LMFDB - Embedded newform 1440.2.bj.a.593.21

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			593.21
Character	$\chi$	$=$	1440.593
Dual form			1440.2.bj.a.17.21

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.03368 + 0.929591i) q^{5} +(2.49469 - 2.49469i) q^{7} +O(q^{10})$	$q+(2.03368 + 0.929591i) q^{5} +(2.49469 - 2.49469i) q^{7} +3.92878 q^{11} +(4.55591 - 4.55591i) q^{13} +(-1.88566 - 1.88566i) q^{17} -4.61555 q^{19} +(-0.741221 + 0.741221i) q^{23} +(3.27172 + 3.78098i) q^{25} -4.35885i q^{29} -9.67119 q^{31} +(7.39245 - 2.75436i) q^{35} +(-5.39704 - 5.39704i) q^{37} +6.33584i q^{41} +(-0.206110 + 0.206110i) q^{43} +(3.48081 + 3.48081i) q^{47} -5.44695i q^{49} +(1.01974 + 1.01974i) q^{53} +(7.98989 + 3.65216i) q^{55} +0.531064i q^{59} -3.00356i q^{61} +(13.5004 - 5.03014i) q^{65} +(1.28660 + 1.28660i) q^{67} +7.61692i q^{71} +(-0.509262 - 0.509262i) q^{73} +(9.80109 - 9.80109i) q^{77} -1.31920i q^{79} +(9.85533 + 9.85533i) q^{83} +(-2.08194 - 5.58774i) q^{85} -2.91798 q^{89} -22.7312i q^{91} +(-9.38656 - 4.29057i) q^{95} +(-8.11369 + 8.11369i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	2.03368	+	0.929591i	0.909490	+	0.415726i
$5$	2.03368	+	0.929591i	0.909490	+	0.415726i
$6$		0			0
$7$	2.49469	−	2.49469i	0.942904	−	0.942904i	−0.0555517	−	0.998456i	$-0.517692\pi$
$7$	2.49469	−	2.49469i	0.942904	−	0.942904i	0.998456	+	0.0555517i	$0.0176918\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	3.92878			1.18457			0.592286	−	0.805728i	$-0.298226\pi$
$11$	3.92878			1.18457			0.592286	+	0.805728i	$0.298226\pi$
$12$		0			0
$13$	4.55591	−	4.55591i	1.26358	−	1.26358i	0.314237	−	0.949345i	$-0.398251\pi$
$13$	4.55591	−	4.55591i	1.26358	−	1.26358i	0.949345	−	0.314237i	$-0.101749\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−1.88566	−	1.88566i	−0.457341	−	0.457341i	0.440441	−	0.897782i	$-0.354822\pi$
$17$	−1.88566	−	1.88566i	−0.457341	−	0.457341i	−0.897782	+	0.440441i	$0.854822\pi$
$18$		0			0
$19$	−4.61555			−1.05888			−0.529440	−	0.848347i	$-0.677598\pi$
$19$	−4.61555			−1.05888			−0.529440	+	0.848347i	$0.677598\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−0.741221	+	0.741221i	−0.154555	+	0.154555i	−0.780149	−	0.625594i	$-0.784857\pi$
$23$	−0.741221	+	0.741221i	−0.154555	+	0.154555i	0.625594	+	0.780149i	$0.284857\pi$
$24$		0			0
$25$	3.27172	+	3.78098i	0.654344	+	0.756197i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	4.35885i		−	0.809418i	−0.914446	−	0.404709i	$-0.867373\pi$
$29$		−	4.35885i		−	0.809418i	0.914446	−	0.404709i	$-0.132627\pi$
$30$		0			0
$31$	−9.67119			−1.73700			−0.868499	−	0.495691i	$-0.834915\pi$
$31$	−9.67119			−1.73700			−0.868499	+	0.495691i	$0.834915\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	7.39245	−	2.75436i	1.24955	−	0.465572i
$36$		0			0
$37$	−5.39704	−	5.39704i	−0.887267	−	0.887267i	0.106992	−	0.994260i	$-0.465878\pi$
$37$	−5.39704	−	5.39704i	−0.887267	−	0.887267i	−0.994260	+	0.106992i	$0.965878\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			6.33584i			0.989492i	0.869038	+	0.494746i	$0.164739\pi$
$41$			6.33584i			0.989492i	−0.869038	+	0.494746i	$0.835261\pi$
$42$		0			0
$43$	−0.206110	+	0.206110i	−0.0314315	+	0.0314315i	−0.722648	−	0.691216i	$-0.757075\pi$
$43$	−0.206110	+	0.206110i	−0.0314315	+	0.0314315i	0.691216	+	0.722648i	$0.257075\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	3.48081	+	3.48081i	0.507729	+	0.507729i	0.913829	−	0.406100i	$-0.133112\pi$
$47$	3.48081	+	3.48081i	0.507729	+	0.507729i	−0.406100	+	0.913829i	$0.633112\pi$
$48$		0			0
$49$		−	5.44695i		−	0.778136i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	1.01974	+	1.01974i	0.140073	+	0.140073i	0.773666	−	0.633593i	$-0.218421\pi$
$53$	1.01974	+	1.01974i	0.140073	+	0.140073i	−0.633593	+	0.773666i	$0.718421\pi$
$54$		0			0
$55$	7.98989	+	3.65216i	1.07736	+	0.492457i
$56$		0			0
$57$		0			0
$58$		0			0
$59$			0.531064i			0.0691386i	0.999402	+	0.0345693i	$0.0110059\pi$
$59$			0.531064i			0.0691386i	−0.999402	+	0.0345693i	$0.988994\pi$
$60$		0			0
$61$		−	3.00356i		−	0.384566i	−0.981340	−	0.192283i	$-0.938411\pi$
$61$		−	3.00356i		−	0.384566i	0.981340	−	0.192283i	$-0.0615892\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	13.5004	−	5.03014i	1.67452	−	0.623912i
$66$		0			0
$67$	1.28660	+	1.28660i	0.157183	+	0.157183i	0.781317	−	0.624134i	$-0.214548\pi$
$67$	1.28660	+	1.28660i	0.157183	+	0.157183i	−0.624134	+	0.781317i	$0.714548\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			7.61692i			0.903962i	0.892028	+	0.451981i	$0.149282\pi$
$71$			7.61692i			0.903962i	−0.892028	+	0.451981i	$0.850718\pi$
$72$		0			0
$73$	−0.509262	−	0.509262i	−0.0596047	−	0.0596047i	0.676676	−	0.736281i	$-0.263420\pi$
$73$	−0.509262	−	0.509262i	−0.0596047	−	0.0596047i	−0.736281	+	0.676676i	$0.763420\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	9.80109	−	9.80109i	1.11694	−	1.11694i
$78$		0			0
$79$		−	1.31920i		−	0.148422i	−0.997243	−	0.0742111i	$-0.976356\pi$
$79$		−	1.31920i		−	0.148422i	0.997243	−	0.0742111i	$-0.0236439\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	9.85533	+	9.85533i	1.08176	+	1.08176i	0.996345	+	0.0854172i	$0.0272223\pi$
$83$	9.85533	+	9.85533i	1.08176	+	1.08176i	0.0854172	+	0.996345i	$0.472778\pi$
$84$		0			0
$85$	−2.08194	−	5.58774i	−0.225819	−	0.606075i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−2.91798			−0.309306			−0.154653	−	0.987969i	$-0.549426\pi$
$89$	−2.91798			−0.309306			−0.154653	+	0.987969i	$0.549426\pi$
$90$		0			0
$91$		−	22.7312i		−	2.38287i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−9.38656	−	4.29057i	−0.963041	−	0.440204i
$96$		0			0
$97$	−8.11369	+	8.11369i	−0.823820	+	0.823820i	−0.986654	−	0.162834i	$-0.947937\pi$
$97$	−8.11369	+	8.11369i	−0.823820	+	0.823820i	0.162834	+	0.986654i	$0.447937\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	11.0269			1.09721			0.548607	−	0.836080i	$-0.315158\pi$
$101$	11.0269			1.09721			0.548607	+	0.836080i	$0.315158\pi$
$102$		0			0
$103$	2.89128	+	2.89128i	0.284887	+	0.284887i	0.835054	−	0.550168i	$-0.185436\pi$
$103$	2.89128	+	2.89128i	0.284887	+	0.284887i	−0.550168	+	0.835054i	$0.685436\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	5.23891	−	5.23891i	0.506465	−	0.506465i	−0.406975	−	0.913439i	$-0.633416\pi$
$107$	5.23891	−	5.23891i	0.506465	−	0.506465i	0.913439	+	0.406975i	$0.133416\pi$
$108$		0			0
$109$	5.31464			0.509050			0.254525	−	0.967066i	$-0.418081\pi$
$109$	5.31464			0.509050			0.254525	+	0.967066i	$0.418081\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−10.3559	+	10.3559i	−0.974204	+	0.974204i	−0.999676	−	0.0254720i	$-0.991891\pi$
$113$	−10.3559	+	10.3559i	−0.974204	+	0.974204i	0.0254720	+	0.999676i	$0.491891\pi$
$114$		0			0
$115$	−2.19644	+	0.818375i	−0.204819	+	0.0763138i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−9.40829			−0.862457
$120$		0			0
$121$	4.43532			0.403211
$122$		0			0
$123$		0			0
$124$		0			0
$125$	3.13887	+	10.7307i	0.280749	+	0.959781i
$126$		0			0
$127$	1.10872	−	1.10872i	0.0983826	−	0.0983826i	−0.656202	−	0.754585i	$-0.727838\pi$
$127$	1.10872	−	1.10872i	0.0983826	−	0.0983826i	0.754585	+	0.656202i	$0.227838\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	0.996979			0.0871065			0.0435532	−	0.999051i	$-0.486132\pi$
$131$	0.996979			0.0871065			0.0435532	+	0.999051i	$0.486132\pi$
$132$		0			0
$133$	−11.5144	+	11.5144i	−0.998422	+	0.998422i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	1.03232	+	1.03232i	0.0881970	+	0.0881970i	0.749829	−	0.661632i	$-0.230136\pi$
$137$	1.03232	+	1.03232i	0.0881970	+	0.0881970i	−0.661632	+	0.749829i	$0.730136\pi$
$138$		0			0
$139$	16.9830			1.44048			0.720239	−	0.693726i	$-0.244032\pi$
$139$	16.9830			1.44048			0.720239	+	0.693726i	$0.244032\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	17.8992	−	17.8992i	1.49680	−	1.49680i
$144$		0			0
$145$	4.05194	−	8.86451i	0.336496	−	0.736157i
$146$		0			0
$147$		0			0
$148$		0			0
$149$			6.48772i			0.531495i	0.964043	+	0.265747i	$0.0856187\pi$
$149$			6.48772i			0.531495i	−0.964043	+	0.265747i	$0.914381\pi$
$150$		0			0
$151$	−4.21210			−0.342776			−0.171388	−	0.985204i	$-0.554825\pi$
$151$	−4.21210			−0.342776			−0.171388	+	0.985204i	$0.554825\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−19.6681	−	8.99025i	−1.57978	−	0.722114i
$156$		0			0
$157$	−0.0368875	−	0.0368875i	−0.00294395	−	0.00294395i	0.705633	−	0.708577i	$-0.250663\pi$
$157$	−0.0368875	−	0.0368875i	−0.00294395	−	0.00294395i	−0.708577	+	0.705633i	$0.750663\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			3.69823i			0.291461i
$162$		0			0
$163$	−1.33256	+	1.33256i	−0.104374	+	0.104374i	−0.757365	−	0.652991i	$-0.773514\pi$
$163$	−1.33256	+	1.33256i	−0.104374	+	0.104374i	0.652991	+	0.757365i	$0.273514\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	15.0291	+	15.0291i	1.16299	+	1.16299i	0.983818	+	0.179169i	$0.0573410\pi$
$167$	15.0291	+	15.0291i	1.16299	+	1.16299i	0.179169	+	0.983818i	$0.442659\pi$
$168$		0			0
$169$		−	28.5126i		−	2.19328i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−2.16963	−	2.16963i	−0.164954	−	0.164954i	0.619804	−	0.784757i	$-0.287212\pi$
$173$	−2.16963	−	2.16963i	−0.164954	−	0.164954i	−0.784757	+	0.619804i	$0.787212\pi$
$174$		0			0
$175$	17.5943	+	1.27045i	1.33000	+	0.0960371i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	14.2973i		−	1.06863i	−0.845285	−	0.534316i	$-0.820569\pi$
$179$		−	14.2973i		−	1.06863i	0.845285	−	0.534316i	$-0.179431\pi$
$180$		0			0
$181$			11.0562i			0.821800i	0.911680	+	0.410900i	$0.134785\pi$
$181$			11.0562i			0.821800i	−0.911680	+	0.410900i	$0.865215\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−5.95882	−	15.9929i	−0.438101	−	1.17582i
$186$		0			0
$187$	−7.40836	−	7.40836i	−0.541753	−	0.541753i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	12.7129i		−	0.919873i	−0.887952	−	0.459937i	$-0.847872\pi$
$191$		−	12.7129i		−	0.919873i	0.887952	−	0.459937i	$-0.152128\pi$
$192$		0			0
$193$	16.5832	+	16.5832i	1.19368	+	1.19368i	0.976025	+	0.217659i	$0.0698420\pi$
$193$	16.5832	+	16.5832i	1.19368	+	1.19368i	0.217659	+	0.976025i	$0.430158\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−9.96244	+	9.96244i	−0.709794	+	0.709794i	−0.966492	−	0.256698i	$-0.917366\pi$
$197$	−9.96244	+	9.96244i	−0.709794	+	0.709794i	0.256698	+	0.966492i	$0.417366\pi$
$198$		0			0
$199$		−	15.6722i		−	1.11097i	−0.831525	−	0.555487i	$-0.812532\pi$
$199$		−	15.6722i		−	1.11097i	0.831525	−	0.555487i	$-0.187468\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−10.8740	−	10.8740i	−0.763203	−	0.763203i
$204$		0			0
$205$	−5.88974	+	12.8851i	−0.411357	+	0.899933i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−18.1335			−1.25432
$210$		0			0
$211$			22.1294i			1.52345i	0.647901	+	0.761725i	$0.275647\pi$
$211$			22.1294i			1.52345i	−0.647901	+	0.761725i	$0.724353\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−0.610761	+	0.227564i	−0.0416535	+	0.0155198i
$216$		0			0
$217$	−24.1266	+	24.1266i	−1.63782	+	1.63782i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−17.1818			−1.15577
$222$		0			0
$223$	3.17650	+	3.17650i	0.212714	+	0.212714i	0.805420	−	0.592705i	$-0.201940\pi$
$223$	3.17650	+	3.17650i	0.212714	+	0.212714i	−0.592705	+	0.805420i	$0.701940\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−18.4920	+	18.4920i	−1.22735	+	1.22735i	−0.262393	+	0.964961i	$0.584512\pi$
$227$	−18.4920	+	18.4920i	−1.22735	+	1.22735i	−0.964961	+	0.262393i	$0.915488\pi$
$228$		0			0
$229$	19.2316			1.27086			0.635428	−	0.772160i	$-0.280824\pi$
$229$	19.2316			1.27086			0.635428	+	0.772160i	$0.280824\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	13.7302	−	13.7302i	0.899495	−	0.899495i	−0.0958959	−	0.995391i	$-0.530572\pi$
$233$	13.7302	−	13.7302i	0.899495	−	0.899495i	0.995391	+	0.0958959i	$0.0305716\pi$
$234$		0			0
$235$	3.84313	+	10.3146i	0.250698	+	0.672850i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−26.7732			−1.73182			−0.865908	−	0.500203i	$-0.833259\pi$
$239$	−26.7732			−1.73182			−0.865908	+	0.500203i	$0.833259\pi$
$240$		0			0
$241$	−17.6357			−1.13601			−0.568007	−	0.823023i	$-0.692286\pi$
$241$	−17.6357			−1.13601			−0.568007	+	0.823023i	$0.692286\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	5.06344	−	11.0774i	0.323491	−	0.707707i
$246$		0			0
$247$	−21.0280	+	21.0280i	−1.33798	+	1.33798i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−13.0522			−0.823846			−0.411923	−	0.911219i	$-0.635143\pi$
$251$	−13.0522			−0.823846			−0.411923	+	0.911219i	$0.635143\pi$
$252$		0			0
$253$	−2.91209	+	2.91209i	−0.183082	+	0.183082i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	9.03127	+	9.03127i	0.563355	+	0.563355i	0.930259	−	0.366904i	$-0.119582\pi$
$257$	9.03127	+	9.03127i	0.563355	+	0.563355i	−0.366904	+	0.930259i	$0.619582\pi$
$258$		0			0
$259$	−26.9279			−1.67322
$260$		0			0
$261$		0			0
$262$		0			0
$263$	2.83442	−	2.83442i	0.174778	−	0.174778i	−0.614297	−	0.789075i	$-0.710560\pi$
$263$	2.83442	−	2.83442i	0.174778	−	0.174778i	0.789075	+	0.614297i	$0.210560\pi$
$264$		0			0
$265$	1.12589	+	3.02178i	0.0691629	+	0.185627i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			9.89975i			0.603599i	0.953371	+	0.301799i	$0.0975873\pi$
$269$			9.89975i			0.603599i	−0.953371	+	0.301799i	$0.902413\pi$
$270$		0			0
$271$	−16.6235			−1.00981			−0.504904	−	0.863176i	$-0.668472\pi$
$271$	−16.6235			−1.00981			−0.504904	+	0.863176i	$0.668472\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	12.8539	+	14.8547i	0.775118	+	0.895769i
$276$		0			0
$277$	−6.42695	−	6.42695i	−0.386158	−	0.386158i	0.487156	−	0.873315i	$-0.338034\pi$
$277$	−6.42695	−	6.42695i	−0.386158	−	0.386158i	−0.873315	+	0.487156i	$0.838034\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			27.0270i			1.61230i	0.591714	+	0.806148i	$0.298451\pi$
$281$			27.0270i			1.61230i	−0.591714	+	0.806148i	$0.701549\pi$
$282$		0			0
$283$	18.1481	−	18.1481i	1.07879	−	1.07879i	0.0821722	−	0.996618i	$-0.473814\pi$
$283$	18.1481	−	18.1481i	1.07879	−	1.07879i	0.996618	−	0.0821722i	$-0.0261857\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	15.8060	+	15.8060i	0.932996	+	0.932996i
$288$		0			0
$289$		−	9.88854i		−	0.581679i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−11.4905	−	11.4905i	−0.671281	−	0.671281i	0.286730	−	0.958011i	$-0.407432\pi$
$293$	−11.4905	−	11.4905i	−0.671281	−	0.671281i	−0.958011	+	0.286730i	$0.907432\pi$
$294$		0			0
$295$	−0.493672	+	1.08001i	−0.0287427	+	0.0628809i
$296$		0			0
$297$		0			0
$298$		0			0
$299$			6.75387i			0.390586i
$300$		0			0
$301$			1.02836i			0.0592738i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	2.79208	−	6.10828i	0.159874	−	0.349759i
$306$		0			0
$307$	17.8635	+	17.8635i	1.01953	+	1.01953i	0.999806	+	0.0197205i	$0.00627765\pi$
$307$	17.8635	+	17.8635i	1.01953	+	1.01953i	0.0197205	+	0.999806i	$0.493722\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	9.93768i		−	0.563514i	−0.959486	−	0.281757i	$-0.909083\pi$
$311$		−	9.93768i		−	0.563514i	0.959486	−	0.281757i	$-0.0909173\pi$
$312$		0			0
$313$	5.20616	+	5.20616i	0.294269	+	0.294269i	0.838764	−	0.544495i	$-0.183279\pi$
$313$	5.20616	+	5.20616i	0.294269	+	0.294269i	−0.544495	+	0.838764i	$0.683279\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−13.2883	+	13.2883i	−0.746346	+	0.746346i	−0.973791	−	0.227445i	$-0.926963\pi$
$317$	−13.2883	+	13.2883i	−0.746346	+	0.746346i	0.227445	+	0.973791i	$0.426963\pi$
$318$		0			0
$319$		−	17.1250i		−	0.958813i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	8.70338	+	8.70338i	0.484269	+	0.484269i
$324$		0			0
$325$	32.1315	+	2.32015i	1.78233	+	0.128699i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	17.3671			0.957479
$330$		0			0
$331$			8.60834i			0.473157i	0.971612	+	0.236579i	$0.0760261\pi$
$331$			8.60834i			0.473157i	−0.971612	+	0.236579i	$0.923974\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	1.42052	+	3.81255i	0.0776114	+	0.208302i
$336$		0			0
$337$	1.29928	−	1.29928i	0.0707764	−	0.0707764i	−0.670832	−	0.741609i	$-0.734063\pi$
$337$	1.29928	−	1.29928i	0.0707764	−	0.0707764i	0.741609	+	0.670832i	$0.234063\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−37.9960			−2.05760
$342$		0			0
$343$	3.87437	+	3.87437i	0.209196	+	0.209196i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−18.3626	+	18.3626i	−0.985756	+	0.985756i	−0.999900	−	0.0141438i	$-0.995498\pi$
$347$	−18.3626	+	18.3626i	−0.985756	+	0.985756i	0.0141438	+	0.999900i	$0.495498\pi$
$348$		0			0
$349$	−31.1787			−1.66896			−0.834478	−	0.551041i	$-0.814231\pi$
$349$	−31.1787			−1.66896			−0.834478	+	0.551041i	$0.814231\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	9.04629	−	9.04629i	0.481485	−	0.481485i	−0.424120	−	0.905606i	$-0.639417\pi$
$353$	9.04629	−	9.04629i	0.481485	−	0.481485i	0.905606	+	0.424120i	$0.139417\pi$
$354$		0			0
$355$	−7.08062	+	15.4904i	−0.375800	+	0.822144i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−13.1073			−0.691776			−0.345888	−	0.938276i	$-0.612422\pi$
$359$	−13.1073			−0.691776			−0.345888	+	0.938276i	$0.612422\pi$
$360$		0			0
$361$	2.30330			0.121226
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−0.562272	−	1.50908i	−0.0294307	−	0.0789890i
$366$		0			0
$367$	−3.44746	+	3.44746i	−0.179956	+	0.179956i	−0.791337	−	0.611381i	$-0.790614\pi$
$367$	−3.44746	+	3.44746i	−0.179956	+	0.179956i	0.611381	+	0.791337i	$0.290614\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	5.08789			0.264150
$372$		0			0
$373$	7.24984	−	7.24984i	0.375382	−	0.375382i	−0.494051	−	0.869433i	$-0.664484\pi$
$373$	7.24984	−	7.24984i	0.375382	−	0.375382i	0.869433	+	0.494051i	$0.164484\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−19.8585	−	19.8585i	−1.02277	−	1.02277i
$378$		0			0
$379$	29.1664			1.49818			0.749088	−	0.662470i	$-0.230492\pi$
$379$	29.1664			1.49818			0.749088	+	0.662470i	$0.230492\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−12.2895	+	12.2895i	−0.627965	+	0.627965i	−0.947556	−	0.319591i	$-0.896455\pi$
$383$	−12.2895	+	12.2895i	−0.627965	+	0.627965i	0.319591	+	0.947556i	$0.396455\pi$
$384$		0			0
$385$	29.0433	−	10.8213i	1.48018	−	0.551504i
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	31.4290i		−	1.59351i	−0.604301	−	0.796756i	$-0.706548\pi$
$389$		−	31.4290i		−	1.59351i	0.604301	−	0.796756i	$-0.293452\pi$
$390$		0			0
$391$	2.79539			0.141369
$392$		0			0
$393$		0			0
$394$		0			0
$395$	1.22632	−	2.68284i	0.0617029	−	0.134988i
$396$		0			0
$397$	14.9362	+	14.9362i	0.749627	+	0.749627i	0.974409	−	0.224782i	$-0.0721671\pi$
$397$	14.9362	+	14.9362i	0.749627	+	0.749627i	−0.224782	+	0.974409i	$0.572167\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	27.0477i		−	1.35070i	−0.737499	−	0.675349i	$-0.763993\pi$
$401$		−	27.0477i		−	1.35070i	0.737499	−	0.675349i	$-0.236007\pi$
$402$		0			0
$403$	−44.0611	+	44.0611i	−2.19484	+	2.19484i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−21.2038	−	21.2038i	−1.05103	−	1.05103i
$408$		0			0
$409$		−	4.66540i		−	0.230689i	−0.993326	−	0.115345i	$-0.963203\pi$
$409$		−	4.66540i		−	0.230689i	0.993326	−	0.115345i	$-0.0367973\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	1.32484	+	1.32484i	0.0651911	+	0.0651911i
$414$		0			0
$415$	10.8812	+	29.2040i	0.534136	+	1.43357i
$416$		0			0
$417$		0			0
$418$		0			0
$419$			2.43646i			0.119029i	0.998227	+	0.0595144i	$0.0189552\pi$
$419$			2.43646i			0.119029i	−0.998227	+	0.0595144i	$0.981045\pi$
$420$		0			0
$421$		−	9.62136i		−	0.468916i	−0.972126	−	0.234458i	$-0.924668\pi$
$421$		−	9.62136i		−	0.468916i	0.972126	−	0.234458i	$-0.0753316\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	0.960298	−	13.2990i	0.0465813	−	0.645098i
$426$		0			0
$427$	−7.49294	−	7.49294i	−0.362609	−	0.362609i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			26.4447i			1.27380i	0.770948	+	0.636898i	$0.219783\pi$
$431$			26.4447i			1.27380i	−0.770948	+	0.636898i	$0.780217\pi$
$432$		0			0
$433$	5.22850	+	5.22850i	0.251266	+	0.251266i	0.821490	−	0.570224i	$-0.193143\pi$
$433$	5.22850	+	5.22850i	0.251266	+	0.251266i	−0.570224	+	0.821490i	$0.693143\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	3.42114	−	3.42114i	0.163655	−	0.163655i
$438$		0			0
$439$		−	37.3742i		−	1.78377i	−0.452258	−	0.891887i	$-0.649381\pi$
$439$		−	37.3742i		−	1.78377i	0.452258	−	0.891887i	$-0.350619\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−18.0711	−	18.0711i	−0.858585	−	0.858585i	0.132587	−	0.991171i	$-0.457672\pi$
$443$	−18.0711	−	18.0711i	−0.858585	−	0.858585i	−0.991171	+	0.132587i	$0.957672\pi$
$444$		0			0
$445$	−5.93425	−	2.71253i	−0.281310	−	0.128586i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−24.4570			−1.15420			−0.577098	−	0.816675i	$-0.695815\pi$
$449$	−24.4570			−1.15420			−0.577098	+	0.816675i	$0.695815\pi$
$450$		0			0
$451$			24.8921i			1.17212i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	21.1307	−	46.2279i	0.990621	−	2.16720i
$456$		0			0
$457$	5.76322	−	5.76322i	0.269592	−	0.269592i	−0.559344	−	0.828936i	$-0.688947\pi$
$457$	5.76322	−	5.76322i	0.269592	−	0.269592i	0.828936	+	0.559344i	$0.188947\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−10.6204			−0.494639			−0.247320	−	0.968934i	$-0.579550\pi$
$461$	−10.6204			−0.494639			−0.247320	+	0.968934i	$0.579550\pi$
$462$		0			0
$463$	−13.7984	−	13.7984i	−0.641268	−	0.641268i	0.309599	−	0.950867i	$-0.399805\pi$
$463$	−13.7984	−	13.7984i	−0.641268	−	0.641268i	−0.950867	+	0.309599i	$0.899805\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	11.6749	−	11.6749i	0.540248	−	0.540248i	−0.383354	−	0.923602i	$-0.625231\pi$
$467$	11.6749	−	11.6749i	0.540248	−	0.540248i	0.923602	+	0.383354i	$0.125231\pi$
$468$		0			0
$469$	6.41933			0.296417
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−0.809762	+	0.809762i	−0.0372329	+	0.0372329i
$474$		0			0
$475$	−15.1008	−	17.4513i	−0.692872	−	0.800721i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	5.10716			0.233352			0.116676	−	0.993170i	$-0.462776\pi$
$479$	5.10716			0.233352			0.116676	+	0.993170i	$0.462776\pi$
$480$		0			0
$481$	−49.1768			−2.24227
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−24.0431	+	8.95824i	−1.09174	+	0.406773i
$486$		0			0
$487$	−3.80542	+	3.80542i	−0.172440	+	0.172440i	−0.788050	−	0.615611i	$-0.788909\pi$
$487$	−3.80542	+	3.80542i	−0.172440	+	0.172440i	0.615611	+	0.788050i	$0.288909\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−19.3774			−0.874492			−0.437246	−	0.899342i	$-0.644046\pi$
$491$	−19.3774			−0.874492			−0.437246	+	0.899342i	$0.644046\pi$
$492$		0			0
$493$	−8.21932	+	8.21932i	−0.370180	+	0.370180i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	19.0018	+	19.0018i	0.852349	+	0.852349i
$498$		0			0
$499$	21.5723			0.965707			0.482853	−	0.875701i	$-0.339600\pi$
$499$	21.5723			0.965707			0.482853	+	0.875701i	$0.339600\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−0.650945	+	0.650945i	−0.0290242	+	0.0290242i	−0.721470	−	0.692446i	$-0.756533\pi$
$503$	−0.650945	+	0.650945i	−0.0290242	+	0.0290242i	0.692446	+	0.721470i	$0.256533\pi$
$504$		0			0
$505$	22.4251	+	10.2505i	0.997906	+	0.456140i
$506$		0			0
$507$		0			0
$508$		0			0
$509$			9.29371i			0.411937i	0.978559	+	0.205968i	$0.0660344\pi$
$509$			9.29371i			0.411937i	−0.978559	+	0.205968i	$0.933966\pi$
$510$		0			0
$511$	−2.54090			−0.112403
$512$		0			0
$513$		0			0
$514$		0			0
$515$	3.19224	+	8.56766i	0.140667	+	0.377536i
$516$		0			0
$517$	13.6753	+	13.6753i	0.601441	+	0.601441i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			37.4945i			1.64266i	0.570451	+	0.821331i	$0.306768\pi$
$521$			37.4945i			1.64266i	−0.570451	+	0.821331i	$0.693232\pi$
$522$		0			0
$523$	18.0635	−	18.0635i	0.789861	−	0.789861i	−0.191610	−	0.981471i	$-0.561371\pi$
$523$	18.0635	−	18.0635i	0.789861	−	0.789861i	0.981471	+	0.191610i	$0.0613710\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	18.2366	+	18.2366i	0.794400	+	0.794400i
$528$		0			0
$529$			21.9012i			0.952225i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	28.8655	+	28.8655i	1.25030	+	1.25030i
$534$		0			0
$535$	15.5243	−	5.78423i	0.671175	−	0.250074i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	21.3999i		−	0.921758i
$540$		0			0
$541$		−	29.6103i		−	1.27305i	−0.771258	−	0.636523i	$-0.780372\pi$
$541$		−	29.6103i		−	1.27305i	0.771258	−	0.636523i	$-0.219628\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	10.8083	+	4.94044i	0.462976	+	0.211625i
$546$		0			0
$547$	−13.8891	−	13.8891i	−0.593855	−	0.593855i	0.344816	−	0.938670i	$-0.387941\pi$
$547$	−13.8891	−	13.8891i	−0.593855	−	0.593855i	−0.938670	+	0.344816i	$0.887941\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			20.1185i			0.857076i
$552$		0			0
$553$	−3.29101	−	3.29101i	−0.139948	−	0.139948i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	16.0818	−	16.0818i	0.681408	−	0.681408i	−0.278910	−	0.960317i	$-0.589973\pi$
$557$	16.0818	−	16.0818i	0.681408	−	0.681408i	0.960317	+	0.278910i	$0.0899730\pi$
$558$		0			0
$559$			1.87804i			0.0794326i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	11.6731	+	11.6731i	0.491962	+	0.491962i	0.908924	−	0.416962i	$-0.136905\pi$
$563$	11.6731	+	11.6731i	0.491962	+	0.491962i	−0.416962	+	0.908924i	$0.636905\pi$
$564$		0			0
$565$	−30.6874	+	11.4339i	−1.29103	+	0.481027i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−26.5009			−1.11098			−0.555488	−	0.831524i	$-0.687469\pi$
$569$	−26.5009			−1.11098			−0.555488	+	0.831524i	$0.687469\pi$
$570$		0			0
$571$		−	42.2873i		−	1.76967i	−0.465904	−	0.884835i	$-0.654271\pi$
$571$		−	42.2873i		−	1.76967i	0.465904	−	0.884835i	$-0.345729\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−5.22761	−	0.377476i	−0.218006	−	0.0157418i
$576$		0			0
$577$	10.2682	−	10.2682i	0.427473	−	0.427473i	−0.460294	−	0.887767i	$-0.652256\pi$
$577$	10.2682	−	10.2682i	0.427473	−	0.427473i	0.887767	+	0.460294i	$0.152256\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	49.1720			2.04000
$582$		0			0
$583$	4.00635	+	4.00635i	0.165926	+	0.165926i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	18.7596	−	18.7596i	0.774290	−	0.774290i	−0.204563	−	0.978853i	$-0.565577\pi$
$587$	18.7596	−	18.7596i	0.774290	−	0.774290i	0.978853	+	0.204563i	$0.0655773\pi$
$588$		0			0
$589$	44.6379			1.83927
$590$		0			0
$591$		0			0
$592$		0			0
$593$	1.79755	−	1.79755i	0.0738164	−	0.0738164i	−0.669235	−	0.743051i	$-0.733378\pi$
$593$	1.79755	−	1.79755i	0.0738164	−	0.0738164i	0.743051	+	0.669235i	$0.233378\pi$
$594$		0			0
$595$	−19.1335	−	8.74586i	−0.784396	−	0.358545i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	34.7082			1.41814			0.709069	−	0.705139i	$-0.249115\pi$
$599$	34.7082			1.41814			0.709069	+	0.705139i	$0.249115\pi$
$600$		0			0
$601$	−8.66540			−0.353469			−0.176735	−	0.984259i	$-0.556553\pi$
$601$	−8.66540			−0.353469			−0.176735	+	0.984259i	$0.556553\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	9.02002	+	4.12303i	0.366716	+	0.167625i
$606$		0			0
$607$	5.31702	−	5.31702i	0.215811	−	0.215811i	−0.590919	−	0.806731i	$-0.701235\pi$
$607$	5.31702	−	5.31702i	0.215811	−	0.215811i	0.806731	+	0.590919i	$0.201235\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	31.7165			1.28311
$612$		0			0
$613$	0.848748	−	0.848748i	0.0342806	−	0.0342806i	−0.689759	−	0.724039i	$-0.742283\pi$
$613$	0.848748	−	0.848748i	0.0342806	−	0.0342806i	0.724039	+	0.689759i	$0.242283\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	16.6187	+	16.6187i	0.669043	+	0.669043i	0.957494	−	0.288452i	$-0.0931405\pi$
$617$	16.6187	+	16.6187i	0.669043	+	0.669043i	−0.288452	+	0.957494i	$0.593141\pi$
$618$		0			0
$619$	−6.21940			−0.249979			−0.124989	−	0.992158i	$-0.539890\pi$
$619$	−6.21940			−0.249979			−0.124989	+	0.992158i	$0.539890\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−7.27946	+	7.27946i	−0.291646	+	0.291646i
$624$		0			0
$625$	−3.59168	+	24.7407i	−0.143667	+	0.989626i
$626$		0			0
$627$		0			0
$628$		0			0
$629$			20.3540i			0.811567i
$630$		0			0
$631$	−12.0019			−0.477790			−0.238895	−	0.971045i	$-0.576785\pi$
$631$	−12.0019			−0.477790			−0.238895	+	0.971045i	$0.576785\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	3.28543	−	1.22412i	0.130378	−	0.0485778i
$636$		0			0
$637$	−24.8158	−	24.8158i	−0.983239	−	0.983239i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	6.27061i		−	0.247674i	−0.992303	−	0.123837i	$-0.960480\pi$
$641$		−	6.27061i		−	0.247674i	0.992303	−	0.123837i	$-0.0395200\pi$
$642$		0			0
$643$	15.3358	−	15.3358i	0.604787	−	0.604787i	−0.336792	−	0.941579i	$-0.609342\pi$
$643$	15.3358	−	15.3358i	0.604787	−	0.604787i	0.941579	+	0.336792i	$0.109342\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−28.9583	−	28.9583i	−1.13847	−	1.13847i	−0.988725	−	0.149741i	$-0.952156\pi$
$647$	−28.9583	−	28.9583i	−1.13847	−	1.13847i	−0.149741	−	0.988725i	$-0.547844\pi$
$648$		0			0
$649$			2.08643i			0.0818997i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	2.21463	+	2.21463i	0.0866651	+	0.0866651i	0.749110	−	0.662445i	$-0.230481\pi$
$653$	2.21463	+	2.21463i	0.0866651	+	0.0866651i	−0.662445	+	0.749110i	$0.730481\pi$
$654$		0			0
$655$	2.02754	+	0.926783i	0.0792225	+	0.0362124i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			37.2920i			1.45269i	0.687330	+	0.726345i	$0.258782\pi$
$659$			37.2920i			1.45269i	−0.687330	+	0.726345i	$0.741218\pi$
$660$		0			0
$661$		−	6.19442i		−	0.240935i	−0.992717	−	0.120467i	$-0.961561\pi$
$661$		−	6.19442i		−	0.240935i	0.992717	−	0.120467i	$-0.0384393\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−34.1202	+	12.7129i	−1.32312	+	0.492985i
$666$		0			0
$667$	3.23087	+	3.23087i	0.125100	+	0.125100i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	11.8003i		−	0.455546i
$672$		0			0
$673$	−27.8727	−	27.8727i	−1.07441	−	1.07441i	−0.996999	−	0.0774133i	$-0.975334\pi$
$673$	−27.8727	−	27.8727i	−1.07441	−	1.07441i	−0.0774133	−	0.996999i	$-0.524666\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	10.0200	−	10.0200i	0.385098	−	0.385098i	−0.487837	−	0.872935i	$-0.662214\pi$
$677$	10.0200	−	10.0200i	0.385098	−	0.385098i	0.872935	+	0.487837i	$0.162214\pi$
$678$		0			0
$679$			40.4823i			1.55357i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−11.5229	−	11.5229i	−0.440911	−	0.440911i	0.451407	−	0.892318i	$-0.350922\pi$
$683$	−11.5229	−	11.5229i	−0.440911	−	0.440911i	−0.892318	+	0.451407i	$0.850922\pi$
$684$		0			0
$685$	1.13977	+	3.05904i	0.0435485	+	0.116880i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	9.29173			0.353987
$690$		0			0
$691$		−	18.3907i		−	0.699614i	−0.936822	−	0.349807i	$-0.886247\pi$
$691$		−	18.3907i		−	0.699614i	0.936822	−	0.349807i	$-0.113753\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	34.5380	+	15.7872i	1.31010	+	0.598844i
$696$		0			0
$697$	11.9473	−	11.9473i	0.452535	−	0.452535i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	52.0764			1.96690			0.983449	−	0.181186i	$-0.0579937\pi$
$701$	52.0764			1.96690			0.983449	+	0.181186i	$0.0579937\pi$
$702$		0			0
$703$	24.9103	+	24.9103i	0.939509	+	0.939509i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	27.5086	−	27.5086i	1.03457	−	1.03457i
$708$		0			0
$709$	−49.8546			−1.87233			−0.936165	−	0.351561i	$-0.885651\pi$
$709$	−49.8546			−1.87233			−0.936165	+	0.351561i	$0.885651\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	7.16849	−	7.16849i	0.268462	−	0.268462i
$714$		0			0
$715$	53.0401	−	19.7623i	1.98359	−	0.739068i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−21.3548			−0.796400			−0.398200	−	0.917299i	$-0.630365\pi$
$719$	−21.3548			−0.796400			−0.398200	+	0.917299i	$0.630365\pi$
$720$		0			0
$721$	14.4257			0.537242
$722$		0			0
$723$		0			0
$724$		0			0
$725$	16.4807	−	14.2609i	0.612079	−	0.529638i
$726$		0			0
$727$	−13.1166	+	13.1166i	−0.486469	+	0.486469i	−0.907190	−	0.420721i	$-0.861777\pi$
$727$	−13.1166	+	13.1166i	−0.486469	+	0.486469i	0.420721	+	0.907190i	$0.361777\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	0.777309			0.0287498
$732$		0			0
$733$	−3.40526	+	3.40526i	−0.125776	+	0.125776i	−0.767193	−	0.641417i	$-0.778347\pi$
$733$	−3.40526	+	3.40526i	−0.125776	+	0.125776i	0.641417	+	0.767193i	$0.278347\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	5.05477	+	5.05477i	0.186195	+	0.186195i
$738$		0			0
$739$	−6.84745			−0.251887			−0.125944	−	0.992037i	$-0.540196\pi$
$739$	−6.84745			−0.251887			−0.125944	+	0.992037i	$0.540196\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−9.83167	+	9.83167i	−0.360689	+	0.360689i	−0.864067	−	0.503377i	$-0.832091\pi$
$743$	−9.83167	+	9.83167i	−0.360689	+	0.360689i	0.503377	+	0.864067i	$0.332091\pi$
$744$		0			0
$745$	−6.03092	+	13.1940i	−0.220956	+	0.483389i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		−	26.1389i		−	0.955095i
$750$		0			0
$751$	36.1038			1.31745			0.658723	−	0.752385i	$-0.271097\pi$
$751$	36.1038			1.31745			0.658723	+	0.752385i	$0.271097\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−8.56606	−	3.91553i	−0.311751	−	0.142501i
$756$		0			0
$757$	−26.0495	−	26.0495i	−0.946785	−	0.946785i	0.0518693	−	0.998654i	$-0.483482\pi$
$757$	−26.0495	−	26.0495i	−0.946785	−	0.946785i	−0.998654	+	0.0518693i	$0.983482\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	29.4006i		−	1.06577i	−0.846188	−	0.532885i	$-0.821108\pi$
$761$		−	29.4006i		−	1.06577i	0.846188	−	0.532885i	$-0.178892\pi$
$762$		0			0
$763$	13.2584	−	13.2584i	0.479986	−	0.479986i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	2.41948	+	2.41948i	0.0873623	+	0.0873623i
$768$		0			0
$769$		−	39.4234i		−	1.42164i	−0.703372	−	0.710822i	$-0.748323\pi$
$769$		−	39.4234i		−	1.42164i	0.703372	−	0.710822i	$-0.251677\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	2.76896	+	2.76896i	0.0995925	+	0.0995925i	0.755147	−	0.655555i	$-0.227565\pi$
$773$	2.76896	+	2.76896i	0.0995925	+	0.0995925i	−0.655555	+	0.755147i	$0.727565\pi$
$774$		0			0
$775$	−31.6414	−	36.5666i	−1.13659	−	1.31351i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	29.2434i		−	1.04775i
$780$		0			0
$781$			29.9252i			1.07081i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−0.0407272	−	0.109308i	−0.00145362	−	0.00390136i
$786$		0			0
$787$	−20.1445	−	20.1445i	−0.718072	−	0.718072i	0.250138	−	0.968210i	$-0.419524\pi$
$787$	−20.1445	−	20.1445i	−0.718072	−	0.718072i	−0.968210	+	0.250138i	$0.919524\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			51.6696i			1.83716i
$792$		0			0
$793$	−13.6839	−	13.6839i	−0.485931	−	0.485931i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−27.4953	+	27.4953i	−0.973933	+	0.973933i	−0.999669	−	0.0257360i	$-0.991807\pi$
$797$	−27.4953	+	27.4953i	−0.973933	+	0.973933i	0.0257360	+	0.999669i	$0.491807\pi$
$798$		0			0
$799$		−	13.1273i		−	0.464410i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−2.00078	−	2.00078i	−0.0706060	−	0.0706060i
$804$		0			0
$805$	−3.43784	+	7.52102i	−0.121168	+	0.265081i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	25.2679			0.888372			0.444186	−	0.895935i	$-0.353493\pi$
$809$	25.2679			0.888372			0.444186	+	0.895935i	$0.353493\pi$
$810$		0			0
$811$		−	25.9061i		−	0.909687i	−0.890571	−	0.454844i	$-0.849695\pi$
$811$		−	25.9061i		−	0.909687i	0.890571	−	0.454844i	$-0.150305\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−3.94873	+	1.47126i	−0.138318	+	0.0515361i
$816$		0			0
$817$	0.951312	−	0.951312i	0.0332822	−	0.0332822i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−14.0659			−0.490904			−0.245452	−	0.969409i	$-0.578936\pi$
$821$	−14.0659			−0.490904			−0.245452	+	0.969409i	$0.578936\pi$
$822$		0			0
$823$	18.9660	+	18.9660i	0.661112	+	0.661112i	0.955642	−	0.294530i	$-0.0951632\pi$
$823$	18.9660	+	18.9660i	0.661112	+	0.661112i	−0.294530	+	0.955642i	$0.595163\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−5.96783	+	5.96783i	−0.207522	+	0.207522i	−0.803213	−	0.595692i	$-0.796878\pi$
$827$	−5.96783	+	5.96783i	−0.207522	+	0.207522i	0.595692	+	0.803213i	$0.296878\pi$
$828$		0			0
$829$	−8.93872			−0.310454			−0.155227	−	0.987879i	$-0.549611\pi$
$829$	−8.93872			−0.310454			−0.155227	+	0.987879i	$0.549611\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−10.2711	+	10.2711i	−0.355873	+	0.355873i
$834$		0			0
$835$	16.5935	+	44.5354i	0.574242	+	1.54121i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−7.98961			−0.275832			−0.137916	−	0.990444i	$-0.544040\pi$
$839$	−7.98961			−0.275832			−0.137916	+	0.990444i	$0.544040\pi$
$840$		0			0
$841$	10.0005			0.344843
$842$		0			0
$843$		0			0
$844$		0			0
$845$	26.5051	−	57.9855i	0.911802	−	1.99476i
$846$		0			0
$847$	11.0647	−	11.0647i	0.380189	−	0.380189i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	8.00079			0.274264
$852$		0			0
$853$	17.2928	−	17.2928i	0.592094	−	0.592094i	−0.346103	−	0.938197i	$-0.612495\pi$
$853$	17.2928	−	17.2928i	0.592094	−	0.592094i	0.938197	+	0.346103i	$0.112495\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	31.0209	+	31.0209i	1.05965	+	1.05965i	0.998104	+	0.0615490i	$0.0196040\pi$
$857$	31.0209	+	31.0209i	1.05965	+	1.05965i	0.0615490	+	0.998104i	$0.480396\pi$
$858$		0			0
$859$	−30.7443			−1.04898			−0.524492	−	0.851416i	$-0.675745\pi$
$859$	−30.7443			−1.04898			−0.524492	+	0.851416i	$0.675745\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	26.9037	−	26.9037i	0.915812	−	0.915812i	−0.0809092	−	0.996721i	$-0.525782\pi$
$863$	26.9037	−	26.9037i	0.915812	−	0.915812i	0.996721	+	0.0809092i	$0.0257824\pi$
$864$		0			0
$865$	−2.39546	−	6.42919i	−0.0814482	−	0.218599i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	5.18287i		−	0.175817i
$870$		0			0
$871$	11.7233			0.397228
$872$		0			0
$873$		0			0
$874$		0			0
$875$	34.6002	+	18.9392i	1.16970	+	0.640262i
$876$		0			0
$877$	26.8633	+	26.8633i	0.907111	+	0.907111i	0.996038	−	0.0889274i	$-0.0283439\pi$
$877$	26.8633	+	26.8633i	0.907111	+	0.907111i	−0.0889274	+	0.996038i	$0.528344\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			5.85067i			0.197114i	0.995131	+	0.0985571i	$0.0314227\pi$
$881$			5.85067i			0.197114i	−0.995131	+	0.0985571i	$0.968577\pi$
$882$		0			0
$883$	−14.4524	+	14.4524i	−0.486362	+	0.486362i	−0.907156	−	0.420794i	$-0.861751\pi$
$883$	−14.4524	+	14.4524i	−0.486362	+	0.486362i	0.420794	+	0.907156i	$0.361751\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−23.4185	−	23.4185i	−0.786317	−	0.786317i	0.194571	−	0.980888i	$-0.437668\pi$
$887$	−23.4185	−	23.4185i	−0.786317	−	0.786317i	−0.980888	+	0.194571i	$0.937668\pi$
$888$		0			0
$889$		−	5.53180i		−	0.185531i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−16.0659	−	16.0659i	−0.537623	−	0.537623i
$894$		0			0
$895$	13.2906	−	29.0762i	0.444257	−	0.971909i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			42.1552i			1.40596i
$900$		0			0
$901$		−	3.84579i		−	0.128122i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−10.2777	+	22.4848i	−0.341643	+	0.747419i
$906$		0			0
$907$	−15.8605	−	15.8605i	−0.526640	−	0.526640i	0.392929	−	0.919569i	$-0.371462\pi$
$907$	−15.8605	−	15.8605i	−0.526640	−	0.526640i	−0.919569	+	0.392929i	$0.871462\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	48.6371i		−	1.61142i	−0.592310	−	0.805710i	$-0.701784\pi$
$911$		−	48.6371i		−	1.61142i	0.592310	−	0.805710i	$-0.298216\pi$
$912$		0			0
$913$	38.7194	+	38.7194i	1.28143	+	1.28143i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	2.48715	−	2.48715i	0.0821330	−	0.0821330i
$918$		0			0
$919$		−	28.6171i		−	0.943990i	−0.881601	−	0.471995i	$-0.843534\pi$
$919$		−	28.6171i		−	0.943990i	0.881601	−	0.471995i	$-0.156466\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	34.7020	+	34.7020i	1.14223	+	1.14223i
$924$		0			0
$925$	2.74851	−	38.0637i	0.0903704	−	1.25153i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−27.8635			−0.914171			−0.457086	−	0.889423i	$-0.651107\pi$
$929$	−27.8635			−0.914171			−0.457086	+	0.889423i	$0.651107\pi$
$930$		0			0
$931$			25.1407i			0.823953i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−8.17950	−	21.9530i	−0.267498	−	0.717939i
$936$		0			0
$937$	−2.53180	+	2.53180i	−0.0827105	+	0.0827105i	−0.747252	−	0.664541i	$-0.768627\pi$
$937$	−2.53180	+	2.53180i	−0.0827105	+	0.0827105i	0.664541	+	0.747252i	$0.268627\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	28.8111			0.939216			0.469608	−	0.882875i	$-0.344395\pi$
$941$	28.8111			0.939216			0.469608	+	0.882875i	$0.344395\pi$
$942$		0			0
$943$	−4.69626	−	4.69626i	−0.152931	−	0.152931i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	28.0862	−	28.0862i	0.912678	−	0.912678i	−0.0838038	−	0.996482i	$-0.526707\pi$
$947$	28.0862	−	28.0862i	0.912678	−	0.912678i	0.996482	+	0.0838038i	$0.0267069\pi$
$948$		0			0
$949$	−4.64031			−0.150631
$950$		0			0
$951$		0			0
$952$		0			0
$953$	23.3080	−	23.3080i	0.755020	−	0.755020i	−0.220392	−	0.975411i	$-0.570734\pi$
$953$	23.3080	−	23.3080i	0.755020	−	0.755020i	0.975411	+	0.220392i	$0.0707336\pi$
$954$		0			0
$955$	11.8178	−	25.8540i	0.382415	−	0.836616i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	5.15063			0.166323
$960$		0			0
$961$	62.5320			2.01716
$962$		0			0
$963$		0			0
$964$		0			0
$965$	18.3093	+	49.1405i	0.589398	+	1.58189i
$966$		0			0
$967$	−27.4623	+	27.4623i	−0.883128	+	0.883128i	−0.993851	−	0.110723i	$-0.964683\pi$
$967$	−27.4623	+	27.4623i	−0.883128	+	0.883128i	0.110723	+	0.993851i	$0.464683\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−41.1632			−1.32099			−0.660496	−	0.750830i	$-0.729654\pi$
$971$	−41.1632			−1.32099			−0.660496	+	0.750830i	$0.729654\pi$
$972$		0			0
$973$	42.3673	−	42.3673i	1.35823	−	1.35823i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−8.98437	−	8.98437i	−0.287435	−	0.287435i	0.548630	−	0.836065i	$-0.315150\pi$
$977$	−8.98437	−	8.98437i	−0.287435	−	0.287435i	−0.836065	+	0.548630i	$0.815150\pi$
$978$		0			0
$979$	−11.4641			−0.366395
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−37.2602	+	37.2602i	−1.18842	+	1.18842i	−0.210910	+	0.977505i	$0.567643\pi$
$983$	−37.2602	+	37.2602i	−1.18842	+	1.18842i	−0.977505	+	0.210910i	$0.932357\pi$
$984$		0			0
$985$	−29.5214	+	10.9994i	−0.940631	+	0.350471i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	0.305546i		−	0.00971581i
$990$		0			0
$991$	−20.4634			−0.650040			−0.325020	−	0.945707i	$-0.605371\pi$
$991$	−20.4634			−0.650040			−0.325020	+	0.945707i	$0.605371\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	14.5687	−	31.8723i	0.461860	−	1.01042i
$996$		0			0
$997$	23.1571	+	23.1571i	0.733392	+	0.733392i	0.971290	−	0.237898i	$-0.0764584\pi$
$997$	23.1571	+	23.1571i	0.733392	+	0.733392i	−0.237898	+	0.971290i	$0.576458\pi$
$998$		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.593.21		48
3.2	odd	2	inner	1440.2.bj.a.593.3		48
4.3	odd	2		360.2.x.a.53.21	yes	48
5.2	odd	4	inner	1440.2.bj.a.17.22		48
8.3	odd	2		360.2.x.a.53.16	yes	48
8.5	even	2	inner	1440.2.bj.a.593.4		48
12.11	even	2		360.2.x.a.53.4	✓	48
15.2	even	4	inner	1440.2.bj.a.17.4		48
20.7	even	4		360.2.x.a.197.9	yes	48
24.5	odd	2	inner	1440.2.bj.a.593.22		48
24.11	even	2		360.2.x.a.53.9	yes	48
40.27	even	4		360.2.x.a.197.4	yes	48
40.37	odd	4	inner	1440.2.bj.a.17.3		48
60.47	odd	4		360.2.x.a.197.16	yes	48
120.77	even	4	inner	1440.2.bj.a.17.21		48
120.107	odd	4		360.2.x.a.197.21	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
360.2.x.a.53.4	✓	48	12.11	even	2
360.2.x.a.53.9	yes	48	24.11	even	2
360.2.x.a.53.16	yes	48	8.3	odd	2
360.2.x.a.53.21	yes	48	4.3	odd	2
360.2.x.a.197.4	yes	48	40.27	even	4
360.2.x.a.197.9	yes	48	20.7	even	4
360.2.x.a.197.16	yes	48	60.47	odd	4
360.2.x.a.197.21	yes	48	120.107	odd	4
1440.2.bj.a.17.3		48	40.37	odd	4	inner
1440.2.bj.a.17.4		48	15.2	even	4	inner
1440.2.bj.a.17.21		48	120.77	even	4	inner
1440.2.bj.a.17.22		48	5.2	odd	4	inner
1440.2.bj.a.593.3		48	3.2	odd	2	inner
1440.2.bj.a.593.4		48	8.5	even	2	inner
1440.2.bj.a.593.21		48	1.1	even	1	trivial
1440.2.bj.a.593.22		48	24.5	odd	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:	\( N \)	\(=\)	\( 1440 = 2^{5} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1440.bj (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(2.03368 + 0.929591i) q^{5} +(2.49469 - 2.49469i) q^{7} +O(q^{10})\)	\(q+(2.03368 + 0.929591i) q^{5} +(2.49469 - 2.49469i) q^{7} +3.92878 q^{11} +(4.55591 - 4.55591i) q^{13} +(-1.88566 - 1.88566i) q^{17} -4.61555 q^{19} +(-0.741221 + 0.741221i) q^{23} +(3.27172 + 3.78098i) q^{25} -4.35885i q^{29} -9.67119 q^{31} +(7.39245 - 2.75436i) q^{35} +(-5.39704 - 5.39704i) q^{37} +6.33584i q^{41} +(-0.206110 + 0.206110i) q^{43} +(3.48081 + 3.48081i) q^{47} -5.44695i q^{49} +(1.01974 + 1.01974i) q^{53} +(7.98989 + 3.65216i) q^{55} +0.531064i q^{59} -3.00356i q^{61} +(13.5004 - 5.03014i) q^{65} +(1.28660 + 1.28660i) q^{67} +7.61692i q^{71} +(-0.509262 - 0.509262i) q^{73} +(9.80109 - 9.80109i) q^{77} -1.31920i q^{79} +(9.85533 + 9.85533i) q^{83} +(-2.08194 - 5.58774i) q^{85} -2.91798 q^{89} -22.7312i q^{91} +(-9.38656 - 4.29057i) q^{95} +(-8.11369 + 8.11369i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q+O(q^{10}) \) 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

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