LMFDB - Embedded newform 5520.2.be.a.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			$0.476829 - 0.476829i$ of defining polynomial
Character	$\chi$	$=$	5520.1471
Dual form			5520.2.be.a.1471.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} +1.00000i q^{5} -3.79952 q^{7} -1.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} +1.00000i q^{5} -3.79952 q^{7} -1.00000 q^{9} -3.69013 q^{11} +2.60963 q^{13} +1.00000 q^{15} -1.60151i q^{17} +8.20535 q^{19} +3.79952i q^{21} +(-1.19801 - 4.64379i) q^{23} -1.00000 q^{25} +1.00000i q^{27} +0.706420 q^{29} -6.46598i q^{31} +3.69013i q^{33} -3.79952i q^{35} +5.01787i q^{37} -2.60963i q^{39} -10.6396 q^{41} +5.94134 q^{43} -1.00000i q^{45} -4.46270i q^{47} +7.43635 q^{49} -1.60151 q^{51} +7.25579i q^{53} -3.69013i q^{55} -8.20535i q^{57} +6.04730i q^{59} -3.36597i q^{61} +3.79952 q^{63} +2.60963i q^{65} -13.5383 q^{67} +(-4.64379 + 1.19801i) q^{69} -8.69292i q^{71} -0.516947 q^{73} +1.00000i q^{75} +14.0207 q^{77} +3.23175 q^{79} +1.00000 q^{81} -4.10586 q^{83} +1.60151 q^{85} -0.706420i q^{87} +5.11034i q^{89} -9.91534 q^{91} -6.46598 q^{93} +8.20535i q^{95} +19.4018i q^{97} +3.69013 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$	−3.79952			−1.43608			−0.718042	−	0.696000i	$-0.754961\pi$
$7$	−3.79952			−1.43608			−0.718042	+	0.696000i	$0.754961\pi$
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	−3.69013			−1.11262			−0.556308	−	0.830976i	$-0.687783\pi$
$11$	−3.69013			−1.11262			−0.556308	+	0.830976i	$0.687783\pi$
$12$		0			0
$13$	2.60963			0.723781			0.361891	−	0.932221i	$-0.382131\pi$
$13$	2.60963			0.723781			0.361891	+	0.932221i	$0.382131\pi$
$14$		0			0
$15$	1.00000			0.258199
$16$		0			0
$17$		−	1.60151i		−	0.388424i	−0.980960	−	0.194212i	$-0.937785\pi$
$17$		−	1.60151i		−	0.388424i	0.980960	−	0.194212i	$-0.0622150\pi$
$18$		0			0
$19$	8.20535			1.88244			0.941218	−	0.337801i	$-0.109683\pi$
$19$	8.20535			1.88244			0.941218	+	0.337801i	$0.109683\pi$
$20$		0			0
$21$			3.79952i			0.829123i
$22$		0			0
$23$	−1.19801	−	4.64379i	−0.249802	−	0.968297i
$23$	−1.19801	−	4.64379i	−0.249802	−	0.968297i
$24$		0			0
$25$	−1.00000			−0.200000
$26$		0			0
$27$			1.00000i			0.192450i
$28$		0			0
$29$	0.706420			0.131179			0.0655894	−	0.997847i	$-0.479107\pi$
$29$	0.706420			0.131179			0.0655894	+	0.997847i	$0.479107\pi$
$30$		0			0
$31$		−	6.46598i		−	1.16132i	−0.814145	−	0.580662i	$-0.802794\pi$
$31$		−	6.46598i		−	1.16132i	0.814145	−	0.580662i	$-0.197206\pi$
$32$		0			0
$33$			3.69013i			0.642369i
$34$		0			0
$35$		−	3.79952i		−	0.642236i
$36$		0			0
$37$			5.01787i			0.824933i	0.910973	+	0.412466i	$0.135333\pi$
$37$			5.01787i			0.824933i	−0.910973	+	0.412466i	$0.864667\pi$
$38$		0			0
$39$		−	2.60963i		−	0.417875i
$40$		0			0
$41$	−10.6396			−1.66162			−0.830812	−	0.556553i	$-0.812124\pi$
$41$	−10.6396			−1.66162			−0.830812	+	0.556553i	$0.812124\pi$
$42$		0			0
$43$	5.94134			0.906045			0.453023	−	0.891499i	$-0.350346\pi$
$43$	5.94134			0.906045			0.453023	+	0.891499i	$0.350346\pi$
$44$		0			0
$45$		−	1.00000i		−	0.149071i
$46$		0			0
$47$		−	4.46270i		−	0.650952i	−0.945550	−	0.325476i	$-0.894476\pi$
$47$		−	4.46270i		−	0.650952i	0.945550	−	0.325476i	$-0.105524\pi$
$48$		0			0
$49$	7.43635			1.06234
$50$		0			0
$51$	−1.60151			−0.224257
$52$		0			0
$53$			7.25579i			0.996660i	0.866987	+	0.498330i	$0.166053\pi$
$53$			7.25579i			0.996660i	−0.866987	+	0.498330i	$0.833947\pi$
$54$		0			0
$55$		−	3.69013i		−	0.497577i
$56$		0			0
$57$		−	8.20535i		−	1.08682i
$58$		0			0
$59$			6.04730i			0.787291i	0.919262	+	0.393646i	$0.128786\pi$
$59$			6.04730i			0.787291i	−0.919262	+	0.393646i	$0.871214\pi$
$60$		0			0
$61$		−	3.36597i		−	0.430968i	−0.976507	−	0.215484i	$-0.930867\pi$
$61$		−	3.36597i		−	0.430968i	0.976507	−	0.215484i	$-0.0691329\pi$
$62$		0			0
$63$	3.79952			0.478695
$64$		0			0
$65$			2.60963i			0.323685i
$66$		0			0
$67$	−13.5383			−1.65397			−0.826984	−	0.562226i	$-0.809945\pi$
$67$	−13.5383			−1.65397			−0.826984	+	0.562226i	$0.809945\pi$
$68$		0			0
$69$	−4.64379	+	1.19801i	−0.559047	+	0.144223i
$70$		0			0
$71$		−	8.69292i		−	1.03166i	−0.856691	−	0.515830i	$-0.827484\pi$
$71$		−	8.69292i		−	1.03166i	0.856691	−	0.515830i	$-0.172516\pi$
$72$		0			0
$73$	−0.516947			−0.0605040			−0.0302520	−	0.999542i	$-0.509631\pi$
$73$	−0.516947			−0.0605040			−0.0302520	+	0.999542i	$0.509631\pi$
$74$		0			0
$75$			1.00000i			0.115470i
$76$		0			0
$77$	14.0207			1.59781
$78$		0			0
$79$	3.23175			0.363600			0.181800	−	0.983336i	$-0.441808\pi$
$79$	3.23175			0.363600			0.181800	+	0.983336i	$0.441808\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$	−4.10586			−0.450677			−0.225338	−	0.974281i	$-0.572349\pi$
$83$	−4.10586			−0.450677			−0.225338	+	0.974281i	$0.572349\pi$
$84$		0			0
$85$	1.60151			0.173709
$86$		0			0
$87$		−	0.706420i		−	0.0757361i
$88$		0			0
$89$			5.11034i			0.541695i	0.962622	+	0.270848i	$0.0873040\pi$
$89$			5.11034i			0.541695i	−0.962622	+	0.270848i	$0.912696\pi$
$90$		0			0
$91$	−9.91534			−1.03941
$92$		0			0
$93$	−6.46598			−0.670491
$94$		0			0
$95$			8.20535i			0.841851i
$96$		0			0
$97$			19.4018i			1.96996i	0.172675	+	0.984979i	$0.444759\pi$
$97$			19.4018i			1.96996i	−0.172675	+	0.984979i	$0.555241\pi$
$98$		0			0
$99$	3.69013			0.370872
$100$		0			0
$101$	13.1514			1.30861			0.654305	−	0.756231i	$-0.272961\pi$
$101$	13.1514			1.30861			0.654305	+	0.756231i	$0.272961\pi$
$102$		0			0
$103$	−5.37041			−0.529162			−0.264581	−	0.964363i	$-0.585234\pi$
$103$	−5.37041			−0.529162			−0.264581	+	0.964363i	$0.585234\pi$
$104$		0			0
$105$	−3.79952			−0.370795
$106$		0			0
$107$	−17.9219			−1.73258			−0.866290	−	0.499542i	$-0.833502\pi$
$107$	−17.9219			−1.73258			−0.866290	+	0.499542i	$0.833502\pi$
$108$		0			0
$109$			1.27328i			0.121958i	0.998139	+	0.0609792i	$0.0194223\pi$
$109$			1.27328i			0.121958i	−0.998139	+	0.0609792i	$0.980578\pi$
$110$		0			0
$111$	5.01787			0.476275
$112$		0			0
$113$			3.32431i			0.312725i	0.987700	+	0.156362i	$0.0499768\pi$
$113$			3.32431i			0.312725i	−0.987700	+	0.156362i	$0.950023\pi$
$114$		0			0
$115$	4.64379	−	1.19801i	0.433036	−	0.111715i
$116$		0			0
$117$	−2.60963			−0.241260
$118$		0			0
$119$			6.08498i			0.557810i
$120$		0			0
$121$	2.61707			0.237915
$122$		0			0
$123$			10.6396i			0.959339i
$124$		0			0
$125$		−	1.00000i		−	0.0894427i
$126$		0			0
$127$			15.7160i			1.39457i	0.716793	+	0.697286i	$0.245609\pi$
$127$			15.7160i			1.39457i	−0.716793	+	0.697286i	$0.754391\pi$
$128$		0			0
$129$		−	5.94134i		−	0.523106i
$130$		0			0
$131$		−	10.5445i		−	0.921280i	−0.887587	−	0.460640i	$-0.847620\pi$
$131$		−	10.5445i		−	0.921280i	0.887587	−	0.460640i	$-0.152380\pi$
$132$		0			0
$133$	−31.1764			−2.70333
$134$		0			0
$135$	−1.00000			−0.0860663
$136$		0			0
$137$		−	1.67864i		−	0.143416i	−0.997426	−	0.0717081i	$-0.977155\pi$
$137$		−	1.67864i		−	0.143416i	0.997426	−	0.0717081i	$-0.0228450\pi$
$138$		0			0
$139$			13.7161i			1.16338i	0.813410	+	0.581691i	$0.197609\pi$
$139$			13.7161i			1.16338i	−0.813410	+	0.581691i	$0.802391\pi$
$140$		0			0
$141$	−4.46270			−0.375827
$142$		0			0
$143$	−9.62988			−0.805291
$144$		0			0
$145$			0.706420i			0.0586650i
$146$		0			0
$147$		−	7.43635i		−	0.613340i
$148$		0			0
$149$			22.3923i			1.83445i	0.398368	+	0.917226i	$0.369577\pi$
$149$			22.3923i			1.83445i	−0.398368	+	0.917226i	$0.630423\pi$
$150$		0			0
$151$			22.9206i			1.86526i	0.360840	+	0.932628i	$0.382490\pi$
$151$			22.9206i			1.86526i	−0.360840	+	0.932628i	$0.617510\pi$
$152$		0			0
$153$			1.60151i			0.129475i
$154$		0			0
$155$	6.46598			0.519360
$156$		0			0
$157$			14.5054i			1.15765i	0.815450	+	0.578827i	$0.196489\pi$
$157$			14.5054i			1.15765i	−0.815450	+	0.578827i	$0.803511\pi$
$158$		0			0
$159$	7.25579			0.575422
$160$		0			0
$161$	4.55185	+	17.6442i	0.358736	+	1.39056i
$162$		0			0
$163$			6.70939i			0.525520i	0.964861	+	0.262760i	$0.0846327\pi$
$163$			6.70939i			0.525520i	−0.964861	+	0.262760i	$0.915367\pi$
$164$		0			0
$165$	−3.69013			−0.287276
$166$		0			0
$167$			5.17703i			0.400611i	0.979734	+	0.200305i	$0.0641934\pi$
$167$			5.17703i			0.400611i	−0.979734	+	0.200305i	$0.935807\pi$
$168$		0			0
$169$	−6.18983			−0.476141
$170$		0			0
$171$	−8.20535			−0.627478
$172$		0			0
$173$	14.0638			1.06925			0.534624	−	0.845090i	$-0.320453\pi$
$173$	14.0638			1.06925			0.534624	+	0.845090i	$0.320453\pi$
$174$		0			0
$175$	3.79952			0.287217
$176$		0			0
$177$	6.04730			0.454543
$178$		0			0
$179$			11.9606i			0.893975i	0.894540	+	0.446988i	$0.147503\pi$
$179$			11.9606i			0.893975i	−0.894540	+	0.446988i	$0.852497\pi$
$180$		0			0
$181$		−	1.90385i		−	0.141512i	−0.997494	−	0.0707561i	$-0.977459\pi$
$181$		−	1.90385i		−	0.141512i	0.997494	−	0.0707561i	$-0.0225412\pi$
$182$		0			0
$183$	−3.36597			−0.248819
$184$		0			0
$185$	−5.01787			−0.368921
$186$		0			0
$187$			5.90980i			0.432167i
$188$		0			0
$189$		−	3.79952i		−	0.276374i
$190$		0			0
$191$	−16.6906			−1.20769			−0.603845	−	0.797102i	$-0.706365\pi$
$191$	−16.6906			−1.20769			−0.603845	+	0.797102i	$0.706365\pi$
$192$		0			0
$193$	−11.9024			−0.856751			−0.428376	−	0.903601i	$-0.640914\pi$
$193$	−11.9024			−0.856751			−0.428376	+	0.903601i	$0.640914\pi$
$194$		0			0
$195$	2.60963			0.186879
$196$		0			0
$197$	15.3515			1.09375			0.546876	−	0.837214i	$-0.315817\pi$
$197$	15.3515			1.09375			0.546876	+	0.837214i	$0.315817\pi$
$198$		0			0
$199$	17.4361			1.23601			0.618006	−	0.786173i	$-0.287941\pi$
$199$	17.4361			1.23601			0.618006	+	0.786173i	$0.287941\pi$
$200$		0			0
$201$			13.5383i			0.954919i
$202$		0			0
$203$	−2.68406			−0.188384
$204$		0			0
$205$		−	10.6396i		−	0.743101i
$206$		0			0
$207$	1.19801	+	4.64379i	0.0832672	+	0.322766i
$208$		0			0
$209$	−30.2788			−2.09443
$210$		0			0
$211$		−	7.11509i		−	0.489823i	−0.969545	−	0.244911i	$-0.921241\pi$
$211$		−	7.11509i		−	0.489823i	0.969545	−	0.244911i	$-0.0787589\pi$
$212$		0			0
$213$	−8.69292			−0.595629
$214$		0			0
$215$			5.94134i			0.405196i
$216$		0			0
$217$			24.5676i			1.66776i
$218$		0			0
$219$			0.516947i			0.0349320i
$220$		0			0
$221$		−	4.17936i		−	0.281134i
$222$		0			0
$223$			2.92403i			0.195808i	0.995196	+	0.0979038i	$0.0312137\pi$
$223$			2.92403i			0.195808i	−0.995196	+	0.0979038i	$0.968786\pi$
$224$		0			0
$225$	1.00000			0.0666667
$226$		0			0
$227$	−8.86711			−0.588531			−0.294265	−	0.955724i	$-0.595075\pi$
$227$	−8.86711			−0.588531			−0.294265	+	0.955724i	$0.595075\pi$
$228$		0			0
$229$		−	7.81117i		−	0.516177i	−0.966121	−	0.258088i	$-0.916907\pi$
$229$		−	7.81117i		−	0.516177i	0.966121	−	0.258088i	$-0.0830926\pi$
$230$		0			0
$231$		−	14.0207i		−	0.922496i
$232$		0			0
$233$	−5.09372			−0.333701			−0.166850	−	0.985982i	$-0.553360\pi$
$233$	−5.09372			−0.333701			−0.166850	+	0.985982i	$0.553360\pi$
$234$		0			0
$235$	4.46270			0.291114
$236$		0			0
$237$		−	3.23175i		−	0.209925i
$238$		0			0
$239$		−	19.5551i		−	1.26491i	−0.774595	−	0.632457i	$-0.782046\pi$
$239$		−	19.5551i		−	1.26491i	0.774595	−	0.632457i	$-0.217954\pi$
$240$		0			0
$241$			8.39986i			0.541083i	0.962708	+	0.270541i	$0.0872027\pi$
$241$			8.39986i			0.541083i	−0.962708	+	0.270541i	$0.912797\pi$
$242$		0			0
$243$		−	1.00000i		−	0.0641500i
$244$		0			0
$245$			7.43635i			0.475091i
$246$		0			0
$247$	21.4129			1.36247
$248$		0			0
$249$			4.10586i			0.260198i
$250$		0			0
$251$	−7.84091			−0.494914			−0.247457	−	0.968899i	$-0.579595\pi$
$251$	−7.84091			−0.494914			−0.247457	+	0.968899i	$0.579595\pi$
$252$		0			0
$253$	4.42080	+	17.1362i	0.277933	+	1.07734i
$254$		0			0
$255$		−	1.60151i		−	0.100291i
$256$		0			0
$257$	−5.18308			−0.323312			−0.161656	−	0.986847i	$-0.551683\pi$
$257$	−5.18308			−0.323312			−0.161656	+	0.986847i	$0.551683\pi$
$258$		0			0
$259$		−	19.0655i		−	1.18467i
$260$		0			0
$261$	−0.706420			−0.0437263
$262$		0			0
$263$	−6.41083			−0.395309			−0.197654	−	0.980272i	$-0.563332\pi$
$263$	−6.41083			−0.395309			−0.197654	+	0.980272i	$0.563332\pi$
$264$		0			0
$265$	−7.25579			−0.445720
$266$		0			0
$267$	5.11034			0.312748
$268$		0			0
$269$	28.1418			1.71584			0.857919	−	0.513785i	$-0.171757\pi$
$269$	28.1418			1.71584			0.857919	+	0.513785i	$0.171757\pi$
$270$		0			0
$271$		−	23.8196i		−	1.44694i	−0.690356	−	0.723470i	$-0.742546\pi$
$271$		−	23.8196i		−	1.44694i	0.690356	−	0.723470i	$-0.257454\pi$
$272$		0			0
$273$			9.91534i			0.600104i
$274$		0			0
$275$	3.69013			0.222523
$276$		0			0
$277$	−22.1168			−1.32887			−0.664436	−	0.747345i	$-0.731328\pi$
$277$	−22.1168			−1.32887			−0.664436	+	0.747345i	$0.731328\pi$
$278$		0			0
$279$			6.46598i			0.387108i
$280$		0			0
$281$			10.5452i			0.629075i	0.949245	+	0.314538i	$0.101849\pi$
$281$			10.5452i			0.629075i	−0.949245	+	0.314538i	$0.898151\pi$
$282$		0			0
$283$	−13.3512			−0.793648			−0.396824	−	0.917895i	$-0.629888\pi$
$283$	−13.3512			−0.793648			−0.396824	+	0.917895i	$0.629888\pi$
$284$		0			0
$285$	8.20535			0.486043
$286$		0			0
$287$	40.4253			2.38623
$288$		0			0
$289$	14.4352			0.849127
$290$		0			0
$291$	19.4018			1.13736
$292$		0			0
$293$		−	13.0456i		−	0.762129i	−0.924548	−	0.381065i	$-0.875557\pi$
$293$		−	13.0456i		−	0.762129i	0.924548	−	0.381065i	$-0.124443\pi$
$294$		0			0
$295$	−6.04730			−0.352087
$296$		0			0
$297$		−	3.69013i		−	0.214123i
$298$		0			0
$299$	−3.12635	−	12.1186i	−0.180802	−	0.700835i
$300$		0			0
$301$	−22.5742			−1.30116
$302$		0			0
$303$		−	13.1514i		−	0.755526i
$304$		0			0
$305$	3.36597			0.192735
$306$		0			0
$307$		−	24.4269i		−	1.39412i	−0.717015	−	0.697058i	$-0.754492\pi$
$307$		−	24.4269i		−	1.39412i	0.717015	−	0.697058i	$-0.245508\pi$
$308$		0			0
$309$			5.37041i			0.305512i
$310$		0			0
$311$		−	2.74305i		−	0.155544i	−0.996971	−	0.0777721i	$-0.975219\pi$
$311$		−	2.74305i		−	0.155544i	0.996971	−	0.0777721i	$-0.0247807\pi$
$312$		0			0
$313$		−	3.88307i		−	0.219484i	−0.993960	−	0.109742i	$-0.964998\pi$
$313$		−	3.88307i		−	0.219484i	0.993960	−	0.109742i	$-0.0350025\pi$
$314$		0			0
$315$			3.79952i			0.214079i
$316$		0			0
$317$	−26.5346			−1.49033			−0.745166	−	0.666879i	$-0.767630\pi$
$317$	−26.5346			−1.49033			−0.745166	+	0.666879i	$0.767630\pi$
$318$		0			0
$319$	−2.60678			−0.145952
$320$		0			0
$321$			17.9219i			1.00031i
$322$		0			0
$323$		−	13.1410i		−	0.731183i
$324$		0			0
$325$	−2.60963			−0.144756
$326$		0			0
$327$	1.27328			0.0704127
$328$		0			0
$329$			16.9561i			0.934821i
$330$		0			0
$331$			28.5918i			1.57155i	0.618515	+	0.785773i	$0.287735\pi$
$331$			28.5918i			1.57155i	−0.618515	+	0.785773i	$0.712265\pi$
$332$		0			0
$333$		−	5.01787i		−	0.274978i
$334$		0			0
$335$		−	13.5383i		−	0.739677i
$336$		0			0
$337$		−	6.76479i		−	0.368501i	−0.982879	−	0.184251i	$-0.941014\pi$
$337$		−	6.76479i		−	0.368501i	0.982879	−	0.184251i	$-0.0589858\pi$
$338$		0			0
$339$	3.32431			0.180552
$340$		0			0
$341$			23.8603i			1.29211i
$342$		0			0
$343$	−1.65793			−0.0895200
$344$		0			0
$345$	−1.19801	−	4.64379i	−0.0644985	−	0.250013i
$346$		0			0
$347$			29.0461i			1.55927i	0.626231	+	0.779637i	$0.284597\pi$
$347$			29.0461i			1.55927i	−0.626231	+	0.779637i	$0.715403\pi$
$348$		0			0
$349$	−20.4208			−1.09310			−0.546550	−	0.837427i	$-0.684059\pi$
$349$	−20.4208			−1.09310			−0.546550	+	0.837427i	$0.684059\pi$
$350$		0			0
$351$			2.60963i			0.139292i
$352$		0			0
$353$	−18.6709			−0.993751			−0.496876	−	0.867822i	$-0.665519\pi$
$353$	−18.6709			−0.993751			−0.496876	+	0.867822i	$0.665519\pi$
$354$		0			0
$355$	8.69292			0.461372
$356$		0			0
$357$	6.08498			0.322052
$358$		0			0
$359$	−5.96281			−0.314705			−0.157353	−	0.987542i	$-0.550296\pi$
$359$	−5.96281			−0.314705			−0.157353	+	0.987542i	$0.550296\pi$
$360$		0			0
$361$	48.3277			2.54356
$362$		0			0
$363$		−	2.61707i		−	0.137360i
$364$		0			0
$365$		−	0.516947i		−	0.0270582i
$366$		0			0
$367$	23.4914			1.22624			0.613121	−	0.789989i	$-0.289914\pi$
$367$	23.4914			1.22624			0.613121	+	0.789989i	$0.289914\pi$
$368$		0			0
$369$	10.6396			0.553875
$370$		0			0
$371$		−	27.5685i		−	1.43129i
$372$		0			0
$373$		−	20.8948i		−	1.08189i	−0.841057	−	0.540946i	$-0.818066\pi$
$373$		−	20.8948i		−	1.08189i	0.841057	−	0.540946i	$-0.181934\pi$
$374$		0			0
$375$	−1.00000			−0.0516398
$376$		0			0
$377$	1.84349			0.0949447
$378$		0			0
$379$	−8.21337			−0.421892			−0.210946	−	0.977498i	$-0.567654\pi$
$379$	−8.21337			−0.421892			−0.210946	+	0.977498i	$0.567654\pi$
$380$		0			0
$381$	15.7160			0.805156
$382$		0			0
$383$	−7.84776			−0.401002			−0.200501	−	0.979694i	$-0.564257\pi$
$383$	−7.84776			−0.401002			−0.200501	+	0.979694i	$0.564257\pi$
$384$		0			0
$385$			14.0207i			0.714562i
$386$		0			0
$387$	−5.94134			−0.302015
$388$		0			0
$389$			25.0255i			1.26884i	0.772988	+	0.634421i	$0.218761\pi$
$389$			25.0255i			1.26884i	−0.772988	+	0.634421i	$0.781239\pi$
$390$		0			0
$391$	−7.43709	+	1.91862i	−0.376110	+	0.0970290i
$392$		0			0
$393$	−10.5445			−0.531901
$394$		0			0
$395$			3.23175i			0.162607i
$396$		0			0
$397$	19.5249			0.979928			0.489964	−	0.871743i	$-0.337010\pi$
$397$	19.5249			0.979928			0.489964	+	0.871743i	$0.337010\pi$
$398$		0			0
$399$			31.1764i			1.56077i
$400$		0			0
$401$		−	27.8201i		−	1.38927i	−0.719363	−	0.694634i	$-0.755566\pi$
$401$		−	27.8201i		−	1.38927i	0.719363	−	0.694634i	$-0.244434\pi$
$402$		0			0
$403$		−	16.8738i		−	0.840545i
$404$		0			0
$405$			1.00000i			0.0496904i
$406$		0			0
$407$		−	18.5166i		−	0.917834i
$408$		0			0
$409$	−18.0574			−0.892882			−0.446441	−	0.894813i	$-0.647309\pi$
$409$	−18.0574			−0.892882			−0.446441	+	0.894813i	$0.647309\pi$
$410$		0			0
$411$	−1.67864			−0.0828014
$412$		0			0
$413$		−	22.9768i		−	1.13062i
$414$		0			0
$415$		−	4.10586i		−	0.201549i
$416$		0			0
$417$	13.7161			0.671679
$418$		0			0
$419$	11.4005			0.556952			0.278476	−	0.960443i	$-0.410171\pi$
$419$	11.4005			0.556952			0.278476	+	0.960443i	$0.410171\pi$
$420$		0			0
$421$		−	20.5611i		−	1.00209i	−0.865422	−	0.501044i	$-0.832949\pi$
$421$		−	20.5611i		−	1.00209i	0.865422	−	0.501044i	$-0.167051\pi$
$422$		0			0
$423$			4.46270i			0.216984i
$424$		0			0
$425$			1.60151i			0.0776848i
$426$		0			0
$427$			12.7891i			0.618906i
$428$		0			0
$429$			9.62988i			0.464935i
$430$		0			0
$431$	3.10368			0.149499			0.0747496	−	0.997202i	$-0.476184\pi$
$431$	3.10368			0.149499			0.0747496	+	0.997202i	$0.476184\pi$
$432$		0			0
$433$			9.48758i			0.455944i	0.973668	+	0.227972i	$0.0732095\pi$
$433$			9.48758i			0.455944i	−0.973668	+	0.227972i	$0.926791\pi$
$434$		0			0
$435$	0.706420			0.0338702
$436$		0			0
$437$	−9.83005	−	38.1039i	−0.470235	−	1.82276i
$438$		0			0
$439$			4.16012i			0.198551i	0.995060	+	0.0992757i	$0.0316526\pi$
$439$			4.16012i			0.198551i	−0.995060	+	0.0992757i	$0.968347\pi$
$440$		0			0
$441$	−7.43635			−0.354112
$442$		0			0
$443$			31.5136i			1.49726i	0.662990	+	0.748629i	$0.269287\pi$
$443$			31.5136i			1.49726i	−0.662990	+	0.748629i	$0.730713\pi$
$444$		0			0
$445$	−5.11034			−0.242254
$446$		0			0
$447$	22.3923			1.05912
$448$		0			0
$449$	−15.5729			−0.734930			−0.367465	−	0.930037i	$-0.619774\pi$
$449$	−15.5729			−0.734930			−0.367465	+	0.930037i	$0.619774\pi$
$450$		0			0
$451$	39.2615			1.84875
$452$		0			0
$453$	22.9206			1.07691
$454$		0			0
$455$		−	9.91534i		−	0.464838i
$456$		0			0
$457$			30.9828i			1.44931i	0.689110	+	0.724657i	$0.258002\pi$
$457$			30.9828i			1.44931i	−0.689110	+	0.724657i	$0.741998\pi$
$458$		0			0
$459$	1.60151			0.0747523
$460$		0			0
$461$	15.7446			0.733301			0.366651	−	0.930359i	$-0.380504\pi$
$461$	15.7446			0.733301			0.366651	+	0.930359i	$0.380504\pi$
$462$		0			0
$463$		−	1.71750i		−	0.0798191i	−0.999203	−	0.0399096i	$-0.987293\pi$
$463$		−	1.71750i		−	0.0798191i	0.999203	−	0.0399096i	$-0.0127070\pi$
$464$		0			0
$465$		−	6.46598i		−	0.299853i
$466$		0			0
$467$	−35.3495			−1.63578			−0.817891	−	0.575373i	$-0.804857\pi$
$467$	−35.3495			−1.63578			−0.817891	+	0.575373i	$0.804857\pi$
$468$		0			0
$469$	51.4391			2.37524
$470$		0			0
$471$	14.5054			0.668372
$472$		0			0
$473$	−21.9243			−1.00808
$474$		0			0
$475$	−8.20535			−0.376487
$476$		0			0
$477$		−	7.25579i		−	0.332220i
$478$		0			0
$479$	−27.0253			−1.23482			−0.617408	−	0.786643i	$-0.711817\pi$
$479$	−27.0253			−1.23482			−0.617408	+	0.786643i	$0.711817\pi$
$480$		0			0
$481$			13.0948i			0.597071i
$482$		0			0
$483$	17.6442	−	4.55185i	0.802838	−	0.207116i
$484$		0			0
$485$	−19.4018			−0.880992
$486$		0			0
$487$			42.0999i			1.90773i	0.300237	+	0.953865i	$0.402934\pi$
$487$			42.0999i			1.90773i	−0.300237	+	0.953865i	$0.597066\pi$
$488$		0			0
$489$	6.70939			0.303409
$490$		0			0
$491$			14.3694i			0.648480i	0.945975	+	0.324240i	$0.105109\pi$
$491$			14.3694i			0.648480i	−0.945975	+	0.324240i	$0.894891\pi$
$492$		0			0
$493$		−	1.13134i		−	0.0509530i
$494$		0			0
$495$			3.69013i			0.165859i
$496$		0			0
$497$			33.0289i			1.48155i
$498$		0			0
$499$			11.1409i			0.498734i	0.968409	+	0.249367i	$0.0802226\pi$
$499$			11.1409i			0.498734i	−0.968409	+	0.249367i	$0.919777\pi$
$500$		0			0
$501$	5.17703			0.231293
$502$		0			0
$503$	−15.8161			−0.705207			−0.352603	−	0.935773i	$-0.614703\pi$
$503$	−15.8161			−0.705207			−0.352603	+	0.935773i	$0.614703\pi$
$504$		0			0
$505$			13.1514i			0.585228i
$506$		0			0
$507$			6.18983i			0.274900i
$508$		0			0
$509$	−31.3061			−1.38762			−0.693809	−	0.720159i	$-0.744069\pi$
$509$	−31.3061			−1.38762			−0.693809	+	0.720159i	$0.744069\pi$
$510$		0			0
$511$	1.96415			0.0868888
$512$		0			0
$513$			8.20535i			0.362275i
$514$		0			0
$515$		−	5.37041i		−	0.236648i
$516$		0			0
$517$			16.4679i			0.724259i
$518$		0			0
$519$		−	14.0638i		−	0.617331i
$520$		0			0
$521$			23.5026i			1.02967i	0.857290	+	0.514833i	$0.172146\pi$
$521$			23.5026i			1.02967i	−0.857290	+	0.514833i	$0.827854\pi$
$522$		0			0
$523$	35.4909			1.55191			0.775954	−	0.630789i	$-0.217269\pi$
$523$	35.4909			1.55191			0.775954	+	0.630789i	$0.217269\pi$
$524$		0			0
$525$		−	3.79952i		−	0.165825i
$526$		0			0
$527$	−10.3554			−0.451087
$528$		0			0
$529$	−20.1296	+	11.1266i	−0.875198	+	0.483764i
$530$		0			0
$531$		−	6.04730i		−	0.262430i
$532$		0			0
$533$	−27.7654			−1.20265
$534$		0			0
$535$		−	17.9219i		−	0.774833i
$536$		0			0
$537$	11.9606			0.516137
$538$		0			0
$539$	−27.4411			−1.18197
$540$		0			0
$541$	31.7968			1.36705			0.683525	−	0.729927i	$-0.260446\pi$
$541$	31.7968			1.36705			0.683525	+	0.729927i	$0.260446\pi$
$542$		0			0
$543$	−1.90385			−0.0817022
$544$		0			0
$545$	−1.27328			−0.0545415
$546$		0			0
$547$		−	21.8699i		−	0.935090i	−0.883969	−	0.467545i	$-0.845139\pi$
$547$		−	21.8699i		−	0.935090i	0.883969	−	0.467545i	$-0.154861\pi$
$548$		0			0
$549$			3.36597i			0.143656i
$550$		0			0
$551$	5.79642			0.246936
$552$		0			0
$553$	−12.2791			−0.522160
$554$		0			0
$555$			5.01787i			0.212997i
$556$		0			0
$557$			1.91446i			0.0811182i	0.999177	+	0.0405591i	$0.0129139\pi$
$557$			1.91446i			0.0811182i	−0.999177	+	0.0405591i	$0.987086\pi$
$558$		0			0
$559$	15.5047			0.655779
$560$		0			0
$561$	5.90980			0.249512
$562$		0			0
$563$	−33.0420			−1.39255			−0.696277	−	0.717773i	$-0.745162\pi$
$563$	−33.0420			−1.39255			−0.696277	+	0.717773i	$0.745162\pi$
$564$		0			0
$565$	−3.32431			−0.139855
$566$		0			0
$567$	−3.79952			−0.159565
$568$		0			0
$569$			10.1677i			0.426250i	0.977025	+	0.213125i	$0.0683642\pi$
$569$			10.1677i			0.426250i	−0.977025	+	0.213125i	$0.931636\pi$
$570$		0			0
$571$	22.6137			0.946355			0.473177	−	0.880967i	$-0.343107\pi$
$571$	22.6137			0.946355			0.473177	+	0.880967i	$0.343107\pi$
$572$		0			0
$573$			16.6906i			0.697260i
$574$		0			0
$575$	1.19801	+	4.64379i	0.0499603	+	0.193659i
$576$		0			0
$577$	−10.0069			−0.416591			−0.208295	−	0.978066i	$-0.566792\pi$
$577$	−10.0069			−0.416591			−0.208295	+	0.978066i	$0.566792\pi$
$578$		0			0
$579$			11.9024i			0.494646i
$580$		0			0
$581$	15.6003			0.647210
$582$		0			0
$583$		−	26.7748i		−	1.10890i
$584$		0			0
$585$		−	2.60963i		−	0.107895i
$586$		0			0
$587$		−	20.6747i		−	0.853336i	−0.904408	−	0.426668i	$-0.859687\pi$
$587$		−	20.6747i		−	0.853336i	0.904408	−	0.426668i	$-0.140313\pi$
$588$		0			0
$589$		−	53.0556i		−	2.18612i
$590$		0			0
$591$		−	15.3515i		−	0.631478i
$592$		0			0
$593$	−34.2092			−1.40480			−0.702402	−	0.711780i	$-0.747889\pi$
$593$	−34.2092			−1.40480			−0.702402	+	0.711780i	$0.747889\pi$
$594$		0			0
$595$	−6.08498			−0.249460
$596$		0			0
$597$		−	17.4361i		−	0.713612i
$598$		0			0
$599$			29.2726i			1.19605i	0.801479	+	0.598023i	$0.204047\pi$
$599$			29.2726i			1.19605i	−0.801479	+	0.598023i	$0.795953\pi$
$600$		0			0
$601$	32.7471			1.33578			0.667891	−	0.744259i	$-0.267197\pi$
$601$	32.7471			1.33578			0.667891	+	0.744259i	$0.267197\pi$
$602$		0			0
$603$	13.5383			0.551323
$604$		0			0
$605$			2.61707i			0.106399i
$606$		0			0
$607$			29.9681i			1.21637i	0.793796	+	0.608185i	$0.208102\pi$
$607$			29.9681i			1.21637i	−0.793796	+	0.608185i	$0.791898\pi$
$608$		0			0
$609$			2.68406i			0.108763i
$610$		0			0
$611$		−	11.6460i		−	0.471146i
$612$		0			0
$613$			12.6769i			0.512014i	0.966675	+	0.256007i	$0.0824070\pi$
$613$			12.6769i			0.512014i	−0.966675	+	0.256007i	$0.917593\pi$
$614$		0			0
$615$	−10.6396			−0.429029
$616$		0			0
$617$			6.68220i			0.269015i	0.990913	+	0.134508i	$0.0429453\pi$
$617$			6.68220i			0.269015i	−0.990913	+	0.134508i	$0.957055\pi$
$618$		0			0
$619$	12.1774			0.489451			0.244725	−	0.969592i	$-0.421302\pi$
$619$	12.1774			0.489451			0.244725	+	0.969592i	$0.421302\pi$
$620$		0			0
$621$	4.64379	−	1.19801i	0.186349	−	0.0480743i
$622$		0			0
$623$		−	19.4169i		−	0.777920i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$			30.2788i			1.20922i
$628$		0			0
$629$	8.03619			0.320424
$630$		0			0
$631$	20.6325			0.821365			0.410683	−	0.911778i	$-0.365290\pi$
$631$	20.6325			0.821365			0.410683	+	0.911778i	$0.365290\pi$
$632$		0			0
$633$	−7.11509			−0.282799
$634$		0			0
$635$	−15.7160			−0.623671
$636$		0			0
$637$	19.4061			0.768899
$638$		0			0
$639$			8.69292i			0.343887i
$640$		0			0
$641$		−	18.8025i		−	0.742655i	−0.928502	−	0.371328i	$-0.878903\pi$
$641$		−	18.8025i		−	0.742655i	0.928502	−	0.371328i	$-0.121097\pi$
$642$		0			0
$643$	−6.17101			−0.243361			−0.121680	−	0.992569i	$-0.538828\pi$
$643$	−6.17101			−0.243361			−0.121680	+	0.992569i	$0.538828\pi$
$644$		0			0
$645$	5.94134			0.233940
$646$		0			0
$647$			11.0800i			0.435601i	0.975993	+	0.217801i	$0.0698882\pi$
$647$			11.0800i			0.435601i	−0.975993	+	0.217801i	$0.930112\pi$
$648$		0			0
$649$		−	22.3153i		−	0.875953i
$650$		0			0
$651$	24.5676			0.962881
$652$		0			0
$653$	−14.6281			−0.572441			−0.286221	−	0.958164i	$-0.592399\pi$
$653$	−14.6281			−0.572441			−0.286221	+	0.958164i	$0.592399\pi$
$654$		0			0
$655$	10.5445			0.412009
$656$		0			0
$657$	0.516947			0.0201680
$658$		0			0
$659$	19.1684			0.746694			0.373347	−	0.927692i	$-0.378210\pi$
$659$	19.1684			0.746694			0.373347	+	0.927692i	$0.378210\pi$
$660$		0			0
$661$			32.1112i			1.24898i	0.781033	+	0.624490i	$0.214693\pi$
$661$			32.1112i			1.24898i	−0.781033	+	0.624490i	$0.785307\pi$
$662$		0			0
$663$	−4.17936			−0.162313
$664$		0			0
$665$		−	31.1764i		−	1.20897i
$666$		0			0
$667$	−0.846295	−	3.28046i	−0.0327687	−	0.127020i
$668$		0			0
$669$	2.92403			0.113050
$670$		0			0
$671$			12.4209i			0.479502i
$672$		0			0
$673$	0.150913			0.00581729			0.00290864	−	0.999996i	$-0.499074\pi$
$673$	0.150913			0.00581729			0.00290864	+	0.999996i	$0.499074\pi$
$674$		0			0
$675$		−	1.00000i		−	0.0384900i
$676$		0			0
$677$		−	41.2608i		−	1.58578i	−0.609365	−	0.792890i	$-0.708575\pi$
$677$		−	41.2608i		−	1.58578i	0.609365	−	0.792890i	$-0.291425\pi$
$678$		0			0
$679$		−	73.7177i		−	2.82902i
$680$		0			0
$681$			8.86711i			0.339788i
$682$		0			0
$683$		−	20.7486i		−	0.793925i	−0.917835	−	0.396963i	$-0.870064\pi$
$683$		−	20.7486i		−	0.793925i	0.917835	−	0.396963i	$-0.129936\pi$
$684$		0			0
$685$	1.67864			0.0641377
$686$		0			0
$687$	−7.81117			−0.298015
$688$		0			0
$689$			18.9349i			0.721364i
$690$		0			0
$691$		−	4.48560i		−	0.170640i	−0.996354	−	0.0853202i	$-0.972809\pi$
$691$		−	4.48560i		−	0.170640i	0.996354	−	0.0853202i	$-0.0271913\pi$
$692$		0			0
$693$	−14.0207			−0.532603
$694$		0			0
$695$	−13.7161			−0.520280
$696$		0			0
$697$			17.0394i			0.645415i
$698$		0			0
$699$			5.09372i			0.192662i
$700$		0			0
$701$		−	1.49486i		−	0.0564601i	−0.999601	−	0.0282301i	$-0.991013\pi$
$701$		−	1.49486i		−	0.0564601i	0.999601	−	0.0282301i	$-0.00898710\pi$
$702$		0			0
$703$			41.1734i			1.55288i
$704$		0			0
$705$		−	4.46270i		−	0.168075i
$706$		0			0
$707$	−49.9689			−1.87927
$708$		0			0
$709$		−	48.1753i		−	1.80926i	−0.426194	−	0.904632i	$-0.640146\pi$
$709$		−	48.1753i		−	1.80926i	0.426194	−	0.904632i	$-0.359854\pi$
$710$		0			0
$711$	−3.23175			−0.121200
$712$		0			0
$713$	−30.0267	+	7.74629i	−1.12451	+	0.290101i
$714$		0			0
$715$		−	9.62988i		−	0.360137i
$716$		0			0
$717$	−19.5551			−0.730299
$718$		0			0
$719$			13.1017i			0.488610i	0.969698	+	0.244305i	$0.0785599\pi$
$719$			13.1017i			0.488610i	−0.969698	+	0.244305i	$0.921440\pi$
$720$		0			0
$721$	20.4050			0.759921
$722$		0			0
$723$	8.39986			0.312394
$724$		0			0
$725$	−0.706420			−0.0262358
$726$		0			0
$727$	−7.30395			−0.270888			−0.135444	−	0.990785i	$-0.543246\pi$
$727$	−7.30395			−0.270888			−0.135444	+	0.990785i	$0.543246\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$		−	9.51513i		−	0.351930i
$732$		0			0
$733$		−	17.8028i		−	0.657563i	−0.944406	−	0.328781i	$-0.893362\pi$
$733$		−	17.8028i		−	0.657563i	0.944406	−	0.328781i	$-0.106638\pi$
$734$		0			0
$735$	7.43635			0.274294
$736$		0			0
$737$	49.9581			1.84023
$738$		0			0
$739$			30.6415i			1.12716i	0.826060	+	0.563582i	$0.190577\pi$
$739$			30.6415i			1.12716i	−0.826060	+	0.563582i	$0.809423\pi$
$740$		0			0
$741$		−	21.4129i		−	0.786623i
$742$		0			0
$743$	23.7573			0.871572			0.435786	−	0.900050i	$-0.356470\pi$
$743$	23.7573			0.871572			0.435786	+	0.900050i	$0.356470\pi$
$744$		0			0
$745$	−22.3923			−0.820392
$746$		0			0
$747$	4.10586			0.150226
$748$		0			0
$749$	68.0948			2.48813
$750$		0			0
$751$	−11.6741			−0.425996			−0.212998	−	0.977053i	$-0.568323\pi$
$751$	−11.6741			−0.425996			−0.212998	+	0.977053i	$0.568323\pi$
$752$		0			0
$753$			7.84091i			0.285739i
$754$		0			0
$755$	−22.9206			−0.834167
$756$		0			0
$757$			16.7159i			0.607551i	0.952744	+	0.303776i	$0.0982473\pi$
$757$			16.7159i			0.607551i	−0.952744	+	0.303776i	$0.901753\pi$
$758$		0			0
$759$	17.1362	−	4.42080i	0.622004	−	0.160465i
$760$		0			0
$761$	25.0602			0.908430			0.454215	−	0.890892i	$-0.349920\pi$
$761$	25.0602			0.908430			0.454215	+	0.890892i	$0.349920\pi$
$762$		0			0
$763$		−	4.83786i		−	0.175142i
$764$		0			0
$765$	−1.60151			−0.0579029
$766$		0			0
$767$			15.7812i			0.569826i
$768$		0			0
$769$			21.2483i			0.766234i	0.923700	+	0.383117i	$0.125149\pi$
$769$			21.2483i			0.766234i	−0.923700	+	0.383117i	$0.874851\pi$
$770$		0			0
$771$			5.18308i			0.186664i
$772$		0			0
$773$			19.8965i			0.715628i	0.933793	+	0.357814i	$0.116478\pi$
$773$			19.8965i			0.715628i	−0.933793	+	0.357814i	$0.883522\pi$
$774$		0			0
$775$			6.46598i			0.232265i
$776$		0			0
$777$	−19.0655			−0.683971
$778$		0			0
$779$	−87.3015			−3.12790
$780$		0			0
$781$			32.0780i			1.14784i
$782$		0			0
$783$			0.706420i			0.0252454i
$784$		0			0
$785$	−14.5054			−0.517719
$786$		0			0
$787$	9.36272			0.333745			0.166873	−	0.985978i	$-0.446633\pi$
$787$	9.36272			0.333745			0.166873	+	0.985978i	$0.446633\pi$
$788$		0			0
$789$			6.41083i			0.228232i
$790$		0			0
$791$		−	12.6308i		−	0.449099i
$792$		0			0
$793$		−	8.78393i		−	0.311926i
$794$		0			0
$795$			7.25579i			0.257336i
$796$		0			0
$797$		−	43.4609i		−	1.53946i	−0.638367	−	0.769732i	$-0.720390\pi$
$797$		−	43.4609i		−	1.53946i	0.638367	−	0.769732i	$-0.279610\pi$
$798$		0			0
$799$	−7.14708			−0.252845
$800$		0			0
$801$		−	5.11034i		−	0.180565i
$802$		0			0
$803$	1.90760			0.0673178
$804$		0			0
$805$	−17.6442	+	4.55185i	−0.621875	+	0.160432i
$806$		0			0
$807$		−	28.1418i		−	0.990640i
$808$		0			0
$809$	42.4500			1.49246			0.746232	−	0.665686i	$-0.231861\pi$
$809$	42.4500			1.49246			0.746232	+	0.665686i	$0.231861\pi$
$810$		0			0
$811$		−	38.4953i		−	1.35175i	−0.737015	−	0.675876i	$-0.763765\pi$
$811$		−	38.4953i		−	1.35175i	0.737015	−	0.675876i	$-0.236235\pi$
$812$		0			0
$813$	−23.8196			−0.835391
$814$		0			0
$815$	−6.70939			−0.235020
$816$		0			0
$817$	48.7507			1.70557
$818$		0			0
$819$	9.91534			0.346470
$820$		0			0
$821$	33.0615			1.15386			0.576928	−	0.816795i	$-0.304251\pi$
$821$	33.0615			1.15386			0.576928	+	0.816795i	$0.304251\pi$
$822$		0			0
$823$			50.4029i			1.75693i	0.477802	+	0.878467i	$0.341434\pi$
$823$			50.4029i			1.75693i	−0.477802	+	0.878467i	$0.658566\pi$
$824$		0			0
$825$		−	3.69013i		−	0.128474i
$826$		0			0
$827$	−26.6674			−0.927318			−0.463659	−	0.886014i	$-0.653464\pi$
$827$	−26.6674			−0.927318			−0.463659	+	0.886014i	$0.653464\pi$
$828$		0			0
$829$	44.4837			1.54498			0.772492	−	0.635025i	$-0.219010\pi$
$829$	44.4837			1.54498			0.772492	+	0.635025i	$0.219010\pi$
$830$		0			0
$831$			22.1168i			0.767224i
$832$		0			0
$833$		−	11.9094i		−	0.412637i
$834$		0			0
$835$	−5.17703			−0.179159
$836$		0			0
$837$	6.46598			0.223497
$838$		0			0
$839$	22.8945			0.790404			0.395202	−	0.918594i	$-0.370675\pi$
$839$	22.8945			0.790404			0.395202	+	0.918594i	$0.370675\pi$
$840$		0			0
$841$	−28.5010			−0.982792
$842$		0			0
$843$	10.5452			0.363197
$844$		0			0
$845$		−	6.18983i		−	0.212937i
$846$		0			0
$847$	−9.94360			−0.341666
$848$		0			0
$849$			13.3512i			0.458213i
$850$		0			0
$851$	23.3019	−	6.01144i	0.798780	−	0.206070i
$852$		0			0
$853$	−27.5273			−0.942516			−0.471258	−	0.881995i	$-0.656200\pi$
$853$	−27.5273			−0.942516			−0.471258	+	0.881995i	$0.656200\pi$
$854$		0			0
$855$		−	8.20535i		−	0.280617i
$856$		0			0
$857$	27.4676			0.938276			0.469138	−	0.883125i	$-0.344565\pi$
$857$	27.4676			0.938276			0.469138	+	0.883125i	$0.344565\pi$
$858$		0			0
$859$		−	26.7023i		−	0.911071i	−0.890218	−	0.455536i	$-0.849448\pi$
$859$		−	26.7023i		−	0.911071i	0.890218	−	0.455536i	$-0.150552\pi$
$860$		0			0
$861$		−	40.4253i		−	1.37769i
$862$		0			0
$863$			45.7892i			1.55868i	0.626600	+	0.779341i	$0.284446\pi$
$863$			45.7892i			1.55868i	−0.626600	+	0.779341i	$0.715554\pi$
$864$		0			0
$865$			14.0638i			0.478182i
$866$		0			0
$867$		−	14.4352i		−	0.490243i
$868$		0			0
$869$	−11.9256			−0.404547
$870$		0			0
$871$	−35.3300			−1.19711
$872$		0			0
$873$		−	19.4018i		−	0.656653i
$874$		0			0
$875$			3.79952i			0.128447i
$876$		0			0
$877$	−49.1607			−1.66004			−0.830019	−	0.557735i	$-0.811671\pi$
$877$	−49.1607			−1.66004			−0.830019	+	0.557735i	$0.811671\pi$
$878$		0			0
$879$	−13.0456			−0.440016
$880$		0			0
$881$		−	20.3938i		−	0.687085i	−0.939137	−	0.343543i	$-0.888373\pi$
$881$		−	20.3938i		−	0.687085i	0.939137	−	0.343543i	$-0.111627\pi$
$882$		0			0
$883$			49.8470i			1.67749i	0.544527	+	0.838743i	$0.316709\pi$
$883$			49.8470i			1.67749i	−0.544527	+	0.838743i	$0.683291\pi$
$884$		0			0
$885$			6.04730i			0.203278i
$886$		0			0
$887$		−	30.4376i		−	1.02200i	−0.859582	−	0.510998i	$-0.829276\pi$
$887$		−	30.4376i		−	1.02200i	0.859582	−	0.510998i	$-0.170724\pi$
$888$		0			0
$889$		−	59.7133i		−	2.00272i
$890$		0			0
$891$	−3.69013			−0.123624
$892$		0			0
$893$		−	36.6180i		−	1.22537i
$894$		0			0
$895$	−11.9606			−0.399798
$896$		0			0
$897$	−12.1186	+	3.12635i	−0.404627	+	0.104386i
$898$		0			0
$899$		−	4.56770i		−	0.152341i
$900$		0			0
$901$	11.6203			0.387127
$902$		0			0
$903$			22.5742i			0.751223i
$904$		0			0
$905$	1.90385			0.0632862
$906$		0			0
$907$	−10.3539			−0.343795			−0.171898	−	0.985115i	$-0.554990\pi$
$907$	−10.3539			−0.343795			−0.171898	+	0.985115i	$0.554990\pi$
$908$		0			0
$909$	−13.1514			−0.436203
$910$		0			0
$911$	−40.1853			−1.33140			−0.665699	−	0.746220i	$-0.731867\pi$
$911$	−40.1853			−1.33140			−0.665699	+	0.746220i	$0.731867\pi$
$912$		0			0
$913$	15.1512			0.501430
$914$		0			0
$915$		−	3.36597i		−	0.111275i
$916$		0			0
$917$			40.0641i			1.32303i
$918$		0			0
$919$	−36.2326			−1.19520			−0.597601	−	0.801793i	$-0.703879\pi$
$919$	−36.2326			−1.19520			−0.597601	+	0.801793i	$0.703879\pi$
$920$		0			0
$921$	−24.4269			−0.804893
$922$		0			0
$923$		−	22.6853i		−	0.746696i
$924$		0			0
$925$		−	5.01787i		−	0.164987i
$926$		0			0
$927$	5.37041			0.176387
$928$		0			0
$929$	−58.9418			−1.93382			−0.966909	−	0.255121i	$-0.917885\pi$
$929$	−58.9418			−1.93382			−0.966909	+	0.255121i	$0.917885\pi$
$930$		0			0
$931$	61.0179			1.99978
$932$		0			0
$933$	−2.74305			−0.0898035
$934$		0			0
$935$	−5.90980			−0.193271
$936$		0			0
$937$		−	60.0374i		−	1.96134i	−0.195681	−	0.980668i	$-0.562692\pi$
$937$		−	60.0374i		−	1.96134i	0.195681	−	0.980668i	$-0.437308\pi$
$938$		0			0
$939$	−3.88307			−0.126719
$940$		0			0
$941$		−	54.3566i		−	1.77198i	−0.463709	−	0.885988i	$-0.653482\pi$
$941$		−	54.3566i		−	1.77198i	0.463709	−	0.885988i	$-0.346518\pi$
$942$		0			0
$943$	12.7463	+	49.4080i	0.415076	+	1.60895i
$944$		0			0
$945$	3.79952			0.123598
$946$		0			0
$947$			11.9091i			0.386995i	0.981101	+	0.193498i	$0.0619832\pi$
$947$			11.9091i			0.386995i	−0.981101	+	0.193498i	$0.938017\pi$
$948$		0			0
$949$	−1.34904			−0.0437917
$950$		0			0
$951$			26.5346i			0.860444i
$952$		0			0
$953$		−	8.43091i		−	0.273104i	−0.990633	−	0.136552i	$-0.956398\pi$
$953$		−	8.43091i		−	0.273104i	0.990633	−	0.136552i	$-0.0436021\pi$
$954$		0			0
$955$		−	16.6906i		−	0.540096i
$956$		0			0
$957$			2.60678i			0.0842653i
$958$		0			0
$959$			6.37804i			0.205958i
$960$		0			0
$961$	−10.8089			−0.348675
$962$		0			0
$963$	17.9219			0.577526
$964$		0			0
$965$		−	11.9024i		−	0.383151i
$966$		0			0
$967$			1.93945i			0.0623686i	0.999514	+	0.0311843i	$0.00992788\pi$
$967$			1.93945i			0.0623686i	−0.999514	+	0.0311843i	$0.990072\pi$
$968$		0			0
$969$	−13.1410			−0.422149
$970$		0			0
$971$	7.00608			0.224836			0.112418	−	0.993661i	$-0.464140\pi$
$971$	7.00608			0.224836			0.112418	+	0.993661i	$0.464140\pi$
$972$		0			0
$973$		−	52.1145i		−	1.67071i
$974$		0			0
$975$			2.60963i			0.0835750i
$976$		0			0
$977$			18.8261i			0.602302i	0.953576	+	0.301151i	$0.0973708\pi$
$977$			18.8261i			0.602302i	−0.953576	+	0.301151i	$0.902629\pi$
$978$		0			0
$979$		−	18.8578i		−	0.602699i
$980$		0			0
$981$		−	1.27328i		−	0.0406528i
$982$		0			0
$983$	46.2604			1.47548			0.737740	−	0.675085i	$-0.235893\pi$
$983$	46.2604			1.47548			0.737740	+	0.675085i	$0.235893\pi$
$984$		0			0
$985$			15.3515i			0.489141i
$986$		0			0
$987$	16.9561			0.539719
$988$		0			0
$989$	−7.11776	−	27.5903i	−0.226332	−	0.877321i
$990$		0			0
$991$			10.0493i			0.319228i	0.987180	+	0.159614i	$0.0510249\pi$
$991$			10.0493i			0.319228i	−0.987180	+	0.159614i	$0.948975\pi$
$992$		0			0
$993$	28.5918			0.907333
$994$		0			0
$995$			17.4361i			0.552761i
$996$		0			0
$997$	−34.7460			−1.10042			−0.550209	−	0.835027i	$-0.685452\pi$
$997$	−34.7460			−1.10042			−0.550209	+	0.835027i	$0.685452\pi$
$998$		0			0
$999$	−5.01787			−0.158758

Twists

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

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

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)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} +1.00000i q^{5} -3.79952 q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} +1.00000i q^{5} -3.79952 q^{7} -1.00000 q^{9} -3.69013 q^{11} +2.60963 q^{13} +1.00000 q^{15} -1.60151i q^{17} +8.20535 q^{19} +3.79952i q^{21} +(-1.19801 - 4.64379i) q^{23} -1.00000 q^{25} +1.00000i q^{27} +0.706420 q^{29} -6.46598i q^{31} +3.69013i q^{33} -3.79952i q^{35} +5.01787i q^{37} -2.60963i q^{39} -10.6396 q^{41} +5.94134 q^{43} -1.00000i q^{45} -4.46270i q^{47} +7.43635 q^{49} -1.60151 q^{51} +7.25579i q^{53} -3.69013i q^{55} -8.20535i q^{57} +6.04730i q^{59} -3.36597i q^{61} +3.79952 q^{63} +2.60963i q^{65} -13.5383 q^{67} +(-4.64379 + 1.19801i) q^{69} -8.69292i q^{71} -0.516947 q^{73} +1.00000i q^{75} +14.0207 q^{77} +3.23175 q^{79} +1.00000 q^{81} -4.10586 q^{83} +1.60151 q^{85} -0.706420i q^{87} +5.11034i q^{89} -9.91534 q^{91} -6.46598 q^{93} +8.20535i q^{95} +19.4018i q^{97} +3.69013 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 8 q^{7} - 16 q^{9}+O(q^{10}) \) 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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