LMFDB - Embedded newform 3360.2.d.a.2911.10

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3360,2,Mod(2911,3360)] 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("3360.2911"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3360, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 0, 0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3360 = 2^{5} \cdot 3 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3360.d (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,-16,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$26.8297350792$
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} + 24x^{14} + 238x^{12} + 1262x^{10} + 3861x^{8} + 6834x^{6} + 6589x^{4} + 2916x^{2} + 324 $ x^16 + 24x^14 + 238x^12 + 1262x^10 + 3861x^8 + 6834x^6 + 6589x^4 + 2916*x^2 + 324
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{14} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2911.10
Root			$1.95785i$ of defining polynomial
Character	$\chi$	$=$	3360.2911
Dual form			3360.2.d.a.2911.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +1.00000i q^{5} +(-1.95785 + 1.77956i) q^{7} +1.00000 q^{9} -0.0547787i q^{11} +6.64493i q^{13} -1.00000i q^{15} +7.35292i q^{17} +1.62072 q^{19} +(1.95785 - 1.77956i) q^{21} +1.66636i q^{23} -1.00000 q^{25} -1.00000 q^{27} +3.58889 q^{29} +5.02299 q^{31} +0.0547787i q^{33} +(-1.77956 - 1.95785i) q^{35} -5.91302 q^{37} -6.64493i q^{39} -3.00981i q^{41} -10.1493i q^{43} +1.00000i q^{45} -0.427413 q^{47} +(0.666361 - 6.96821i) q^{49} -7.35292i q^{51} -5.43410 q^{53} +0.0547787 q^{55} -1.62072 q^{57} +7.77369 q^{59} +10.5379i q^{61} +(-1.95785 + 1.77956i) q^{63} -6.64493 q^{65} -4.32681i q^{67} -1.66636i q^{69} +11.0089i q^{71} +10.4414i q^{73} +1.00000 q^{75} +(0.0974818 + 0.107249i) q^{77} +1.40079i q^{79} +1.00000 q^{81} -14.4307 q^{83} -7.35292 q^{85} -3.58889 q^{87} +5.35419i q^{89} +(-11.8250 - 13.0098i) q^{91} -5.02299 q^{93} +1.62072i q^{95} -1.94030i q^{97} -0.0547787i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 16 q^{3} + 16 q^{9} + 16 q^{19} - 16 q^{25} - 16 q^{27} + 8 q^{29} - 8 q^{31} - 8 q^{37} + 16 q^{47} - 16 q^{49} - 48 q^{53} + 8 q^{55} - 16 q^{57} + 16 q^{59} - 8 q^{65} + 16 q^{75} + 16 q^{77}+ \cdots + 8 q^{93}+O(q^{100}) $ 16 * q - 16 * q^3 + 16 * q^9 + 16 * q^19 - 16 * q^25 - 16 * q^27 + 8 * q^29 - 8 * q^31 - 8 * q^37 + 16 * q^47 - 16 * q^49 - 48 * q^53 + 8 * q^55 - 16 * q^57 + 16 * q^59 - 8 * q^65 + 16 * q^75 + 16 * q^77 + 16 * q^81 - 24 * q^83 + 8 * q^85 - 8 * q^87 + 24 * q^91 + 8 * q^93

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3360\mathbb{Z}\right)^\times$.

$n$	$421$	$1121$	$1471$	$1921$	$2017$
$\chi(n)$	$1$	$1$	$-1$	$-1$	$1$

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
$3$	−1.00000			−0.577350
$3$	−1.00000			−0.577350
$4$		0			0
$5$			1.00000i			0.447214i
$5$			1.00000i			0.447214i
$6$		0			0
$7$	−1.95785	+	1.77956i	−0.739998	+	0.672609i
$7$	−1.95785	+	1.77956i	−0.739998	+	0.672609i
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$		−	0.0547787i		−	0.0165164i	−0.999966	−	0.00825820i	$-0.997371\pi$
$11$		−	0.0547787i		−	0.0165164i	0.999966	−	0.00825820i	$-0.00262870\pi$
$12$		0			0
$13$			6.64493i			1.84297i	0.388413	+	0.921486i	$0.373023\pi$
$13$			6.64493i			1.84297i	−0.388413	+	0.921486i	$0.626977\pi$
$14$		0			0
$15$		−	1.00000i		−	0.258199i
$16$		0			0
$17$			7.35292i			1.78335i	0.452681	+	0.891673i	$0.350468\pi$
$17$			7.35292i			1.78335i	−0.452681	+	0.891673i	$0.649532\pi$
$18$		0			0
$19$	1.62072			0.371819			0.185909	−	0.982567i	$-0.440477\pi$
$19$	1.62072			0.371819			0.185909	+	0.982567i	$0.440477\pi$
$20$		0			0
$21$	1.95785	−	1.77956i	0.427238	−	0.388331i
$22$		0			0
$23$			1.66636i			0.347460i	0.984793	+	0.173730i	$0.0555820\pi$
$23$			1.66636i			0.347460i	−0.984793	+	0.173730i	$0.944418\pi$
$24$		0			0
$25$	−1.00000			−0.200000
$26$		0			0
$27$	−1.00000			−0.192450
$28$		0			0
$29$	3.58889			0.666440			0.333220	−	0.942849i	$-0.391865\pi$
$29$	3.58889			0.666440			0.333220	+	0.942849i	$0.391865\pi$
$30$		0			0
$31$	5.02299			0.902156			0.451078	−	0.892485i	$-0.351040\pi$
$31$	5.02299			0.902156			0.451078	+	0.892485i	$0.351040\pi$
$32$		0			0
$33$			0.0547787i			0.00953575i
$34$		0			0
$35$	−1.77956	−	1.95785i	−0.300800	−	0.330937i
$36$		0			0
$37$	−5.91302			−0.972095			−0.486048	−	0.873932i	$-0.661562\pi$
$37$	−5.91302			−0.972095			−0.486048	+	0.873932i	$0.661562\pi$
$38$		0			0
$39$		−	6.64493i		−	1.06404i
$40$		0			0
$41$		−	3.00981i		−	0.470053i	−0.971989	−	0.235026i	$-0.924482\pi$
$41$		−	3.00981i		−	0.470053i	0.971989	−	0.235026i	$-0.0755176\pi$
$42$		0			0
$43$		−	10.1493i		−	1.54775i	−0.633340	−	0.773874i	$-0.718316\pi$
$43$		−	10.1493i		−	1.54775i	0.633340	−	0.773874i	$-0.281684\pi$
$44$		0			0
$45$			1.00000i			0.149071i
$46$		0			0
$47$	−0.427413			−0.0623447			−0.0311723	−	0.999514i	$-0.509924\pi$
$47$	−0.427413			−0.0623447			−0.0311723	+	0.999514i	$0.509924\pi$
$48$		0			0
$49$	0.666361	−	6.96821i	0.0951944	−	0.995459i
$50$		0			0
$51$		−	7.35292i		−	1.02961i
$52$		0			0
$53$	−5.43410			−0.746431			−0.373215	−	0.927745i	$-0.621745\pi$
$53$	−5.43410			−0.746431			−0.373215	+	0.927745i	$0.621745\pi$
$54$		0			0
$55$	0.0547787			0.00738636
$56$		0			0
$57$	−1.62072			−0.214670
$58$		0			0
$59$	7.77369			1.01205			0.506024	−	0.862519i	$-0.331115\pi$
$59$	7.77369			1.01205			0.506024	+	0.862519i	$0.331115\pi$
$60$		0			0
$61$			10.5379i			1.34924i	0.738163	+	0.674622i	$0.235694\pi$
$61$			10.5379i			1.34924i	−0.738163	+	0.674622i	$0.764306\pi$
$62$		0			0
$63$	−1.95785	+	1.77956i	−0.246666	+	0.224203i
$64$		0			0
$65$	−6.64493			−0.824202
$66$		0			0
$67$		−	4.32681i		−	0.528604i	−0.964440	−	0.264302i	$-0.914858\pi$
$67$		−	4.32681i		−	0.528604i	0.964440	−	0.264302i	$-0.0851416\pi$
$68$		0			0
$69$		−	1.66636i		−	0.200606i
$70$		0			0
$71$			11.0089i			1.30652i	0.757134	+	0.653260i	$0.226599\pi$
$71$			11.0089i			1.30652i	−0.757134	+	0.653260i	$0.773401\pi$
$72$		0			0
$73$			10.4414i			1.22207i	0.791602	+	0.611037i	$0.209248\pi$
$73$			10.4414i			1.22207i	−0.791602	+	0.611037i	$0.790752\pi$
$74$		0			0
$75$	1.00000			0.115470
$76$		0			0
$77$	0.0974818	+	0.107249i	0.0111091	+	0.0122221i
$78$		0			0
$79$			1.40079i			0.157601i	0.996890	+	0.0788007i	$0.0251091\pi$
$79$			1.40079i			0.157601i	−0.996890	+	0.0788007i	$0.974891\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$	−14.4307			−1.58397			−0.791987	−	0.610538i	$-0.790953\pi$
$83$	−14.4307			−1.58397			−0.791987	+	0.610538i	$0.790953\pi$
$84$		0			0
$85$	−7.35292			−0.797536
$86$		0			0
$87$	−3.58889			−0.384770
$88$		0			0
$89$			5.35419i			0.567543i	0.958892	+	0.283772i	$0.0915858\pi$
$89$			5.35419i			0.567543i	−0.958892	+	0.283772i	$0.908414\pi$
$90$		0			0
$91$	−11.8250	−	13.0098i	−1.23960	−	1.36380i
$92$		0			0
$93$	−5.02299			−0.520860
$94$		0			0
$95$			1.62072i			0.166282i
$96$		0			0
$97$		−	1.94030i		−	0.197007i	−0.995137	−	0.0985036i	$-0.968594\pi$
$97$		−	1.94030i		−	0.197007i	0.995137	−	0.0985036i	$-0.0314056\pi$
$98$		0			0
$99$		−	0.0547787i		−	0.00550547i
$100$		0			0
$101$		−	14.2561i		−	1.41854i	−0.704937	−	0.709269i	$-0.749025\pi$
$101$		−	14.2561i		−	1.41854i	0.704937	−	0.709269i	$-0.250975\pi$
$102$		0			0
$103$	−0.0842981			−0.00830614			−0.00415307	−	0.999991i	$-0.501322\pi$
$103$	−0.0842981			−0.00830614			−0.00415307	+	0.999991i	$0.501322\pi$
$104$		0			0
$105$	1.77956	+	1.95785i	0.173667	+	0.191067i
$106$		0			0
$107$			6.12527i			0.592153i	0.955164	+	0.296076i	$0.0956783\pi$
$107$			6.12527i			0.592153i	−0.955164	+	0.296076i	$0.904322\pi$
$108$		0			0
$109$	4.57858			0.438548			0.219274	−	0.975663i	$-0.429631\pi$
$109$	4.57858			0.438548			0.219274	+	0.975663i	$0.429631\pi$
$110$		0			0
$111$	5.91302			0.561239
$112$		0			0
$113$	8.90168			0.837399			0.418700	−	0.908125i	$-0.362486\pi$
$113$	8.90168			0.837399			0.418700	+	0.908125i	$0.362486\pi$
$114$		0			0
$115$	−1.66636			−0.155389
$116$		0			0
$117$			6.64493i			0.614324i
$118$		0			0
$119$	−13.0849	−	14.3959i	−1.19949	−	1.31967i
$120$		0			0
$121$	10.9970			0.999727
$122$		0			0
$123$			3.00981i			0.271385i
$124$		0			0
$125$		−	1.00000i		−	0.0894427i
$126$		0			0
$127$		−	6.98114i		−	0.619476i	−0.950822	−	0.309738i	$-0.899759\pi$
$127$		−	6.98114i		−	0.619476i	0.950822	−	0.309738i	$-0.100241\pi$
$128$		0			0
$129$			10.1493i			0.893593i
$130$		0			0
$131$	−19.5621			−1.70915			−0.854576	−	0.519326i	$-0.826183\pi$
$131$	−19.5621			−1.70915			−0.854576	+	0.519326i	$0.826183\pi$
$132$		0			0
$133$	−3.17313	+	2.88416i	−0.275145	+	0.250089i
$134$		0			0
$135$		−	1.00000i		−	0.0860663i
$136$		0			0
$137$	−3.50640			−0.299572			−0.149786	−	0.988718i	$-0.547858\pi$
$137$	−3.50640			−0.299572			−0.149786	+	0.988718i	$0.547858\pi$
$138$		0			0
$139$	−11.0184			−0.934571			−0.467286	−	0.884106i	$-0.654768\pi$
$139$	−11.0184			−0.934571			−0.467286	+	0.884106i	$0.654768\pi$
$140$		0			0
$141$	0.427413			0.0359947
$142$		0			0
$143$	0.364001			0.0304393
$144$		0			0
$145$			3.58889i			0.298041i
$146$		0			0
$147$	−0.666361	+	6.96821i	−0.0549605	+	0.574728i
$148$		0			0
$149$	−10.7737			−0.882618			−0.441309	−	0.897355i	$-0.645486\pi$
$149$	−10.7737			−0.882618			−0.441309	+	0.897355i	$0.645486\pi$
$150$		0			0
$151$			10.1954i			0.829690i	0.909892	+	0.414845i	$0.136164\pi$
$151$			10.1954i			0.829690i	−0.909892	+	0.414845i	$0.863836\pi$
$152$		0			0
$153$			7.35292i			0.594448i
$154$		0			0
$155$			5.02299i			0.403456i
$156$		0			0
$157$		−	12.1922i		−	0.973045i	−0.873668	−	0.486523i	$-0.838265\pi$
$157$		−	12.1922i		−	0.973045i	0.873668	−	0.486523i	$-0.161735\pi$
$158$		0			0
$159$	5.43410			0.430952
$160$		0			0
$161$	−2.96538	−	3.26249i	−0.233705	−	0.257120i
$162$		0			0
$163$			4.84283i			0.379320i	0.981850	+	0.189660i	$0.0607385\pi$
$163$			4.84283i			0.379320i	−0.981850	+	0.189660i	$0.939262\pi$
$164$		0			0
$165$	−0.0547787			−0.00426452
$166$		0			0
$167$	−1.57068			−0.121543			−0.0607716	−	0.998152i	$-0.519356\pi$
$167$	−1.57068			−0.121543			−0.0607716	+	0.998152i	$0.519356\pi$
$168$		0			0
$169$	−31.1550			−2.39654
$170$		0			0
$171$	1.62072			0.123940
$172$		0			0
$173$			10.7321i			0.815949i	0.912993	+	0.407974i	$0.133765\pi$
$173$			10.7321i			0.815949i	−0.912993	+	0.407974i	$0.866235\pi$
$174$		0			0
$175$	1.95785	−	1.77956i	0.148000	−	0.134522i
$176$		0			0
$177$	−7.77369			−0.584306
$178$		0			0
$179$		−	25.8788i		−	1.93428i	−0.254253	−	0.967138i	$-0.581829\pi$
$179$		−	25.8788i		−	1.93428i	0.254253	−	0.967138i	$-0.418171\pi$
$180$		0			0
$181$			11.6102i			0.862978i	0.902118	+	0.431489i	$0.142012\pi$
$181$			11.6102i			0.862978i	−0.902118	+	0.431489i	$0.857988\pi$
$182$		0			0
$183$		−	10.5379i		−	0.778987i
$184$		0			0
$185$		−	5.91302i		−	0.434734i
$186$		0			0
$187$	0.402784			0.0294545
$188$		0			0
$189$	1.95785	−	1.77956i	0.142413	−	0.129444i
$190$		0			0
$191$		−	4.94711i		−	0.357960i	−0.983853	−	0.178980i	$-0.942720\pi$
$191$		−	4.94711i		−	0.357960i	0.983853	−	0.178980i	$-0.0572798\pi$
$192$		0			0
$193$	9.22316			0.663898			0.331949	−	0.943297i	$-0.392294\pi$
$193$	9.22316			0.663898			0.331949	+	0.943297i	$0.392294\pi$
$194$		0			0
$195$	6.64493			0.475853
$196$		0			0
$197$	−0.892469			−0.0635858			−0.0317929	−	0.999494i	$-0.510122\pi$
$197$	−0.892469			−0.0635858			−0.0317929	+	0.999494i	$0.510122\pi$
$198$		0			0
$199$	1.34704			0.0954894			0.0477447	−	0.998860i	$-0.484797\pi$
$199$	1.34704			0.0954894			0.0477447	+	0.998860i	$0.484797\pi$
$200$		0			0
$201$			4.32681i			0.305190i
$202$		0			0
$203$	−7.02652	+	6.38663i	−0.493165	+	0.448254i
$204$		0			0
$205$	3.00981			0.210214
$206$		0			0
$207$			1.66636i			0.115820i
$208$		0			0
$209$		−	0.0887810i		−	0.00614111i
$210$		0			0
$211$		−	22.5220i		−	1.55048i	−0.631666	−	0.775241i	$-0.717628\pi$
$211$		−	22.5220i		−	1.55048i	0.631666	−	0.775241i	$-0.282372\pi$
$212$		0			0
$213$		−	11.0089i		−	0.754319i
$214$		0			0
$215$	10.1493			0.692174
$216$		0			0
$217$	−9.83427	+	8.93869i	−0.667593	+	0.606798i
$218$		0			0
$219$		−	10.4414i		−	0.705565i
$220$		0			0
$221$	−48.8596			−3.28665
$222$		0			0
$223$	24.7362			1.65646			0.828228	−	0.560391i	$-0.189349\pi$
$223$	24.7362			1.65646			0.828228	+	0.560391i	$0.189349\pi$
$224$		0			0
$225$	−1.00000			−0.0666667
$226$		0			0
$227$	−22.4528			−1.49025			−0.745123	−	0.666927i	$-0.767609\pi$
$227$	−22.4528			−1.49025			−0.745123	+	0.666927i	$0.767609\pi$
$228$		0			0
$229$			1.32608i			0.0876296i	0.999040	+	0.0438148i	$0.0139511\pi$
$229$			1.32608i			0.0876296i	−0.999040	+	0.0438148i	$0.986049\pi$
$230$		0			0
$231$	−0.0974818	−	0.107249i	−0.00641383	−	0.00705644i
$232$		0			0
$233$	−18.5339			−1.21419			−0.607097	−	0.794628i	$-0.707666\pi$
$233$	−18.5339			−1.21419			−0.607097	+	0.794628i	$0.707666\pi$
$234$		0			0
$235$		−	0.427413i		−	0.0278814i
$236$		0			0
$237$		−	1.40079i		−	0.0909912i
$238$		0			0
$239$		−	1.18656i		−	0.0767520i	−0.999263	−	0.0383760i	$-0.987782\pi$
$239$		−	1.18656i		−	0.0767520i	0.999263	−	0.0383760i	$-0.0122185\pi$
$240$		0			0
$241$			15.6276i			1.00667i	0.864093	+	0.503333i	$0.167893\pi$
$241$			15.6276i			1.00667i	−0.864093	+	0.503333i	$0.832107\pi$
$242$		0			0
$243$	−1.00000			−0.0641500
$244$		0			0
$245$	6.96821	+	0.666361i	0.445183	+	0.0425722i
$246$		0			0
$247$			10.7696i			0.685251i
$248$		0			0
$249$	14.4307			0.914508
$250$		0			0
$251$	−6.44794			−0.406990			−0.203495	−	0.979076i	$-0.565230\pi$
$251$	−6.44794			−0.406990			−0.203495	+	0.979076i	$0.565230\pi$
$252$		0			0
$253$	0.0912811			0.00573879
$254$		0			0
$255$	7.35292			0.460458
$256$		0			0
$257$		−	9.67705i		−	0.603638i	−0.953365	−	0.301819i	$-0.902406\pi$
$257$		−	9.67705i		−	0.603638i	0.953365	−	0.301819i	$-0.0975938\pi$
$258$		0			0
$259$	11.5768	−	10.5226i	0.719349	−	0.653840i
$260$		0			0
$261$	3.58889			0.222147
$262$		0			0
$263$			18.1405i			1.11859i	0.828968	+	0.559296i	$0.188929\pi$
$263$			18.1405i			1.11859i	−0.828968	+	0.559296i	$0.811071\pi$
$264$		0			0
$265$		−	5.43410i		−	0.333814i
$266$		0			0
$267$		−	5.35419i		−	0.327671i
$268$		0			0
$269$		−	20.3092i		−	1.23827i	−0.785284	−	0.619136i	$-0.787483\pi$
$269$		−	20.3092i		−	1.23827i	0.785284	−	0.619136i	$-0.212517\pi$
$270$		0			0
$271$	−18.0333			−1.09544			−0.547722	−	0.836661i	$-0.684505\pi$
$271$	−18.0333			−1.09544			−0.547722	+	0.836661i	$0.684505\pi$
$272$		0			0
$273$	11.8250	+	13.0098i	0.715683	+	0.787387i
$274$		0			0
$275$			0.0547787i			0.00330328i
$276$		0			0
$277$	24.0475			1.44487			0.722437	−	0.691437i	$-0.243022\pi$
$277$	24.0475			1.44487			0.722437	+	0.691437i	$0.243022\pi$
$278$		0			0
$279$	5.02299			0.300719
$280$		0			0
$281$	21.0356			1.25488			0.627438	−	0.778667i	$-0.284104\pi$
$281$	21.0356			1.25488			0.627438	+	0.778667i	$0.284104\pi$
$282$		0			0
$283$	−9.42870			−0.560478			−0.280239	−	0.959930i	$-0.590414\pi$
$283$	−9.42870			−0.560478			−0.280239	+	0.959930i	$0.590414\pi$
$284$		0			0
$285$		−	1.62072i		−	0.0960032i
$286$		0			0
$287$	5.35612	+	5.89275i	0.316162	+	0.347838i
$288$		0			0
$289$	−37.0655			−2.18032
$290$		0			0
$291$			1.94030i			0.113742i
$292$		0			0
$293$			9.13240i			0.533521i	0.963763	+	0.266760i	$0.0859532\pi$
$293$			9.13240i			0.533521i	−0.963763	+	0.266760i	$0.914047\pi$
$294$		0			0
$295$			7.77369i			0.452602i
$296$		0			0
$297$			0.0547787i			0.00317858i
$298$		0			0
$299$	−11.0728			−0.640359
$300$		0			0
$301$	18.0612	+	19.8707i	1.04103	+	1.14533i
$302$		0			0
$303$			14.2561i			0.818994i
$304$		0			0
$305$	−10.5379			−0.603401
$306$		0			0
$307$	8.81510			0.503105			0.251552	−	0.967844i	$-0.419059\pi$
$307$	8.81510			0.503105			0.251552	+	0.967844i	$0.419059\pi$
$308$		0			0
$309$	0.0842981			0.00479555
$310$		0			0
$311$	−23.0061			−1.30456			−0.652279	−	0.757979i	$-0.726187\pi$
$311$	−23.0061			−1.30456			−0.652279	+	0.757979i	$0.726187\pi$
$312$		0			0
$313$			24.7012i			1.39619i	0.716004	+	0.698096i	$0.245969\pi$
$313$			24.7012i			1.39619i	−0.716004	+	0.698096i	$0.754031\pi$
$314$		0			0
$315$	−1.77956	−	1.95785i	−0.100267	−	0.110312i
$316$		0			0
$317$	−23.4990			−1.31984			−0.659918	−	0.751337i	$-0.729409\pi$
$317$	−23.4990			−1.31984			−0.659918	+	0.751337i	$0.729409\pi$
$318$		0			0
$319$		−	0.196595i		−	0.0110072i
$320$		0			0
$321$		−	6.12527i		−	0.341879i
$322$		0			0
$323$			11.9170i			0.663081i
$324$		0			0
$325$		−	6.64493i		−	0.368594i
$326$		0			0
$327$	−4.57858			−0.253196
$328$		0			0
$329$	0.836812	−	0.760606i	0.0461349	−	0.0419336i
$330$		0			0
$331$			9.44377i			0.519077i	0.965733	+	0.259538i	$0.0835704\pi$
$331$			9.44377i			0.519077i	−0.965733	+	0.259538i	$0.916430\pi$
$332$		0			0
$333$	−5.91302			−0.324032
$334$		0			0
$335$	4.32681			0.236399
$336$		0			0
$337$	31.9066			1.73806			0.869030	−	0.494759i	$-0.164744\pi$
$337$	31.9066			1.73806			0.869030	+	0.494759i	$0.164744\pi$
$338$		0			0
$339$	−8.90168			−0.483473
$340$		0			0
$341$		−	0.275153i		−	0.0149004i
$342$		0			0
$343$	11.0957	+	14.8285i	0.599111	+	0.800666i
$344$		0			0
$345$	1.66636			0.0897138
$346$		0			0
$347$		−	17.2632i		−	0.926737i	−0.886166	−	0.463368i	$-0.846641\pi$
$347$		−	17.2632i		−	0.926737i	0.886166	−	0.463368i	$-0.153359\pi$
$348$		0			0
$349$		−	0.377047i		−	0.0201829i	−0.999949	−	0.0100914i	$-0.996788\pi$
$349$		−	0.377047i		−	0.0201829i	0.999949	−	0.0100914i	$-0.00321226\pi$
$350$		0			0
$351$		−	6.64493i		−	0.354680i
$352$		0			0
$353$		−	27.8904i		−	1.48446i	−0.670148	−	0.742228i	$-0.733769\pi$
$353$		−	27.8904i		−	1.48446i	0.670148	−	0.742228i	$-0.266231\pi$
$354$		0			0
$355$	−11.0089			−0.584293
$356$		0			0
$357$	13.0849	+	14.3959i	0.692528	+	0.761913i
$358$		0			0
$359$			30.9526i			1.63362i	0.576909	+	0.816809i	$0.304259\pi$
$359$			30.9526i			1.63362i	−0.576909	+	0.816809i	$0.695741\pi$
$360$		0			0
$361$	−16.3733			−0.861751
$362$		0			0
$363$	−10.9970			−0.577193
$364$		0			0
$365$	−10.4414			−0.546529
$366$		0			0
$367$	3.65582			0.190832			0.0954162	−	0.995437i	$-0.469582\pi$
$367$	3.65582			0.190832			0.0954162	+	0.995437i	$0.469582\pi$
$368$		0			0
$369$		−	3.00981i		−	0.156684i
$370$		0			0
$371$	10.6392	−	9.67028i	0.552357	−	0.502056i
$372$		0			0
$373$	35.2205			1.82365			0.911824	−	0.410580i	$-0.134674\pi$
$373$	35.2205			1.82365			0.911824	+	0.410580i	$0.134674\pi$
$374$		0			0
$375$			1.00000i			0.0516398i
$376$		0			0
$377$			23.8479i			1.22823i
$378$		0			0
$379$		−	17.0902i		−	0.877864i	−0.898520	−	0.438932i	$-0.855357\pi$
$379$		−	17.0902i		−	0.877864i	0.898520	−	0.438932i	$-0.144643\pi$
$380$		0			0
$381$			6.98114i			0.357654i
$382$		0			0
$383$	−10.3006			−0.526338			−0.263169	−	0.964750i	$-0.584768\pi$
$383$	−10.3006			−0.526338			−0.263169	+	0.964750i	$0.584768\pi$
$384$		0			0
$385$	−0.107249	+	0.0974818i	−0.00546589	+	0.00496813i
$386$		0			0
$387$		−	10.1493i		−	0.515916i
$388$		0			0
$389$	−11.0019			−0.557818			−0.278909	−	0.960318i	$-0.589973\pi$
$389$	−11.0019			−0.557818			−0.278909	+	0.960318i	$0.589973\pi$
$390$		0			0
$391$	−12.2526			−0.619642
$392$		0			0
$393$	19.5621			0.986780
$394$		0			0
$395$	−1.40079			−0.0704815
$396$		0			0
$397$		−	15.7177i		−	0.788848i	−0.918928	−	0.394424i	$-0.870944\pi$
$397$		−	15.7177i		−	0.788848i	0.918928	−	0.394424i	$-0.129056\pi$
$398$		0			0
$399$	3.17313	−	2.88416i	0.158855	−	0.144389i
$400$		0			0
$401$	18.0169			0.899721			0.449860	−	0.893099i	$-0.351474\pi$
$401$	18.0169			0.899721			0.449860	+	0.893099i	$0.351474\pi$
$402$		0			0
$403$			33.3774i			1.66265i
$404$		0			0
$405$			1.00000i			0.0496904i
$406$		0			0
$407$			0.323908i			0.0160555i
$408$		0			0
$409$		−	33.9737i		−	1.67989i	−0.542672	−	0.839944i	$-0.682587\pi$
$409$		−	33.9737i		−	1.67989i	0.542672	−	0.839944i	$-0.317413\pi$
$410$		0			0
$411$	3.50640			0.172958
$412$		0			0
$413$	−15.2197	+	13.8337i	−0.748914	+	0.680712i
$414$		0			0
$415$		−	14.4307i		−	0.708375i
$416$		0			0
$417$	11.0184			0.539575
$418$		0			0
$419$	7.48988			0.365904			0.182952	−	0.983122i	$-0.441435\pi$
$419$	7.48988			0.365904			0.182952	+	0.983122i	$0.441435\pi$
$420$		0			0
$421$	37.6198			1.83348			0.916738	−	0.399488i	$-0.130812\pi$
$421$	37.6198			1.83348			0.916738	+	0.399488i	$0.130812\pi$
$422$		0			0
$423$	−0.427413			−0.0207816
$424$		0			0
$425$		−	7.35292i		−	0.356669i
$426$		0			0
$427$	−18.7529	−	20.6317i	−0.907514	−	0.998439i
$428$		0			0
$429$	−0.364001			−0.0175741
$430$		0			0
$431$			28.9124i			1.39266i	0.717722	+	0.696330i	$0.245185\pi$
$431$			28.9124i			1.39266i	−0.717722	+	0.696330i	$0.754815\pi$
$432$		0			0
$433$		−	15.0641i		−	0.723934i	−0.932191	−	0.361967i	$-0.882105\pi$
$433$		−	15.0641i		−	0.723934i	0.932191	−	0.361967i	$-0.117895\pi$
$434$		0			0
$435$		−	3.58889i		−	0.172074i
$436$		0			0
$437$			2.70070i			0.129192i
$438$		0			0
$439$	36.0767			1.72185			0.860924	−	0.508734i	$-0.169886\pi$
$439$	36.0767			1.72185			0.860924	+	0.508734i	$0.169886\pi$
$440$		0			0
$441$	0.666361	−	6.96821i	0.0317315	−	0.331820i
$442$		0			0
$443$			32.6080i			1.54925i	0.632420	+	0.774626i	$0.282062\pi$
$443$			32.6080i			1.54925i	−0.632420	+	0.774626i	$0.717938\pi$
$444$		0			0
$445$	−5.35419			−0.253813
$446$		0			0
$447$	10.7737			0.509580
$448$		0			0
$449$	−28.4627			−1.34324			−0.671620	−	0.740896i	$-0.734401\pi$
$449$	−28.4627			−1.34324			−0.671620	+	0.740896i	$0.734401\pi$
$450$		0			0
$451$	−0.164873			−0.00776358
$452$		0			0
$453$		−	10.1954i		−	0.479022i
$454$		0			0
$455$	13.0098	−	11.8250i	0.609908	−	0.554365i
$456$		0			0
$457$	−17.5368			−0.820336			−0.410168	−	0.912010i	$-0.634530\pi$
$457$	−17.5368			−0.820336			−0.410168	+	0.912010i	$0.634530\pi$
$458$		0			0
$459$		−	7.35292i		−	0.343205i
$460$		0			0
$461$			11.5411i			0.537520i	0.963207	+	0.268760i	$0.0866139\pi$
$461$			11.5411i			0.537520i	−0.963207	+	0.268760i	$0.913386\pi$
$462$		0			0
$463$		−	15.2442i		−	0.708460i	−0.935158	−	0.354230i	$-0.884743\pi$
$463$		−	15.2442i		−	0.708460i	0.935158	−	0.354230i	$-0.115257\pi$
$464$		0			0
$465$		−	5.02299i		−	0.232936i
$466$		0			0
$467$	−18.9478			−0.876799			−0.438400	−	0.898780i	$-0.644455\pi$
$467$	−18.9478			−0.876799			−0.438400	+	0.898780i	$0.644455\pi$
$468$		0			0
$469$	7.69980	+	8.47125i	0.355544	+	0.391166i
$470$		0			0
$471$			12.1922i			0.561788i
$472$		0			0
$473$	−0.555963			−0.0255632
$474$		0			0
$475$	−1.62072			−0.0743637
$476$		0			0
$477$	−5.43410			−0.248810
$478$		0			0
$479$	35.9630			1.64319			0.821596	−	0.570070i	$-0.193084\pi$
$479$	35.9630			1.64319			0.821596	+	0.570070i	$0.193084\pi$
$480$		0			0
$481$		−	39.2916i		−	1.79154i
$482$		0			0
$483$	2.96538	+	3.26249i	0.134930	+	0.148448i
$484$		0			0
$485$	1.94030			0.0881043
$486$		0			0
$487$		−	27.1574i		−	1.23062i	−0.788286	−	0.615309i	$-0.789031\pi$
$487$		−	27.1574i		−	1.23062i	0.788286	−	0.615309i	$-0.210969\pi$
$488$		0			0
$489$		−	4.84283i		−	0.219000i
$490$		0			0
$491$			7.18909i			0.324439i	0.986755	+	0.162220i	$0.0518653\pi$
$491$			7.18909i			0.324439i	−0.986755	+	0.162220i	$0.948135\pi$
$492$		0			0
$493$			26.3888i			1.18849i
$494$		0			0
$495$	0.0547787			0.00246212
$496$		0			0
$497$	−19.5910	−	21.5538i	−0.878777	−	0.966822i
$498$		0			0
$499$			12.2704i			0.549298i	0.961544	+	0.274649i	$0.0885617\pi$
$499$			12.2704i			0.549298i	−0.961544	+	0.274649i	$0.911438\pi$
$500$		0			0
$501$	1.57068			0.0701730
$502$		0			0
$503$	−1.52736			−0.0681018			−0.0340509	−	0.999420i	$-0.510841\pi$
$503$	−1.52736			−0.0681018			−0.0340509	+	0.999420i	$0.510841\pi$
$504$		0			0
$505$	14.2561			0.634390
$506$		0			0
$507$	31.1550			1.38364
$508$		0			0
$509$		−	34.5490i		−	1.53136i	−0.643223	−	0.765679i	$-0.722403\pi$
$509$		−	34.5490i		−	1.53136i	0.643223	−	0.765679i	$-0.277597\pi$
$510$		0			0
$511$	−18.5811	−	20.4427i	−0.821979	−	0.904333i
$512$		0			0
$513$	−1.62072			−0.0715565
$514$		0			0
$515$		−	0.0842981i		−	0.00371462i
$516$		0			0
$517$			0.0234132i			0.00102971i
$518$		0			0
$519$		−	10.7321i		−	0.471088i
$520$		0			0
$521$		−	0.668573i		−	0.0292907i	−0.999893	−	0.0146454i	$-0.995338\pi$
$521$		−	0.668573i		−	0.0292907i	0.999893	−	0.0146454i	$-0.00466193\pi$
$522$		0			0
$523$	−28.4949			−1.24599			−0.622997	−	0.782224i	$-0.714085\pi$
$523$	−28.4949			−1.24599			−0.622997	+	0.782224i	$0.714085\pi$
$524$		0			0
$525$	−1.95785	+	1.77956i	−0.0854476	+	0.0776662i
$526$		0			0
$527$			36.9336i			1.60885i
$528$		0			0
$529$	20.2232			0.879271
$530$		0			0
$531$	7.77369			0.337349
$532$		0			0
$533$	19.9999			0.866294
$534$		0			0
$535$	−6.12527			−0.264819
$536$		0			0
$537$			25.8788i			1.11675i
$538$		0			0
$539$	−0.381710	−	0.0365024i	−0.0164414	−	0.00157227i
$540$		0			0
$541$	−21.7710			−0.936007			−0.468004	−	0.883727i	$-0.655027\pi$
$541$	−21.7710			−0.936007			−0.468004	+	0.883727i	$0.655027\pi$
$542$		0			0
$543$		−	11.6102i		−	0.498241i
$544$		0			0
$545$			4.57858i			0.196125i
$546$		0			0
$547$		−	40.8185i		−	1.74527i	−0.488369	−	0.872637i	$-0.662408\pi$
$547$		−	40.8185i		−	1.74527i	0.488369	−	0.872637i	$-0.337592\pi$
$548$		0			0
$549$			10.5379i			0.449748i
$550$		0			0
$551$	5.81659			0.247795
$552$		0			0
$553$	−2.49279	−	2.74254i	−0.106004	−	0.116625i
$554$		0			0
$555$			5.91302i			0.250994i
$556$		0			0
$557$	13.0790			0.554175			0.277087	−	0.960845i	$-0.410631\pi$
$557$	13.0790			0.554175			0.277087	+	0.960845i	$0.410631\pi$
$558$		0			0
$559$	67.4411			2.85245
$560$		0			0
$561$	−0.402784			−0.0170055
$562$		0			0
$563$	−10.5812			−0.445945			−0.222972	−	0.974825i	$-0.571576\pi$
$563$	−10.5812			−0.445945			−0.222972	+	0.974825i	$0.571576\pi$
$564$		0			0
$565$			8.90168i			0.374496i
$566$		0			0
$567$	−1.95785	+	1.77956i	−0.0822220	+	0.0747343i
$568$		0			0
$569$	−37.1235			−1.55630			−0.778149	−	0.628079i	$-0.783841\pi$
$569$	−37.1235			−1.55630			−0.778149	+	0.628079i	$0.783841\pi$
$570$		0			0
$571$			39.1766i			1.63949i	0.572730	+	0.819744i	$0.305884\pi$
$571$			39.1766i			1.63949i	−0.572730	+	0.819744i	$0.694116\pi$
$572$		0			0
$573$			4.94711i			0.206668i
$574$		0			0
$575$		−	1.66636i		−	0.0694920i
$576$		0			0
$577$		−	38.0773i		−	1.58518i	−0.609757	−	0.792589i	$-0.708733\pi$
$577$		−	38.0773i		−	1.58518i	0.609757	−	0.792589i	$-0.291267\pi$
$578$		0			0
$579$	−9.22316			−0.383302
$580$		0			0
$581$	28.2531	−	25.6802i	1.17214	−	1.06540i
$582$		0			0
$583$			0.297673i			0.0123284i
$584$		0			0
$585$	−6.64493			−0.274734
$586$		0			0
$587$	−21.0591			−0.869204			−0.434602	−	0.900623i	$-0.643111\pi$
$587$	−21.0591			−0.869204			−0.434602	+	0.900623i	$0.643111\pi$
$588$		0			0
$589$	8.14086			0.335438
$590$		0			0
$591$	0.892469			0.0367113
$592$		0			0
$593$			18.6425i			0.765555i	0.923841	+	0.382777i	$0.125032\pi$
$593$			18.6425i			0.765555i	−0.923841	+	0.382777i	$0.874968\pi$
$594$		0			0
$595$	14.3959	−	13.0849i	0.590175	−	0.536430i
$596$		0			0
$597$	−1.34704			−0.0551309
$598$		0			0
$599$			6.32820i			0.258563i	0.991608	+	0.129282i	$0.0412671\pi$
$599$			6.32820i			0.258563i	−0.991608	+	0.129282i	$0.958733\pi$
$600$		0			0
$601$			1.87173i			0.0763493i	0.999271	+	0.0381746i	$0.0121543\pi$
$601$			1.87173i			0.0763493i	−0.999271	+	0.0381746i	$0.987846\pi$
$602$		0			0
$603$		−	4.32681i		−	0.176201i
$604$		0			0
$605$			10.9970i			0.447092i
$606$		0			0
$607$	−13.8087			−0.560476			−0.280238	−	0.959931i	$-0.590413\pi$
$607$	−13.8087			−0.560476			−0.280238	+	0.959931i	$0.590413\pi$
$608$		0			0
$609$	7.02652	−	6.38663i	0.284729	−	0.258799i
$610$		0			0
$611$		−	2.84013i		−	0.114899i
$612$		0			0
$613$	−6.46613			−0.261164			−0.130582	−	0.991437i	$-0.541685\pi$
$613$	−6.46613			−0.261164			−0.130582	+	0.991437i	$0.541685\pi$
$614$		0			0
$615$	−3.00981			−0.121367
$616$		0			0
$617$	14.5896			0.587356			0.293678	−	0.955904i	$-0.405121\pi$
$617$	14.5896			0.587356			0.293678	+	0.955904i	$0.405121\pi$
$618$		0			0
$619$	27.2821			1.09656			0.548280	−	0.836295i	$-0.315283\pi$
$619$	27.2821			1.09656			0.548280	+	0.836295i	$0.315283\pi$
$620$		0			0
$621$		−	1.66636i		−	0.0668687i
$622$		0			0
$623$	−9.52809	−	10.4827i	−0.381735	−	0.419981i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$			0.0887810i			0.00354557i
$628$		0			0
$629$		−	43.4780i		−	1.73358i
$630$		0			0
$631$			33.6743i			1.34055i	0.742112	+	0.670276i	$0.233824\pi$
$631$			33.6743i			1.34055i	−0.742112	+	0.670276i	$0.766176\pi$
$632$		0			0
$633$			22.5220i			0.895171i
$634$		0			0
$635$	6.98114			0.277038
$636$		0			0
$637$	46.3032	+	4.42792i	1.83460	+	0.175440i
$638$		0			0
$639$			11.0089i			0.435506i
$640$		0			0
$641$	31.3470			1.23813			0.619066	−	0.785339i	$-0.287511\pi$
$641$	31.3470			1.23813			0.619066	+	0.785339i	$0.287511\pi$
$642$		0			0
$643$	2.88031			0.113588			0.0567942	−	0.998386i	$-0.481912\pi$
$643$	2.88031			0.113588			0.0567942	+	0.998386i	$0.481912\pi$
$644$		0			0
$645$	−10.1493			−0.399627
$646$		0			0
$647$	25.5739			1.00542			0.502708	−	0.864456i	$-0.332337\pi$
$647$	25.5739			1.00542			0.502708	+	0.864456i	$0.332337\pi$
$648$		0			0
$649$		−	0.425833i		−	0.0167154i
$650$		0			0
$651$	9.83427	−	8.93869i	0.385435	−	0.350335i
$652$		0			0
$653$	20.4799			0.801442			0.400721	−	0.916200i	$-0.368760\pi$
$653$	20.4799			0.801442			0.400721	+	0.916200i	$0.368760\pi$
$654$		0			0
$655$		−	19.5621i		−	0.764356i
$656$		0			0
$657$			10.4414i			0.407358i
$658$		0			0
$659$			3.35843i			0.130826i	0.997858	+	0.0654129i	$0.0208365\pi$
$659$			3.35843i			0.130826i	−0.997858	+	0.0654129i	$0.979164\pi$
$660$		0			0
$661$			28.4284i			1.10574i	0.833269	+	0.552868i	$0.186467\pi$
$661$			28.4284i			1.10574i	−0.833269	+	0.552868i	$0.813533\pi$
$662$		0			0
$663$	48.8596			1.89755
$664$		0			0
$665$	−2.88416	−	3.17313i	−0.111843	−	0.123049i
$666$		0			0
$667$			5.98039i			0.231562i
$668$		0			0
$669$	−24.7362			−0.956355
$670$		0			0
$671$	0.577255			0.0222847
$672$		0			0
$673$	23.6111			0.910141			0.455071	−	0.890455i	$-0.349614\pi$
$673$	23.6111			0.910141			0.455071	+	0.890455i	$0.349614\pi$
$674$		0			0
$675$	1.00000			0.0384900
$676$		0			0
$677$		−	22.5289i		−	0.865856i	−0.901429	−	0.432928i	$-0.857480\pi$
$677$		−	22.5289i		−	0.865856i	0.901429	−	0.432928i	$-0.142520\pi$
$678$		0			0
$679$	3.45287	+	3.79881i	0.132509	+	0.145785i
$680$		0			0
$681$	22.4528			0.860394
$682$		0			0
$683$			14.9074i			0.570416i	0.958466	+	0.285208i	$0.0920627\pi$
$683$			14.9074i			0.570416i	−0.958466	+	0.285208i	$0.907937\pi$
$684$		0			0
$685$		−	3.50640i		−	0.133973i
$686$		0			0
$687$		−	1.32608i		−	0.0505930i
$688$		0			0
$689$		−	36.1092i		−	1.37565i
$690$		0			0
$691$	51.0222			1.94098			0.970488	−	0.241150i	$-0.0775247\pi$
$691$	51.0222			1.94098			0.970488	+	0.241150i	$0.0775247\pi$
$692$		0			0
$693$	0.0974818	+	0.107249i	0.00370303	+	0.00407404i
$694$		0			0
$695$		−	11.0184i		−	0.417953i
$696$		0			0
$697$	22.1309			0.838266
$698$		0			0
$699$	18.5339			0.701015
$700$		0			0
$701$	18.4947			0.698536			0.349268	−	0.937023i	$-0.386430\pi$
$701$	18.4947			0.698536			0.349268	+	0.937023i	$0.386430\pi$
$702$		0			0
$703$	−9.58336			−0.361443
$704$		0			0
$705$			0.427413i			0.0160973i
$706$		0			0
$707$	25.3696	+	27.9114i	0.954122	+	1.04972i
$708$		0			0
$709$	−21.8260			−0.819692			−0.409846	−	0.912155i	$-0.634418\pi$
$709$	−21.8260			−0.819692			−0.409846	+	0.912155i	$0.634418\pi$
$710$		0			0
$711$			1.40079i			0.0525338i
$712$		0			0
$713$			8.37011i			0.313463i
$714$		0			0
$715$			0.364001i			0.0136129i
$716$		0			0
$717$			1.18656i			0.0443128i
$718$		0			0
$719$	−30.1254			−1.12349			−0.561744	−	0.827311i	$-0.689869\pi$
$719$	−30.1254			−1.12349			−0.561744	+	0.827311i	$0.689869\pi$
$720$		0			0
$721$	0.165043	−	0.150013i	0.00614653	−	0.00558678i
$722$		0			0
$723$		−	15.6276i		−	0.581198i
$724$		0			0
$725$	−3.58889			−0.133288
$726$		0			0
$727$	2.30337			0.0854272			0.0427136	−	0.999087i	$-0.486400\pi$
$727$	2.30337			0.0854272			0.0427136	+	0.999087i	$0.486400\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$	74.6267			2.76017
$732$		0			0
$733$			50.7568i			1.87475i	0.348326	+	0.937373i	$0.386750\pi$
$733$			50.7568i			1.87475i	−0.348326	+	0.937373i	$0.613250\pi$
$734$		0			0
$735$	−6.96821	−	0.666361i	−0.257026	−	0.0245791i
$736$		0			0
$737$	−0.237017			−0.00873064
$738$		0			0
$739$		−	23.5081i		−	0.864759i	−0.901692	−	0.432379i	$-0.857674\pi$
$739$		−	23.5081i		−	0.864759i	0.901692	−	0.432379i	$-0.142326\pi$
$740$		0			0
$741$		−	10.7696i		−	0.395630i
$742$		0			0
$743$		−	12.2540i		−	0.449555i	−0.974410	−	0.224777i	$-0.927835\pi$
$743$		−	12.2540i		−	0.449555i	0.974410	−	0.224777i	$-0.0721654\pi$
$744$		0			0
$745$		−	10.7737i		−	0.394719i
$746$		0			0
$747$	−14.4307			−0.527991
$748$		0			0
$749$	−10.9003	−	11.9924i	−0.398287	−	0.438192i
$750$		0			0
$751$			24.9341i			0.909860i	0.890527	+	0.454930i	$0.150336\pi$
$751$			24.9341i			0.909860i	−0.890527	+	0.454930i	$0.849664\pi$
$752$		0			0
$753$	6.44794			0.234976
$754$		0			0
$755$	−10.1954			−0.371049
$756$		0			0
$757$	27.0418			0.982852			0.491426	−	0.870919i	$-0.336476\pi$
$757$	27.0418			0.982852			0.491426	+	0.870919i	$0.336476\pi$
$758$		0			0
$759$	−0.0912811			−0.00331329
$760$		0			0
$761$			26.9591i			0.977268i	0.872489	+	0.488634i	$0.162505\pi$
$761$			26.9591i			0.977268i	−0.872489	+	0.488634i	$0.837495\pi$
$762$		0			0
$763$	−8.96417	+	8.14783i	−0.324525	+	0.294971i
$764$		0			0
$765$	−7.35292			−0.265845
$766$		0			0
$767$			51.6556i			1.86517i
$768$		0			0
$769$		−	4.20598i		−	0.151671i	−0.997120	−	0.0758357i	$-0.975838\pi$
$769$		−	4.20598i		−	0.151671i	0.997120	−	0.0758357i	$-0.0241625\pi$
$770$		0			0
$771$			9.67705i			0.348511i
$772$		0			0
$773$		−	32.2738i		−	1.16081i	−0.814328	−	0.580404i	$-0.802895\pi$
$773$		−	32.2738i		−	1.16081i	0.814328	−	0.580404i	$-0.197105\pi$
$774$		0			0
$775$	−5.02299			−0.180431
$776$		0			0
$777$	−11.5768	+	10.5226i	−0.415316	+	0.377495i
$778$		0			0
$779$		−	4.87805i		−	0.174774i
$780$		0			0
$781$	0.603055			0.0215790
$782$		0			0
$783$	−3.58889			−0.128257
$784$		0			0
$785$	12.1922			0.435159
$786$		0			0
$787$	51.9732			1.85265			0.926323	−	0.376731i	$-0.122952\pi$
$787$	51.9732			1.85265			0.926323	+	0.376731i	$0.122952\pi$
$788$		0			0
$789$		−	18.1405i		−	0.645820i
$790$		0			0
$791$	−17.4282	+	15.8410i	−0.619674	+	0.563242i
$792$		0			0
$793$	−70.0238			−2.48662
$794$		0			0
$795$			5.43410i			0.192728i
$796$		0			0
$797$		−	4.95467i		−	0.175504i	−0.996142	−	0.0877518i	$-0.972032\pi$
$797$		−	4.95467i		−	0.175504i	0.996142	−	0.0877518i	$-0.0279682\pi$
$798$		0			0
$799$		−	3.14274i		−	0.111182i
$800$		0			0
$801$			5.35419i			0.189181i
$802$		0			0
$803$	0.571967			0.0201843
$804$		0			0
$805$	3.26249	−	2.96538i	0.114988	−	0.104516i
$806$		0			0
$807$			20.3092i			0.714917i
$808$		0			0
$809$	12.9559			0.455504			0.227752	−	0.973719i	$-0.426862\pi$
$809$	12.9559			0.455504			0.227752	+	0.973719i	$0.426862\pi$
$810$		0			0
$811$	−25.2095			−0.885226			−0.442613	−	0.896713i	$-0.645948\pi$
$811$	−25.2095			−0.885226			−0.442613	+	0.896713i	$0.645948\pi$
$812$		0			0
$813$	18.0333			0.632455
$814$		0			0
$815$	−4.84283			−0.169637
$816$		0			0
$817$		−	16.4491i		−	0.575481i
$818$		0			0
$819$	−11.8250	−	13.0098i	−0.413200	−	0.454598i
$820$		0			0
$821$	−12.6458			−0.441340			−0.220670	−	0.975349i	$-0.570824\pi$
$821$	−12.6458			−0.441340			−0.220670	+	0.975349i	$0.570824\pi$
$822$		0			0
$823$			16.5072i			0.575406i	0.957720	+	0.287703i	$0.0928916\pi$
$823$			16.5072i			0.575406i	−0.957720	+	0.287703i	$0.907108\pi$
$824$		0			0
$825$		−	0.0547787i		−	0.00190715i
$826$		0			0
$827$		−	36.5249i		−	1.27010i	−0.772473	−	0.635048i	$-0.780980\pi$
$827$		−	36.5249i		−	1.27010i	0.772473	−	0.635048i	$-0.219020\pi$
$828$		0			0
$829$			56.6040i			1.96594i	0.183768	+	0.982970i	$0.441171\pi$
$829$			56.6040i			1.96594i	−0.183768	+	0.982970i	$0.558829\pi$
$830$		0			0
$831$	−24.0475			−0.834198
$832$		0			0
$833$	51.2367	+	4.89970i	1.77525	+	0.169764i
$834$		0			0
$835$		−	1.57068i		−	0.0543558i
$836$		0			0
$837$	−5.02299			−0.173620
$838$		0			0
$839$	−44.9455			−1.55169			−0.775846	−	0.630922i	$-0.782677\pi$
$839$	−44.9455			−1.55169			−0.775846	+	0.630922i	$0.782677\pi$
$840$		0			0
$841$	−16.1199			−0.555857
$842$		0			0
$843$	−21.0356			−0.724503
$844$		0			0
$845$		−	31.1550i		−	1.07177i
$846$		0			0
$847$	−21.5305	+	19.5698i	−0.739796	+	0.672425i
$848$		0			0
$849$	9.42870			0.323592
$850$		0			0
$851$		−	9.85323i		−	0.337764i
$852$		0			0
$853$		−	35.5340i		−	1.21666i	−0.793683	−	0.608331i	$-0.791839\pi$
$853$		−	35.5340i		−	1.21666i	0.793683	−	0.608331i	$-0.208161\pi$
$854$		0			0
$855$			1.62072i			0.0554275i
$856$		0			0
$857$			24.9839i			0.853433i	0.904385	+	0.426716i	$0.140330\pi$
$857$			24.9839i			0.853433i	−0.904385	+	0.426716i	$0.859670\pi$
$858$		0			0
$859$	55.6512			1.89879			0.949397	−	0.314079i	$-0.101696\pi$
$859$	55.6512			1.89879			0.949397	+	0.314079i	$0.101696\pi$
$860$		0			0
$861$	−5.35612	−	5.89275i	−0.182536	−	0.200824i
$862$		0			0
$863$			32.4473i			1.10452i	0.833673	+	0.552259i	$0.186234\pi$
$863$			32.4473i			1.10452i	−0.833673	+	0.552259i	$0.813766\pi$
$864$		0			0
$865$	−10.7321			−0.364903
$866$		0			0
$867$	37.0655			1.25881
$868$		0			0
$869$	0.0767336			0.00260301
$870$		0			0
$871$	28.7513			0.974202
$872$		0			0
$873$		−	1.94030i		−	0.0656691i
$874$		0			0
$875$	1.77956	+	1.95785i	0.0601600	+	0.0661874i
$876$		0			0
$877$	7.53109			0.254307			0.127153	−	0.991883i	$-0.459416\pi$
$877$	7.53109			0.254307			0.127153	+	0.991883i	$0.459416\pi$
$878$		0			0
$879$		−	9.13240i		−	0.308028i
$880$		0			0
$881$		−	32.1091i		−	1.08178i	−0.841092	−	0.540892i	$-0.818087\pi$
$881$		−	32.1091i		−	1.08178i	0.841092	−	0.540892i	$-0.181913\pi$
$882$		0			0
$883$			56.8788i			1.91413i	0.289881	+	0.957063i	$0.406384\pi$
$883$			56.8788i			1.91413i	−0.289881	+	0.957063i	$0.593616\pi$
$884$		0			0
$885$		−	7.77369i		−	0.261310i
$886$		0			0
$887$	6.14903			0.206464			0.103232	−	0.994657i	$-0.467082\pi$
$887$	6.14903			0.206464			0.103232	+	0.994657i	$0.467082\pi$
$888$		0			0
$889$	12.4233	+	13.6680i	0.416665	+	0.458411i
$890$		0			0
$891$		−	0.0547787i		−	0.00183516i
$892$		0			0
$893$	−0.692718			−0.0231809
$894$		0			0
$895$	25.8788			0.865034
$896$		0			0
$897$	11.0728			0.369711
$898$		0			0
$899$	18.0270			0.601233
$900$		0			0
$901$		−	39.9565i		−	1.33114i
$902$		0			0
$903$	−18.0612	−	19.8707i	−0.601038	−	0.661257i
$904$		0			0
$905$	−11.6102			−0.385936
$906$		0			0
$907$			10.1426i			0.336778i	0.985721	+	0.168389i	$0.0538565\pi$
$907$			10.1426i			0.336778i	−0.985721	+	0.168389i	$0.946144\pi$
$908$		0			0
$909$		−	14.2561i		−	0.472846i
$910$		0			0
$911$		−	6.33583i		−	0.209916i	−0.994477	−	0.104958i	$-0.966529\pi$
$911$		−	6.33583i		−	0.209916i	0.994477	−	0.104958i	$-0.0334707\pi$
$912$		0			0
$913$			0.790495i			0.0261616i
$914$		0			0
$915$	10.5379			0.348373
$916$		0			0
$917$	38.2998	−	34.8119i	1.26477	−	1.14959i
$918$		0			0
$919$		−	43.0635i		−	1.42054i	−0.703932	−	0.710268i	$-0.748574\pi$
$919$		−	43.0635i		−	1.42054i	0.703932	−	0.710268i	$-0.251426\pi$
$920$		0			0
$921$	−8.81510			−0.290468
$922$		0			0
$923$	−73.1535			−2.40788
$924$		0			0
$925$	5.91302			0.194419
$926$		0			0
$927$	−0.0842981			−0.00276871
$928$		0			0
$929$			22.9113i			0.751696i	0.926681	+	0.375848i	$0.122649\pi$
$929$			22.9113i			0.751696i	−0.926681	+	0.375848i	$0.877351\pi$
$930$		0			0
$931$	1.07998	−	11.2935i	0.0353950	−	0.370130i
$932$		0			0
$933$	23.0061			0.753187
$934$		0			0
$935$			0.402784i			0.0131724i
$936$		0			0
$937$		−	20.3954i		−	0.666288i	−0.942876	−	0.333144i	$-0.891891\pi$
$937$		−	20.3954i		−	0.666288i	0.942876	−	0.333144i	$-0.108109\pi$
$938$		0			0
$939$		−	24.7012i		−	0.806092i
$940$		0			0
$941$			16.7464i			0.545917i	0.962026	+	0.272958i	$0.0880021\pi$
$941$			16.7464i			0.545917i	−0.962026	+	0.272958i	$0.911998\pi$
$942$		0			0
$943$	5.01542			0.163325
$944$		0			0
$945$	1.77956	+	1.95785i	0.0578890	+	0.0636889i
$946$		0			0
$947$			50.9723i			1.65638i	0.560450	+	0.828188i	$0.310628\pi$
$947$			50.9723i			1.65638i	−0.560450	+	0.828188i	$0.689372\pi$
$948$		0			0
$949$	−69.3824			−2.25225
$950$		0			0
$951$	23.4990			0.762008
$952$		0			0
$953$	27.2791			0.883658			0.441829	−	0.897099i	$-0.354330\pi$
$953$	27.2791			0.883658			0.441829	+	0.897099i	$0.354330\pi$
$954$		0			0
$955$	4.94711			0.160085
$956$		0			0
$957$			0.196595i			0.00635501i
$958$		0			0
$959$	6.86501	−	6.23983i	0.221683	−	0.201495i
$960$		0			0
$961$	−5.76957			−0.186115
$962$		0			0
$963$			6.12527i			0.197384i
$964$		0			0
$965$			9.22316i			0.296904i
$966$		0			0
$967$			12.7562i			0.410212i	0.978740	+	0.205106i	$0.0657539\pi$
$967$			12.7562i			0.410212i	−0.978740	+	0.205106i	$0.934246\pi$
$968$		0			0
$969$		−	11.9170i		−	0.382830i
$970$		0			0
$971$	−36.4726			−1.17046			−0.585231	−	0.810867i	$-0.698996\pi$
$971$	−36.4726			−1.17046			−0.585231	+	0.810867i	$0.698996\pi$
$972$		0			0
$973$	21.5725	−	19.6079i	0.691581	−	0.628601i
$974$		0			0
$975$			6.64493i			0.212808i
$976$		0			0
$977$	−3.13940			−0.100438			−0.0502191	−	0.998738i	$-0.515992\pi$
$977$	−3.13940			−0.100438			−0.0502191	+	0.998738i	$0.515992\pi$
$978$		0			0
$979$	0.293296			0.00937378
$980$		0			0
$981$	4.57858			0.146183
$982$		0			0
$983$	1.37231			0.0437699			0.0218850	−	0.999760i	$-0.493033\pi$
$983$	1.37231			0.0437699			0.0218850	+	0.999760i	$0.493033\pi$
$984$		0			0
$985$		−	0.892469i		−	0.0284364i
$986$		0			0
$987$	−0.836812	+	0.760606i	−0.0266360	+	0.0242104i
$988$		0			0
$989$	16.9123			0.537781
$990$		0			0
$991$			13.9694i			0.443753i	0.975075	+	0.221876i	$0.0712182\pi$
$991$			13.9694i			0.443753i	−0.975075	+	0.221876i	$0.928782\pi$
$992$		0			0
$993$		−	9.44377i		−	0.299689i
$994$		0			0
$995$			1.34704i			0.0427042i
$996$		0			0
$997$			19.4384i			0.615621i	0.951448	+	0.307811i	$0.0995963\pi$
$997$			19.4384i			0.615621i	−0.951448	+	0.307811i	$0.900404\pi$
$998$		0			0
$999$	5.91302			0.187080

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3360.2.d.a.2911.10	yes	16
4.3	odd	2		3360.2.d.d.2911.15	yes	16
7.6	odd	2		3360.2.d.d.2911.7	yes	16
28.27	even	2	inner	3360.2.d.a.2911.2	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3360.2.d.a.2911.2	✓	16	28.27	even	2	inner
3360.2.d.a.2911.10	yes	16	1.1	even	1	trivial
3360.2.d.d.2911.7	yes	16	7.6	odd	2
3360.2.d.d.2911.15	yes	16	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3360 = 2^{5} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3360.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.00000i q^{5} +(-1.95785 + 1.77956i) q^{7} +1.00000 q^{9} -0.0547787i q^{11} +6.64493i q^{13} -1.00000i q^{15} +7.35292i q^{17} +1.62072 q^{19} +(1.95785 - 1.77956i) q^{21} +1.66636i q^{23} -1.00000 q^{25} -1.00000 q^{27} +3.58889 q^{29} +5.02299 q^{31} +0.0547787i q^{33} +(-1.77956 - 1.95785i) q^{35} -5.91302 q^{37} -6.64493i q^{39} -3.00981i q^{41} -10.1493i q^{43} +1.00000i q^{45} -0.427413 q^{47} +(0.666361 - 6.96821i) q^{49} -7.35292i q^{51} -5.43410 q^{53} +0.0547787 q^{55} -1.62072 q^{57} +7.77369 q^{59} +10.5379i q^{61} +(-1.95785 + 1.77956i) q^{63} -6.64493 q^{65} -4.32681i q^{67} -1.66636i q^{69} +11.0089i q^{71} +10.4414i q^{73} +1.00000 q^{75} +(0.0974818 + 0.107249i) q^{77} +1.40079i q^{79} +1.00000 q^{81} -14.4307 q^{83} -7.35292 q^{85} -3.58889 q^{87} +5.35419i q^{89} +(-11.8250 - 13.0098i) q^{91} -5.02299 q^{93} +1.62072i q^{95} -1.94030i q^{97} -0.0547787i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{3} + 16 q^{9} + 16 q^{19} - 16 q^{25} - 16 q^{27} + 8 q^{29} - 8 q^{31} - 8 q^{37} + 16 q^{47} - 16 q^{49} - 48 q^{53} + 8 q^{55} - 16 q^{57} + 16 q^{59} - 8 q^{65} + 16 q^{75} + 16 q^{77}+ \cdots + 8 q^{93}+O(q^{100}) \) 16 * q - 16 * q^3 + 16 * q^9 + 16 * q^19 - 16 * q^25 - 16 * q^27 + 8 * q^29 - 8 * q^31 - 8 * q^37 + 16 * q^47 - 16 * q^49 - 48 * q^53 + 8 * q^55 - 16 * q^57 + 16 * q^59 - 8 * q^65 + 16 * q^75 + 16 * q^77 + 16 * q^81 - 24 * q^83 + 8 * q^85 - 8 * q^87 + 24 * q^91 + 8 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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