LMFDB - Embedded newform 6660.2.f.c.5329.14

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6660,2,Mod(5329,6660)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6660, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("6660.5329"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 6660 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6660.f (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [18,0,0,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$53.1803677462$
Analytic rank:	$0$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} + 26 x^{16} + 283 x^{14} + 1674 x^{12} + 5841 x^{10} + 12196 x^{8} + 14736 x^{6} + 9408 x^{4} + \cdots + 144 $ x^18 + 26x^16 + 283x^14 + 1674x^12 + 5841x^10 + 12196x^8 + 14736x^6 + 9408x^4 + 2592x^2 + 144
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 740)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			5329.14
Root			$1.45428i$ of defining polynomial
Character	$\chi$	$=$	6660.5329
Dual form			6660.2.f.c.5329.13

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.07541 + 1.96049i) q^{5} -2.78997i q^{7} +4.91898 q^{11} +0.675823i q^{13} +0.868398i q^{17} -3.32168 q^{19} +0.927380i q^{23} +(-2.68701 + 4.21663i) q^{25} +3.82597 q^{29} -0.988046 q^{31} +(5.46970 - 3.00035i) q^{35} +1.00000i q^{37} +4.10171 q^{41} -6.26649i q^{43} +0.511352i q^{47} -0.783946 q^{49} +12.7869i q^{53} +(5.28990 + 9.64360i) q^{55} +6.90064 q^{59} +3.39134 q^{61} +(-1.32494 + 0.726784i) q^{65} -1.78220i q^{67} +3.84596 q^{71} -1.62628i q^{73} -13.7238i q^{77} +12.2262 q^{79} +13.1605i q^{83} +(-1.70248 + 0.933880i) q^{85} -1.89884 q^{89} +1.88553 q^{91} +(-3.57215 - 6.51210i) q^{95} +18.3123i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q - 2 q^{5} + 4 q^{19} - 2 q^{25} + 4 q^{29} + 8 q^{31} + 2 q^{35} + 4 q^{41} + 6 q^{49} - 6 q^{55} - 8 q^{59} + 12 q^{65} + 24 q^{71} - 24 q^{79} + 36 q^{91} - 24 q^{95}+O(q^{100}) $ 18 * q - 2 * q^5 + 4 * q^19 - 2 * q^25 + 4 * q^29 + 8 * q^31 + 2 * q^35 + 4 * q^41 + 6 * q^49 - 6 * q^55 - 8 * q^59 + 12 * q^65 + 24 * q^71 - 24 * q^79 + 36 * q^91 - 24 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/6660\mathbb{Z}\right)^\times$.

$n$	$3331$	$3701$	$3961$	$3997$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$5$	1.07541	+	1.96049i	0.480936	+	0.876756i
$5$	1.07541	+	1.96049i	0.480936	+	0.876756i
$6$		0			0
$7$		−	2.78997i		−	1.05451i	−0.849707	−	0.527255i	$-0.823221\pi$
$7$		−	2.78997i		−	1.05451i	0.849707	−	0.527255i	$-0.176779\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	4.91898			1.48313			0.741565	−	0.670881i	$-0.234084\pi$
$11$	4.91898			1.48313			0.741565	+	0.670881i	$0.234084\pi$
$12$		0			0
$13$			0.675823i			0.187440i	0.995599	+	0.0937198i	$0.0298758\pi$
$13$			0.675823i			0.187440i	−0.995599	+	0.0937198i	$0.970124\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			0.868398i			0.210618i	0.994440	+	0.105309i	$0.0335831\pi$
$17$			0.868398i			0.210618i	−0.994440	+	0.105309i	$0.966417\pi$
$18$		0			0
$19$	−3.32168			−0.762045			−0.381022	−	0.924566i	$-0.624428\pi$
$19$	−3.32168			−0.762045			−0.381022	+	0.924566i	$0.624428\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			0.927380i			0.193372i	0.995315	+	0.0966861i	$0.0308243\pi$
$23$			0.927380i			0.193372i	−0.995315	+	0.0966861i	$0.969176\pi$
$24$		0			0
$25$	−2.68701	+	4.21663i	−0.537401	+	0.843327i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	3.82597			0.710466			0.355233	−	0.934778i	$-0.384402\pi$
$29$	3.82597			0.710466			0.355233	+	0.934778i	$0.384402\pi$
$30$		0			0
$31$	−0.988046			−0.177458			−0.0887292	−	0.996056i	$-0.528281\pi$
$31$	−0.988046			−0.177458			−0.0887292	+	0.996056i	$0.528281\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	5.46970	−	3.00035i	0.924548	−	0.507152i
$36$		0			0
$37$			1.00000i			0.164399i
$37$			1.00000i			0.164399i
$38$		0			0
$39$		0			0
$40$		0			0
$41$	4.10171			0.640580			0.320290	−	0.947320i	$-0.396220\pi$
$41$	4.10171			0.640580			0.320290	+	0.947320i	$0.396220\pi$
$42$		0			0
$43$		−	6.26649i		−	0.955631i	−0.878460	−	0.477816i	$-0.841429\pi$
$43$		−	6.26649i		−	0.955631i	0.878460	−	0.477816i	$-0.158571\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			0.511352i			0.0745883i	0.999304	+	0.0372942i	$0.0118739\pi$
$47$			0.511352i			0.0745883i	−0.999304	+	0.0372942i	$0.988126\pi$
$48$		0			0
$49$	−0.783946			−0.111992
$50$		0			0
$51$		0			0
$52$		0			0
$53$			12.7869i			1.75642i	0.478279	+	0.878208i	$0.341261\pi$
$53$			12.7869i			1.75642i	−0.478279	+	0.878208i	$0.658739\pi$
$54$		0			0
$55$	5.28990	+	9.64360i	0.713290	+	1.30034i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	6.90064			0.898386			0.449193	−	0.893435i	$-0.351711\pi$
$59$	6.90064			0.898386			0.449193	+	0.893435i	$0.351711\pi$
$60$		0			0
$61$	3.39134			0.434217			0.217109	−	0.976147i	$-0.430337\pi$
$61$	3.39134			0.434217			0.217109	+	0.976147i	$0.430337\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−1.32494	+	0.726784i	−0.164339	+	0.0901464i
$66$		0			0
$67$		−	1.78220i		−	0.217731i	−0.994057	−	0.108865i	$-0.965278\pi$
$67$		−	1.78220i		−	0.217731i	0.994057	−	0.108865i	$-0.0347217\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	3.84596			0.456431			0.228216	−	0.973611i	$-0.426711\pi$
$71$	3.84596			0.456431			0.228216	+	0.973611i	$0.426711\pi$
$72$		0			0
$73$		−	1.62628i		−	0.190341i	−0.995461	−	0.0951706i	$-0.969660\pi$
$73$		−	1.62628i		−	0.190341i	0.995461	−	0.0951706i	$-0.0303397\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	13.7238i		−	1.56398i
$78$		0			0
$79$	12.2262			1.37555			0.687775	−	0.725924i	$-0.258588\pi$
$79$	12.2262			1.37555			0.687775	+	0.725924i	$0.258588\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			13.1605i			1.44455i	0.691606	+	0.722275i	$0.256904\pi$
$83$			13.1605i			1.44455i	−0.691606	+	0.722275i	$0.743096\pi$
$84$		0			0
$85$	−1.70248	+	0.933880i	−0.184660	+	0.101294i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−1.89884			−0.201277			−0.100638	−	0.994923i	$-0.532089\pi$
$89$	−1.89884			−0.201277			−0.100638	+	0.994923i	$0.532089\pi$
$90$		0			0
$91$	1.88553			0.197657
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−3.57215	−	6.51210i	−0.366495	−	0.668127i
$96$		0			0
$97$			18.3123i			1.85934i	0.368399	+	0.929668i	$0.379906\pi$
$97$			18.3123i			1.85934i	−0.368399	+	0.929668i	$0.620094\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−10.3130			−1.02618			−0.513090	−	0.858335i	$-0.671499\pi$
$101$	−10.3130			−1.02618			−0.513090	+	0.858335i	$0.671499\pi$
$102$		0			0
$103$		−	1.89570i		−	0.186789i	−0.995629	−	0.0933945i	$-0.970228\pi$
$103$		−	1.89570i		−	0.186789i	0.995629	−	0.0933945i	$-0.0297718\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	10.1682i		−	0.982997i	−0.870879	−	0.491498i	$-0.836449\pi$
$107$		−	10.1682i		−	0.982997i	0.870879	−	0.491498i	$-0.163551\pi$
$108$		0			0
$109$	4.60538			0.441116			0.220558	−	0.975374i	$-0.429212\pi$
$109$	4.60538			0.441116			0.220558	+	0.975374i	$0.429212\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			10.4343i			0.981576i	0.871279	+	0.490788i	$0.163291\pi$
$113$			10.4343i			0.981576i	−0.871279	+	0.490788i	$0.836709\pi$
$114$		0			0
$115$	−1.81812	+	0.997310i	−0.169540	+	0.0929996i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	2.42281			0.222098
$120$		0			0
$121$	13.1964			1.19967
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−11.1563	−	0.733247i	−0.997847	−	0.0655836i
$126$		0			0
$127$		−	19.7053i		−	1.74856i	−0.485419	−	0.874282i	$-0.661333\pi$
$127$		−	19.7053i		−	1.74856i	0.485419	−	0.874282i	$-0.338667\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	15.6435			1.36678			0.683388	−	0.730055i	$-0.260506\pi$
$131$	15.6435			1.36678			0.683388	+	0.730055i	$0.260506\pi$
$132$		0			0
$133$			9.26738i			0.803584i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	6.48900i		−	0.554393i	−0.960813	−	0.277197i	$-0.910595\pi$
$137$		−	6.48900i		−	0.554393i	0.960813	−	0.277197i	$-0.0894053\pi$
$138$		0			0
$139$	−9.82436			−0.833291			−0.416646	−	0.909069i	$-0.636794\pi$
$139$	−9.82436			−0.833291			−0.416646	+	0.909069i	$0.636794\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			3.32436i			0.277997i
$144$		0			0
$145$	4.11447	+	7.50077i	0.341688	+	0.622905i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−12.5182			−1.02553			−0.512767	−	0.858528i	$-0.671380\pi$
$149$	−12.5182			−1.02553			−0.512767	+	0.858528i	$0.671380\pi$
$150$		0			0
$151$	4.32437			0.351912			0.175956	−	0.984398i	$-0.443698\pi$
$151$	4.32437			0.351912			0.175956	+	0.984398i	$0.443698\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−1.06255	−	1.93705i	−0.0853461	−	0.155588i
$156$		0			0
$157$		−	11.1777i		−	0.892078i	−0.895013	−	0.446039i	$-0.852834\pi$
$157$		−	11.1777i		−	0.892078i	0.895013	−	0.446039i	$-0.147166\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	2.58737			0.203913
$162$		0			0
$163$		−	1.72565i		−	0.135163i	−0.997714	−	0.0675817i	$-0.978472\pi$
$163$		−	1.72565i		−	0.135163i	0.997714	−	0.0675817i	$-0.0215283\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	7.70763i		−	0.596434i	−0.954498	−	0.298217i	$-0.903608\pi$
$167$		−	7.70763i		−	0.596434i	0.954498	−	0.298217i	$-0.0963920\pi$
$168$		0			0
$169$	12.5433			0.964866
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	13.0830i		−	0.994683i	−0.867555	−	0.497341i	$-0.834310\pi$
$173$		−	13.0830i		−	0.994683i	0.867555	−	0.497341i	$-0.165690\pi$
$174$		0			0
$175$	11.7643	+	7.49667i	0.889297	+	0.566695i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−15.2412			−1.13918			−0.569591	−	0.821928i	$-0.692898\pi$
$179$	−15.2412			−1.13918			−0.569591	+	0.821928i	$0.692898\pi$
$180$		0			0
$181$	23.7209			1.76316			0.881580	−	0.472034i	$-0.156480\pi$
$181$	23.7209			1.76316			0.881580	+	0.472034i	$0.156480\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−1.96049	+	1.07541i	−0.144138	+	0.0790654i
$186$		0			0
$187$			4.27164i			0.312373i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−12.9125			−0.934313			−0.467156	−	0.884175i	$-0.654721\pi$
$191$	−12.9125			−0.934313			−0.467156	+	0.884175i	$0.654721\pi$
$192$		0			0
$193$		−	21.1198i		−	1.52023i	−0.649787	−	0.760117i	$-0.725142\pi$
$193$		−	21.1198i		−	1.52023i	0.649787	−	0.760117i	$-0.274858\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	8.77563i		−	0.625238i	−0.949879	−	0.312619i	$-0.898794\pi$
$197$		−	8.77563i		−	0.625238i	0.949879	−	0.312619i	$-0.101206\pi$
$198$		0			0
$199$	11.7954			0.836152			0.418076	−	0.908412i	$-0.362704\pi$
$199$	11.7954			0.836152			0.418076	+	0.908412i	$0.362704\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		−	10.6744i		−	0.749193i
$204$		0			0
$205$	4.41100	+	8.04135i	0.308078	+	0.561632i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−16.3393			−1.13021
$210$		0			0
$211$	−6.44067			−0.443394			−0.221697	−	0.975116i	$-0.571160\pi$
$211$	−6.44067			−0.443394			−0.221697	+	0.975116i	$0.571160\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	12.2854	−	6.73902i	0.837855	−	0.459598i
$216$		0			0
$217$			2.75662i			0.187132i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−0.586883			−0.0394781
$222$		0			0
$223$			14.0754i			0.942559i	0.881984	+	0.471280i	$0.156208\pi$
$223$			14.0754i			0.942559i	−0.881984	+	0.471280i	$0.843792\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			13.8099i			0.916593i	0.888799	+	0.458297i	$0.151540\pi$
$227$			13.8099i			0.916593i	−0.888799	+	0.458297i	$0.848460\pi$
$228$		0			0
$229$	3.58722			0.237050			0.118525	−	0.992951i	$-0.462183\pi$
$229$	3.58722			0.237050			0.118525	+	0.992951i	$0.462183\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			12.1433i			0.795536i	0.917486	+	0.397768i	$0.130215\pi$
$233$			12.1433i			0.795536i	−0.917486	+	0.397768i	$0.869785\pi$
$234$		0			0
$235$	−1.00250	+	0.549911i	−0.0653958	+	0.0358722i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	12.7427			0.824258			0.412129	−	0.911126i	$-0.364785\pi$
$239$	12.7427			0.824258			0.412129	+	0.911126i	$0.364785\pi$
$240$		0			0
$241$	−3.23919			−0.208654			−0.104327	−	0.994543i	$-0.533269\pi$
$241$	−3.23919			−0.208654			−0.104327	+	0.994543i	$0.533269\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−0.843060	−	1.53691i	−0.0538611	−	0.0981899i
$246$		0			0
$247$		−	2.24486i		−	0.142837i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	17.0744			1.07773			0.538865	−	0.842392i	$-0.318853\pi$
$251$	17.0744			1.07773			0.538865	+	0.842392i	$0.318853\pi$
$252$		0			0
$253$			4.56177i			0.286796i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			11.7859i			0.735182i	0.929988	+	0.367591i	$0.119817\pi$
$257$			11.7859i			0.735182i	−0.929988	+	0.367591i	$0.880183\pi$
$258$		0			0
$259$	2.78997			0.173360
$260$		0			0
$261$		0			0
$262$		0			0
$263$			22.8428i			1.40855i	0.709927	+	0.704275i	$0.248728\pi$
$263$			22.8428i			1.40855i	−0.709927	+	0.704275i	$0.751272\pi$
$264$		0			0
$265$	−25.0685	+	13.7511i	−1.53995	+	0.844723i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	9.99679			0.609515			0.304758	−	0.952430i	$-0.401425\pi$
$269$	9.99679			0.609515			0.304758	+	0.952430i	$0.401425\pi$
$270$		0			0
$271$	19.3145			1.17327			0.586637	−	0.809850i	$-0.300452\pi$
$271$	19.3145			1.17327			0.586637	+	0.809850i	$0.300452\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−13.2173	+	20.7416i	−0.797036	+	1.25076i
$276$		0			0
$277$			24.9065i			1.49649i	0.663423	+	0.748244i	$0.269103\pi$
$277$			24.9065i			1.49649i	−0.663423	+	0.748244i	$0.730897\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	25.6321			1.52908			0.764542	−	0.644574i	$-0.222965\pi$
$281$	25.6321			1.52908			0.764542	+	0.644574i	$0.222965\pi$
$282$		0			0
$283$			9.83221i			0.584464i	0.956347	+	0.292232i	$0.0943980\pi$
$283$			9.83221i			0.584464i	−0.956347	+	0.292232i	$0.905602\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	11.4437i		−	0.675498i
$288$		0			0
$289$	16.2459			0.955640
$290$		0			0
$291$		0			0
$292$		0			0
$293$			30.8182i			1.80042i	0.435456	+	0.900210i	$0.356587\pi$
$293$			30.8182i			1.80042i	−0.435456	+	0.900210i	$0.643413\pi$
$294$		0			0
$295$	7.42098	+	13.5286i	0.432066	+	0.787665i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−0.626745			−0.0362456
$300$		0			0
$301$	−17.4833			−1.00772
$302$		0			0
$303$		0			0
$304$		0			0
$305$	3.64707	+	6.64868i	0.208831	+	0.380702i
$306$		0			0
$307$		−	13.9375i		−	0.795453i	−0.917504	−	0.397726i	$-0.869799\pi$
$307$		−	13.9375i		−	0.795453i	0.917504	−	0.397726i	$-0.130201\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	20.6100			1.16869			0.584343	−	0.811507i	$-0.301352\pi$
$311$	20.6100			1.16869			0.584343	+	0.811507i	$0.301352\pi$
$312$		0			0
$313$		−	13.4418i		−	0.759778i	−0.925032	−	0.379889i	$-0.875962\pi$
$313$		−	13.4418i		−	0.759778i	0.925032	−	0.379889i	$-0.124038\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	4.62447i		−	0.259736i	−0.991531	−	0.129868i	$-0.958545\pi$
$317$		−	4.62447i		−	0.259736i	0.991531	−	0.129868i	$-0.0414553\pi$
$318$		0			0
$319$	18.8199			1.05371
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	2.88454i		−	0.160500i
$324$		0			0
$325$	−2.84970	−	1.81594i	−0.158073	−	0.100730i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	1.42666			0.0786542
$330$		0			0
$331$	25.4286			1.39768			0.698842	−	0.715276i	$-0.253699\pi$
$331$	25.4286			1.39768			0.698842	+	0.715276i	$0.253699\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	3.49398	−	1.91659i	0.190897	−	0.104715i
$336$		0			0
$337$		−	4.62875i		−	0.252144i	−0.992021	−	0.126072i	$-0.959763\pi$
$337$		−	4.62875i		−	0.252144i	0.992021	−	0.126072i	$-0.0402370\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−4.86018			−0.263194
$342$		0			0
$343$		−	17.3426i		−	0.936413i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	23.7317i		−	1.27398i	−0.770871	−	0.636992i	$-0.780178\pi$
$347$		−	23.7317i		−	1.27398i	0.770871	−	0.636992i	$-0.219822\pi$
$348$		0			0
$349$	−33.4558			−1.79085			−0.895424	−	0.445214i	$-0.853128\pi$
$349$	−33.4558			−1.79085			−0.895424	+	0.445214i	$0.853128\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			27.9064i			1.48531i	0.669676	+	0.742653i	$0.266433\pi$
$353$			27.9064i			1.48531i	−0.669676	+	0.742653i	$0.733567\pi$
$354$		0			0
$355$	4.13596	+	7.53995i	0.219514	+	0.400179i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	0.671752			0.0354537			0.0177269	−	0.999843i	$-0.494357\pi$
$359$	0.671752			0.0354537			0.0177269	+	0.999843i	$0.494357\pi$
$360$		0			0
$361$	−7.96647			−0.419288
$362$		0			0
$363$		0			0
$364$		0			0
$365$	3.18829	−	1.74891i	0.166883	−	0.0915419i
$366$		0			0
$367$		−	10.2877i		−	0.537015i	−0.963278	−	0.268507i	$-0.913470\pi$
$367$		−	10.2877i		−	0.537015i	0.963278	−	0.268507i	$-0.0865304\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	35.6751			1.85216
$372$		0			0
$373$			7.79225i			0.403468i	0.979440	+	0.201734i	$0.0646576\pi$
$373$			7.79225i			0.403468i	−0.979440	+	0.201734i	$0.935342\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			2.58568i			0.133169i
$378$		0			0
$379$	5.02261			0.257994			0.128997	−	0.991645i	$-0.458824\pi$
$379$	5.02261			0.257994			0.128997	+	0.991645i	$0.458824\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	6.41421i		−	0.327751i	−0.986481	−	0.163875i	$-0.947600\pi$
$383$		−	6.41421i		−	0.327751i	0.986481	−	0.163875i	$-0.0523995\pi$
$384$		0			0
$385$	26.9054	−	14.7587i	1.37122	−	0.752172i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	1.26890			0.0643360			0.0321680	−	0.999482i	$-0.489759\pi$
$389$	1.26890			0.0643360			0.0321680	+	0.999482i	$0.489759\pi$
$390$		0			0
$391$	−0.805336			−0.0407276
$392$		0			0
$393$		0			0
$394$		0			0
$395$	13.1481	+	23.9692i	0.661551	+	1.20602i
$396$		0			0
$397$			22.9803i			1.15335i	0.816974	+	0.576675i	$0.195650\pi$
$397$			22.9803i			1.15335i	−0.816974	+	0.576675i	$0.804350\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	13.1103			0.654696			0.327348	−	0.944904i	$-0.393845\pi$
$401$	13.1103			0.654696			0.327348	+	0.944904i	$0.393845\pi$
$402$		0			0
$403$		−	0.667744i		−	0.0332627i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			4.91898i			0.243825i
$408$		0			0
$409$	−0.384530			−0.0190138			−0.00950689	−	0.999955i	$-0.503026\pi$
$409$	−0.384530			−0.0190138			−0.00950689	+	0.999955i	$0.503026\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	19.2526i		−	0.947358i
$414$		0			0
$415$	−25.8009	+	14.1529i	−1.26652	+	0.694736i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	29.1304			1.42312			0.711558	−	0.702628i	$-0.247990\pi$
$419$	29.1304			1.42312			0.711558	+	0.702628i	$0.247990\pi$
$420$		0			0
$421$	−35.4562			−1.72803			−0.864015	−	0.503466i	$-0.832058\pi$
$421$	−35.4562			−1.72803			−0.864015	+	0.503466i	$0.832058\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−3.66172	−	2.33339i	−0.177619	−	0.113186i
$426$		0			0
$427$		−	9.46176i		−	0.457886i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−21.7142			−1.04594			−0.522969	−	0.852352i	$-0.675176\pi$
$431$	−21.7142			−1.04594			−0.522969	+	0.852352i	$0.675176\pi$
$432$		0			0
$433$		−	0.987472i		−	0.0474549i	−0.999718	−	0.0237274i	$-0.992447\pi$
$433$		−	0.987472i		−	0.0474549i	0.999718	−	0.0237274i	$-0.00755339\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	3.08046i		−	0.147358i
$438$		0			0
$439$	−14.7974			−0.706243			−0.353121	−	0.935578i	$-0.614880\pi$
$439$	−14.7974			−0.706243			−0.353121	+	0.935578i	$0.614880\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			36.7175i			1.74450i	0.489057	+	0.872252i	$0.337341\pi$
$443$			36.7175i			1.74450i	−0.489057	+	0.872252i	$0.662659\pi$
$444$		0			0
$445$	−2.04202	−	3.72265i	−0.0968012	−	0.176471i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−12.1035			−0.571200			−0.285600	−	0.958349i	$-0.592193\pi$
$449$	−12.1035			−0.571200			−0.285600	+	0.958349i	$0.592193\pi$
$450$		0			0
$451$	20.1763			0.950063
$452$		0			0
$453$		0			0
$454$		0			0
$455$	2.02771	+	3.69655i	0.0950603	+	0.173297i
$456$		0			0
$457$		−	9.06950i		−	0.424253i	−0.977242	−	0.212127i	$-0.931961\pi$
$457$		−	9.06950i		−	0.424253i	0.977242	−	0.212127i	$-0.0680389\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−29.5187			−1.37482			−0.687411	−	0.726268i	$-0.741253\pi$
$461$	−29.5187			−1.37482			−0.687411	+	0.726268i	$0.741253\pi$
$462$		0			0
$463$			34.5147i			1.60403i	0.597301	+	0.802017i	$0.296240\pi$
$463$			34.5147i			1.60403i	−0.597301	+	0.802017i	$0.703760\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			39.0409i			1.80660i	0.429014	+	0.903298i	$0.358861\pi$
$467$			39.0409i			1.80660i	−0.429014	+	0.903298i	$0.641139\pi$
$468$		0			0
$469$	−4.97230			−0.229599
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	30.8248i		−	1.41733i
$474$		0			0
$475$	8.92536	−	14.0063i	0.409524	−	0.642653i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	6.22321			0.284346			0.142173	−	0.989842i	$-0.454591\pi$
$479$	6.22321			0.284346			0.142173	+	0.989842i	$0.454591\pi$
$480$		0			0
$481$	−0.675823			−0.0308149
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−35.9011	+	19.6932i	−1.63018	+	0.894221i
$486$		0			0
$487$			33.2302i			1.50580i	0.658133	+	0.752902i	$0.271346\pi$
$487$			33.2302i			1.50580i	−0.658133	+	0.752902i	$0.728654\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−28.5311			−1.28759			−0.643795	−	0.765198i	$-0.722641\pi$
$491$	−28.5311			−1.28759			−0.643795	+	0.765198i	$0.722641\pi$
$492$		0			0
$493$			3.32247i			0.149636i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	10.7301i		−	0.481312i
$498$		0			0
$499$	−7.23410			−0.323843			−0.161921	−	0.986804i	$-0.551769\pi$
$499$	−7.23410			−0.323843			−0.161921	+	0.986804i	$0.551769\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	32.0994i		−	1.43124i	−0.698488	−	0.715621i	$-0.746144\pi$
$503$		−	32.0994i		−	1.43124i	0.698488	−	0.715621i	$-0.253856\pi$
$504$		0			0
$505$	−11.0906	−	20.2184i	−0.493526	−	0.899708i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	24.9450			1.10567			0.552835	−	0.833291i	$-0.313546\pi$
$509$	24.9450			1.10567			0.552835	+	0.833291i	$0.313546\pi$
$510$		0			0
$511$	−4.53727			−0.200717
$512$		0			0
$513$		0			0
$514$		0			0
$515$	3.71650	−	2.03865i	0.163768	−	0.0898336i
$516$		0			0
$517$			2.51533i			0.110624i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−11.5767			−0.507186			−0.253593	−	0.967311i	$-0.581612\pi$
$521$	−11.5767			−0.507186			−0.253593	+	0.967311i	$0.581612\pi$
$522$		0			0
$523$			33.4820i			1.46407i	0.681268	+	0.732034i	$0.261429\pi$
$523$			33.4820i			1.46407i	−0.681268	+	0.732034i	$0.738571\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	0.858017i		−	0.0373758i
$528$		0			0
$529$	22.1400			0.962607
$530$		0			0
$531$		0			0
$532$		0			0
$533$			2.77203i			0.120070i
$534$		0			0
$535$	19.9346	−	10.9349i	0.861848	−	0.472758i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−3.85622			−0.166099
$540$		0			0
$541$	3.43850			0.147833			0.0739163	−	0.997264i	$-0.476450\pi$
$541$	3.43850			0.147833			0.0739163	+	0.997264i	$0.476450\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	4.95265	+	9.02879i	0.212148	+	0.386751i
$546$		0			0
$547$			2.34563i			0.100292i	0.998742	+	0.0501459i	$0.0159686\pi$
$547$			2.34563i			0.100292i	−0.998742	+	0.0501459i	$0.984031\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−12.7086			−0.541406
$552$		0			0
$553$		−	34.1106i		−	1.45053i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			3.33175i			0.141171i	0.997506	+	0.0705853i	$0.0224867\pi$
$557$			3.33175i			0.141171i	−0.997506	+	0.0705853i	$0.977513\pi$
$558$		0			0
$559$	4.23504			0.179123
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	44.2561i		−	1.86517i	−0.360945	−	0.932587i	$-0.617546\pi$
$563$		−	44.2561i		−	1.86517i	0.360945	−	0.932587i	$-0.382454\pi$
$564$		0			0
$565$	−20.4563	+	11.2211i	−0.860602	+	0.472075i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	34.4028			1.44224			0.721121	−	0.692809i	$-0.243627\pi$
$569$	34.4028			1.44224			0.721121	+	0.692809i	$0.243627\pi$
$570$		0			0
$571$	13.5998			0.569132			0.284566	−	0.958656i	$-0.408151\pi$
$571$	13.5998			0.569132			0.284566	+	0.958656i	$0.408151\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−3.91042	−	2.49188i	−0.163076	−	0.103918i
$576$		0			0
$577$			23.3215i			0.970886i	0.874268	+	0.485443i	$0.161342\pi$
$577$			23.3215i			0.970886i	−0.874268	+	0.485443i	$0.838658\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	36.7174			1.52329
$582$		0			0
$583$			62.8986i			2.60499i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			11.6458i			0.480675i	0.970689	+	0.240337i	$0.0772581\pi$
$587$			11.6458i			0.480675i	−0.970689	+	0.240337i	$0.922742\pi$
$588$		0			0
$589$	3.28197			0.135231
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	24.5494i		−	1.00812i	−0.863667	−	0.504062i	$-0.831838\pi$
$593$		−	24.5494i		−	1.00812i	0.863667	−	0.504062i	$-0.168162\pi$
$594$		0			0
$595$	2.60550	+	4.74988i	0.106815	+	0.194726i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−12.7917			−0.522654			−0.261327	−	0.965250i	$-0.584160\pi$
$599$	−12.7917			−0.522654			−0.261327	+	0.965250i	$0.584160\pi$
$600$		0			0
$601$	−24.3206			−0.992057			−0.496029	−	0.868306i	$-0.665209\pi$
$601$	−24.3206			−0.992057			−0.496029	+	0.868306i	$0.665209\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	14.1915	+	25.8714i	0.576966	+	1.05182i
$606$		0			0
$607$		−	33.2276i		−	1.34867i	−0.738427	−	0.674333i	$-0.764431\pi$
$607$		−	33.2276i		−	1.34867i	0.738427	−	0.674333i	$-0.235569\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−0.345583			−0.0139808
$612$		0			0
$613$		−	41.8241i		−	1.68926i	−0.535352	−	0.844629i	$-0.679821\pi$
$613$		−	41.8241i		−	1.68926i	0.535352	−	0.844629i	$-0.320179\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	28.4110i		−	1.14378i	−0.820330	−	0.571891i	$-0.806210\pi$
$617$		−	28.4110i		−	1.14378i	0.820330	−	0.571891i	$-0.193790\pi$
$618$		0			0
$619$	−32.1759			−1.29326			−0.646629	−	0.762805i	$-0.723822\pi$
$619$	−32.1759			−1.29326			−0.646629	+	0.762805i	$0.723822\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			5.29771i			0.212248i
$624$		0			0
$625$	−10.5600	−	22.6602i	−0.422400	−	0.906410i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−0.868398			−0.0346253
$630$		0			0
$631$	25.7684			1.02583			0.512913	−	0.858441i	$-0.328567\pi$
$631$	25.7684			1.02583			0.512913	+	0.858441i	$0.328567\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	38.6320	−	21.1912i	1.53306	−	0.840947i
$636$		0			0
$637$		−	0.529809i		−	0.0209918i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−28.3807			−1.12097			−0.560486	−	0.828164i	$-0.689386\pi$
$641$	−28.3807			−1.12097			−0.560486	+	0.828164i	$0.689386\pi$
$642$		0			0
$643$			39.1936i			1.54565i	0.634621	+	0.772823i	$0.281156\pi$
$643$			39.1936i			1.54565i	−0.634621	+	0.772823i	$0.718844\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	43.4658i		−	1.70882i	−0.519603	−	0.854408i	$-0.673920\pi$
$647$		−	43.4658i		−	1.70882i	0.519603	−	0.854408i	$-0.326080\pi$
$648$		0			0
$649$	33.9441			1.33242
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	5.37315i		−	0.210268i	−0.994458	−	0.105134i	$-0.966473\pi$
$653$		−	5.37315i		−	0.210268i	0.994458	−	0.105134i	$-0.0335271\pi$
$654$		0			0
$655$	16.8231	+	30.6688i	0.657332	+	1.19833i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−25.6633			−0.999700			−0.499850	−	0.866112i	$-0.666611\pi$
$659$	−25.6633			−0.999700			−0.499850	+	0.866112i	$0.666611\pi$
$660$		0			0
$661$	4.62652			0.179951			0.0899754	−	0.995944i	$-0.471321\pi$
$661$	4.62652			0.179951			0.0899754	+	0.995944i	$0.471321\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−18.1686	+	9.96619i	−0.704547	+	0.386472i
$666$		0			0
$667$			3.54813i			0.137384i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	16.6820			0.644000
$672$		0			0
$673$		−	15.4924i		−	0.597187i	−0.954380	−	0.298593i	$-0.903483\pi$
$673$		−	15.4924i		−	0.597187i	0.954380	−	0.298593i	$-0.0965174\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	6.46888i		−	0.248619i	−0.992243	−	0.124310i	$-0.960328\pi$
$677$		−	6.46888i		−	0.248619i	0.992243	−	0.124310i	$-0.0396716\pi$
$678$		0			0
$679$	51.0909			1.96069
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	18.2745i		−	0.699255i	−0.936889	−	0.349627i	$-0.886308\pi$
$683$		−	18.2745i		−	0.699255i	0.936889	−	0.349627i	$-0.113692\pi$
$684$		0			0
$685$	12.7216	−	6.97831i	0.486067	−	0.266628i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−8.64168			−0.329222
$690$		0			0
$691$	31.5509			1.20025			0.600126	−	0.799906i	$-0.295117\pi$
$691$	31.5509			1.20025			0.600126	+	0.799906i	$0.295117\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−10.5652	−	19.2605i	−0.400760	−	0.730593i
$696$		0			0
$697$			3.56192i			0.134917i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−46.0106			−1.73780			−0.868899	−	0.494989i	$-0.835172\pi$
$701$	−46.0106			−1.73780			−0.868899	+	0.494989i	$0.835172\pi$
$702$		0			0
$703$		−	3.32168i		−	0.125279i
$704$		0			0
$705$		0			0
$706$		0			0
$707$			28.7729i			1.08212i
$708$		0			0
$709$	−44.7147			−1.67930			−0.839649	−	0.543130i	$-0.817239\pi$
$709$	−44.7147			−1.67930			−0.839649	+	0.543130i	$0.817239\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	0.916295i		−	0.0343155i
$714$		0			0
$715$	−6.51736	+	3.57504i	−0.243736	+	0.133699i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−50.7140			−1.89131			−0.945657	−	0.325165i	$-0.894580\pi$
$719$	−50.7140			−1.89131			−0.945657	+	0.325165i	$0.894580\pi$
$720$		0			0
$721$	−5.28896			−0.196971
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−10.2804	+	16.1327i	−0.381805	+	0.599155i
$726$		0			0
$727$			10.5836i			0.392524i	0.980551	+	0.196262i	$0.0628803\pi$
$727$			10.5836i			0.392524i	−0.980551	+	0.196262i	$0.937120\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	5.44181			0.201273
$732$		0			0
$733$			26.3914i			0.974790i	0.873182	+	0.487395i	$0.162053\pi$
$733$			26.3914i			0.974790i	−0.873182	+	0.487395i	$0.837947\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	8.76663i		−	0.322923i
$738$		0			0
$739$	36.3461			1.33701			0.668507	−	0.743706i	$-0.266934\pi$
$739$	36.3461			1.33701			0.668507	+	0.743706i	$0.266934\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			24.4625i			0.897444i	0.893671	+	0.448722i	$0.148121\pi$
$743$			24.4625i			0.897444i	−0.893671	+	0.448722i	$0.851879\pi$
$744$		0			0
$745$	−13.4622	−	24.5418i	−0.493216	−	0.899143i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−28.3690			−1.03658
$750$		0			0
$751$	41.7141			1.52217			0.761084	−	0.648653i	$-0.224667\pi$
$751$	41.7141			1.52217			0.761084	+	0.648653i	$0.224667\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	4.65045	+	8.47787i	0.169247	+	0.308541i
$756$		0			0
$757$		−	42.0026i		−	1.52661i	−0.646039	−	0.763305i	$-0.723576\pi$
$757$		−	42.0026i		−	1.52661i	0.646039	−	0.763305i	$-0.276424\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	35.3022			1.27970			0.639852	−	0.768498i	$-0.278995\pi$
$761$	35.3022			1.27970			0.639852	+	0.768498i	$0.278995\pi$
$762$		0			0
$763$		−	12.8489i		−	0.465161i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			4.66361i			0.168393i
$768$		0			0
$769$	6.93348			0.250028			0.125014	−	0.992155i	$-0.460102\pi$
$769$	6.93348			0.250028			0.125014	+	0.992155i	$0.460102\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			8.53154i			0.306858i	0.988160	+	0.153429i	$0.0490317\pi$
$773$			8.53154i			0.306858i	−0.988160	+	0.153429i	$0.950968\pi$
$774$		0			0
$775$	2.65489	−	4.16623i	0.0953663	−	0.149655i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−13.6246			−0.488151
$780$		0			0
$781$	18.9182			0.676947
$782$		0			0
$783$		0			0
$784$		0			0
$785$	21.9137	−	12.0206i	0.782135	−	0.429032i
$786$		0			0
$787$		−	23.1035i		−	0.823550i	−0.911286	−	0.411775i	$-0.864909\pi$
$787$		−	23.1035i		−	0.823550i	0.911286	−	0.411775i	$-0.135091\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	29.1114			1.03508
$792$		0			0
$793$			2.29195i			0.0813895i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	28.9895i		−	1.02686i	−0.858131	−	0.513430i	$-0.828374\pi$
$797$		−	28.9895i		−	1.02686i	0.858131	−	0.513430i	$-0.171626\pi$
$798$		0			0
$799$	−0.444057			−0.0157096
$800$		0			0
$801$		0			0
$802$		0			0
$803$		−	7.99963i		−	0.282301i
$804$		0			0
$805$	2.78247	+	5.07249i	0.0980691	+	0.178782i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	26.1018			0.917692			0.458846	−	0.888516i	$-0.348263\pi$
$809$	26.1018			0.917692			0.458846	+	0.888516i	$0.348263\pi$
$810$		0			0
$811$	−49.3488			−1.73287			−0.866436	−	0.499288i	$-0.833595\pi$
$811$	−49.3488			−1.73287			−0.866436	+	0.499288i	$0.833595\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	3.38311	−	1.85577i	0.118505	−	0.0650049i
$816$		0			0
$817$			20.8153i			0.728234i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−16.4815			−0.575209			−0.287604	−	0.957749i	$-0.592859\pi$
$821$	−16.4815			−0.575209			−0.287604	+	0.957749i	$0.592859\pi$
$822$		0			0
$823$			1.21506i			0.0423544i	0.999776	+	0.0211772i	$0.00674142\pi$
$823$			1.21506i			0.0423544i	−0.999776	+	0.0211772i	$0.993259\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			4.80215i			0.166987i	0.996508	+	0.0834936i	$0.0266078\pi$
$827$			4.80215i			0.166987i	−0.996508	+	0.0834936i	$0.973392\pi$
$828$		0			0
$829$	27.8739			0.968100			0.484050	−	0.875040i	$-0.339165\pi$
$829$	27.8739			0.968100			0.484050	+	0.875040i	$0.339165\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	0.680777i		−	0.0235875i
$834$		0			0
$835$	15.1107	−	8.28883i	0.522927	−	0.286847i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	27.4872			0.948962			0.474481	−	0.880266i	$-0.342636\pi$
$839$	27.4872			0.948962			0.474481	+	0.880266i	$0.342636\pi$
$840$		0			0
$841$	−14.3619			−0.495239
$842$		0			0
$843$		0			0
$844$		0			0
$845$	13.4891	+	24.5909i	0.464039	+	0.845952i
$846$		0			0
$847$		−	36.8176i		−	1.26507i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−0.927380			−0.0317902
$852$		0			0
$853$		−	55.1393i		−	1.88793i	−0.330040	−	0.943967i	$-0.607062\pi$
$853$		−	55.1393i		−	1.88793i	0.330040	−	0.943967i	$-0.392938\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	49.8754i		−	1.70371i	−0.523778	−	0.851855i	$-0.675478\pi$
$857$		−	49.8754i		−	1.70371i	0.523778	−	0.851855i	$-0.324522\pi$
$858$		0			0
$859$	15.5687			0.531196			0.265598	−	0.964084i	$-0.414431\pi$
$859$	15.5687			0.531196			0.265598	+	0.964084i	$0.414431\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	27.2265i		−	0.926802i	−0.886149	−	0.463401i	$-0.846629\pi$
$863$		−	27.2265i		−	0.926802i	0.886149	−	0.463401i	$-0.153371\pi$
$864$		0			0
$865$	25.6490	−	14.0695i	0.872094	−	0.478379i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	60.1403			2.04012
$870$		0			0
$871$	1.20445			0.0408113
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−2.04574	+	31.1257i	−0.0691586	+	1.05224i
$876$		0			0
$877$			25.4262i			0.858580i	0.903167	+	0.429290i	$0.141236\pi$
$877$			25.4262i			0.858580i	−0.903167	+	0.429290i	$0.858764\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−3.12728			−0.105361			−0.0526804	−	0.998611i	$-0.516776\pi$
$881$	−3.12728			−0.105361			−0.0526804	+	0.998611i	$0.516776\pi$
$882$		0			0
$883$			16.3438i			0.550012i	0.961442	+	0.275006i	$0.0886799\pi$
$883$			16.3438i			0.550012i	−0.961442	+	0.275006i	$0.911320\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			31.4746i			1.05681i	0.848991	+	0.528407i	$0.177211\pi$
$887$			31.4746i			1.05681i	−0.848991	+	0.528407i	$0.822789\pi$
$888$		0			0
$889$	−54.9773			−1.84388
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	1.69855i		−	0.0568396i
$894$		0			0
$895$	−16.3905	−	29.8802i	−0.547873	−	0.998784i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−3.78024			−0.126078
$900$		0			0
$901$	−11.1041			−0.369932
$902$		0			0
$903$		0			0
$904$		0			0
$905$	25.5096	+	46.5045i	0.847967	+	1.54586i
$906$		0			0
$907$			41.0365i			1.36259i	0.732007	+	0.681297i	$0.238584\pi$
$907$			41.0365i			1.36259i	−0.732007	+	0.681297i	$0.761416\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	15.4822			0.512949			0.256474	−	0.966551i	$-0.417439\pi$
$911$	15.4822			0.512949			0.256474	+	0.966551i	$0.417439\pi$
$912$		0			0
$913$			64.7362i			2.14246i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	43.6448i		−	1.44128i
$918$		0			0
$919$	27.1760			0.896455			0.448227	−	0.893920i	$-0.352055\pi$
$919$	27.1760			0.896455			0.448227	+	0.893920i	$0.352055\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			2.59919i			0.0855533i
$924$		0			0
$925$	−4.21663	−	2.68701i	−0.138642	−	0.0883482i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−11.3917			−0.373749			−0.186874	−	0.982384i	$-0.559836\pi$
$929$	−11.3917			−0.373749			−0.186874	+	0.982384i	$0.559836\pi$
$930$		0			0
$931$	2.60401			0.0853431
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−8.37448	+	4.59374i	−0.273875	+	0.150231i
$936$		0			0
$937$		−	42.7886i		−	1.39784i	−0.715199	−	0.698921i	$-0.753664\pi$
$937$		−	42.7886i		−	1.39784i	0.715199	−	0.698921i	$-0.246336\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−7.24333			−0.236126			−0.118063	−	0.993006i	$-0.537668\pi$
$941$	−7.24333			−0.236126			−0.118063	+	0.993006i	$0.537668\pi$
$942$		0			0
$943$			3.80385i			0.123870i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			4.38167i			0.142385i	0.997463	+	0.0711925i	$0.0226805\pi$
$947$			4.38167i			0.142385i	−0.997463	+	0.0711925i	$0.977320\pi$
$948$		0			0
$949$	1.09907			0.0356775
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	44.9223i		−	1.45518i	−0.686014	−	0.727589i	$-0.740641\pi$
$953$		−	44.9223i		−	1.45518i	0.686014	−	0.727589i	$-0.259359\pi$
$954$		0			0
$955$	−13.8861	−	25.3147i	−0.449345	−	0.819164i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−18.1041			−0.584613
$960$		0			0
$961$	−30.0238			−0.968509
$962$		0			0
$963$		0			0
$964$		0			0
$965$	41.4050	−	22.7123i	1.33287	−	0.731135i
$966$		0			0
$967$		−	0.112424i		−	0.00361532i	−0.999998	−	0.00180766i	$-0.999425\pi$
$967$		−	0.112424i		−	0.00361532i	0.999998	−	0.00180766i	$-0.000575396\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−42.1108			−1.35140			−0.675700	−	0.737177i	$-0.736159\pi$
$971$	−42.1108			−1.35140			−0.675700	+	0.737177i	$0.736159\pi$
$972$		0			0
$973$			27.4097i			0.878714i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	50.0879i		−	1.60245i	−0.598360	−	0.801227i	$-0.704181\pi$
$977$		−	50.0879i		−	1.60245i	0.598360	−	0.801227i	$-0.295819\pi$
$978$		0			0
$979$	−9.34037			−0.298520
$980$		0			0
$981$		0			0
$982$		0			0
$983$			12.2414i			0.390440i	0.980759	+	0.195220i	$0.0625421\pi$
$983$			12.2414i			0.390440i	−0.980759	+	0.195220i	$0.937458\pi$
$984$		0			0
$985$	17.2045	−	9.43736i	0.548181	−	0.300699i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	5.81142			0.184793
$990$		0			0
$991$	−26.7437			−0.849541			−0.424770	−	0.905301i	$-0.639645\pi$
$991$	−26.7437			−0.849541			−0.424770	+	0.905301i	$0.639645\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	12.6848	+	23.1247i	0.402136	+	0.733101i
$996$		0			0
$997$		−	44.9308i		−	1.42297i	−0.702701	−	0.711486i	$-0.748023\pi$
$997$		−	44.9308i		−	1.42297i	0.702701	−	0.711486i	$-0.251977\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6660.2.f.c.5329.14		18
3.2	odd	2		740.2.d.a.149.9	✓	18
5.4	even	2	inner	6660.2.f.c.5329.13		18
15.2	even	4		3700.2.a.o.1.5		9
15.8	even	4		3700.2.a.p.1.5		9
15.14	odd	2		740.2.d.a.149.10	yes	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
740.2.d.a.149.9	✓	18	3.2	odd	2
740.2.d.a.149.10	yes	18	15.14	odd	2
3700.2.a.o.1.5		9	15.2	even	4
3700.2.a.p.1.5		9	15.8	even	4
6660.2.f.c.5329.13		18	5.4	even	2	inner
6660.2.f.c.5329.14		18	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 6660 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6660.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(1.07541 + 1.96049i) q^{5} -2.78997i q^{7} +4.91898 q^{11} +0.675823i q^{13} +0.868398i q^{17} -3.32168 q^{19} +0.927380i q^{23} +(-2.68701 + 4.21663i) q^{25} +3.82597 q^{29} -0.988046 q^{31} +(5.46970 - 3.00035i) q^{35} +1.00000i q^{37} +4.10171 q^{41} -6.26649i q^{43} +0.511352i q^{47} -0.783946 q^{49} +12.7869i q^{53} +(5.28990 + 9.64360i) q^{55} +6.90064 q^{59} +3.39134 q^{61} +(-1.32494 + 0.726784i) q^{65} -1.78220i q^{67} +3.84596 q^{71} -1.62628i q^{73} -13.7238i q^{77} +12.2262 q^{79} +13.1605i q^{83} +(-1.70248 + 0.933880i) q^{85} -1.89884 q^{89} +1.88553 q^{91} +(-3.57215 - 6.51210i) q^{95} +18.3123i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q - 2 q^{5} + 4 q^{19} - 2 q^{25} + 4 q^{29} + 8 q^{31} + 2 q^{35} + 4 q^{41} + 6 q^{49} - 6 q^{55} - 8 q^{59} + 12 q^{65} + 24 q^{71} - 24 q^{79} + 36 q^{91} - 24 q^{95}+O(q^{100}) \) 18 * q - 2 * q^5 + 4 * q^19 - 2 * q^25 + 4 * q^29 + 8 * q^31 + 2 * q^35 + 4 * q^41 + 6 * q^49 - 6 * q^55 - 8 * q^59 + 12 * q^65 + 24 * q^71 - 24 * q^79 + 36 * q^91 - 24 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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