LMFDB - Embedded newform 5520.2.be.b.1471.3

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			1471.3
Root			$1.30491 + 1.30491i$ of defining polynomial
Character	$\chi$	$=$	5520.1471
Dual form			5520.2.be.b.1471.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} -1.00000i q^{5} -0.482745 q^{7} -1.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} -1.00000i q^{5} -0.482745 q^{7} -1.00000 q^{9} -1.10189 q^{11} +2.25517 q^{13} -1.00000 q^{15} -6.03534i q^{17} +1.48254 q^{19} +0.482745i q^{21} +(4.55260 - 1.50793i) q^{23} -1.00000 q^{25} +1.00000i q^{27} -7.68679 q^{29} +5.12465i q^{31} +1.10189i q^{33} +0.482745i q^{35} -5.83765i q^{37} -2.25517i q^{39} +1.27767 q^{41} +8.46629 q^{43} +1.00000i q^{45} -11.2632i q^{47} -6.76696 q^{49} -6.03534 q^{51} +8.81467i q^{53} +1.10189i q^{55} -1.48254i q^{57} -8.08002i q^{59} +0.0392191i q^{61} +0.482745 q^{63} -2.25517i q^{65} +8.39370 q^{67} +(-1.50793 - 4.55260i) q^{69} +1.25598i q^{71} -3.47480 q^{73} +1.00000i q^{75} +0.531931 q^{77} +3.61831 q^{79} +1.00000 q^{81} -9.12164 q^{83} -6.03534 q^{85} +7.68679i q^{87} +6.85106i q^{89} -1.08867 q^{91} +5.12465 q^{93} -1.48254i q^{95} +0.330286i q^{97} +1.10189 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$	−0.482745			−0.182460			−0.0912302	−	0.995830i	$-0.529080\pi$
$7$	−0.482745			−0.182460			−0.0912302	+	0.995830i	$0.529080\pi$
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	−1.10189			−0.332232			−0.166116	−	0.986106i	$-0.553123\pi$
$11$	−1.10189			−0.332232			−0.166116	+	0.986106i	$0.553123\pi$
$12$		0			0
$13$	2.25517			0.625471			0.312735	−	0.949840i	$-0.398755\pi$
$13$	2.25517			0.625471			0.312735	+	0.949840i	$0.398755\pi$
$14$		0			0
$15$	−1.00000			−0.258199
$16$		0			0
$17$		−	6.03534i		−	1.46379i	−0.681420	−	0.731893i	$-0.738637\pi$
$17$		−	6.03534i		−	1.46379i	0.681420	−	0.731893i	$-0.261363\pi$
$18$		0			0
$19$	1.48254			0.340118			0.170059	−	0.985434i	$-0.445604\pi$
$19$	1.48254			0.340118			0.170059	+	0.985434i	$0.445604\pi$
$20$		0			0
$21$			0.482745i			0.105344i
$22$		0			0
$23$	4.55260	−	1.50793i	0.949283	−	0.314424i
$23$	4.55260	−	1.50793i	0.949283	−	0.314424i
$24$		0			0
$25$	−1.00000			−0.200000
$26$		0			0
$27$			1.00000i			0.192450i
$28$		0			0
$29$	−7.68679			−1.42740			−0.713700	−	0.700451i	$-0.752982\pi$
$29$	−7.68679			−1.42740			−0.713700	+	0.700451i	$0.752982\pi$
$30$		0			0
$31$			5.12465i			0.920414i	0.887812	+	0.460207i	$0.152225\pi$
$31$			5.12465i			0.920414i	−0.887812	+	0.460207i	$0.847775\pi$
$32$		0			0
$33$			1.10189i			0.191814i
$34$		0			0
$35$			0.482745i			0.0815987i
$36$		0			0
$37$		−	5.83765i		−	0.959705i	−0.877349	−	0.479852i	$-0.840690\pi$
$37$		−	5.83765i		−	0.959705i	0.877349	−	0.479852i	$-0.159310\pi$
$38$		0			0
$39$		−	2.25517i		−	0.361116i
$40$		0			0
$41$	1.27767			0.199538			0.0997689	−	0.995011i	$-0.468190\pi$
$41$	1.27767			0.199538			0.0997689	+	0.995011i	$0.468190\pi$
$42$		0			0
$43$	8.46629			1.29110			0.645548	−	0.763720i	$-0.276629\pi$
$43$	8.46629			1.29110			0.645548	+	0.763720i	$0.276629\pi$
$44$		0			0
$45$			1.00000i			0.149071i
$46$		0			0
$47$		−	11.2632i		−	1.64290i	−0.570278	−	0.821452i	$-0.693164\pi$
$47$		−	11.2632i		−	1.64290i	0.570278	−	0.821452i	$-0.306836\pi$
$48$		0			0
$49$	−6.76696			−0.966708
$50$		0			0
$51$	−6.03534			−0.845117
$52$		0			0
$53$			8.81467i			1.21079i	0.795926	+	0.605394i	$0.206985\pi$
$53$			8.81467i			1.21079i	−0.795926	+	0.605394i	$0.793015\pi$
$54$		0			0
$55$			1.10189i			0.148579i
$56$		0			0
$57$		−	1.48254i		−	0.196367i
$58$		0			0
$59$		−	8.08002i		−	1.05193i	−0.850507	−	0.525964i	$-0.823705\pi$
$59$		−	8.08002i		−	1.05193i	0.850507	−	0.525964i	$-0.176295\pi$
$60$		0			0
$61$			0.0392191i			0.00502149i	0.999997	+	0.00251075i	$0.000799197\pi$
$61$			0.0392191i			0.00502149i	−0.999997	+	0.00251075i	$0.999201\pi$
$62$		0			0
$63$	0.482745			0.0608201
$64$		0			0
$65$		−	2.25517i		−	0.279719i
$66$		0			0
$67$	8.39370			1.02545			0.512727	−	0.858552i	$-0.328635\pi$
$67$	8.39370			1.02545			0.512727	+	0.858552i	$0.328635\pi$
$68$		0			0
$69$	−1.50793	−	4.55260i	−0.181533	−	0.548069i
$70$		0			0
$71$			1.25598i			0.149058i	0.997219	+	0.0745289i	$0.0237453\pi$
$71$			1.25598i			0.149058i	−0.997219	+	0.0745289i	$0.976255\pi$
$72$		0			0
$73$	−3.47480			−0.406694			−0.203347	−	0.979107i	$-0.565182\pi$
$73$	−3.47480			−0.406694			−0.203347	+	0.979107i	$0.565182\pi$
$74$		0			0
$75$			1.00000i			0.115470i
$76$		0			0
$77$	0.531931			0.0606192
$78$		0			0
$79$	3.61831			0.407092			0.203546	−	0.979065i	$-0.434753\pi$
$79$	3.61831			0.407092			0.203546	+	0.979065i	$0.434753\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$	−9.12164			−1.00123			−0.500615	−	0.865670i	$-0.666893\pi$
$83$	−9.12164			−1.00123			−0.500615	+	0.865670i	$0.666893\pi$
$84$		0			0
$85$	−6.03534			−0.654625
$86$		0			0
$87$			7.68679i			0.824110i
$88$		0			0
$89$			6.85106i			0.726211i	0.931748	+	0.363105i	$0.118283\pi$
$89$			6.85106i			0.726211i	−0.931748	+	0.363105i	$0.881717\pi$
$90$		0			0
$91$	−1.08867			−0.114124
$92$		0			0
$93$	5.12465			0.531401
$94$		0			0
$95$		−	1.48254i		−	0.152105i
$96$		0			0
$97$			0.330286i			0.0335355i	0.999859	+	0.0167677i	$0.00533759\pi$
$97$			0.330286i			0.0335355i	−0.999859	+	0.0167677i	$0.994662\pi$
$98$		0			0
$99$	1.10189			0.110744
$100$		0			0
$101$	−3.89119			−0.387188			−0.193594	−	0.981082i	$-0.562014\pi$
$101$	−3.89119			−0.387188			−0.193594	+	0.981082i	$0.562014\pi$
$102$		0			0
$103$	−11.2588			−1.10937			−0.554683	−	0.832062i	$-0.687160\pi$
$103$	−11.2588			−1.10937			−0.554683	+	0.832062i	$0.687160\pi$
$104$		0			0
$105$	0.482745			0.0471111
$106$		0			0
$107$	0.801493			0.0774833			0.0387416	−	0.999249i	$-0.487665\pi$
$107$	0.801493			0.0774833			0.0387416	+	0.999249i	$0.487665\pi$
$108$		0			0
$109$		−	1.25885i		−	0.120576i	−0.998181	−	0.0602880i	$-0.980798\pi$
$109$		−	1.25885i		−	0.120576i	0.998181	−	0.0602880i	$-0.0192019\pi$
$110$		0			0
$111$	−5.83765			−0.554086
$112$		0			0
$113$		−	8.70667i		−	0.819054i	−0.912298	−	0.409527i	$-0.865694\pi$
$113$		−	8.70667i		−	0.819054i	0.912298	−	0.409527i	$-0.134306\pi$
$114$		0			0
$115$	−1.50793	−	4.55260i	−0.140615	−	0.424532i
$116$		0			0
$117$	−2.25517			−0.208490
$118$		0			0
$119$			2.91353i			0.267083i
$120$		0			0
$121$	−9.78584			−0.889622
$122$		0			0
$123$		−	1.27767i		−	0.115203i
$124$		0			0
$125$			1.00000i			0.0894427i
$126$		0			0
$127$			5.51836i			0.489676i	0.969564	+	0.244838i	$0.0787347\pi$
$127$			5.51836i			0.489676i	−0.969564	+	0.244838i	$0.921265\pi$
$128$		0			0
$129$		−	8.46629i		−	0.745415i
$130$		0			0
$131$		−	14.5197i		−	1.26859i	−0.773089	−	0.634297i	$-0.781289\pi$
$131$		−	14.5197i		−	1.26859i	0.773089	−	0.634297i	$-0.218711\pi$
$132$		0			0
$133$	−0.715687			−0.0620580
$134$		0			0
$135$	1.00000			0.0860663
$136$		0			0
$137$		−	4.50448i		−	0.384844i	−0.981312	−	0.192422i	$-0.938366\pi$
$137$		−	4.50448i		−	0.384844i	0.981312	−	0.192422i	$-0.0616342\pi$
$138$		0			0
$139$		−	2.13377i		−	0.180984i	−0.995897	−	0.0904921i	$-0.971156\pi$
$139$		−	2.13377i		−	0.180984i	0.995897	−	0.0904921i	$-0.0288440\pi$
$140$		0			0
$141$	−11.2632			−0.948531
$142$		0			0
$143$	−2.48494			−0.207801
$144$		0			0
$145$			7.68679i			0.638353i
$146$		0			0
$147$			6.76696i			0.558129i
$148$		0			0
$149$		−	11.9315i		−	0.977468i	−0.872433	−	0.488734i	$-0.837459\pi$
$149$		−	11.9315i		−	0.977468i	0.872433	−	0.488734i	$-0.162541\pi$
$150$		0			0
$151$		−	9.63555i		−	0.784130i	−0.919937	−	0.392065i	$-0.871761\pi$
$151$		−	9.63555i		−	0.784130i	0.919937	−	0.392065i	$-0.128239\pi$
$152$		0			0
$153$			6.03534i			0.487929i
$154$		0			0
$155$	5.12465			0.411622
$156$		0			0
$157$			0.935197i			0.0746368i	0.999303	+	0.0373184i	$0.0118816\pi$
$157$			0.935197i			0.0746368i	−0.999303	+	0.0373184i	$0.988118\pi$
$158$		0			0
$159$	8.81467			0.699049
$160$		0			0
$161$	−2.19774	+	0.727943i	−0.173206	+	0.0573700i
$162$		0			0
$163$		−	13.8165i		−	1.08220i	−0.840960	−	0.541098i	$-0.818009\pi$
$163$		−	13.8165i		−	1.08220i	0.840960	−	0.541098i	$-0.181991\pi$
$164$		0			0
$165$	1.10189			0.0857820
$166$		0			0
$167$		−	6.69307i		−	0.517925i	−0.965887	−	0.258963i	$-0.916619\pi$
$167$		−	6.69307i		−	0.517925i	0.965887	−	0.258963i	$-0.0833807\pi$
$168$		0			0
$169$	−7.91423			−0.608787
$170$		0			0
$171$	−1.48254			−0.113373
$172$		0			0
$173$	−24.6878			−1.87698			−0.938490	−	0.345305i	$-0.887775\pi$
$173$	−24.6878			−1.87698			−0.938490	+	0.345305i	$0.887775\pi$
$174$		0			0
$175$	0.482745			0.0364921
$176$		0			0
$177$	−8.08002			−0.607331
$178$		0			0
$179$			4.65679i			0.348065i	0.984740	+	0.174032i	$0.0556797\pi$
$179$			4.65679i			0.348065i	−0.984740	+	0.174032i	$0.944320\pi$
$180$		0			0
$181$		−	10.4913i		−	0.779809i	−0.920855	−	0.389905i	$-0.872508\pi$
$181$		−	10.4913i		−	0.779809i	0.920855	−	0.389905i	$-0.127492\pi$
$182$		0			0
$183$	0.0392191			0.00289916
$184$		0			0
$185$	−5.83765			−0.429193
$186$		0			0
$187$			6.65028i			0.486317i
$188$		0			0
$189$		−	0.482745i		−	0.0351145i
$190$		0			0
$191$	10.1702			0.735886			0.367943	−	0.929848i	$-0.380062\pi$
$191$	10.1702			0.735886			0.367943	+	0.929848i	$0.380062\pi$
$192$		0			0
$193$	−13.3593			−0.961626			−0.480813	−	0.876823i	$-0.659658\pi$
$193$	−13.3593			−0.961626			−0.480813	+	0.876823i	$0.659658\pi$
$194$		0			0
$195$	−2.25517			−0.161496
$196$		0			0
$197$	−2.65615			−0.189243			−0.0946216	−	0.995513i	$-0.530164\pi$
$197$	−2.65615			−0.189243			−0.0946216	+	0.995513i	$0.530164\pi$
$198$		0			0
$199$	−8.43038			−0.597614			−0.298807	−	0.954314i	$-0.596589\pi$
$199$	−8.43038			−0.597614			−0.298807	+	0.954314i	$0.596589\pi$
$200$		0			0
$201$		−	8.39370i		−	0.592046i
$202$		0			0
$203$	3.71075			0.260444
$204$		0			0
$205$		−	1.27767i		−	0.0892360i
$206$		0			0
$207$	−4.55260	+	1.50793i	−0.316428	+	0.104808i
$208$		0			0
$209$	−1.63359			−0.112998
$210$		0			0
$211$			0.771524i			0.0531139i	0.999647	+	0.0265570i	$0.00845434\pi$
$211$			0.771524i			0.0531139i	−0.999647	+	0.0265570i	$0.991546\pi$
$212$		0			0
$213$	1.25598			0.0860586
$214$		0			0
$215$		−	8.46629i		−	0.577396i
$216$		0			0
$217$		−	2.47390i		−	0.167939i
$218$		0			0
$219$			3.47480i			0.234805i
$220$		0			0
$221$		−	13.6107i		−	0.915555i
$222$		0			0
$223$			19.0727i			1.27720i	0.769538	+	0.638601i	$0.220487\pi$
$223$			19.0727i			1.27720i	−0.769538	+	0.638601i	$0.779513\pi$
$224$		0			0
$225$	1.00000			0.0666667
$226$		0			0
$227$	6.95607			0.461691			0.230845	−	0.972990i	$-0.425851\pi$
$227$	6.95607			0.461691			0.230845	+	0.972990i	$0.425851\pi$
$228$		0			0
$229$		−	1.27163i		−	0.0840316i	−0.999117	−	0.0420158i	$-0.986622\pi$
$229$		−	1.27163i		−	0.0840316i	0.999117	−	0.0420158i	$-0.0133780\pi$
$230$		0			0
$231$		−	0.531931i		−	0.0349985i
$232$		0			0
$233$	−9.90098			−0.648635			−0.324317	−	0.945948i	$-0.605135\pi$
$233$	−9.90098			−0.648635			−0.324317	+	0.945948i	$0.605135\pi$
$234$		0			0
$235$	−11.2632			−0.734729
$236$		0			0
$237$		−	3.61831i		−	0.235035i
$238$		0			0
$239$			10.3420i			0.668968i	0.942401	+	0.334484i	$0.108562\pi$
$239$			10.3420i			0.668968i	−0.942401	+	0.334484i	$0.891438\pi$
$240$		0			0
$241$		−	10.9888i		−	0.707852i	−0.935273	−	0.353926i	$-0.884847\pi$
$241$		−	10.9888i		−	0.707852i	0.935273	−	0.353926i	$-0.115153\pi$
$242$		0			0
$243$		−	1.00000i		−	0.0641500i
$244$		0			0
$245$			6.76696i			0.432325i
$246$		0			0
$247$	3.34337			0.212734
$248$		0			0
$249$			9.12164i			0.578061i
$250$		0			0
$251$	−8.22629			−0.519239			−0.259619	−	0.965711i	$-0.583597\pi$
$251$	−8.22629			−0.519239			−0.259619	+	0.965711i	$0.583597\pi$
$252$		0			0
$253$	−5.01646	+	1.66157i	−0.315382	+	0.104462i
$254$		0			0
$255$			6.03534i			0.377948i
$256$		0			0
$257$	−30.1958			−1.88357			−0.941783	−	0.336223i	$-0.890851\pi$
$257$	−30.1958			−1.88357			−0.941783	+	0.336223i	$0.890851\pi$
$258$		0			0
$259$			2.81810i			0.175108i
$260$		0			0
$261$	7.68679			0.475800
$262$		0			0
$263$	19.7029			1.21493			0.607466	−	0.794346i	$-0.292186\pi$
$263$	19.7029			1.21493			0.607466	+	0.794346i	$0.292186\pi$
$264$		0			0
$265$	8.81467			0.541481
$266$		0			0
$267$	6.85106			0.419278
$268$		0			0
$269$	12.1993			0.743805			0.371902	−	0.928272i	$-0.378706\pi$
$269$	12.1993			0.743805			0.371902	+	0.928272i	$0.378706\pi$
$270$		0			0
$271$			17.7526i			1.07840i	0.842179	+	0.539198i	$0.181273\pi$
$271$			17.7526i			1.07840i	−0.842179	+	0.539198i	$0.818727\pi$
$272$		0			0
$273$			1.08867i			0.0658893i
$274$		0			0
$275$	1.10189			0.0664465
$276$		0			0
$277$	−17.0510			−1.02449			−0.512246	−	0.858839i	$-0.671187\pi$
$277$	−17.0510			−1.02449			−0.512246	+	0.858839i	$0.671187\pi$
$278$		0			0
$279$		−	5.12465i		−	0.306805i
$280$		0			0
$281$			19.4880i			1.16256i	0.813705	+	0.581278i	$0.197447\pi$
$281$			19.4880i			1.16256i	−0.813705	+	0.581278i	$0.802553\pi$
$282$		0			0
$283$	−4.52826			−0.269177			−0.134589	−	0.990902i	$-0.542971\pi$
$283$	−4.52826			−0.269177			−0.134589	+	0.990902i	$0.542971\pi$
$284$		0			0
$285$	−1.48254			−0.0878180
$286$		0			0
$287$	−0.616786			−0.0364077
$288$		0			0
$289$	−19.4254			−1.14267
$290$		0			0
$291$	0.330286			0.0193617
$292$		0			0
$293$			14.3433i			0.837942i	0.908000	+	0.418971i	$0.137609\pi$
$293$			14.3433i			0.837942i	−0.908000	+	0.418971i	$0.862391\pi$
$294$		0			0
$295$	−8.08002			−0.470437
$296$		0			0
$297$		−	1.10189i		−	0.0639381i
$298$		0			0
$299$	10.2669	−	3.40062i	0.593748	−	0.196663i
$300$		0			0
$301$	−4.08705			−0.235574
$302$		0			0
$303$			3.89119i			0.223543i
$304$		0			0
$305$	0.0392191			0.00224568
$306$		0			0
$307$			20.1488i			1.14995i	0.818170	+	0.574976i	$0.194989\pi$
$307$			20.1488i			1.14995i	−0.818170	+	0.574976i	$0.805011\pi$
$308$		0			0
$309$			11.2588i			0.640493i
$310$		0			0
$311$			7.50388i			0.425506i	0.977106	+	0.212753i	$0.0682430\pi$
$311$			7.50388i			0.425506i	−0.977106	+	0.212753i	$0.931757\pi$
$312$		0			0
$313$			27.7341i			1.56763i	0.620997	+	0.783813i	$0.286728\pi$
$313$			27.7341i			1.56763i	−0.620997	+	0.783813i	$0.713272\pi$
$314$		0			0
$315$		−	0.482745i		−	0.0271996i
$316$		0			0
$317$	23.7305			1.33284			0.666419	−	0.745577i	$-0.267826\pi$
$317$	23.7305			1.33284			0.666419	+	0.745577i	$0.267826\pi$
$318$		0			0
$319$	8.46999			0.474228
$320$		0			0
$321$		−	0.801493i		−	0.0447350i
$322$		0			0
$323$		−	8.94763i		−	0.497859i
$324$		0			0
$325$	−2.25517			−0.125094
$326$		0			0
$327$	−1.25885			−0.0696146
$328$		0			0
$329$			5.43724i			0.299765i
$330$		0			0
$331$			14.4675i			0.795205i	0.917558	+	0.397602i	$0.130158\pi$
$331$			14.4675i			0.795205i	−0.917558	+	0.397602i	$0.869842\pi$
$332$		0			0
$333$			5.83765i			0.319902i
$334$		0			0
$335$		−	8.39370i		−	0.458597i
$336$		0			0
$337$		−	2.09258i		−	0.113990i	−0.998374	−	0.0569950i	$-0.981848\pi$
$337$		−	2.09258i		−	0.113990i	0.998374	−	0.0569950i	$-0.0181519\pi$
$338$		0			0
$339$	−8.70667			−0.472881
$340$		0			0
$341$		−	5.64680i		−	0.305791i
$342$		0			0
$343$	6.64593			0.358846
$344$		0			0
$345$	−4.55260	+	1.50793i	−0.245104	+	0.0811840i
$346$		0			0
$347$		−	35.3150i		−	1.89581i	−0.318552	−	0.947905i	$-0.603196\pi$
$347$		−	35.3150i		−	1.89581i	0.318552	−	0.947905i	$-0.396804\pi$
$348$		0			0
$349$	25.4994			1.36495			0.682477	−	0.730907i	$-0.260903\pi$
$349$	25.4994			1.36495			0.682477	+	0.730907i	$0.260903\pi$
$350$		0			0
$351$			2.25517i			0.120372i
$352$		0			0
$353$	−22.8722			−1.21737			−0.608683	−	0.793414i	$-0.708302\pi$
$353$	−22.8722			−1.21737			−0.608683	+	0.793414i	$0.708302\pi$
$354$		0			0
$355$	1.25598			0.0666607
$356$		0			0
$357$	2.91353			0.154200
$358$		0			0
$359$	−29.5592			−1.56008			−0.780038	−	0.625732i	$-0.784800\pi$
$359$	−29.5592			−1.56008			−0.780038	+	0.625732i	$0.784800\pi$
$360$		0			0
$361$	−16.8021			−0.884320
$362$		0			0
$363$			9.78584i			0.513623i
$364$		0			0
$365$			3.47480i			0.181879i
$366$		0			0
$367$	13.9579			0.728598			0.364299	−	0.931282i	$-0.381309\pi$
$367$	13.9579			0.728598			0.364299	+	0.931282i	$0.381309\pi$
$368$		0			0
$369$	−1.27767			−0.0665126
$370$		0			0
$371$		−	4.25524i		−	0.220921i
$372$		0			0
$373$		−	18.8723i		−	0.977170i	−0.872516	−	0.488585i	$-0.837513\pi$
$373$		−	18.8723i		−	0.977170i	0.872516	−	0.488585i	$-0.162487\pi$
$374$		0			0
$375$	1.00000			0.0516398
$376$		0			0
$377$	−17.3350			−0.892797
$378$		0			0
$379$	−15.7908			−0.811121			−0.405560	−	0.914068i	$-0.632924\pi$
$379$	−15.7908			−0.811121			−0.405560	+	0.914068i	$0.632924\pi$
$380$		0			0
$381$	5.51836			0.282714
$382$		0			0
$383$	−12.4419			−0.635749			−0.317875	−	0.948133i	$-0.602969\pi$
$383$	−12.4419			−0.635749			−0.317875	+	0.948133i	$0.602969\pi$
$384$		0			0
$385$		−	0.531931i		−	0.0271097i
$386$		0			0
$387$	−8.46629			−0.430365
$388$		0			0
$389$		−	13.3784i		−	0.678312i	−0.940730	−	0.339156i	$-0.889858\pi$
$389$		−	13.3784i		−	0.678312i	0.940730	−	0.339156i	$-0.110142\pi$
$390$		0			0
$391$	−9.10085	−	27.4765i	−0.460250	−	1.38955i
$392$		0			0
$393$	−14.5197			−0.732424
$394$		0			0
$395$		−	3.61831i		−	0.182057i
$396$		0			0
$397$	22.8329			1.14595			0.572976	−	0.819572i	$-0.305789\pi$
$397$	22.8329			1.14595			0.572976	+	0.819572i	$0.305789\pi$
$398$		0			0
$399$			0.715687i			0.0358292i
$400$		0			0
$401$		−	24.0495i		−	1.20098i	−0.799634	−	0.600488i	$-0.794973\pi$
$401$		−	24.0495i		−	1.20098i	0.799634	−	0.600488i	$-0.205027\pi$
$402$		0			0
$403$			11.5569i			0.575692i
$404$		0			0
$405$		−	1.00000i		−	0.0496904i
$406$		0			0
$407$			6.43245i			0.318845i
$408$		0			0
$409$	−19.5572			−0.967042			−0.483521	−	0.875333i	$-0.660642\pi$
$409$	−19.5572			−0.967042			−0.483521	+	0.875333i	$0.660642\pi$
$410$		0			0
$411$	−4.50448			−0.222190
$412$		0			0
$413$			3.90059i			0.191935i
$414$		0			0
$415$			9.12164i			0.447764i
$416$		0			0
$417$	−2.13377			−0.104491
$418$		0			0
$419$	26.0411			1.27219			0.636095	−	0.771611i	$-0.280549\pi$
$419$	26.0411			1.27219			0.636095	+	0.771611i	$0.280549\pi$
$420$		0			0
$421$		−	37.9967i		−	1.85185i	−0.377711	−	0.925924i	$-0.623289\pi$
$421$		−	37.9967i		−	1.85185i	0.377711	−	0.925924i	$-0.376711\pi$
$422$		0			0
$423$			11.2632i			0.547635i
$424$		0			0
$425$			6.03534i			0.292757i
$426$		0			0
$427$		−	0.0189328i		−	0.000916224i
$428$		0			0
$429$			2.48494i			0.119974i
$430$		0			0
$431$	−41.2449			−1.98670			−0.993348	−	0.115152i	$-0.963265\pi$
$431$	−41.2449			−1.98670			−0.993348	+	0.115152i	$0.963265\pi$
$432$		0			0
$433$			2.54638i			0.122371i	0.998126	+	0.0611856i	$0.0194882\pi$
$433$			2.54638i			0.122371i	−0.998126	+	0.0611856i	$0.980512\pi$
$434$		0			0
$435$	7.68679			0.368553
$436$		0			0
$437$	6.74940	−	2.23556i	0.322868	−	0.106941i
$438$		0			0
$439$		−	9.22800i		−	0.440428i	−0.975452	−	0.220214i	$-0.929324\pi$
$439$		−	9.22800i		−	0.440428i	0.975452	−	0.220214i	$-0.0706756\pi$
$440$		0			0
$441$	6.76696			0.322236
$442$		0			0
$443$		−	10.8713i		−	0.516511i	−0.966077	−	0.258256i	$-0.916852\pi$
$443$		−	10.8713i		−	0.516511i	0.966077	−	0.258256i	$-0.0831477\pi$
$444$		0			0
$445$	6.85106			0.324771
$446$		0			0
$447$	−11.9315			−0.564341
$448$		0			0
$449$	−12.3548			−0.583061			−0.291530	−	0.956562i	$-0.594164\pi$
$449$	−12.3548			−0.583061			−0.291530	+	0.956562i	$0.594164\pi$
$450$		0			0
$451$	−1.40785			−0.0662929
$452$		0			0
$453$	−9.63555			−0.452718
$454$		0			0
$455$			1.08867i			0.0510376i
$456$		0			0
$457$		−	36.0224i		−	1.68506i	−0.538651	−	0.842529i	$-0.681066\pi$
$457$		−	36.0224i		−	1.68506i	0.538651	−	0.842529i	$-0.318934\pi$
$458$		0			0
$459$	6.03534			0.281706
$460$		0			0
$461$	−1.41328			−0.0658229			−0.0329115	−	0.999458i	$-0.510478\pi$
$461$	−1.41328			−0.0658229			−0.0329115	+	0.999458i	$0.510478\pi$
$462$		0			0
$463$			22.3322i			1.03787i	0.854815	+	0.518934i	$0.173671\pi$
$463$			22.3322i			1.03787i	−0.854815	+	0.518934i	$0.826329\pi$
$464$		0			0
$465$		−	5.12465i		−	0.237650i
$466$		0			0
$467$	13.6777			0.632930			0.316465	−	0.948604i	$-0.397504\pi$
$467$	13.6777			0.632930			0.316465	+	0.948604i	$0.397504\pi$
$468$		0			0
$469$	−4.05201			−0.187105
$470$		0			0
$471$	0.935197			0.0430916
$472$		0			0
$473$	−9.32891			−0.428944
$474$		0			0
$475$	−1.48254			−0.0680235
$476$		0			0
$477$		−	8.81467i		−	0.403596i
$478$		0			0
$479$	19.9153			0.909954			0.454977	−	0.890503i	$-0.349647\pi$
$479$	19.9153			0.909954			0.454977	+	0.890503i	$0.349647\pi$
$480$		0			0
$481$		−	13.1649i		−	0.600267i
$482$		0			0
$483$	0.727943	+	2.19774i	0.0331226	+	0.100001i
$484$		0			0
$485$	0.330286			0.0149975
$486$		0			0
$487$			36.4307i			1.65083i	0.564524	+	0.825417i	$0.309060\pi$
$487$			36.4307i			1.65083i	−0.564524	+	0.825417i	$0.690940\pi$
$488$		0			0
$489$	−13.8165			−0.624806
$490$		0			0
$491$		−	18.4791i		−	0.833948i	−0.908918	−	0.416974i	$-0.863091\pi$
$491$		−	18.4791i		−	0.833948i	0.908918	−	0.416974i	$-0.136909\pi$
$492$		0			0
$493$			46.3924i			2.08941i
$494$		0			0
$495$		−	1.10189i		−	0.0495263i
$496$		0			0
$497$		−	0.606320i		−	0.0271971i
$498$		0			0
$499$			11.1690i			0.499993i	0.968247	+	0.249997i	$0.0804295\pi$
$499$			11.1690i			0.499993i	−0.968247	+	0.249997i	$0.919571\pi$
$500$		0			0
$501$	−6.69307			−0.299024
$502$		0			0
$503$	6.27100			0.279610			0.139805	−	0.990179i	$-0.455352\pi$
$503$	6.27100			0.279610			0.139805	+	0.990179i	$0.455352\pi$
$504$		0			0
$505$			3.89119i			0.173156i
$506$		0			0
$507$			7.91423i			0.351483i
$508$		0			0
$509$	39.1536			1.73545			0.867726	−	0.497043i	$-0.165581\pi$
$509$	39.1536			1.73545			0.867726	+	0.497043i	$0.165581\pi$
$510$		0			0
$511$	1.67744			0.0742056
$512$		0			0
$513$			1.48254i			0.0654557i
$514$		0			0
$515$			11.2588i			0.496124i
$516$		0			0
$517$			12.4108i			0.545826i
$518$		0			0
$519$			24.6878i			1.08368i
$520$		0			0
$521$		−	37.4123i		−	1.63906i	−0.573035	−	0.819531i	$-0.694234\pi$
$521$		−	37.4123i		−	1.63906i	0.573035	−	0.819531i	$-0.305766\pi$
$522$		0			0
$523$	14.7986			0.647098			0.323549	−	0.946211i	$-0.395124\pi$
$523$	14.7986			0.647098			0.323549	+	0.946211i	$0.395124\pi$
$524$		0			0
$525$		−	0.482745i		−	0.0210687i
$526$		0			0
$527$	30.9290			1.34729
$528$		0			0
$529$	18.4523	−	13.7300i	0.802275	−	0.596955i
$530$		0			0
$531$			8.08002i			0.350643i
$532$		0			0
$533$	2.88135			0.124805
$534$		0			0
$535$		−	0.801493i		−	0.0346516i
$536$		0			0
$537$	4.65679			0.200955
$538$		0			0
$539$	7.45644			0.321172
$540$		0			0
$541$	1.29246			0.0555670			0.0277835	−	0.999614i	$-0.491155\pi$
$541$	1.29246			0.0555670			0.0277835	+	0.999614i	$0.491155\pi$
$542$		0			0
$543$	−10.4913			−0.450223
$544$		0			0
$545$	−1.25885			−0.0539232
$546$		0			0
$547$		−	12.2259i		−	0.522741i	−0.965239	−	0.261370i	$-0.915826\pi$
$547$		−	12.2259i		−	0.522741i	0.965239	−	0.261370i	$-0.0841744\pi$
$548$		0			0
$549$		−	0.0392191i		−	0.00167383i
$550$		0			0
$551$	−11.3960			−0.485484
$552$		0			0
$553$	−1.74672			−0.0742781
$554$		0			0
$555$			5.83765i			0.247795i
$556$		0			0
$557$			17.1020i			0.724637i	0.932054	+	0.362318i	$0.118015\pi$
$557$			17.1020i			0.724637i	−0.932054	+	0.362318i	$0.881985\pi$
$558$		0			0
$559$	19.0929			0.807543
$560$		0			0
$561$	6.65028			0.280775
$562$		0			0
$563$	−23.9801			−1.01064			−0.505321	−	0.862931i	$-0.668626\pi$
$563$	−23.9801			−1.01064			−0.505321	+	0.862931i	$0.668626\pi$
$564$		0			0
$565$	−8.70667			−0.366292
$566$		0			0
$567$	−0.482745			−0.0202734
$568$		0			0
$569$		−	2.83032i		−	0.118653i	−0.998239	−	0.0593265i	$-0.981105\pi$
$569$		−	2.83032i		−	0.118653i	0.998239	−	0.0593265i	$-0.0188953\pi$
$570$		0			0
$571$	7.44436			0.311537			0.155768	−	0.987794i	$-0.450215\pi$
$571$	7.44436			0.311537			0.155768	+	0.987794i	$0.450215\pi$
$572$		0			0
$573$		−	10.1702i		−	0.424864i
$574$		0			0
$575$	−4.55260	+	1.50793i	−0.189857	+	0.0628849i
$576$		0			0
$577$	−10.4658			−0.435695			−0.217848	−	0.975983i	$-0.569904\pi$
$577$	−10.4658			−0.435695			−0.217848	+	0.975983i	$0.569904\pi$
$578$		0			0
$579$			13.3593i			0.555195i
$580$		0			0
$581$	4.40342			0.182685
$582$		0			0
$583$		−	9.71280i		−	0.402263i
$584$		0			0
$585$			2.25517i			0.0932396i
$586$		0			0
$587$		−	5.27783i		−	0.217839i	−0.994051	−	0.108920i	$-0.965261\pi$
$587$		−	5.27783i		−	0.217839i	0.994051	−	0.108920i	$-0.0347391\pi$
$588$		0			0
$589$			7.59749i			0.313049i
$590$		0			0
$591$			2.65615i			0.109260i
$592$		0			0
$593$	18.1506			0.745357			0.372679	−	0.927960i	$-0.378439\pi$
$593$	18.1506			0.745357			0.372679	+	0.927960i	$0.378439\pi$
$594$		0			0
$595$	2.91353			0.119443
$596$		0			0
$597$			8.43038i			0.345033i
$598$		0			0
$599$		−	15.0499i		−	0.614921i	−0.951561	−	0.307460i	$-0.900521\pi$
$599$		−	15.0499i		−	0.614921i	0.951561	−	0.307460i	$-0.0994792\pi$
$600$		0			0
$601$	17.8769			0.729213			0.364607	−	0.931162i	$-0.381203\pi$
$601$	17.8769			0.729213			0.364607	+	0.931162i	$0.381203\pi$
$602$		0			0
$603$	−8.39370			−0.341818
$604$		0			0
$605$			9.78584i			0.397851i
$606$		0			0
$607$		−	16.8849i		−	0.685336i	−0.939457	−	0.342668i	$-0.888669\pi$
$607$		−	16.8849i		−	0.685336i	0.939457	−	0.342668i	$-0.111331\pi$
$608$		0			0
$609$		−	3.71075i		−	0.150367i
$610$		0			0
$611$		−	25.4003i		−	1.02759i
$612$		0			0
$613$			25.0275i			1.01085i	0.862871	+	0.505425i	$0.168664\pi$
$613$			25.0275i			1.01085i	−0.862871	+	0.505425i	$0.831336\pi$
$614$		0			0
$615$	−1.27767			−0.0515204
$616$		0			0
$617$		−	32.3884i		−	1.30391i	−0.758259	−	0.651953i	$-0.773950\pi$
$617$		−	32.3884i		−	1.30391i	0.758259	−	0.651953i	$-0.226050\pi$
$618$		0			0
$619$	−34.1700			−1.37341			−0.686703	−	0.726938i	$-0.740943\pi$
$619$	−34.1700			−1.37341			−0.686703	+	0.726938i	$0.740943\pi$
$620$		0			0
$621$	1.50793	+	4.55260i	0.0605110	+	0.182690i
$622$		0			0
$623$		−	3.30731i		−	0.132505i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$			1.63359i			0.0652394i
$628$		0			0
$629$	−35.2323			−1.40480
$630$		0			0
$631$	−8.12167			−0.323319			−0.161659	−	0.986847i	$-0.551685\pi$
$631$	−8.12167			−0.323319			−0.161659	+	0.986847i	$0.551685\pi$
$632$		0			0
$633$	0.771524			0.0306653
$634$		0			0
$635$	5.51836			0.218990
$636$		0			0
$637$	−15.2606			−0.604648
$638$		0			0
$639$		−	1.25598i		−	0.0496860i
$640$		0			0
$641$		−	32.8333i		−	1.29684i	−0.761285	−	0.648418i	$-0.775431\pi$
$641$		−	32.8333i		−	1.29684i	0.761285	−	0.648418i	$-0.224569\pi$
$642$		0			0
$643$	11.4559			0.451776			0.225888	−	0.974153i	$-0.427472\pi$
$643$	11.4559			0.451776			0.225888	+	0.974153i	$0.427472\pi$
$644$		0			0
$645$	−8.46629			−0.333360
$646$		0			0
$647$		−	11.8499i		−	0.465867i	−0.972493	−	0.232933i	$-0.925168\pi$
$647$		−	11.8499i		−	0.465867i	0.972493	−	0.232933i	$-0.0748324\pi$
$648$		0			0
$649$			8.90329i			0.349485i
$650$		0			0
$651$	−2.47390			−0.0969597
$652$		0			0
$653$	19.4675			0.761821			0.380911	−	0.924612i	$-0.375611\pi$
$653$	19.4675			0.761821			0.380911	+	0.924612i	$0.375611\pi$
$654$		0			0
$655$	−14.5197			−0.567333
$656$		0			0
$657$	3.47480			0.135565
$658$		0			0
$659$	38.7108			1.50796			0.753980	−	0.656897i	$-0.228131\pi$
$659$	38.7108			1.50796			0.753980	+	0.656897i	$0.228131\pi$
$660$		0			0
$661$			23.6248i			0.918900i	0.888204	+	0.459450i	$0.151953\pi$
$661$			23.6248i			0.918900i	−0.888204	+	0.459450i	$0.848047\pi$
$662$		0			0
$663$	−13.6107			−0.528596
$664$		0			0
$665$			0.715687i			0.0277532i
$666$		0			0
$667$	−34.9949	+	11.5911i	−1.35501	+	0.448809i
$668$		0			0
$669$	19.0727			0.737393
$670$		0			0
$671$		−	0.0432152i		−	0.00166830i
$672$		0			0
$673$	−41.6743			−1.60643			−0.803213	−	0.595692i	$-0.796878\pi$
$673$	−41.6743			−1.60643			−0.803213	+	0.595692i	$0.796878\pi$
$674$		0			0
$675$		−	1.00000i		−	0.0384900i
$676$		0			0
$677$		−	2.61727i		−	0.100590i	−0.998734	−	0.0502948i	$-0.983984\pi$
$677$		−	2.61727i		−	0.100590i	0.998734	−	0.0502948i	$-0.0160161\pi$
$678$		0			0
$679$		−	0.159444i		−	0.00611890i
$680$		0			0
$681$		−	6.95607i		−	0.266557i
$682$		0			0
$683$		−	17.4757i		−	0.668688i	−0.942451	−	0.334344i	$-0.891485\pi$
$683$		−	17.4757i		−	0.668688i	0.942451	−	0.334344i	$-0.108515\pi$
$684$		0			0
$685$	−4.50448			−0.172107
$686$		0			0
$687$	−1.27163			−0.0485157
$688$		0			0
$689$			19.8786i			0.757313i
$690$		0			0
$691$			33.9569i			1.29178i	0.763430	+	0.645890i	$0.223514\pi$
$691$			33.9569i			1.29178i	−0.763430	+	0.645890i	$0.776486\pi$
$692$		0			0
$693$	−0.531931			−0.0202064
$694$		0			0
$695$	−2.13377			−0.0809386
$696$		0			0
$697$		−	7.71115i		−	0.292081i
$698$		0			0
$699$			9.90098i			0.374489i
$700$		0			0
$701$		−	17.9294i		−	0.677183i	−0.940933	−	0.338592i	$-0.890049\pi$
$701$		−	17.9294i		−	0.677183i	0.940933	−	0.338592i	$-0.109951\pi$
$702$		0			0
$703$		−	8.65455i		−	0.326412i
$704$		0			0
$705$			11.2632i			0.424196i
$706$		0			0
$707$	1.87845			0.0706464
$708$		0			0
$709$		−	23.2328i		−	0.872526i	−0.899819	−	0.436263i	$-0.856302\pi$
$709$		−	23.2328i		−	0.872526i	0.899819	−	0.436263i	$-0.143698\pi$
$710$		0			0
$711$	−3.61831			−0.135697
$712$		0			0
$713$	7.72759	+	23.3305i	0.289401	+	0.873733i
$714$		0			0
$715$			2.48494i			0.0929317i
$716$		0			0
$717$	10.3420			0.386229
$718$		0			0
$719$		−	34.5036i		−	1.28677i	−0.765544	−	0.643384i	$-0.777530\pi$
$719$		−	34.5036i		−	1.28677i	0.765544	−	0.643384i	$-0.222470\pi$
$720$		0			0
$721$	5.43514			0.202415
$722$		0			0
$723$	−10.9888			−0.408678
$724$		0			0
$725$	7.68679			0.285480
$726$		0			0
$727$	45.7234			1.69579			0.847893	−	0.530168i	$-0.177871\pi$
$727$	45.7234			1.69579			0.847893	+	0.530168i	$0.177871\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$		−	51.0969i		−	1.88989i
$732$		0			0
$733$			4.56648i			0.168667i	0.996438	+	0.0843333i	$0.0268760\pi$
$733$			4.56648i			0.168667i	−0.996438	+	0.0843333i	$0.973124\pi$
$734$		0			0
$735$	6.76696			0.249603
$736$		0			0
$737$	−9.24893			−0.340689
$738$		0			0
$739$			8.71551i			0.320605i	0.987068	+	0.160303i	$0.0512470\pi$
$739$			8.71551i			0.320605i	−0.987068	+	0.160303i	$0.948753\pi$
$740$		0			0
$741$		−	3.34337i		−	0.122822i
$742$		0			0
$743$	39.8622			1.46240			0.731202	−	0.682161i	$-0.238960\pi$
$743$	39.8622			1.46240			0.731202	+	0.682161i	$0.238960\pi$
$744$		0			0
$745$	−11.9315			−0.437137
$746$		0			0
$747$	9.12164			0.333743
$748$		0			0
$749$	−0.386917			−0.0141376
$750$		0			0
$751$	6.71643			0.245086			0.122543	−	0.992463i	$-0.460895\pi$
$751$	6.71643			0.245086			0.122543	+	0.992463i	$0.460895\pi$
$752$		0			0
$753$			8.22629i			0.299783i
$754$		0			0
$755$	−9.63555			−0.350674
$756$		0			0
$757$			39.8926i			1.44992i	0.688790	+	0.724961i	$0.258142\pi$
$757$			39.8926i			1.44992i	−0.688790	+	0.724961i	$0.741858\pi$
$758$		0			0
$759$	1.66157	+	5.01646i	0.0603111	+	0.182086i
$760$		0			0
$761$	2.24308			0.0813116			0.0406558	−	0.999173i	$-0.487055\pi$
$761$	2.24308			0.0813116			0.0406558	+	0.999173i	$0.487055\pi$
$762$		0			0
$763$			0.607703i			0.0220003i
$764$		0			0
$765$	6.03534			0.218208
$766$		0			0
$767$		−	18.2218i		−	0.657950i
$768$		0			0
$769$		−	26.5172i		−	0.956233i	−0.878296	−	0.478117i	$-0.841320\pi$
$769$		−	26.5172i		−	0.956233i	0.878296	−	0.478117i	$-0.158680\pi$
$770$		0			0
$771$			30.1958i			1.08748i
$772$		0			0
$773$			49.8502i			1.79299i	0.443057	+	0.896493i	$0.353894\pi$
$773$			49.8502i			1.79299i	−0.443057	+	0.896493i	$0.646106\pi$
$774$		0			0
$775$		−	5.12465i		−	0.184083i
$776$		0			0
$777$	2.81810			0.101099
$778$		0			0
$779$	1.89419			0.0678663
$780$		0			0
$781$		−	1.38396i		−	0.0495218i
$782$		0			0
$783$		−	7.68679i		−	0.274703i
$784$		0			0
$785$	0.935197			0.0333786
$786$		0			0
$787$	−25.8578			−0.921732			−0.460866	−	0.887470i	$-0.652461\pi$
$787$	−25.8578			−0.921732			−0.460866	+	0.887470i	$0.652461\pi$
$788$		0			0
$789$		−	19.7029i		−	0.701441i
$790$		0			0
$791$			4.20310i			0.149445i
$792$		0			0
$793$			0.0884456i			0.00314080i
$794$		0			0
$795$		−	8.81467i		−	0.312624i
$796$		0			0
$797$		−	5.23771i		−	0.185529i	−0.995688	−	0.0927647i	$-0.970430\pi$
$797$		−	5.23771i		−	0.185529i	0.995688	−	0.0927647i	$-0.0295704\pi$
$798$		0			0
$799$	−67.9772			−2.40486
$800$		0			0
$801$		−	6.85106i		−	0.242070i
$802$		0			0
$803$	3.82884			0.135117
$804$		0			0
$805$	0.727943	+	2.19774i	0.0256566	+	0.0774603i
$806$		0			0
$807$		−	12.1993i		−	0.429436i
$808$		0			0
$809$	−11.3522			−0.399123			−0.199562	−	0.979885i	$-0.563952\pi$
$809$	−11.3522			−0.399123			−0.199562	+	0.979885i	$0.563952\pi$
$810$		0			0
$811$			21.9307i			0.770092i	0.922897	+	0.385046i	$0.125814\pi$
$811$			21.9307i			0.770092i	−0.922897	+	0.385046i	$0.874186\pi$
$812$		0			0
$813$	17.7526			0.622612
$814$		0			0
$815$	−13.8165			−0.483973
$816$		0			0
$817$	12.5516			0.439125
$818$		0			0
$819$	1.08867			0.0380412
$820$		0			0
$821$	18.5030			0.645758			0.322879	−	0.946440i	$-0.395349\pi$
$821$	18.5030			0.645758			0.322879	+	0.946440i	$0.395349\pi$
$822$		0			0
$823$			29.3898i			1.02446i	0.858848	+	0.512231i	$0.171181\pi$
$823$			29.3898i			1.02446i	−0.858848	+	0.512231i	$0.828819\pi$
$824$		0			0
$825$		−	1.10189i		−	0.0383629i
$826$		0			0
$827$	30.4155			1.05765			0.528826	−	0.848730i	$-0.322632\pi$
$827$	30.4155			1.05765			0.528826	+	0.848730i	$0.322632\pi$
$828$		0			0
$829$	7.00083			0.243149			0.121574	−	0.992582i	$-0.461206\pi$
$829$	7.00083			0.243149			0.121574	+	0.992582i	$0.461206\pi$
$830$		0			0
$831$			17.0510i			0.591491i
$832$		0			0
$833$			40.8409i			1.41505i
$834$		0			0
$835$	−6.69307			−0.231623
$836$		0			0
$837$	−5.12465			−0.177134
$838$		0			0
$839$	17.9514			0.619750			0.309875	−	0.950777i	$-0.399713\pi$
$839$	17.9514			0.619750			0.309875	+	0.950777i	$0.399713\pi$
$840$		0			0
$841$	30.0867			1.03747
$842$		0			0
$843$	19.4880			0.671202
$844$		0			0
$845$			7.91423i			0.272258i
$846$		0			0
$847$	4.72406			0.162321
$848$		0			0
$849$			4.52826i			0.155410i
$850$		0			0
$851$	−8.80275	−	26.5765i	−0.301754	−	0.911031i
$852$		0			0
$853$	28.2089			0.965853			0.482926	−	0.875661i	$-0.339574\pi$
$853$	28.2089			0.965853			0.482926	+	0.875661i	$0.339574\pi$
$854$		0			0
$855$			1.48254i			0.0507017i
$856$		0			0
$857$	44.0494			1.50470			0.752349	−	0.658765i	$-0.228921\pi$
$857$	44.0494			1.50470			0.752349	+	0.658765i	$0.228921\pi$
$858$		0			0
$859$		−	14.9359i		−	0.509607i	−0.966993	−	0.254804i	$-0.917989\pi$
$859$		−	14.9359i		−	0.509607i	0.966993	−	0.254804i	$-0.0820108\pi$
$860$		0			0
$861$			0.616786i			0.0210200i
$862$		0			0
$863$		−	50.8360i		−	1.73048i	−0.501361	−	0.865238i	$-0.667167\pi$
$863$		−	50.8360i		−	1.73048i	0.501361	−	0.865238i	$-0.332833\pi$
$864$		0			0
$865$			24.6878i			0.839411i
$866$		0			0
$867$			19.4254i			0.659720i
$868$		0			0
$869$	−3.98698			−0.135249
$870$		0			0
$871$	18.9292			0.641391
$872$		0			0
$873$		−	0.330286i		−	0.0111785i
$874$		0			0
$875$		−	0.482745i		−	0.0163197i
$876$		0			0
$877$	32.5651			1.09965			0.549823	−	0.835281i	$-0.314695\pi$
$877$	32.5651			1.09965			0.549823	+	0.835281i	$0.314695\pi$
$878$		0			0
$879$	14.3433			0.483786
$880$		0			0
$881$		−	3.91908i		−	0.132037i	−0.997818	−	0.0660187i	$-0.978970\pi$
$881$		−	3.91908i		−	0.132037i	0.997818	−	0.0660187i	$-0.0210297\pi$
$882$		0			0
$883$		−	4.11783i		−	0.138576i	−0.997597	−	0.0692880i	$-0.977927\pi$
$883$		−	4.11783i		−	0.138576i	0.997597	−	0.0692880i	$-0.0220727\pi$
$884$		0			0
$885$			8.08002i			0.271607i
$886$		0			0
$887$			53.8097i			1.80675i	0.428850	+	0.903376i	$0.358919\pi$
$887$			53.8097i			1.80675i	−0.428850	+	0.903376i	$0.641081\pi$
$888$		0			0
$889$		−	2.66396i		−	0.0893464i
$890$		0			0
$891$	−1.10189			−0.0369147
$892$		0			0
$893$		−	16.6981i		−	0.558781i
$894$		0			0
$895$	4.65679			0.155659
$896$		0			0
$897$	−3.40062	−	10.2669i	−0.113543	−	0.342801i
$898$		0			0
$899$		−	39.3921i		−	1.31380i
$900$		0			0
$901$	53.1996			1.77234
$902$		0			0
$903$			4.08705i			0.136009i
$904$		0			0
$905$	−10.4913			−0.348741
$906$		0			0
$907$	−40.1782			−1.33409			−0.667047	−	0.745016i	$-0.732442\pi$
$907$	−40.1782			−1.33409			−0.667047	+	0.745016i	$0.732442\pi$
$908$		0			0
$909$	3.89119			0.129063
$910$		0			0
$911$	13.6169			0.451147			0.225574	−	0.974226i	$-0.427574\pi$
$911$	13.6169			0.451147			0.225574	+	0.974226i	$0.427574\pi$
$912$		0			0
$913$	10.0510			0.332641
$914$		0			0
$915$		−	0.0392191i		−	0.00129654i
$916$		0			0
$917$			7.00932i			0.231468i
$918$		0			0
$919$	−37.9380			−1.25146			−0.625729	−	0.780040i	$-0.715198\pi$
$919$	−37.9380			−1.25146			−0.625729	+	0.780040i	$0.715198\pi$
$920$		0			0
$921$	20.1488			0.663925
$922$		0			0
$923$			2.83245i			0.0932313i
$924$		0			0
$925$			5.83765i			0.191941i
$926$		0			0
$927$	11.2588			0.369789
$928$		0			0
$929$	4.09450			0.134336			0.0671681	−	0.997742i	$-0.478604\pi$
$929$	4.09450			0.134336			0.0671681	+	0.997742i	$0.478604\pi$
$930$		0			0
$931$	−10.0323			−0.328794
$932$		0			0
$933$	7.50388			0.245666
$934$		0			0
$935$	6.65028			0.217488
$936$		0			0
$937$		−	20.6199i		−	0.673621i	−0.941572	−	0.336811i	$-0.890652\pi$
$937$		−	20.6199i		−	0.673621i	0.941572	−	0.336811i	$-0.109348\pi$
$938$		0			0
$939$	27.7341			0.905069
$940$		0			0
$941$			8.99208i			0.293133i	0.989201	+	0.146567i	$0.0468223\pi$
$941$			8.99208i			0.293133i	−0.989201	+	0.146567i	$0.953178\pi$
$942$		0			0
$943$	5.81670	−	1.92662i	0.189418	−	0.0627395i
$944$		0			0
$945$	−0.482745			−0.0157037
$946$		0			0
$947$		−	12.5823i		−	0.408870i	−0.978880	−	0.204435i	$-0.934464\pi$
$947$		−	12.5823i		−	0.408870i	0.978880	−	0.204435i	$-0.0655357\pi$
$948$		0			0
$949$	−7.83624			−0.254375
$950$		0			0
$951$		−	23.7305i		−	0.769515i
$952$		0			0
$953$			18.1150i			0.586801i	0.955990	+	0.293401i	$0.0947870\pi$
$953$			18.1150i			0.586801i	−0.955990	+	0.293401i	$0.905213\pi$
$954$		0			0
$955$		−	10.1702i		−	0.329098i
$956$		0			0
$957$		−	8.46999i		−	0.273796i
$958$		0			0
$959$			2.17451i			0.0702187i
$960$		0			0
$961$	4.73797			0.152838
$962$		0			0
$963$	−0.801493			−0.0258278
$964$		0			0
$965$			13.3593i			0.430052i
$966$		0			0
$967$		−	42.6245i		−	1.37071i	−0.728209	−	0.685355i	$-0.759647\pi$
$967$		−	42.6245i		−	1.37071i	0.728209	−	0.685355i	$-0.240353\pi$
$968$		0			0
$969$	−8.94763			−0.287439
$970$		0			0
$971$	28.0477			0.900094			0.450047	−	0.893005i	$-0.351407\pi$
$971$	28.0477			0.900094			0.450047	+	0.893005i	$0.351407\pi$
$972$		0			0
$973$			1.03007i			0.0330224i
$974$		0			0
$975$			2.25517i			0.0722231i
$976$		0			0
$977$		−	24.7913i		−	0.793145i	−0.918003	−	0.396573i	$-0.870199\pi$
$977$		−	24.7913i		−	0.793145i	0.918003	−	0.396573i	$-0.129801\pi$
$978$		0			0
$979$		−	7.54911i		−	0.241271i
$980$		0			0
$981$			1.25885i			0.0401920i
$982$		0			0
$983$	−45.0003			−1.43529			−0.717644	−	0.696411i	$-0.754779\pi$
$983$	−45.0003			−1.43529			−0.717644	+	0.696411i	$0.754779\pi$
$984$		0			0
$985$			2.65615i			0.0846321i
$986$		0			0
$987$	5.43724			0.173069
$988$		0			0
$989$	38.5436	−	12.7665i	1.22562	−	0.405952i
$990$		0			0
$991$			47.7981i			1.51836i	0.650883	+	0.759178i	$0.274399\pi$
$991$			47.7981i			1.51836i	−0.650883	+	0.759178i	$0.725601\pi$
$992$		0			0
$993$	14.4675			0.459112
$994$		0			0
$995$			8.43038i			0.267261i
$996$		0			0
$997$	25.9172			0.820807			0.410404	−	0.911904i	$-0.365388\pi$
$997$	25.9172			0.820807			0.410404	+	0.911904i	$0.365388\pi$
$998$		0			0
$999$	5.83765			0.184695

Twists

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

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

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} -1.00000i q^{5} -0.482745 q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} -1.00000i q^{5} -0.482745 q^{7} -1.00000 q^{9} -1.10189 q^{11} +2.25517 q^{13} -1.00000 q^{15} -6.03534i q^{17} +1.48254 q^{19} +0.482745i q^{21} +(4.55260 - 1.50793i) q^{23} -1.00000 q^{25} +1.00000i q^{27} -7.68679 q^{29} +5.12465i q^{31} +1.10189i q^{33} +0.482745i q^{35} -5.83765i q^{37} -2.25517i q^{39} +1.27767 q^{41} +8.46629 q^{43} +1.00000i q^{45} -11.2632i q^{47} -6.76696 q^{49} -6.03534 q^{51} +8.81467i q^{53} +1.10189i q^{55} -1.48254i q^{57} -8.08002i q^{59} +0.0392191i q^{61} +0.482745 q^{63} -2.25517i q^{65} +8.39370 q^{67} +(-1.50793 - 4.55260i) q^{69} +1.25598i q^{71} -3.47480 q^{73} +1.00000i q^{75} +0.531931 q^{77} +3.61831 q^{79} +1.00000 q^{81} -9.12164 q^{83} -6.03534 q^{85} +7.68679i q^{87} +6.85106i q^{89} -1.08867 q^{91} +5.12465 q^{93} -1.48254i q^{95} +0.330286i q^{97} +1.10189 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 8 q^{7} - 16 q^{9}+O(q^{10}) \) 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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