LMFDB - Embedded newform 5520.2.be.b.1471.2

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			1471.2
Root			$-0.373815 - 0.373815i$ of defining polynomial
Character	$\chi$	$=$	5520.1471
Dual form			5520.2.be.b.1471.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} -1.00000i q^{5} -1.61153 q^{7} -1.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} -1.00000i q^{5} -1.61153 q^{7} -1.00000 q^{9} -1.15035 q^{11} +7.10687 q^{13} -1.00000 q^{15} +1.79275i q^{17} -5.93632 q^{19} +1.61153i q^{21} +(-4.40428 + 1.89798i) q^{23} -1.00000 q^{25} +1.00000i q^{27} -0.0365806 q^{29} -4.39344i q^{31} +1.15035i q^{33} +1.61153i q^{35} +10.1186i q^{37} -7.10687i q^{39} +5.43873 q^{41} +9.60250 q^{43} +1.00000i q^{45} +9.16374i q^{47} -4.40297 q^{49} +1.79275 q^{51} +1.95476i q^{53} +1.15035i q^{55} +5.93632i q^{57} +5.29905i q^{59} -9.53475i q^{61} +1.61153 q^{63} -7.10687i q^{65} +2.33657 q^{67} +(1.89798 + 4.40428i) q^{69} +11.3842i q^{71} -1.61161 q^{73} +1.00000i q^{75} +1.85383 q^{77} +5.33892 q^{79} +1.00000 q^{81} +0.722088 q^{83} +1.79275 q^{85} +0.0365806i q^{87} -2.09024i q^{89} -11.4529 q^{91} -4.39344 q^{93} +5.93632i q^{95} +1.93236i q^{97} +1.15035 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$	−1.61153			−0.609101			−0.304551	−	0.952496i	$-0.598506\pi$
$7$	−1.61153			−0.609101			−0.304551	+	0.952496i	$0.598506\pi$
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	−1.15035			−0.346844			−0.173422	−	0.984848i	$-0.555482\pi$
$11$	−1.15035			−0.346844			−0.173422	+	0.984848i	$0.555482\pi$
$12$		0			0
$13$	7.10687			1.97109			0.985545	−	0.169414i	$-0.0541876\pi$
$13$	7.10687			1.97109			0.985545	+	0.169414i	$0.0541876\pi$
$14$		0			0
$15$	−1.00000			−0.258199
$16$		0			0
$17$			1.79275i			0.434805i	0.976082	+	0.217403i	$0.0697584\pi$
$17$			1.79275i			0.434805i	−0.976082	+	0.217403i	$0.930242\pi$
$18$		0			0
$19$	−5.93632			−1.36188			−0.680942	−	0.732337i	$-0.738429\pi$
$19$	−5.93632			−1.36188			−0.680942	+	0.732337i	$0.738429\pi$
$20$		0			0
$21$			1.61153i			0.351665i
$22$		0			0
$23$	−4.40428	+	1.89798i	−0.918356	+	0.395756i
$23$	−4.40428	+	1.89798i	−0.918356	+	0.395756i
$24$		0			0
$25$	−1.00000			−0.200000
$26$		0			0
$27$			1.00000i			0.192450i
$28$		0			0
$29$	−0.0365806			−0.00679285			−0.00339643	−	0.999994i	$-0.501081\pi$
$29$	−0.0365806			−0.00679285			−0.00339643	+	0.999994i	$0.501081\pi$
$30$		0			0
$31$		−	4.39344i		−	0.789085i	−0.918878	−	0.394542i	$-0.870903\pi$
$31$		−	4.39344i		−	0.789085i	0.918878	−	0.394542i	$-0.129097\pi$
$32$		0			0
$33$			1.15035i			0.200251i
$34$		0			0
$35$			1.61153i			0.272398i
$36$		0			0
$37$			10.1186i			1.66349i	0.555158	+	0.831745i	$0.312658\pi$
$37$			10.1186i			1.66349i	−0.555158	+	0.831745i	$0.687342\pi$
$38$		0			0
$39$		−	7.10687i		−	1.13801i
$40$		0			0
$41$	5.43873			0.849387			0.424693	−	0.905337i	$-0.360382\pi$
$41$	5.43873			0.849387			0.424693	+	0.905337i	$0.360382\pi$
$42$		0			0
$43$	9.60250			1.46437			0.732183	−	0.681107i	$-0.238501\pi$
$43$	9.60250			1.46437			0.732183	+	0.681107i	$0.238501\pi$
$44$		0			0
$45$			1.00000i			0.149071i
$46$		0			0
$47$			9.16374i			1.33667i	0.743861	+	0.668334i	$0.232992\pi$
$47$			9.16374i			1.33667i	−0.743861	+	0.668334i	$0.767008\pi$
$48$		0			0
$49$	−4.40297			−0.628995
$50$		0			0
$51$	1.79275			0.251035
$52$		0			0
$53$			1.95476i			0.268507i	0.990947	+	0.134253i	$0.0428636\pi$
$53$			1.95476i			0.268507i	−0.990947	+	0.134253i	$0.957136\pi$
$54$		0			0
$55$			1.15035i			0.155113i
$56$		0			0
$57$			5.93632i			0.786284i
$58$		0			0
$59$			5.29905i			0.689877i	0.938625	+	0.344938i	$0.112100\pi$
$59$			5.29905i			0.689877i	−0.938625	+	0.344938i	$0.887900\pi$
$60$		0			0
$61$		−	9.53475i		−	1.22080i	−0.792093	−	0.610400i	$-0.791009\pi$
$61$		−	9.53475i		−	1.22080i	0.792093	−	0.610400i	$-0.208991\pi$
$62$		0			0
$63$	1.61153			0.203034
$64$		0			0
$65$		−	7.10687i		−	0.881498i
$66$		0			0
$67$	2.33657			0.285457			0.142729	−	0.989762i	$-0.454412\pi$
$67$	2.33657			0.285457			0.142729	+	0.989762i	$0.454412\pi$
$68$		0			0
$69$	1.89798	+	4.40428i	0.228490	+	0.530213i
$70$		0			0
$71$			11.3842i			1.35106i	0.737334	+	0.675528i	$0.236084\pi$
$71$			11.3842i			1.35106i	−0.737334	+	0.675528i	$0.763916\pi$
$72$		0			0
$73$	−1.61161			−0.188624			−0.0943122	−	0.995543i	$-0.530065\pi$
$73$	−1.61161			−0.188624			−0.0943122	+	0.995543i	$0.530065\pi$
$74$		0			0
$75$			1.00000i			0.115470i
$76$		0			0
$77$	1.85383			0.211263
$78$		0			0
$79$	5.33892			0.600675			0.300338	−	0.953833i	$-0.402901\pi$
$79$	5.33892			0.600675			0.300338	+	0.953833i	$0.402901\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$	0.722088			0.0792594			0.0396297	−	0.999214i	$-0.487382\pi$
$83$	0.722088			0.0792594			0.0396297	+	0.999214i	$0.487382\pi$
$84$		0			0
$85$	1.79275			0.194451
$86$		0			0
$87$			0.0365806i			0.00392186i
$88$		0			0
$89$		−	2.09024i		−	0.221565i	−0.993845	−	0.110782i	$-0.964664\pi$
$89$		−	2.09024i		−	0.221565i	0.993845	−	0.110782i	$-0.0353356\pi$
$90$		0			0
$91$	−11.4529			−1.20059
$92$		0			0
$93$	−4.39344			−0.455578
$94$		0			0
$95$			5.93632i			0.609053i
$96$		0			0
$97$			1.93236i			0.196201i	0.995177	+	0.0981005i	$0.0312767\pi$
$97$			1.93236i			0.196201i	−0.995177	+	0.0981005i	$0.968723\pi$
$98$		0			0
$99$	1.15035			0.115615
$100$		0			0
$101$	15.0107			1.49362			0.746808	−	0.665040i	$-0.231585\pi$
$101$	15.0107			1.49362			0.746808	+	0.665040i	$0.231585\pi$
$102$		0			0
$103$	−3.04742			−0.300271			−0.150136	−	0.988665i	$-0.547971\pi$
$103$	−3.04742			−0.300271			−0.150136	+	0.988665i	$0.547971\pi$
$104$		0			0
$105$	1.61153			0.157269
$106$		0			0
$107$	−12.6032			−1.21840			−0.609199	−	0.793017i	$-0.708509\pi$
$107$	−12.6032			−1.21840			−0.609199	+	0.793017i	$0.708509\pi$
$108$		0			0
$109$			15.0300i			1.43961i	0.694174	+	0.719807i	$0.255770\pi$
$109$			15.0300i			1.43961i	−0.694174	+	0.719807i	$0.744230\pi$
$110$		0			0
$111$	10.1186			0.960417
$112$		0			0
$113$		−	8.39445i		−	0.789684i	−0.918749	−	0.394842i	$-0.870799\pi$
$113$		−	8.39445i		−	0.789684i	0.918749	−	0.394842i	$-0.129201\pi$
$114$		0			0
$115$	1.89798	+	4.40428i	0.176988	+	0.410701i
$116$		0			0
$117$	−7.10687			−0.657030
$118$		0			0
$119$		−	2.88907i		−	0.264840i
$120$		0			0
$121$	−9.67669			−0.879699
$122$		0			0
$123$		−	5.43873i		−	0.490394i
$124$		0			0
$125$			1.00000i			0.0894427i
$126$		0			0
$127$		−	18.1343i		−	1.60916i	−0.593845	−	0.804580i	$-0.702391\pi$
$127$		−	18.1343i		−	1.60916i	0.593845	−	0.804580i	$-0.297609\pi$
$128$		0			0
$129$		−	9.60250i		−	0.845453i
$130$		0			0
$131$			1.77034i			0.154676i	0.997005	+	0.0773378i	$0.0246420\pi$
$131$			1.77034i			0.154676i	−0.997005	+	0.0773378i	$0.975358\pi$
$132$		0			0
$133$	9.56656			0.829526
$134$		0			0
$135$	1.00000			0.0860663
$136$		0			0
$137$			17.7574i			1.51712i	0.651603	+	0.758560i	$0.274097\pi$
$137$			17.7574i			1.51712i	−0.651603	+	0.758560i	$0.725903\pi$
$138$		0			0
$139$			13.6548i			1.15819i	0.815261	+	0.579094i	$0.196594\pi$
$139$			13.6548i			1.15819i	−0.815261	+	0.579094i	$0.803406\pi$
$140$		0			0
$141$	9.16374			0.771726
$142$		0			0
$143$	−8.17540			−0.683661
$144$		0			0
$145$			0.0365806i			0.00303786i
$146$		0			0
$147$			4.40297i			0.363151i
$148$		0			0
$149$			9.38262i			0.768654i	0.923197	+	0.384327i	$0.125566\pi$
$149$			9.38262i			0.768654i	−0.923197	+	0.384327i	$0.874434\pi$
$150$		0			0
$151$		−	13.1263i		−	1.06820i	−0.845421	−	0.534100i	$-0.820651\pi$
$151$		−	13.1263i		−	1.06820i	0.845421	−	0.534100i	$-0.179349\pi$
$152$		0			0
$153$		−	1.79275i		−	0.144935i
$154$		0			0
$155$	−4.39344			−0.352889
$156$		0			0
$157$		−	2.75711i		−	0.220041i	−0.993929	−	0.110021i	$-0.964908\pi$
$157$		−	2.75711i		−	0.220041i	0.993929	−	0.110021i	$-0.0350917\pi$
$158$		0			0
$159$	1.95476			0.155022
$160$		0			0
$161$	7.09763	−	3.05865i	0.559372	−	0.241056i
$162$		0			0
$163$		−	7.13282i		−	0.558686i	−0.960191	−	0.279343i	$-0.909883\pi$
$163$		−	7.13282i		−	0.558686i	0.960191	−	0.279343i	$-0.0901166\pi$
$164$		0			0
$165$	1.15035			0.0895548
$166$		0			0
$167$		−	0.264942i		−	0.0205018i	−0.999947	−	0.0102509i	$-0.996737\pi$
$167$		−	0.264942i		−	0.0205018i	0.999947	−	0.0102509i	$-0.00326303\pi$
$168$		0			0
$169$	37.5075			2.88520
$170$		0			0
$171$	5.93632			0.453961
$172$		0			0
$173$	−9.93924			−0.755666			−0.377833	−	0.925874i	$-0.623331\pi$
$173$	−9.93924			−0.755666			−0.377833	+	0.925874i	$0.623331\pi$
$174$		0			0
$175$	1.61153			0.121820
$176$		0			0
$177$	5.29905			0.398301
$178$		0			0
$179$			5.00824i			0.374333i	0.982328	+	0.187167i	$0.0599304\pi$
$179$			5.00824i			0.374333i	−0.982328	+	0.187167i	$0.940070\pi$
$180$		0			0
$181$			1.45486i			0.108139i	0.998537	+	0.0540695i	$0.0172193\pi$
$181$			1.45486i			0.108139i	−0.998537	+	0.0540695i	$0.982781\pi$
$182$		0			0
$183$	−9.53475			−0.704829
$184$		0			0
$185$	10.1186			0.743936
$186$		0			0
$187$		−	2.06229i		−	0.150810i
$188$		0			0
$189$		−	1.61153i		−	0.117222i
$190$		0			0
$191$	21.9857			1.59083			0.795416	−	0.606064i	$-0.207252\pi$
$191$	21.9857			1.59083			0.795416	+	0.606064i	$0.207252\pi$
$192$		0			0
$193$	13.5269			0.973686			0.486843	−	0.873490i	$-0.338148\pi$
$193$	13.5269			0.973686			0.486843	+	0.873490i	$0.338148\pi$
$194$		0			0
$195$	−7.10687			−0.508933
$196$		0			0
$197$	14.6237			1.04190			0.520948	−	0.853588i	$-0.325579\pi$
$197$	14.6237			1.04190			0.520948	+	0.853588i	$0.325579\pi$
$198$		0			0
$199$	−3.24225			−0.229837			−0.114919	−	0.993375i	$-0.536661\pi$
$199$	−3.24225			−0.229837			−0.114919	+	0.993375i	$0.536661\pi$
$200$		0			0
$201$		−	2.33657i		−	0.164809i
$202$		0			0
$203$	0.0589508			0.00413754
$204$		0			0
$205$		−	5.43873i		−	0.379857i
$206$		0			0
$207$	4.40428	−	1.89798i	0.306119	−	0.131919i
$208$		0			0
$209$	6.82885			0.472362
$210$		0			0
$211$			25.4246i			1.75030i	0.483850	+	0.875151i	$0.339238\pi$
$211$			25.4246i			1.75030i	−0.483850	+	0.875151i	$0.660762\pi$
$212$		0			0
$213$	11.3842			0.780033
$214$		0			0
$215$		−	9.60250i		−	0.654885i
$216$		0			0
$217$			7.08016i			0.480633i
$218$		0			0
$219$			1.61161i			0.108902i
$220$		0			0
$221$			12.7408i			0.857040i
$222$		0			0
$223$			18.2691i			1.22339i	0.791095	+	0.611694i	$0.209511\pi$
$223$			18.2691i			1.22339i	−0.791095	+	0.611694i	$0.790489\pi$
$224$		0			0
$225$	1.00000			0.0666667
$226$		0			0
$227$	23.8464			1.58274			0.791372	−	0.611335i	$-0.209367\pi$
$227$	23.8464			1.58274			0.791372	+	0.611335i	$0.209367\pi$
$228$		0			0
$229$			3.95960i			0.261658i	0.991405	+	0.130829i	$0.0417639\pi$
$229$			3.95960i			0.261658i	−0.991405	+	0.130829i	$0.958236\pi$
$230$		0			0
$231$		−	1.85383i		−	0.121973i
$232$		0			0
$233$	26.6401			1.74525			0.872625	−	0.488392i	$-0.162416\pi$
$233$	26.6401			1.74525			0.872625	+	0.488392i	$0.162416\pi$
$234$		0			0
$235$	9.16374			0.597776
$236$		0			0
$237$		−	5.33892i		−	0.346800i
$238$		0			0
$239$			15.7541i			1.01905i	0.860457	+	0.509523i	$0.170178\pi$
$239$			15.7541i			1.01905i	−0.860457	+	0.509523i	$0.829822\pi$
$240$		0			0
$241$			2.31154i			0.148899i	0.997225	+	0.0744497i	$0.0237200\pi$
$241$			2.31154i			0.148899i	−0.997225	+	0.0744497i	$0.976280\pi$
$242$		0			0
$243$		−	1.00000i		−	0.0641500i
$244$		0			0
$245$			4.40297i			0.281295i
$246$		0			0
$247$	−42.1886			−2.68440
$248$		0			0
$249$		−	0.722088i		−	0.0457605i
$250$		0			0
$251$	21.6173			1.36447			0.682235	−	0.731133i	$-0.261008\pi$
$251$	21.6173			1.36447			0.682235	+	0.731133i	$0.261008\pi$
$252$		0			0
$253$	5.06647	−	2.18335i	0.318526	−	0.137266i
$254$		0			0
$255$		−	1.79275i		−	0.112266i
$256$		0			0
$257$	4.71865			0.294341			0.147170	−	0.989111i	$-0.452983\pi$
$257$	4.71865			0.294341			0.147170	+	0.989111i	$0.452983\pi$
$258$		0			0
$259$		−	16.3065i		−	1.01323i
$260$		0			0
$261$	0.0365806			0.00226428
$262$		0			0
$263$	4.44240			0.273930			0.136965	−	0.990576i	$-0.456265\pi$
$263$	4.44240			0.273930			0.136965	+	0.990576i	$0.456265\pi$
$264$		0			0
$265$	1.95476			0.120080
$266$		0			0
$267$	−2.09024			−0.127920
$268$		0			0
$269$	−9.10119			−0.554909			−0.277455	−	0.960739i	$-0.589491\pi$
$269$	−9.10119			−0.554909			−0.277455	+	0.960739i	$0.589491\pi$
$270$		0			0
$271$		−	22.9892i		−	1.39650i	−0.715856	−	0.698248i	$-0.753963\pi$
$271$		−	22.9892i		−	1.39650i	0.715856	−	0.698248i	$-0.246037\pi$
$272$		0			0
$273$			11.4529i			0.693163i
$274$		0			0
$275$	1.15035			0.0693688
$276$		0			0
$277$	18.6686			1.12169			0.560843	−	0.827922i	$-0.310477\pi$
$277$	18.6686			1.12169			0.560843	+	0.827922i	$0.310477\pi$
$278$		0			0
$279$			4.39344i			0.263028i
$280$		0			0
$281$			23.4639i			1.39974i	0.714271	+	0.699869i	$0.246758\pi$
$281$			23.4639i			1.39974i	−0.714271	+	0.699869i	$0.753242\pi$
$282$		0			0
$283$	13.4760			0.801068			0.400534	−	0.916282i	$-0.368825\pi$
$283$	13.4760			0.801068			0.400534	+	0.916282i	$0.368825\pi$
$284$		0			0
$285$	5.93632			0.351637
$286$		0			0
$287$	−8.76468			−0.517363
$288$		0			0
$289$	13.7861			0.810944
$290$		0			0
$291$	1.93236			0.113277
$292$		0			0
$293$		−	26.8108i		−	1.56630i	−0.621831	−	0.783151i	$-0.713611\pi$
$293$		−	26.8108i		−	1.56630i	0.621831	−	0.783151i	$-0.286389\pi$
$294$		0			0
$295$	5.29905			0.308522
$296$		0			0
$297$		−	1.15035i		−	0.0667502i
$298$		0			0
$299$	−31.3006	+	13.4887i	−1.81016	+	0.780071i
$300$		0			0
$301$	−15.4747			−0.891948
$302$		0			0
$303$		−	15.0107i		−	0.862340i
$304$		0			0
$305$	−9.53475			−0.545958
$306$		0			0
$307$		−	1.71461i		−	0.0978580i	−0.998802	−	0.0489290i	$-0.984419\pi$
$307$		−	1.71461i		−	0.0978580i	0.998802	−	0.0489290i	$-0.0155808\pi$
$308$		0			0
$309$			3.04742i			0.173362i
$310$		0			0
$311$		−	1.97438i		−	0.111957i	−0.998432	−	0.0559785i	$-0.982172\pi$
$311$		−	1.97438i		−	0.111957i	0.998432	−	0.0559785i	$-0.0178278\pi$
$312$		0			0
$313$			8.62500i			0.487514i	0.969836	+	0.243757i	$0.0783799\pi$
$313$			8.62500i			0.487514i	−0.969836	+	0.243757i	$0.921620\pi$
$314$		0			0
$315$		−	1.61153i		−	0.0907995i
$316$		0			0
$317$	−22.2686			−1.25073			−0.625366	−	0.780332i	$-0.715050\pi$
$317$	−22.2686			−1.25073			−0.625366	+	0.780332i	$0.715050\pi$
$318$		0			0
$319$	0.0420806			0.00235606
$320$		0			0
$321$			12.6032i			0.703442i
$322$		0			0
$323$		−	10.6423i		−	0.592154i
$324$		0			0
$325$	−7.10687			−0.394218
$326$		0			0
$327$	15.0300			0.831162
$328$		0			0
$329$		−	14.7676i		−	0.814167i
$330$		0			0
$331$		−	13.5388i		−	0.744162i	−0.928200	−	0.372081i	$-0.878644\pi$
$331$		−	13.5388i		−	0.744162i	0.928200	−	0.372081i	$-0.121356\pi$
$332$		0			0
$333$		−	10.1186i		−	0.554497i
$334$		0			0
$335$		−	2.33657i		−	0.127660i
$336$		0			0
$337$			18.2890i			0.996267i	0.867100	+	0.498134i	$0.165981\pi$
$337$			18.2890i			0.996267i	−0.867100	+	0.498134i	$0.834019\pi$
$338$		0			0
$339$	−8.39445			−0.455924
$340$		0			0
$341$			5.05400i			0.273689i
$342$		0			0
$343$	18.3762			0.992223
$344$		0			0
$345$	4.40428	−	1.89798i	0.237118	−	0.102184i
$346$		0			0
$347$		−	15.4805i		−	0.831037i	−0.909585	−	0.415519i	$-0.863600\pi$
$347$		−	15.4805i		−	0.831037i	0.909585	−	0.415519i	$-0.136400\pi$
$348$		0			0
$349$	−26.2209			−1.40357			−0.701785	−	0.712388i	$-0.747614\pi$
$349$	−26.2209			−1.40357			−0.701785	+	0.712388i	$0.747614\pi$
$350$		0			0
$351$			7.10687i			0.379336i
$352$		0			0
$353$	22.9065			1.21919			0.609596	−	0.792713i	$-0.291332\pi$
$353$	22.9065			1.21919			0.609596	+	0.792713i	$0.291332\pi$
$354$		0			0
$355$	11.3842			0.604211
$356$		0			0
$357$	−2.88907			−0.152906
$358$		0			0
$359$	6.73100			0.355248			0.177624	−	0.984098i	$-0.443159\pi$
$359$	6.73100			0.355248			0.177624	+	0.984098i	$0.443159\pi$
$360$		0			0
$361$	16.2398			0.854729
$362$		0			0
$363$			9.67669i			0.507895i
$364$		0			0
$365$			1.61161i			0.0843554i
$366$		0			0
$367$	9.41638			0.491531			0.245765	−	0.969329i	$-0.420961\pi$
$367$	9.41638			0.491531			0.245765	+	0.969329i	$0.420961\pi$
$368$		0			0
$369$	−5.43873			−0.283129
$370$		0			0
$371$		−	3.15015i		−	0.163548i
$372$		0			0
$373$		−	32.7090i		−	1.69361i	−0.531904	−	0.846805i	$-0.678523\pi$
$373$		−	32.7090i		−	1.69361i	0.531904	−	0.846805i	$-0.321477\pi$
$374$		0			0
$375$	1.00000			0.0516398
$376$		0			0
$377$	−0.259974			−0.0133893
$378$		0			0
$379$	−12.0214			−0.617496			−0.308748	−	0.951144i	$-0.599910\pi$
$379$	−12.0214			−0.617496			−0.308748	+	0.951144i	$0.599910\pi$
$380$		0			0
$381$	−18.1343			−0.929049
$382$		0			0
$383$	−28.7465			−1.46888			−0.734438	−	0.678676i	$-0.762554\pi$
$383$	−28.7465			−1.46888			−0.734438	+	0.678676i	$0.762554\pi$
$384$		0			0
$385$		−	1.85383i		−	0.0944798i
$386$		0			0
$387$	−9.60250			−0.488122
$388$		0			0
$389$		−	8.14172i		−	0.412801i	−0.978468	−	0.206401i	$-0.933825\pi$
$389$		−	8.14172i		−	0.412801i	0.978468	−	0.206401i	$-0.0661750\pi$
$390$		0			0
$391$	−3.40260	−	7.89576i	−0.172077	−	0.399306i
$392$		0			0
$393$	1.77034			0.0893021
$394$		0			0
$395$		−	5.33892i		−	0.268630i
$396$		0			0
$397$	−8.41445			−0.422309			−0.211154	−	0.977453i	$-0.567722\pi$
$397$	−8.41445			−0.422309			−0.211154	+	0.977453i	$0.567722\pi$
$398$		0			0
$399$		−	9.56656i		−	0.478927i
$400$		0			0
$401$			24.7496i			1.23594i	0.786203	+	0.617968i	$0.212044\pi$
$401$			24.7496i			1.23594i	−0.786203	+	0.617968i	$0.787956\pi$
$402$		0			0
$403$		−	31.2236i		−	1.55536i
$404$		0			0
$405$		−	1.00000i		−	0.0496904i
$406$		0			0
$407$		−	11.6400i		−	0.576972i
$408$		0			0
$409$	21.2723			1.05185			0.525923	−	0.850532i	$-0.323720\pi$
$409$	21.2723			1.05185			0.525923	+	0.850532i	$0.323720\pi$
$410$		0			0
$411$	17.7574			0.875910
$412$		0			0
$413$		−	8.53957i		−	0.420205i
$414$		0			0
$415$		−	0.722088i		−	0.0354459i
$416$		0			0
$417$	13.6548			0.668680
$418$		0			0
$419$	−7.56247			−0.369451			−0.184726	−	0.982790i	$-0.559140\pi$
$419$	−7.56247			−0.369451			−0.184726	+	0.982790i	$0.559140\pi$
$420$		0			0
$421$			26.6686i			1.29975i	0.760042	+	0.649874i	$0.225178\pi$
$421$			26.6686i			1.29975i	−0.760042	+	0.649874i	$0.774822\pi$
$422$		0			0
$423$		−	9.16374i		−	0.445556i
$424$		0			0
$425$		−	1.79275i		−	0.0869610i
$426$		0			0
$427$			15.3656i			0.743591i
$428$		0			0
$429$			8.17540i			0.394712i
$430$		0			0
$431$	−19.3147			−0.930356			−0.465178	−	0.885217i	$-0.654010\pi$
$431$	−19.3147			−0.930356			−0.465178	+	0.885217i	$0.654010\pi$
$432$		0			0
$433$		−	15.6094i		−	0.750141i	−0.926996	−	0.375071i	$-0.877618\pi$
$433$		−	15.6094i		−	0.750141i	0.926996	−	0.375071i	$-0.122382\pi$
$434$		0			0
$435$	0.0365806			0.00175391
$436$		0			0
$437$	26.1452	−	11.2670i	1.25069	−	0.538974i
$438$		0			0
$439$		−	12.5249i		−	0.597779i	−0.954288	−	0.298889i	$-0.903384\pi$
$439$		−	12.5249i		−	0.597779i	0.954288	−	0.298889i	$-0.0966161\pi$
$440$		0			0
$441$	4.40297			0.209665
$442$		0			0
$443$			37.7262i			1.79242i	0.443625	+	0.896212i	$0.353692\pi$
$443$			37.7262i			1.79242i	−0.443625	+	0.896212i	$0.646308\pi$
$444$		0			0
$445$	−2.09024			−0.0990867
$446$		0			0
$447$	9.38262			0.443783
$448$		0			0
$449$	9.98001			0.470986			0.235493	−	0.971876i	$-0.424330\pi$
$449$	9.98001			0.470986			0.235493	+	0.971876i	$0.424330\pi$
$450$		0			0
$451$	−6.25645			−0.294605
$452$		0			0
$453$	−13.1263			−0.616726
$454$		0			0
$455$			11.4529i			0.536922i
$456$		0			0
$457$		−	39.0608i		−	1.82719i	−0.406628	−	0.913594i	$-0.633295\pi$
$457$		−	39.0608i		−	1.82719i	0.406628	−	0.913594i	$-0.366705\pi$
$458$		0			0
$459$	−1.79275			−0.0836783
$460$		0			0
$461$	−38.5061			−1.79341			−0.896705	−	0.442629i	$-0.854046\pi$
$461$	−38.5061			−1.79341			−0.896705	+	0.442629i	$0.854046\pi$
$462$		0			0
$463$			7.15257i			0.332408i	0.986091	+	0.166204i	$0.0531510\pi$
$463$			7.15257i			0.332408i	−0.986091	+	0.166204i	$0.946849\pi$
$464$		0			0
$465$			4.39344i			0.203741i
$466$		0			0
$467$	−21.2146			−0.981692			−0.490846	−	0.871246i	$-0.663312\pi$
$467$	−21.2146			−0.981692			−0.490846	+	0.871246i	$0.663312\pi$
$468$		0			0
$469$	−3.76545			−0.173872
$470$		0			0
$471$	−2.75711			−0.127041
$472$		0			0
$473$	−11.0462			−0.507907
$474$		0			0
$475$	5.93632			0.272377
$476$		0			0
$477$		−	1.95476i		−	0.0895023i
$478$		0			0
$479$	−29.8774			−1.36513			−0.682566	−	0.730824i	$-0.739136\pi$
$479$	−29.8774			−1.36513			−0.682566	+	0.730824i	$0.739136\pi$
$480$		0			0
$481$			71.9117i			3.27889i
$482$		0			0
$483$	−3.05865	−	7.09763i	−0.139174	−	0.322953i
$484$		0			0
$485$	1.93236			0.0877438
$486$		0			0
$487$			1.40199i			0.0635301i	0.999495	+	0.0317650i	$0.0101128\pi$
$487$			1.40199i			0.0635301i	−0.999495	+	0.0317650i	$0.989887\pi$
$488$		0			0
$489$	−7.13282			−0.322557
$490$		0			0
$491$			29.6976i			1.34023i	0.742255	+	0.670117i	$0.233756\pi$
$491$			29.6976i			1.34023i	−0.742255	+	0.670117i	$0.766244\pi$
$492$		0			0
$493$		−	0.0655798i		−	0.00295357i
$494$		0			0
$495$		−	1.15035i		−	0.0517045i
$496$		0			0
$497$		−	18.3460i		−	0.822930i
$498$		0			0
$499$		−	8.06976i		−	0.361252i	−0.983552	−	0.180626i	$-0.942188\pi$
$499$		−	8.06976i		−	0.361252i	0.983552	−	0.180626i	$-0.0578124\pi$
$500$		0			0
$501$	−0.264942			−0.0118367
$502$		0			0
$503$	22.4638			1.00161			0.500807	−	0.865559i	$-0.333037\pi$
$503$	22.4638			1.00161			0.500807	+	0.865559i	$0.333037\pi$
$504$		0			0
$505$		−	15.0107i		−	0.667965i
$506$		0			0
$507$		−	37.5075i		−	1.66577i
$508$		0			0
$509$	1.54110			0.0683082			0.0341541	−	0.999417i	$-0.489126\pi$
$509$	1.54110			0.0683082			0.0341541	+	0.999417i	$0.489126\pi$
$510$		0			0
$511$	2.59715			0.114891
$512$		0			0
$513$		−	5.93632i		−	0.262095i
$514$		0			0
$515$			3.04742i			0.134285i
$516$		0			0
$517$		−	10.5415i		−	0.463616i
$518$		0			0
$519$			9.93924i			0.436284i
$520$		0			0
$521$			2.26345i			0.0991637i	0.998770	+	0.0495818i	$0.0157889\pi$
$521$			2.26345i			0.0991637i	−0.998770	+	0.0495818i	$0.984211\pi$
$522$		0			0
$523$	3.43328			0.150127			0.0750635	−	0.997179i	$-0.476084\pi$
$523$	3.43328			0.150127			0.0750635	+	0.997179i	$0.476084\pi$
$524$		0			0
$525$		−	1.61153i		−	0.0703330i
$526$		0			0
$527$	7.87632			0.343098
$528$		0			0
$529$	15.7953	−	16.7185i	0.686754	−	0.726890i
$530$		0			0
$531$		−	5.29905i		−	0.229959i
$532$		0			0
$533$	38.6523			1.67422
$534$		0			0
$535$			12.6032i			0.544884i
$536$		0			0
$537$	5.00824			0.216121
$538$		0			0
$539$	5.06496			0.218163
$540$		0			0
$541$	−5.62058			−0.241647			−0.120824	−	0.992674i	$-0.538554\pi$
$541$	−5.62058			−0.241647			−0.120824	+	0.992674i	$0.538554\pi$
$542$		0			0
$543$	1.45486			0.0624341
$544$		0			0
$545$	15.0300			0.643815
$546$		0			0
$547$		−	24.6582i		−	1.05431i	−0.849770	−	0.527154i	$-0.823259\pi$
$547$		−	24.6582i		−	1.05431i	0.849770	−	0.527154i	$-0.176741\pi$
$548$		0			0
$549$			9.53475i			0.406933i
$550$		0			0
$551$	0.217154			0.00925108
$552$		0			0
$553$	−8.60383			−0.365872
$554$		0			0
$555$		−	10.1186i		−	0.429511i
$556$		0			0
$557$			33.0462i			1.40021i	0.714039	+	0.700106i	$0.246864\pi$
$557$			33.0462i			1.40021i	−0.714039	+	0.700106i	$0.753136\pi$
$558$		0			0
$559$	68.2436			2.88640
$560$		0			0
$561$	−2.06229			−0.0870700
$562$		0			0
$563$	−15.8634			−0.668563			−0.334281	−	0.942473i	$-0.608494\pi$
$563$	−15.8634			−0.668563			−0.334281	+	0.942473i	$0.608494\pi$
$564$		0			0
$565$	−8.39445			−0.353157
$566$		0			0
$567$	−1.61153			−0.0676779
$568$		0			0
$569$			16.6195i			0.696724i	0.937360	+	0.348362i	$0.113262\pi$
$569$			16.6195i			0.696724i	−0.937360	+	0.348362i	$0.886738\pi$
$570$		0			0
$571$	−6.96762			−0.291586			−0.145793	−	0.989315i	$-0.546573\pi$
$571$	−6.96762			−0.291586			−0.145793	+	0.989315i	$0.546573\pi$
$572$		0			0
$573$		−	21.9857i		−	0.918467i
$574$		0			0
$575$	4.40428	−	1.89798i	0.183671	−	0.0791513i
$576$		0			0
$577$	20.7254			0.862811			0.431406	−	0.902158i	$-0.358018\pi$
$577$	20.7254			0.862811			0.431406	+	0.902158i	$0.358018\pi$
$578$		0			0
$579$		−	13.5269i		−	0.562158i
$580$		0			0
$581$	−1.16367			−0.0482770
$582$		0			0
$583$		−	2.24866i		−	0.0931300i
$584$		0			0
$585$			7.10687i			0.293833i
$586$		0			0
$587$			32.8221i			1.35471i	0.735655	+	0.677357i	$0.236875\pi$
$587$			32.8221i			1.35471i	−0.735655	+	0.677357i	$0.763125\pi$
$588$		0			0
$589$			26.0808i			1.07464i
$590$		0			0
$591$		−	14.6237i		−	0.601539i
$592$		0			0
$593$	−4.92783			−0.202362			−0.101181	−	0.994868i	$-0.532262\pi$
$593$	−4.92783			−0.202362			−0.101181	+	0.994868i	$0.532262\pi$
$594$		0			0
$595$	−2.88907			−0.118440
$596$		0			0
$597$			3.24225i			0.132696i
$598$		0			0
$599$		−	19.6282i		−	0.801987i	−0.916081	−	0.400994i	$-0.868665\pi$
$599$		−	19.6282i		−	0.801987i	0.916081	−	0.400994i	$-0.131335\pi$
$600$		0			0
$601$	11.1852			0.456253			0.228126	−	0.973632i	$-0.426740\pi$
$601$	11.1852			0.456253			0.228126	+	0.973632i	$0.426740\pi$
$602$		0			0
$603$	−2.33657			−0.0951524
$604$		0			0
$605$			9.67669i			0.393413i
$606$		0			0
$607$		−	12.9158i		−	0.524238i	−0.965036	−	0.262119i	$-0.915579\pi$
$607$		−	12.9158i		−	0.524238i	0.965036	−	0.262119i	$-0.0844213\pi$
$608$		0			0
$609$		−	0.0589508i		−	0.00238881i
$610$		0			0
$611$			65.1254i			2.63469i
$612$		0			0
$613$			24.6992i			0.997591i	0.866720	+	0.498795i	$0.166224\pi$
$613$			24.6992i			0.997591i	−0.866720	+	0.498795i	$0.833776\pi$
$614$		0			0
$615$	−5.43873			−0.219311
$616$		0			0
$617$			44.9940i			1.81139i	0.423928	+	0.905696i	$0.360651\pi$
$617$			44.9940i			1.81139i	−0.423928	+	0.905696i	$0.639349\pi$
$618$		0			0
$619$	36.3335			1.46037			0.730183	−	0.683252i	$-0.239435\pi$
$619$	36.3335			1.46037			0.730183	+	0.683252i	$0.239435\pi$
$620$		0			0
$621$	−1.89798	−	4.40428i	−0.0761633	−	0.176738i
$622$		0			0
$623$			3.36848i			0.134955i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$		−	6.82885i		−	0.272718i
$628$		0			0
$629$	−18.1401			−0.723294
$630$		0			0
$631$	17.8111			0.709049			0.354525	−	0.935047i	$-0.384643\pi$
$631$	17.8111			0.709049			0.354525	+	0.935047i	$0.384643\pi$
$632$		0			0
$633$	25.4246			1.01054
$634$		0			0
$635$	−18.1343			−0.719638
$636$		0			0
$637$	−31.2913			−1.23981
$638$		0			0
$639$		−	11.3842i		−	0.450352i
$640$		0			0
$641$			34.0713i			1.34574i	0.739762	+	0.672868i	$0.234938\pi$
$641$			34.0713i			1.34574i	−0.739762	+	0.672868i	$0.765062\pi$
$642$		0			0
$643$	−24.0327			−0.947756			−0.473878	−	0.880590i	$-0.657146\pi$
$643$	−24.0327			−0.947756			−0.473878	+	0.880590i	$0.657146\pi$
$644$		0			0
$645$	−9.60250			−0.378098
$646$		0			0
$647$			29.5150i			1.16036i	0.814490	+	0.580178i	$0.197017\pi$
$647$			29.5150i			1.16036i	−0.814490	+	0.580178i	$0.802983\pi$
$648$		0			0
$649$		−	6.09577i		−	0.239280i
$650$		0			0
$651$	7.08016			0.277493
$652$		0			0
$653$	−20.6782			−0.809202			−0.404601	−	0.914493i	$-0.632590\pi$
$653$	−20.6782			−0.809202			−0.404601	+	0.914493i	$0.632590\pi$
$654$		0			0
$655$	1.77034			0.0691731
$656$		0			0
$657$	1.61161			0.0628748
$658$		0			0
$659$	24.6283			0.959382			0.479691	−	0.877438i	$-0.340749\pi$
$659$	24.6283			0.959382			0.479691	+	0.877438i	$0.340749\pi$
$660$		0			0
$661$			15.4778i			0.602016i	0.953622	+	0.301008i	$0.0973231\pi$
$661$			15.4778i			0.602016i	−0.953622	+	0.301008i	$0.902677\pi$
$662$		0			0
$663$	12.7408			0.494812
$664$		0			0
$665$		−	9.56656i		−	0.370975i
$666$		0			0
$667$	0.161111	−	0.0694293i	0.00623825	−	0.00268831i
$668$		0			0
$669$	18.2691			0.706323
$670$		0			0
$671$			10.9683i			0.423427i
$672$		0			0
$673$	9.60160			0.370115			0.185057	−	0.982728i	$-0.440753\pi$
$673$	9.60160			0.370115			0.185057	+	0.982728i	$0.440753\pi$
$674$		0			0
$675$		−	1.00000i		−	0.0384900i
$676$		0			0
$677$		−	45.5851i		−	1.75198i	−0.482331	−	0.875989i	$-0.660210\pi$
$677$		−	45.5851i		−	1.75198i	0.482331	−	0.875989i	$-0.339790\pi$
$678$		0			0
$679$		−	3.11405i		−	0.119506i
$680$		0			0
$681$		−	23.8464i		−	0.913798i
$682$		0			0
$683$			27.2853i			1.04404i	0.852932	+	0.522022i	$0.174822\pi$
$683$			27.2853i			1.04404i	−0.852932	+	0.522022i	$0.825178\pi$
$684$		0			0
$685$	17.7574			0.678477
$686$		0			0
$687$	3.95960			0.151068
$688$		0			0
$689$			13.8922i			0.529251i
$690$		0			0
$691$			19.1940i			0.730173i	0.930974	+	0.365087i	$0.118961\pi$
$691$			19.1940i			0.730173i	−0.930974	+	0.365087i	$0.881039\pi$
$692$		0			0
$693$	−1.85383			−0.0704211
$694$		0			0
$695$	13.6548			0.517957
$696$		0			0
$697$			9.75027i			0.369318i
$698$		0			0
$699$		−	26.6401i		−	1.00762i
$700$		0			0
$701$		−	22.8391i		−	0.862622i	−0.902203	−	0.431311i	$-0.858051\pi$
$701$		−	22.8391i		−	0.862622i	0.902203	−	0.431311i	$-0.141949\pi$
$702$		0			0
$703$		−	60.0673i		−	2.26548i
$704$		0			0
$705$		−	9.16374i		−	0.345126i
$706$		0			0
$707$	−24.1901			−0.909764
$708$		0			0
$709$		−	1.45458i		−	0.0546278i	−0.999627	−	0.0273139i	$-0.991305\pi$
$709$		−	1.45458i		−	0.0546278i	0.999627	−	0.0273139i	$-0.00869536\pi$
$710$		0			0
$711$	−5.33892			−0.200225
$712$		0			0
$713$	8.33866	+	19.3499i	0.312285	+	0.724660i
$714$		0			0
$715$			8.17540i			0.305742i
$716$		0			0
$717$	15.7541			0.588346
$718$		0			0
$719$			29.2388i			1.09042i	0.838298	+	0.545212i	$0.183551\pi$
$719$			29.2388i			1.09042i	−0.838298	+	0.545212i	$0.816449\pi$
$720$		0			0
$721$	4.91101			0.182896
$722$		0			0
$723$	2.31154			0.0859671
$724$		0			0
$725$	0.0365806			0.00135857
$726$		0			0
$727$	−40.7426			−1.51106			−0.755529	−	0.655115i	$-0.772620\pi$
$727$	−40.7426			−1.51106			−0.755529	+	0.655115i	$0.772620\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$			17.2149i			0.636714i
$732$		0			0
$733$		−	25.3692i		−	0.937033i	−0.883455	−	0.468516i	$-0.844789\pi$
$733$		−	25.3692i		−	0.937033i	0.883455	−	0.468516i	$-0.155211\pi$
$734$		0			0
$735$	4.40297			0.162406
$736$		0			0
$737$	−2.68787			−0.0990091
$738$		0			0
$739$		−	8.67928i		−	0.319273i	−0.987176	−	0.159636i	$-0.948968\pi$
$739$		−	8.67928i		−	0.319273i	0.987176	−	0.159636i	$-0.0510321\pi$
$740$		0			0
$741$			42.1886i			1.54984i
$742$		0			0
$743$	18.8741			0.692424			0.346212	−	0.938156i	$-0.387468\pi$
$743$	18.8741			0.692424			0.346212	+	0.938156i	$0.387468\pi$
$744$		0			0
$745$	9.38262			0.343753
$746$		0			0
$747$	−0.722088			−0.0264198
$748$		0			0
$749$	20.3105			0.742128
$750$		0			0
$751$	−11.9864			−0.437390			−0.218695	−	0.975793i	$-0.570180\pi$
$751$	−11.9864			−0.437390			−0.218695	+	0.975793i	$0.570180\pi$
$752$		0			0
$753$		−	21.6173i		−	0.787777i
$754$		0			0
$755$	−13.1263			−0.477714
$756$		0			0
$757$		−	9.99273i		−	0.363192i	−0.983373	−	0.181596i	$-0.941874\pi$
$757$		−	9.99273i		−	0.363192i	0.983373	−	0.181596i	$-0.0581263\pi$
$758$		0			0
$759$	−2.18335	−	5.06647i	−0.0792504	−	0.183901i
$760$		0			0
$761$	−13.8964			−0.503746			−0.251873	−	0.967760i	$-0.581046\pi$
$761$	−13.8964			−0.503746			−0.251873	+	0.967760i	$0.581046\pi$
$762$		0			0
$763$		−	24.2213i		−	0.876871i
$764$		0			0
$765$	−1.79275			−0.0648169
$766$		0			0
$767$			37.6596i			1.35981i
$768$		0			0
$769$		−	8.90960i		−	0.321288i	−0.987012	−	0.160644i	$-0.948643\pi$
$769$		−	8.90960i		−	0.321288i	0.987012	−	0.160644i	$-0.0513572\pi$
$770$		0			0
$771$		−	4.71865i		−	0.169938i
$772$		0			0
$773$		−	11.6166i		−	0.417821i	−0.977935	−	0.208910i	$-0.933008\pi$
$773$		−	11.6166i		−	0.417821i	0.977935	−	0.208910i	$-0.0669917\pi$
$774$		0			0
$775$			4.39344i			0.157817i
$776$		0			0
$777$	−16.3065			−0.584991
$778$		0			0
$779$	−32.2860			−1.15677
$780$		0			0
$781$		−	13.0958i		−	0.468606i
$782$		0			0
$783$		−	0.0365806i		−	0.00130729i
$784$		0			0
$785$	−2.75711			−0.0984054
$786$		0			0
$787$	32.1690			1.14670			0.573350	−	0.819310i	$-0.305643\pi$
$787$	32.1690			1.14670			0.573350	+	0.819310i	$0.305643\pi$
$788$		0			0
$789$		−	4.44240i		−	0.158154i
$790$		0			0
$791$			13.5279i			0.480998i
$792$		0			0
$793$		−	67.7622i		−	2.40631i
$794$		0			0
$795$		−	1.95476i		−	0.0693282i
$796$		0			0
$797$			12.5397i			0.444178i	0.975026	+	0.222089i	$0.0712876\pi$
$797$			12.5397i			0.444178i	−0.975026	+	0.222089i	$0.928712\pi$
$798$		0			0
$799$	−16.4283			−0.581190
$800$		0			0
$801$			2.09024i			0.0738549i
$802$		0			0
$803$	1.85392			0.0654233
$804$		0			0
$805$	−3.05865	−	7.09763i	−0.107803	−	0.250159i
$806$		0			0
$807$			9.10119i			0.320377i
$808$		0			0
$809$	−26.7323			−0.939857			−0.469928	−	0.882705i	$-0.655720\pi$
$809$	−26.7323			−0.939857			−0.469928	+	0.882705i	$0.655720\pi$
$810$		0			0
$811$		−	4.43929i		−	0.155885i	−0.996958	−	0.0779424i	$-0.975165\pi$
$811$		−	4.43929i		−	0.155885i	0.996958	−	0.0779424i	$-0.0248350\pi$
$812$		0			0
$813$	−22.9892			−0.806267
$814$		0			0
$815$	−7.13282			−0.249852
$816$		0			0
$817$	−57.0034			−1.99430
$818$		0			0
$819$	11.4529			0.400198
$820$		0			0
$821$	−49.1591			−1.71567			−0.857833	−	0.513929i	$-0.828189\pi$
$821$	−49.1591			−1.71567			−0.857833	+	0.513929i	$0.828189\pi$
$822$		0			0
$823$			33.2711i			1.15976i	0.814702	+	0.579879i	$0.196900\pi$
$823$			33.2711i			1.15976i	−0.814702	+	0.579879i	$0.803100\pi$
$824$		0			0
$825$		−	1.15035i		−	0.0400501i
$826$		0			0
$827$	−34.1610			−1.18790			−0.593948	−	0.804504i	$-0.702431\pi$
$827$	−34.1610			−1.18790			−0.593948	+	0.804504i	$0.702431\pi$
$828$		0			0
$829$	−26.3351			−0.914656			−0.457328	−	0.889298i	$-0.651194\pi$
$829$	−26.3351			−0.914656			−0.457328	+	0.889298i	$0.651194\pi$
$830$		0			0
$831$		−	18.6686i		−	0.647606i
$832$		0			0
$833$		−	7.89341i		−	0.273490i
$834$		0			0
$835$	−0.264942			−0.00916870
$836$		0			0
$837$	4.39344			0.151859
$838$		0			0
$839$	−21.8376			−0.753917			−0.376959	−	0.926230i	$-0.623030\pi$
$839$	−21.8376			−0.753917			−0.376959	+	0.926230i	$0.623030\pi$
$840$		0			0
$841$	−28.9987			−0.999954
$842$		0			0
$843$	23.4639			0.808139
$844$		0			0
$845$		−	37.5075i		−	1.29030i
$846$		0			0
$847$	15.5943			0.535826
$848$		0			0
$849$		−	13.4760i		−	0.462497i
$850$		0			0
$851$	−19.2049	−	44.5652i	−0.658337	−	1.52768i
$852$		0			0
$853$	1.88065			0.0643921			0.0321961	−	0.999482i	$-0.489750\pi$
$853$	1.88065			0.0643921			0.0321961	+	0.999482i	$0.489750\pi$
$854$		0			0
$855$		−	5.93632i		−	0.203018i
$856$		0			0
$857$	−20.9391			−0.715267			−0.357634	−	0.933862i	$-0.616416\pi$
$857$	−20.9391			−0.715267			−0.357634	+	0.933862i	$0.616416\pi$
$858$		0			0
$859$		−	23.2118i		−	0.791977i	−0.918255	−	0.395989i	$-0.870402\pi$
$859$		−	23.2118i		−	0.791977i	0.918255	−	0.395989i	$-0.129598\pi$
$860$		0			0
$861$			8.76468i			0.298700i
$862$		0			0
$863$		−	6.73403i		−	0.229229i	−0.993410	−	0.114614i	$-0.963437\pi$
$863$		−	6.73403i		−	0.229229i	0.993410	−	0.114614i	$-0.0365633\pi$
$864$		0			0
$865$			9.93924i			0.337944i
$866$		0			0
$867$		−	13.7861i		−	0.468199i
$868$		0			0
$869$	−6.14163			−0.208341
$870$		0			0
$871$	16.6057			0.562662
$872$		0			0
$873$		−	1.93236i		−	0.0654003i
$874$		0			0
$875$		−	1.61153i		−	0.0544797i
$876$		0			0
$877$	−22.5879			−0.762738			−0.381369	−	0.924423i	$-0.624547\pi$
$877$	−22.5879			−0.762738			−0.381369	+	0.924423i	$0.624547\pi$
$878$		0			0
$879$	−26.8108			−0.904305
$880$		0			0
$881$			13.5991i			0.458166i	0.973407	+	0.229083i	$0.0735727\pi$
$881$			13.5991i			0.458166i	−0.973407	+	0.229083i	$0.926427\pi$
$882$		0			0
$883$			10.4786i			0.352634i	0.984333	+	0.176317i	$0.0564183\pi$
$883$			10.4786i			0.352634i	−0.984333	+	0.176317i	$0.943582\pi$
$884$		0			0
$885$		−	5.29905i		−	0.178125i
$886$		0			0
$887$			3.87472i			0.130100i	0.997882	+	0.0650502i	$0.0207207\pi$
$887$			3.87472i			0.130100i	−0.997882	+	0.0650502i	$0.979279\pi$
$888$		0			0
$889$			29.2240i			0.980141i
$890$		0			0
$891$	−1.15035			−0.0385382
$892$		0			0
$893$		−	54.3988i		−	1.82039i
$894$		0			0
$895$	5.00824			0.167407
$896$		0			0
$897$	13.4887	+	31.3006i	0.450374	+	1.04510i
$898$		0			0
$899$			0.160715i			0.00536014i
$900$		0			0
$901$	−3.50439			−0.116748
$902$		0			0
$903$			15.4747i			0.514966i
$904$		0			0
$905$	1.45486			0.0483613
$906$		0			0
$907$	−2.27157			−0.0754264			−0.0377132	−	0.999289i	$-0.512007\pi$
$907$	−2.27157			−0.0754264			−0.0377132	+	0.999289i	$0.512007\pi$
$908$		0			0
$909$	−15.0107			−0.497872
$910$		0			0
$911$	21.8453			0.723766			0.361883	−	0.932224i	$-0.382134\pi$
$911$	21.8453			0.723766			0.361883	+	0.932224i	$0.382134\pi$
$912$		0			0
$913$	−0.830655			−0.0274907
$914$		0			0
$915$			9.53475i			0.315209i
$916$		0			0
$917$		−	2.85296i		−	0.0942132i
$918$		0			0
$919$	30.9530			1.02104			0.510522	−	0.859865i	$-0.329452\pi$
$919$	30.9530			1.02104			0.510522	+	0.859865i	$0.329452\pi$
$920$		0			0
$921$	−1.71461			−0.0564983
$922$		0			0
$923$			80.9060i			2.66305i
$924$		0			0
$925$		−	10.1186i		−	0.332698i
$926$		0			0
$927$	3.04742			0.100090
$928$		0			0
$929$	−35.7703			−1.17359			−0.586793	−	0.809737i	$-0.699610\pi$
$929$	−35.7703			−1.17359			−0.586793	+	0.809737i	$0.699610\pi$
$930$		0			0
$931$	26.1374			0.856619
$932$		0			0
$933$	−1.97438			−0.0646384
$934$		0			0
$935$	−2.06229			−0.0674441
$936$		0			0
$937$			43.2380i			1.41252i	0.707950	+	0.706262i	$0.249620\pi$
$937$			43.2380i			1.41252i	−0.707950	+	0.706262i	$0.750380\pi$
$938$		0			0
$939$	8.62500			0.281466
$940$		0			0
$941$		−	14.6294i		−	0.476906i	−0.971154	−	0.238453i	$-0.923360\pi$
$941$		−	14.6294i		−	0.476906i	0.971154	−	0.238453i	$-0.0766404\pi$
$942$		0			0
$943$	−23.9537	+	10.3226i	−0.780039	+	0.336150i
$944$		0			0
$945$	−1.61153			−0.0524231
$946$		0			0
$947$			37.7617i			1.22709i	0.789660	+	0.613545i	$0.210257\pi$
$947$			37.7617i			1.22709i	−0.789660	+	0.613545i	$0.789743\pi$
$948$		0			0
$949$	−11.4535			−0.371796
$950$		0			0
$951$			22.2686i			0.722110i
$952$		0			0
$953$		−	8.72070i		−	0.282491i	−0.989975	−	0.141246i	$-0.954889\pi$
$953$		−	8.72070i		−	0.282491i	0.989975	−	0.141246i	$-0.0451107\pi$
$954$		0			0
$955$		−	21.9857i		−	0.711442i
$956$		0			0
$957$		−	0.0420806i		−	0.00136027i
$958$		0			0
$959$		−	28.6167i		−	0.924080i
$960$		0			0
$961$	11.6977			0.377345
$962$		0			0
$963$	12.6032			0.406133
$964$		0			0
$965$		−	13.5269i		−	0.435446i
$966$		0			0
$967$			42.8147i			1.37683i	0.725317	+	0.688415i	$0.241693\pi$
$967$			42.8147i			1.37683i	−0.725317	+	0.688415i	$0.758307\pi$
$968$		0			0
$969$	−10.6423			−0.341880
$970$		0			0
$971$	−19.9783			−0.641133			−0.320566	−	0.947226i	$-0.603873\pi$
$971$	−19.9783			−0.641133			−0.320566	+	0.947226i	$0.603873\pi$
$972$		0			0
$973$		−	22.0052i		−	0.705454i
$974$		0			0
$975$			7.10687i			0.227602i
$976$		0			0
$977$			27.7007i			0.886224i	0.896466	+	0.443112i	$0.146126\pi$
$977$			27.7007i			0.886224i	−0.896466	+	0.443112i	$0.853874\pi$
$978$		0			0
$979$			2.40451i			0.0768484i
$980$		0			0
$981$		−	15.0300i		−	0.479871i
$982$		0			0
$983$	−45.6168			−1.45495			−0.727475	−	0.686134i	$-0.759306\pi$
$983$	−45.6168			−1.45495			−0.727475	+	0.686134i	$0.759306\pi$
$984$		0			0
$985$		−	14.6237i		−	0.465950i
$986$		0			0
$987$	−14.7676			−0.470059
$988$		0			0
$989$	−42.2921	+	18.2254i	−1.34481	+	0.579533i
$990$		0			0
$991$			1.50770i			0.0478936i	0.999713	+	0.0239468i	$0.00762323\pi$
$991$			1.50770i			0.0478936i	−0.999713	+	0.0239468i	$0.992377\pi$
$992$		0			0
$993$	−13.5388			−0.429642
$994$		0			0
$995$			3.24225i			0.102786i
$996$		0			0
$997$	−42.1215			−1.33400			−0.667000	−	0.745057i	$-0.732422\pi$
$997$	−42.1215			−1.33400			−0.667000	+	0.745057i	$0.732422\pi$
$998$		0			0
$999$	−10.1186			−0.320139

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5520.2.be.b.1471.2	yes	16
4.3	odd	2		5520.2.be.a.1471.15	yes	16
23.22	odd	2		5520.2.be.a.1471.7	✓	16
92.91	even	2	inner	5520.2.be.b.1471.10	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5520.2.be.a.1471.7	✓	16	23.22	odd	2
5520.2.be.a.1471.15	yes	16	4.3	odd	2
5520.2.be.b.1471.2	yes	16	1.1	even	1	trivial
5520.2.be.b.1471.10	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)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} -1.00000i q^{5} -1.61153 q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} -1.00000i q^{5} -1.61153 q^{7} -1.00000 q^{9} -1.15035 q^{11} +7.10687 q^{13} -1.00000 q^{15} +1.79275i q^{17} -5.93632 q^{19} +1.61153i q^{21} +(-4.40428 + 1.89798i) q^{23} -1.00000 q^{25} +1.00000i q^{27} -0.0365806 q^{29} -4.39344i q^{31} +1.15035i q^{33} +1.61153i q^{35} +10.1186i q^{37} -7.10687i q^{39} +5.43873 q^{41} +9.60250 q^{43} +1.00000i q^{45} +9.16374i q^{47} -4.40297 q^{49} +1.79275 q^{51} +1.95476i q^{53} +1.15035i q^{55} +5.93632i q^{57} +5.29905i q^{59} -9.53475i q^{61} +1.61153 q^{63} -7.10687i q^{65} +2.33657 q^{67} +(1.89798 + 4.40428i) q^{69} +11.3842i q^{71} -1.61161 q^{73} +1.00000i q^{75} +1.85383 q^{77} +5.33892 q^{79} +1.00000 q^{81} +0.722088 q^{83} +1.79275 q^{85} +0.0365806i q^{87} -2.09024i q^{89} -11.4529 q^{91} -4.39344 q^{93} +5.93632i q^{95} +1.93236i q^{97} +1.15035 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 8 q^{7} - 16 q^{9}+O(q^{10}) \) 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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