LMFDB - Embedded newform 1575.2.b.b.251.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1575,2,Mod(251,1575)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1575, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1575.251");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1575 = 3^{2} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1575.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$12.5764383184$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 4x^{2} + 1 $ x^4 + 4*x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 315)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			251.4
Root			$1.93185i$ of defining polynomial
Character	$\chi$	$=$	1575.251
Dual form			1575.2.b.b.251.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.93185i q^{2} -1.73205 q^{4} +(-1.00000 + 2.44949i) q^{7} +0.517638i q^{8} +O(q^{10})$	$q+1.93185i q^{2} -1.73205 q^{4} +(-1.00000 + 2.44949i) q^{7} +0.517638i q^{8} -5.27792i q^{11} -6.69213i q^{13} +(-4.73205 - 1.93185i) q^{14} -4.46410 q^{16} +3.46410 q^{17} -6.69213i q^{19} +10.1962 q^{22} -1.41421i q^{23} +12.9282 q^{26} +(1.73205 - 4.24264i) q^{28} +1.41421i q^{29} -1.79315i q^{31} -7.58871i q^{32} +6.69213i q^{34} -5.46410 q^{37} +12.9282 q^{38} +10.3923 q^{41} -8.92820 q^{43} +9.14162i q^{44} +2.73205 q^{46} -2.53590 q^{47} +(-5.00000 - 4.89898i) q^{49} +11.5911i q^{52} -4.52004i q^{53} +(-1.26795 - 0.517638i) q^{56} -2.73205 q^{58} +2.53590 q^{59} -3.58630i q^{61} +3.46410 q^{62} +5.73205 q^{64} -2.92820 q^{67} -6.00000 q^{68} +9.41902i q^{71} +6.69213i q^{73} -10.5558i q^{74} +11.5911i q^{76} +(12.9282 + 5.27792i) q^{77} +2.92820 q^{79} +20.0764i q^{82} +2.53590 q^{83} -17.2480i q^{86} +2.73205 q^{88} -0.928203 q^{89} +(16.3923 + 6.69213i) q^{91} +2.44949i q^{92} -4.89898i q^{94} -10.2784i q^{97} +(9.46410 - 9.65926i) q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{7}+O(q^{10}) $ 4 * q - 4 * q^7	$ 4 q - 4 q^{7} - 12 q^{14} - 4 q^{16} + 20 q^{22} + 24 q^{26} - 8 q^{37} + 24 q^{38} - 8 q^{43} + 4 q^{46} - 24 q^{47} - 20 q^{49} - 12 q^{56} - 4 q^{58} + 24 q^{59} + 16 q^{64} + 16 q^{67} - 24 q^{68} + 24 q^{77} - 16 q^{79} + 24 q^{83} + 4 q^{88} + 24 q^{89} + 24 q^{91} + 24 q^{98}+O(q^{100}) $ 4 * q - 4 * q^7 - 12 * q^14 - 4 * q^16 + 20 * q^22 + 24 * q^26 - 8 * q^37 + 24 * q^38 - 8 * q^43 + 4 * q^46 - 24 * q^47 - 20 * q^49 - 12 * q^56 - 4 * q^58 + 24 * q^59 + 16 * q^64 + 16 * q^67 - 24 * q^68 + 24 * q^77 - 16 * q^79 + 24 * q^83 + 4 * q^88 + 24 * q^89 + 24 * q^91 + 24 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1575\mathbb{Z}\right)^\times$.

$n$	$127$	$451$	$1226$
$\chi(n)$	$1$	$-1$	$-1$

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 $

$2$			1.93185i			1.36603i	0.730406	+	0.683013i	$0.239331\pi$
$2$			1.93185i			1.36603i	−0.730406	+	0.683013i	$0.760669\pi$
$3$		0			0
$3$		0			0
$4$	−1.73205			−0.866025
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−1.00000	+	2.44949i	−0.377964	+	0.925820i
$7$	−1.00000	+	2.44949i	−0.377964	+	0.925820i
$8$			0.517638i			0.183013i
$9$		0			0
$10$		0			0
$11$		−	5.27792i		−	1.59135i	−0.605723	−	0.795676i	$-0.707116\pi$
$11$		−	5.27792i		−	1.59135i	0.605723	−	0.795676i	$-0.292884\pi$
$12$		0			0
$13$		−	6.69213i		−	1.85606i	−0.372502	−	0.928032i	$-0.621500\pi$
$13$		−	6.69213i		−	1.85606i	0.372502	−	0.928032i	$-0.378500\pi$
$14$	−4.73205	−	1.93185i	−1.26469	−	0.516309i
$15$		0			0
$16$	−4.46410			−1.11603
$17$	3.46410			0.840168			0.420084	−	0.907485i	$-0.362001\pi$
$17$	3.46410			0.840168			0.420084	+	0.907485i	$0.362001\pi$
$18$		0			0
$19$		−	6.69213i		−	1.53528i	−0.640881	−	0.767640i	$-0.721431\pi$
$19$		−	6.69213i		−	1.53528i	0.640881	−	0.767640i	$-0.278569\pi$
$20$		0			0
$21$		0			0
$22$	10.1962			2.17383
$23$		−	1.41421i		−	0.294884i	−0.989071	−	0.147442i	$-0.952896\pi$
$23$		−	1.41421i		−	0.294884i	0.989071	−	0.147442i	$-0.0471040\pi$
$24$		0			0
$25$		0			0
$26$	12.9282			2.53543
$27$		0			0
$28$	1.73205	−	4.24264i	0.327327	−	0.801784i
$29$			1.41421i			0.262613i	0.991342	+	0.131306i	$0.0419172\pi$
$29$			1.41421i			0.262613i	−0.991342	+	0.131306i	$0.958083\pi$
$30$		0			0
$31$		−	1.79315i		−	0.322059i	−0.986950	−	0.161030i	$-0.948519\pi$
$31$		−	1.79315i		−	0.322059i	0.986950	−	0.161030i	$-0.0514815\pi$
$32$		−	7.58871i		−	1.34151i
$33$		0			0
$34$			6.69213i			1.14769i
$35$		0			0
$36$		0			0
$37$	−5.46410			−0.898293			−0.449146	−	0.893458i	$-0.648272\pi$
$37$	−5.46410			−0.898293			−0.449146	+	0.893458i	$0.648272\pi$
$38$	12.9282			2.09723
$39$		0			0
$40$		0			0
$41$	10.3923			1.62301			0.811503	−	0.584349i	$-0.198650\pi$
$41$	10.3923			1.62301			0.811503	+	0.584349i	$0.198650\pi$
$42$		0			0
$43$	−8.92820			−1.36154			−0.680769	−	0.732498i	$-0.738354\pi$
$43$	−8.92820			−1.36154			−0.680769	+	0.732498i	$0.738354\pi$
$44$			9.14162i			1.37815i
$45$		0			0
$46$	2.73205			0.402819
$47$	−2.53590			−0.369899			−0.184949	−	0.982748i	$-0.559212\pi$
$47$	−2.53590			−0.369899			−0.184949	+	0.982748i	$0.559212\pi$
$48$		0			0
$49$	−5.00000	−	4.89898i	−0.714286	−	0.699854i
$50$		0			0
$51$		0			0
$52$			11.5911i			1.60740i
$53$		−	4.52004i		−	0.620876i	−0.950594	−	0.310438i	$-0.899524\pi$
$53$		−	4.52004i		−	0.620876i	0.950594	−	0.310438i	$-0.100476\pi$
$54$		0			0
$55$		0			0
$56$	−1.26795	−	0.517638i	−0.169437	−	0.0691723i
$57$		0			0
$58$	−2.73205			−0.358736
$59$	2.53590			0.330146			0.165073	−	0.986281i	$-0.447214\pi$
$59$	2.53590			0.330146			0.165073	+	0.986281i	$0.447214\pi$
$60$		0			0
$61$		−	3.58630i		−	0.459179i	−0.973288	−	0.229589i	$-0.926262\pi$
$61$		−	3.58630i		−	0.459179i	0.973288	−	0.229589i	$-0.0737383\pi$
$62$	3.46410			0.439941
$63$		0			0
$64$	5.73205			0.716506
$65$		0			0
$66$		0			0
$67$	−2.92820			−0.357737			−0.178868	−	0.983873i	$-0.557244\pi$
$67$	−2.92820			−0.357737			−0.178868	+	0.983873i	$0.557244\pi$
$68$	−6.00000			−0.727607
$69$		0			0
$70$		0			0
$71$			9.41902i			1.11783i	0.829224	+	0.558916i	$0.188783\pi$
$71$			9.41902i			1.11783i	−0.829224	+	0.558916i	$0.811217\pi$
$72$		0			0
$73$			6.69213i			0.783255i	0.920124	+	0.391627i	$0.128088\pi$
$73$			6.69213i			0.783255i	−0.920124	+	0.391627i	$0.871912\pi$
$74$		−	10.5558i		−	1.22709i
$75$		0			0
$76$			11.5911i			1.32959i
$77$	12.9282	+	5.27792i	1.47331	+	0.601474i
$78$		0			0
$79$	2.92820			0.329449			0.164724	−	0.986340i	$-0.447327\pi$
$79$	2.92820			0.329449			0.164724	+	0.986340i	$0.447327\pi$
$80$		0			0
$81$		0			0
$82$			20.0764i			2.21707i
$83$	2.53590			0.278351			0.139176	−	0.990268i	$-0.455555\pi$
$83$	2.53590			0.278351			0.139176	+	0.990268i	$0.455555\pi$
$84$		0			0
$85$		0			0
$86$		−	17.2480i		−	1.85990i
$87$		0			0
$88$	2.73205			0.291238
$89$	−0.928203			−0.0983893			−0.0491947	−	0.998789i	$-0.515665\pi$
$89$	−0.928203			−0.0983893			−0.0491947	+	0.998789i	$0.515665\pi$
$90$		0			0
$91$	16.3923	+	6.69213i	1.71838	+	0.701526i
$92$			2.44949i			0.255377i
$93$		0			0
$94$		−	4.89898i		−	0.505291i
$95$		0			0
$96$		0			0
$97$		−	10.2784i		−	1.04362i	−0.853063	−	0.521808i	$-0.825258\pi$
$97$		−	10.2784i		−	1.04362i	0.853063	−	0.521808i	$-0.174742\pi$
$98$	9.46410	−	9.65926i	0.956019	−	0.975732i
$99$		0			0
$100$		0			0
$101$	10.3923			1.03407			0.517036	−	0.855963i	$-0.327035\pi$
$101$	10.3923			1.03407			0.517036	+	0.855963i	$0.327035\pi$
$102$		0			0
$103$			3.58630i			0.353369i	0.984268	+	0.176684i	$0.0565372\pi$
$103$			3.58630i			0.353369i	−0.984268	+	0.176684i	$0.943463\pi$
$104$	3.46410			0.339683
$105$		0			0
$106$	8.73205			0.848132
$107$		−	16.1112i		−	1.55752i	−0.627320	−	0.778762i	$-0.715848\pi$
$107$		−	16.1112i		−	1.55752i	0.627320	−	0.778762i	$-0.284152\pi$
$108$		0			0
$109$	8.00000			0.766261			0.383131	−	0.923694i	$-0.374846\pi$
$109$	8.00000			0.766261			0.383131	+	0.923694i	$0.374846\pi$
$110$		0			0
$111$		0			0
$112$	4.46410	−	10.9348i	0.421818	−	1.03324i
$113$			15.0759i			1.41822i	0.705098	+	0.709110i	$0.250903\pi$
$113$			15.0759i			1.41822i	−0.705098	+	0.709110i	$0.749097\pi$
$114$		0			0
$115$		0			0
$116$		−	2.44949i		−	0.227429i
$117$		0			0
$118$			4.89898i			0.450988i
$119$	−3.46410	+	8.48528i	−0.317554	+	0.777844i
$120$		0			0
$121$	−16.8564			−1.53240
$122$	6.92820			0.627250
$123$		0			0
$124$			3.10583i			0.278912i
$125$		0			0
$126$		0			0
$127$	10.9282			0.969721			0.484861	−	0.874591i	$-0.338870\pi$
$127$	10.9282			0.969721			0.484861	+	0.874591i	$0.338870\pi$
$128$		−	4.10394i		−	0.362740i
$129$		0			0
$130$		0			0
$131$	−9.46410			−0.826882			−0.413441	−	0.910531i	$-0.635673\pi$
$131$	−9.46410			−0.826882			−0.413441	+	0.910531i	$0.635673\pi$
$132$		0			0
$133$	16.3923	+	6.69213i	1.42139	+	0.580281i
$134$		−	5.65685i		−	0.488678i
$135$		0			0
$136$			1.79315i			0.153761i
$137$		−	6.79367i		−	0.580422i	−0.956963	−	0.290211i	$-0.906274\pi$
$137$		−	6.79367i		−	0.580422i	0.956963	−	0.290211i	$-0.0937255\pi$
$138$		0			0
$139$		−	8.00481i		−	0.678959i	−0.940613	−	0.339479i	$-0.889749\pi$
$139$		−	8.00481i		−	0.678959i	0.940613	−	0.339479i	$-0.110251\pi$
$140$		0			0
$141$		0			0
$142$	−18.1962			−1.52699
$143$	−35.3205			−2.95365
$144$		0			0
$145$		0			0
$146$	−12.9282			−1.06995
$147$		0			0
$148$	9.46410			0.777944
$149$			5.00052i			0.409658i	0.978798	+	0.204829i	$0.0656639\pi$
$149$			5.00052i			0.409658i	−0.978798	+	0.204829i	$0.934336\pi$
$150$		0			0
$151$	−7.46410			−0.607420			−0.303710	−	0.952765i	$-0.598225\pi$
$151$	−7.46410			−0.607420			−0.303710	+	0.952765i	$0.598225\pi$
$152$	3.46410			0.280976
$153$		0			0
$154$	−10.1962	+	24.9754i	−0.821629	+	2.01257i
$155$		0			0
$156$		0			0
$157$		−	11.5911i		−	0.925071i	−0.886601	−	0.462536i	$-0.846940\pi$
$157$		−	11.5911i		−	0.925071i	0.886601	−	0.462536i	$-0.153060\pi$
$158$			5.65685i			0.450035i
$159$		0			0
$160$		0			0
$161$	3.46410	+	1.41421i	0.273009	+	0.111456i
$162$		0			0
$163$	4.00000			0.313304			0.156652	−	0.987654i	$-0.449930\pi$
$163$	4.00000			0.313304			0.156652	+	0.987654i	$0.449930\pi$
$164$	−18.0000			−1.40556
$165$		0			0
$166$			4.89898i			0.380235i
$167$	12.0000			0.928588			0.464294	−	0.885681i	$-0.346308\pi$
$167$	12.0000			0.928588			0.464294	+	0.885681i	$0.346308\pi$
$168$		0			0
$169$	−31.7846			−2.44497
$170$		0			0
$171$		0			0
$172$	15.4641			1.17913
$173$	−6.00000			−0.456172			−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000			−0.456172			−0.228086	+	0.973641i	$0.573247\pi$
$174$		0			0
$175$		0			0
$176$			23.5612i			1.77599i
$177$		0			0
$178$		−	1.79315i		−	0.134402i
$179$		−	3.96524i		−	0.296376i	−0.988959	−	0.148188i	$-0.952656\pi$
$179$		−	3.96524i		−	0.296376i	0.988959	−	0.148188i	$-0.0473441\pi$
$180$		0			0
$181$			21.8695i			1.62555i	0.582578	+	0.812775i	$0.302044\pi$
$181$			21.8695i			1.62555i	−0.582578	+	0.812775i	$0.697956\pi$
$182$	−12.9282	+	31.6675i	−0.958302	+	2.34735i
$183$		0			0
$184$	0.732051			0.0539675
$185$		0			0
$186$		0			0
$187$		−	18.2832i		−	1.33700i
$188$	4.39230			0.320342
$189$		0			0
$190$		0			0
$191$		−	1.69161i		−	0.122401i	−0.998125	−	0.0612005i	$-0.980507\pi$
$191$		−	1.69161i		−	0.122401i	0.998125	−	0.0612005i	$-0.0194929\pi$
$192$		0			0
$193$	−3.60770			−0.259688			−0.129844	−	0.991534i	$-0.541448\pi$
$193$	−3.60770			−0.259688			−0.129844	+	0.991534i	$0.541448\pi$
$194$	19.8564			1.42561
$195$		0			0
$196$	8.66025	+	8.48528i	0.618590	+	0.606092i
$197$		−	3.20736i		−	0.228515i	−0.993451	−	0.114258i	$-0.963551\pi$
$197$		−	3.20736i		−	0.228515i	0.993451	−	0.114258i	$-0.0364489\pi$
$198$		0			0
$199$		−	10.2784i		−	0.728619i	−0.931278	−	0.364309i	$-0.881305\pi$
$199$		−	10.2784i		−	0.728619i	0.931278	−	0.364309i	$-0.118695\pi$
$200$		0			0
$201$		0			0
$202$			20.0764i			1.41257i
$203$	−3.46410	−	1.41421i	−0.243132	−	0.0992583i
$204$		0			0
$205$		0			0
$206$	−6.92820			−0.482711
$207$		0			0
$208$			29.8744i			2.07141i
$209$	−35.3205			−2.44317
$210$		0			0
$211$	18.3923			1.26618			0.633089	−	0.774079i	$-0.281787\pi$
$211$	18.3923			1.26618			0.633089	+	0.774079i	$0.281787\pi$
$212$			7.82894i			0.537694i
$213$		0			0
$214$	31.1244			2.12762
$215$		0			0
$216$		0			0
$217$	4.39230	+	1.79315i	0.298169	+	0.121727i
$218$			15.4548i			1.04673i
$219$		0			0
$220$		0			0
$221$		−	23.1822i		−	1.55940i
$222$		0			0
$223$			4.89898i			0.328060i	0.986455	+	0.164030i	$0.0524494\pi$
$223$			4.89898i			0.328060i	−0.986455	+	0.164030i	$0.947551\pi$
$224$	18.5885	+	7.58871i	1.24199	+	0.507042i
$225$		0			0
$226$	−29.1244			−1.93732
$227$	−23.3205			−1.54784			−0.773918	−	0.633286i	$-0.781706\pi$
$227$	−23.3205			−1.54784			−0.773918	+	0.633286i	$0.781706\pi$
$228$		0			0
$229$		−	14.6969i		−	0.971201i	−0.874181	−	0.485601i	$-0.838601\pi$
$229$		−	14.6969i		−	0.971201i	0.874181	−	0.485601i	$-0.161399\pi$
$230$		0			0
$231$		0			0
$232$	−0.732051			−0.0480615
$233$			18.6622i			1.22260i	0.791399	+	0.611300i	$0.209353\pi$
$233$			18.6622i			1.22260i	−0.791399	+	0.611300i	$0.790647\pi$
$234$		0			0
$235$		0			0
$236$	−4.39230			−0.285915
$237$		0			0
$238$	−16.3923	−	6.69213i	−1.06256	−	0.433786i
$239$		−	18.6622i		−	1.20716i	−0.797304	−	0.603578i	$-0.793741\pi$
$239$		−	18.6622i		−	1.20716i	0.797304	−	0.603578i	$-0.206259\pi$
$240$		0			0
$241$			18.2832i			1.17773i	0.808232	+	0.588864i	$0.200424\pi$
$241$			18.2832i			1.17773i	−0.808232	+	0.588864i	$0.799576\pi$
$242$		−	32.5641i		−	2.09330i
$243$		0			0
$244$			6.21166i			0.397661i
$245$		0			0
$246$		0			0
$247$	−44.7846			−2.84958
$248$	0.928203			0.0589410
$249$		0			0
$250$		0			0
$251$	−23.3205			−1.47198			−0.735989	−	0.676994i	$-0.763282\pi$
$251$	−23.3205			−1.47198			−0.735989	+	0.676994i	$0.763282\pi$
$252$		0			0
$253$	−7.46410			−0.469264
$254$			21.1117i			1.32466i
$255$		0			0
$256$	19.3923			1.21202
$257$	8.53590			0.532455			0.266227	−	0.963910i	$-0.414223\pi$
$257$	8.53590			0.532455			0.266227	+	0.963910i	$0.414223\pi$
$258$		0			0
$259$	5.46410	−	13.3843i	0.339523	−	0.831658i
$260$		0			0
$261$		0			0
$262$		−	18.2832i		−	1.12954i
$263$			2.17209i			0.133937i	0.997755	+	0.0669684i	$0.0213327\pi$
$263$			2.17209i			0.133937i	−0.997755	+	0.0669684i	$0.978667\pi$
$264$		0			0
$265$		0			0
$266$	−12.9282	+	31.6675i	−0.792679	+	1.94166i
$267$		0			0
$268$	5.07180			0.309809
$269$	15.4641			0.942863			0.471431	−	0.881903i	$-0.343737\pi$
$269$	15.4641			0.942863			0.471431	+	0.881903i	$0.343737\pi$
$270$		0			0
$271$		−	17.8028i		−	1.08144i	−0.841202	−	0.540721i	$-0.818152\pi$
$271$		−	17.8028i		−	1.08144i	0.841202	−	0.540721i	$-0.181848\pi$
$272$	−15.4641			−0.937649
$273$		0			0
$274$	13.1244			0.792871
$275$		0			0
$276$		0			0
$277$	22.0000			1.32185			0.660926	−	0.750451i	$-0.270164\pi$
$277$	22.0000			1.32185			0.660926	+	0.750451i	$0.270164\pi$
$278$	15.4641			0.927475
$279$		0			0
$280$		0			0
$281$			6.31319i			0.376614i	0.982110	+	0.188307i	$0.0602999\pi$
$281$			6.31319i			0.376614i	−0.982110	+	0.188307i	$0.939700\pi$
$282$		0			0
$283$		−	24.4949i		−	1.45607i	−0.685540	−	0.728035i	$-0.740434\pi$
$283$		−	24.4949i		−	1.45607i	0.685540	−	0.728035i	$-0.259566\pi$
$284$		−	16.3142i		−	0.968071i
$285$		0			0
$286$		−	68.2340i		−	4.03476i
$287$	−10.3923	+	25.4558i	−0.613438	+	1.50261i
$288$		0			0
$289$	−5.00000			−0.294118
$290$		0			0
$291$		0			0
$292$		−	11.5911i		−	0.678318i
$293$	−10.3923			−0.607125			−0.303562	−	0.952812i	$-0.598176\pi$
$293$	−10.3923			−0.607125			−0.303562	+	0.952812i	$0.598176\pi$
$294$		0			0
$295$		0			0
$296$		−	2.82843i		−	0.164399i
$297$		0			0
$298$	−9.66025			−0.559603
$299$	−9.46410			−0.547323
$300$		0			0
$301$	8.92820	−	21.8695i	0.514613	−	1.26054i
$302$		−	14.4195i		−	0.829751i
$303$		0			0
$304$			29.8744i			1.71341i
$305$		0			0
$306$		0			0
$307$		−	14.6969i		−	0.838799i	−0.907802	−	0.419399i	$-0.862241\pi$
$307$		−	14.6969i		−	0.838799i	0.907802	−	0.419399i	$-0.137759\pi$
$308$	−22.3923	−	9.14162i	−1.27592	−	0.520892i
$309$		0			0
$310$		0			0
$311$	7.60770			0.431393			0.215696	−	0.976460i	$-0.430798\pi$
$311$	7.60770			0.431393			0.215696	+	0.976460i	$0.430798\pi$
$312$		0			0
$313$		−	15.1774i		−	0.857878i	−0.903333	−	0.428939i	$-0.858888\pi$
$313$		−	15.1774i		−	0.857878i	0.903333	−	0.428939i	$-0.141112\pi$
$314$	22.3923			1.26367
$315$		0			0
$316$	−5.07180			−0.285311
$317$			12.4505i			0.699291i	0.936882	+	0.349645i	$0.113698\pi$
$317$			12.4505i			0.699291i	−0.936882	+	0.349645i	$0.886302\pi$
$318$		0			0
$319$	7.46410			0.417909
$320$		0			0
$321$		0			0
$322$	−2.73205	+	6.69213i	−0.152251	+	0.372938i
$323$		−	23.1822i		−	1.28989i
$324$		0			0
$325$		0			0
$326$			7.72741i			0.427981i
$327$		0			0
$328$			5.37945i			0.297031i
$329$	2.53590	−	6.21166i	0.139809	−	0.342460i
$330$		0			0
$331$	−26.3923			−1.45065			−0.725326	−	0.688405i	$-0.758311\pi$
$331$	−26.3923			−1.45065			−0.725326	+	0.688405i	$0.758311\pi$
$332$	−4.39230			−0.241059
$333$		0			0
$334$			23.1822i			1.26847i
$335$		0			0
$336$		0			0
$337$	8.39230			0.457158			0.228579	−	0.973525i	$-0.426592\pi$
$337$	8.39230			0.457158			0.228579	+	0.973525i	$0.426592\pi$
$338$		−	61.4032i		−	3.33989i
$339$		0			0
$340$		0			0
$341$	−9.46410			−0.512510
$342$		0			0
$343$	17.0000	−	7.34847i	0.917914	−	0.396780i
$344$		−	4.62158i		−	0.249179i
$345$		0			0
$346$		−	11.5911i		−	0.623142i
$347$		−	8.58682i		−	0.460965i	−0.973077	−	0.230482i	$-0.925970\pi$
$347$		−	8.58682i		−	0.460965i	0.973077	−	0.230482i	$-0.0740304\pi$
$348$		0			0
$349$		0			0		1.00000			$0$
$349$		0			0		−1.00000			$\pi$
$350$		0			0
$351$		0			0
$352$	−40.0526			−2.13481
$353$	30.0000			1.59674			0.798369	−	0.602168i	$-0.205696\pi$
$353$	30.0000			1.59674			0.798369	+	0.602168i	$0.205696\pi$
$354$		0			0
$355$		0			0
$356$	1.60770			0.0852077
$357$		0			0
$358$	7.66025			0.404857
$359$			25.0769i			1.32351i	0.749721	+	0.661754i	$0.230188\pi$
$359$			25.0769i			1.32351i	−0.749721	+	0.661754i	$0.769812\pi$
$360$		0			0
$361$	−25.7846			−1.35708
$362$	−42.2487			−2.22054
$363$		0			0
$364$	−28.3923	−	11.5911i	−1.48816	−	0.607539i
$365$		0			0
$366$		0			0
$367$			12.0716i			0.630132i	0.949070	+	0.315066i	$0.102027\pi$
$367$			12.0716i			0.630132i	−0.949070	+	0.315066i	$0.897973\pi$
$368$			6.31319i			0.329098i
$369$		0			0
$370$		0			0
$371$	11.0718	+	4.52004i	0.574819	+	0.234669i
$372$		0			0
$373$	−20.9282			−1.08362			−0.541811	−	0.840501i	$-0.682261\pi$
$373$	−20.9282			−1.08362			−0.541811	+	0.840501i	$0.682261\pi$
$374$	35.3205			1.82638
$375$		0			0
$376$		−	1.31268i		−	0.0676962i
$377$	9.46410			0.487426
$378$		0			0
$379$	4.53590			0.232993			0.116497	−	0.993191i	$-0.462834\pi$
$379$	4.53590			0.232993			0.116497	+	0.993191i	$0.462834\pi$
$380$		0			0
$381$		0			0
$382$	3.26795			0.167203
$383$	17.0718			0.872328			0.436164	−	0.899867i	$-0.356337\pi$
$383$	17.0718			0.872328			0.436164	+	0.899867i	$0.356337\pi$
$384$		0			0
$385$		0			0
$386$		−	6.96953i		−	0.354740i
$387$		0			0
$388$			17.8028i			0.903799i
$389$		−	0.859411i		−	0.0435739i	−0.999763	−	0.0217869i	$-0.993064\pi$
$389$		−	0.859411i		−	0.0435739i	0.999763	−	0.0217869i	$-0.00693554\pi$
$390$		0			0
$391$		−	4.89898i		−	0.247752i
$392$	2.53590	−	2.58819i	0.128082	−	0.130723i
$393$		0			0
$394$	6.19615			0.312158
$395$		0			0
$396$		0			0
$397$			21.3891i			1.07349i	0.843746	+	0.536743i	$0.180346\pi$
$397$			21.3891i			1.07349i	−0.843746	+	0.536743i	$0.819654\pi$
$398$	19.8564			0.995312
$399$		0			0
$400$		0			0
$401$		−	18.1817i		−	0.907951i	−0.891014	−	0.453975i	$-0.850005\pi$
$401$		−	18.1817i		−	0.907951i	0.891014	−	0.453975i	$-0.149995\pi$
$402$		0			0
$403$	−12.0000			−0.597763
$404$	−18.0000			−0.895533
$405$		0			0
$406$	2.73205	−	6.69213i	0.135589	−	0.332125i
$407$			28.8391i			1.42950i
$408$		0			0
$409$			2.27362i			0.112423i	0.998419	+	0.0562117i	$0.0179022\pi$
$409$			2.27362i			0.112423i	−0.998419	+	0.0562117i	$0.982098\pi$
$410$		0			0
$411$		0			0
$412$		−	6.21166i		−	0.306026i
$413$	−2.53590	+	6.21166i	−0.124783	+	0.305656i
$414$		0			0
$415$		0			0
$416$	−50.7846			−2.48992
$417$		0			0
$418$		−	68.2340i		−	3.33743i
$419$	37.8564			1.84941			0.924703	−	0.380689i	$-0.124313\pi$
$419$	37.8564			1.84941			0.924703	+	0.380689i	$0.124313\pi$
$420$		0			0
$421$	27.8564			1.35764			0.678819	−	0.734306i	$-0.262492\pi$
$421$	27.8564			1.35764			0.678819	+	0.734306i	$0.262492\pi$
$422$			35.5312i			1.72963i
$423$		0			0
$424$	2.33975			0.113628
$425$		0			0
$426$		0			0
$427$	8.78461	+	3.58630i	0.425117	+	0.173553i
$428$			27.9053i			1.34886i
$429$		0			0
$430$		0			0
$431$			21.4906i			1.03517i	0.855633	+	0.517583i	$0.173168\pi$
$431$			21.4906i			1.03517i	−0.855633	+	0.517583i	$0.826832\pi$
$432$		0			0
$433$		−	24.9754i		−	1.20024i	−0.799910	−	0.600120i	$-0.795120\pi$
$433$		−	24.9754i		−	1.20024i	0.799910	−	0.600120i	$-0.204880\pi$
$434$	−3.46410	+	8.48528i	−0.166282	+	0.407307i
$435$		0			0
$436$	−13.8564			−0.663602
$437$	−9.46410			−0.452729
$438$		0			0
$439$			33.4607i			1.59699i	0.602002	+	0.798495i	$0.294370\pi$
$439$			33.4607i			1.59699i	−0.602002	+	0.798495i	$0.705630\pi$
$440$		0			0
$441$		0			0
$442$	44.7846			2.13019
$443$			15.5563i			0.739104i	0.929210	+	0.369552i	$0.120489\pi$
$443$			15.5563i			0.739104i	−0.929210	+	0.369552i	$0.879511\pi$
$444$		0			0
$445$		0			0
$446$	−9.46410			−0.448138
$447$		0			0
$448$	−5.73205	+	14.0406i	−0.270814	+	0.663356i
$449$		−	26.6670i		−	1.25849i	−0.777206	−	0.629246i	$-0.783364\pi$
$449$		−	26.6670i		−	1.25849i	0.777206	−	0.629246i	$-0.216636\pi$
$450$		0			0
$451$		−	54.8497i		−	2.58277i
$452$		−	26.1122i		−	1.22821i
$453$		0			0
$454$		−	45.0518i		−	2.11438i
$455$		0			0
$456$		0			0
$457$	22.2487			1.04075			0.520375	−	0.853938i	$-0.325792\pi$
$457$	22.2487			1.04075			0.520375	+	0.853938i	$0.325792\pi$
$458$	28.3923			1.32669
$459$		0			0
$460$		0			0
$461$	19.8564			0.924805			0.462403	−	0.886670i	$-0.346988\pi$
$461$	19.8564			0.924805			0.462403	+	0.886670i	$0.346988\pi$
$462$		0			0
$463$	−8.00000			−0.371792			−0.185896	−	0.982569i	$-0.559519\pi$
$463$	−8.00000			−0.371792			−0.185896	+	0.982569i	$0.559519\pi$
$464$		−	6.31319i		−	0.293083i
$465$		0			0
$466$	−36.0526			−1.67010
$467$	7.60770			0.352042			0.176021	−	0.984386i	$-0.443677\pi$
$467$	7.60770			0.352042			0.176021	+	0.984386i	$0.443677\pi$
$468$		0			0
$469$	2.92820	−	7.17260i	0.135212	−	0.331200i
$470$		0			0
$471$		0			0
$472$			1.31268i			0.0604209i
$473$			47.1223i			2.16669i
$474$		0			0
$475$		0			0
$476$	6.00000	−	14.6969i	0.275010	−	0.673633i
$477$		0			0
$478$	36.0526			1.64901
$479$	−21.4641			−0.980720			−0.490360	−	0.871520i	$-0.663135\pi$
$479$	−21.4641			−0.980720			−0.490360	+	0.871520i	$0.663135\pi$
$480$		0			0
$481$			36.5665i			1.66729i
$482$	−35.3205			−1.60881
$483$		0			0
$484$	29.1962			1.32710
$485$		0			0
$486$		0			0
$487$	−15.8564			−0.718522			−0.359261	−	0.933237i	$-0.616971\pi$
$487$	−15.8564			−0.718522			−0.359261	+	0.933237i	$0.616971\pi$
$488$	1.85641			0.0840356
$489$		0			0
$490$		0			0
$491$		−	1.69161i		−	0.0763415i	−0.999271	−	0.0381708i	$-0.987847\pi$
$491$		−	1.69161i		−	0.0763415i	0.999271	−	0.0381708i	$-0.0121531\pi$
$492$		0			0
$493$			4.89898i			0.220639i
$494$		−	86.5172i		−	3.89259i
$495$		0			0
$496$			8.00481i			0.359426i
$497$	−23.0718	−	9.41902i	−1.03491	−	0.422501i
$498$		0			0
$499$	−38.3923			−1.71868			−0.859338	−	0.511408i	$-0.829124\pi$
$499$	−38.3923			−1.71868			−0.859338	+	0.511408i	$0.829124\pi$
$500$		0			0
$501$		0			0
$502$		−	45.0518i		−	2.01076i
$503$	−6.92820			−0.308913			−0.154457	−	0.988000i	$-0.549363\pi$
$503$	−6.92820			−0.308913			−0.154457	+	0.988000i	$0.549363\pi$
$504$		0			0
$505$		0			0
$506$		−	14.4195i		−	0.641027i
$507$		0			0
$508$	−18.9282			−0.839803
$509$	−34.3923			−1.52441			−0.762206	−	0.647334i	$-0.775884\pi$
$509$	−34.3923			−1.52441			−0.762206	+	0.647334i	$0.775884\pi$
$510$		0			0
$511$	−16.3923	−	6.69213i	−0.725153	−	0.296042i
$512$			29.2552i			1.29291i
$513$		0			0
$514$			16.4901i			0.727347i
$515$		0			0
$516$		0			0
$517$			13.3843i			0.588639i
$518$	25.8564	+	10.5558i	1.13607	+	0.463797i
$519$		0			0
$520$		0			0
$521$	10.3923			0.455295			0.227648	−	0.973744i	$-0.426897\pi$
$521$	10.3923			0.455295			0.227648	+	0.973744i	$0.426897\pi$
$522$		0			0
$523$		−	10.7589i		−	0.470454i	−0.971940	−	0.235227i	$-0.924417\pi$
$523$		−	10.7589i		−	0.470454i	0.971940	−	0.235227i	$-0.0755834\pi$
$524$	16.3923			0.716101
$525$		0			0
$526$	−4.19615			−0.182961
$527$		−	6.21166i		−	0.270584i
$528$		0			0
$529$	21.0000			0.913043
$530$		0			0
$531$		0			0
$532$	−28.3923	−	11.5911i	−1.23096	−	0.502538i
$533$		−	69.5467i		−	3.01240i
$534$		0			0
$535$		0			0
$536$		−	1.51575i		−	0.0654704i
$537$		0			0
$538$			29.8744i			1.28797i
$539$	−25.8564	+	26.3896i	−1.11371	+	1.13668i
$540$		0			0
$541$	15.8564			0.681720			0.340860	−	0.940114i	$-0.389282\pi$
$541$	15.8564			0.681720			0.340860	+	0.940114i	$0.389282\pi$
$542$	34.3923			1.47728
$543$		0			0
$544$		−	26.2880i		−	1.12709i
$545$		0			0
$546$		0			0
$547$	2.14359			0.0916534			0.0458267	−	0.998949i	$-0.485408\pi$
$547$	2.14359			0.0916534			0.0458267	+	0.998949i	$0.485408\pi$
$548$			11.7670i			0.502660i
$549$		0			0
$550$		0			0
$551$	9.46410			0.403184
$552$		0			0
$553$	−2.92820	+	7.17260i	−0.124520	+	0.305010i
$554$			42.5007i			1.80568i
$555$		0			0
$556$			13.8647i			0.587996i
$557$		−	33.5622i		−	1.42208i	−0.703154	−	0.711038i	$-0.748225\pi$
$557$		−	33.5622i		−	1.42208i	0.703154	−	0.711038i	$-0.251775\pi$
$558$		0			0
$559$			59.7487i			2.52710i
$560$		0			0
$561$		0			0
$562$	−12.1962			−0.514464
$563$	−17.0718			−0.719490			−0.359745	−	0.933051i	$-0.617136\pi$
$563$	−17.0718			−0.719490			−0.359745	+	0.933051i	$0.617136\pi$
$564$		0			0
$565$		0			0
$566$	47.3205			1.98903
$567$		0			0
$568$	−4.87564			−0.204577
$569$			36.6680i			1.53720i	0.639728	+	0.768602i	$0.279047\pi$
$569$			36.6680i			1.53720i	−0.639728	+	0.768602i	$0.720953\pi$
$570$		0			0
$571$	−4.00000			−0.167395			−0.0836974	−	0.996491i	$-0.526673\pi$
$571$	−4.00000			−0.167395			−0.0836974	+	0.996491i	$0.526673\pi$
$572$	61.1769			2.55794
$573$		0			0
$574$	−49.1769	−	20.0764i	−2.05260	−	0.837972i
$575$		0			0
$576$		0			0
$577$			17.8028i			0.741139i	0.928805	+	0.370569i	$0.120837\pi$
$577$			17.8028i			0.741139i	−0.928805	+	0.370569i	$0.879163\pi$
$578$		−	9.65926i		−	0.401772i
$579$		0			0
$580$		0			0
$581$	−2.53590	+	6.21166i	−0.105207	+	0.257703i
$582$		0			0
$583$	−23.8564			−0.988031
$584$	−3.46410			−0.143346
$585$		0			0
$586$		−	20.0764i		−	0.829348i
$587$	−39.7128			−1.63912			−0.819562	−	0.572991i	$-0.805783\pi$
$587$	−39.7128			−1.63912			−0.819562	+	0.572991i	$0.805783\pi$
$588$		0			0
$589$	−12.0000			−0.494451
$590$		0			0
$591$		0			0
$592$	24.3923			1.00252
$593$	−45.7128			−1.87720			−0.938600	−	0.345007i	$-0.887877\pi$
$593$	−45.7128			−1.87720			−0.938600	+	0.345007i	$0.887877\pi$
$594$		0			0
$595$		0			0
$596$		−	8.66115i		−	0.354774i
$597$		0			0
$598$		−	18.2832i		−	0.747657i
$599$			37.5002i			1.53222i	0.642711	+	0.766109i	$0.277810\pi$
$599$			37.5002i			1.53222i	−0.642711	+	0.766109i	$0.722190\pi$
$600$		0			0
$601$			19.5959i			0.799334i	0.916660	+	0.399667i	$0.130874\pi$
$601$			19.5959i			0.799334i	−0.916660	+	0.399667i	$0.869126\pi$
$602$	42.2487	+	17.2480i	1.72193	+	0.702975i
$603$		0			0
$604$	12.9282			0.526041
$605$		0			0
$606$		0			0
$607$		−	18.2832i		−	0.742094i	−0.928614	−	0.371047i	$-0.878999\pi$
$607$		−	18.2832i		−	0.742094i	0.928614	−	0.371047i	$-0.121001\pi$
$608$	−50.7846			−2.05959
$609$		0			0
$610$		0			0
$611$			16.9706i			0.686555i
$612$		0			0
$613$	15.0718			0.608744			0.304372	−	0.952553i	$-0.401553\pi$
$613$	15.0718			0.608744			0.304372	+	0.952553i	$0.401553\pi$
$614$	28.3923			1.14582
$615$		0			0
$616$	−2.73205	+	6.69213i	−0.110077	+	0.269634i
$617$			1.69161i			0.0681019i	0.999420	+	0.0340509i	$0.0108408\pi$
$617$			1.69161i			0.0681019i	−0.999420	+	0.0340509i	$0.989159\pi$
$618$		0			0
$619$		−	41.9459i		−	1.68595i	−0.537953	−	0.842975i	$-0.680802\pi$
$619$		−	41.9459i		−	1.68595i	0.537953	−	0.842975i	$-0.319198\pi$
$620$		0			0
$621$		0			0
$622$			14.6969i			0.589294i
$623$	0.928203	−	2.27362i	0.0371877	−	0.0910908i
$624$		0			0
$625$		0			0
$626$	29.3205			1.17188
$627$		0			0
$628$			20.0764i			0.801135i
$629$	−18.9282			−0.754717
$630$		0			0
$631$	16.7846			0.668185			0.334092	−	0.942540i	$-0.391570\pi$
$631$	16.7846			0.668185			0.334092	+	0.942540i	$0.391570\pi$
$632$			1.51575i			0.0602933i
$633$		0			0
$634$	−24.0526			−0.955249
$635$		0			0
$636$		0			0
$637$	−32.7846	+	33.4607i	−1.29897	+	1.32576i
$638$			14.4195i			0.570875i
$639$		0			0
$640$		0			0
$641$			8.93855i			0.353051i	0.984296	+	0.176526i	$0.0564859\pi$
$641$			8.93855i			0.353051i	−0.984296	+	0.176526i	$0.943514\pi$
$642$		0			0
$643$		−	11.1106i		−	0.438161i	−0.975707	−	0.219080i	$-0.929694\pi$
$643$		−	11.1106i		−	0.438161i	0.975707	−	0.219080i	$-0.0703057\pi$
$644$	−6.00000	−	2.44949i	−0.236433	−	0.0965234i
$645$		0			0
$646$	44.7846			1.76203
$647$	8.78461			0.345359			0.172679	−	0.984978i	$-0.444758\pi$
$647$	8.78461			0.345359			0.172679	+	0.984978i	$0.444758\pi$
$648$		0			0
$649$		−	13.3843i		−	0.525378i
$650$		0			0
$651$		0			0
$652$	−6.92820			−0.271329
$653$		−	36.1875i		−	1.41613i	−0.706148	−	0.708064i	$-0.749569\pi$
$653$		−	36.1875i		−	1.41613i	0.706148	−	0.708064i	$-0.250431\pi$
$654$		0			0
$655$		0			0
$656$	−46.3923			−1.81132
$657$		0			0
$658$	12.0000	+	4.89898i	0.467809	+	0.190982i
$659$		−	3.96524i		−	0.154464i	−0.997013	−	0.0772319i	$-0.975392\pi$
$659$		−	3.96524i		−	0.154464i	0.997013	−	0.0772319i	$-0.0246082\pi$
$660$		0			0
$661$			32.9802i			1.28278i	0.767215	+	0.641390i	$0.221642\pi$
$661$			32.9802i			1.28278i	−0.767215	+	0.641390i	$0.778358\pi$
$662$		−	50.9860i		−	1.98163i
$663$		0			0
$664$			1.31268i			0.0509418i
$665$		0			0
$666$		0			0
$667$	2.00000			0.0774403
$668$	−20.7846			−0.804181
$669$		0			0
$670$		0			0
$671$	−18.9282			−0.730715
$672$		0			0
$673$	44.3923			1.71120			0.855599	−	0.517640i	$-0.173189\pi$
$673$	44.3923			1.71120			0.855599	+	0.517640i	$0.173189\pi$
$674$			16.2127i			0.624489i
$675$		0			0
$676$	55.0526			2.11741
$677$	−6.67949			−0.256714			−0.128357	−	0.991728i	$-0.540970\pi$
$677$	−6.67949			−0.256714			−0.128357	+	0.991728i	$0.540970\pi$
$678$		0			0
$679$	25.1769	+	10.2784i	0.966201	+	0.394450i
$680$		0			0
$681$		0			0
$682$		−	18.2832i		−	0.700101i
$683$			33.8396i			1.29484i	0.762135	+	0.647418i	$0.224151\pi$
$683$			33.8396i			1.29484i	−0.762135	+	0.647418i	$0.775849\pi$
$684$		0			0
$685$		0			0
$686$	14.1962	+	32.8415i	0.542012	+	1.25389i
$687$		0			0
$688$	39.8564			1.51951
$689$	−30.2487			−1.15238
$690$		0			0
$691$			41.9459i			1.59570i	0.602857	+	0.797849i	$0.294029\pi$
$691$			41.9459i			1.59570i	−0.602857	+	0.797849i	$0.705971\pi$
$692$	10.3923			0.395056
$693$		0			0
$694$	16.5885			0.629689
$695$		0			0
$696$		0			0
$697$	36.0000			1.36360
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	32.5269i		−	1.22852i	−0.789102	−	0.614262i	$-0.789454\pi$
$701$		−	32.5269i		−	1.22852i	0.789102	−	0.614262i	$-0.210546\pi$
$702$		0			0
$703$			36.5665i			1.37913i
$704$		−	30.2533i		−	1.14021i
$705$		0			0
$706$			57.9555i			2.18119i
$707$	−10.3923	+	25.4558i	−0.390843	+	0.957366i
$708$		0			0
$709$	37.5692			1.41094			0.705471	−	0.708739i	$-0.250736\pi$
$709$	37.5692			1.41094			0.705471	+	0.708739i	$0.250736\pi$
$710$		0			0
$711$		0			0
$712$		−	0.480473i		−	0.0180065i
$713$	−2.53590			−0.0949701
$714$		0			0
$715$		0			0
$716$			6.86800i			0.256669i
$717$		0			0
$718$	−48.4449			−1.80795
$719$	16.3923			0.611330			0.305665	−	0.952139i	$-0.401121\pi$
$719$	16.3923			0.611330			0.305665	+	0.952139i	$0.401121\pi$
$720$		0			0
$721$	−8.78461	−	3.58630i	−0.327156	−	0.133561i
$722$		−	49.8120i		−	1.85381i
$723$		0			0
$724$		−	37.8792i		−	1.40777i
$725$		0			0
$726$		0			0
$727$			19.5959i			0.726772i	0.931639	+	0.363386i	$0.118379\pi$
$727$			19.5959i			0.726772i	−0.931639	+	0.363386i	$0.881621\pi$
$728$	−3.46410	+	8.48528i	−0.128388	+	0.314485i
$729$		0			0
$730$		0			0
$731$	−30.9282			−1.14392
$732$		0			0
$733$			27.2490i			1.00646i	0.864151	+	0.503232i	$0.167856\pi$
$733$			27.2490i			1.00646i	−0.864151	+	0.503232i	$0.832144\pi$
$734$	−23.3205			−0.860776
$735$		0			0
$736$	−10.7321			−0.395589
$737$			15.4548i			0.569285i
$738$		0			0
$739$	38.9282			1.43200			0.715999	−	0.698102i	$-0.245972\pi$
$739$	38.9282			1.43200			0.715999	+	0.698102i	$0.245972\pi$
$740$		0			0
$741$		0			0
$742$	−8.73205	+	21.3891i	−0.320564	+	0.785217i
$743$		−	16.1112i		−	0.591061i	−0.955333	−	0.295530i	$-0.904504\pi$
$743$		−	16.1112i		−	0.591061i	0.955333	−	0.295530i	$-0.0954964\pi$
$744$		0			0
$745$		0			0
$746$		−	40.4302i		−	1.48025i
$747$		0			0
$748$			31.6675i			1.15788i
$749$	39.4641	+	16.1112i	1.44199	+	0.588689i
$750$		0			0
$751$	−5.60770			−0.204628			−0.102314	−	0.994752i	$-0.532625\pi$
$751$	−5.60770			−0.204628			−0.102314	+	0.994752i	$0.532625\pi$
$752$	11.3205			0.412816
$753$		0			0
$754$			18.2832i			0.665836i
$755$		0			0
$756$		0			0
$757$	−45.1769			−1.64198			−0.820991	−	0.570940i	$-0.806579\pi$
$757$	−45.1769			−1.64198			−0.820991	+	0.570940i	$0.806579\pi$
$758$			8.76268i			0.318275i
$759$		0			0
$760$		0			0
$761$	47.5692			1.72438			0.862191	−	0.506583i	$-0.169091\pi$
$761$	47.5692			1.72438			0.862191	+	0.506583i	$0.169091\pi$
$762$		0			0
$763$	−8.00000	+	19.5959i	−0.289619	+	0.709420i
$764$			2.92996i			0.106002i
$765$		0			0
$766$			32.9802i			1.19162i
$767$		−	16.9706i		−	0.612772i
$768$		0			0
$769$		−	39.1918i		−	1.41329i	−0.707566	−	0.706647i	$-0.750207\pi$
$769$		−	39.1918i		−	1.41329i	0.707566	−	0.706647i	$-0.249793\pi$
$770$		0			0
$771$		0			0
$772$	6.24871			0.224896
$773$	21.7128			0.780956			0.390478	−	0.920612i	$-0.372310\pi$
$773$	21.7128			0.780956			0.390478	+	0.920612i	$0.372310\pi$
$774$		0			0
$775$		0			0
$776$	5.32051			0.190995
$777$		0			0
$778$	1.66025			0.0595230
$779$		−	69.5467i		−	2.49177i
$780$		0			0
$781$	49.7128			1.77886
$782$	9.46410			0.338436
$783$		0			0
$784$	22.3205	+	21.8695i	0.797161	+	0.781055i
$785$		0			0
$786$		0			0
$787$		−	2.62536i		−	0.0935838i	−0.998905	−	0.0467919i	$-0.985100\pi$
$787$		−	2.62536i		−	0.0935838i	0.998905	−	0.0467919i	$-0.0148998\pi$
$788$			5.55532i			0.197900i
$789$		0			0
$790$		0			0
$791$	−36.9282	−	15.0759i	−1.31302	−	0.536036i
$792$		0			0
$793$	−24.0000			−0.852265
$794$	−41.3205			−1.46641
$795$		0			0
$796$			17.8028i			0.631002i
$797$	42.4974			1.50534			0.752668	−	0.658400i	$-0.228767\pi$
$797$	42.4974			1.50534			0.752668	+	0.658400i	$0.228767\pi$
$798$		0			0
$799$	−8.78461			−0.310777
$800$		0			0
$801$		0			0
$802$	35.1244			1.24028
$803$	35.3205			1.24643
$804$		0			0
$805$		0			0
$806$		−	23.1822i		−	0.816559i
$807$		0			0
$808$			5.37945i			0.189248i
$809$		−	2.17209i		−	0.0763666i	−0.999271	−	0.0381833i	$-0.987843\pi$
$809$		−	2.17209i		−	0.0763666i	0.999271	−	0.0381833i	$-0.0121571\pi$
$810$		0			0
$811$		−	24.9754i		−	0.877004i	−0.898730	−	0.438502i	$-0.855509\pi$
$811$		−	24.9754i		−	0.877004i	0.898730	−	0.438502i	$-0.144491\pi$
$812$	6.00000	+	2.44949i	0.210559	+	0.0859602i
$813$		0			0
$814$	−55.7128			−1.95273
$815$		0			0
$816$		0			0
$817$			59.7487i			2.09034i
$818$	−4.39230			−0.153573
$819$		0			0
$820$		0			0
$821$			47.7787i			1.66749i	0.552152	+	0.833743i	$0.313807\pi$
$821$			47.7787i			1.66749i	−0.552152	+	0.833743i	$0.686193\pi$
$822$		0			0
$823$	24.7846			0.863937			0.431969	−	0.901889i	$-0.357819\pi$
$823$	24.7846			0.863937			0.431969	+	0.901889i	$0.357819\pi$
$824$	−1.85641			−0.0646710
$825$		0			0
$826$	−12.0000	−	4.89898i	−0.417533	−	0.170457i
$827$		−	31.7690i		−	1.10472i	−0.833606	−	0.552359i	$-0.813728\pi$
$827$		−	31.7690i		−	1.10472i	0.833606	−	0.552359i	$-0.186272\pi$
$828$		0			0
$829$			45.0518i			1.56471i	0.622831	+	0.782356i	$0.285982\pi$
$829$			45.0518i			1.56471i	−0.622831	+	0.782356i	$0.714018\pi$
$830$		0			0
$831$		0			0
$832$		−	38.3596i		−	1.32988i
$833$	−17.3205	−	16.9706i	−0.600120	−	0.587995i
$834$		0			0
$835$		0			0
$836$	61.1769			2.11585
$837$		0			0
$838$			73.1330i			2.52634i
$839$	6.92820			0.239188			0.119594	−	0.992823i	$-0.461841\pi$
$839$	6.92820			0.239188			0.119594	+	0.992823i	$0.461841\pi$
$840$		0			0
$841$	27.0000			0.931034
$842$			53.8144i			1.85457i
$843$		0			0
$844$	−31.8564			−1.09654
$845$		0			0
$846$		0			0
$847$	16.8564	−	41.2896i	0.579193	−	1.41873i
$848$			20.1779i			0.692913i
$849$		0			0
$850$		0			0
$851$			7.72741i			0.264892i
$852$		0			0
$853$		−	8.96575i		−	0.306982i	−0.988150	−	0.153491i	$-0.950948\pi$
$853$		−	8.96575i		−	0.306982i	0.988150	−	0.153491i	$-0.0490515\pi$
$854$	−6.92820	+	16.9706i	−0.237078	+	0.580721i
$855$		0			0
$856$	8.33975			0.285047
$857$	50.7846			1.73477			0.867385	−	0.497638i	$-0.165799\pi$
$857$	50.7846			1.73477			0.867385	+	0.497638i	$0.165799\pi$
$858$		0			0
$859$			37.0470i			1.26403i	0.774958	+	0.632013i	$0.217771\pi$
$859$			37.0470i			1.26403i	−0.774958	+	0.632013i	$0.782229\pi$
$860$		0			0
$861$		0			0
$862$	−41.5167			−1.41406
$863$			25.3543i			0.863071i	0.902096	+	0.431535i	$0.142028\pi$
$863$			25.3543i			0.863071i	−0.902096	+	0.431535i	$0.857972\pi$
$864$		0			0
$865$		0			0
$866$	48.2487			1.63956
$867$		0			0
$868$	−7.60770	−	3.10583i	−0.258222	−	0.105419i
$869$		−	15.4548i		−	0.524269i
$870$		0			0
$871$			19.5959i			0.663982i
$872$			4.14110i			0.140236i
$873$		0			0
$874$		−	18.2832i		−	0.618440i
$875$		0			0
$876$		0			0
$877$	22.0000			0.742887			0.371444	−	0.928456i	$-0.378863\pi$
$877$	22.0000			0.742887			0.371444	+	0.928456i	$0.378863\pi$
$878$	−64.6410			−2.18153
$879$		0			0
$880$		0			0
$881$	−9.21539			−0.310474			−0.155237	−	0.987877i	$-0.549614\pi$
$881$	−9.21539			−0.310474			−0.155237	+	0.987877i	$0.549614\pi$
$882$		0			0
$883$	−39.8564			−1.34127			−0.670637	−	0.741785i	$-0.733979\pi$
$883$	−39.8564			−1.34127			−0.670637	+	0.741785i	$0.733979\pi$
$884$			40.1528i			1.35048i
$885$		0			0
$886$	−30.0526			−1.00964
$887$	−30.2487			−1.01565			−0.507826	−	0.861460i	$-0.669551\pi$
$887$	−30.2487			−1.01565			−0.507826	+	0.861460i	$0.669551\pi$
$888$		0			0
$889$	−10.9282	+	26.7685i	−0.366520	+	0.897787i
$890$		0			0
$891$		0			0
$892$		−	8.48528i		−	0.284108i
$893$			16.9706i			0.567898i
$894$		0			0
$895$		0			0
$896$	10.0526	+	4.10394i	0.335832	+	0.137103i
$897$		0			0
$898$	51.5167			1.71913
$899$	2.53590			0.0845769
$900$		0			0
$901$		−	15.6579i		−	0.521640i
$902$	105.962			3.52813
$903$		0			0
$904$	−7.80385			−0.259552
$905$		0			0
$906$		0			0
$907$	30.7846			1.02219			0.511093	−	0.859525i	$-0.329241\pi$
$907$	30.7846			1.02219			0.511093	+	0.859525i	$0.329241\pi$
$908$	40.3923			1.34047
$909$		0			0
$910$		0			0
$911$		−	33.3591i		−	1.10524i	−0.833434	−	0.552618i	$-0.813629\pi$
$911$		−	33.3591i		−	1.10524i	0.833434	−	0.552618i	$-0.186371\pi$
$912$		0			0
$913$		−	13.3843i		−	0.442954i
$914$			42.9812i			1.42169i
$915$		0			0
$916$			25.4558i			0.841085i
$917$	9.46410	−	23.1822i	0.312532	−	0.765544i
$918$		0			0
$919$	−42.1051			−1.38892			−0.694460	−	0.719531i	$-0.744357\pi$
$919$	−42.1051			−1.38892			−0.694460	+	0.719531i	$0.744357\pi$
$920$		0			0
$921$		0			0
$922$			38.3596i			1.26331i
$923$	63.0333			2.07477
$924$		0			0
$925$		0			0
$926$		−	15.4548i		−	0.507877i
$927$		0			0
$928$	10.7321			0.352297
$929$	23.0718			0.756961			0.378481	−	0.925609i	$-0.376447\pi$
$929$	23.0718			0.756961			0.378481	+	0.925609i	$0.376447\pi$
$930$		0			0
$931$	−32.7846	+	33.4607i	−1.07447	+	1.09663i
$932$		−	32.3238i		−	1.05880i
$933$		0			0
$934$			14.6969i			0.480899i
$935$		0			0
$936$		0			0
$937$		−	15.1774i		−	0.495824i	−0.968783	−	0.247912i	$-0.920256\pi$
$937$		−	15.1774i		−	0.495824i	0.968783	−	0.247912i	$-0.0797445\pi$
$938$	13.8564	+	5.65685i	0.452428	+	0.184703i
$939$		0			0
$940$		0			0
$941$	0.928203			0.0302586			0.0151293	−	0.999886i	$-0.495184\pi$
$941$	0.928203			0.0302586			0.0151293	+	0.999886i	$0.495184\pi$
$942$		0			0
$943$		−	14.6969i		−	0.478598i
$944$	−11.3205			−0.368451
$945$		0			0
$946$	−91.0333			−2.95975
$947$			19.4944i			0.633482i	0.948512	+	0.316741i	$0.102589\pi$
$947$			19.4944i			0.633482i	−0.948512	+	0.316741i	$0.897411\pi$
$948$		0			0
$949$	44.7846			1.45377
$950$		0			0
$951$		0			0
$952$	−4.39230	−	1.79315i	−0.142355	−	0.0581164i
$953$		−	6.79367i		−	0.220068i	−0.993928	−	0.110034i	$-0.964904\pi$
$953$		−	6.79367i		−	0.220068i	0.993928	−	0.110034i	$-0.0350960\pi$
$954$		0			0
$955$		0			0
$956$			32.3238i			1.04543i
$957$		0			0
$958$		−	41.4655i		−	1.33969i
$959$	16.6410	+	6.79367i	0.537366	+	0.219379i
$960$		0			0
$961$	27.7846			0.896278
$962$	−70.6410			−2.27756
$963$		0			0
$964$		−	31.6675i		−	1.01994i
$965$		0			0
$966$		0			0
$967$	12.7846			0.411125			0.205563	−	0.978644i	$-0.434098\pi$
$967$	12.7846			0.411125			0.205563	+	0.978644i	$0.434098\pi$
$968$		−	8.72552i		−	0.280449i
$969$		0			0
$970$		0			0
$971$	5.07180			0.162762			0.0813809	−	0.996683i	$-0.474067\pi$
$971$	5.07180			0.162762			0.0813809	+	0.996683i	$0.474067\pi$
$972$		0			0
$973$	19.6077	+	8.00481i	0.628594	+	0.256622i
$974$		−	30.6322i		−	0.981520i
$975$		0			0
$976$			16.0096i			0.512455i
$977$			31.0855i			0.994513i	0.867604	+	0.497256i	$0.165659\pi$
$977$			31.0855i			0.994513i	−0.867604	+	0.497256i	$0.834341\pi$
$978$		0			0
$979$			4.89898i			0.156572i
$980$		0			0
$981$		0			0
$982$	3.26795			0.104284
$983$	2.53590			0.0808826			0.0404413	−	0.999182i	$-0.487124\pi$
$983$	2.53590			0.0808826			0.0404413	+	0.999182i	$0.487124\pi$
$984$		0			0
$985$		0			0
$986$	−9.46410			−0.301398
$987$		0			0
$988$	77.5692			2.46781
$989$			12.6264i			0.401496i
$990$		0			0
$991$	−21.3205			−0.677268			−0.338634	−	0.940918i	$-0.609965\pi$
$991$	−21.3205			−0.677268			−0.338634	+	0.940918i	$0.609965\pi$
$992$	−13.6077			−0.432045
$993$		0			0
$994$	18.1962	−	44.5713i	0.577147	−	1.41372i
$995$		0			0
$996$		0			0
$997$			33.8124i			1.07085i	0.844583	+	0.535424i	$0.179848\pi$
$997$			33.8124i			1.07085i	−0.844583	+	0.535424i	$0.820152\pi$
$998$		−	74.1682i		−	2.34775i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1575.2.b.b.251.4		4
3.2	odd	2		1575.2.b.c.251.1		4
5.2	odd	4		1575.2.g.c.1574.1		8
5.3	odd	4		1575.2.g.c.1574.8		8
5.4	even	2		315.2.b.a.251.1	✓	4
7.6	odd	2		1575.2.b.c.251.4		4
15.2	even	4		1575.2.g.a.1574.7		8
15.8	even	4		1575.2.g.a.1574.2		8
15.14	odd	2		315.2.b.b.251.4	yes	4
20.19	odd	2		5040.2.f.a.881.4		4
21.20	even	2	inner	1575.2.b.b.251.1		4
35.13	even	4		1575.2.g.a.1574.8		8
35.27	even	4		1575.2.g.a.1574.1		8
35.34	odd	2		315.2.b.b.251.1	yes	4
60.59	even	2		5040.2.f.c.881.3		4
105.62	odd	4		1575.2.g.c.1574.7		8
105.83	odd	4		1575.2.g.c.1574.2		8
105.104	even	2		315.2.b.a.251.4	yes	4
140.139	even	2		5040.2.f.c.881.2		4
420.419	odd	2		5040.2.f.a.881.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
315.2.b.a.251.1	✓	4	5.4	even	2
315.2.b.a.251.4	yes	4	105.104	even	2
315.2.b.b.251.1	yes	4	35.34	odd	2
315.2.b.b.251.4	yes	4	15.14	odd	2
1575.2.b.b.251.1		4	21.20	even	2	inner
1575.2.b.b.251.4		4	1.1	even	1	trivial
1575.2.b.c.251.1		4	3.2	odd	2
1575.2.b.c.251.4		4	7.6	odd	2
1575.2.g.a.1574.1		8	35.27	even	4
1575.2.g.a.1574.2		8	15.8	even	4
1575.2.g.a.1574.7		8	15.2	even	4
1575.2.g.a.1574.8		8	35.13	even	4
1575.2.g.c.1574.1		8	5.2	odd	4
1575.2.g.c.1574.2		8	105.83	odd	4
1575.2.g.c.1574.7		8	105.62	odd	4
1575.2.g.c.1574.8		8	5.3	odd	4
5040.2.f.a.881.1		4	420.419	odd	2
5040.2.f.a.881.4		4	20.19	odd	2
5040.2.f.c.881.2		4	140.139	even	2
5040.2.f.c.881.3		4	60.59	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1575.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.93185i q^{2} -1.73205 q^{4} +(-1.00000 + 2.44949i) q^{7} +0.517638i q^{8} +O(q^{10})\)	\(q+1.93185i q^{2} -1.73205 q^{4} +(-1.00000 + 2.44949i) q^{7} +0.517638i q^{8} -5.27792i q^{11} -6.69213i q^{13} +(-4.73205 - 1.93185i) q^{14} -4.46410 q^{16} +3.46410 q^{17} -6.69213i q^{19} +10.1962 q^{22} -1.41421i q^{23} +12.9282 q^{26} +(1.73205 - 4.24264i) q^{28} +1.41421i q^{29} -1.79315i q^{31} -7.58871i q^{32} +6.69213i q^{34} -5.46410 q^{37} +12.9282 q^{38} +10.3923 q^{41} -8.92820 q^{43} +9.14162i q^{44} +2.73205 q^{46} -2.53590 q^{47} +(-5.00000 - 4.89898i) q^{49} +11.5911i q^{52} -4.52004i q^{53} +(-1.26795 - 0.517638i) q^{56} -2.73205 q^{58} +2.53590 q^{59} -3.58630i q^{61} +3.46410 q^{62} +5.73205 q^{64} -2.92820 q^{67} -6.00000 q^{68} +9.41902i q^{71} +6.69213i q^{73} -10.5558i q^{74} +11.5911i q^{76} +(12.9282 + 5.27792i) q^{77} +2.92820 q^{79} +20.0764i q^{82} +2.53590 q^{83} -17.2480i q^{86} +2.73205 q^{88} -0.928203 q^{89} +(16.3923 + 6.69213i) q^{91} +2.44949i q^{92} -4.89898i q^{94} -10.2784i q^{97} +(9.46410 - 9.65926i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{7}+O(q^{10}) \) 4 * q - 4 * q^7	\( 4 q - 4 q^{7} - 12 q^{14} - 4 q^{16} + 20 q^{22} + 24 q^{26} - 8 q^{37} + 24 q^{38} - 8 q^{43} + 4 q^{46} - 24 q^{47} - 20 q^{49} - 12 q^{56} - 4 q^{58} + 24 q^{59} + 16 q^{64} + 16 q^{67} - 24 q^{68} + 24 q^{77} - 16 q^{79} + 24 q^{83} + 4 q^{88} + 24 q^{89} + 24 q^{91} + 24 q^{98}+O(q^{100}) \) 4 * q - 4 * q^7 - 12 * q^14 - 4 * q^16 + 20 * q^22 + 24 * q^26 - 8 * q^37 + 24 * q^38 - 8 * q^43 + 4 * q^46 - 24 * q^47 - 20 * q^49 - 12 * q^56 - 4 * q^58 + 24 * q^59 + 16 * q^64 + 16 * q^67 - 24 * q^68 + 24 * q^77 - 16 * q^79 + 24 * q^83 + 4 * q^88 + 24 * q^89 + 24 * q^91 + 24 * q^98

Introduction

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