LMFDB - Embedded newform 3024.2.bf.c.1711.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3024,2,Mod(1711,3024)]

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, 1]))

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

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

chi := DirichletCharacter("3024.1711");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3024 = 2^{4} \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3024.bf (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:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1008)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			1711.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	3024.1711
Dual form			3024.2.bf.c.2287.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.00000 + 1.73205i) q^{7} +O(q^{10})$	$q+(2.00000 + 1.73205i) q^{7} -3.46410i q^{11} +(1.50000 + 0.866025i) q^{13} +(-1.50000 - 0.866025i) q^{17} +(2.50000 + 4.33013i) q^{19} -3.46410i q^{23} +5.00000 q^{25} +(1.50000 + 2.59808i) q^{29} +(0.500000 + 0.866025i) q^{31} +(-3.50000 - 6.06218i) q^{37} +(-1.50000 - 0.866025i) q^{41} +(1.50000 - 0.866025i) q^{43} +(4.50000 - 7.79423i) q^{47} +(1.00000 + 6.92820i) q^{49} +(-4.50000 + 7.79423i) q^{53} +(7.50000 + 12.9904i) q^{59} +(1.50000 + 0.866025i) q^{61} +(13.5000 - 7.79423i) q^{67} -10.3923i q^{71} +(1.50000 + 0.866025i) q^{73} +(6.00000 - 6.92820i) q^{77} +(-1.50000 - 0.866025i) q^{79} +(-4.50000 - 7.79423i) q^{83} +(-1.50000 + 0.866025i) q^{89} +(1.50000 + 4.33013i) q^{91} +(1.50000 - 0.866025i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{7}+O(q^{10}) $ 2 * q + 4 * q^7	$ 2 q + 4 q^{7} + 3 q^{13} - 3 q^{17} + 5 q^{19} + 10 q^{25} + 3 q^{29} + q^{31} - 7 q^{37} - 3 q^{41} + 3 q^{43} + 9 q^{47} + 2 q^{49} - 9 q^{53} + 15 q^{59} + 3 q^{61} + 27 q^{67} + 3 q^{73} + 12 q^{77} - 3 q^{79} - 9 q^{83} - 3 q^{89} + 3 q^{91} + 3 q^{97}+O(q^{100}) $ 2 * q + 4 * q^7 + 3 * q^13 - 3 * q^17 + 5 * q^19 + 10 * q^25 + 3 * q^29 + q^31 - 7 * q^37 - 3 * q^41 + 3 * q^43 + 9 * q^47 + 2 * q^49 - 9 * q^53 + 15 * q^59 + 3 * q^61 + 27 * q^67 + 3 * q^73 + 12 * q^77 - 3 * q^79 - 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 $\left(\mathbb{Z}/3024\mathbb{Z}\right)^\times$.

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

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		1.00000			$0$
$5$		0			0		−1.00000			$\pi$
$6$		0			0
$7$	2.00000	+	1.73205i	0.755929	+	0.654654i
$7$	2.00000	+	1.73205i	0.755929	+	0.654654i
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	3.46410i		−	1.04447i	−0.852803	−	0.522233i	$-0.825099\pi$
$11$		−	3.46410i		−	1.04447i	0.852803	−	0.522233i	$-0.174901\pi$
$12$		0			0
$13$	1.50000	+	0.866025i	0.416025	+	0.240192i	0.693375	−	0.720577i	$-0.256123\pi$
$13$	1.50000	+	0.866025i	0.416025	+	0.240192i	−0.277350	+	0.960769i	$0.589456\pi$
$14$		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	$0.0277634\pi$
$19$	2.50000	+	4.33013i	0.573539	+	0.993399i	−0.422659	+	0.906289i	$0.638903\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	3.46410i		−	0.722315i	−0.932505	−	0.361158i	$-0.882382\pi$
$23$		−	3.46410i		−	0.722315i	0.932505	−	0.361158i	$-0.117618\pi$
$24$		0			0
$25$	5.00000			1.00000
$26$		0			0
$27$		0			0
$28$		0			0
$29$	1.50000	+	2.59808i	0.278543	+	0.482451i	0.971023	−	0.238987i	$-0.0768152\pi$
$29$	1.50000	+	2.59808i	0.278543	+	0.482451i	−0.692480	+	0.721437i	$0.743482\pi$
$30$		0			0
$31$	0.500000	+	0.866025i	0.0898027	+	0.155543i	0.907428	−	0.420208i	$-0.138043\pi$
$31$	0.500000	+	0.866025i	0.0898027	+	0.155543i	−0.817625	+	0.575751i	$0.804710\pi$
$32$		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	$-0.376517\pi$
$41$	−1.50000	−	0.866025i	−0.234261	−	0.135250i	−0.612536	+	0.790443i	$0.709851\pi$
$42$		0			0
$43$	1.50000	−	0.866025i	0.228748	−	0.132068i	−0.381246	−	0.924473i	$-0.624505\pi$
$43$	1.50000	−	0.866025i	0.228748	−	0.132068i	0.609994	+	0.792406i	$0.291172\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	4.50000	−	7.79423i	0.656392	−	1.13691i	−0.325150	−	0.945662i	$-0.605415\pi$
$47$	4.50000	−	7.79423i	0.656392	−	1.13691i	0.981543	−	0.191243i	$-0.0612518\pi$
$48$		0			0
$49$	1.00000	+	6.92820i	0.142857	+	0.989743i
$50$		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
$59$	7.50000	+	12.9904i	0.976417	+	1.69120i	0.675178	+	0.737655i	$0.264067\pi$
$59$	7.50000	+	12.9904i	0.976417	+	1.69120i	0.301239	+	0.953549i	$0.402600\pi$
$60$		0			0
$61$	1.50000	+	0.866025i	0.192055	+	0.110883i	0.592944	−	0.805243i	$-0.297965\pi$
$61$	1.50000	+	0.866025i	0.192055	+	0.110883i	−0.400889	+	0.916127i	$0.631299\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	13.5000	−	7.79423i	1.64929	−	0.952217i	0.671932	−	0.740613i	$-0.265465\pi$
$67$	13.5000	−	7.79423i	1.64929	−	0.952217i	0.977356	−	0.211604i	$-0.0678686\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	10.3923i		−	1.23334i	−0.787222	−	0.616670i	$-0.788481\pi$
$71$		−	10.3923i		−	1.23334i	0.787222	−	0.616670i	$-0.211519\pi$
$72$		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
$77$	6.00000	−	6.92820i	0.683763	−	0.789542i
$78$		0			0
$79$	−1.50000	−	0.866025i	−0.168763	−	0.0974355i	0.413239	−	0.910622i	$-0.364397\pi$
$79$	−1.50000	−	0.866025i	−0.168763	−	0.0974355i	−0.582003	+	0.813187i	$0.697731\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−4.50000	−	7.79423i	−0.493939	−	0.855528i	0.506036	−	0.862512i	$-0.331110\pi$
$83$	−4.50000	−	7.79423i	−0.493939	−	0.855528i	−0.999976	+	0.00698436i	$0.997777\pi$
$84$		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	$-0.638641\pi$
$97$	1.50000	−	0.866025i	0.152302	−	0.0879316i	0.574214	+	0.818705i	$0.305308\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			13.8564i			1.37876i	0.724398	+	0.689382i	$0.242118\pi$
$101$			13.8564i			1.37876i	−0.724398	+	0.689382i	$0.757882\pi$
$102$		0			0
$103$	8.00000			0.788263			0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000			0.788263			0.394132	+	0.919054i	$0.371045\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−1.50000	+	0.866025i	−0.145010	+	0.0837218i	−0.570750	−	0.821124i	$-0.693347\pi$
$107$	−1.50000	+	0.866025i	−0.145010	+	0.0837218i	0.425739	+	0.904846i	$0.360014\pi$
$108$		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	−0.996262	−	0.0863794i	$-0.972470\pi$
$113$	−4.50000	+	7.79423i	−0.423324	+	0.733219i	0.572938	+	0.819599i	$0.305804\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−1.50000	−	4.33013i	−0.137505	−	0.396942i
$120$		0			0
$121$	−1.00000			−0.0909091
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$			10.3923i			0.922168i	0.887357	+	0.461084i	$0.152539\pi$
$127$			10.3923i			0.922168i	−0.887357	+	0.461084i	$0.847461\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	12.0000			1.04844			0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000			1.04844			0.524222	+	0.851581i	$0.324356\pi$
$132$		0			0
$133$	−2.50000	+	12.9904i	−0.216777	+	1.12641i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	18.0000			1.53784			0.768922	−	0.639343i	$-0.220793\pi$
$137$	18.0000			1.53784			0.768922	+	0.639343i	$0.220793\pi$
$138$		0			0
$139$	−2.50000	+	4.33013i	−0.212047	+	0.367277i	−0.952355	−	0.304991i	$-0.901346\pi$
$139$	−2.50000	+	4.33013i	−0.212047	+	0.367277i	0.740308	+	0.672268i	$0.234680\pi$
$140$		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
$149$	6.00000			0.491539			0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000			0.491539			0.245770	+	0.969328i	$0.420959\pi$
$150$		0			0
$151$		−	17.3205i		−	1.40952i	−0.709444	−	0.704761i	$-0.751054\pi$
$151$		−	17.3205i		−	1.40952i	0.709444	−	0.704761i	$-0.248946\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−10.5000	+	6.06218i	−0.837991	+	0.483814i	−0.856581	−	0.516013i	$-0.827416\pi$
$157$	−10.5000	+	6.06218i	−0.837991	+	0.483814i	0.0185897	+	0.999827i	$0.494082\pi$
$158$		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	0.132689	−	0.991158i	$-0.457639\pi$
$163$	13.5000	−	7.79423i	1.05740	−	0.610491i	0.924712	+	0.380667i	$0.124305\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	10.5000	−	18.1865i	0.812514	−	1.40732i	−0.0985846	−	0.995129i	$-0.531432\pi$
$167$	10.5000	−	18.1865i	0.812514	−	1.40732i	0.911099	−	0.412188i	$-0.135235\pi$
$168$		0			0
$169$	−5.00000	−	8.66025i	−0.384615	−	0.666173i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	16.5000	+	9.52628i	1.25447	+	0.724270i	0.971994	−	0.235004i	$-0.0755104\pi$
$173$	16.5000	+	9.52628i	1.25447	+	0.724270i	0.282477	+	0.959274i	$0.408844\pi$
$174$		0			0
$175$	10.0000	+	8.66025i	0.755929	+	0.654654i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−10.5000	−	6.06218i	−0.784807	−	0.453108i	0.0533243	−	0.998577i	$-0.483018\pi$
$179$	−10.5000	−	6.06218i	−0.784807	−	0.453108i	−0.838131	+	0.545469i	$0.816352\pi$
$180$		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
$187$	−3.00000	+	5.19615i	−0.219382	+	0.379980i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	1.50000	+	0.866025i	0.108536	+	0.0626634i	0.553285	−	0.832992i	$-0.313374\pi$
$191$	1.50000	+	0.866025i	0.108536	+	0.0626634i	−0.444749	+	0.895655i	$0.646707\pi$
$192$		0			0
$193$	8.50000	+	14.7224i	0.611843	+	1.05974i	0.990930	+	0.134382i	$0.0429051\pi$
$193$	8.50000	+	14.7224i	0.611843	+	1.05974i	−0.379086	+	0.925361i	$0.623762\pi$
$194$		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	−0.883203	−	0.468990i	$-0.844618\pi$
$199$	−0.500000	+	0.866025i	−0.0354441	+	0.0613909i	0.847759	+	0.530381i	$0.177951\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−1.50000	+	7.79423i	−0.105279	+	0.547048i
$204$		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	0.815801	−	0.578333i	$-0.196297\pi$
$211$	10.5000	+	6.06218i	0.722850	+	0.417338i	−0.0929509	+	0.995671i	$0.529630\pi$
$212$		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
$221$	−1.50000	−	2.59808i	−0.100901	−	0.174766i
$222$		0			0
$223$	2.50000	+	4.33013i	0.167412	+	0.289967i	0.937509	−	0.347960i	$-0.113126\pi$
$223$	2.50000	+	4.33013i	0.167412	+	0.289967i	−0.770097	+	0.637927i	$0.779792\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	12.0000			0.796468			0.398234	−	0.917284i	$-0.369623\pi$
$227$	12.0000			0.796468			0.398234	+	0.917284i	$0.369623\pi$
$228$		0			0
$229$		−	13.8564i		−	0.915657i	−0.889041	−	0.457829i	$-0.848627\pi$
$229$		−	13.8564i		−	0.915657i	0.889041	−	0.457829i	$-0.151373\pi$
$230$		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	0.973336	−	0.229385i	$-0.0736716\pi$
$239$	19.5000	+	11.2583i	1.26135	+	0.728241i	0.288014	+	0.957626i	$0.407005\pi$
$240$		0			0
$241$		−	20.7846i		−	1.33885i	−0.742878	−	0.669427i	$-0.766540\pi$
$241$		−	20.7846i		−	1.33885i	0.742878	−	0.669427i	$-0.233460\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$			8.66025i			0.551039i
$248$		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
$257$		−	27.7128i		−	1.72868i	−0.502910	−	0.864339i	$-0.667737\pi$
$257$		−	27.7128i		−	1.72868i	0.502910	−	0.864339i	$-0.332263\pi$
$258$		0			0
$259$	3.50000	−	18.1865i	0.217479	−	1.13006i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			31.1769i			1.92245i	0.275764	+	0.961225i	$0.411069\pi$
$263$			31.1769i			1.92245i	−0.275764	+	0.961225i	$0.588931\pi$
$264$		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	0.880812	−	0.473466i	$-0.156997\pi$
$271$	0.500000	+	0.866025i	0.0303728	+	0.0526073i	−0.850439	+	0.526073i	$0.823664\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		−	17.3205i		−	1.04447i
$276$		0			0
$277$	−14.0000			−0.841178			−0.420589	−	0.907251i	$-0.638177\pi$
$277$	−14.0000			−0.841178			−0.420589	+	0.907251i	$0.638177\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	7.50000	+	12.9904i	0.447412	+	0.774941i	0.998217	−	0.0596933i	$-0.0190123\pi$
$281$	7.50000	+	12.9904i	0.447412	+	0.774941i	−0.550804	+	0.834634i	$0.685679\pi$
$282$		0			0
$283$	−15.5000	−	26.8468i	−0.921379	−	1.59588i	−0.797283	−	0.603606i	$-0.793730\pi$
$283$	−15.5000	−	26.8468i	−0.921379	−	1.59588i	−0.124096	−	0.992270i	$-0.539603\pi$
$284$		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	$-0.0590674\pi$
$293$	22.5000	+	12.9904i	1.31446	+	0.758906i	0.331632	+	0.943409i	$0.392401\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	3.00000	−	5.19615i	0.173494	−	0.300501i
$300$		0			0
$301$	4.50000	+	0.866025i	0.259376	+	0.0499169i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−8.00000			−0.456584			−0.228292	−	0.973593i	$-0.573314\pi$
$307$	−8.00000			−0.456584			−0.228292	+	0.973593i	$0.573314\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	1.50000	+	2.59808i	0.0850572	+	0.147323i	0.905416	−	0.424526i	$-0.139559\pi$
$311$	1.50000	+	2.59808i	0.0850572	+	0.147323i	−0.820358	+	0.571850i	$0.806226\pi$
$312$		0			0
$313$	25.5000	+	14.7224i	1.44135	+	0.832161i	0.997940	−	0.0641600i	$-0.0204368\pi$
$313$	25.5000	+	14.7224i	1.44135	+	0.832161i	0.443406	+	0.896321i	$0.353770\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−10.5000	+	18.1865i	−0.589739	+	1.02146i	0.404528	+	0.914526i	$0.367436\pi$
$317$	−10.5000	+	18.1865i	−0.589739	+	1.02146i	−0.994266	+	0.106932i	$0.965897\pi$
$318$		0			0
$319$	9.00000	−	5.19615i	0.503903	−	0.290929i
$320$		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
$329$	22.5000	−	7.79423i	1.24047	−	0.429710i
$330$		0			0
$331$	4.50000	+	2.59808i	0.247342	+	0.142803i	0.618547	−	0.785748i	$-0.287722\pi$
$331$	4.50000	+	2.59808i	0.247342	+	0.142803i	−0.371204	+	0.928551i	$0.621055\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−5.50000	+	9.52628i	−0.299604	+	0.518930i	−0.976045	−	0.217567i	$-0.930188\pi$
$337$	−5.50000	+	9.52628i	−0.299604	+	0.518930i	0.676441	+	0.736497i	$0.263521\pi$
$338$		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
$347$	−19.5000	+	11.2583i	−1.04681	+	0.604379i	−0.921756	−	0.387770i	$-0.873245\pi$
$347$	−19.5000	+	11.2583i	−1.04681	+	0.604379i	−0.125059	+	0.992149i	$0.539912\pi$
$348$		0			0
$349$	−22.5000	+	12.9904i	−1.20440	+	0.695359i	−0.961530	−	0.274700i	$-0.911421\pi$
$349$	−22.5000	+	12.9904i	−1.20440	+	0.695359i	−0.242867	+	0.970059i	$0.578088\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			20.7846i			1.10625i	0.833097	+	0.553127i	$0.186565\pi$
$353$			20.7846i			1.10625i	−0.833097	+	0.553127i	$0.813435\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	10.5000	−	6.06218i	0.554169	−	0.319950i	−0.196633	−	0.980477i	$-0.563001\pi$
$359$	10.5000	−	6.06218i	0.554169	−	0.319950i	0.750802	+	0.660528i	$0.229667\pi$
$360$		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
$367$	−8.00000			−0.417597			−0.208798	−	0.977959i	$-0.566955\pi$
$367$	−8.00000			−0.417597			−0.208798	+	0.977959i	$0.566955\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−22.5000	+	7.79423i	−1.16814	+	0.404656i
$372$		0			0
$373$	−10.0000			−0.517780			−0.258890	−	0.965907i	$-0.583357\pi$
$373$	−10.0000			−0.517780			−0.258890	+	0.965907i	$0.583357\pi$
$374$		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	$-0.786001\pi$
$379$		−	24.2487i		−	1.24557i	0.782392	−	0.622786i	$-0.213999\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	24.0000			1.22634			0.613171	−	0.789950i	$-0.289894\pi$
$383$	24.0000			1.22634			0.613171	+	0.789950i	$0.289894\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−18.0000			−0.912636			−0.456318	−	0.889817i	$-0.650832\pi$
$389$	−18.0000			−0.912636			−0.456318	+	0.889817i	$0.650832\pi$
$390$		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
$401$	−30.0000			−1.49813			−0.749064	−	0.662497i	$-0.769497\pi$
$401$	−30.0000			−1.49813			−0.749064	+	0.662497i	$0.769497\pi$
$402$		0			0
$403$			1.73205i			0.0862796i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−21.0000	+	12.1244i	−1.04093	+	0.600982i
$408$		0			0
$409$	25.5000	−	14.7224i	1.26089	−	0.727977i	0.287646	−	0.957737i	$-0.407127\pi$
$409$	25.5000	−	14.7224i	1.26089	−	0.727977i	0.973247	+	0.229759i	$0.0737939\pi$
$410$		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	$-0.952743\pi$
$419$	−7.50000	+	12.9904i	−0.366399	+	0.634622i	0.622601	+	0.782540i	$0.286076\pi$
$420$		0			0
$421$	2.50000	+	4.33013i	0.121843	+	0.211037i	0.920494	−	0.390756i	$-0.127786\pi$
$421$	2.50000	+	4.33013i	0.121843	+	0.211037i	−0.798652	+	0.601793i	$0.794453\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−7.50000	−	4.33013i	−0.363803	−	0.210042i
$426$		0			0
$427$	1.50000	+	4.33013i	0.0725901	+	0.209550i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−22.5000	−	12.9904i	−1.08379	−	0.625725i	−0.151871	−	0.988400i	$-0.548530\pi$
$431$	−22.5000	−	12.9904i	−1.08379	−	0.625725i	−0.931915	+	0.362676i	$0.881863\pi$
$432$		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
$437$	15.0000	−	8.66025i	0.717547	−	0.414276i
$438$		0			0
$439$	−12.5000	+	21.6506i	−0.596592	+	1.03333i	0.396728	+	0.917936i	$0.370146\pi$
$439$	−12.5000	+	21.6506i	−0.596592	+	1.03333i	−0.993320	+	0.115392i	$0.963188\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−22.5000	−	12.9904i	−1.06901	−	0.617192i	−0.141097	−	0.989996i	$-0.545063\pi$
$443$	−22.5000	−	12.9904i	−1.06901	−	0.617192i	−0.927910	+	0.372804i	$0.878396\pi$
$444$		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
$457$	0.500000	−	0.866025i	0.0233890	−	0.0405110i	−0.854094	−	0.520119i	$-0.825888\pi$
$457$	0.500000	−	0.866025i	0.0233890	−	0.0405110i	0.877483	+	0.479608i	$0.159221\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−19.5000	+	11.2583i	−0.908206	+	0.524353i	−0.879853	−	0.475245i	$-0.842359\pi$
$461$	−19.5000	+	11.2583i	−0.908206	+	0.524353i	−0.0283522	+	0.999598i	$0.509026\pi$
$462$		0			0
$463$	−31.5000	−	18.1865i	−1.46393	−	0.845200i	−0.464739	−	0.885448i	$-0.653852\pi$
$463$	−31.5000	−	18.1865i	−1.46393	−	0.845200i	−0.999190	+	0.0402476i	$0.987185\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	7.50000	+	12.9904i	0.347059	+	0.601123i	0.985726	−	0.168360i	$-0.0538472\pi$
$467$	7.50000	+	12.9904i	0.347059	+	0.601123i	−0.638667	+	0.769483i	$0.720514\pi$
$468$		0			0
$469$	40.5000	+	7.79423i	1.87012	+	0.359904i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−3.00000	−	5.19615i	−0.137940	−	0.238919i
$474$		0			0
$475$	12.5000	+	21.6506i	0.573539	+	0.993399i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		0			0
$481$		−	12.1244i		−	0.552823i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−13.5000	−	7.79423i	−0.611743	−	0.353190i	0.161904	−	0.986807i	$-0.448236\pi$
$487$	−13.5000	−	7.79423i	−0.611743	−	0.353190i	−0.773647	+	0.633616i	$0.781570\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	7.50000	+	4.33013i	0.338470	+	0.195416i	0.659595	−	0.751621i	$-0.270728\pi$
$491$	7.50000	+	4.33013i	0.338470	+	0.195416i	−0.321125	+	0.947037i	$0.604061\pi$
$492$		0			0
$493$		−	5.19615i		−	0.234023i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	18.0000	−	20.7846i	0.807410	−	0.932317i
$498$		0			0
$499$			3.46410i			0.155074i	0.996989	+	0.0775372i	$0.0247057\pi$
$499$			3.46410i			0.155074i	−0.996989	+	0.0775372i	$0.975294\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−24.0000			−1.07011			−0.535054	−	0.844818i	$-0.679709\pi$
$503$	−24.0000			−1.07011			−0.535054	+	0.844818i	$0.679709\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	34.6410i		−	1.53544i	−0.640788	−	0.767718i	$-0.721392\pi$
$509$		−	34.6410i		−	1.53544i	0.640788	−	0.767718i	$-0.278608\pi$
$510$		0			0
$511$	1.50000	+	4.33013i	0.0663561	+	0.191554i
$512$		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	0.972421	−	0.233233i	$-0.0749303\pi$
$523$	6.50000	+	11.2583i	0.284225	+	0.492292i	−0.688196	+	0.725525i	$0.741597\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	1.73205i		−	0.0754493i
$528$		0			0
$529$	11.0000			0.478261
$530$		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	−0.331017	−	0.943625i	$-0.607392\pi$
$547$	7.50000	−	4.33013i	0.320677	−	0.185143i	0.651694	+	0.758482i	$0.274059\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−7.50000	+	12.9904i	−0.319511	+	0.553409i
$552$		0			0
$553$	−1.50000	−	4.33013i	−0.0637865	−	0.184136i
$554$		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
$563$	−10.5000	−	18.1865i	−0.442522	−	0.766471i	0.555354	−	0.831614i	$-0.312583\pi$
$563$	−10.5000	−	18.1865i	−0.442522	−	0.766471i	−0.997876	+	0.0651433i	$0.979250\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	1.50000	−	2.59808i	0.0628833	−	0.108917i	−0.832870	−	0.553469i	$-0.813304\pi$
$569$	1.50000	−	2.59808i	0.0628833	−	0.108917i	0.895753	+	0.444552i	$0.146637\pi$
$570$		0			0
$571$	−34.5000	+	19.9186i	−1.44378	+	0.833567i	−0.998099	−	0.0616252i	$-0.980372\pi$
$571$	−34.5000	+	19.9186i	−1.44378	+	0.833567i	−0.445681	+	0.895192i	$0.647038\pi$
$572$		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
$581$	4.50000	−	23.3827i	0.186691	−	0.970077i
$582$		0			0
$583$	27.0000	+	15.5885i	1.11823	+	0.645608i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	13.5000	+	23.3827i	0.557205	+	0.965107i	0.997728	+	0.0673658i	$0.0214594\pi$
$587$	13.5000	+	23.3827i	0.557205	+	0.965107i	−0.440524	+	0.897741i	$0.645207\pi$
$588$		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
$599$	−31.5000	+	18.1865i	−1.28706	+	0.743082i	−0.978128	−	0.208004i	$-0.933303\pi$
$599$	−31.5000	+	18.1865i	−1.28706	+	0.743082i	−0.308927	+	0.951086i	$0.599970\pi$
$600$		0			0
$601$	31.5000	−	18.1865i	1.28491	−	0.741844i	0.307170	−	0.951655i	$-0.400618\pi$
$601$	31.5000	−	18.1865i	1.28491	−	0.741844i	0.977742	+	0.209811i	$0.0672847\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−32.0000			−1.29884			−0.649420	−	0.760430i	$-0.724988\pi$
$607$	−32.0000			−1.29884			−0.649420	+	0.760430i	$0.724988\pi$
$608$		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	−0.455211	−	0.890384i	$-0.650436\pi$
$617$	13.5000	−	23.3827i	0.543490	−	0.941351i	0.998700	−	0.0509678i	$-0.0162306\pi$
$618$		0			0
$619$	−28.0000			−1.12542			−0.562708	−	0.826656i	$-0.690240\pi$
$619$	−28.0000			−1.12542			−0.562708	+	0.826656i	$0.690240\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−4.50000	−	0.866025i	−0.180289	−	0.0346966i
$624$		0			0
$625$	25.0000			1.00000
$626$		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	$-0.786903\pi$
$631$		−	31.1769i		−	1.24113i	0.784154	−	0.620567i	$-0.213097\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−4.50000	+	11.2583i	−0.178296	+	0.446071i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−6.00000			−0.236986			−0.118493	−	0.992955i	$-0.537806\pi$
$641$	−6.00000			−0.236986			−0.118493	+	0.992955i	$0.537806\pi$
$642$		0			0
$643$	−6.50000	+	11.2583i	−0.256335	+	0.443985i	−0.965257	−	0.261301i	$-0.915848\pi$
$643$	−6.50000	+	11.2583i	−0.256335	+	0.443985i	0.708922	+	0.705287i	$0.249182\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−7.50000	+	12.9904i	−0.294855	+	0.510705i	−0.974951	−	0.222419i	$-0.928605\pi$
$647$	−7.50000	+	12.9904i	−0.294855	+	0.510705i	0.680096	+	0.733123i	$0.261938\pi$
$648$		0			0
$649$	45.0000	−	25.9808i	1.76640	−	1.01983i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−30.0000			−1.17399			−0.586995	−	0.809590i	$-0.699689\pi$
$653$	−30.0000			−1.17399			−0.586995	+	0.809590i	$0.699689\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−19.5000	+	11.2583i	−0.759612	+	0.438562i	−0.829156	−	0.559017i	$-0.811179\pi$
$659$	−19.5000	+	11.2583i	−0.759612	+	0.438562i	0.0695443	+	0.997579i	$0.477845\pi$
$660$		0			0
$661$	7.50000	−	4.33013i	0.291716	−	0.168422i	−0.346999	−	0.937865i	$-0.612799\pi$
$661$	7.50000	−	4.33013i	0.291716	−	0.168422i	0.638716	+	0.769443i	$0.279466\pi$
$662$		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	0.790650	−	0.612268i	$-0.209743\pi$
$673$	−3.50000	−	6.06218i	−0.134915	−	0.233680i	−0.925565	+	0.378589i	$0.876409\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−1.50000	−	0.866025i	−0.0576497	−	0.0332841i	0.470898	−	0.882188i	$-0.343930\pi$
$677$	−1.50000	−	0.866025i	−0.0576497	−	0.0332841i	−0.528548	+	0.848904i	$0.677263\pi$
$678$		0			0
$679$	4.50000	+	0.866025i	0.172694	+	0.0332350i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	37.5000	+	21.6506i	1.43490	+	0.828439i	0.997489	−	0.0708242i	$-0.0225629\pi$
$683$	37.5000	+	21.6506i	1.43490	+	0.828439i	0.437409	+	0.899263i	$0.355896\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−13.5000	+	7.79423i	−0.514309	+	0.296936i
$690$		0			0
$691$	−2.50000	+	4.33013i	−0.0951045	+	0.164726i	−0.909652	−	0.415371i	$-0.863652\pi$
$691$	−2.50000	+	4.33013i	−0.0951045	+	0.164726i	0.814548	+	0.580097i	$0.196985\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	1.50000	+	2.59808i	0.0568166	+	0.0984092i
$698$		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
$707$	−24.0000	+	27.7128i	−0.902613	+	1.04225i
$708$		0			0
$709$	18.5000	−	32.0429i	0.694782	−	1.20340i	−0.275472	−	0.961309i	$-0.588834\pi$
$709$	18.5000	−	32.0429i	0.694782	−	1.20340i	0.970254	−	0.242089i	$-0.0778325\pi$
$710$		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	$0.00127506\pi$
$719$	13.5000	+	23.3827i	0.503465	+	0.872027i	−0.496527	+	0.868021i	$0.665392\pi$
$720$		0			0
$721$	16.0000	+	13.8564i	0.595871	+	0.516040i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	7.50000	+	12.9904i	0.278543	+	0.482451i
$726$		0			0
$727$	−17.5000	−	30.3109i	−0.649039	−	1.12417i	−0.983353	−	0.181707i	$-0.941838\pi$
$727$	−17.5000	−	30.3109i	−0.649039	−	1.12417i	0.334314	−	0.942462i	$-0.391496\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−3.00000			−0.110959
$732$		0			0
$733$		−	20.7846i		−	0.767697i	−0.923396	−	0.383849i	$-0.874598\pi$
$733$		−	20.7846i		−	0.767697i	0.923396	−	0.383849i	$-0.125402\pi$
$734$		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	0.230705	−	0.973024i	$-0.425897\pi$
$739$	−13.5000	−	7.79423i	−0.496606	−	0.286715i	−0.727311	+	0.686308i	$0.759230\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−10.5000	−	6.06218i	−0.385208	−	0.222400i	0.294874	−	0.955536i	$-0.404722\pi$
$743$	−10.5000	−	6.06218i	−0.385208	−	0.222400i	−0.680082	+	0.733136i	$0.738056\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−4.50000	−	0.866025i	−0.164426	−	0.0316439i
$750$		0			0
$751$		−	17.3205i		−	0.632034i	−0.948753	−	0.316017i	$-0.897654\pi$
$751$		−	17.3205i		−	0.632034i	0.948753	−	0.316017i	$-0.102346\pi$
$752$		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
$761$		−	48.4974i		−	1.75803i	−0.476794	−	0.879015i	$-0.658201\pi$
$761$		−	48.4974i		−	1.75803i	0.476794	−	0.879015i	$-0.341799\pi$
$762$		0			0
$763$	−27.5000	+	9.52628i	−0.995567	+	0.344874i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			25.9808i			0.938111i
$768$		0			0
$769$	−4.50000	−	2.59808i	−0.162274	−	0.0936890i	0.416664	−	0.909061i	$-0.363199\pi$
$769$	−4.50000	−	2.59808i	−0.162274	−	0.0936890i	−0.578938	+	0.815372i	$0.696533\pi$
$770$		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
$779$		−	8.66025i		−	0.310286i
$780$		0			0
$781$	−36.0000			−1.28818
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	6.50000	+	11.2583i	0.231700	+	0.401316i	0.958308	−	0.285736i	$-0.0922379\pi$
$787$	6.50000	+	11.2583i	0.231700	+	0.401316i	−0.726609	+	0.687052i	$0.758905\pi$
$788$		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	$-0.135272\pi$
$797$	28.5000	+	16.4545i	1.00952	+	0.582848i	0.0984702	+	0.995140i	$0.468605\pi$
$798$		0			0
$799$	−13.5000	+	7.79423i	−0.477596	+	0.275740i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	3.00000	−	5.19615i	0.105868	−	0.183368i
$804$		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	$-0.866228\pi$
$811$	−52.0000			−1.82597			−0.912983	+	0.407997i	$0.866228\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	7.50000	+	4.33013i	0.262392	+	0.151492i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−16.5000	+	28.5788i	−0.575854	+	0.997408i	0.420094	+	0.907480i	$0.361997\pi$
$821$	−16.5000	+	28.5788i	−0.575854	+	0.997408i	−0.995948	+	0.0899279i	$0.971336\pi$
$822$		0			0
$823$	−10.5000	+	6.06218i	−0.366007	+	0.211314i	−0.671713	−	0.740812i	$-0.734441\pi$
$823$	−10.5000	+	6.06218i	−0.366007	+	0.211314i	0.305706	+	0.952126i	$0.401108\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	17.3205i		−	0.602293i	−0.953578	−	0.301147i	$-0.902631\pi$
$827$		−	17.3205i		−	0.602293i	0.953578	−	0.301147i	$-0.0973693\pi$
$828$		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
$833$	4.50000	−	11.2583i	0.155916	−	0.390078i
$834$		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	$-0.0123355\pi$
$839$	13.5000	+	23.3827i	0.466072	+	0.807260i	−0.533177	+	0.846003i	$0.679002\pi$
$840$		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
$851$	−21.0000	+	12.1244i	−0.719871	+	0.415618i
$852$		0			0
$853$	−46.5000	+	26.8468i	−1.59213	+	0.919216i	−0.599189	+	0.800608i	$0.704510\pi$
$853$	−46.5000	+	26.8468i	−1.59213	+	0.919216i	−0.992941	+	0.118609i	$0.962157\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			6.92820i			0.236663i	0.992974	+	0.118331i	$0.0377545\pi$
$857$			6.92820i			0.236663i	−0.992974	+	0.118331i	$0.962245\pi$
$858$		0			0
$859$	4.00000			0.136478			0.0682391	−	0.997669i	$-0.478262\pi$
$859$	4.00000			0.136478			0.0682391	+	0.997669i	$0.478262\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	22.5000	−	12.9904i	0.765909	−	0.442198i	−0.0655043	−	0.997852i	$-0.520866\pi$
$863$	22.5000	−	12.9904i	0.765909	−	0.442198i	0.831413	+	0.555655i	$0.187532\pi$
$864$		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
$871$	27.0000			0.914860
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−46.0000			−1.55331			−0.776655	−	0.629926i	$-0.783085\pi$
$877$	−46.0000			−1.55331			−0.776655	+	0.629926i	$0.783085\pi$
$878$		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	$0.175783\pi$
$883$			31.1769i			1.04919i	−0.851353	+	0.524593i	$0.824217\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−12.0000			−0.402921			−0.201460	−	0.979497i	$-0.564569\pi$
$887$	−12.0000			−0.402921			−0.201460	+	0.979497i	$0.564569\pi$
$888$		0			0
$889$	−18.0000	+	20.7846i	−0.603701	+	0.697093i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	45.0000			1.50587
$894$		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
$907$		−	38.1051i		−	1.26526i	−0.774454	−	0.632630i	$-0.781975\pi$
$907$		−	38.1051i		−	1.26526i	0.774454	−	0.632630i	$-0.218025\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−49.5000	+	28.5788i	−1.64001	+	0.946859i	−0.659180	+	0.751985i	$0.729096\pi$
$911$	−49.5000	+	28.5788i	−1.64001	+	0.946859i	−0.980828	+	0.194874i	$0.937570\pi$
$912$		0			0
$913$	−27.0000	+	15.5885i	−0.893570	+	0.515903i
$914$		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	−0.142618	−	0.989778i	$-0.545552\pi$
$919$	19.5000	−	11.2583i	0.643246	−	0.371378i	0.785864	+	0.618400i	$0.212219\pi$
$920$		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
$929$	−7.50000	−	4.33013i	−0.246067	−	0.142067i	0.371895	−	0.928275i	$-0.378708\pi$
$929$	−7.50000	−	4.33013i	−0.246067	−	0.142067i	−0.617962	+	0.786208i	$0.712041\pi$
$930$		0			0
$931$	−27.5000	+	21.6506i	−0.901276	+	0.709571i
$932$		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
$941$	−25.5000	+	14.7224i	−0.831276	+	0.479938i	−0.854289	−	0.519798i	$-0.826007\pi$
$941$	−25.5000	+	14.7224i	−0.831276	+	0.479938i	0.0230132	+	0.999735i	$0.492674\pi$
$942$		0			0
$943$	−3.00000	+	5.19615i	−0.0976934	+	0.169210i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−4.50000	−	2.59808i	−0.146230	−	0.0844261i	0.425100	−	0.905147i	$-0.360239\pi$
$947$	−4.50000	−	2.59808i	−0.146230	−	0.0844261i	−0.571330	+	0.820720i	$0.693572\pi$
$948$		0			0
$949$	1.50000	+	2.59808i	0.0486921	+	0.0843371i
$950$		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
$959$	36.0000	+	31.1769i	1.16250	+	1.00676i
$960$		0			0
$961$	15.0000	−	25.9808i	0.483871	−	0.838089i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−13.5000	−	7.79423i	−0.434131	−	0.250645i	0.266974	−	0.963704i	$-0.413976\pi$
$967$	−13.5000	−	7.79423i	−0.434131	−	0.250645i	−0.701105	+	0.713058i	$0.747310\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	7.50000	+	12.9904i	0.240686	+	0.416881i	0.960910	−	0.276861i	$-0.0892941\pi$
$971$	7.50000	+	12.9904i	0.240686	+	0.416881i	−0.720224	+	0.693742i	$0.755961\pi$
$972$		0			0
$973$	−12.5000	+	4.33013i	−0.400732	+	0.138817i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	25.5000	+	44.1673i	0.815817	+	1.41304i	0.908740	+	0.417364i	$0.137046\pi$
$977$	25.5000	+	44.1673i	0.815817	+	1.41304i	−0.0929223	+	0.995673i	$0.529621\pi$
$978$		0			0
$979$	3.00000	+	5.19615i	0.0958804	+	0.166070i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−12.0000			−0.382741			−0.191370	−	0.981518i	$-0.561293\pi$
$983$	−12.0000			−0.382741			−0.191370	+	0.981518i	$0.561293\pi$
$984$		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	−0.0921957	−	0.995741i	$-0.529389\pi$
$991$	−31.5000	−	18.1865i	−1.00063	−	0.577714i	−0.908435	+	0.418027i	$0.862722\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$			55.4256i			1.75535i	0.479259	+	0.877674i	$0.340906\pi$
$997$			55.4256i			1.75535i	−0.479259	+	0.877674i	$0.659094\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3024.2.bf.c.1711.1		2
3.2	odd	2		1008.2.bf.c.31.1	yes	2
4.3	odd	2		3024.2.bf.b.1711.1		2
7.5	odd	6		3024.2.cz.a.1279.1		2
9.2	odd	6		1008.2.cz.d.367.1	yes	2
9.7	even	3		3024.2.cz.b.2719.1		2
12.11	even	2		1008.2.bf.b.31.1	✓	2
21.5	even	6		1008.2.cz.a.607.1	yes	2
28.19	even	6		3024.2.cz.b.1279.1		2
36.7	odd	6		3024.2.cz.a.2719.1		2
36.11	even	6		1008.2.cz.a.367.1	yes	2
63.47	even	6		1008.2.bf.b.943.1	yes	2
63.61	odd	6		3024.2.bf.b.2287.1		2
84.47	odd	6		1008.2.cz.d.607.1	yes	2
252.47	odd	6		1008.2.bf.c.943.1	yes	2
252.187	even	6	inner	3024.2.bf.c.2287.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1008.2.bf.b.31.1	✓	2	12.11	even	2
1008.2.bf.b.943.1	yes	2	63.47	even	6
1008.2.bf.c.31.1	yes	2	3.2	odd	2
1008.2.bf.c.943.1	yes	2	252.47	odd	6
1008.2.cz.a.367.1	yes	2	36.11	even	6
1008.2.cz.a.607.1	yes	2	21.5	even	6
1008.2.cz.d.367.1	yes	2	9.2	odd	6
1008.2.cz.d.607.1	yes	2	84.47	odd	6
3024.2.bf.b.1711.1		2	4.3	odd	2
3024.2.bf.b.2287.1		2	63.61	odd	6
3024.2.bf.c.1711.1		2	1.1	even	1	trivial
3024.2.bf.c.2287.1		2	252.187	even	6	inner
3024.2.cz.a.1279.1		2	7.5	odd	6
3024.2.cz.a.2719.1		2	36.7	odd	6
3024.2.cz.b.1279.1		2	28.19	even	6
3024.2.cz.b.2719.1		2	9.7	even	3

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.bf (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(2.00000 + 1.73205i) q^{7} +O(q^{10})\)	\(q+(2.00000 + 1.73205i) q^{7} -3.46410i q^{11} +(1.50000 + 0.866025i) q^{13} +(-1.50000 - 0.866025i) q^{17} +(2.50000 + 4.33013i) q^{19} -3.46410i q^{23} +5.00000 q^{25} +(1.50000 + 2.59808i) q^{29} +(0.500000 + 0.866025i) q^{31} +(-3.50000 - 6.06218i) q^{37} +(-1.50000 - 0.866025i) q^{41} +(1.50000 - 0.866025i) q^{43} +(4.50000 - 7.79423i) q^{47} +(1.00000 + 6.92820i) q^{49} +(-4.50000 + 7.79423i) q^{53} +(7.50000 + 12.9904i) q^{59} +(1.50000 + 0.866025i) q^{61} +(13.5000 - 7.79423i) q^{67} -10.3923i q^{71} +(1.50000 + 0.866025i) q^{73} +(6.00000 - 6.92820i) q^{77} +(-1.50000 - 0.866025i) q^{79} +(-4.50000 - 7.79423i) q^{83} +(-1.50000 + 0.866025i) q^{89} +(1.50000 + 4.33013i) q^{91} +(1.50000 - 0.866025i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{7}+O(q^{10}) \) 2 * q + 4 * q^7	\( 2 q + 4 q^{7} + 3 q^{13} - 3 q^{17} + 5 q^{19} + 10 q^{25} + 3 q^{29} + q^{31} - 7 q^{37} - 3 q^{41} + 3 q^{43} + 9 q^{47} + 2 q^{49} - 9 q^{53} + 15 q^{59} + 3 q^{61} + 27 q^{67} + 3 q^{73} + 12 q^{77} - 3 q^{79} - 9 q^{83} - 3 q^{89} + 3 q^{91} + 3 q^{97}+O(q^{100}) \) 2 * q + 4 * q^7 + 3 * q^13 - 3 * q^17 + 5 * q^19 + 10 * q^25 + 3 * q^29 + q^31 - 7 * q^37 - 3 * q^41 + 3 * q^43 + 9 * q^47 + 2 * q^49 - 9 * q^53 + 15 * q^59 + 3 * q^61 + 27 * q^67 + 3 * q^73 + 12 * q^77 - 3 * q^79 - 9 * q^83 - 3 * q^89 + 3 * q^91 + 3 * q^97

Introduction

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