LMFDB - Embedded newform 1440.2.bj.a.593.18

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1440,2,Mod(17,1440)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1440, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 2, 2, 1]))

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

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

chi := DirichletCharacter("1440.17");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1440 = 2^{5} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1440.bj (of order $4$, degree $2$, not 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:	$11.4984578911$
Analytic rank:	$0$
Dimension:	$48$
Relative dimension:	$24$ over $\Q(i)$
Twist minimal:	no (minimal twist has level 360)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			593.18
Character	$\chi$	$=$	1440.593
Dual form			1440.2.bj.a.17.18

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.42597 - 1.72238i) q^{5} +(-3.11972 + 3.11972i) q^{7} +O(q^{10})$	$q+(1.42597 - 1.72238i) q^{5} +(-3.11972 + 3.11972i) q^{7} +1.17794 q^{11} +(2.15100 - 2.15100i) q^{13} +(1.33507 + 1.33507i) q^{17} -0.322444 q^{19} +(4.71347 - 4.71347i) q^{23} +(-0.933209 - 4.91214i) q^{25} +6.63043i q^{29} +0.0675826 q^{31} +(0.924721 + 9.82198i) q^{35} +(7.60938 + 7.60938i) q^{37} -3.19684i q^{41} +(6.70505 - 6.70505i) q^{43} +(7.34279 + 7.34279i) q^{47} -12.4653i q^{49} +(5.73432 + 5.73432i) q^{53} +(1.67971 - 2.02887i) q^{55} +8.68448i q^{59} -12.5620i q^{61} +(-0.637580 - 6.77210i) q^{65} +(-1.87740 - 1.87740i) q^{67} -4.18221i q^{71} +(3.97893 + 3.97893i) q^{73} +(-3.67485 + 3.67485i) q^{77} -9.66644i q^{79} +(0.585119 + 0.585119i) q^{83} +(4.20329 - 0.395732i) q^{85} -0.557322 q^{89} +13.4210i q^{91} +(-0.459796 + 0.555372i) q^{95} +(-10.5772 + 10.5772i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q+O(q^{10}) $ 48 * q	$ 48 q - 32 q^{31} - 32 q^{97}+O(q^{100}) $ 48 * q - 32 * q^31 - 32 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1440\mathbb{Z}\right)^\times$.

$n$	$577$	$641$	$901$	$991$
$\chi(n)$	$e\left(\frac{3}{4}\right)$	$-1$	$-1$	$1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	1.42597	−	1.72238i	0.637714	−	0.770273i
$5$	1.42597	−	1.72238i	0.637714	−	0.770273i
$6$		0			0
$7$	−3.11972	+	3.11972i	−1.17914	+	1.17914i	−0.199179	+	0.979963i	$0.563828\pi$
$7$	−3.11972	+	3.11972i	−1.17914	+	1.17914i	−0.979963	+	0.199179i	$0.936172\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	1.17794			0.355163			0.177581	−	0.984106i	$-0.443173\pi$
$11$	1.17794			0.355163			0.177581	+	0.984106i	$0.443173\pi$
$12$		0			0
$13$	2.15100	−	2.15100i	0.596579	−	0.596579i	−0.342822	−	0.939401i	$-0.611383\pi$
$13$	2.15100	−	2.15100i	0.596579	−	0.596579i	0.939401	+	0.342822i	$0.111383\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	1.33507	+	1.33507i	0.323803	+	0.323803i	0.850224	−	0.526421i	$-0.176466\pi$
$17$	1.33507	+	1.33507i	0.323803	+	0.323803i	−0.526421	+	0.850224i	$0.676466\pi$
$18$		0			0
$19$	−0.322444			−0.0739738			−0.0369869	−	0.999316i	$-0.511776\pi$
$19$	−0.322444			−0.0739738			−0.0369869	+	0.999316i	$0.511776\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	4.71347	−	4.71347i	0.982826	−	0.982826i	−0.0170294	−	0.999855i	$-0.505421\pi$
$23$	4.71347	−	4.71347i	0.982826	−	0.982826i	0.999855	+	0.0170294i	$0.00542088\pi$
$24$		0			0
$25$	−0.933209	−	4.91214i	−0.186642	−	0.982428i
$26$		0			0
$27$		0			0
$28$		0			0
$29$			6.63043i			1.23124i	0.788043	+	0.615620i	$0.211094\pi$
$29$			6.63043i			1.23124i	−0.788043	+	0.615620i	$0.788906\pi$
$30$		0			0
$31$	0.0675826			0.0121382			0.00606909	−	0.999982i	$-0.498068\pi$
$31$	0.0675826			0.0121382			0.00606909	+	0.999982i	$0.498068\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	0.924721	+	9.82198i	0.156306	+	1.66022i
$36$		0			0
$37$	7.60938	+	7.60938i	1.25097	+	1.25097i	0.955284	+	0.295690i	$0.0955496\pi$
$37$	7.60938	+	7.60938i	1.25097	+	1.25097i	0.295690	+	0.955284i	$0.404450\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	3.19684i		−	0.499262i	−0.968341	−	0.249631i	$-0.919691\pi$
$41$		−	3.19684i		−	0.499262i	0.968341	−	0.249631i	$-0.0803093\pi$
$42$		0			0
$43$	6.70505	−	6.70505i	1.02251	−	1.02251i	0.0227703	−	0.999741i	$-0.492751\pi$
$43$	6.70505	−	6.70505i	1.02251	−	1.02251i	0.999741	−	0.0227703i	$-0.00724865\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	7.34279	+	7.34279i	1.07106	+	1.07106i	0.997274	+	0.0737817i	$0.0235068\pi$
$47$	7.34279	+	7.34279i	1.07106	+	1.07106i	0.0737817	+	0.997274i	$0.476493\pi$
$48$		0			0
$49$		−	12.4653i		−	1.78075i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	5.73432	+	5.73432i	0.787669	+	0.787669i	0.981112	−	0.193442i	$-0.0619653\pi$
$53$	5.73432	+	5.73432i	0.787669	+	0.787669i	−0.193442	+	0.981112i	$0.561965\pi$
$54$		0			0
$55$	1.67971	−	2.02887i	0.226492	−	0.273572i
$56$		0			0
$57$		0			0
$58$		0			0
$59$			8.68448i			1.13062i	0.824877	+	0.565312i	$0.191244\pi$
$59$			8.68448i			1.13062i	−0.824877	+	0.565312i	$0.808756\pi$
$60$		0			0
$61$		−	12.5620i		−	1.60840i	−0.594359	−	0.804199i	$-0.702594\pi$
$61$		−	12.5620i		−	1.60840i	0.594359	−	0.804199i	$-0.297406\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−0.637580	−	6.77210i	−0.0790821	−	0.839976i
$66$		0			0
$67$	−1.87740	−	1.87740i	−0.229360	−	0.229360i	0.583065	−	0.812425i	$-0.301853\pi$
$67$	−1.87740	−	1.87740i	−0.229360	−	0.229360i	−0.812425	+	0.583065i	$0.801853\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	4.18221i		−	0.496337i	−0.968717	−	0.248168i	$-0.920171\pi$
$71$		−	4.18221i		−	0.496337i	0.968717	−	0.248168i	$-0.0798286\pi$
$72$		0			0
$73$	3.97893	+	3.97893i	0.465699	+	0.465699i	0.900518	−	0.434819i	$-0.143188\pi$
$73$	3.97893	+	3.97893i	0.465699	+	0.465699i	−0.434819	+	0.900518i	$0.643188\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−3.67485	+	3.67485i	−0.418788	+	0.418788i
$78$		0			0
$79$		−	9.66644i		−	1.08756i	−0.839228	−	0.543780i	$-0.816993\pi$
$79$		−	9.66644i		−	1.08756i	0.839228	−	0.543780i	$-0.183007\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	0.585119	+	0.585119i	0.0642251	+	0.0642251i	0.738490	−	0.674265i	$-0.235539\pi$
$83$	0.585119	+	0.585119i	0.0642251	+	0.0642251i	−0.674265	+	0.738490i	$0.735539\pi$
$84$		0			0
$85$	4.20329	−	0.395732i	0.455911	−	0.0429231i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−0.557322			−0.0590760			−0.0295380	−	0.999564i	$-0.509404\pi$
$89$	−0.557322			−0.0590760			−0.0295380	+	0.999564i	$0.509404\pi$
$90$		0			0
$91$			13.4210i			1.40690i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−0.459796	+	0.555372i	−0.0471741	+	0.0569800i
$96$		0			0
$97$	−10.5772	+	10.5772i	−1.07395	+	1.07395i	−0.0769138	+	0.997038i	$0.524507\pi$
$97$	−10.5772	+	10.5772i	−1.07395	+	1.07395i	−0.997038	+	0.0769138i	$0.975493\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	9.52870			0.948141			0.474071	−	0.880487i	$-0.342784\pi$
$101$	9.52870			0.948141			0.474071	+	0.880487i	$0.342784\pi$
$102$		0			0
$103$	0.162450	+	0.162450i	0.0160066	+	0.0160066i	0.715065	−	0.699058i	$-0.246397\pi$
$103$	0.162450	+	0.162450i	0.0160066	+	0.0160066i	−0.699058	+	0.715065i	$0.746397\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	11.7955	−	11.7955i	1.14032	−	1.14032i	0.151925	−	0.988392i	$-0.451453\pi$
$107$	11.7955	−	11.7955i	1.14032	−	1.14032i	0.988392	−	0.151925i	$-0.0485471\pi$
$108$		0			0
$109$	−10.8601			−1.04021			−0.520105	−	0.854102i	$-0.674107\pi$
$109$	−10.8601			−1.04021			−0.520105	+	0.854102i	$0.674107\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	8.74476	−	8.74476i	0.822637	−	0.822637i	−0.163848	−	0.986486i	$-0.552391\pi$
$113$	8.74476	−	8.74476i	0.822637	−	0.822637i	0.986486	+	0.163848i	$0.0523907\pi$
$114$		0			0
$115$	−1.39713	−	14.8397i	−0.130283	−	1.38381i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−8.33011			−0.763620
$120$		0			0
$121$	−9.61245			−0.873859
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−9.79132	−	5.39723i	−0.875762	−	0.482743i
$126$		0			0
$127$	3.83755	−	3.83755i	0.340528	−	0.340528i	−0.516038	−	0.856566i	$-0.672594\pi$
$127$	3.83755	−	3.83755i	0.340528	−	0.340528i	0.856566	+	0.516038i	$0.172594\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−1.85715			−0.162260			−0.0811298	−	0.996704i	$-0.525853\pi$
$131$	−1.85715			−0.162260			−0.0811298	+	0.996704i	$0.525853\pi$
$132$		0			0
$133$	1.00593	−	1.00593i	0.0872256	−	0.0872256i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	1.89240	+	1.89240i	0.161678	+	0.161678i	0.783310	−	0.621631i	$-0.213530\pi$
$137$	1.89240	+	1.89240i	0.161678	+	0.161678i	−0.621631	+	0.783310i	$0.713530\pi$
$138$		0			0
$139$	18.4370			1.56380			0.781902	−	0.623401i	$-0.214250\pi$
$139$	18.4370			1.56380			0.781902	+	0.623401i	$0.214250\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	2.53375	−	2.53375i	0.211883	−	0.211883i
$144$		0			0
$145$	11.4201	+	9.45481i	0.948392	+	0.785179i
$146$		0			0
$147$		0			0
$148$		0			0
$149$			1.24573i			0.102054i	0.998697	+	0.0510272i	$0.0162495\pi$
$149$			1.24573i			0.102054i	−0.998697	+	0.0510272i	$0.983750\pi$
$150$		0			0
$151$	−16.7587			−1.36380			−0.681902	−	0.731444i	$-0.738847\pi$
$151$	−16.7587			−1.36380			−0.681902	+	0.731444i	$0.738847\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	0.0963708	−	0.116403i	0.00774069	−	0.00934972i
$156$		0			0
$157$	6.90784	+	6.90784i	0.551306	+	0.551306i	0.926818	−	0.375512i	$-0.122533\pi$
$157$	6.90784	+	6.90784i	0.551306	+	0.551306i	−0.375512	+	0.926818i	$0.622533\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			29.4094i			2.31778i
$162$		0			0
$163$	−6.07821	+	6.07821i	−0.476082	+	0.476082i	−0.903876	−	0.427794i	$-0.859291\pi$
$163$	−6.07821	+	6.07821i	−0.476082	+	0.476082i	0.427794	+	0.903876i	$0.359291\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−2.82159	−	2.82159i	−0.218341	−	0.218341i	0.589458	−	0.807799i	$-0.299341\pi$
$167$	−2.82159	−	2.82159i	−0.218341	−	0.218341i	−0.807799	+	0.589458i	$0.799341\pi$
$168$		0			0
$169$			3.74643i			0.288187i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	7.27921	+	7.27921i	0.553428	+	0.553428i	0.927428	−	0.374001i	$-0.122014\pi$
$173$	7.27921	+	7.27921i	0.553428	+	0.553428i	−0.374001	+	0.927428i	$0.622014\pi$
$174$		0			0
$175$	18.2358	+	12.4131i	1.37850	+	0.938345i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			9.56604i			0.714999i	0.933913	+	0.357500i	$0.116371\pi$
$179$			9.56604i			0.714999i	−0.933913	+	0.357500i	$0.883629\pi$
$180$		0			0
$181$			4.44857i			0.330659i	0.986238	+	0.165330i	$0.0528688\pi$
$181$			4.44857i			0.330659i	−0.986238	+	0.165330i	$0.947131\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	23.9570	−	2.25551i	1.76136	−	0.165828i
$186$		0			0
$187$	1.57264	+	1.57264i	0.115003	+	0.115003i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			3.16704i			0.229159i	0.993414	+	0.114579i	$0.0365521\pi$
$191$			3.16704i			0.229159i	−0.993414	+	0.114579i	$0.963448\pi$
$192$		0			0
$193$	1.07726	+	1.07726i	0.0775426	+	0.0775426i	0.744814	−	0.667272i	$-0.232538\pi$
$193$	1.07726	+	1.07726i	0.0775426	+	0.0775426i	−0.667272	+	0.744814i	$0.732538\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−5.59062	+	5.59062i	−0.398315	+	0.398315i	−0.877638	−	0.479323i	$-0.840882\pi$
$197$	−5.59062	+	5.59062i	−0.398315	+	0.398315i	0.479323	+	0.877638i	$0.340882\pi$
$198$		0			0
$199$		−	15.7707i		−	1.11796i	−0.829182	−	0.558978i	$-0.811194\pi$
$199$		−	15.7707i		−	1.11796i	0.829182	−	0.558978i	$-0.188806\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−20.6851	−	20.6851i	−1.45181	−	1.45181i
$204$		0			0
$205$	−5.50618	−	4.55860i	−0.384568	−	0.318386i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−0.379820			−0.0262727
$210$		0			0
$211$			10.9274i			0.752273i	0.926564	+	0.376137i	$0.122748\pi$
$211$			10.9274i			0.752273i	−0.926564	+	0.376137i	$0.877252\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−1.98746	−	21.1099i	−0.135543	−	1.43968i
$216$		0			0
$217$	−0.210839	+	0.210839i	−0.0143127	+	0.0143127i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	5.74348			0.386348
$222$		0			0
$223$	−0.947865	−	0.947865i	−0.0634737	−	0.0634737i	0.674657	−	0.738131i	$-0.264291\pi$
$223$	−0.947865	−	0.947865i	−0.0634737	−	0.0634737i	−0.738131	+	0.674657i	$0.764291\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−4.87410	+	4.87410i	−0.323506	+	0.323506i	−0.850110	−	0.526605i	$-0.823465\pi$
$227$	−4.87410	+	4.87410i	−0.323506	+	0.323506i	0.526605	+	0.850110i	$0.323465\pi$
$228$		0			0
$229$	−17.1740			−1.13489			−0.567444	−	0.823412i	$-0.692068\pi$
$229$	−17.1740			−1.13489			−0.567444	+	0.823412i	$0.692068\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−17.1696	+	17.1696i	−1.12482	+	1.12482i	−0.133811	+	0.991007i	$0.542722\pi$
$233$	−17.1696	+	17.1696i	−1.12482	+	1.12482i	−0.991007	+	0.133811i	$0.957278\pi$
$234$		0			0
$235$	23.1177	−	2.17649i	1.50803	−	0.141978i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−19.8347			−1.28300			−0.641501	−	0.767122i	$-0.721688\pi$
$239$	−19.8347			−1.28300			−0.641501	+	0.767122i	$0.721688\pi$
$240$		0			0
$241$	−2.89615			−0.186557			−0.0932787	−	0.995640i	$-0.529735\pi$
$241$	−2.89615			−0.186557			−0.0932787	+	0.995640i	$0.529735\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−21.4700	−	17.7751i	−1.37167	−	1.13561i
$246$		0			0
$247$	−0.693576	+	0.693576i	−0.0441312	+	0.0441312i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	8.93270			0.563827			0.281914	−	0.959440i	$-0.409031\pi$
$251$	8.93270			0.563827			0.281914	+	0.959440i	$0.409031\pi$
$252$		0			0
$253$	5.55219	−	5.55219i	0.349063	−	0.349063i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−12.4301	−	12.4301i	−0.775367	−	0.775367i	0.203672	−	0.979039i	$-0.434712\pi$
$257$	−12.4301	−	12.4301i	−0.775367	−	0.775367i	−0.979039	+	0.203672i	$0.934712\pi$
$258$		0			0
$259$	−47.4782			−2.95015
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−12.1529	+	12.1529i	−0.749383	+	0.749383i	−0.974363	−	0.224981i	$-0.927768\pi$
$263$	−12.1529	+	12.1529i	−0.749383	+	0.749383i	0.224981	+	0.974363i	$0.427768\pi$
$264$		0			0
$265$	18.0537	−	1.69972i	1.10903	−	0.104413i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	19.0912i		−	1.16401i	−0.813185	−	0.582005i	$-0.802268\pi$
$269$		−	19.0912i		−	1.16401i	0.813185	−	0.582005i	$-0.197732\pi$
$270$		0			0
$271$	−13.6087			−0.826670			−0.413335	−	0.910579i	$-0.635636\pi$
$271$	−13.6087			−0.826670			−0.413335	+	0.910579i	$0.635636\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−1.09927	−	5.78622i	−0.0662882	−	0.348922i
$276$		0			0
$277$	−13.9103	−	13.9103i	−0.835788	−	0.835788i	0.152514	−	0.988301i	$-0.451263\pi$
$277$	−13.9103	−	13.9103i	−0.835788	−	0.835788i	−0.988301	+	0.152514i	$0.951263\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	13.9435i		−	0.831799i	−0.909411	−	0.415899i	$-0.863467\pi$
$281$		−	13.9435i		−	0.831799i	0.909411	−	0.415899i	$-0.136533\pi$
$282$		0			0
$283$	−14.3526	+	14.3526i	−0.853174	+	0.853174i	−0.990523	−	0.137349i	$-0.956142\pi$
$283$	−14.3526	+	14.3526i	−0.853174	+	0.853174i	0.137349	+	0.990523i	$0.456142\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	9.97323	+	9.97323i	0.588701	+	0.588701i
$288$		0			0
$289$		−	13.4352i		−	0.790303i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−12.4180	−	12.4180i	−0.725465	−	0.725465i	0.244248	−	0.969713i	$-0.421459\pi$
$293$	−12.4180	−	12.4180i	−0.725465	−	0.725465i	−0.969713	+	0.244248i	$0.921459\pi$
$294$		0			0
$295$	14.9580	+	12.3838i	0.870889	+	0.721014i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		−	20.2773i		−	1.17267i
$300$		0			0
$301$			41.8357i			2.41137i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−21.6366	−	17.9131i	−1.23891	−	1.02570i
$306$		0			0
$307$	−14.9940	−	14.9940i	−0.855753	−	0.855753i	0.135082	−	0.990834i	$-0.456870\pi$
$307$	−14.9940	−	14.9940i	−0.855753	−	0.855753i	−0.990834	+	0.135082i	$0.956870\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			32.7316i			1.85604i	0.372534	+	0.928019i	$0.378489\pi$
$311$			32.7316i			1.85604i	−0.372534	+	0.928019i	$0.621511\pi$
$312$		0			0
$313$	8.25286	+	8.25286i	0.466479	+	0.466479i	0.900772	−	0.434293i	$-0.143002\pi$
$313$	8.25286	+	8.25286i	0.466479	+	0.466479i	−0.434293	+	0.900772i	$0.643002\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	8.30940	−	8.30940i	0.466702	−	0.466702i	−0.434142	−	0.900844i	$-0.642948\pi$
$317$	8.30940	−	8.30940i	0.466702	−	0.466702i	0.900844	+	0.434142i	$0.142948\pi$
$318$		0			0
$319$			7.81027i			0.437291i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−0.430487	−	0.430487i	−0.0239529	−	0.0239529i
$324$		0			0
$325$	−12.5733	−	8.55867i	−0.697443	−	0.474749i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−45.8149			−2.52586
$330$		0			0
$331$		−	7.43429i		−	0.408626i	−0.978906	−	0.204313i	$-0.934504\pi$
$331$		−	7.43429i		−	0.408626i	0.978906	−	0.204313i	$-0.0654960\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−5.91071	+	0.556482i	−0.322936	+	0.0304038i
$336$		0			0
$337$	−3.88868	+	3.88868i	−0.211830	+	0.211830i	−0.805044	−	0.593215i	$-0.797859\pi$
$337$	−3.88868	+	3.88868i	−0.211830	+	0.211830i	0.593215	+	0.805044i	$0.297859\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	0.0796083			0.00431103
$342$		0			0
$343$	17.0501	+	17.0501i	0.920620	+	0.920620i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	20.7432	−	20.7432i	1.11355	−	1.11355i	0.120889	−	0.992666i	$-0.461426\pi$
$347$	20.7432	−	20.7432i	1.11355	−	1.11355i	0.992666	−	0.120889i	$-0.0385744\pi$
$348$		0			0
$349$	−3.04055			−0.162757			−0.0813783	−	0.996683i	$-0.525932\pi$
$349$	−3.04055			−0.162757			−0.0813783	+	0.996683i	$0.525932\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	3.46489	−	3.46489i	0.184417	−	0.184417i	−0.608860	−	0.793278i	$-0.708373\pi$
$353$	3.46489	−	3.46489i	0.184417	−	0.184417i	0.793278	+	0.608860i	$0.208373\pi$
$354$		0			0
$355$	−7.20337	−	5.96371i	−0.382315	−	0.316521i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	8.63299			0.455632			0.227816	−	0.973704i	$-0.426842\pi$
$359$	8.63299			0.455632			0.227816	+	0.973704i	$0.426842\pi$
$360$		0			0
$361$	−18.8960			−0.994528
$362$		0			0
$363$		0			0
$364$		0			0
$365$	12.5271	−	1.17940i	0.655698	−	0.0617327i
$366$		0			0
$367$	−15.3844	+	15.3844i	−0.803060	+	0.803060i	−0.983573	−	0.180512i	$-0.942224\pi$
$367$	−15.3844	+	15.3844i	−0.803060	+	0.803060i	0.180512	+	0.983573i	$0.442224\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−35.7789			−1.85755
$372$		0			0
$373$	7.40616	−	7.40616i	0.383476	−	0.383476i	−0.488877	−	0.872353i	$-0.662593\pi$
$373$	7.40616	−	7.40616i	0.383476	−	0.383476i	0.872353	+	0.488877i	$0.162593\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	14.2620	+	14.2620i	0.734532	+	0.734532i
$378$		0			0
$379$	−6.54952			−0.336426			−0.168213	−	0.985751i	$-0.553800\pi$
$379$	−6.54952			−0.336426			−0.168213	+	0.985751i	$0.553800\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	14.8778	−	14.8778i	0.760222	−	0.760222i	−0.216140	−	0.976362i	$-0.569347\pi$
$383$	14.8778	−	14.8778i	0.760222	−	0.760222i	0.976362	+	0.216140i	$0.0693468\pi$
$384$		0			0
$385$	1.08927	+	11.5697i	0.0555142	+	0.589648i
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	0.254115i		−	0.0128842i	−0.999979	−	0.00644208i	$-0.997949\pi$
$389$		−	0.254115i		−	0.0128842i	0.999979	−	0.00644208i	$-0.00205059\pi$
$390$		0			0
$391$	12.5857			0.636484
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−16.6493	−	13.7841i	−0.837718	−	0.693552i
$396$		0			0
$397$	20.6584	+	20.6584i	1.03682	+	1.03682i	0.999296	+	0.0375215i	$0.0119463\pi$
$397$	20.6584	+	20.6584i	1.03682	+	1.03682i	0.0375215	+	0.999296i	$0.488054\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			20.6863i			1.03303i	0.856279	+	0.516513i	$0.172770\pi$
$401$			20.6863i			1.03303i	−0.856279	+	0.516513i	$0.827230\pi$
$402$		0			0
$403$	0.145370	−	0.145370i	0.00724139	−	0.00724139i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	8.96340	+	8.96340i	0.444299	+	0.444299i
$408$		0			0
$409$		−	26.9787i		−	1.33401i	−0.745053	−	0.667005i	$-0.767576\pi$
$409$		−	26.9787i		−	1.33401i	0.745053	−	0.667005i	$-0.232424\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−27.0931	−	27.0931i	−1.33317	−	1.33317i
$414$		0			0
$415$	1.84216	−	0.173436i	0.0904282	−	0.00851364i
$416$		0			0
$417$		0			0
$418$		0			0
$419$			34.3778i			1.67946i	0.543001	+	0.839732i	$0.317288\pi$
$419$			34.3778i			1.67946i	−0.543001	+	0.839732i	$0.682712\pi$
$420$		0			0
$421$		−	4.05773i		−	0.197762i	−0.995099	−	0.0988809i	$-0.968474\pi$
$421$		−	4.05773i		−	0.197762i	0.995099	−	0.0988809i	$-0.0315263\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	5.31217	−	7.80398i	0.257678	−	0.378549i
$426$		0			0
$427$	39.1899	+	39.1899i	1.89653	+	1.89653i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	16.0853i		−	0.774800i	−0.921912	−	0.387400i	$-0.873373\pi$
$431$		−	16.0853i		−	0.774800i	0.921912	−	0.387400i	$-0.126627\pi$
$432$		0			0
$433$	−3.04812	−	3.04812i	−0.146483	−	0.146483i	0.630062	−	0.776545i	$-0.283030\pi$
$433$	−3.04812	−	3.04812i	−0.146483	−	0.146483i	−0.776545	+	0.630062i	$0.783030\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−1.51983	+	1.51983i	−0.0727033	+	0.0727033i
$438$		0			0
$439$			4.78470i			0.228361i	0.993460	+	0.114181i	$0.0364243\pi$
$439$			4.78470i			0.228361i	−0.993460	+	0.114181i	$0.963576\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	11.2608	+	11.2608i	0.535019	+	0.535019i	0.922062	−	0.387043i	$-0.126503\pi$
$443$	11.2608	+	11.2608i	0.535019	+	0.535019i	−0.387043	+	0.922062i	$0.626503\pi$
$444$		0			0
$445$	−0.794726	+	0.959922i	−0.0376736	+	0.0455047i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	7.49133			0.353538			0.176769	−	0.984252i	$-0.443435\pi$
$449$	7.49133			0.353538			0.176769	+	0.984252i	$0.443435\pi$
$450$		0			0
$451$		−	3.76569i		−	0.177319i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	23.1161	+	19.1380i	1.08370	+	0.897202i
$456$		0			0
$457$	−14.2198	+	14.2198i	−0.665173	+	0.665173i	−0.956595	−	0.291422i	$-0.905872\pi$
$457$	−14.2198	+	14.2198i	−0.665173	+	0.665173i	0.291422	+	0.956595i	$0.405872\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	19.4366			0.905254			0.452627	−	0.891700i	$-0.350487\pi$
$461$	19.4366			0.905254			0.452627	+	0.891700i	$0.350487\pi$
$462$		0			0
$463$	2.18789	+	2.18789i	0.101680	+	0.101680i	0.756117	−	0.654437i	$-0.227094\pi$
$463$	2.18789	+	2.18789i	0.101680	+	0.101680i	−0.654437	+	0.756117i	$0.727094\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−2.06992	+	2.06992i	−0.0957845	+	0.0957845i	−0.753375	−	0.657591i	$-0.771576\pi$
$467$	−2.06992	+	2.06992i	−0.0957845	+	0.0957845i	0.657591	+	0.753375i	$0.271576\pi$
$468$		0			0
$469$	11.7139			0.540897
$470$		0			0
$471$		0			0
$472$		0			0
$473$	7.89816	−	7.89816i	0.363158	−	0.363158i
$474$		0			0
$475$	0.300908	+	1.58389i	0.0138066	+	0.0726739i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−27.5481			−1.25871			−0.629353	−	0.777120i	$-0.716680\pi$
$479$	−27.5481			−1.25871			−0.629353	+	0.777120i	$0.716680\pi$
$480$		0			0
$481$	32.7355			1.49261
$482$		0			0
$483$		0			0
$484$		0			0
$485$	3.13520	+	33.3008i	0.142362	+	1.51211i
$486$		0			0
$487$	−27.7336	+	27.7336i	−1.25673	+	1.25673i	−0.304083	+	0.952646i	$0.598350\pi$
$487$	−27.7336	+	27.7336i	−1.25673	+	1.25673i	−0.952646	+	0.304083i	$0.901650\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	2.58679			0.116740			0.0583700	−	0.998295i	$-0.481410\pi$
$491$	2.58679			0.116740			0.0583700	+	0.998295i	$0.481410\pi$
$492$		0			0
$493$	−8.85213	+	8.85213i	−0.398680	+	0.398680i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	13.0473	+	13.0473i	0.585252	+	0.585252i
$498$		0			0
$499$	−25.5951			−1.14579			−0.572897	−	0.819627i	$-0.694181\pi$
$499$	−25.5951			−1.14579			−0.572897	+	0.819627i	$0.694181\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	9.39909	−	9.39909i	0.419085	−	0.419085i	−0.465804	−	0.884888i	$-0.654235\pi$
$503$	9.39909	−	9.39909i	0.419085	−	0.419085i	0.884888	+	0.465804i	$0.154235\pi$
$504$		0			0
$505$	13.5877	−	16.4121i	0.604643	−	0.730328i
$506$		0			0
$507$		0			0
$508$		0			0
$509$			11.5075i			0.510059i	0.966933	+	0.255030i	$0.0820852\pi$
$509$			11.5075i			0.510059i	−0.966933	+	0.255030i	$0.917915\pi$
$510$		0			0
$511$	−24.8263			−1.09825
$512$		0			0
$513$		0			0
$514$		0			0
$515$	0.511449	−	0.0481519i	0.0225371	−	0.00212183i
$516$		0			0
$517$	8.64938	+	8.64938i	0.380399	+	0.380399i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	8.12668i		−	0.356036i	−0.984027	−	0.178018i	$-0.943031\pi$
$521$		−	8.12668i		−	0.356036i	0.984027	−	0.178018i	$-0.0569686\pi$
$522$		0			0
$523$	23.2646	−	23.2646i	1.01729	−	1.01729i	0.0174440	−	0.999848i	$-0.494447\pi$
$523$	23.2646	−	23.2646i	1.01729	−	1.01729i	0.999848	−	0.0174440i	$-0.00555287\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	0.0902278	+	0.0902278i	0.00393038	+	0.00393038i
$528$		0			0
$529$		−	21.4335i		−	0.931892i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−6.87638	−	6.87638i	−0.297849	−	0.297849i
$534$		0			0
$535$	−3.49633	−	37.1365i	−0.151160	−	1.60555i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	14.6834i		−	0.632458i
$540$		0			0
$541$		−	21.6496i		−	0.930787i	−0.885104	−	0.465393i	$-0.845913\pi$
$541$		−	21.6496i		−	0.930787i	0.885104	−	0.465393i	$-0.154087\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−15.4862	+	18.7053i	−0.663357	+	0.801246i
$546$		0			0
$547$	−15.4574	−	15.4574i	−0.660912	−	0.660912i	0.294683	−	0.955595i	$-0.404786\pi$
$547$	−15.4574	−	15.4574i	−0.660912	−	0.660912i	−0.955595	+	0.294683i	$0.904786\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	2.13794i		−	0.0910795i
$552$		0			0
$553$	30.1566	+	30.1566i	1.28239	+	1.28239i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	10.9318	−	10.9318i	0.463194	−	0.463194i	−0.436507	−	0.899701i	$-0.643785\pi$
$557$	10.9318	−	10.9318i	0.463194	−	0.463194i	0.899701	+	0.436507i	$0.143785\pi$
$558$		0			0
$559$		−	28.8451i		−	1.22002i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	13.9005	+	13.9005i	0.585835	+	0.585835i	0.936501	−	0.350666i	$-0.114045\pi$
$563$	13.9005	+	13.9005i	0.585835	+	0.585835i	−0.350666	+	0.936501i	$0.614045\pi$
$564$		0			0
$565$	−2.59205	−	27.5316i	−0.109048	−	1.15826i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−3.03632			−0.127289			−0.0636446	−	0.997973i	$-0.520272\pi$
$569$	−3.03632			−0.127289			−0.0636446	+	0.997973i	$0.520272\pi$
$570$		0			0
$571$		−	1.26119i		−	0.0527792i	−0.999652	−	0.0263896i	$-0.991599\pi$
$571$		−	1.26119i		−	0.0527792i	0.999652	−	0.0263896i	$-0.00840105\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−27.5519	−	18.7546i	−1.14899	−	0.782119i
$576$		0			0
$577$	−5.10444	+	5.10444i	−0.212501	+	0.212501i	−0.805329	−	0.592828i	$-0.798011\pi$
$577$	−5.10444	+	5.10444i	−0.212501	+	0.212501i	0.592828	+	0.805329i	$0.298011\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−3.65081			−0.151461
$582$		0			0
$583$	6.75469	+	6.75469i	0.279751	+	0.279751i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	2.16591	−	2.16591i	0.0893966	−	0.0893966i	−0.660994	−	0.750391i	$-0.729865\pi$
$587$	2.16591	−	2.16591i	0.0893966	−	0.0893966i	0.750391	+	0.660994i	$0.229865\pi$
$588$		0			0
$589$	−0.0217916			−0.000897908
$590$		0			0
$591$		0			0
$592$		0			0
$593$	14.8495	−	14.8495i	0.609795	−	0.609795i	−0.333097	−	0.942892i	$-0.608094\pi$
$593$	14.8495	−	14.8495i	0.609795	−	0.609795i	0.942892	+	0.333097i	$0.108094\pi$
$594$		0			0
$595$	−11.8785	+	14.3476i	−0.486971	+	0.588196i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−20.1979			−0.825262			−0.412631	−	0.910898i	$-0.635390\pi$
$599$	−20.1979			−0.825262			−0.412631	+	0.910898i	$0.635390\pi$
$600$		0			0
$601$	−30.9787			−1.26365			−0.631823	−	0.775113i	$-0.717693\pi$
$601$	−30.9787			−1.26365			−0.631823	+	0.775113i	$0.717693\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−13.7071	+	16.5563i	−0.557272	+	0.673110i
$606$		0			0
$607$	7.72619	−	7.72619i	0.313596	−	0.313596i	−0.532705	−	0.846301i	$-0.678824\pi$
$607$	7.72619	−	7.72619i	0.313596	−	0.313596i	0.846301	+	0.532705i	$0.178824\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	31.5886			1.27794
$612$		0			0
$613$	−7.56398	+	7.56398i	−0.305506	+	0.305506i	−0.843164	−	0.537657i	$-0.819309\pi$
$613$	−7.56398	+	7.56398i	−0.305506	+	0.305506i	0.537657	+	0.843164i	$0.319309\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	20.1379	+	20.1379i	0.810722	+	0.810722i	0.984742	−	0.174020i	$-0.0556757\pi$
$617$	20.1379	+	20.1379i	0.810722	+	0.810722i	−0.174020	+	0.984742i	$0.555676\pi$
$618$		0			0
$619$	26.5448			1.06692			0.533462	−	0.845824i	$-0.320891\pi$
$619$	26.5448			1.06692			0.533462	+	0.845824i	$0.320891\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	1.73869	−	1.73869i	0.0696591	−	0.0696591i
$624$		0			0
$625$	−23.2582	+	9.16811i	−0.930330	+	0.366724i
$626$		0			0
$627$		0			0
$628$		0			0
$629$			20.3182i			0.810139i
$630$		0			0
$631$	38.2805			1.52392			0.761961	−	0.647623i	$-0.224237\pi$
$631$	38.2805			1.52392			0.761961	+	0.647623i	$0.224237\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−1.13749	−	12.0820i	−0.0451401	−	0.479458i
$636$		0			0
$637$	−26.8128	−	26.8128i	−1.06236	−	1.06236i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	24.4013i		−	0.963792i	−0.876228	−	0.481896i	$-0.839948\pi$
$641$		−	24.4013i		−	0.963792i	0.876228	−	0.481896i	$-0.160052\pi$
$642$		0			0
$643$	−1.08062	+	1.08062i	−0.0426157	+	0.0426157i	−0.728094	−	0.685478i	$-0.759593\pi$
$643$	−1.08062	+	1.08062i	−0.0426157	+	0.0426157i	0.685478	+	0.728094i	$0.259593\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−22.0445	−	22.0445i	−0.866659	−	0.866659i	0.125442	−	0.992101i	$-0.459965\pi$
$647$	−22.0445	−	22.0445i	−0.866659	−	0.866659i	−0.992101	+	0.125442i	$0.959965\pi$
$648$		0			0
$649$			10.2298i			0.401556i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	20.9241	+	20.9241i	0.818823	+	0.818823i	0.985937	−	0.167115i	$-0.0534450\pi$
$653$	20.9241	+	20.9241i	0.818823	+	0.818823i	−0.167115	+	0.985937i	$0.553445\pi$
$654$		0			0
$655$	−2.64824	+	3.19872i	−0.103475	+	0.124984i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			7.22520i			0.281454i	0.990048	+	0.140727i	$0.0449440\pi$
$659$			7.22520i			0.281454i	−0.990048	+	0.140727i	$0.955056\pi$
$660$		0			0
$661$			20.3836i			0.792830i	0.918071	+	0.396415i	$0.129746\pi$
$661$			20.3836i			0.792830i	−0.918071	+	0.396415i	$0.870254\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−0.298171	−	3.16704i	−0.0115626	−	0.122813i
$666$		0			0
$667$	31.2523	+	31.2523i	1.21009	+	1.21009i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	14.7973i		−	0.571244i
$672$		0			0
$673$	−19.4517	−	19.4517i	−0.749807	−	0.749807i	0.224636	−	0.974443i	$-0.427881\pi$
$673$	−19.4517	−	19.4517i	−0.749807	−	0.749807i	−0.974443	+	0.224636i	$0.927881\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	24.8145	−	24.8145i	0.953698	−	0.953698i	−0.0452768	−	0.998974i	$-0.514417\pi$
$677$	24.8145	−	24.8145i	0.953698	−	0.953698i	0.998974	+	0.0452768i	$0.0144170\pi$
$678$		0			0
$679$		−	65.9957i		−	2.53268i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	7.85886	+	7.85886i	0.300711	+	0.300711i	0.841292	−	0.540581i	$-0.181796\pi$
$683$	7.85886	+	7.85886i	0.300711	+	0.300711i	−0.540581	+	0.841292i	$0.681796\pi$
$684$		0			0
$685$	5.95794	−	0.560928i	0.227641	−	0.0214320i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	24.6690			0.939814
$690$		0			0
$691$		−	49.6748i		−	1.88972i	−0.327479	−	0.944858i	$-0.606199\pi$
$691$		−	49.6748i		−	1.88972i	0.327479	−	0.944858i	$-0.393801\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	26.2906	−	31.7556i	0.997260	−	1.20456i
$696$		0			0
$697$	4.26802	−	4.26802i	0.161663	−	0.161663i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−30.5703			−1.15462			−0.577312	−	0.816524i	$-0.695898\pi$
$701$	−30.5703			−1.15462			−0.577312	+	0.816524i	$0.695898\pi$
$702$		0			0
$703$	−2.45360	−	2.45360i	−0.0925393	−	0.0925393i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−29.7269	+	29.7269i	−1.11799	+	1.11799i
$708$		0			0
$709$	−11.1384			−0.418310			−0.209155	−	0.977882i	$-0.567071\pi$
$709$	−11.1384			−0.418310			−0.209155	+	0.977882i	$0.567071\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	0.318548	−	0.318548i	0.0119297	−	0.0119297i
$714$		0			0
$715$	−0.751033	−	7.97714i	−0.0280870	−	0.298328i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	41.0316			1.53022			0.765111	−	0.643899i	$-0.222684\pi$
$719$	41.0316			1.53022			0.765111	+	0.643899i	$0.222684\pi$
$720$		0			0
$721$	−1.01359			−0.0377482
$722$		0			0
$723$		0			0
$724$		0			0
$725$	32.5696	−	6.18758i	1.20961	−	0.229801i
$726$		0			0
$727$	4.35975	−	4.35975i	0.161694	−	0.161694i	−0.621623	−	0.783317i	$-0.713526\pi$
$727$	4.35975	−	4.35975i	0.161694	−	0.161694i	0.783317	+	0.621623i	$0.213526\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	17.9035			0.662185
$732$		0			0
$733$	28.3786	−	28.3786i	1.04819	−	1.04819i	0.0494094	−	0.998779i	$-0.484266\pi$
$733$	28.3786	−	28.3786i	1.04819	−	1.04819i	0.998779	−	0.0494094i	$-0.0157339\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−2.21146	−	2.21146i	−0.0814603	−	0.0814603i
$738$		0			0
$739$	6.53862			0.240527			0.120264	−	0.992742i	$-0.461626\pi$
$739$	6.53862			0.240527			0.120264	+	0.992742i	$0.461626\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−20.0409	+	20.0409i	−0.735230	+	0.735230i	−0.971651	−	0.236421i	$-0.924026\pi$
$743$	−20.0409	+	20.0409i	−0.735230	+	0.735230i	0.236421	+	0.971651i	$0.424026\pi$
$744$		0			0
$745$	2.14563	+	1.77638i	0.0786098	+	0.0650816i
$746$		0			0
$747$		0			0
$748$		0			0
$749$			73.5974i			2.68919i
$750$		0			0
$751$	−7.89337			−0.288033			−0.144017	−	0.989575i	$-0.546002\pi$
$751$	−7.89337			−0.288033			−0.144017	+	0.989575i	$0.546002\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−23.8974	+	28.8649i	−0.869717	+	1.05050i
$756$		0			0
$757$	5.76852	+	5.76852i	0.209660	+	0.209660i	0.804123	−	0.594463i	$-0.202635\pi$
$757$	5.76852	+	5.76852i	0.209660	+	0.209660i	−0.594463	+	0.804123i	$0.702635\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	11.7806i		−	0.427045i	−0.976938	−	0.213522i	$-0.931506\pi$
$761$		−	11.7806i		−	0.427045i	0.976938	−	0.213522i	$-0.0684936\pi$
$762$		0			0
$763$	33.8805	−	33.8805i	1.22656	−	1.22656i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	18.6803	+	18.6803i	0.674506	+	0.674506i
$768$		0			0
$769$			34.7788i			1.25416i	0.778956	+	0.627078i	$0.215749\pi$
$769$			34.7788i			1.25416i	−0.778956	+	0.627078i	$0.784251\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	27.6201	+	27.6201i	0.993426	+	0.993426i	0.999979	−	0.00655287i	$-0.00208586\pi$
$773$	27.6201	+	27.6201i	0.993426	+	0.993426i	−0.00655287	+	0.999979i	$0.502086\pi$
$774$		0			0
$775$	−0.0630687	−	0.331975i	−0.00226549	−	0.0119249i
$776$		0			0
$777$		0			0
$778$		0			0
$779$			1.03080i			0.0369323i
$780$		0			0
$781$		−	4.92640i		−	0.176280i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	21.7483	−	2.04756i	0.776232	−	0.0730807i
$786$		0			0
$787$	18.2310	+	18.2310i	0.649864	+	0.649864i	0.952960	−	0.303096i	$-0.0980203\pi$
$787$	18.2310	+	18.2310i	0.649864	+	0.649864i	−0.303096	+	0.952960i	$0.598020\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			54.5623i			1.94001i
$792$		0			0
$793$	−27.0208	−	27.0208i	−0.959537	−	0.959537i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−11.4983	+	11.4983i	−0.407290	+	0.407290i	−0.880793	−	0.473502i	$-0.842990\pi$
$797$	−11.4983	+	11.4983i	−0.407290	+	0.407290i	0.473502	+	0.880793i	$0.342990\pi$
$798$		0			0
$799$			19.6064i			0.693623i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	4.68695	+	4.68695i	0.165399	+	0.165399i
$804$		0			0
$805$	50.6542	+	41.9369i	1.78533	+	1.47808i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	17.8481			0.627506			0.313753	−	0.949505i	$-0.398414\pi$
$809$	17.8481			0.627506			0.313753	+	0.949505i	$0.398414\pi$
$810$		0			0
$811$			41.5242i			1.45811i	0.684454	+	0.729056i	$0.260041\pi$
$811$			41.5242i			1.45811i	−0.684454	+	0.729056i	$0.739959\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	1.80165	+	19.1364i	0.0631091	+	0.670317i
$816$		0			0
$817$	−2.16201	+	2.16201i	−0.0756390	+	0.0756390i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−28.9849			−1.01158			−0.505790	−	0.862657i	$-0.668799\pi$
$821$	−28.9849			−1.01158			−0.505790	+	0.862657i	$0.668799\pi$
$822$		0			0
$823$	−20.6586	−	20.6586i	−0.720112	−	0.720112i	0.248516	−	0.968628i	$-0.420057\pi$
$823$	−20.6586	−	20.6586i	−0.720112	−	0.720112i	−0.968628	+	0.248516i	$0.920057\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−25.4075	+	25.4075i	−0.883504	+	0.883504i	−0.993889	−	0.110385i	$-0.964792\pi$
$827$	−25.4075	+	25.4075i	−0.883504	+	0.883504i	0.110385	+	0.993889i	$0.464792\pi$
$828$		0			0
$829$	24.7882			0.860931			0.430465	−	0.902607i	$-0.358349\pi$
$829$	24.7882			0.860931			0.430465	+	0.902607i	$0.358349\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	16.6421	−	16.6421i	0.576614	−	0.576614i
$834$		0			0
$835$	−8.88336	+	0.836351i	−0.307421	+	0.0289431i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	52.8553			1.82477			0.912383	−	0.409338i	$-0.134240\pi$
$839$	52.8553			1.82477			0.912383	+	0.409338i	$0.134240\pi$
$840$		0			0
$841$	−14.9627			−0.515954
$842$		0			0
$843$		0			0
$844$		0			0
$845$	6.45279	+	5.34230i	0.221983	+	0.183781i
$846$		0			0
$847$	29.9881	−	29.9881i	1.03040	−	1.03040i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	71.7331			2.45898
$852$		0			0
$853$	7.39491	−	7.39491i	0.253197	−	0.253197i	−0.569083	−	0.822280i	$-0.692702\pi$
$853$	7.39491	−	7.39491i	0.253197	−	0.253197i	0.822280	+	0.569083i	$0.192702\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−6.82308	−	6.82308i	−0.233072	−	0.233072i	0.580902	−	0.813974i	$-0.302700\pi$
$857$	−6.82308	−	6.82308i	−0.233072	−	0.233072i	−0.813974	+	0.580902i	$0.802700\pi$
$858$		0			0
$859$	43.6736			1.49012			0.745061	−	0.666996i	$-0.232420\pi$
$859$	43.6736			1.49012			0.745061	+	0.666996i	$0.232420\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	13.1338	−	13.1338i	0.447080	−	0.447080i	−0.447303	−	0.894383i	$-0.647615\pi$
$863$	13.1338	−	13.1338i	0.447080	−	0.447080i	0.894383	+	0.447303i	$0.147615\pi$
$864$		0			0
$865$	22.9175	−	2.15764i	0.779219	−	0.0733620i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	11.3865i		−	0.386261i
$870$		0			0
$871$	−8.07654			−0.273663
$872$		0			0
$873$		0			0
$874$		0			0
$875$	47.3840	−	13.7083i	1.60187	−	0.463426i
$876$		0			0
$877$	−28.9437	−	28.9437i	−0.977360	−	0.977360i	0.0223891	−	0.999749i	$-0.492873\pi$
$877$	−28.9437	−	28.9437i	−0.977360	−	0.977360i	−0.999749	+	0.0223891i	$0.992873\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			7.23302i			0.243687i	0.992549	+	0.121843i	$0.0388805\pi$
$881$			7.23302i			0.243687i	−0.992549	+	0.121843i	$0.961119\pi$
$882$		0			0
$883$	−7.01625	+	7.01625i	−0.236116	+	0.236116i	−0.815239	−	0.579124i	$-0.803395\pi$
$883$	−7.01625	+	7.01625i	−0.236116	+	0.236116i	0.579124	+	0.815239i	$0.303395\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−15.2597	−	15.2597i	−0.512371	−	0.512371i	0.402881	−	0.915252i	$-0.368009\pi$
$887$	−15.2597	−	15.2597i	−0.512371	−	0.512371i	−0.915252	+	0.402881i	$0.868009\pi$
$888$		0			0
$889$			23.9441i			0.803061i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−2.36764	−	2.36764i	−0.0792301	−	0.0792301i
$894$		0			0
$895$	16.4764	+	13.6409i	0.550745	+	0.455965i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			0.448102i			0.0149450i
$900$		0			0
$901$			15.3115i			0.510100i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	7.66214	+	6.34353i	0.254698	+	0.210866i
$906$		0			0
$907$	−36.0510	−	36.0510i	−1.19706	−	1.19706i	−0.975044	−	0.222011i	$-0.928738\pi$
$907$	−36.0510	−	36.0510i	−1.19706	−	1.19706i	−0.222011	−	0.975044i	$-0.571262\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	24.0944i		−	0.798282i	−0.916890	−	0.399141i	$-0.869308\pi$
$911$		−	24.0944i		−	0.798282i	0.916890	−	0.399141i	$-0.130692\pi$
$912$		0			0
$913$	0.689236	+	0.689236i	0.0228104	+	0.0228104i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	5.79377	−	5.79377i	0.191327	−	0.191327i
$918$		0			0
$919$			47.1031i			1.55379i	0.629632	+	0.776893i	$0.283206\pi$
$919$			47.1031i			1.55379i	−0.629632	+	0.776893i	$0.716794\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−8.99592	−	8.99592i	−0.296104	−	0.296104i
$924$		0			0
$925$	30.2772	−	44.4795i	0.995508	−	1.46248i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−34.0301			−1.11649			−0.558246	−	0.829675i	$-0.688526\pi$
$929$	−34.0301			−1.11649			−0.558246	+	0.829675i	$0.688526\pi$
$930$		0			0
$931$			4.01936i			0.131729i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	4.95123	−	0.466149i	0.161923	−	0.0152447i
$936$		0			0
$937$	26.9441	−	26.9441i	0.880227	−	0.880227i	−0.113330	−	0.993557i	$-0.536152\pi$
$937$	26.9441	−	26.9441i	0.880227	−	0.880227i	0.993557	+	0.113330i	$0.0361519\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−19.1115			−0.623018			−0.311509	−	0.950243i	$-0.600834\pi$
$941$	−19.1115			−0.623018			−0.311509	+	0.950243i	$0.600834\pi$
$942$		0			0
$943$	−15.0682	−	15.0682i	−0.490687	−	0.490687i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−26.6896	+	26.6896i	−0.867294	+	0.867294i	−0.992172	−	0.124878i	$-0.960146\pi$
$947$	−26.6896	+	26.6896i	−0.867294	+	0.867294i	0.124878	+	0.992172i	$0.460146\pi$
$948$		0			0
$949$	17.1173			0.555652
$950$		0			0
$951$		0			0
$952$		0			0
$953$	18.0251	−	18.0251i	0.583890	−	0.583890i	−0.352080	−	0.935970i	$-0.614525\pi$
$953$	18.0251	−	18.0251i	0.583890	−	0.583890i	0.935970	+	0.352080i	$0.114525\pi$
$954$		0			0
$955$	5.45486	+	4.51611i	0.176515	+	0.146138i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−11.8075			−0.381284
$960$		0			0
$961$	−30.9954			−0.999853
$962$		0			0
$963$		0			0
$964$		0			0
$965$	3.39159	−	0.319311i	0.109179	−	0.0102790i
$966$		0			0
$967$	−20.9523	+	20.9523i	−0.673779	+	0.673779i	−0.958585	−	0.284806i	$-0.908071\pi$
$967$	−20.9523	+	20.9523i	−0.673779	+	0.673779i	0.284806	+	0.958585i	$0.408071\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	6.93592			0.222584			0.111292	−	0.993788i	$-0.464501\pi$
$971$	6.93592			0.222584			0.111292	+	0.993788i	$0.464501\pi$
$972$		0			0
$973$	−57.5182	+	57.5182i	−1.84395	+	1.84395i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−6.98111	−	6.98111i	−0.223346	−	0.223346i	0.586560	−	0.809906i	$-0.300482\pi$
$977$	−6.98111	−	6.98111i	−0.223346	−	0.223346i	−0.809906	+	0.586560i	$0.800482\pi$
$978$		0			0
$979$	−0.656493			−0.0209816
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−21.0079	+	21.0079i	−0.670047	+	0.670047i	−0.957727	−	0.287680i	$-0.907116\pi$
$983$	−21.0079	+	21.0079i	−0.670047	+	0.670047i	0.287680	+	0.957727i	$0.407116\pi$
$984$		0			0
$985$	1.65712	+	17.6013i	0.0528004	+	0.560823i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	63.2081i		−	2.00990i
$990$		0			0
$991$	−6.65846			−0.211513			−0.105756	−	0.994392i	$-0.533726\pi$
$991$	−6.65846			−0.211513			−0.105756	+	0.994392i	$0.533726\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−27.1632	−	22.4886i	−0.861132	−	0.712936i
$996$		0			0
$997$	9.19447	+	9.19447i	0.291192	+	0.291192i	0.837551	−	0.546359i	$-0.183987\pi$
$997$	9.19447	+	9.19447i	0.291192	+	0.291192i	−0.546359	+	0.837551i	$0.683987\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1440.2.bj.a.593.18		48
3.2	odd	2	inner	1440.2.bj.a.593.7		48
4.3	odd	2		360.2.x.a.53.14	yes	48
5.2	odd	4	inner	1440.2.bj.a.17.17		48
8.3	odd	2		360.2.x.a.53.23	yes	48
8.5	even	2	inner	1440.2.bj.a.593.8		48
12.11	even	2		360.2.x.a.53.11	yes	48
15.2	even	4	inner	1440.2.bj.a.17.8		48
20.7	even	4		360.2.x.a.197.2	yes	48
24.5	odd	2	inner	1440.2.bj.a.593.17		48
24.11	even	2		360.2.x.a.53.2	✓	48
40.27	even	4		360.2.x.a.197.11	yes	48
40.37	odd	4	inner	1440.2.bj.a.17.7		48
60.47	odd	4		360.2.x.a.197.23	yes	48
120.77	even	4	inner	1440.2.bj.a.17.18		48
120.107	odd	4		360.2.x.a.197.14	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
360.2.x.a.53.2	✓	48	24.11	even	2
360.2.x.a.53.11	yes	48	12.11	even	2
360.2.x.a.53.14	yes	48	4.3	odd	2
360.2.x.a.53.23	yes	48	8.3	odd	2
360.2.x.a.197.2	yes	48	20.7	even	4
360.2.x.a.197.11	yes	48	40.27	even	4
360.2.x.a.197.14	yes	48	120.107	odd	4
360.2.x.a.197.23	yes	48	60.47	odd	4
1440.2.bj.a.17.7		48	40.37	odd	4	inner
1440.2.bj.a.17.8		48	15.2	even	4	inner
1440.2.bj.a.17.17		48	5.2	odd	4	inner
1440.2.bj.a.17.18		48	120.77	even	4	inner
1440.2.bj.a.593.7		48	3.2	odd	2	inner
1440.2.bj.a.593.8		48	8.5	even	2	inner
1440.2.bj.a.593.17		48	24.5	odd	2	inner
1440.2.bj.a.593.18		48	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1440 = 2^{5} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1440.bj (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.42597 - 1.72238i) q^{5} +(-3.11972 + 3.11972i) q^{7} +O(q^{10})\)	\(q+(1.42597 - 1.72238i) q^{5} +(-3.11972 + 3.11972i) q^{7} +1.17794 q^{11} +(2.15100 - 2.15100i) q^{13} +(1.33507 + 1.33507i) q^{17} -0.322444 q^{19} +(4.71347 - 4.71347i) q^{23} +(-0.933209 - 4.91214i) q^{25} +6.63043i q^{29} +0.0675826 q^{31} +(0.924721 + 9.82198i) q^{35} +(7.60938 + 7.60938i) q^{37} -3.19684i q^{41} +(6.70505 - 6.70505i) q^{43} +(7.34279 + 7.34279i) q^{47} -12.4653i q^{49} +(5.73432 + 5.73432i) q^{53} +(1.67971 - 2.02887i) q^{55} +8.68448i q^{59} -12.5620i q^{61} +(-0.637580 - 6.77210i) q^{65} +(-1.87740 - 1.87740i) q^{67} -4.18221i q^{71} +(3.97893 + 3.97893i) q^{73} +(-3.67485 + 3.67485i) q^{77} -9.66644i q^{79} +(0.585119 + 0.585119i) q^{83} +(4.20329 - 0.395732i) q^{85} -0.557322 q^{89} +13.4210i q^{91} +(-0.459796 + 0.555372i) q^{95} +(-10.5772 + 10.5772i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q+O(q^{10}) \) 48 * q	\( 48 q - 32 q^{31} - 32 q^{97}+O(q^{100}) \) 48 * q - 32 * q^31 - 32 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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