LMFDB - Embedded newform 900.4.j.c.557.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [900,4,Mod(557,900)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(900, base_ring=CyclotomicField(4))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("900.557");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	900.j (of order $4$, degree $2$, 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:	$53.1017190052$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 2737x^{12} + 5811553x^{8} - 4597108992x^{4} + 2821109907456 $ x^16 - 2737x^12 + 5811553x^8 - 4597108992*x^4 + 2821109907456
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{20}\cdot 3^{16} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			557.7
Root			$5.33268 + 1.42889i$ of defining polynomial
Character	$\chi$	$=$	900.557
Dual form			900.4.j.c.593.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(17.1974 + 17.1974i) q^{7} +O(q^{10})$	$q+(17.1974 + 17.1974i) q^{7} -68.0587i q^{11} +(-33.1700 + 33.1700i) q^{13} +(15.5885 - 15.5885i) q^{17} -40.1248i q^{19} +(-52.1777 - 52.1777i) q^{23} +97.9337 q^{29} -206.374 q^{31} +(-238.008 - 238.008i) q^{37} +43.1323i q^{41} +(255.205 - 255.205i) q^{43} +(134.884 - 134.884i) q^{47} +248.499i q^{49} +(458.126 + 458.126i) q^{53} -748.646 q^{59} +240.747 q^{61} +(133.546 + 133.546i) q^{67} -899.793i q^{71} +(26.5368 - 26.5368i) q^{73} +(1170.43 - 1170.43i) q^{77} +112.250i q^{79} +(644.539 + 644.539i) q^{83} +554.367 q^{89} -1140.87 q^{91} +(-1242.09 - 1242.09i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q - 1568 q^{31} - 4240 q^{61} - 14208 q^{91}+O(q^{100}) $ 16 * q - 1568 * q^31 - 4240 * q^61 - 14208 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/900\mathbb{Z}\right)^\times$.

$n$	$101$	$451$	$577$
$\chi(n)$	$-1$	$1$	$e\left(\frac{1}{4}\right)$

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
$5$		0			0
$5$		0			0
$6$		0			0
$7$	17.1974	+	17.1974i	0.928571	+	0.928571i	0.997614	−	0.0690429i	$-0.0219945\pi$
$7$	17.1974	+	17.1974i	0.928571	+	0.928571i	−0.0690429	+	0.997614i	$0.521995\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	68.0587i		−	1.86550i	−0.360527	−	0.932749i	$-0.617403\pi$
$11$		−	68.0587i		−	1.86550i	0.360527	−	0.932749i	$-0.382597\pi$
$12$		0			0
$13$	−33.1700	+	33.1700i	−0.707669	+	0.707669i	−0.966045	−	0.258375i	$-0.916813\pi$
$13$	−33.1700	+	33.1700i	−0.707669	+	0.707669i	0.258375	+	0.966045i	$0.416813\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	15.5885	−	15.5885i	0.222397	−	0.222397i	−0.587110	−	0.809507i	$-0.699734\pi$
$17$	15.5885	−	15.5885i	0.222397	−	0.222397i	0.809507	+	0.587110i	$0.199734\pi$
$18$		0			0
$19$		−	40.1248i		−	0.484487i	−0.970215	−	0.242244i	$-0.922117\pi$
$19$		−	40.1248i		−	0.484487i	0.970215	−	0.242244i	$-0.0778833\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−52.1777	−	52.1777i	−0.473034	−	0.473034i	0.429861	−	0.902895i	$-0.358563\pi$
$23$	−52.1777	−	52.1777i	−0.473034	−	0.473034i	−0.902895	+	0.429861i	$0.858563\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		0			0
$28$		0			0
$29$	97.9337			0.627097			0.313549	−	0.949572i	$-0.398482\pi$
$29$	97.9337			0.627097			0.313549	+	0.949572i	$0.398482\pi$
$30$		0			0
$31$	−206.374			−1.19568			−0.597838	−	0.801617i	$-0.703973\pi$
$31$	−206.374			−1.19568			−0.597838	+	0.801617i	$0.703973\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−238.008	−	238.008i	−1.05752	−	1.05752i	−0.998241	−	0.0592797i	$-0.981120\pi$
$37$	−238.008	−	238.008i	−1.05752	−	1.05752i	−0.0592797	−	0.998241i	$-0.518880\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			43.1323i			0.164296i	0.996620	+	0.0821480i	$0.0261780\pi$
$41$			43.1323i			0.164296i	−0.996620	+	0.0821480i	$0.973822\pi$
$42$		0			0
$43$	255.205	−	255.205i	0.905080	−	0.905080i	−0.0907896	−	0.995870i	$-0.528939\pi$
$43$	255.205	−	255.205i	0.905080	−	0.905080i	0.995870	+	0.0907896i	$0.0289391\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	134.884	−	134.884i	0.418613	−	0.418613i	−0.466112	−	0.884726i	$-0.654346\pi$
$47$	134.884	−	134.884i	0.418613	−	0.418613i	0.884726	+	0.466112i	$0.154346\pi$
$48$		0			0
$49$			248.499i			0.724487i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	458.126	+	458.126i	1.18733	+	1.18733i	0.977803	+	0.209526i	$0.0671921\pi$
$53$	458.126	+	458.126i	1.18733	+	1.18733i	0.209526	+	0.977803i	$0.432808\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−748.646			−1.65196			−0.825978	−	0.563702i	$-0.809376\pi$
$59$	−748.646			−1.65196			−0.825978	+	0.563702i	$0.809376\pi$
$60$		0			0
$61$	240.747			0.505320			0.252660	−	0.967555i	$-0.418695\pi$
$61$	240.747			0.505320			0.252660	+	0.967555i	$0.418695\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	133.546	+	133.546i	0.243511	+	0.243511i	0.818301	−	0.574790i	$-0.194916\pi$
$67$	133.546	+	133.546i	0.243511	+	0.243511i	−0.574790	+	0.818301i	$0.694916\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	899.793i		−	1.50402i	−0.659149	−	0.752012i	$-0.729083\pi$
$71$		−	899.793i		−	1.50402i	0.659149	−	0.752012i	$-0.270917\pi$
$72$		0			0
$73$	26.5368	−	26.5368i	0.0425466	−	0.0425466i	−0.685513	−	0.728060i	$-0.740422\pi$
$73$	26.5368	−	26.5368i	0.0425466	−	0.0425466i	0.728060	+	0.685513i	$0.240422\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	1170.43	−	1170.43i	1.73225	−	1.73225i
$78$		0			0
$79$			112.250i			0.159862i	0.996800	+	0.0799308i	$0.0254699\pi$
$79$			112.250i			0.159862i	−0.996800	+	0.0799308i	$0.974530\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	644.539	+	644.539i	0.852378	+	0.852378i	0.990426	−	0.138048i	$-0.0440828\pi$
$83$	644.539	+	644.539i	0.852378	+	0.852378i	−0.138048	+	0.990426i	$0.544083\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	554.367			0.660256			0.330128	−	0.943936i	$-0.392908\pi$
$89$	554.367			0.660256			0.330128	+	0.943936i	$0.392908\pi$
$90$		0			0
$91$	−1140.87			−1.31424
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−1242.09	−	1242.09i	−1.30016	−	1.30016i	−0.928282	−	0.371877i	$-0.878714\pi$
$97$	−1242.09	−	1242.09i	−1.30016	−	1.30016i	−0.371877	−	0.928282i	$-0.621286\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	1570.48i		−	1.54722i	−0.633664	−	0.773608i	$-0.718450\pi$
$101$		−	1570.48i		−	1.54722i	0.633664	−	0.773608i	$-0.281550\pi$
$102$		0			0
$103$	479.591	−	479.591i	0.458791	−	0.458791i	−0.439468	−	0.898258i	$-0.644833\pi$
$103$	479.591	−	479.591i	0.458791	−	0.458791i	0.898258	+	0.439468i	$0.144833\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	226.469	−	226.469i	0.204613	−	0.204613i	−0.597360	−	0.801973i	$-0.703784\pi$
$107$	226.469	−	226.469i	0.204613	−	0.204613i	0.801973	+	0.597360i	$0.203784\pi$
$108$		0			0
$109$		−	686.744i		−	0.603469i	−0.953392	−	0.301734i	$-0.902434\pi$
$109$		−	686.744i		−	0.603469i	0.953392	−	0.301734i	$-0.0975656\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−1012.38	−	1012.38i	−0.842799	−	0.842799i	0.146423	−	0.989222i	$-0.453224\pi$
$113$	−1012.38	−	1012.38i	−0.842799	−	0.842799i	−0.989222	+	0.146423i	$0.953224\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$	536.161			0.413023
$120$		0			0
$121$	−3300.99			−2.48008
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−1633.19	−	1633.19i	−1.14112	−	1.14112i	−0.988245	−	0.152877i	$-0.951146\pi$
$127$	−1633.19	−	1633.19i	−1.14112	−	1.14112i	−0.152877	−	0.988245i	$-0.548854\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	2377.29i		−	1.58553i	−0.609525	−	0.792767i	$-0.708640\pi$
$131$		−	2377.29i		−	1.58553i	0.609525	−	0.792767i	$-0.291360\pi$
$132$		0			0
$133$	690.041	−	690.041i	0.449881	−	0.449881i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	1396.90	−	1396.90i	0.871133	−	0.871133i	−0.121463	−	0.992596i	$-0.538758\pi$
$137$	1396.90	−	1396.90i	0.871133	−	0.871133i	0.992596	+	0.121463i	$0.0387585\pi$
$138$		0			0
$139$			1837.74i			1.12140i	0.828018	+	0.560702i	$0.189469\pi$
$139$			1837.74i			1.12140i	−0.828018	+	0.560702i	$0.810531\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	2257.51	+	2257.51i	1.32016	+	1.32016i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−3072.72			−1.68944			−0.844722	−	0.535205i	$-0.820235\pi$
$149$	−3072.72			−1.68944			−0.844722	+	0.535205i	$0.820235\pi$
$150$		0			0
$151$	26.6170			0.0143448			0.00717239	−	0.999974i	$-0.497717\pi$
$151$	26.6170			0.0143448			0.00717239	+	0.999974i	$0.497717\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	1063.37	+	1063.37i	0.540551	+	0.540551i	0.923690	−	0.383140i	$-0.125157\pi$
$157$	1063.37	+	1063.37i	0.540551	+	0.540551i	−0.383140	+	0.923690i	$0.625157\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	1794.64i		−	0.878492i
$162$		0			0
$163$	175.090	−	175.090i	0.0841355	−	0.0841355i	−0.663787	−	0.747922i	$-0.731052\pi$
$163$	175.090	−	175.090i	0.0841355	−	0.0841355i	0.747922	+	0.663787i	$0.231052\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	2498.04	−	2498.04i	1.15751	−	1.15751i	0.172502	−	0.985009i	$-0.444815\pi$
$167$	2498.04	−	2498.04i	1.15751	−	1.15751i	0.985009	−	0.172502i	$-0.0551852\pi$
$168$		0			0
$169$		−	3.49740i		−	0.00159190i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	2089.28	+	2089.28i	0.918178	+	0.918178i	0.996897	−	0.0787191i	$-0.0250830\pi$
$173$	2089.28	+	2089.28i	0.918178	+	0.918178i	−0.0787191	+	0.996897i	$0.525083\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	1103.97			0.460975			0.230487	−	0.973075i	$-0.425968\pi$
$179$	1103.97			0.460975			0.230487	+	0.973075i	$0.425968\pi$
$180$		0			0
$181$	3144.23			1.29121			0.645604	−	0.763672i	$-0.276605\pi$
$181$	3144.23			1.29121			0.645604	+	0.763672i	$0.276605\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−1060.93	−	1060.93i	−0.414882	−	0.414882i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			423.014i			0.160252i	0.996785	+	0.0801262i	$0.0255323\pi$
$191$			423.014i			0.160252i	−0.996785	+	0.0801262i	$0.974468\pi$
$192$		0			0
$193$	1430.90	−	1430.90i	0.533672	−	0.533672i	−0.387991	−	0.921663i	$-0.626831\pi$
$193$	1430.90	−	1430.90i	0.533672	−	0.533672i	0.921663	+	0.387991i	$0.126831\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	2074.56	−	2074.56i	0.750286	−	0.750286i	−0.224246	−	0.974533i	$-0.571992\pi$
$197$	2074.56	−	2074.56i	0.750286	−	0.750286i	0.974533	+	0.224246i	$0.0719920\pi$
$198$		0			0
$199$		−	1940.36i		−	0.691200i	−0.938382	−	0.345600i	$-0.887675\pi$
$199$		−	1940.36i		−	0.691200i	0.938382	−	0.345600i	$-0.112325\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	1684.20	+	1684.20i	0.582304	+	0.582304i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−2730.84			−0.903810
$210$		0			0
$211$	−2047.37			−0.667994			−0.333997	−	0.942574i	$-0.608398\pi$
$211$	−2047.37			−0.667994			−0.333997	+	0.942574i	$0.608398\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−3549.10	−	3549.10i	−1.11027	−	1.11027i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			1034.14i			0.314768i
$222$		0			0
$223$	1662.75	−	1662.75i	0.499308	−	0.499308i	−0.411915	−	0.911222i	$-0.635140\pi$
$223$	1662.75	−	1662.75i	0.499308	−	0.499308i	0.911222	+	0.411915i	$0.135140\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−997.661	+	997.661i	−0.291705	+	0.291705i	−0.837754	−	0.546048i	$-0.816131\pi$
$227$	−997.661	+	997.661i	−0.291705	+	0.291705i	0.546048	+	0.837754i	$0.316131\pi$
$228$		0			0
$229$			6260.72i			1.80664i	0.428969	+	0.903319i	$0.358877\pi$
$229$			6260.72i			1.80664i	−0.428969	+	0.903319i	$0.641123\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−3341.12	−	3341.12i	−0.939416	−	0.939416i	0.0588511	−	0.998267i	$-0.481256\pi$
$233$	−3341.12	−	3341.12i	−0.939416	−	0.939416i	−0.998267	+	0.0588511i	$0.981256\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	5610.87			1.51857			0.759283	−	0.650761i	$-0.225550\pi$
$239$	5610.87			1.51857			0.759283	+	0.650761i	$0.225550\pi$
$240$		0			0
$241$	−6156.97			−1.64566			−0.822832	−	0.568284i	$-0.807607\pi$
$241$	−6156.97			−1.64566			−0.822832	+	0.568284i	$0.807607\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	1330.94	+	1330.94i	0.342857	+	0.342857i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	667.146i		−	0.167768i	−0.996476	−	0.0838842i	$-0.973267\pi$
$251$		−	667.146i		−	0.167768i	0.996476	−	0.0838842i	$-0.0267326\pi$
$252$		0			0
$253$	−3551.14	+	3551.14i	−0.882445	+	0.882445i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−4982.22	+	4982.22i	−1.20927	+	1.20927i	−0.238006	+	0.971264i	$0.576494\pi$
$257$	−4982.22	+	4982.22i	−1.20927	+	1.20927i	−0.971264	+	0.238006i	$0.923506\pi$
$258$		0			0
$259$		−	8186.23i		−	1.96397i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−3235.01	−	3235.01i	−0.758477	−	0.758477i	0.217568	−	0.976045i	$-0.430188\pi$
$263$	−3235.01	−	3235.01i	−0.758477	−	0.758477i	−0.976045	+	0.217568i	$0.930188\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−4228.13			−0.958341			−0.479171	−	0.877722i	$-0.659063\pi$
$269$	−4228.13			−0.958341			−0.479171	+	0.877722i	$0.659063\pi$
$270$		0			0
$271$	−620.742			−0.139142			−0.0695708	−	0.997577i	$-0.522163\pi$
$271$	−620.742			−0.139142			−0.0695708	+	0.997577i	$0.522163\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	2607.17	+	2607.17i	0.565522	+	0.565522i	0.930871	−	0.365349i	$-0.119050\pi$
$277$	2607.17	+	2607.17i	0.565522	+	0.565522i	−0.365349	+	0.930871i	$0.619050\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	3852.28i		−	0.817821i	−0.912574	−	0.408911i	$-0.865909\pi$
$281$		−	3852.28i		−	0.817821i	0.912574	−	0.408911i	$-0.134091\pi$
$282$		0			0
$283$	1023.43	−	1023.43i	0.214970	−	0.214970i	−0.591405	−	0.806375i	$-0.701426\pi$
$283$	1023.43	−	1023.43i	0.214970	−	0.214970i	0.806375	+	0.591405i	$0.201426\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−741.762	+	741.762i	−0.152560	+	0.152560i
$288$		0			0
$289$			4427.00i			0.901079i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	1308.11	+	1308.11i	0.260820	+	0.260820i	0.825387	−	0.564567i	$-0.190957\pi$
$293$	1308.11	+	1308.11i	0.260820	+	0.260820i	−0.564567	+	0.825387i	$0.690957\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	3461.47			0.669504
$300$		0			0
$301$	8777.72			1.68086
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	5352.58	+	5352.58i	0.995075	+	0.995075i	0.999988	−	0.00491325i	$-0.00156394\pi$
$307$	5352.58	+	5352.58i	0.995075	+	0.995075i	−0.00491325	+	0.999988i	$0.501564\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			513.190i			0.0935703i	0.998905	+	0.0467851i	$0.0148976\pi$
$311$			513.190i			0.0935703i	−0.998905	+	0.0467851i	$0.985102\pi$
$312$		0			0
$313$	−7279.45	+	7279.45i	−1.31456	+	1.31456i	−0.396553	+	0.918012i	$0.629794\pi$
$313$	−7279.45	+	7279.45i	−1.31456	+	1.31456i	−0.918012	+	0.396553i	$0.870206\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	2703.71	−	2703.71i	0.479039	−	0.479039i	−0.425785	−	0.904824i	$-0.640002\pi$
$317$	2703.71	−	2703.71i	0.479039	−	0.479039i	0.904824	+	0.425785i	$0.140002\pi$
$318$		0			0
$319$		−	6665.24i		−	1.16985i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−625.483	−	625.483i	−0.107749	−	0.107749i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	4639.29			0.777424
$330$		0			0
$331$	8902.72			1.47836			0.739181	−	0.673507i	$-0.235213\pi$
$331$	8902.72			1.47836			0.739181	+	0.673507i	$0.235213\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	1287.01	+	1287.01i	0.208035	+	0.208035i	0.803432	−	0.595397i	$-0.203005\pi$
$337$	1287.01	+	1287.01i	0.208035	+	0.208035i	−0.595397	+	0.803432i	$0.703005\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			14045.6i			2.23053i
$342$		0			0
$343$	1625.17	−	1625.17i	0.255833	−	0.255833i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−6023.43	+	6023.43i	−0.931858	+	0.931858i	−0.997822	−	0.0659641i	$-0.978988\pi$
$347$	−6023.43	+	6023.43i	−0.931858	+	0.931858i	0.0659641	+	0.997822i	$0.478988\pi$
$348$		0			0
$349$			1223.49i			0.187657i	0.995588	+	0.0938283i	$0.0299105\pi$
$349$			1223.49i			0.187657i	−0.995588	+	0.0938283i	$0.970090\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−7157.70	−	7157.70i	−1.07922	−	1.07922i	−0.996579	−	0.0826438i	$-0.973664\pi$
$353$	−7157.70	−	7157.70i	−1.07922	−	1.07922i	−0.0826438	−	0.996579i	$-0.526336\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−3146.58			−0.462591			−0.231296	−	0.972883i	$-0.574296\pi$
$359$	−3146.58			−0.462591			−0.231296	+	0.972883i	$0.574296\pi$
$360$		0			0
$361$	5249.00			0.765272
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	3322.08	+	3322.08i	0.472510	+	0.472510i	0.902726	−	0.430216i	$-0.141563\pi$
$367$	3322.08	+	3322.08i	0.472510	+	0.472510i	−0.430216	+	0.902726i	$0.641563\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			15757.1i			2.20504i
$372$		0			0
$373$	857.196	−	857.196i	0.118992	−	0.118992i	−0.645103	−	0.764095i	$-0.723186\pi$
$373$	857.196	−	857.196i	0.118992	−	0.118992i	0.764095	+	0.645103i	$0.223186\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−3248.46	+	3248.46i	−0.443778	+	0.443778i
$378$		0			0
$379$			8818.58i			1.19520i	0.801795	+	0.597599i	$0.203878\pi$
$379$			8818.58i			1.19520i	−0.801795	+	0.597599i	$0.796122\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	9459.12	+	9459.12i	1.26198	+	1.26198i	0.950132	+	0.311849i	$0.100948\pi$
$383$	9459.12	+	9459.12i	1.26198	+	1.26198i	0.311849	+	0.950132i	$0.399052\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−6779.73			−0.883665			−0.441833	−	0.897098i	$-0.645671\pi$
$389$	−6779.73			−0.883665			−0.441833	+	0.897098i	$0.645671\pi$
$390$		0			0
$391$	−1626.74			−0.210403
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−5125.23	−	5125.23i	−0.647929	−	0.647929i	0.304563	−	0.952492i	$-0.401490\pi$
$397$	−5125.23	−	5125.23i	−0.647929	−	0.647929i	−0.952492	+	0.304563i	$0.901490\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			10286.3i			1.28098i	0.767967	+	0.640489i	$0.221268\pi$
$401$			10286.3i			1.28098i	−0.767967	+	0.640489i	$0.778732\pi$
$402$		0			0
$403$	6845.44	−	6845.44i	0.846143	−	0.846143i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−16198.5	+	16198.5i	−1.97280	+	1.97280i
$408$		0			0
$409$		−	9317.45i		−	1.12645i	−0.826303	−	0.563225i	$-0.809560\pi$
$409$		−	9317.45i		−	1.12645i	0.826303	−	0.563225i	$-0.190440\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−12874.7	−	12874.7i	−1.53396	−	1.53396i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	8038.75			0.937276			0.468638	−	0.883390i	$-0.344745\pi$
$419$	8038.75			0.937276			0.468638	+	0.883390i	$0.344745\pi$
$420$		0			0
$421$	−44.4784			−0.00514903			−0.00257452	−	0.999997i	$-0.500819\pi$
$421$	−44.4784			−0.00514903			−0.00257452	+	0.999997i	$0.500819\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	4140.21	+	4140.21i	0.469225	+	0.469225i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	2623.86i		−	0.293241i	−0.989193	−	0.146621i	$-0.953160\pi$
$431$		−	2623.86i		−	0.293241i	0.989193	−	0.146621i	$-0.0468397\pi$
$432$		0			0
$433$	−9478.92	+	9478.92i	−1.05203	+	1.05203i	−0.0534567	+	0.998570i	$0.517024\pi$
$433$	−9478.92	+	9478.92i	−1.05203	+	1.05203i	−0.998570	+	0.0534567i	$0.982976\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−2093.62	+	2093.62i	−0.229179	+	0.229179i
$438$		0			0
$439$			3465.56i			0.376771i	0.982095	+	0.188385i	$0.0603254\pi$
$439$			3465.56i			0.376771i	−0.982095	+	0.188385i	$0.939675\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	3292.18	+	3292.18i	0.353084	+	0.353084i	0.861256	−	0.508172i	$-0.169678\pi$
$443$	3292.18	+	3292.18i	0.353084	+	0.353084i	−0.508172	+	0.861256i	$0.669678\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−1289.09			−0.135492			−0.0677458	−	0.997703i	$-0.521581\pi$
$449$	−1289.09			−0.135492			−0.0677458	+	0.997703i	$0.521581\pi$
$450$		0			0
$451$	2935.53			0.306494
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	12282.9	+	12282.9i	1.25727	+	1.25727i	0.952393	+	0.304874i	$0.0986143\pi$
$457$	12282.9	+	12282.9i	1.25727	+	1.25727i	0.304874	+	0.952393i	$0.401386\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			4084.26i			0.412631i	0.978486	+	0.206315i	$0.0661473\pi$
$461$			4084.26i			0.412631i	−0.978486	+	0.206315i	$0.933853\pi$
$462$		0			0
$463$	−8233.08	+	8233.08i	−0.826401	+	0.826401i	−0.987017	−	0.160616i	$-0.948652\pi$
$463$	−8233.08	+	8233.08i	−0.826401	+	0.826401i	0.160616	+	0.987017i	$0.448652\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−6727.08	+	6727.08i	−0.666579	+	0.666579i	−0.956922	−	0.290344i	$-0.906230\pi$
$467$	−6727.08	+	6727.08i	−0.666579	+	0.666579i	0.290344	+	0.956922i	$0.406230\pi$
$468$		0			0
$469$			4593.28i			0.452235i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−17369.0	−	17369.0i	−1.68843	−	1.68843i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$	12678.2			1.20936			0.604680	−	0.796469i	$-0.293301\pi$
$479$	12678.2			1.20936			0.604680	+	0.796469i	$0.293301\pi$
$480$		0			0
$481$	15789.5			1.49675
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	10238.6	+	10238.6i	0.952676	+	0.952676i	0.998930	−	0.0462535i	$-0.0147282\pi$
$487$	10238.6	+	10238.6i	0.952676	+	0.952676i	−0.0462535	+	0.998930i	$0.514728\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	9285.31i		−	0.853442i	−0.904383	−	0.426721i	$-0.859669\pi$
$491$		−	9285.31i		−	0.853442i	0.904383	−	0.426721i	$-0.140331\pi$
$492$		0			0
$493$	1526.63	−	1526.63i	0.139465	−	0.139465i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	15474.1	−	15474.1i	1.39659	−	1.39659i
$498$		0			0
$499$		−	22119.1i		−	1.98434i	−0.124902	−	0.992169i	$-0.539862\pi$
$499$		−	22119.1i		−	1.98434i	0.124902	−	0.992169i	$-0.460138\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	5422.19	+	5422.19i	0.480643	+	0.480643i	0.905337	−	0.424694i	$-0.139618\pi$
$503$	5422.19	+	5422.19i	0.480643	+	0.480643i	−0.424694	+	0.905337i	$0.639618\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−5503.78			−0.479274			−0.239637	−	0.970863i	$-0.577028\pi$
$509$	−5503.78			−0.479274			−0.239637	+	0.970863i	$0.577028\pi$
$510$		0			0
$511$	912.728			0.0790151
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−9180.02	−	9180.02i	−0.780922	−	0.780922i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	2600.71i		−	0.218693i	−0.994004	−	0.109347i	$-0.965124\pi$
$521$		−	2600.71i		−	0.218693i	0.994004	−	0.109347i	$-0.0348758\pi$
$522$		0			0
$523$	−15064.6	+	15064.6i	−1.25952	+	1.25952i	−0.308194	+	0.951323i	$0.599725\pi$
$523$	−15064.6	+	15064.6i	−1.25952	+	1.25952i	−0.951323	+	0.308194i	$0.900275\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−3217.06	+	3217.06i	−0.265915	+	0.265915i
$528$		0			0
$529$		−	6721.98i		−	0.552477i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−1430.70	−	1430.70i	−0.116267	−	0.116267i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	16912.5			1.35153
$540$		0			0
$541$	−5918.22			−0.470322			−0.235161	−	0.971956i	$-0.575562\pi$
$541$	−5918.22			−0.470322			−0.235161	+	0.971956i	$0.575562\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	4172.22	+	4172.22i	0.326126	+	0.326126i	0.851111	−	0.524985i	$-0.175929\pi$
$547$	4172.22	+	4172.22i	0.326126	+	0.326126i	−0.524985	+	0.851111i	$0.675929\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	3929.57i		−	0.303821i
$552$		0			0
$553$	−1930.40	+	1930.40i	−0.148443	+	0.148443i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−6458.33	+	6458.33i	−0.491290	+	0.491290i	−0.908712	−	0.417423i	$-0.862933\pi$
$557$	−6458.33	+	6458.33i	−0.491290	+	0.491290i	0.417423	+	0.908712i	$0.362933\pi$
$558$		0			0
$559$			16930.3i			1.28100i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−4911.83	−	4911.83i	−0.367689	−	0.367689i	0.498945	−	0.866634i	$-0.333721\pi$
$563$	−4911.83	−	4911.83i	−0.367689	−	0.367689i	−0.866634	+	0.498945i	$0.833721\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−11416.2			−0.841112			−0.420556	−	0.907267i	$-0.638165\pi$
$569$	−11416.2			−0.841112			−0.420556	+	0.907267i	$0.638165\pi$
$570$		0			0
$571$	−6061.11			−0.444220			−0.222110	−	0.975022i	$-0.571294\pi$
$571$	−6061.11			−0.444220			−0.222110	+	0.975022i	$0.571294\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−5236.02	−	5236.02i	−0.377779	−	0.377779i	0.492522	−	0.870300i	$-0.336075\pi$
$577$	−5236.02	−	5236.02i	−0.377779	−	0.377779i	−0.870300	+	0.492522i	$0.836075\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$			22168.8i			1.58299i
$582$		0			0
$583$	31179.5	−	31179.5i	2.21496	−	2.21496i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−13825.2	+	13825.2i	−0.972107	+	0.972107i	−0.999621	−	0.0275148i	$-0.991241\pi$
$587$	−13825.2	+	13825.2i	−0.972107	+	0.972107i	0.0275148	+	0.999621i	$0.491241\pi$
$588$		0			0
$589$			8280.73i			0.579289i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	14374.2	+	14374.2i	0.995409	+	0.995409i	0.999990	−	0.00458055i	$-0.00145804\pi$
$593$	14374.2	+	14374.2i	0.995409	+	0.995409i	−0.00458055	+	0.999990i	$0.501458\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−16921.3			−1.15423			−0.577117	−	0.816661i	$-0.695823\pi$
$599$	−16921.3			−1.15423			−0.577117	+	0.816661i	$0.695823\pi$
$600$		0			0
$601$	−3618.50			−0.245593			−0.122797	−	0.992432i	$-0.539186\pi$
$601$	−3618.50			−0.245593			−0.122797	+	0.992432i	$0.539186\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−10332.9	−	10332.9i	−0.690940	−	0.690940i	0.271499	−	0.962439i	$-0.412481\pi$
$607$	−10332.9	−	10332.9i	−0.690940	−	0.690940i	−0.962439	+	0.271499i	$0.912481\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			8948.19i			0.592480i
$612$		0			0
$613$	−4237.25	+	4237.25i	−0.279186	+	0.279186i	−0.832784	−	0.553598i	$-0.813254\pi$
$613$	−4237.25	+	4237.25i	−0.279186	+	0.279186i	0.553598	+	0.832784i	$0.313254\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	12490.7	−	12490.7i	0.815001	−	0.815001i	−0.170378	−	0.985379i	$-0.554499\pi$
$617$	12490.7	−	12490.7i	0.815001	−	0.815001i	0.985379	+	0.170378i	$0.0544989\pi$
$618$		0			0
$619$		−	9473.67i		−	0.615152i	−0.951524	−	0.307576i	$-0.900482\pi$
$619$		−	9473.67i		−	0.615152i	0.951524	−	0.307576i	$-0.0995178\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	9533.65	+	9533.65i	0.613094	+	0.613094i
$624$		0			0
$625$		0			0
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−7420.36			−0.470380
$630$		0			0
$631$	−10901.1			−0.687743			−0.343871	−	0.939017i	$-0.611738\pi$
$631$	−10901.1			−0.687743			−0.343871	+	0.939017i	$0.611738\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−8242.72	−	8242.72i	−0.512697	−	0.512697i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	12046.3i		−	0.742277i	−0.928578	−	0.371138i	$-0.878968\pi$
$641$		−	12046.3i		−	0.742277i	0.928578	−	0.371138i	$-0.121032\pi$
$642$		0			0
$643$	17654.7	−	17654.7i	1.08279	−	1.08279i	0.0865385	−	0.996249i	$-0.472419\pi$
$643$	17654.7	−	17654.7i	1.08279	−	1.08279i	0.996249	−	0.0865385i	$-0.0275806\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	16838.5	−	16838.5i	1.02317	−	1.02317i	0.0234444	−	0.999725i	$-0.492537\pi$
$647$	16838.5	−	16838.5i	1.02317	−	1.02317i	0.999725	−	0.0234444i	$-0.00746326\pi$
$648$		0			0
$649$			50951.9i			3.08172i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−6615.12	−	6615.12i	−0.396431	−	0.396431i	0.480541	−	0.876972i	$-0.340440\pi$
$653$	−6615.12	−	6615.12i	−0.396431	−	0.396431i	−0.876972	+	0.480541i	$0.840440\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	11271.6			0.666281			0.333141	−	0.942877i	$-0.391892\pi$
$659$	11271.6			0.666281			0.333141	+	0.942877i	$0.391892\pi$
$660$		0			0
$661$	−3545.53			−0.208631			−0.104315	−	0.994544i	$-0.533265\pi$
$661$	−3545.53			−0.208631			−0.104315	+	0.994544i	$0.533265\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−5109.95	−	5109.95i	−0.296639	−	0.296639i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	16384.9i		−	0.942673i
$672$		0			0
$673$	−6803.80	+	6803.80i	−0.389698	+	0.389698i	−0.874580	−	0.484881i	$-0.838863\pi$
$673$	−6803.80	+	6803.80i	−0.389698	+	0.389698i	0.484881	+	0.874580i	$0.338863\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	1928.26	−	1928.26i	0.109467	−	0.109467i	−0.650252	−	0.759719i	$-0.725337\pi$
$677$	1928.26	−	1928.26i	0.109467	−	0.109467i	0.759719	+	0.650252i	$0.225337\pi$
$678$		0			0
$679$		−	42721.5i		−	2.41458i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	17254.6	+	17254.6i	0.966663	+	0.966663i	0.999462	−	0.0327991i	$-0.0104421\pi$
$683$	17254.6	+	17254.6i	0.966663	+	0.966663i	−0.0327991	+	0.999462i	$0.510442\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−30392.1			−1.68047
$690$		0			0
$691$	10518.9			0.579099			0.289550	−	0.957163i	$-0.406495\pi$
$691$	10518.9			0.579099			0.289550	+	0.957163i	$0.406495\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	672.366	+	672.366i	0.0365390	+	0.0365390i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	25618.7i		−	1.38032i	−0.723656	−	0.690161i	$-0.757540\pi$
$701$		−	25618.7i		−	1.38032i	0.723656	−	0.690161i	$-0.242460\pi$
$702$		0			0
$703$	−9550.02	+	9550.02i	−0.512355	+	0.512355i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	27008.2	−	27008.2i	1.43670	−	1.43670i
$708$		0			0
$709$		−	2999.25i		−	0.158871i	−0.996840	−	0.0794353i	$-0.974688\pi$
$709$		−	2999.25i		−	0.158871i	0.996840	−	0.0794353i	$-0.0253117\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	10768.1	+	10768.1i	0.565596	+	0.565596i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	28513.8			1.47898			0.739489	−	0.673169i	$-0.235067\pi$
$719$	28513.8			1.47898			0.739489	+	0.673169i	$0.235067\pi$
$720$		0			0
$721$	16495.4			0.852040
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	6492.49	+	6492.49i	0.331215	+	0.331215i	0.853048	−	0.521833i	$-0.174752\pi$
$727$	6492.49	+	6492.49i	0.331215	+	0.331215i	−0.521833	+	0.853048i	$0.674752\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	7956.52i		−	0.402575i
$732$		0			0
$733$	−4941.80	+	4941.80i	−0.249017	+	0.249017i	−0.820567	−	0.571550i	$-0.806342\pi$
$733$	−4941.80	+	4941.80i	−0.249017	+	0.249017i	0.571550	+	0.820567i	$0.306342\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	9088.97	−	9088.97i	0.454269	−	0.454269i
$738$		0			0
$739$		−	10897.6i		−	0.542456i	−0.962515	−	0.271228i	$-0.912570\pi$
$739$		−	10897.6i		−	0.542456i	0.962515	−	0.271228i	$-0.0874297\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	6564.51	+	6564.51i	0.324130	+	0.324130i	0.850349	−	0.526219i	$-0.176391\pi$
$743$	6564.51	+	6564.51i	0.324130	+	0.324130i	−0.526219	+	0.850349i	$0.676391\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$	7789.36			0.379996
$750$		0			0
$751$	−23953.7			−1.16389			−0.581946	−	0.813227i	$-0.697709\pi$
$751$	−23953.7			−1.16389			−0.581946	+	0.813227i	$0.697709\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−17525.0	−	17525.0i	−0.841423	−	0.841423i	0.147621	−	0.989044i	$-0.452838\pi$
$757$	−17525.0	−	17525.0i	−0.841423	−	0.841423i	−0.989044	+	0.147621i	$0.952838\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	35017.3i		−	1.66803i	−0.551738	−	0.834017i	$-0.686035\pi$
$761$		−	35017.3i		−	1.66803i	0.551738	−	0.834017i	$-0.313965\pi$
$762$		0			0
$763$	11810.2	−	11810.2i	0.560363	−	0.560363i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	24832.6	−	24832.6i	1.16904	−	1.16904i
$768$		0			0
$769$			10872.9i			0.509866i	0.966959	+	0.254933i	$0.0820534\pi$
$769$			10872.9i			0.509866i	−0.966959	+	0.254933i	$0.917947\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−15022.0	−	15022.0i	−0.698969	−	0.698969i	0.265219	−	0.964188i	$-0.414556\pi$
$773$	−15022.0	−	15022.0i	−0.698969	−	0.698969i	−0.964188	+	0.265219i	$0.914556\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	1730.67			0.0795993
$780$		0			0
$781$	−61238.7			−2.80575
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	6050.27	+	6050.27i	0.274039	+	0.274039i	0.830724	−	0.556685i	$-0.187927\pi$
$787$	6050.27	+	6050.27i	0.274039	+	0.274039i	−0.556685	+	0.830724i	$0.687927\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	34820.4i		−	1.56520i
$792$		0			0
$793$	−7985.58	+	7985.58i	−0.357599	+	0.357599i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−1067.74	+	1067.74i	−0.0474547	+	0.0474547i	−0.730436	−	0.682981i	$-0.760683\pi$
$797$	−1067.74	+	1067.74i	−0.0474547	+	0.0474547i	0.682981	+	0.730436i	$0.260683\pi$
$798$		0			0
$799$		−	4205.26i		−	0.186197i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−1806.06	−	1806.06i	−0.0793706	−	0.0793706i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−15823.6			−0.687674			−0.343837	−	0.939029i	$-0.611727\pi$
$809$	−15823.6			−0.687674			−0.343837	+	0.939029i	$0.611727\pi$
$810$		0			0
$811$	−14955.1			−0.647525			−0.323763	−	0.946138i	$-0.604948\pi$
$811$	−14955.1			−0.647525			−0.323763	+	0.946138i	$0.604948\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−10240.1	−	10240.1i	−0.438500	−	0.438500i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	37969.7i		−	1.61407i	−0.590502	−	0.807036i	$-0.701070\pi$
$821$		−	37969.7i		−	1.61407i	0.590502	−	0.807036i	$-0.298930\pi$
$822$		0			0
$823$	25075.4	−	25075.4i	1.06206	−	1.06206i	0.0641133	−	0.997943i	$-0.479578\pi$
$823$	25075.4	−	25075.4i	1.06206	−	1.06206i	0.997943	−	0.0641133i	$-0.0204219\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−20449.4	+	20449.4i	−0.859847	+	0.859847i	−0.991320	−	0.131473i	$-0.958029\pi$
$827$	−20449.4	+	20449.4i	−0.859847	+	0.859847i	0.131473	+	0.991320i	$0.458029\pi$
$828$		0			0
$829$			15970.0i			0.669074i	0.942383	+	0.334537i	$0.108580\pi$
$829$			15970.0i			0.669074i	−0.942383	+	0.334537i	$0.891420\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	3873.72	+	3873.72i	0.161124	+	0.161124i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−10980.8			−0.451846			−0.225923	−	0.974145i	$-0.572540\pi$
$839$	−10980.8			−0.451846			−0.225923	+	0.974145i	$0.572540\pi$
$840$		0			0
$841$	−14798.0			−0.606749
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−56768.3	−	56768.3i	−2.30293	−	2.30293i
$848$		0			0
$849$		0			0
$850$		0			0
$851$			24837.4i			1.00049i
$852$		0			0
$853$	11094.3	−	11094.3i	0.445324	−	0.445324i	−0.448472	−	0.893797i	$-0.648032\pi$
$853$	11094.3	−	11094.3i	0.445324	−	0.445324i	0.893797	+	0.448472i	$0.148032\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	3869.43	−	3869.43i	0.154233	−	0.154233i	−0.625773	−	0.780005i	$-0.715216\pi$
$857$	3869.43	−	3869.43i	0.154233	−	0.154233i	0.780005	+	0.625773i	$0.215216\pi$
$858$		0			0
$859$			47137.7i			1.87232i	0.351581	+	0.936158i	$0.385644\pi$
$859$			47137.7i			1.87232i	−0.351581	+	0.936158i	$0.614356\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	14620.2	+	14620.2i	0.576684	+	0.576684i	0.933988	−	0.357304i	$-0.116304\pi$
$863$	14620.2	+	14620.2i	0.576684	+	0.576684i	−0.357304	+	0.933988i	$0.616304\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	7639.56			0.298221
$870$		0			0
$871$	−8859.44			−0.344651
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	13450.6	+	13450.6i	0.517895	+	0.517895i	0.916934	−	0.399039i	$-0.130656\pi$
$877$	13450.6	+	13450.6i	0.517895	+	0.517895i	−0.399039	+	0.916934i	$0.630656\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			37873.1i			1.44833i	0.689627	+	0.724165i	$0.257775\pi$
$881$			37873.1i			1.44833i	−0.689627	+	0.724165i	$0.742225\pi$
$882$		0			0
$883$	−20138.8	+	20138.8i	−0.767524	+	0.767524i	−0.977670	−	0.210146i	$-0.932606\pi$
$883$	−20138.8	+	20138.8i	−0.767524	+	0.767524i	0.210146	+	0.977670i	$0.432606\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	4734.41	−	4734.41i	0.179217	−	0.179217i	−0.611797	−	0.791015i	$-0.709553\pi$
$887$	4734.41	−	4734.41i	0.179217	−	0.179217i	0.791015	+	0.611797i	$0.209553\pi$
$888$		0			0
$889$		−	56173.3i		−	2.11923i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−5412.18	−	5412.18i	−0.202813	−	0.202813i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−20211.0			−0.749805
$900$		0			0
$901$	14283.0			0.528118
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	11503.0	+	11503.0i	0.421115	+	0.421115i	0.885588	−	0.464472i	$-0.153756\pi$
$907$	11503.0	+	11503.0i	0.421115	+	0.421115i	−0.464472	+	0.885588i	$0.653756\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	9526.76i		−	0.346472i	−0.984880	−	0.173236i	$-0.944578\pi$
$911$		−	9526.76i		−	0.346472i	0.984880	−	0.173236i	$-0.0554223\pi$
$912$		0			0
$913$	43866.5	−	43866.5i	1.59011	−	1.59011i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	40883.1	−	40883.1i	1.47228	−	1.47228i
$918$		0			0
$919$			26085.9i			0.936337i	0.883639	+	0.468169i	$0.155086\pi$
$919$			26085.9i			0.936337i	−0.883639	+	0.468169i	$0.844914\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	29846.1	+	29846.1i	1.06435	+	1.06435i
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−21975.3			−0.776089			−0.388045	−	0.921641i	$-0.626849\pi$
$929$	−21975.3			−0.776089			−0.388045	+	0.921641i	$0.626849\pi$
$930$		0			0
$931$	9970.97			0.351005
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−24027.4	−	24027.4i	−0.837717	−	0.837717i	0.150841	−	0.988558i	$-0.451802\pi$
$937$	−24027.4	−	24027.4i	−0.837717	−	0.837717i	−0.988558	+	0.150841i	$0.951802\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	31436.8i		−	1.08907i	−0.838739	−	0.544533i	$-0.816707\pi$
$941$		−	31436.8i		−	1.08907i	0.838739	−	0.544533i	$-0.183293\pi$
$942$		0			0
$943$	2250.54	−	2250.54i	0.0777176	−	0.0777176i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	1211.31	−	1211.31i	0.0415651	−	0.0415651i	−0.686019	−	0.727584i	$-0.740643\pi$
$947$	1211.31	−	1211.31i	0.0415651	−	0.0415651i	0.727584	+	0.686019i	$0.240643\pi$
$948$		0			0
$949$			1760.45i			0.0602179i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	12579.0	+	12579.0i	0.427568	+	0.427568i	0.887799	−	0.460231i	$-0.152233\pi$
$953$	12579.0	+	12579.0i	0.427568	+	0.427568i	−0.460231	+	0.887799i	$0.652233\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$	48046.0			1.61782
$960$		0			0
$961$	12799.4			0.429639
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	20901.6	+	20901.6i	0.695086	+	0.695086i	0.963347	−	0.268260i	$-0.0864487\pi$
$967$	20901.6	+	20901.6i	0.695086	+	0.695086i	−0.268260	+	0.963347i	$0.586449\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	8890.52i		−	0.293831i	−0.989149	−	0.146916i	$-0.953065\pi$
$971$		−	8890.52i		−	0.293831i	0.989149	−	0.146916i	$-0.0469346\pi$
$972$		0			0
$973$	−31604.3	+	31604.3i	−1.04130	+	1.04130i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	13518.6	−	13518.6i	0.442679	−	0.442679i	−0.450232	−	0.892911i	$-0.648659\pi$
$977$	13518.6	−	13518.6i	0.442679	−	0.442679i	0.892911	+	0.450232i	$0.148659\pi$
$978$		0			0
$979$		−	37729.5i		−	1.23171i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	17408.8	+	17408.8i	0.564857	+	0.564857i	0.930683	−	0.365826i	$-0.119214\pi$
$983$	17408.8	+	17408.8i	0.564857	+	0.564857i	−0.365826	+	0.930683i	$0.619214\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−26632.0			−0.856269
$990$		0			0
$991$	40862.4			1.30983			0.654913	−	0.755705i	$-0.272705\pi$
$991$	40862.4			1.30983			0.654913	+	0.755705i	$0.272705\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	30705.5	+	30705.5i	0.975379	+	0.975379i	0.999704	−	0.0243248i	$-0.00774357\pi$
$997$	30705.5	+	30705.5i	0.975379	+	0.975379i	−0.0243248	+	0.999704i	$0.507744\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	900.4.j.c.557.7	yes	16
3.2	odd	2	inner	900.4.j.c.557.8	yes	16
5.2	odd	4	inner	900.4.j.c.593.1	yes	16
5.3	odd	4	inner	900.4.j.c.593.7	yes	16
5.4	even	2	inner	900.4.j.c.557.1	✓	16
15.2	even	4	inner	900.4.j.c.593.2	yes	16
15.8	even	4	inner	900.4.j.c.593.8	yes	16
15.14	odd	2	inner	900.4.j.c.557.2	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
900.4.j.c.557.1	✓	16	5.4	even	2	inner
900.4.j.c.557.2	yes	16	15.14	odd	2	inner
900.4.j.c.557.7	yes	16	1.1	even	1	trivial
900.4.j.c.557.8	yes	16	3.2	odd	2	inner
900.4.j.c.593.1	yes	16	5.2	odd	4	inner
900.4.j.c.593.2	yes	16	15.2	even	4	inner
900.4.j.c.593.7	yes	16	5.3	odd	4	inner
900.4.j.c.593.8	yes	16	15.8	even	4	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:	\( N \)	\(=\)	\( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	900.j (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(17.1974 + 17.1974i) q^{7} +O(q^{10})\)	\(q+(17.1974 + 17.1974i) q^{7} -68.0587i q^{11} +(-33.1700 + 33.1700i) q^{13} +(15.5885 - 15.5885i) q^{17} -40.1248i q^{19} +(-52.1777 - 52.1777i) q^{23} +97.9337 q^{29} -206.374 q^{31} +(-238.008 - 238.008i) q^{37} +43.1323i q^{41} +(255.205 - 255.205i) q^{43} +(134.884 - 134.884i) q^{47} +248.499i q^{49} +(458.126 + 458.126i) q^{53} -748.646 q^{59} +240.747 q^{61} +(133.546 + 133.546i) q^{67} -899.793i q^{71} +(26.5368 - 26.5368i) q^{73} +(1170.43 - 1170.43i) q^{77} +112.250i q^{79} +(644.539 + 644.539i) q^{83} +554.367 q^{89} -1140.87 q^{91} +(-1242.09 - 1242.09i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q - 1568 q^{31} - 4240 q^{61} - 14208 q^{91}+O(q^{100}) \) 16 * q - 1568 * q^31 - 4240 * q^61 - 14208 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more