LMFDB - Embedded newform 84.4.k.a.17.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [84,4,Mod(5,84)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(84, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 3, 5]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("84.5");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 84 = 2^{2} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	84.k (of order $6$, degree $2$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$4.95616044048$
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			17.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	84.17
Dual form			84.4.k.a.5.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-4.50000 - 2.59808i) q^{3} +(-8.50000 + 16.4545i) q^{7} +(13.5000 + 23.3827i) q^{9} +O(q^{10})$	$q+(-4.50000 - 2.59808i) q^{3} +(-8.50000 + 16.4545i) q^{7} +(13.5000 + 23.3827i) q^{9} +91.7987i q^{13} +(-109.500 + 63.2199i) q^{19} +(81.0000 - 51.9615i) q^{21} +(62.5000 - 108.253i) q^{25} -140.296i q^{27} +(-163.500 - 94.3968i) q^{31} +(161.500 + 279.726i) q^{37} +(238.500 - 413.094i) q^{39} -71.0000 q^{43} +(-198.500 - 279.726i) q^{49} +657.000 q^{57} +(-810.000 + 467.654i) q^{61} +(-499.500 + 23.3827i) q^{63} +(63.5000 - 109.985i) q^{67} +(1054.50 + 608.816i) q^{73} +(-562.500 + 324.760i) q^{75} +(693.500 + 1201.18i) q^{79} +(-364.500 + 631.333i) q^{81} +(-1510.50 - 780.289i) q^{91} +(490.500 + 849.571i) q^{93} -1371.78i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 9 q^{3} - 17 q^{7} + 27 q^{9}+O(q^{10}) $ 2 * q - 9 * q^3 - 17 * q^7 + 27 * q^9	$ 2 q - 9 q^{3} - 17 q^{7} + 27 q^{9} - 219 q^{19} + 162 q^{21} + 125 q^{25} - 327 q^{31} + 323 q^{37} + 477 q^{39} - 142 q^{43} - 397 q^{49} + 1314 q^{57} - 1620 q^{61} - 999 q^{63} + 127 q^{67} + 2109 q^{73} - 1125 q^{75} + 1387 q^{79} - 729 q^{81} - 3021 q^{91} + 981 q^{93}+O(q^{100}) $ 2 * q - 9 * q^3 - 17 * q^7 + 27 * q^9 - 219 * q^19 + 162 * q^21 + 125 * q^25 - 327 * q^31 + 323 * q^37 + 477 * q^39 - 142 * q^43 - 397 * q^49 + 1314 * q^57 - 1620 * q^61 - 999 * q^63 + 127 * q^67 + 2109 * q^73 - 1125 * q^75 + 1387 * q^79 - 729 * q^81 - 3021 * q^91 + 981 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/84\mathbb{Z}\right)^\times$.

$n$	$29$	$43$	$73$
$\chi(n)$	$-1$	$1$	$e\left(\frac{1}{6}\right)$

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		0.866025	−	0.500000i	$-0.166667\pi$
$5$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$6$		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		0.500000	−	0.866025i	$-0.333333\pi$
$11$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$12$		0			0
$13$			91.7987i			1.95849i	0.202679	+	0.979245i	$0.435035\pi$
$13$			91.7987i			1.95849i	−0.202679	+	0.979245i	$0.564965\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$17$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$18$		0			0
$19$	−109.500	+	63.2199i	−1.32216	+	0.763349i	−0.984073	−	0.177766i	$-0.943113\pi$
$19$	−109.500	+	63.2199i	−1.32216	+	0.763349i	−0.338086	+	0.941115i	$0.609780\pi$
$20$		0			0
$21$	81.0000	−	51.9615i	0.841698	−	0.539949i
$22$		0			0
$23$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$23$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$24$		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	$-0.517529\pi$
$31$	−163.500	−	94.3968i	−0.947273	−	0.546908i	−0.892233	+	0.451576i	$0.850862\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	161.500	+	279.726i	0.717579	+	1.24288i	0.961956	+	0.273204i	$0.0880833\pi$
$37$	161.500	+	279.726i	0.717579	+	1.24288i	−0.244377	+	0.969680i	$0.578583\pi$
$38$		0			0
$39$	238.500	−	413.094i	0.979245	−	1.69610i
$40$		0			0
$41$		0			0			−	1.00000i	$-0.5\pi$
$41$		0			0				1.00000i	$0.5\pi$
$42$		0			0
$43$	−71.0000			−0.251800			−0.125900	−	0.992043i	$-0.540182\pi$
$43$	−71.0000			−0.251800			−0.125900	+	0.992043i	$0.540182\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$47$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$48$		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	$-0.333333\pi$
$53$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	657.000			1.52670
$58$		0			0
$59$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$59$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$60$		0			0
$61$	−810.000	+	467.654i	−1.70016	+	0.981589i	−0.754578	+	0.656210i	$0.772158\pi$
$61$	−810.000	+	467.654i	−1.70016	+	0.981589i	−0.945584	+	0.325379i	$0.894508\pi$
$62$		0			0
$63$	−499.500	+	23.3827i	−0.998906	+	0.0467610i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	63.5000	−	109.985i	0.115787	−	0.200550i	−0.802307	−	0.596912i	$-0.796394\pi$
$67$	63.5000	−	109.985i	0.115787	−	0.200550i	0.918094	+	0.396362i	$0.129728\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$	1054.50	+	608.816i	1.69068	+	0.976117i	0.953966	+	0.299916i	$0.0969588\pi$
$73$	1054.50	+	608.816i	1.69068	+	0.976117i	0.736718	+	0.676200i	$0.236375\pi$
$74$		0			0
$75$	−562.500	+	324.760i	−0.866025	+	0.500000i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	693.500	+	1201.18i	0.987656	+	1.71067i	0.629480	+	0.777017i	$0.283268\pi$
$79$	693.500	+	1201.18i	0.987656	+	1.71067i	0.358177	+	0.933654i	$0.383399\pi$
$80$		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		0.866025	−	0.500000i	$-0.166667\pi$
$89$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$90$		0			0
$91$	−1510.50	−	780.289i	−1.74004	−	0.898863i
$92$		0			0
$93$	490.500	+	849.571i	0.546908	+	0.947273i
$94$		0			0
$95$		0			0
$96$		0			0
$97$		−	1371.78i		−	1.43591i	−0.696088	−	0.717957i	$-0.745078\pi$
$97$		−	1371.78i		−	1.43591i	0.696088	−	0.717957i	$-0.254922\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$101$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$102$		0			0
$103$	919.500	−	530.874i	0.879622	−	0.507850i	0.00908799	−	0.999959i	$-0.497107\pi$
$103$	919.500	−	530.874i	0.879622	−	0.507850i	0.870534	+	0.492109i	$0.163774\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$107$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$108$		0			0
$109$	1106.50	−	1916.51i	0.972325	−	1.68412i	0.283833	−	0.958874i	$-0.408394\pi$
$109$	1106.50	−	1916.51i	0.972325	−	1.68412i	0.688493	−	0.725243i	$-0.258273\pi$
$110$		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
$117$	−2146.50	+	1239.28i	−1.69610	+	0.979245i
$118$		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
$127$	2267.00			1.58397			0.791983	−	0.610543i	$-0.209049\pi$
$127$	2267.00			1.58397			0.791983	+	0.610543i	$0.209049\pi$
$128$		0			0
$129$	319.500	+	184.463i	0.218065	+	0.125900i
$130$		0			0
$131$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$131$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$132$		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	$-0.333333\pi$
$137$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$138$		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	$-0.666667\pi$
$149$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$150$		0			0
$151$	874.000	−	1513.81i	0.471027	−	0.815843i	−0.528424	−	0.848981i	$-0.677217\pi$
$151$	874.000	−	1513.81i	0.471027	−	0.815843i	0.999451	+	0.0331378i	$0.0105500\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−702.000	−	405.300i	−0.356852	−	0.206028i	0.310847	−	0.950460i	$-0.399387\pi$
$157$	−702.000	−	405.300i	−0.356852	−	0.206028i	−0.667699	+	0.744432i	$0.732721\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	1700.00	+	2944.49i	0.816897	+	1.41491i	0.907957	+	0.419062i	$0.137641\pi$
$163$	1700.00	+	2944.49i	0.816897	+	1.41491i	−0.0910600	+	0.995845i	$0.529026\pi$
$164$		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
$169$	−6230.00			−2.83569
$170$		0			0
$171$	−2956.50	−	1706.94i	−1.32216	−	0.763349i
$172$		0			0
$173$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$173$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$174$		0			0
$175$	1250.00	+	1948.56i	0.539949	+	0.841698i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$179$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$180$		0			0
$181$			1279.99i			0.525639i	0.964845	+	0.262819i	$0.0846523\pi$
$181$			1279.99i			0.525639i	−0.964845	+	0.262819i	$0.915348\pi$
$182$		0			0
$183$	4860.00			1.96318
$184$		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		−0.500000	−	0.866025i	$-0.666667\pi$
$191$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$192$		0			0
$193$	−1980.50	+	3430.33i	−0.738650	+	1.27938i	0.214453	+	0.976734i	$0.431203\pi$
$193$	−1980.50	+	3430.33i	−0.738650	+	1.27938i	−0.953103	+	0.302646i	$0.902130\pi$
$194$		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	0.778879	−	0.627175i	$-0.215789\pi$
$199$	1755.00	+	1013.25i	0.625169	+	0.360942i	−0.153710	+	0.988116i	$0.549122\pi$
$200$		0			0
$201$	−571.500	+	329.956i	−0.200550	+	0.115787i
$202$		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
$211$	−6032.00			−1.96806			−0.984028	−	0.178011i	$-0.943034\pi$
$211$	−6032.00			−1.96806			−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
$219$	−3163.50	−	5479.34i	−0.976117	−	1.69068i
$220$		0			0
$221$		0			0
$222$		0			0
$223$			5830.08i			1.75072i	0.483469	+	0.875362i	$0.339377\pi$
$223$			5830.08i			1.75072i	−0.483469	+	0.875362i	$0.660623\pi$
$224$		0			0
$225$	3375.00			1.00000
$226$		0			0
$227$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$227$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$228$		0			0
$229$	5644.50	−	3258.85i	1.62882	−	0.940398i	0.644370	−	0.764714i	$-0.277120\pi$
$229$	5644.50	−	3258.85i	1.62882	−	0.940398i	0.984447	−	0.175684i	$-0.0562138\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$233$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		−	7207.06i		−	1.97531i
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$		0			0
$241$	−1080.00	−	623.538i	−0.288668	−	0.166662i	0.348673	−	0.937244i	$-0.386632\pi$
$241$	−1080.00	−	623.538i	−0.288668	−	0.166662i	−0.637341	+	0.770582i	$0.719966\pi$
$242$		0			0
$243$	3280.50	−	1894.00i	0.866025	−	0.500000i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−5803.50	−	10052.0i	−1.49501	−	2.58944i
$248$		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		0.866025	−	0.500000i	$-0.166667\pi$
$257$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$258$		0			0
$259$	−5975.50	+	279.726i	−1.43359	+	0.0671094i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$263$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$269$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$270$		0			0
$271$	7695.00	−	4442.71i	1.72486	−	0.995850i	0.816928	−	0.576739i	$-0.195675\pi$
$271$	7695.00	−	4442.71i	1.72486	−	0.995850i	0.907935	−	0.419111i	$-0.137658\pi$
$272$		0			0
$273$	4770.00	+	7435.69i	1.05749	+	1.64846i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−4598.50	+	7964.84i	−0.997462	+	1.72766i	−0.437074	+	0.899425i	$0.643985\pi$
$277$	−4598.50	+	7964.84i	−0.997462	+	1.72766i	−0.560388	+	0.828230i	$0.689348\pi$
$278$		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	0.588137	−	0.808761i	$-0.299862\pi$
$283$	865.500	+	499.697i	0.181797	+	0.104961i	−0.406340	+	0.913722i	$0.633195\pi$
$284$		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
$291$	−3564.00	+	6173.03i	−0.717957	+	1.24354i
$292$		0			0
$293$		0			0			−	1.00000i	$-0.5\pi$
$293$		0			0				1.00000i	$0.5\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	603.500	−	1168.27i	0.115565	−	0.223714i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$			10009.5i			1.86083i	0.366513	+	0.930413i	$0.380552\pi$
$307$			10009.5i			1.86083i	−0.366513	+	0.930413i	$0.619448\pi$
$308$		0			0
$309$	−5517.00			−1.01570
$310$		0			0
$311$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$311$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$312$		0			0
$313$	5455.50	−	3149.73i	0.985186	−	0.568797i	0.0813539	−	0.996685i	$-0.474076\pi$
$313$	5455.50	−	3149.73i	0.985186	−	0.568797i	0.903832	+	0.427888i	$0.140742\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$317$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	9937.50	+	5737.42i	1.69610	+	0.979245i
$326$		0			0
$327$	−9958.50	+	5749.54i	−1.68412	+	0.972325i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−5445.50	−	9431.88i	−0.904265	−	1.56623i	−0.821901	−	0.569631i	$-0.807086\pi$
$331$	−5445.50	−	9431.88i	−0.904265	−	1.56623i	−0.0823644	−	0.996602i	$-0.526247\pi$
$332$		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		0.500000	−	0.866025i	$-0.333333\pi$
$347$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$348$		0			0
$349$		−	5300.08i		−	0.812913i	−0.913670	−	0.406456i	$-0.866764\pi$
$349$		−	5300.08i		−	0.812913i	0.913670	−	0.406456i	$-0.133236\pi$
$350$		0			0
$351$	12879.0			1.95849
$352$		0			0
$353$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$353$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$359$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$360$		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	−0.308646	−	0.951177i	$-0.599876\pi$
$367$	−9046.50	−	5223.00i	−1.28671	−	0.742884i	−0.978066	+	0.208293i	$0.933209\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	6300.50	+	10912.8i	0.874605	+	1.51486i	0.857183	+	0.515011i	$0.172212\pi$
$373$	6300.50	+	10912.8i	0.874605	+	1.51486i	0.0174213	+	0.999848i	$0.494454\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	6103.00			0.827151			0.413575	−	0.910470i	$-0.364280\pi$
$379$	6103.00			0.827151			0.413575	+	0.910470i	$0.364280\pi$
$380$		0			0
$381$	−10201.5	−	5889.84i	−1.37176	−	0.791983i
$382$		0			0
$383$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$383$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−958.500	−	1660.17i	−0.125900	−	0.218065i
$388$		0			0
$389$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$389$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	5938.50	−	3428.59i	0.750742	−	0.433441i	−0.0752196	−	0.997167i	$-0.523966\pi$
$397$	5938.50	−	3428.59i	0.750742	−	0.433441i	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	$-0.666667\pi$
$401$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$402$		0			0
$403$	8665.50	−	15009.1i	1.07111	−	1.85523i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	25.5000	+	14.7224i	0.00308287	+	0.00177990i	0.501541	−	0.865134i	$-0.332767\pi$
$409$	25.5000	+	14.7224i	0.00308287	+	0.00177990i	−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	$-0.626350\pi$
$421$	−6679.00			−0.773194			−0.386597	+	0.922249i	$0.626350\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−810.000	−	17303.2i	−0.0918001	−	1.96103i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$431$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$432$		0			0
$433$		−	6673.59i		−	0.740675i	−0.928897	−	0.370338i	$-0.879242\pi$
$433$		−	6673.59i		−	0.740675i	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	−0.911985	−	0.410224i	$-0.865450\pi$
$439$	−9315.00	+	5378.02i	−1.01271	+	0.584690i	−0.100728	+	0.994914i	$0.532117\pi$
$440$		0			0
$441$	3861.00	−	8417.77i	0.416910	−	0.908948i
$442$		0			0
$443$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$443$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$444$		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
$453$	−7866.00	+	4541.44i	−0.815843	+	0.471027i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	3248.50	+	5626.57i	0.332513	+	0.575929i	0.983004	−	0.183585i	$-0.0587702\pi$
$457$	3248.50	+	5626.57i	0.332513	+	0.575929i	−0.650491	+	0.759514i	$0.725437\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0			−	1.00000i	$-0.5\pi$
$461$		0			0				1.00000i	$0.5\pi$
$462$		0			0
$463$	11969.0			1.20140			0.600698	−	0.799476i	$-0.294889\pi$
$463$	11969.0			1.20140			0.600698	+	0.799476i	$0.294889\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$467$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$468$		0			0
$469$	1270.00	+	1979.73i	0.125039	+	0.194916i
$470$		0			0
$471$	2106.00	+	3647.70i	0.206028	+	0.356852i
$472$		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	$-0.833333\pi$
$479$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$480$		0			0
$481$	−25678.5	+	14825.5i	−2.43418	+	1.40537i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−3051.50	+	5285.35i	−0.283936	+	0.491791i	−0.972351	−	0.233526i	$-0.924974\pi$
$487$	−3051.50	+	5285.35i	−0.283936	+	0.491791i	0.688415	+	0.725317i	$0.258307\pi$
$488$		0			0
$489$		−	17666.9i		−	1.63379i
$490$		0			0
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	10871.5	+	18830.0i	0.975301	+	1.68927i	0.678938	+	0.734195i	$0.262440\pi$
$499$	10871.5	+	18830.0i	0.975301	+	1.68927i	0.296363	+	0.955075i	$0.404226\pi$
$500$		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
$507$	28035.0	+	16186.0i	2.45578	+	1.41784i
$508$		0			0
$509$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$509$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$510$		0			0
$511$	−18981.0	+	12176.3i	−1.64319	+	1.05411i
$512$		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		−0.866025	−	0.500000i	$-0.833333\pi$
$521$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$522$		0			0
$523$	79.5000	−	45.8993i	0.00664683	−	0.00383755i	−0.496673	−	0.867938i	$-0.665445\pi$
$523$	79.5000	−	45.8993i	0.00664683	−	0.00383755i	0.503320	+	0.864100i	$0.332112\pi$
$524$		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	$-0.190581\pi$
$541$	−944.500	−	1635.92i	−0.0750596	−	0.130007i	−0.901112	+	0.433586i	$0.857248\pi$
$542$		0			0
$543$	3325.50	−	5759.93i	0.262819	−	0.455216i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	1640.00			0.128193			0.0640963	−	0.997944i	$-0.479584\pi$
$547$	1640.00			0.128193			0.0640963	+	0.997944i	$0.479584\pi$
$548$		0			0
$549$	−21870.0	−	12626.7i	−1.70016	−	0.981589i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−25659.5	+	1201.18i	−1.97315	+	0.0923675i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$557$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$558$		0			0
$559$		−	6517.71i		−	0.493148i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$563$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$564$		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	$-0.666667\pi$
$569$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$570$		0			0
$571$	11970.5	−	20733.5i	0.877320	−	1.51956i	0.0230498	−	0.999734i	$-0.492662\pi$
$571$	11970.5	−	20733.5i	0.877320	−	1.51956i	0.854270	−	0.519829i	$-0.174004\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	22516.5	+	12999.9i	1.62457	+	0.937943i	0.985677	+	0.168644i	$0.0539387\pi$
$577$	22516.5	+	12999.9i	1.62457	+	0.937943i	0.638888	+	0.769300i	$0.279395\pi$
$578$		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		0.866025	−	0.500000i	$-0.166667\pi$
$593$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−5265.00	−	9119.25i	−0.360942	−	0.625169i
$598$		0			0
$599$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$599$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$600$		0			0
$601$		−	23817.4i		−	1.61653i	−0.588820	−	0.808264i	$-0.700407\pi$
$601$		−	23817.4i		−	1.61653i	0.588820	−	0.808264i	$-0.299593\pi$
$602$		0			0
$603$	3429.00			0.231575
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−25351.5	+	14636.7i	−1.69520	+	0.978723i	−0.745007	+	0.667056i	$0.767554\pi$
$607$	−25351.5	+	14636.7i	−1.69520	+	0.978723i	−0.950191	+	0.311667i	$0.899113\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−8695.00	+	15060.2i	−0.572900	+	0.992292i	0.423366	+	0.905959i	$0.360848\pi$
$613$	−8695.00	+	15060.2i	−0.572900	+	0.992292i	−0.996266	+	0.0863334i	$0.972485\pi$
$614$		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	$-0.499618\pi$
$619$	−13309.5	−	7684.24i	−0.864223	−	0.498959i	−0.865424	+	0.501040i	$0.832951\pi$
$620$		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
$631$	1892.00			0.119365			0.0596825	−	0.998217i	$-0.480991\pi$
$631$	1892.00			0.119365			0.0596825	+	0.998217i	$0.480991\pi$
$632$		0			0
$633$	27144.0	+	15671.6i	1.70439	+	0.984028i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	25678.5	−	18222.0i	1.59720	−	1.13341i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$641$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$642$		0			0
$643$		−	26315.0i		−	1.61394i	−0.590592	−	0.806971i	$-0.701106\pi$
$643$		−	26315.0i		−	1.61394i	0.590592	−	0.806971i	$-0.298894\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$647$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$648$		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	$-0.666667\pi$
$653$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$			32876.1i			1.95223i
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$	3616.50	+	2087.99i	0.212807	+	0.122864i	0.602615	−	0.798032i	$-0.294125\pi$
$661$	3616.50	+	2087.99i	0.212807	+	0.122864i	−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	$-0.591492\pi$
$673$	−9899.00			−0.566981			−0.283491	+	0.958975i	$0.591492\pi$
$674$		0			0
$675$	−15187.5	−	8768.51i	−0.866025	−	0.500000i
$676$		0			0
$677$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$677$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$678$		0			0
$679$	22572.0	+	11660.2i	1.27575	+	0.659022i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$683$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−33867.0			−1.88080
$688$		0			0
$689$		0			0
$690$		0			0
$691$	26161.5	−	15104.3i	1.44028	−	0.831543i	0.442408	−	0.896814i	$-0.354124\pi$
$691$	26161.5	−	15104.3i	1.44028	−	0.831543i	0.997868	+	0.0652705i	$0.0207910\pi$
$692$		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	0.728948	+	0.684569i	$0.240010\pi$
$709$	18073.0	+	31303.4i	0.957328	+	1.65814i	0.228381	+	0.973572i	$0.426657\pi$
$710$		0			0
$711$	−18724.5	+	32431.8i	−0.987656	+	1.71067i
$712$		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	$-0.166667\pi$
$719$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$720$		0			0
$721$	919.500	+	19642.3i	0.0474951	+	1.01459i
$722$		0			0
$723$	3240.00	+	5611.84i	0.166662	+	0.288668i
$724$		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
$733$	4615.50	−	2664.76i	0.232575	−	0.134277i	−0.379184	−	0.925321i	$-0.623795\pi$
$733$	4615.50	−	2664.76i	0.232575	−	0.134277i	0.611759	+	0.791044i	$0.290462\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−3023.50	+	5236.86i	−0.150502	+	0.260678i	−0.931412	−	0.363966i	$-0.881422\pi$
$739$	−3023.50	+	5236.86i	−0.150502	+	0.260678i	0.780910	+	0.624644i	$0.214756\pi$
$740$		0			0
$741$			60311.7i			2.99002i
$742$		0			0
$743$		0			0		1.00000			$0$
$743$		0			0		−1.00000			$\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−8784.50	−	15215.2i	−0.426832	−	0.739295i	0.569757	−	0.821813i	$-0.307037\pi$
$751$	−8784.50	−	15215.2i	−0.426832	−	0.739295i	−0.996590	+	0.0825179i	$0.973704\pi$
$752$		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		0.866025	−	0.500000i	$-0.166667\pi$
$761$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$762$		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
$769$			25189.2i			1.18120i	0.806963	+	0.590602i	$0.201110\pi$
$769$			25189.2i			1.18120i	−0.806963	+	0.590602i	$0.798890\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$773$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$774$		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	$-0.274338\pi$
$787$	7047.00	+	4068.59i	0.319185	+	0.184281i	−0.331844	+	0.943334i	$0.607671\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−42930.0	−	74356.9i	−1.92243	−	3.32975i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0			−	1.00000i	$-0.5\pi$
$797$		0			0				1.00000i	$0.5\pi$
$798$		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	$-0.333333\pi$
$809$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$810$		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
$817$	7774.50	−	4488.61i	0.332920	−	0.192211i
$818$		0			0
$819$	−2146.50	−	45853.4i	−0.0915809	−	1.95635i
$820$		0			0
$821$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$821$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$822$		0			0
$823$	6110.00	−	10582.8i	0.258786	−	0.448231i	−0.707131	−	0.707083i	$-0.750011\pi$
$823$	6110.00	−	10582.8i	0.258786	−	0.448231i	0.965917	+	0.258852i	$0.0833441\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		1.00000			$0$
$827$		0			0		−1.00000			$\pi$
$828$		0			0
$829$	−32104.5	−	18535.5i	−1.34504	−	0.776557i	−0.357495	−	0.933915i	$-0.616369\pi$
$829$	−32104.5	−	18535.5i	−1.34504	−	0.776557i	−0.987542	+	0.157358i	$0.949702\pi$
$830$		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
$853$		−	49133.1i		−	1.97220i	−0.166159	−	0.986099i	$-0.553137\pi$
$853$		−	49133.1i		−	1.97220i	0.166159	−	0.986099i	$-0.446863\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$857$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$858$		0			0
$859$	34155.0	−	19719.4i	1.35664	−	0.783256i	0.367470	−	0.930035i	$-0.380224\pi$
$859$	34155.0	−	19719.4i	1.35664	−	0.783256i	0.989169	+	0.146779i	$0.0468906\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$863$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		−	25528.7i		−	1.00000i
$868$		0			0
$869$		0			0
$870$		0			0
$871$	10096.5	+	5829.22i	0.392775	+	0.226769i
$872$		0			0
$873$	32076.0	−	18519.1i	1.24354	−	0.717957i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−25075.0	−	43431.2i	−0.965476	−	1.67225i	−0.708330	−	0.705881i	$-0.750551\pi$
$877$	−25075.0	−	43431.2i	−0.965476	−	1.67225i	−0.257146	−	0.966373i	$-0.582782\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0			−	1.00000i	$-0.5\pi$
$881$		0			0				1.00000i	$0.5\pi$
$882$		0			0
$883$	−52109.0			−1.98597			−0.992983	−	0.118260i	$-0.962269\pi$
$883$	−52109.0			−1.98597			−0.992983	+	0.118260i	$0.962269\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$887$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$888$		0			0
$889$	−19269.5	+	37302.3i	−0.726972	+	1.40729i
$890$		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
$903$	−5751.00	+	3689.27i	−0.211939	+	0.135959i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−24723.5	+	42822.4i	−0.905105	+	1.56769i	−0.0843291	+	0.996438i	$0.526875\pi$
$907$	−24723.5	+	42822.4i	−0.905105	+	1.56769i	−0.820776	+	0.571250i	$0.806459\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	23408.5	+	40544.7i	0.840234	+	1.45533i	0.889697	+	0.456552i	$0.150916\pi$
$919$	23408.5	+	40544.7i	0.840234	+	1.45533i	−0.0494625	+	0.998776i	$0.515751\pi$
$920$		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		0.866025	−	0.500000i	$-0.166667\pi$
$929$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$930$		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
$937$		−	40840.0i		−	1.42389i	−0.702235	−	0.711945i	$-0.747814\pi$
$937$		−	40840.0i		−	1.42389i	0.702235	−	0.711945i	$-0.252186\pi$
$938$		0			0
$939$	−32733.0			−1.13759
$940$		0			0
$941$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$941$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$947$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$948$		0			0
$949$	−55888.5	+	96801.7i	−1.91171	+	3.31119i
$950$		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
$967$	−3907.00			−0.129928			−0.0649641	−	0.997888i	$-0.520693\pi$
$967$	−3907.00			−0.129928			−0.0649641	+	0.997888i	$0.520693\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$971$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$972$		0			0
$973$	−53380.5	−	27575.1i	−1.75879	−	0.908548i
$974$		0			0
$975$	−29812.5	−	51636.8i	−0.979245	−	1.69610i
$976$		0			0
$977$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$977$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	59751.0			1.94465
$982$		0			0
$983$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$983$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−7020.50	+	12159.9i	−0.225039	+	0.389779i	−0.956331	−	0.292286i	$-0.905584\pi$
$991$	−7020.50	+	12159.9i	−0.225039	+	0.389779i	0.731292	+	0.682064i	$0.238918\pi$
$992$		0			0
$993$			56591.3i			1.80853i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−2536.50	−	1464.45i	−0.0805735	−	0.0465191i	0.459172	−	0.888347i	$-0.348146\pi$
$997$	−2536.50	−	1464.45i	−0.0805735	−	0.0465191i	−0.539745	+	0.841828i	$0.681480\pi$
$998$		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	84.4.k.a.17.1	yes	2
3.2	odd	2	CM	84.4.k.a.17.1	yes	2
4.3	odd	2		336.4.bc.b.17.1		2
7.2	even	3		588.4.k.b.509.1		2
7.3	odd	6		588.4.f.a.293.1		2
7.4	even	3		588.4.f.a.293.2		2
7.5	odd	6	inner	84.4.k.a.5.1	✓	2
7.6	odd	2		588.4.k.b.521.1		2
12.11	even	2		336.4.bc.b.17.1		2
21.2	odd	6		588.4.k.b.509.1		2
21.5	even	6	inner	84.4.k.a.5.1	✓	2
21.11	odd	6		588.4.f.a.293.2		2
21.17	even	6		588.4.f.a.293.1		2
21.20	even	2		588.4.k.b.521.1		2
28.19	even	6		336.4.bc.b.257.1		2
84.47	odd	6		336.4.bc.b.257.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.4.k.a.5.1	✓	2	7.5	odd	6	inner
84.4.k.a.5.1	✓	2	21.5	even	6	inner
84.4.k.a.17.1	yes	2	1.1	even	1	trivial
84.4.k.a.17.1	yes	2	3.2	odd	2	CM
336.4.bc.b.17.1		2	4.3	odd	2
336.4.bc.b.17.1		2	12.11	even	2
336.4.bc.b.257.1		2	28.19	even	6
336.4.bc.b.257.1		2	84.47	odd	6
588.4.f.a.293.1		2	7.3	odd	6
588.4.f.a.293.1		2	21.17	even	6
588.4.f.a.293.2		2	7.4	even	3
588.4.f.a.293.2		2	21.11	odd	6
588.4.k.b.509.1		2	7.2	even	3
588.4.k.b.509.1		2	21.2	odd	6
588.4.k.b.521.1		2	7.6	odd	2
588.4.k.b.521.1		2	21.20	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 84 = 2^{2} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	84.k (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-4.50000 - 2.59808i) q^{3} +(-8.50000 + 16.4545i) q^{7} +(13.5000 + 23.3827i) q^{9} +O(q^{10})\)	\(q+(-4.50000 - 2.59808i) q^{3} +(-8.50000 + 16.4545i) q^{7} +(13.5000 + 23.3827i) q^{9} +91.7987i q^{13} +(-109.500 + 63.2199i) q^{19} +(81.0000 - 51.9615i) q^{21} +(62.5000 - 108.253i) q^{25} -140.296i q^{27} +(-163.500 - 94.3968i) q^{31} +(161.500 + 279.726i) q^{37} +(238.500 - 413.094i) q^{39} -71.0000 q^{43} +(-198.500 - 279.726i) q^{49} +657.000 q^{57} +(-810.000 + 467.654i) q^{61} +(-499.500 + 23.3827i) q^{63} +(63.5000 - 109.985i) q^{67} +(1054.50 + 608.816i) q^{73} +(-562.500 + 324.760i) q^{75} +(693.500 + 1201.18i) q^{79} +(-364.500 + 631.333i) q^{81} +(-1510.50 - 780.289i) q^{91} +(490.500 + 849.571i) q^{93} -1371.78i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 9 q^{3} - 17 q^{7} + 27 q^{9}+O(q^{10}) \) 2 * q - 9 * q^3 - 17 * q^7 + 27 * q^9	\( 2 q - 9 q^{3} - 17 q^{7} + 27 q^{9} - 219 q^{19} + 162 q^{21} + 125 q^{25} - 327 q^{31} + 323 q^{37} + 477 q^{39} - 142 q^{43} - 397 q^{49} + 1314 q^{57} - 1620 q^{61} - 999 q^{63} + 127 q^{67} + 2109 q^{73} - 1125 q^{75} + 1387 q^{79} - 729 q^{81} - 3021 q^{91} + 981 q^{93}+O(q^{100}) \) 2 * q - 9 * q^3 - 17 * q^7 + 27 * q^9 - 219 * q^19 + 162 * q^21 + 125 * q^25 - 327 * q^31 + 323 * q^37 + 477 * q^39 - 142 * q^43 - 397 * q^49 + 1314 * q^57 - 1620 * q^61 - 999 * q^63 + 127 * q^67 + 2109 * q^73 - 1125 * q^75 + 1387 * q^79 - 729 * q^81 - 3021 * q^91 + 981 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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