LMFDB - Embedded newform 672.2.bi.a.17.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [672,2,Mod(17,672)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(672, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 3, 3, 1]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("672.17");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 672 = 2^{5} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	672.bi (of order $6$, degree $2$, not 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:	$5.36594701583$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{-3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 2x^{2} + 4 $ x^4 + 2*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 168)
Sato-Tate group:	$\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label			17.2
Root			$0.707107 - 1.22474i$ of defining polynomial
Character	$\chi$	$=$	672.17
Dual form			672.2.bi.a.593.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.50000 - 0.866025i) q^{3} +(0.621320 - 0.358719i) q^{5} +(2.62132 + 0.358719i) q^{7} +(1.50000 + 2.59808i) q^{9} +O(q^{10})$	$q+(-1.50000 - 0.866025i) q^{3} +(0.621320 - 0.358719i) q^{5} +(2.62132 + 0.358719i) q^{7} +(1.50000 + 2.59808i) q^{9} +(-2.91421 + 5.04757i) q^{11} -1.24264 q^{15} +(-3.62132 - 2.80821i) q^{21} +(-2.24264 + 3.88437i) q^{25} -5.19615i q^{27} +7.58579 q^{29} +(9.62132 + 5.55487i) q^{31} +(8.74264 - 5.04757i) q^{33} +(1.75736 - 0.717439i) q^{35} +(1.86396 + 1.07616i) q^{45} +(6.74264 + 1.88064i) q^{49} +(-2.03553 + 3.52565i) q^{53} +4.18154i q^{55} +(12.9853 + 7.49706i) q^{59} +(3.00000 + 7.34847i) q^{63} +(-8.48528 - 4.89898i) q^{73} +(6.72792 - 3.88437i) q^{75} +(-9.44975 + 12.1859i) q^{77} +(-8.86396 - 15.3528i) q^{79} +(-4.50000 + 7.79423i) q^{81} -13.5592i q^{83} +(-11.3787 - 6.56948i) q^{87} +(-9.62132 - 16.6646i) q^{93} +8.06591i q^{97} -17.4853 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 6 q^{3} - 6 q^{5} + 2 q^{7} + 6 q^{9}+O(q^{10}) $ 4 * q - 6 * q^3 - 6 * q^5 + 2 * q^7 + 6 * q^9	$ 4 q - 6 q^{3} - 6 q^{5} + 2 q^{7} + 6 q^{9} - 6 q^{11} + 12 q^{15} - 6 q^{21} + 8 q^{25} + 36 q^{29} + 30 q^{31} + 18 q^{33} + 24 q^{35} - 18 q^{45} + 10 q^{49} + 6 q^{53} + 18 q^{59} + 12 q^{63} - 24 q^{75} - 18 q^{77} - 10 q^{79} - 18 q^{81} - 54 q^{87} - 30 q^{93} - 36 q^{99}+O(q^{100}) $ 4 * q - 6 * q^3 - 6 * q^5 + 2 * q^7 + 6 * q^9 - 6 * q^11 + 12 * q^15 - 6 * q^21 + 8 * q^25 + 36 * q^29 + 30 * q^31 + 18 * q^33 + 24 * q^35 - 18 * q^45 + 10 * q^49 + 6 * q^53 + 18 * q^59 + 12 * q^63 - 24 * q^75 - 18 * q^77 - 10 * q^79 - 18 * q^81 - 54 * q^87 - 30 * q^93 - 36 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/672\mathbb{Z}\right)^\times$.

$n$	$127$	$421$	$449$	$577$
$\chi(n)$	$1$	$-1$	$-1$	$e\left(\frac{1}{6}\right)$

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
$3$	−1.50000	−	0.866025i	−0.866025	−	0.500000i
$3$	−1.50000	−	0.866025i	−0.866025	−	0.500000i
$4$		0			0
$5$	0.621320	−	0.358719i	0.277863	−	0.160424i	−0.354593	−	0.935021i	$-0.615380\pi$
$5$	0.621320	−	0.358719i	0.277863	−	0.160424i	0.632456	+	0.774597i	$0.282047\pi$
$6$		0			0
$7$	2.62132	+	0.358719i	0.990766	+	0.135583i
$7$	2.62132	+	0.358719i	0.990766	+	0.135583i
$8$		0			0
$9$	1.50000	+	2.59808i	0.500000	+	0.866025i
$10$		0			0
$11$	−2.91421	+	5.04757i	−0.878668	+	1.52190i	−0.0258656	+	0.999665i	$0.508234\pi$
$11$	−2.91421	+	5.04757i	−0.878668	+	1.52190i	−0.852803	+	0.522233i	$0.825099\pi$
$12$		0			0
$13$		0			0			−	1.00000i	$-0.5\pi$
$13$		0			0				1.00000i	$0.5\pi$
$14$		0			0
$15$	−1.24264			−0.320848
$16$		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
$19$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$19$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$20$		0			0
$21$	−3.62132	−	2.80821i	−0.790237	−	0.612801i
$22$		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
$25$	−2.24264	+	3.88437i	−0.448528	+	0.776874i
$26$		0			0
$27$		−	5.19615i		−	1.00000i
$28$		0			0
$29$	7.58579			1.40865			0.704323	−	0.709880i	$-0.251251\pi$
$29$	7.58579			1.40865			0.704323	+	0.709880i	$0.251251\pi$
$30$		0			0
$31$	9.62132	+	5.55487i	1.72804	+	0.997684i	0.898027	+	0.439941i	$0.145001\pi$
$31$	9.62132	+	5.55487i	1.72804	+	0.997684i	0.830014	+	0.557743i	$0.188333\pi$
$32$		0			0
$33$	8.74264	−	5.04757i	1.52190	−	0.878668i
$34$		0			0
$35$	1.75736	−	0.717439i	0.297048	−	0.121269i
$36$		0			0
$37$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$37$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0			−	1.00000i	$-0.5\pi$
$41$		0			0				1.00000i	$0.5\pi$
$42$		0			0
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$		0			0
$45$	1.86396	+	1.07616i	0.277863	+	0.160424i
$46$		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
$49$	6.74264	+	1.88064i	0.963234	+	0.268662i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−2.03553	+	3.52565i	−0.279602	+	0.484285i	−0.971286	−	0.237915i	$-0.923536\pi$
$53$	−2.03553	+	3.52565i	−0.279602	+	0.484285i	0.691684	+	0.722200i	$0.256869\pi$
$54$		0			0
$55$			4.18154i			0.563839i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	12.9853	+	7.49706i	1.69054	+	0.976034i	0.954080	+	0.299552i	$0.0968372\pi$
$59$	12.9853	+	7.49706i	1.69054	+	0.976034i	0.736460	+	0.676481i	$0.236496\pi$
$60$		0			0
$61$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$61$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$62$		0			0
$63$	3.00000	+	7.34847i	0.377964	+	0.925820i
$64$		0			0
$65$		0			0
$66$		0			0
$67$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$67$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$	−8.48528	−	4.89898i	−0.993127	−	0.573382i	−0.0869195	−	0.996215i	$-0.527702\pi$
$73$	−8.48528	−	4.89898i	−0.993127	−	0.573382i	−0.906208	+	0.422833i	$0.861036\pi$
$74$		0			0
$75$	6.72792	−	3.88437i	0.776874	−	0.448528i
$76$		0			0
$77$	−9.44975	+	12.1859i	−1.07690	+	1.38871i
$78$		0			0
$79$	−8.86396	−	15.3528i	−0.997274	−	1.72733i	−0.562544	−	0.826767i	$-0.690177\pi$
$79$	−8.86396	−	15.3528i	−0.997274	−	1.72733i	−0.434730	−	0.900561i	$-0.643156\pi$
$80$		0			0
$81$	−4.50000	+	7.79423i	−0.500000	+	0.866025i
$82$		0			0
$83$		−	13.5592i		−	1.48832i	−0.668002	−	0.744160i	$-0.732850\pi$
$83$		−	13.5592i		−	1.48832i	0.668002	−	0.744160i	$-0.267150\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−11.3787	−	6.56948i	−1.21992	−	0.704323i
$88$		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
$91$		0			0
$92$		0			0
$93$	−9.62132	−	16.6646i	−0.997684	−	1.72804i
$94$		0			0
$95$		0			0
$96$		0			0
$97$			8.06591i			0.818969i	0.912317	+	0.409484i	$0.134291\pi$
$97$			8.06591i			0.818969i	−0.912317	+	0.409484i	$0.865709\pi$
$98$		0			0
$99$	−17.4853			−1.75734
$100$		0			0
$101$	3.00000	+	1.73205i	0.298511	+	0.172345i	0.641774	−	0.766894i	$-0.278199\pi$
$101$	3.00000	+	1.73205i	0.298511	+	0.172345i	−0.343263	+	0.939239i	$0.611532\pi$
$102$		0			0
$103$	12.7279	−	7.34847i	1.25412	−	0.724066i	0.282194	−	0.959357i	$-0.408938\pi$
$103$	12.7279	−	7.34847i	1.25412	−	0.724066i	0.971925	+	0.235291i	$0.0756043\pi$
$104$		0			0
$105$	−3.25736	−	0.445759i	−0.317886	−	0.0435017i
$106$		0			0
$107$	4.67157	+	8.09140i	0.451618	+	0.782225i	0.998487	−	0.0549930i	$-0.0175137\pi$
$107$	4.67157	+	8.09140i	0.451618	+	0.782225i	−0.546869	+	0.837218i	$0.684180\pi$
$108$		0			0
$109$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$109$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−11.4853	−	19.8931i	−1.04412	−	1.80846i
$122$		0			0
$123$		0			0
$124$		0			0
$125$			6.80511i			0.608668i
$126$		0			0
$127$	−15.2426			−1.35257			−0.676283	−	0.736642i	$-0.736410\pi$
$127$	−15.2426			−1.35257			−0.676283	+	0.736642i	$0.736410\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−15.4706	+	8.93193i	−1.35167	+	0.780387i	−0.988483	−	0.151330i	$-0.951644\pi$
$131$	−15.4706	+	8.93193i	−1.35167	+	0.780387i	−0.363186	+	0.931717i	$0.618311\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	−1.86396	−	3.22848i	−0.160424	−	0.277863i
$136$		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
$139$		0			0			−	1.00000i	$-0.5\pi$
$139$		0			0				1.00000i	$0.5\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$	4.71320	−	2.72117i	0.391410	−	0.225981i
$146$		0			0
$147$	−8.48528	−	8.66025i	−0.699854	−	0.714286i
$148$		0			0
$149$	1.41421	+	2.44949i	0.115857	+	0.200670i	0.918122	−	0.396298i	$-0.129705\pi$
$149$	1.41421	+	2.44949i	0.115857	+	0.200670i	−0.802265	+	0.596968i	$0.796372\pi$
$150$		0			0
$151$	−10.1066	+	17.5051i	−0.822464	+	1.42455i	0.0813788	+	0.996683i	$0.474068\pi$
$151$	−10.1066	+	17.5051i	−0.822464	+	1.42455i	−0.903842	+	0.427865i	$0.859266\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	7.97056			0.640211
$156$		0			0
$157$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$157$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$158$		0			0
$159$	6.10660	−	3.52565i	0.484285	−	0.279602i
$160$		0			0
$161$		0			0
$162$		0			0
$163$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$163$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$164$		0			0
$165$	3.62132	−	6.27231i	0.281919	−	0.488299i
$166$		0			0
$167$		0			0			−	1.00000i	$-0.5\pi$
$167$		0			0				1.00000i	$0.5\pi$
$168$		0			0
$169$	−13.0000			−1.00000
$170$		0			0
$171$		0			0
$172$		0			0
$173$	15.0000	−	8.66025i	1.14043	−	0.658427i	0.193892	−	0.981023i	$-0.437889\pi$
$173$	15.0000	−	8.66025i	1.14043	−	0.658427i	0.946537	+	0.322596i	$0.104555\pi$
$174$		0			0
$175$	−7.27208	+	9.37769i	−0.549717	+	0.708887i
$176$		0			0
$177$	−12.9853	−	22.4912i	−0.976034	−	1.69054i
$178$		0			0
$179$	−5.65685	+	9.79796i	−0.422813	+	0.732334i	−0.996213	−	0.0869415i	$-0.972291\pi$
$179$	−5.65685	+	9.79796i	−0.422813	+	0.732334i	0.573400	+	0.819275i	$0.305624\pi$
$180$		0			0
$181$		0			0			−	1.00000i	$-0.5\pi$
$181$		0			0				1.00000i	$0.5\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	1.86396	−	13.6208i	0.135583	−	0.990766i
$190$		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
$193$	2.25736	−	3.90986i	0.162488	−	0.281438i	−0.773272	−	0.634074i	$-0.781381\pi$
$193$	2.25736	−	3.90986i	0.162488	−	0.281438i	0.935760	+	0.352636i	$0.114715\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	14.1421			1.00759			0.503793	−	0.863825i	$-0.331938\pi$
$197$	14.1421			1.00759			0.503793	+	0.863825i	$0.331938\pi$
$198$		0			0
$199$	−21.2132	−	12.2474i	−1.50376	−	0.868199i	−0.999990	−	0.00436292i	$-0.998611\pi$
$199$	−21.2132	−	12.2474i	−1.50376	−	0.868199i	−0.503774	−	0.863836i	$-0.668055\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	19.8848	+	2.72117i	1.39564	+	0.190989i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	23.2279	+	18.0125i	1.57681	+	1.22277i
$218$		0			0
$219$	8.48528	+	14.6969i	0.573382	+	0.993127i
$220$		0			0
$221$		0			0
$222$		0			0
$223$		−	15.1682i		−	1.01574i	−0.861435	−	0.507869i	$-0.830434\pi$
$223$		−	15.1682i		−	1.01574i	0.861435	−	0.507869i	$-0.169566\pi$
$224$		0			0
$225$	−13.4558			−0.897056
$226$		0			0
$227$	−16.7132	−	9.64937i	−1.10929	−	0.640451i	−0.170648	−	0.985332i	$-0.554586\pi$
$227$	−16.7132	−	9.64937i	−1.10929	−	0.640451i	−0.938647	+	0.344881i	$0.887919\pi$
$228$		0			0
$229$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$229$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$230$		0			0
$231$	24.7279	−	10.0951i	1.62698	−	0.664211i
$232$		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
$237$			30.7057i			1.99455i
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$		0			0
$241$	5.22792	+	3.01834i	0.336760	+	0.194429i	0.658838	−	0.752285i	$-0.271048\pi$
$241$	5.22792	+	3.01834i	0.336760	+	0.194429i	−0.322078	+	0.946713i	$0.604381\pi$
$242$		0			0
$243$	13.5000	−	7.79423i	0.866025	−	0.500000i
$244$		0			0
$245$	4.86396	−	1.25024i	0.310747	−	0.0798748i
$246$		0			0
$247$		0			0
$248$		0			0
$249$	−11.7426	+	20.3389i	−0.744160	+	1.28892i
$250$		0			0
$251$			10.6895i			0.674714i	0.941377	+	0.337357i	$0.109533\pi$
$251$			10.6895i			0.674714i	−0.941377	+	0.337357i	$0.890467\pi$
$252$		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
$259$		0			0
$260$		0			0
$261$	11.3787	+	19.7085i	0.704323	+	1.21992i
$262$		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
$265$			2.92074i			0.179420i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−27.8345	−	16.0703i	−1.69710	−	0.979822i	−0.948487	−	0.316815i	$-0.897387\pi$
$269$	−27.8345	−	16.0703i	−1.69710	−	0.979822i	−0.748614	−	0.663007i	$-0.769280\pi$
$270$		0			0
$271$	27.1066	−	15.6500i	1.64661	−	0.950670i	0.668202	−	0.743980i	$-0.267064\pi$
$271$	27.1066	−	15.6500i	1.64661	−	0.950670i	0.978406	−	0.206691i	$-0.0662693\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−13.0711	−	22.6398i	−0.788215	−	1.36523i
$276$		0			0
$277$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$277$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$278$		0			0
$279$			33.3292i			1.99537i
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$		0			0
$283$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$283$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	8.50000	+	14.7224i	0.500000	+	0.866025i
$290$		0			0
$291$	6.98528	−	12.0989i	0.409484	−	0.709248i
$292$		0			0
$293$		−	27.8359i		−	1.62619i	−0.582130	−	0.813095i	$-0.697781\pi$
$293$		−	27.8359i		−	1.62619i	0.582130	−	0.813095i	$-0.302219\pi$
$294$		0			0
$295$	10.7574			0.626318
$296$		0			0
$297$	26.2279	+	15.1427i	1.52190	+	0.878668i
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$	−3.00000	−	5.19615i	−0.172345	−	0.298511i
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0			−	1.00000i	$-0.5\pi$
$307$		0			0				1.00000i	$0.5\pi$
$308$		0			0
$309$	−25.4558			−1.44813
$310$		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
$313$	29.7426	−	17.1719i	1.68115	−	0.970614i	0.720257	−	0.693708i	$-0.244024\pi$
$313$	29.7426	−	17.1719i	1.68115	−	0.970614i	0.960897	−	0.276907i	$-0.0893093\pi$
$314$		0			0
$315$	4.50000	+	3.48960i	0.253546	+	0.196616i
$316$		0			0
$317$	0.278175	+	0.481813i	0.0156238	+	0.0270613i	0.873732	−	0.486408i	$-0.161693\pi$
$317$	0.278175	+	0.481813i	0.0156238	+	0.0270613i	−0.858108	+	0.513470i	$0.828360\pi$
$318$		0			0
$319$	−22.1066	+	38.2898i	−1.23773	+	2.14381i
$320$		0			0
$321$		−	16.1828i		−	0.903236i
$322$		0			0
$323$		0			0
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$331$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−14.4558			−0.787460			−0.393730	−	0.919226i	$-0.628816\pi$
$337$	−14.4558			−0.787460			−0.393730	+	0.919226i	$0.628816\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−56.0772	+	32.3762i	−3.03675	+	1.75327i
$342$		0			0
$343$	17.0000	+	7.34847i	0.917914	+	0.396780i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	14.1421	−	24.4949i	0.759190	−	1.31495i	−0.184075	−	0.982912i	$-0.558929\pi$
$347$	14.1421	−	24.4949i	0.759190	−	1.31495i	0.943264	−	0.332043i	$-0.107738\pi$
$348$		0			0
$349$		0			0			−	1.00000i	$-0.5\pi$
$349$		0			0				1.00000i	$0.5\pi$
$350$		0			0
$351$		0			0
$352$		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
$361$	9.50000	−	16.4545i	0.500000	−	0.866025i
$362$		0			0
$363$			39.7862i			2.08823i
$364$		0			0
$365$	−7.02944			−0.367938
$366$		0			0
$367$	30.6213	+	17.6792i	1.59842	+	0.922848i	0.991792	+	0.127862i	$0.0408116\pi$
$367$	30.6213	+	17.6792i	1.59842	+	0.922848i	0.606628	+	0.794986i	$0.292522\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−6.60051	+	8.51167i	−0.342681	+	0.441904i
$372$		0			0
$373$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$373$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$374$		0			0
$375$	5.89340	−	10.2077i	0.304334	−	0.527122i
$376$		0			0
$377$		0			0
$378$		0			0
$379$		0			0		1.00000			$0$
$379$		0			0		−1.00000			$\pi$
$380$		0			0
$381$	22.8640	+	13.2005i	1.17136	+	0.676283i
$382$		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
$385$	−1.50000	+	10.9612i	−0.0764471	+	0.558632i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	15.5563	−	26.9444i	0.788738	−	1.36613i	−0.138002	−	0.990432i	$-0.544068\pi$
$389$	15.5563	−	26.9444i	0.788738	−	1.36613i	0.926740	−	0.375703i	$-0.122599\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$	30.9411			1.56077
$394$		0			0
$395$	−11.0147	−	6.35935i	−0.554211	−	0.319974i
$396$		0			0
$397$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$397$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$398$		0			0
$399$		0			0
$400$		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
$403$		0			0
$404$		0			0
$405$			6.45695i			0.320848i
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−24.4706	−	14.1281i	−1.20999	−	0.698589i	−0.247234	−	0.968956i	$-0.579522\pi$
$409$	−24.4706	−	14.1281i	−1.20999	−	0.698589i	−0.962757	+	0.270367i	$0.912855\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	31.3492	+	24.3103i	1.54260	+	1.19623i
$414$		0			0
$415$	−4.86396	−	8.42463i	−0.238762	−	0.413549i
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	10.3923i		−	0.507697i	−0.967244	−	0.253849i	$-0.918303\pi$
$419$		−	10.3923i		−	0.507697i	0.967244	−	0.253849i	$-0.0816965\pi$
$420$		0			0
$421$		0			0		1.00000			$0$
$421$		0			0		−1.00000			$\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$		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
$433$			39.1918i			1.88344i	0.336399	+	0.941720i	$0.390791\pi$
$433$			39.1918i			1.88344i	−0.336399	+	0.941720i	$0.609209\pi$
$434$		0			0
$435$	−9.42641			−0.451962
$436$		0			0
$437$		0			0
$438$		0			0
$439$	−14.8934	+	8.59871i	−0.710823	+	0.410394i	−0.811366	−	0.584539i	$-0.801275\pi$
$439$	−14.8934	+	8.59871i	−0.710823	+	0.410394i	0.100543	+	0.994933i	$0.467942\pi$
$440$		0			0
$441$	5.22792	+	20.3389i	0.248949	+	0.968517i
$442$		0			0
$443$	−6.42893	−	11.1352i	−0.305448	−	0.529051i	0.671913	−	0.740630i	$-0.265473\pi$
$443$	−6.42893	−	11.1352i	−0.305448	−	0.529051i	−0.977361	+	0.211579i	$0.932139\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		−	4.89898i		−	0.231714i
$448$		0			0
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	30.3198	−	17.5051i	1.42455	−	0.822464i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	1.01472	+	1.75754i	0.0474665	+	0.0822145i	0.888783	−	0.458329i	$-0.151552\pi$
$457$	1.01472	+	1.75754i	0.0474665	+	0.0822145i	−0.841316	+	0.540544i	$0.818219\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	38.1051i		−	1.77473i	−0.461065	−	0.887366i	$-0.652533\pi$
$461$		−	38.1051i		−	1.77473i	0.461065	−	0.887366i	$-0.347467\pi$
$462$		0			0
$463$	26.0000			1.20832			0.604161	−	0.796862i	$-0.293508\pi$
$463$	26.0000			1.20832			0.604161	+	0.796862i	$0.293508\pi$
$464$		0			0
$465$	−11.9558	−	6.90271i	−0.554439	−	0.320106i
$466$		0			0
$467$	−15.0000	+	8.66025i	−0.694117	+	0.400749i	−0.805153	−	0.593068i	$-0.797917\pi$
$467$	−15.0000	+	8.66025i	−0.694117	+	0.400749i	0.111035	+	0.993816i	$0.464583\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−12.2132			−0.559204
$478$		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
$481$		0			0
$482$		0			0
$483$		0			0
$484$		0			0
$485$	2.89340	+	5.01151i	0.131382	+	0.227561i
$486$		0			0
$487$	19.5919	−	33.9341i	0.887793	−	1.53770i	0.0453143	−	0.998973i	$-0.485571\pi$
$487$	19.5919	−	33.9341i	0.887793	−	1.53770i	0.842479	−	0.538730i	$-0.181096\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	44.3137			1.99985			0.999925	−	0.0122607i	$-0.00390281\pi$
$491$	44.3137			1.99985			0.999925	+	0.0122607i	$0.00390281\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$	−10.8640	+	6.27231i	−0.488299	+	0.281919i
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$499$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$500$		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
$505$	2.48528			0.110594
$506$		0			0
$507$	19.5000	+	11.2583i	0.866025	+	0.500000i
$508$		0			0
$509$	21.6213	−	12.4831i	0.958348	−	0.553303i	0.0626839	−	0.998033i	$-0.480034\pi$
$509$	21.6213	−	12.4831i	0.958348	−	0.553303i	0.895664	+	0.444731i	$0.146701\pi$
$510$		0			0
$511$	−20.4853	−	15.8856i	−0.906215	−	0.702739i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	5.27208	−	9.13151i	0.232316	−	0.402382i
$516$		0			0
$517$		0			0
$518$		0			0
$519$	−30.0000			−1.31685
$520$		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
$523$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$523$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$524$		0			0
$525$	19.0294	−	7.76874i	0.830513	−	0.339055i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−11.5000	+	19.9186i	−0.500000	+	0.866025i
$530$		0			0
$531$			44.9823i			1.95207i
$532$		0			0
$533$		0			0
$534$		0			0
$535$	5.80509	+	3.35157i	0.250976	+	0.144901i
$536$		0			0
$537$	16.9706	−	9.79796i	0.732334	−	0.422813i
$538$		0			0
$539$	−29.1421	+	28.5533i	−1.25524	+	1.22988i
$540$		0			0
$541$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$541$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0		1.00000			$0$
$547$		0			0		−1.00000			$\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−17.7279	−	43.4244i	−0.753868	−	1.84659i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−23.0355	+	39.8987i	−0.976047	+	1.69056i	−0.299611	+	0.954062i	$0.596857\pi$
$557$	−23.0355	+	39.8987i	−0.976047	+	1.69056i	−0.676436	+	0.736501i	$0.736477\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−37.7132	−	21.7737i	−1.58942	−	0.917653i	−0.993402	−	0.114684i	$-0.963415\pi$
$563$	−37.7132	−	21.7737i	−1.58942	−	0.917653i	−0.596020	−	0.802970i	$-0.703252\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−14.5919	+	18.8169i	−0.612801	+	0.790237i
$568$		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
$571$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$571$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−15.7721	−	9.10601i	−0.656600	−	0.379088i	0.134380	−	0.990930i	$-0.457096\pi$
$577$	−15.7721	−	9.10601i	−0.656600	−	0.379088i	−0.790980	+	0.611842i	$0.790429\pi$
$578$		0			0
$579$	−6.77208	+	3.90986i	−0.281438	+	0.162488i
$580$		0			0
$581$	4.86396	−	35.5431i	0.201791	−	1.47458i
$582$		0			0
$583$	−11.8640	−	20.5490i	−0.491355	−	0.851052i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	47.8521i		−	1.97507i	−0.157409	−	0.987534i	$-0.550314\pi$
$587$		−	47.8521i		−	1.97507i	0.157409	−	0.987534i	$-0.449686\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$	−21.2132	−	12.2474i	−0.872595	−	0.503793i
$592$		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
$597$	21.2132	+	36.7423i	0.868199	+	1.50376i
$598$		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
$601$		−	26.2269i		−	1.06982i	−0.844909	−	0.534910i	$-0.820346\pi$
$601$		−	26.2269i		−	1.06982i	0.844909	−	0.534910i	$-0.179654\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−14.2721	−	8.23999i	−0.580242	−	0.335003i
$606$		0			0
$607$	−2.59188	+	1.49642i	−0.105201	+	0.0607380i	−0.551678	−	0.834058i	$-0.686012\pi$
$607$	−2.59188	+	1.49642i	−0.105201	+	0.0607380i	0.446476	+	0.894795i	$0.352679\pi$
$608$		0			0
$609$	−27.4706	−	21.3025i	−1.11316	−	0.863220i
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$613$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		1.00000			$0$
$617$		0			0		−1.00000			$\pi$
$618$		0			0
$619$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$619$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−8.77208	−	15.1937i	−0.350883	−	0.607747i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−29.2426			−1.16413			−0.582066	−	0.813142i	$-0.697755\pi$
$631$	−29.2426			−1.16413			−0.582066	+	0.813142i	$0.697755\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−9.47056	+	5.46783i	−0.375828	+	0.216984i
$636$		0			0
$637$		0			0
$638$		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
$643$		0			0			−	1.00000i	$-0.5\pi$
$643$		0			0				1.00000i	$0.5\pi$
$644$		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
$649$	−75.6838	+	43.6960i	−2.97085	+	1.71522i
$650$		0			0
$651$	−19.2426	−	47.1347i	−0.754179	−	1.84735i
$652$		0			0
$653$	−19.5208	−	33.8110i	−0.763909	−	1.32313i	−0.940822	−	0.338902i	$-0.889945\pi$
$653$	−19.5208	−	33.8110i	−0.763909	−	1.32313i	0.176913	−	0.984226i	$-0.443389\pi$
$654$		0			0
$655$	−6.40812	+	11.0992i	−0.250386	+	0.433681i
$656$		0			0
$657$		−	29.3939i		−	1.14676i
$658$		0			0
$659$	45.2548			1.76288			0.881439	−	0.472298i	$-0.156575\pi$
$659$	45.2548			1.76288			0.881439	+	0.472298i	$0.156575\pi$
$660$		0			0
$661$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$661$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−13.1360	+	22.7523i	−0.507869	+	0.879654i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	16.9411			0.653032			0.326516	−	0.945192i	$-0.394125\pi$
$673$	16.9411			0.653032			0.326516	+	0.945192i	$0.394125\pi$
$674$		0			0
$675$	20.1838	+	11.6531i	0.776874	+	0.448528i
$676$		0			0
$677$	−20.3787	+	11.7656i	−0.783216	+	0.452190i	−0.837569	−	0.546332i	$-0.816024\pi$
$677$	−20.3787	+	11.7656i	−0.783216	+	0.452190i	0.0543526	+	0.998522i	$0.482690\pi$
$678$		0			0
$679$	−2.89340	+	21.1433i	−0.111038	+	0.811407i
$680$		0			0
$681$	16.7132	+	28.9481i	0.640451	+	1.10929i
$682$		0			0
$683$	−23.9142	+	41.4206i	−0.915052	+	1.58492i	−0.108227	+	0.994126i	$0.534517\pi$
$683$	−23.9142	+	41.4206i	−0.915052	+	1.58492i	−0.806825	+	0.590790i	$0.798816\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$691$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$692$		0			0
$693$	−45.8345	−	6.27231i	−1.74111	−	0.238265i
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−14.6152			−0.552009			−0.276005	−	0.961156i	$-0.589011\pi$
$701$	−14.6152			−0.552009			−0.276005	+	0.961156i	$0.589011\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$	7.24264	+	5.61642i	0.272388	+	0.211227i
$708$		0			0
$709$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$709$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$710$		0			0
$711$	26.5919	−	46.0585i	0.997274	−	1.72733i
$712$		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
$721$	36.0000	−	14.6969i	1.34071	−	0.547343i
$722$		0			0
$723$	−5.22792	−	9.05503i	−0.194429	−	0.336760i
$724$		0			0
$725$	−17.0122	+	29.4660i	−0.631817	+	1.09434i
$726$		0			0
$727$		−	25.2123i		−	0.935074i	−0.883974	−	0.467537i	$-0.845142\pi$
$727$		−	25.2123i		−	0.935074i	0.883974	−	0.467537i	$-0.154858\pi$
$728$		0			0
$729$	−27.0000			−1.00000
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$733$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$734$		0			0
$735$	−8.37868	−	2.33696i	−0.309052	−	0.0861999i
$736$		0			0
$737$		0			0
$738$		0			0
$739$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$739$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		1.00000			$0$
$743$		0			0		−1.00000			$\pi$
$744$		0			0
$745$	1.75736	+	1.01461i	0.0643847	+	0.0371725i
$746$		0			0
$747$	35.2279	−	20.3389i	1.28892	−	0.744160i
$748$		0			0
$749$	9.34315	+	22.8859i	0.341391	+	0.836234i
$750$		0			0
$751$	20.8345	+	36.0865i	0.760263	+	1.31681i	0.942715	+	0.333599i	$0.108263\pi$
$751$	20.8345	+	36.0865i	0.760263	+	1.31681i	−0.182453	+	0.983215i	$0.558404\pi$
$752$		0			0
$753$	9.25736	−	16.0342i	0.337357	−	0.584319i
$754$		0			0
$755$			14.5017i			0.527772i
$756$		0			0
$757$		0			0		1.00000			$0$
$757$		0			0		−1.00000			$\pi$
$758$		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
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$		−	1.97824i		−	0.0713370i	−0.999364	−	0.0356685i	$-0.988644\pi$
$769$		−	1.97824i		−	0.0713370i	0.999364	−	0.0356685i	$-0.0113561\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	45.0000	+	25.9808i	1.61854	+	0.934463i	0.987299	+	0.158874i	$0.0507865\pi$
$773$	45.0000	+	25.9808i	1.61854	+	0.934463i	0.631239	+	0.775589i	$0.282547\pi$
$774$		0			0
$775$	−43.1543	+	24.9152i	−1.55015	+	0.894979i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		−	39.4169i		−	1.40865i
$784$		0			0
$785$		0			0
$786$		0			0
$787$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$787$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$	2.52944	−	4.38111i	0.0897099	−	0.155382i
$796$		0			0
$797$		−	37.8801i		−	1.34178i	−0.741557	−	0.670890i	$-0.765912\pi$
$797$		−	37.8801i		−	1.34178i	0.741557	−	0.670890i	$-0.234088\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$	49.4558	−	28.5533i	1.74526	−	1.00763i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	27.8345	+	48.2108i	0.979822	+	1.69710i
$808$		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
$811$		0			0			−	1.00000i	$-0.5\pi$
$811$		0			0				1.00000i	$0.5\pi$
$812$		0			0
$813$	−54.2132			−1.90134
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	1.47918	+	2.56202i	0.0516239	+	0.0894152i	0.890683	−	0.454626i	$-0.150227\pi$
$821$	1.47918	+	2.56202i	0.0516239	+	0.0894152i	−0.839059	+	0.544041i	$0.816894\pi$
$822$		0			0
$823$	23.0000	−	39.8372i	0.801730	−	1.38864i	−0.116747	−	0.993162i	$-0.537247\pi$
$823$	23.0000	−	39.8372i	0.801730	−	1.38864i	0.918477	−	0.395475i	$-0.129420\pi$
$824$		0			0
$825$			45.2795i			1.57643i
$826$		0			0
$827$	−37.2843			−1.29650			−0.648251	−	0.761427i	$-0.724499\pi$
$827$	−37.2843			−1.29650			−0.648251	+	0.761427i	$0.724499\pi$
$828$		0			0
$829$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$829$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	28.8640	−	49.9938i	0.997684	−	1.72804i
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	28.5442			0.984281
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−8.07716	+	4.66335i	−0.277863	+	0.160424i
$846$		0			0
$847$	−22.9706	−	56.2662i	−0.789278	−	1.93333i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$		0			0			−	1.00000i	$-0.5\pi$
$853$		0			0				1.00000i	$0.5\pi$
$854$		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
$859$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$859$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$860$		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
$865$	6.21320	−	10.7616i	0.211255	−	0.365905i
$866$		0			0
$867$		−	29.4449i		−	1.00000i
$868$		0			0
$869$	103.326			3.50509
$870$		0			0
$871$		0			0
$872$		0			0
$873$	−20.9558	+	12.0989i	−0.709248	+	0.409484i
$874$		0			0
$875$	−2.44113	+	17.8384i	−0.0825251	+	0.603047i
$876$		0			0
$877$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$877$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$878$		0			0
$879$	−24.1066	+	41.7539i	−0.813095	+	1.40832i
$880$		0			0
$881$		0			0			−	1.00000i	$-0.5\pi$
$881$		0			0				1.00000i	$0.5\pi$
$882$		0			0
$883$		0			0		1.00000			$0$
$883$		0			0		−1.00000			$\pi$
$884$		0			0
$885$	−16.1360	−	9.31615i	−0.542407	−	0.313159i
$886$		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
$889$	−39.9558	−	5.46783i	−1.34008	−	0.183385i
$890$		0			0
$891$	−26.2279	−	45.4281i	−0.878668	−	1.52190i
$892$		0			0
$893$		0			0
$894$		0			0
$895$			8.11689i			0.271318i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	72.9853	+	42.1381i	2.43420	+	1.40538i
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$907$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$908$		0			0
$909$			10.3923i			0.344691i
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$		0			0
$913$	68.4411	+	39.5145i	2.26507	+	1.30774i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−43.7574	+	17.8639i	−1.44500	+	0.589917i
$918$		0			0
$919$	25.0000	+	43.3013i	0.824674	+	1.42838i	0.902168	+	0.431384i	$0.141975\pi$
$919$	25.0000	+	43.3013i	0.824674	+	1.42838i	−0.0774944	+	0.996993i	$0.524692\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$	38.1838	+	22.0454i	1.25412	+	0.724066i
$928$		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
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		−	40.4315i		−	1.32084i	−0.750896	−	0.660420i	$-0.770378\pi$
$937$		−	40.4315i		−	1.32084i	0.750896	−	0.660420i	$-0.229622\pi$
$938$		0			0
$939$	−59.4853			−1.94123
$940$		0			0
$941$	52.5624	+	30.3469i	1.71349	+	0.989282i	0.929752	+	0.368186i	$0.120021\pi$
$941$	52.5624	+	30.3469i	1.71349	+	0.989282i	0.783735	+	0.621096i	$0.213312\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$	−3.72792	−	9.13151i	−0.121269	−	0.297048i
$946$		0			0
$947$	28.2843	+	48.9898i	0.919115	+	1.59195i	0.800762	+	0.598983i	$0.204428\pi$
$947$	28.2843	+	48.9898i	0.919115	+	1.59195i	0.118354	+	0.992972i	$0.462238\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		−	0.963625i		−	0.0312477i
$952$		0			0
$953$		0			0		1.00000			$0$
$953$		0			0		−1.00000			$\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	66.3198	−	38.2898i	2.14381	−	1.23773i
$958$		0			0
$959$		0			0
$960$		0			0
$961$	46.2132	+	80.0436i	1.49075	+	2.58205i
$962$		0			0
$963$	−14.0147	+	24.2742i	−0.451618	+	0.782225i
$964$		0			0
$965$		−	3.23903i		−	0.104268i
$966$		0			0
$967$	26.7574			0.860459			0.430229	−	0.902720i	$-0.358433\pi$
$967$	26.7574			0.860459			0.430229	+	0.902720i	$0.358433\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−45.1690	+	26.0784i	−1.44954	+	0.836894i	−0.998454	−	0.0555842i	$-0.982298\pi$
$971$	−45.1690	+	26.0784i	−1.44954	+	0.836894i	−0.451090	+	0.892479i	$0.648965\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		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
$981$		0			0
$982$		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
$985$	8.78680	−	5.07306i	0.279971	−	0.161641i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	3.89340	−	6.74356i	0.123678	−	0.214216i	−0.797537	−	0.603269i	$-0.793864\pi$
$991$	3.89340	−	6.74356i	0.123678	−	0.214216i	0.921215	+	0.389053i	$0.127198\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−17.5736			−0.557120
$996$		0			0
$997$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$997$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	672.2.bi.a.17.2		4
3.2	odd	2		672.2.bi.b.17.1		4
4.3	odd	2		168.2.ba.b.101.2	yes	4
7.5	odd	6	inner	672.2.bi.a.593.2		4
8.3	odd	2		168.2.ba.a.101.1	yes	4
8.5	even	2		672.2.bi.b.17.1		4
12.11	even	2		168.2.ba.a.101.1	yes	4
21.5	even	6		672.2.bi.b.593.1		4
24.5	odd	2	CM	672.2.bi.a.17.2		4
24.11	even	2		168.2.ba.b.101.2	yes	4
28.19	even	6		168.2.ba.b.5.2	yes	4
56.5	odd	6		672.2.bi.b.593.1		4
56.19	even	6		168.2.ba.a.5.1	✓	4
84.47	odd	6		168.2.ba.a.5.1	✓	4
168.5	even	6	inner	672.2.bi.a.593.2		4
168.131	odd	6		168.2.ba.b.5.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.2.ba.a.5.1	✓	4	56.19	even	6
168.2.ba.a.5.1	✓	4	84.47	odd	6
168.2.ba.a.101.1	yes	4	8.3	odd	2
168.2.ba.a.101.1	yes	4	12.11	even	2
168.2.ba.b.5.2	yes	4	28.19	even	6
168.2.ba.b.5.2	yes	4	168.131	odd	6
168.2.ba.b.101.2	yes	4	4.3	odd	2
168.2.ba.b.101.2	yes	4	24.11	even	2
672.2.bi.a.17.2		4	1.1	even	1	trivial
672.2.bi.a.17.2		4	24.5	odd	2	CM
672.2.bi.a.593.2		4	7.5	odd	6	inner
672.2.bi.a.593.2		4	168.5	even	6	inner
672.2.bi.b.17.1		4	3.2	odd	2
672.2.bi.b.17.1		4	8.5	even	2
672.2.bi.b.593.1		4	21.5	even	6
672.2.bi.b.593.1		4	56.5	odd	6

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:	\( N \)	\(=\)	\( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	672.bi (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.50000 - 0.866025i) q^{3} +(0.621320 - 0.358719i) q^{5} +(2.62132 + 0.358719i) q^{7} +(1.50000 + 2.59808i) q^{9} +O(q^{10})\)	\(q+(-1.50000 - 0.866025i) q^{3} +(0.621320 - 0.358719i) q^{5} +(2.62132 + 0.358719i) q^{7} +(1.50000 + 2.59808i) q^{9} +(-2.91421 + 5.04757i) q^{11} -1.24264 q^{15} +(-3.62132 - 2.80821i) q^{21} +(-2.24264 + 3.88437i) q^{25} -5.19615i q^{27} +7.58579 q^{29} +(9.62132 + 5.55487i) q^{31} +(8.74264 - 5.04757i) q^{33} +(1.75736 - 0.717439i) q^{35} +(1.86396 + 1.07616i) q^{45} +(6.74264 + 1.88064i) q^{49} +(-2.03553 + 3.52565i) q^{53} +4.18154i q^{55} +(12.9853 + 7.49706i) q^{59} +(3.00000 + 7.34847i) q^{63} +(-8.48528 - 4.89898i) q^{73} +(6.72792 - 3.88437i) q^{75} +(-9.44975 + 12.1859i) q^{77} +(-8.86396 - 15.3528i) q^{79} +(-4.50000 + 7.79423i) q^{81} -13.5592i q^{83} +(-11.3787 - 6.56948i) q^{87} +(-9.62132 - 16.6646i) q^{93} +8.06591i q^{97} -17.4853 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 6 q^{3} - 6 q^{5} + 2 q^{7} + 6 q^{9}+O(q^{10}) \) 4 * q - 6 * q^3 - 6 * q^5 + 2 * q^7 + 6 * q^9	\( 4 q - 6 q^{3} - 6 q^{5} + 2 q^{7} + 6 q^{9} - 6 q^{11} + 12 q^{15} - 6 q^{21} + 8 q^{25} + 36 q^{29} + 30 q^{31} + 18 q^{33} + 24 q^{35} - 18 q^{45} + 10 q^{49} + 6 q^{53} + 18 q^{59} + 12 q^{63} - 24 q^{75} - 18 q^{77} - 10 q^{79} - 18 q^{81} - 54 q^{87} - 30 q^{93} - 36 q^{99}+O(q^{100}) \) 4 * q - 6 * q^3 - 6 * q^5 + 2 * q^7 + 6 * q^9 - 6 * q^11 + 12 * q^15 - 6 * q^21 + 8 * q^25 + 36 * q^29 + 30 * q^31 + 18 * q^33 + 24 * q^35 - 18 * q^45 + 10 * q^49 + 6 * q^53 + 18 * q^59 + 12 * q^63 - 24 * q^75 - 18 * q^77 - 10 * q^79 - 18 * q^81 - 54 * q^87 - 30 * q^93 - 36 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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