LMFDB - Embedded newform 1176.2.q.l.361.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1176,2,Mod(361,1176)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1176, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 0, 4]))

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

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

chi := DirichletCharacter("1176.361");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1176 = 2^{3} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1176.q (of order $3$, 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:	$9.39040727770$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{-19})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 4x^{2} - 5x + 25 $ x^4 - x^3 - 4x^2 - 5x + 25
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 168)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			361.2
Root			$2.13746 + 0.656712i$ of defining polynomial
Character	$\chi$	$=$	1176.361
Dual form			1176.2.q.l.961.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.500000 + 0.866025i) q^{3} +(1.63746 + 2.83616i) q^{5} +(-0.500000 - 0.866025i) q^{9} +O(q^{10})$	\(q+(-0.500000 + 0.866025i) q^{3} +(1.63746 + 2.83616i) q^{5} +(-0.500000 - 0.866025i) q^{9} +(-1.63746 + 2.83616i) q^{11} -6.27492 q^{13} -3.27492 q^{15} +(-2.00000 + 3.46410i) q^{17} +(-3.13746 - 5.43424i) q^{19} +(-2.00000 - 3.46410i) q^{23} +(-2.86254 + 4.95807i) q^{25} +1.00000 q^{27} +5.27492 q^{29} +(-0.500000 + 0.866025i) q^{31} +(-1.63746 - 2.83616i) q^{33} +(1.13746 + 1.97014i) q^{37} +(3.13746 - 5.43424i) q^{39} +4.54983 q^{41} +0.274917 q^{43} +(1.63746 - 2.83616i) q^{45} +(-3.00000 - 5.19615i) q^{47} +(-2.00000 - 3.46410i) q^{51} +(-4.63746 + 8.03231i) q^{53} -10.7251 q^{55} +6.27492 q^{57} +(-0.637459 + 1.10411i) q^{59} +(5.00000 + 8.66025i) q^{61} +(-10.2749 - 17.7967i) q^{65} +(0.137459 - 0.238085i) q^{67} +4.00000 q^{69} +2.00000 q^{71} +(-2.13746 + 3.70219i) q^{73} +(-2.86254 - 4.95807i) q^{75} +(-5.77492 - 10.0025i) q^{79} +(-0.500000 + 0.866025i) q^{81} -7.27492 q^{83} -13.0997 q^{85} +(-2.63746 + 4.56821i) q^{87} +(-5.27492 - 9.13642i) q^{89} +(-0.500000 - 0.866025i) q^{93} +(10.2749 - 17.7967i) q^{95} -8.72508 q^{97} +3.27492 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{3} - q^{5} - 2 q^{9}+O(q^{10}) $ 4 * q - 2 * q^3 - q^5 - 2 * q^9	$ 4 q - 2 q^{3} - q^{5} - 2 q^{9} + q^{11} - 10 q^{13} + 2 q^{15} - 8 q^{17} - 5 q^{19} - 8 q^{23} - 19 q^{25} + 4 q^{27} + 6 q^{29} - 2 q^{31} + q^{33} - 3 q^{37} + 5 q^{39} - 12 q^{41} - 14 q^{43} - q^{45} - 12 q^{47} - 8 q^{51} - 11 q^{53} - 58 q^{55} + 10 q^{57} + 5 q^{59} + 20 q^{61} - 26 q^{65} - 7 q^{67} + 16 q^{69} + 8 q^{71} - q^{73} - 19 q^{75} - 8 q^{79} - 2 q^{81} - 14 q^{83} + 8 q^{85} - 3 q^{87} - 6 q^{89} - 2 q^{93} + 26 q^{95} - 50 q^{97} - 2 q^{99}+O(q^{100}) $ 4 * q - 2 * q^3 - q^5 - 2 * q^9 + q^11 - 10 * q^13 + 2 * q^15 - 8 * q^17 - 5 * q^19 - 8 * q^23 - 19 * q^25 + 4 * q^27 + 6 * q^29 - 2 * q^31 + q^33 - 3 * q^37 + 5 * q^39 - 12 * q^41 - 14 * q^43 - q^45 - 12 * q^47 - 8 * q^51 - 11 * q^53 - 58 * q^55 + 10 * q^57 + 5 * q^59 + 20 * q^61 - 26 * q^65 - 7 * q^67 + 16 * q^69 + 8 * q^71 - q^73 - 19 * q^75 - 8 * q^79 - 2 * q^81 - 14 * q^83 + 8 * q^85 - 3 * q^87 - 6 * q^89 - 2 * q^93 + 26 * q^95 - 50 * q^97 - 2 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1176\mathbb{Z}\right)^\times$.

$n$	$295$	$589$	$785$	$1081$
$\chi(n)$	$1$	$1$	$1$	$e\left(\frac{2}{3}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	−0.500000	+	0.866025i	−0.288675	+	0.500000i
$3$	−0.500000	+	0.866025i	−0.288675	+	0.500000i
$4$		0			0
$5$	1.63746	+	2.83616i	0.732294	+	1.26837i	0.955901	+	0.293691i	$0.0948835\pi$
$5$	1.63746	+	2.83616i	0.732294	+	1.26837i	−0.223607	+	0.974679i	$0.571783\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$	−0.500000	−	0.866025i	−0.166667	−	0.288675i
$10$		0			0
$11$	−1.63746	+	2.83616i	−0.493712	+	0.855135i	−0.999974	−	0.00724520i	$-0.997694\pi$
$11$	−1.63746	+	2.83616i	−0.493712	+	0.855135i	0.506261	+	0.862380i	$0.331027\pi$
$12$		0			0
$13$	−6.27492			−1.74035			−0.870174	−	0.492744i	$-0.835994\pi$
$13$	−6.27492			−1.74035			−0.870174	+	0.492744i	$0.835994\pi$
$14$		0			0
$15$	−3.27492			−0.845580
$16$		0			0
$17$	−2.00000	+	3.46410i	−0.485071	+	0.840168i	−0.999853	−	0.0171533i	$-0.994540\pi$
$17$	−2.00000	+	3.46410i	−0.485071	+	0.840168i	0.514782	+	0.857321i	$0.327873\pi$
$18$		0			0
$19$	−3.13746	−	5.43424i	−0.719782	−	1.24670i	−0.961086	−	0.276250i	$-0.910908\pi$
$19$	−3.13746	−	5.43424i	−0.719782	−	1.24670i	0.241303	−	0.970450i	$-0.422425\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−2.00000	−	3.46410i	−0.417029	−	0.722315i	0.578610	−	0.815604i	$-0.303595\pi$
$23$	−2.00000	−	3.46410i	−0.417029	−	0.722315i	−0.995639	+	0.0932891i	$0.970262\pi$
$24$		0			0
$25$	−2.86254	+	4.95807i	−0.572508	+	0.991613i
$26$		0			0
$27$	1.00000			0.192450
$28$		0			0
$29$	5.27492			0.979528			0.489764	−	0.871855i	$-0.337083\pi$
$29$	5.27492			0.979528			0.489764	+	0.871855i	$0.337083\pi$
$30$		0			0
$31$	−0.500000	+	0.866025i	−0.0898027	+	0.155543i	−0.907428	−	0.420208i	$-0.861957\pi$
$31$	−0.500000	+	0.866025i	−0.0898027	+	0.155543i	0.817625	+	0.575751i	$0.195290\pi$
$32$		0			0
$33$	−1.63746	−	2.83616i	−0.285045	−	0.493712i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	1.13746	+	1.97014i	0.186997	+	0.323888i	0.944248	−	0.329236i	$-0.106791\pi$
$37$	1.13746	+	1.97014i	0.186997	+	0.323888i	−0.757251	+	0.653124i	$0.773458\pi$
$38$		0			0
$39$	3.13746	−	5.43424i	0.502395	−	0.870174i
$40$		0			0
$41$	4.54983			0.710565			0.355282	−	0.934759i	$-0.384385\pi$
$41$	4.54983			0.710565			0.355282	+	0.934759i	$0.384385\pi$
$42$		0			0
$43$	0.274917			0.0419245			0.0209622	−	0.999780i	$-0.493327\pi$
$43$	0.274917			0.0419245			0.0209622	+	0.999780i	$0.493327\pi$
$44$		0			0
$45$	1.63746	−	2.83616i	0.244098	−	0.422790i
$46$		0			0
$47$	−3.00000	−	5.19615i	−0.437595	−	0.757937i	0.559908	−	0.828554i	$-0.310836\pi$
$47$	−3.00000	−	5.19615i	−0.437595	−	0.757937i	−0.997503	+	0.0706177i	$0.977503\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$	−2.00000	−	3.46410i	−0.280056	−	0.485071i
$52$		0			0
$53$	−4.63746	+	8.03231i	−0.637004	+	1.10332i	0.349083	+	0.937092i	$0.386493\pi$
$53$	−4.63746	+	8.03231i	−0.637004	+	1.10332i	−0.986087	+	0.166231i	$0.946840\pi$
$54$		0			0
$55$	−10.7251			−1.44617
$56$		0			0
$57$	6.27492			0.831133
$58$		0			0
$59$	−0.637459	+	1.10411i	−0.0829900	+	0.143743i	−0.904533	−	0.426404i	$-0.859780\pi$
$59$	−0.637459	+	1.10411i	−0.0829900	+	0.143743i	0.821543	+	0.570147i	$0.193114\pi$
$60$		0			0
$61$	5.00000	+	8.66025i	0.640184	+	1.10883i	0.985391	+	0.170305i	$0.0544754\pi$
$61$	5.00000	+	8.66025i	0.640184	+	1.10883i	−0.345207	+	0.938527i	$0.612191\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−10.2749	−	17.7967i	−1.27445	−	2.20741i
$66$		0			0
$67$	0.137459	−	0.238085i	0.0167932	−	0.0290867i	−0.857507	−	0.514473i	$-0.827988\pi$
$67$	0.137459	−	0.238085i	0.0167932	−	0.0290867i	0.874300	+	0.485386i	$0.161321\pi$
$68$		0			0
$69$	4.00000			0.481543
$70$		0			0
$71$	2.00000			0.237356			0.118678	−	0.992933i	$-0.462134\pi$
$71$	2.00000			0.237356			0.118678	+	0.992933i	$0.462134\pi$
$72$		0			0
$73$	−2.13746	+	3.70219i	−0.250171	+	0.433308i	−0.963573	−	0.267447i	$-0.913820\pi$
$73$	−2.13746	+	3.70219i	−0.250171	+	0.433308i	0.713402	+	0.700755i	$0.247153\pi$
$74$		0			0
$75$	−2.86254	−	4.95807i	−0.330538	−	0.572508i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−5.77492	−	10.0025i	−0.649729	−	1.12536i	−0.983188	−	0.182599i	$-0.941549\pi$
$79$	−5.77492	−	10.0025i	−0.649729	−	1.12536i	0.333459	−	0.942765i	$-0.391784\pi$
$80$		0			0
$81$	−0.500000	+	0.866025i	−0.0555556	+	0.0962250i
$82$		0			0
$83$	−7.27492			−0.798526			−0.399263	−	0.916836i	$-0.630734\pi$
$83$	−7.27492			−0.798526			−0.399263	+	0.916836i	$0.630734\pi$
$84$		0			0
$85$	−13.0997			−1.42086
$86$		0			0
$87$	−2.63746	+	4.56821i	−0.282765	+	0.489764i
$88$		0			0
$89$	−5.27492	−	9.13642i	−0.559140	−	0.968459i	−0.997568	−	0.0696929i	$-0.977798\pi$
$89$	−5.27492	−	9.13642i	−0.559140	−	0.968459i	0.438428	−	0.898766i	$-0.355535\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	−0.500000	−	0.866025i	−0.0518476	−	0.0898027i
$94$		0			0
$95$	10.2749	−	17.7967i	1.05418	−	1.82590i
$96$		0			0
$97$	−8.72508			−0.885898			−0.442949	−	0.896547i	$-0.646068\pi$
$97$	−8.72508			−0.885898			−0.442949	+	0.896547i	$0.646068\pi$
$98$		0			0
$99$	3.27492			0.329142
$100$		0			0
$101$	−3.00000	+	5.19615i	−0.298511	+	0.517036i	−0.975796	−	0.218685i	$-0.929823\pi$
$101$	−3.00000	+	5.19615i	−0.298511	+	0.517036i	0.677284	+	0.735721i	$0.263157\pi$
$102$		0			0
$103$	5.41238	+	9.37451i	0.533297	+	0.923698i	0.999244	+	0.0388850i	$0.0123806\pi$
$103$	5.41238	+	9.37451i	0.533297	+	0.923698i	−0.465946	+	0.884813i	$0.654286\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	7.91238	+	13.7046i	0.764918	+	1.32488i	0.940290	+	0.340375i	$0.110554\pi$
$107$	7.91238	+	13.7046i	0.764918	+	1.32488i	−0.175372	+	0.984502i	$0.556113\pi$
$108$		0			0
$109$	−8.41238	+	14.5707i	−0.805759	+	1.39562i	0.110018	+	0.993930i	$0.464909\pi$
$109$	−8.41238	+	14.5707i	−0.805759	+	1.39562i	−0.915777	+	0.401687i	$0.868424\pi$
$110$		0			0
$111$	−2.27492			−0.215926
$112$		0			0
$113$	−4.54983			−0.428012			−0.214006	−	0.976832i	$-0.568651\pi$
$113$	−4.54983			−0.428012			−0.214006	+	0.976832i	$0.568651\pi$
$114$		0			0
$115$	6.54983	−	11.3446i	0.610775	−	1.05789i
$116$		0			0
$117$	3.13746	+	5.43424i	0.290058	+	0.502395i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	0.137459	+	0.238085i	0.0124962	+	0.0216441i
$122$		0			0
$123$	−2.27492	+	3.94027i	−0.205122	+	0.355282i
$124$		0			0
$125$	−2.37459			−0.212389
$126$		0			0
$127$	−6.45017			−0.572360			−0.286180	−	0.958176i	$-0.592385\pi$
$127$	−6.45017			−0.572360			−0.286180	+	0.958176i	$0.592385\pi$
$128$		0			0
$129$	−0.137459	+	0.238085i	−0.0121026	+	0.0209622i
$130$		0			0
$131$	3.63746	+	6.30026i	0.317806	+	0.550457i	0.980030	−	0.198850i	$-0.0637205\pi$
$131$	3.63746	+	6.30026i	0.317806	+	0.550457i	−0.662224	+	0.749306i	$0.730387\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	1.63746	+	2.83616i	0.140930	+	0.244098i
$136$		0			0
$137$	−0.725083	+	1.25588i	−0.0619480	+	0.107297i	−0.895336	−	0.445391i	$-0.853065\pi$
$137$	−0.725083	+	1.25588i	−0.0619480	+	0.107297i	0.833388	+	0.552688i	$0.186398\pi$
$138$		0			0
$139$	−8.27492			−0.701869			−0.350935	−	0.936400i	$-0.614136\pi$
$139$	−8.27492			−0.701869			−0.350935	+	0.936400i	$0.614136\pi$
$140$		0			0
$141$	6.00000			0.505291
$142$		0			0
$143$	10.2749	−	17.7967i	0.859232	−	1.48823i
$144$		0			0
$145$	8.63746	+	14.9605i	0.717302	+	1.24240i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	7.27492	+	12.6005i	0.595984	+	1.03228i	0.993407	+	0.114640i	$0.0365713\pi$
$149$	7.27492	+	12.6005i	0.595984	+	1.03228i	−0.397423	+	0.917636i	$0.630095\pi$
$150$		0			0
$151$	−11.1873	+	19.3770i	−0.910409	+	1.57687i	−0.0969217	+	0.995292i	$0.530900\pi$
$151$	−11.1873	+	19.3770i	−0.910409	+	1.57687i	−0.813487	+	0.581583i	$0.802434\pi$
$152$		0			0
$153$	4.00000			0.323381
$154$		0			0
$155$	−3.27492			−0.263048
$156$		0			0
$157$	−0.274917	+	0.476171i	−0.0219408	+	0.0380026i	−0.876787	−	0.480878i	$-0.840318\pi$
$157$	−0.274917	+	0.476171i	−0.0219408	+	0.0380026i	0.854847	+	0.518881i	$0.173651\pi$
$158$		0			0
$159$	−4.63746	−	8.03231i	−0.367774	−	0.637004i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	6.00000	+	10.3923i	0.469956	+	0.813988i	0.999410	−	0.0343508i	$-0.0109363\pi$
$163$	6.00000	+	10.3923i	0.469956	+	0.813988i	−0.529454	+	0.848339i	$0.677603\pi$
$164$		0			0
$165$	5.36254	−	9.28819i	0.417473	−	0.723085i
$166$		0			0
$167$	6.00000			0.464294			0.232147	−	0.972681i	$-0.425425\pi$
$167$	6.00000			0.464294			0.232147	+	0.972681i	$0.425425\pi$
$168$		0			0
$169$	26.3746			2.02881
$170$		0			0
$171$	−3.13746	+	5.43424i	−0.239927	+	0.415567i
$172$		0			0
$173$	−11.2749	−	19.5287i	−0.857216	−	1.48474i	−0.874574	−	0.484892i	$-0.838859\pi$
$173$	−11.2749	−	19.5287i	−0.857216	−	1.48474i	0.0173577	−	0.999849i	$-0.494475\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−0.637459	−	1.10411i	−0.0479143	−	0.0829900i
$178$		0			0
$179$	8.27492	−	14.3326i	0.618496	−	1.07127i	−0.371264	−	0.928527i	$-0.621075\pi$
$179$	8.27492	−	14.3326i	0.618496	−	1.07127i	0.989760	−	0.142740i	$-0.0455912\pi$
$180$		0			0
$181$	18.8248			1.39923			0.699616	−	0.714519i	$-0.253354\pi$
$181$	18.8248			1.39923			0.699616	+	0.714519i	$0.253354\pi$
$182$		0			0
$183$	−10.0000			−0.739221
$184$		0			0
$185$	−3.72508	+	6.45203i	−0.273874	+	0.474363i
$186$		0			0
$187$	−6.54983	−	11.3446i	−0.478971	−	0.829603i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−2.27492	−	3.94027i	−0.164607	−	0.285108i	0.771909	−	0.635734i	$-0.219302\pi$
$191$	−2.27492	−	3.94027i	−0.164607	−	0.285108i	−0.936516	+	0.350626i	$0.885969\pi$
$192$		0			0
$193$	−0.225083	+	0.389855i	−0.0162018	+	0.0280624i	−0.874013	−	0.485903i	$-0.838491\pi$
$193$	−0.225083	+	0.389855i	−0.0162018	+	0.0280624i	0.857811	+	0.513966i	$0.171824\pi$
$194$		0			0
$195$	20.5498			1.47160
$196$		0			0
$197$	−1.45017			−0.103320			−0.0516600	−	0.998665i	$-0.516451\pi$
$197$	−1.45017			−0.103320			−0.0516600	+	0.998665i	$0.516451\pi$
$198$		0			0
$199$	−2.54983	+	4.41644i	−0.180753	+	0.313073i	−0.942137	−	0.335228i	$-0.891187\pi$
$199$	−2.54983	+	4.41644i	−0.180753	+	0.313073i	0.761384	+	0.648301i	$0.224520\pi$
$200$		0			0
$201$	0.137459	+	0.238085i	0.00969558	+	0.0167932i
$202$		0			0
$203$		0			0
$204$		0			0
$205$	7.45017	+	12.9041i	0.520342	+	0.901259i
$206$		0			0
$207$	−2.00000	+	3.46410i	−0.139010	+	0.240772i
$208$		0			0
$209$	20.5498			1.42146
$210$		0			0
$211$	−27.6495			−1.90347			−0.951735	−	0.306921i	$-0.900701\pi$
$211$	−27.6495			−1.90347			−0.951735	+	0.306921i	$0.900701\pi$
$212$		0			0
$213$	−1.00000	+	1.73205i	−0.0685189	+	0.118678i
$214$		0			0
$215$	0.450166	+	0.779710i	0.0307010	+	0.0531758i
$216$		0			0
$217$		0			0
$218$		0			0
$219$	−2.13746	−	3.70219i	−0.144436	−	0.250171i
$220$		0			0
$221$	12.5498	−	21.7370i	0.844193	−	1.46219i
$222$		0			0
$223$	1.27492			0.0853748			0.0426874	−	0.999088i	$-0.486408\pi$
$223$	1.27492			0.0853748			0.0426874	+	0.999088i	$0.486408\pi$
$224$		0			0
$225$	5.72508			0.381672
$226$		0			0
$227$	−5.63746	+	9.76436i	−0.374171	+	0.648084i	−0.990203	−	0.139638i	$-0.955406\pi$
$227$	−5.63746	+	9.76436i	−0.374171	+	0.648084i	0.616031	+	0.787722i	$0.288739\pi$
$228$		0			0
$229$	9.13746	+	15.8265i	0.603820	+	1.04585i	0.992237	+	0.124363i	$0.0396889\pi$
$229$	9.13746	+	15.8265i	0.603820	+	1.04585i	−0.388416	+	0.921484i	$0.626978\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	0.274917	+	0.476171i	0.0180104	+	0.0311950i	0.874890	−	0.484321i	$-0.160933\pi$
$233$	0.274917	+	0.476171i	0.0180104	+	0.0311950i	−0.856880	+	0.515516i	$0.827600\pi$
$234$		0			0
$235$	9.82475	−	17.0170i	0.640896	−	1.11006i
$236$		0			0
$237$	11.5498			0.750242
$238$		0			0
$239$	15.4502			0.999388			0.499694	−	0.866202i	$-0.333446\pi$
$239$	15.4502			0.999388			0.499694	+	0.866202i	$0.333446\pi$
$240$		0			0
$241$	4.91238	−	8.50848i	0.316434	−	0.548080i	−0.663307	−	0.748347i	$-0.730848\pi$
$241$	4.91238	−	8.50848i	0.316434	−	0.548080i	0.979741	+	0.200267i	$0.0641811\pi$
$242$		0			0
$243$	−0.500000	−	0.866025i	−0.0320750	−	0.0555556i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	19.6873	+	34.0994i	1.25267	+	2.16969i
$248$		0			0
$249$	3.63746	−	6.30026i	0.230515	−	0.399263i
$250$		0			0
$251$	18.3746			1.15979			0.579897	−	0.814690i	$-0.303093\pi$
$251$	18.3746			1.15979			0.579897	+	0.814690i	$0.303093\pi$
$252$		0			0
$253$	13.0997			0.823569
$254$		0			0
$255$	6.54983	−	11.3446i	0.410167	−	0.710429i
$256$		0			0
$257$	5.54983	+	9.61260i	0.346189	+	0.599617i	0.985569	−	0.169274i	$-0.0541422\pi$
$257$	5.54983	+	9.61260i	0.346189	+	0.599617i	−0.639380	+	0.768891i	$0.720809\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−2.63746	−	4.56821i	−0.163255	−	0.282765i
$262$		0			0
$263$	4.72508	−	8.18408i	0.291361	−	0.504652i	−0.682771	−	0.730633i	$-0.739225\pi$
$263$	4.72508	−	8.18408i	0.291361	−	0.504652i	0.974132	+	0.225980i	$0.0725586\pi$
$264$		0			0
$265$	−30.3746			−1.86590
$266$		0			0
$267$	10.5498			0.645639
$268$		0			0
$269$	10.3625	−	17.9484i	0.631815	−	1.09434i	−0.355365	−	0.934728i	$-0.615643\pi$
$269$	10.3625	−	17.9484i	0.631815	−	1.09434i	0.987180	−	0.159609i	$-0.0510232\pi$
$270$		0			0
$271$	0.637459	+	1.10411i	0.0387229	+	0.0670699i	0.884737	−	0.466090i	$-0.154338\pi$
$271$	0.637459	+	1.10411i	0.0387229	+	0.0670699i	−0.846014	+	0.533160i	$0.821004\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−9.37459	−	16.2373i	−0.565309	−	0.979144i
$276$		0			0
$277$	13.4124	−	23.2309i	0.805872	−	1.39581i	−0.109830	−	0.993950i	$-0.535030\pi$
$277$	13.4124	−	23.2309i	0.805872	−	1.39581i	0.915701	−	0.401860i	$-0.131636\pi$
$278$		0			0
$279$	1.00000			0.0598684
$280$		0			0
$281$	−26.5498			−1.58383			−0.791915	−	0.610631i	$-0.790916\pi$
$281$	−26.5498			−1.58383			−0.791915	+	0.610631i	$0.790916\pi$
$282$		0			0
$283$	−12.9622	+	22.4512i	−0.770523	+	1.33459i	0.166753	+	0.985999i	$0.446672\pi$
$283$	−12.9622	+	22.4512i	−0.770523	+	1.33459i	−0.937276	+	0.348587i	$0.886662\pi$
$284$		0			0
$285$	10.2749	+	17.7967i	0.608634	+	1.05418i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	0.500000	+	0.866025i	0.0294118	+	0.0509427i
$290$		0			0
$291$	4.36254	−	7.55614i	0.255737	−	0.442949i
$292$		0			0
$293$	27.8248			1.62554			0.812770	−	0.582585i	$-0.197959\pi$
$293$	27.8248			1.62554			0.812770	+	0.582585i	$0.197959\pi$
$294$		0			0
$295$	−4.17525			−0.243092
$296$		0			0
$297$	−1.63746	+	2.83616i	−0.0950150	+	0.164571i
$298$		0			0
$299$	12.5498	+	21.7370i	0.725776	+	1.25708i
$300$		0			0
$301$		0			0
$302$		0			0
$303$	−3.00000	−	5.19615i	−0.172345	−	0.298511i
$304$		0			0
$305$	−16.3746	+	28.3616i	−0.937606	+	1.62398i
$306$		0			0
$307$	−11.3746			−0.649182			−0.324591	−	0.945854i	$-0.605227\pi$
$307$	−11.3746			−0.649182			−0.324591	+	0.945854i	$0.605227\pi$
$308$		0			0
$309$	−10.8248			−0.615799
$310$		0			0
$311$	−2.27492	+	3.94027i	−0.128999	+	0.223432i	−0.923289	−	0.384106i	$-0.874510\pi$
$311$	−2.27492	+	3.94027i	−0.128999	+	0.223432i	0.794290	+	0.607539i	$0.207843\pi$
$312$		0			0
$313$	−9.77492	−	16.9307i	−0.552511	−	0.956977i	−0.998093	−	0.0617357i	$-0.980336\pi$
$313$	−9.77492	−	16.9307i	−0.552511	−	0.956977i	0.445582	−	0.895241i	$-0.352997\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	13.9124	+	24.0969i	0.781397	+	1.35342i	0.931128	+	0.364692i	$0.118826\pi$
$317$	13.9124	+	24.0969i	0.781397	+	1.35342i	−0.149731	+	0.988727i	$0.547841\pi$
$318$		0			0
$319$	−8.63746	+	14.9605i	−0.483605	+	0.837628i
$320$		0			0
$321$	−15.8248			−0.883252
$322$		0			0
$323$	25.0997			1.39658
$324$		0			0
$325$	17.9622	−	31.1115i	0.996364	−	1.72575i
$326$		0			0
$327$	−8.41238	−	14.5707i	−0.465205	−	0.805759i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−0.587624	−	1.01779i	−0.0322987	−	0.0559431i	0.849424	−	0.527711i	$-0.176949\pi$
$331$	−0.587624	−	1.01779i	−0.0322987	−	0.0559431i	−0.881723	+	0.471768i	$0.843616\pi$
$332$		0			0
$333$	1.13746	−	1.97014i	0.0623323	−	0.107963i
$334$		0			0
$335$	0.900331			0.0491903
$336$		0			0
$337$	−24.0997			−1.31279			−0.656396	−	0.754416i	$-0.727920\pi$
$337$	−24.0997			−1.31279			−0.656396	+	0.754416i	$0.727920\pi$
$338$		0			0
$339$	2.27492	−	3.94027i	0.123557	−	0.214006i
$340$		0			0
$341$	−1.63746	−	2.83616i	−0.0886734	−	0.153587i
$342$		0			0
$343$		0			0
$344$		0			0
$345$	6.54983	+	11.3446i	0.352631	+	0.610775i
$346$		0			0
$347$	−15.0997	+	26.1534i	−0.810593	+	1.40399i	0.101857	+	0.994799i	$0.467522\pi$
$347$	−15.0997	+	26.1534i	−0.810593	+	1.40399i	−0.912450	+	0.409189i	$0.865812\pi$
$348$		0			0
$349$	6.00000			0.321173			0.160586	−	0.987022i	$-0.448662\pi$
$349$	6.00000			0.321173			0.160586	+	0.987022i	$0.448662\pi$
$350$		0			0
$351$	−6.27492			−0.334930
$352$		0			0
$353$	−10.2749	+	17.7967i	−0.546879	+	0.947222i	0.451607	+	0.892217i	$0.350851\pi$
$353$	−10.2749	+	17.7967i	−0.546879	+	0.947222i	−0.998486	+	0.0550049i	$0.982483\pi$
$354$		0			0
$355$	3.27492	+	5.67232i	0.173815	+	0.301056i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	9.82475	+	17.0170i	0.518531	+	0.898121i	0.999768	+	0.0215311i	$0.00685409\pi$
$359$	9.82475	+	17.0170i	0.518531	+	0.898121i	−0.481238	+	0.876590i	$0.659813\pi$
$360$		0			0
$361$	−10.1873	+	17.6449i	−0.536173	+	0.928679i
$362$		0			0
$363$	−0.274917			−0.0144294
$364$		0			0
$365$	−14.0000			−0.732793
$366$		0			0
$367$	11.0498	−	19.1389i	0.576797	−	0.999041i	−0.419047	−	0.907964i	$-0.637636\pi$
$367$	11.0498	−	19.1389i	0.576797	−	0.999041i	0.995844	−	0.0910767i	$-0.0290308\pi$
$368$		0			0
$369$	−2.27492	−	3.94027i	−0.118427	−	0.205122i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	3.13746	+	5.43424i	0.162451	+	0.281374i	0.935747	−	0.352671i	$-0.114727\pi$
$373$	3.13746	+	5.43424i	0.162451	+	0.281374i	−0.773296	+	0.634045i	$0.781393\pi$
$374$		0			0
$375$	1.18729	−	2.05645i	0.0613115	−	0.106195i
$376$		0			0
$377$	−33.0997			−1.70472
$378$		0			0
$379$	13.1752			0.676767			0.338384	−	0.941008i	$-0.390120\pi$
$379$	13.1752			0.676767			0.338384	+	0.941008i	$0.390120\pi$
$380$		0			0
$381$	3.22508	−	5.58601i	0.165226	−	0.286180i
$382$		0			0
$383$	5.27492	+	9.13642i	0.269536	+	0.466849i	0.968742	−	0.248070i	$-0.0797965\pi$
$383$	5.27492	+	9.13642i	0.269536	+	0.466849i	−0.699206	+	0.714920i	$0.746463\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−0.137459	−	0.238085i	−0.00698741	−	0.0121026i
$388$		0			0
$389$	1.00000	−	1.73205i	0.0507020	−	0.0878185i	−0.839561	−	0.543266i	$-0.817187\pi$
$389$	1.00000	−	1.73205i	0.0507020	−	0.0878185i	0.890263	+	0.455448i	$0.150521\pi$
$390$		0			0
$391$	16.0000			0.809155
$392$		0			0
$393$	−7.27492			−0.366971
$394$		0			0
$395$	18.9124	−	32.7572i	0.951585	−	1.64819i
$396$		0			0
$397$	−1.68729	−	2.92248i	−0.0846828	−	0.146675i	0.820573	−	0.571541i	$-0.193654\pi$
$397$	−1.68729	−	2.92248i	−0.0846828	−	0.146675i	−0.905256	+	0.424867i	$0.860321\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−12.0000	−	20.7846i	−0.599251	−	1.03793i	−0.992932	−	0.118686i	$-0.962132\pi$
$401$	−12.0000	−	20.7846i	−0.599251	−	1.03793i	0.393680	−	0.919247i	$-0.371202\pi$
$402$		0			0
$403$	3.13746	−	5.43424i	0.156288	−	0.270699i
$404$		0			0
$405$	−3.27492			−0.162732
$406$		0			0
$407$	−7.45017			−0.369291
$408$		0			0
$409$	5.22508	−	9.05011i	0.258364	−	0.447499i	−0.707440	−	0.706773i	$-0.750150\pi$
$409$	5.22508	−	9.05011i	0.258364	−	0.447499i	0.965804	+	0.259274i	$0.0834834\pi$
$410$		0			0
$411$	−0.725083	−	1.25588i	−0.0357657	−	0.0619480i
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−11.9124	−	20.6328i	−0.584756	−	1.01283i
$416$		0			0
$417$	4.13746	−	7.16629i	0.202612	−	0.350935i
$418$		0			0
$419$	−28.5498			−1.39475			−0.697375	−	0.716706i	$-0.745649\pi$
$419$	−28.5498			−1.39475			−0.697375	+	0.716706i	$0.745649\pi$
$420$		0			0
$421$	8.82475			0.430092			0.215046	−	0.976604i	$-0.431010\pi$
$421$	8.82475			0.430092			0.215046	+	0.976604i	$0.431010\pi$
$422$		0			0
$423$	−3.00000	+	5.19615i	−0.145865	+	0.252646i
$424$		0			0
$425$	−11.4502	−	19.8323i	−0.555415	−	0.962006i
$426$		0			0
$427$		0			0
$428$		0			0
$429$	10.2749	+	17.7967i	0.496078	+	0.859232i
$430$		0			0
$431$	8.82475	−	15.2849i	0.425073	−	0.736249i	−0.571354	−	0.820704i	$-0.693582\pi$
$431$	8.82475	−	15.2849i	0.425073	−	0.736249i	0.996427	+	0.0844552i	$0.0269150\pi$
$432$		0			0
$433$	−3.17525			−0.152593			−0.0762963	−	0.997085i	$-0.524310\pi$
$433$	−3.17525			−0.152593			−0.0762963	+	0.997085i	$0.524310\pi$
$434$		0			0
$435$	−17.2749			−0.828269
$436$		0			0
$437$	−12.5498	+	21.7370i	−0.600340	+	1.03982i
$438$		0			0
$439$	8.63746	+	14.9605i	0.412243	+	0.714027i	0.995135	−	0.0985236i	$-0.0314120\pi$
$439$	8.63746	+	14.9605i	0.412243	+	0.714027i	−0.582891	+	0.812550i	$0.698079\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	3.18729	+	5.52055i	0.151433	+	0.262289i	0.931754	−	0.363089i	$-0.118278\pi$
$443$	3.18729	+	5.52055i	0.151433	+	0.262289i	−0.780322	+	0.625378i	$0.784945\pi$
$444$		0			0
$445$	17.2749	−	29.9210i	0.818910	−	1.41839i
$446$		0			0
$447$	−14.5498			−0.688184
$448$		0			0
$449$	20.5498			0.969807			0.484903	−	0.874568i	$-0.338855\pi$
$449$	20.5498			0.969807			0.484903	+	0.874568i	$0.338855\pi$
$450$		0			0
$451$	−7.45017	+	12.9041i	−0.350815	+	0.607629i
$452$		0			0
$453$	−11.1873	−	19.3770i	−0.525625	−	0.910409i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−18.3248	−	31.7394i	−0.857196	−	1.48471i	−0.874593	−	0.484858i	$-0.838871\pi$
$457$	−18.3248	−	31.7394i	−0.857196	−	1.48471i	0.0173972	−	0.999849i	$-0.494462\pi$
$458$		0			0
$459$	−2.00000	+	3.46410i	−0.0933520	+	0.161690i
$460$		0			0
$461$	3.64950			0.169974			0.0849872	−	0.996382i	$-0.472915\pi$
$461$	3.64950			0.169974			0.0849872	+	0.996382i	$0.472915\pi$
$462$		0			0
$463$	13.1752			0.612306			0.306153	−	0.951982i	$-0.400958\pi$
$463$	13.1752			0.612306			0.306153	+	0.951982i	$0.400958\pi$
$464$		0			0
$465$	1.63746	−	2.83616i	0.0759353	−	0.131524i
$466$		0			0
$467$	20.2749	+	35.1172i	0.938211	+	1.62503i	0.768805	+	0.639483i	$0.220852\pi$
$467$	20.2749	+	35.1172i	0.938211	+	1.62503i	0.169406	+	0.985546i	$0.445815\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−0.274917	−	0.476171i	−0.0126675	−	0.0219408i
$472$		0			0
$473$	−0.450166	+	0.779710i	−0.0206986	+	0.0358511i
$474$		0			0
$475$	35.9244			1.64833
$476$		0			0
$477$	9.27492			0.424669
$478$		0			0
$479$	−2.72508	+	4.71998i	−0.124512	+	0.215661i	−0.921542	−	0.388278i	$-0.873070\pi$
$479$	−2.72508	+	4.71998i	−0.124512	+	0.215661i	0.797030	+	0.603940i	$0.206403\pi$
$480$		0			0
$481$	−7.13746	−	12.3624i	−0.325440	−	0.563679i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−14.2870	−	24.7457i	−0.648738	−	1.12365i
$486$		0			0
$487$	−0.500000	+	0.866025i	−0.0226572	+	0.0392434i	−0.877132	−	0.480250i	$-0.840546\pi$
$487$	−0.500000	+	0.866025i	−0.0226572	+	0.0392434i	0.854475	+	0.519493i	$0.173879\pi$
$488$		0			0
$489$	−12.0000			−0.542659
$490$		0			0
$491$	36.9244			1.66638			0.833188	−	0.552990i	$-0.186513\pi$
$491$	36.9244			1.66638			0.833188	+	0.552990i	$0.186513\pi$
$492$		0			0
$493$	−10.5498	+	18.2728i	−0.475141	+	0.822968i
$494$		0			0
$495$	5.36254	+	9.28819i	0.241028	+	0.417473i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−16.1375	−	27.9509i	−0.722412	−	1.25125i	−0.960030	−	0.279896i	$-0.909700\pi$
$499$	−16.1375	−	27.9509i	−0.722412	−	1.25125i	0.237619	−	0.971359i	$-0.423633\pi$
$500$		0			0
$501$	−3.00000	+	5.19615i	−0.134030	+	0.232147i
$502$		0			0
$503$	37.6495			1.67871			0.839354	−	0.543585i	$-0.182933\pi$
$503$	37.6495			1.67871			0.839354	+	0.543585i	$0.182933\pi$
$504$		0			0
$505$	−19.6495			−0.874391
$506$		0			0
$507$	−13.1873	+	22.8411i	−0.585668	+	1.01441i
$508$		0			0
$509$	−5.63746	−	9.76436i	−0.249876	−	0.432798i	0.713615	−	0.700538i	$-0.247057\pi$
$509$	−5.63746	−	9.76436i	−0.249876	−	0.432798i	−0.963491	+	0.267740i	$0.913723\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	−3.13746	−	5.43424i	−0.138522	−	0.239927i
$514$		0			0
$515$	−17.7251	+	30.7007i	−0.781060	+	1.35284i
$516$		0			0
$517$	19.6495			0.864184
$518$		0			0
$519$	22.5498			0.989828
$520$		0			0
$521$	−7.27492	+	12.6005i	−0.318720	+	0.552039i	−0.980221	−	0.197905i	$-0.936586\pi$
$521$	−7.27492	+	12.6005i	−0.318720	+	0.552039i	0.661501	+	0.749944i	$0.269920\pi$
$522$		0			0
$523$	−8.86254	−	15.3504i	−0.387532	−	0.671225i	0.604585	−	0.796541i	$-0.293339\pi$
$523$	−8.86254	−	15.3504i	−0.387532	−	0.671225i	−0.992117	+	0.125316i	$0.960006\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−2.00000	−	3.46410i	−0.0871214	−	0.150899i
$528$		0			0
$529$	3.50000	−	6.06218i	0.152174	−	0.263573i
$530$		0			0
$531$	1.27492			0.0553267
$532$		0			0
$533$	−28.5498			−1.23663
$534$		0			0
$535$	−25.9124	+	44.8816i	−1.12029	+	1.94040i
$536$		0			0
$537$	8.27492	+	14.3326i	0.357089	+	0.618496i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	4.13746	+	7.16629i	0.177883	+	0.308103i	0.941155	−	0.337974i	$-0.109742\pi$
$541$	4.13746	+	7.16629i	0.177883	+	0.308103i	−0.763272	+	0.646077i	$0.776408\pi$
$542$		0			0
$543$	−9.41238	+	16.3027i	−0.403924	+	0.699616i
$544$		0			0
$545$	−55.0997			−2.36021
$546$		0			0
$547$	−28.0000			−1.19719			−0.598597	−	0.801050i	$-0.704275\pi$
$547$	−28.0000			−1.19719			−0.598597	+	0.801050i	$0.704275\pi$
$548$		0			0
$549$	5.00000	−	8.66025i	0.213395	−	0.369611i
$550$		0			0
$551$	−16.5498	−	28.6652i	−0.705047	−	1.22118i
$552$		0			0
$553$		0			0
$554$		0			0
$555$	−3.72508	−	6.45203i	−0.158121	−	0.273874i
$556$		0			0
$557$	14.9124	−	25.8290i	0.631858	−	1.09441i	−0.355314	−	0.934747i	$-0.615626\pi$
$557$	14.9124	−	25.8290i	0.631858	−	1.09441i	0.987172	−	0.159663i	$-0.0510406\pi$
$558$		0			0
$559$	−1.72508			−0.0729632
$560$		0			0
$561$	13.0997			0.553068
$562$		0			0
$563$	7.63746	−	13.2285i	0.321881	−	0.557513i	−0.658996	−	0.752147i	$-0.729018\pi$
$563$	7.63746	−	13.2285i	0.321881	−	0.557513i	0.980876	+	0.194633i	$0.0623517\pi$
$564$		0			0
$565$	−7.45017	−	12.9041i	−0.313431	−	0.542878i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−5.72508	−	9.91613i	−0.240008	−	0.415706i	0.720708	−	0.693238i	$-0.243817\pi$
$569$	−5.72508	−	9.91613i	−0.240008	−	0.415706i	−0.960716	+	0.277532i	$0.910483\pi$
$570$		0			0
$571$	−4.13746	+	7.16629i	−0.173147	+	0.299900i	−0.939519	−	0.342498i	$-0.888727\pi$
$571$	−4.13746	+	7.16629i	−0.173147	+	0.299900i	0.766371	+	0.642398i	$0.222060\pi$
$572$		0			0
$573$	4.54983			0.190072
$574$		0			0
$575$	22.9003			0.955010
$576$		0			0
$577$	−12.5000	+	21.6506i	−0.520382	+	0.901328i	0.479337	+	0.877631i	$0.340877\pi$
$577$	−12.5000	+	21.6506i	−0.520382	+	0.901328i	−0.999719	+	0.0236970i	$0.992456\pi$
$578$		0			0
$579$	−0.225083	−	0.389855i	−0.00935412	−	0.0162018i
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−15.1873	−	26.3052i	−0.628993	−	1.08945i
$584$		0			0
$585$	−10.2749	+	17.7967i	−0.424816	+	0.735802i
$586$		0			0
$587$	−9.27492			−0.382817			−0.191408	−	0.981510i	$-0.561305\pi$
$587$	−9.27492			−0.382817			−0.191408	+	0.981510i	$0.561305\pi$
$588$		0			0
$589$	6.27492			0.258553
$590$		0			0
$591$	0.725083	−	1.25588i	0.0298259	−	0.0516600i
$592$		0			0
$593$	−0.274917	−	0.476171i	−0.0112895	−	0.0195540i	0.860325	−	0.509745i	$-0.170260\pi$
$593$	−0.274917	−	0.476171i	−0.0112895	−	0.0195540i	−0.871615	+	0.490191i	$0.836927\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−2.54983	−	4.41644i	−0.104358	−	0.180753i
$598$		0			0
$599$	−11.2749	+	19.5287i	−0.460681	+	0.797922i	−0.998995	−	0.0448219i	$-0.985728\pi$
$599$	−11.2749	+	19.5287i	−0.460681	+	0.797922i	0.538314	+	0.842744i	$0.319061\pi$
$600$		0			0
$601$	−4.09967			−0.167229			−0.0836145	−	0.996498i	$-0.526646\pi$
$601$	−4.09967			−0.167229			−0.0836145	+	0.996498i	$0.526646\pi$
$602$		0			0
$603$	−0.274917			−0.0111955
$604$		0			0
$605$	−0.450166	+	0.779710i	−0.0183018	+	0.0316997i
$606$		0			0
$607$	−3.50000	−	6.06218i	−0.142061	−	0.246056i	0.786212	−	0.617957i	$-0.212039\pi$
$607$	−3.50000	−	6.06218i	−0.142061	−	0.246056i	−0.928272	+	0.371901i	$0.878706\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	18.8248	+	32.6054i	0.761568	+	1.31907i
$612$		0			0
$613$	4.27492	−	7.40437i	0.172662	−	0.299060i	−0.766688	−	0.642020i	$-0.778096\pi$
$613$	4.27492	−	7.40437i	0.172662	−	0.299060i	0.939350	+	0.342961i	$0.111430\pi$
$614$		0			0
$615$	−14.9003			−0.600839
$616$		0			0
$617$	−6.00000			−0.241551			−0.120775	−	0.992680i	$-0.538538\pi$
$617$	−6.00000			−0.241551			−0.120775	+	0.992680i	$0.538538\pi$
$618$		0			0
$619$	14.4124	−	24.9630i	0.579282	−	1.00335i	−0.416280	−	0.909237i	$-0.636666\pi$
$619$	14.4124	−	24.9630i	0.579282	−	1.00335i	0.995562	−	0.0941097i	$-0.0300004\pi$
$620$		0			0
$621$	−2.00000	−	3.46410i	−0.0802572	−	0.139010i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	10.4244	+	18.0556i	0.416977	+	0.722225i
$626$		0			0
$627$	−10.2749	+	17.7967i	−0.410341	+	0.710731i
$628$		0			0
$629$	−9.09967			−0.362828
$630$		0			0
$631$	19.8248			0.789211			0.394605	−	0.918851i	$-0.370881\pi$
$631$	19.8248			0.789211			0.394605	+	0.918851i	$0.370881\pi$
$632$		0			0
$633$	13.8248	−	23.9452i	0.549485	−	0.951735i
$634$		0			0
$635$	−10.5619	−	18.2937i	−0.419135	−	0.725964i
$636$		0			0
$637$		0			0
$638$		0			0
$639$	−1.00000	−	1.73205i	−0.0395594	−	0.0685189i
$640$		0			0
$641$	1.82475	−	3.16056i	0.0720734	−	0.124835i	−0.827736	−	0.561117i	$-0.810372\pi$
$641$	1.82475	−	3.16056i	0.0720734	−	0.124835i	0.899810	+	0.436282i	$0.143705\pi$
$642$		0			0
$643$	−5.37459			−0.211953			−0.105976	−	0.994369i	$-0.533797\pi$
$643$	−5.37459			−0.211953			−0.105976	+	0.994369i	$0.533797\pi$
$644$		0			0
$645$	−0.900331			−0.0354505
$646$		0			0
$647$	−17.0000	+	29.4449i	−0.668339	+	1.15760i	0.310029	+	0.950727i	$0.399661\pi$
$647$	−17.0000	+	29.4449i	−0.668339	+	1.15760i	−0.978368	+	0.206870i	$0.933672\pi$
$648$		0			0
$649$	−2.08762	−	3.61587i	−0.0819464	−	0.141935i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	3.46221	+	5.99672i	0.135487	+	0.234670i	0.925783	−	0.378055i	$-0.123407\pi$
$653$	3.46221	+	5.99672i	0.135487	+	0.234670i	−0.790297	+	0.612725i	$0.790074\pi$
$654$		0			0
$655$	−11.9124	+	20.6328i	−0.465455	+	0.806192i
$656$		0			0
$657$	4.27492			0.166780
$658$		0			0
$659$	−42.1993			−1.64385			−0.821926	−	0.569594i	$-0.807101\pi$
$659$	−42.1993			−1.64385			−0.821926	+	0.569594i	$0.807101\pi$
$660$		0			0
$661$	3.58762	−	6.21395i	0.139542	−	0.241695i	−0.787781	−	0.615955i	$-0.788770\pi$
$661$	3.58762	−	6.21395i	0.139542	−	0.241695i	0.927323	+	0.374261i	$0.122104\pi$
$662$		0			0
$663$	12.5498	+	21.7370i	0.487395	+	0.844193i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−10.5498	−	18.2728i	−0.408491	−	0.707528i
$668$		0			0
$669$	−0.637459	+	1.10411i	−0.0246456	+	0.0426874i
$670$		0			0
$671$	−32.7492			−1.26427
$672$		0			0
$673$	41.5498			1.60163			0.800814	−	0.598913i	$-0.204400\pi$
$673$	41.5498			1.60163			0.800814	+	0.598913i	$0.204400\pi$
$674$		0			0
$675$	−2.86254	+	4.95807i	−0.110179	+	0.190836i
$676$		0			0
$677$	−6.63746	−	11.4964i	−0.255098	−	0.441843i	0.709824	−	0.704379i	$-0.248774\pi$
$677$	−6.63746	−	11.4964i	−0.255098	−	0.441843i	−0.964922	+	0.262536i	$0.915441\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	−5.63746	−	9.76436i	−0.216028	−	0.374171i
$682$		0			0
$683$	−6.08762	+	10.5441i	−0.232936	+	0.403458i	−0.958671	−	0.284517i	$-0.908167\pi$
$683$	−6.08762	+	10.5441i	−0.232936	+	0.403458i	0.725735	+	0.687975i	$0.241500\pi$
$684$		0			0
$685$	−4.74917			−0.181457
$686$		0			0
$687$	−18.2749			−0.697232
$688$		0			0
$689$	29.0997	−	50.4021i	1.10861	−	1.92017i
$690$		0			0
$691$	−5.41238	−	9.37451i	−0.205896	−	0.356623i	0.744522	−	0.667598i	$-0.232678\pi$
$691$	−5.41238	−	9.37451i	−0.205896	−	0.356623i	−0.950418	+	0.310975i	$0.899344\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−13.5498	−	23.4690i	−0.513975	−	0.890230i
$696$		0			0
$697$	−9.09967	+	15.7611i	−0.344675	+	0.596994i
$698$		0			0
$699$	−0.549834			−0.0207966
$700$		0			0
$701$	−12.9244			−0.488149			−0.244074	−	0.969757i	$-0.578484\pi$
$701$	−12.9244			−0.488149			−0.244074	+	0.969757i	$0.578484\pi$
$702$		0			0
$703$	7.13746	−	12.3624i	0.269194	−	0.466258i
$704$		0			0
$705$	9.82475	+	17.0170i	0.370022	+	0.640896i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	14.0997	+	24.4213i	0.529524	+	0.917163i	0.999407	+	0.0344340i	$0.0109628\pi$
$709$	14.0997	+	24.4213i	0.529524	+	0.917163i	−0.469883	+	0.882729i	$0.655704\pi$
$710$		0			0
$711$	−5.77492	+	10.0025i	−0.216576	+	0.375121i
$712$		0			0
$713$	4.00000			0.149801
$714$		0			0
$715$	67.2990			2.51684
$716$		0			0
$717$	−7.72508	+	13.3802i	−0.288499	+	0.499694i
$718$		0			0
$719$	14.0997	+	24.4213i	0.525829	+	0.910762i	0.999547	+	0.0300860i	$0.00957813\pi$
$719$	14.0997	+	24.4213i	0.525829	+	0.910762i	−0.473718	+	0.880676i	$0.657089\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	4.91238	+	8.50848i	0.182693	+	0.316434i
$724$		0			0
$725$	−15.0997	+	26.1534i	−0.560788	+	0.971313i
$726$		0			0
$727$	−31.5498			−1.17012			−0.585059	−	0.810991i	$-0.698929\pi$
$727$	−31.5498			−1.17012			−0.585059	+	0.810991i	$0.698929\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$	−0.549834	+	0.952341i	−0.0203364	+	0.0352236i
$732$		0			0
$733$	2.96221	+	5.13070i	0.109412	+	0.189507i	0.915532	−	0.402245i	$-0.131770\pi$
$733$	2.96221	+	5.13070i	0.109412	+	0.189507i	−0.806120	+	0.591752i	$0.798437\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	0.450166	+	0.779710i	0.0165821	+	0.0287210i
$738$		0			0
$739$	−0.687293	+	1.19043i	−0.0252825	+	0.0437905i	−0.878390	−	0.477945i	$-0.841382\pi$
$739$	−0.687293	+	1.19043i	−0.0252825	+	0.0437905i	0.853107	+	0.521735i	$0.174715\pi$
$740$		0			0
$741$	−39.3746			−1.44646
$742$		0			0
$743$	−44.1993			−1.62152			−0.810758	−	0.585381i	$-0.800945\pi$
$743$	−44.1993			−1.62152			−0.810758	+	0.585381i	$0.800945\pi$
$744$		0			0
$745$	−23.8248	+	41.2657i	−0.872871	+	1.51186i
$746$		0			0
$747$	3.63746	+	6.30026i	0.133088	+	0.230515i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	5.22508	+	9.05011i	0.190666	+	0.330243i	0.945471	−	0.325706i	$-0.105602\pi$
$751$	5.22508	+	9.05011i	0.190666	+	0.330243i	−0.754805	+	0.655949i	$0.772269\pi$
$752$		0			0
$753$	−9.18729	+	15.9129i	−0.334804	+	0.579897i
$754$		0			0
$755$	−73.2749			−2.66675
$756$		0			0
$757$	−22.0000			−0.799604			−0.399802	−	0.916602i	$-0.630921\pi$
$757$	−22.0000			−0.799604			−0.399802	+	0.916602i	$0.630921\pi$
$758$		0			0
$759$	−6.54983	+	11.3446i	−0.237744	+	0.411785i
$760$		0			0
$761$	−12.5498	−	21.7370i	−0.454931	−	0.787964i	0.543753	−	0.839245i	$-0.317003\pi$
$761$	−12.5498	−	21.7370i	−0.454931	−	0.787964i	−0.998684	+	0.0512814i	$0.983669\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	6.54983	+	11.3446i	0.236810	+	0.410167i
$766$		0			0
$767$	4.00000	−	6.92820i	0.144432	−	0.250163i
$768$		0			0
$769$	32.6495			1.17737			0.588686	−	0.808362i	$-0.299646\pi$
$769$	32.6495			1.17737			0.588686	+	0.808362i	$0.299646\pi$
$770$		0			0
$771$	−11.0997			−0.399745
$772$		0			0
$773$	1.54983	−	2.68439i	0.0557437	−	0.0965509i	−0.836807	−	0.547498i	$-0.815580\pi$
$773$	1.54983	−	2.68439i	0.0557437	−	0.0965509i	0.892551	+	0.450947i	$0.148914\pi$
$774$		0			0
$775$	−2.86254	−	4.95807i	−0.102826	−	0.178099i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−14.2749	−	24.7249i	−0.511452	−	0.885861i
$780$		0			0
$781$	−3.27492	+	5.67232i	−0.117186	+	0.202972i
$782$		0			0
$783$	5.27492			0.188510
$784$		0			0
$785$	−1.80066			−0.0642684
$786$		0			0
$787$	1.27492	−	2.20822i	0.0454459	−	0.0787146i	−0.842408	−	0.538841i	$-0.818862\pi$
$787$	1.27492	−	2.20822i	0.0454459	−	0.0787146i	0.887854	+	0.460126i	$0.152196\pi$
$788$		0			0
$789$	4.72508	+	8.18408i	0.168217	+	0.291361i
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−31.3746	−	54.3424i	−1.11414	−	1.92975i
$794$

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:	\( N \)	\(=\)	\( 1176 = 2^{3} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1176.q (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.500000 + 0.866025i) q^{3} +(1.63746 + 2.83616i) q^{5} +(-0.500000 - 0.866025i) q^{9} +O(q^{10})\)	\(q+(-0.500000 + 0.866025i) q^{3} +(1.63746 + 2.83616i) q^{5} +(-0.500000 - 0.866025i) q^{9} +(-1.63746 + 2.83616i) q^{11} -6.27492 q^{13} -3.27492 q^{15} +(-2.00000 + 3.46410i) q^{17} +(-3.13746 - 5.43424i) q^{19} +(-2.00000 - 3.46410i) q^{23} +(-2.86254 + 4.95807i) q^{25} +1.00000 q^{27} +5.27492 q^{29} +(-0.500000 + 0.866025i) q^{31} +(-1.63746 - 2.83616i) q^{33} +(1.13746 + 1.97014i) q^{37} +(3.13746 - 5.43424i) q^{39} +4.54983 q^{41} +0.274917 q^{43} +(1.63746 - 2.83616i) q^{45} +(-3.00000 - 5.19615i) q^{47} +(-2.00000 - 3.46410i) q^{51} +(-4.63746 + 8.03231i) q^{53} -10.7251 q^{55} +6.27492 q^{57} +(-0.637459 + 1.10411i) q^{59} +(5.00000 + 8.66025i) q^{61} +(-10.2749 - 17.7967i) q^{65} +(0.137459 - 0.238085i) q^{67} +4.00000 q^{69} +2.00000 q^{71} +(-2.13746 + 3.70219i) q^{73} +(-2.86254 - 4.95807i) q^{75} +(-5.77492 - 10.0025i) q^{79} +(-0.500000 + 0.866025i) q^{81} -7.27492 q^{83} -13.0997 q^{85} +(-2.63746 + 4.56821i) q^{87} +(-5.27492 - 9.13642i) q^{89} +(-0.500000 - 0.866025i) q^{93} +(10.2749 - 17.7967i) q^{95} -8.72508 q^{97} +3.27492 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{3} - q^{5} - 2 q^{9}+O(q^{10}) \) 4 * q - 2 * q^3 - q^5 - 2 * q^9	\( 4 q - 2 q^{3} - q^{5} - 2 q^{9} + q^{11} - 10 q^{13} + 2 q^{15} - 8 q^{17} - 5 q^{19} - 8 q^{23} - 19 q^{25} + 4 q^{27} + 6 q^{29} - 2 q^{31} + q^{33} - 3 q^{37} + 5 q^{39} - 12 q^{41} - 14 q^{43} - q^{45} - 12 q^{47} - 8 q^{51} - 11 q^{53} - 58 q^{55} + 10 q^{57} + 5 q^{59} + 20 q^{61} - 26 q^{65} - 7 q^{67} + 16 q^{69} + 8 q^{71} - q^{73} - 19 q^{75} - 8 q^{79} - 2 q^{81} - 14 q^{83} + 8 q^{85} - 3 q^{87} - 6 q^{89} - 2 q^{93} + 26 q^{95} - 50 q^{97} - 2 q^{99}+O(q^{100}) \) 4 * q - 2 * q^3 - q^5 - 2 * q^9 + q^11 - 10 * q^13 + 2 * q^15 - 8 * q^17 - 5 * q^19 - 8 * q^23 - 19 * q^25 + 4 * q^27 + 6 * q^29 - 2 * q^31 + q^33 - 3 * q^37 + 5 * q^39 - 12 * q^41 - 14 * q^43 - q^45 - 12 * q^47 - 8 * q^51 - 11 * q^53 - 58 * q^55 + 10 * q^57 + 5 * q^59 + 20 * q^61 - 26 * q^65 - 7 * q^67 + 16 * q^69 + 8 * q^71 - q^73 - 19 * q^75 - 8 * q^79 - 2 * q^81 - 14 * q^83 + 8 * q^85 - 3 * q^87 - 6 * q^89 - 2 * q^93 + 26 * q^95 - 50 * q^97 - 2 * q^99

Introduction

L-functions

Modular forms

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