LMFDB - Embedded newform 900.6.j.b.557.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [900,6,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, 6, names="a")

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

chi := DirichletCharacter("900.557");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
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:	$144.345437832$
Analytic rank:	$0$
Dimension:	$20$
Relative dimension:	$10$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} + 222025 x^{16} + 11247583920 x^{12} + 151104106237945 x^{8} + \cdots + 67\!\cdots\!76 $ x^20 + 222025x^16 + 11247583920x^12 + 151104106237945x^8 + 21346152874807825x^4 + 672057978863947776
Coefficient ring:	$\Z[a_1, \ldots, a_{49}]$
Coefficient ring index:	$ 2^{46}\cdot 3^{24}\cdot 5^{4} $
Twist minimal:	no (minimal twist has level 180)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			557.8
Root			$-10.2322 + 10.2322i$ of defining polynomial
Character	$\chi$	$=$	900.557
Dual form			900.6.j.b.593.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(41.6720 + 41.6720i) q^{7} +O(q^{10})$	$q+(41.6720 + 41.6720i) q^{7} +548.792i q^{11} +(188.730 - 188.730i) q^{13} +(-443.937 + 443.937i) q^{17} -1937.60i q^{19} +(637.976 + 637.976i) q^{23} +6379.14 q^{29} +3654.47 q^{31} +(6164.72 + 6164.72i) q^{37} +7467.59i q^{41} +(-15004.7 + 15004.7i) q^{43} +(1711.56 - 1711.56i) q^{47} -13333.9i q^{49} +(-22583.7 - 22583.7i) q^{53} +23360.2 q^{59} +9537.66 q^{61} +(-20971.6 - 20971.6i) q^{67} +9739.04i q^{71} +(-50230.1 + 50230.1i) q^{73} +(-22869.2 + 22869.2i) q^{77} -27253.3i q^{79} +(10124.1 + 10124.1i) q^{83} -1148.18 q^{89} +15729.5 q^{91} +(56149.2 + 56149.2i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 152 q^{7}+O(q^{10}) $ 20 * q + 152 * q^7	$ 20 q + 152 q^{7} - 252 q^{13} + 9440 q^{31} - 29988 q^{37} - 6720 q^{43} + 67200 q^{61} - 139552 q^{67} - 134828 q^{73} - 220560 q^{91} - 207132 q^{97}+O(q^{100}) $ 20 * q + 152 * q^7 - 252 * q^13 + 9440 * q^31 - 29988 * q^37 - 6720 * q^43 + 67200 * q^61 - 139552 * q^67 - 134828 * q^73 - 220560 * q^91 - 207132 * q^97

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
$7$	41.6720	+	41.6720i	0.321439	+	0.321439i	0.849319	−	0.527880i	$-0.177013\pi$
$7$	41.6720	+	41.6720i	0.321439	+	0.321439i	−0.527880	+	0.849319i	$0.677013\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$			548.792i			1.36750i	0.729718	+	0.683748i	$0.239651\pi$
$11$			548.792i			1.36750i	−0.729718	+	0.683748i	$0.760349\pi$
$12$		0			0
$13$	188.730	−	188.730i	0.309729	−	0.309729i	−0.535075	−	0.844804i	$-0.679717\pi$
$13$	188.730	−	188.730i	0.309729	−	0.309729i	0.844804	+	0.535075i	$0.179717\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−443.937	+	443.937i	−0.372563	+	0.372563i	−0.868410	−	0.495847i	$-0.834858\pi$
$17$	−443.937	+	443.937i	−0.372563	+	0.372563i	0.495847	+	0.868410i	$0.334858\pi$
$18$		0			0
$19$		−	1937.60i		−	1.23134i	−0.788002	−	0.615672i	$-0.788885\pi$
$19$		−	1937.60i		−	1.23134i	0.788002	−	0.615672i	$-0.211115\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	637.976	+	637.976i	0.251469	+	0.251469i	0.821573	−	0.570104i	$-0.193097\pi$
$23$	637.976	+	637.976i	0.251469	+	0.251469i	−0.570104	+	0.821573i	$0.693097\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		0			0
$28$		0			0
$29$	6379.14			1.40853			0.704266	−	0.709936i	$-0.251276\pi$
$29$	6379.14			1.40853			0.704266	+	0.709936i	$0.251276\pi$
$30$		0			0
$31$	3654.47			0.682999			0.341500	−	0.939882i	$-0.389065\pi$
$31$	3654.47			0.682999			0.341500	+	0.939882i	$0.389065\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	6164.72	+	6164.72i	0.740303	+	0.740303i	0.972636	−	0.232334i	$-0.0746361\pi$
$37$	6164.72	+	6164.72i	0.740303	+	0.740303i	−0.232334	+	0.972636i	$0.574636\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			7467.59i			0.693779i	0.937906	+	0.346889i	$0.112762\pi$
$41$			7467.59i			0.693779i	−0.937906	+	0.346889i	$0.887238\pi$
$42$		0			0
$43$	−15004.7	+	15004.7i	−1.23753	+	1.23753i	−0.276526	+	0.961007i	$0.589183\pi$
$43$	−15004.7	+	15004.7i	−1.23753	+	1.23753i	−0.961007	+	0.276526i	$0.910817\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	1711.56	−	1711.56i	0.113018	−	0.113018i	−0.648336	−	0.761354i	$-0.724535\pi$
$47$	1711.56	−	1711.56i	0.113018	−	0.113018i	0.761354	+	0.648336i	$0.224535\pi$
$48$		0			0
$49$		−	13333.9i		−	0.793354i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−22583.7	−	22583.7i	−1.10435	−	1.10435i	−0.993880	−	0.110466i	$-0.964766\pi$
$53$	−22583.7	−	22583.7i	−1.10435	−	1.10435i	−0.110466	−	0.993880i	$-0.535234\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	23360.2			0.873668			0.436834	−	0.899542i	$-0.356100\pi$
$59$	23360.2			0.873668			0.436834	+	0.899542i	$0.356100\pi$
$60$		0			0
$61$	9537.66			0.328184			0.164092	−	0.986445i	$-0.447531\pi$
$61$	9537.66			0.328184			0.164092	+	0.986445i	$0.447531\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−20971.6	−	20971.6i	−0.570749	−	0.570749i	0.361589	−	0.932338i	$-0.382234\pi$
$67$	−20971.6	−	20971.6i	−0.570749	−	0.570749i	−0.932338	+	0.361589i	$0.882234\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			9739.04i			0.229282i	0.993407	+	0.114641i	$0.0365718\pi$
$71$			9739.04i			0.229282i	−0.993407	+	0.114641i	$0.963428\pi$
$72$		0			0
$73$	−50230.1	+	50230.1i	−1.10321	+	1.10321i	−0.109185	+	0.994021i	$0.534824\pi$
$73$	−50230.1	+	50230.1i	−1.10321	+	1.10321i	−0.994021	+	0.109185i	$0.965176\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−22869.2	+	22869.2i	−0.439567	+	0.439567i
$78$		0			0
$79$		−	27253.3i		−	0.491304i	−0.969358	−	0.245652i	$-0.920998\pi$
$79$		−	27253.3i		−	0.491304i	0.969358	−	0.245652i	$-0.0790021\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	10124.1	+	10124.1i	0.161310	+	0.161310i	0.783147	−	0.621837i	$-0.213613\pi$
$83$	10124.1	+	10124.1i	0.161310	+	0.161310i	−0.621837	+	0.783147i	$0.713613\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−1148.18			−0.0153651			−0.00768253	−	0.999970i	$-0.502445\pi$
$89$	−1148.18			−0.0153651			−0.00768253	+	0.999970i	$0.502445\pi$
$90$		0			0
$91$	15729.5			0.199118
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	56149.2	+	56149.2i	0.605919	+	0.605919i	0.941877	−	0.335958i	$-0.109060\pi$
$97$	56149.2	+	56149.2i	0.605919	+	0.605919i	−0.335958	+	0.941877i	$0.609060\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			12048.7i			0.117527i	0.998272	+	0.0587633i	$0.0187157\pi$
$101$			12048.7i			0.117527i	−0.998272	+	0.0587633i	$0.981284\pi$
$102$		0			0
$103$	101655.	−	101655.i	0.944136	−	0.944136i	−0.0543841	−	0.998520i	$-0.517320\pi$
$103$	101655.	−	101655.i	0.944136	−	0.944136i	0.998520	+	0.0543841i	$0.0173195\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−1852.69	+	1852.69i	−0.0156439	+	0.0156439i	−0.714885	−	0.699242i	$-0.753521\pi$
$107$	−1852.69	+	1852.69i	−0.0156439	+	0.0156439i	0.699242	+	0.714885i	$0.253521\pi$
$108$		0			0
$109$			56962.5i			0.459222i	0.973282	+	0.229611i	$0.0737454\pi$
$109$			56962.5i			0.459222i	−0.973282	+	0.229611i	$0.926255\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	167364.	+	167364.i	1.23301	+	1.23301i	0.962803	+	0.270204i	$0.0870911\pi$
$113$	167364.	+	167364.i	1.23301	+	1.23301i	0.270204	+	0.962803i	$0.412909\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−36999.5			−0.239512
$120$		0			0
$121$	−140121.			−0.870044
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−201861.	−	201861.i	−1.11056	−	1.11056i	−0.993074	−	0.117490i	$-0.962515\pi$
$127$	−201861.	−	201861.i	−1.11056	−	1.11056i	−0.117490	−	0.993074i	$-0.537485\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	52956.9i		−	0.269615i	−0.990872	−	0.134808i	$-0.956958\pi$
$131$		−	52956.9i		−	0.269615i	0.990872	−	0.134808i	$-0.0430416\pi$
$132$		0			0
$133$	80743.5	−	80743.5i	0.395803	−	0.395803i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−175389.	+	175389.i	−0.798366	+	0.798366i	−0.982838	−	0.184472i	$-0.940943\pi$
$137$	−175389.	+	175389.i	−0.798366	+	0.798366i	0.184472	+	0.982838i	$0.440943\pi$
$138$		0			0
$139$			403672.i			1.77211i	0.463577	+	0.886057i	$0.346566\pi$
$139$			403672.i			1.77211i	−0.463577	+	0.886057i	$0.653434\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	103573.	+	103573.i	0.423554	+	0.423554i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	81620.6			0.301186			0.150593	−	0.988596i	$-0.451882\pi$
$149$	81620.6			0.301186			0.150593	+	0.988596i	$0.451882\pi$
$150$		0			0
$151$	−94840.7			−0.338495			−0.169248	−	0.985574i	$-0.554134\pi$
$151$	−94840.7			−0.338495			−0.169248	+	0.985574i	$0.554134\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	23926.8	+	23926.8i	0.0774702	+	0.0774702i	0.744780	−	0.667310i	$-0.232554\pi$
$157$	23926.8	+	23926.8i	0.0774702	+	0.0774702i	−0.667310	+	0.744780i	$0.732554\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			53171.5i			0.161664i
$162$		0			0
$163$	−277403.	+	277403.i	−0.817792	+	0.817792i	−0.985788	−	0.167996i	$-0.946271\pi$
$163$	−277403.	+	277403.i	−0.817792	+	0.817792i	0.167996	+	0.985788i	$0.446271\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	383047.	−	383047.i	1.06282	−	1.06282i	0.0649332	−	0.997890i	$-0.479317\pi$
$167$	383047.	−	383047.i	1.06282	−	1.06282i	0.997890	−	0.0649332i	$-0.0206834\pi$
$168$		0			0
$169$			300055.i			0.808135i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	349911.	+	349911.i	0.888879	+	0.888879i	0.994415	−	0.105536i	$-0.0336559\pi$
$173$	349911.	+	349911.i	0.888879	+	0.888879i	−0.105536	+	0.994415i	$0.533656\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	169256.			0.394832			0.197416	−	0.980320i	$-0.436745\pi$
$179$	169256.			0.394832			0.197416	+	0.980320i	$0.436745\pi$
$180$		0			0
$181$	−597205.			−1.35496			−0.677481	−	0.735540i	$-0.736928\pi$
$181$	−597205.			−1.35496			−0.677481	+	0.735540i	$0.736928\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−243629.	−	243629.i	−0.509478	−	0.509478i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			509082.i			1.00973i	0.863199	+	0.504864i	$0.168457\pi$
$191$			509082.i			1.00973i	−0.863199	+	0.504864i	$0.831543\pi$
$192$		0			0
$193$	−351450.	+	351450.i	−0.679158	+	0.679158i	−0.959810	−	0.280652i	$-0.909449\pi$
$193$	−351450.	+	351450.i	−0.679158	+	0.679158i	0.280652	+	0.959810i	$0.409449\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−663169.	+	663169.i	−1.21747	+	1.21747i	−0.248957	+	0.968515i	$0.580088\pi$
$197$	−663169.	+	663169.i	−1.21747	+	1.21747i	−0.968515	+	0.248957i	$0.919912\pi$
$198$		0			0
$199$			164344.i			0.294186i	0.989123	+	0.147093i	$0.0469916\pi$
$199$			164344.i			0.294186i	−0.989123	+	0.147093i	$0.953008\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	265831.	+	265831.i	0.452758	+	0.452758i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	1.06334e6			1.68386
$210$		0			0
$211$	980204.			1.51569			0.757845	−	0.652435i	$-0.226252\pi$
$211$	980204.			1.51569			0.757845	+	0.652435i	$0.226252\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	152289.	+	152289.i	0.219543	+	0.219543i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			167568.i			0.230787i
$222$		0			0
$223$	−741022.	+	741022.i	−0.997859	+	0.997859i	−0.999998	−	0.00213917i	$-0.999319\pi$
$223$	−741022.	+	741022.i	−0.997859	+	0.997859i	0.00213917	+	0.999998i	$0.499319\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−1.09189e6	+	1.09189e6i	−1.40641	+	1.40641i	−0.629032	+	0.777379i	$0.716549\pi$
$227$	−1.09189e6	+	1.09189e6i	−1.40641	+	1.40641i	−0.777379	+	0.629032i	$0.783451\pi$
$228$		0			0
$229$		−	1.39479e6i		−	1.75760i	−0.477189	−	0.878801i	$-0.658344\pi$
$229$		−	1.39479e6i		−	1.75760i	0.477189	−	0.878801i	$-0.341656\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	213807.	+	213807.i	0.258007	+	0.258007i	0.824243	−	0.566236i	$-0.191601\pi$
$233$	213807.	+	213807.i	0.258007	+	0.258007i	−0.566236	+	0.824243i	$0.691601\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	1.03350e6			1.17035			0.585175	−	0.810907i	$-0.301026\pi$
$239$	1.03350e6			1.17035			0.585175	+	0.810907i	$0.301026\pi$
$240$		0			0
$241$	−549240.			−0.609143			−0.304572	−	0.952489i	$-0.598513\pi$
$241$	−549240.			−0.609143			−0.304572	+	0.952489i	$0.598513\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−365683.	−	365683.i	−0.381384	−	0.381384i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			1.87559e6i			1.87911i	0.342394	+	0.939556i	$0.388762\pi$
$251$			1.87559e6i			1.87911i	−0.342394	+	0.939556i	$0.611238\pi$
$252$		0			0
$253$	−350116.	+	350116.i	−0.343883	+	0.343883i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	433397.	−	433397.i	0.409311	−	0.409311i	−0.472187	−	0.881498i	$-0.656535\pi$
$257$	433397.	−	433397.i	0.409311	−	0.409311i	0.881498	+	0.472187i	$0.156535\pi$
$258$		0			0
$259$			513792.i			0.475925i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	365463.	+	365463.i	0.325802	+	0.325802i	0.850988	−	0.525186i	$-0.176004\pi$
$263$	365463.	+	365463.i	0.325802	+	0.325802i	−0.525186	+	0.850988i	$0.676004\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$	868002.			0.731375			0.365688	−	0.930738i	$-0.380834\pi$
$269$	868002.			0.731375			0.365688	+	0.930738i	$0.380834\pi$
$270$		0			0
$271$	636519.			0.526488			0.263244	−	0.964729i	$-0.415208\pi$
$271$	636519.			0.526488			0.263244	+	0.964729i	$0.415208\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−186918.	−	186918.i	−0.146370	−	0.146370i	0.630124	−	0.776494i	$-0.283004\pi$
$277$	−186918.	−	186918.i	−0.146370	−	0.146370i	−0.776494	+	0.630124i	$0.783004\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	1.87230e6i		−	1.41453i	−0.706951	−	0.707263i	$-0.749930\pi$
$281$		−	1.87230e6i		−	1.41453i	0.706951	−	0.707263i	$-0.250070\pi$
$282$		0			0
$283$	−1.36204e6	+	1.36204e6i	−1.01094	+	1.01094i	−0.0109987	+	0.999940i	$0.503501\pi$
$283$	−1.36204e6	+	1.36204e6i	−1.01094	+	1.01094i	−0.999940	+	0.0109987i	$0.996499\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−311189.	+	311189.i	−0.223008	+	0.223008i
$288$		0			0
$289$			1.02570e6i			0.722394i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	1.32204e6	+	1.32204e6i	0.899653	+	0.899653i	0.995405	−	0.0957518i	$-0.0305255\pi$
$293$	1.32204e6	+	1.32204e6i	0.899653	+	0.899653i	−0.0957518	+	0.995405i	$0.530526\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	240810.			0.155775
$300$		0			0
$301$	−1.25055e6			−0.795583
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−1.24646e6	−	1.24646e6i	−0.754801	−	0.754801i	0.220570	−	0.975371i	$-0.429208\pi$
$307$	−1.24646e6	−	1.24646e6i	−0.754801	−	0.754801i	−0.975371	+	0.220570i	$0.929208\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			1.19609e6i			0.701233i	0.936519	+	0.350617i	$0.114028\pi$
$311$			1.19609e6i			0.701233i	−0.936519	+	0.350617i	$0.885972\pi$
$312$		0			0
$313$	−1.14772e6	+	1.14772e6i	−0.662177	+	0.662177i	−0.955893	−	0.293715i	$-0.905108\pi$
$313$	−1.14772e6	+	1.14772e6i	−0.662177	+	0.662177i	0.293715	+	0.955893i	$0.405108\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−2.07386e6	+	2.07386e6i	−1.15913	+	1.15913i	−0.174466	+	0.984663i	$0.555820\pi$
$317$	−2.07386e6	+	2.07386e6i	−1.15913	+	1.15913i	−0.984663	+	0.174466i	$0.944180\pi$
$318$		0			0
$319$			3.50082e6i			1.92616i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	860172.	+	860172.i	0.458753	+	0.458753i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	142648.			0.0726567
$330$		0			0
$331$	−2.54441e6			−1.27649			−0.638244	−	0.769834i	$-0.720339\pi$
$331$	−2.54441e6			−1.27649			−0.638244	+	0.769834i	$0.720339\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−448987.	−	448987.i	−0.215357	−	0.215357i	0.591182	−	0.806538i	$-0.298662\pi$
$337$	−448987.	−	448987.i	−0.215357	−	0.215357i	−0.806538	+	0.591182i	$0.798662\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			2.00554e6i			0.933998i
$342$		0			0
$343$	1.25603e6	−	1.25603e6i	0.576454	−	0.576454i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	906901.	−	906901.i	0.404330	−	0.404330i	−0.475426	−	0.879756i	$-0.657706\pi$
$347$	906901.	−	906901.i	0.404330	−	0.404330i	0.879756	+	0.475426i	$0.157706\pi$
$348$		0			0
$349$			2.00518e6i			0.881229i	0.897696	+	0.440615i	$0.145239\pi$
$349$			2.00518e6i			0.881229i	−0.897696	+	0.440615i	$0.854761\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	2.59761e6	+	2.59761e6i	1.10953	+	1.10953i	0.993213	+	0.116313i	$0.0371076\pi$
$353$	2.59761e6	+	2.59761e6i	1.10953	+	1.10953i	0.116313	+	0.993213i	$0.462892\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−1.40601e6			−0.575775			−0.287887	−	0.957664i	$-0.592953\pi$
$359$	−1.40601e6			−0.575775			−0.287887	+	0.957664i	$0.592953\pi$
$360$		0			0
$361$	−1.27819e6			−0.516210
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−1.14303e6	−	1.14303e6i	−0.442990	−	0.442990i	0.450026	−	0.893016i	$-0.351415\pi$
$367$	−1.14303e6	−	1.14303e6i	−0.442990	−	0.442990i	−0.893016	+	0.450026i	$0.851415\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		−	1.88221e6i		−	0.709960i
$372$		0			0
$373$	2.42084e6	−	2.42084e6i	0.900934	−	0.900934i	−0.0945827	−	0.995517i	$-0.530152\pi$
$373$	2.42084e6	−	2.42084e6i	0.900934	−	0.900934i	0.995517	+	0.0945827i	$0.0301517\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	1.20393e6	−	1.20393e6i	0.436264	−	0.436264i
$378$		0			0
$379$		−	2.28431e6i		−	0.816877i	−0.912786	−	0.408438i	$-0.866074\pi$
$379$		−	2.28431e6i		−	0.816877i	0.912786	−	0.408438i	$-0.133926\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	2.61482e6	+	2.61482e6i	0.910845	+	0.910845i	0.996339	−	0.0854937i	$-0.0272467\pi$
$383$	2.61482e6	+	2.61482e6i	0.910845	+	0.910845i	−0.0854937	+	0.996339i	$0.527247\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	4.86718e6			1.63081			0.815405	−	0.578890i	$-0.196514\pi$
$389$	4.86718e6			1.63081			0.815405	+	0.578890i	$0.196514\pi$
$390$		0			0
$391$	−566443.			−0.187376
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	1.80699e6	+	1.80699e6i	0.575412	+	0.575412i	0.933636	−	0.358224i	$-0.116618\pi$
$397$	1.80699e6	+	1.80699e6i	0.575412	+	0.575412i	−0.358224	+	0.933636i	$0.616618\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			2.66719e6i			0.828310i	0.910206	+	0.414155i	$0.135923\pi$
$401$			2.66719e6i			0.828310i	−0.910206	+	0.414155i	$0.864077\pi$
$402$		0			0
$403$	689708.	−	689708.i	0.211545	−	0.211545i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−3.38315e6	+	3.38315e6i	−1.01236	+	1.01236i
$408$		0			0
$409$			5.22116e6i			1.54333i	0.636030	+	0.771665i	$0.280576\pi$
$409$			5.22116e6i			1.54333i	−0.636030	+	0.771665i	$0.719424\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	973465.	+	973465.i	0.280831	+	0.280831i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−20170.3			−0.00561276			−0.00280638	−	0.999996i	$-0.500893\pi$
$419$	−20170.3			−0.00561276			−0.00280638	+	0.999996i	$0.500893\pi$
$420$		0			0
$421$	5.75452e6			1.58235			0.791177	−	0.611587i	$-0.209469\pi$
$421$	5.75452e6			1.58235			0.791177	+	0.611587i	$0.209469\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	397453.	+	397453.i	0.105491	+	0.105491i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	1.91156e6i		−	0.495674i	−0.968802	−	0.247837i	$-0.920280\pi$
$431$		−	1.91156e6i		−	0.495674i	0.968802	−	0.247837i	$-0.0797197\pi$
$432$		0			0
$433$	−4.44587e6	+	4.44587e6i	−1.13956	+	1.13956i	−0.151031	+	0.988529i	$0.548259\pi$
$433$	−4.44587e6	+	4.44587e6i	−1.13956	+	1.13956i	−0.988529	+	0.151031i	$0.951741\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	1.23614e6	−	1.23614e6i	0.309645	−	0.309645i
$438$		0			0
$439$		−	3.59277e6i		−	0.889751i	−0.895592	−	0.444876i	$-0.853248\pi$
$439$		−	3.59277e6i		−	0.889751i	0.895592	−	0.444876i	$-0.146752\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−1.16335e6	−	1.16335e6i	−0.281645	−	0.281645i	0.552120	−	0.833765i	$-0.313819\pi$
$443$	−1.16335e6	−	1.16335e6i	−0.281645	−	0.281645i	−0.833765	+	0.552120i	$0.813819\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−2.80340e6			−0.656249			−0.328125	−	0.944634i	$-0.606417\pi$
$449$	−2.80340e6			−0.656249			−0.328125	+	0.944634i	$0.606417\pi$
$450$		0			0
$451$	−4.09815e6			−0.948739
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−1.81493e6	−	1.81493e6i	−0.406508	−	0.406508i	0.474011	−	0.880519i	$-0.342806\pi$
$457$	−1.81493e6	−	1.81493e6i	−0.406508	−	0.406508i	−0.880519	+	0.474011i	$0.842806\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	4.79711e6i		−	1.05130i	−0.850700	−	0.525651i	$-0.823822\pi$
$461$		−	4.79711e6i		−	1.05130i	0.850700	−	0.525651i	$-0.176178\pi$
$462$		0			0
$463$	40543.6	−	40543.6i	0.00878962	−	0.00878962i	−0.702698	−	0.711488i	$-0.748022\pi$
$463$	40543.6	−	40543.6i	0.00878962	−	0.00878962i	0.711488	+	0.702698i	$0.248022\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	139814.	−	139814.i	0.0296659	−	0.0296659i	−0.692118	−	0.721784i	$-0.743322\pi$
$467$	139814.	−	139814.i	0.0296659	−	0.0296659i	0.721784	+	0.692118i	$0.243322\pi$
$468$		0			0
$469$		−	1.74786e6i		−	0.366922i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−8.23446e6	−	8.23446e6i	−1.69232	−	1.69232i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$	2.61055e6			0.519868			0.259934	−	0.965626i	$-0.416299\pi$
$479$	2.61055e6			0.519868			0.259934	+	0.965626i	$0.416299\pi$
$480$		0			0
$481$	2.32694e6			0.458587
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−871965.	−	871965.i	−0.166601	−	0.166601i	0.618883	−	0.785483i	$-0.287586\pi$
$487$	−871965.	−	871965.i	−0.166601	−	0.166601i	−0.785483	+	0.618883i	$0.787586\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			9.40542e6i			1.76066i	0.474366	+	0.880328i	$0.342677\pi$
$491$			9.40542e6i			1.76066i	−0.474366	+	0.880328i	$0.657323\pi$
$492$		0			0
$493$	−2.83194e6	+	2.83194e6i	−0.524766	+	0.524766i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−405845.	+	405845.i	−0.0737003	+	0.0737003i
$498$		0			0
$499$		−	7.73872e6i		−	1.39129i	−0.718385	−	0.695645i	$-0.755119\pi$
$499$		−	7.73872e6i		−	1.39129i	0.718385	−	0.695645i	$-0.244881\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−2.58199e6	−	2.58199e6i	−0.455024	−	0.455024i	0.441994	−	0.897018i	$-0.354271\pi$
$503$	−2.58199e6	−	2.58199e6i	−0.455024	−	0.455024i	−0.897018	+	0.441994i	$0.854271\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−8.53660e6			−1.46046			−0.730231	−	0.683200i	$-0.760588\pi$
$509$	−8.53660e6			−1.46046			−0.730231	+	0.683200i	$0.760588\pi$
$510$		0			0
$511$	−4.18637e6			−0.709228
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$	939289.	+	939289.i	0.154551	+	0.154551i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			2.10034e6i			0.338997i	0.985530	+	0.169499i	$0.0542148\pi$
$521$			2.10034e6i			0.338997i	−0.985530	+	0.169499i	$0.945785\pi$
$522$		0			0
$523$	−3.83985e6	+	3.83985e6i	−0.613847	+	0.613847i	−0.943946	−	0.330100i	$-0.892918\pi$
$523$	−3.83985e6	+	3.83985e6i	−0.613847	+	0.613847i	0.330100	+	0.943946i	$0.392918\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−1.62236e6	+	1.62236e6i	−0.254460	+	0.254460i
$528$		0			0
$529$		−	5.62232e6i		−	0.873526i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	1.40936e6	+	1.40936e6i	0.214884	+	0.214884i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	7.31753e6			1.08491
$540$		0			0
$541$	1.26259e7			1.85468			0.927338	−	0.374225i	$-0.122091\pi$
$541$	1.26259e7			1.85468			0.927338	+	0.374225i	$0.122091\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	6.17833e6	+	6.17833e6i	0.882883	+	0.882883i	0.993827	−	0.110944i	$-0.0353874\pi$
$547$	6.17833e6	+	6.17833e6i	0.882883	+	0.882883i	−0.110944	+	0.993827i	$0.535387\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	1.23602e7i		−	1.73439i
$552$		0			0
$553$	1.13570e6	−	1.13570e6i	0.157925	−	0.157925i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	1.02029e7	−	1.02029e7i	1.39343	−	1.39343i	0.575933	−	0.817497i	$-0.304639\pi$
$557$	1.02029e7	−	1.02029e7i	1.39343	−	1.39343i	0.817497	−	0.575933i	$-0.195361\pi$
$558$		0			0
$559$			5.66368e6i			0.766600i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−7.07749e6	−	7.07749e6i	−0.941040	−	0.941040i	0.0573158	−	0.998356i	$-0.481746\pi$
$563$	−7.07749e6	−	7.07749e6i	−0.941040	−	0.941040i	−0.998356	+	0.0573158i	$0.981746\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	5.21572e6			0.675358			0.337679	−	0.941261i	$-0.390358\pi$
$569$	5.21572e6			0.675358			0.337679	+	0.941261i	$0.390358\pi$
$570$		0			0
$571$	6.38516e6			0.819562			0.409781	−	0.912184i	$-0.365605\pi$
$571$	6.38516e6			0.819562			0.409781	+	0.912184i	$0.365605\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	162190.	+	162190.i	0.0202808	+	0.0202808i	0.717174	−	0.696894i	$-0.245435\pi$
$577$	162190.	+	162190.i	0.0202808	+	0.0202808i	−0.696894	+	0.717174i	$0.745435\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$			843780.i			0.103702i
$582$		0			0
$583$	1.23937e7	−	1.23937e7i	1.51019	−	1.51019i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−7.58398e6	+	7.58398e6i	−0.908452	+	0.908452i	−0.996147	−	0.0876955i	$-0.972050\pi$
$587$	−7.58398e6	+	7.58398e6i	−0.908452	+	0.908452i	0.0876955	+	0.996147i	$0.472050\pi$
$588$		0			0
$589$		−	7.08089e6i		−	0.841008i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	1.93347e6	+	1.93347e6i	0.225788	+	0.225788i	0.810930	−	0.585143i	$-0.198961\pi$
$593$	1.93347e6	+	1.93347e6i	0.225788	+	0.225788i	−0.585143	+	0.810930i	$0.698961\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−1.36234e7			−1.55138			−0.775689	−	0.631115i	$-0.782598\pi$
$599$	−1.36234e7			−1.55138			−0.775689	+	0.631115i	$0.782598\pi$
$600$		0			0
$601$	−2.67406e6			−0.301984			−0.150992	−	0.988535i	$-0.548247\pi$
$601$	−2.67406e6			−0.301984			−0.150992	+	0.988535i	$0.548247\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$	5.65519e6	+	5.65519e6i	0.622982	+	0.622982i	0.946293	−	0.323311i	$-0.104796\pi$
$607$	5.65519e6	+	5.65519e6i	0.622982	+	0.622982i	−0.323311	+	0.946293i	$0.604796\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	646044.i		−	0.0700099i
$612$		0			0
$613$	9.13086e6	−	9.13086e6i	0.981433	−	0.981433i	−0.0183978	−	0.999831i	$-0.505857\pi$
$613$	9.13086e6	−	9.13086e6i	0.981433	−	0.981433i	0.999831	+	0.0183978i	$0.00585652\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	1.09542e7	−	1.09542e7i	1.15843	−	1.15843i	0.173613	−	0.984814i	$-0.444456\pi$
$617$	1.09542e7	−	1.09542e7i	1.15843	−	1.15843i	0.984814	−	0.173613i	$-0.0555442\pi$
$618$		0			0
$619$		−	5.05105e6i		−	0.529852i	−0.964269	−	0.264926i	$-0.914652\pi$
$619$		−	5.05105e6i		−	0.529852i	0.964269	−	0.264926i	$-0.0853476\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−47846.9	−	47846.9i	−0.00493894	−	0.00493894i
$624$		0			0
$625$		0			0
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−5.47350e6			−0.551618
$630$		0			0
$631$	−1.33711e7			−1.33688			−0.668441	−	0.743765i	$-0.733038\pi$
$631$	−1.33711e7			−1.33688			−0.668441	+	0.743765i	$0.733038\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−2.51650e6	−	2.51650e6i	−0.245725	−	0.245725i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	7.24493e6i		−	0.696449i	−0.937411	−	0.348224i	$-0.886785\pi$
$641$		−	7.24493e6i		−	0.696449i	0.937411	−	0.348224i	$-0.113215\pi$
$642$		0			0
$643$	3.42624e6	−	3.42624e6i	0.326807	−	0.326807i	−0.524564	−	0.851371i	$-0.675772\pi$
$643$	3.42624e6	−	3.42624e6i	0.326807	−	0.326807i	0.851371	+	0.524564i	$0.175772\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	4.94740e6	−	4.94740e6i	0.464640	−	0.464640i	−0.435533	−	0.900173i	$-0.643440\pi$
$647$	4.94740e6	−	4.94740e6i	0.464640	−	0.464640i	0.900173	+	0.435533i	$0.143440\pi$
$648$		0			0
$649$			1.28199e7i			1.19474i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−3.11972e6	−	3.11972e6i	−0.286307	−	0.286307i	0.549311	−	0.835618i	$-0.314890\pi$
$653$	−3.11972e6	−	3.11972e6i	−0.286307	−	0.286307i	−0.835618	+	0.549311i	$0.814890\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−1.31180e7			−1.17667			−0.588334	−	0.808618i	$-0.700216\pi$
$659$	−1.31180e7			−1.17667			−0.588334	+	0.808618i	$0.700216\pi$
$660$		0			0
$661$	−6.06793e6			−0.540178			−0.270089	−	0.962835i	$-0.587053\pi$
$661$	−6.06793e6			−0.540178			−0.270089	+	0.962835i	$0.587053\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	4.06974e6	+	4.06974e6i	0.354203	+	0.354203i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			5.23419e6i			0.448790i
$672$		0			0
$673$	5.74358e6	−	5.74358e6i	0.488816	−	0.488816i	−0.419117	−	0.907932i	$-0.637660\pi$
$673$	5.74358e6	−	5.74358e6i	0.488816	−	0.488816i	0.907932	+	0.419117i	$0.137660\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	1.50516e6	−	1.50516e6i	0.126215	−	0.126215i	−0.641178	−	0.767392i	$-0.721554\pi$
$677$	1.50516e6	−	1.50516e6i	0.126215	−	0.126215i	0.767392	+	0.641178i	$0.221554\pi$
$678$		0			0
$679$			4.67970e6i			0.389532i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−1.32297e7	−	1.32297e7i	−1.08517	−	1.08517i	−0.996018	−	0.0891530i	$-0.971584\pi$
$683$	−1.32297e7	−	1.32297e7i	−1.08517	−	1.08517i	−0.0891530	−	0.996018i	$-0.528416\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−8.52443e6			−0.684097
$690$		0			0
$691$	1.28544e7			1.02413			0.512065	−	0.858946i	$-0.328881\pi$
$691$	1.28544e7			1.02413			0.512065	+	0.858946i	$0.328881\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−3.31514e6	−	3.31514e6i	−0.258476	−	0.258476i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	1.69514e7i		−	1.30290i	−0.758693	−	0.651448i	$-0.774162\pi$
$701$		−	1.69514e7i		−	1.30290i	0.758693	−	0.651448i	$-0.225838\pi$
$702$		0			0
$703$	1.19448e7	−	1.19448e7i	0.911568	−	0.911568i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−502093.	+	502093.i	−0.0377777	+	0.0377777i
$708$		0			0
$709$			1.20746e7i			0.902109i	0.892497	+	0.451054i	$0.148952\pi$
$709$			1.20746e7i			0.902109i	−0.892497	+	0.451054i	$0.851048\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	2.33147e6	+	2.33147e6i	0.171753	+	0.171753i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−2.40715e7			−1.73653			−0.868263	−	0.496105i	$-0.834763\pi$
$719$	−2.40715e7			−1.73653			−0.868263	+	0.496105i	$0.834763\pi$
$720$		0			0
$721$	8.47231e6			0.606965
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−7.81809e6	−	7.81809e6i	−0.548611	−	0.548611i	0.377428	−	0.926039i	$-0.376809\pi$
$727$	−7.81809e6	−	7.81809e6i	−0.548611	−	0.548611i	−0.926039	+	0.377428i	$0.876809\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	1.33223e7i		−	0.922116i
$732$		0			0
$733$	−4.81067e6	+	4.81067e6i	−0.330709	+	0.330709i	−0.852856	−	0.522147i	$-0.825131\pi$
$733$	−4.81067e6	+	4.81067e6i	−0.330709	+	0.330709i	0.522147	+	0.852856i	$0.325131\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	1.15091e7	−	1.15091e7i	0.780496	−	0.780496i
$738$		0			0
$739$		−	2.44076e7i		−	1.64405i	−0.569452	−	0.822024i	$-0.692845\pi$
$739$		−	2.44076e7i		−	1.64405i	0.569452	−	0.822024i	$-0.307155\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−1.03889e6	−	1.03889e6i	−0.0690394	−	0.0690394i	0.671744	−	0.740783i	$-0.265545\pi$
$743$	−1.03889e6	−	1.03889e6i	−0.0690394	−	0.0690394i	−0.740783	+	0.671744i	$0.765545\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−154411.			−0.0100571
$750$		0			0
$751$	−9.84080e6			−0.636693			−0.318347	−	0.947974i	$-0.603128\pi$
$751$	−9.84080e6			−0.636693			−0.318347	+	0.947974i	$0.603128\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	1.14055e7	+	1.14055e7i	0.723395	+	0.723395i	0.969295	−	0.245900i	$-0.0790835\pi$
$757$	1.14055e7	+	1.14055e7i	0.723395	+	0.723395i	−0.245900	+	0.969295i	$0.579084\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	6.23184e6i		−	0.390081i	−0.980795	−	0.195040i	$-0.937516\pi$
$761$		−	6.23184e6i		−	0.390081i	0.980795	−	0.195040i	$-0.0624838\pi$
$762$		0			0
$763$	−2.37374e6	+	2.37374e6i	−0.147612	+	0.147612i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	4.40877e6	−	4.40877e6i	0.270601	−	0.270601i
$768$		0			0
$769$		−	1.25766e7i		−	0.766916i	−0.923558	−	0.383458i	$-0.874733\pi$
$769$		−	1.25766e7i		−	0.766916i	0.923558	−	0.383458i	$-0.125267\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−6.61711e6	−	6.61711e6i	−0.398308	−	0.398308i	0.479328	−	0.877636i	$-0.340881\pi$
$773$	−6.61711e6	−	6.61711e6i	−0.398308	−	0.398308i	−0.877636	+	0.479328i	$0.840881\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	1.44692e7			0.854281
$780$		0			0
$781$	−5.34471e6			−0.313542
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−6.35497e6	−	6.35497e6i	−0.365743	−	0.365743i	0.500179	−	0.865922i	$-0.333268\pi$
$787$	−6.35497e6	−	6.35497e6i	−0.365743	−	0.365743i	−0.865922	+	0.500179i	$0.833268\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			1.39488e7i			0.792674i
$792$		0			0
$793$	1.80004e6	−	1.80004e6i	0.101648	−	0.101648i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	5.71910e6	−	5.71910e6i	0.318920	−	0.318920i	−0.529432	−	0.848352i	$-0.677595\pi$
$797$	5.71910e6	−	5.71910e6i	0.318920	−	0.318920i	0.848352	+	0.529432i	$0.177595\pi$
$798$		0			0
$799$			1.51965e6i			0.0842124i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−2.75659e7	−	2.75659e7i	−1.50863	−	1.50863i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	2.11072e6			0.113386			0.0566930	−	0.998392i	$-0.481944\pi$
$809$	2.11072e6			0.113386			0.0566930	+	0.998392i	$0.481944\pi$
$810$		0			0
$811$	2.58090e7			1.37790			0.688952	−	0.724807i	$-0.258071\pi$
$811$	2.58090e7			1.37790			0.688952	+	0.724807i	$0.258071\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	2.90731e7	+	2.90731e7i	1.52383	+	1.52383i
$818$		0			0
$819$		0			0
$820$		0			0
$821$			7.00135e6i			0.362513i	0.983436	+	0.181257i	$0.0580165\pi$
$821$			7.00135e6i			0.362513i	−0.983436	+	0.181257i	$0.941984\pi$
$822$		0			0
$823$	1.50268e7	−	1.50268e7i	0.773335	−	0.773335i	−0.205353	−	0.978688i	$-0.565834\pi$
$823$	1.50268e7	−	1.50268e7i	0.773335	−	0.773335i	0.978688	+	0.205353i	$0.0658344\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−1.88338e7	+	1.88338e7i	−0.957580	+	0.957580i	−0.999136	−	0.0415557i	$-0.986769\pi$
$827$	−1.88338e7	+	1.88338e7i	−0.957580	+	0.957580i	0.0415557	+	0.999136i	$0.486769\pi$
$828$		0			0
$829$		−	2.32214e7i		−	1.17355i	−0.809749	−	0.586776i	$-0.800397\pi$
$829$		−	2.32214e7i		−	1.17355i	0.809749	−	0.586776i	$-0.199603\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	5.91941e6	+	5.91941e6i	0.295574	+	0.295574i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−1.20015e7			−0.588614			−0.294307	−	0.955711i	$-0.595089\pi$
$839$	−1.20015e7			−0.588614			−0.294307	+	0.955711i	$0.595089\pi$
$840$		0			0
$841$	2.01822e7			0.983964
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−5.83914e6	−	5.83914e6i	−0.279666	−	0.279666i
$848$		0			0
$849$		0			0
$850$		0			0
$851$			7.86590e6i			0.372327i
$852$		0			0
$853$	−1.21377e7	+	1.21377e7i	−0.571167	+	0.571167i	−0.932454	−	0.361288i	$-0.882337\pi$
$853$	−1.21377e7	+	1.21377e7i	−0.571167	+	0.571167i	0.361288	+	0.932454i	$0.382337\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−9.49913e6	+	9.49913e6i	−0.441806	+	0.441806i	−0.892619	−	0.450812i	$-0.851134\pi$
$857$	−9.49913e6	+	9.49913e6i	−0.441806	+	0.441806i	0.450812	+	0.892619i	$0.351134\pi$
$858$		0			0
$859$		−	1.87864e7i		−	0.868680i	−0.900749	−	0.434340i	$-0.856982\pi$
$859$		−	1.87864e7i		−	0.868680i	0.900749	−	0.434340i	$-0.143018\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−3.61679e6	−	3.61679e6i	−0.165309	−	0.165309i	0.619605	−	0.784914i	$-0.287293\pi$
$863$	−3.61679e6	−	3.61679e6i	−0.165309	−	0.165309i	−0.784914	+	0.619605i	$0.787293\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	1.49564e7			0.671857
$870$		0			0
$871$	−7.91594e6			−0.353555
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	7.92683e6	+	7.92683e6i	0.348017	+	0.348017i	0.859371	−	0.511353i	$-0.170856\pi$
$877$	7.92683e6	+	7.92683e6i	0.348017	+	0.348017i	−0.511353	+	0.859371i	$0.670856\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	9.65725e6i		−	0.419193i	−0.977788	−	0.209596i	$-0.932785\pi$
$881$		−	9.65725e6i		−	0.419193i	0.977788	−	0.209596i	$-0.0672150\pi$
$882$		0			0
$883$	−1.93601e7	+	1.93601e7i	−0.835613	+	0.835613i	−0.988278	−	0.152665i	$-0.951215\pi$
$883$	−1.93601e7	+	1.93601e7i	−0.835613	+	0.835613i	0.152665	+	0.988278i	$0.451215\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	801763.	−	801763.i	0.0342166	−	0.0342166i	−0.689791	−	0.724008i	$-0.742298\pi$
$887$	801763.	−	801763.i	0.0342166	−	0.0342166i	0.724008	+	0.689791i	$0.242298\pi$
$888$		0			0
$889$		−	1.68239e7i		−	0.713958i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−3.31631e6	−	3.31631e6i	−0.139164	−	0.139164i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$	2.33124e7			0.962027
$900$		0			0
$901$	2.00515e7			0.822876
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−2.98378e7	−	2.98378e7i	−1.20434	−	1.20434i	−0.972834	−	0.231505i	$-0.925635\pi$
$907$	−2.98378e7	−	2.98378e7i	−1.20434	−	1.20434i	−0.231505	−	0.972834i	$-0.574365\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			2.77013e7i			1.10587i	0.833225	+	0.552935i	$0.186492\pi$
$911$			2.77013e7i			1.10587i	−0.833225	+	0.552935i	$0.813508\pi$
$912$		0			0
$913$	−5.55601e6	+	5.55601e6i	−0.220590	+	0.220590i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	2.20682e6	−	2.20682e6i	0.0866649	−	0.0866649i
$918$		0			0
$919$			2.59952e7i			1.01532i	0.861557	+	0.507661i	$0.169490\pi$
$919$			2.59952e7i			1.01532i	−0.861557	+	0.507661i	$0.830510\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	1.83805e6	+	1.83805e6i	0.0710155	+	0.0710155i
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	3.24949e7			1.23531			0.617656	−	0.786449i	$-0.288083\pi$
$929$	3.24949e7			1.23531			0.617656	+	0.786449i	$0.288083\pi$
$930$		0			0
$931$	−2.58357e7			−0.976892
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	3.63663e7	+	3.63663e7i	1.35316	+	1.35316i	0.882099	+	0.471065i	$0.156130\pi$
$937$	3.63663e7	+	3.63663e7i	1.35316	+	1.35316i	0.471065	+	0.882099i	$0.343870\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	4.69675e7i		−	1.72911i	−0.502535	−	0.864557i	$-0.667599\pi$
$941$		−	4.69675e7i		−	1.72911i	0.502535	−	0.864557i	$-0.332401\pi$
$942$		0			0
$943$	−4.76415e6	+	4.76415e6i	−0.174464	+	0.174464i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	2.77823e7	−	2.77823e7i	1.00668	−	1.00668i	0.00670649	−	0.999978i	$-0.497865\pi$
$947$	2.77823e7	−	2.77823e7i	1.00668	−	1.00668i	0.999978	−	0.00670649i	$-0.00213476\pi$
$948$		0			0
$949$			1.89598e7i			0.683391i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	1.53030e7	+	1.53030e7i	0.545814	+	0.545814i	0.925227	−	0.379413i	$-0.123874\pi$
$953$	1.53030e7	+	1.53030e7i	0.545814	+	0.545814i	−0.379413	+	0.925227i	$0.623874\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−1.46176e7			−0.513252
$960$		0			0
$961$	−1.52740e7			−0.533512
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−3.30532e7	−	3.30532e7i	−1.13670	−	1.13670i	−0.989037	−	0.147665i	$-0.952824\pi$
$967$	−3.30532e7	−	3.30532e7i	−1.13670	−	1.13670i	−0.147665	−	0.989037i	$-0.547176\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			5.31354e7i			1.80857i	0.426928	+	0.904286i	$0.359596\pi$
$971$			5.31354e7i			1.80857i	−0.426928	+	0.904286i	$0.640404\pi$
$972$		0			0
$973$	−1.68218e7	+	1.68218e7i	−0.569627	+	0.569627i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	1.51300e7	−	1.51300e7i	0.507111	−	0.507111i	−0.406527	−	0.913639i	$-0.633260\pi$
$977$	1.51300e7	−	1.51300e7i	0.507111	−	0.507111i	0.913639	+	0.406527i	$0.133260\pi$
$978$		0			0
$979$		−	630111.i		−	0.0210117i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	1.89261e7	+	1.89261e7i	0.624709	+	0.624709i	0.946732	−	0.322023i	$-0.104363\pi$
$983$	1.89261e7	+	1.89261e7i	0.624709	+	0.624709i	−0.322023	+	0.946732i	$0.604363\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−1.91453e7			−0.622403
$990$		0			0
$991$	−8.98192e6			−0.290526			−0.145263	−	0.989393i	$-0.546403\pi$
$991$	−8.98192e6			−0.290526			−0.145263	+	0.989393i	$0.546403\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−3.75482e6	−	3.75482e6i	−0.119633	−	0.119633i	0.644756	−	0.764389i	$-0.276959\pi$
$997$	−3.75482e6	−	3.75482e6i	−0.119633	−	0.119633i	−0.764389	+	0.644756i	$0.776959\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	900.6.j.b.557.8		20
3.2	odd	2	inner	900.6.j.b.557.7		20
5.2	odd	4		180.6.j.a.53.10	yes	20
5.3	odd	4	inner	900.6.j.b.593.8		20
5.4	even	2		180.6.j.a.17.1	✓	20
15.2	even	4		180.6.j.a.53.1	yes	20
15.8	even	4	inner	900.6.j.b.593.7		20
15.14	odd	2		180.6.j.a.17.10	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
180.6.j.a.17.1	✓	20	5.4	even	2
180.6.j.a.17.10	yes	20	15.14	odd	2
180.6.j.a.53.1	yes	20	15.2	even	4
180.6.j.a.53.10	yes	20	5.2	odd	4
900.6.j.b.557.7		20	3.2	odd	2	inner
900.6.j.b.557.8		20	1.1	even	1	trivial
900.6.j.b.593.7		20	15.8	even	4	inner
900.6.j.b.593.8		20	5.3	odd	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:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	900.j (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(41.6720 + 41.6720i) q^{7} +O(q^{10})\)	\(q+(41.6720 + 41.6720i) q^{7} +548.792i q^{11} +(188.730 - 188.730i) q^{13} +(-443.937 + 443.937i) q^{17} -1937.60i q^{19} +(637.976 + 637.976i) q^{23} +6379.14 q^{29} +3654.47 q^{31} +(6164.72 + 6164.72i) q^{37} +7467.59i q^{41} +(-15004.7 + 15004.7i) q^{43} +(1711.56 - 1711.56i) q^{47} -13333.9i q^{49} +(-22583.7 - 22583.7i) q^{53} +23360.2 q^{59} +9537.66 q^{61} +(-20971.6 - 20971.6i) q^{67} +9739.04i q^{71} +(-50230.1 + 50230.1i) q^{73} +(-22869.2 + 22869.2i) q^{77} -27253.3i q^{79} +(10124.1 + 10124.1i) q^{83} -1148.18 q^{89} +15729.5 q^{91} +(56149.2 + 56149.2i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 152 q^{7}+O(q^{10}) \) 20 * q + 152 * q^7	\( 20 q + 152 q^{7} - 252 q^{13} + 9440 q^{31} - 29988 q^{37} - 6720 q^{43} + 67200 q^{61} - 139552 q^{67} - 134828 q^{73} - 220560 q^{91} - 207132 q^{97}+O(q^{100}) \) 20 * q + 152 * q^7 - 252 * q^13 + 9440 * q^31 - 29988 * q^37 - 6720 * q^43 + 67200 * q^61 - 139552 * q^67 - 134828 * q^73 - 220560 * q^91 - 207132 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more