LMFDB - Embedded newform 4140.2.f.c.829.12

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			829.12
Root			$-0.511427i$ of defining polynomial
Character	$\chi$	$=$	4140.829
Dual form			4140.2.f.c.829.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.81604 + 1.30461i) q^{5} +3.73844i q^{7} +O(q^{10})$	$q+(1.81604 + 1.30461i) q^{5} +3.73844i q^{7} +5.75306 q^{11} -4.85436i q^{13} +5.12538i q^{17} +8.11472 q^{19} +1.00000i q^{23} +(1.59598 + 4.73844i) q^{25} -0.516159 q^{29} +6.47182 q^{31} +(-4.87721 + 6.78915i) q^{35} -7.07725i q^{37} -4.42825 q^{41} +1.54416i q^{43} -10.0837i q^{47} -6.97595 q^{49} +2.22668i q^{53} +(10.4478 + 7.50550i) q^{55} -7.93996 q^{59} +11.8410 q^{61} +(6.33305 - 8.81570i) q^{65} -8.86898i q^{67} +5.93161 q^{71} -14.3312i q^{73} +21.5075i q^{77} -17.2275 q^{79} +3.73460i q^{83} +(-6.68662 + 9.30788i) q^{85} +11.6613 q^{89} +18.1477 q^{91} +(14.7366 + 10.5866i) q^{95} -0.693347i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	1.81604	+	1.30461i	0.812157	+	0.583440i
$5$	1.81604	+	1.30461i	0.812157	+	0.583440i
$6$		0			0
$7$			3.73844i			1.41300i	0.707714	+	0.706499i	$0.249727\pi$
$7$			3.73844i			1.41300i	−0.707714	+	0.706499i	$0.750273\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	5.75306			1.73461			0.867306	−	0.497775i	$-0.165849\pi$
$11$	5.75306			1.73461			0.867306	+	0.497775i	$0.165849\pi$
$12$		0			0
$13$		−	4.85436i		−	1.34636i	−0.739480	−	0.673178i	$-0.764929\pi$
$13$		−	4.85436i		−	1.34636i	0.739480	−	0.673178i	$-0.235071\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			5.12538i			1.24309i	0.783380	+	0.621544i	$0.213494\pi$
$17$			5.12538i			1.24309i	−0.783380	+	0.621544i	$0.786506\pi$
$18$		0			0
$19$	8.11472			1.86165			0.930823	−	0.365471i	$-0.119092\pi$
$19$	8.11472			1.86165			0.930823	+	0.365471i	$0.119092\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			1.00000i			0.208514i
$23$			1.00000i			0.208514i
$24$		0			0
$25$	1.59598	+	4.73844i	0.319197	+	0.947689i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−0.516159			−0.0958483			−0.0479241	−	0.998851i	$-0.515261\pi$
$29$	−0.516159			−0.0958483			−0.0479241	+	0.998851i	$0.515261\pi$
$30$		0			0
$31$	6.47182			1.16237			0.581186	−	0.813771i	$-0.302589\pi$
$31$	6.47182			1.16237			0.581186	+	0.813771i	$0.302589\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−4.87721	+	6.78915i	−0.824399	+	1.14758i
$36$		0			0
$37$		−	7.07725i		−	1.16349i	−0.813370	−	0.581747i	$-0.802370\pi$
$37$		−	7.07725i		−	1.16349i	0.813370	−	0.581747i	$-0.197630\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−4.42825			−0.691576			−0.345788	−	0.938313i	$-0.612388\pi$
$41$	−4.42825			−0.691576			−0.345788	+	0.938313i	$0.612388\pi$
$42$		0			0
$43$			1.54416i			0.235483i	0.993044	+	0.117741i	$0.0375654\pi$
$43$			1.54416i			0.235483i	−0.993044	+	0.117741i	$0.962435\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	10.0837i		−	1.47086i	−0.677602	−	0.735428i	$-0.736981\pi$
$47$		−	10.0837i		−	1.47086i	0.677602	−	0.735428i	$-0.263019\pi$
$48$		0			0
$49$	−6.97595			−0.996565
$50$		0			0
$51$		0			0
$52$		0			0
$53$			2.22668i			0.305858i	0.988237	+	0.152929i	$0.0488706\pi$
$53$			2.22668i			0.305858i	−0.988237	+	0.152929i	$0.951129\pi$
$54$		0			0
$55$	10.4478	+	7.50550i	1.40878	+	1.01204i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−7.93996			−1.03369			−0.516847	−	0.856078i	$-0.672894\pi$
$59$	−7.93996			−1.03369			−0.516847	+	0.856078i	$0.672894\pi$
$60$		0			0
$61$	11.8410			1.51608			0.758040	−	0.652208i	$-0.226157\pi$
$61$	11.8410			1.51608			0.758040	+	0.652208i	$0.226157\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	6.33305	−	8.81570i	0.785518	−	1.09345i
$66$		0			0
$67$		−	8.86898i		−	1.08352i	−0.840534	−	0.541759i	$-0.817759\pi$
$67$		−	8.86898i		−	1.08352i	0.840534	−	0.541759i	$-0.182241\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	5.93161			0.703953			0.351976	−	0.936009i	$-0.385510\pi$
$71$	5.93161			0.703953			0.351976	+	0.936009i	$0.385510\pi$
$72$		0			0
$73$		−	14.3312i		−	1.67735i	−0.544636	−	0.838673i	$-0.683332\pi$
$73$		−	14.3312i		−	1.67735i	0.544636	−	0.838673i	$-0.316668\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			21.5075i			2.45101i
$78$		0			0
$79$	−17.2275			−1.93825			−0.969124	−	0.246574i	$-0.920695\pi$
$79$	−17.2275			−1.93825			−0.969124	+	0.246574i	$0.920695\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			3.73460i			0.409926i	0.978770	+	0.204963i	$0.0657074\pi$
$83$			3.73460i			0.409926i	−0.978770	+	0.204963i	$0.934293\pi$
$84$		0			0
$85$	−6.68662	+	9.30788i	−0.725266	+	1.00958i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	11.6613			1.23610			0.618048	−	0.786140i	$-0.287924\pi$
$89$	11.6613			1.23610			0.618048	+	0.786140i	$0.287924\pi$
$90$		0			0
$91$	18.1477			1.90240
$92$		0			0
$93$		0			0
$94$		0			0
$95$	14.7366	+	10.5866i	1.51195	+	1.08616i
$96$		0			0
$97$		−	0.693347i		−	0.0703987i	−0.999380	−	0.0351993i	$-0.988793\pi$
$97$		−	0.693347i		−	0.0703987i	0.999380	−	0.0351993i	$-0.0112066\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−14.5828			−1.45104			−0.725519	−	0.688202i	$-0.758400\pi$
$101$	−14.5828			−1.45104			−0.725519	+	0.688202i	$0.758400\pi$
$102$		0			0
$103$		−	8.56001i		−	0.843443i	−0.906725	−	0.421721i	$-0.861426\pi$
$103$		−	8.56001i		−	0.843443i	0.906725	−	0.421721i	$-0.138574\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			8.73965i			0.844893i	0.906388	+	0.422447i	$0.138829\pi$
$107$			8.73965i			0.844893i	−0.906388	+	0.422447i	$0.861171\pi$
$108$		0			0
$109$	−3.63592			−0.348258			−0.174129	−	0.984723i	$-0.555711\pi$
$109$	−3.63592			−0.348258			−0.174129	+	0.984723i	$0.555711\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			4.27239i			0.401912i	0.979600	+	0.200956i	$0.0644049\pi$
$113$			4.27239i			0.401912i	−0.979600	+	0.200956i	$0.935595\pi$
$114$		0			0
$115$	−1.30461	+	1.81604i	−0.121656	+	0.169346i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−19.1609			−1.75648
$120$		0			0
$121$	22.0977			2.00888
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−3.28346	+	10.6873i	−0.293681	+	0.955903i
$126$		0			0
$127$		−	1.05696i		−	0.0937897i	−0.998900	−	0.0468949i	$-0.985067\pi$
$127$		−	1.05696i		−	0.0937897i	0.998900	−	0.0468949i	$-0.0149326\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−1.34886			−0.117850			−0.0589251	−	0.998262i	$-0.518767\pi$
$131$	−1.34886			−0.117850			−0.0589251	+	0.998262i	$0.518767\pi$
$132$		0			0
$133$			30.3364i			2.63050i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			7.10784i			0.607264i	0.952789	+	0.303632i	$0.0981993\pi$
$137$			7.10784i			0.607264i	−0.952789	+	0.303632i	$0.901801\pi$
$138$		0			0
$139$	−15.2920			−1.29705			−0.648524	−	0.761194i	$-0.724613\pi$
$139$	−15.2920			−1.29705			−0.648524	+	0.761194i	$0.724613\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	27.9274i		−	2.33541i
$144$		0			0
$145$	−0.937364	−	0.673386i	−0.0778438	−	0.0559217i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−7.33459			−0.600873			−0.300436	−	0.953802i	$-0.597132\pi$
$149$	−7.33459			−0.600873			−0.300436	+	0.953802i	$0.597132\pi$
$150$		0			0
$151$	0.611024			0.0497245			0.0248622	−	0.999691i	$-0.492085\pi$
$151$	0.611024			0.0497245			0.0248622	+	0.999691i	$0.492085\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	11.7531	+	8.44320i	0.944028	+	0.678174i
$156$		0			0
$157$			9.39699i			0.749962i	0.927033	+	0.374981i	$0.122351\pi$
$157$			9.39699i			0.749962i	−0.927033	+	0.374981i	$0.877649\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−3.73844			−0.294631
$162$		0			0
$163$		−	13.0616i		−	1.02306i	−0.859266	−	0.511530i	$-0.829079\pi$
$163$		−	13.0616i		−	1.02306i	0.859266	−	0.511530i	$-0.170921\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			6.72953i			0.520747i	0.965508	+	0.260373i	$0.0838457\pi$
$167$			6.72953i			0.520747i	−0.965508	+	0.260373i	$0.916154\pi$
$168$		0			0
$169$	−10.5648			−0.812677
$170$		0			0
$171$		0			0
$172$		0			0
$173$			6.92868i			0.526778i	0.964690	+	0.263389i	$0.0848403\pi$
$173$			6.92868i			0.526778i	−0.964690	+	0.263389i	$0.915160\pi$
$174$		0			0
$175$	−17.7144	+	5.96649i	−1.33908	+	0.451024i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−14.7568			−1.10297			−0.551486	−	0.834184i	$-0.685939\pi$
$179$	−14.7568			−1.10297			−0.551486	+	0.834184i	$0.685939\pi$
$180$		0			0
$181$	−7.33909			−0.545510			−0.272755	−	0.962083i	$-0.587935\pi$
$181$	−7.33909			−0.545510			−0.272755	+	0.962083i	$0.587935\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	9.23306	−	12.8526i	0.678828	−	0.944939i
$186$		0			0
$187$			29.4866i			2.15627i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−17.4999			−1.26625			−0.633125	−	0.774050i	$-0.718228\pi$
$191$	−17.4999			−1.26625			−0.633125	+	0.774050i	$0.718228\pi$
$192$		0			0
$193$			18.9248i			1.36224i	0.732174	+	0.681118i	$0.238506\pi$
$193$			18.9248i			1.36224i	−0.732174	+	0.681118i	$0.761494\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			22.8234i			1.62610i	0.582196	+	0.813048i	$0.302194\pi$
$197$			22.8234i			1.62610i	−0.582196	+	0.813048i	$0.697806\pi$
$198$		0			0
$199$	20.6778			1.46581			0.732905	−	0.680331i	$-0.238164\pi$
$199$	20.6778			1.46581			0.732905	+	0.680331i	$0.238164\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		−	1.92963i		−	0.135433i
$204$		0			0
$205$	−8.04187	−	5.77714i	−0.561668	−	0.403493i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	46.6845			3.22923
$210$		0			0
$211$	−9.00429			−0.619881			−0.309941	−	0.950756i	$-0.600309\pi$
$211$	−9.00429			−0.619881			−0.309941	+	0.950756i	$0.600309\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−2.01453	+	2.80426i	−0.137390	+	0.191249i
$216$		0			0
$217$			24.1945i			1.64243i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	24.8804			1.67364
$222$		0			0
$223$		−	10.1914i		−	0.682468i	−0.939978	−	0.341234i	$-0.889155\pi$
$223$		−	10.1914i		−	0.682468i	0.939978	−	0.341234i	$-0.110845\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			14.5207i			0.963775i	0.876233	+	0.481887i	$0.160049\pi$
$227$			14.5207i			0.963775i	−0.876233	+	0.481887i	$0.839951\pi$
$228$		0			0
$229$	−13.5731			−0.896938			−0.448469	−	0.893798i	$-0.648031\pi$
$229$	−13.5731			−0.896938			−0.448469	+	0.893798i	$0.648031\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	3.14626i		−	0.206118i	−0.994675	−	0.103059i	$-0.967137\pi$
$233$		−	3.14626i		−	0.206118i	0.994675	−	0.103059i	$-0.0328631\pi$
$234$		0			0
$235$	13.1553	−	18.3124i	0.858156	−	1.19457i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	29.7050			1.92145			0.960727	−	0.277495i	$-0.0895041\pi$
$239$	29.7050			1.92145			0.960727	+	0.277495i	$0.0895041\pi$
$240$		0			0
$241$	0.943044			0.0607468			0.0303734	−	0.999539i	$-0.490330\pi$
$241$	0.943044			0.0607468			0.0303734	+	0.999539i	$0.490330\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−12.6686	−	9.10090i	−0.809367	−	0.581435i
$246$		0			0
$247$		−	39.3918i		−	2.50644i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−10.5360			−0.665025			−0.332513	−	0.943099i	$-0.607896\pi$
$251$	−10.5360			−0.665025			−0.332513	+	0.943099i	$0.607896\pi$
$252$		0			0
$253$			5.75306i			0.361692i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	9.05621i		−	0.564911i	−0.959280	−	0.282455i	$-0.908851\pi$
$257$		−	9.05621i		−	0.564911i	0.959280	−	0.282455i	$-0.0911489\pi$
$258$		0			0
$259$	26.4579			1.64401
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	8.39011i		−	0.517356i	−0.965964	−	0.258678i	$-0.916713\pi$
$263$		−	8.39011i		−	0.517356i	0.965964	−	0.258678i	$-0.0832869\pi$
$264$		0			0
$265$	−2.90495	+	4.04373i	−0.178450	+	0.248404i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	13.0712			0.796967			0.398484	−	0.917175i	$-0.369537\pi$
$269$	13.0712			0.796967			0.398484	+	0.917175i	$0.369537\pi$
$270$		0			0
$271$	−22.1865			−1.34773			−0.673867	−	0.738853i	$-0.735368\pi$
$271$	−22.1865			−1.34773			−0.673867	+	0.738853i	$0.735368\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	9.18178	+	27.2605i	0.553682	+	1.64387i
$276$		0			0
$277$			20.3257i			1.22125i	0.791919	+	0.610626i	$0.209082\pi$
$277$			20.3257i			1.22125i	−0.791919	+	0.610626i	$0.790918\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	4.09199			0.244108			0.122054	−	0.992523i	$-0.461052\pi$
$281$	4.09199			0.244108			0.122054	+	0.992523i	$0.461052\pi$
$282$		0			0
$283$		−	17.4827i		−	1.03924i	−0.854397	−	0.519620i	$-0.826073\pi$
$283$		−	17.4827i		−	1.03924i	0.854397	−	0.519620i	$-0.173927\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	16.5548i		−	0.977196i
$288$		0			0
$289$	−9.26952			−0.545266
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	9.66599i		−	0.564693i	−0.959312	−	0.282346i	$-0.908887\pi$
$293$		−	9.66599i		−	0.564693i	0.959312	−	0.282346i	$-0.0911128\pi$
$294$		0			0
$295$	−14.4193	−	10.3585i	−0.839521	−	0.603098i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	4.85436			0.280735
$300$		0			0
$301$	−5.77277			−0.332737
$302$		0			0
$303$		0			0
$304$		0			0
$305$	21.5036	+	15.4479i	1.23129	+	0.884542i
$306$		0			0
$307$			21.6323i			1.23462i	0.786720	+	0.617311i	$0.211778\pi$
$307$			21.6323i			1.23462i	−0.786720	+	0.617311i	$0.788222\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−20.7941			−1.17913			−0.589563	−	0.807722i	$-0.700700\pi$
$311$	−20.7941			−1.17913			−0.589563	+	0.807722i	$0.700700\pi$
$312$		0			0
$313$		−	2.69092i		−	0.152100i	−0.997104	−	0.0760498i	$-0.975769\pi$
$313$		−	2.69092i		−	0.152100i	0.997104	−	0.0760498i	$-0.0242308\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	25.1643i		−	1.41337i	−0.707528	−	0.706685i	$-0.750190\pi$
$317$		−	25.1643i		−	1.41337i	0.707528	−	0.706685i	$-0.249810\pi$
$318$		0			0
$319$	−2.96949			−0.166260
$320$		0			0
$321$		0			0
$322$		0			0
$323$			41.5910i			2.31419i
$324$		0			0
$325$	23.0021	−	7.74747i	1.27593	−	0.429752i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	37.6973			2.07832
$330$		0			0
$331$	4.28039			0.235272			0.117636	−	0.993057i	$-0.462468\pi$
$331$	4.28039			0.235272			0.117636	+	0.993057i	$0.462468\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	11.5706	−	16.1064i	0.632167	−	0.879986i
$336$		0			0
$337$		−	3.90650i		−	0.212801i	−0.994323	−	0.106400i	$-0.966067\pi$
$337$		−	3.90650i		−	0.212801i	0.994323	−	0.106400i	$-0.0339325\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	37.2327			2.01627
$342$		0			0
$343$			0.0898947i			0.00485386i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	8.81595i		−	0.473265i	−0.971599	−	0.236633i	$-0.923956\pi$
$347$		−	8.81595i		−	0.473265i	0.971599	−	0.236633i	$-0.0760438\pi$
$348$		0			0
$349$	−4.56125			−0.244158			−0.122079	−	0.992520i	$-0.538956\pi$
$349$	−4.56125			−0.244158			−0.122079	+	0.992520i	$0.538956\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			22.9617i			1.22213i	0.791580	+	0.611065i	$0.209259\pi$
$353$			22.9617i			1.22213i	−0.791580	+	0.611065i	$0.790741\pi$
$354$		0			0
$355$	10.7720	+	7.73844i	0.571720	+	0.410714i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−0.995658			−0.0525488			−0.0262744	−	0.999655i	$-0.508364\pi$
$359$	−0.995658			−0.0525488			−0.0262744	+	0.999655i	$0.508364\pi$
$360$		0			0
$361$	46.8487			2.46572
$362$		0			0
$363$		0			0
$364$		0			0
$365$	18.6967	−	26.0261i	0.978630	−	1.36227i
$366$		0			0
$367$		−	21.9773i		−	1.14720i	−0.819134	−	0.573602i	$-0.805546\pi$
$367$		−	21.9773i		−	1.14720i	0.819134	−	0.573602i	$-0.194454\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−8.32431			−0.432177
$372$		0			0
$373$		−	14.0638i		−	0.728198i	−0.931360	−	0.364099i	$-0.881377\pi$
$373$		−	14.0638i		−	0.728198i	0.931360	−	0.364099i	$-0.118623\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			2.50562i			0.129046i
$378$		0			0
$379$	−22.3420			−1.14763			−0.573816	−	0.818984i	$-0.694538\pi$
$379$	−22.3420			−1.14763			−0.573816	+	0.818984i	$0.694538\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	23.7994i		−	1.21609i	−0.793902	−	0.608046i	$-0.791953\pi$
$383$		−	23.7994i		−	1.21609i	0.793902	−	0.608046i	$-0.208047\pi$
$384$		0			0
$385$	−28.0589	+	39.0584i	−1.43001	+	1.99060i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	23.9689			1.21527			0.607635	−	0.794217i	$-0.292118\pi$
$389$	23.9689			1.21527			0.607635	+	0.794217i	$0.292118\pi$
$390$		0			0
$391$	−5.12538			−0.259202
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−31.2858	−	22.4752i	−1.57416	−	1.13085i
$396$		0			0
$397$			26.9753i			1.35385i	0.736051	+	0.676926i	$0.236688\pi$
$397$			26.9753i			1.35385i	−0.736051	+	0.676926i	$0.763312\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−5.77901			−0.288590			−0.144295	−	0.989535i	$-0.546091\pi$
$401$	−5.77901			−0.288590			−0.144295	+	0.989535i	$0.546091\pi$
$402$		0			0
$403$		−	31.4165i		−	1.56497i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	40.7159i		−	2.01821i
$408$		0			0
$409$	28.8143			1.42477			0.712387	−	0.701787i	$-0.247614\pi$
$409$	28.8143			1.42477			0.712387	+	0.701787i	$0.247614\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	29.6831i		−	1.46061i
$414$		0			0
$415$	−4.87220	+	6.78217i	−0.239167	+	0.332924i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−12.6524			−0.618109			−0.309054	−	0.951044i	$-0.600013\pi$
$419$	−12.6524			−0.618109			−0.309054	+	0.951044i	$0.600013\pi$
$420$		0			0
$421$	1.65723			0.0807684			0.0403842	−	0.999184i	$-0.487142\pi$
$421$	1.65723			0.0807684			0.0403842	+	0.999184i	$0.487142\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−24.2863	+	8.18002i	−1.17806	+	0.396789i
$426$		0			0
$427$			44.2668i			2.14222i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	2.27737			0.109697			0.0548486	−	0.998495i	$-0.482532\pi$
$431$	2.27737			0.109697			0.0548486	+	0.998495i	$0.482532\pi$
$432$		0			0
$433$		−	17.3036i		−	0.831559i	−0.909465	−	0.415780i	$-0.863509\pi$
$433$		−	17.3036i		−	0.831559i	0.909465	−	0.415780i	$-0.136491\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			8.11472i			0.388180i
$438$		0			0
$439$	−24.2985			−1.15970			−0.579851	−	0.814722i	$-0.696889\pi$
$439$	−24.2985			−1.15970			−0.579851	+	0.814722i	$0.696889\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			30.0989i			1.43004i	0.699104	+	0.715020i	$0.253583\pi$
$443$			30.0989i			1.43004i	−0.699104	+	0.715020i	$0.746417\pi$
$444$		0			0
$445$	21.1774	+	15.2135i	1.00390	+	0.721187i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−40.4316			−1.90809			−0.954043	−	0.299670i	$-0.903124\pi$
$449$	−40.4316			−1.90809			−0.954043	+	0.299670i	$0.903124\pi$
$450$		0			0
$451$	−25.4760			−1.19962
$452$		0			0
$453$		0			0
$454$		0			0
$455$	32.9570	+	23.6757i	1.54505	+	1.10994i
$456$		0			0
$457$			1.16559i			0.0545239i	0.999628	+	0.0272619i	$0.00867882\pi$
$457$			1.16559i			0.0545239i	−0.999628	+	0.0272619i	$0.991321\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−17.7593			−0.827131			−0.413566	−	0.910474i	$-0.635717\pi$
$461$	−17.7593			−0.827131			−0.413566	+	0.910474i	$0.635717\pi$
$462$		0			0
$463$			35.1094i			1.63167i	0.578282	+	0.815837i	$0.303723\pi$
$463$			35.1094i			1.63167i	−0.578282	+	0.815837i	$0.696277\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	3.09161i		−	0.143063i	−0.997438	−	0.0715314i	$-0.977211\pi$
$467$		−	3.09161i		−	0.143063i	0.997438	−	0.0715314i	$-0.0227886\pi$
$468$		0			0
$469$	33.1562			1.53101
$470$		0			0
$471$		0			0
$472$		0			0
$473$			8.88367i			0.408472i
$474$		0			0
$475$	12.9510	+	38.4512i	0.594231	+	1.76426i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	6.03828			0.275896			0.137948	−	0.990439i	$-0.455949\pi$
$479$	6.03828			0.275896			0.137948	+	0.990439i	$0.455949\pi$
$480$		0			0
$481$	−34.3555			−1.56648
$482$		0			0
$483$		0			0
$484$		0			0
$485$	0.904547	−	1.25914i	0.0410734	−	0.0571747i
$486$		0			0
$487$			11.4117i			0.517111i	0.965996	+	0.258556i	$0.0832466\pi$
$487$			11.4117i			0.517111i	−0.965996	+	0.258556i	$0.916753\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−33.1654			−1.49673			−0.748366	−	0.663286i	$-0.769161\pi$
$491$	−33.1654			−1.49673			−0.748366	+	0.663286i	$0.769161\pi$
$492$		0			0
$493$		−	2.64551i		−	0.119148i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			22.1750i			0.994684i
$498$		0			0
$499$	6.47182			0.289718			0.144859	−	0.989452i	$-0.453727\pi$
$499$	6.47182			0.289718			0.144859	+	0.989452i	$0.453727\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	40.3818i		−	1.80054i	−0.435335	−	0.900269i	$-0.643370\pi$
$503$		−	40.3818i		−	1.80054i	0.435335	−	0.900269i	$-0.356630\pi$
$504$		0			0
$505$	−26.4828	−	19.0248i	−1.17847	−	0.846593i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−30.3982			−1.34737			−0.673687	−	0.739017i	$-0.735291\pi$
$509$	−30.3982			−1.34737			−0.673687	+	0.739017i	$0.735291\pi$
$510$		0			0
$511$	53.5765			2.37009
$512$		0			0
$513$		0			0
$514$		0			0
$515$	11.1675	−	15.5453i	0.492098	−	0.685008i
$516$		0			0
$517$		−	58.0120i		−	2.55137i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−9.57007			−0.419273			−0.209636	−	0.977779i	$-0.567228\pi$
$521$	−9.57007			−0.419273			−0.209636	+	0.977779i	$0.567228\pi$
$522$		0			0
$523$			5.83229i			0.255028i	0.991837	+	0.127514i	$0.0406998\pi$
$523$			5.83229i			0.255028i	−0.991837	+	0.127514i	$0.959300\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			33.1705i			1.44493i
$528$		0			0
$529$	−1.00000			−0.0434783
$530$		0			0
$531$		0			0
$532$		0			0
$533$			21.4963i			0.931108i
$534$		0			0
$535$	−11.4018	+	15.8715i	−0.492944	+	0.686186i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−40.1331			−1.72865
$540$		0			0
$541$	−9.61788			−0.413505			−0.206753	−	0.978393i	$-0.566290\pi$
$541$	−9.61788			−0.413505			−0.206753	+	0.978393i	$0.566290\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−6.60296	−	4.74345i	−0.282840	−	0.203187i
$546$		0			0
$547$		−	16.3724i		−	0.700034i	−0.936743	−	0.350017i	$-0.886176\pi$
$547$		−	16.3724i		−	0.700034i	0.936743	−	0.350017i	$-0.113824\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−4.18849			−0.178436
$552$		0			0
$553$		−	64.4041i		−	2.73874i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	24.5284i		−	1.03930i	−0.854378	−	0.519652i	$-0.826062\pi$
$557$		−	24.5284i		−	1.03930i	0.854378	−	0.519652i	$-0.173938\pi$
$558$		0			0
$559$	7.49593			0.317044
$560$		0			0
$561$		0			0
$562$		0			0
$563$			29.6330i			1.24888i	0.781072	+	0.624441i	$0.214673\pi$
$563$			29.6330i			1.24888i	−0.781072	+	0.624441i	$0.785327\pi$
$564$		0			0
$565$	−5.57380	+	7.75881i	−0.234492	+	0.326416i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	15.9981			0.670675			0.335338	−	0.942098i	$-0.391150\pi$
$569$	15.9981			0.670675			0.335338	+	0.942098i	$0.391150\pi$
$570$		0			0
$571$	31.3753			1.31301			0.656507	−	0.754320i	$-0.272033\pi$
$571$	31.3753			1.31301			0.656507	+	0.754320i	$0.272033\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−4.73844	+	1.59598i	−0.197607	+	0.0665571i
$576$		0			0
$577$			6.90050i			0.287271i	0.989631	+	0.143636i	$0.0458793\pi$
$577$			6.90050i			0.287271i	−0.989631	+	0.143636i	$0.954121\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−13.9616			−0.579224
$582$		0			0
$583$			12.8102i			0.530545i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			16.5329i			0.682387i	0.939993	+	0.341193i	$0.110831\pi$
$587$			16.5329i			0.682387i	−0.939993	+	0.341193i	$0.889169\pi$
$588$		0			0
$589$	52.5170			2.16393
$590$		0			0
$591$		0			0
$592$		0			0
$593$			22.8397i			0.937912i	0.883222	+	0.468956i	$0.155370\pi$
$593$			22.8397i			0.937912i	−0.883222	+	0.468956i	$0.844630\pi$
$594$		0			0
$595$	−34.7970	−	24.9976i	−1.42654	−	1.02480i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	6.34845			0.259391			0.129695	−	0.991554i	$-0.458600\pi$
$599$	6.34845			0.259391			0.129695	+	0.991554i	$0.458600\pi$
$600$		0			0
$601$	18.1387			0.739893			0.369946	−	0.929053i	$-0.379376\pi$
$601$	18.1387			0.739893			0.369946	+	0.929053i	$0.379376\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	40.1302	+	28.8289i	1.63153	+	1.17206i
$606$		0			0
$607$		−	19.6033i		−	0.795674i	−0.917456	−	0.397837i	$-0.869761\pi$
$607$		−	19.6033i		−	0.795674i	0.917456	−	0.397837i	$-0.130239\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−48.9498			−1.98030
$612$		0			0
$613$			23.9282i			0.966449i	0.875496	+	0.483225i	$0.160535\pi$
$613$			23.9282i			0.966449i	−0.875496	+	0.483225i	$0.839465\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			13.6672i			0.550222i	0.961413	+	0.275111i	$0.0887145\pi$
$617$			13.6672i			0.550222i	−0.961413	+	0.275111i	$0.911285\pi$
$618$		0			0
$619$	−7.40212			−0.297516			−0.148758	−	0.988874i	$-0.547528\pi$
$619$	−7.40212			−0.297516			−0.148758	+	0.988874i	$0.547528\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			43.5951i			1.74660i
$624$		0			0
$625$	−19.9057	+	15.1249i	−0.796227	+	0.604998i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	36.2736			1.44632
$630$		0			0
$631$	−33.2320			−1.32294			−0.661472	−	0.749970i	$-0.730068\pi$
$631$	−33.2320			−1.32294			−0.661472	+	0.749970i	$0.730068\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	1.37892	−	1.91947i	0.0547206	−	0.0761719i
$636$		0			0
$637$			33.8638i			1.34173i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−42.2581			−1.66910			−0.834548	−	0.550935i	$-0.814271\pi$
$641$	−42.2581			−1.66910			−0.834548	+	0.550935i	$0.814271\pi$
$642$		0			0
$643$		−	14.5129i		−	0.572334i	−0.958180	−	0.286167i	$-0.907619\pi$
$643$		−	14.5129i		−	0.572334i	0.958180	−	0.286167i	$-0.0923812\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	9.38824i		−	0.369090i	−0.982824	−	0.184545i	$-0.940919\pi$
$647$		−	9.38824i		−	0.369090i	0.982824	−	0.184545i	$-0.0590811\pi$
$648$		0			0
$649$	−45.6790			−1.79306
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	29.0164i		−	1.13550i	−0.823202	−	0.567749i	$-0.807814\pi$
$653$		−	29.0164i		−	1.13550i	0.823202	−	0.567749i	$-0.192186\pi$
$654$		0			0
$655$	−2.44957	−	1.75973i	−0.0957128	−	0.0687584i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	0.913493			0.0355846			0.0177923	−	0.999842i	$-0.494336\pi$
$659$	0.913493			0.0355846			0.0177923	+	0.999842i	$0.494336\pi$
$660$		0			0
$661$	5.60114			0.217859			0.108930	−	0.994049i	$-0.465258\pi$
$661$	5.60114			0.217859			0.108930	+	0.994049i	$0.465258\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−39.5772	+	55.0921i	−1.53474	+	2.13638i
$666$		0			0
$667$		−	0.516159i		−	0.0199858i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	68.1218			2.62981
$672$		0			0
$673$		−	8.76449i		−	0.337846i	−0.985629	−	0.168923i	$-0.945971\pi$
$673$		−	8.76449i		−	0.337846i	0.985629	−	0.168923i	$-0.0540290\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	3.16274i		−	0.121554i	−0.998151	−	0.0607770i	$-0.980642\pi$
$677$		−	3.16274i		−	0.121554i	0.998151	−	0.0607770i	$-0.0193579\pi$
$678$		0			0
$679$	2.59204			0.0994732
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	35.7646i		−	1.36849i	−0.729251	−	0.684247i	$-0.760131\pi$
$683$		−	35.7646i		−	1.36849i	0.729251	−	0.684247i	$-0.239869\pi$
$684$		0			0
$685$	−9.27296	+	12.9081i	−0.354302	+	0.493193i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	10.8091			0.411794
$690$		0			0
$691$	27.6141			1.05049			0.525246	−	0.850951i	$-0.323973\pi$
$691$	27.6141			1.05049			0.525246	+	0.850951i	$0.323973\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−27.7708	−	19.9501i	−1.05341	−	0.756749i
$696$		0			0
$697$		−	22.6965i		−	0.859690i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	8.44634			0.319014			0.159507	−	0.987197i	$-0.449010\pi$
$701$	8.44634			0.319014			0.159507	+	0.987197i	$0.449010\pi$
$702$		0			0
$703$		−	57.4299i		−	2.16601i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	54.5168i		−	2.05032i
$708$		0			0
$709$	−28.5887			−1.07367			−0.536835	−	0.843687i	$-0.680380\pi$
$709$	−28.5887			−1.07367			−0.536835	+	0.843687i	$0.680380\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			6.47182i			0.242371i
$714$		0			0
$715$	36.4344	−	50.7172i	1.36257	−	1.89672i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−15.4763			−0.577168			−0.288584	−	0.957455i	$-0.593185\pi$
$719$	−15.4763			−0.577168			−0.288584	+	0.957455i	$0.593185\pi$
$720$		0			0
$721$	32.0011			1.19178
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−0.823781	−	2.44579i	−0.0305944	−	0.0908343i
$726$		0			0
$727$		−	9.94521i		−	0.368847i	−0.982847	−	0.184424i	$-0.940958\pi$
$727$		−	9.94521i		−	0.368847i	0.982847	−	0.184424i	$-0.0590418\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−7.91443			−0.292726
$732$		0			0
$733$		−	9.73749i		−	0.359662i	−0.983697	−	0.179831i	$-0.942445\pi$
$733$		−	9.73749i		−	0.359662i	0.983697	−	0.179831i	$-0.0575551\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	51.0237i		−	1.87948i
$738$		0			0
$739$	45.8475			1.68653			0.843264	−	0.537500i	$-0.180631\pi$
$739$	45.8475			1.68653			0.843264	+	0.537500i	$0.180631\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	23.7465i		−	0.871176i	−0.900146	−	0.435588i	$-0.856540\pi$
$743$		−	23.7465i		−	0.871176i	0.900146	−	0.435588i	$-0.143460\pi$
$744$		0			0
$745$	−13.3199	−	9.56878i	−0.488003	−	0.350573i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−32.6727			−1.19383
$750$		0			0
$751$	−18.1883			−0.663700			−0.331850	−	0.943332i	$-0.607673\pi$
$751$	−18.1883			−0.663700			−0.331850	+	0.943332i	$0.607673\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	1.10964	+	0.797149i	0.0403840	+	0.0290112i
$756$		0			0
$757$		−	6.40698i		−	0.232866i	−0.993199	−	0.116433i	$-0.962854\pi$
$757$		−	6.40698i		−	0.232866i	0.993199	−	0.116433i	$-0.0371460\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−18.6546			−0.676230			−0.338115	−	0.941105i	$-0.609789\pi$
$761$	−18.6546			−0.676230			−0.338115	+	0.941105i	$0.609789\pi$
$762$		0			0
$763$		−	13.5927i		−	0.492087i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			38.5434i			1.39172i
$768$		0			0
$769$	−34.5037			−1.24423			−0.622117	−	0.782924i	$-0.713727\pi$
$769$	−34.5037			−1.24423			−0.622117	+	0.782924i	$0.713727\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			11.8163i			0.425003i	0.977161	+	0.212501i	$0.0681610\pi$
$773$			11.8163i			0.425003i	−0.977161	+	0.212501i	$0.931839\pi$
$774$		0			0
$775$	10.3289	+	30.6663i	0.371025	+	1.10157i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−35.9340			−1.28747
$780$		0			0
$781$	34.1249			1.22109
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−12.2594	+	17.0653i	−0.437557	+	0.609086i
$786$		0			0
$787$		−	46.6006i		−	1.66113i	−0.556921	−	0.830566i	$-0.688017\pi$
$787$		−	46.6006i		−	1.66113i	0.556921	−	0.830566i	$-0.311983\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−15.9721			−0.567901
$792$		0			0
$793$		−	57.4803i		−	2.04119i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	1.37291i		−	0.0486309i	−0.999704	−	0.0243155i	$-0.992259\pi$
$797$		−	1.37291i		−	0.0486309i	0.999704	−	0.0243155i	$-0.00774062\pi$
$798$		0			0
$799$	51.6827			1.82840
$800$		0			0
$801$		0			0
$802$		0			0
$803$		−	82.4485i		−	2.90954i
$804$		0			0
$805$	−6.78915	−	4.87721i	−0.239286	−	0.171899i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−8.09883			−0.284740			−0.142370	−	0.989814i	$-0.545472\pi$
$809$	−8.09883			−0.284740			−0.142370	+	0.989814i	$0.545472\pi$
$810$		0			0
$811$	−33.8860			−1.18990			−0.594950	−	0.803763i	$-0.702828\pi$
$811$	−33.8860			−1.18990			−0.594950	+	0.803763i	$0.702828\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	17.0402	−	23.7203i	0.596893	−	0.830885i
$816$		0			0
$817$			12.5305i			0.438386i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−23.1137			−0.806675			−0.403337	−	0.915051i	$-0.632150\pi$
$821$	−23.1137			−0.806675			−0.403337	+	0.915051i	$0.632150\pi$
$822$		0			0
$823$		−	7.33401i		−	0.255648i	−0.991797	−	0.127824i	$-0.959201\pi$
$823$		−	7.33401i		−	0.255648i	0.991797	−	0.127824i	$-0.0407992\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	32.1879i		−	1.11928i	−0.828735	−	0.559641i	$-0.810939\pi$
$827$		−	32.1879i		−	1.11928i	0.828735	−	0.559641i	$-0.189061\pi$
$828$		0			0
$829$	4.90900			0.170497			0.0852483	−	0.996360i	$-0.472832\pi$
$829$	4.90900			0.170497			0.0852483	+	0.996360i	$0.472832\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	35.7544i		−	1.23882i
$834$		0			0
$835$	−8.77942	+	12.2211i	−0.303824	+	0.422928i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	48.5591			1.67645			0.838223	−	0.545327i	$-0.183595\pi$
$839$	48.5591			1.67645			0.838223	+	0.545327i	$0.183595\pi$
$840$		0			0
$841$	−28.7336			−0.990813
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−19.1861	−	13.7829i	−0.660021	−	0.474148i
$846$		0			0
$847$			82.6110i			2.83855i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	7.07725			0.242605
$852$		0			0
$853$		−	8.65980i		−	0.296506i	−0.988949	−	0.148253i	$-0.952635\pi$
$853$		−	8.65980i		−	0.296506i	0.988949	−	0.148253i	$-0.0473650\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			7.44895i			0.254451i	0.991874	+	0.127226i	$0.0406073\pi$
$857$			7.44895i			0.254451i	−0.991874	+	0.127226i	$0.959393\pi$
$858$		0			0
$859$	2.02819			0.0692010			0.0346005	−	0.999401i	$-0.488984\pi$
$859$	2.02819			0.0692010			0.0346005	+	0.999401i	$0.488984\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	53.1631i		−	1.80969i	−0.425738	−	0.904847i	$-0.639985\pi$
$863$		−	53.1631i		−	1.80969i	0.425738	−	0.904847i	$-0.360015\pi$
$864$		0			0
$865$	−9.03923	+	12.5827i	−0.307343	+	0.427826i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−99.1110			−3.36211
$870$		0			0
$871$	−43.0532			−1.45880
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−39.9539	−	12.2750i	−1.35069	−	0.414971i
$876$		0			0
$877$		−	22.7737i		−	0.769012i	−0.923122	−	0.384506i	$-0.874372\pi$
$877$		−	22.7737i		−	0.769012i	0.923122	−	0.384506i	$-0.125628\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	44.1628			1.48788			0.743941	−	0.668246i	$-0.232954\pi$
$881$	44.1628			1.48788			0.743941	+	0.668246i	$0.232954\pi$
$882$		0			0
$883$		−	37.9129i		−	1.27587i	−0.770091	−	0.637935i	$-0.779789\pi$
$883$		−	37.9129i		−	1.27587i	0.770091	−	0.637935i	$-0.220211\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			12.3606i			0.415030i	0.978232	+	0.207515i	$0.0665375\pi$
$887$			12.3606i			0.415030i	−0.978232	+	0.207515i	$0.933462\pi$
$888$		0			0
$889$	3.95137			0.132525
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	81.8263i		−	2.73821i
$894$		0			0
$895$	−26.7988	−	19.2518i	−0.895786	−	0.643518i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−3.34049			−0.111411
$900$		0			0
$901$	−11.4126			−0.380208
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−13.3281	−	9.57465i	−0.443040	−	0.318272i
$906$		0			0
$907$			9.25394i			0.307272i	0.988128	+	0.153636i	$0.0490983\pi$
$907$			9.25394i			0.307272i	−0.988128	+	0.153636i	$0.950902\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	25.3592			0.840188			0.420094	−	0.907481i	$-0.361997\pi$
$911$	25.3592			0.840188			0.420094	+	0.907481i	$0.361997\pi$
$912$		0			0
$913$			21.4854i			0.711062i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	5.04262i		−	0.166522i
$918$		0			0
$919$	−44.3171			−1.46189			−0.730943	−	0.682438i	$-0.760920\pi$
$919$	−44.3171			−1.46189			−0.730943	+	0.682438i	$0.760920\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	28.7942i		−	0.947772i
$924$		0			0
$925$	33.5352	−	11.2952i	1.10263	−	0.371383i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	56.3282			1.84807			0.924034	−	0.382309i	$-0.124871\pi$
$929$	56.3282			1.84807			0.924034	+	0.382309i	$0.124871\pi$
$930$		0			0
$931$	−56.6079			−1.85525
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−38.4685	+	53.5488i	−1.25806	+	1.75123i
$936$		0			0
$937$		−	6.34844i		−	0.207394i	−0.994609	−	0.103697i	$-0.966933\pi$
$937$		−	6.34844i		−	0.207394i	0.994609	−	0.103697i	$-0.0330673\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−8.20087			−0.267341			−0.133670	−	0.991026i	$-0.542676\pi$
$941$	−8.20087			−0.267341			−0.133670	+	0.991026i	$0.542676\pi$
$942$		0			0
$943$		−	4.42825i		−	0.144204i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			22.8385i			0.742152i	0.928603	+	0.371076i	$0.121011\pi$
$947$			22.8385i			0.742152i	−0.928603	+	0.371076i	$0.878989\pi$
$948$		0			0
$949$	−69.5690			−2.25831
$950$		0			0
$951$		0			0
$952$		0			0
$953$			16.9081i			0.547706i	0.961772	+	0.273853i	$0.0882982\pi$
$953$			16.9081i			0.547706i	−0.961772	+	0.273853i	$0.911702\pi$
$954$		0			0
$955$	−31.7805	−	22.8306i	−1.02839	−	0.738780i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−26.5723			−0.858063
$960$		0			0
$961$	10.8844			0.351110
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−24.6895	+	34.3681i	−0.794782	+	1.10635i
$966$		0			0
$967$			2.36769i			0.0761398i	0.999275	+	0.0380699i	$0.0121210\pi$
$967$			2.36769i			0.0761398i	−0.999275	+	0.0380699i	$0.987879\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	39.2027			1.25808			0.629038	−	0.777375i	$-0.283449\pi$
$971$	39.2027			1.25808			0.629038	+	0.777375i	$0.283449\pi$
$972$		0			0
$973$		−	57.1682i		−	1.83273i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			36.8003i			1.17734i	0.808372	+	0.588672i	$0.200349\pi$
$977$			36.8003i			1.17734i	−0.808372	+	0.588672i	$0.799651\pi$
$978$		0			0
$979$	67.0882			2.14415
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	10.3521i		−	0.330180i	−0.986278	−	0.165090i	$-0.947209\pi$
$983$		−	10.3521i		−	0.330180i	0.986278	−	0.165090i	$-0.0527915\pi$
$984$		0			0
$985$	−29.7756	+	41.4481i	−0.948729	+	1.32065i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−1.54416			−0.0491016
$990$		0			0
$991$	57.8425			1.83743			0.918714	−	0.394924i	$-0.129229\pi$
$991$	57.8425			1.83743			0.918714	+	0.394924i	$0.129229\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	37.5517	+	26.9765i	1.19047	+	0.855212i
$996$		0			0
$997$		−	37.1071i		−	1.17520i	−0.809153	−	0.587598i	$-0.800074\pi$
$997$		−	37.1071i		−	1.17520i	0.809153	−	0.587598i	$-0.199926\pi$
$998$		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.829.12		14
3.2	odd	2		1380.2.f.b.829.9	yes	14
5.4	even	2	inner	4140.2.f.c.829.11		14
15.2	even	4		6900.2.a.bd.1.2		7
15.8	even	4		6900.2.a.bc.1.6		7
15.14	odd	2		1380.2.f.b.829.2	✓	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1380.2.f.b.829.2	✓	14	15.14	odd	2
1380.2.f.b.829.9	yes	14	3.2	odd	2
4140.2.f.c.829.11		14	5.4	even	2	inner
4140.2.f.c.829.12		14	1.1	even	1	trivial
6900.2.a.bc.1.6		7	15.8	even	4
6900.2.a.bd.1.2		7	15.2	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)

\(f(q)\)	\(=\)	\(q+(1.81604 + 1.30461i) q^{5} +3.73844i q^{7} +O(q^{10})\)	\(q+(1.81604 + 1.30461i) q^{5} +3.73844i q^{7} +5.75306 q^{11} -4.85436i q^{13} +5.12538i q^{17} +8.11472 q^{19} +1.00000i q^{23} +(1.59598 + 4.73844i) q^{25} -0.516159 q^{29} +6.47182 q^{31} +(-4.87721 + 6.78915i) q^{35} -7.07725i q^{37} -4.42825 q^{41} +1.54416i q^{43} -10.0837i q^{47} -6.97595 q^{49} +2.22668i q^{53} +(10.4478 + 7.50550i) q^{55} -7.93996 q^{59} +11.8410 q^{61} +(6.33305 - 8.81570i) q^{65} -8.86898i q^{67} +5.93161 q^{71} -14.3312i q^{73} +21.5075i q^{77} -17.2275 q^{79} +3.73460i q^{83} +(-6.68662 + 9.30788i) q^{85} +11.6613 q^{89} +18.1477 q^{91} +(14.7366 + 10.5866i) q^{95} -0.693347i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q+O(q^{10}) \) 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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