LMFDB - Embedded newform 3360.2.ba.f.2591.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3360,2,Mod(2591,3360)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3360, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 1, 0, 0]))

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

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

chi := DirichletCharacter("3360.2591");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3360 = 2^{5} \cdot 3 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3360.ba (of order $2$, degree $1$, 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:	$26.8297350792$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2591.2
Character	$\chi$	$=$	3360.2591
Dual form			3360.2.ba.f.2591.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.66651 + 0.471960i) q^{3} +1.00000i q^{5} -1.00000i q^{7} +(2.55451 - 1.57305i) q^{9} +O(q^{10})$	\(q+(-1.66651 + 0.471960i) q^{3} +1.00000i q^{5} -1.00000i q^{7} +(2.55451 - 1.57305i) q^{9} -1.34929 q^{11} -1.04742 q^{13} +(-0.471960 - 1.66651i) q^{15} +2.41361i q^{17} -4.57590i q^{19} +(0.471960 + 1.66651i) q^{21} +1.77710 q^{23} -1.00000 q^{25} +(-3.51470 + 3.82713i) q^{27} +6.95143i q^{29} -8.21898i q^{31} +(2.24861 - 0.636812i) q^{33} +1.00000 q^{35} +0.0319329 q^{37} +(1.74554 - 0.494341i) q^{39} +1.69209i q^{41} +5.81637i q^{43} +(1.57305 + 2.55451i) q^{45} -0.170505 q^{47} -1.00000 q^{49} +(-1.13913 - 4.02231i) q^{51} +4.15753i q^{53} -1.34929i q^{55} +(2.15964 + 7.62579i) q^{57} -5.02939 q^{59} +1.19597 q^{61} +(-1.57305 - 2.55451i) q^{63} -1.04742i q^{65} +3.46592i q^{67} +(-2.96156 + 0.838721i) q^{69} +0.0796653 q^{71} +4.05850 q^{73} +(1.66651 - 0.471960i) q^{75} +1.34929i q^{77} +5.71319i q^{79} +(4.05102 - 8.03674i) q^{81} -0.528155 q^{83} -2.41361 q^{85} +(-3.28079 - 11.5846i) q^{87} +2.77107i q^{89} +1.04742i q^{91} +(3.87903 + 13.6970i) q^{93} +4.57590 q^{95} +13.4213 q^{97} +(-3.44678 + 2.12251i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 4 q^{9}+O(q^{10}) $ 24 * q + 4 * q^9	$ 24 q + 4 q^{9} + 8 q^{11} + 8 q^{13} + 4 q^{15} - 4 q^{21} - 24 q^{25} + 8 q^{33} + 24 q^{35} + 8 q^{37} + 36 q^{39} - 24 q^{49} - 12 q^{51} + 8 q^{57} + 48 q^{59} + 56 q^{61} - 88 q^{71} - 40 q^{73} + 44 q^{81} + 24 q^{83} + 32 q^{87} - 48 q^{93} - 24 q^{95} - 8 q^{97} - 52 q^{99}+O(q^{100}) $ 24 * q + 4 * q^9 + 8 * q^11 + 8 * q^13 + 4 * q^15 - 4 * q^21 - 24 * q^25 + 8 * q^33 + 24 * q^35 + 8 * q^37 + 36 * q^39 - 24 * q^49 - 12 * q^51 + 8 * q^57 + 48 * q^59 + 56 * q^61 - 88 * q^71 - 40 * q^73 + 44 * q^81 + 24 * q^83 + 32 * q^87 - 48 * q^93 - 24 * q^95 - 8 * q^97 - 52 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3360\mathbb{Z}\right)^\times$.

$n$	$421$	$1121$	$1471$	$1921$	$2017$
$\chi(n)$	$1$	$-1$	$-1$	$1$	$1$

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
$3$	−1.66651	+	0.471960i	−0.962160	+	0.272486i
$3$	−1.66651	+	0.471960i	−0.962160	+	0.272486i
$4$		0			0
$5$			1.00000i			0.447214i
$5$			1.00000i			0.447214i
$6$		0			0
$7$		−	1.00000i		−	0.377964i
$7$		−	1.00000i		−	0.377964i
$8$		0			0
$9$	2.55451	−	1.57305i	0.851503	−	0.524350i
$10$		0			0
$11$	−1.34929			−0.406827			−0.203414	−	0.979093i	$-0.565204\pi$
$11$	−1.34929			−0.406827			−0.203414	+	0.979093i	$0.565204\pi$
$12$		0			0
$13$	−1.04742			−0.290502			−0.145251	−	0.989395i	$-0.546399\pi$
$13$	−1.04742			−0.290502			−0.145251	+	0.989395i	$0.546399\pi$
$14$		0			0
$15$	−0.471960	−	1.66651i	−0.121859	−	0.430291i
$16$		0			0
$17$			2.41361i			0.585387i	0.956206	+	0.292694i	$0.0945516\pi$
$17$			2.41361i			0.585387i	−0.956206	+	0.292694i	$0.905448\pi$
$18$		0			0
$19$		−	4.57590i		−	1.04978i	−0.851169	−	0.524892i	$-0.824106\pi$
$19$		−	4.57590i		−	1.04978i	0.851169	−	0.524892i	$-0.175894\pi$
$20$		0			0
$21$	0.471960	+	1.66651i	0.102990	+	0.363662i
$22$		0			0
$23$	1.77710			0.370552			0.185276	−	0.982687i	$-0.440682\pi$
$23$	1.77710			0.370552			0.185276	+	0.982687i	$0.440682\pi$
$24$		0			0
$25$	−1.00000			−0.200000
$26$		0			0
$27$	−3.51470	+	3.82713i	−0.676403	+	0.736531i
$28$		0			0
$29$			6.95143i			1.29085i	0.763825	+	0.645424i	$0.223319\pi$
$29$			6.95143i			1.29085i	−0.763825	+	0.645424i	$0.776681\pi$
$30$		0			0
$31$		−	8.21898i		−	1.47617i	−0.674706	−	0.738086i	$-0.735730\pi$
$31$		−	8.21898i		−	1.47617i	0.674706	−	0.738086i	$-0.264270\pi$
$32$		0			0
$33$	2.24861	−	0.636812i	0.391433	−	0.110855i
$34$		0			0
$35$	1.00000			0.169031
$36$		0			0
$37$	0.0319329			0.00524973			0.00262487	−	0.999997i	$-0.499164\pi$
$37$	0.0319329			0.00524973			0.00262487	+	0.999997i	$0.499164\pi$
$38$		0			0
$39$	1.74554	−	0.494341i	0.279510	−	0.0791579i
$40$		0			0
$41$			1.69209i			0.264261i	0.991232	+	0.132130i	$0.0421818\pi$
$41$			1.69209i			0.264261i	−0.991232	+	0.132130i	$0.957818\pi$
$42$		0			0
$43$			5.81637i			0.886989i	0.896277	+	0.443494i	$0.146261\pi$
$43$			5.81637i			0.886989i	−0.896277	+	0.443494i	$0.853739\pi$
$44$		0			0
$45$	1.57305	+	2.55451i	0.234497	+	0.380804i
$46$		0			0
$47$	−0.170505			−0.0248707			−0.0124353	−	0.999923i	$-0.503958\pi$
$47$	−0.170505			−0.0248707			−0.0124353	+	0.999923i	$0.503958\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	−1.13913	−	4.02231i	−0.159510	−	0.563236i
$52$		0			0
$53$			4.15753i			0.571081i	0.958367	+	0.285541i	$0.0921731\pi$
$53$			4.15753i			0.571081i	−0.958367	+	0.285541i	$0.907827\pi$
$54$		0			0
$55$		−	1.34929i		−	0.181939i
$56$		0			0
$57$	2.15964	+	7.62579i	0.286052	+	1.01006i
$58$		0			0
$59$	−5.02939			−0.654771			−0.327385	−	0.944891i	$-0.606168\pi$
$59$	−5.02939			−0.654771			−0.327385	+	0.944891i	$0.606168\pi$
$60$		0			0
$61$	1.19597			0.153128			0.0765640	−	0.997065i	$-0.475605\pi$
$61$	1.19597			0.153128			0.0765640	+	0.997065i	$0.475605\pi$
$62$		0			0
$63$	−1.57305	−	2.55451i	−0.198186	−	0.321838i
$64$		0			0
$65$		−	1.04742i		−	0.129917i
$66$		0			0
$67$			3.46592i			0.423430i	0.977331	+	0.211715i	$0.0679049\pi$
$67$			3.46592i			0.423430i	−0.977331	+	0.211715i	$0.932095\pi$
$68$		0			0
$69$	−2.96156	+	0.838721i	−0.356530	+	0.100970i
$70$		0			0
$71$	0.0796653			0.00945453			0.00472727	−	0.999989i	$-0.498495\pi$
$71$	0.0796653			0.00945453			0.00472727	+	0.999989i	$0.498495\pi$
$72$		0			0
$73$	4.05850			0.475012			0.237506	−	0.971386i	$-0.423670\pi$
$73$	4.05850			0.475012			0.237506	+	0.971386i	$0.423670\pi$
$74$		0			0
$75$	1.66651	−	0.471960i	0.192432	−	0.0544972i
$76$		0			0
$77$			1.34929i			0.153766i
$78$		0			0
$79$			5.71319i			0.642784i	0.946946	+	0.321392i	$0.104151\pi$
$79$			5.71319i			0.642784i	−0.946946	+	0.321392i	$0.895849\pi$
$80$		0			0
$81$	4.05102	−	8.03674i	0.450114	−	0.892971i
$82$		0			0
$83$	−0.528155			−0.0579725			−0.0289863	−	0.999580i	$-0.509228\pi$
$83$	−0.528155			−0.0579725			−0.0289863	+	0.999580i	$0.509228\pi$
$84$		0			0
$85$	−2.41361			−0.261793
$86$		0			0
$87$	−3.28079	−	11.5846i	−0.351738	−	1.24200i
$88$		0			0
$89$			2.77107i			0.293733i	0.989156	+	0.146866i	$0.0469187\pi$
$89$			2.77107i			0.293733i	−0.989156	+	0.146866i	$0.953081\pi$
$90$		0			0
$91$			1.04742i			0.109800i
$92$		0			0
$93$	3.87903	+	13.6970i	0.402237	+	1.42031i
$94$		0			0
$95$	4.57590			0.469478
$96$		0			0
$97$	13.4213			1.36272			0.681361	−	0.731947i	$-0.261388\pi$
$97$	13.4213			1.36272			0.681361	+	0.731947i	$0.261388\pi$
$98$		0			0
$99$	−3.44678	+	2.12251i	−0.346414	+	0.213320i
$100$		0			0
$101$		−	2.66023i		−	0.264703i	−0.991203	−	0.132351i	$-0.957747\pi$
$101$		−	2.66023i		−	0.264703i	0.991203	−	0.132351i	$-0.0422528\pi$
$102$		0			0
$103$			4.03479i			0.397559i	0.980044	+	0.198780i	$0.0636978\pi$
$103$			4.03479i			0.397559i	−0.980044	+	0.198780i	$0.936302\pi$
$104$		0			0
$105$	−1.66651	+	0.471960i	−0.162635	+	0.0460586i
$106$		0			0
$107$	−13.8801			−1.34184			−0.670919	−	0.741531i	$-0.734100\pi$
$107$	−13.8801			−1.34184			−0.670919	+	0.741531i	$0.734100\pi$
$108$		0			0
$109$	−14.3679			−1.37619			−0.688097	−	0.725619i	$-0.741554\pi$
$109$	−14.3679			−1.37619			−0.688097	+	0.725619i	$0.741554\pi$
$110$		0			0
$111$	−0.0532164	+	0.0150710i	−0.00505108	+	0.00143048i
$112$		0			0
$113$			9.26936i			0.871988i	0.899950	+	0.435994i	$0.143603\pi$
$113$			9.26936i			0.871988i	−0.899950	+	0.435994i	$0.856397\pi$
$114$		0			0
$115$			1.77710i			0.165716i
$116$		0			0
$117$	−2.67565	+	1.64765i	−0.247364	+	0.152325i
$118$		0			0
$119$	2.41361			0.221256
$120$		0			0
$121$	−9.17941			−0.834492
$122$		0			0
$123$	−0.798600	−	2.81989i	−0.0720074	−	0.254261i
$124$		0			0
$125$		−	1.00000i		−	0.0894427i
$126$		0			0
$127$		−	8.61781i		−	0.764707i	−0.924016	−	0.382353i	$-0.875114\pi$
$127$		−	8.61781i		−	0.764707i	0.924016	−	0.382353i	$-0.124886\pi$
$128$		0			0
$129$	−2.74509	−	9.69304i	−0.241692	−	0.853425i
$130$		0			0
$131$	−6.20065			−0.541754			−0.270877	−	0.962614i	$-0.587314\pi$
$131$	−6.20065			−0.541754			−0.270877	+	0.962614i	$0.587314\pi$
$132$		0			0
$133$	−4.57590			−0.396781
$134$		0			0
$135$	−3.82713	−	3.51470i	−0.329387	−	0.302497i
$136$		0			0
$137$			3.99936i			0.341689i	0.985298	+	0.170844i	$0.0546495\pi$
$137$			3.99936i			0.341689i	−0.985298	+	0.170844i	$0.945350\pi$
$138$		0			0
$139$		−	6.87663i		−	0.583268i	−0.956530	−	0.291634i	$-0.905801\pi$
$139$		−	6.87663i		−	0.583268i	0.956530	−	0.291634i	$-0.0941989\pi$
$140$		0			0
$141$	0.284148	−	0.0804714i	0.0239296	−	0.00677691i
$142$		0			0
$143$	1.41328			0.118184
$144$		0			0
$145$	−6.95143			−0.577285
$146$		0			0
$147$	1.66651	−	0.471960i	0.137451	−	0.0389266i
$148$		0			0
$149$			14.5454i			1.19161i	0.803131	+	0.595803i	$0.203166\pi$
$149$			14.5454i			1.19161i	−0.803131	+	0.595803i	$0.796834\pi$
$150$		0			0
$151$		−	1.43228i		−	0.116557i	−0.998300	−	0.0582785i	$-0.981439\pi$
$151$		−	1.43228i		−	0.116557i	0.998300	−	0.0582785i	$-0.0185611\pi$
$152$		0			0
$153$	3.79674	+	6.16560i	0.306948	+	0.498459i
$154$		0			0
$155$	8.21898			0.660165
$156$		0			0
$157$	−0.974472			−0.0777713			−0.0388857	−	0.999244i	$-0.512381\pi$
$157$	−0.974472			−0.0777713			−0.0388857	+	0.999244i	$0.512381\pi$
$158$		0			0
$159$	−1.96219	−	6.92857i	−0.155612	−	0.549471i
$160$		0			0
$161$		−	1.77710i		−	0.140055i
$162$		0			0
$163$			17.2404i			1.35037i	0.737646	+	0.675187i	$0.235937\pi$
$163$			17.2404i			1.35037i	−0.737646	+	0.675187i	$0.764063\pi$
$164$		0			0
$165$	0.636812	+	2.24861i	0.0495757	+	0.175054i
$166$		0			0
$167$	−24.3773			−1.88638			−0.943188	−	0.332260i	$-0.892189\pi$
$167$	−24.3773			−1.88638			−0.943188	+	0.332260i	$0.892189\pi$
$168$		0			0
$169$	−11.9029			−0.915608
$170$		0			0
$171$	−7.19813	−	11.6892i	−0.550455	−	0.893894i
$172$		0			0
$173$			8.74701i			0.665023i	0.943099	+	0.332511i	$0.107896\pi$
$173$			8.74701i			0.665023i	−0.943099	+	0.332511i	$0.892104\pi$
$174$		0			0
$175$			1.00000i			0.0755929i
$176$		0			0
$177$	8.38153	−	2.37367i	0.629994	−	0.178416i
$178$		0			0
$179$	−7.41791			−0.554441			−0.277220	−	0.960806i	$-0.589413\pi$
$179$	−7.41791			−0.554441			−0.277220	+	0.960806i	$0.589413\pi$
$180$		0			0
$181$	5.37874			0.399799			0.199899	−	0.979816i	$-0.435938\pi$
$181$	5.37874			0.399799			0.199899	+	0.979816i	$0.435938\pi$
$182$		0			0
$183$	−1.99309	+	0.564449i	−0.147334	+	0.0417253i
$184$		0			0
$185$			0.0319329i			0.00234775i
$186$		0			0
$187$		−	3.25667i		−	0.238151i
$188$		0			0
$189$	3.82713	+	3.51470i	0.278383	+	0.255656i
$190$		0			0
$191$	−11.6398			−0.842230			−0.421115	−	0.907007i	$-0.638361\pi$
$191$	−11.6398			−0.842230			−0.421115	+	0.907007i	$0.638361\pi$
$192$		0			0
$193$	−25.7251			−1.85174			−0.925868	−	0.377847i	$-0.876664\pi$
$193$	−25.7251			−1.85174			−0.925868	+	0.377847i	$0.876664\pi$
$194$		0			0
$195$	0.494341	+	1.74554i	0.0354005	+	0.125001i
$196$		0			0
$197$			7.46357i			0.531757i	0.964007	+	0.265879i	$0.0856621\pi$
$197$			7.46357i			0.531757i	−0.964007	+	0.265879i	$0.914338\pi$
$198$		0			0
$199$		−	14.3162i		−	1.01485i	−0.861696	−	0.507425i	$-0.830597\pi$
$199$		−	14.3162i		−	1.01485i	0.861696	−	0.507425i	$-0.169403\pi$
$200$		0			0
$201$	−1.63578	−	5.77600i	−0.115379	−	0.407407i
$202$		0			0
$203$	6.95143			0.487895
$204$		0			0
$205$	−1.69209			−0.118181
$206$		0			0
$207$	4.53962	−	2.79547i	0.315526	−	0.194299i
$208$		0			0
$209$			6.17424i			0.427081i
$210$		0			0
$211$			22.3763i			1.54045i	0.637774	+	0.770224i	$0.279855\pi$
$211$			22.3763i			1.54045i	−0.637774	+	0.770224i	$0.720145\pi$
$212$		0			0
$213$	−0.132763	+	0.0375988i	−0.00909677	+	0.00257623i
$214$		0			0
$215$	−5.81637			−0.396673
$216$		0			0
$217$	−8.21898			−0.557941
$218$		0			0
$219$	−6.76353	+	1.91545i	−0.457037	+	0.129434i
$220$		0			0
$221$		−	2.52807i		−	0.170056i
$222$		0			0
$223$			0.199005i			0.0133263i	0.999978	+	0.00666317i	$0.00212097\pi$
$223$			0.199005i			0.0133263i	−0.999978	+	0.00666317i	$0.997879\pi$
$224$		0			0
$225$	−2.55451	+	1.57305i	−0.170301	+	0.104870i
$226$		0			0
$227$	12.7580			0.846779			0.423390	−	0.905948i	$-0.360840\pi$
$227$	12.7580			0.846779			0.423390	+	0.905948i	$0.360840\pi$
$228$		0			0
$229$	−19.5996			−1.29518			−0.647589	−	0.761990i	$-0.724223\pi$
$229$	−19.5996			−1.29518			−0.647589	+	0.761990i	$0.724223\pi$
$230$		0			0
$231$	−0.636812	−	2.24861i	−0.0418992	−	0.147948i
$232$		0			0
$233$			20.9745i			1.37408i	0.726618	+	0.687042i	$0.241091\pi$
$233$			20.9745i			1.37408i	−0.726618	+	0.687042i	$0.758909\pi$
$234$		0			0
$235$		−	0.170505i		−	0.0111225i
$236$		0			0
$237$	−2.69639	−	9.52108i	−0.175150	−	0.618461i
$238$		0			0
$239$	−0.776995			−0.0502596			−0.0251298	−	0.999684i	$-0.508000\pi$
$239$	−0.776995			−0.0502596			−0.0251298	+	0.999684i	$0.508000\pi$
$240$		0			0
$241$	−10.7549			−0.692782			−0.346391	−	0.938090i	$-0.612593\pi$
$241$	−10.7549			−0.692782			−0.346391	+	0.938090i	$0.612593\pi$
$242$		0			0
$243$	−2.95805	+	15.3052i	−0.189759	+	0.981831i
$244$		0			0
$245$		−	1.00000i		−	0.0638877i
$246$		0			0
$247$			4.79290i			0.304965i
$248$		0			0
$249$	0.880175	−	0.249268i	0.0557788	−	0.0157967i
$250$		0			0
$251$	−27.1753			−1.71529			−0.857644	−	0.514245i	$-0.828072\pi$
$251$	−27.1753			−1.71529			−0.857644	+	0.514245i	$0.828072\pi$
$252$		0			0
$253$	−2.39783			−0.150750
$254$		0			0
$255$	4.02231	−	1.13913i	0.251887	−	0.0713350i
$256$		0			0
$257$		−	6.46607i		−	0.403342i	−0.979453	−	0.201671i	$-0.935363\pi$
$257$		−	6.46607i		−	0.403342i	0.979453	−	0.201671i	$-0.0646372\pi$
$258$		0			0
$259$		−	0.0319329i		−	0.00198421i
$260$		0			0
$261$	10.9350	+	17.7575i	0.676856	+	1.09916i
$262$		0			0
$263$	−20.1847			−1.24464			−0.622320	−	0.782763i	$-0.713810\pi$
$263$	−20.1847			−1.24464			−0.622320	+	0.782763i	$0.713810\pi$
$264$		0			0
$265$	−4.15753			−0.255395
$266$		0			0
$267$	−1.30783	−	4.61801i	−0.0800380	−	0.282618i
$268$		0			0
$269$		−	27.3928i		−	1.67017i	−0.550121	−	0.835085i	$-0.685418\pi$
$269$		−	27.3928i		−	1.67017i	0.550121	−	0.835085i	$-0.314582\pi$
$270$		0			0
$271$			3.08258i			0.187253i	0.995607	+	0.0936267i	$0.0298460\pi$
$271$			3.08258i			0.187253i	−0.995607	+	0.0936267i	$0.970154\pi$
$272$		0			0
$273$	−0.494341	−	1.74554i	−0.0299189	−	0.105645i
$274$		0			0
$275$	1.34929			0.0813654
$276$		0			0
$277$	−1.31055			−0.0787431			−0.0393715	−	0.999225i	$-0.512536\pi$
$277$	−1.31055			−0.0787431			−0.0393715	+	0.999225i	$0.512536\pi$
$278$		0			0
$279$	−12.9289	−	20.9955i	−0.774032	−	1.25697i
$280$		0			0
$281$		−	5.17085i		−	0.308467i	−0.988034	−	0.154233i	$-0.950709\pi$
$281$		−	5.17085i		−	0.308467i	0.988034	−	0.154233i	$-0.0492908\pi$
$282$		0			0
$283$			25.9862i			1.54472i	0.635185	+	0.772360i	$0.280924\pi$
$283$			25.9862i			1.54472i	−0.635185	+	0.772360i	$0.719076\pi$
$284$		0			0
$285$	−7.62579	+	2.15964i	−0.451713	+	0.127926i
$286$		0			0
$287$	1.69209			0.0998812
$288$		0			0
$289$	11.1745			0.657322
$290$		0			0
$291$	−22.3667	+	6.33429i	−1.31116	+	0.371323i
$292$		0			0
$293$		−	9.87907i		−	0.577141i	−0.957459	−	0.288571i	$-0.906820\pi$
$293$		−	9.87907i		−	0.577141i	0.957459	−	0.288571i	$-0.0931800\pi$
$294$		0			0
$295$		−	5.02939i		−	0.292822i
$296$		0			0
$297$	4.74235	−	5.16392i	0.275179	−	0.299641i
$298$		0			0
$299$	−1.86138			−0.107646
$300$		0			0
$301$	5.81637			0.335250
$302$		0			0
$303$	1.25552	+	4.43330i	0.0721279	+	0.254687i
$304$		0			0
$305$			1.19597i			0.0684809i
$306$		0			0
$307$			30.4222i			1.73629i	0.496315	+	0.868143i	$0.334686\pi$
$307$			30.4222i			1.73629i	−0.496315	+	0.868143i	$0.665314\pi$
$308$		0			0
$309$	−1.90426	−	6.72401i	−0.108329	−	0.382516i
$310$		0			0
$311$	−13.6254			−0.772628			−0.386314	−	0.922367i	$-0.626252\pi$
$311$	−13.6254			−0.772628			−0.386314	+	0.922367i	$0.626252\pi$
$312$		0			0
$313$	2.63736			0.149072			0.0745362	−	0.997218i	$-0.476252\pi$
$313$	2.63736			0.149072			0.0745362	+	0.997218i	$0.476252\pi$
$314$		0			0
$315$	2.55451	−	1.57305i	0.143930	−	0.0886314i
$316$		0			0
$317$		−	15.2006i		−	0.853750i	−0.904311	−	0.426875i	$-0.859614\pi$
$317$		−	15.2006i		−	0.853750i	0.904311	−	0.426875i	$-0.140386\pi$
$318$		0			0
$319$		−	9.37951i		−	0.525152i
$320$		0			0
$321$	23.1313	−	6.55084i	1.29106	−	0.365632i
$322$		0			0
$323$	11.0445			0.614531
$324$		0			0
$325$	1.04742			0.0581005
$326$		0			0
$327$	23.9442	−	6.78106i	1.32412	−	0.374994i
$328$		0			0
$329$			0.170505i			0.00940023i
$330$		0			0
$331$			25.2879i			1.38995i	0.719033	+	0.694976i	$0.244585\pi$
$331$			25.2879i			1.38995i	−0.719033	+	0.694976i	$0.755415\pi$
$332$		0			0
$333$	0.0815728	−	0.0502320i	0.00447016	−	0.00275270i
$334$		0			0
$335$	−3.46592			−0.189364
$336$		0			0
$337$	9.52237			0.518717			0.259358	−	0.965781i	$-0.416489\pi$
$337$	9.52237			0.518717			0.259358	+	0.965781i	$0.416489\pi$
$338$		0			0
$339$	−4.37476	−	15.4475i	−0.237605	−	0.838992i
$340$		0			0
$341$			11.0898i			0.600547i
$342$		0			0
$343$			1.00000i			0.0539949i
$344$		0			0
$345$	−0.838721	−	2.96156i	−0.0451552	−	0.159445i
$346$		0			0
$347$	−25.5359			−1.37084			−0.685419	−	0.728149i	$-0.740381\pi$
$347$	−25.5359			−1.37084			−0.685419	+	0.728149i	$0.740381\pi$
$348$		0			0
$349$	−11.1822			−0.598569			−0.299284	−	0.954164i	$-0.596748\pi$
$349$	−11.1822			−0.598569			−0.299284	+	0.954164i	$0.596748\pi$
$350$		0			0
$351$	3.68137	−	4.00862i	0.196497	−	0.213964i
$352$		0			0
$353$			16.1670i			0.860481i	0.902714	+	0.430241i	$0.141571\pi$
$353$			16.1670i			0.860481i	−0.902714	+	0.430241i	$0.858429\pi$
$354$		0			0
$355$			0.0796653i			0.00422819i
$356$		0			0
$357$	−4.02231	+	1.13913i	−0.212883	+	0.0602891i
$358$		0			0
$359$	32.3870			1.70932			0.854659	−	0.519189i	$-0.173766\pi$
$359$	32.3870			1.70932			0.854659	+	0.519189i	$0.173766\pi$
$360$		0			0
$361$	−1.93890			−0.102047
$362$		0			0
$363$	15.2976	−	4.33231i	0.802914	−	0.227387i
$364$		0			0
$365$			4.05850i			0.212432i
$366$		0			0
$367$			11.6055i			0.605801i	0.953022	+	0.302900i	$0.0979550\pi$
$367$			11.6055i			0.605801i	−0.953022	+	0.302900i	$0.902045\pi$
$368$		0			0
$369$	2.66175	+	4.32247i	0.138565	+	0.225019i
$370$		0			0
$371$	4.15753			0.215848
$372$		0			0
$373$	31.7339			1.64312			0.821559	−	0.570123i	$-0.193105\pi$
$373$	31.7339			1.64312			0.821559	+	0.570123i	$0.193105\pi$
$374$		0			0
$375$	0.471960	+	1.66651i	0.0243719	+	0.0860582i
$376$		0			0
$377$		−	7.28108i		−	0.374994i
$378$		0			0
$379$			7.67125i			0.394046i	0.980399	+	0.197023i	$0.0631273\pi$
$379$			7.67125i			0.394046i	−0.980399	+	0.197023i	$0.936873\pi$
$380$		0			0
$381$	4.06726	+	14.3617i	0.208372	+	0.735770i
$382$		0			0
$383$	25.7705			1.31681			0.658406	−	0.752663i	$-0.271231\pi$
$383$	25.7705			1.31681			0.658406	+	0.752663i	$0.271231\pi$
$384$		0			0
$385$	−1.34929			−0.0687663
$386$		0			0
$387$	9.14945	+	14.8580i	0.465093	+	0.755273i
$388$		0			0
$389$		−	26.1207i		−	1.32437i	−0.749340	−	0.662185i	$-0.769629\pi$
$389$		−	26.1207i		−	1.32437i	0.749340	−	0.662185i	$-0.230371\pi$
$390$		0			0
$391$			4.28924i			0.216916i
$392$		0			0
$393$	10.3335	−	2.92646i	0.521254	−	0.147620i
$394$		0			0
$395$	−5.71319			−0.287462
$396$		0			0
$397$	−25.8215			−1.29595			−0.647973	−	0.761664i	$-0.724383\pi$
$397$	−25.8215			−1.29595			−0.647973	+	0.761664i	$0.724383\pi$
$398$		0			0
$399$	7.62579	−	2.15964i	0.381767	−	0.108117i
$400$		0			0
$401$			6.94268i			0.346701i	0.984860	+	0.173351i	$0.0554594\pi$
$401$			6.94268i			0.346701i	−0.984860	+	0.173351i	$0.944541\pi$
$402$		0			0
$403$			8.60874i			0.428832i
$404$		0			0
$405$	8.03674	+	4.05102i	0.399349	+	0.201297i
$406$		0			0
$407$	−0.0430868			−0.00213573
$408$		0			0
$409$	−28.5981			−1.41408			−0.707042	−	0.707171i	$-0.749971\pi$
$409$	−28.5981			−1.41408			−0.707042	+	0.707171i	$0.749971\pi$
$410$		0			0
$411$	−1.88754	−	6.66498i	−0.0931054	−	0.328759i
$412$		0			0
$413$			5.02939i			0.247480i
$414$		0			0
$415$		−	0.528155i		−	0.0259261i
$416$		0			0
$417$	3.24549	+	11.4600i	0.158932	+	0.561197i
$418$		0			0
$419$	17.1179			0.836263			0.418132	−	0.908387i	$-0.362685\pi$
$419$	17.1179			0.836263			0.418132	+	0.908387i	$0.362685\pi$
$420$		0			0
$421$	29.2800			1.42702			0.713510	−	0.700645i	$-0.247104\pi$
$421$	29.2800			1.42702			0.713510	+	0.700645i	$0.247104\pi$
$422$		0			0
$423$	−0.435556	+	0.268213i	−0.0211774	+	0.0130409i
$424$		0			0
$425$		−	2.41361i		−	0.117077i
$426$		0			0
$427$		−	1.19597i		−	0.0578770i
$428$		0			0
$429$	−2.35524	+	0.667011i	−0.113712	+	0.0322036i
$430$		0			0
$431$	38.2192			1.84096			0.920478	−	0.390795i	$-0.127800\pi$
$431$	38.2192			1.84096			0.920478	+	0.390795i	$0.127800\pi$
$432$		0			0
$433$	23.1152			1.11085			0.555423	−	0.831568i	$-0.312556\pi$
$433$	23.1152			1.11085			0.555423	+	0.831568i	$0.312556\pi$
$434$		0			0
$435$	11.5846	−	3.28079i	0.555440	−	0.157302i
$436$		0			0
$437$		−	8.13185i		−	0.388999i
$438$		0			0
$439$		−	28.7138i		−	1.37044i	−0.728338	−	0.685218i	$-0.759707\pi$
$439$		−	28.7138i		−	1.37044i	0.728338	−	0.685218i	$-0.240293\pi$
$440$		0			0
$441$	−2.55451	+	1.57305i	−0.121643	+	0.0749072i
$442$		0			0
$443$	11.4179			0.542480			0.271240	−	0.962512i	$-0.412566\pi$
$443$	11.4179			0.542480			0.271240	+	0.962512i	$0.412566\pi$
$444$		0			0
$445$	−2.77107			−0.131361
$446$		0			0
$447$	−6.86484	−	24.2401i	−0.324696	−	1.14652i
$448$		0			0
$449$			17.0039i			0.802463i	0.915977	+	0.401232i	$0.131418\pi$
$449$			17.0039i			0.802463i	−0.915977	+	0.401232i	$0.868582\pi$
$450$		0			0
$451$		−	2.28313i		−	0.107508i
$452$		0			0
$453$	0.675977	+	2.38690i	0.0317602	+	0.112146i
$454$		0			0
$455$	−1.04742			−0.0491039
$456$		0			0
$457$	−32.8700			−1.53760			−0.768798	−	0.639492i	$-0.779145\pi$
$457$	−32.8700			−1.53760			−0.768798	+	0.639492i	$0.779145\pi$
$458$		0			0
$459$	−9.23721	−	8.48312i	−0.431156	−	0.395958i
$460$		0			0
$461$		−	27.1434i		−	1.26419i	−0.774889	−	0.632097i	$-0.782194\pi$
$461$		−	27.1434i		−	1.26419i	0.774889	−	0.632097i	$-0.217806\pi$
$462$		0			0
$463$			18.2611i			0.848663i	0.905507	+	0.424332i	$0.139491\pi$
$463$			18.2611i			0.848663i	−0.905507	+	0.424332i	$0.860509\pi$
$464$		0			0
$465$	−13.6970	+	3.87903i	−0.635184	+	0.179886i
$466$		0			0
$467$	−11.2099			−0.518733			−0.259366	−	0.965779i	$-0.583514\pi$
$467$	−11.2099			−0.518733			−0.259366	+	0.965779i	$0.583514\pi$
$468$		0			0
$469$	3.46592			0.160042
$470$		0			0
$471$	1.62397	−	0.459912i	0.0748285	−	0.0211916i
$472$		0			0
$473$		−	7.84799i		−	0.360851i
$474$		0			0
$475$			4.57590i			0.209957i
$476$		0			0
$477$	6.54001	+	10.6205i	0.299447	+	0.486277i
$478$		0			0
$479$	−3.28189			−0.149953			−0.0749767	−	0.997185i	$-0.523888\pi$
$479$	−3.28189			−0.149953			−0.0749767	+	0.997185i	$0.523888\pi$
$480$		0			0
$481$	−0.0334472			−0.00152506
$482$		0			0
$483$	0.838721	+	2.96156i	0.0381631	+	0.134756i
$484$		0			0
$485$			13.4213i			0.609428i
$486$		0			0
$487$		−	0.0226211i		−	0.00102506i	−1.00000		0.000512530i	$-0.999837\pi$
$487$		−	0.0226211i		−	0.00102506i	1.00000		0.000512530i	$-0.000163143\pi$
$488$		0			0
$489$	−8.13679	−	28.7313i	−0.367958	−	1.29928i
$490$		0			0
$491$	−3.39585			−0.153252			−0.0766262	−	0.997060i	$-0.524415\pi$
$491$	−3.39585			−0.153252			−0.0766262	+	0.997060i	$0.524415\pi$
$492$		0			0
$493$	−16.7781			−0.755646
$494$		0			0
$495$	−2.12251	−	3.44678i	−0.0953996	−	0.154921i
$496$		0			0
$497$		−	0.0796653i		−	0.00357348i
$498$		0			0
$499$			6.04713i			0.270707i	0.990797	+	0.135353i	$0.0432170\pi$
$499$			6.04713i			0.270707i	−0.990797	+	0.135353i	$0.956783\pi$
$500$		0			0
$501$	40.6251	−	11.5051i	1.81499	−	0.514011i
$502$		0			0
$503$	−37.4048			−1.66780			−0.833900	−	0.551916i	$-0.813897\pi$
$503$	−37.4048			−1.66780			−0.833900	+	0.551916i	$0.813897\pi$
$504$		0			0
$505$	2.66023			0.118379
$506$		0			0
$507$	19.8363	−	5.61769i	0.880961	−	0.249491i
$508$		0			0
$509$		−	13.8789i		−	0.615172i	−0.951520	−	0.307586i	$-0.900479\pi$
$509$		−	13.8789i		−	0.615172i	0.951520	−	0.307586i	$-0.0995211\pi$
$510$		0			0
$511$		−	4.05850i		−	0.179538i
$512$		0			0
$513$	17.5126	+	16.0829i	0.773199	+	0.710078i
$514$		0			0
$515$	−4.03479			−0.177794
$516$		0			0
$517$	0.230061			0.0101181
$518$		0			0
$519$	−4.12823	−	14.5770i	−0.181209	−	0.639858i
$520$		0			0
$521$			13.2093i			0.578708i	0.957222	+	0.289354i	$0.0934405\pi$
$521$			13.2093i			0.578708i	−0.957222	+	0.289354i	$0.906560\pi$
$522$		0			0
$523$			5.47398i			0.239360i	0.992813	+	0.119680i	$0.0381869\pi$
$523$			5.47398i			0.239360i	−0.992813	+	0.119680i	$0.961813\pi$
$524$		0			0
$525$	−0.471960	−	1.66651i	−0.0205980	−	0.0727324i
$526$		0			0
$527$	19.8375			0.864133
$528$		0			0
$529$	−19.8419			−0.862691
$530$		0			0
$531$	−12.8476	+	7.91149i	−0.557539	+	0.343329i
$532$		0			0
$533$		−	1.77234i		−	0.0767684i
$534$		0			0
$535$		−	13.8801i		−	0.600088i
$536$		0			0
$537$	12.3620	−	3.50095i	0.533460	−	0.151077i
$538$		0			0
$539$	1.34929			0.0581182
$540$		0			0
$541$	25.2798			1.08686			0.543432	−	0.839453i	$-0.317125\pi$
$541$	25.2798			1.08686			0.543432	+	0.839453i	$0.317125\pi$
$542$		0			0
$543$	−8.96373	+	2.53855i	−0.384670	+	0.108940i
$544$		0			0
$545$		−	14.3679i		−	0.615453i
$546$		0			0
$547$		−	25.9816i		−	1.11089i	−0.831553	−	0.555446i	$-0.812548\pi$
$547$		−	25.9816i		−	1.11089i	0.831553	−	0.555446i	$-0.187452\pi$
$548$		0			0
$549$	3.05511	−	1.88132i	0.130389	−	0.0802927i
$550$		0			0
$551$	31.8091			1.35511
$552$		0			0
$553$	5.71319			0.242949
$554$		0			0
$555$	−0.0150710	−	0.0532164i	−0.000639730	−	0.00225891i
$556$		0			0
$557$		−	43.6588i		−	1.84988i	−0.380112	−	0.924941i	$-0.624114\pi$
$557$		−	43.6588i		−	1.84988i	0.380112	−	0.924941i	$-0.375886\pi$
$558$		0			0
$559$		−	6.09219i		−	0.257672i
$560$		0			0
$561$	1.53702	+	5.42728i	0.0648930	+	0.229140i
$562$		0			0
$563$	30.3789			1.28032			0.640159	−	0.768242i	$-0.278868\pi$
$563$	30.3789			1.28032			0.640159	+	0.768242i	$0.278868\pi$
$564$		0			0
$565$	−9.26936			−0.389965
$566$		0			0
$567$	−8.03674	−	4.05102i	−0.337511	−	0.170127i
$568$		0			0
$569$			16.6690i			0.698799i	0.936974	+	0.349400i	$0.113614\pi$
$569$			16.6690i			0.698799i	−0.936974	+	0.349400i	$0.886386\pi$
$570$		0			0
$571$		−	22.4915i		−	0.941239i	−0.882336	−	0.470619i	$-0.844031\pi$
$571$		−	22.4915i		−	0.941239i	0.882336	−	0.470619i	$-0.155969\pi$
$572$		0			0
$573$	19.3979	−	5.49354i	0.810359	−	0.229496i
$574$		0			0
$575$	−1.77710			−0.0741103
$576$		0			0
$577$	31.3145			1.30364			0.651820	−	0.758374i	$-0.274006\pi$
$577$	31.3145			1.30364			0.651820	+	0.758374i	$0.274006\pi$
$578$		0			0
$579$	42.8712	−	12.1412i	1.78167	−	0.504572i
$580$		0			0
$581$			0.528155i			0.0219115i
$582$		0			0
$583$		−	5.60973i		−	0.232331i
$584$		0			0
$585$	−1.64765	−	2.67565i	−0.0681218	−	0.110624i
$586$		0			0
$587$	−32.4958			−1.34124			−0.670622	−	0.741799i	$-0.733973\pi$
$587$	−32.4958			−1.34124			−0.670622	+	0.741799i	$0.733973\pi$
$588$		0			0
$589$	−37.6093			−1.54966
$590$		0			0
$591$	−3.52250	−	12.4381i	−0.144896	−	0.511635i
$592$		0			0
$593$			32.0254i			1.31513i	0.753400	+	0.657563i	$0.228412\pi$
$593$			32.0254i			1.31513i	−0.753400	+	0.657563i	$0.771588\pi$
$594$		0			0
$595$			2.41361i			0.0989485i
$596$		0			0
$597$	6.75668	+	23.8581i	0.276532	+	0.976448i
$598$		0			0
$599$	16.0824			0.657108			0.328554	−	0.944485i	$-0.393439\pi$
$599$	16.0824			0.657108			0.328554	+	0.944485i	$0.393439\pi$
$600$		0			0
$601$	−10.3019			−0.420224			−0.210112	−	0.977677i	$-0.567383\pi$
$601$	−10.3019			−0.420224			−0.210112	+	0.977677i	$0.567383\pi$
$602$		0			0
$603$	5.45208	+	8.85373i	0.222026	+	0.360552i
$604$		0			0
$605$		−	9.17941i		−	0.373196i
$606$		0			0
$607$			20.6506i			0.838181i	0.907944	+	0.419091i	$0.137651\pi$
$607$			20.6506i			0.838181i	−0.907944	+	0.419091i	$0.862349\pi$
$608$		0			0
$609$	−11.5846	+	3.28079i	−0.469433	+	0.132945i
$610$		0			0
$611$	0.178590			0.00722499
$612$		0			0
$613$	31.2236			1.26111			0.630555	−	0.776144i	$-0.282827\pi$
$613$	31.2236			1.26111			0.630555	+	0.776144i	$0.282827\pi$
$614$		0			0
$615$	2.81989	−	0.798600i	0.113709	−	0.0322027i
$616$		0			0
$617$			47.2258i			1.90124i	0.310357	+	0.950620i	$0.399552\pi$
$617$			47.2258i			1.90124i	−0.310357	+	0.950620i	$0.600448\pi$
$618$		0			0
$619$			15.2975i			0.614858i	0.951571	+	0.307429i	$0.0994686\pi$
$619$			15.2975i			0.614858i	−0.951571	+	0.307429i	$0.900531\pi$
$620$		0			0
$621$	−6.24598	+	6.80120i	−0.250642	+	0.272923i
$622$		0			0
$623$	2.77107			0.111020
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	−2.91399	−	10.2894i	−0.116374	−	0.410920i
$628$		0			0
$629$			0.0770736i			0.00307313i
$630$		0			0
$631$			14.9732i			0.596073i	0.954554	+	0.298037i	$0.0963318\pi$
$631$			14.9732i			0.596073i	−0.954554	+	0.298037i	$0.903668\pi$
$632$		0			0
$633$	−10.5607	−	37.2903i	−0.419750	−	1.48216i
$634$		0			0
$635$	8.61781			0.341987
$636$		0			0
$637$	1.04742			0.0415003
$638$		0			0
$639$	0.203506	−	0.125318i	0.00805056	−	0.00495749i
$640$		0			0
$641$		−	32.0554i		−	1.26611i	−0.774106	−	0.633056i	$-0.781800\pi$
$641$		−	32.0554i		−	1.26611i	0.774106	−	0.633056i	$-0.218200\pi$
$642$		0			0
$643$			21.6946i			0.855552i	0.903885	+	0.427776i	$0.140703\pi$
$643$			21.6946i			0.855552i	−0.903885	+	0.427776i	$0.859297\pi$
$644$		0			0
$645$	9.69304	−	2.74509i	0.381663	−	0.108088i
$646$		0			0
$647$	20.8733			0.820615			0.410307	−	0.911947i	$-0.365421\pi$
$647$	20.8733			0.820615			0.410307	+	0.911947i	$0.365421\pi$
$648$		0			0
$649$	6.78612			0.266379
$650$		0			0
$651$	13.6970	−	3.87903i	0.536828	−	0.152031i
$652$		0			0
$653$		−	4.17224i		−	0.163272i	−0.996662	−	0.0816362i	$-0.973985\pi$
$653$		−	4.17224i		−	0.163272i	0.996662	−	0.0816362i	$-0.0260146\pi$
$654$		0			0
$655$		−	6.20065i		−	0.242280i
$656$		0			0
$657$	10.3675	−	6.38423i	0.404474	−	0.249073i
$658$		0			0
$659$	12.4147			0.483607			0.241804	−	0.970325i	$-0.422261\pi$
$659$	12.4147			0.483607			0.241804	+	0.970325i	$0.422261\pi$
$660$		0			0
$661$	7.39478			0.287623			0.143812	−	0.989605i	$-0.454064\pi$
$661$	7.39478			0.287623			0.143812	+	0.989605i	$0.454064\pi$
$662$		0			0
$663$	1.19315	+	4.21305i	0.0463380	+	0.163621i
$664$		0			0
$665$		−	4.57590i		−	0.177446i
$666$		0			0
$667$			12.3534i			0.478326i
$668$		0			0
$669$	−0.0939221	−	0.331643i	−0.00363124	−	0.0128221i
$670$		0			0
$671$	−1.61371			−0.0622966
$672$		0			0
$673$	24.1638			0.931448			0.465724	−	0.884930i	$-0.345794\pi$
$673$	24.1638			0.931448			0.465724	+	0.884930i	$0.345794\pi$
$674$		0			0
$675$	3.51470	−	3.82713i	0.135281	−	0.147306i
$676$		0			0
$677$		−	38.9172i		−	1.49571i	−0.663863	−	0.747854i	$-0.731084\pi$
$677$		−	38.9172i		−	1.49571i	0.663863	−	0.747854i	$-0.268916\pi$
$678$		0			0
$679$		−	13.4213i		−	0.515061i
$680$		0			0
$681$	−21.2614	+	6.02127i	−0.814737	+	0.230736i
$682$		0			0
$683$	15.6393			0.598423			0.299211	−	0.954187i	$-0.403276\pi$
$683$	15.6393			0.598423			0.299211	+	0.954187i	$0.403276\pi$
$684$		0			0
$685$	−3.99936			−0.152808
$686$		0			0
$687$	32.6629	−	9.25022i	1.24617	−	0.352918i
$688$		0			0
$689$		−	4.35469i		−	0.165900i
$690$		0			0
$691$		−	7.05814i		−	0.268504i	−0.990947	−	0.134252i	$-0.957137\pi$
$691$		−	7.05814i		−	0.268504i	0.990947	−	0.134252i	$-0.0428632\pi$
$692$		0			0
$693$	2.12251	+	3.44678i	0.0806274	+	0.130932i
$694$		0			0
$695$	6.87663			0.260845
$696$		0			0
$697$	−4.08406			−0.154695
$698$		0			0
$699$	−9.89911	−	34.9542i	−0.374419	−	1.32209i
$700$		0			0
$701$		−	31.6166i		−	1.19414i	−0.802188	−	0.597071i	$-0.796331\pi$
$701$		−	31.6166i		−	1.19414i	0.802188	−	0.597071i	$-0.203669\pi$
$702$		0			0
$703$		−	0.146122i		−	0.00551109i
$704$		0			0
$705$	0.0804714	+	0.284148i	0.00303073	+	0.0107016i
$706$		0			0
$707$	−2.66023			−0.100048
$708$		0			0
$709$	−29.6084			−1.11197			−0.555983	−	0.831194i	$-0.687658\pi$
$709$	−29.6084			−1.11197			−0.555983	+	0.831194i	$0.687658\pi$
$710$		0			0
$711$	8.98713	+	14.5944i	0.337044	+	0.547332i
$712$		0			0
$713$		−	14.6060i		−	0.546998i
$714$		0			0
$715$			1.41328i			0.0528536i
$716$		0			0
$717$	1.29487	−	0.366710i	0.0483578	−	0.0136950i
$718$		0			0
$719$	35.9233			1.33971			0.669857	−	0.742490i	$-0.266355\pi$
$719$	35.9233			1.33971			0.669857	+	0.742490i	$0.266355\pi$
$720$		0			0
$721$	4.03479			0.150263
$722$		0			0
$723$	17.9231	−	5.07586i	0.666567	−	0.188773i
$724$		0			0
$725$		−	6.95143i		−	0.258170i
$726$		0			0
$727$		−	3.99271i		−	0.148081i	−0.997255	−	0.0740407i	$-0.976411\pi$
$727$		−	3.99271i		−	0.148081i	0.997255	−	0.0740407i	$-0.0235895\pi$
$728$		0			0
$729$	−2.29383	−	26.9024i	−0.0849568	−	0.996385i
$730$		0			0
$731$	−14.0385			−0.519232
$732$		0			0
$733$	−42.3972			−1.56598			−0.782989	−	0.622036i	$-0.786306\pi$
$733$	−42.3972			−1.56598			−0.782989	+	0.622036i	$0.786306\pi$
$734$		0			0
$735$	0.471960	+	1.66651i	0.0174085	+	0.0614701i
$736$		0			0
$737$		−	4.67655i		−	0.172263i
$738$		0			0
$739$		−	51.3151i		−	1.88766i	−0.330436	−	0.943828i	$-0.607196\pi$
$739$		−	51.3151i		−	1.88766i	0.330436	−	0.943828i	$-0.392804\pi$
$740$		0			0
$741$	−2.26206	−	7.98741i	−0.0830987	−	0.293425i
$742$		0			0
$743$	−23.8953			−0.876633			−0.438317	−	0.898821i	$-0.644425\pi$
$743$	−23.8953			−0.876633			−0.438317	+	0.898821i	$0.644425\pi$
$744$		0			0
$745$	−14.5454			−0.532902
$746$		0			0
$747$	−1.34918	+	0.830814i	−0.0493637	+	0.0303979i
$748$		0			0
$749$			13.8801i			0.507167i
$750$		0			0
$751$			50.6446i			1.84805i	0.382333	+	0.924025i	$0.375121\pi$
$751$			50.6446i			1.84805i	−0.382333	+	0.924025i	$0.624879\pi$
$752$		0			0
$753$	45.2878	−	12.8256i	1.65038	−	0.467392i
$754$		0			0
$755$	1.43228			0.0521259
$756$		0			0
$757$	−31.0817			−1.12968			−0.564841	−	0.825200i	$-0.691062\pi$
$757$	−31.0817			−1.12968			−0.564841	+	0.825200i	$0.691062\pi$
$758$		0			0
$759$	3.99601	−	1.13168i	0.145046	−	0.0410774i
$760$		0			0
$761$			8.12199i			0.294422i	0.989105	+	0.147211i	$0.0470296\pi$
$761$			8.12199i			0.294422i	−0.989105	+	0.147211i	$0.952970\pi$
$762$		0			0
$763$			14.3679i			0.520152i
$764$		0			0
$765$	−6.16560	+	3.79674i	−0.222918	+	0.137271i
$766$		0			0
$767$	5.26789			0.190213
$768$		0			0
$769$	30.8483			1.11242			0.556209	−	0.831043i	$-0.312255\pi$
$769$	30.8483			1.11242			0.556209	+	0.831043i	$0.312255\pi$
$770$		0			0
$771$	3.05172	+	10.7758i	0.109905	+	0.388080i
$772$		0			0
$773$		−	18.5166i		−	0.665996i	−0.942928	−	0.332998i	$-0.891940\pi$
$773$		−	18.5166i		−	0.665996i	0.942928	−	0.332998i	$-0.108060\pi$
$774$		0			0
$775$			8.21898i			0.295235i
$776$		0			0
$777$	0.0150710	+	0.0532164i	0.000540670	+	0.00190913i
$778$		0			0
$779$	7.74286			0.277417
$780$		0			0
$781$	−0.107492			−0.00384636
$782$		0			0
$783$	−26.6040	−	24.4322i	−0.950750	−	0.873134i
$784$		0			0
$785$		−	0.974472i		−	0.0347804i
$786$		0			0
$787$			20.8287i			0.742464i	0.928540	+	0.371232i	$0.121065\pi$
$787$			20.8287i			0.742464i	−0.928540	+	0.371232i	$0.878935\pi$
$788$		0			0
$789$	33.6379	−	9.52634i	1.19754	−	0.339147i
$790$		0			0
$791$	9.26936			0.329580
$792$		0			0
$793$	−1.25268			−0.0444841
$794$		0			0
$795$	6.92857	−	1.96219i	0.245731	−	0.0695917i
$796$		0			0
$797$		−	28.7237i		−	1.01744i	−0.860931	−	0.508722i	$-0.830118\pi$
$797$		−	28.7237i		−	1.01744i	0.860931	−	0.508722i	$-0.169882\pi$
$798$		0			0
$799$		−	0.411533i		−	0.0145590i
$800$		0			0
$801$	4.35903	+	7.07871i	0.154019	+	0.250114i
$802$		0			0
$803$	−5.47611			−0.193248
$804$		0			0
$805$	1.77710			0.0626347
$806$		0			0
$807$	12.9283	+	45.6504i	0.455098	+	1.60697i
$808$		0			0
$809$		−	12.8124i		−	0.450461i	−0.974306	−	0.225230i	$-0.927687\pi$
$809$		−	12.8124i		−	0.450461i	0.974306	−	0.225230i	$-0.0723135\pi$
$810$		0			0
$811$			5.50626i			0.193351i	0.995316	+	0.0966755i	$0.0308209\pi$
$811$			5.50626i			0.193351i	−0.995316	+	0.0966755i	$0.969179\pi$
$812$		0			0
$813$	−1.45485	−	5.13715i	−0.0510239	−	0.180168i
$814$		0			0
$815$	−17.2404			−0.603906
$816$		0			0
$817$	26.6152			0.931147
$818$		0			0
$819$	1.64765	+	2.67565i	0.0575735	+	0.0934947i
$820$		0			0
$821$			32.9362i			1.14948i	0.818336	+	0.574740i	$0.194897\pi$
$821$			32.9362i			1.14948i	−0.818336	+	0.574740i	$0.805103\pi$
$822$		0			0
$823$			27.9239i			0.973365i	0.873579	+	0.486683i	$0.161793\pi$
$823$			27.9239i			0.973365i	−0.873579	+	0.486683i	$0.838207\pi$
$824$		0			0
$825$	−2.24861	+	0.636812i	−0.0782865	+	0.0221709i
$826$		0			0
$827$	28.3794			0.986847			0.493423	−	0.869789i	$-0.335745\pi$
$827$	28.3794			0.986847			0.493423	+	0.869789i	$0.335745\pi$
$828$		0			0
$829$	6.04197			0.209846			0.104923	−	0.994480i	$-0.466540\pi$
$829$	6.04197			0.209846			0.104923	+	0.994480i	$0.466540\pi$
$830$		0			0
$831$	2.18404	−	0.618525i	0.0757634	−	0.0214564i
$832$		0			0
$833$		−	2.41361i		−	0.0836268i
$834$		0			0
$835$		−	24.3773i		−	0.843613i
$836$		0			0
$837$	31.4551	+	28.8872i	1.08725	+	0.998488i
$838$		0			0
$839$	−23.8802			−0.824435			−0.412218	−	0.911085i	$-0.635246\pi$
$839$	−23.8802			−0.824435			−0.412218	+	0.911085i	$0.635246\pi$
$840$		0			0
$841$	−19.3224			−0.666288
$842$		0			0
$843$	2.44043	+	8.61727i	0.0840530	+	0.296794i
$844$		0			0
$845$		−	11.9029i		−	0.409472i
$846$		0			0
$847$			9.17941i			0.315408i
$848$		0			0
$849$	−12.2644	−	43.3063i	−0.420915	−	1.48627i
$850$		0			0
$851$	0.0567480			0.00194530
$852$		0			0
$853$	54.5041			1.86618			0.933092	−	0.359639i	$-0.117100\pi$
$853$	54.5041			1.86618			0.933092	+	0.359639i	$0.117100\pi$
$854$		0			0
$855$	11.6892	−	7.19813i	0.399762	−	0.246171i
$856$		0			0
$857$		−	41.6961i		−	1.42431i	−0.702021	−	0.712156i	$-0.747719\pi$
$857$		−	41.6961i		−	1.42431i	0.702021	−	0.712156i	$-0.252281\pi$
$858$		0			0
$859$			23.5895i			0.804863i	0.915450	+	0.402432i	$0.131835\pi$
$859$			23.5895i			0.804863i	−0.915450	+	0.402432i	$0.868165\pi$
$860$		0			0
$861$	−2.81989	+	0.798600i	−0.0961016	+	0.0272162i
$862$		0			0
$863$	0.949882			0.0323344			0.0161672	−	0.999869i	$-0.494854\pi$
$863$	0.949882			0.0323344			0.0161672	+	0.999869i	$0.494854\pi$
$864$		0			0
$865$	−8.74701			−0.297407
$866$		0			0
$867$	−18.6224	+	5.27390i	−0.632448	+	0.179111i
$868$		0			0
$869$		−	7.70876i		−	0.261502i
$870$		0			0
$871$		−	3.63028i		−	0.123007i
$872$		0			0
$873$	34.2847	−	21.1123i	1.16036	−	0.714544i
$874$		0			0
$875$	−1.00000			−0.0338062
$876$		0			0
$877$	−5.06229			−0.170941			−0.0854707	−	0.996341i	$-0.527239\pi$
$877$	−5.06229			−0.170941			−0.0854707	+	0.996341i	$0.527239\pi$
$878$		0			0
$879$	4.66252	+	16.4636i	0.157263	+	0.555302i
$880$		0			0
$881$		−	26.4137i		−	0.889899i	−0.895556	−	0.444949i	$-0.853222\pi$
$881$		−	26.4137i		−	0.889899i	0.895556	−	0.444949i	$-0.146778\pi$
$882$		0			0
$883$			26.2714i			0.884104i	0.896989	+	0.442052i	$0.145749\pi$
$883$			26.2714i			0.884104i	−0.896989	+	0.442052i	$0.854251\pi$
$884$		0			0
$885$	2.37367	+	8.38153i	0.0797900	+	0.281742i
$886$		0			0
$887$	−12.4533			−0.418139			−0.209070	−	0.977901i	$-0.567044\pi$
$887$	−12.4533			−0.418139			−0.209070	+	0.977901i	$0.567044\pi$
$888$		0			0
$889$	−8.61781			−0.289032
$890$		0			0
$891$	−5.46602	+	10.8439i	−0.183118	+	0.363285i
$892$		0			0
$893$			0.780213i			0.0261088i
$894$		0			0
$895$		−	7.41791i		−	0.247953i
$896$		0			0
$897$	3.10200	−	0.878495i	0.103573	−	0.0293321i
$898$		0			0
$899$	57.1337			1.90551
$900$		0			0
$901$	−10.0347			−0.334304
$902$		0			0
$903$	−9.69304	+	2.74509i	−0.322564	+	0.0913510i
$904$		0			0
$905$			5.37874i			0.178795i
$906$		0			0
$907$		−	39.4693i		−	1.31056i	−0.755387	−	0.655279i	$-0.772551\pi$
$907$		−	39.4693i		−	1.31056i	0.755387	−	0.655279i	$-0.227449\pi$
$908$		0			0
$909$	−4.18468	−	6.79558i	−0.138797	−	0.225395i
$910$		0			0
$911$	50.3720			1.66890			0.834450	−	0.551084i	$-0.185786\pi$
$911$	50.3720			1.66890			0.834450	+	0.551084i	$0.185786\pi$
$912$		0			0
$913$	0.712635			0.0235848
$914$		0			0
$915$	−0.564449	−	1.99309i	−0.0186601	−	0.0658896i
$916$		0			0
$917$			6.20065i			0.204764i
$918$		0			0
$919$		−	36.1710i		−	1.19317i	−0.802550	−	0.596585i	$-0.796524\pi$
$919$		−	36.1710i		−	1.19317i	0.802550	−	0.596585i	$-0.203476\pi$
$920$		0			0
$921$	−14.3580	−	50.6988i	−0.473114	−	1.67058i
$922$		0			0
$923$	−0.0834431			−0.00274656
$924$		0			0
$925$	−0.0319329			−0.00104995
$926$		0			0
$927$	6.34693	+	10.3069i	0.208460	+	0.338523i
$928$		0			0
$929$			56.7512i			1.86195i	0.365087	+	0.930973i	$0.381039\pi$
$929$			56.7512i			1.86195i	−0.365087	+	0.930973i	$0.618961\pi$
$930$		0			0
$931$			4.57590i			0.149969i
$932$		0			0
$933$	22.7069	−	6.43066i	0.743392	−	0.210530i
$934$		0			0
$935$	3.25667			0.106505
$936$		0			0
$937$	−17.9272			−0.585655			−0.292827	−	0.956165i	$-0.594596\pi$
$937$	−17.9272			−0.585655			−0.292827	+	0.956165i	$0.594596\pi$
$938$		0			0
$939$	−4.39519	+	1.24473i	−0.143431	+	0.0406202i
$940$		0			0
$941$			32.3153i			1.05345i	0.850036	+	0.526725i	$0.176580\pi$
$941$			32.3153i			1.05345i	−0.850036	+	0.526725i	$0.823420\pi$
$942$		0			0
$943$			3.00703i			0.0979222i
$944$		0			0
$945$	−3.51470	+	3.82713i	−0.114333	+	0.124497i
$946$		0			0
$947$	17.0807			0.555047			0.277524	−	0.960719i	$-0.410486\pi$
$947$	17.0807			0.555047			0.277524	+	0.960719i	$0.410486\pi$
$948$		0			0
$949$	−4.25096			−0.137992
$950$		0			0
$951$	7.17407	+	25.3319i	0.232635	+	0.821444i
$952$		0			0
$953$			28.8100i			0.933246i	0.884456	+	0.466623i	$0.154529\pi$
$953$			28.8100i			0.933246i	−0.884456	+	0.466623i	$0.845471\pi$
$954$		0			0
$955$		−	11.6398i		−	0.376657i
$956$		0			0
$957$	4.42675	+	15.6311i	0.143097	+	0.505280i
$958$		0			0
$959$	3.99936			0.129146
$960$		0			0
$961$	−36.5517			−1.17909
$962$		0			0
$963$	−35.4568	+	21.8341i	−1.14258	+	0.703593i
$964$		0			0
$965$		−	25.7251i		−	0.828121i
$966$		0			0
$967$			44.5466i			1.43252i	0.697832	+	0.716262i	$0.254148\pi$
$967$			44.5466i			1.43252i	−0.697832	+	0.716262i	$0.745852\pi$
$968$		0			0
$969$	−18.4057	+	5.21254i	−0.591277	+	0.167451i
$970$		0			0
$971$	−4.97433			−0.159634			−0.0798169	−	0.996810i	$-0.525434\pi$
$971$	−4.97433			−0.159634			−0.0798169	+	0.996810i	$0.525434\pi$
$972$		0			0
$973$	−6.87663			−0.220455
$974$		0			0
$975$	−1.74554	+	0.494341i	−0.0559019	+	0.0158316i
$976$		0			0
$977$			0.855078i			0.0273564i	0.999906	+	0.0136782i	$0.00435403\pi$
$977$			0.855078i			0.0273564i	−0.999906	+	0.0136782i	$0.995646\pi$
$978$		0			0
$979$		−	3.73898i		−	0.119498i
$980$		0			0
$981$	−36.7029	+	22.6014i	−1.17183	+	0.721608i
$982$		0			0
$983$	−49.2592			−1.57112			−0.785562	−	0.618783i	$-0.787626\pi$
$983$	−49.2592			−1.57112			−0.785562	+	0.618783i	$0.787626\pi$
$984$		0			0
$985$	−7.46357			−0.237809
$986$		0			0
$987$	−0.0804714	−	0.284148i	−0.00256143	−	0.00904452i
$988$		0			0
$989$			10.3363i			0.328675i
$990$		0			0
$991$			5.27346i			0.167517i	0.996486	+	0.0837584i	$0.0266924\pi$
$991$			5.27346i			0.167517i	−0.996486	+	0.0837584i	$0.973308\pi$
$992$		0			0
$993$	−11.9349	−	42.1426i	−0.378742	−	1.33736i
$994$		0			0
$995$	14.3162			0.453855
$996$		0			0
$997$	45.0575			1.42698			0.713492	−	0.700663i	$-0.247112\pi$
$997$	45.0575			1.42698			0.713492	+	0.700663i	$0.247112\pi$
$998$		0			0
$999$	−0.112234	+	0.122211i	−0.00355094	+	0.00386659i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3360.2.ba.f.2591.2	yes	24
3.2	odd	2		3360.2.ba.e.2591.24	yes	24
4.3	odd	2		3360.2.ba.e.2591.23	✓	24
12.11	even	2	inner	3360.2.ba.f.2591.1	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3360.2.ba.e.2591.23	✓	24	4.3	odd	2
3360.2.ba.e.2591.24	yes	24	3.2	odd	2
3360.2.ba.f.2591.1	yes	24	12.11	even	2	inner
3360.2.ba.f.2591.2	yes	24	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 3360 = 2^{5} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3360.ba (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.66651 + 0.471960i) q^{3} +1.00000i q^{5} -1.00000i q^{7} +(2.55451 - 1.57305i) q^{9} +O(q^{10})\)	\(q+(-1.66651 + 0.471960i) q^{3} +1.00000i q^{5} -1.00000i q^{7} +(2.55451 - 1.57305i) q^{9} -1.34929 q^{11} -1.04742 q^{13} +(-0.471960 - 1.66651i) q^{15} +2.41361i q^{17} -4.57590i q^{19} +(0.471960 + 1.66651i) q^{21} +1.77710 q^{23} -1.00000 q^{25} +(-3.51470 + 3.82713i) q^{27} +6.95143i q^{29} -8.21898i q^{31} +(2.24861 - 0.636812i) q^{33} +1.00000 q^{35} +0.0319329 q^{37} +(1.74554 - 0.494341i) q^{39} +1.69209i q^{41} +5.81637i q^{43} +(1.57305 + 2.55451i) q^{45} -0.170505 q^{47} -1.00000 q^{49} +(-1.13913 - 4.02231i) q^{51} +4.15753i q^{53} -1.34929i q^{55} +(2.15964 + 7.62579i) q^{57} -5.02939 q^{59} +1.19597 q^{61} +(-1.57305 - 2.55451i) q^{63} -1.04742i q^{65} +3.46592i q^{67} +(-2.96156 + 0.838721i) q^{69} +0.0796653 q^{71} +4.05850 q^{73} +(1.66651 - 0.471960i) q^{75} +1.34929i q^{77} +5.71319i q^{79} +(4.05102 - 8.03674i) q^{81} -0.528155 q^{83} -2.41361 q^{85} +(-3.28079 - 11.5846i) q^{87} +2.77107i q^{89} +1.04742i q^{91} +(3.87903 + 13.6970i) q^{93} +4.57590 q^{95} +13.4213 q^{97} +(-3.44678 + 2.12251i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 4 q^{9}+O(q^{10}) \) 24 * q + 4 * q^9	\( 24 q + 4 q^{9} + 8 q^{11} + 8 q^{13} + 4 q^{15} - 4 q^{21} - 24 q^{25} + 8 q^{33} + 24 q^{35} + 8 q^{37} + 36 q^{39} - 24 q^{49} - 12 q^{51} + 8 q^{57} + 48 q^{59} + 56 q^{61} - 88 q^{71} - 40 q^{73} + 44 q^{81} + 24 q^{83} + 32 q^{87} - 48 q^{93} - 24 q^{95} - 8 q^{97} - 52 q^{99}+O(q^{100}) \) 24 * q + 4 * q^9 + 8 * q^11 + 8 * q^13 + 4 * q^15 - 4 * q^21 - 24 * q^25 + 8 * q^33 + 24 * q^35 + 8 * q^37 + 36 * q^39 - 24 * q^49 - 12 * q^51 + 8 * q^57 + 48 * q^59 + 56 * q^61 - 88 * q^71 - 40 * q^73 + 44 * q^81 + 24 * q^83 + 32 * q^87 - 48 * q^93 - 24 * q^95 - 8 * q^97 - 52 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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