LMFDB - Embedded newform 1456.2.r.f.625.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1456,2,Mod(417,1456)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1456, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("1456.417");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1456 = 2^{4} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1456.r (of order $3$, degree $2$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$11.6262185343$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 182)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			625.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	1456.625
Dual form			1456.2.r.f.417.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.50000 + 0.866025i) q^{7} +(1.50000 - 2.59808i) q^{9} +(0.500000 + 0.866025i) q^{11} -1.00000 q^{13} +(0.500000 + 0.866025i) q^{17} +(2.50000 - 4.33013i) q^{19} +(1.00000 - 1.73205i) q^{23} +(2.50000 + 4.33013i) q^{25} -5.00000 q^{29} +(-4.00000 - 6.92820i) q^{31} +(4.00000 - 6.92820i) q^{37} +6.00000 q^{43} +(5.50000 - 9.52628i) q^{47} +(5.50000 - 4.33013i) q^{49} +(-2.50000 - 4.33013i) q^{53} +(-1.50000 - 2.59808i) q^{59} +(-0.500000 + 0.866025i) q^{61} +(-1.50000 + 7.79423i) q^{63} +(-3.50000 - 6.06218i) q^{67} +9.00000 q^{71} +(1.00000 + 1.73205i) q^{73} +(-2.00000 - 1.73205i) q^{77} +(-7.00000 + 12.1244i) q^{79} +(-4.50000 - 7.79423i) q^{81} +8.00000 q^{83} +(8.00000 - 13.8564i) q^{89} +(2.50000 - 0.866025i) q^{91} +2.00000 q^{97} +3.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 5 q^{7} + 3 q^{9} + q^{11} - 2 q^{13} + q^{17} + 5 q^{19} + 2 q^{23} + 5 q^{25} - 10 q^{29} - 8 q^{31} + 8 q^{37} + 12 q^{43} + 11 q^{47} + 11 q^{49} - 5 q^{53} - 3 q^{59} - q^{61} - 3 q^{63} - 7 q^{67}+ \cdots + 6 q^{99}+O(q^{100}) $ 2 * q - 5 * q^7 + 3 * q^9 + q^11 - 2 * q^13 + q^17 + 5 * q^19 + 2 * q^23 + 5 * q^25 - 10 * q^29 - 8 * q^31 + 8 * q^37 + 12 * q^43 + 11 * q^47 + 11 * q^49 - 5 * q^53 - 3 * q^59 - q^61 - 3 * q^63 - 7 * q^67 + 18 * q^71 + 2 * q^73 - 4 * q^77 - 14 * q^79 - 9 * q^81 + 16 * q^83 + 16 * q^89 + 5 * q^91 + 4 * q^97 + 6 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1456\mathbb{Z}\right)^\times$.

$n$	$561$	$911$	$1093$	$1249$
$\chi(n)$	$1$	$1$	$1$	$e\left(\frac{1}{3}\right)$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$3$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$4$		0			0
$5$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$5$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$6$		0			0
$7$	−2.50000	+	0.866025i	−0.944911	+	0.327327i
$7$	−2.50000	+	0.866025i	−0.944911	+	0.327327i
$8$		0			0
$9$	1.50000	−	2.59808i	0.500000	−	0.866025i
$10$		0			0
$11$	0.500000	+	0.866025i	0.150756	+	0.261116i	0.931505	−	0.363727i	$-0.118496\pi$
$11$	0.500000	+	0.866025i	0.150756	+	0.261116i	−0.780750	+	0.624844i	$0.785163\pi$
$12$		0			0
$13$	−1.00000			−0.277350
$13$	−1.00000			−0.277350
$14$		0			0
$15$		0			0
$16$		0			0
$17$	0.500000	+	0.866025i	0.121268	+	0.210042i	0.920268	−	0.391289i	$-0.127971\pi$
$17$	0.500000	+	0.866025i	0.121268	+	0.210042i	−0.799000	+	0.601331i	$0.794637\pi$
$18$		0			0
$19$	2.50000	−	4.33013i	0.573539	−	0.993399i	−0.422659	−	0.906289i	$-0.638903\pi$
$19$	2.50000	−	4.33013i	0.573539	−	0.993399i	0.996199	−	0.0871106i	$-0.0277634\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	1.00000	−	1.73205i	0.208514	−	0.361158i	−0.742732	−	0.669588i	$-0.766471\pi$
$23$	1.00000	−	1.73205i	0.208514	−	0.361158i	0.951247	+	0.308431i	$0.0998038\pi$
$24$		0			0
$25$	2.50000	+	4.33013i	0.500000	+	0.866025i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−5.00000			−0.928477			−0.464238	−	0.885710i	$-0.653672\pi$
$29$	−5.00000			−0.928477			−0.464238	+	0.885710i	$0.653672\pi$
$30$		0			0
$31$	−4.00000	−	6.92820i	−0.718421	−	1.24434i	−0.961625	−	0.274367i	$-0.911532\pi$
$31$	−4.00000	−	6.92820i	−0.718421	−	1.24434i	0.243204	−	0.969975i	$-0.421802\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	4.00000	−	6.92820i	0.657596	−	1.13899i	−0.323640	−	0.946180i	$-0.604907\pi$
$37$	4.00000	−	6.92820i	0.657596	−	1.13899i	0.981236	−	0.192809i	$-0.0617599\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0			−	1.00000i	$-0.5\pi$
$41$		0			0				1.00000i	$0.5\pi$
$42$		0			0
$43$	6.00000			0.914991			0.457496	−	0.889212i	$-0.348747\pi$
$43$	6.00000			0.914991			0.457496	+	0.889212i	$0.348747\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	5.50000	−	9.52628i	0.802257	−	1.38955i	−0.115870	−	0.993264i	$-0.536965\pi$
$47$	5.50000	−	9.52628i	0.802257	−	1.38955i	0.918127	−	0.396286i	$-0.129701\pi$
$48$		0			0
$49$	5.50000	−	4.33013i	0.785714	−	0.618590i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−2.50000	−	4.33013i	−0.343401	−	0.594789i	0.641661	−	0.766989i	$-0.278246\pi$
$53$	−2.50000	−	4.33013i	−0.343401	−	0.594789i	−0.985062	+	0.172200i	$0.944912\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−1.50000	−	2.59808i	−0.195283	−	0.338241i	0.751710	−	0.659494i	$-0.229229\pi$
$59$	−1.50000	−	2.59808i	−0.195283	−	0.338241i	−0.946993	+	0.321253i	$0.895896\pi$
$60$		0			0
$61$	−0.500000	+	0.866025i	−0.0640184	+	0.110883i	−0.896258	−	0.443533i	$-0.853725\pi$
$61$	−0.500000	+	0.866025i	−0.0640184	+	0.110883i	0.832240	+	0.554416i	$0.187058\pi$
$62$		0			0
$63$	−1.50000	+	7.79423i	−0.188982	+	0.981981i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−3.50000	−	6.06218i	−0.427593	−	0.740613i	0.569066	−	0.822292i	$-0.307305\pi$
$67$	−3.50000	−	6.06218i	−0.427593	−	0.740613i	−0.996659	+	0.0816792i	$0.973972\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	9.00000			1.06810			0.534052	−	0.845452i	$-0.320669\pi$
$71$	9.00000			1.06810			0.534052	+	0.845452i	$0.320669\pi$
$72$		0			0
$73$	1.00000	+	1.73205i	0.117041	+	0.202721i	0.918594	−	0.395203i	$-0.129326\pi$
$73$	1.00000	+	1.73205i	0.117041	+	0.202721i	−0.801553	+	0.597924i	$0.795992\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−2.00000	−	1.73205i	−0.227921	−	0.197386i
$78$		0			0
$79$	−7.00000	+	12.1244i	−0.787562	+	1.36410i	0.139895	+	0.990166i	$0.455323\pi$
$79$	−7.00000	+	12.1244i	−0.787562	+	1.36410i	−0.927457	+	0.373930i	$0.878010\pi$
$80$		0			0
$81$	−4.50000	−	7.79423i	−0.500000	−	0.866025i
$82$		0			0
$83$	8.00000			0.878114			0.439057	−	0.898459i	$-0.355313\pi$
$83$	8.00000			0.878114			0.439057	+	0.898459i	$0.355313\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	8.00000	−	13.8564i	0.847998	−	1.46878i	−0.0349934	−	0.999388i	$-0.511141\pi$
$89$	8.00000	−	13.8564i	0.847998	−	1.46878i	0.882992	−	0.469389i	$-0.155526\pi$
$90$		0			0
$91$	2.50000	−	0.866025i	0.262071	−	0.0907841i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	2.00000			0.203069			0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000			0.203069			0.101535	+	0.994832i	$0.467625\pi$
$98$		0			0
$99$	3.00000			0.301511
$100$		0			0
$101$	−9.00000	−	15.5885i	−0.895533	−	1.55111i	−0.833143	−	0.553058i	$-0.813461\pi$
$101$	−9.00000	−	15.5885i	−0.895533	−	1.55111i	−0.0623905	−	0.998052i	$-0.519872\pi$
$102$		0			0
$103$	3.00000	−	5.19615i	0.295599	−	0.511992i	−0.679525	−	0.733652i	$-0.737814\pi$
$103$	3.00000	−	5.19615i	0.295599	−	0.511992i	0.975124	+	0.221660i	$0.0711475\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−4.00000	+	6.92820i	−0.386695	+	0.669775i	−0.992003	−	0.126217i	$-0.959717\pi$
$107$	−4.00000	+	6.92820i	−0.386695	+	0.669775i	0.605308	+	0.795991i	$0.293050\pi$
$108$		0			0
$109$	6.00000	+	10.3923i	0.574696	+	0.995402i	0.996075	+	0.0885176i	$0.0282129\pi$
$109$	6.00000	+	10.3923i	0.574696	+	0.995402i	−0.421379	+	0.906885i	$0.638454\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	1.00000			0.0940721			0.0470360	−	0.998893i	$-0.485022\pi$
$113$	1.00000			0.0940721			0.0470360	+	0.998893i	$0.485022\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−1.50000	+	2.59808i	−0.138675	+	0.240192i
$118$		0			0
$119$	−2.00000	−	1.73205i	−0.183340	−	0.158777i
$120$		0			0
$121$	5.00000	−	8.66025i	0.454545	−	0.787296i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	6.00000			0.532414			0.266207	−	0.963916i	$-0.414230\pi$
$127$	6.00000			0.532414			0.266207	+	0.963916i	$0.414230\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−5.00000	+	8.66025i	−0.436852	+	0.756650i	−0.997445	−	0.0714417i	$-0.977240\pi$
$131$	−5.00000	+	8.66025i	−0.436852	+	0.756650i	0.560593	+	0.828092i	$0.310573\pi$
$132$		0			0
$133$	−2.50000	+	12.9904i	−0.216777	+	1.12641i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	6.00000	+	10.3923i	0.512615	+	0.887875i	0.999893	+	0.0146279i	$0.00465636\pi$
$137$	6.00000	+	10.3923i	0.512615	+	0.887875i	−0.487278	+	0.873247i	$0.662010\pi$
$138$		0			0
$139$	−16.0000			−1.35710			−0.678551	−	0.734553i	$-0.737392\pi$
$139$	−16.0000			−1.35710			−0.678551	+	0.734553i	$0.737392\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−0.500000	−	0.866025i	−0.0418121	−	0.0724207i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−5.00000	+	8.66025i	−0.409616	+	0.709476i	−0.994847	−	0.101391i	$-0.967671\pi$
$149$	−5.00000	+	8.66025i	−0.409616	+	0.709476i	0.585231	+	0.810867i	$0.301004\pi$
$150$		0			0
$151$	−7.50000	−	12.9904i	−0.610341	−	1.05714i	−0.991183	−	0.132502i	$-0.957699\pi$
$151$	−7.50000	−	12.9904i	−0.610341	−	1.05714i	0.380841	−	0.924640i	$-0.375634\pi$
$152$		0			0
$153$	3.00000			0.242536
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−3.50000	−	6.06218i	−0.279330	−	0.483814i	0.691888	−	0.722005i	$-0.256779\pi$
$157$	−3.50000	−	6.06218i	−0.279330	−	0.483814i	−0.971219	+	0.238190i	$0.923446\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−1.00000	+	5.19615i	−0.0788110	+	0.409514i
$162$		0			0
$163$	3.50000	−	6.06218i	0.274141	−	0.474826i	−0.695777	−	0.718258i	$-0.744940\pi$
$163$	3.50000	−	6.06218i	0.274141	−	0.474826i	0.969918	+	0.243432i	$0.0782731\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	19.0000			1.47026			0.735132	−	0.677924i	$-0.237120\pi$
$167$	19.0000			1.47026			0.735132	+	0.677924i	$0.237120\pi$
$168$		0			0
$169$	1.00000			0.0769231
$170$		0			0
$171$	−7.50000	−	12.9904i	−0.573539	−	0.993399i
$172$		0			0
$173$	−7.50000	+	12.9904i	−0.570214	+	0.987640i	0.426329	+	0.904568i	$0.359807\pi$
$173$	−7.50000	+	12.9904i	−0.570214	+	0.987640i	−0.996544	+	0.0830722i	$0.973527\pi$
$174$		0			0
$175$	−10.0000	−	8.66025i	−0.755929	−	0.654654i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−1.00000	−	1.73205i	−0.0747435	−	0.129460i	0.826231	−	0.563331i	$-0.190480\pi$
$179$	−1.00000	−	1.73205i	−0.0747435	−	0.129460i	−0.900975	+	0.433872i	$0.857147\pi$
$180$		0			0
$181$	−23.0000			−1.70958			−0.854788	−	0.518977i	$-0.826313\pi$
$181$	−23.0000			−1.70958			−0.854788	+	0.518977i	$0.826313\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−0.500000	+	0.866025i	−0.0365636	+	0.0633300i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−2.00000	+	3.46410i	−0.144715	+	0.250654i	−0.929267	−	0.369410i	$-0.879560\pi$
$191$	−2.00000	+	3.46410i	−0.144715	+	0.250654i	0.784552	+	0.620063i	$0.212893\pi$
$192$		0			0
$193$	−2.00000	−	3.46410i	−0.143963	−	0.249351i	0.785022	−	0.619467i	$-0.212651\pi$
$193$	−2.00000	−	3.46410i	−0.143963	−	0.249351i	−0.928986	+	0.370116i	$0.879318\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	18.0000			1.28245			0.641223	−	0.767354i	$-0.278427\pi$
$197$	18.0000			1.28245			0.641223	+	0.767354i	$0.278427\pi$
$198$		0			0
$199$	5.00000	+	8.66025i	0.354441	+	0.613909i	0.987022	−	0.160585i	$-0.0513380\pi$
$199$	5.00000	+	8.66025i	0.354441	+	0.613909i	−0.632581	+	0.774494i	$0.718005\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	12.5000	−	4.33013i	0.877328	−	0.303915i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−3.00000	−	5.19615i	−0.208514	−	0.361158i
$208$		0			0
$209$	5.00000			0.345857
$210$		0			0
$211$	2.00000			0.137686			0.0688428	−	0.997628i	$-0.478069\pi$
$211$	2.00000			0.137686			0.0688428	+	0.997628i	$0.478069\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	16.0000	+	13.8564i	1.08615	+	0.940634i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−0.500000	−	0.866025i	−0.0336336	−	0.0582552i
$222$		0			0
$223$	15.0000			1.00447			0.502237	−	0.864730i	$-0.332510\pi$
$223$	15.0000			1.00447			0.502237	+	0.864730i	$0.332510\pi$
$224$		0			0
$225$	15.0000			1.00000
$226$		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
$229$	1.00000	−	1.73205i	0.0660819	−	0.114457i	−0.831092	−	0.556136i	$-0.812283\pi$
$229$	1.00000	−	1.73205i	0.0660819	−	0.114457i	0.897173	+	0.441679i	$0.145617\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−10.5000	+	18.1865i	−0.687878	+	1.19144i	0.284645	+	0.958633i	$0.408124\pi$
$233$	−10.5000	+	18.1865i	−0.687878	+	1.19144i	−0.972523	+	0.232806i	$0.925209\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−9.00000			−0.582162			−0.291081	−	0.956698i	$-0.594015\pi$
$239$	−9.00000			−0.582162			−0.291081	+	0.956698i	$0.594015\pi$
$240$		0			0
$241$	−6.00000	−	10.3923i	−0.386494	−	0.669427i	0.605481	−	0.795860i	$-0.292981\pi$
$241$	−6.00000	−	10.3923i	−0.386494	−	0.669427i	−0.991975	+	0.126432i	$0.959647\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−2.50000	+	4.33013i	−0.159071	+	0.275519i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−26.0000			−1.64111			−0.820553	−	0.571571i	$-0.806334\pi$
$251$	−26.0000			−1.64111			−0.820553	+	0.571571i	$0.806334\pi$
$252$		0			0
$253$	2.00000			0.125739
$254$		0			0
$255$		0			0
$256$		0			0
$257$	7.00000	−	12.1244i	0.436648	−	0.756297i	−0.560781	−	0.827964i	$-0.689499\pi$
$257$	7.00000	−	12.1244i	0.436648	−	0.756297i	0.997429	+	0.0716680i	$0.0228322\pi$
$258$		0			0
$259$	−4.00000	+	20.7846i	−0.248548	+	1.29149i
$260$		0			0
$261$	−7.50000	+	12.9904i	−0.464238	+	0.804084i
$262$		0			0
$263$	12.0000	+	20.7846i	0.739952	+	1.28163i	0.952517	+	0.304487i	$0.0984850\pi$
$263$	12.0000	+	20.7846i	0.739952	+	1.28163i	−0.212565	+	0.977147i	$0.568182\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$	2.50000	+	4.33013i	0.152428	+	0.264013i	0.932119	−	0.362151i	$-0.117958\pi$
$269$	2.50000	+	4.33013i	0.152428	+	0.264013i	−0.779692	+	0.626164i	$0.784624\pi$
$270$		0			0
$271$	−9.50000	+	16.4545i	−0.577084	+	0.999539i	0.418728	+	0.908112i	$0.362476\pi$
$271$	−9.50000	+	16.4545i	−0.577084	+	0.999539i	−0.995812	+	0.0914269i	$0.970857\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−2.50000	+	4.33013i	−0.150756	+	0.261116i
$276$		0			0
$277$	−11.5000	−	19.9186i	−0.690968	−	1.19679i	−0.971521	−	0.236953i	$-0.923851\pi$
$277$	−11.5000	−	19.9186i	−0.690968	−	1.19679i	0.280553	−	0.959839i	$-0.409482\pi$
$278$		0			0
$279$	−24.0000			−1.43684
$280$		0			0
$281$	20.0000			1.19310			0.596550	−	0.802576i	$-0.296538\pi$
$281$	20.0000			1.19310			0.596550	+	0.802576i	$0.296538\pi$
$282$		0			0
$283$	−3.00000	−	5.19615i	−0.178331	−	0.308879i	0.762978	−	0.646425i	$-0.223737\pi$
$283$	−3.00000	−	5.19615i	−0.178331	−	0.308879i	−0.941309	+	0.337546i	$0.890403\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	8.00000	−	13.8564i	0.470588	−	0.815083i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−18.0000			−1.05157			−0.525786	−	0.850617i	$-0.676229\pi$
$293$	−18.0000			−1.05157			−0.525786	+	0.850617i	$0.676229\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−1.00000	+	1.73205i	−0.0578315	+	0.100167i
$300$		0			0
$301$	−15.0000	+	5.19615i	−0.864586	+	0.299501i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	7.00000			0.399511			0.199756	−	0.979846i	$-0.435985\pi$
$307$	7.00000			0.399511			0.199756	+	0.979846i	$0.435985\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	16.0000	+	27.7128i	0.907277	+	1.57145i	0.817832	+	0.575458i	$0.195176\pi$
$311$	16.0000	+	27.7128i	0.907277	+	1.57145i	0.0894452	+	0.995992i	$0.471491\pi$
$312$		0			0
$313$	−9.00000	+	15.5885i	−0.508710	+	0.881112i	0.491239	+	0.871025i	$0.336544\pi$
$313$	−9.00000	+	15.5885i	−0.508710	+	0.881112i	−0.999949	+	0.0100869i	$0.996789\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−13.0000	+	22.5167i	−0.730153	+	1.26466i	0.226665	+	0.973973i	$0.427218\pi$
$317$	−13.0000	+	22.5167i	−0.730153	+	1.26466i	−0.956818	+	0.290689i	$0.906116\pi$
$318$		0			0
$319$	−2.50000	−	4.33013i	−0.139973	−	0.242441i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	5.00000			0.278207
$324$		0			0
$325$	−2.50000	−	4.33013i	−0.138675	−	0.240192i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−5.50000	+	28.5788i	−0.303225	+	1.57560i
$330$		0			0
$331$	−6.00000	+	10.3923i	−0.329790	+	0.571213i	−0.982470	−	0.186421i	$-0.940311\pi$
$331$	−6.00000	+	10.3923i	−0.329790	+	0.571213i	0.652680	+	0.757634i	$0.273645\pi$
$332$		0			0
$333$	−12.0000	−	20.7846i	−0.657596	−	1.13899i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	15.0000			0.817102			0.408551	−	0.912735i	$-0.366034\pi$
$337$	15.0000			0.817102			0.408551	+	0.912735i	$0.366034\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	4.00000	−	6.92820i	0.216612	−	0.375183i
$342$		0			0
$343$	−10.0000	+	15.5885i	−0.539949	+	0.841698i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	2.00000	+	3.46410i	0.107366	+	0.185963i	0.914702	−	0.404128i	$-0.132425\pi$
$347$	2.00000	+	3.46410i	0.107366	+	0.185963i	−0.807337	+	0.590091i	$0.799092\pi$
$348$		0			0
$349$	−34.0000			−1.81998			−0.909989	−	0.414632i	$-0.863910\pi$
$349$	−34.0000			−1.81998			−0.909989	+	0.414632i	$0.863910\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	9.00000	+	15.5885i	0.479022	+	0.829690i	0.999711	−	0.0240566i	$-0.00765819\pi$
$353$	9.00000	+	15.5885i	0.479022	+	0.829690i	−0.520689	+	0.853746i	$0.674325\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−8.00000	+	13.8564i	−0.422224	+	0.731313i	−0.996157	−	0.0875892i	$-0.972084\pi$
$359$	−8.00000	+	13.8564i	−0.422224	+	0.731313i	0.573933	+	0.818902i	$0.305417\pi$
$360$		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
$367$	−5.00000	−	8.66025i	−0.260998	−	0.452062i	0.705509	−	0.708700i	$-0.250718\pi$
$367$	−5.00000	−	8.66025i	−0.260998	−	0.452062i	−0.966507	+	0.256639i	$0.917385\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	10.0000	+	8.66025i	0.519174	+	0.449618i
$372$		0			0
$373$	−15.5000	+	26.8468i	−0.802560	+	1.39007i	0.115367	+	0.993323i	$0.463196\pi$
$373$	−15.5000	+	26.8468i	−0.802560	+	1.39007i	−0.917926	+	0.396751i	$0.870138\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	5.00000			0.257513
$378$		0			0
$379$	4.00000			0.205466			0.102733	−	0.994709i	$-0.467241\pi$
$379$	4.00000			0.205466			0.102733	+	0.994709i	$0.467241\pi$
$380$		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
$387$	9.00000	−	15.5885i	0.457496	−	0.792406i
$388$		0			0
$389$	9.50000	+	16.4545i	0.481669	+	0.834275i	0.999779	−	0.0210389i	$-0.00669738\pi$
$389$	9.50000	+	16.4545i	0.481669	+	0.834275i	−0.518110	+	0.855314i	$0.673364\pi$
$390$		0			0
$391$	2.00000			0.101144
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	9.00000	−	15.5885i	0.451697	−	0.782362i	−0.546795	−	0.837267i	$-0.684152\pi$
$397$	9.00000	−	15.5885i	0.451697	−	0.782362i	0.998492	+	0.0549046i	$0.0174855\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−19.0000	+	32.9090i	−0.948815	+	1.64340i	−0.200888	+	0.979614i	$0.564383\pi$
$401$	−19.0000	+	32.9090i	−0.948815	+	1.64340i	−0.747927	+	0.663781i	$0.768951\pi$
$402$		0			0
$403$	4.00000	+	6.92820i	0.199254	+	0.345118i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	8.00000			0.396545
$408$		0			0
$409$	−18.0000	−	31.1769i	−0.890043	−	1.54160i	−0.839823	−	0.542861i	$-0.817341\pi$
$409$	−18.0000	−	31.1769i	−0.890043	−	1.54160i	−0.0502202	−	0.998738i	$-0.515992\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	6.00000	+	5.19615i	0.295241	+	0.255686i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−6.00000			−0.293119			−0.146560	−	0.989202i	$-0.546820\pi$
$419$	−6.00000			−0.293119			−0.146560	+	0.989202i	$0.546820\pi$
$420$		0			0
$421$	−2.00000			−0.0974740			−0.0487370	−	0.998812i	$-0.515520\pi$
$421$	−2.00000			−0.0974740			−0.0487370	+	0.998812i	$0.515520\pi$
$422$		0			0
$423$	−16.5000	−	28.5788i	−0.802257	−	1.38955i
$424$		0			0
$425$	−2.50000	+	4.33013i	−0.121268	+	0.210042i
$426$		0			0
$427$	0.500000	−	2.59808i	0.0241967	−	0.125730i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$431$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$432$		0			0
$433$	5.00000			0.240285			0.120142	−	0.992757i	$-0.461665\pi$
$433$	5.00000			0.240285			0.120142	+	0.992757i	$0.461665\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−5.00000	−	8.66025i	−0.239182	−	0.414276i
$438$		0			0
$439$	11.0000	−	19.0526i	0.525001	−	0.909329i	−0.474575	−	0.880215i	$-0.657398\pi$
$439$	11.0000	−	19.0526i	0.525001	−	0.909329i	0.999576	−	0.0291138i	$-0.00926853\pi$
$440$		0			0
$441$	−3.00000	−	20.7846i	−0.142857	−	0.989743i
$442$		0			0
$443$	6.00000	−	10.3923i	0.285069	−	0.493753i	−0.687557	−	0.726130i	$-0.741317\pi$
$443$	6.00000	−	10.3923i	0.285069	−	0.493753i	0.972626	+	0.232377i	$0.0746503\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	36.0000			1.69895			0.849473	−	0.527633i	$-0.176920\pi$
$449$	36.0000			1.69895			0.849473	+	0.527633i	$0.176920\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	11.0000	−	19.0526i	0.514558	−	0.891241i	−0.485299	−	0.874348i	$-0.661289\pi$
$457$	11.0000	−	19.0526i	0.514558	−	0.891241i	0.999857	−	0.0168929i	$-0.00537742\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	36.0000			1.67669			0.838344	−	0.545142i	$-0.183524\pi$
$461$	36.0000			1.67669			0.838344	+	0.545142i	$0.183524\pi$
$462$		0			0
$463$	−24.0000			−1.11537			−0.557687	−	0.830051i	$-0.688311\pi$
$463$	−24.0000			−1.11537			−0.557687	+	0.830051i	$0.688311\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−21.0000	+	36.3731i	−0.971764	+	1.68314i	−0.281539	+	0.959550i	$0.590845\pi$
$467$	−21.0000	+	36.3731i	−0.971764	+	1.68314i	−0.690225	+	0.723595i	$0.742488\pi$
$468$		0			0
$469$	14.0000	+	12.1244i	0.646460	+	0.559851i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	3.00000	+	5.19615i	0.137940	+	0.238919i
$474$		0			0
$475$	25.0000			1.14708
$476$		0			0
$477$	−15.0000			−0.686803
$478$		0			0
$479$	−18.5000	−	32.0429i	−0.845287	−	1.46408i	−0.885372	−	0.464883i	$-0.846096\pi$
$479$	−18.5000	−	32.0429i	−0.845287	−	1.46408i	0.0400855	−	0.999196i	$-0.487237\pi$
$480$		0			0
$481$	−4.00000	+	6.92820i	−0.182384	+	0.315899i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	18.5000	+	32.0429i	0.838315	+	1.45200i	0.891303	+	0.453409i	$0.149792\pi$
$487$	18.5000	+	32.0429i	0.838315	+	1.45200i	−0.0529875	+	0.998595i	$0.516874\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	26.0000			1.17336			0.586682	−	0.809818i	$-0.300434\pi$
$491$	26.0000			1.17336			0.586682	+	0.809818i	$0.300434\pi$
$492$		0			0
$493$	−2.50000	−	4.33013i	−0.112594	−	0.195019i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−22.5000	+	7.79423i	−1.00926	+	0.349619i
$498$		0			0
$499$	−16.0000	+	27.7128i	−0.716258	+	1.24060i	0.246214	+	0.969216i	$0.420813\pi$
$499$	−16.0000	+	27.7128i	−0.716258	+	1.24060i	−0.962472	+	0.271380i	$0.912520\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−20.0000			−0.891756			−0.445878	−	0.895094i	$-0.647108\pi$
$503$	−20.0000			−0.891756			−0.445878	+	0.895094i	$0.647108\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$	6.00000	−	10.3923i	0.265945	−	0.460631i	−0.701866	−	0.712309i	$-0.747649\pi$
$509$	6.00000	−	10.3923i	0.265945	−	0.460631i	0.967811	+	0.251679i	$0.0809826\pi$
$510$		0			0
$511$	−4.00000	−	3.46410i	−0.176950	−	0.153243i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$	11.0000			0.483779
$518$		0			0
$519$		0			0
$520$		0			0
$521$	7.00000	+	12.1244i	0.306676	+	0.531178i	0.977633	−	0.210318i	$-0.0674500\pi$
$521$	7.00000	+	12.1244i	0.306676	+	0.531178i	−0.670957	+	0.741496i	$0.734117\pi$
$522$		0			0
$523$	13.0000	−	22.5167i	0.568450	−	0.984585i	−0.428269	−	0.903651i	$-0.640876\pi$
$523$	13.0000	−	22.5167i	0.568450	−	0.984585i	0.996719	−	0.0809336i	$-0.0257902\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	4.00000	−	6.92820i	0.174243	−	0.301797i
$528$		0			0
$529$	9.50000	+	16.4545i	0.413043	+	0.715412i
$530$		0			0
$531$	−9.00000			−0.390567
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	6.50000	+	2.59808i	0.279975	+	0.111907i
$540$		0			0
$541$	12.0000	−	20.7846i	0.515920	−	0.893600i	−0.483909	−	0.875118i	$-0.660783\pi$
$541$	12.0000	−	20.7846i	0.515920	−	0.893600i	0.999829	−	0.0184818i	$-0.00588327\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	38.0000			1.62476			0.812381	−	0.583127i	$-0.198171\pi$
$547$	38.0000			1.62476			0.812381	+	0.583127i	$0.198171\pi$
$548$		0			0
$549$	1.50000	+	2.59808i	0.0640184	+	0.110883i
$550$		0			0
$551$	−12.5000	+	21.6506i	−0.532518	+	0.922348i
$552$		0			0
$553$	7.00000	−	36.3731i	0.297670	−	1.54674i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	15.0000	+	25.9808i	0.635570	+	1.10084i	0.986394	+	0.164399i	$0.0525683\pi$
$557$	15.0000	+	25.9808i	0.635570	+	1.10084i	−0.350824	+	0.936442i	$0.614098\pi$
$558$		0			0
$559$	−6.00000			−0.253773
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−2.00000	−	3.46410i	−0.0842900	−	0.145994i	0.820798	−	0.571218i	$-0.193529\pi$
$563$	−2.00000	−	3.46410i	−0.0842900	−	0.145994i	−0.905088	+	0.425223i	$0.860196\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	18.0000	+	15.5885i	0.755929	+	0.654654i
$568$		0			0
$569$	7.50000	−	12.9904i	0.314416	−	0.544585i	−0.664897	−	0.746935i	$-0.731525\pi$
$569$	7.50000	−	12.9904i	0.314416	−	0.544585i	0.979313	+	0.202350i	$0.0648579\pi$
$570$		0			0
$571$	−22.0000	−	38.1051i	−0.920671	−	1.59465i	−0.798379	−	0.602155i	$-0.794309\pi$
$571$	−22.0000	−	38.1051i	−0.920671	−	1.59465i	−0.122292	−	0.992494i	$-0.539025\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	10.0000			0.417029
$576$		0			0
$577$	−7.00000	−	12.1244i	−0.291414	−	0.504744i	0.682730	−	0.730670i	$-0.260792\pi$
$577$	−7.00000	−	12.1244i	−0.291414	−	0.504744i	−0.974144	+	0.225927i	$0.927459\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−20.0000	+	6.92820i	−0.829740	+	0.287430i
$582$		0			0
$583$	2.50000	−	4.33013i	0.103539	−	0.179336i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−33.0000			−1.36206			−0.681028	−	0.732257i	$-0.738467\pi$
$587$	−33.0000			−1.36206			−0.681028	+	0.732257i	$0.738467\pi$
$588$		0			0
$589$	−40.0000			−1.64817
$590$		0			0
$591$		0			0
$592$		0			0
$593$	8.00000	−	13.8564i	0.328521	−	0.569014i	−0.653698	−	0.756756i	$-0.726783\pi$
$593$	8.00000	−	13.8564i	0.328521	−	0.569014i	0.982219	+	0.187741i	$0.0601166\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$599$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$600$		0			0
$601$	17.0000			0.693444			0.346722	−	0.937968i	$-0.387295\pi$
$601$	17.0000			0.693444			0.346722	+	0.937968i	$0.387295\pi$
$602$		0			0
$603$	−21.0000			−0.855186
$604$		0			0
$605$		0			0
$606$		0			0
$607$	11.0000	−	19.0526i	0.446476	−	0.773320i	−0.551678	−	0.834058i	$-0.686012\pi$
$607$	11.0000	−	19.0526i	0.446476	−	0.773320i	0.998154	+	0.0607380i	$0.0193454\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−5.50000	+	9.52628i	−0.222506	+	0.385392i
$612$		0			0
$613$	−8.00000	−	13.8564i	−0.323117	−	0.559655i	0.658012	−	0.753007i	$-0.271397\pi$
$613$	−8.00000	−	13.8564i	−0.323117	−	0.559655i	−0.981129	+	0.193352i	$0.938064\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−18.0000			−0.724653			−0.362326	−	0.932051i	$-0.618017\pi$
$617$	−18.0000			−0.724653			−0.362326	+	0.932051i	$0.618017\pi$
$618$		0			0
$619$	−18.0000	−	31.1769i	−0.723481	−	1.25311i	−0.959596	−	0.281381i	$-0.909208\pi$
$619$	−18.0000	−	31.1769i	−0.723481	−	1.25311i	0.236115	−	0.971725i	$-0.424126\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−8.00000	+	41.5692i	−0.320513	+	1.66544i
$624$		0			0
$625$	−12.5000	+	21.6506i	−0.500000	+	0.866025i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	8.00000			0.318981
$630$		0			0
$631$	8.00000			0.318475			0.159237	−	0.987240i	$-0.449096\pi$
$631$	8.00000			0.318475			0.159237	+	0.987240i	$0.449096\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−5.50000	+	4.33013i	−0.217918	+	0.171566i
$638$		0			0
$639$	13.5000	−	23.3827i	0.534052	−	0.925005i
$640$		0			0
$641$	−9.00000	−	15.5885i	−0.355479	−	0.615707i	0.631721	−	0.775196i	$-0.282349\pi$
$641$	−9.00000	−	15.5885i	−0.355479	−	0.615707i	−0.987200	+	0.159489i	$0.949015\pi$
$642$		0			0
$643$	−21.0000			−0.828159			−0.414080	−	0.910241i	$-0.635896\pi$
$643$	−21.0000			−0.828159			−0.414080	+	0.910241i	$0.635896\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−9.00000	−	15.5885i	−0.353827	−	0.612845i	0.633090	−	0.774078i	$-0.281786\pi$
$647$	−9.00000	−	15.5885i	−0.353827	−	0.612845i	−0.986916	+	0.161233i	$0.948453\pi$
$648$		0			0
$649$	1.50000	−	2.59808i	0.0588802	−	0.101983i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	3.00000	−	5.19615i	0.117399	−	0.203341i	−0.801337	−	0.598213i	$-0.795878\pi$
$653$	3.00000	−	5.19615i	0.117399	−	0.203341i	0.918736	+	0.394872i	$0.129211\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	6.00000			0.234082
$658$		0			0
$659$	40.0000			1.55818			0.779089	−	0.626913i	$-0.215682\pi$
$659$	40.0000			1.55818			0.779089	+	0.626913i	$0.215682\pi$
$660$		0			0
$661$	16.0000	+	27.7128i	0.622328	+	1.07790i	0.989051	+	0.147573i	$0.0471463\pi$
$661$	16.0000	+	27.7128i	0.622328	+	1.07790i	−0.366723	+	0.930330i	$0.619520\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−5.00000	+	8.66025i	−0.193601	+	0.335326i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−1.00000			−0.0386046
$672$		0			0
$673$	6.00000			0.231283			0.115642	−	0.993291i	$-0.463108\pi$
$673$	6.00000			0.231283			0.115642	+	0.993291i	$0.463108\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	13.5000	−	23.3827i	0.518847	−	0.898670i	−0.480913	−	0.876768i	$-0.659695\pi$
$677$	13.5000	−	23.3827i	0.518847	−	0.898670i	0.999760	−	0.0219013i	$-0.00697196\pi$
$678$		0			0
$679$	−5.00000	+	1.73205i	−0.191882	+	0.0664700i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−24.0000	−	41.5692i	−0.918334	−	1.59060i	−0.801945	−	0.597398i	$-0.796201\pi$
$683$	−24.0000	−	41.5692i	−0.918334	−	1.59060i	−0.116390	−	0.993204i	$-0.537132\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$	2.50000	+	4.33013i	0.0952424	+	0.164965i
$690$		0			0
$691$	7.50000	−	12.9904i	0.285313	−	0.494177i	−0.687372	−	0.726306i	$-0.741236\pi$
$691$	7.50000	−	12.9904i	0.285313	−	0.494177i	0.972685	+	0.232128i	$0.0745690\pi$
$692$		0			0
$693$	−7.50000	+	2.59808i	−0.284901	+	0.0986928i
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$	30.0000			1.13308			0.566542	−	0.824033i	$-0.308281\pi$
$701$	30.0000			1.13308			0.566542	+	0.824033i	$0.308281\pi$
$702$		0			0
$703$	−20.0000	−	34.6410i	−0.754314	−	1.30651i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	36.0000	+	31.1769i	1.35392	+	1.17253i
$708$		0			0
$709$	−1.00000	+	1.73205i	−0.0375558	+	0.0650485i	−0.884192	−	0.467123i	$-0.845291\pi$
$709$	−1.00000	+	1.73205i	−0.0375558	+	0.0650485i	0.846637	+	0.532172i	$0.178624\pi$
$710$		0			0
$711$	21.0000	+	36.3731i	0.787562	+	1.36410i
$712$		0			0
$713$	−16.0000			−0.599205
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	15.0000	−	25.9808i	0.559406	−	0.968919i	−0.438141	−	0.898906i	$-0.644363\pi$
$719$	15.0000	−	25.9808i	0.559406	−	0.968919i	0.997546	−	0.0700124i	$-0.0223039\pi$
$720$		0			0
$721$	−3.00000	+	15.5885i	−0.111726	+	0.580544i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−12.5000	−	21.6506i	−0.464238	−	0.804084i
$726$		0			0
$727$	4.00000			0.148352			0.0741759	−	0.997245i	$-0.476367\pi$
$727$	4.00000			0.148352			0.0741759	+	0.997245i	$0.476367\pi$
$728$		0			0
$729$	−27.0000			−1.00000
$730$		0			0
$731$	3.00000	+	5.19615i	0.110959	+	0.192187i
$732$		0			0
$733$	−13.0000	+	22.5167i	−0.480166	+	0.831672i	−0.999741	−	0.0227529i	$-0.992757\pi$
$733$	−13.0000	+	22.5167i	−0.480166	+	0.831672i	0.519575	+	0.854425i	$0.326090\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	3.50000	−	6.06218i	0.128924	−	0.223303i
$738$		0			0
$739$	−2.00000	−	3.46410i	−0.0735712	−	0.127429i	0.826893	−	0.562360i	$-0.190106\pi$
$739$	−2.00000	−	3.46410i	−0.0735712	−	0.127429i	−0.900464	+	0.434930i	$0.856773\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	27.0000			0.990534			0.495267	−	0.868741i	$-0.335070\pi$
$743$	27.0000			0.990534			0.495267	+	0.868741i	$0.335070\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	12.0000	−	20.7846i	0.439057	−	0.760469i
$748$		0			0
$749$	4.00000	−	20.7846i	0.146157	−	0.759453i
$750$		0			0
$751$	14.0000	−	24.2487i	0.510867	−	0.884848i	−0.489053	−	0.872254i	$-0.662658\pi$
$751$	14.0000	−	24.2487i	0.510867	−	0.884848i	0.999921	−	0.0125942i	$-0.00400897\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	33.0000			1.19941			0.599703	−	0.800223i	$-0.295286\pi$
$757$	33.0000			1.19941			0.599703	+	0.800223i	$0.295286\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	24.0000	−	41.5692i	0.869999	−	1.50688i	0.00800331	−	0.999968i	$-0.497452\pi$
$761$	24.0000	−	41.5692i	0.869999	−	1.50688i	0.861996	−	0.506915i	$-0.169214\pi$
$762$		0			0
$763$	−24.0000	−	20.7846i	−0.868858	−	0.752453i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	1.50000	+	2.59808i	0.0541619	+	0.0938111i
$768$		0			0
$769$	−46.0000			−1.65880			−0.829401	−	0.558653i	$-0.811318\pi$
$769$	−46.0000			−1.65880			−0.829401	+	0.558653i	$0.811318\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−5.00000	−	8.66025i	−0.179838	−	0.311488i	0.761987	−	0.647592i	$-0.224224\pi$
$773$	−5.00000	−	8.66025i	−0.179838	−	0.311488i	−0.941825	+	0.336104i	$0.890891\pi$
$774$		0			0
$775$	20.0000	−	34.6410i	0.718421	−	1.24434i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$	4.50000	+	7.79423i	0.161023	+	0.278899i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	5.50000	+	9.52628i	0.196054	+	0.339575i	0.947245	−	0.320509i	$-0.103854\pi$
$787$	5.50000	+	9.52628i	0.196054	+	0.339575i	−0.751192	+	0.660084i	$0.770521\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−2.50000	+	0.866025i	−0.0888898	+	0.0307923i
$792$		0			0
$793$	0.500000	−	0.866025i	0.0177555	−	0.0307535i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	54.0000			1.91278			0.956389	−	0.292096i	$-0.0943526\pi$
$797$	54.0000			1.91278			0.956389	+	0.292096i	$0.0943526\pi$
$798$		0			0
$799$	11.0000			0.389152
$800$		0			0
$801$	−24.0000	−	41.5692i	−0.847998	−	1.46878i
$802$		0			0
$803$	−1.00000	+	1.73205i	−0.0352892	+	0.0611227i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−4.50000	−	7.79423i	−0.158212	−	0.274030i	0.776012	−	0.630718i	$-0.217239\pi$
$809$	−4.50000	−	7.79423i	−0.158212	−	0.274030i	−0.934224	+	0.356687i	$0.883906\pi$
$810$		0			0
$811$	20.0000			0.702295			0.351147	−	0.936320i	$-0.385792\pi$
$811$	20.0000			0.702295			0.351147	+	0.936320i	$0.385792\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	15.0000	−	25.9808i	0.524784	−	0.908952i
$818$		0			0
$819$	1.50000	−	7.79423i	0.0524142	−	0.272352i
$820$		0			0
$821$	6.00000	−	10.3923i	0.209401	−	0.362694i	−0.742125	−	0.670262i	$-0.766182\pi$
$821$	6.00000	−	10.3923i	0.209401	−	0.362694i	0.951526	+	0.307568i	$0.0995151\pi$
$822$		0			0
$823$	18.0000	+	31.1769i	0.627441	+	1.08676i	0.988063	+	0.154047i	$0.0492308\pi$
$823$	18.0000	+	31.1769i	0.627441	+	1.08676i	−0.360623	+	0.932712i	$0.617436\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	33.0000			1.14752			0.573761	−	0.819023i	$-0.305484\pi$
$827$	33.0000			1.14752			0.573761	+	0.819023i	$0.305484\pi$
$828$		0			0
$829$	−13.5000	−	23.3827i	−0.468874	−	0.812114i	0.530493	−	0.847690i	$-0.322007\pi$
$829$	−13.5000	−	23.3827i	−0.468874	−	0.812114i	−0.999367	+	0.0355753i	$0.988674\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	6.50000	+	2.59808i	0.225212	+	0.0900180i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$	37.0000			1.27738			0.638691	−	0.769463i	$-0.279476\pi$
$839$	37.0000			1.27738			0.638691	+	0.769463i	$0.279476\pi$
$840$		0			0
$841$	−4.00000			−0.137931
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−5.00000	+	25.9808i	−0.171802	+	0.892710i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−8.00000	−	13.8564i	−0.274236	−	0.474991i
$852$		0			0
$853$	38.0000			1.30110			0.650548	−	0.759465i	$-0.274539\pi$
$853$	38.0000			1.30110			0.650548	+	0.759465i	$0.274539\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	21.5000	+	37.2391i	0.734426	+	1.27206i	0.954975	+	0.296687i	$0.0958819\pi$
$857$	21.5000	+	37.2391i	0.734426	+	1.27206i	−0.220549	+	0.975376i	$0.570785\pi$
$858$		0			0
$859$	−20.0000	+	34.6410i	−0.682391	+	1.18194i	0.291858	+	0.956462i	$0.405727\pi$
$859$	−20.0000	+	34.6410i	−0.682391	+	1.18194i	−0.974249	+	0.225475i	$0.927607\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−6.00000	+	10.3923i	−0.204242	+	0.353758i	−0.949891	−	0.312581i	$-0.898806\pi$
$863$	−6.00000	+	10.3923i	−0.204242	+	0.353758i	0.745649	+	0.666339i	$0.232140\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−14.0000			−0.474917
$870$		0			0
$871$	3.50000	+	6.06218i	0.118593	+	0.205409i
$872$		0			0
$873$	3.00000	−	5.19615i	0.101535	−	0.175863i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	16.0000	−	27.7128i	0.540282	−	0.935795i	−0.458606	−	0.888640i	$-0.651651\pi$
$877$	16.0000	−	27.7128i	0.540282	−	0.935795i	0.998888	−	0.0471555i	$-0.0150156\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	42.0000			1.41502			0.707508	−	0.706705i	$-0.249819\pi$
$881$	42.0000			1.41502			0.707508	+	0.706705i	$0.249819\pi$
$882$		0			0
$883$	−24.0000			−0.807664			−0.403832	−	0.914833i	$-0.632322\pi$
$883$	−24.0000			−0.807664			−0.403832	+	0.914833i	$0.632322\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	9.00000	−	15.5885i	0.302190	−	0.523409i	−0.674441	−	0.738328i	$-0.735615\pi$
$887$	9.00000	−	15.5885i	0.302190	−	0.523409i	0.976632	+	0.214919i	$0.0689488\pi$
$888$		0			0
$889$	−15.0000	+	5.19615i	−0.503084	+	0.174273i
$890$		0			0
$891$	4.50000	−	7.79423i	0.150756	−	0.261116i
$892$		0			0
$893$	−27.5000	−	47.6314i	−0.920252	−	1.59392i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$	20.0000	+	34.6410i	0.667037	+	1.15534i
$900$		0			0
$901$	2.50000	−	4.33013i	0.0832871	−	0.144257i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−23.0000	−	39.8372i	−0.763702	−	1.32277i	−0.940930	−	0.338602i	$-0.890046\pi$
$907$	−23.0000	−	39.8372i	−0.763702	−	1.32277i	0.177227	−	0.984170i	$-0.443287\pi$
$908$		0			0
$909$	−54.0000			−1.79107
$910$		0			0
$911$	−34.0000			−1.12647			−0.563235	−	0.826297i	$-0.690443\pi$
$911$	−34.0000			−1.12647			−0.563235	+	0.826297i	$0.690443\pi$
$912$		0			0
$913$	4.00000	+	6.92820i	0.132381	+	0.229290i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	5.00000	−	25.9808i	0.165115	−	0.857960i
$918$		0			0
$919$	−19.0000	+	32.9090i	−0.626752	+	1.08557i	0.361447	+	0.932393i	$0.382283\pi$
$919$	−19.0000	+	32.9090i	−0.626752	+	1.08557i	−0.988199	+	0.153174i	$0.951051\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−9.00000			−0.296239
$924$		0			0
$925$	40.0000			1.31519
$926$		0			0
$927$	−9.00000	−	15.5885i	−0.295599	−	0.511992i
$928$		0			0
$929$	−23.0000	+	39.8372i	−0.754606	+	1.30702i	0.190965	+	0.981597i	$0.438838\pi$
$929$	−23.0000	+	39.8372i	−0.754606	+	1.30702i	−0.945570	+	0.325418i	$0.894495\pi$
$930$		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
$937$	−41.0000			−1.33941			−0.669706	−	0.742627i	$-0.733580\pi$
$937$	−41.0000			−1.33941			−0.669706	+	0.742627i	$0.733580\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−6.00000	−	10.3923i	−0.195594	−	0.338779i	0.751501	−	0.659732i	$-0.229330\pi$
$941$	−6.00000	−	10.3923i	−0.195594	−	0.338779i	−0.947095	+	0.320953i	$0.895997\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−3.50000	+	6.06218i	−0.113735	+	0.196994i	−0.917273	−	0.398258i	$-0.869615\pi$
$947$	−3.50000	+	6.06218i	−0.113735	+	0.196994i	0.803539	+	0.595253i	$0.202948\pi$
$948$		0			0
$949$	−1.00000	−	1.73205i	−0.0324614	−	0.0562247i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	9.00000			0.291539			0.145769	−	0.989319i	$-0.453434\pi$
$953$	9.00000			0.291539			0.145769	+	0.989319i	$0.453434\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−24.0000	−	20.7846i	−0.775000	−	0.671170i
$960$		0			0
$961$	−16.5000	+	28.5788i	−0.532258	+	0.921898i
$962$		0			0
$963$	12.0000	+	20.7846i	0.386695	+	0.669775i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−23.0000			−0.739630			−0.369815	−	0.929105i	$-0.620579\pi$
$967$	−23.0000			−0.739630			−0.369815	+	0.929105i	$0.620579\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−30.0000	+	51.9615i	−0.962746	+	1.66752i	−0.247193	+	0.968966i	$0.579508\pi$
$971$	−30.0000	+	51.9615i	−0.962746	+	1.66752i	−0.715553	+	0.698558i	$0.753825\pi$
$972$		0			0
$973$	40.0000	−	13.8564i	1.28234	−	0.444216i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−5.00000	−	8.66025i	−0.159964	−	0.277066i	0.774891	−	0.632094i	$-0.217805\pi$
$977$	−5.00000	−	8.66025i	−0.159964	−	0.277066i	−0.934856	+	0.355028i	$0.884471\pi$
$978$		0			0
$979$	16.0000			0.511362
$980$		0			0
$981$	36.0000			1.14939
$982$		0			0
$983$	6.50000	+	11.2583i	0.207318	+	0.359085i	0.950869	−	0.309594i	$-0.100193\pi$
$983$	6.50000	+	11.2583i	0.207318	+	0.359085i	−0.743551	+	0.668679i	$0.766860\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	6.00000	−	10.3923i	0.190789	−	0.330456i
$990$		0			0
$991$	−14.0000	−	24.2487i	−0.444725	−	0.770286i	0.553308	−	0.832977i	$-0.313365\pi$
$991$	−14.0000	−	24.2487i	−0.444725	−	0.770286i	−0.998033	+	0.0626908i	$0.980032\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	7.50000	+	12.9904i	0.237527	+	0.411409i	0.960004	−	0.279986i	$-0.0903297\pi$
$997$	7.50000	+	12.9904i	0.237527	+	0.411409i	−0.722477	+	0.691395i	$0.756996\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1456.2.r.f.625.1		2
4.3	odd	2		182.2.f.a.79.1	yes	2
7.4	even	3	inner	1456.2.r.f.417.1		2
12.11	even	2		1638.2.j.c.1171.1		2
28.3	even	6		1274.2.f.q.1145.1		2
28.11	odd	6		182.2.f.a.53.1	✓	2
28.19	even	6		1274.2.a.f.1.1		1
28.23	odd	6		1274.2.a.e.1.1		1
28.27	even	2		1274.2.f.q.79.1		2
84.11	even	6		1638.2.j.c.235.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
182.2.f.a.53.1	✓	2	28.11	odd	6
182.2.f.a.79.1	yes	2	4.3	odd	2
1274.2.a.e.1.1		1	28.23	odd	6
1274.2.a.f.1.1		1	28.19	even	6
1274.2.f.q.79.1		2	28.27	even	2
1274.2.f.q.1145.1		2	28.3	even	6
1456.2.r.f.417.1		2	7.4	even	3	inner
1456.2.r.f.625.1		2	1.1	even	1	trivial
1638.2.j.c.235.1		2	84.11	even	6
1638.2.j.c.1171.1		2	12.11	even	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:	\( N \)	\(=\)	\( 1456 = 2^{4} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1456.r (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-2.50000 + 0.866025i) q^{7} +(1.50000 - 2.59808i) q^{9} +(0.500000 + 0.866025i) q^{11} -1.00000 q^{13} +(0.500000 + 0.866025i) q^{17} +(2.50000 - 4.33013i) q^{19} +(1.00000 - 1.73205i) q^{23} +(2.50000 + 4.33013i) q^{25} -5.00000 q^{29} +(-4.00000 - 6.92820i) q^{31} +(4.00000 - 6.92820i) q^{37} +6.00000 q^{43} +(5.50000 - 9.52628i) q^{47} +(5.50000 - 4.33013i) q^{49} +(-2.50000 - 4.33013i) q^{53} +(-1.50000 - 2.59808i) q^{59} +(-0.500000 + 0.866025i) q^{61} +(-1.50000 + 7.79423i) q^{63} +(-3.50000 - 6.06218i) q^{67} +9.00000 q^{71} +(1.00000 + 1.73205i) q^{73} +(-2.00000 - 1.73205i) q^{77} +(-7.00000 + 12.1244i) q^{79} +(-4.50000 - 7.79423i) q^{81} +8.00000 q^{83} +(8.00000 - 13.8564i) q^{89} +(2.50000 - 0.866025i) q^{91} +2.00000 q^{97} +3.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 5 q^{7} + 3 q^{9} + q^{11} - 2 q^{13} + q^{17} + 5 q^{19} + 2 q^{23} + 5 q^{25} - 10 q^{29} - 8 q^{31} + 8 q^{37} + 12 q^{43} + 11 q^{47} + 11 q^{49} - 5 q^{53} - 3 q^{59} - q^{61} - 3 q^{63} - 7 q^{67}+ \cdots + 6 q^{99}+O(q^{100}) \) 2 * q - 5 * q^7 + 3 * q^9 + q^11 - 2 * q^13 + q^17 + 5 * q^19 + 2 * q^23 + 5 * q^25 - 10 * q^29 - 8 * q^31 + 8 * q^37 + 12 * q^43 + 11 * q^47 + 11 * q^49 - 5 * q^53 - 3 * q^59 - q^61 - 3 * q^63 - 7 * q^67 + 18 * q^71 + 2 * q^73 - 4 * q^77 - 14 * q^79 - 9 * q^81 + 16 * q^83 + 16 * q^89 + 5 * q^91 + 4 * q^97 + 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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