LMFDB - Embedded newform 4140.2.f.b.829.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4140,2,Mod(829,4140)] 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("4140.829"); S:= CuspForms(chi, 2); N := Newforms(S);

sage: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")

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:traces = [12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$33.0580664368$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 24x^{10} + 188x^{8} + 530x^{6} + 508x^{4} + 80x^{2} + 1 $ x^12 + 24x^10 + 188x^8 + 530x^6 + 508x^4 + 80*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 460)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			829.4
Root			$3.16223i$ of defining polynomial
Character	$\chi$	$=$	4140.829
Dual form			4140.2.f.b.829.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.59013 + 1.57210i) q^{5} +2.43185i q^{7} +0.884969 q^{11} +5.10522i q^{13} +0.366626i q^{17} +2.79847 q^{19} -1.00000i q^{23} +(0.0570016 - 4.99968i) q^{25} +8.02431 q^{29} +7.24179 q^{31} +(-3.82311 - 3.86694i) q^{35} +3.10036i q^{37} +3.47185 q^{41} +8.56841i q^{43} +5.25528i q^{47} +1.08612 q^{49} +11.6413i q^{53} +(-1.40721 + 1.39126i) q^{55} -9.33209 q^{59} +5.46699 q^{61} +(-8.02592 - 8.11795i) q^{65} -1.49020i q^{67} -8.29949 q^{71} -10.2409i q^{73} +2.15211i q^{77} -6.06522 q^{79} -16.2520i q^{83} +(-0.576373 - 0.582981i) q^{85} -17.6033 q^{89} -12.4151 q^{91} +(-4.44992 + 4.39948i) q^{95} +6.55618i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 4 q^{11} - 8 q^{19} + 8 q^{25} + 10 q^{29} + 18 q^{31} + 10 q^{35} + 2 q^{41} - 38 q^{49} + 16 q^{55} - 22 q^{59} - 8 q^{61} - 38 q^{65} + 34 q^{71} - 20 q^{79} + 6 q^{85} - 48 q^{89} - 8 q^{91}+ \cdots - 12 q^{95}+O(q^{100}) $ 12 * q - 4 * q^11 - 8 * q^19 + 8 * q^25 + 10 * q^29 + 18 * q^31 + 10 * q^35 + 2 * q^41 - 38 * q^49 + 16 * q^55 - 22 * q^59 - 8 * q^61 - 38 * q^65 + 34 * q^71 - 20 * q^79 + 6 * q^85 - 48 * q^89 - 8 * q^91 - 12 * q^95

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
$5$	−1.59013	+	1.57210i	−0.711126	+	0.703065i
$5$	−1.59013	+	1.57210i	−0.711126	+	0.703065i
$6$		0			0
$7$			2.43185i			0.919152i	0.888138	+	0.459576i	$0.151999\pi$
$7$			2.43185i			0.919152i	−0.888138	+	0.459576i	$0.848001\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	0.884969			0.266828			0.133414	−	0.991060i	$-0.457406\pi$
$11$	0.884969			0.266828			0.133414	+	0.991060i	$0.457406\pi$
$12$		0			0
$13$			5.10522i			1.41593i	0.706245	+	0.707967i	$0.250388\pi$
$13$			5.10522i			1.41593i	−0.706245	+	0.707967i	$0.749612\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			0.366626i			0.0889198i	0.999011	+	0.0444599i	$0.0141567\pi$
$17$			0.366626i			0.0889198i	−0.999011	+	0.0444599i	$0.985843\pi$
$18$		0			0
$19$	2.79847			0.642014			0.321007	−	0.947077i	$-0.395979\pi$
$19$	2.79847			0.642014			0.321007	+	0.947077i	$0.395979\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	1.00000i		−	0.208514i
$23$		−	1.00000i		−	0.208514i
$24$		0			0
$25$	0.0570016	−	4.99968i	0.0114003	−	0.999935i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	8.02431			1.49008			0.745038	−	0.667022i	$-0.232431\pi$
$29$	8.02431			1.49008			0.745038	+	0.667022i	$0.232431\pi$
$30$		0			0
$31$	7.24179			1.30066			0.650332	−	0.759650i	$-0.274630\pi$
$31$	7.24179			1.30066			0.650332	+	0.759650i	$0.274630\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−3.82311	−	3.86694i	−0.646223	−	0.653633i
$36$		0			0
$37$			3.10036i			0.509697i	0.966981	+	0.254848i	$0.0820256\pi$
$37$			3.10036i			0.509697i	−0.966981	+	0.254848i	$0.917974\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	3.47185			0.542212			0.271106	−	0.962550i	$-0.412611\pi$
$41$	3.47185			0.542212			0.271106	+	0.962550i	$0.412611\pi$
$42$		0			0
$43$			8.56841i			1.30667i	0.757069	+	0.653335i	$0.226631\pi$
$43$			8.56841i			1.30667i	−0.757069	+	0.653335i	$0.773369\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			5.25528i			0.766561i	0.923632	+	0.383281i	$0.125206\pi$
$47$			5.25528i			0.766561i	−0.923632	+	0.383281i	$0.874794\pi$
$48$		0			0
$49$	1.08612			0.155160
$50$		0			0
$51$		0			0
$52$		0			0
$53$			11.6413i			1.59905i	0.600632	+	0.799526i	$0.294916\pi$
$53$			11.6413i			1.59905i	−0.600632	+	0.799526i	$0.705084\pi$
$54$		0			0
$55$	−1.40721	+	1.39126i	−0.189749	+	0.187598i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−9.33209			−1.21493			−0.607467	−	0.794345i	$-0.707814\pi$
$59$	−9.33209			−1.21493			−0.607467	+	0.794345i	$0.707814\pi$
$60$		0			0
$61$	5.46699			0.699976			0.349988	−	0.936754i	$-0.386186\pi$
$61$	5.46699			0.699976			0.349988	+	0.936754i	$0.386186\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−8.02592	−	8.11795i	−0.995493	−	1.00691i
$66$		0			0
$67$		−	1.49020i		−	0.182057i	−0.995848	−	0.0910285i	$-0.970985\pi$
$67$		−	1.49020i		−	0.182057i	0.995848	−	0.0910285i	$-0.0290154\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−8.29949			−0.984969			−0.492484	−	0.870321i	$-0.663911\pi$
$71$	−8.29949			−0.984969			−0.492484	+	0.870321i	$0.663911\pi$
$72$		0			0
$73$		−	10.2409i		−	1.19860i	−0.800523	−	0.599302i	$-0.795445\pi$
$73$		−	10.2409i		−	1.19860i	0.800523	−	0.599302i	$-0.204555\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			2.15211i			0.245256i
$78$		0			0
$79$	−6.06522			−0.682391			−0.341195	−	0.939992i	$-0.610832\pi$
$79$	−6.06522			−0.682391			−0.341195	+	0.939992i	$0.610832\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	16.2520i		−	1.78389i	−0.452148	−	0.891943i	$-0.649342\pi$
$83$		−	16.2520i		−	1.78389i	0.452148	−	0.891943i	$-0.350658\pi$
$84$		0			0
$85$	−0.576373	−	0.582981i	−0.0625164	−	0.0632332i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−17.6033			−1.86594			−0.932972	−	0.359949i	$-0.882794\pi$
$89$	−17.6033			−1.86594			−0.932972	+	0.359949i	$0.882794\pi$
$90$		0			0
$91$	−12.4151			−1.30146
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−4.44992	+	4.39948i	−0.456553	+	0.451377i
$96$		0			0
$97$			6.55618i			0.665679i	0.942984	+	0.332839i	$0.108007\pi$
$97$			6.55618i			0.665679i	−0.942984	+	0.332839i	$0.891993\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−13.4912			−1.34243			−0.671213	−	0.741265i	$-0.734226\pi$
$101$	−13.4912			−1.34243			−0.671213	+	0.741265i	$0.734226\pi$
$102$		0			0
$103$			12.5460i			1.23619i	0.786103	+	0.618096i	$0.212096\pi$
$103$			12.5460i			1.23619i	−0.786103	+	0.618096i	$0.787904\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			1.46269i			0.141404i	0.997497	+	0.0707020i	$0.0225239\pi$
$107$			1.46269i			0.141404i	−0.997497	+	0.0707020i	$0.977476\pi$
$108$		0			0
$109$	19.2173			1.84069			0.920343	−	0.391113i	$-0.127910\pi$
$109$	19.2173			1.84069			0.920343	+	0.391113i	$0.127910\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			0.834901i			0.0785409i	0.999229	+	0.0392705i	$0.0125034\pi$
$113$			0.834901i			0.0785409i	−0.999229	+	0.0392705i	$0.987497\pi$
$114$		0			0
$115$	1.57210	+	1.59013i	0.146599	+	0.148280i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−0.891578			−0.0817308
$120$		0			0
$121$	−10.2168			−0.928803
$122$		0			0
$123$		0			0
$124$		0			0
$125$	7.76935	+	8.03973i	0.694912	+	0.719095i
$126$		0			0
$127$		−	7.87453i		−	0.698751i	−0.936983	−	0.349376i	$-0.886394\pi$
$127$		−	7.87453i		−	0.698751i	0.936983	−	0.349376i	$-0.113606\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	12.8740			1.12481			0.562403	−	0.826863i	$-0.309877\pi$
$131$	12.8740			1.12481			0.562403	+	0.826863i	$0.309877\pi$
$132$		0			0
$133$			6.80546i			0.590108i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			3.26675i			0.279097i	0.990215	+	0.139549i	$0.0445651\pi$
$137$			3.26675i			0.279097i	−0.990215	+	0.139549i	$0.955435\pi$
$138$		0			0
$139$	12.5578			1.06514			0.532570	−	0.846386i	$-0.321226\pi$
$139$	12.5578			1.06514			0.532570	+	0.846386i	$0.321226\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			4.51797i			0.377811i
$144$		0			0
$145$	−12.7597	+	12.6150i	−1.05963	+	1.04762i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−0.188265			−0.0154233			−0.00771164	−	0.999970i	$-0.502455\pi$
$149$	−0.188265			−0.0154233			−0.00771164	+	0.999970i	$0.502455\pi$
$150$		0			0
$151$	−18.6708			−1.51941			−0.759704	−	0.650269i	$-0.774656\pi$
$151$	−18.6708			−1.51941			−0.759704	+	0.650269i	$0.774656\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−11.5154	+	11.3848i	−0.924936	+	0.914451i
$156$		0			0
$157$		−	20.0277i		−	1.59839i	−0.601074	−	0.799193i	$-0.705260\pi$
$157$		−	20.0277i		−	1.59839i	0.601074	−	0.799193i	$-0.294740\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	2.43185			0.191656
$162$		0			0
$163$			3.35607i			0.262867i	0.991325	+	0.131434i	$0.0419580\pi$
$163$			3.35607i			0.262867i	−0.991325	+	0.131434i	$0.958042\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	2.57745i		−	0.199449i	−0.995015	−	0.0997246i	$-0.968204\pi$
$167$		−	2.57745i		−	0.199449i	0.995015	−	0.0997246i	$-0.0317962\pi$
$168$		0			0
$169$	−13.0633			−1.00487
$170$		0			0
$171$		0			0
$172$		0			0
$173$			19.4182i			1.47634i	0.674615	+	0.738170i	$0.264310\pi$
$173$			19.4182i			1.47634i	−0.674615	+	0.738170i	$0.735690\pi$
$174$		0			0
$175$	12.1584	+	0.138619i	0.919092	+	0.0104786i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−20.3628			−1.52199			−0.760993	−	0.648760i	$-0.775288\pi$
$179$	−20.3628			−1.52199			−0.760993	+	0.648760i	$0.775288\pi$
$180$		0			0
$181$	−22.5629			−1.67709			−0.838543	−	0.544835i	$-0.816592\pi$
$181$	−22.5629			−1.67709			−0.838543	+	0.544835i	$0.816592\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−4.87408	−	4.92997i	−0.358350	−	0.362459i
$186$		0			0
$187$			0.324453i			0.0237263i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	16.6554			1.20514			0.602571	−	0.798065i	$-0.294143\pi$
$191$	16.6554			1.20514			0.602571	+	0.798065i	$0.294143\pi$
$192$		0			0
$193$			14.5214i			1.04527i	0.852556	+	0.522636i	$0.175051\pi$
$193$			14.5214i			1.04527i	−0.852556	+	0.522636i	$0.824949\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	13.3479i		−	0.950995i	−0.879717	−	0.475498i	$-0.842268\pi$
$197$		−	13.3479i		−	0.950995i	0.879717	−	0.475498i	$-0.157732\pi$
$198$		0			0
$199$	−18.7856			−1.33168			−0.665838	−	0.746097i	$-0.731926\pi$
$199$	−18.7856			−1.33168			−0.665838	+	0.746097i	$0.731926\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$			19.5139i			1.36961i
$204$		0			0
$205$	−5.52068	+	5.45810i	−0.385581	+	0.381210i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	2.47656			0.171307
$210$		0			0
$211$	17.1078			1.17775			0.588874	−	0.808225i	$-0.299571\pi$
$211$	17.1078			1.17775			0.588874	+	0.808225i	$0.299571\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−13.4704	−	13.6249i	−0.918674	−	0.929207i
$216$		0			0
$217$			17.6109i			1.19551i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−1.87171			−0.125905
$222$		0			0
$223$		−	13.5184i		−	0.905262i	−0.891698	−	0.452631i	$-0.850485\pi$
$223$		−	13.5184i		−	0.905262i	0.891698	−	0.452631i	$-0.149515\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			8.29581i			0.550612i	0.961357	+	0.275306i	$0.0887792\pi$
$227$			8.29581i			0.550612i	−0.961357	+	0.275306i	$0.911221\pi$
$228$		0			0
$229$	6.22318			0.411239			0.205620	−	0.978632i	$-0.434079\pi$
$229$	6.22318			0.411239			0.205620	+	0.978632i	$0.434079\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			13.6347i			0.893242i	0.894723	+	0.446621i	$0.147373\pi$
$233$			13.6347i			0.893242i	−0.894723	+	0.446621i	$0.852627\pi$
$234$		0			0
$235$	−8.26183	−	8.35656i	−0.538942	−	0.545122i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	11.5281			0.745693			0.372847	−	0.927893i	$-0.378382\pi$
$239$	11.5281			0.745693			0.372847	+	0.927893i	$0.378382\pi$
$240$		0			0
$241$	−17.5952			−1.13341			−0.566704	−	0.823921i	$-0.691782\pi$
$241$	−17.5952			−1.13341			−0.566704	+	0.823921i	$0.691782\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−1.72707	+	1.70749i	−0.110338	+	0.109087i
$246$		0			0
$247$			14.2868i			0.909049i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−12.5471			−0.791968			−0.395984	−	0.918257i	$-0.629596\pi$
$251$	−12.5471			−0.791968			−0.395984	+	0.918257i	$0.629596\pi$
$252$		0			0
$253$		−	0.884969i		−	0.0556375i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			4.88072i			0.304451i	0.988346	+	0.152225i	$0.0486439\pi$
$257$			4.88072i			0.304451i	−0.988346	+	0.152225i	$0.951356\pi$
$258$		0			0
$259$	−7.53961			−0.468489
$260$		0			0
$261$		0			0
$262$		0			0
$263$			11.6045i			0.715566i	0.933805	+	0.357783i	$0.116467\pi$
$263$			11.6045i			0.715566i	−0.933805	+	0.357783i	$0.883533\pi$
$264$		0			0
$265$	−18.3012	−	18.5111i	−1.12424	−	1.13713i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−6.04327			−0.368465			−0.184232	−	0.982883i	$-0.558980\pi$
$269$	−6.04327			−0.368465			−0.184232	+	0.982883i	$0.558980\pi$
$270$		0			0
$271$	3.93401			0.238974			0.119487	−	0.992836i	$-0.461875\pi$
$271$	3.93401			0.238974			0.119487	+	0.992836i	$0.461875\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	0.0504447	−	4.42456i	0.00304193	−	0.266811i
$276$		0			0
$277$		−	7.92174i		−	0.475971i	−0.971269	−	0.237986i	$-0.923513\pi$
$277$		−	7.92174i		−	0.475971i	0.971269	−	0.237986i	$-0.0764871\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	6.00419			0.358180			0.179090	−	0.983833i	$-0.442685\pi$
$281$	6.00419			0.358180			0.179090	+	0.983833i	$0.442685\pi$
$282$		0			0
$283$			7.65299i			0.454923i	0.973787	+	0.227461i	$0.0730426\pi$
$283$			7.65299i			0.454923i	−0.973787	+	0.227461i	$0.926957\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			8.44301i			0.498375i
$288$		0			0
$289$	16.8656			0.992093
$290$		0			0
$291$		0			0
$292$		0			0
$293$			13.5480i			0.791485i	0.918361	+	0.395743i	$0.129513\pi$
$293$			13.5480i			0.791485i	−0.918361	+	0.395743i	$0.870487\pi$
$294$		0			0
$295$	14.8392	−	14.6710i	0.863971	−	0.854177i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	5.10522			0.295243
$300$		0			0
$301$	−20.8371			−1.20103
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−8.69320	+	8.59466i	−0.497771	+	0.492129i
$306$		0			0
$307$		−	10.0870i		−	0.575697i	−0.957676	−	0.287848i	$-0.907060\pi$
$307$		−	10.0870i		−	0.575697i	0.957676	−	0.287848i	$-0.0929399\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	28.2481			1.60180			0.800902	−	0.598795i	$-0.204354\pi$
$311$	28.2481			1.60180			0.800902	+	0.598795i	$0.204354\pi$
$312$		0			0
$313$		−	14.4504i		−	0.816786i	−0.912806	−	0.408393i	$-0.866089\pi$
$313$		−	14.4504i		−	0.816786i	0.912806	−	0.408393i	$-0.133911\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			14.7835i			0.830324i	0.909748	+	0.415162i	$0.136275\pi$
$317$			14.7835i			0.830324i	−0.909748	+	0.415162i	$0.863725\pi$
$318$		0			0
$319$	7.10127			0.397595
$320$		0			0
$321$		0			0
$322$		0			0
$323$			1.02599i			0.0570877i
$324$		0			0
$325$	25.5245	+	0.291006i	1.41584	+	0.0161421i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−12.7800			−0.704586
$330$		0			0
$331$	−20.1494			−1.10751			−0.553757	−	0.832679i	$-0.686806\pi$
$331$	−20.1494			−1.10751			−0.553757	+	0.832679i	$0.686806\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	2.34275	+	2.36961i	0.127998	+	0.129465i
$336$		0			0
$337$		−	24.0660i		−	1.31096i	−0.755214	−	0.655479i	$-0.772467\pi$
$337$		−	24.0660i		−	1.31096i	0.755214	−	0.655479i	$-0.227533\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	6.40876			0.347054
$342$		0			0
$343$			19.6642i			1.06177i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	26.3788i		−	1.41609i	−0.706168	−	0.708045i	$-0.749578\pi$
$347$		−	26.3788i		−	1.41609i	0.706168	−	0.708045i	$-0.250422\pi$
$348$		0			0
$349$	−0.535518			−0.0286656			−0.0143328	−	0.999897i	$-0.504562\pi$
$349$	−0.535518			−0.0286656			−0.0143328	+	0.999897i	$0.504562\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			5.27606i			0.280816i	0.990094	+	0.140408i	$0.0448415\pi$
$353$			5.27606i			0.280816i	−0.990094	+	0.140408i	$0.955159\pi$
$354$		0			0
$355$	13.1972	−	13.0476i	0.700437	−	0.692497i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	20.4351			1.07852			0.539261	−	0.842139i	$-0.318704\pi$
$359$	20.4351			1.07852			0.539261	+	0.842139i	$0.318704\pi$
$360$		0			0
$361$	−11.1685			−0.587818
$362$		0			0
$363$		0			0
$364$		0			0
$365$	16.0997	+	16.2843i	0.842696	+	0.852359i
$366$		0			0
$367$		−	12.4382i		−	0.649268i	−0.945840	−	0.324634i	$-0.894759\pi$
$367$		−	12.4382i		−	0.649268i	0.945840	−	0.324634i	$-0.105241\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−28.3098			−1.46977
$372$		0			0
$373$			2.99999i			0.155334i	0.996979	+	0.0776669i	$0.0247471\pi$
$373$			2.99999i			0.155334i	−0.996979	+	0.0776669i	$0.975253\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			40.9659i			2.10985i
$378$		0			0
$379$	3.02165			0.155212			0.0776059	−	0.996984i	$-0.475272\pi$
$379$	3.02165			0.155212			0.0776059	+	0.996984i	$0.475272\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	8.15719i		−	0.416813i	−0.978042	−	0.208406i	$-0.933172\pi$
$383$		−	8.15719i		−	0.416813i	0.978042	−	0.208406i	$-0.0668277\pi$
$384$		0			0
$385$	−3.38333	−	3.42213i	−0.172431	−	0.174408i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	11.7789			0.597214			0.298607	−	0.954376i	$-0.403478\pi$
$389$	11.7789			0.597214			0.298607	+	0.954376i	$0.403478\pi$
$390$		0			0
$391$	0.366626			0.0185411
$392$		0			0
$393$		0			0
$394$		0			0
$395$	9.64447	−	9.53514i	0.485266	−	0.479765i
$396$		0			0
$397$			16.7004i			0.838167i	0.907948	+	0.419083i	$0.137649\pi$
$397$			16.7004i			0.838167i	−0.907948	+	0.419083i	$0.862351\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−9.99936			−0.499344			−0.249672	−	0.968330i	$-0.580323\pi$
$401$	−9.99936			−0.499344			−0.249672	+	0.968330i	$0.580323\pi$
$402$		0			0
$403$			36.9710i			1.84165i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			2.74373i			0.136002i
$408$		0			0
$409$	−2.20305			−0.108934			−0.0544668	−	0.998516i	$-0.517346\pi$
$409$	−2.20305			−0.108934			−0.0544668	+	0.998516i	$0.517346\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	22.6942i		−	1.11671i
$414$		0			0
$415$	25.5497	+	25.8427i	1.25419	+	1.26857i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	4.12814			0.201673			0.100836	−	0.994903i	$-0.467848\pi$
$419$	4.12814			0.201673			0.100836	+	0.994903i	$0.467848\pi$
$420$		0			0
$421$	−15.7354			−0.766899			−0.383449	−	0.923562i	$-0.625264\pi$
$421$	−15.7354			−0.766899			−0.383449	+	0.923562i	$0.625264\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	1.83301	+	0.0208983i	0.0889140	+	0.00101372i
$426$		0			0
$427$			13.2949i			0.643385i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	1.35158			0.0651034			0.0325517	−	0.999470i	$-0.489637\pi$
$431$	1.35158			0.0651034			0.0325517	+	0.999470i	$0.489637\pi$
$432$		0			0
$433$			39.4114i			1.89399i	0.321244	+	0.946996i	$0.395899\pi$
$433$			39.4114i			1.89399i	−0.321244	+	0.946996i	$0.604101\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	2.79847i		−	0.133869i
$438$		0			0
$439$	−10.1059			−0.482326			−0.241163	−	0.970485i	$-0.577529\pi$
$439$	−10.1059			−0.482326			−0.241163	+	0.970485i	$0.577529\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	6.25526i		−	0.297197i	−0.988898	−	0.148598i	$-0.952524\pi$
$443$		−	6.25526i		−	0.297197i	0.988898	−	0.148598i	$-0.0474761\pi$
$444$		0			0
$445$	27.9914	−	27.6741i	1.32692	−	1.31188i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	2.28779			0.107968			0.0539839	−	0.998542i	$-0.482808\pi$
$449$	2.28779			0.107968			0.0539839	+	0.998542i	$0.482808\pi$
$450$		0			0
$451$	3.07248			0.144677
$452$		0			0
$453$		0			0
$454$		0			0
$455$	19.7416	−	19.5178i	0.925501	−	0.915010i
$456$		0			0
$457$			14.3062i			0.669217i	0.942357	+	0.334609i	$0.108604\pi$
$457$			14.3062i			0.669217i	−0.942357	+	0.334609i	$0.891396\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	2.38995			0.111311			0.0556556	−	0.998450i	$-0.482275\pi$
$461$	2.38995			0.111311			0.0556556	+	0.998450i	$0.482275\pi$
$462$		0			0
$463$		−	9.52232i		−	0.442540i	−0.975213	−	0.221270i	$-0.928980\pi$
$463$		−	9.52232i		−	0.442540i	0.975213	−	0.221270i	$-0.0710202\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	6.14501i		−	0.284357i	−0.989841	−	0.142178i	$-0.954589\pi$
$467$		−	6.14501i		−	0.284357i	0.989841	−	0.142178i	$-0.0454107\pi$
$468$		0			0
$469$	3.62394			0.167338
$470$		0			0
$471$		0			0
$472$		0			0
$473$			7.58278i			0.348657i
$474$		0			0
$475$	0.159518	−	13.9915i	0.00731917	−	0.641972i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−1.40986			−0.0644180			−0.0322090	−	0.999481i	$-0.510254\pi$
$479$	−1.40986			−0.0644180			−0.0322090	+	0.999481i	$0.510254\pi$
$480$		0			0
$481$	−15.8281			−0.721697
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−10.3070	−	10.4251i	−0.468015	−	0.473382i
$486$		0			0
$487$		−	31.2993i		−	1.41831i	−0.705054	−	0.709154i	$-0.749077\pi$
$487$		−	31.2993i		−	1.41831i	0.705054	−	0.709154i	$-0.250923\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	25.9254			1.17000			0.584999	−	0.811034i	$-0.301095\pi$
$491$	25.9254			1.17000			0.584999	+	0.811034i	$0.301095\pi$
$492$		0			0
$493$			2.94192i			0.132497i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	20.1831i		−	0.905336i
$498$		0			0
$499$	8.04562			0.360172			0.180086	−	0.983651i	$-0.442362\pi$
$499$	8.04562			0.360172			0.180086	+	0.983651i	$0.442362\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			5.50490i			0.245451i	0.992441	+	0.122726i	$0.0391635\pi$
$503$			5.50490i			0.245451i	−0.992441	+	0.122726i	$0.960836\pi$
$504$		0			0
$505$	21.4527	−	21.2095i	0.954634	−	0.943812i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−8.26586			−0.366378			−0.183189	−	0.983078i	$-0.558642\pi$
$509$	−8.26586			−0.366378			−0.183189	+	0.983078i	$0.558642\pi$
$510$		0			0
$511$	24.9043			1.10170
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−19.7235	−	19.9497i	−0.869122	−	0.879088i
$516$		0			0
$517$			4.65076i			0.204540i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−11.9776			−0.524747			−0.262374	−	0.964966i	$-0.584505\pi$
$521$	−11.9776			−0.524747			−0.262374	+	0.964966i	$0.584505\pi$
$522$		0			0
$523$		−	0.711546i		−	0.0311137i	−0.999879	−	0.0155569i	$-0.995048\pi$
$523$		−	0.711546i		−	0.0311137i	0.999879	−	0.0155569i	$-0.00495210\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			2.65503i			0.115655i
$528$		0			0
$529$	−1.00000			−0.0434783
$530$		0			0
$531$		0			0
$532$		0			0
$533$			17.7246i			0.767737i
$534$		0			0
$535$	−2.29950	−	2.32587i	−0.0994161	−	0.100556i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	0.961182			0.0414011
$540$		0			0
$541$	34.9533			1.50276			0.751381	−	0.659869i	$-0.229388\pi$
$541$	34.9533			1.50276			0.751381	+	0.659869i	$0.229388\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−30.5580	+	30.2115i	−1.30896	+	1.29412i
$546$		0			0
$547$		−	17.8478i		−	0.763118i	−0.924345	−	0.381559i	$-0.875387\pi$
$547$		−	17.8478i		−	0.763118i	0.924345	−	0.381559i	$-0.124613\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	22.4558			0.956650
$552$		0			0
$553$		−	14.7497i		−	0.627221i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			3.29661i			0.139682i	0.997558	+	0.0698410i	$0.0222492\pi$
$557$			3.29661i			0.139682i	−0.997558	+	0.0698410i	$0.977751\pi$
$558$		0			0
$559$	−43.7437			−1.85016
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	4.44153i		−	0.187188i	−0.995610	−	0.0935941i	$-0.970164\pi$
$563$		−	4.44153i		−	0.187188i	0.995610	−	0.0935941i	$-0.0298356\pi$
$564$		0			0
$565$	−1.31255	−	1.32760i	−0.0552193	−	0.0558525i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−6.00088			−0.251570			−0.125785	−	0.992058i	$-0.540145\pi$
$569$	−6.00088			−0.251570			−0.125785	+	0.992058i	$0.540145\pi$
$570$		0			0
$571$	18.1609			0.760008			0.380004	−	0.924985i	$-0.375923\pi$
$571$	18.1609			0.760008			0.380004	+	0.924985i	$0.375923\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−4.99968	−	0.0570016i	−0.208501	−	0.00237713i
$576$		0			0
$577$			10.0474i			0.418279i	0.977886	+	0.209140i	$0.0670663\pi$
$577$			10.0474i			0.418279i	−0.977886	+	0.209140i	$0.932934\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	39.5223			1.63966
$582$		0			0
$583$			10.3022i			0.426672i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	42.3593i		−	1.74836i	−0.485605	−	0.874178i	$-0.661401\pi$
$587$		−	42.3593i		−	1.74836i	0.485605	−	0.874178i	$-0.338599\pi$
$588$		0			0
$589$	20.2660			0.835044
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	19.6741i		−	0.807917i	−0.914777	−	0.403958i	$-0.867634\pi$
$593$		−	19.6741i		−	0.807917i	0.914777	−	0.403958i	$-0.132366\pi$
$594$		0			0
$595$	1.41772	−	1.40165i	0.0581209	−	0.0574620i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−18.2740			−0.746655			−0.373328	−	0.927700i	$-0.621783\pi$
$599$	−18.2740			−0.746655			−0.373328	+	0.927700i	$0.621783\pi$
$600$		0			0
$601$	−8.17793			−0.333585			−0.166792	−	0.985992i	$-0.553341\pi$
$601$	−8.17793			−0.333585			−0.166792	+	0.985992i	$0.553341\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	16.2460	−	16.0619i	0.660496	−	0.653008i
$606$		0			0
$607$		−	21.7927i		−	0.884538i	−0.896883	−	0.442269i	$-0.854174\pi$
$607$		−	21.7927i		−	0.884538i	0.896883	−	0.442269i	$-0.145826\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−26.8294			−1.08540
$612$		0			0
$613$		−	15.2379i		−	0.615455i	−0.951475	−	0.307727i	$-0.900432\pi$
$613$		−	15.2379i		−	0.615455i	0.951475	−	0.307727i	$-0.0995685\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			20.2945i			0.817024i	0.912753	+	0.408512i	$0.133952\pi$
$617$			20.2945i			0.817024i	−0.912753	+	0.408512i	$0.866048\pi$
$618$		0			0
$619$	−2.08510			−0.0838073			−0.0419037	−	0.999122i	$-0.513342\pi$
$619$	−2.08510			−0.0838073			−0.0419037	+	0.999122i	$0.513342\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	42.8085i		−	1.71509i
$624$		0			0
$625$	−24.9935	−	0.569979i	−0.999740	−	0.0227992i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−1.13667			−0.0453221
$630$		0			0
$631$	39.0512			1.55461			0.777303	−	0.629127i	$-0.216587\pi$
$631$	39.0512			1.55461			0.777303	+	0.629127i	$0.216587\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	12.3795	+	12.5215i	0.491267	+	0.496900i
$636$		0			0
$637$			5.54488i			0.219696i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	21.8361			0.862475			0.431238	−	0.902238i	$-0.358077\pi$
$641$	21.8361			0.862475			0.431238	+	0.902238i	$0.358077\pi$
$642$		0			0
$643$		−	16.3727i		−	0.645676i	−0.946454	−	0.322838i	$-0.895363\pi$
$643$		−	16.3727i		−	0.645676i	0.946454	−	0.322838i	$-0.104637\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	20.9991i		−	0.825560i	−0.910831	−	0.412780i	$-0.864558\pi$
$647$		−	20.9991i		−	0.825560i	0.910831	−	0.412780i	$-0.135442\pi$
$648$		0			0
$649$	−8.25861			−0.324179
$650$		0			0
$651$		0			0
$652$		0			0
$653$			23.2756i			0.910846i	0.890275	+	0.455423i	$0.150512\pi$
$653$			23.2756i			0.910846i	−0.890275	+	0.455423i	$0.849488\pi$
$654$		0			0
$655$	−20.4713	+	20.2392i	−0.799878	+	0.790811i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−2.09639			−0.0816637			−0.0408318	−	0.999166i	$-0.513001\pi$
$659$	−2.09639			−0.0816637			−0.0408318	+	0.999166i	$0.513001\pi$
$660$		0			0
$661$	−31.3184			−1.21814			−0.609072	−	0.793115i	$-0.708458\pi$
$661$	−31.3184			−1.21814			−0.609072	+	0.793115i	$0.708458\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−10.6989	−	10.8215i	−0.414884	−	0.419641i
$666$		0			0
$667$		−	8.02431i		−	0.310703i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	4.83812			0.186773
$672$		0			0
$673$			44.3377i			1.70909i	0.519377	+	0.854545i	$0.326164\pi$
$673$			44.3377i			1.70909i	−0.519377	+	0.854545i	$0.673836\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			3.03772i			0.116749i	0.998295	+	0.0583746i	$0.0185918\pi$
$677$			3.03772i			0.116749i	−0.998295	+	0.0583746i	$0.981408\pi$
$678$		0			0
$679$	−15.9436			−0.611860
$680$		0			0
$681$		0			0
$682$		0			0
$683$			22.6750i			0.867635i	0.901001	+	0.433817i	$0.142834\pi$
$683$			22.6750i			0.867635i	−0.901001	+	0.433817i	$0.857166\pi$
$684$		0			0
$685$	−5.13566	−	5.19454i	−0.196223	−	0.198473i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−59.4313			−2.26415
$690$		0			0
$691$	−18.2394			−0.693860			−0.346930	−	0.937891i	$-0.612776\pi$
$691$	−18.2394			−0.693860			−0.346930	+	0.937891i	$0.612776\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−19.9685	+	19.7421i	−0.757448	+	0.748862i
$696$		0			0
$697$			1.27287i			0.0482134i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−17.4315			−0.658378			−0.329189	−	0.944264i	$-0.606775\pi$
$701$	−17.4315			−0.658378			−0.329189	+	0.944264i	$0.606775\pi$
$702$		0			0
$703$			8.67629i			0.327232i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	32.8086i		−	1.23389i
$708$		0			0
$709$	32.4207			1.21758			0.608792	−	0.793330i	$-0.291654\pi$
$709$	32.4207			1.21758			0.608792	+	0.793330i	$0.291654\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	7.24179i		−	0.271207i
$714$		0			0
$715$	−7.10270	−	7.18414i	−0.265626	−	0.268671i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	0.494269			0.0184331			0.00921656	−	0.999958i	$-0.497066\pi$
$719$	0.494269			0.0184331			0.00921656	+	0.999958i	$0.497066\pi$
$720$		0			0
$721$	−30.5099			−1.13625
$722$		0			0
$723$		0			0
$724$		0			0
$725$	0.457399	−	40.1189i	0.0169874	−	1.48998i
$726$		0			0
$727$			4.25774i			0.157911i	0.996878	+	0.0789555i	$0.0251585\pi$
$727$			4.25774i			0.157911i	−0.996878	+	0.0789555i	$0.974842\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−3.14140			−0.116189
$732$		0			0
$733$			14.8525i			0.548589i	0.961646	+	0.274295i	$0.0884443\pi$
$733$			14.8525i			0.548589i	−0.961646	+	0.274295i	$0.911556\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	1.31878i		−	0.0485780i
$738$		0			0
$739$	42.1389			1.55010			0.775052	−	0.631898i	$-0.217724\pi$
$739$	42.1389			1.55010			0.775052	+	0.631898i	$0.217724\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	3.97014i		−	0.145650i	−0.997345	−	0.0728252i	$-0.976798\pi$
$743$		−	3.97014i		−	0.145650i	0.997345	−	0.0728252i	$-0.0232015\pi$
$744$		0			0
$745$	0.299365	−	0.295971i	0.0109679	−	0.0108436i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−3.55705			−0.129972
$750$		0			0
$751$	51.5382			1.88066			0.940328	−	0.340269i	$-0.110518\pi$
$751$	51.5382			1.88066			0.940328	+	0.340269i	$0.110518\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	29.6889	−	29.3524i	1.08049	−	1.06824i
$756$		0			0
$757$			39.9808i			1.45313i	0.687099	+	0.726564i	$0.258884\pi$
$757$			39.9808i			1.45313i	−0.687099	+	0.726564i	$0.741116\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	20.6683			0.749226			0.374613	−	0.927181i	$-0.377776\pi$
$761$	20.6683			0.749226			0.374613	+	0.927181i	$0.377776\pi$
$762$		0			0
$763$			46.7336i			1.69187i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	47.6424i		−	1.72027i
$768$		0			0
$769$	0.0670558			0.00241809			0.00120905	−	0.999999i	$-0.499615\pi$
$769$	0.0670558			0.00241809			0.00120905	+	0.999999i	$0.499615\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	21.2074i		−	0.762778i	−0.924415	−	0.381389i	$-0.875446\pi$
$773$		−	21.2074i		−	0.762778i	0.924415	−	0.381389i	$-0.124554\pi$
$774$		0			0
$775$	0.412794	−	36.2066i	0.0148280	−	1.30058i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	9.71588			0.348108
$780$		0			0
$781$	−7.34480			−0.262817
$782$		0			0
$783$		0			0
$784$		0			0
$785$	31.4856	+	31.8466i	1.12377	+	1.13665i
$786$		0			0
$787$		−	19.1047i		−	0.681010i	−0.940243	−	0.340505i	$-0.889402\pi$
$787$		−	19.1047i		−	0.681010i	0.940243	−	0.340505i	$-0.110598\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−2.03035			−0.0721910
$792$		0			0
$793$			27.9102i			0.991121i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	51.0751i		−	1.80917i	−0.426290	−	0.904587i	$-0.640180\pi$
$797$		−	51.0751i		−	1.80917i	0.426290	−	0.904587i	$-0.359820\pi$
$798$		0			0
$799$	−1.92672			−0.0681625
$800$		0			0
$801$		0			0
$802$		0			0
$803$		−	9.06286i		−	0.319822i
$804$		0			0
$805$	−3.86694	+	3.82311i	−0.136292	+	0.134747i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−5.02800			−0.176775			−0.0883876	−	0.996086i	$-0.528171\pi$
$809$	−5.02800			−0.176775			−0.0883876	+	0.996086i	$0.528171\pi$
$810$		0			0
$811$	51.3571			1.80339			0.901696	−	0.432371i	$-0.142323\pi$
$811$	51.3571			1.80339			0.901696	+	0.432371i	$0.142323\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−5.27607	−	5.33657i	−0.184813	−	0.186932i
$816$		0			0
$817$			23.9785i			0.838900i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−13.7483			−0.479818			−0.239909	−	0.970795i	$-0.577118\pi$
$821$	−13.7483			−0.479818			−0.239909	+	0.970795i	$0.577118\pi$
$822$		0			0
$823$		−	9.07220i		−	0.316237i	−0.987420	−	0.158119i	$-0.949457\pi$
$823$		−	9.07220i		−	0.316237i	0.987420	−	0.158119i	$-0.0505428\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	48.2010i		−	1.67611i	−0.545583	−	0.838057i	$-0.683692\pi$
$827$		−	48.2010i		−	1.67611i	0.545583	−	0.838057i	$-0.316308\pi$
$828$		0			0
$829$	−41.0601			−1.42608			−0.713038	−	0.701125i	$-0.752681\pi$
$829$	−41.0601			−1.42608			−0.713038	+	0.701125i	$0.752681\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			0.398199i			0.0137968i
$834$		0			0
$835$	4.05201	+	4.09847i	0.140226	+	0.141834i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	29.3970			1.01490			0.507449	−	0.861682i	$-0.330588\pi$
$839$	29.3970			1.01490			0.507449	+	0.861682i	$0.330588\pi$
$840$		0			0
$841$	35.3895			1.22033
$842$		0			0
$843$		0			0
$844$		0			0
$845$	20.7723	−	20.5368i	0.714590	−	0.706489i
$846$		0			0
$847$		−	24.8458i		−	0.853711i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	3.10036			0.106279
$852$		0			0
$853$		−	1.39913i		−	0.0479053i	−0.999713	−	0.0239527i	$-0.992375\pi$
$853$		−	1.39913i		−	0.0479053i	0.999713	−	0.0239527i	$-0.00762510\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	3.68808i		−	0.125982i	−0.998014	−	0.0629912i	$-0.979936\pi$
$857$		−	3.68808i		−	0.125982i	0.998014	−	0.0629912i	$-0.0200640\pi$
$858$		0			0
$859$	−16.0281			−0.546871			−0.273435	−	0.961890i	$-0.588160\pi$
$859$	−16.0281			−0.546871			−0.273435	+	0.961890i	$0.588160\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			35.0979i			1.19475i	0.801963	+	0.597374i	$0.203789\pi$
$863$			35.0979i			1.19475i	−0.801963	+	0.597374i	$0.796211\pi$
$864$		0			0
$865$	−30.5274	−	30.8774i	−1.03796	−	1.04986i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−5.36753			−0.182081
$870$		0			0
$871$	7.60781			0.257781
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−19.5514	+	18.8939i	−0.660957	+	0.638729i
$876$		0			0
$877$		−	53.2491i		−	1.79809i	−0.437853	−	0.899046i	$-0.644261\pi$
$877$		−	53.2491i		−	1.79809i	0.437853	−	0.899046i	$-0.355739\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−40.8449			−1.37610			−0.688050	−	0.725663i	$-0.741533\pi$
$881$	−40.8449			−1.37610			−0.688050	+	0.725663i	$0.741533\pi$
$882$		0			0
$883$		−	14.9288i		−	0.502393i	−0.967936	−	0.251196i	$-0.919176\pi$
$883$		−	14.9288i		−	0.502393i	0.967936	−	0.251196i	$-0.0808240\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			34.9975i			1.17510i	0.809188	+	0.587550i	$0.199908\pi$
$887$			34.9975i			1.17510i	−0.809188	+	0.587550i	$0.800092\pi$
$888$		0			0
$889$	19.1496			0.642259
$890$		0			0
$891$		0			0
$892$		0			0
$893$			14.7068i			0.492143i
$894$		0			0
$895$	32.3794	−	32.0123i	1.08232	−	1.07005i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	58.1104			1.93809
$900$		0			0
$901$	−4.26799			−0.142187
$902$		0			0
$903$		0			0
$904$		0			0
$905$	35.8778	−	35.4711i	1.19262	−	1.17910i
$906$		0			0
$907$			16.3920i			0.544288i	0.962257	+	0.272144i	$0.0877327\pi$
$907$			16.3920i			0.544288i	−0.962257	+	0.272144i	$0.912267\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	44.1111			1.46147			0.730733	−	0.682663i	$-0.239178\pi$
$911$	44.1111			1.46147			0.730733	+	0.682663i	$0.239178\pi$
$912$		0			0
$913$		−	14.3825i		−	0.475991i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			31.3076i			1.03387i
$918$		0			0
$919$	−28.5629			−0.942203			−0.471101	−	0.882079i	$-0.656143\pi$
$919$	−28.5629			−0.942203			−0.471101	+	0.882079i	$0.656143\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	42.3708i		−	1.39465i
$924$		0			0
$925$	15.5008	+	0.176726i	0.509664	+	0.00581071i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	16.2273			0.532400			0.266200	−	0.963918i	$-0.414232\pi$
$929$	16.2273			0.532400			0.266200	+	0.963918i	$0.414232\pi$
$930$		0			0
$931$	3.03948			0.0996148
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−0.510072	−	0.515920i	−0.0166811	−	0.0168724i
$936$		0			0
$937$			5.65418i			0.184714i	0.995726	+	0.0923571i	$0.0294401\pi$
$937$			5.65418i			0.184714i	−0.995726	+	0.0923571i	$0.970560\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	25.6246			0.835338			0.417669	−	0.908599i	$-0.362847\pi$
$941$	25.6246			0.835338			0.417669	+	0.908599i	$0.362847\pi$
$942$		0			0
$943$		−	3.47185i		−	0.113059i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			0.248718i			0.00808223i	0.999992	+	0.00404112i	$0.00128633\pi$
$947$			0.248718i			0.00808223i	−0.999992	+	0.00404112i	$0.998714\pi$
$948$		0			0
$949$	52.2820			1.69715
$950$		0			0
$951$		0			0
$952$		0			0
$953$			39.1737i			1.26896i	0.772939	+	0.634481i	$0.218786\pi$
$953$			39.1737i			1.26896i	−0.772939	+	0.634481i	$0.781214\pi$
$954$		0			0
$955$	−26.4842	+	26.1840i	−0.857008	+	0.847293i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−7.94423			−0.256533
$960$		0			0
$961$	21.4435			0.691726
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−22.8291	−	23.0908i	−0.734894	−	0.743320i
$966$		0			0
$967$		−	2.04114i		−	0.0656386i	−0.999461	−	0.0328193i	$-0.989551\pi$
$967$		−	2.04114i		−	0.0656386i	0.999461	−	0.0328193i	$-0.0104486\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	13.6713			0.438734			0.219367	−	0.975642i	$-0.429601\pi$
$971$	13.6713			0.438734			0.219367	+	0.975642i	$0.429601\pi$
$972$		0			0
$973$			30.5387i			0.979025i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			45.8777i			1.46776i	0.679281	+	0.733878i	$0.262292\pi$
$977$			45.8777i			1.46776i	−0.679281	+	0.733878i	$0.737708\pi$
$978$		0			0
$979$	−15.5784			−0.497887
$980$		0			0
$981$		0			0
$982$		0			0
$983$			59.6356i			1.90208i	0.309070	+	0.951039i	$0.399982\pi$
$983$			59.6356i			1.90208i	−0.309070	+	0.951039i	$0.600018\pi$
$984$		0			0
$985$	20.9842	+	21.2248i	0.668611	+	0.676277i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	8.56841			0.272460
$990$		0			0
$991$	−42.3396			−1.34496			−0.672480	−	0.740115i	$-0.734771\pi$
$991$	−42.3396			−1.34496			−0.672480	+	0.740115i	$0.734771\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	29.8715	−	29.5328i	0.946989	−	0.936254i
$996$		0			0
$997$			7.02379i			0.222446i	0.993795	+	0.111223i	$0.0354767\pi$
$997$			7.02379i			0.222446i	−0.993795	+	0.111223i	$0.964523\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4140.2.f.b.829.4		12
3.2	odd	2		460.2.c.a.369.12	yes	12
5.4	even	2	inner	4140.2.f.b.829.3		12
12.11	even	2		1840.2.e.f.369.1		12
15.2	even	4		2300.2.a.o.1.6		6
15.8	even	4		2300.2.a.n.1.1		6
15.14	odd	2		460.2.c.a.369.1	✓	12
60.23	odd	4		9200.2.a.cy.1.6		6
60.47	odd	4		9200.2.a.cx.1.1		6
60.59	even	2		1840.2.e.f.369.12		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
460.2.c.a.369.1	✓	12	15.14	odd	2
460.2.c.a.369.12	yes	12	3.2	odd	2
1840.2.e.f.369.1		12	12.11	even	2
1840.2.e.f.369.12		12	60.59	even	2
2300.2.a.n.1.1		6	15.8	even	4
2300.2.a.o.1.6		6	15.2	even	4
4140.2.f.b.829.3		12	5.4	even	2	inner
4140.2.f.b.829.4		12	1.1	even	1	trivial
9200.2.a.cx.1.1		6	60.47	odd	4
9200.2.a.cy.1.6		6	60.23	odd	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 $\F_{q}$
Belyi maps

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)

\(f(q)\)	\(=\)	\(q+(-1.59013 + 1.57210i) q^{5} +2.43185i q^{7} +0.884969 q^{11} +5.10522i q^{13} +0.366626i q^{17} +2.79847 q^{19} -1.00000i q^{23} +(0.0570016 - 4.99968i) q^{25} +8.02431 q^{29} +7.24179 q^{31} +(-3.82311 - 3.86694i) q^{35} +3.10036i q^{37} +3.47185 q^{41} +8.56841i q^{43} +5.25528i q^{47} +1.08612 q^{49} +11.6413i q^{53} +(-1.40721 + 1.39126i) q^{55} -9.33209 q^{59} +5.46699 q^{61} +(-8.02592 - 8.11795i) q^{65} -1.49020i q^{67} -8.29949 q^{71} -10.2409i q^{73} +2.15211i q^{77} -6.06522 q^{79} -16.2520i q^{83} +(-0.576373 - 0.582981i) q^{85} -17.6033 q^{89} -12.4151 q^{91} +(-4.44992 + 4.39948i) q^{95} +6.55618i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 4 q^{11} - 8 q^{19} + 8 q^{25} + 10 q^{29} + 18 q^{31} + 10 q^{35} + 2 q^{41} - 38 q^{49} + 16 q^{55} - 22 q^{59} - 8 q^{61} - 38 q^{65} + 34 q^{71} - 20 q^{79} + 6 q^{85} - 48 q^{89} - 8 q^{91}+ \cdots - 12 q^{95}+O(q^{100}) \) 12 * q - 4 * q^11 - 8 * q^19 + 8 * q^25 + 10 * q^29 + 18 * q^31 + 10 * q^35 + 2 * q^41 - 38 * q^49 + 16 * q^55 - 22 * q^59 - 8 * q^61 - 38 * q^65 + 34 * q^71 - 20 * q^79 + 6 * q^85 - 48 * q^89 - 8 * q^91 - 12 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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