LMFDB - Embedded newform 5520.2.be.b.1471.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5520,2,Mod(1471,5520)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5520, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("5520.1471");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5520 = 2^{4} \cdot 3 \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5520.be (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:	$44.0774219157$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 4 x^{15} + 8 x^{14} + 28 x^{13} + 373 x^{12} - 920 x^{11} + 1088 x^{10} - 168 x^{9} + 16460 x^{8} + \cdots + 64 $ x^16 - 4x^15 + 8x^14 + 28x^13 + 373x^12 - 920x^11 + 1088x^10 - 168x^9 + 16460x^8 - 45408x^7 + 62624x^6 - 18048x^5 + 2160x^4 - 1664x^3 + 6272x^2 - 896*x + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1471.6
Root			$3.68710 + 3.68710i$ of defining polynomial
Character	$\chi$	$=$	5520.1471
Dual form			5520.2.be.b.1471.14

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} -1.00000i q^{5} +1.58474 q^{7} -1.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} -1.00000i q^{5} +1.58474 q^{7} -1.00000 q^{9} -2.94482 q^{11} -1.23011 q^{13} -1.00000 q^{15} -1.25392i q^{17} +5.26714 q^{19} -1.58474i q^{21} +(1.83865 - 4.42937i) q^{23} -1.00000 q^{25} +1.00000i q^{27} +7.54806 q^{29} -0.690205i q^{31} +2.94482i q^{33} -1.58474i q^{35} +4.83340i q^{37} +1.23011i q^{39} +6.98783 q^{41} +4.06681 q^{43} +1.00000i q^{45} +8.95368i q^{47} -4.48861 q^{49} -1.25392 q^{51} -10.8852i q^{53} +2.94482i q^{55} -5.26714i q^{57} +2.33680i q^{59} -5.28658i q^{61} -1.58474 q^{63} +1.23011i q^{65} -9.49359 q^{67} +(-4.42937 - 1.83865i) q^{69} -13.5262i q^{71} -9.51828 q^{73} +1.00000i q^{75} -4.66677 q^{77} -3.71307 q^{79} +1.00000 q^{81} +16.5130 q^{83} -1.25392 q^{85} -7.54806i q^{87} -12.2406i q^{89} -1.94940 q^{91} -0.690205 q^{93} -5.26714i q^{95} -7.70591i q^{97} +2.94482 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 8 q^{7} - 16 q^{9}+O(q^{10}) $ 16 * q + 8 * q^7 - 16 * q^9	$ 16 q + 8 q^{7} - 16 q^{9} - 8 q^{11} + 8 q^{13} - 16 q^{15} - 12 q^{23} - 16 q^{25} - 4 q^{29} + 4 q^{41} + 20 q^{49} + 4 q^{51} - 8 q^{63} + 16 q^{67} + 40 q^{73} + 24 q^{77} - 32 q^{79} + 16 q^{81} + 4 q^{85} + 48 q^{91} + 8 q^{99}+O(q^{100}) $ 16 * q + 8 * q^7 - 16 * q^9 - 8 * q^11 + 8 * q^13 - 16 * q^15 - 12 * q^23 - 16 * q^25 - 4 * q^29 + 4 * q^41 + 20 * q^49 + 4 * q^51 - 8 * q^63 + 16 * q^67 + 40 * q^73 + 24 * q^77 - 32 * q^79 + 16 * q^81 + 4 * q^85 + 48 * q^91 + 8 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5520\mathbb{Z}\right)^\times$.

$n$	$1201$	$1381$	$1841$	$4417$	$4831$
$\chi(n)$	$-1$	$1$	$1$	$1$	$-1$

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
$3$		−	1.00000i		−	0.577350i
$3$		−	1.00000i		−	0.577350i
$4$		0			0
$5$		−	1.00000i		−	0.447214i
$5$		−	1.00000i		−	0.447214i
$6$		0			0
$7$	1.58474			0.598974			0.299487	−	0.954100i	$-0.403184\pi$
$7$	1.58474			0.598974			0.299487	+	0.954100i	$0.403184\pi$
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	−2.94482			−0.887898			−0.443949	−	0.896052i	$-0.646423\pi$
$11$	−2.94482			−0.887898			−0.443949	+	0.896052i	$0.646423\pi$
$12$		0			0
$13$	−1.23011			−0.341172			−0.170586	−	0.985343i	$-0.554566\pi$
$13$	−1.23011			−0.341172			−0.170586	+	0.985343i	$0.554566\pi$
$14$		0			0
$15$	−1.00000			−0.258199
$16$		0			0
$17$		−	1.25392i		−	0.304120i	−0.988371	−	0.152060i	$-0.951409\pi$
$17$		−	1.25392i		−	0.304120i	0.988371	−	0.152060i	$-0.0485906\pi$
$18$		0			0
$19$	5.26714			1.20836			0.604182	−	0.796846i	$-0.293500\pi$
$19$	5.26714			1.20836			0.604182	+	0.796846i	$0.293500\pi$
$20$		0			0
$21$		−	1.58474i		−	0.345818i
$22$		0			0
$23$	1.83865	−	4.42937i	0.383386	−	0.923588i
$23$	1.83865	−	4.42937i	0.383386	−	0.923588i
$24$		0			0
$25$	−1.00000			−0.200000
$26$		0			0
$27$			1.00000i			0.192450i
$28$		0			0
$29$	7.54806			1.40164			0.700820	−	0.713338i	$-0.252818\pi$
$29$	7.54806			1.40164			0.700820	+	0.713338i	$0.252818\pi$
$30$		0			0
$31$		−	0.690205i		−	0.123965i	−0.998077	−	0.0619823i	$-0.980258\pi$
$31$		−	0.690205i		−	0.123965i	0.998077	−	0.0619823i	$-0.0197422\pi$
$32$		0			0
$33$			2.94482i			0.512628i
$34$		0			0
$35$		−	1.58474i		−	0.267869i
$36$		0			0
$37$			4.83340i			0.794606i	0.917688	+	0.397303i	$0.130054\pi$
$37$			4.83340i			0.794606i	−0.917688	+	0.397303i	$0.869946\pi$
$38$		0			0
$39$			1.23011i			0.196976i
$40$		0			0
$41$	6.98783			1.09132			0.545658	−	0.838008i	$-0.316280\pi$
$41$	6.98783			1.09132			0.545658	+	0.838008i	$0.316280\pi$
$42$		0			0
$43$	4.06681			0.620183			0.310091	−	0.950707i	$-0.399640\pi$
$43$	4.06681			0.620183			0.310091	+	0.950707i	$0.399640\pi$
$44$		0			0
$45$			1.00000i			0.149071i
$46$		0			0
$47$			8.95368i			1.30603i	0.757346	+	0.653014i	$0.226496\pi$
$47$			8.95368i			1.30603i	−0.757346	+	0.653014i	$0.773504\pi$
$48$		0			0
$49$	−4.48861			−0.641230
$50$		0			0
$51$	−1.25392			−0.175583
$52$		0			0
$53$		−	10.8852i		−	1.49520i	−0.664148	−	0.747601i	$-0.731206\pi$
$53$		−	10.8852i		−	1.49520i	0.664148	−	0.747601i	$-0.268794\pi$
$54$		0			0
$55$			2.94482i			0.397080i
$56$		0			0
$57$		−	5.26714i		−	0.697649i
$58$		0			0
$59$			2.33680i			0.304226i	0.988363	+	0.152113i	$0.0486077\pi$
$59$			2.33680i			0.304226i	−0.988363	+	0.152113i	$0.951392\pi$
$60$		0			0
$61$		−	5.28658i		−	0.676877i	−0.940989	−	0.338438i	$-0.890101\pi$
$61$		−	5.28658i		−	0.676877i	0.940989	−	0.338438i	$-0.109899\pi$
$62$		0			0
$63$	−1.58474			−0.199658
$64$		0			0
$65$			1.23011i			0.152577i
$66$		0			0
$67$	−9.49359			−1.15983			−0.579913	−	0.814678i	$-0.696913\pi$
$67$	−9.49359			−1.15983			−0.579913	+	0.814678i	$0.696913\pi$
$68$		0			0
$69$	−4.42937	−	1.83865i	−0.533234	−	0.221348i
$70$		0			0
$71$		−	13.5262i		−	1.60527i	−0.596473	−	0.802633i	$-0.703432\pi$
$71$		−	13.5262i		−	1.60527i	0.596473	−	0.802633i	$-0.296568\pi$
$72$		0			0
$73$	−9.51828			−1.11403			−0.557015	−	0.830502i	$-0.688054\pi$
$73$	−9.51828			−1.11403			−0.557015	+	0.830502i	$0.688054\pi$
$74$		0			0
$75$			1.00000i			0.115470i
$76$		0			0
$77$	−4.66677			−0.531828
$78$		0			0
$79$	−3.71307			−0.417753			−0.208876	−	0.977942i	$-0.566981\pi$
$79$	−3.71307			−0.417753			−0.208876	+	0.977942i	$0.566981\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$	16.5130			1.81253			0.906267	−	0.422705i	$-0.138919\pi$
$83$	16.5130			1.81253			0.906267	+	0.422705i	$0.138919\pi$
$84$		0			0
$85$	−1.25392			−0.136006
$86$		0			0
$87$		−	7.54806i		−	0.809237i
$88$		0			0
$89$		−	12.2406i		−	1.29750i	−0.761003	−	0.648748i	$-0.775293\pi$
$89$		−	12.2406i		−	1.29750i	0.761003	−	0.648748i	$-0.224707\pi$
$90$		0			0
$91$	−1.94940			−0.204353
$92$		0			0
$93$	−0.690205			−0.0715710
$94$		0			0
$95$		−	5.26714i		−	0.540397i
$96$		0			0
$97$		−	7.70591i		−	0.782417i	−0.920302	−	0.391208i	$-0.872057\pi$
$97$		−	7.70591i		−	0.782417i	0.920302	−	0.391208i	$-0.127943\pi$
$98$		0			0
$99$	2.94482			0.295966
$100$		0			0
$101$	−9.43609			−0.938926			−0.469463	−	0.882952i	$-0.655552\pi$
$101$	−9.43609			−0.938926			−0.469463	+	0.882952i	$0.655552\pi$
$102$		0			0
$103$	7.07692			0.697310			0.348655	−	0.937251i	$-0.386639\pi$
$103$	7.07692			0.697310			0.348655	+	0.937251i	$0.386639\pi$
$104$		0			0
$105$	−1.58474			−0.154654
$106$		0			0
$107$	6.20879			0.600227			0.300113	−	0.953904i	$-0.402976\pi$
$107$	6.20879			0.600227			0.300113	+	0.953904i	$0.402976\pi$
$108$		0			0
$109$		−	5.46182i		−	0.523147i	−0.965184	−	0.261574i	$-0.915759\pi$
$109$		−	5.46182i		−	0.523147i	0.965184	−	0.261574i	$-0.0842414\pi$
$110$		0			0
$111$	4.83340			0.458766
$112$		0			0
$113$		−	1.98548i		−	0.186778i	−0.995630	−	0.0933891i	$-0.970230\pi$
$113$		−	1.98548i		−	0.186778i	0.995630	−	0.0933891i	$-0.0297700\pi$
$114$		0			0
$115$	−4.42937	−	1.83865i	−0.413041	−	0.171455i
$116$		0			0
$117$	1.23011			0.113724
$118$		0			0
$119$		−	1.98713i		−	0.182160i
$120$		0			0
$121$	−2.32801			−0.211638
$122$		0			0
$123$		−	6.98783i		−	0.630071i
$124$		0			0
$125$			1.00000i			0.0894427i
$126$		0			0
$127$			4.35495i			0.386439i	0.981156	+	0.193220i	$0.0618929\pi$
$127$			4.35495i			0.386439i	−0.981156	+	0.193220i	$0.938107\pi$
$128$		0			0
$129$		−	4.06681i		−	0.358063i
$130$		0			0
$131$			1.92541i			0.168224i	0.996456	+	0.0841118i	$0.0268053\pi$
$131$			1.92541i			0.168224i	−0.996456	+	0.0841118i	$0.973195\pi$
$132$		0			0
$133$	8.34702			0.723779
$134$		0			0
$135$	1.00000			0.0860663
$136$		0			0
$137$		−	11.9282i		−	1.01910i	−0.860442	−	0.509548i	$-0.829813\pi$
$137$		−	11.9282i		−	1.01910i	0.860442	−	0.509548i	$-0.170187\pi$
$138$		0			0
$139$		−	1.31703i		−	0.111709i	−0.998439	−	0.0558545i	$-0.982212\pi$
$139$		−	1.31703i		−	0.111709i	0.998439	−	0.0558545i	$-0.0177883\pi$
$140$		0			0
$141$	8.95368			0.754036
$142$		0			0
$143$	3.62246			0.302925
$144$		0			0
$145$		−	7.54806i		−	0.626832i
$146$		0			0
$147$			4.48861i			0.370214i
$148$		0			0
$149$			7.50615i			0.614928i	0.951560	+	0.307464i	$0.0994804\pi$
$149$			7.50615i			0.614928i	−0.951560	+	0.307464i	$0.900520\pi$
$150$		0			0
$151$		−	4.78698i		−	0.389559i	−0.980847	−	0.194780i	$-0.937601\pi$
$151$		−	4.78698i		−	0.389559i	0.980847	−	0.194780i	$-0.0623992\pi$
$152$		0			0
$153$			1.25392i			0.101373i
$154$		0			0
$155$	−0.690205			−0.0554386
$156$		0			0
$157$		−	6.01182i		−	0.479795i	−0.970798	−	0.239898i	$-0.922886\pi$
$157$		−	6.01182i		−	0.479795i	0.970798	−	0.239898i	$-0.0771139\pi$
$158$		0			0
$159$	−10.8852			−0.863255
$160$		0			0
$161$	2.91378	−	7.01939i	0.229638	−	0.553206i
$162$		0			0
$163$			9.41003i			0.737051i	0.929618	+	0.368525i	$0.120137\pi$
$163$			9.41003i			0.737051i	−0.929618	+	0.368525i	$0.879863\pi$
$164$		0			0
$165$	2.94482			0.229254
$166$		0			0
$167$		−	2.39043i		−	0.184977i	−0.995714	−	0.0924884i	$-0.970518\pi$
$167$		−	2.39043i		−	0.184977i	0.995714	−	0.0924884i	$-0.0294821\pi$
$168$		0			0
$169$	−11.4868			−0.883602
$170$		0			0
$171$	−5.26714			−0.402788
$172$		0			0
$173$	7.24943			0.551164			0.275582	−	0.961278i	$-0.411129\pi$
$173$	7.24943			0.551164			0.275582	+	0.961278i	$0.411129\pi$
$174$		0			0
$175$	−1.58474			−0.119795
$176$		0			0
$177$	2.33680			0.175645
$178$		0			0
$179$		−	12.2088i		−	0.912532i	−0.889843	−	0.456266i	$-0.849187\pi$
$179$		−	12.2088i		−	0.912532i	0.889843	−	0.456266i	$-0.150813\pi$
$180$		0			0
$181$		−	16.9328i		−	1.25860i	−0.777161	−	0.629302i	$-0.783341\pi$
$181$		−	16.9328i		−	1.25860i	0.777161	−	0.629302i	$-0.216659\pi$
$182$		0			0
$183$	−5.28658			−0.390795
$184$		0			0
$185$	4.83340			0.355359
$186$		0			0
$187$			3.69256i			0.270027i
$188$		0			0
$189$			1.58474i			0.115273i
$190$		0			0
$191$	−6.46417			−0.467731			−0.233865	−	0.972269i	$-0.575138\pi$
$191$	−6.46417			−0.467731			−0.233865	+	0.972269i	$0.575138\pi$
$192$		0			0
$193$	−21.5952			−1.55445			−0.777227	−	0.629220i	$-0.783375\pi$
$193$	−21.5952			−1.55445			−0.777227	+	0.629220i	$0.783375\pi$
$194$		0			0
$195$	1.23011			0.0880901
$196$		0			0
$197$	−4.11055			−0.292865			−0.146432	−	0.989221i	$-0.546779\pi$
$197$	−4.11055			−0.292865			−0.146432	+	0.989221i	$0.546779\pi$
$198$		0			0
$199$	−3.25603			−0.230814			−0.115407	−	0.993318i	$-0.536817\pi$
$199$	−3.25603			−0.230814			−0.115407	+	0.993318i	$0.536817\pi$
$200$		0			0
$201$			9.49359i			0.669626i
$202$		0			0
$203$	11.9617			0.839546
$204$		0			0
$205$		−	6.98783i		−	0.488051i
$206$		0			0
$207$	−1.83865	+	4.42937i	−0.127795	+	0.307863i
$208$		0			0
$209$	−15.5108			−1.07290
$210$		0			0
$211$			18.1011i			1.24613i	0.782169	+	0.623066i	$0.214113\pi$
$211$			18.1011i			1.24613i	−0.782169	+	0.623066i	$0.785887\pi$
$212$		0			0
$213$	−13.5262			−0.926801
$214$		0			0
$215$		−	4.06681i		−	0.277354i
$216$		0			0
$217$		−	1.09379i		−	0.0742516i
$218$		0			0
$219$			9.51828i			0.643186i
$220$		0			0
$221$			1.54246i			0.103757i
$222$		0			0
$223$		−	2.72334i		−	0.182368i	−0.995834	−	0.0911842i	$-0.970935\pi$
$223$		−	2.72334i		−	0.182368i	0.995834	−	0.0911842i	$-0.0290652\pi$
$224$		0			0
$225$	1.00000			0.0666667
$226$		0			0
$227$	21.7426			1.44311			0.721554	−	0.692358i	$-0.243428\pi$
$227$	21.7426			1.44311			0.721554	+	0.692358i	$0.243428\pi$
$228$		0			0
$229$			1.81560i			0.119978i	0.998199	+	0.0599891i	$0.0191066\pi$
$229$			1.81560i			0.119978i	−0.998199	+	0.0599891i	$0.980893\pi$
$230$		0			0
$231$			4.66677i			0.307051i
$232$		0			0
$233$	−25.1093			−1.64496			−0.822482	−	0.568792i	$-0.807411\pi$
$233$	−25.1093			−1.64496			−0.822482	+	0.568792i	$0.807411\pi$
$234$		0			0
$235$	8.95368			0.584074
$236$		0			0
$237$			3.71307i			0.241190i
$238$		0			0
$239$			25.5463i			1.65245i	0.563338	+	0.826226i	$0.309517\pi$
$239$			25.5463i			1.65245i	−0.563338	+	0.826226i	$0.690483\pi$
$240$		0			0
$241$		−	17.7500i		−	1.14338i	−0.820471	−	0.571689i	$-0.806289\pi$
$241$		−	17.7500i		−	1.14338i	0.820471	−	0.571689i	$-0.193711\pi$
$242$		0			0
$243$		−	1.00000i		−	0.0641500i
$244$		0			0
$245$			4.48861i			0.286767i
$246$		0			0
$247$	−6.47916			−0.412259
$248$		0			0
$249$		−	16.5130i		−	1.04647i
$250$		0			0
$251$	18.8514			1.18989			0.594946	−	0.803766i	$-0.297174\pi$
$251$	18.8514			1.18989			0.594946	+	0.803766i	$0.297174\pi$
$252$		0			0
$253$	−5.41451	+	13.0437i	−0.340407	+	0.820052i
$254$		0			0
$255$			1.25392i			0.0785233i
$256$		0			0
$257$	18.1245			1.13058			0.565289	−	0.824893i	$-0.308765\pi$
$257$	18.1245			1.13058			0.565289	+	0.824893i	$0.308765\pi$
$258$		0			0
$259$			7.65967i			0.475948i
$260$		0			0
$261$	−7.54806			−0.467213
$262$		0			0
$263$	−29.0810			−1.79321			−0.896606	−	0.442829i	$-0.853975\pi$
$263$	−29.0810			−1.79321			−0.896606	+	0.442829i	$0.853975\pi$
$264$		0			0
$265$	−10.8852			−0.668675
$266$		0			0
$267$	−12.2406			−0.749110
$268$		0			0
$269$	−21.2290			−1.29435			−0.647176	−	0.762340i	$-0.724050\pi$
$269$	−21.2290			−1.29435			−0.647176	+	0.762340i	$0.724050\pi$
$270$		0			0
$271$			9.18826i			0.558147i	0.960270	+	0.279073i	$0.0900273\pi$
$271$			9.18826i			0.558147i	−0.960270	+	0.279073i	$0.909973\pi$
$272$		0			0
$273$			1.94940i			0.117983i
$274$		0			0
$275$	2.94482			0.177580
$276$		0			0
$277$	23.9814			1.44090			0.720451	−	0.693505i	$-0.243935\pi$
$277$	23.9814			1.44090			0.720451	+	0.693505i	$0.243935\pi$
$278$		0			0
$279$			0.690205i			0.0413215i
$280$		0			0
$281$		−	18.5638i		−	1.10742i	−0.832709	−	0.553711i	$-0.813211\pi$
$281$		−	18.5638i		−	1.10742i	0.832709	−	0.553711i	$-0.186789\pi$
$282$		0			0
$283$	12.8548			0.764138			0.382069	−	0.924134i	$-0.375212\pi$
$283$	12.8548			0.764138			0.382069	+	0.924134i	$0.375212\pi$
$284$		0			0
$285$	−5.26714			−0.311998
$286$		0			0
$287$	11.0739			0.653670
$288$		0			0
$289$	15.4277			0.907511
$290$		0			0
$291$	−7.70591			−0.451728
$292$		0			0
$293$		−	25.3576i		−	1.48141i	−0.671831	−	0.740705i	$-0.734492\pi$
$293$		−	25.3576i		−	1.48141i	0.671831	−	0.740705i	$-0.265508\pi$
$294$		0			0
$295$	2.33680			0.136054
$296$		0			0
$297$		−	2.94482i		−	0.170876i
$298$		0			0
$299$	−2.26175	+	5.44862i	−0.130800	+	0.315102i
$300$		0			0
$301$	6.44483			0.371474
$302$		0			0
$303$			9.43609i			0.542089i
$304$		0			0
$305$	−5.28658			−0.302709
$306$		0			0
$307$			12.7630i			0.728422i	0.931316	+	0.364211i	$0.118661\pi$
$307$			12.7630i			0.728422i	−0.931316	+	0.364211i	$0.881339\pi$
$308$		0			0
$309$		−	7.07692i		−	0.402592i
$310$		0			0
$311$		−	14.7842i		−	0.838333i	−0.907909	−	0.419166i	$-0.862322\pi$
$311$		−	14.7842i		−	0.838333i	0.907909	−	0.419166i	$-0.137678\pi$
$312$		0			0
$313$			1.48723i			0.0840632i	0.999116	+	0.0420316i	$0.0133830\pi$
$313$			1.48723i			0.0840632i	−0.999116	+	0.0420316i	$0.986617\pi$
$314$		0			0
$315$			1.58474i			0.0892898i
$316$		0			0
$317$	0.771575			0.0433360			0.0216680	−	0.999765i	$-0.493102\pi$
$317$	0.771575			0.0433360			0.0216680	+	0.999765i	$0.493102\pi$
$318$		0			0
$319$	−22.2277			−1.24451
$320$		0			0
$321$		−	6.20879i		−	0.346541i
$322$		0			0
$323$		−	6.60455i		−	0.367487i
$324$		0			0
$325$	1.23011			0.0682343
$326$		0			0
$327$	−5.46182			−0.302039
$328$		0			0
$329$			14.1892i			0.782277i
$330$		0			0
$331$		−	33.8360i		−	1.85980i	−0.367817	−	0.929898i	$-0.619895\pi$
$331$		−	33.8360i		−	1.85980i	0.367817	−	0.929898i	$-0.380105\pi$
$332$		0			0
$333$		−	4.83340i		−	0.264869i
$334$		0			0
$335$			9.49359i			0.518690i
$336$		0			0
$337$		−	25.3943i		−	1.38331i	−0.722226	−	0.691657i	$-0.756881\pi$
$337$		−	25.3943i		−	1.38331i	0.722226	−	0.691657i	$-0.243119\pi$
$338$		0			0
$339$	−1.98548			−0.107836
$340$		0			0
$341$			2.03253i			0.110068i
$342$		0			0
$343$	−18.2064			−0.983054
$344$		0			0
$345$	−1.83865	+	4.42937i	−0.0989898	+	0.238469i
$346$		0			0
$347$		−	7.66039i		−	0.411231i	−0.978633	−	0.205616i	$-0.934080\pi$
$347$		−	7.66039i		−	0.411231i	0.978633	−	0.205616i	$-0.0659197\pi$
$348$		0			0
$349$	15.8624			0.849093			0.424546	−	0.905406i	$-0.360434\pi$
$349$	15.8624			0.849093			0.424546	+	0.905406i	$0.360434\pi$
$350$		0			0
$351$		−	1.23011i		−	0.0656585i
$352$		0			0
$353$	−26.7036			−1.42129			−0.710643	−	0.703552i	$-0.751596\pi$
$353$	−26.7036			−1.42129			−0.710643	+	0.703552i	$0.751596\pi$
$354$		0			0
$355$	−13.5262			−0.717897
$356$		0			0
$357$	−1.98713			−0.105170
$358$		0			0
$359$	17.3015			0.913137			0.456569	−	0.889688i	$-0.349078\pi$
$359$	17.3015			0.913137			0.456569	+	0.889688i	$0.349078\pi$
$360$		0			0
$361$	8.74271			0.460143
$362$		0			0
$363$			2.32801i			0.122189i
$364$		0			0
$365$			9.51828i			0.498210i
$366$		0			0
$367$	19.1050			0.997271			0.498635	−	0.866812i	$-0.333835\pi$
$367$	19.1050			0.997271			0.498635	+	0.866812i	$0.333835\pi$
$368$		0			0
$369$	−6.98783			−0.363772
$370$		0			0
$371$		−	17.2502i		−	0.895587i
$372$		0			0
$373$			29.5261i			1.52880i	0.644740	+	0.764402i	$0.276966\pi$
$373$			29.5261i			1.52880i	−0.644740	+	0.764402i	$0.723034\pi$
$374$		0			0
$375$	1.00000			0.0516398
$376$		0			0
$377$	−9.28496			−0.478200
$378$		0			0
$379$	−29.9649			−1.53920			−0.769598	−	0.638529i	$-0.779543\pi$
$379$	−29.9649			−1.53920			−0.769598	+	0.638529i	$0.779543\pi$
$380$		0			0
$381$	4.35495			0.223111
$382$		0			0
$383$	6.78342			0.346616			0.173308	−	0.984868i	$-0.444554\pi$
$383$	6.78342			0.346616			0.173308	+	0.984868i	$0.444554\pi$
$384$		0			0
$385$			4.66677i			0.237841i
$386$		0			0
$387$	−4.06681			−0.206728
$388$		0			0
$389$		−	8.20216i		−	0.415866i	−0.978143	−	0.207933i	$-0.933326\pi$
$389$		−	8.20216i		−	0.415866i	0.978143	−	0.207933i	$-0.0666736\pi$
$390$		0			0
$391$	−5.55407	−	2.30552i	−0.280881	−	0.116595i
$392$		0			0
$393$	1.92541			0.0971240
$394$		0			0
$395$			3.71307i			0.186825i
$396$		0			0
$397$	−28.2572			−1.41819			−0.709095	−	0.705113i	$-0.750896\pi$
$397$	−28.2572			−1.41819			−0.709095	+	0.705113i	$0.750896\pi$
$398$		0			0
$399$		−	8.34702i		−	0.417874i
$400$		0			0
$401$		−	17.1193i		−	0.854899i	−0.904039	−	0.427449i	$-0.859412\pi$
$401$		−	17.1193i		−	0.854899i	0.904039	−	0.427449i	$-0.140588\pi$
$402$		0			0
$403$			0.849030i			0.0422932i
$404$		0			0
$405$		−	1.00000i		−	0.0496904i
$406$		0			0
$407$		−	14.2335i		−	0.705529i
$408$		0			0
$409$	22.8130			1.12803			0.564016	−	0.825764i	$-0.309256\pi$
$409$	22.8130			1.12803			0.564016	+	0.825764i	$0.309256\pi$
$410$		0			0
$411$	−11.9282			−0.588375
$412$		0			0
$413$			3.70322i			0.182223i
$414$		0			0
$415$		−	16.5130i		−	0.810590i
$416$		0			0
$417$	−1.31703			−0.0644952
$418$		0			0
$419$	4.54342			0.221960			0.110980	−	0.993823i	$-0.464601\pi$
$419$	4.54342			0.221960			0.110980	+	0.993823i	$0.464601\pi$
$420$		0			0
$421$		−	2.02280i		−	0.0985853i	−0.998784	−	0.0492926i	$-0.984303\pi$
$421$		−	2.02280i		−	0.0985853i	0.998784	−	0.0492926i	$-0.0156967\pi$
$422$		0			0
$423$		−	8.95368i		−	0.435343i
$424$		0			0
$425$			1.25392i			0.0608239i
$426$		0			0
$427$		−	8.37783i		−	0.405432i
$428$		0			0
$429$		−	3.62246i		−	0.174894i
$430$		0			0
$431$	10.5534			0.508339			0.254169	−	0.967160i	$-0.418198\pi$
$431$	10.5534			0.508339			0.254169	+	0.967160i	$0.418198\pi$
$432$		0			0
$433$		−	33.9550i		−	1.63177i	−0.578211	−	0.815887i	$-0.696249\pi$
$433$		−	33.9550i		−	1.63177i	0.578211	−	0.815887i	$-0.303751\pi$
$434$		0			0
$435$	−7.54806			−0.361902
$436$		0			0
$437$	9.68444	−	23.3301i	0.463270	−	1.11603i
$438$		0			0
$439$			2.99231i			0.142815i	0.997447	+	0.0714076i	$0.0227491\pi$
$439$			2.99231i			0.142815i	−0.997447	+	0.0714076i	$0.977251\pi$
$440$		0			0
$441$	4.48861			0.213743
$442$		0			0
$443$		−	20.9670i		−	0.996174i	−0.867127	−	0.498087i	$-0.834036\pi$
$443$		−	20.9670i		−	0.996174i	0.867127	−	0.498087i	$-0.165964\pi$
$444$		0			0
$445$	−12.2406			−0.580258
$446$		0			0
$447$	7.50615			0.355029
$448$		0			0
$449$	11.8249			0.558053			0.279026	−	0.960283i	$-0.409988\pi$
$449$	11.8249			0.558053			0.279026	+	0.960283i	$0.409988\pi$
$450$		0			0
$451$	−20.5779			−0.968977
$452$		0			0
$453$	−4.78698			−0.224912
$454$		0			0
$455$			1.94940i			0.0913894i
$456$		0			0
$457$			10.8771i			0.508810i	0.967098	+	0.254405i	$0.0818796\pi$
$457$			10.8771i			0.508810i	−0.967098	+	0.254405i	$0.918120\pi$
$458$		0			0
$459$	1.25392			0.0585278
$460$		0			0
$461$	15.2626			0.710849			0.355425	−	0.934705i	$-0.384336\pi$
$461$	15.2626			0.710849			0.355425	+	0.934705i	$0.384336\pi$
$462$		0			0
$463$			38.1730i			1.77405i	0.461721	+	0.887025i	$0.347232\pi$
$463$			38.1730i			1.77405i	−0.461721	+	0.887025i	$0.652768\pi$
$464$		0			0
$465$			0.690205i			0.0320075i
$466$		0			0
$467$	4.21856			0.195212			0.0976059	−	0.995225i	$-0.468882\pi$
$467$	4.21856			0.195212			0.0976059	+	0.995225i	$0.468882\pi$
$468$		0			0
$469$	−15.0448			−0.694706
$470$		0			0
$471$	−6.01182			−0.277010
$472$		0			0
$473$	−11.9760			−0.550659
$474$		0			0
$475$	−5.26714			−0.241673
$476$		0			0
$477$			10.8852i			0.498401i
$478$		0			0
$479$	−0.347762			−0.0158897			−0.00794483	−	0.999968i	$-0.502529\pi$
$479$	−0.347762			−0.0158897			−0.00794483	+	0.999968i	$0.502529\pi$
$480$		0			0
$481$		−	5.94562i		−	0.271097i
$482$		0			0
$483$	−7.01939	−	2.91378i	−0.319393	−	0.132582i
$484$		0			0
$485$	−7.70591			−0.349907
$486$		0			0
$487$		−	12.4417i		−	0.563789i	−0.959445	−	0.281894i	$-0.909037\pi$
$487$		−	12.4417i		−	0.563789i	0.959445	−	0.281894i	$-0.0909628\pi$
$488$		0			0
$489$	9.41003			0.425536
$490$		0			0
$491$			4.12588i			0.186198i	0.995657	+	0.0930991i	$0.0296774\pi$
$491$			4.12588i			0.186198i	−0.995657	+	0.0930991i	$0.970323\pi$
$492$		0			0
$493$		−	9.46464i		−	0.426266i
$494$		0			0
$495$		−	2.94482i		−	0.132360i
$496$		0			0
$497$		−	21.4355i		−	0.961514i
$498$		0			0
$499$		−	24.3075i		−	1.08815i	−0.839035	−	0.544077i	$-0.816880\pi$
$499$		−	24.3075i		−	1.08815i	0.839035	−	0.544077i	$-0.183120\pi$
$500$		0			0
$501$	−2.39043			−0.106796
$502$		0			0
$503$	−11.9762			−0.533991			−0.266995	−	0.963698i	$-0.586031\pi$
$503$	−11.9762			−0.533991			−0.266995	+	0.963698i	$0.586031\pi$
$504$		0			0
$505$			9.43609i			0.419900i
$506$		0			0
$507$			11.4868i			0.510148i
$508$		0			0
$509$	43.4081			1.92403			0.962015	−	0.272997i	$-0.0880150\pi$
$509$	43.4081			1.92403			0.962015	+	0.272997i	$0.0880150\pi$
$510$		0			0
$511$	−15.0840			−0.667276
$512$		0			0
$513$			5.26714i			0.232550i
$514$		0			0
$515$		−	7.07692i		−	0.311846i
$516$		0			0
$517$		−	26.3670i		−	1.15962i
$518$		0			0
$519$		−	7.24943i		−	0.318215i
$520$		0			0
$521$			30.1561i			1.32116i	0.750754	+	0.660582i	$0.229690\pi$
$521$			30.1561i			1.32116i	−0.750754	+	0.660582i	$0.770310\pi$
$522$		0			0
$523$	−25.0503			−1.09537			−0.547686	−	0.836684i	$-0.684491\pi$
$523$	−25.0503			−1.09537			−0.547686	+	0.836684i	$0.684491\pi$
$524$		0			0
$525$			1.58474i			0.0691636i
$526$		0			0
$527$	−0.865460			−0.0377000
$528$		0			0
$529$	−16.2387	−	16.2882i	−0.706031	−	0.708181i
$530$		0			0
$531$		−	2.33680i		−	0.101409i
$532$		0			0
$533$	−8.59581			−0.372326
$534$		0			0
$535$		−	6.20879i		−	0.268430i
$536$		0			0
$537$	−12.2088			−0.526851
$538$		0			0
$539$	13.2182			0.569346
$540$		0			0
$541$	38.9949			1.67652			0.838260	−	0.545271i	$-0.183573\pi$
$541$	38.9949			1.67652			0.838260	+	0.545271i	$0.183573\pi$
$542$		0			0
$543$	−16.9328			−0.726656
$544$		0			0
$545$	−5.46182			−0.233959
$546$		0			0
$547$			35.9994i			1.53922i	0.638512	+	0.769612i	$0.279550\pi$
$547$			35.9994i			1.53922i	−0.638512	+	0.769612i	$0.720450\pi$
$548$		0			0
$549$			5.28658i			0.225626i
$550$		0			0
$551$	39.7567			1.69369
$552$		0			0
$553$	−5.88424			−0.250223
$554$		0			0
$555$		−	4.83340i		−	0.205166i
$556$		0			0
$557$			22.1601i			0.938954i	0.882945	+	0.469477i	$0.155558\pi$
$557$			22.1601i			0.938954i	−0.882945	+	0.469477i	$0.844442\pi$
$558$		0			0
$559$	−5.00263			−0.211589
$560$		0			0
$561$	3.69256			0.155900
$562$		0			0
$563$	−2.91544			−0.122871			−0.0614356	−	0.998111i	$-0.519568\pi$
$563$	−2.91544			−0.122871			−0.0614356	+	0.998111i	$0.519568\pi$
$564$		0			0
$565$	−1.98548			−0.0835297
$566$		0			0
$567$	1.58474			0.0665527
$568$		0			0
$569$		−	14.9836i		−	0.628144i	−0.949399	−	0.314072i	$-0.898307\pi$
$569$		−	14.9836i		−	0.628144i	0.949399	−	0.314072i	$-0.101693\pi$
$570$		0			0
$571$	−3.15642			−0.132092			−0.0660461	−	0.997817i	$-0.521038\pi$
$571$	−3.15642			−0.132092			−0.0660461	+	0.997817i	$0.521038\pi$
$572$		0			0
$573$			6.46417i			0.270044i
$574$		0			0
$575$	−1.83865	+	4.42937i	−0.0766772	+	0.184718i
$576$		0			0
$577$	−6.25189			−0.260270			−0.130135	−	0.991496i	$-0.541541\pi$
$577$	−6.25189			−0.260270			−0.130135	+	0.991496i	$0.541541\pi$
$578$		0			0
$579$			21.5952i			0.897465i
$580$		0			0
$581$	26.1687			1.08566
$582$		0			0
$583$			32.0551i			1.32759i
$584$		0			0
$585$		−	1.23011i		−	0.0508589i
$586$		0			0
$587$		−	3.22099i		−	0.132945i	−0.997788	−	0.0664723i	$-0.978826\pi$
$587$		−	3.22099i		−	0.132945i	0.997788	−	0.0664723i	$-0.0211744\pi$
$588$		0			0
$589$		−	3.63541i		−	0.149794i
$590$		0			0
$591$			4.11055i			0.169085i
$592$		0			0
$593$	−18.3093			−0.751873			−0.375937	−	0.926645i	$-0.622679\pi$
$593$	−18.3093			−0.751873			−0.375937	+	0.926645i	$0.622679\pi$
$594$		0			0
$595$	−1.98713			−0.0814643
$596$		0			0
$597$			3.25603i			0.133261i
$598$		0			0
$599$		−	5.79992i		−	0.236978i	−0.992955	−	0.118489i	$-0.962195\pi$
$599$		−	5.79992i		−	0.236978i	0.992955	−	0.118489i	$-0.0378051\pi$
$600$		0			0
$601$	−12.1653			−0.496233			−0.248116	−	0.968730i	$-0.579812\pi$
$601$	−12.1653			−0.496233			−0.248116	+	0.968730i	$0.579812\pi$
$602$		0			0
$603$	9.49359			0.386609
$604$		0			0
$605$			2.32801i			0.0946472i
$606$		0			0
$607$			33.5799i			1.36296i	0.731834	+	0.681482i	$0.238664\pi$
$607$			33.5799i			1.36296i	−0.731834	+	0.681482i	$0.761336\pi$
$608$		0			0
$609$		−	11.9617i		−	0.484712i
$610$		0			0
$611$		−	11.0140i		−	0.445580i
$612$		0			0
$613$			24.5388i			0.991113i	0.868576	+	0.495556i	$0.165036\pi$
$613$			24.5388i			0.991113i	−0.868576	+	0.495556i	$0.834964\pi$
$614$		0			0
$615$	−6.98783			−0.281776
$616$		0			0
$617$			10.9240i			0.439782i	0.975524	+	0.219891i	$0.0705701\pi$
$617$			10.9240i			0.439782i	−0.975524	+	0.219891i	$0.929430\pi$
$618$		0			0
$619$	29.8447			1.19956			0.599780	−	0.800165i	$-0.295255\pi$
$619$	29.8447			1.19956			0.599780	+	0.800165i	$0.295255\pi$
$620$		0			0
$621$	4.42937	+	1.83865i	0.177745	+	0.0737826i
$622$		0			0
$623$		−	19.3981i		−	0.777167i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$			15.5108i			0.619441i
$628$		0			0
$629$	6.06068			0.241655
$630$		0			0
$631$	−4.26198			−0.169667			−0.0848333	−	0.996395i	$-0.527036\pi$
$631$	−4.26198			−0.169667			−0.0848333	+	0.996395i	$0.527036\pi$
$632$		0			0
$633$	18.1011			0.719455
$634$		0			0
$635$	4.35495			0.172821
$636$		0			0
$637$	5.52149			0.218769
$638$		0			0
$639$			13.5262i			0.535089i
$640$		0			0
$641$		−	43.5638i		−	1.72067i	−0.509731	−	0.860334i	$-0.670255\pi$
$641$		−	43.5638i		−	1.72067i	0.509731	−	0.860334i	$-0.329745\pi$
$642$		0			0
$643$	−11.2217			−0.442542			−0.221271	−	0.975212i	$-0.571021\pi$
$643$	−11.2217			−0.442542			−0.221271	+	0.975212i	$0.571021\pi$
$644$		0			0
$645$	−4.06681			−0.160131
$646$		0			0
$647$			36.7686i			1.44552i	0.691098	+	0.722761i	$0.257127\pi$
$647$			36.7686i			1.44552i	−0.691098	+	0.722761i	$0.742873\pi$
$648$		0			0
$649$		−	6.88147i		−	0.270121i
$650$		0			0
$651$	−1.09379			−0.0428692
$652$		0			0
$653$	−31.9401			−1.24991			−0.624957	−	0.780659i	$-0.714883\pi$
$653$	−31.9401			−1.24991			−0.624957	+	0.780659i	$0.714883\pi$
$654$		0			0
$655$	1.92541			0.0752319
$656$		0			0
$657$	9.51828			0.371344
$658$		0			0
$659$	43.5902			1.69803			0.849016	−	0.528367i	$-0.177196\pi$
$659$	43.5902			1.69803			0.849016	+	0.528367i	$0.177196\pi$
$660$		0			0
$661$			10.4393i			0.406042i	0.979174	+	0.203021i	$0.0650759\pi$
$661$			10.4393i			0.406042i	−0.979174	+	0.203021i	$0.934924\pi$
$662$		0			0
$663$	1.54246			0.0599041
$664$		0			0
$665$		−	8.34702i		−	0.323684i
$666$		0			0
$667$	13.8783	−	33.4332i	0.537369	−	1.29454i
$668$		0			0
$669$	−2.72334			−0.105290
$670$		0			0
$671$			15.5680i			0.600997i
$672$		0			0
$673$	2.83519			0.109288			0.0546442	−	0.998506i	$-0.482598\pi$
$673$	2.83519			0.109288			0.0546442	+	0.998506i	$0.482598\pi$
$674$		0			0
$675$		−	1.00000i		−	0.0384900i
$676$		0			0
$677$			36.3830i			1.39831i	0.714969	+	0.699156i	$0.246441\pi$
$677$			36.3830i			1.39831i	−0.714969	+	0.699156i	$0.753559\pi$
$678$		0			0
$679$		−	12.2118i		−	0.468647i
$680$		0			0
$681$		−	21.7426i		−	0.833179i
$682$		0			0
$683$			2.81755i			0.107811i	0.998546	+	0.0539053i	$0.0171669\pi$
$683$			2.81755i			0.107811i	−0.998546	+	0.0539053i	$0.982833\pi$
$684$		0			0
$685$	−11.9282			−0.455754
$686$		0			0
$687$	1.81560			0.0692695
$688$		0			0
$689$			13.3901i			0.510120i
$690$		0			0
$691$		−	49.1990i		−	1.87162i	−0.352506	−	0.935809i	$-0.614670\pi$
$691$		−	49.1990i		−	1.87162i	0.352506	−	0.935809i	$-0.385330\pi$
$692$		0			0
$693$	4.66677			0.177276
$694$		0			0
$695$	−1.31703			−0.0499578
$696$		0			0
$697$		−	8.76216i		−	0.331890i
$698$		0			0
$699$			25.1093i			0.949720i
$700$		0			0
$701$			15.5212i			0.586228i	0.956078	+	0.293114i	$0.0946915\pi$
$701$			15.5212i			0.586228i	−0.956078	+	0.293114i	$0.905308\pi$
$702$		0			0
$703$			25.4582i			0.960173i
$704$		0			0
$705$		−	8.95368i		−	0.337215i
$706$		0			0
$707$	−14.9537			−0.562392
$708$		0			0
$709$		−	1.54150i		−	0.0578923i	−0.999581	−	0.0289462i	$-0.990785\pi$
$709$		−	1.54150i		−	0.0578923i	0.999581	−	0.0289462i	$-0.00921514\pi$
$710$		0			0
$711$	3.71307			0.139251
$712$		0			0
$713$	−3.05718	−	1.26905i	−0.114492	−	0.0475263i
$714$		0			0
$715$		−	3.62246i		−	0.135472i
$716$		0			0
$717$	25.5463			0.954044
$718$		0			0
$719$			2.00464i			0.0747606i	0.999301	+	0.0373803i	$0.0119013\pi$
$719$			2.00464i			0.0747606i	−0.999301	+	0.0373803i	$0.988099\pi$
$720$		0			0
$721$	11.2151			0.417671
$722$		0			0
$723$	−17.7500			−0.660129
$724$		0			0
$725$	−7.54806			−0.280328
$726$		0			0
$727$	−17.3038			−0.641761			−0.320881	−	0.947120i	$-0.603979\pi$
$727$	−17.3038			−0.641761			−0.320881	+	0.947120i	$0.603979\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$		−	5.09944i		−	0.188610i
$732$		0			0
$733$			26.4881i			0.978358i	0.872183	+	0.489179i	$0.162704\pi$
$733$			26.4881i			0.978358i	−0.872183	+	0.489179i	$0.837296\pi$
$734$		0			0
$735$	4.48861			0.165565
$736$		0			0
$737$	27.9569			1.02981
$738$		0			0
$739$		−	7.11302i		−	0.261657i	−0.991405	−	0.130828i	$-0.958236\pi$
$739$		−	7.11302i		−	0.261657i	0.991405	−	0.130828i	$-0.0417637\pi$
$740$		0			0
$741$			6.47916i			0.238018i
$742$		0			0
$743$	38.7611			1.42201			0.711003	−	0.703189i	$-0.248241\pi$
$743$	38.7611			1.42201			0.711003	+	0.703189i	$0.248241\pi$
$744$		0			0
$745$	7.50615			0.275004
$746$		0			0
$747$	−16.5130			−0.604178
$748$		0			0
$749$	9.83930			0.359520
$750$		0			0
$751$	26.4017			0.963412			0.481706	−	0.876333i	$-0.340017\pi$
$751$	26.4017			0.963412			0.481706	+	0.876333i	$0.340017\pi$
$752$		0			0
$753$		−	18.8514i		−	0.686984i
$754$		0			0
$755$	−4.78698			−0.174216
$756$		0			0
$757$			19.6012i			0.712419i	0.934406	+	0.356210i	$0.115931\pi$
$757$			19.6012i			0.712419i	−0.934406	+	0.356210i	$0.884069\pi$
$758$		0			0
$759$	13.0437	+	5.41451i	0.473457	+	0.196534i
$760$		0			0
$761$	−1.90461			−0.0690421			−0.0345211	−	0.999404i	$-0.510991\pi$
$761$	−1.90461			−0.0690421			−0.0345211	+	0.999404i	$0.510991\pi$
$762$		0			0
$763$		−	8.65554i		−	0.313352i
$764$		0			0
$765$	1.25392			0.0453355
$766$		0			0
$767$		−	2.87453i		−	0.103793i
$768$		0			0
$769$			24.1496i			0.870858i	0.900223	+	0.435429i	$0.143403\pi$
$769$			24.1496i			0.870858i	−0.900223	+	0.435429i	$0.856597\pi$
$770$		0			0
$771$		−	18.1245i		−	0.652739i
$772$		0			0
$773$			35.9153i			1.29179i	0.763428	+	0.645893i	$0.223515\pi$
$773$			35.9153i			1.29179i	−0.763428	+	0.645893i	$0.776485\pi$
$774$		0			0
$775$			0.690205i			0.0247929i
$776$		0			0
$777$	7.65967			0.274789
$778$		0			0
$779$	36.8058			1.31871
$780$		0			0
$781$			39.8323i			1.42531i
$782$		0			0
$783$			7.54806i			0.269746i
$784$		0			0
$785$	−6.01182			−0.214571
$786$		0			0
$787$	−25.6536			−0.914452			−0.457226	−	0.889351i	$-0.651157\pi$
$787$	−25.6536			−0.914452			−0.457226	+	0.889351i	$0.651157\pi$
$788$		0			0
$789$			29.0810i			1.03531i
$790$		0			0
$791$		−	3.14646i		−	0.111875i
$792$		0			0
$793$			6.50308i			0.230931i
$794$		0			0
$795$			10.8852i			0.386059i
$796$		0			0
$797$			11.6082i			0.411185i	0.978638	+	0.205592i	$0.0659121\pi$
$797$			11.6082i			0.411185i	−0.978638	+	0.205592i	$0.934088\pi$
$798$		0			0
$799$	11.2272			0.397189
$800$		0			0
$801$			12.2406i			0.432499i
$802$		0			0
$803$	28.0297			0.989145
$804$		0			0
$805$	−7.01939	−	2.91378i	−0.247401	−	0.102697i
$806$		0			0
$807$			21.2290i			0.747295i
$808$		0			0
$809$	−33.1262			−1.16465			−0.582327	−	0.812955i	$-0.697858\pi$
$809$	−33.1262			−1.16465			−0.582327	+	0.812955i	$0.697858\pi$
$810$		0			0
$811$			45.0060i			1.58037i	0.612867	+	0.790186i	$0.290016\pi$
$811$			45.0060i			1.58037i	−0.612867	+	0.790186i	$0.709984\pi$
$812$		0			0
$813$	9.18826			0.322246
$814$		0			0
$815$	9.41003			0.329619
$816$		0			0
$817$	21.4204			0.749407
$818$		0			0
$819$	1.94940			0.0681177
$820$		0			0
$821$	26.5182			0.925493			0.462746	−	0.886491i	$-0.346864\pi$
$821$	26.5182			0.925493			0.462746	+	0.886491i	$0.346864\pi$
$822$		0			0
$823$			10.3626i			0.361216i	0.983555	+	0.180608i	$0.0578066\pi$
$823$			10.3626i			0.361216i	−0.983555	+	0.180608i	$0.942193\pi$
$824$		0			0
$825$		−	2.94482i		−	0.102526i
$826$		0			0
$827$	47.8592			1.66423			0.832114	−	0.554605i	$-0.187130\pi$
$827$	47.8592			1.66423			0.832114	+	0.554605i	$0.187130\pi$
$828$		0			0
$829$	−10.3355			−0.358968			−0.179484	−	0.983761i	$-0.557443\pi$
$829$	−10.3355			−0.358968			−0.179484	+	0.983761i	$0.557443\pi$
$830$		0			0
$831$		−	23.9814i		−	0.831906i
$832$		0			0
$833$			5.62834i			0.195011i
$834$		0			0
$835$	−2.39043			−0.0827241
$836$		0			0
$837$	0.690205			0.0238570
$838$		0			0
$839$	−6.62580			−0.228748			−0.114374	−	0.993438i	$-0.536486\pi$
$839$	−6.62580			−0.228748			−0.114374	+	0.993438i	$0.536486\pi$
$840$		0			0
$841$	27.9732			0.964594
$842$		0			0
$843$	−18.5638			−0.639370
$844$		0			0
$845$			11.4868i			0.395159i
$846$		0			0
$847$	−3.68929			−0.126766
$848$		0			0
$849$		−	12.8548i		−	0.441175i
$850$		0			0
$851$	21.4089	+	8.88695i	0.733889	+	0.304641i
$852$		0			0
$853$	−44.4991			−1.52362			−0.761810	−	0.647800i	$-0.775689\pi$
$853$	−44.4991			−1.52362			−0.761810	+	0.647800i	$0.775689\pi$
$854$		0			0
$855$			5.26714i			0.180132i
$856$		0			0
$857$	9.12506			0.311706			0.155853	−	0.987780i	$-0.450187\pi$
$857$	9.12506			0.311706			0.155853	+	0.987780i	$0.450187\pi$
$858$		0			0
$859$		−	30.9281i		−	1.05525i	−0.849476	−	0.527627i	$-0.823082\pi$
$859$		−	30.9281i		−	1.05525i	0.849476	−	0.527627i	$-0.176918\pi$
$860$		0			0
$861$		−	11.0739i		−	0.377397i
$862$		0			0
$863$		−	8.46438i		−	0.288131i	−0.989568	−	0.144065i	$-0.953982\pi$
$863$		−	8.46438i		−	0.288131i	0.989568	−	0.144065i	$-0.0460176\pi$
$864$		0			0
$865$		−	7.24943i		−	0.246488i
$866$		0			0
$867$		−	15.4277i		−	0.523952i
$868$		0			0
$869$	10.9343			0.370922
$870$		0			0
$871$	11.6782			0.395700
$872$		0			0
$873$			7.70591i			0.260806i
$874$		0			0
$875$			1.58474i			0.0535739i
$876$		0			0
$877$	20.0279			0.676295			0.338147	−	0.941093i	$-0.390200\pi$
$877$	20.0279			0.676295			0.338147	+	0.941093i	$0.390200\pi$
$878$		0			0
$879$	−25.3576			−0.855292
$880$		0			0
$881$		−	38.0341i		−	1.28140i	−0.767790	−	0.640701i	$-0.778644\pi$
$881$		−	38.0341i		−	1.28140i	0.767790	−	0.640701i	$-0.221356\pi$
$882$		0			0
$883$		−	14.9644i		−	0.503592i	−0.967780	−	0.251796i	$-0.918979\pi$
$883$		−	14.9644i		−	0.503592i	0.967780	−	0.251796i	$-0.0810211\pi$
$884$		0			0
$885$		−	2.33680i		−	0.0785508i
$886$		0			0
$887$		−	0.728725i		−	0.0244682i	−0.999925	−	0.0122341i	$-0.996106\pi$
$887$		−	0.728725i		−	0.0244682i	0.999925	−	0.0122341i	$-0.00389433\pi$
$888$		0			0
$889$			6.90144i			0.231467i
$890$		0			0
$891$	−2.94482			−0.0986553
$892$		0			0
$893$			47.1602i			1.57816i
$894$		0			0
$895$	−12.2088			−0.408097
$896$		0			0
$897$	5.44862	+	2.26175i	0.181924	+	0.0755176i
$898$		0			0
$899$		−	5.20971i		−	0.173754i
$900$		0			0
$901$	−13.6492			−0.454720
$902$		0			0
$903$		−	6.44483i		−	0.214470i
$904$		0			0
$905$	−16.9328			−0.562865
$906$		0			0
$907$	−3.84373			−0.127629			−0.0638145	−	0.997962i	$-0.520327\pi$
$907$	−3.84373			−0.127629			−0.0638145	+	0.997962i	$0.520327\pi$
$908$		0			0
$909$	9.43609			0.312975
$910$		0			0
$911$	−55.8963			−1.85193			−0.925964	−	0.377611i	$-0.876746\pi$
$911$	−55.8963			−1.85193			−0.925964	+	0.377611i	$0.876746\pi$
$912$		0			0
$913$	−48.6278			−1.60935
$914$		0			0
$915$			5.28658i			0.174769i
$916$		0			0
$917$			3.05126i			0.100762i
$918$		0			0
$919$	−0.643465			−0.0212260			−0.0106130	−	0.999944i	$-0.503378\pi$
$919$	−0.643465			−0.0212260			−0.0106130	+	0.999944i	$0.503378\pi$
$920$		0			0
$921$	12.7630			0.420555
$922$		0			0
$923$			16.6388i			0.547672i
$924$		0			0
$925$		−	4.83340i		−	0.158921i
$926$		0			0
$927$	−7.07692			−0.232437
$928$		0			0
$929$	−1.92764			−0.0632440			−0.0316220	−	0.999500i	$-0.510067\pi$
$929$	−1.92764			−0.0632440			−0.0316220	+	0.999500i	$0.510067\pi$
$930$		0			0
$931$	−23.6421			−0.774839
$932$		0			0
$933$	−14.7842			−0.484012
$934$		0			0
$935$	3.69256			0.120760
$936$		0			0
$937$		−	18.6541i		−	0.609404i	−0.952448	−	0.304702i	$-0.901443\pi$
$937$		−	18.6541i		−	0.609404i	0.952448	−	0.304702i	$-0.0985568\pi$
$938$		0			0
$939$	1.48723			0.0485339
$940$		0			0
$941$		−	20.7327i		−	0.675868i	−0.941170	−	0.337934i	$-0.890272\pi$
$941$		−	20.7327i		−	0.675868i	0.941170	−	0.337934i	$-0.109728\pi$
$942$		0			0
$943$	12.8482	−	30.9517i	0.418395	−	1.00793i
$944$		0			0
$945$	1.58474			0.0515515
$946$		0			0
$947$		−	34.9977i		−	1.13727i	−0.822589	−	0.568636i	$-0.807471\pi$
$947$		−	34.9977i		−	1.13727i	0.822589	−	0.568636i	$-0.192529\pi$
$948$		0			0
$949$	11.7086			0.380076
$950$		0			0
$951$		−	0.771575i		−	0.0250200i
$952$		0			0
$953$			42.1023i			1.36383i	0.731433	+	0.681914i	$0.238852\pi$
$953$			42.1023i			1.36383i	−0.731433	+	0.681914i	$0.761148\pi$
$954$		0			0
$955$			6.46417i			0.209176i
$956$		0			0
$957$			22.2277i			0.718520i
$958$		0			0
$959$		−	18.9031i		−	0.610412i
$960$		0			0
$961$	30.5236			0.984633
$962$		0			0
$963$	−6.20879			−0.200076
$964$		0			0
$965$			21.5952i			0.695173i
$966$		0			0
$967$			21.6032i			0.694711i	0.937734	+	0.347355i	$0.112920\pi$
$967$			21.6032i			0.694711i	−0.937734	+	0.347355i	$0.887080\pi$
$968$		0			0
$969$	−6.60455			−0.212169
$970$		0			0
$971$	−38.8071			−1.24538			−0.622690	−	0.782469i	$-0.713960\pi$
$971$	−38.8071			−1.24538			−0.622690	+	0.782469i	$0.713960\pi$
$972$		0			0
$973$		−	2.08715i		−	0.0669108i
$974$		0			0
$975$		−	1.23011i		−	0.0393951i
$976$		0			0
$977$		−	33.0657i		−	1.05787i	−0.848664	−	0.528933i	$-0.822592\pi$
$977$		−	33.0657i		−	1.05787i	0.848664	−	0.528933i	$-0.177408\pi$
$978$		0			0
$979$			36.0463i			1.15204i
$980$		0			0
$981$			5.46182i			0.174382i
$982$		0			0
$983$	19.4381			0.619980			0.309990	−	0.950740i	$-0.399674\pi$
$983$	19.4381			0.619980			0.309990	+	0.950740i	$0.399674\pi$
$984$		0			0
$985$			4.11055i			0.130973i
$986$		0			0
$987$	14.1892			0.451648
$988$		0			0
$989$	7.47746	−	18.0134i	0.237769	−	0.572794i
$990$		0			0
$991$			1.60244i			0.0509033i	0.999676	+	0.0254517i	$0.00810239\pi$
$991$			1.60244i			0.0509033i	−0.999676	+	0.0254517i	$0.991898\pi$
$992$		0			0
$993$	−33.8360			−1.07375
$994$		0			0
$995$			3.25603i			0.103223i
$996$		0			0
$997$	10.4207			0.330026			0.165013	−	0.986291i	$-0.447233\pi$
$997$	10.4207			0.330026			0.165013	+	0.986291i	$0.447233\pi$
$998$		0			0
$999$	−4.83340			−0.152922

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5520.2.be.b.1471.6	yes	16
4.3	odd	2		5520.2.be.a.1471.11	yes	16
23.22	odd	2		5520.2.be.a.1471.3	✓	16
92.91	even	2	inner	5520.2.be.b.1471.14	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5520.2.be.a.1471.3	✓	16	23.22	odd	2
5520.2.be.a.1471.11	yes	16	4.3	odd	2
5520.2.be.b.1471.6	yes	16	1.1	even	1	trivial
5520.2.be.b.1471.14	yes	16	92.91	even	2	inner

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:	\( N \)	\(=\)	\( 5520 = 2^{4} \cdot 3 \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5520.be (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} -1.00000i q^{5} +1.58474 q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} -1.00000i q^{5} +1.58474 q^{7} -1.00000 q^{9} -2.94482 q^{11} -1.23011 q^{13} -1.00000 q^{15} -1.25392i q^{17} +5.26714 q^{19} -1.58474i q^{21} +(1.83865 - 4.42937i) q^{23} -1.00000 q^{25} +1.00000i q^{27} +7.54806 q^{29} -0.690205i q^{31} +2.94482i q^{33} -1.58474i q^{35} +4.83340i q^{37} +1.23011i q^{39} +6.98783 q^{41} +4.06681 q^{43} +1.00000i q^{45} +8.95368i q^{47} -4.48861 q^{49} -1.25392 q^{51} -10.8852i q^{53} +2.94482i q^{55} -5.26714i q^{57} +2.33680i q^{59} -5.28658i q^{61} -1.58474 q^{63} +1.23011i q^{65} -9.49359 q^{67} +(-4.42937 - 1.83865i) q^{69} -13.5262i q^{71} -9.51828 q^{73} +1.00000i q^{75} -4.66677 q^{77} -3.71307 q^{79} +1.00000 q^{81} +16.5130 q^{83} -1.25392 q^{85} -7.54806i q^{87} -12.2406i q^{89} -1.94940 q^{91} -0.690205 q^{93} -5.26714i q^{95} -7.70591i q^{97} +2.94482 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 8 q^{7} - 16 q^{9}+O(q^{10}) \) 16 * q + 8 * q^7 - 16 * q^9	\( 16 q + 8 q^{7} - 16 q^{9} - 8 q^{11} + 8 q^{13} - 16 q^{15} - 12 q^{23} - 16 q^{25} - 4 q^{29} + 4 q^{41} + 20 q^{49} + 4 q^{51} - 8 q^{63} + 16 q^{67} + 40 q^{73} + 24 q^{77} - 32 q^{79} + 16 q^{81} + 4 q^{85} + 48 q^{91} + 8 q^{99}+O(q^{100}) \) 16 * q + 8 * q^7 - 16 * q^9 - 8 * q^11 + 8 * q^13 - 16 * q^15 - 12 * q^23 - 16 * q^25 - 4 * q^29 + 4 * q^41 + 20 * q^49 + 4 * q^51 - 8 * q^63 + 16 * q^67 + 40 * q^73 + 24 * q^77 - 32 * q^79 + 16 * q^81 + 4 * q^85 + 48 * q^91 + 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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