LMFDB - Embedded newform 336.4.bj.d.191.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [336,4,Mod(95,336)] 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("336.95"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(336, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 0, 3, 4])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 336 = 2^{4} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	336.bj (of order $6$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,9,0,0,0,-17] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$19.8246417619$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label			191.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	336.191
Dual form			336.4.bj.d.95.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(4.50000 - 2.59808i) q^{3} +(-8.50000 - 16.4545i) q^{7} +(13.5000 - 23.3827i) q^{9} +19.0000 q^{13} +(-109.500 - 63.2199i) q^{19} +(-81.0000 - 51.9615i) q^{21} +(-62.5000 - 108.253i) q^{25} -140.296i q^{27} +(163.500 - 94.3968i) q^{31} +(-161.500 + 279.726i) q^{37} +(85.5000 - 49.3634i) q^{39} -559.452i q^{43} +(-198.500 + 279.726i) q^{49} -657.000 q^{57} +(-91.0000 + 157.617i) q^{61} +(-499.500 - 23.3827i) q^{63} +(943.500 - 544.730i) q^{67} +(-135.500 - 234.693i) q^{73} +(-562.500 - 324.760i) q^{75} +(190.500 + 109.985i) q^{79} +(-364.500 - 631.333i) q^{81} +(-161.500 - 312.635i) q^{91} +(490.500 - 849.571i) q^{93} +1330.00 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 9 q^{3} - 17 q^{7} + 27 q^{9} + 38 q^{13} - 219 q^{19} - 162 q^{21} - 125 q^{25} + 327 q^{31} - 323 q^{37} + 171 q^{39} - 397 q^{49} - 1314 q^{57} - 182 q^{61} - 999 q^{63} + 1887 q^{67} - 271 q^{73}+ \cdots + 2660 q^{97}+O(q^{100}) $ 2 * q + 9 * q^3 - 17 * q^7 + 27 * q^9 + 38 * q^13 - 219 * q^19 - 162 * q^21 - 125 * q^25 + 327 * q^31 - 323 * q^37 + 171 * q^39 - 397 * q^49 - 1314 * q^57 - 182 * q^61 - 999 * q^63 + 1887 * q^67 - 271 * q^73 - 1125 * q^75 + 381 * q^79 - 729 * q^81 - 323 * q^91 + 981 * q^93 + 2660 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/336\mathbb{Z}\right)^\times$.

$n$	$85$	$113$	$127$	$241$
$\chi(n)$	$1$	$-1$	$-1$	$e\left(\frac{1}{3}\right)$

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
$3$	4.50000	−	2.59808i	0.866025	−	0.500000i
$3$	4.50000	−	2.59808i	0.866025	−	0.500000i
$4$		0			0
$5$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$5$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$6$		0			0
$7$	−8.50000	−	16.4545i	−0.458957	−	0.888459i
$7$	−8.50000	−	16.4545i	−0.458957	−	0.888459i
$8$		0			0
$9$	13.5000	−	23.3827i	0.500000	−	0.866025i
$10$		0			0
$11$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$11$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$12$		0			0
$13$	19.0000			0.405358			0.202679	−	0.979245i	$-0.435035\pi$
$13$	19.0000			0.405358			0.202679	+	0.979245i	$0.435035\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$17$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$18$		0			0
$19$	−109.500	−	63.2199i	−1.32216	−	0.763349i	−0.338086	−	0.941115i	$-0.609780\pi$
$19$	−109.500	−	63.2199i	−1.32216	−	0.763349i	−0.984073	+	0.177766i	$0.943113\pi$
$20$		0			0
$21$	−81.0000	−	51.9615i	−0.841698	−	0.539949i
$22$		0			0
$23$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$23$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$24$		0			0
$25$	−62.5000	−	108.253i	−0.500000	−	0.866025i
$26$		0			0
$27$		−	140.296i		−	1.00000i
$28$		0			0
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$	163.500	−	94.3968i	0.947273	−	0.546908i	0.0550403	−	0.998484i	$-0.482471\pi$
$31$	163.500	−	94.3968i	0.947273	−	0.546908i	0.892233	+	0.451576i	$0.149138\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−161.500	+	279.726i	−0.717579	+	1.24288i	0.244377	+	0.969680i	$0.421417\pi$
$37$	−161.500	+	279.726i	−0.717579	+	1.24288i	−0.961956	+	0.273204i	$0.911917\pi$
$38$		0			0
$39$	85.5000	−	49.3634i	0.351050	−	0.202679i
$40$		0			0
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$		−	559.452i		−	1.98409i	−0.125900	−	0.992043i	$-0.540182\pi$
$43$		−	559.452i		−	1.98409i	0.125900	−	0.992043i	$-0.459818\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$47$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$48$		0			0
$49$	−198.500	+	279.726i	−0.578717	+	0.815528i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$53$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−657.000			−1.52670
$58$		0			0
$59$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$59$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$60$		0			0
$61$	−91.0000	+	157.617i	−0.191006	+	0.330832i	−0.945584	−	0.325379i	$-0.894508\pi$
$61$	−91.0000	+	157.617i	−0.191006	+	0.330832i	0.754578	+	0.656210i	$0.227842\pi$
$62$		0			0
$63$	−499.500	−	23.3827i	−0.998906	−	0.0467610i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	943.500	−	544.730i	1.72040	−	0.993274i	0.802307	−	0.596912i	$-0.203606\pi$
$67$	943.500	−	544.730i	1.72040	−	0.993274i	0.918094	−	0.396362i	$-0.129728\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$		0			0
$73$	−135.500	−	234.693i	−0.217248	−	0.376284i	0.736718	−	0.676200i	$-0.236375\pi$
$73$	−135.500	−	234.693i	−0.217248	−	0.376284i	−0.953966	+	0.299916i	$0.903041\pi$
$74$		0			0
$75$	−562.500	−	324.760i	−0.866025	−	0.500000i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	190.500	+	109.985i	0.271303	+	0.156637i	0.629480	−	0.777017i	$-0.283268\pi$
$79$	190.500	+	109.985i	0.271303	+	0.156637i	−0.358177	+	0.933654i	$0.616601\pi$
$80$		0			0
$81$	−364.500	−	631.333i	−0.500000	−	0.866025i
$82$		0			0
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$89$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$90$		0			0
$91$	−161.500	−	312.635i	−0.186042	−	0.360144i
$92$		0			0
$93$	490.500	−	849.571i	0.546908	−	0.947273i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	1330.00			1.39218			0.696088	−	0.717957i	$-0.254922\pi$
$97$	1330.00			1.39218			0.696088	+	0.717957i	$0.254922\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$101$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$102$		0			0
$103$	−919.500	−	530.874i	−0.879622	−	0.507850i	−0.00908799	−	0.999959i	$-0.502893\pi$
$103$	−919.500	−	530.874i	−0.879622	−	0.507850i	−0.870534	+	0.492109i	$0.836226\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$107$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$108$		0			0
$109$	1106.50	+	1916.51i	0.972325	+	1.68412i	0.688493	+	0.725243i	$0.258273\pi$
$109$	1106.50	+	1916.51i	0.972325	+	1.68412i	0.283833	+	0.958874i	$0.408394\pi$
$110$		0			0
$111$			1678.36i			1.43516i
$112$		0			0
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	256.500	−	444.271i	0.202679	−	0.351050i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	665.500	−	1152.68i	0.500000	−	0.866025i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$			1747.64i			1.22109i	0.791983	+	0.610543i	$0.209049\pi$
$127$			1747.64i			1.22109i	−0.791983	+	0.610543i	$0.790951\pi$
$128$		0			0
$129$	−1453.50	−	2517.54i	−0.992043	−	1.71827i
$130$		0			0
$131$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$131$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$132$		0			0
$133$	−109.500	+	2339.13i	−0.0713899	+	1.52503i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$137$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$138$		0			0
$139$			3244.13i			1.97959i	0.142484	+	0.989797i	$0.454491\pi$
$139$			3244.13i			1.97959i	−0.142484	+	0.989797i	$0.545509\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−166.500	+	1774.49i	−0.0934196	+	0.995627i
$148$		0			0
$149$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$149$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$150$		0			0
$151$	2835.00	−	1636.79i	1.52787	−	0.882119i	0.528424	−	0.848981i	$-0.322783\pi$
$151$	2835.00	−	1636.79i	1.52787	−	0.882119i	0.999451	−	0.0331378i	$-0.0105500\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	1925.00	+	3334.20i	0.978546	+	1.69489i	0.667699	+	0.744432i	$0.267279\pi$
$157$	1925.00	+	3334.20i	0.978546	+	1.69489i	0.310847	+	0.950460i	$0.399387\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	2079.00	+	1200.31i	0.999017	+	0.576783i	0.907957	−	0.419062i	$-0.137641\pi$
$163$	2079.00	+	1200.31i	0.999017	+	0.576783i	0.0910600	+	0.995845i	$0.470974\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0			−	1.00000i	$-0.5\pi$
$167$		0			0				1.00000i	$0.5\pi$
$168$		0			0
$169$	−1836.00			−0.835685
$170$		0			0
$171$	−2956.50	+	1706.94i	−1.32216	+	0.763349i
$172$		0			0
$173$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$173$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$174$		0			0
$175$	−1250.00	+	1948.56i	−0.539949	+	0.841698i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$179$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$180$		0			0
$181$	4699.00			1.92969			0.964845	−	0.262819i	$-0.0846523\pi$
$181$	4699.00			1.92969			0.964845	+	0.262819i	$0.0846523\pi$
$182$		0			0
$183$			945.700i			0.382012i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	−2308.50	+	1192.52i	−0.888459	+	0.458957i
$190$		0			0
$191$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$191$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$192$		0			0
$193$	1980.50	+	3430.33i	0.738650	+	1.27938i	0.953103	+	0.302646i	$0.0978698\pi$
$193$	1980.50	+	3430.33i	0.738650	+	1.27938i	−0.214453	+	0.976734i	$0.568797\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$	1755.00	−	1013.25i	0.625169	−	0.360942i	−0.153710	−	0.988116i	$-0.549122\pi$
$199$	1755.00	−	1013.25i	0.625169	−	0.360942i	0.778879	+	0.627175i	$0.215789\pi$
$200$		0			0
$201$	2830.50	−	4902.57i	0.993274	−	1.72040i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$			1091.19i			0.356023i	0.984028	+	0.178011i	$0.0569664\pi$
$211$			1091.19i			0.356023i	−0.984028	+	0.178011i	$0.943034\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−2943.00	−	1887.94i	−0.920663	−	0.590606i
$218$		0			0
$219$	−1219.50	−	704.079i	−0.376284	−	0.217248i
$220$		0			0
$221$		0			0
$222$		0			0
$223$		−	5830.08i		−	1.75072i	−0.483469	−	0.875362i	$-0.660623\pi$
$223$		−	5830.08i		−	1.75072i	0.483469	−	0.875362i	$-0.339377\pi$
$224$		0			0
$225$	−3375.00			−1.00000
$226$		0			0
$227$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$227$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$228$		0			0
$229$	1178.50	−	2041.22i	0.340076	−	0.589029i	−0.644370	−	0.764714i	$-0.722880\pi$
$229$	1178.50	−	2041.22i	0.340076	−	0.589029i	0.984447	+	0.175684i	$0.0562138\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$233$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	1143.00			0.313274
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$	−3689.00	−	6389.54i	−0.986014	−	1.70783i	−0.637341	−	0.770582i	$-0.719966\pi$
$241$	−3689.00	−	6389.54i	−0.986014	−	1.70783i	−0.348673	−	0.937244i	$-0.613368\pi$
$242$		0			0
$243$	−3280.50	−	1894.00i	−0.866025	−	0.500000i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−2080.50	−	1201.18i	−0.535948	−	0.309429i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$257$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$258$		0			0
$259$	5975.50	+	279.726i	1.43359	+	0.0671094i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$263$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$269$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$270$		0			0
$271$	−7695.00	−	4442.71i	−1.72486	−	0.995850i	−0.907935	−	0.419111i	$-0.862342\pi$
$271$	−7695.00	−	4442.71i	−1.72486	−	0.995850i	−0.816928	−	0.576739i	$-0.804325\pi$
$272$		0			0
$273$	−1539.00	−	987.269i	−0.341189	−	0.218873i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	4598.50	+	7964.84i	0.997462	+	1.72766i	0.560388	+	0.828230i	$0.310652\pi$
$277$	4598.50	+	7964.84i	0.997462	+	1.72766i	0.437074	+	0.899425i	$0.356015\pi$
$278$		0			0
$279$		−	5097.43i		−	1.09382i
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$		0			0
$283$	865.500	−	499.697i	0.181797	−	0.104961i	−0.406340	−	0.913722i	$-0.633195\pi$
$283$	865.500	−	499.697i	0.181797	−	0.104961i	0.588137	+	0.808761i	$0.299862\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−2456.50	+	4254.78i	−0.500000	+	0.866025i
$290$		0			0
$291$	5985.00	−	3455.44i	1.20566	−	0.696088i
$292$		0			0
$293$		0			0		1.00000			$0$
$293$		0			0		−1.00000			$\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	−9205.50	+	4755.35i	−1.76278	+	0.910610i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		−	10009.5i		−	1.86083i	−0.366513	−	0.930413i	$-0.619448\pi$
$307$		−	10009.5i		−	1.86083i	0.366513	−	0.930413i	$-0.380552\pi$
$308$		0			0
$309$	−5517.00			−1.01570
$310$		0			0
$311$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$311$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$312$		0			0
$313$	4554.50	−	7888.63i	0.822478	−	1.42457i	−0.0813539	−	0.996685i	$-0.525924\pi$
$313$	4554.50	−	7888.63i	0.822478	−	1.42457i	0.903832	−	0.427888i	$-0.140742\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$317$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−1187.50	−	2056.81i	−0.202679	−	0.351050i
$326$		0			0
$327$	9958.50	+	5749.54i	1.68412	+	0.972325i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−4453.50	−	2571.23i	−0.739536	−	0.426971i	0.0823644	−	0.996602i	$-0.473753\pi$
$331$	−4453.50	−	2571.23i	−0.739536	−	0.426971i	−0.821901	+	0.569631i	$0.807086\pi$
$332$		0			0
$333$	4360.50	+	7552.61i	0.717579	+	1.24288i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−12293.0			−1.98707			−0.993535	−	0.113529i	$-0.963785\pi$
$337$	−12293.0			−1.98707			−0.993535	+	0.113529i	$0.963785\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	6290.00	+	888.542i	0.990169	+	0.139874i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$347$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$348$		0			0
$349$	−11914.0			−1.82734			−0.913670	−	0.406456i	$-0.866764\pi$
$349$	−11914.0			−1.82734			−0.913670	+	0.406456i	$0.866764\pi$
$350$		0			0
$351$		−	2665.63i		−	0.405358i
$352$		0			0
$353$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$353$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$359$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$360$		0			0
$361$	4564.00	+	7905.08i	0.665403	+	1.15251i
$362$		0			0
$363$		−	6916.08i		−	1.00000i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−9046.50	+	5223.00i	−1.28671	+	0.742884i	−0.978066	−	0.208293i	$-0.933209\pi$
$367$	−9046.50	+	5223.00i	−1.28671	+	0.742884i	−0.308646	+	0.951177i	$0.599876\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	6300.50	−	10912.8i	0.874605	−	1.51486i	0.0174213	−	0.999848i	$-0.494454\pi$
$373$	6300.50	−	10912.8i	0.874605	−	1.51486i	0.857183	−	0.515011i	$-0.172212\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$			13435.5i			1.82094i	0.413575	+	0.910470i	$0.364280\pi$
$379$			13435.5i			1.82094i	−0.413575	+	0.910470i	$0.635720\pi$
$380$		0			0
$381$	4540.50	+	7864.38i	0.610543	+	1.05749i
$382$		0			0
$383$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$383$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−13081.5	−	7552.61i	−1.71827	−	0.992043i
$388$		0			0
$389$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$389$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	7128.50	−	12346.9i	0.901182	−	1.56089i	0.0752196	−	0.997167i	$-0.476034\pi$
$397$	7128.50	−	12346.9i	0.901182	−	1.56089i	0.825962	−	0.563726i	$-0.190632\pi$
$398$		0			0
$399$	5584.50	+	10810.6i	0.700688	+	1.35641i
$400$		0			0
$401$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$401$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$402$		0			0
$403$	3106.50	−	1793.54i	0.383985	−	0.221694i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−8271.50	−	14326.7i	−0.999998	−	1.73205i	−0.501541	−	0.865134i	$-0.667233\pi$
$409$	−8271.50	−	14326.7i	−0.999998	−	1.73205i	−0.498458	−	0.866914i	$-0.666100\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	8428.50	+	14598.6i	0.989797	+	1.71438i
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$	6679.00			0.773194			0.386597	−	0.922249i	$-0.373650\pi$
$421$	6679.00			0.773194			0.386597	+	0.922249i	$0.373650\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	3367.00	+	157.617i	0.381594	+	0.0178632i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$431$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$432$		0			0
$433$	16739.0			1.85779			0.928897	−	0.370338i	$-0.120758\pi$
$433$	16739.0			1.85779			0.928897	+	0.370338i	$0.120758\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	−9315.00	−	5378.02i	−1.01271	−	0.584690i	−0.100728	−	0.994914i	$-0.532117\pi$
$439$	−9315.00	−	5378.02i	−1.01271	−	0.584690i	−0.911985	+	0.410224i	$0.865450\pi$
$440$		0			0
$441$	3861.00	+	8417.77i	0.416910	+	0.908948i
$442$		0			0
$443$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$443$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	8505.00	−	14731.1i	0.882119	−	1.52787i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	3248.50	−	5626.57i	0.332513	−	0.575929i	−0.650491	−	0.759514i	$-0.725437\pi$
$457$	3248.50	−	5626.57i	0.332513	−	0.575929i	0.983004	+	0.183585i	$0.0587702\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$			15929.7i			1.59895i	0.600698	+	0.799476i	$0.294889\pi$
$463$			15929.7i			1.59895i	−0.600698	+	0.799476i	$0.705111\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$467$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$468$		0			0
$469$	−16983.0	−	10894.6i	−1.67207	−	1.07264i
$470$		0			0
$471$	17325.0	+	10002.6i	1.69489	+	0.978546i
$472$		0			0
$473$		0			0
$474$		0			0
$475$			15805.0i			1.52670i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$479$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$480$		0			0
$481$	−3068.50	+	5314.80i	−0.290876	+	0.503813i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−17848.5	+	10304.8i	−1.66077	+	0.958843i	−0.688415	+	0.725317i	$0.741693\pi$
$487$	−17848.5	+	10304.8i	−1.66077	+	0.958843i	−0.972351	+	0.233526i	$0.924974\pi$
$488$		0			0
$489$	12474.0			1.15357
$490$		0			0
$491$		0			0			−	1.00000i	$-0.5\pi$
$491$		0			0				1.00000i	$0.5\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	4264.50	+	2462.11i	0.382576	+	0.220880i	0.678938	−	0.734195i	$-0.262440\pi$
$499$	4264.50	+	2462.11i	0.382576	+	0.220880i	−0.296363	+	0.955075i	$0.595774\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−8262.00	+	4770.07i	−0.723724	+	0.417843i
$508$		0			0
$509$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$509$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$510$		0			0
$511$	−2710.00	+	4224.47i	−0.234605	+	0.365714i
$512$		0			0
$513$	−8869.50	+	15362.4i	−0.763349	+	1.32216i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$521$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$522$		0			0
$523$	−79.5000	−	45.8993i	−0.00664683	−	0.00383755i	0.496673	−	0.867938i	$-0.334555\pi$
$523$	−79.5000	−	45.8993i	−0.00664683	−	0.00383755i	−0.503320	+	0.864100i	$0.667888\pi$
$524$		0			0
$525$	−562.500	+	12016.1i	−0.0467610	+	0.998906i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	6083.50	+	10536.9i	0.500000	+	0.866025i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	944.500	−	1635.92i	0.0750596	−	0.130007i	−0.826053	−	0.563593i	$-0.809419\pi$
$541$	944.500	−	1635.92i	0.0750596	−	0.130007i	0.901112	+	0.433586i	$0.142752\pi$
$542$		0			0
$543$	21145.5	−	12208.4i	1.67116	−	0.964845i
$544$		0			0
$545$		0			0
$546$		0			0
$547$			25533.9i			1.99589i	0.0640963	+	0.997944i	$0.479584\pi$
$547$			25533.9i			1.99589i	−0.0640963	+	0.997944i	$0.520416\pi$
$548$		0			0
$549$	2457.00	+	4255.65i	0.191006	+	0.330832i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	190.500	−	4069.45i	0.0146490	−	0.312931i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$557$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$558$		0			0
$559$		−	10629.6i		−	0.804265i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$563$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−7290.00	+	11364.0i	−0.539949	+	0.841698i
$568$		0			0
$569$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$569$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$570$		0			0
$571$	11341.5	−	6548.02i	0.831221	−	0.479905i	−0.0230498	−	0.999734i	$-0.507338\pi$
$571$	11341.5	−	6548.02i	0.831221	−	0.479905i	0.854270	+	0.519829i	$0.174004\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	4806.50	+	8325.10i	0.346789	+	0.600656i	0.985677	−	0.168644i	$-0.0539387\pi$
$577$	4806.50	+	8325.10i	0.346789	+	0.600656i	−0.638888	+	0.769300i	$0.720605\pi$
$578$		0			0
$579$	17824.5	+	10291.0i	1.27938	+	0.738650i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0			−	1.00000i	$-0.5\pi$
$587$		0			0				1.00000i	$0.5\pi$
$588$		0			0
$589$	−23871.0			−1.66993
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$593$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	5265.00	−	9119.25i	0.360942	−	0.625169i
$598$		0			0
$599$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$599$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$600$		0			0
$601$	17351.0			1.17764			0.588820	−	0.808264i	$-0.299593\pi$
$601$	17351.0			1.17764			0.588820	+	0.808264i	$0.299593\pi$
$602$		0			0
$603$		−	29415.4i		−	1.98655i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−25351.5	−	14636.7i	−1.69520	−	0.978723i	−0.950191	−	0.311667i	$-0.899113\pi$
$607$	−25351.5	−	14636.7i	−1.69520	−	0.978723i	−0.745007	−	0.667056i	$-0.767554\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−8695.00	−	15060.2i	−0.572900	−	0.992292i	−0.996266	−	0.0863334i	$-0.972485\pi$
$613$	−8695.00	−	15060.2i	−0.572900	−	0.992292i	0.423366	−	0.905959i	$-0.360848\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		1.00000			$0$
$617$		0			0		−1.00000			$\pi$
$618$		0			0
$619$	13309.5	−	7684.24i	0.864223	−	0.498959i	−0.00120126	−	0.999999i	$-0.500382\pi$
$619$	13309.5	−	7684.24i	0.864223	−	0.498959i	0.865424	+	0.501040i	$0.167049\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−7812.50	+	13531.6i	−0.500000	+	0.866025i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$		−	31644.6i		−	1.99643i	−0.0596825	−	0.998217i	$-0.519009\pi$
$631$		−	31644.6i		−	1.99643i	0.0596825	−	0.998217i	$-0.480991\pi$
$632$		0			0
$633$	2835.00	+	4910.36i	0.178011	+	0.308325i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−3771.50	+	5314.80i	−0.234588	+	0.330581i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$641$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$642$		0			0
$643$			26315.0i			1.61394i	0.590592	+	0.806971i	$0.298894\pi$
$643$			26315.0i			1.61394i	−0.590592	+	0.806971i	$0.701106\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$647$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	−18148.5	−	849.571i	−1.09262	−	0.0511479i
$652$		0			0
$653$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$653$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−7317.00			−0.434495
$658$		0			0
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$	−16865.5	−	29211.9i	−0.992423	−	1.71893i	−0.602615	−	0.798032i	$-0.705875\pi$
$661$	−16865.5	−	29211.9i	−0.992423	−	1.71893i	−0.389808	−	0.920896i	$-0.627459\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−15147.0	−	26235.4i	−0.875362	−	1.51617i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	9899.00			0.566981			0.283491	−	0.958975i	$-0.408508\pi$
$673$	9899.00			0.566981			0.283491	+	0.958975i	$0.408508\pi$
$674$		0			0
$675$	−15187.5	+	8768.51i	−0.866025	+	0.500000i
$676$		0			0
$677$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$677$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$678$		0			0
$679$	−11305.0	−	21884.5i	−0.638949	−	1.23689i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$683$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		−	12247.3i		−	0.680153i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−26161.5	−	15104.3i	−1.44028	−	0.831543i	−0.442408	−	0.896814i	$-0.645876\pi$
$691$	−26161.5	−	15104.3i	−1.44028	−	0.831543i	−0.997868	+	0.0652705i	$0.979209\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$	35368.5	−	20420.0i	1.89751	−	1.09553i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	18073.0	−	31303.4i	0.957328	−	1.65814i	0.228381	−	0.973572i	$-0.426657\pi$
$709$	18073.0	−	31303.4i	0.957328	−	1.65814i	0.728948	−	0.684569i	$-0.240010\pi$
$710$		0			0
$711$	5143.50	−	2969.60i	0.271303	−	0.156637i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$719$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$720$		0			0
$721$	−919.500	+	19642.3i	−0.0474951	+	1.01459i
$722$		0			0
$723$	−33201.0	−	19168.6i	−1.70783	−	0.986014i
$724$		0			0
$725$		0			0
$726$		0			0
$727$		−	9510.69i		−	0.485188i	−0.970128	−	0.242594i	$-0.922002\pi$
$727$		−	9510.69i		−	0.485188i	0.970128	−	0.242594i	$-0.0779984\pi$
$728$		0			0
$729$	−19683.0			−1.00000
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−19665.5	+	34061.6i	−0.990944	+	1.71637i	−0.379184	+	0.925321i	$0.623795\pi$
$733$	−19665.5	+	34061.6i	−0.990944	+	1.71637i	−0.611759	+	0.791044i	$0.709538\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	34399.5	−	19860.6i	1.71232	−	0.988610i	0.780910	−	0.624644i	$-0.214756\pi$
$739$	34399.5	−	19860.6i	1.71232	−	0.988610i	0.931412	−	0.363966i	$-0.118578\pi$
$740$		0			0
$741$	−12483.0			−0.618859
$742$		0			0
$743$		0			0			−	1.00000i	$-0.5\pi$
$743$		0			0				1.00000i	$0.5\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	32236.5	+	18611.8i	1.56635	+	0.904331i	0.996590	+	0.0825179i	$0.0262962\pi$
$751$	32236.5	+	18611.8i	1.56635	+	0.904331i	0.569757	+	0.821813i	$0.307037\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	41470.0			1.99109			0.995543	−	0.0943039i	$-0.0300625\pi$
$757$	41470.0			1.99109			0.995543	+	0.0943039i	$0.0300625\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$761$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$762$		0			0
$763$	22130.0	−	34497.3i	1.05001	−	1.63681i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−34417.0			−1.61393			−0.806963	−	0.590602i	$-0.798890\pi$
$769$	−34417.0			−1.61393			−0.806963	+	0.590602i	$0.798890\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$773$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$774$		0			0
$775$	−20437.5	−	11799.6i	−0.947273	−	0.546908i
$776$		0			0
$777$	27616.5	−	14266.0i	1.27508	−	0.658676i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−7047.00	+	4068.59i	−0.319185	+	0.184281i	−0.651029	−	0.759053i	$-0.725662\pi$
$787$	−7047.00	+	4068.59i	−0.319185	+	0.184281i	0.331844	+	0.943334i	$0.392329\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−1729.00	+	2994.72i	−0.0774257	+	0.134105i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		1.00000			$0$
$797$		0			0		−1.00000			$\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$809$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$810$		0			0
$811$			24162.1i			1.04617i	0.852280	+	0.523087i	$0.175220\pi$
$811$			24162.1i			1.04617i	−0.852280	+	0.523087i	$0.824780\pi$
$812$		0			0
$813$	−46170.0			−1.99170
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−35368.5	+	61260.0i	−1.51455	+	2.62328i
$818$		0			0
$819$	−9490.50	−	444.271i	−0.404914	−	0.0189549i
$820$		0			0
$821$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$821$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$822$		0			0
$823$	−39501.0	+	22805.9i	−1.67305	+	0.965935i	−0.707131	+	0.707083i	$0.750011\pi$
$823$	−39501.0	+	22805.9i	−1.67305	+	0.965935i	−0.965917	+	0.258852i	$0.916656\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0			−	1.00000i	$-0.5\pi$
$827$		0			0				1.00000i	$0.5\pi$
$828$		0			0
$829$	15038.5	+	26047.4i	0.630047	+	1.09127i	0.987542	+	0.157358i	$0.0502976\pi$
$829$	15038.5	+	26047.4i	0.630047	+	1.09127i	−0.357495	+	0.933915i	$0.616369\pi$
$830$		0			0
$831$	41386.5	+	23894.5i	1.72766	+	0.997462i
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−13243.5	−	22938.4i	−0.546908	−	0.947273i
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	24389.0			1.00000
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−24623.5	−	1152.68i	−0.998906	−	0.0467610i
$848$		0			0
$849$	2596.50	−	4497.27i	0.104961	−	0.181797i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	8279.00			0.332318			0.166159	−	0.986099i	$-0.446863\pi$
$853$	8279.00			0.332318			0.166159	+	0.986099i	$0.446863\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$857$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$858$		0			0
$859$	34155.0	+	19719.4i	1.35664	+	0.783256i	0.989169	−	0.146779i	$-0.0468906\pi$
$859$	34155.0	+	19719.4i	1.35664	+	0.783256i	0.367470	+	0.930035i	$0.380224\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$863$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$			25528.7i			1.00000i
$868$		0			0
$869$		0			0
$870$		0			0
$871$	17926.5	−	10349.9i	0.697378	−	0.402631i
$872$		0			0
$873$	17955.0	−	31099.0i	0.696088	−	1.20566i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−25075.0	+	43431.2i	−0.965476	+	1.67225i	−0.257146	+	0.966373i	$0.582782\pi$
$877$	−25075.0	+	43431.2i	−0.965476	+	1.67225i	−0.708330	+	0.705881i	$0.750551\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$		0			0
$883$		−	6205.94i		−	0.236519i	−0.992983	−	0.118260i	$-0.962269\pi$
$883$		−	6205.94i		−	0.236519i	0.992983	−	0.118260i	$-0.0377315\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$887$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$888$		0			0
$889$	28756.5	−	14854.9i	1.08488	−	0.560426i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$	−29070.0	+	45315.6i	−1.07131	+	1.67000i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−20116.5	+	11614.3i	−0.736447	+	0.425188i	−0.820776	−	0.571250i	$-0.806459\pi$
$907$	−20116.5	+	11614.3i	−0.736447	+	0.425188i	0.0843291	+	0.996438i	$0.473125\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0			−	1.00000i	$-0.5\pi$
$911$		0			0				1.00000i	$0.5\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	26164.5	+	15106.1i	0.939159	+	0.542224i	0.889697	−	0.456552i	$-0.150916\pi$
$919$	26164.5	+	15106.1i	0.939159	+	0.542224i	0.0494625	+	0.998776i	$0.484249\pi$
$920$		0			0
$921$	−26005.5	−	45042.8i	−0.930413	−	1.61152i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	40375.0			1.43516
$926$		0			0
$927$	−24826.5	+	14333.6i	−0.879622	+	0.507850i
$928$		0			0
$929$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$929$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$930$		0			0
$931$	39420.0	−	18080.9i	1.38769	−	0.636495i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	40283.0			1.40447			0.702235	−	0.711945i	$-0.252186\pi$
$937$	40283.0			1.40447			0.702235	+	0.711945i	$0.252186\pi$
$938$		0			0
$939$		−	47331.8i		−	1.64496i
$940$		0			0
$941$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$941$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$947$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$948$		0			0
$949$	−2574.50	−	4459.16i	−0.0880630	−	0.152530i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		1.00000			$0$
$953$		0			0		−1.00000			$\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	2926.00	−	5067.98i	0.0982176	−	0.170118i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$			60013.8i			1.99578i	0.0649641	+	0.997888i	$0.479307\pi$
$967$			60013.8i			1.99578i	−0.0649641	+	0.997888i	$0.520693\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$971$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$972$		0			0
$973$	53380.5	−	27575.1i	1.75879	−	0.908548i
$974$		0			0
$975$	−10687.5	−	6170.43i	−0.351050	−	0.202679i
$976$		0			0
$977$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$977$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	59751.0			1.94465
$982$		0			0
$983$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$983$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−52648.5	+	30396.6i	−1.68762	+	0.974350i	−0.731292	+	0.682064i	$0.761082\pi$
$991$	−52648.5	+	30396.6i	−1.68762	+	0.974350i	−0.956331	+	0.292286i	$0.905584\pi$
$992$		0			0
$993$	−26721.0			−0.853943
$994$		0			0
$995$		0			0
$996$		0			0
$997$	31446.5	+	54466.9i	0.998917	+	1.73018i	0.539745	+	0.841828i	$0.318520\pi$
$997$	31446.5	+	54466.9i	0.998917	+	1.73018i	0.459172	+	0.888347i	$0.348146\pi$
$998$		0			0
$999$	39244.5	+	22657.8i	1.24288	+	0.717579i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	336.4.bj.d.191.1	yes	2
3.2	odd	2	CM	336.4.bj.d.191.1	yes	2
4.3	odd	2		336.4.bj.a.191.1	yes	2
7.4	even	3		336.4.bj.a.95.1	✓	2
12.11	even	2		336.4.bj.a.191.1	yes	2
21.11	odd	6		336.4.bj.a.95.1	✓	2
28.11	odd	6	inner	336.4.bj.d.95.1	yes	2
84.11	even	6	inner	336.4.bj.d.95.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
336.4.bj.a.95.1	✓	2	7.4	even	3
336.4.bj.a.95.1	✓	2	21.11	odd	6
336.4.bj.a.191.1	yes	2	4.3	odd	2
336.4.bj.a.191.1	yes	2	12.11	even	2
336.4.bj.d.95.1	yes	2	28.11	odd	6	inner
336.4.bj.d.95.1	yes	2	84.11	even	6	inner
336.4.bj.d.191.1	yes	2	1.1	even	1	trivial
336.4.bj.d.191.1	yes	2	3.2	odd	2	CM

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:	\( N \)	\(=\)	\( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	336.bj (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(4.50000 - 2.59808i) q^{3} +(-8.50000 - 16.4545i) q^{7} +(13.5000 - 23.3827i) q^{9} +19.0000 q^{13} +(-109.500 - 63.2199i) q^{19} +(-81.0000 - 51.9615i) q^{21} +(-62.5000 - 108.253i) q^{25} -140.296i q^{27} +(163.500 - 94.3968i) q^{31} +(-161.500 + 279.726i) q^{37} +(85.5000 - 49.3634i) q^{39} -559.452i q^{43} +(-198.500 + 279.726i) q^{49} -657.000 q^{57} +(-91.0000 + 157.617i) q^{61} +(-499.500 - 23.3827i) q^{63} +(943.500 - 544.730i) q^{67} +(-135.500 - 234.693i) q^{73} +(-562.500 - 324.760i) q^{75} +(190.500 + 109.985i) q^{79} +(-364.500 - 631.333i) q^{81} +(-161.500 - 312.635i) q^{91} +(490.500 - 849.571i) q^{93} +1330.00 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 9 q^{3} - 17 q^{7} + 27 q^{9} + 38 q^{13} - 219 q^{19} - 162 q^{21} - 125 q^{25} + 327 q^{31} - 323 q^{37} + 171 q^{39} - 397 q^{49} - 1314 q^{57} - 182 q^{61} - 999 q^{63} + 1887 q^{67} - 271 q^{73}+ \cdots + 2660 q^{97}+O(q^{100}) \) 2 * q + 9 * q^3 - 17 * q^7 + 27 * q^9 + 38 * q^13 - 219 * q^19 - 162 * q^21 - 125 * q^25 + 327 * q^31 - 323 * q^37 + 171 * q^39 - 397 * q^49 - 1314 * q^57 - 182 * q^61 - 999 * q^63 + 1887 * q^67 - 271 * q^73 - 1125 * q^75 + 381 * q^79 - 729 * q^81 - 323 * q^91 + 981 * q^93 + 2660 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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