LMFDB - Embedded newform 672.2.bi.a.17.1

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			17.1
Root			$-0.707107 + 1.22474i$ of defining polynomial
Character	$\chi$	$=$	672.17
Dual form			672.2.bi.a.593.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.50000 - 0.866025i) q^{3} +(-3.62132 + 2.09077i) q^{5} +(-1.62132 - 2.09077i) q^{7} +(1.50000 + 2.59808i) q^{9} +O(q^{10})$	$q+(-1.50000 - 0.866025i) q^{3} +(-3.62132 + 2.09077i) q^{5} +(-1.62132 - 2.09077i) q^{7} +(1.50000 + 2.59808i) q^{9} +(-0.0857864 + 0.148586i) q^{11} +7.24264 q^{15} +(0.621320 + 4.54026i) q^{21} +(6.24264 - 10.8126i) q^{25} -5.19615i q^{27} +10.4142 q^{29} +(5.37868 + 3.10538i) q^{31} +(0.257359 - 0.148586i) q^{33} +(10.2426 + 4.18154i) q^{35} +(-10.8640 - 6.27231i) q^{45} +(-1.74264 + 6.77962i) q^{49} +(5.03553 - 8.72180i) q^{53} -0.717439i q^{55} +(-3.98528 - 2.30090i) q^{59} +(3.00000 - 7.34847i) q^{63} +(8.48528 + 4.89898i) q^{73} +(-18.7279 + 10.8126i) q^{75} +(0.449747 - 0.0615465i) q^{77} +(3.86396 + 6.69258i) q^{79} +(-4.50000 + 7.79423i) q^{81} -3.76127i q^{83} +(-15.6213 - 9.01897i) q^{87} +(-5.37868 - 9.31615i) q^{93} -11.5300i q^{97} -0.514719 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−3.62132	+	2.09077i	−1.61950	+	0.935021i	−0.632456	+	0.774597i	$0.717953\pi$
$5$	−3.62132	+	2.09077i	−1.61950	+	0.935021i	−0.987048	+	0.160424i	$0.948714\pi$
$6$		0			0
$7$	−1.62132	−	2.09077i	−0.612801	−	0.790237i
$7$	−1.62132	−	2.09077i	−0.612801	−	0.790237i
$8$		0			0
$9$	1.50000	+	2.59808i	0.500000	+	0.866025i
$10$		0			0
$11$	−0.0857864	+	0.148586i	−0.0258656	+	0.0448005i	−0.878668	−	0.477432i	$-0.841568\pi$
$11$	−0.0857864	+	0.148586i	−0.0258656	+	0.0448005i	0.852803	+	0.522233i	$0.174901\pi$
$12$		0			0
$13$		0			0			−	1.00000i	$-0.5\pi$
$13$		0			0				1.00000i	$0.5\pi$
$14$		0			0
$15$	7.24264			1.87004
$16$		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
$21$	0.621320	+	4.54026i	0.135583	+	0.990766i
$22$		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
$25$	6.24264	−	10.8126i	1.24853	−	2.16251i
$26$		0			0
$27$		−	5.19615i		−	1.00000i
$28$		0			0
$29$	10.4142			1.93387			0.966935	−	0.255021i	$-0.0820825\pi$
$29$	10.4142			1.93387			0.966935	+	0.255021i	$0.0820825\pi$
$30$		0			0
$31$	5.37868	+	3.10538i	0.966039	+	0.557743i	0.898027	−	0.439941i	$-0.145001\pi$
$31$	5.37868	+	3.10538i	0.966039	+	0.557743i	0.0680129	+	0.997684i	$0.478334\pi$
$32$		0			0
$33$	0.257359	−	0.148586i	0.0448005	−	0.0258656i
$34$		0			0
$35$	10.2426	+	4.18154i	1.73132	+	0.706809i
$36$		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
$45$	−10.8640	−	6.27231i	−1.61950	−	0.935021i
$46$		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
$49$	−1.74264	+	6.77962i	−0.248949	+	0.968517i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	5.03553	−	8.72180i	0.691684	−	1.19803i	−0.279602	−	0.960116i	$-0.590203\pi$
$53$	5.03553	−	8.72180i	0.691684	−	1.19803i	0.971286	−	0.237915i	$-0.0764641\pi$
$54$		0			0
$55$		−	0.717439i		−	0.0967394i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−3.98528	−	2.30090i	−0.518839	−	0.299552i	0.217620	−	0.976034i	$-0.430171\pi$
$59$	−3.98528	−	2.30090i	−0.518839	−	0.299552i	−0.736460	+	0.676481i	$0.763504\pi$
$60$		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	0.906208	−	0.422833i	$-0.138964\pi$
$73$	8.48528	+	4.89898i	0.993127	+	0.573382i	0.0869195	+	0.996215i	$0.472298\pi$
$74$		0			0
$75$	−18.7279	+	10.8126i	−2.16251	+	1.24853i
$76$		0			0
$77$	0.449747	−	0.0615465i	0.0512535	−	0.00701388i
$78$		0			0
$79$	3.86396	+	6.69258i	0.434730	+	0.752974i	0.997274	−	0.0737937i	$-0.0235106\pi$
$79$	3.86396	+	6.69258i	0.434730	+	0.752974i	−0.562544	+	0.826767i	$0.690177\pi$
$80$		0			0
$81$	−4.50000	+	7.79423i	−0.500000	+	0.866025i
$82$		0			0
$83$		−	3.76127i		−	0.412854i	−0.978462	−	0.206427i	$-0.933816\pi$
$83$		−	3.76127i		−	0.412854i	0.978462	−	0.206427i	$-0.0661835\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−15.6213	−	9.01897i	−1.67478	−	0.966935i
$88$		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
$93$	−5.37868	−	9.31615i	−0.557743	−	0.966039i
$94$		0			0
$95$		0			0
$96$		0			0
$97$		−	11.5300i		−	1.17070i	−0.810782	−	0.585348i	$-0.800958\pi$
$97$		−	11.5300i		−	1.17070i	0.810782	−	0.585348i	$-0.199042\pi$
$98$		0			0
$99$	−0.514719			−0.0517312
$100$		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	−0.971925	−	0.235291i	$-0.924396\pi$
$103$	−12.7279	+	7.34847i	−1.25412	+	0.724066i	−0.282194	+	0.959357i	$0.591062\pi$
$104$		0			0
$105$	−11.7426	−	15.1427i	−1.14596	−	1.47778i
$106$		0			0
$107$	10.3284	+	17.8894i	0.998487	+	1.72943i	0.546869	+	0.837218i	$0.315820\pi$
$107$	10.3284	+	17.8894i	0.998487	+	1.72943i	0.451618	+	0.892211i	$0.350847\pi$
$108$		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
$121$	5.48528	+	9.50079i	0.498662	+	0.863708i
$122$		0			0
$123$		0			0
$124$		0			0
$125$			31.3000i			2.79956i
$126$		0			0
$127$	−6.75736			−0.599619			−0.299809	−	0.953999i	$-0.596923\pi$
$127$	−6.75736			−0.599619			−0.299809	+	0.953999i	$0.596923\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	18.4706	−	10.6640i	1.61378	−	0.931717i	0.625297	−	0.780387i	$-0.284978\pi$
$131$	18.4706	−	10.6640i	1.61378	−	0.931717i	0.988483	−	0.151330i	$-0.0483556\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	10.8640	+	18.8169i	0.935021	+	1.61950i
$136$		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
$145$	−37.7132	+	21.7737i	−3.13191	+	1.80821i
$146$		0			0
$147$	8.48528	−	8.66025i	0.699854	−	0.714286i
$148$		0			0
$149$	−1.41421	−	2.44949i	−0.115857	−	0.200670i	0.802265	−	0.596968i	$-0.203628\pi$
$149$	−1.41421	−	2.44949i	−0.115857	−	0.200670i	−0.918122	+	0.396298i	$0.870295\pi$
$150$		0			0
$151$	11.1066	−	19.2372i	0.903842	−	1.56550i	0.0813788	−	0.996683i	$-0.474068\pi$
$151$	11.1066	−	19.2372i	0.903842	−	1.56550i	0.822464	−	0.568818i	$-0.192599\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−25.9706			−2.08601
$156$		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
$159$	−15.1066	+	8.72180i	−1.19803	+	0.691684i
$160$		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
$165$	−0.621320	+	1.07616i	−0.0483697	+	0.0837788i
$166$		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
$175$	−32.7279	+	4.47871i	−2.47400	+	0.338559i
$176$		0			0
$177$	3.98528	+	6.90271i	0.299552	+	0.518839i
$178$		0			0
$179$	5.65685	−	9.79796i	0.422813	−	0.732334i	−0.573400	−	0.819275i	$-0.694376\pi$
$179$	5.65685	−	9.79796i	0.422813	−	0.732334i	0.996213	+	0.0869415i	$0.0277093\pi$
$180$		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
$189$	−10.8640	+	8.42463i	−0.790237	+	0.612801i
$190$		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
$193$	10.7426	−	18.6068i	0.773272	−	1.33935i	−0.162488	−	0.986710i	$-0.551952\pi$
$193$	10.7426	−	18.6068i	0.773272	−	1.33935i	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	$-0.668062\pi$
$197$	−14.1421			−1.00759			−0.503793	+	0.863825i	$0.668062\pi$
$198$		0			0
$199$	21.2132	+	12.2474i	1.50376	+	0.868199i	0.999990	+	0.00436292i	$0.00138876\pi$
$199$	21.2132	+	12.2474i	1.50376	+	0.868199i	0.503774	+	0.863836i	$0.331945\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−16.8848	−	21.7737i	−1.18508	−	1.52822i
$204$		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
$217$	−2.22792	−	16.2804i	−0.151241	−	1.10519i
$218$		0			0
$219$	−8.48528	−	14.6969i	−0.573382	−	0.993127i
$220$		0			0
$221$		0			0
$222$		0			0
$223$		−	29.8651i		−	1.99992i	−0.00910984	−	0.999959i	$-0.502900\pi$
$223$		−	29.8651i		−	1.99992i	0.00910984	−	0.999959i	$-0.497100\pi$
$224$		0			0
$225$	37.4558			2.49706
$226$		0			0
$227$	25.7132	+	14.8455i	1.70665	+	0.985332i	0.938647	+	0.344881i	$0.112081\pi$
$227$	25.7132	+	14.8455i	1.70665	+	0.985332i	0.767999	+	0.640451i	$0.221253\pi$
$228$		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
$231$	−0.727922	−	0.297173i	−0.0478938	−	0.0195525i
$232$		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
$237$		−	13.3852i		−	0.869459i
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$		0			0
$241$	−20.2279	−	11.6786i	−1.30300	−	0.752285i	−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−20.2279	−	11.6786i	−1.30300	−	0.752285i	−0.980917	+	0.194429i	$0.937715\pi$
$242$		0			0
$243$	13.5000	−	7.79423i	0.866025	−	0.500000i
$244$		0			0
$245$	−7.86396	−	28.1946i	−0.502410	−	1.80129i
$246$		0			0
$247$		0			0
$248$		0			0
$249$	−3.25736	+	5.64191i	−0.206427	+	0.357542i
$250$		0			0
$251$			20.4874i			1.29316i	0.762848	+	0.646578i	$0.223800\pi$
$251$			20.4874i			1.29316i	−0.762848	+	0.646578i	$0.776200\pi$
$252$		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
$261$	15.6213	+	27.0569i	0.966935	+	1.67478i
$262$		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
$265$			42.1126i			2.58696i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	18.8345	+	10.8741i	1.14836	+	0.663007i	0.948487	−	0.316815i	$-0.102613\pi$
$269$	18.8345	+	10.8741i	1.14836	+	0.663007i	0.199874	+	0.979822i	$0.435947\pi$
$270$		0			0
$271$	5.89340	−	3.40256i	0.357998	−	0.206691i	−0.310204	−	0.950670i	$-0.600397\pi$
$271$	5.89340	−	3.40256i	0.357998	−	0.206691i	0.668202	+	0.743980i	$0.267064\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	1.07107	+	1.85514i	0.0645878	+	0.111869i
$276$		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
$279$			18.6323i			1.11549i
$280$		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
$291$	−9.98528	+	17.2950i	−0.585348	+	1.01385i
$292$		0			0
$293$		−	3.34101i		−	0.195184i	−0.995227	−	0.0975919i	$-0.968886\pi$
$293$		−	3.34101i		−	0.195184i	0.995227	−	0.0975919i	$-0.0311140\pi$
$294$		0			0
$295$	19.2426			1.12035
$296$		0			0
$297$	0.772078	+	0.445759i	0.0448005	+	0.0258656i
$298$		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
$313$	21.2574	−	12.2729i	1.20154	−	0.693708i	0.240640	−	0.970614i	$-0.422643\pi$
$313$	21.2574	−	12.2729i	1.20154	−	0.693708i	0.960897	+	0.276907i	$0.0893093\pi$
$314$		0			0
$315$	4.50000	+	32.8835i	0.253546	+	1.85277i
$316$		0			0
$317$	−15.2782	−	26.4626i	−0.858108	−	1.48629i	−0.873732	−	0.486408i	$-0.838307\pi$
$317$	−15.2782	−	26.4626i	−0.858108	−	1.48629i	0.0156238	−	0.999878i	$-0.495027\pi$
$318$		0			0
$319$	−0.893398	+	1.54741i	−0.0500207	+	0.0866384i
$320$		0			0
$321$		−	35.7787i		−	1.99697i
$322$		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
$337$	36.4558			1.98588			0.992938	−	0.118633i	$-0.0378512\pi$
$337$	36.4558			1.98588			0.992938	+	0.118633i	$0.0378512\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−0.922836	+	0.532799i	−0.0499743	+	0.0288527i
$342$		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	$0.441071\pi$
$347$	−14.1421	+	24.4949i	−0.759190	+	1.31495i	−0.943264	+	0.332043i	$0.892262\pi$
$348$		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
$363$		−	19.0016i		−	0.997324i
$364$		0			0
$365$	−40.9706			−2.14450
$366$		0			0
$367$	26.3787	+	15.2297i	1.37696	+	0.794986i	0.991792	−	0.127862i	$-0.0408116\pi$
$367$	26.3787	+	15.2297i	1.37696	+	0.794986i	0.385164	+	0.922848i	$0.374145\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−26.3995	+	3.61269i	−1.37059	+	0.187561i
$372$		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
$375$	27.1066	−	46.9500i	1.39978	−	2.42449i
$376$		0			0
$377$		0			0
$378$		0			0
$379$		0			0		1.00000			$0$
$379$		0			0		−1.00000			$\pi$
$380$		0			0
$381$	10.1360	+	5.85204i	0.519285	+	0.299809i
$382$		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	+	1.16320i	−0.0764471	+	0.0592821i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−15.5563	+	26.9444i	−0.788738	+	1.36613i	0.138002	+	0.990432i	$0.455932\pi$
$389$	−15.5563	+	26.9444i	−0.788738	+	1.36613i	−0.926740	+	0.375703i	$0.877401\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$	−36.9411			−1.86343
$394$		0			0
$395$	−27.9853	−	16.1573i	−1.40809	−	0.812962i
$396$		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
$405$		−	37.6339i		−	1.87004i
$406$		0			0
$407$		0			0
$408$		0			0
$409$	9.47056	+	5.46783i	0.468289	+	0.270367i	0.715523	−	0.698589i	$-0.246188\pi$
$409$	9.47056	+	5.46783i	0.468289	+	0.270367i	−0.247234	+	0.968956i	$0.579522\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	1.65076	+	12.0628i	0.0812285	+	0.593572i
$414$		0			0
$415$	7.86396	+	13.6208i	0.386027	+	0.668618i
$416$		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	$-0.609209\pi$
$433$		−	39.1918i		−	1.88344i	0.336399	−	0.941720i	$-0.390791\pi$
$434$		0			0
$435$	75.4264			3.61642
$436$		0			0
$437$		0			0
$438$		0			0
$439$	−36.1066	+	20.8462i	−1.72327	+	0.994933i	−0.811366	+	0.584539i	$0.801275\pi$
$439$	−36.1066	+	20.8462i	−1.72327	+	0.994933i	−0.911908	+	0.410394i	$0.865391\pi$
$440$		0			0
$441$	−20.2279	+	5.64191i	−0.963234	+	0.268662i
$442$		0			0
$443$	−20.5711	−	35.6301i	−0.977361	−	1.69284i	−0.671913	−	0.740630i	$-0.734527\pi$
$443$	−20.5711	−	35.6301i	−0.977361	−	1.69284i	−0.305448	−	0.952209i	$-0.598806\pi$
$444$		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
$453$	−33.3198	+	19.2372i	−1.56550	+	0.903842i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	17.9853	+	31.1514i	0.841316	+	1.45720i	0.888783	+	0.458329i	$0.151552\pi$
$457$	17.9853	+	31.1514i	0.841316	+	1.45720i	−0.0474665	+	0.998873i	$0.515115\pi$
$458$		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
$465$	38.9558	+	22.4912i	1.80653	+	1.04300i
$466$		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
$477$	30.2132			1.38337
$478$		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
$485$	24.1066	+	41.7539i	1.09462	+	1.89595i
$486$		0			0
$487$	−18.5919	+	32.2021i	−0.842479	+	1.45922i	0.0453143	+	0.998973i	$0.485571\pi$
$487$	−18.5919	+	32.2021i	−0.842479	+	1.45922i	−0.887793	+	0.460243i	$0.847762\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	21.6863			0.978689			0.489344	−	0.872091i	$-0.337236\pi$
$491$	21.6863			0.978689			0.489344	+	0.872091i	$0.337236\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$	1.86396	−	1.07616i	0.0837788	−	0.0483697i
$496$		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
$505$	−14.4853			−0.644587
$506$		0			0
$507$	19.5000	+	11.2583i	0.866025	+	0.500000i
$508$		0			0
$509$	17.3787	−	10.0336i	0.770296	−	0.444731i	−0.0626839	−	0.998033i	$-0.519966\pi$
$509$	17.3787	−	10.0336i	0.770296	−	0.444731i	0.832980	+	0.553303i	$0.186633\pi$
$510$		0			0
$511$	−3.51472	−	25.6836i	−0.155482	−	1.13618i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	30.7279	−	53.2223i	1.35403	−	2.34526i
$516$		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
$525$	52.9706	+	21.6251i	2.31182	+	0.943799i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−11.5000	+	19.9186i	−0.500000	+	0.866025i
$530$		0			0
$531$		−	13.8054i		−	0.599104i
$532$		0			0
$533$		0			0
$534$		0			0
$535$	−74.8051	−	43.1887i	−3.23411	−	1.86721i
$536$		0			0
$537$	−16.9706	+	9.79796i	−0.732334	+	0.422813i
$538$		0			0
$539$	−0.857864	−	0.840532i	−0.0369508	−	0.0362043i
$540$		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
$553$	7.72792	−	18.9295i	0.328625	−	0.804963i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−15.9645	+	27.6513i	−0.676436	+	1.17162i	0.299611	+	0.954062i	$0.403143\pi$
$557$	−15.9645	+	27.6513i	−0.676436	+	1.17162i	−0.976047	+	0.217560i	$0.930190\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$	4.71320	+	2.72117i	0.198638	+	0.114684i	0.596020	−	0.802970i	$-0.296748\pi$
$563$	4.71320	+	2.72117i	0.198638	+	0.114684i	−0.397382	+	0.917653i	$0.630081\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	23.5919	−	3.22848i	0.990766	−	0.135583i
$568$		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
$577$	−41.2279	−	23.8030i	−1.71634	−	0.990930i	−0.925361	−	0.379088i	$-0.876238\pi$
$577$	−41.2279	−	23.8030i	−1.71634	−	0.990930i	−0.790980	−	0.611842i	$-0.790429\pi$
$578$		0			0
$579$	−32.2279	+	18.6068i	−1.33935	+	0.773272i
$580$		0			0
$581$	−7.86396	+	6.09823i	−0.326252	+	0.252997i
$582$		0			0
$583$	0.863961	+	1.49642i	0.0357816	+	0.0619756i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			30.5316i			1.26017i	0.776525	+	0.630087i	$0.216981\pi$
$587$			30.5316i			1.26017i	−0.776525	+	0.630087i	$0.783019\pi$
$588$		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
$601$			22.7628i			0.928516i	0.885700	+	0.464258i	$0.153679\pi$
$601$			22.7628i			0.928516i	−0.885700	+	0.464258i	$0.846321\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−39.7279	−	22.9369i	−1.61517	−	0.932519i
$606$		0			0
$607$	35.5919	−	20.5490i	1.44463	−	0.834058i	0.446476	−	0.894795i	$-0.352679\pi$
$607$	35.5919	−	20.5490i	1.44463	−	0.834058i	0.998154	+	0.0607380i	$0.0193454\pi$
$608$		0			0
$609$	6.47056	+	47.2832i	0.262200	+	1.91601i
$610$		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
$625$	−34.2279	−	59.2845i	−1.36912	−	2.37138i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−20.7574			−0.826337			−0.413169	−	0.910654i	$-0.635578\pi$
$631$	−20.7574			−0.826337			−0.413169	+	0.910654i	$0.635578\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	24.4706	−	14.1281i	0.971085	−	0.560656i
$636$		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
$649$	0.683766	−	0.394773i	0.0268402	−	0.0154962i
$650$		0			0
$651$	−10.7574	+	26.3500i	−0.421614	+	1.03274i
$652$		0			0
$653$	4.52082	+	7.83028i	0.176913	+	0.306423i	0.940822	−	0.338902i	$-0.110055\pi$
$653$	4.52082	+	7.83028i	0.176913	+	0.306423i	−0.763909	+	0.645325i	$0.776722\pi$
$654$		0			0
$655$	−44.5919	+	77.2354i	−1.74235	+	3.01784i
$656$		0			0
$657$			29.3939i			1.14676i
$658$		0			0
$659$	−45.2548			−1.76288			−0.881439	−	0.472298i	$-0.843425\pi$
$659$	−45.2548			−1.76288			−0.881439	+	0.472298i	$0.843425\pi$
$660$		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
$669$	−25.8640	+	44.7977i	−0.999959	+	1.73198i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−50.9411			−1.96364			−0.981818	−	0.189824i	$-0.939208\pi$
$673$	−50.9411			−1.96364			−0.981818	+	0.189824i	$0.939208\pi$
$674$		0			0
$675$	−56.1838	−	32.4377i	−2.16251	−	1.24853i
$676$		0			0
$677$	−24.6213	+	14.2151i	−0.946274	+	0.546332i	−0.891922	−	0.452190i	$-0.850643\pi$
$677$	−24.6213	+	14.2151i	−0.946274	+	0.546332i	−0.0543526	+	0.998522i	$0.517310\pi$
$678$		0			0
$679$	−24.1066	+	18.6938i	−0.925126	+	0.717404i
$680$		0			0
$681$	−25.7132	−	44.5366i	−0.985332	−	1.70665i
$682$		0			0
$683$	−21.0858	+	36.5217i	−0.806825	+	1.39746i	0.108227	+	0.994126i	$0.465483\pi$
$683$	−21.0858	+	36.5217i	−0.806825	+	1.39746i	−0.915052	+	0.403336i	$0.867851\pi$
$684$		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
$693$	0.834524	+	1.07616i	0.0317009	+	0.0408799i
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−51.3848			−1.94078			−0.970388	−	0.241551i	$-0.922344\pi$
$701$	−51.3848			−1.94078			−0.970388	+	0.241551i	$0.922344\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−1.24264	−	9.08052i	−0.0467343	−	0.341508i
$708$		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
$711$	−11.5919	+	20.0777i	−0.434730	+	0.752974i
$712$		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
$723$	20.2279	+	35.0358i	0.752285	+	1.30300i
$724$		0			0
$725$	65.0122	−	112.604i	2.41449	−	4.18202i
$726$		0			0
$727$			28.6764i			1.06355i	0.846886	+	0.531775i	$0.178475\pi$
$727$			28.6764i			1.06355i	−0.846886	+	0.531775i	$0.821525\pi$
$728$		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
$735$	−12.6213	+	49.1023i	−0.465544	+	1.81117i
$736$		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
$745$	10.2426	+	5.91359i	0.375261	+	0.216657i
$746$		0			0
$747$	9.77208	−	5.64191i	0.357542	−	0.206427i
$748$		0			0
$749$	20.6569	−	50.5988i	0.754785	−	1.84884i
$750$		0			0
$751$	−25.8345	−	44.7467i	−0.942715	−	1.63283i	−0.760263	−	0.649616i	$-0.774930\pi$
$751$	−25.8345	−	44.7467i	−0.942715	−	1.63283i	−0.182453	−	0.983215i	$-0.558404\pi$
$752$		0			0
$753$	17.7426	−	30.7312i	0.646578	−	1.11991i
$754$		0			0
$755$			92.8854i			3.38045i
$756$		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
$769$			47.0116i			1.69528i	0.530572	+	0.847640i	$0.321977\pi$
$769$			47.0116i			1.69528i	−0.530572	+	0.847640i	$0.678023\pi$
$770$		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
$775$	67.1543	−	38.7716i	2.41225	−	1.39272i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		−	54.1138i		−	1.93387i
$784$		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
$795$	36.4706	−	63.1689i	1.29348	−	2.24037i
$796$		0			0
$797$			55.2006i			1.95530i	0.210230	+	0.977652i	$0.432579\pi$
$797$			55.2006i			1.95530i	−0.210230	+	0.977652i	$0.567421\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−1.45584	+	0.840532i	−0.0513756	+	0.0296617i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−18.8345	−	32.6224i	−0.663007	−	1.14836i
$808$		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
$813$	−11.7868			−0.413381
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	25.5208	+	44.2033i	0.890683	+	1.54271i	0.839059	+	0.544041i	$0.183106\pi$
$821$	25.5208	+	44.2033i	0.890683	+	1.54271i	0.0516239	+	0.998667i	$0.483560\pi$
$822$		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
$825$		−	3.71029i		−	0.129176i
$826$		0			0
$827$	19.2843			0.670580			0.335290	−	0.942115i	$-0.391166\pi$
$827$	19.2843			0.670580			0.335290	+	0.942115i	$0.391166\pi$
$828$		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
$837$	16.1360	−	27.9484i	0.557743	−	0.966039i
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	79.4558			2.73986
$842$		0			0
$843$		0			0
$844$		0			0
$845$	47.0772	−	27.1800i	1.61950	−	0.935021i
$846$		0			0
$847$	10.9706	−	26.8723i	0.376953	−	0.923342i
$848$		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
$865$	−36.2132	+	62.7231i	−1.23129	+	2.13265i
$866$		0			0
$867$		−	29.4449i		−	1.00000i
$868$		0			0
$869$	−1.32590			−0.0449781
$870$		0			0
$871$		0			0
$872$		0			0
$873$	29.9558	−	17.2950i	1.01385	−	0.585348i
$874$		0			0
$875$	65.4411	−	50.7473i	2.21231	−	1.71557i
$876$		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
$879$	−2.89340	+	5.01151i	−0.0975919	+	0.169034i
$880$		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
$885$	−28.8640	−	16.6646i	−0.970251	−	0.560175i
$886$		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
$889$	10.9558	+	14.1281i	0.367447	+	0.473841i
$890$		0			0
$891$	−0.772078	−	1.33728i	−0.0258656	−	0.0448005i
$892$		0			0
$893$		0			0
$894$		0			0
$895$			47.3087i			1.58136i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	56.0147	+	32.3401i	1.86820	+	1.07860i
$900$		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
$913$	0.558875	+	0.322666i	0.0184960	+	0.0106787i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−52.2426	−	21.3280i	−1.72520	−	0.704312i
$918$		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
$937$		−	60.0274i		−	1.96101i	−0.196492	−	0.980505i	$-0.562955\pi$
$937$		−	60.0274i		−	1.96101i	0.196492	−	0.980505i	$-0.437045\pi$
$938$		0			0
$939$	−42.5147			−1.38742
$940$		0			0
$941$	−19.5624	−	11.2944i	−0.637718	−	0.368186i	0.146017	−	0.989282i	$-0.453354\pi$
$941$	−19.5624	−	11.2944i	−0.637718	−	0.368186i	−0.783735	+	0.621096i	$0.786688\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$	21.7279	−	53.2223i	0.706809	−	1.73132i
$946$		0			0
$947$	−28.2843	−	48.9898i	−0.919115	−	1.59195i	−0.800762	−	0.598983i	$-0.795572\pi$
$947$	−28.2843	−	48.9898i	−0.919115	−	1.59195i	−0.118354	−	0.992972i	$-0.537762\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$			52.9251i			1.71622i
$952$		0			0
$953$		0			0		1.00000			$0$
$953$		0			0		−1.00000			$\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	2.68019	−	1.54741i	0.0866384	−	0.0500207i
$958$		0			0
$959$		0			0
$960$		0			0
$961$	3.78680	+	6.55892i	0.122155	+	0.211578i
$962$		0			0
$963$	−30.9853	+	53.6681i	−0.998487	+	1.72943i
$964$		0			0
$965$			89.8416i			2.89210i
$966$		0			0
$967$	35.2426			1.13333			0.566663	−	0.823949i	$-0.308234\pi$
$967$	35.2426			1.13333			0.566663	+	0.823949i	$0.308234\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	48.1690	−	27.8104i	1.54582	−	0.892479i	0.547364	−	0.836894i	$-0.315631\pi$
$971$	48.1690	−	27.8104i	1.54582	−	0.892479i	0.998454	−	0.0555842i	$-0.0177021\pi$
$972$		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
$985$	51.2132	−	29.5680i	1.63179	−	0.942113i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	25.1066	−	43.4859i	0.797537	−	1.38138i	−0.123678	−	0.992322i	$-0.539469\pi$
$991$	25.1066	−	43.4859i	0.797537	−	1.38138i	0.921215	−	0.389053i	$-0.127198\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−102.426			−3.24714
$996$		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.17.1		4
3.2	odd	2		672.2.bi.b.17.2		4
4.3	odd	2		168.2.ba.b.101.1	yes	4
7.5	odd	6	inner	672.2.bi.a.593.1		4
8.3	odd	2		168.2.ba.a.101.2	yes	4
8.5	even	2		672.2.bi.b.17.2		4
12.11	even	2		168.2.ba.a.101.2	yes	4
21.5	even	6		672.2.bi.b.593.2		4
24.5	odd	2	CM	672.2.bi.a.17.1		4
24.11	even	2		168.2.ba.b.101.1	yes	4
28.19	even	6		168.2.ba.b.5.1	yes	4
56.5	odd	6		672.2.bi.b.593.2		4
56.19	even	6		168.2.ba.a.5.2	✓	4
84.47	odd	6		168.2.ba.a.5.2	✓	4
168.5	even	6	inner	672.2.bi.a.593.1		4
168.131	odd	6		168.2.ba.b.5.1	yes	4

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

\(f(q)\)	\(=\)	\(q+(-1.50000 - 0.866025i) q^{3} +(-3.62132 + 2.09077i) q^{5} +(-1.62132 - 2.09077i) q^{7} +(1.50000 + 2.59808i) q^{9} +O(q^{10})\)	\(q+(-1.50000 - 0.866025i) q^{3} +(-3.62132 + 2.09077i) q^{5} +(-1.62132 - 2.09077i) q^{7} +(1.50000 + 2.59808i) q^{9} +(-0.0857864 + 0.148586i) q^{11} +7.24264 q^{15} +(0.621320 + 4.54026i) q^{21} +(6.24264 - 10.8126i) q^{25} -5.19615i q^{27} +10.4142 q^{29} +(5.37868 + 3.10538i) q^{31} +(0.257359 - 0.148586i) q^{33} +(10.2426 + 4.18154i) q^{35} +(-10.8640 - 6.27231i) q^{45} +(-1.74264 + 6.77962i) q^{49} +(5.03553 - 8.72180i) q^{53} -0.717439i q^{55} +(-3.98528 - 2.30090i) q^{59} +(3.00000 - 7.34847i) q^{63} +(8.48528 + 4.89898i) q^{73} +(-18.7279 + 10.8126i) q^{75} +(0.449747 - 0.0615465i) q^{77} +(3.86396 + 6.69258i) q^{79} +(-4.50000 + 7.79423i) q^{81} -3.76127i q^{83} +(-15.6213 - 9.01897i) q^{87} +(-5.37868 - 9.31615i) q^{93} -11.5300i q^{97} -0.514719 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 6 q^{3} - 6 q^{5} + 2 q^{7} + 6 q^{9}+O(q^{10}) \) 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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