LMFDB - Embedded newform 1080.2.q.a.721.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1080,2,Mod(361,1080)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1080, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("1080.361");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1080 = 2^{3} \cdot 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1080.q (of order $3$, degree $2$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$8.62384341830$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 360)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			721.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	1080.721
Dual form			1080.2.q.a.361.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.500000 - 0.866025i) q^{5} +O(q^{10})$	$q+(-0.500000 - 0.866025i) q^{5} +(-2.50000 + 4.33013i) q^{11} -3.00000 q^{17} +5.00000 q^{19} +(3.00000 + 5.19615i) q^{23} +(-0.500000 + 0.866025i) q^{25} +(-5.00000 + 8.66025i) q^{29} +(1.00000 + 1.73205i) q^{31} +4.00000 q^{37} +(-1.50000 - 2.59808i) q^{41} +(-1.50000 + 2.59808i) q^{43} +(2.00000 - 3.46410i) q^{47} +(3.50000 + 6.06218i) q^{49} +6.00000 q^{53} +5.00000 q^{55} +(-1.50000 - 2.59808i) q^{59} +(-1.00000 + 1.73205i) q^{61} +(5.50000 + 9.52628i) q^{67} +14.0000 q^{71} -15.0000 q^{73} +(-5.00000 + 8.66025i) q^{79} +(-6.00000 + 10.3923i) q^{83} +(1.50000 + 2.59808i) q^{85} -14.0000 q^{89} +(-2.50000 - 4.33013i) q^{95} +(6.50000 - 11.2583i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{5}+O(q^{10}) $ 2 * q - q^5	$ 2 q - q^{5} - 5 q^{11} - 6 q^{17} + 10 q^{19} + 6 q^{23} - q^{25} - 10 q^{29} + 2 q^{31} + 8 q^{37} - 3 q^{41} - 3 q^{43} + 4 q^{47} + 7 q^{49} + 12 q^{53} + 10 q^{55} - 3 q^{59} - 2 q^{61} + 11 q^{67} + 28 q^{71} - 30 q^{73} - 10 q^{79} - 12 q^{83} + 3 q^{85} - 28 q^{89} - 5 q^{95} + 13 q^{97}+O(q^{100}) $ 2 * q - q^5 - 5 * q^11 - 6 * q^17 + 10 * q^19 + 6 * q^23 - q^25 - 10 * q^29 + 2 * q^31 + 8 * q^37 - 3 * q^41 - 3 * q^43 + 4 * q^47 + 7 * q^49 + 12 * q^53 + 10 * q^55 - 3 * q^59 - 2 * q^61 + 11 * q^67 + 28 * q^71 - 30 * q^73 - 10 * q^79 - 12 * q^83 + 3 * q^85 - 28 * q^89 - 5 * q^95 + 13 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1080\mathbb{Z}\right)^\times$.

$n$	$217$	$271$	$541$	$1001$
$\chi(n)$	$1$	$1$	$1$	$e\left(\frac{2}{3}\right)$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−0.500000	−	0.866025i	−0.223607	−	0.387298i
$5$	−0.500000	−	0.866025i	−0.223607	−	0.387298i
$6$		0			0
$7$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$7$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−2.50000	+	4.33013i	−0.753778	+	1.30558i	0.192201	+	0.981356i	$0.438437\pi$
$11$	−2.50000	+	4.33013i	−0.753778	+	1.30558i	−0.945979	+	0.324227i	$0.894896\pi$
$12$		0			0
$13$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$13$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−3.00000			−0.727607			−0.363803	−	0.931476i	$-0.618522\pi$
$17$	−3.00000			−0.727607			−0.363803	+	0.931476i	$0.618522\pi$
$18$		0			0
$19$	5.00000			1.14708			0.573539	−	0.819178i	$-0.305570\pi$
$19$	5.00000			1.14708			0.573539	+	0.819178i	$0.305570\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	3.00000	+	5.19615i	0.625543	+	1.08347i	0.988436	+	0.151642i	$0.0484560\pi$
$23$	3.00000	+	5.19615i	0.625543	+	1.08347i	−0.362892	+	0.931831i	$0.618211\pi$
$24$		0			0
$25$	−0.500000	+	0.866025i	−0.100000	+	0.173205i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−5.00000	+	8.66025i	−0.928477	+	1.60817i	−0.142605	+	0.989780i	$0.545548\pi$
$29$	−5.00000	+	8.66025i	−0.928477	+	1.60817i	−0.785872	+	0.618389i	$0.787786\pi$
$30$		0			0
$31$	1.00000	+	1.73205i	0.179605	+	0.311086i	0.941745	−	0.336327i	$-0.109185\pi$
$31$	1.00000	+	1.73205i	0.179605	+	0.311086i	−0.762140	+	0.647412i	$0.775851\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	4.00000			0.657596			0.328798	−	0.944400i	$-0.393356\pi$
$37$	4.00000			0.657596			0.328798	+	0.944400i	$0.393356\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−1.50000	−	2.59808i	−0.234261	−	0.405751i	0.724797	−	0.688963i	$-0.241934\pi$
$41$	−1.50000	−	2.59808i	−0.234261	−	0.405751i	−0.959058	+	0.283211i	$0.908600\pi$
$42$		0			0
$43$	−1.50000	+	2.59808i	−0.228748	+	0.396203i	−0.957437	−	0.288641i	$-0.906796\pi$
$43$	−1.50000	+	2.59808i	−0.228748	+	0.396203i	0.728689	+	0.684844i	$0.240130\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	2.00000	−	3.46410i	0.291730	−	0.505291i	−0.682489	−	0.730896i	$-0.739102\pi$
$47$	2.00000	−	3.46410i	0.291730	−	0.505291i	0.974219	+	0.225605i	$0.0724358\pi$
$48$		0			0
$49$	3.50000	+	6.06218i	0.500000	+	0.866025i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	6.00000			0.824163			0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000			0.824163			0.412082	+	0.911147i	$0.364802\pi$
$54$		0			0
$55$	5.00000			0.674200
$56$		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
$61$	−1.00000	+	1.73205i	−0.128037	+	0.221766i	−0.922916	−	0.385002i	$-0.874201\pi$
$61$	−1.00000	+	1.73205i	−0.128037	+	0.221766i	0.794879	+	0.606768i	$0.207534\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	5.50000	+	9.52628i	0.671932	+	1.16382i	0.977356	+	0.211604i	$0.0678686\pi$
$67$	5.50000	+	9.52628i	0.671932	+	1.16382i	−0.305424	+	0.952217i	$0.598798\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	14.0000			1.66149			0.830747	−	0.556650i	$-0.187914\pi$
$71$	14.0000			1.66149			0.830747	+	0.556650i	$0.187914\pi$
$72$		0			0
$73$	−15.0000			−1.75562			−0.877809	−	0.479012i	$-0.840995\pi$
$73$	−15.0000			−1.75562			−0.877809	+	0.479012i	$0.840995\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−5.00000	+	8.66025i	−0.562544	+	0.974355i	0.434730	+	0.900561i	$0.356844\pi$
$79$	−5.00000	+	8.66025i	−0.562544	+	0.974355i	−0.997274	+	0.0737937i	$0.976489\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−6.00000	+	10.3923i	−0.658586	+	1.14070i	0.322396	+	0.946605i	$0.395512\pi$
$83$	−6.00000	+	10.3923i	−0.658586	+	1.14070i	−0.980982	+	0.194099i	$0.937822\pi$
$84$		0			0
$85$	1.50000	+	2.59808i	0.162698	+	0.281801i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−14.0000			−1.48400			−0.741999	−	0.670402i	$-0.766122\pi$
$89$	−14.0000			−1.48400			−0.741999	+	0.670402i	$0.766122\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−2.50000	−	4.33013i	−0.256495	−	0.444262i
$96$		0			0
$97$	6.50000	−	11.2583i	0.659975	−	1.14311i	−0.320647	−	0.947199i	$-0.603900\pi$
$97$	6.50000	−	11.2583i	0.659975	−	1.14311i	0.980622	−	0.195911i	$-0.0627665\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−6.00000	+	10.3923i	−0.597022	+	1.03407i	0.396236	+	0.918149i	$0.370316\pi$
$101$	−6.00000	+	10.3923i	−0.597022	+	1.03407i	−0.993258	+	0.115924i	$0.963017\pi$
$102$		0			0
$103$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$103$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	17.0000			1.64345			0.821726	−	0.569883i	$-0.193011\pi$
$107$	17.0000			1.64345			0.821726	+	0.569883i	$0.193011\pi$
$108$		0			0
$109$	−12.0000			−1.14939			−0.574696	−	0.818367i	$-0.694880\pi$
$109$	−12.0000			−1.14939			−0.574696	+	0.818367i	$0.694880\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−3.00000	−	5.19615i	−0.282216	−	0.488813i	0.689714	−	0.724082i	$-0.257736\pi$
$113$	−3.00000	−	5.19615i	−0.282216	−	0.488813i	−0.971930	+	0.235269i	$0.924403\pi$
$114$		0			0
$115$	3.00000	−	5.19615i	0.279751	−	0.484544i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−7.00000	−	12.1244i	−0.636364	−	1.10221i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	1.00000			0.0894427
$126$		0			0
$127$	10.0000			0.887357			0.443678	−	0.896186i	$-0.353673\pi$
$127$	10.0000			0.887357			0.443678	+	0.896186i	$0.353673\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−2.00000	−	3.46410i	−0.174741	−	0.302660i	0.765331	−	0.643637i	$-0.222575\pi$
$131$	−2.00000	−	3.46410i	−0.174741	−	0.302660i	−0.940072	+	0.340977i	$0.889242\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	3.50000	−	6.06218i	0.299025	−	0.517927i	−0.676888	−	0.736086i	$-0.736672\pi$
$137$	3.50000	−	6.06218i	0.299025	−	0.517927i	0.975913	+	0.218159i	$0.0700052\pi$
$138$		0			0
$139$	−3.50000	−	6.06218i	−0.296866	−	0.514187i	0.678551	−	0.734553i	$-0.262608\pi$
$139$	−3.50000	−	6.06218i	−0.296866	−	0.514187i	−0.975417	+	0.220366i	$0.929275\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$	10.0000			0.830455
$146$		0			0
$147$		0			0
$148$		0			0
$149$	2.00000	+	3.46410i	0.163846	+	0.283790i	0.936245	−	0.351348i	$-0.114277\pi$
$149$	2.00000	+	3.46410i	0.163846	+	0.283790i	−0.772399	+	0.635138i	$0.780943\pi$
$150$		0			0
$151$	−11.0000	+	19.0526i	−0.895167	+	1.55048i	−0.0615699	+	0.998103i	$0.519611\pi$
$151$	−11.0000	+	19.0526i	−0.895167	+	1.55048i	−0.833597	+	0.552372i	$0.813723\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	1.00000	−	1.73205i	0.0803219	−	0.139122i
$156$		0			0
$157$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$157$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	16.0000			1.25322			0.626608	−	0.779334i	$-0.284443\pi$
$163$	16.0000			1.25322			0.626608	+	0.779334i	$0.284443\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−11.0000	−	19.0526i	−0.851206	−	1.47433i	−0.880121	−	0.474749i	$-0.842539\pi$
$167$	−11.0000	−	19.0526i	−0.851206	−	1.47433i	0.0289155	−	0.999582i	$-0.490795\pi$
$168$		0			0
$169$	6.50000	−	11.2583i	0.500000	−	0.866025i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−1.00000	+	1.73205i	−0.0760286	+	0.131685i	−0.901533	−	0.432710i	$-0.857557\pi$
$173$	−1.00000	+	1.73205i	−0.0760286	+	0.131685i	0.825505	+	0.564396i	$0.190891\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$		0			0
$181$	−4.00000			−0.297318			−0.148659	−	0.988889i	$-0.547496\pi$
$181$	−4.00000			−0.297318			−0.148659	+	0.988889i	$0.547496\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−2.00000	−	3.46410i	−0.147043	−	0.254686i
$186$		0			0
$187$	7.50000	−	12.9904i	0.548454	−	0.949951i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	6.00000	−	10.3923i	0.434145	−	0.751961i	−0.563081	−	0.826402i	$-0.690384\pi$
$191$	6.00000	−	10.3923i	0.434145	−	0.751961i	0.997225	+	0.0744412i	$0.0237173\pi$
$192$		0			0
$193$	−9.50000	−	16.4545i	−0.683825	−	1.18442i	−0.973805	−	0.227387i	$-0.926982\pi$
$193$	−9.50000	−	16.4545i	−0.683825	−	1.18442i	0.289980	−	0.957033i	$-0.406351\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	8.00000			0.569976			0.284988	−	0.958531i	$-0.408010\pi$
$197$	8.00000			0.569976			0.284988	+	0.958531i	$0.408010\pi$
$198$		0			0
$199$	−24.0000			−1.70131			−0.850657	−	0.525720i	$-0.823796\pi$
$199$	−24.0000			−1.70131			−0.850657	+	0.525720i	$0.823796\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−1.50000	+	2.59808i	−0.104765	+	0.181458i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−12.5000	+	21.6506i	−0.864643	+	1.49761i
$210$		0			0
$211$	−6.00000	−	10.3923i	−0.413057	−	0.715436i	0.582165	−	0.813070i	$-0.302206\pi$
$211$	−6.00000	−	10.3923i	−0.413057	−	0.715436i	−0.995222	+	0.0976347i	$0.968872\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	3.00000			0.204598
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−1.00000	+	1.73205i	−0.0669650	+	0.115987i	−0.897564	−	0.440884i	$-0.854665\pi$
$223$	−1.00000	+	1.73205i	−0.0669650	+	0.115987i	0.830599	+	0.556871i	$0.187998\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	6.50000	−	11.2583i	0.431420	−	0.747242i	−0.565576	−	0.824696i	$-0.691346\pi$
$227$	6.50000	−	11.2583i	0.431420	−	0.747242i	0.996996	+	0.0774548i	$0.0246793\pi$
$228$		0			0
$229$	8.00000	+	13.8564i	0.528655	+	0.915657i	0.999442	+	0.0334101i	$0.0106368\pi$
$229$	8.00000	+	13.8564i	0.528655	+	0.915657i	−0.470787	+	0.882247i	$0.656030\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	11.0000			0.720634			0.360317	−	0.932830i	$-0.382669\pi$
$233$	11.0000			0.720634			0.360317	+	0.932830i	$0.382669\pi$
$234$		0			0
$235$	−4.00000			−0.260931
$236$		0			0
$237$		0			0
$238$		0			0
$239$	3.00000	+	5.19615i	0.194054	+	0.336111i	0.946590	−	0.322440i	$-0.104503\pi$
$239$	3.00000	+	5.19615i	0.194054	+	0.336111i	−0.752536	+	0.658551i	$0.771170\pi$
$240$		0			0
$241$	11.5000	−	19.9186i	0.740780	−	1.28307i	−0.211360	−	0.977408i	$-0.567789\pi$
$241$	11.5000	−	19.9186i	0.740780	−	1.28307i	0.952141	−	0.305661i	$-0.0988773\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	3.50000	−	6.06218i	0.223607	−	0.387298i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−11.0000			−0.694314			−0.347157	−	0.937807i	$-0.612853\pi$
$251$	−11.0000			−0.694314			−0.347157	+	0.937807i	$0.612853\pi$
$252$		0			0
$253$	−30.0000			−1.88608
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−15.5000	−	26.8468i	−0.966863	−	1.67466i	−0.704523	−	0.709681i	$-0.748839\pi$
$257$	−15.5000	−	26.8468i	−0.966863	−	1.67466i	−0.262341	−	0.964975i	$-0.584494\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−6.00000	+	10.3923i	−0.369976	+	0.640817i	−0.989561	−	0.144112i	$-0.953967\pi$
$263$	−6.00000	+	10.3923i	−0.369976	+	0.640817i	0.619586	+	0.784929i	$0.287301\pi$
$264$		0			0
$265$	−3.00000	−	5.19615i	−0.184289	−	0.319197i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	30.0000			1.82913			0.914566	−	0.404436i	$-0.132532\pi$
$269$	30.0000			1.82913			0.914566	+	0.404436i	$0.132532\pi$
$270$		0			0
$271$	24.0000			1.45790			0.728948	−	0.684569i	$-0.240010\pi$
$271$	24.0000			1.45790			0.728948	+	0.684569i	$0.240010\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−2.50000	−	4.33013i	−0.150756	−	0.261116i
$276$		0			0
$277$	7.00000	−	12.1244i	0.420589	−	0.728482i	−0.575408	−	0.817867i	$-0.695157\pi$
$277$	7.00000	−	12.1244i	0.420589	−	0.728482i	0.995997	+	0.0893846i	$0.0284900\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	5.00000	−	8.66025i	0.298275	−	0.516627i	−0.677466	−	0.735554i	$-0.736922\pi$
$281$	5.00000	−	8.66025i	0.298275	−	0.516627i	0.975741	+	0.218926i	$0.0702554\pi$
$282$		0			0
$283$	2.00000	+	3.46410i	0.118888	+	0.205919i	0.919327	−	0.393494i	$-0.128734\pi$
$283$	2.00000	+	3.46410i	0.118888	+	0.205919i	−0.800439	+	0.599414i	$0.795400\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−8.00000			−0.470588
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−5.00000	−	8.66025i	−0.292103	−	0.505937i	0.682204	−	0.731162i	$-0.261022\pi$
$293$	−5.00000	−	8.66025i	−0.292103	−	0.505937i	−0.974307	+	0.225225i	$0.927688\pi$
$294$		0			0
$295$	−1.50000	+	2.59808i	−0.0873334	+	0.151266i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	2.00000			0.114520
$306$		0			0
$307$	−7.00000			−0.399511			−0.199756	−	0.979846i	$-0.564015\pi$
$307$	−7.00000			−0.399511			−0.199756	+	0.979846i	$0.564015\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	5.00000	+	8.66025i	0.283524	+	0.491078i	0.972250	−	0.233944i	$-0.0751631\pi$
$311$	5.00000	+	8.66025i	0.283524	+	0.491078i	−0.688726	+	0.725022i	$0.741830\pi$
$312$		0			0
$313$	−3.50000	+	6.06218i	−0.197832	+	0.342655i	−0.947825	−	0.318791i	$-0.896723\pi$
$313$	−3.50000	+	6.06218i	−0.197832	+	0.342655i	0.749993	+	0.661445i	$0.230057\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	14.0000	−	24.2487i	0.786318	−	1.36194i	−0.141890	−	0.989882i	$-0.545318\pi$
$317$	14.0000	−	24.2487i	0.786318	−	1.36194i	0.928208	−	0.372061i	$-0.121349\pi$
$318$		0			0
$319$	−25.0000	−	43.3013i	−1.39973	−	2.42441i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−15.0000			−0.834622
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−10.0000	+	17.3205i	−0.549650	+	0.952021i	0.448649	+	0.893708i	$0.351905\pi$
$331$	−10.0000	+	17.3205i	−0.549650	+	0.952021i	−0.998298	+	0.0583130i	$0.981428\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	5.50000	−	9.52628i	0.300497	−	0.520476i
$336$		0			0
$337$	15.5000	+	26.8468i	0.844339	+	1.46244i	0.886194	+	0.463314i	$0.153340\pi$
$337$	15.5000	+	26.8468i	0.844339	+	1.46244i	−0.0418554	+	0.999124i	$0.513327\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−10.0000			−0.541530
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−1.50000	−	2.59808i	−0.0805242	−	0.139472i	0.822951	−	0.568112i	$-0.192326\pi$
$347$	−1.50000	−	2.59808i	−0.0805242	−	0.139472i	−0.903475	+	0.428640i	$0.858993\pi$
$348$		0			0
$349$	−16.0000	+	27.7128i	−0.856460	+	1.48343i	0.0188232	+	0.999823i	$0.494008\pi$
$349$	−16.0000	+	27.7128i	−0.856460	+	1.48343i	−0.875284	+	0.483610i	$0.839325\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	12.5000	−	21.6506i	0.665308	−	1.15235i	−0.313894	−	0.949458i	$-0.601634\pi$
$353$	12.5000	−	21.6506i	0.665308	−	1.15235i	0.979202	−	0.202889i	$-0.0650330\pi$
$354$		0			0
$355$	−7.00000	−	12.1244i	−0.371521	−	0.643494i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	12.0000			0.633336			0.316668	−	0.948536i	$-0.397436\pi$
$359$	12.0000			0.633336			0.316668	+	0.948536i	$0.397436\pi$
$360$		0			0
$361$	6.00000			0.315789
$362$		0			0
$363$		0			0
$364$		0			0
$365$	7.50000	+	12.9904i	0.392568	+	0.679948i
$366$		0			0
$367$	4.00000	−	6.92820i	0.208798	−	0.361649i	−0.742538	−	0.669804i	$-0.766378\pi$
$367$	4.00000	−	6.92820i	0.208798	−	0.361649i	0.951336	+	0.308155i	$0.0997115\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	13.0000	+	22.5167i	0.673114	+	1.16587i	0.977016	+	0.213165i	$0.0683772\pi$
$373$	13.0000	+	22.5167i	0.673114	+	1.16587i	−0.303902	+	0.952703i	$0.598289\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−35.0000			−1.79783			−0.898915	−	0.438124i	$-0.855643\pi$
$379$	−35.0000			−1.79783			−0.898915	+	0.438124i	$0.855643\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	6.00000	+	10.3923i	0.306586	+	0.531022i	0.977613	−	0.210411i	$-0.0674801\pi$
$383$	6.00000	+	10.3923i	0.306586	+	0.531022i	−0.671027	+	0.741433i	$0.734147\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−12.0000	+	20.7846i	−0.608424	+	1.05382i	0.383076	+	0.923717i	$0.374865\pi$
$389$	−12.0000	+	20.7846i	−0.608424	+	1.05382i	−0.991500	+	0.130105i	$0.958469\pi$
$390$		0			0
$391$	−9.00000	−	15.5885i	−0.455150	−	0.788342i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	10.0000			0.503155
$396$		0			0
$397$	8.00000			0.401508			0.200754	−	0.979642i	$-0.435661\pi$
$397$	8.00000			0.401508			0.200754	+	0.979642i	$0.435661\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−3.50000	−	6.06218i	−0.174782	−	0.302731i	0.765304	−	0.643669i	$-0.222589\pi$
$401$	−3.50000	−	6.06218i	−0.174782	−	0.302731i	−0.940086	+	0.340938i	$0.889255\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−10.0000	+	17.3205i	−0.495682	+	0.858546i
$408$		0			0
$409$	3.50000	+	6.06218i	0.173064	+	0.299755i	0.939490	−	0.342578i	$-0.111300\pi$
$409$	3.50000	+	6.06218i	0.173064	+	0.299755i	−0.766426	+	0.642333i	$0.777967\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	12.0000			0.589057
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$419$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$420$		0			0
$421$	−19.0000	+	32.9090i	−0.926003	+	1.60388i	−0.136064	+	0.990700i	$0.543445\pi$
$421$	−19.0000	+	32.9090i	−0.926003	+	1.60388i	−0.789940	+	0.613185i	$0.789888\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	1.50000	−	2.59808i	0.0727607	−	0.126025i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−28.0000			−1.34871			−0.674356	−	0.738406i	$-0.735579\pi$
$431$	−28.0000			−1.34871			−0.674356	+	0.738406i	$0.735579\pi$
$432$		0			0
$433$	27.0000			1.29754			0.648769	−	0.760986i	$-0.275284\pi$
$433$	27.0000			1.29754			0.648769	+	0.760986i	$0.275284\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	15.0000	+	25.9808i	0.717547	+	1.24283i
$438$		0			0
$439$	7.00000	−	12.1244i	0.334092	−	0.578664i	−0.649218	−	0.760602i	$-0.724904\pi$
$439$	7.00000	−	12.1244i	0.334092	−	0.578664i	0.983310	+	0.181938i	$0.0582371\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−14.5000	+	25.1147i	−0.688916	+	1.19324i	0.283273	+	0.959039i	$0.408580\pi$
$443$	−14.5000	+	25.1147i	−0.688916	+	1.19324i	−0.972189	+	0.234198i	$0.924754\pi$
$444$		0			0
$445$	7.00000	+	12.1244i	0.331832	+	0.574750i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−9.00000			−0.424736			−0.212368	−	0.977190i	$-0.568118\pi$
$449$	−9.00000			−0.424736			−0.212368	+	0.977190i	$0.568118\pi$
$450$		0			0
$451$	15.0000			0.706322
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	12.5000	−	21.6506i	0.584725	−	1.01277i	−0.410184	−	0.912003i	$-0.634536\pi$
$457$	12.5000	−	21.6506i	0.584725	−	1.01277i	0.994910	−	0.100771i	$-0.0321310\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	2.00000	−	3.46410i	0.0931493	−	0.161339i	−0.815685	−	0.578496i	$-0.803640\pi$
$461$	2.00000	−	3.46410i	0.0931493	−	0.161339i	0.908835	+	0.417156i	$0.136973\pi$
$462$		0			0
$463$	2.00000	+	3.46410i	0.0929479	+	0.160990i	0.908750	−	0.417340i	$-0.137038\pi$
$463$	2.00000	+	3.46410i	0.0929479	+	0.160990i	−0.815802	+	0.578331i	$0.803704\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	11.0000			0.509019			0.254510	−	0.967070i	$-0.418086\pi$
$467$	11.0000			0.509019			0.254510	+	0.967070i	$0.418086\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−7.50000	−	12.9904i	−0.344850	−	0.597298i
$474$		0			0
$475$	−2.50000	+	4.33013i	−0.114708	+	0.198680i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	13.0000	−	22.5167i	0.593985	−	1.02881i	−0.399704	−	0.916644i	$-0.630887\pi$
$479$	13.0000	−	22.5167i	0.593985	−	1.02881i	0.993689	−	0.112168i	$-0.0357796\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−13.0000			−0.590300
$486$		0			0
$487$	34.0000			1.54069			0.770344	−	0.637629i	$-0.220085\pi$
$487$	34.0000			1.54069			0.770344	+	0.637629i	$0.220085\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	7.50000	+	12.9904i	0.338470	+	0.586248i	0.984145	−	0.177365i	$-0.0567572\pi$
$491$	7.50000	+	12.9904i	0.338470	+	0.586248i	−0.645675	+	0.763612i	$0.723424\pi$
$492$		0			0
$493$	15.0000	−	25.9808i	0.675566	−	1.17011i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	11.5000	+	19.9186i	0.514811	+	0.891678i	0.999852	+	0.0171872i	$0.00547113\pi$
$499$	11.5000	+	19.9186i	0.514811	+	0.891678i	−0.485042	+	0.874491i	$0.661196\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−34.0000			−1.51599			−0.757993	−	0.652263i	$-0.773820\pi$
$503$	−34.0000			−1.51599			−0.757993	+	0.652263i	$0.773820\pi$
$504$		0			0
$505$	12.0000			0.533993
$506$		0			0
$507$		0			0
$508$		0			0
$509$	22.0000	+	38.1051i	0.975133	+	1.68898i	0.679496	+	0.733679i	$0.262199\pi$
$509$	22.0000	+	38.1051i	0.975133	+	1.68898i	0.295637	+	0.955300i	$0.404468\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$	10.0000	+	17.3205i	0.439799	+	0.761755i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	27.0000			1.18289			0.591446	−	0.806345i	$-0.298557\pi$
$521$	27.0000			1.18289			0.591446	+	0.806345i	$0.298557\pi$
$522$		0			0
$523$	−4.00000			−0.174908			−0.0874539	−	0.996169i	$-0.527873\pi$
$523$	−4.00000			−0.174908			−0.0874539	+	0.996169i	$0.527873\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−3.00000	−	5.19615i	−0.130682	−	0.226348i
$528$		0			0
$529$	−6.50000	+	11.2583i	−0.282609	+	0.489493i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$	−8.50000	−	14.7224i	−0.367487	−	0.636506i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−35.0000			−1.50756
$540$		0			0
$541$	24.0000			1.03184			0.515920	−	0.856637i	$-0.327450\pi$
$541$	24.0000			1.03184			0.515920	+	0.856637i	$0.327450\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	6.00000	+	10.3923i	0.257012	+	0.445157i
$546$		0			0
$547$	4.50000	−	7.79423i	0.192406	−	0.333257i	−0.753641	−	0.657286i	$-0.771704\pi$
$547$	4.50000	−	7.79423i	0.192406	−	0.333257i	0.946047	+	0.324029i	$0.105038\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−25.0000	+	43.3013i	−1.06504	+	1.84470i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0			−	1.00000i	$-0.5\pi$
$557$		0			0				1.00000i	$0.5\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$	2.50000	+	4.33013i	0.105362	+	0.182493i	0.913886	−	0.405970i	$-0.133066\pi$
$563$	2.50000	+	4.33013i	0.105362	+	0.182493i	−0.808524	+	0.588463i	$0.799733\pi$
$564$		0			0
$565$	−3.00000	+	5.19615i	−0.126211	+	0.218604i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−7.50000	+	12.9904i	−0.314416	+	0.544585i	−0.979313	−	0.202350i	$-0.935142\pi$
$569$	−7.50000	+	12.9904i	−0.314416	+	0.544585i	0.664897	+	0.746935i	$0.268475\pi$
$570$		0			0
$571$	1.50000	+	2.59808i	0.0627730	+	0.108726i	0.895704	−	0.444651i	$-0.146672\pi$
$571$	1.50000	+	2.59808i	0.0627730	+	0.108726i	−0.832931	+	0.553377i	$0.813339\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−6.00000			−0.250217
$576$		0			0
$577$	−31.0000			−1.29055			−0.645273	−	0.763952i	$-0.723257\pi$
$577$	−31.0000			−1.29055			−0.645273	+	0.763952i	$0.723257\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−15.0000	+	25.9808i	−0.621237	+	1.07601i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−8.50000	+	14.7224i	−0.350833	+	0.607660i	−0.986396	−	0.164389i	$-0.947435\pi$
$587$	−8.50000	+	14.7224i	−0.350833	+	0.607660i	0.635563	+	0.772049i	$0.280768\pi$
$588$		0			0
$589$	5.00000	+	8.66025i	0.206021	+	0.356840i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−30.0000			−1.23195			−0.615976	−	0.787765i	$-0.711238\pi$
$593$	−30.0000			−1.23195			−0.615976	+	0.787765i	$0.711238\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	8.00000	+	13.8564i	0.326871	+	0.566157i	0.981889	−	0.189456i	$-0.0606724\pi$
$599$	8.00000	+	13.8564i	0.326871	+	0.566157i	−0.655018	+	0.755613i	$0.727339\pi$
$600$		0			0
$601$	0.500000	−	0.866025i	0.0203954	−	0.0353259i	−0.855648	−	0.517559i	$-0.826841\pi$
$601$	0.500000	−	0.866025i	0.0203954	−	0.0353259i	0.876043	+	0.482233i	$0.160174\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−7.00000	+	12.1244i	−0.284590	+	0.492925i
$606$		0			0
$607$	16.0000	+	27.7128i	0.649420	+	1.12483i	0.983262	+	0.182199i	$0.0583216\pi$
$607$	16.0000	+	27.7128i	0.649420	+	1.12483i	−0.333842	+	0.942629i	$0.608345\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−2.00000			−0.0807792			−0.0403896	−	0.999184i	$-0.512860\pi$
$613$	−2.00000			−0.0807792			−0.0403896	+	0.999184i	$0.512860\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	12.5000	+	21.6506i	0.503231	+	0.871622i	0.999993	+	0.00373492i	$0.00118887\pi$
$617$	12.5000	+	21.6506i	0.503231	+	0.871622i	−0.496762	+	0.867887i	$0.665478\pi$
$618$		0			0
$619$	9.50000	−	16.4545i	0.381837	−	0.661361i	−0.609488	−	0.792796i	$-0.708625\pi$
$619$	9.50000	−	16.4545i	0.381837	−	0.661361i	0.991325	+	0.131434i	$0.0419582\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−0.500000	−	0.866025i	−0.0200000	−	0.0346410i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−12.0000			−0.478471
$630$		0			0
$631$	2.00000			0.0796187			0.0398094	−	0.999207i	$-0.487325\pi$
$631$	2.00000			0.0796187			0.0398094	+	0.999207i	$0.487325\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−5.00000	−	8.66025i	−0.198419	−	0.343672i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	16.5000	−	28.5788i	0.651711	−	1.12880i	−0.330997	−	0.943632i	$-0.607385\pi$
$641$	16.5000	−	28.5788i	0.651711	−	1.12880i	0.982708	−	0.185164i	$-0.0592817\pi$
$642$		0			0
$643$	20.5000	+	35.5070i	0.808441	+	1.40026i	0.913943	+	0.405842i	$0.133022\pi$
$643$	20.5000	+	35.5070i	0.808441	+	1.40026i	−0.105502	+	0.994419i	$0.533645\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−30.0000			−1.17942			−0.589711	−	0.807614i	$-0.700758\pi$
$647$	−30.0000			−1.17942			−0.589711	+	0.807614i	$0.700758\pi$
$648$		0			0
$649$	15.0000			0.588802
$650$		0			0
$651$		0			0
$652$		0			0
$653$	17.0000	+	29.4449i	0.665261	+	1.15227i	0.979214	+	0.202828i	$0.0650132\pi$
$653$	17.0000	+	29.4449i	0.665261	+	1.15227i	−0.313953	+	0.949439i	$0.601653\pi$
$654$		0			0
$655$	−2.00000	+	3.46410i	−0.0781465	+	0.135354i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	22.0000	−	38.1051i	0.856998	−	1.48436i	−0.0177803	−	0.999842i	$-0.505660\pi$
$659$	22.0000	−	38.1051i	0.856998	−	1.48436i	0.874779	−	0.484523i	$-0.161007\pi$
$660$		0			0
$661$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$661$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−60.0000			−2.32321
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−5.00000	−	8.66025i	−0.193023	−	0.334325i
$672$		0			0
$673$	13.0000	−	22.5167i	0.501113	−	0.867953i	−0.498886	−	0.866668i	$-0.666257\pi$
$673$	13.0000	−	22.5167i	0.501113	−	0.867953i	0.999999	−	0.00128586i	$-0.000409302\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	14.0000	−	24.2487i	0.538064	−	0.931954i	−0.460945	−	0.887429i	$-0.652489\pi$
$677$	14.0000	−	24.2487i	0.538064	−	0.931954i	0.999008	−	0.0445248i	$-0.0141774\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	13.0000			0.497431			0.248716	−	0.968577i	$-0.419992\pi$
$683$	13.0000			0.497431			0.248716	+	0.968577i	$0.419992\pi$
$684$		0			0
$685$	−7.00000			−0.267456
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	4.00000	−	6.92820i	0.152167	−	0.263561i	−0.779857	−	0.625958i	$-0.784708\pi$
$691$	4.00000	−	6.92820i	0.152167	−	0.263561i	0.932024	+	0.362397i	$0.118041\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−3.50000	+	6.06218i	−0.132763	+	0.229952i
$696$		0			0
$697$	4.50000	+	7.79423i	0.170450	+	0.295227i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	6.00000			0.226617			0.113308	−	0.993560i	$-0.463855\pi$
$701$	6.00000			0.226617			0.113308	+	0.993560i	$0.463855\pi$
$702$		0			0
$703$	20.0000			0.754314
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	23.0000	−	39.8372i	0.863783	−	1.49612i	−0.00446726	−	0.999990i	$-0.501422\pi$
$709$	23.0000	−	39.8372i	0.863783	−	1.49612i	0.868250	−	0.496126i	$-0.165245\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−6.00000	+	10.3923i	−0.224702	+	0.389195i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−5.00000	−	8.66025i	−0.185695	−	0.321634i
$726$		0			0
$727$	−8.00000	+	13.8564i	−0.296704	+	0.513906i	−0.975380	−	0.220532i	$-0.929221\pi$
$727$	−8.00000	+	13.8564i	−0.296704	+	0.513906i	0.678676	+	0.734438i	$0.262554\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	4.50000	−	7.79423i	0.166439	−	0.288280i
$732$		0			0
$733$	−14.0000	−	24.2487i	−0.517102	−	0.895647i	−0.999803	−	0.0198613i	$-0.993678\pi$
$733$	−14.0000	−	24.2487i	−0.517102	−	0.895647i	0.482701	−	0.875785i	$-0.339656\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−55.0000			−2.02595
$738$		0			0
$739$	5.00000			0.183928			0.0919640	−	0.995762i	$-0.470686\pi$
$739$	5.00000			0.183928			0.0919640	+	0.995762i	$0.470686\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	17.0000	+	29.4449i	0.623670	+	1.08023i	0.988797	+	0.149270i	$0.0476922\pi$
$743$	17.0000	+	29.4449i	0.623670	+	1.08023i	−0.365127	+	0.930958i	$0.618974\pi$
$744$		0			0
$745$	2.00000	−	3.46410i	0.0732743	−	0.126915i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−10.0000	−	17.3205i	−0.364905	−	0.632034i	0.623856	−	0.781540i	$-0.285565\pi$
$751$	−10.0000	−	17.3205i	−0.364905	−	0.632034i	−0.988761	+	0.149505i	$0.952232\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	22.0000			0.800662
$756$		0			0
$757$	−26.0000			−0.944986			−0.472493	−	0.881334i	$-0.656646\pi$
$757$	−26.0000			−0.944986			−0.472493	+	0.881334i	$0.656646\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	5.00000	+	8.66025i	0.181250	+	0.313934i	0.942306	−	0.334752i	$-0.108652\pi$
$761$	5.00000	+	8.66025i	0.181250	+	0.313934i	−0.761057	+	0.648686i	$0.775319\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	27.0000	+	46.7654i	0.973645	+	1.68640i	0.684336	+	0.729167i	$0.260092\pi$
$769$	27.0000	+	46.7654i	0.973645	+	1.68640i	0.289309	+	0.957236i	$0.406575\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	30.0000			1.07903			0.539513	−	0.841978i	$-0.318609\pi$
$773$	30.0000			1.07903			0.539513	+	0.841978i	$0.318609\pi$
$774$		0			0
$775$	−2.00000			−0.0718421
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−7.50000	−	12.9904i	−0.268715	−	0.465429i
$780$		0			0
$781$	−35.0000	+	60.6218i	−1.25240	+	2.16922i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−14.0000	−	24.2487i	−0.499046	−	0.864373i	0.500953	−	0.865474i	$-0.332983\pi$
$787$	−14.0000	−	24.2487i	−0.499046	−	0.864373i	−0.999999	+	0.00110111i	$0.999650\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$	16.0000	+	27.7128i	0.566749	+	0.981638i	0.996885	+	0.0788739i	$0.0251324\pi$
$797$	16.0000	+	27.7128i	0.566749	+	0.981638i	−0.430136	+	0.902764i	$0.641534\pi$
$798$		0			0
$799$	−6.00000	+	10.3923i	−0.212265	+	0.367653i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	37.5000	−	64.9519i	1.32335	−	2.29210i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	15.0000			0.527372			0.263686	−	0.964609i	$-0.415062\pi$
$809$	15.0000			0.527372			0.263686	+	0.964609i	$0.415062\pi$
$810$		0			0
$811$	35.0000			1.22902			0.614508	−	0.788911i	$-0.289355\pi$
$811$	35.0000			1.22902			0.614508	+	0.788911i	$0.289355\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−8.00000	−	13.8564i	−0.280228	−	0.485369i
$816$		0			0
$817$	−7.50000	+	12.9904i	−0.262392	+	0.454476i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	2.00000	−	3.46410i	0.0698005	−	0.120898i	−0.829013	−	0.559229i	$-0.811097\pi$
$821$	2.00000	−	3.46410i	0.0698005	−	0.120898i	0.898813	+	0.438331i	$0.144430\pi$
$822$		0			0
$823$	−6.00000	−	10.3923i	−0.209147	−	0.362253i	0.742299	−	0.670069i	$-0.233735\pi$
$823$	−6.00000	−	10.3923i	−0.209147	−	0.362253i	−0.951446	+	0.307816i	$0.900402\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	12.0000			0.417281			0.208640	−	0.977992i	$-0.433096\pi$
$827$	12.0000			0.417281			0.208640	+	0.977992i	$0.433096\pi$
$828$		0			0
$829$	40.0000			1.38926			0.694629	−	0.719368i	$-0.255569\pi$
$829$	40.0000			1.38926			0.694629	+	0.719368i	$0.255569\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−10.5000	−	18.1865i	−0.363803	−	0.630126i
$834$		0			0
$835$	−11.0000	+	19.0526i	−0.380671	+	0.659341i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	7.00000	−	12.1244i	0.241667	−	0.418579i	−0.719522	−	0.694469i	$-0.755639\pi$
$839$	7.00000	−	12.1244i	0.241667	−	0.418579i	0.961189	+	0.275890i	$0.0889726\pi$
$840$		0			0
$841$	−35.5000	−	61.4878i	−1.22414	−	2.12027i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−13.0000			−0.447214
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$	12.0000	+	20.7846i	0.411355	+	0.712487i
$852$		0			0
$853$	−16.0000	+	27.7128i	−0.547830	+	0.948869i	0.450593	+	0.892729i	$0.351212\pi$
$853$	−16.0000	+	27.7128i	−0.547830	+	0.948869i	−0.998423	+	0.0561393i	$0.982121\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−1.00000	+	1.73205i	−0.0341593	+	0.0591657i	−0.882600	−	0.470125i	$-0.844209\pi$
$857$	−1.00000	+	1.73205i	−0.0341593	+	0.0591657i	0.848440	+	0.529291i	$0.177542\pi$
$858$		0			0
$859$	5.50000	+	9.52628i	0.187658	+	0.325032i	0.944469	−	0.328601i	$-0.106577\pi$
$859$	5.50000	+	9.52628i	0.187658	+	0.325032i	−0.756811	+	0.653633i	$0.773244\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	26.0000			0.885050			0.442525	−	0.896756i	$-0.354083\pi$
$863$	26.0000			0.885050			0.442525	+	0.896756i	$0.354083\pi$
$864$		0			0
$865$	2.00000			0.0680020
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−25.0000	−	43.3013i	−0.848067	−	1.46889i
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−11.0000	−	19.0526i	−0.371444	−	0.643359i	0.618344	−	0.785907i	$-0.287804\pi$
$877$	−11.0000	−	19.0526i	−0.371444	−	0.643359i	−0.989788	+	0.142548i	$0.954470\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	2.00000			0.0673817			0.0336909	−	0.999432i	$-0.489274\pi$
$881$	2.00000			0.0673817			0.0336909	+	0.999432i	$0.489274\pi$
$882$		0			0
$883$	−47.0000			−1.58168			−0.790838	−	0.612026i	$-0.790355\pi$
$883$	−47.0000			−1.58168			−0.790838	+	0.612026i	$0.790355\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−3.00000	−	5.19615i	−0.100730	−	0.174470i	0.811256	−	0.584692i	$-0.198785\pi$
$887$	−3.00000	−	5.19615i	−0.100730	−	0.174470i	−0.911986	+	0.410222i	$0.865451\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	10.0000	−	17.3205i	0.334637	−	0.579609i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−20.0000			−0.667037
$900$		0			0
$901$	−18.0000			−0.599667
$902$		0			0
$903$		0			0
$904$		0			0
$905$	2.00000	+	3.46410i	0.0664822	+	0.115151i
$906$		0			0
$907$	−6.50000	+	11.2583i	−0.215829	+	0.373827i	−0.953529	−	0.301302i	$-0.902579\pi$
$907$	−6.50000	+	11.2583i	−0.215829	+	0.373827i	0.737700	+	0.675129i	$0.235912\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	3.00000	−	5.19615i	0.0993944	−	0.172156i	−0.812040	−	0.583602i	$-0.801643\pi$
$911$	3.00000	−	5.19615i	0.0993944	−	0.172156i	0.911434	+	0.411446i	$0.134976\pi$
$912$		0			0
$913$	−30.0000	−	51.9615i	−0.992855	−	1.71968i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	14.0000			0.461817			0.230909	−	0.972975i	$-0.425830\pi$
$919$	14.0000			0.461817			0.230909	+	0.972975i	$0.425830\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−2.00000	+	3.46410i	−0.0657596	+	0.113899i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	15.0000	−	25.9808i	0.492134	−	0.852401i	−0.507825	−	0.861460i	$-0.669550\pi$
$929$	15.0000	−	25.9808i	0.492134	−	0.852401i	0.999959	+	0.00905914i	$0.00288365\pi$
$930$		0			0
$931$	17.5000	+	30.3109i	0.573539	+	0.993399i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−15.0000			−0.490552
$936$		0			0
$937$	−42.0000			−1.37208			−0.686040	−	0.727564i	$-0.740653\pi$
$937$	−42.0000			−1.37208			−0.686040	+	0.727564i	$0.740653\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−8.00000	−	13.8564i	−0.260793	−	0.451706i	0.705660	−	0.708550i	$-0.250651\pi$
$941$	−8.00000	−	13.8564i	−0.260793	−	0.451706i	−0.966453	+	0.256844i	$0.917317\pi$
$942$		0			0
$943$	9.00000	−	15.5885i	0.293080	−	0.507630i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−10.5000	+	18.1865i	−0.341204	+	0.590983i	−0.984657	−	0.174503i	$-0.944168\pi$
$947$	−10.5000	+	18.1865i	−0.341204	+	0.590983i	0.643452	+	0.765486i	$0.277501\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	13.0000			0.421111			0.210556	−	0.977582i	$-0.432473\pi$
$953$	13.0000			0.421111			0.210556	+	0.977582i	$0.432473\pi$
$954$		0			0
$955$	−12.0000			−0.388311
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	13.5000	−	23.3827i	0.435484	−	0.754280i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−9.50000	+	16.4545i	−0.305816	+	0.529689i
$966$		0			0
$967$	3.00000	+	5.19615i	0.0964735	+	0.167097i	0.910223	−	0.414119i	$-0.135910\pi$
$967$	3.00000	+	5.19615i	0.0964735	+	0.167097i	−0.813749	+	0.581216i	$0.802577\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−12.0000			−0.385098			−0.192549	−	0.981287i	$-0.561675\pi$
$971$	−12.0000			−0.385098			−0.192549	+	0.981287i	$0.561675\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	11.5000	+	19.9186i	0.367918	+	0.637252i	0.989240	−	0.146303i	$-0.0467375\pi$
$977$	11.5000	+	19.9186i	0.367918	+	0.637252i	−0.621322	+	0.783555i	$0.713404\pi$
$978$		0			0
$979$	35.0000	−	60.6218i	1.11860	−	1.93748i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	26.0000	−	45.0333i	0.829271	−	1.43634i	−0.0693395	−	0.997593i	$-0.522089\pi$
$983$	26.0000	−	45.0333i	0.829271	−	1.43634i	0.898611	−	0.438747i	$-0.144577\pi$
$984$		0			0
$985$	−4.00000	−	6.92820i	−0.127451	−	0.220751i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−18.0000			−0.572367
$990$		0			0
$991$	28.0000			0.889449			0.444725	−	0.895667i	$-0.353302\pi$
$991$	28.0000			0.889449			0.444725	+	0.895667i	$0.353302\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	12.0000	+	20.7846i	0.380426	+	0.658916i
$996$		0			0
$997$	21.0000	−	36.3731i	0.665077	−	1.15195i	−0.314188	−	0.949361i	$-0.601732\pi$
$997$	21.0000	−	36.3731i	0.665077	−	1.15195i	0.979265	−	0.202586i	$-0.0649345\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1080.2.q.a.721.1		2
3.2	odd	2		360.2.q.a.241.1	yes	2
4.3	odd	2		2160.2.q.c.721.1		2
9.2	odd	6		3240.2.a.b.1.1		1
9.4	even	3	inner	1080.2.q.a.361.1		2
9.5	odd	6		360.2.q.a.121.1	✓	2
9.7	even	3		3240.2.a.f.1.1		1
12.11	even	2		720.2.q.e.241.1		2
36.7	odd	6		6480.2.a.q.1.1		1
36.11	even	6		6480.2.a.e.1.1		1
36.23	even	6		720.2.q.e.481.1		2
36.31	odd	6		2160.2.q.c.1441.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
360.2.q.a.121.1	✓	2	9.5	odd	6
360.2.q.a.241.1	yes	2	3.2	odd	2
720.2.q.e.241.1		2	12.11	even	2
720.2.q.e.481.1		2	36.23	even	6
1080.2.q.a.361.1		2	9.4	even	3	inner
1080.2.q.a.721.1		2	1.1	even	1	trivial
2160.2.q.c.721.1		2	4.3	odd	2
2160.2.q.c.1441.1		2	36.31	odd	6
3240.2.a.b.1.1		1	9.2	odd	6
3240.2.a.f.1.1		1	9.7	even	3
6480.2.a.e.1.1		1	36.11	even	6
6480.2.a.q.1.1		1	36.7	odd	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1080 = 2^{3} \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1080.q (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.500000 - 0.866025i) q^{5} +O(q^{10})\)	\(q+(-0.500000 - 0.866025i) q^{5} +(-2.50000 + 4.33013i) q^{11} -3.00000 q^{17} +5.00000 q^{19} +(3.00000 + 5.19615i) q^{23} +(-0.500000 + 0.866025i) q^{25} +(-5.00000 + 8.66025i) q^{29} +(1.00000 + 1.73205i) q^{31} +4.00000 q^{37} +(-1.50000 - 2.59808i) q^{41} +(-1.50000 + 2.59808i) q^{43} +(2.00000 - 3.46410i) q^{47} +(3.50000 + 6.06218i) q^{49} +6.00000 q^{53} +5.00000 q^{55} +(-1.50000 - 2.59808i) q^{59} +(-1.00000 + 1.73205i) q^{61} +(5.50000 + 9.52628i) q^{67} +14.0000 q^{71} -15.0000 q^{73} +(-5.00000 + 8.66025i) q^{79} +(-6.00000 + 10.3923i) q^{83} +(1.50000 + 2.59808i) q^{85} -14.0000 q^{89} +(-2.50000 - 4.33013i) q^{95} +(6.50000 - 11.2583i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{5}+O(q^{10}) \) 2 * q - q^5	\( 2 q - q^{5} - 5 q^{11} - 6 q^{17} + 10 q^{19} + 6 q^{23} - q^{25} - 10 q^{29} + 2 q^{31} + 8 q^{37} - 3 q^{41} - 3 q^{43} + 4 q^{47} + 7 q^{49} + 12 q^{53} + 10 q^{55} - 3 q^{59} - 2 q^{61} + 11 q^{67} + 28 q^{71} - 30 q^{73} - 10 q^{79} - 12 q^{83} + 3 q^{85} - 28 q^{89} - 5 q^{95} + 13 q^{97}+O(q^{100}) \) 2 * q - q^5 - 5 * q^11 - 6 * q^17 + 10 * q^19 + 6 * q^23 - q^25 - 10 * q^29 + 2 * q^31 + 8 * q^37 - 3 * q^41 - 3 * q^43 + 4 * q^47 + 7 * q^49 + 12 * q^53 + 10 * q^55 - 3 * q^59 - 2 * q^61 + 11 * q^67 + 28 * q^71 - 30 * q^73 - 10 * q^79 - 12 * q^83 + 3 * q^85 - 28 * q^89 - 5 * q^95 + 13 * q^97

Introduction

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