LMFDB - Embedded newform 3024.2.cz.a.2719.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3024,2,Mod(1279,3024)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3024, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 0, 2, 5]))

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

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

chi := DirichletCharacter("3024.1279");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3024 = 2^{4} \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3024.cz (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:	$24.1467615712$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1008)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			2719.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	3024.2719
Dual form			3024.2.cz.a.1279.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.500000 + 2.59808i) q^{7} +O(q^{10})$	$q+(-0.500000 + 2.59808i) q^{7} +(3.00000 - 1.73205i) q^{11} +(-1.50000 + 0.866025i) q^{13} +(-1.50000 - 0.866025i) q^{17} +(-2.50000 - 4.33013i) q^{19} +(-3.00000 - 1.73205i) q^{23} +(-2.50000 - 4.33013i) q^{25} +(1.50000 - 2.59808i) q^{29} +1.00000 q^{31} +(-3.50000 - 6.06218i) q^{37} +(1.50000 - 0.866025i) q^{41} +(1.50000 + 0.866025i) q^{43} +9.00000 q^{47} +(-6.50000 - 2.59808i) q^{49} +(-4.50000 + 7.79423i) q^{53} +15.0000 q^{59} -1.73205i q^{61} -15.5885i q^{67} +10.3923i q^{71} +(1.50000 + 0.866025i) q^{73} +(3.00000 + 8.66025i) q^{77} -1.73205i q^{79} +(4.50000 - 7.79423i) q^{83} +(-1.50000 + 0.866025i) q^{89} +(-1.50000 - 4.33013i) q^{91} +(-1.50000 - 0.866025i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{7}+O(q^{10}) $ 2 * q - q^7	$ 2 q - q^{7} + 6 q^{11} - 3 q^{13} - 3 q^{17} - 5 q^{19} - 6 q^{23} - 5 q^{25} + 3 q^{29} + 2 q^{31} - 7 q^{37} + 3 q^{41} + 3 q^{43} + 18 q^{47} - 13 q^{49} - 9 q^{53} + 30 q^{59} + 3 q^{73} + 6 q^{77} + 9 q^{83} - 3 q^{89} - 3 q^{91} - 3 q^{97}+O(q^{100}) $ 2 * q - q^7 + 6 * q^11 - 3 * q^13 - 3 * q^17 - 5 * q^19 - 6 * q^23 - 5 * q^25 + 3 * q^29 + 2 * q^31 - 7 * q^37 + 3 * q^41 + 3 * q^43 + 18 * q^47 - 13 * q^49 - 9 * q^53 + 30 * q^59 + 3 * q^73 + 6 * q^77 + 9 * q^83 - 3 * q^89 - 3 * q^91 - 3 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3024\mathbb{Z}\right)^\times$.

$n$	$757$	$785$	$1135$	$2593$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$	$-1$	$e\left(\frac{1}{6}\right)$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$5$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$6$		0			0
$7$	−0.500000	+	2.59808i	−0.188982	+	0.981981i
$7$	−0.500000	+	2.59808i	−0.188982	+	0.981981i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	3.00000	−	1.73205i	0.904534	−	0.522233i	0.0258656	−	0.999665i	$-0.491766\pi$
$11$	3.00000	−	1.73205i	0.904534	−	0.522233i	0.878668	+	0.477432i	$0.158432\pi$
$12$		0			0
$13$	−1.50000	+	0.866025i	−0.416025	+	0.240192i	−0.693375	−	0.720577i	$-0.743877\pi$
$13$	−1.50000	+	0.866025i	−0.416025	+	0.240192i	0.277350	+	0.960769i	$0.410544\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−1.50000	−	0.866025i	−0.363803	−	0.210042i	0.306944	−	0.951727i	$-0.400693\pi$
$17$	−1.50000	−	0.866025i	−0.363803	−	0.210042i	−0.670748	+	0.741685i	$0.734027\pi$
$18$		0			0
$19$	−2.50000	−	4.33013i	−0.573539	−	0.993399i	−0.996199	−	0.0871106i	$-0.972237\pi$
$19$	−2.50000	−	4.33013i	−0.573539	−	0.993399i	0.422659	−	0.906289i	$-0.361097\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−3.00000	−	1.73205i	−0.625543	−	0.361158i	0.153481	−	0.988152i	$-0.450952\pi$
$23$	−3.00000	−	1.73205i	−0.625543	−	0.361158i	−0.779024	+	0.626994i	$0.784285\pi$
$24$		0			0
$25$	−2.50000	−	4.33013i	−0.500000	−	0.866025i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	1.50000	−	2.59808i	0.278543	−	0.482451i	−0.692480	−	0.721437i	$-0.743482\pi$
$29$	1.50000	−	2.59808i	0.278543	−	0.482451i	0.971023	+	0.238987i	$0.0768152\pi$
$30$		0			0
$31$	1.00000			0.179605			0.0898027	−	0.995960i	$-0.471376\pi$
$31$	1.00000			0.179605			0.0898027	+	0.995960i	$0.471376\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−3.50000	−	6.06218i	−0.575396	−	0.996616i	−0.995998	−	0.0893706i	$-0.971514\pi$
$37$	−3.50000	−	6.06218i	−0.575396	−	0.996616i	0.420602	−	0.907245i	$-0.361819\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	1.50000	−	0.866025i	0.234261	−	0.135250i	−0.378275	−	0.925693i	$-0.623483\pi$
$41$	1.50000	−	0.866025i	0.234261	−	0.135250i	0.612536	+	0.790443i	$0.290149\pi$
$42$		0			0
$43$	1.50000	+	0.866025i	0.228748	+	0.132068i	0.609994	−	0.792406i	$-0.291172\pi$
$43$	1.50000	+	0.866025i	0.228748	+	0.132068i	−0.381246	+	0.924473i	$0.624505\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	9.00000			1.31278			0.656392	−	0.754420i	$-0.272082\pi$
$47$	9.00000			1.31278			0.656392	+	0.754420i	$0.272082\pi$
$48$		0			0
$49$	−6.50000	−	2.59808i	−0.928571	−	0.371154i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−4.50000	+	7.79423i	−0.618123	+	1.07062i	0.371706	+	0.928351i	$0.378773\pi$
$53$	−4.50000	+	7.79423i	−0.618123	+	1.07062i	−0.989828	+	0.142269i	$0.954560\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	15.0000			1.95283			0.976417	−	0.215894i	$-0.0692665\pi$
$59$	15.0000			1.95283			0.976417	+	0.215894i	$0.0692665\pi$
$60$		0			0
$61$		−	1.73205i		−	0.221766i	−0.993833	−	0.110883i	$-0.964632\pi$
$61$		−	1.73205i		−	0.221766i	0.993833	−	0.110883i	$-0.0353679\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$		−	15.5885i		−	1.90443i	−0.305424	−	0.952217i	$-0.598798\pi$
$67$		−	15.5885i		−	1.90443i	0.305424	−	0.952217i	$-0.401202\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			10.3923i			1.23334i	0.787222	+	0.616670i	$0.211519\pi$
$71$			10.3923i			1.23334i	−0.787222	+	0.616670i	$0.788481\pi$
$72$		0			0
$73$	1.50000	+	0.866025i	0.175562	+	0.101361i	0.585206	−	0.810885i	$-0.301014\pi$
$73$	1.50000	+	0.866025i	0.175562	+	0.101361i	−0.409644	+	0.912245i	$0.634347\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	3.00000	+	8.66025i	0.341882	+	0.986928i
$78$		0			0
$79$		−	1.73205i		−	0.194871i	−0.995242	−	0.0974355i	$-0.968936\pi$
$79$		−	1.73205i		−	0.194871i	0.995242	−	0.0974355i	$-0.0310640\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	4.50000	−	7.79423i	0.493939	−	0.855528i	−0.506036	−	0.862512i	$-0.668890\pi$
$83$	4.50000	−	7.79423i	0.493939	−	0.855528i	0.999976	+	0.00698436i	$0.00222321\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−1.50000	+	0.866025i	−0.159000	+	0.0917985i	−0.577389	−	0.816469i	$-0.695928\pi$
$89$	−1.50000	+	0.866025i	−0.159000	+	0.0917985i	0.418389	+	0.908268i	$0.362595\pi$
$90$		0			0
$91$	−1.50000	−	4.33013i	−0.157243	−	0.453921i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−1.50000	−	0.866025i	−0.152302	−	0.0879316i	0.421912	−	0.906637i	$-0.361359\pi$
$97$	−1.50000	−	0.866025i	−0.152302	−	0.0879316i	−0.574214	+	0.818705i	$0.694692\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	12.0000	−	6.92820i	1.19404	−	0.689382i	0.234823	−	0.972038i	$-0.424549\pi$
$101$	12.0000	−	6.92820i	1.19404	−	0.689382i	0.959221	+	0.282656i	$0.0912155\pi$
$102$		0			0
$103$	4.00000	−	6.92820i	0.394132	−	0.682656i	−0.598858	−	0.800855i	$-0.704379\pi$
$103$	4.00000	−	6.92820i	0.394132	−	0.682656i	0.992990	+	0.118199i	$0.0377120\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	1.50000	−	0.866025i	0.145010	−	0.0837218i	−0.425739	−	0.904846i	$-0.639986\pi$
$107$	1.50000	−	0.866025i	0.145010	−	0.0837218i	0.570750	+	0.821124i	$0.306653\pi$
$108$		0			0
$109$	−5.50000	+	9.52628i	−0.526804	+	0.912452i	0.472708	+	0.881219i	$0.343277\pi$
$109$	−5.50000	+	9.52628i	−0.526804	+	0.912452i	−0.999512	+	0.0312328i	$0.990057\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−4.50000	−	7.79423i	−0.423324	−	0.733219i	0.572938	−	0.819599i	$-0.305804\pi$
$113$	−4.50000	−	7.79423i	−0.423324	−	0.733219i	−0.996262	+	0.0863794i	$0.972470\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$	3.00000	−	3.46410i	0.275010	−	0.317554i
$120$		0			0
$121$	0.500000	−	0.866025i	0.0454545	−	0.0787296i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$		−	10.3923i		−	0.922168i	−0.887357	−	0.461084i	$-0.847461\pi$
$127$		−	10.3923i		−	0.922168i	0.887357	−	0.461084i	$-0.152539\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	6.00000	−	10.3923i	0.524222	−	0.907980i	−0.475380	−	0.879781i	$-0.657689\pi$
$131$	6.00000	−	10.3923i	0.524222	−	0.907980i	0.999602	−	0.0281993i	$-0.00897729\pi$
$132$		0			0
$133$	12.5000	−	4.33013i	1.08389	−	0.375470i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−9.00000	−	15.5885i	−0.768922	−	1.33181i	−0.938148	−	0.346235i	$-0.887460\pi$
$137$	−9.00000	−	15.5885i	−0.768922	−	1.33181i	0.169226	−	0.985577i	$-0.445873\pi$
$138$		0			0
$139$	2.50000	+	4.33013i	0.212047	+	0.367277i	0.952355	−	0.304991i	$-0.0986536\pi$
$139$	2.50000	+	4.33013i	0.212047	+	0.367277i	−0.740308	+	0.672268i	$0.765320\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−3.00000	+	5.19615i	−0.250873	+	0.434524i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−3.00000	+	5.19615i	−0.245770	+	0.425685i	−0.962348	−	0.271821i	$-0.912374\pi$
$149$	−3.00000	+	5.19615i	−0.245770	+	0.425685i	0.716578	+	0.697507i	$0.245707\pi$
$150$		0			0
$151$	15.0000	−	8.66025i	1.22068	−	0.704761i	0.255619	−	0.966778i	$-0.417721\pi$
$151$	15.0000	−	8.66025i	1.22068	−	0.704761i	0.965064	+	0.262016i	$0.0843873\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$		−	12.1244i		−	0.967629i	−0.875171	−	0.483814i	$-0.839251\pi$
$157$		−	12.1244i		−	0.967629i	0.875171	−	0.483814i	$-0.160749\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	6.00000	−	6.92820i	0.472866	−	0.546019i
$162$		0			0
$163$	−13.5000	+	7.79423i	−1.05740	+	0.610491i	−0.924712	−	0.380667i	$-0.875695\pi$
$163$	−13.5000	+	7.79423i	−1.05740	+	0.610491i	−0.132689	+	0.991158i	$0.542361\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−10.5000	−	18.1865i	−0.812514	−	1.40732i	−0.911099	−	0.412188i	$-0.864765\pi$
$167$	−10.5000	−	18.1865i	−0.812514	−	1.40732i	0.0985846	−	0.995129i	$-0.468568\pi$
$168$		0			0
$169$	−5.00000	+	8.66025i	−0.384615	+	0.666173i
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	19.0526i		−	1.44854i	−0.689517	−	0.724270i	$-0.742177\pi$
$173$		−	19.0526i		−	1.44854i	0.689517	−	0.724270i	$-0.257823\pi$
$174$		0			0
$175$	12.5000	−	4.33013i	0.944911	−	0.327327i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	10.5000	+	6.06218i	0.784807	+	0.453108i	0.838131	−	0.545469i	$-0.183648\pi$
$179$	10.5000	+	6.06218i	0.784807	+	0.453108i	−0.0533243	+	0.998577i	$0.516982\pi$
$180$		0			0
$181$			13.8564i			1.02994i	0.857209	+	0.514969i	$0.172197\pi$
$181$			13.8564i			1.02994i	−0.857209	+	0.514969i	$0.827803\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−6.00000			−0.438763
$188$		0			0
$189$		0			0
$190$		0			0
$191$			1.73205i			0.125327i	0.998035	+	0.0626634i	$0.0199595\pi$
$191$			1.73205i			0.125327i	−0.998035	+	0.0626634i	$0.980041\pi$
$192$		0			0
$193$	−17.0000			−1.22369			−0.611843	−	0.790979i	$-0.709572\pi$
$193$	−17.0000			−1.22369			−0.611843	+	0.790979i	$0.709572\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−6.00000			−0.427482			−0.213741	−	0.976890i	$-0.568565\pi$
$197$	−6.00000			−0.427482			−0.213741	+	0.976890i	$0.568565\pi$
$198$		0			0
$199$	0.500000	−	0.866025i	0.0354441	−	0.0613909i	−0.847759	−	0.530381i	$-0.822049\pi$
$199$	0.500000	−	0.866025i	0.0354441	−	0.0613909i	0.883203	+	0.468990i	$0.155382\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	6.00000	+	5.19615i	0.421117	+	0.364698i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−15.0000	−	8.66025i	−1.03757	−	0.599042i
$210$		0			0
$211$	10.5000	−	6.06218i	0.722850	−	0.417338i	−0.0929509	−	0.995671i	$-0.529630\pi$
$211$	10.5000	−	6.06218i	0.722850	−	0.417338i	0.815801	+	0.578333i	$0.196297\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−0.500000	+	2.59808i	−0.0339422	+	0.176369i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	3.00000			0.201802
$222$		0			0
$223$	−2.50000	+	4.33013i	−0.167412	+	0.289967i	−0.937509	−	0.347960i	$-0.886874\pi$
$223$	−2.50000	+	4.33013i	−0.167412	+	0.289967i	0.770097	+	0.637927i	$0.220208\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	6.00000	+	10.3923i	0.398234	+	0.689761i	0.993508	−	0.113761i	$-0.0362899\pi$
$227$	6.00000	+	10.3923i	0.398234	+	0.689761i	−0.595274	+	0.803523i	$0.702957\pi$
$228$		0			0
$229$	12.0000	+	6.92820i	0.792982	+	0.457829i	0.841011	−	0.541017i	$-0.181961\pi$
$229$	12.0000	+	6.92820i	0.792982	+	0.457829i	−0.0480291	+	0.998846i	$0.515294\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−10.5000	−	18.1865i	−0.687878	−	1.19144i	−0.972523	−	0.232806i	$-0.925209\pi$
$233$	−10.5000	−	18.1865i	−0.687878	−	1.19144i	0.284645	−	0.958633i	$-0.408124\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	19.5000	−	11.2583i	1.26135	−	0.728241i	0.288014	−	0.957626i	$-0.407005\pi$
$239$	19.5000	−	11.2583i	1.26135	−	0.728241i	0.973336	+	0.229385i	$0.0736716\pi$
$240$		0			0
$241$	−18.0000	+	10.3923i	−1.15948	+	0.669427i	−0.951180	−	0.308637i	$-0.900127\pi$
$241$	−18.0000	+	10.3923i	−1.15948	+	0.669427i	−0.208302	+	0.978065i	$0.566794\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	7.50000	+	4.33013i	0.477214	+	0.275519i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$		0			0
$253$	−12.0000			−0.754434
$254$		0			0
$255$		0			0
$256$		0			0
$257$	24.0000	+	13.8564i	1.49708	+	0.864339i	0.999994	−	0.00336324i	$-0.00107055\pi$
$257$	24.0000	+	13.8564i	1.49708	+	0.864339i	0.497085	+	0.867702i	$0.334404\pi$
$258$		0			0
$259$	17.5000	−	6.06218i	1.08740	−	0.376685i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−27.0000	+	15.5885i	−1.66489	+	0.961225i	−0.694564	+	0.719431i	$0.744403\pi$
$263$	−27.0000	+	15.5885i	−1.66489	+	0.961225i	−0.970328	+	0.241794i	$0.922264\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−19.5000	−	11.2583i	−1.18894	−	0.686433i	−0.230871	−	0.972984i	$-0.574158\pi$
$269$	−19.5000	−	11.2583i	−1.18894	−	0.686433i	−0.958065	+	0.286552i	$0.907491\pi$
$270$		0			0
$271$	−0.500000	−	0.866025i	−0.0303728	−	0.0526073i	0.850439	−	0.526073i	$-0.176336\pi$
$271$	−0.500000	−	0.866025i	−0.0303728	−	0.0526073i	−0.880812	+	0.473466i	$0.843003\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−15.0000	−	8.66025i	−0.904534	−	0.522233i
$276$		0			0
$277$	7.00000	+	12.1244i	0.420589	+	0.728482i	0.995997	−	0.0893846i	$-0.0284900\pi$
$277$	7.00000	+	12.1244i	0.420589	+	0.728482i	−0.575408	+	0.817867i	$0.695157\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	7.50000	−	12.9904i	0.447412	−	0.774941i	−0.550804	−	0.834634i	$-0.685679\pi$
$281$	7.50000	−	12.9904i	0.447412	−	0.774941i	0.998217	+	0.0596933i	$0.0190123\pi$
$282$		0			0
$283$	−31.0000			−1.84276			−0.921379	−	0.388664i	$-0.872937\pi$
$283$	−31.0000			−1.84276			−0.921379	+	0.388664i	$0.872937\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	1.50000	+	4.33013i	0.0885422	+	0.255599i
$288$		0			0
$289$	−7.00000	−	12.1244i	−0.411765	−	0.713197i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−22.5000	+	12.9904i	−1.31446	+	0.758906i	−0.982832	−	0.184503i	$-0.940933\pi$
$293$	−22.5000	+	12.9904i	−1.31446	+	0.758906i	−0.331632	+	0.943409i	$0.607599\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	6.00000			0.346989
$300$		0			0
$301$	−3.00000	+	3.46410i	−0.172917	+	0.199667i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	8.00000			0.456584			0.228292	−	0.973593i	$-0.426686\pi$
$307$	8.00000			0.456584			0.228292	+	0.973593i	$0.426686\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	3.00000			0.170114			0.0850572	−	0.996376i	$-0.472893\pi$
$311$	3.00000			0.170114			0.0850572	+	0.996376i	$0.472893\pi$
$312$		0			0
$313$		−	29.4449i		−	1.66432i	−0.554534	−	0.832161i	$-0.687103\pi$
$313$		−	29.4449i		−	1.66432i	0.554534	−	0.832161i	$-0.312897\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	21.0000			1.17948			0.589739	−	0.807594i	$-0.299231\pi$
$317$	21.0000			1.17948			0.589739	+	0.807594i	$0.299231\pi$
$318$		0			0
$319$		−	10.3923i		−	0.581857i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			8.66025i			0.481869i
$324$		0			0
$325$	7.50000	+	4.33013i	0.416025	+	0.240192i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−4.50000	+	23.3827i	−0.248093	+	1.28913i
$330$		0			0
$331$			5.19615i			0.285606i	0.989751	+	0.142803i	$0.0456116\pi$
$331$			5.19615i			0.285606i	−0.989751	+	0.142803i	$0.954388\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−5.50000	−	9.52628i	−0.299604	−	0.518930i	0.676441	−	0.736497i	$-0.263521\pi$
$337$	−5.50000	−	9.52628i	−0.299604	−	0.518930i	−0.976045	+	0.217567i	$0.930188\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	3.00000	−	1.73205i	0.162459	−	0.0937958i
$342$		0			0
$343$	10.0000	−	15.5885i	0.539949	−	0.841698i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			22.5167i			1.20876i	0.796697	+	0.604379i	$0.206579\pi$
$347$			22.5167i			1.20876i	−0.796697	+	0.604379i	$0.793421\pi$
$348$		0			0
$349$	22.5000	+	12.9904i	1.20440	+	0.695359i	0.961530	−	0.274700i	$-0.0885786\pi$
$349$	22.5000	+	12.9904i	1.20440	+	0.695359i	0.242867	+	0.970059i	$0.421912\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	18.0000	−	10.3923i	0.958043	−	0.553127i	0.0624731	−	0.998047i	$-0.480101\pi$
$353$	18.0000	−	10.3923i	0.958043	−	0.553127i	0.895570	+	0.444920i	$0.146768\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−10.5000	+	6.06218i	−0.554169	+	0.319950i	−0.750802	−	0.660528i	$-0.770333\pi$
$359$	−10.5000	+	6.06218i	−0.554169	+	0.319950i	0.196633	+	0.980477i	$0.436999\pi$
$360$		0			0
$361$	−3.00000	+	5.19615i	−0.157895	+	0.273482i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−4.00000	−	6.92820i	−0.208798	−	0.361649i	0.742538	−	0.669804i	$-0.233622\pi$
$367$	−4.00000	−	6.92820i	−0.208798	−	0.361649i	−0.951336	+	0.308155i	$0.900289\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−18.0000	−	15.5885i	−0.934513	−	0.809312i
$372$		0			0
$373$	5.00000	−	8.66025i	0.258890	−	0.448411i	−0.707055	−	0.707159i	$-0.749977\pi$
$373$	5.00000	−	8.66025i	0.258890	−	0.448411i	0.965945	+	0.258748i	$0.0833099\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			5.19615i			0.267615i
$378$		0			0
$379$			24.2487i			1.24557i	0.782392	+	0.622786i	$0.213999\pi$
$379$			24.2487i			1.24557i	−0.782392	+	0.622786i	$0.786001\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	12.0000	−	20.7846i	0.613171	−	1.06204i	−0.377531	−	0.925997i	$-0.623227\pi$
$383$	12.0000	−	20.7846i	0.613171	−	1.06204i	0.990702	−	0.136047i	$-0.0434398\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	9.00000	+	15.5885i	0.456318	+	0.790366i	0.998763	−	0.0497253i	$-0.0158346\pi$
$389$	9.00000	+	15.5885i	0.456318	+	0.790366i	−0.542445	+	0.840091i	$0.682501\pi$
$390$		0			0
$391$	3.00000	+	5.19615i	0.151717	+	0.262781i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	13.5000	−	7.79423i	0.677546	−	0.391181i	−0.121384	−	0.992606i	$-0.538733\pi$
$397$	13.5000	−	7.79423i	0.677546	−	0.391181i	0.798930	+	0.601424i	$0.205400\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	15.0000	−	25.9808i	0.749064	−	1.29742i	−0.199207	−	0.979957i	$-0.563837\pi$
$401$	15.0000	−	25.9808i	0.749064	−	1.29742i	0.948272	−	0.317460i	$-0.102830\pi$
$402$		0			0
$403$	−1.50000	+	0.866025i	−0.0747203	+	0.0431398i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−21.0000	−	12.1244i	−1.04093	−	0.600982i
$408$		0			0
$409$			29.4449i			1.45595i	0.685601	+	0.727977i	$0.259539\pi$
$409$			29.4449i			1.45595i	−0.685601	+	0.727977i	$0.740461\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−7.50000	+	38.9711i	−0.369051	+	1.91764i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	7.50000	+	12.9904i	0.366399	+	0.634622i	0.989000	−	0.147918i	$-0.0472572\pi$
$419$	7.50000	+	12.9904i	0.366399	+	0.634622i	−0.622601	+	0.782540i	$0.713924\pi$
$420$		0			0
$421$	2.50000	−	4.33013i	0.121843	−	0.211037i	−0.798652	−	0.601793i	$-0.794453\pi$
$421$	2.50000	−	4.33013i	0.121843	−	0.211037i	0.920494	+	0.390756i	$0.127786\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			8.66025i			0.420084i
$426$		0			0
$427$	4.50000	+	0.866025i	0.217770	+	0.0419099i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	22.5000	+	12.9904i	1.08379	+	0.625725i	0.931915	−	0.362676i	$-0.118137\pi$
$431$	22.5000	+	12.9904i	1.08379	+	0.625725i	0.151871	+	0.988400i	$0.451470\pi$
$432$		0			0
$433$		−	13.8564i		−	0.665896i	−0.942945	−	0.332948i	$-0.891957\pi$
$433$		−	13.8564i		−	0.665896i	0.942945	−	0.332948i	$-0.108043\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			17.3205i			0.828552i
$438$		0			0
$439$	−25.0000			−1.19318			−0.596592	−	0.802544i	$-0.703479\pi$
$439$	−25.0000			−1.19318			−0.596592	+	0.802544i	$0.703479\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	25.9808i		−	1.23438i	−0.786813	−	0.617192i	$-0.788270\pi$
$443$		−	25.9808i		−	1.23438i	0.786813	−	0.617192i	$-0.211730\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−18.0000			−0.849473			−0.424736	−	0.905317i	$-0.639633\pi$
$449$	−18.0000			−0.849473			−0.424736	+	0.905317i	$0.639633\pi$
$450$		0			0
$451$	3.00000	−	5.19615i	0.141264	−	0.244677i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−1.00000			−0.0467780			−0.0233890	−	0.999726i	$-0.507446\pi$
$457$	−1.00000			−0.0467780			−0.0233890	+	0.999726i	$0.507446\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	19.5000	+	11.2583i	0.908206	+	0.524353i	0.879853	−	0.475245i	$-0.157641\pi$
$461$	19.5000	+	11.2583i	0.908206	+	0.524353i	0.0283522	+	0.999598i	$0.490974\pi$
$462$		0			0
$463$	−31.5000	+	18.1865i	−1.46393	+	0.845200i	−0.999190	−	0.0402476i	$-0.987185\pi$
$463$	−31.5000	+	18.1865i	−1.46393	+	0.845200i	−0.464739	+	0.885448i	$0.653852\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−7.50000	−	12.9904i	−0.347059	−	0.601123i	0.638667	−	0.769483i	$-0.279486\pi$
$467$	−7.50000	−	12.9904i	−0.347059	−	0.601123i	−0.985726	+	0.168360i	$0.946153\pi$
$468$		0			0
$469$	40.5000	+	7.79423i	1.87012	+	0.359904i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	6.00000			0.275880
$474$		0			0
$475$	−12.5000	+	21.6506i	−0.573539	+	0.993399i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$479$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$480$		0			0
$481$	10.5000	+	6.06218i	0.478759	+	0.276412i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	13.5000	+	7.79423i	0.611743	+	0.353190i	0.773647	−	0.633616i	$-0.218430\pi$
$487$	13.5000	+	7.79423i	0.611743	+	0.353190i	−0.161904	+	0.986807i	$0.551764\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	7.50000	−	4.33013i	0.338470	−	0.195416i	−0.321125	−	0.947037i	$-0.604061\pi$
$491$	7.50000	−	4.33013i	0.338470	−	0.195416i	0.659595	+	0.751621i	$0.270728\pi$
$492$		0			0
$493$	−4.50000	+	2.59808i	−0.202670	+	0.117011i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−27.0000	−	5.19615i	−1.21112	−	0.233079i
$498$		0			0
$499$	3.00000	+	1.73205i	0.134298	+	0.0775372i	0.565644	−	0.824650i	$-0.308628\pi$
$499$	3.00000	+	1.73205i	0.134298	+	0.0775372i	−0.431346	+	0.902187i	$0.641961\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	24.0000			1.07011			0.535054	−	0.844818i	$-0.320291\pi$
$503$	24.0000			1.07011			0.535054	+	0.844818i	$0.320291\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$	30.0000	+	17.3205i	1.32973	+	0.767718i	0.985257	−	0.171080i	$-0.0547255\pi$
$509$	30.0000	+	17.3205i	1.32973	+	0.767718i	0.344469	+	0.938798i	$0.388059\pi$
$510$		0			0
$511$	−3.00000	+	3.46410i	−0.132712	+	0.153243i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$	27.0000	−	15.5885i	1.18746	−	0.685580i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−19.5000	−	11.2583i	−0.854311	−	0.493236i	0.00779240	−	0.999970i	$-0.497520\pi$
$521$	−19.5000	−	11.2583i	−0.854311	−	0.493236i	−0.862103	+	0.506733i	$0.830853\pi$
$522$		0			0
$523$	−6.50000	−	11.2583i	−0.284225	−	0.492292i	0.688196	−	0.725525i	$-0.258403\pi$
$523$	−6.50000	−	11.2583i	−0.284225	−	0.492292i	−0.972421	+	0.233233i	$0.925070\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−1.50000	−	0.866025i	−0.0653410	−	0.0377247i
$528$		0			0
$529$	−5.50000	−	9.52628i	−0.239130	−	0.414186i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−1.50000	+	2.59808i	−0.0649722	+	0.112535i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−24.0000	+	3.46410i	−1.03375	+	0.149209i
$540$		0			0
$541$	−21.5000	−	37.2391i	−0.924357	−	1.60103i	−0.792592	−	0.609753i	$-0.791269\pi$
$541$	−21.5000	−	37.2391i	−0.924357	−	1.60103i	−0.131765	−	0.991281i	$-0.542065\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	7.50000	+	4.33013i	0.320677	+	0.185143i	0.651694	−	0.758482i	$-0.274059\pi$
$547$	7.50000	+	4.33013i	0.320677	+	0.185143i	−0.331017	+	0.943625i	$0.607392\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−15.0000			−0.639021
$552$		0			0
$553$	4.50000	+	0.866025i	0.191359	+	0.0368271i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−16.5000	+	28.5788i	−0.699127	+	1.21092i	0.269642	+	0.962961i	$0.413095\pi$
$557$	−16.5000	+	28.5788i	−0.699127	+	1.21092i	−0.968769	+	0.247964i	$0.920239\pi$
$558$		0			0
$559$	−3.00000			−0.126886
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−21.0000			−0.885044			−0.442522	−	0.896758i	$-0.645916\pi$
$563$	−21.0000			−0.885044			−0.442522	+	0.896758i	$0.645916\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−3.00000			−0.125767			−0.0628833	−	0.998021i	$-0.520030\pi$
$569$	−3.00000			−0.125767			−0.0628833	+	0.998021i	$0.520030\pi$
$570$		0			0
$571$			39.8372i			1.66713i	0.552419	+	0.833567i	$0.313705\pi$
$571$			39.8372i			1.66713i	−0.552419	+	0.833567i	$0.686295\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			17.3205i			0.722315i
$576$		0			0
$577$	−22.5000	−	12.9904i	−0.936687	−	0.540797i	−0.0477669	−	0.998859i	$-0.515210\pi$
$577$	−22.5000	−	12.9904i	−0.936687	−	0.540797i	−0.888920	+	0.458062i	$0.848544\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	18.0000	+	15.5885i	0.746766	+	0.646718i
$582$		0			0
$583$			31.1769i			1.29122i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−13.5000	+	23.3827i	−0.557205	+	0.965107i	0.440524	+	0.897741i	$0.354793\pi$
$587$	−13.5000	+	23.3827i	−0.557205	+	0.965107i	−0.997728	+	0.0673658i	$0.978541\pi$
$588$		0			0
$589$	−2.50000	−	4.33013i	−0.103011	−	0.178420i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−7.50000	+	4.33013i	−0.307988	+	0.177817i	−0.646026	−	0.763316i	$-0.723570\pi$
$593$	−7.50000	+	4.33013i	−0.307988	+	0.177817i	0.338038	+	0.941133i	$0.390237\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$			36.3731i			1.48616i	0.669201	+	0.743082i	$0.266637\pi$
$599$			36.3731i			1.48616i	−0.669201	+	0.743082i	$0.733363\pi$
$600$		0			0
$601$	−31.5000	−	18.1865i	−1.28491	−	0.741844i	−0.307170	−	0.951655i	$-0.599382\pi$
$601$	−31.5000	−	18.1865i	−1.28491	−	0.741844i	−0.977742	+	0.209811i	$0.932715\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−16.0000	+	27.7128i	−0.649420	+	1.12483i	0.333842	+	0.942629i	$0.391655\pi$
$607$	−16.0000	+	27.7128i	−0.649420	+	1.12483i	−0.983262	+	0.182199i	$0.941678\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−13.5000	+	7.79423i	−0.546152	+	0.315321i
$612$		0			0
$613$	−17.5000	+	30.3109i	−0.706818	+	1.22425i	0.259213	+	0.965820i	$0.416537\pi$
$613$	−17.5000	+	30.3109i	−0.706818	+	1.22425i	−0.966031	+	0.258425i	$0.916796\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	13.5000	+	23.3827i	0.543490	+	0.941351i	0.998700	+	0.0509678i	$0.0162306\pi$
$617$	13.5000	+	23.3827i	0.543490	+	0.941351i	−0.455211	+	0.890384i	$0.650436\pi$
$618$		0			0
$619$	−14.0000	−	24.2487i	−0.562708	−	0.974638i	−0.997259	−	0.0739910i	$-0.976426\pi$
$619$	−14.0000	−	24.2487i	−0.562708	−	0.974638i	0.434551	−	0.900647i	$-0.356907\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−1.50000	−	4.33013i	−0.0600962	−	0.173483i
$624$		0			0
$625$	−12.5000	+	21.6506i	−0.500000	+	0.866025i
$626$		0			0
$627$		0			0
$628$		0			0
$629$			12.1244i			0.483430i
$630$		0			0
$631$			31.1769i			1.24113i	0.784154	+	0.620567i	$0.213097\pi$
$631$			31.1769i			1.24113i	−0.784154	+	0.620567i	$0.786903\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	12.0000	−	1.73205i	0.475457	−	0.0686264i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	3.00000	+	5.19615i	0.118493	+	0.205236i	0.919171	−	0.393860i	$-0.128860\pi$
$641$	3.00000	+	5.19615i	0.118493	+	0.205236i	−0.800678	+	0.599095i	$0.795527\pi$
$642$		0			0
$643$	6.50000	+	11.2583i	0.256335	+	0.443985i	0.965257	−	0.261301i	$-0.0841516\pi$
$643$	6.50000	+	11.2583i	0.256335	+	0.443985i	−0.708922	+	0.705287i	$0.750818\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	7.50000	−	12.9904i	0.294855	−	0.510705i	−0.680096	−	0.733123i	$-0.738062\pi$
$647$	7.50000	−	12.9904i	0.294855	−	0.510705i	0.974951	+	0.222419i	$0.0713952\pi$
$648$		0			0
$649$	45.0000	−	25.9808i	1.76640	−	1.01983i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	15.0000	−	25.9808i	0.586995	−	1.01671i	−0.407628	−	0.913148i	$-0.633644\pi$
$653$	15.0000	−	25.9808i	0.586995	−	1.01671i	0.994623	−	0.103558i	$-0.0330227\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−19.5000	−	11.2583i	−0.759612	−	0.438562i	0.0695443	−	0.997579i	$-0.477845\pi$
$659$	−19.5000	−	11.2583i	−0.759612	−	0.438562i	−0.829156	+	0.559017i	$0.811179\pi$
$660$		0			0
$661$			8.66025i			0.336845i	0.985715	+	0.168422i	$0.0538673\pi$
$661$			8.66025i			0.336845i	−0.985715	+	0.168422i	$0.946133\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−9.00000	+	5.19615i	−0.348481	+	0.201196i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−3.00000	−	5.19615i	−0.115814	−	0.200595i
$672$		0			0
$673$	−3.50000	+	6.06218i	−0.134915	+	0.233680i	−0.925565	−	0.378589i	$-0.876409\pi$
$673$	−3.50000	+	6.06218i	−0.134915	+	0.233680i	0.790650	+	0.612268i	$0.209743\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			1.73205i			0.0665681i	0.999446	+	0.0332841i	$0.0105966\pi$
$677$			1.73205i			0.0665681i	−0.999446	+	0.0332841i	$0.989403\pi$
$678$		0			0
$679$	3.00000	−	3.46410i	0.115129	−	0.132940i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−37.5000	−	21.6506i	−1.43490	−	0.828439i	−0.437409	−	0.899263i	$-0.644104\pi$
$683$	−37.5000	−	21.6506i	−1.43490	−	0.828439i	−0.997489	+	0.0708242i	$0.977437\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	15.5885i		−	0.593873i
$690$		0			0
$691$	−5.00000			−0.190209			−0.0951045	−	0.995467i	$-0.530319\pi$
$691$	−5.00000			−0.190209			−0.0951045	+	0.995467i	$0.530319\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−3.00000			−0.113633
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−18.0000			−0.679851			−0.339925	−	0.940452i	$-0.610402\pi$
$701$	−18.0000			−0.679851			−0.339925	+	0.940452i	$0.610402\pi$
$702$		0			0
$703$	−17.5000	+	30.3109i	−0.660025	+	1.14320i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	12.0000	+	34.6410i	0.451306	+	1.30281i
$708$		0			0
$709$	−37.0000			−1.38956			−0.694782	−	0.719220i	$-0.744499\pi$
$709$	−37.0000			−1.38956			−0.694782	+	0.719220i	$0.744499\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−3.00000	−	1.73205i	−0.112351	−	0.0648658i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−13.5000	−	23.3827i	−0.503465	−	0.872027i	−0.999992	−	0.00400572i	$-0.998725\pi$
$719$	−13.5000	−	23.3827i	−0.503465	−	0.872027i	0.496527	−	0.868021i	$-0.334608\pi$
$720$		0			0
$721$	16.0000	+	13.8564i	0.595871	+	0.516040i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−15.0000			−0.557086
$726$		0			0
$727$	17.5000	−	30.3109i	0.649039	−	1.12417i	−0.334314	−	0.942462i	$-0.608504\pi$
$727$	17.5000	−	30.3109i	0.649039	−	1.12417i	0.983353	−	0.181707i	$-0.0581622\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−1.50000	−	2.59808i	−0.0554795	−	0.0960933i
$732$		0			0
$733$	18.0000	+	10.3923i	0.664845	+	0.383849i	0.794121	−	0.607760i	$-0.207932\pi$
$733$	18.0000	+	10.3923i	0.664845	+	0.383849i	−0.129275	+	0.991609i	$0.541265\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−27.0000	−	46.7654i	−0.994558	−	1.72262i
$738$		0			0
$739$	13.5000	+	7.79423i	0.496606	+	0.286715i	0.727311	−	0.686308i	$-0.240770\pi$
$739$	13.5000	+	7.79423i	0.496606	+	0.286715i	−0.230705	+	0.973024i	$0.574103\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−10.5000	+	6.06218i	−0.385208	+	0.222400i	−0.680082	−	0.733136i	$-0.738056\pi$
$743$	−10.5000	+	6.06218i	−0.385208	+	0.222400i	0.294874	+	0.955536i	$0.404722\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$	1.50000	+	4.33013i	0.0548088	+	0.158219i
$750$		0			0
$751$	−15.0000	−	8.66025i	−0.547358	−	0.316017i	0.200698	−	0.979653i	$-0.435679\pi$
$751$	−15.0000	−	8.66025i	−0.547358	−	0.316017i	−0.748056	+	0.663636i	$0.769012\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	2.00000			0.0726912			0.0363456	−	0.999339i	$-0.488428\pi$
$757$	2.00000			0.0726912			0.0363456	+	0.999339i	$0.488428\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	42.0000	+	24.2487i	1.52250	+	0.879015i	0.999646	+	0.0265919i	$0.00846546\pi$
$761$	42.0000	+	24.2487i	1.52250	+	0.879015i	0.522852	+	0.852423i	$0.324868\pi$
$762$		0			0
$763$	−22.0000	−	19.0526i	−0.796453	−	0.689749i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−22.5000	+	12.9904i	−0.812428	+	0.469055i
$768$		0			0
$769$	4.50000	−	2.59808i	0.162274	−	0.0936890i	−0.416664	−	0.909061i	$-0.636801\pi$
$769$	4.50000	−	2.59808i	0.162274	−	0.0936890i	0.578938	+	0.815372i	$0.303467\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	28.5000	+	16.4545i	1.02507	+	0.591827i	0.915570	−	0.402160i	$-0.131740\pi$
$773$	28.5000	+	16.4545i	1.02507	+	0.591827i	0.109504	+	0.993986i	$0.465074\pi$
$774$		0			0
$775$	−2.50000	−	4.33013i	−0.0898027	−	0.155543i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−7.50000	−	4.33013i	−0.268715	−	0.155143i
$780$		0			0
$781$	18.0000	+	31.1769i	0.644091	+	1.11560i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	13.0000			0.463400			0.231700	−	0.972787i	$-0.425571\pi$
$787$	13.0000			0.463400			0.231700	+	0.972787i	$0.425571\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	22.5000	−	7.79423i	0.800008	−	0.277131i
$792$		0			0
$793$	1.50000	+	2.59808i	0.0532666	+	0.0922604i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−28.5000	+	16.4545i	−1.00952	+	0.582848i	−0.911052	−	0.412292i	$-0.864728\pi$
$797$	−28.5000	+	16.4545i	−1.00952	+	0.582848i	−0.0984702	+	0.995140i	$0.531395\pi$
$798$		0			0
$799$	−13.5000	−	7.79423i	−0.477596	−	0.275740i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	6.00000			0.211735
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−4.50000	+	7.79423i	−0.158212	+	0.274030i	−0.934224	−	0.356687i	$-0.883906\pi$
$809$	−4.50000	+	7.79423i	−0.158212	+	0.274030i	0.776012	+	0.630718i	$0.217239\pi$
$810$		0			0
$811$	52.0000			1.82597			0.912983	−	0.407997i	$-0.133772\pi$
$811$	52.0000			1.82597			0.912983	+	0.407997i	$0.133772\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		−	8.66025i		−	0.302984i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	33.0000			1.15171			0.575854	−	0.817553i	$-0.304670\pi$
$821$	33.0000			1.15171			0.575854	+	0.817553i	$0.304670\pi$
$822$		0			0
$823$			12.1244i			0.422628i	0.977418	+	0.211314i	$0.0677743\pi$
$823$			12.1244i			0.422628i	−0.977418	+	0.211314i	$0.932226\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			17.3205i			0.602293i	0.953578	+	0.301147i	$0.0973693\pi$
$827$			17.3205i			0.602293i	−0.953578	+	0.301147i	$0.902631\pi$
$828$		0			0
$829$	13.5000	+	7.79423i	0.468874	+	0.270705i	0.715768	−	0.698338i	$-0.246077\pi$
$829$	13.5000	+	7.79423i	0.468874	+	0.270705i	−0.246894	+	0.969042i	$0.579410\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	7.50000	+	9.52628i	0.259860	+	0.330066i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−13.5000	+	23.3827i	−0.466072	+	0.807260i	−0.999249	−	0.0387435i	$-0.987664\pi$
$839$	−13.5000	+	23.3827i	−0.466072	+	0.807260i	0.533177	+	0.846003i	$0.320998\pi$
$840$		0			0
$841$	10.0000	+	17.3205i	0.344828	+	0.597259i
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	2.00000	+	1.73205i	0.0687208	+	0.0595140i
$848$		0			0
$849$		0			0
$850$		0			0
$851$			24.2487i			0.831235i
$852$		0			0
$853$	46.5000	+	26.8468i	1.59213	+	0.919216i	0.992941	+	0.118609i	$0.0378434\pi$
$853$	46.5000	+	26.8468i	1.59213	+	0.919216i	0.599189	+	0.800608i	$0.295490\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	6.00000	−	3.46410i	0.204956	−	0.118331i	−0.394009	−	0.919107i	$-0.628912\pi$
$857$	6.00000	−	3.46410i	0.204956	−	0.118331i	0.598965	+	0.800775i	$0.295579\pi$
$858$		0			0
$859$	2.00000	−	3.46410i	0.0682391	−	0.118194i	−0.829887	−	0.557931i	$-0.811595\pi$
$859$	2.00000	−	3.46410i	0.0682391	−	0.118194i	0.898126	+	0.439738i	$0.144929\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−22.5000	+	12.9904i	−0.765909	+	0.442198i	−0.831413	−	0.555655i	$-0.812468\pi$
$863$	−22.5000	+	12.9904i	−0.765909	+	0.442198i	0.0655043	+	0.997852i	$0.479134\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−3.00000	−	5.19615i	−0.101768	−	0.176267i
$870$		0			0
$871$	13.5000	+	23.3827i	0.457430	+	0.792292i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	23.0000	−	39.8372i	0.776655	−	1.34521i	−0.157205	−	0.987566i	$-0.550248\pi$
$877$	23.0000	−	39.8372i	0.776655	−	1.34521i	0.933860	−	0.357640i	$-0.116418\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	20.7846i		−	0.700251i	−0.936703	−	0.350126i	$-0.886139\pi$
$881$		−	20.7846i		−	0.700251i	0.936703	−	0.350126i	$-0.113861\pi$
$882$		0			0
$883$		−	31.1769i		−	1.04919i	−0.851353	−	0.524593i	$-0.824217\pi$
$883$		−	31.1769i		−	1.04919i	0.851353	−	0.524593i	$-0.175783\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−6.00000	+	10.3923i	−0.201460	+	0.348939i	−0.948999	−	0.315279i	$-0.897902\pi$
$887$	−6.00000	+	10.3923i	−0.201460	+	0.348939i	0.747539	+	0.664218i	$0.231235\pi$
$888$		0			0
$889$	27.0000	+	5.19615i	0.905551	+	0.174273i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−22.5000	−	38.9711i	−0.752934	−	1.30412i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$	1.50000	−	2.59808i	0.0500278	−	0.0866507i
$900$		0			0
$901$	13.5000	−	7.79423i	0.449750	−	0.259663i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	33.0000	−	19.0526i	1.09575	−	0.632630i	0.160646	−	0.987012i	$-0.448642\pi$
$907$	33.0000	−	19.0526i	1.09575	−	0.632630i	0.935101	+	0.354382i	$0.115309\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−49.5000	−	28.5788i	−1.64001	−	0.946859i	−0.980828	−	0.194874i	$-0.937570\pi$
$911$	−49.5000	−	28.5788i	−1.64001	−	0.946859i	−0.659180	−	0.751985i	$-0.729096\pi$
$912$		0			0
$913$		−	31.1769i		−	1.03181i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	24.0000	+	20.7846i	0.792550	+	0.686368i
$918$		0			0
$919$	−19.5000	+	11.2583i	−0.643246	+	0.371378i	−0.785864	−	0.618400i	$-0.787781\pi$
$919$	−19.5000	+	11.2583i	−0.643246	+	0.371378i	0.142618	+	0.989778i	$0.454448\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−9.00000	−	15.5885i	−0.296239	−	0.513100i
$924$		0			0
$925$	−17.5000	+	30.3109i	−0.575396	+	0.996616i
$926$		0			0
$927$		0			0
$928$		0			0
$929$			8.66025i			0.284134i	0.989857	+	0.142067i	$0.0453748\pi$
$929$			8.66025i			0.284134i	−0.989857	+	0.142067i	$0.954625\pi$
$930$		0			0
$931$	5.00000	+	34.6410i	0.163868	+	1.13531i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$			48.4974i			1.58434i	0.610299	+	0.792171i	$0.291049\pi$
$937$			48.4974i			1.58434i	−0.610299	+	0.792171i	$0.708951\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	29.4449i		−	0.959875i	−0.877303	−	0.479938i	$-0.840659\pi$
$941$		−	29.4449i		−	0.959875i	0.877303	−	0.479938i	$-0.159341\pi$
$942$		0			0
$943$	−6.00000			−0.195387
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	5.19615i		−	0.168852i	−0.996430	−	0.0844261i	$-0.973094\pi$
$947$		−	5.19615i		−	0.168852i	0.996430	−	0.0844261i	$-0.0269057\pi$
$948$		0			0
$949$	−3.00000			−0.0973841
$950$		0			0
$951$		0			0
$952$		0			0
$953$	18.0000			0.583077			0.291539	−	0.956559i	$-0.405833\pi$
$953$	18.0000			0.583077			0.291539	+	0.956559i	$0.405833\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$	45.0000	−	15.5885i	1.45313	−	0.503378i
$960$		0			0
$961$	−30.0000			−0.967742
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−13.5000	+	7.79423i	−0.434131	+	0.250645i	−0.701105	−	0.713058i	$-0.747310\pi$
$967$	−13.5000	+	7.79423i	−0.434131	+	0.250645i	0.266974	+	0.963704i	$0.413976\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−7.50000	−	12.9904i	−0.240686	−	0.416881i	0.720224	−	0.693742i	$-0.244039\pi$
$971$	−7.50000	−	12.9904i	−0.240686	−	0.416881i	−0.960910	+	0.276861i	$0.910706\pi$
$972$		0			0
$973$	−12.5000	+	4.33013i	−0.400732	+	0.138817i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−51.0000			−1.63163			−0.815817	−	0.578310i	$-0.803713\pi$
$977$	−51.0000			−1.63163			−0.815817	+	0.578310i	$0.803713\pi$
$978$		0			0
$979$	−3.00000	+	5.19615i	−0.0958804	+	0.166070i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−6.00000	−	10.3923i	−0.191370	−	0.331463i	0.754334	−	0.656490i	$-0.227960\pi$
$983$	−6.00000	−	10.3923i	−0.191370	−	0.331463i	−0.945705	+	0.325027i	$0.894626\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−3.00000	−	5.19615i	−0.0953945	−	0.165228i
$990$		0			0
$991$	31.5000	+	18.1865i	1.00063	+	0.577714i	0.908435	−	0.418027i	$-0.137278\pi$
$991$	31.5000	+	18.1865i	1.00063	+	0.577714i	0.0921957	+	0.995741i	$0.470611\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	48.0000	−	27.7128i	1.52018	−	0.877674i	0.520458	−	0.853887i	$-0.325761\pi$
$997$	48.0000	−	27.7128i	1.52018	−	0.877674i	0.999717	−	0.0237864i	$-0.00757217\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3024.2.cz.a.2719.1		2
3.2	odd	2		1008.2.cz.a.367.1	yes	2
4.3	odd	2		3024.2.cz.b.2719.1		2
7.5	odd	6		3024.2.bf.c.2287.1		2
9.4	even	3		3024.2.bf.b.1711.1		2
9.5	odd	6		1008.2.bf.b.31.1	✓	2
12.11	even	2		1008.2.cz.d.367.1	yes	2
21.5	even	6		1008.2.bf.c.943.1	yes	2
28.19	even	6		3024.2.bf.b.2287.1		2
36.23	even	6		1008.2.bf.c.31.1	yes	2
36.31	odd	6		3024.2.bf.c.1711.1		2
63.5	even	6		1008.2.cz.d.607.1	yes	2
63.40	odd	6		3024.2.cz.b.1279.1		2
84.47	odd	6		1008.2.bf.b.943.1	yes	2
252.103	even	6	inner	3024.2.cz.a.1279.1		2
252.131	odd	6		1008.2.cz.a.607.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1008.2.bf.b.31.1	✓	2	9.5	odd	6
1008.2.bf.b.943.1	yes	2	84.47	odd	6
1008.2.bf.c.31.1	yes	2	36.23	even	6
1008.2.bf.c.943.1	yes	2	21.5	even	6
1008.2.cz.a.367.1	yes	2	3.2	odd	2
1008.2.cz.a.607.1	yes	2	252.131	odd	6
1008.2.cz.d.367.1	yes	2	12.11	even	2
1008.2.cz.d.607.1	yes	2	63.5	even	6
3024.2.bf.b.1711.1		2	9.4	even	3
3024.2.bf.b.2287.1		2	28.19	even	6
3024.2.bf.c.1711.1		2	36.31	odd	6
3024.2.bf.c.2287.1		2	7.5	odd	6
3024.2.cz.a.1279.1		2	252.103	even	6	inner
3024.2.cz.a.2719.1		2	1.1	even	1	trivial
3024.2.cz.b.1279.1		2	63.40	odd	6
3024.2.cz.b.2719.1		2	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3024 = 2^{4} \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3024.cz (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.500000 + 2.59808i) q^{7} +O(q^{10})\)	\(q+(-0.500000 + 2.59808i) q^{7} +(3.00000 - 1.73205i) q^{11} +(-1.50000 + 0.866025i) q^{13} +(-1.50000 - 0.866025i) q^{17} +(-2.50000 - 4.33013i) q^{19} +(-3.00000 - 1.73205i) q^{23} +(-2.50000 - 4.33013i) q^{25} +(1.50000 - 2.59808i) q^{29} +1.00000 q^{31} +(-3.50000 - 6.06218i) q^{37} +(1.50000 - 0.866025i) q^{41} +(1.50000 + 0.866025i) q^{43} +9.00000 q^{47} +(-6.50000 - 2.59808i) q^{49} +(-4.50000 + 7.79423i) q^{53} +15.0000 q^{59} -1.73205i q^{61} -15.5885i q^{67} +10.3923i q^{71} +(1.50000 + 0.866025i) q^{73} +(3.00000 + 8.66025i) q^{77} -1.73205i q^{79} +(4.50000 - 7.79423i) q^{83} +(-1.50000 + 0.866025i) q^{89} +(-1.50000 - 4.33013i) q^{91} +(-1.50000 - 0.866025i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{7}+O(q^{10}) \) 2 * q - q^7	\( 2 q - q^{7} + 6 q^{11} - 3 q^{13} - 3 q^{17} - 5 q^{19} - 6 q^{23} - 5 q^{25} + 3 q^{29} + 2 q^{31} - 7 q^{37} + 3 q^{41} + 3 q^{43} + 18 q^{47} - 13 q^{49} - 9 q^{53} + 30 q^{59} + 3 q^{73} + 6 q^{77} + 9 q^{83} - 3 q^{89} - 3 q^{91} - 3 q^{97}+O(q^{100}) \) 2 * q - q^7 + 6 * q^11 - 3 * q^13 - 3 * q^17 - 5 * q^19 - 6 * q^23 - 5 * q^25 + 3 * q^29 + 2 * q^31 - 7 * q^37 + 3 * q^41 + 3 * q^43 + 18 * q^47 - 13 * q^49 - 9 * q^53 + 30 * q^59 + 3 * q^73 + 6 * q^77 + 9 * q^83 - 3 * q^89 - 3 * q^91 - 3 * q^97

Introduction

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Varieties

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Representations

Groups

Database

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