LMFDB - Embedded newform 675.4.b.f.649.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [675,4,Mod(649,675)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(675, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("675.649");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 675 = 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	675.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$39.8262892539$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 135)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			649.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	675.649
Dual form			675.4.b.f.649.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} +7.00000 q^{4} -6.00000i q^{7} -15.0000i q^{8} +O(q^{10})$	$q-1.00000i q^{2} +7.00000 q^{4} -6.00000i q^{7} -15.0000i q^{8} +47.0000 q^{11} +5.00000i q^{13} -6.00000 q^{14} +41.0000 q^{16} +131.000i q^{17} +56.0000 q^{19} -47.0000i q^{22} +3.00000i q^{23} +5.00000 q^{26} -42.0000i q^{28} -157.000 q^{29} +225.000 q^{31} -161.000i q^{32} +131.000 q^{34} -70.0000i q^{37} -56.0000i q^{38} -140.000 q^{41} -397.000i q^{43} +329.000 q^{44} +3.00000 q^{46} +347.000i q^{47} +307.000 q^{49} +35.0000i q^{52} +4.00000i q^{53} -90.0000 q^{56} +157.000i q^{58} +748.000 q^{59} -338.000 q^{61} -225.000i q^{62} +167.000 q^{64} +492.000i q^{67} +917.000i q^{68} -32.0000 q^{71} -970.000i q^{73} -70.0000 q^{74} +392.000 q^{76} -282.000i q^{77} +1257.00 q^{79} +140.000i q^{82} -102.000i q^{83} -397.000 q^{86} -705.000i q^{88} -1488.00 q^{89} +30.0000 q^{91} +21.0000i q^{92} +347.000 q^{94} +974.000i q^{97} -307.000i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 14 q^{4}+O(q^{10}) $ 2 * q + 14 * q^4	$ 2 q + 14 q^{4} + 94 q^{11} - 12 q^{14} + 82 q^{16} + 112 q^{19} + 10 q^{26} - 314 q^{29} + 450 q^{31} + 262 q^{34} - 280 q^{41} + 658 q^{44} + 6 q^{46} + 614 q^{49} - 180 q^{56} + 1496 q^{59} - 676 q^{61} + 334 q^{64} - 64 q^{71} - 140 q^{74} + 784 q^{76} + 2514 q^{79} - 794 q^{86} - 2976 q^{89} + 60 q^{91} + 694 q^{94}+O(q^{100}) $ 2 * q + 14 * q^4 + 94 * q^11 - 12 * q^14 + 82 * q^16 + 112 * q^19 + 10 * q^26 - 314 * q^29 + 450 * q^31 + 262 * q^34 - 280 * q^41 + 658 * q^44 + 6 * q^46 + 614 * q^49 - 180 * q^56 + 1496 * q^59 - 676 * q^61 + 334 * q^64 - 64 * q^71 - 140 * q^74 + 784 * q^76 + 2514 * q^79 - 794 * q^86 - 2976 * q^89 + 60 * q^91 + 694 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/675\mathbb{Z}\right)^\times$.

$n$	$326$	$352$
$\chi(n)$	$1$	$-1$

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 $

$2$	− 1.00000i	− 0.353553i	−0.984251	−	0.176777i	$-0.943433\pi$
$2$	− 1.00000i	− 0.353553i	0.984251	−	0.176777i	$-0.0565670\pi$
$3$	0	0
$3$	0	0
$4$	7.00000	0.875000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 6.00000i	− 0.323970i	−0.986793	−	0.161985i	$-0.948210\pi$
$7$	− 6.00000i	− 0.323970i	0.986793	−	0.161985i	$-0.0517895\pi$
$8$	− 15.0000i	− 0.662913i
$9$	0	0
$10$	0	0
$11$	47.0000	1.28828	0.644138	−	0.764909i	$-0.277216\pi$
$11$	47.0000	1.28828	0.644138	+	0.764909i	$0.277216\pi$
$12$	0	0
$13$	5.00000i	0.106673i	0.998577	+	0.0533366i	$0.0169856\pi$
$13$	5.00000i	0.106673i	−0.998577	+	0.0533366i	$0.983014\pi$
$14$	−6.00000	−0.114541
$15$	0	0
$16$	41.0000	0.640625
$17$	131.000i	1.86895i	0.356027	+	0.934475i	$0.384131\pi$
$17$	131.000i	1.86895i	−0.356027	+	0.934475i	$0.615869\pi$
$18$	0	0
$19$	56.0000	0.676173	0.338086	−	0.941115i	$-0.390220\pi$
$19$	56.0000	0.676173	0.338086	+	0.941115i	$0.390220\pi$
$20$	0	0
$21$	0	0
$22$	− 47.0000i	− 0.455474i
$23$	3.00000i	0.0271975i	0.999908	+	0.0135988i	$0.00432876\pi$
$23$	3.00000i	0.0271975i	−0.999908	+	0.0135988i	$0.995671\pi$
$24$	0	0
$25$	0	0
$26$	5.00000	0.0377146
$27$	0	0
$28$	− 42.0000i	− 0.283473i
$29$	−157.000	−1.00532	−0.502658	−	0.864485i	$-0.667645\pi$
$29$	−157.000	−1.00532	−0.502658	+	0.864485i	$0.667645\pi$
$30$	0	0
$31$	225.000	1.30359	0.651793	−	0.758397i	$-0.274017\pi$
$31$	225.000	1.30359	0.651793	+	0.758397i	$0.274017\pi$
$32$	− 161.000i	− 0.889408i
$33$	0	0
$34$	131.000	0.660774
$35$	0	0
$36$	0	0
$37$	− 70.0000i	− 0.311025i	−0.987834	−	0.155513i	$-0.950297\pi$
$37$	− 70.0000i	− 0.311025i	0.987834	−	0.155513i	$-0.0497029\pi$
$38$	− 56.0000i	− 0.239063i
$39$	0	0
$40$	0	0
$41$	−140.000	−0.533276	−0.266638	−	0.963797i	$-0.585913\pi$
$41$	−140.000	−0.533276	−0.266638	+	0.963797i	$0.585913\pi$
$42$	0	0
$43$	− 397.000i	− 1.40795i	−0.710224	−	0.703976i	$-0.751406\pi$
$43$	− 397.000i	− 1.40795i	0.710224	−	0.703976i	$-0.248594\pi$
$44$	329.000	1.12724
$45$	0	0
$46$	3.00000	0.00961578
$47$	347.000i	1.07692i	0.842652	+	0.538459i	$0.180993\pi$
$47$	347.000i	1.07692i	−0.842652	+	0.538459i	$0.819007\pi$
$48$	0	0
$49$	307.000	0.895044
$50$	0	0
$51$	0	0
$52$	35.0000i	0.0933390i
$53$	4.00000i	0.0103668i	0.999987	+	0.00518342i	$0.00164994\pi$
$53$	4.00000i	0.0103668i	−0.999987	+	0.00518342i	$0.998350\pi$
$54$	0	0
$55$	0	0
$56$	−90.0000	−0.214763
$57$	0	0
$58$	157.000i	0.355433i
$59$	748.000	1.65053	0.825265	−	0.564745i	$-0.191026\pi$
$59$	748.000	1.65053	0.825265	+	0.564745i	$0.191026\pi$
$60$	0	0
$61$	−338.000	−0.709450	−0.354725	−	0.934971i	$-0.615426\pi$
$61$	−338.000	−0.709450	−0.354725	+	0.934971i	$0.615426\pi$
$62$	− 225.000i	− 0.460888i
$63$	0	0
$64$	167.000	0.326172
$65$	0	0
$66$	0	0
$67$	492.000i	0.897125i	0.893751	+	0.448562i	$0.148064\pi$
$67$	492.000i	0.897125i	−0.893751	+	0.448562i	$0.851936\pi$
$68$	917.000i	1.63533i
$69$	0	0
$70$	0	0
$71$	−32.0000	−0.0534888	−0.0267444	−	0.999642i	$-0.508514\pi$
$71$	−32.0000	−0.0534888	−0.0267444	+	0.999642i	$0.508514\pi$
$72$	0	0
$73$	− 970.000i	− 1.55520i	−0.628757	−	0.777602i	$-0.716436\pi$
$73$	− 970.000i	− 1.55520i	0.628757	−	0.777602i	$-0.283564\pi$
$74$	−70.0000	−0.109964
$75$	0	0
$76$	392.000	0.591651
$77$	− 282.000i	− 0.417362i
$78$	0	0
$79$	1257.00	1.79017	0.895086	−	0.445894i	$-0.147114\pi$
$79$	1257.00	1.79017	0.895086	+	0.445894i	$0.147114\pi$
$80$	0	0
$81$	0	0
$82$	140.000i	0.188542i
$83$	− 102.000i	− 0.134891i	−0.997723	−	0.0674455i	$-0.978515\pi$
$83$	− 102.000i	− 0.134891i	0.997723	−	0.0674455i	$-0.0214849\pi$
$84$	0	0
$85$	0	0
$86$	−397.000	−0.497786
$87$	0	0
$88$	− 705.000i	− 0.854014i
$89$	−1488.00	−1.77222	−0.886111	−	0.463474i	$-0.846603\pi$
$89$	−1488.00	−1.77222	−0.886111	+	0.463474i	$0.846603\pi$
$90$	0	0
$91$	30.0000	0.0345588
$92$	21.0000i	0.0237978i
$93$	0	0
$94$	347.000	0.380748
$95$	0	0
$96$	0	0
$97$	974.000i	1.01953i	0.860313	+	0.509767i	$0.170268\pi$
$97$	974.000i	1.01953i	−0.860313	+	0.509767i	$0.829732\pi$
$98$	− 307.000i	− 0.316446i
$99$	0	0
$100$	0	0
$101$	1335.00	1.31522	0.657611	−	0.753357i	$-0.271567\pi$
$101$	1335.00	1.31522	0.657611	+	0.753357i	$0.271567\pi$
$102$	0	0
$103$	− 686.000i	− 0.656248i	−0.944635	−	0.328124i	$-0.893584\pi$
$103$	− 686.000i	− 0.656248i	0.944635	−	0.328124i	$-0.106416\pi$
$104$	75.0000	0.0707150
$105$	0	0
$106$	4.00000	0.00366523
$107$	1098.00i	0.992034i	0.868313	+	0.496017i	$0.165205\pi$
$107$	1098.00i	0.992034i	−0.868313	+	0.496017i	$0.834795\pi$
$108$	0	0
$109$	700.000	0.615118	0.307559	−	0.951529i	$-0.400488\pi$
$109$	700.000	0.615118	0.307559	+	0.951529i	$0.400488\pi$
$110$	0	0
$111$	0	0
$112$	− 246.000i	− 0.207543i
$113$	− 1055.00i	− 0.878284i	−0.898418	−	0.439142i	$-0.855283\pi$
$113$	− 1055.00i	− 0.878284i	0.898418	−	0.439142i	$-0.144717\pi$
$114$	0	0
$115$	0	0
$116$	−1099.00	−0.879652
$117$	0	0
$118$	− 748.000i	− 0.583551i
$119$	786.000	0.605483
$120$	0	0
$121$	878.000	0.659654
$122$	338.000i	0.250829i
$123$	0	0
$124$	1575.00	1.14064
$125$	0	0
$126$	0	0
$127$	− 1646.00i	− 1.15007i	−0.818129	−	0.575035i	$-0.804988\pi$
$127$	− 1646.00i	− 1.15007i	0.818129	−	0.575035i	$-0.195012\pi$
$128$	− 1455.00i	− 1.00473i
$129$	0	0
$130$	0	0
$131$	1833.00	1.22252	0.611259	−	0.791430i	$-0.290663\pi$
$131$	1833.00	1.22252	0.611259	+	0.791430i	$0.290663\pi$
$132$	0	0
$133$	− 336.000i	− 0.219059i
$134$	492.000	0.317182
$135$	0	0
$136$	1965.00	1.23895
$137$	− 1098.00i	− 0.684733i	−0.939566	−	0.342367i	$-0.888771\pi$
$137$	− 1098.00i	− 0.684733i	0.939566	−	0.342367i	$-0.111229\pi$
$138$	0	0
$139$	1042.00	0.635837	0.317918	−	0.948118i	$-0.397016\pi$
$139$	1042.00	0.635837	0.317918	+	0.948118i	$0.397016\pi$
$140$	0	0
$141$	0	0
$142$	32.0000i	0.0189111i
$143$	235.000i	0.137424i
$144$	0	0
$145$	0	0
$146$	−970.000	−0.549848
$147$	0	0
$148$	− 490.000i	− 0.272147i
$149$	−2941.00	−1.61702	−0.808510	−	0.588482i	$-0.799726\pi$
$149$	−2941.00	−1.61702	−0.808510	+	0.588482i	$0.799726\pi$
$150$	0	0
$151$	511.000	0.275395	0.137697	−	0.990474i	$-0.456030\pi$
$151$	511.000	0.275395	0.137697	+	0.990474i	$0.456030\pi$
$152$	− 840.000i	− 0.448243i
$153$	0	0
$154$	−282.000	−0.147560
$155$	0	0
$156$	0	0
$157$	− 571.000i	− 0.290260i	−0.989413	−	0.145130i	$-0.953640\pi$
$157$	− 571.000i	− 0.290260i	0.989413	−	0.145130i	$-0.0463600\pi$
$158$	− 1257.00i	− 0.632921i
$159$	0	0
$160$	0	0
$161$	18.0000	0.00881117
$162$	0	0
$163$	− 713.000i	− 0.342616i	−0.985217	−	0.171308i	$-0.945201\pi$
$163$	− 713.000i	− 0.342616i	0.985217	−	0.171308i	$-0.0547994\pi$
$164$	−980.000	−0.466617
$165$	0	0
$166$	−102.000	−0.0476912
$167$	− 1596.00i	− 0.739534i	−0.929125	−	0.369767i	$-0.879437\pi$
$167$	− 1596.00i	− 0.739534i	0.929125	−	0.369767i	$-0.120563\pi$
$168$	0	0
$169$	2172.00	0.988621
$170$	0	0
$171$	0	0
$172$	− 2779.00i	− 1.23196i
$173$	4134.00i	1.81678i	0.418129	+	0.908388i	$0.362686\pi$
$173$	4134.00i	1.81678i	−0.418129	+	0.908388i	$0.637314\pi$
$174$	0	0
$175$	0	0
$176$	1927.00	0.825302
$177$	0	0
$178$	1488.00i	0.626575i
$179$	1828.00	0.763302	0.381651	−	0.924306i	$-0.375356\pi$
$179$	1828.00	0.763302	0.381651	+	0.924306i	$0.375356\pi$
$180$	0	0
$181$	−520.000	−0.213543	−0.106772	−	0.994284i	$-0.534051\pi$
$181$	−520.000	−0.213543	−0.106772	+	0.994284i	$0.534051\pi$
$182$	− 30.0000i	− 0.0122184i
$183$	0	0
$184$	45.0000	0.0180296
$185$	0	0
$186$	0	0
$187$	6157.00i	2.40772i
$188$	2429.00i	0.942303i
$189$	0	0
$190$	0	0
$191$	−4826.00	−1.82826	−0.914129	−	0.405424i	$-0.867124\pi$
$191$	−4826.00	−1.82826	−0.914129	+	0.405424i	$0.867124\pi$
$192$	0	0
$193$	− 1670.00i	− 0.622846i	−0.950271	−	0.311423i	$-0.899194\pi$
$193$	− 1670.00i	− 0.622846i	0.950271	−	0.311423i	$-0.100806\pi$
$194$	974.000	0.360459
$195$	0	0
$196$	2149.00	0.783163
$197$	1380.00i	0.499091i	0.968363	+	0.249546i	$0.0802812\pi$
$197$	1380.00i	0.499091i	−0.968363	+	0.249546i	$0.919719\pi$
$198$	0	0
$199$	−4357.00	−1.55206	−0.776029	−	0.630697i	$-0.782769\pi$
$199$	−4357.00	−1.55206	−0.776029	+	0.630697i	$0.782769\pi$
$200$	0	0
$201$	0	0
$202$	− 1335.00i	− 0.465001i
$203$	942.000i	0.325692i
$204$	0	0
$205$	0	0
$206$	−686.000	−0.232019
$207$	0	0
$208$	205.000i	0.0683375i
$209$	2632.00	0.871097
$210$	0	0
$211$	−4162.00	−1.35793	−0.678967	−	0.734169i	$-0.737572\pi$
$211$	−4162.00	−1.35793	−0.678967	+	0.734169i	$0.737572\pi$
$212$	28.0000i	0.00907098i
$213$	0	0
$214$	1098.00	0.350737
$215$	0	0
$216$	0	0
$217$	− 1350.00i	− 0.422322i
$218$	− 700.000i	− 0.217477i
$219$	0	0
$220$	0	0
$221$	−655.000	−0.199367
$222$	0	0
$223$	5956.00i	1.78853i	0.447533	+	0.894267i	$0.352303\pi$
$223$	5956.00i	1.78853i	−0.447533	+	0.894267i	$0.647697\pi$
$224$	−966.000	−0.288141
$225$	0	0
$226$	−1055.00	−0.310520
$227$	− 4940.00i	− 1.44440i	−0.691683	−	0.722201i	$-0.743130\pi$
$227$	− 4940.00i	− 1.44440i	0.691683	−	0.722201i	$-0.256870\pi$
$228$	0	0
$229$	−4344.00	−1.25354	−0.626768	−	0.779206i	$-0.715622\pi$
$229$	−4344.00	−1.25354	−0.626768	+	0.779206i	$0.715622\pi$
$230$	0	0
$231$	0	0
$232$	2355.00i	0.666437i
$233$	− 5202.00i	− 1.46264i	−0.682036	−	0.731318i	$-0.738905\pi$
$233$	− 5202.00i	− 1.46264i	0.682036	−	0.731318i	$-0.261095\pi$
$234$	0	0
$235$	0	0
$236$	5236.00	1.44421
$237$	0	0
$238$	− 786.000i	− 0.214071i
$239$	1546.00	0.418420	0.209210	−	0.977871i	$-0.432911\pi$
$239$	1546.00	0.418420	0.209210	+	0.977871i	$0.432911\pi$
$240$	0	0
$241$	−3659.00	−0.977995	−0.488998	−	0.872285i	$-0.662637\pi$
$241$	−3659.00	−0.977995	−0.488998	+	0.872285i	$0.662637\pi$
$242$	− 878.000i	− 0.233223i
$243$	0	0
$244$	−2366.00	−0.620769
$245$	0	0
$246$	0	0
$247$	280.000i	0.0721294i
$248$	− 3375.00i	− 0.864164i
$249$	0	0
$250$	0	0
$251$	−1221.00	−0.307047	−0.153524	−	0.988145i	$-0.549062\pi$
$251$	−1221.00	−0.307047	−0.153524	+	0.988145i	$0.549062\pi$
$252$	0	0
$253$	141.000i	0.0350379i
$254$	−1646.00	−0.406611
$255$	0	0
$256$	−119.000	−0.0290527
$257$	6255.00i	1.51820i	0.650977	+	0.759098i	$0.274360\pi$
$257$	6255.00i	1.51820i	−0.650977	+	0.759098i	$0.725640\pi$
$258$	0	0
$259$	−420.000	−0.100763
$260$	0	0
$261$	0	0
$262$	− 1833.00i	− 0.432226i
$263$	836.000i	0.196007i	0.995186	+	0.0980037i	$0.0312457\pi$
$263$	836.000i	0.196007i	−0.995186	+	0.0980037i	$0.968754\pi$
$264$	0	0
$265$	0	0
$266$	−336.000	−0.0774492
$267$	0	0
$268$	3444.00i	0.784984i
$269$	−2231.00	−0.505675	−0.252837	−	0.967509i	$-0.581364\pi$
$269$	−2231.00	−0.505675	−0.252837	+	0.967509i	$0.581364\pi$
$270$	0	0
$271$	−4832.00	−1.08311	−0.541556	−	0.840665i	$-0.682164\pi$
$271$	−4832.00	−1.08311	−0.541556	+	0.840665i	$0.682164\pi$
$272$	5371.00i	1.19730i
$273$	0	0
$274$	−1098.00	−0.242090
$275$	0	0
$276$	0	0
$277$	6450.00i	1.39907i	0.714597	+	0.699536i	$0.246610\pi$
$277$	6450.00i	1.39907i	−0.714597	+	0.699536i	$0.753390\pi$
$278$	− 1042.00i	− 0.224802i
$279$	0	0
$280$	0	0
$281$	1050.00	0.222910	0.111455	−	0.993769i	$-0.464449\pi$
$281$	1050.00	0.222910	0.111455	+	0.993769i	$0.464449\pi$
$282$	0	0
$283$	1584.00i	0.332717i	0.986065	+	0.166359i	$0.0532010\pi$
$283$	1584.00i	0.332717i	−0.986065	+	0.166359i	$0.946799\pi$
$284$	−224.000	−0.0468027
$285$	0	0
$286$	235.000	0.0485869
$287$	840.000i	0.172765i
$288$	0	0
$289$	−12248.0	−2.49298
$290$	0	0
$291$	0	0
$292$	− 6790.00i	− 1.36080i
$293$	6594.00i	1.31476i	0.753558	+	0.657382i	$0.228336\pi$
$293$	6594.00i	1.31476i	−0.753558	+	0.657382i	$0.771664\pi$
$294$	0	0
$295$	0	0
$296$	−1050.00	−0.206182
$297$	0	0
$298$	2941.00i	0.571703i
$299$	−15.0000	−0.00290125
$300$	0	0
$301$	−2382.00	−0.456134
$302$	− 511.000i	− 0.0973667i
$303$	0	0
$304$	2296.00	0.433173
$305$	0	0
$306$	0	0
$307$	− 4343.00i	− 0.807388i	−0.914894	−	0.403694i	$-0.867726\pi$
$307$	− 4343.00i	− 0.807388i	0.914894	−	0.403694i	$-0.132274\pi$
$308$	− 1974.00i	− 0.365192i
$309$	0	0
$310$	0	0
$311$	2124.00	0.387270	0.193635	−	0.981074i	$-0.437972\pi$
$311$	2124.00	0.387270	0.193635	+	0.981074i	$0.437972\pi$
$312$	0	0
$313$	7516.00i	1.35728i	0.734470	+	0.678641i	$0.237431\pi$
$313$	7516.00i	1.35728i	−0.734470	+	0.678641i	$0.762569\pi$
$314$	−571.000	−0.102622
$315$	0	0
$316$	8799.00	1.56640
$317$	6880.00i	1.21899i	0.792791	+	0.609494i	$0.208627\pi$
$317$	6880.00i	1.21899i	−0.792791	+	0.609494i	$0.791373\pi$
$318$	0	0
$319$	−7379.00	−1.29512
$320$	0	0
$321$	0	0
$322$	− 18.0000i	− 0.00311522i
$323$	7336.00i	1.26373i
$324$	0	0
$325$	0	0
$326$	−713.000	−0.121133
$327$	0	0
$328$	2100.00i	0.353516i
$329$	2082.00	0.348889
$330$	0	0
$331$	−4986.00	−0.827962	−0.413981	−	0.910286i	$-0.635862\pi$
$331$	−4986.00	−0.827962	−0.413981	+	0.910286i	$0.635862\pi$
$332$	− 714.000i	− 0.118030i
$333$	0	0
$334$	−1596.00	−0.261465
$335$	0	0
$336$	0	0
$337$	904.000i	0.146125i	0.997327	+	0.0730623i	$0.0232772\pi$
$337$	904.000i	0.146125i	−0.997327	+	0.0730623i	$0.976723\pi$
$338$	− 2172.00i	− 0.349530i
$339$	0	0
$340$	0	0
$341$	10575.0	1.67938
$342$	0	0
$343$	− 3900.00i	− 0.613936i
$344$	−5955.00	−0.933349
$345$	0	0
$346$	4134.00	0.642327
$347$	− 8860.00i	− 1.37069i	−0.728218	−	0.685345i	$-0.759651\pi$
$347$	− 8860.00i	− 1.37069i	0.728218	−	0.685345i	$-0.240349\pi$
$348$	0	0
$349$	4454.00	0.683144	0.341572	−	0.939856i	$-0.389041\pi$
$349$	4454.00	0.683144	0.341572	+	0.939856i	$0.389041\pi$
$350$	0	0
$351$	0	0
$352$	− 7567.00i	− 1.14580i
$353$	− 8781.00i	− 1.32398i	−0.749512	−	0.661991i	$-0.769712\pi$
$353$	− 8781.00i	− 1.32398i	0.749512	−	0.661991i	$-0.230288\pi$
$354$	0	0
$355$	0	0
$356$	−10416.0	−1.55069
$357$	0	0
$358$	− 1828.00i	− 0.269868i
$359$	2928.00	0.430457	0.215228	−	0.976564i	$-0.430950\pi$
$359$	2928.00	0.430457	0.215228	+	0.976564i	$0.430950\pi$
$360$	0	0
$361$	−3723.00	−0.542790
$362$	520.000i	0.0754989i
$363$	0	0
$364$	210.000	0.0302390
$365$	0	0
$366$	0	0
$367$	9102.00i	1.29461i	0.762233	+	0.647303i	$0.224103\pi$
$367$	9102.00i	1.29461i	−0.762233	+	0.647303i	$0.775897\pi$
$368$	123.000i	0.0174234i
$369$	0	0
$370$	0	0
$371$	24.0000	0.00335854
$372$	0	0
$373$	8183.00i	1.13592i	0.823055	+	0.567962i	$0.192268\pi$
$373$	8183.00i	1.13592i	−0.823055	+	0.567962i	$0.807732\pi$
$374$	6157.00	0.851259
$375$	0	0
$376$	5205.00	0.713903
$377$	− 785.000i	− 0.107240i
$378$	0	0
$379$	−6136.00	−0.831623	−0.415812	−	0.909451i	$-0.636502\pi$
$379$	−6136.00	−0.831623	−0.415812	+	0.909451i	$0.636502\pi$
$380$	0	0
$381$	0	0
$382$	4826.00i	0.646386i
$383$	5643.00i	0.752856i	0.926446	+	0.376428i	$0.122848\pi$
$383$	5643.00i	0.752856i	−0.926446	+	0.376428i	$0.877152\pi$
$384$	0	0
$385$	0	0
$386$	−1670.00	−0.220209
$387$	0	0
$388$	6818.00i	0.892092i
$389$	8991.00	1.17188	0.585941	−	0.810354i	$-0.300725\pi$
$389$	8991.00	1.17188	0.585941	+	0.810354i	$0.300725\pi$
$390$	0	0
$391$	−393.000	−0.0508309
$392$	− 4605.00i	− 0.593336i
$393$	0	0
$394$	1380.00	0.176455
$395$	0	0
$396$	0	0
$397$	− 12449.0i	− 1.57380i	−0.617082	−	0.786898i	$-0.711686\pi$
$397$	− 12449.0i	− 1.57380i	0.617082	−	0.786898i	$-0.288314\pi$
$398$	4357.00i	0.548735i
$399$	0	0
$400$	0	0
$401$	−8076.00	−1.00573	−0.502863	−	0.864366i	$-0.667720\pi$
$401$	−8076.00	−1.00573	−0.502863	+	0.864366i	$0.667720\pi$
$402$	0	0
$403$	1125.00i	0.139058i
$404$	9345.00	1.15082
$405$	0	0
$406$	942.000	0.115149
$407$	− 3290.00i	− 0.400686i
$408$	0	0
$409$	2833.00	0.342501	0.171250	−	0.985228i	$-0.445219\pi$
$409$	2833.00	0.342501	0.171250	+	0.985228i	$0.445219\pi$
$410$	0	0
$411$	0	0
$412$	− 4802.00i	− 0.574217i
$413$	− 4488.00i	− 0.534722i
$414$	0	0
$415$	0	0
$416$	805.000	0.0948759
$417$	0	0
$418$	− 2632.00i	− 0.307979i
$419$	−4777.00	−0.556973	−0.278487	−	0.960440i	$-0.589833\pi$
$419$	−4777.00	−0.556973	−0.278487	+	0.960440i	$0.589833\pi$
$420$	0	0
$421$	−6464.00	−0.748304	−0.374152	−	0.927367i	$-0.622066\pi$
$421$	−6464.00	−0.748304	−0.374152	+	0.927367i	$0.622066\pi$
$422$	4162.00i	0.480102i
$423$	0	0
$424$	60.0000	0.00687231
$425$	0	0
$426$	0	0
$427$	2028.00i	0.229840i
$428$	7686.00i	0.868030i
$429$	0	0
$430$	0	0
$431$	−10680.0	−1.19359	−0.596795	−	0.802394i	$-0.703560\pi$
$431$	−10680.0	−1.19359	−0.596795	+	0.802394i	$0.703560\pi$
$432$	0	0
$433$	− 11566.0i	− 1.28366i	−0.766845	−	0.641832i	$-0.778175\pi$
$433$	− 11566.0i	− 1.28366i	0.766845	−	0.641832i	$-0.221825\pi$
$434$	−1350.00	−0.149314
$435$	0	0
$436$	4900.00	0.538228
$437$	168.000i	0.0183902i
$438$	0	0
$439$	1448.00	0.157424	0.0787122	−	0.996897i	$-0.474919\pi$
$439$	1448.00	0.157424	0.0787122	+	0.996897i	$0.474919\pi$
$440$	0	0
$441$	0	0
$442$	655.000i	0.0704868i
$443$	− 2376.00i	− 0.254824i	−0.991850	−	0.127412i	$-0.959333\pi$
$443$	− 2376.00i	− 0.254824i	0.991850	−	0.127412i	$-0.0406671\pi$
$444$	0	0
$445$	0	0
$446$	5956.00	0.632343
$447$	0	0
$448$	− 1002.00i	− 0.105670i
$449$	−14894.0	−1.56546	−0.782730	−	0.622362i	$-0.786173\pi$
$449$	−14894.0	−1.56546	−0.782730	+	0.622362i	$0.786173\pi$
$450$	0	0
$451$	−6580.00	−0.687007
$452$	− 7385.00i	− 0.768498i
$453$	0	0
$454$	−4940.00	−0.510673
$455$	0	0
$456$	0	0
$457$	16204.0i	1.65862i	0.558786	+	0.829312i	$0.311267\pi$
$457$	16204.0i	1.65862i	−0.558786	+	0.829312i	$0.688733\pi$
$458$	4344.00i	0.443192i
$459$	0	0
$460$	0	0
$461$	5082.00	0.513432	0.256716	−	0.966487i	$-0.417359\pi$
$461$	5082.00	0.513432	0.256716	+	0.966487i	$0.417359\pi$
$462$	0	0
$463$	10326.0i	1.03648i	0.855235	+	0.518240i	$0.173412\pi$
$463$	10326.0i	1.03648i	−0.855235	+	0.518240i	$0.826588\pi$
$464$	−6437.00	−0.644031
$465$	0	0
$466$	−5202.00	−0.517120
$467$	− 4184.00i	− 0.414588i	−0.978279	−	0.207294i	$-0.933534\pi$
$467$	− 4184.00i	− 0.414588i	0.978279	−	0.207294i	$-0.0664656\pi$
$468$	0	0
$469$	2952.00	0.290641
$470$	0	0
$471$	0	0
$472$	− 11220.0i	− 1.09416i
$473$	− 18659.0i	− 1.81383i
$474$	0	0
$475$	0	0
$476$	5502.00	0.529798
$477$	0	0
$478$	− 1546.00i	− 0.147934i
$479$	−15576.0	−1.48577	−0.742887	−	0.669417i	$-0.766544\pi$
$479$	−15576.0	−1.48577	−0.742887	+	0.669417i	$0.766544\pi$
$480$	0	0
$481$	350.000	0.0331780
$482$	3659.00i	0.345774i
$483$	0	0
$484$	6146.00	0.577198
$485$	0	0
$486$	0	0
$487$	10220.0i	0.950949i	0.879729	+	0.475475i	$0.157724\pi$
$487$	10220.0i	0.950949i	−0.879729	+	0.475475i	$0.842276\pi$
$488$	5070.00i	0.470304i
$489$	0	0
$490$	0	0
$491$	2692.00	0.247430	0.123715	−	0.992318i	$-0.460519\pi$
$491$	2692.00	0.247430	0.123715	+	0.992318i	$0.460519\pi$
$492$	0	0
$493$	− 20567.0i	− 1.87889i
$494$	280.000	0.0255016
$495$	0	0
$496$	9225.00	0.835110
$497$	192.000i	0.0173287i
$498$	0	0
$499$	−5764.00	−0.517098	−0.258549	−	0.965998i	$-0.583244\pi$
$499$	−5764.00	−0.517098	−0.258549	+	0.965998i	$0.583244\pi$
$500$	0	0
$501$	0	0
$502$	1221.00i	0.108558i
$503$	− 2437.00i	− 0.216025i	−0.994150	−	0.108012i	$-0.965551\pi$
$503$	− 2437.00i	− 0.216025i	0.994150	−	0.108012i	$-0.0344486\pi$
$504$	0	0
$505$	0	0
$506$	141.000	0.0123878
$507$	0	0
$508$	− 11522.0i	− 1.00631i
$509$	−5849.00	−0.509337	−0.254668	−	0.967028i	$-0.581966\pi$
$509$	−5849.00	−0.509337	−0.254668	+	0.967028i	$0.581966\pi$
$510$	0	0
$511$	−5820.00	−0.503839
$512$	− 11521.0i	− 0.994455i
$513$	0	0
$514$	6255.00	0.536763
$515$	0	0
$516$	0	0
$517$	16309.0i	1.38737i
$518$	420.000i	0.0356250i
$519$	0	0
$520$	0	0
$521$	17032.0	1.43222	0.716109	−	0.697989i	$-0.245921\pi$
$521$	17032.0	1.43222	0.716109	+	0.697989i	$0.245921\pi$
$522$	0	0
$523$	− 4147.00i	− 0.346722i	−0.984858	−	0.173361i	$-0.944537\pi$
$523$	− 4147.00i	− 0.346722i	0.984858	−	0.173361i	$-0.0554627\pi$
$524$	12831.0	1.06970
$525$	0	0
$526$	836.000	0.0692991
$527$	29475.0i	2.43634i
$528$	0	0
$529$	12158.0	0.999260
$530$	0	0
$531$	0	0
$532$	− 2352.00i	− 0.191677i
$533$	− 700.000i	− 0.0568862i
$534$	0	0
$535$	0	0
$536$	7380.00	0.594715
$537$	0	0
$538$	2231.00i	0.178783i
$539$	14429.0	1.15306
$540$	0	0
$541$	−3942.00	−0.313271	−0.156636	−	0.987656i	$-0.550065\pi$
$541$	−3942.00	−0.313271	−0.156636	+	0.987656i	$0.550065\pi$
$542$	4832.00i	0.382938i
$543$	0	0
$544$	21091.0	1.66226
$545$	0	0
$546$	0	0
$547$	− 13751.0i	− 1.07486i	−0.843307	−	0.537432i	$-0.819395\pi$
$547$	− 13751.0i	− 1.07486i	0.843307	−	0.537432i	$-0.180605\pi$
$548$	− 7686.00i	− 0.599142i
$549$	0	0
$550$	0	0
$551$	−8792.00	−0.679767
$552$	0	0
$553$	− 7542.00i	− 0.579961i
$554$	6450.00	0.494647
$555$	0	0
$556$	7294.00	0.556357
$557$	− 7944.00i	− 0.604305i	−0.953260	−	0.302153i	$-0.902295\pi$
$557$	− 7944.00i	− 0.604305i	0.953260	−	0.302153i	$-0.0977052\pi$
$558$	0	0
$559$	1985.00	0.150191
$560$	0	0
$561$	0	0
$562$	− 1050.00i	− 0.0788106i
$563$	− 6702.00i	− 0.501697i	−0.968026	−	0.250849i	$-0.919290\pi$
$563$	− 6702.00i	− 0.501697i	0.968026	−	0.250849i	$-0.0807097\pi$
$564$	0	0
$565$	0	0
$566$	1584.00	0.117633
$567$	0	0
$568$	480.000i	0.0354584i
$569$	−2760.00	−0.203348	−0.101674	−	0.994818i	$-0.532420\pi$
$569$	−2760.00	−0.203348	−0.101674	+	0.994818i	$0.532420\pi$
$570$	0	0
$571$	8930.00	0.654481	0.327241	−	0.944941i	$-0.393881\pi$
$571$	8930.00	0.654481	0.327241	+	0.944941i	$0.393881\pi$
$572$	1645.00i	0.120246i
$573$	0	0
$574$	840.000	0.0610817
$575$	0	0
$576$	0	0
$577$	6944.00i	0.501010i	0.968115	+	0.250505i	$0.0805966\pi$
$577$	6944.00i	0.501010i	−0.968115	+	0.250505i	$0.919403\pi$
$578$	12248.0i	0.881401i
$579$	0	0
$580$	0	0
$581$	−612.000	−0.0437006
$582$	0	0
$583$	188.000i	0.0133553i
$584$	−14550.0	−1.03096
$585$	0	0
$586$	6594.00	0.464839
$587$	4206.00i	0.295741i	0.989007	+	0.147871i	$0.0472419\pi$
$587$	4206.00i	0.295741i	−0.989007	+	0.147871i	$0.952758\pi$
$588$	0	0
$589$	12600.0	0.881450
$590$	0	0
$591$	0	0
$592$	− 2870.00i	− 0.199250i
$593$	− 6571.00i	− 0.455040i	−0.973773	−	0.227520i	$-0.926938\pi$
$593$	− 6571.00i	− 0.455040i	0.973773	−	0.227520i	$-0.0730617\pi$
$594$	0	0
$595$	0	0
$596$	−20587.0	−1.41489
$597$	0	0
$598$	15.0000i	0.00102575i
$599$	−9490.00	−0.647330	−0.323665	−	0.946172i	$-0.604915\pi$
$599$	−9490.00	−0.647330	−0.323665	+	0.946172i	$0.604915\pi$
$600$	0	0
$601$	11861.0	0.805025	0.402513	−	0.915414i	$-0.368137\pi$
$601$	11861.0	0.805025	0.402513	+	0.915414i	$0.368137\pi$
$602$	2382.00i	0.161268i
$603$	0	0
$604$	3577.00	0.240970
$605$	0	0
$606$	0	0
$607$	− 518.000i	− 0.0346375i	−0.999850	−	0.0173188i	$-0.994487\pi$
$607$	− 518.000i	− 0.0346375i	0.999850	−	0.0173188i	$-0.00551301\pi$
$608$	− 9016.00i	− 0.601393i
$609$	0	0
$610$	0	0
$611$	−1735.00	−0.114878
$612$	0	0
$613$	− 15163.0i	− 0.999067i	−0.866295	−	0.499533i	$-0.833505\pi$
$613$	− 15163.0i	− 0.999067i	0.866295	−	0.499533i	$-0.166495\pi$
$614$	−4343.00	−0.285455
$615$	0	0
$616$	−4230.00	−0.276675
$617$	− 19011.0i	− 1.24044i	−0.784426	−	0.620222i	$-0.787042\pi$
$617$	− 19011.0i	− 1.24044i	0.784426	−	0.620222i	$-0.212958\pi$
$618$	0	0
$619$	7906.00	0.513359	0.256679	−	0.966497i	$-0.417372\pi$
$619$	7906.00	0.513359	0.256679	+	0.966497i	$0.417372\pi$
$620$	0	0
$621$	0	0
$622$	− 2124.00i	− 0.136921i
$623$	8928.00i	0.574146i
$624$	0	0
$625$	0	0
$626$	7516.00	0.479872
$627$	0	0
$628$	− 3997.00i	− 0.253977i
$629$	9170.00	0.581291
$630$	0	0
$631$	−3416.00	−0.215513	−0.107757	−	0.994177i	$-0.534367\pi$
$631$	−3416.00	−0.215513	−0.107757	+	0.994177i	$0.534367\pi$
$632$	− 18855.0i	− 1.18673i
$633$	0	0
$634$	6880.00	0.430977
$635$	0	0
$636$	0	0
$637$	1535.00i	0.0954771i
$638$	7379.00i	0.457896i
$639$	0	0
$640$	0	0
$641$	−4830.00	−0.297619	−0.148809	−	0.988866i	$-0.547544\pi$
$641$	−4830.00	−0.297619	−0.148809	+	0.988866i	$0.547544\pi$
$642$	0	0
$643$	− 12549.0i	− 0.769649i	−0.922990	−	0.384824i	$-0.874262\pi$
$643$	− 12549.0i	− 0.769649i	0.922990	−	0.384824i	$-0.125738\pi$
$644$	126.000	0.00770978
$645$	0	0
$646$	7336.00	0.446797
$647$	− 8164.00i	− 0.496074i	−0.968751	−	0.248037i	$-0.920215\pi$
$647$	− 8164.00i	− 0.496074i	0.968751	−	0.248037i	$-0.0797855\pi$
$648$	0	0
$649$	35156.0	2.12634
$650$	0	0
$651$	0	0
$652$	− 4991.00i	− 0.299789i
$653$	23768.0i	1.42437i	0.701992	+	0.712185i	$0.252294\pi$
$653$	23768.0i	1.42437i	−0.701992	+	0.712185i	$0.747706\pi$
$654$	0	0
$655$	0	0
$656$	−5740.00	−0.341630
$657$	0	0
$658$	− 2082.00i	− 0.123351i
$659$	21472.0	1.26924	0.634621	−	0.772824i	$-0.281156\pi$
$659$	21472.0	1.26924	0.634621	+	0.772824i	$0.281156\pi$
$660$	0	0
$661$	12982.0	0.763905	0.381953	−	0.924182i	$-0.375252\pi$
$661$	12982.0	0.763905	0.381953	+	0.924182i	$0.375252\pi$
$662$	4986.00i	0.292729i
$663$	0	0
$664$	−1530.00	−0.0894210
$665$	0	0
$666$	0	0
$667$	− 471.000i	− 0.0273421i
$668$	− 11172.0i	− 0.647092i
$669$	0	0
$670$	0	0
$671$	−15886.0	−0.913968
$672$	0	0
$673$	6006.00i	0.344003i	0.985097	+	0.172002i	$0.0550235\pi$
$673$	6006.00i	0.344003i	−0.985097	+	0.172002i	$0.944977\pi$
$674$	904.000	0.0516629
$675$	0	0
$676$	15204.0	0.865043
$677$	− 1164.00i	− 0.0660800i	−0.999454	−	0.0330400i	$-0.989481\pi$
$677$	− 1164.00i	− 0.0660800i	0.999454	−	0.0330400i	$-0.0105189\pi$
$678$	0	0
$679$	5844.00	0.330298
$680$	0	0
$681$	0	0
$682$	− 10575.0i	− 0.593750i
$683$	26496.0i	1.48439i	0.670182	+	0.742197i	$0.266216\pi$
$683$	26496.0i	1.48439i	−0.670182	+	0.742197i	$0.733784\pi$
$684$	0	0
$685$	0	0
$686$	−3900.00	−0.217059
$687$	0	0
$688$	− 16277.0i	− 0.901969i
$689$	−20.0000	−0.00110586
$690$	0	0
$691$	17110.0	0.941961	0.470981	−	0.882144i	$-0.343900\pi$
$691$	17110.0	0.941961	0.470981	+	0.882144i	$0.343900\pi$
$692$	28938.0i	1.58968i
$693$	0	0
$694$	−8860.00	−0.484612
$695$	0	0
$696$	0	0
$697$	− 18340.0i	− 0.996667i
$698$	− 4454.00i	− 0.241528i
$699$	0	0
$700$	0	0
$701$	−30251.0	−1.62991	−0.814953	−	0.579527i	$-0.803237\pi$
$701$	−30251.0	−1.62991	−0.814953	+	0.579527i	$0.803237\pi$
$702$	0	0
$703$	− 3920.00i	− 0.210307i
$704$	7849.00	0.420199
$705$	0	0
$706$	−8781.00	−0.468098
$707$	− 8010.00i	− 0.426092i
$708$	0	0
$709$	−18820.0	−0.996897	−0.498448	−	0.866919i	$-0.666097\pi$
$709$	−18820.0	−0.996897	−0.498448	+	0.866919i	$0.666097\pi$
$710$	0	0
$711$	0	0
$712$	22320.0i	1.17483i
$713$	675.000i	0.0354543i
$714$	0	0
$715$	0	0
$716$	12796.0	0.667890
$717$	0	0
$718$	− 2928.00i	− 0.152189i
$719$	31890.0	1.65410	0.827049	−	0.562130i	$-0.190018\pi$
$719$	31890.0	1.65410	0.827049	+	0.562130i	$0.190018\pi$
$720$	0	0
$721$	−4116.00	−0.212605
$722$	3723.00i	0.191905i
$723$	0	0
$724$	−3640.00	−0.186850
$725$	0	0
$726$	0	0
$727$	− 11452.0i	− 0.584224i	−0.956384	−	0.292112i	$-0.905642\pi$
$727$	− 11452.0i	− 0.584224i	0.956384	−	0.292112i	$-0.0943581\pi$
$728$	− 450.000i	− 0.0229095i
$729$	0	0
$730$	0	0
$731$	52007.0	2.63139
$732$	0	0
$733$	− 7094.00i	− 0.357466i	−0.983898	−	0.178733i	$-0.942800\pi$
$733$	− 7094.00i	− 0.357466i	0.983898	−	0.178733i	$-0.0571999\pi$
$734$	9102.00	0.457712
$735$	0	0
$736$	483.000	0.0241897
$737$	23124.0i	1.15574i
$738$	0	0
$739$	3200.00	0.159288	0.0796440	−	0.996823i	$-0.474622\pi$
$739$	3200.00	0.159288	0.0796440	+	0.996823i	$0.474622\pi$
$740$	0	0
$741$	0	0
$742$	− 24.0000i	− 0.00118742i
$743$	− 20831.0i	− 1.02855i	−0.857624	−	0.514277i	$-0.828060\pi$
$743$	− 20831.0i	− 1.02855i	0.857624	−	0.514277i	$-0.171940\pi$
$744$	0	0
$745$	0	0
$746$	8183.00	0.401610
$747$	0	0
$748$	43099.0i	2.10676i
$749$	6588.00	0.321389
$750$	0	0
$751$	−15605.0	−0.758235	−0.379118	−	0.925349i	$-0.623772\pi$
$751$	−15605.0	−0.758235	−0.379118	+	0.925349i	$0.623772\pi$
$752$	14227.0i	0.689901i
$753$	0	0
$754$	−785.000	−0.0379151
$755$	0	0
$756$	0	0
$757$	21349.0i	1.02502i	0.858680	+	0.512512i	$0.171285\pi$
$757$	21349.0i	1.02502i	−0.858680	+	0.512512i	$0.828715\pi$
$758$	6136.00i	0.294023i
$759$	0	0
$760$	0	0
$761$	−3702.00	−0.176343	−0.0881717	−	0.996105i	$-0.528102\pi$
$761$	−3702.00	−0.176343	−0.0881717	+	0.996105i	$0.528102\pi$
$762$	0	0
$763$	− 4200.00i	− 0.199279i
$764$	−33782.0	−1.59972
$765$	0	0
$766$	5643.00	0.266175
$767$	3740.00i	0.176067i
$768$	0	0
$769$	1393.00	0.0653223	0.0326612	−	0.999466i	$-0.489602\pi$
$769$	1393.00	0.0653223	0.0326612	+	0.999466i	$0.489602\pi$
$770$	0	0
$771$	0	0
$772$	− 11690.0i	− 0.544990i
$773$	− 6906.00i	− 0.321334i	−0.987009	−	0.160667i	$-0.948635\pi$
$773$	− 6906.00i	− 0.321334i	0.987009	−	0.160667i	$-0.0513646\pi$
$774$	0	0
$775$	0	0
$776$	14610.0	0.675861
$777$	0	0
$778$	− 8991.00i	− 0.414323i
$779$	−7840.00	−0.360587
$780$	0	0
$781$	−1504.00	−0.0689083
$782$	393.000i	0.0179714i
$783$	0	0
$784$	12587.0	0.573387
$785$	0	0
$786$	0	0
$787$	30493.0i	1.38114i	0.723265	+	0.690571i	$0.242641\pi$
$787$	30493.0i	1.38114i	−0.723265	+	0.690571i	$0.757359\pi$
$788$	9660.00i	0.436705i
$789$	0	0
$790$	0	0
$791$	−6330.00	−0.284537
$792$	0	0
$793$	− 1690.00i	− 0.0756793i
$794$	−12449.0	−0.556421
$795$	0	0
$796$	−30499.0	−1.35805
$797$	− 33488.0i	− 1.48834i	−0.667991	−	0.744169i	$-0.732846\pi$
$797$	− 33488.0i	− 1.48834i	0.667991	−	0.744169i	$-0.267154\pi$
$798$	0	0
$799$	−45457.0	−2.01271
$800$	0	0
$801$	0	0
$802$	8076.00i	0.355578i
$803$	− 45590.0i	− 2.00353i
$804$	0	0
$805$	0	0
$806$	1125.00	0.0491643
$807$	0	0
$808$	− 20025.0i	− 0.871878i
$809$	−15304.0	−0.665093	−0.332546	−	0.943087i	$-0.607908\pi$
$809$	−15304.0	−0.665093	−0.332546	+	0.943087i	$0.607908\pi$
$810$	0	0
$811$	−40122.0	−1.73721	−0.868603	−	0.495509i	$-0.834982\pi$
$811$	−40122.0	−1.73721	−0.868603	+	0.495509i	$0.834982\pi$
$812$	6594.00i	0.284980i
$813$	0	0
$814$	−3290.00	−0.141664
$815$	0	0
$816$	0	0
$817$	− 22232.0i	− 0.952019i
$818$	− 2833.00i	− 0.121092i
$819$	0	0
$820$	0	0
$821$	−25098.0	−1.06690	−0.533451	−	0.845831i	$-0.679105\pi$
$821$	−25098.0	−1.06690	−0.533451	+	0.845831i	$0.679105\pi$
$822$	0	0
$823$	43492.0i	1.84208i	0.389462	+	0.921042i	$0.372661\pi$
$823$	43492.0i	1.84208i	−0.389462	+	0.921042i	$0.627339\pi$
$824$	−10290.0	−0.435035
$825$	0	0
$826$	−4488.00	−0.189053
$827$	− 11206.0i	− 0.471186i	−0.971852	−	0.235593i	$-0.924297\pi$
$827$	− 11206.0i	− 0.471186i	0.971852	−	0.235593i	$-0.0757032\pi$
$828$	0	0
$829$	23964.0	1.00399	0.501993	−	0.864872i	$-0.332600\pi$
$829$	23964.0	1.00399	0.501993	+	0.864872i	$0.332600\pi$
$830$	0	0
$831$	0	0
$832$	835.000i	0.0347938i
$833$	40217.0i	1.67279i
$834$	0	0
$835$	0	0
$836$	18424.0	0.762210
$837$	0	0
$838$	4777.00i	0.196920i
$839$	−34606.0	−1.42399	−0.711997	−	0.702182i	$-0.752209\pi$
$839$	−34606.0	−1.42399	−0.711997	+	0.702182i	$0.752209\pi$
$840$	0	0
$841$	260.000	0.0106605
$842$	6464.00i	0.264566i
$843$	0	0
$844$	−29134.0	−1.18819
$845$	0	0
$846$	0	0
$847$	− 5268.00i	− 0.213708i
$848$	164.000i	0.00664125i
$849$	0	0
$850$	0	0
$851$	210.000	0.00845912
$852$	0	0
$853$	18477.0i	0.741665i	0.928700	+	0.370833i	$0.120928\pi$
$853$	18477.0i	0.741665i	−0.928700	+	0.370833i	$0.879072\pi$
$854$	2028.00	0.0812608
$855$	0	0
$856$	16470.0	0.657632
$857$	41342.0i	1.64786i	0.566692	+	0.823930i	$0.308223\pi$
$857$	41342.0i	1.64786i	−0.566692	+	0.823930i	$0.691777\pi$
$858$	0	0
$859$	−21898.0	−0.869791	−0.434895	−	0.900481i	$-0.643215\pi$
$859$	−21898.0	−0.869791	−0.434895	+	0.900481i	$0.643215\pi$
$860$	0	0
$861$	0	0
$862$	10680.0i	0.421998i
$863$	− 18487.0i	− 0.729206i	−0.931163	−	0.364603i	$-0.881205\pi$
$863$	− 18487.0i	− 0.729206i	0.931163	−	0.364603i	$-0.118795\pi$
$864$	0	0
$865$	0	0
$866$	−11566.0	−0.453844
$867$	0	0
$868$	− 9450.00i	− 0.369532i
$869$	59079.0	2.30623
$870$	0	0
$871$	−2460.00	−0.0956991
$872$	− 10500.0i	− 0.407769i
$873$	0	0
$874$	168.000	0.00650193
$875$	0	0
$876$	0	0
$877$	7593.00i	0.292357i	0.989258	+	0.146179i	$0.0466974\pi$
$877$	7593.00i	0.292357i	−0.989258	+	0.146179i	$0.953303\pi$
$878$	− 1448.00i	− 0.0556579i
$879$	0	0
$880$	0	0
$881$	3038.00	0.116178	0.0580890	−	0.998311i	$-0.481499\pi$
$881$	3038.00	0.116178	0.0580890	+	0.998311i	$0.481499\pi$
$882$	0	0
$883$	16732.0i	0.637686i	0.947808	+	0.318843i	$0.103294\pi$
$883$	16732.0i	0.637686i	−0.947808	+	0.318843i	$0.896706\pi$
$884$	−4585.00	−0.174446
$885$	0	0
$886$	−2376.00	−0.0900940
$887$	8031.00i	0.304007i	0.988380	+	0.152004i	$0.0485726\pi$
$887$	8031.00i	0.304007i	−0.988380	+	0.152004i	$0.951427\pi$
$888$	0	0
$889$	−9876.00	−0.372588
$890$	0	0
$891$	0	0
$892$	41692.0i	1.56497i
$893$	19432.0i	0.728183i
$894$	0	0
$895$	0	0
$896$	−8730.00	−0.325501
$897$	0	0
$898$	14894.0i	0.553474i
$899$	−35325.0	−1.31052
$900$	0	0
$901$	−524.000	−0.0193751
$902$	6580.00i	0.242894i
$903$	0	0
$904$	−15825.0	−0.582225
$905$	0	0
$906$	0	0
$907$	− 38487.0i	− 1.40897i	−0.709717	−	0.704487i	$-0.751177\pi$
$907$	− 38487.0i	− 1.40897i	0.709717	−	0.704487i	$-0.248823\pi$
$908$	− 34580.0i	− 1.26385i
$909$	0	0
$910$	0	0
$911$	5120.00	0.186205	0.0931027	−	0.995657i	$-0.470321\pi$
$911$	5120.00	0.186205	0.0931027	+	0.995657i	$0.470321\pi$
$912$	0	0
$913$	− 4794.00i	− 0.173777i
$914$	16204.0	0.586412
$915$	0	0
$916$	−30408.0	−1.09684
$917$	− 10998.0i	− 0.396059i
$918$	0	0
$919$	−28075.0	−1.00774	−0.503868	−	0.863781i	$-0.668090\pi$
$919$	−28075.0	−1.00774	−0.503868	+	0.863781i	$0.668090\pi$
$920$	0	0
$921$	0	0
$922$	− 5082.00i	− 0.181526i
$923$	− 160.000i	− 0.00570581i
$924$	0	0
$925$	0	0
$926$	10326.0	0.366451
$927$	0	0
$928$	25277.0i	0.894136i
$929$	−12856.0	−0.454028	−0.227014	−	0.973892i	$-0.572896\pi$
$929$	−12856.0	−0.454028	−0.227014	+	0.973892i	$0.572896\pi$
$930$	0	0
$931$	17192.0	0.605204
$932$	− 36414.0i	− 1.27981i
$933$	0	0
$934$	−4184.00	−0.146579
$935$	0	0
$936$	0	0
$937$	− 1374.00i	− 0.0479046i	−0.999713	−	0.0239523i	$-0.992375\pi$
$937$	− 1374.00i	− 0.0479046i	0.999713	−	0.0239523i	$-0.00762499\pi$
$938$	− 2952.00i	− 0.102757i
$939$	0	0
$940$	0	0
$941$	−8543.00	−0.295955	−0.147978	−	0.988991i	$-0.547276\pi$
$941$	−8543.00	−0.295955	−0.147978	+	0.988991i	$0.547276\pi$
$942$	0	0
$943$	− 420.000i	− 0.0145038i
$944$	30668.0	1.05737
$945$	0	0
$946$	−18659.0	−0.641286
$947$	− 13506.0i	− 0.463449i	−0.972781	−	0.231724i	$-0.925563\pi$
$947$	− 13506.0i	− 0.463449i	0.972781	−	0.231724i	$-0.0744368\pi$
$948$	0	0
$949$	4850.00	0.165898
$950$	0	0
$951$	0	0
$952$	− 11790.0i	− 0.401382i
$953$	21775.0i	0.740148i	0.929002	+	0.370074i	$0.120668\pi$
$953$	21775.0i	0.740148i	−0.929002	+	0.370074i	$0.879332\pi$
$954$	0	0
$955$	0	0
$956$	10822.0	0.366118
$957$	0	0
$958$	15576.0i	0.525300i
$959$	−6588.00	−0.221833
$960$	0	0
$961$	20834.0	0.699339
$962$	− 350.000i	− 0.0117302i
$963$	0	0
$964$	−25613.0	−0.855746
$965$	0	0
$966$	0	0
$967$	3854.00i	0.128166i	0.997945	+	0.0640829i	$0.0204122\pi$
$967$	3854.00i	0.128166i	−0.997945	+	0.0640829i	$0.979588\pi$
$968$	− 13170.0i	− 0.437293i
$969$	0	0
$970$	0	0
$971$	−12933.0	−0.427435	−0.213718	−	0.976895i	$-0.568557\pi$
$971$	−12933.0	−0.427435	−0.213718	+	0.976895i	$0.568557\pi$
$972$	0	0
$973$	− 6252.00i	− 0.205992i
$974$	10220.0	0.336211
$975$	0	0
$976$	−13858.0	−0.454492
$977$	− 17521.0i	− 0.573743i	−0.957969	−	0.286871i	$-0.907385\pi$
$977$	− 17521.0i	− 0.573743i	0.957969	−	0.286871i	$-0.0926152\pi$
$978$	0	0
$979$	−69936.0	−2.28311
$980$	0	0
$981$	0	0
$982$	− 2692.00i	− 0.0874798i
$983$	− 12573.0i	− 0.407952i	−0.978976	−	0.203976i	$-0.934614\pi$
$983$	− 12573.0i	− 0.407952i	0.978976	−	0.203976i	$-0.0653864\pi$
$984$	0	0
$985$	0	0
$986$	−20567.0	−0.664287
$987$	0	0
$988$	1960.00i	0.0631133i
$989$	1191.00	0.0382928
$990$	0	0
$991$	8945.00	0.286728	0.143364	−	0.989670i	$-0.454208\pi$
$991$	8945.00	0.286728	0.143364	+	0.989670i	$0.454208\pi$
$992$	− 36225.0i	− 1.15942i
$993$	0	0
$994$	192.000	0.00612663
$995$	0	0
$996$	0	0
$997$	− 58179.0i	− 1.84809i	−0.382282	−	0.924046i	$-0.624862\pi$
$997$	− 58179.0i	− 1.84809i	0.382282	−	0.924046i	$-0.375138\pi$
$998$	5764.00i	0.182822i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	675.4.b.f.649.1		2
3.2	odd	2		675.4.b.e.649.2		2
5.2	odd	4		675.4.a.h.1.1		1
5.3	odd	4		135.4.a.b.1.1	✓	1
5.4	even	2	inner	675.4.b.f.649.2		2
15.2	even	4		675.4.a.c.1.1		1
15.8	even	4		135.4.a.c.1.1	yes	1
15.14	odd	2		675.4.b.e.649.1		2
20.3	even	4		2160.4.a.f.1.1		1
45.13	odd	12		405.4.e.h.136.1		2
45.23	even	12		405.4.e.f.136.1		2
45.38	even	12		405.4.e.f.271.1		2
45.43	odd	12		405.4.e.h.271.1		2
60.23	odd	4		2160.4.a.p.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
135.4.a.b.1.1	✓	1	5.3	odd	4
135.4.a.c.1.1	yes	1	15.8	even	4
405.4.e.f.136.1		2	45.23	even	12
405.4.e.f.271.1		2	45.38	even	12
405.4.e.h.136.1		2	45.13	odd	12
405.4.e.h.271.1		2	45.43	odd	12
675.4.a.c.1.1		1	15.2	even	4
675.4.a.h.1.1		1	5.2	odd	4
675.4.b.e.649.1		2	15.14	odd	2
675.4.b.e.649.2		2	3.2	odd	2
675.4.b.f.649.1		2	1.1	even	1	trivial
675.4.b.f.649.2		2	5.4	even	2	inner
2160.4.a.f.1.1		1	20.3	even	4
2160.4.a.p.1.1		1	60.23	odd	4

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:	\( N \)	\(=\)	\( 675 = 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	675.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} +7.00000 q^{4} -6.00000i q^{7} -15.0000i q^{8} +O(q^{10})\)	\(q-1.00000i q^{2} +7.00000 q^{4} -6.00000i q^{7} -15.0000i q^{8} +47.0000 q^{11} +5.00000i q^{13} -6.00000 q^{14} +41.0000 q^{16} +131.000i q^{17} +56.0000 q^{19} -47.0000i q^{22} +3.00000i q^{23} +5.00000 q^{26} -42.0000i q^{28} -157.000 q^{29} +225.000 q^{31} -161.000i q^{32} +131.000 q^{34} -70.0000i q^{37} -56.0000i q^{38} -140.000 q^{41} -397.000i q^{43} +329.000 q^{44} +3.00000 q^{46} +347.000i q^{47} +307.000 q^{49} +35.0000i q^{52} +4.00000i q^{53} -90.0000 q^{56} +157.000i q^{58} +748.000 q^{59} -338.000 q^{61} -225.000i q^{62} +167.000 q^{64} +492.000i q^{67} +917.000i q^{68} -32.0000 q^{71} -970.000i q^{73} -70.0000 q^{74} +392.000 q^{76} -282.000i q^{77} +1257.00 q^{79} +140.000i q^{82} -102.000i q^{83} -397.000 q^{86} -705.000i q^{88} -1488.00 q^{89} +30.0000 q^{91} +21.0000i q^{92} +347.000 q^{94} +974.000i q^{97} -307.000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 14 q^{4}+O(q^{10}) \) 2 * q + 14 * q^4	\( 2 q + 14 q^{4} + 94 q^{11} - 12 q^{14} + 82 q^{16} + 112 q^{19} + 10 q^{26} - 314 q^{29} + 450 q^{31} + 262 q^{34} - 280 q^{41} + 658 q^{44} + 6 q^{46} + 614 q^{49} - 180 q^{56} + 1496 q^{59} - 676 q^{61} + 334 q^{64} - 64 q^{71} - 140 q^{74} + 784 q^{76} + 2514 q^{79} - 794 q^{86} - 2976 q^{89} + 60 q^{91} + 694 q^{94}+O(q^{100}) \) 2 * q + 14 * q^4 + 94 * q^11 - 12 * q^14 + 82 * q^16 + 112 * q^19 + 10 * q^26 - 314 * q^29 + 450 * q^31 + 262 * q^34 - 280 * q^41 + 658 * q^44 + 6 * q^46 + 614 * q^49 - 180 * q^56 + 1496 * q^59 - 676 * q^61 + 334 * q^64 - 64 * q^71 - 140 * q^74 + 784 * q^76 + 2514 * q^79 - 794 * q^86 - 2976 * q^89 + 60 * q^91 + 694 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more