g.307.4		16
168.83	odd	2		2016.2.p.g.559.6		16
168.125	even	2		504.2.p.g.307.1		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.2.p.a.139.13	✓	16	28.27	even	2
168.2.p.a.139.14	yes	16	4.3	odd	2
168.2.p.a.139.15	yes	16	56.13	odd	2
168.2.p.a.139.16	yes	16	8.5	even	2
504.2.p.g.307.1		16	168.125	even	2
504.2.p.g.307.2		16	24.5	odd	2
504.2.p.g.307.3		16	12.11	even	2
504.2.p.g.307.4		16	84.83	odd	2
672.2.p.a.559.3		16	8.3	odd	2	inner
672.2.p.a.559.6		16	1.1	even	1	trivial
672.2.p.a.559.11		16	7.6	odd	2	inner
672.2.p.a.559.14		16	56.27	even	2	inner
2016.2.p.g.559.5		16	3.2	odd	2
2016.2.p.g.559.6		16	168.83	odd	2
2016.2.p.g.559.11		16	21.20	even	2
2016.2.p.g.559.12		16	24.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	672.p (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} +1.58069 q^{5} +(2.37995 - 1.15578i) q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} +1.58069 q^{5} +(2.37995 - 1.15578i) q^{7} -1.00000 q^{9} +2.26057 q^{11} +0.548664 q^{13} -1.58069i q^{15} +0.433635i q^{17} +6.02255i q^{19} +(-1.15578 - 2.37995i) q^{21} -8.24028i q^{23} -2.50141 q^{25} +1.00000i q^{27} -0.548664i q^{29} +7.50941 q^{31} -2.26057i q^{33} +(3.76198 - 1.82694i) q^{35} -4.21124i q^{37} -0.548664i q^{39} +7.09032i q^{41} +1.82694 q^{43} -1.58069 q^{45} -11.5839 q^{47} +(4.32834 - 5.50141i) q^{49} +0.433635 q^{51} -3.71005i q^{53} +3.57327 q^{55} +6.02255 q^{57} -11.5240i q^{59} +12.1325 q^{61} +(-2.37995 + 1.15578i) q^{63} +0.867270 q^{65} -9.35089 q^{67} -8.24028 q^{69} -1.27953i q^{71} +0.867270i q^{73} +2.50141i q^{75} +(5.38005 - 2.61272i) q^{77} +10.3696i q^{79} +1.00000 q^{81} +6.13554i q^{83} +0.685444i q^{85} -0.548664 q^{87} -7.95759i q^{89} +(1.30579 - 0.634135i) q^{91} -7.50941i q^{93} +9.51981i q^{95} +19.1778i q^{97} -2.26057 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{9}+O(q^{10}) \) 16 * q - 16 * q^9	\( 16 q - 16 q^{9} + 8 q^{11} + 16 q^{25} - 24 q^{35} + 8 q^{43} - 8 q^{49} - 16 q^{57} + 40 q^{67} + 16 q^{81} + 56 q^{91} - 8 q^{99}+O(q^{100}) \) 16 * q - 16 * q^9 + 8 * q^11 + 16 * q^25 - 24 * q^35 + 8 * q^43 - 8 * q^49 - 16 * q^57 + 40 * q^67 + 16 * q^81 + 56 * q^91 - 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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