LMFDB - Embedded newform 4140.2.f.c.829.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4140,2,Mod(829,4140)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4140, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4140.829");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4140.f (of order $2$, degree $1$, 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:	$33.0580664368$
Analytic rank:	$0$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} + 27x^{12} + 283x^{10} + 1441x^{8} + 3596x^{6} + 3740x^{4} + 772x^{2} + 4 $ x^14 + 27x^12 + 283x^10 + 1441x^8 + 3596x^6 + 3740x^4 + 772x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 1380)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			829.2
Root			$-1.52925i$ of defining polynomial
Character	$\chi$	$=$	4140.829
Dual form			4140.2.f.c.829.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.14859 + 0.619333i) q^{5} -1.66138i q^{7} +O(q^{10})$	$q+(-2.14859 + 0.619333i) q^{5} -1.66138i q^{7} +5.96040 q^{11} +3.02956i q^{13} +6.90402i q^{17} -6.93014 q^{19} -1.00000i q^{23} +(4.23285 - 2.66138i) q^{25} +7.66535 q^{29} -5.56664 q^{31} +(1.02895 + 3.56962i) q^{35} -6.17065i q^{37} -6.47918 q^{41} -3.84736i q^{43} -7.29956i q^{47} +4.23981 q^{49} +11.8349i q^{53} +(-12.8064 + 3.69147i) q^{55} -5.53671 q^{59} +1.81587 q^{61} +(-1.87631 - 6.50927i) q^{65} +9.32858i q^{67} -9.14110 q^{71} +8.35232i q^{73} -9.90251i q^{77} -3.89951 q^{79} +12.1427i q^{83} +(-4.27589 - 14.8339i) q^{85} +8.30087 q^{89} +5.03326 q^{91} +(14.8900 - 4.29206i) q^{95} -10.7485i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q+O(q^{10}) $ 14 * q	$ 14 q + 4 q^{19} - 6 q^{25} + 30 q^{29} + 6 q^{31} + 14 q^{35} - 46 q^{41} - 20 q^{49} - 16 q^{55} + 10 q^{59} + 64 q^{61} + 36 q^{65} - 42 q^{71} - 32 q^{79} - 42 q^{85} + 52 q^{89} + 28 q^{91} + 44 q^{95}+O(q^{100}) $ 14 * q + 4 * q^19 - 6 * q^25 + 30 * q^29 + 6 * q^31 + 14 * q^35 - 46 * q^41 - 20 * q^49 - 16 * q^55 + 10 * q^59 + 64 * q^61 + 36 * q^65 - 42 * q^71 - 32 * q^79 - 42 * q^85 + 52 * q^89 + 28 * q^91 + 44 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4140\mathbb{Z}\right)^\times$.

$n$	$461$	$1657$	$2071$	$3961$
$\chi(n)$	$1$	$-1$	$1$	$1$

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			0
$3$		0			0
$4$		0			0
$5$	−2.14859	+	0.619333i	−0.960877	+	0.276974i
$5$	−2.14859	+	0.619333i	−0.960877	+	0.276974i
$6$		0			0
$7$		−	1.66138i		−	0.627943i	−0.949432	−	0.313972i	$-0.898340\pi$
$7$		−	1.66138i		−	0.627943i	0.949432	−	0.313972i	$-0.101660\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	5.96040			1.79713			0.898565	−	0.438841i	$-0.144611\pi$
$11$	5.96040			1.79713			0.898565	+	0.438841i	$0.144611\pi$
$12$		0			0
$13$			3.02956i			0.840249i	0.907467	+	0.420124i	$0.138014\pi$
$13$			3.02956i			0.840249i	−0.907467	+	0.420124i	$0.861986\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			6.90402i			1.67447i	0.546843	+	0.837235i	$0.315829\pi$
$17$			6.90402i			1.67447i	−0.546843	+	0.837235i	$0.684171\pi$
$18$		0			0
$19$	−6.93014			−1.58988			−0.794942	−	0.606686i	$-0.792499\pi$
$19$	−6.93014			−1.58988			−0.794942	+	0.606686i	$0.792499\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	1.00000i		−	0.208514i
$23$		−	1.00000i		−	0.208514i
$24$		0			0
$25$	4.23285	−	2.66138i	0.846571	−	0.532276i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	7.66535			1.42342			0.711710	−	0.702473i	$-0.247921\pi$
$29$	7.66535			1.42342			0.711710	+	0.702473i	$0.247921\pi$
$30$		0			0
$31$	−5.56664			−0.999797			−0.499899	−	0.866084i	$-0.666629\pi$
$31$	−5.56664			−0.999797			−0.499899	+	0.866084i	$0.666629\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	1.02895	+	3.56962i	0.173924	+	0.603377i
$36$		0			0
$37$		−	6.17065i		−	1.01445i	−0.861814	−	0.507225i	$-0.830671\pi$
$37$		−	6.17065i		−	1.01445i	0.861814	−	0.507225i	$-0.169329\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−6.47918			−1.01188			−0.505939	−	0.862569i	$-0.668854\pi$
$41$	−6.47918			−1.01188			−0.505939	+	0.862569i	$0.668854\pi$
$42$		0			0
$43$		−	3.84736i		−	0.586717i	−0.956003	−	0.293358i	$-0.905227\pi$
$43$		−	3.84736i		−	0.586717i	0.956003	−	0.293358i	$-0.0947729\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	7.29956i		−	1.06475i	−0.846509	−	0.532375i	$-0.821300\pi$
$47$		−	7.29956i		−	1.06475i	0.846509	−	0.532375i	$-0.178700\pi$
$48$		0			0
$49$	4.23981			0.605687
$50$		0			0
$51$		0			0
$52$		0			0
$53$			11.8349i			1.62564i	0.582513	+	0.812822i	$0.302070\pi$
$53$			11.8349i			1.62564i	−0.582513	+	0.812822i	$0.697930\pi$
$54$		0			0
$55$	−12.8064	+	3.69147i	−1.72682	+	0.497758i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−5.53671			−0.720818			−0.360409	−	0.932794i	$-0.617363\pi$
$59$	−5.53671			−0.720818			−0.360409	+	0.932794i	$0.617363\pi$
$60$		0			0
$61$	1.81587			0.232499			0.116249	−	0.993220i	$-0.462913\pi$
$61$	1.81587			0.232499			0.116249	+	0.993220i	$0.462913\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−1.87631	−	6.50927i	−0.232727	−	0.807376i
$66$		0			0
$67$			9.32858i			1.13967i	0.821760	+	0.569834i	$0.192992\pi$
$67$			9.32858i			1.13967i	−0.821760	+	0.569834i	$0.807008\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−9.14110			−1.08485			−0.542424	−	0.840105i	$-0.682493\pi$
$71$	−9.14110			−1.08485			−0.542424	+	0.840105i	$0.682493\pi$
$72$		0			0
$73$			8.35232i			0.977566i	0.872406	+	0.488783i	$0.162559\pi$
$73$			8.35232i			0.977566i	−0.872406	+	0.488783i	$0.837441\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	9.90251i		−	1.12850i
$78$		0			0
$79$	−3.89951			−0.438730			−0.219365	−	0.975643i	$-0.570398\pi$
$79$	−3.89951			−0.438730			−0.219365	+	0.975643i	$0.570398\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			12.1427i			1.33283i	0.745580	+	0.666416i	$0.232173\pi$
$83$			12.1427i			1.33283i	−0.745580	+	0.666416i	$0.767827\pi$
$84$		0			0
$85$	−4.27589	−	14.8339i	−0.463785	−	1.60896i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	8.30087			0.879890			0.439945	−	0.898025i	$-0.354998\pi$
$89$	8.30087			0.879890			0.439945	+	0.898025i	$0.354998\pi$
$90$		0			0
$91$	5.03326			0.527629
$92$		0			0
$93$		0			0
$94$		0			0
$95$	14.8900	−	4.29206i	1.52768	−	0.440357i
$96$		0			0
$97$		−	10.7485i		−	1.09135i	−0.837997	−	0.545675i	$-0.816273\pi$
$97$		−	10.7485i		−	1.09135i	0.837997	−	0.545675i	$-0.183727\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	9.86213			0.981319			0.490659	−	0.871352i	$-0.336756\pi$
$101$	9.86213			0.981319			0.490659	+	0.871352i	$0.336756\pi$
$102$		0			0
$103$			7.23171i			0.712562i	0.934379	+	0.356281i	$0.115955\pi$
$103$			7.23171i			0.712562i	−0.934379	+	0.356281i	$0.884045\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			11.6068i			1.12207i	0.827791	+	0.561036i	$0.189597\pi$
$107$			11.6068i			1.12207i	−0.827791	+	0.561036i	$0.810403\pi$
$108$		0			0
$109$	−9.50689			−0.910595			−0.455298	−	0.890339i	$-0.650467\pi$
$109$	−9.50689			−0.910595			−0.455298	+	0.890339i	$0.650467\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			17.9519i			1.68877i	0.535736	+	0.844385i	$0.320034\pi$
$113$			17.9519i			1.68877i	−0.535736	+	0.844385i	$0.679966\pi$
$114$		0			0
$115$	0.619333	+	2.14859i	0.0577531	+	0.200357i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	11.4702			1.05147
$120$		0			0
$121$	24.5264			2.22967
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−7.44637	+	8.33976i	−0.666024	+	0.745930i
$126$		0			0
$127$		−	4.83213i		−	0.428782i	−0.976748	−	0.214391i	$-0.931223\pi$
$127$		−	4.83213i		−	0.428782i	0.976748	−	0.214391i	$-0.0687767\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−10.7210			−0.936701			−0.468351	−	0.883543i	$-0.655152\pi$
$131$	−10.7210			−0.936701			−0.468351	+	0.883543i	$0.655152\pi$
$132$		0			0
$133$			11.5136i			0.998357i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	19.9756i		−	1.70663i	−0.521396	−	0.853315i	$-0.674588\pi$
$137$		−	19.9756i		−	1.70663i	0.521396	−	0.853315i	$-0.325412\pi$
$138$		0			0
$139$	−13.3404			−1.13152			−0.565761	−	0.824570i	$-0.691417\pi$
$139$	−13.3404			−1.13152			−0.565761	+	0.824570i	$0.691417\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			18.0574i			1.51004i
$144$		0			0
$145$	−16.4697	+	4.74741i	−1.36773	+	0.394251i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	10.9871			0.900095			0.450048	−	0.893005i	$-0.351407\pi$
$149$	10.9871			0.900095			0.450048	+	0.893005i	$0.351407\pi$
$150$		0			0
$151$	−5.46730			−0.444922			−0.222461	−	0.974942i	$-0.571409\pi$
$151$	−5.46730			−0.444922			−0.222461	+	0.974942i	$0.571409\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	11.9604	−	3.44760i	0.960683	−	0.276918i
$156$		0			0
$157$			13.5788i			1.08371i	0.840472	+	0.541854i	$0.182278\pi$
$157$			13.5788i			1.08371i	−0.840472	+	0.541854i	$0.817722\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−1.66138			−0.130935
$162$		0			0
$163$			1.26734i			0.0992657i	0.998768	+	0.0496329i	$0.0158051\pi$
$163$			1.26734i			0.0992657i	−0.998768	+	0.0496329i	$0.984195\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			17.0321i			1.31798i	0.752151	+	0.658991i	$0.229017\pi$
$167$			17.0321i			1.31798i	−0.752151	+	0.658991i	$0.770983\pi$
$168$		0			0
$169$	3.82177			0.293982
$170$		0			0
$171$		0			0
$172$		0			0
$173$			10.1017i			0.768018i	0.923329	+	0.384009i	$0.125457\pi$
$173$			10.1017i			0.768018i	−0.923329	+	0.384009i	$0.874543\pi$
$174$		0			0
$175$	−4.42157	−	7.03239i	−0.334239	−	0.531598i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−5.64394			−0.421848			−0.210924	−	0.977502i	$-0.567647\pi$
$179$	−5.64394			−0.421848			−0.210924	+	0.977502i	$0.567647\pi$
$180$		0			0
$181$	12.1133			0.900376			0.450188	−	0.892934i	$-0.351357\pi$
$181$	12.1133			0.900376			0.450188	+	0.892934i	$0.351357\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	3.82169	+	13.2582i	0.280976	+	0.974761i
$186$		0			0
$187$			41.1507i			3.00924i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−11.1718			−0.808363			−0.404182	−	0.914679i	$-0.632444\pi$
$191$	−11.1718			−0.808363			−0.404182	+	0.914679i	$0.632444\pi$
$192$		0			0
$193$			23.0149i			1.65665i	0.560251	+	0.828323i	$0.310704\pi$
$193$			23.0149i			1.65665i	−0.560251	+	0.828323i	$0.689296\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			2.55284i			0.181882i	0.995856	+	0.0909411i	$0.0289875\pi$
$197$			2.55284i			0.181882i	−0.995856	+	0.0909411i	$0.971012\pi$
$198$		0			0
$199$	0.158538			0.0112385			0.00561923	−	0.999984i	$-0.498211\pi$
$199$	0.158538			0.0112385			0.00561923	+	0.999984i	$0.498211\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		−	12.7351i		−	0.893827i
$204$		0			0
$205$	13.9211	−	4.01277i	0.972291	−	0.280264i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−41.3064			−2.85723
$210$		0			0
$211$	1.05770			0.0728151			0.0364076	−	0.999337i	$-0.488409\pi$
$211$	1.05770			0.0728151			0.0364076	+	0.999337i	$0.488409\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	2.38280	+	8.26638i	0.162505	+	0.563763i
$216$		0			0
$217$			9.24831i			0.627816i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−20.9161			−1.40697
$222$		0			0
$223$			15.8649i			1.06239i	0.847250	+	0.531195i	$0.178257\pi$
$223$			15.8649i			1.06239i	−0.847250	+	0.531195i	$0.821743\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			11.9194i			0.791118i	0.918440	+	0.395559i	$0.129449\pi$
$227$			11.9194i			0.791118i	−0.918440	+	0.395559i	$0.870551\pi$
$228$		0			0
$229$	−11.8560			−0.783469			−0.391734	−	0.920078i	$-0.628125\pi$
$229$	−11.8560			−0.783469			−0.391734	+	0.920078i	$0.628125\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			16.5827i			1.08637i	0.839613	+	0.543185i	$0.182782\pi$
$233$			16.5827i			1.08637i	−0.839613	+	0.543185i	$0.817218\pi$
$234$		0			0
$235$	4.52086	+	15.6837i	0.294908	+	1.02309i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−21.7434			−1.40646			−0.703231	−	0.710961i	$-0.748260\pi$
$239$	−21.7434			−1.40646			−0.703231	+	0.710961i	$0.748260\pi$
$240$		0			0
$241$	6.83213			0.440096			0.220048	−	0.975489i	$-0.429379\pi$
$241$	6.83213			0.440096			0.220048	+	0.975489i	$0.429379\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−9.10960	+	2.62585i	−0.581991	+	0.167760i
$246$		0			0
$247$		−	20.9953i		−	1.33590i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−6.41943			−0.405191			−0.202596	−	0.979262i	$-0.564938\pi$
$251$	−6.41943			−0.405191			−0.202596	+	0.979262i	$0.564938\pi$
$252$		0			0
$253$		−	5.96040i		−	0.374727i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			26.7985i			1.67164i	0.549002	+	0.835821i	$0.315008\pi$
$257$			26.7985i			1.67164i	−0.549002	+	0.835821i	$0.684992\pi$
$258$		0			0
$259$	−10.2518			−0.637017
$260$		0			0
$261$		0			0
$262$		0			0
$263$			13.3269i			0.821772i	0.911687	+	0.410886i	$0.134781\pi$
$263$			13.3269i			0.821772i	−0.911687	+	0.410886i	$0.865219\pi$
$264$		0			0
$265$	−7.32972	−	25.4282i	−0.450261	−	1.56204i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−10.4267			−0.635726			−0.317863	−	0.948137i	$-0.602965\pi$
$269$	−10.4267			−0.635726			−0.317863	+	0.948137i	$0.602965\pi$
$270$		0			0
$271$	11.4263			0.694101			0.347051	−	0.937846i	$-0.387183\pi$
$271$	11.4263			0.694101			0.347051	+	0.937846i	$0.387183\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	25.2295	−	15.8629i	1.52140	−	0.956570i
$276$		0			0
$277$			7.32701i			0.440237i	0.975473	+	0.220119i	$0.0706445\pi$
$277$			7.32701i			0.440237i	−0.975473	+	0.220119i	$0.929356\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−21.4028			−1.27678			−0.638392	−	0.769711i	$-0.720400\pi$
$281$	−21.4028			−1.27678			−0.638392	+	0.769711i	$0.720400\pi$
$282$		0			0
$283$		−	12.1753i		−	0.723747i	−0.932227	−	0.361874i	$-0.882137\pi$
$283$		−	12.1753i		−	0.723747i	0.932227	−	0.361874i	$-0.117863\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			10.7644i			0.635402i
$288$		0			0
$289$	−30.6655			−1.80385
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	25.7879i		−	1.50655i	−0.657707	−	0.753274i	$-0.728473\pi$
$293$		−	25.7879i		−	1.50655i	0.657707	−	0.753274i	$-0.271527\pi$
$294$		0			0
$295$	11.8961	−	3.42907i	0.692618	−	0.199648i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	3.02956			0.175204
$300$		0			0
$301$	−6.39193			−0.368425
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−3.90156	+	1.12463i	−0.223403	+	0.0643961i
$306$		0			0
$307$		−	31.9642i		−	1.82429i	−0.409867	−	0.912145i	$-0.634425\pi$
$307$		−	31.9642i		−	1.82429i	0.409867	−	0.912145i	$-0.365575\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	9.15088			0.518899			0.259449	−	0.965757i	$-0.416459\pi$
$311$	9.15088			0.518899			0.259449	+	0.965757i	$0.416459\pi$
$312$		0			0
$313$			15.4343i			0.872398i	0.899850	+	0.436199i	$0.143676\pi$
$313$			15.4343i			0.872398i	−0.899850	+	0.436199i	$0.856324\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	15.2130i		−	0.854446i	−0.904146	−	0.427223i	$-0.859492\pi$
$317$		−	15.2130i		−	0.854446i	0.904146	−	0.427223i	$-0.140508\pi$
$318$		0			0
$319$	45.6886			2.55807
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	47.8458i		−	2.66221i
$324$		0			0
$325$	8.06282	+	12.8237i	0.447245	+	0.711330i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−12.1273			−0.668602
$330$		0			0
$331$	−13.4315			−0.738262			−0.369131	−	0.929377i	$-0.620345\pi$
$331$	−13.4315			−0.738262			−0.369131	+	0.929377i	$0.620345\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−5.77750	−	20.0433i	−0.315658	−	1.09508i
$336$		0			0
$337$			19.0078i			1.03542i	0.855556	+	0.517711i	$0.173216\pi$
$337$			19.0078i			1.03542i	−0.855556	+	0.517711i	$0.826784\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−33.1794			−1.79677
$342$		0			0
$343$		−	18.6736i		−	1.00828i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			2.08009i			0.111665i	0.998440	+	0.0558326i	$0.0177813\pi$
$347$			2.08009i			0.111665i	−0.998440	+	0.0558326i	$0.982219\pi$
$348$		0			0
$349$	23.3947			1.25229			0.626143	−	0.779708i	$-0.284632\pi$
$349$	23.3947			1.25229			0.626143	+	0.779708i	$0.284632\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			13.3513i			0.710617i	0.934749	+	0.355309i	$0.115624\pi$
$353$			13.3513i			0.710617i	−0.934749	+	0.355309i	$0.884376\pi$
$354$		0			0
$355$	19.6404	−	5.66138i	1.04241	−	0.300475i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−8.37614			−0.442076			−0.221038	−	0.975265i	$-0.570944\pi$
$359$	−8.37614			−0.442076			−0.221038	+	0.975265i	$0.570944\pi$
$360$		0			0
$361$	29.0269			1.52773
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−5.17287	−	17.9457i	−0.270760	−	0.939321i
$366$		0			0
$367$		−	12.3672i		−	0.645563i	−0.946474	−	0.322781i	$-0.895382\pi$
$367$		−	12.3672i		−	0.645563i	0.946474	−	0.322781i	$-0.104618\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	19.6622			1.02081
$372$		0			0
$373$		−	19.7379i		−	1.02199i	−0.859584	−	0.510994i	$-0.829277\pi$
$373$		−	19.7379i		−	1.02199i	0.859584	−	0.510994i	$-0.170723\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			23.2226i			1.19603i
$378$		0			0
$379$	28.7192			1.47521			0.737604	−	0.675234i	$-0.235957\pi$
$379$	28.7192			1.47521			0.737604	+	0.675234i	$0.235957\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	11.4405i		−	0.584580i	−0.956330	−	0.292290i	$-0.905583\pi$
$383$		−	11.4405i		−	0.584580i	0.956330	−	0.292290i	$-0.0944173\pi$
$384$		0			0
$385$	6.13295	+	21.2764i	0.312564	+	1.08435i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	31.7345			1.60900			0.804501	−	0.593951i	$-0.202433\pi$
$389$	31.7345			1.60900			0.804501	+	0.593951i	$0.202433\pi$
$390$		0			0
$391$	6.90402			0.349151
$392$		0			0
$393$		0			0
$394$		0			0
$395$	8.37845	−	2.41510i	0.421565	−	0.121517i
$396$		0			0
$397$			4.36456i			0.219051i	0.993984	+	0.109526i	$0.0349331\pi$
$397$			4.36456i			0.219051i	−0.993984	+	0.109526i	$0.965067\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	23.3692			1.16700			0.583501	−	0.812113i	$-0.301683\pi$
$401$	23.3692			1.16700			0.583501	+	0.812113i	$0.301683\pi$
$402$		0			0
$403$		−	16.8645i		−	0.840078i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	36.7796i		−	1.82310i
$408$		0			0
$409$	−14.8242			−0.733011			−0.366505	−	0.930416i	$-0.619446\pi$
$409$	−14.8242			−0.733011			−0.366505	+	0.930416i	$0.619446\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			9.19859i			0.452633i
$414$		0			0
$415$	−7.52036	−	26.0896i	−0.369160	−	1.28069i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	24.7461			1.20893			0.604463	−	0.796633i	$-0.293388\pi$
$419$	24.7461			1.20893			0.604463	+	0.796633i	$0.293388\pi$
$420$		0			0
$421$	13.5591			0.660832			0.330416	−	0.943835i	$-0.392811\pi$
$421$	13.5591			0.660832			0.330416	+	0.943835i	$0.392811\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	18.3742	+	29.2237i	0.891281	+	1.41756i
$426$		0			0
$427$		−	3.01686i		−	0.145996i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−18.7443			−0.902881			−0.451440	−	0.892301i	$-0.649090\pi$
$431$	−18.7443			−0.902881			−0.451440	+	0.892301i	$0.649090\pi$
$432$		0			0
$433$			25.4358i			1.22236i	0.791490	+	0.611182i	$0.209306\pi$
$433$			25.4358i			1.22236i	−0.791490	+	0.611182i	$0.790694\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			6.93014i			0.331514i
$438$		0			0
$439$	−33.2466			−1.58677			−0.793387	−	0.608717i	$-0.791684\pi$
$439$	−33.2466			−1.58677			−0.793387	+	0.608717i	$0.791684\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			23.3935i			1.11146i	0.831364	+	0.555729i	$0.187561\pi$
$443$			23.3935i			1.11146i	−0.831364	+	0.555729i	$0.812439\pi$
$444$		0			0
$445$	−17.8351	+	5.14100i	−0.845467	+	0.243707i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−14.1715			−0.668794			−0.334397	−	0.942432i	$-0.608533\pi$
$449$	−14.1715			−0.668794			−0.334397	+	0.942432i	$0.608533\pi$
$450$		0			0
$451$	−38.6185			−1.81848
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−10.8144	+	3.11726i	−0.506986	+	0.146139i
$456$		0			0
$457$			8.18318i			0.382793i	0.981513	+	0.191397i	$0.0613016\pi$
$457$			8.18318i			0.382793i	−0.981513	+	0.191397i	$0.938698\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−24.7094			−1.15083			−0.575416	−	0.817861i	$-0.695160\pi$
$461$	−24.7094			−1.15083			−0.575416	+	0.817861i	$0.695160\pi$
$462$		0			0
$463$		−	12.2246i		−	0.568124i	−0.958806	−	0.284062i	$-0.908318\pi$
$463$		−	12.2246i		−	0.568124i	0.958806	−	0.284062i	$-0.0916822\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			12.7632i			0.590609i	0.955403	+	0.295305i	$0.0954211\pi$
$467$			12.7632i			0.590609i	−0.955403	+	0.295305i	$0.904579\pi$
$468$		0			0
$469$	15.4983			0.715647
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	22.9318i		−	1.05441i
$474$		0			0
$475$	−29.3343	+	18.4438i	−1.34595	+	0.846257i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	26.1879			1.19655			0.598277	−	0.801289i	$-0.295852\pi$
$479$	26.1879			1.19655			0.598277	+	0.801289i	$0.295852\pi$
$480$		0			0
$481$	18.6944			0.852390
$482$		0			0
$483$		0			0
$484$		0			0
$485$	6.65693	+	23.0942i	0.302276	+	1.04865i
$486$		0			0
$487$		−	11.5012i		−	0.521171i	−0.965451	−	0.260585i	$-0.916084\pi$
$487$		−	11.5012i		−	0.521171i	0.965451	−	0.260585i	$-0.0839156\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	9.70556			0.438006			0.219003	−	0.975724i	$-0.429720\pi$
$491$	9.70556			0.438006			0.219003	+	0.975724i	$0.429720\pi$
$492$		0			0
$493$			52.9217i			2.38347i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			15.1868i			0.681223i
$498$		0			0
$499$	−5.56664			−0.249197			−0.124598	−	0.992207i	$-0.539764\pi$
$499$	−5.56664			−0.249197			−0.124598	+	0.992207i	$0.539764\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	9.77687i		−	0.435929i	−0.975957	−	0.217965i	$-0.930058\pi$
$503$		−	9.77687i		−	0.435929i	0.975957	−	0.217965i	$-0.0699417\pi$
$504$		0			0
$505$	−21.1896	+	6.10794i	−0.942927	+	0.271800i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	9.60259			0.425627			0.212814	−	0.977093i	$-0.431737\pi$
$509$	9.60259			0.425627			0.212814	+	0.977093i	$0.431737\pi$
$510$		0			0
$511$	13.8764			0.613856
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−4.47884	−	15.5380i	−0.197361	−	0.684685i
$516$		0			0
$517$		−	43.5083i		−	1.91349i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−11.9991			−0.525689			−0.262844	−	0.964838i	$-0.584661\pi$
$521$	−11.9991			−0.525689			−0.262844	+	0.964838i	$0.584661\pi$
$522$		0			0
$523$			36.7555i			1.60721i	0.595166	+	0.803603i	$0.297086\pi$
$523$			36.7555i			1.60721i	−0.595166	+	0.803603i	$0.702914\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	38.4322i		−	1.67413i
$528$		0			0
$529$	−1.00000			−0.0434783
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	19.6291i		−	0.850229i
$534$		0			0
$535$	−7.18848	−	24.9382i	−0.310785	−	1.07817i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	25.2710			1.08850
$540$		0			0
$541$	14.7011			0.632048			0.316024	−	0.948751i	$-0.397652\pi$
$541$	14.7011			0.632048			0.316024	+	0.948751i	$0.397652\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	20.4264	−	5.88793i	0.874970	−	0.252211i
$546$		0			0
$547$		−	34.2901i		−	1.46614i	−0.680153	−	0.733070i	$-0.738087\pi$
$547$		−	34.2901i		−	1.46614i	0.680153	−	0.733070i	$-0.261913\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−53.1220			−2.26307
$552$		0			0
$553$			6.47858i			0.275497i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	24.7484i		−	1.04862i	−0.851526	−	0.524312i	$-0.824323\pi$
$557$		−	24.7484i		−	1.04862i	0.851526	−	0.524312i	$-0.175677\pi$
$558$		0			0
$559$	11.6558			0.492988
$560$		0			0
$561$		0			0
$562$		0			0
$563$			29.7457i			1.25363i	0.779168	+	0.626815i	$0.215642\pi$
$563$			29.7457i			1.25363i	−0.779168	+	0.626815i	$0.784358\pi$
$564$		0			0
$565$	−11.1182	−	38.5712i	−0.467746	−	1.62270i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	28.3325			1.18776			0.593881	−	0.804553i	$-0.297595\pi$
$569$	28.3325			1.18776			0.593881	+	0.804553i	$0.297595\pi$
$570$		0			0
$571$	4.93277			0.206430			0.103215	−	0.994659i	$-0.467087\pi$
$571$	4.93277			0.206430			0.103215	+	0.994659i	$0.467087\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−2.66138	−	4.23285i	−0.110987	−	0.176522i
$576$		0			0
$577$		−	12.0318i		−	0.500892i	−0.968131	−	0.250446i	$-0.919423\pi$
$577$		−	12.0318i		−	0.500892i	0.968131	−	0.250446i	$-0.0805773\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	20.1736			0.836943
$582$		0			0
$583$			70.5406i			2.92149i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			34.4921i			1.42364i	0.702362	+	0.711820i	$0.252129\pi$
$587$			34.4921i			1.42364i	−0.702362	+	0.711820i	$0.747871\pi$
$588$		0			0
$589$	38.5776			1.58956
$590$		0			0
$591$		0			0
$592$		0			0
$593$			4.73207i			0.194323i	0.995269	+	0.0971614i	$0.0309763\pi$
$593$			4.73207i			0.194323i	−0.995269	+	0.0971614i	$0.969024\pi$
$594$		0			0
$595$	−24.6447	+	7.10388i	−1.01034	+	0.291231i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	32.1270			1.31267			0.656336	−	0.754468i	$-0.272105\pi$
$599$	32.1270			1.31267			0.656336	+	0.754468i	$0.272105\pi$
$600$		0			0
$601$	−39.7807			−1.62269			−0.811345	−	0.584567i	$-0.801264\pi$
$601$	−39.7807			−1.62269			−0.811345	+	0.584567i	$0.801264\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−52.6971	+	15.1900i	−2.14244	+	0.617562i
$606$		0			0
$607$		−	3.69623i		−	0.150025i	−0.997183	−	0.0750126i	$-0.976100\pi$
$607$		−	3.69623i		−	0.150025i	0.997183	−	0.0750126i	$-0.0238997\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	22.1144			0.894654
$612$		0			0
$613$			9.57318i			0.386657i	0.981134	+	0.193329i	$0.0619284\pi$
$613$			9.57318i			0.386657i	−0.981134	+	0.193329i	$0.938072\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	1.43662i		−	0.0578360i	−0.999582	−	0.0289180i	$-0.990794\pi$
$617$		−	1.43662i		−	0.0578360i	0.999582	−	0.0289180i	$-0.00920617\pi$
$618$		0			0
$619$	−8.66435			−0.348250			−0.174125	−	0.984724i	$-0.555710\pi$
$619$	−8.66435			−0.348250			−0.174125	+	0.984724i	$0.555710\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	13.7909i		−	0.552521i
$624$		0			0
$625$	10.8341	−	22.5305i	0.433364	−	0.901219i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	42.6023			1.69867
$630$		0			0
$631$	43.5855			1.73511			0.867555	−	0.497341i	$-0.165690\pi$
$631$	43.5855			1.73511			0.867555	+	0.497341i	$0.165690\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	2.99270	+	10.3822i	0.118762	+	0.412007i
$636$		0			0
$637$			12.8448i			0.508928i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−17.0520			−0.673513			−0.336756	−	0.941592i	$-0.609330\pi$
$641$	−17.0520			−0.673513			−0.336756	+	0.941592i	$0.609330\pi$
$642$		0			0
$643$		−	47.1068i		−	1.85771i	−0.370443	−	0.928855i	$-0.620794\pi$
$643$		−	47.1068i		−	1.85771i	0.370443	−	0.928855i	$-0.379206\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	29.0224i		−	1.14099i	−0.821301	−	0.570494i	$-0.806752\pi$
$647$		−	29.0224i		−	1.14099i	0.821301	−	0.570494i	$-0.193248\pi$
$648$		0			0
$649$	−33.0010			−1.29540
$650$		0			0
$651$		0			0
$652$		0			0
$653$			35.3644i			1.38392i	0.721938	+	0.691958i	$0.243252\pi$
$653$			35.3644i			1.38392i	−0.721938	+	0.691958i	$0.756748\pi$
$654$		0			0
$655$	23.0351	−	6.63989i	0.900055	−	0.259442i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−37.4688			−1.45958			−0.729789	−	0.683672i	$-0.760382\pi$
$659$	−37.4688			−1.45958			−0.729789	+	0.683672i	$0.760382\pi$
$660$		0			0
$661$	−14.7705			−0.574505			−0.287253	−	0.957855i	$-0.592742\pi$
$661$	−14.7705			−0.574505			−0.287253	+	0.957855i	$0.592742\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−7.13076	−	24.7380i	−0.276519	−	0.959298i
$666$		0			0
$667$		−	7.66535i		−	0.296804i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	10.8233			0.417830
$672$		0			0
$673$			0.709461i			0.0273477i	0.999907	+	0.0136739i	$0.00435266\pi$
$673$			0.709461i			0.0273477i	−0.999907	+	0.0136739i	$0.995647\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	29.6988i		−	1.14142i	−0.821152	−	0.570710i	$-0.806668\pi$
$677$		−	29.6988i		−	1.14142i	0.821152	−	0.570710i	$-0.193332\pi$
$678$		0			0
$679$	−17.8574			−0.685306
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	28.3994i		−	1.08667i	−0.839515	−	0.543336i	$-0.817161\pi$
$683$		−	28.3994i		−	1.08667i	0.839515	−	0.543336i	$-0.182839\pi$
$684$		0			0
$685$	12.3715	+	42.9193i	0.472692	+	1.63986i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−35.8544			−1.36594
$690$		0			0
$691$	−8.25059			−0.313867			−0.156934	−	0.987609i	$-0.550161\pi$
$691$	−8.25059			−0.313867			−0.156934	+	0.987609i	$0.550161\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	28.6631	−	8.26217i	1.08725	−	0.313402i
$696$		0			0
$697$		−	44.7324i		−	1.69436i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	43.5638			1.64538			0.822691	−	0.568489i	$-0.192472\pi$
$701$	43.5638			1.64538			0.822691	+	0.568489i	$0.192472\pi$
$702$		0			0
$703$			42.7635i			1.61286i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	16.3848i		−	0.616212i
$708$		0			0
$709$	26.1345			0.981501			0.490750	−	0.871300i	$-0.336723\pi$
$709$	26.1345			0.981501			0.490750	+	0.871300i	$0.336723\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			5.56664i			0.208472i
$714$		0			0
$715$	−11.1835	−	38.7979i	−0.418241	−	1.45096i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−12.0640			−0.449911			−0.224956	−	0.974369i	$-0.572224\pi$
$719$	−12.0640			−0.449911			−0.224956	+	0.974369i	$0.572224\pi$
$720$		0			0
$721$	12.0146			0.447449
$722$		0			0
$723$		0			0
$724$		0			0
$725$	32.4463	−	20.4004i	1.20503	−	0.757653i
$726$		0			0
$727$		−	17.2201i		−	0.638659i	−0.947644	−	0.319330i	$-0.896542\pi$
$727$		−	17.2201i		−	0.638659i	0.947644	−	0.319330i	$-0.103458\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	26.5622			0.982439
$732$		0			0
$733$		−	23.8445i		−	0.880716i	−0.897822	−	0.440358i	$-0.854852\pi$
$733$		−	23.8445i		−	0.880716i	0.897822	−	0.440358i	$-0.145148\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			55.6021i			2.04813i
$738$		0			0
$739$	−24.4219			−0.898373			−0.449187	−	0.893438i	$-0.648286\pi$
$739$	−24.4219			−0.898373			−0.449187	+	0.893438i	$0.648286\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			4.54176i			0.166621i	0.996524	+	0.0833105i	$0.0265493\pi$
$743$			4.54176i			0.166621i	−0.996524	+	0.0833105i	$0.973451\pi$
$744$		0			0
$745$	−23.6067	+	6.80465i	−0.864881	+	0.249303i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	19.2833			0.704598
$750$		0			0
$751$	−21.2411			−0.775100			−0.387550	−	0.921849i	$-0.626679\pi$
$751$	−21.2411			−0.775100			−0.387550	+	0.921849i	$0.626679\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	11.7470	−	3.38608i	0.427516	−	0.123232i
$756$		0			0
$757$		−	24.1157i		−	0.876499i	−0.898853	−	0.438250i	$-0.855599\pi$
$757$		−	24.1157i		−	0.876499i	0.898853	−	0.438250i	$-0.144401\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−42.0856			−1.52560			−0.762801	−	0.646634i	$-0.776176\pi$
$761$	−42.0856			−1.52560			−0.762801	+	0.646634i	$0.776176\pi$
$762$		0			0
$763$			15.7946i			0.571802i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	16.7738i		−	0.605667i
$768$		0			0
$769$	−50.5912			−1.82436			−0.912182	−	0.409784i	$-0.865604\pi$
$769$	−50.5912			−1.82436			−0.912182	+	0.409784i	$0.865604\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	19.5591i		−	0.703492i	−0.936096	−	0.351746i	$-0.885588\pi$
$773$		−	19.5591i		−	0.703492i	0.936096	−	0.351746i	$-0.114412\pi$
$774$		0			0
$775$	−23.5628	+	14.8149i	−0.846399	+	0.532169i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	44.9016			1.60877
$780$		0			0
$781$	−54.4846			−1.94961
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−8.40982	−	29.1753i	−0.300159	−	1.04131i
$786$		0			0
$787$			18.0313i			0.642745i	0.946953	+	0.321373i	$0.104144\pi$
$787$			18.0313i			0.642745i	−0.946953	+	0.321373i	$0.895856\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	29.8249			1.06045
$792$		0			0
$793$			5.50129i			0.195357i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	40.8298i		−	1.44627i	−0.690709	−	0.723133i	$-0.742701\pi$
$797$		−	40.8298i		−	1.44627i	0.690709	−	0.723133i	$-0.257299\pi$
$798$		0			0
$799$	50.3963			1.78289
$800$		0			0
$801$		0			0
$802$		0			0
$803$			49.7832i			1.75681i
$804$		0			0
$805$	3.56962	−	1.02895i	0.125813	−	0.0362657i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−9.38602			−0.329995			−0.164997	−	0.986294i	$-0.552762\pi$
$809$	−9.38602			−0.329995			−0.164997	+	0.986294i	$0.552762\pi$
$810$		0			0
$811$	41.9927			1.47456			0.737281	−	0.675586i	$-0.236109\pi$
$811$	41.9927			1.47456			0.737281	+	0.675586i	$0.236109\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−0.784906	−	2.72299i	−0.0274940	−	0.0953822i
$816$		0			0
$817$			26.6627i			0.932811i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	32.5658			1.13656			0.568278	−	0.822837i	$-0.307610\pi$
$821$	32.5658			1.13656			0.568278	+	0.822837i	$0.307610\pi$
$822$		0			0
$823$		−	1.81829i		−	0.0633817i	−0.999498	−	0.0316909i	$-0.989911\pi$
$823$		−	1.81829i		−	0.0633817i	0.999498	−	0.0316909i	$-0.0100892\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	13.4447i		−	0.467516i	−0.972295	−	0.233758i	$-0.924898\pi$
$827$		−	13.4447i		−	0.467516i	0.972295	−	0.233758i	$-0.0751024\pi$
$828$		0			0
$829$	6.84911			0.237880			0.118940	−	0.992901i	$-0.462050\pi$
$829$	6.84911			0.237880			0.118940	+	0.992901i	$0.462050\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			29.2717i			1.01421i
$834$		0			0
$835$	−10.5485	−	36.5949i	−0.365047	−	1.26642i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−9.99933			−0.345215			−0.172608	−	0.984991i	$-0.555219\pi$
$839$	−9.99933			−0.345215			−0.172608	+	0.984991i	$0.555219\pi$
$840$		0			0
$841$	29.7576			1.02612
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−8.21140	+	2.36695i	−0.282481	+	0.0814255i
$846$		0			0
$847$		−	40.7477i		−	1.40011i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−6.17065			−0.211527
$852$		0			0
$853$		−	57.0510i		−	1.95339i	−0.214634	−	0.976695i	$-0.568856\pi$
$853$		−	57.0510i		−	1.95339i	0.214634	−	0.976695i	$-0.431144\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			14.2653i			0.487293i	0.969864	+	0.243646i	$0.0783436\pi$
$857$			14.2653i			0.487293i	−0.969864	+	0.243646i	$0.921656\pi$
$858$		0			0
$859$	9.09063			0.310168			0.155084	−	0.987901i	$-0.450435\pi$
$859$	9.09063			0.310168			0.155084	+	0.987901i	$0.450435\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	9.62012i		−	0.327473i	−0.986504	−	0.163736i	$-0.947645\pi$
$863$		−	9.62012i		−	0.327473i	0.986504	−	0.163736i	$-0.0523546\pi$
$864$		0			0
$865$	−6.25631	−	21.7044i	−0.212721	−	0.737971i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−23.2427			−0.788454
$870$		0			0
$871$	−28.2615			−0.957604
$872$		0			0
$873$		0			0
$874$		0			0
$875$	13.8555	+	12.3713i	0.468402	+	0.418225i
$876$		0			0
$877$			23.0770i			0.779255i	0.920973	+	0.389627i	$0.127396\pi$
$877$			23.0770i			0.779255i	−0.920973	+	0.389627i	$0.872604\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	35.9282			1.21045			0.605226	−	0.796054i	$-0.293083\pi$
$881$	35.9282			1.21045			0.605226	+	0.796054i	$0.293083\pi$
$882$		0			0
$883$		−	22.6146i		−	0.761042i	−0.924772	−	0.380521i	$-0.875745\pi$
$883$		−	22.6146i		−	0.761042i	0.924772	−	0.380521i	$-0.124255\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	13.5543i		−	0.455109i	−0.973765	−	0.227555i	$-0.926927\pi$
$887$		−	13.5543i		−	0.455109i	0.973765	−	0.227555i	$-0.0730730\pi$
$888$		0			0
$889$	−8.02801			−0.269251
$890$		0			0
$891$		0			0
$892$		0			0
$893$			50.5869i			1.69283i
$894$		0			0
$895$	12.1265	−	3.49548i	0.405344	−	0.116841i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−42.6702			−1.42313
$900$		0			0
$901$	−81.7081			−2.72209
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−26.0265	+	7.50218i	−0.865151	+	0.249381i
$906$		0			0
$907$			5.24536i			0.174169i	0.996201	+	0.0870847i	$0.0277551\pi$
$907$			5.24536i			0.174169i	−0.996201	+	0.0870847i	$0.972245\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−20.1402			−0.667274			−0.333637	−	0.942702i	$-0.608276\pi$
$911$	−20.1402			−0.667274			−0.333637	+	0.942702i	$0.608276\pi$
$912$		0			0
$913$			72.3753i			2.39527i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			17.8117i			0.588195i
$918$		0			0
$919$	8.31421			0.274260			0.137130	−	0.990553i	$-0.456212\pi$
$919$	8.31421			0.274260			0.137130	+	0.990553i	$0.456212\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	27.6935i		−	0.911542i
$924$		0			0
$925$	−16.4225	−	26.1195i	−0.539967	−	0.858803i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	50.6005			1.66015			0.830075	−	0.557652i	$-0.188298\pi$
$929$	50.6005			1.66015			0.830075	+	0.557652i	$0.188298\pi$
$930$		0			0
$931$	−29.3825			−0.962972
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−25.4860	−	88.4159i	−0.833482	−	2.89151i
$936$		0			0
$937$			0.881787i			0.0288067i	0.999896	+	0.0144034i	$0.00458489\pi$
$937$			0.881787i			0.0288067i	−0.999896	+	0.0144034i	$0.995415\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−5.93501			−0.193476			−0.0967378	−	0.995310i	$-0.530841\pi$
$941$	−5.93501			−0.193476			−0.0967378	+	0.995310i	$0.530841\pi$
$942$		0			0
$943$			6.47918i			0.210991i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	45.3569i		−	1.47390i	−0.675947	−	0.736951i	$-0.736265\pi$
$947$		−	45.3569i		−	1.47390i	0.675947	−	0.736951i	$-0.263735\pi$
$948$		0			0
$949$	−25.3039			−0.821398
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	11.4329i		−	0.370349i	−0.982706	−	0.185174i	$-0.940715\pi$
$953$		−	11.4329i		−	0.370349i	0.982706	−	0.185174i	$-0.0592850\pi$
$954$		0			0
$955$	24.0036	−	6.91907i	0.776738	−	0.223896i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−33.1871			−1.07167
$960$		0			0
$961$	−0.0125560			−0.000405031
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−14.2539	−	49.4494i	−0.458848	−	1.59183i
$966$		0			0
$967$			7.96415i			0.256110i	0.991767	+	0.128055i	$0.0408734\pi$
$967$			7.96415i			0.256110i	−0.991767	+	0.128055i	$0.959127\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	34.9817			1.12262			0.561308	−	0.827607i	$-0.310298\pi$
$971$	34.9817			1.12262			0.561308	+	0.827607i	$0.310298\pi$
$972$		0			0
$973$			22.1636i			0.710531i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			31.2726i			1.00050i	0.865881	+	0.500249i	$0.166758\pi$
$977$			31.2726i			1.00050i	−0.865881	+	0.500249i	$0.833242\pi$
$978$		0			0
$979$	49.4765			1.58128
$980$		0			0
$981$		0			0
$982$		0			0
$983$			54.7829i			1.74730i	0.486551	+	0.873652i	$0.338255\pi$
$983$			54.7829i			1.74730i	−0.486551	+	0.873652i	$0.661745\pi$
$984$		0			0
$985$	−1.58106	−	5.48500i	−0.0503767	−	0.174767i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−3.84736			−0.122339
$990$		0			0
$991$	17.4565			0.554524			0.277262	−	0.960794i	$-0.410573\pi$
$991$	17.4565			0.554524			0.277262	+	0.960794i	$0.410573\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−0.340633	+	0.0981879i	−0.0107988	+	0.00311277i
$996$		0			0
$997$		−	15.7271i		−	0.498082i	−0.968493	−	0.249041i	$-0.919885\pi$
$997$		−	15.7271i		−	0.498082i	0.968493	−	0.249041i	$-0.0801155\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4140.2.f.c.829.2		14
3.2	odd	2		1380.2.f.b.829.7	✓	14
5.4	even	2	inner	4140.2.f.c.829.1		14
15.2	even	4		6900.2.a.bc.1.5		7
15.8	even	4		6900.2.a.bd.1.3		7
15.14	odd	2		1380.2.f.b.829.14	yes	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1380.2.f.b.829.7	✓	14	3.2	odd	2
1380.2.f.b.829.14	yes	14	15.14	odd	2
4140.2.f.c.829.1		14	5.4	even	2	inner
4140.2.f.c.829.2		14	1.1	even	1	trivial
6900.2.a.bc.1.5		7	15.2	even	4
6900.2.a.bd.1.3		7	15.8	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4140.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-2.14859 + 0.619333i) q^{5} -1.66138i q^{7} +O(q^{10})\)	\(q+(-2.14859 + 0.619333i) q^{5} -1.66138i q^{7} +5.96040 q^{11} +3.02956i q^{13} +6.90402i q^{17} -6.93014 q^{19} -1.00000i q^{23} +(4.23285 - 2.66138i) q^{25} +7.66535 q^{29} -5.56664 q^{31} +(1.02895 + 3.56962i) q^{35} -6.17065i q^{37} -6.47918 q^{41} -3.84736i q^{43} -7.29956i q^{47} +4.23981 q^{49} +11.8349i q^{53} +(-12.8064 + 3.69147i) q^{55} -5.53671 q^{59} +1.81587 q^{61} +(-1.87631 - 6.50927i) q^{65} +9.32858i q^{67} -9.14110 q^{71} +8.35232i q^{73} -9.90251i q^{77} -3.89951 q^{79} +12.1427i q^{83} +(-4.27589 - 14.8339i) q^{85} +8.30087 q^{89} +5.03326 q^{91} +(14.8900 - 4.29206i) q^{95} -10.7485i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q+O(q^{10}) \) 14 * q	\( 14 q + 4 q^{19} - 6 q^{25} + 30 q^{29} + 6 q^{31} + 14 q^{35} - 46 q^{41} - 20 q^{49} - 16 q^{55} + 10 q^{59} + 64 q^{61} + 36 q^{65} - 42 q^{71} - 32 q^{79} - 42 q^{85} + 52 q^{89} + 28 q^{91} + 44 q^{95}+O(q^{100}) \) 14 * q + 4 * q^19 - 6 * q^25 + 30 * q^29 + 6 * q^31 + 14 * q^35 - 46 * q^41 - 20 * q^49 - 16 * q^55 + 10 * q^59 + 64 * q^61 + 36 * q^65 - 42 * q^71 - 32 * q^79 - 42 * q^85 + 52 * q^89 + 28 * q^91 + 44 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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