LMFDB - Embedded newform 882.3.b.c.197.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [882,3,Mod(197,882)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(882, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("882.197");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 882 = 2 \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	882.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$24.0327593166$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-2}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 2 $ x^2 + 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 126)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			197.1
Root			$-1.41421i$ of defining polynomial
Character	$\chi$	$=$	882.197
Dual form			882.3.b.c.197.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421i q^{2} -2.00000 q^{4} -1.41421i q^{5} +2.82843i q^{8} +O(q^{10})$	$q-1.41421i q^{2} -2.00000 q^{4} -1.41421i q^{5} +2.82843i q^{8} -2.00000 q^{10} -7.07107i q^{11} +15.0000 q^{13} +4.00000 q^{16} -11.3137i q^{17} -13.0000 q^{19} +2.82843i q^{20} -10.0000 q^{22} +22.6274i q^{23} +23.0000 q^{25} -21.2132i q^{26} -22.6274i q^{29} +3.00000 q^{31} -5.65685i q^{32} -16.0000 q^{34} +17.0000 q^{37} +18.3848i q^{38} +4.00000 q^{40} -80.6102i q^{41} -85.0000 q^{43} +14.1421i q^{44} +32.0000 q^{46} -72.1249i q^{47} -32.5269i q^{50} -30.0000 q^{52} +33.9411i q^{53} -10.0000 q^{55} -32.0000 q^{58} -90.5097i q^{59} -72.0000 q^{61} -4.24264i q^{62} -8.00000 q^{64} -21.2132i q^{65} +43.0000 q^{67} +22.6274i q^{68} +52.3259i q^{71} -95.0000 q^{73} -24.0416i q^{74} +26.0000 q^{76} +69.0000 q^{79} -5.65685i q^{80} -114.000 q^{82} +60.8112i q^{83} -16.0000 q^{85} +120.208i q^{86} +20.0000 q^{88} -135.765i q^{89} -45.2548i q^{92} -102.000 q^{94} +18.3848i q^{95} +16.0000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{4}+O(q^{10}) $ 2 * q - 4 * q^4	$ 2 q - 4 q^{4} - 4 q^{10} + 30 q^{13} + 8 q^{16} - 26 q^{19} - 20 q^{22} + 46 q^{25} + 6 q^{31} - 32 q^{34} + 34 q^{37} + 8 q^{40} - 170 q^{43} + 64 q^{46} - 60 q^{52} - 20 q^{55} - 64 q^{58} - 144 q^{61} - 16 q^{64} + 86 q^{67} - 190 q^{73} + 52 q^{76} + 138 q^{79} - 228 q^{82} - 32 q^{85} + 40 q^{88} - 204 q^{94} + 32 q^{97}+O(q^{100}) $ 2 * q - 4 * q^4 - 4 * q^10 + 30 * q^13 + 8 * q^16 - 26 * q^19 - 20 * q^22 + 46 * q^25 + 6 * q^31 - 32 * q^34 + 34 * q^37 + 8 * q^40 - 170 * q^43 + 64 * q^46 - 60 * q^52 - 20 * q^55 - 64 * q^58 - 144 * q^61 - 16 * q^64 + 86 * q^67 - 190 * q^73 + 52 * q^76 + 138 * q^79 - 228 * q^82 - 32 * q^85 + 40 * q^88 - 204 * q^94 + 32 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/882\mathbb{Z}\right)^\times$.

$n$	$199$	$785$
$\chi(n)$	$1$	$-1$

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 $

$2$	− 1.41421i	− 0.707107i
$2$	− 1.41421i	− 0.707107i
$3$	0	0
$3$	0	0
$4$	−2.00000	−0.500000
$5$	− 1.41421i	− 0.282843i	−0.989949	−	0.141421i	$-0.954833\pi$
$5$	− 1.41421i	− 0.282843i	0.989949	−	0.141421i	$-0.0451672\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	2.82843i	0.353553i
$9$	0	0
$10$	−2.00000	−0.200000
$11$	− 7.07107i	− 0.642824i	−0.946939	−	0.321412i	$-0.895843\pi$
$11$	− 7.07107i	− 0.642824i	0.946939	−	0.321412i	$-0.104157\pi$
$12$	0	0
$13$	15.0000	1.15385	0.576923	−	0.816798i	$-0.304253\pi$
$13$	15.0000	1.15385	0.576923	+	0.816798i	$0.304253\pi$
$14$	0	0
$15$	0	0
$16$	4.00000	0.250000
$17$	− 11.3137i	− 0.665512i	−0.943013	−	0.332756i	$-0.892021\pi$
$17$	− 11.3137i	− 0.665512i	0.943013	−	0.332756i	$-0.107979\pi$
$18$	0	0
$19$	−13.0000	−0.684211	−0.342105	−	0.939662i	$-0.611140\pi$
$19$	−13.0000	−0.684211	−0.342105	+	0.939662i	$0.611140\pi$
$20$	2.82843i	0.141421i
$21$	0	0
$22$	−10.0000	−0.454545
$23$	22.6274i	0.983801i	0.870651	+	0.491900i	$0.163698\pi$
$23$	22.6274i	0.983801i	−0.870651	+	0.491900i	$0.836302\pi$
$24$	0	0
$25$	23.0000	0.920000
$26$	− 21.2132i	− 0.815892i
$27$	0	0
$28$	0	0
$29$	− 22.6274i	− 0.780256i	−0.920761	−	0.390128i	$-0.872431\pi$
$29$	− 22.6274i	− 0.780256i	0.920761	−	0.390128i	$-0.127569\pi$
$30$	0	0
$31$	3.00000	0.0967742	0.0483871	−	0.998829i	$-0.484592\pi$
$31$	3.00000	0.0967742	0.0483871	+	0.998829i	$0.484592\pi$
$32$	− 5.65685i	− 0.176777i
$33$	0	0
$34$	−16.0000	−0.470588
$35$	0	0
$36$	0	0
$37$	17.0000	0.459459	0.229730	−	0.973254i	$-0.426216\pi$
$37$	17.0000	0.459459	0.229730	+	0.973254i	$0.426216\pi$
$38$	18.3848i	0.483810i
$39$	0	0
$40$	4.00000	0.100000
$41$	− 80.6102i	− 1.96610i	−0.183333	−	0.983051i	$-0.558689\pi$
$41$	− 80.6102i	− 1.96610i	0.183333	−	0.983051i	$-0.441311\pi$
$42$	0	0
$43$	−85.0000	−1.97674	−0.988372	−	0.152055i	$-0.951411\pi$
$43$	−85.0000	−1.97674	−0.988372	+	0.152055i	$0.951411\pi$
$44$	14.1421i	0.321412i
$45$	0	0
$46$	32.0000	0.695652
$47$	− 72.1249i	− 1.53457i	−0.641305	−	0.767286i	$-0.721607\pi$
$47$	− 72.1249i	− 1.53457i	0.641305	−	0.767286i	$-0.278393\pi$
$48$	0	0
$49$	0	0
$50$	− 32.5269i	− 0.650538i
$51$	0	0
$52$	−30.0000	−0.576923
$53$	33.9411i	0.640399i	0.947350	+	0.320199i	$0.103750\pi$
$53$	33.9411i	0.640399i	−0.947350	+	0.320199i	$0.896250\pi$
$54$	0	0
$55$	−10.0000	−0.181818
$56$	0	0
$57$	0	0
$58$	−32.0000	−0.551724
$59$	− 90.5097i	− 1.53406i	−0.641610	−	0.767031i	$-0.721733\pi$
$59$	− 90.5097i	− 1.53406i	0.641610	−	0.767031i	$-0.278267\pi$
$60$	0	0
$61$	−72.0000	−1.18033	−0.590164	−	0.807283i	$-0.700937\pi$
$61$	−72.0000	−1.18033	−0.590164	+	0.807283i	$0.700937\pi$
$62$	− 4.24264i	− 0.0684297i
$63$	0	0
$64$	−8.00000	−0.125000
$65$	− 21.2132i	− 0.326357i
$66$	0	0
$67$	43.0000	0.641791	0.320896	−	0.947115i	$-0.396016\pi$
$67$	43.0000	0.641791	0.320896	+	0.947115i	$0.396016\pi$
$68$	22.6274i	0.332756i
$69$	0	0
$70$	0	0
$71$	52.3259i	0.736985i	0.929631	+	0.368492i	$0.120126\pi$
$71$	52.3259i	0.736985i	−0.929631	+	0.368492i	$0.879874\pi$
$72$	0	0
$73$	−95.0000	−1.30137	−0.650685	−	0.759348i	$-0.725518\pi$
$73$	−95.0000	−1.30137	−0.650685	+	0.759348i	$0.725518\pi$
$74$	− 24.0416i	− 0.324887i
$75$	0	0
$76$	26.0000	0.342105
$77$	0	0
$78$	0	0
$79$	69.0000	0.873418	0.436709	−	0.899603i	$-0.356144\pi$
$79$	69.0000	0.873418	0.436709	+	0.899603i	$0.356144\pi$
$80$	− 5.65685i	− 0.0707107i
$81$	0	0
$82$	−114.000	−1.39024
$83$	60.8112i	0.732665i	0.930484	+	0.366332i	$0.119387\pi$
$83$	60.8112i	0.732665i	−0.930484	+	0.366332i	$0.880613\pi$
$84$	0	0
$85$	−16.0000	−0.188235
$86$	120.208i	1.39777i
$87$	0	0
$88$	20.0000	0.227273
$89$	− 135.765i	− 1.52544i	−0.646727	−	0.762722i	$-0.723863\pi$
$89$	− 135.765i	− 1.52544i	0.646727	−	0.762722i	$-0.276137\pi$
$90$	0	0
$91$	0	0
$92$	− 45.2548i	− 0.491900i
$93$	0	0
$94$	−102.000	−1.08511
$95$	18.3848i	0.193524i
$96$	0	0
$97$	16.0000	0.164948	0.0824742	−	0.996593i	$-0.473718\pi$
$97$	16.0000	0.164948	0.0824742	+	0.996593i	$0.473718\pi$
$98$	0	0
$99$	0	0
$100$	−46.0000	−0.460000
$101$	− 69.2965i	− 0.686104i	−0.939316	−	0.343052i	$-0.888539\pi$
$101$	− 69.2965i	− 0.686104i	0.939316	−	0.343052i	$-0.111461\pi$
$102$	0	0
$103$	−61.0000	−0.592233	−0.296117	−	0.955152i	$-0.595692\pi$
$103$	−61.0000	−0.592233	−0.296117	+	0.955152i	$0.595692\pi$
$104$	42.4264i	0.407946i
$105$	0	0
$106$	48.0000	0.452830
$107$	− 169.706i	− 1.58603i	−0.609200	−	0.793017i	$-0.708509\pi$
$107$	− 169.706i	− 1.58603i	0.609200	−	0.793017i	$-0.291491\pi$
$108$	0	0
$109$	−65.0000	−0.596330	−0.298165	−	0.954514i	$-0.596375\pi$
$109$	−65.0000	−0.596330	−0.298165	+	0.954514i	$0.596375\pi$
$110$	14.1421i	0.128565i
$111$	0	0
$112$	0	0
$113$	− 137.179i	− 1.21397i	−0.794713	−	0.606985i	$-0.792379\pi$
$113$	− 137.179i	− 1.21397i	0.794713	−	0.606985i	$-0.207621\pi$
$114$	0	0
$115$	32.0000	0.278261
$116$	45.2548i	0.390128i
$117$	0	0
$118$	−128.000	−1.08475
$119$	0	0
$120$	0	0
$121$	71.0000	0.586777
$122$	101.823i	0.834618i
$123$	0	0
$124$	−6.00000	−0.0483871
$125$	− 67.8823i	− 0.543058i
$126$	0	0
$127$	−171.000	−1.34646	−0.673228	−	0.739435i	$-0.735093\pi$
$127$	−171.000	−1.34646	−0.673228	+	0.739435i	$0.735093\pi$
$128$	11.3137i	0.0883883i
$129$	0	0
$130$	−30.0000	−0.230769
$131$	117.380i	0.896028i	0.894026	+	0.448014i	$0.147869\pi$
$131$	117.380i	0.896028i	−0.894026	+	0.448014i	$0.852131\pi$
$132$	0	0
$133$	0	0
$134$	− 60.8112i	− 0.453815i
$135$	0	0
$136$	32.0000	0.235294
$137$	214.960i	1.56905i	0.620094	+	0.784527i	$0.287094\pi$
$137$	214.960i	1.56905i	−0.620094	+	0.784527i	$0.712906\pi$
$138$	0	0
$139$	−83.0000	−0.597122	−0.298561	−	0.954391i	$-0.596507\pi$
$139$	−83.0000	−0.597122	−0.298561	+	0.954391i	$0.596507\pi$
$140$	0	0
$141$	0	0
$142$	74.0000	0.521127
$143$	− 106.066i	− 0.741720i
$144$	0	0
$145$	−32.0000	−0.220690
$146$	134.350i	0.920207i
$147$	0	0
$148$	−34.0000	−0.229730
$149$	101.823i	0.683378i	0.939813	+	0.341689i	$0.110999\pi$
$149$	101.823i	0.683378i	−0.939813	+	0.341689i	$0.889001\pi$
$150$	0	0
$151$	40.0000	0.264901	0.132450	−	0.991190i	$-0.457715\pi$
$151$	40.0000	0.264901	0.132450	+	0.991190i	$0.457715\pi$
$152$	− 36.7696i	− 0.241905i
$153$	0	0
$154$	0	0
$155$	− 4.24264i	− 0.0273719i
$156$	0	0
$157$	296.000	1.88535	0.942675	−	0.333712i	$-0.108301\pi$
$157$	296.000	1.88535	0.942675	+	0.333712i	$0.108301\pi$
$158$	− 97.5807i	− 0.617600i
$159$	0	0
$160$	−8.00000	−0.0500000
$161$	0	0
$162$	0	0
$163$	128.000	0.785276	0.392638	−	0.919693i	$-0.371563\pi$
$163$	128.000	0.785276	0.392638	+	0.919693i	$0.371563\pi$
$164$	161.220i	0.983051i
$165$	0	0
$166$	86.0000	0.518072
$167$	− 60.8112i	− 0.364139i	−0.983286	−	0.182069i	$-0.941720\pi$
$167$	− 60.8112i	− 0.364139i	0.983286	−	0.182069i	$-0.0582796\pi$
$168$	0	0
$169$	56.0000	0.331361
$170$	22.6274i	0.133102i
$171$	0	0
$172$	170.000	0.988372
$173$	113.137i	0.653972i	0.945029	+	0.326986i	$0.106033\pi$
$173$	113.137i	0.653972i	−0.945029	+	0.326986i	$0.893967\pi$
$174$	0	0
$175$	0	0
$176$	− 28.2843i	− 0.160706i
$177$	0	0
$178$	−192.000	−1.07865
$179$	− 29.6985i	− 0.165913i	−0.996553	−	0.0829567i	$-0.973564\pi$
$179$	− 29.6985i	− 0.165913i	0.996553	−	0.0829567i	$-0.0264363\pi$
$180$	0	0
$181$	81.0000	0.447514	0.223757	−	0.974645i	$-0.428168\pi$
$181$	81.0000	0.447514	0.223757	+	0.974645i	$0.428168\pi$
$182$	0	0
$183$	0	0
$184$	−64.0000	−0.347826
$185$	− 24.0416i	− 0.129955i
$186$	0	0
$187$	−80.0000	−0.427807
$188$	144.250i	0.767286i
$189$	0	0
$190$	26.0000	0.136842
$191$	− 63.6396i	− 0.333192i	−0.986025	−	0.166596i	$-0.946722\pi$
$191$	− 63.6396i	− 0.333192i	0.986025	−	0.166596i	$-0.0532775\pi$
$192$	0	0
$193$	−223.000	−1.15544	−0.577720	−	0.816235i	$-0.696058\pi$
$193$	−223.000	−1.15544	−0.577720	+	0.816235i	$0.696058\pi$
$194$	− 22.6274i	− 0.116636i
$195$	0	0
$196$	0	0
$197$	158.392i	0.804020i	0.915635	+	0.402010i	$0.131688\pi$
$197$	158.392i	0.804020i	−0.915635	+	0.402010i	$0.868312\pi$
$198$	0	0
$199$	−136.000	−0.683417	−0.341709	−	0.939806i	$-0.611006\pi$
$199$	−136.000	−0.683417	−0.341709	+	0.939806i	$0.611006\pi$
$200$	65.0538i	0.325269i
$201$	0	0
$202$	−98.0000	−0.485149
$203$	0	0
$204$	0	0
$205$	−114.000	−0.556098
$206$	86.2670i	0.418772i
$207$	0	0
$208$	60.0000	0.288462
$209$	91.9239i	0.439827i
$210$	0	0
$211$	272.000	1.28910	0.644550	−	0.764562i	$-0.277045\pi$
$211$	272.000	1.28910	0.644550	+	0.764562i	$0.277045\pi$
$212$	− 67.8823i	− 0.320199i
$213$	0	0
$214$	−240.000	−1.12150
$215$	120.208i	0.559108i
$216$	0	0
$217$	0	0
$218$	91.9239i	0.421669i
$219$	0	0
$220$	20.0000	0.0909091
$221$	− 169.706i	− 0.767899i
$222$	0	0
$223$	248.000	1.11211	0.556054	−	0.831146i	$-0.312315\pi$
$223$	248.000	1.11211	0.556054	+	0.831146i	$0.312315\pi$
$224$	0	0
$225$	0	0
$226$	−194.000	−0.858407
$227$	165.463i	0.728912i	0.931221	+	0.364456i	$0.118745\pi$
$227$	165.463i	0.728912i	−0.931221	+	0.364456i	$0.881255\pi$
$228$	0	0
$229$	433.000	1.89083	0.945415	−	0.325869i	$-0.105657\pi$
$229$	433.000	1.89083	0.945415	+	0.325869i	$0.105657\pi$
$230$	− 45.2548i	− 0.196760i
$231$	0	0
$232$	64.0000	0.275862
$233$	− 227.688i	− 0.977203i	−0.872507	−	0.488602i	$-0.837507\pi$
$233$	− 227.688i	− 0.977203i	0.872507	−	0.488602i	$-0.162493\pi$
$234$	0	0
$235$	−102.000	−0.434043
$236$	181.019i	0.767031i
$237$	0	0
$238$	0	0
$239$	343.654i	1.43788i	0.695071	+	0.718941i	$0.255373\pi$
$239$	343.654i	1.43788i	−0.695071	+	0.718941i	$0.744627\pi$
$240$	0	0
$241$	158.000	0.655602	0.327801	−	0.944747i	$-0.393693\pi$
$241$	158.000	0.655602	0.327801	+	0.944747i	$0.393693\pi$
$242$	− 100.409i	− 0.414914i
$243$	0	0
$244$	144.000	0.590164
$245$	0	0
$246$	0	0
$247$	−195.000	−0.789474
$248$	8.48528i	0.0342148i
$249$	0	0
$250$	−96.0000	−0.384000
$251$	− 237.588i	− 0.946565i	−0.880911	−	0.473283i	$-0.843069\pi$
$251$	− 237.588i	− 0.946565i	0.880911	−	0.473283i	$-0.156931\pi$
$252$	0	0
$253$	160.000	0.632411
$254$	241.831i	0.952089i
$255$	0	0
$256$	16.0000	0.0625000
$257$	− 318.198i	− 1.23812i	−0.785342	−	0.619062i	$-0.787513\pi$
$257$	− 318.198i	− 1.23812i	0.785342	−	0.619062i	$-0.212487\pi$
$258$	0	0
$259$	0	0
$260$	42.4264i	0.163178i
$261$	0	0
$262$	166.000	0.633588
$263$	− 169.706i	− 0.645269i	−0.946524	−	0.322634i	$-0.895432\pi$
$263$	− 169.706i	− 0.645269i	0.946524	−	0.322634i	$-0.104568\pi$
$264$	0	0
$265$	48.0000	0.181132
$266$	0	0
$267$	0	0
$268$	−86.0000	−0.320896
$269$	− 485.075i	− 1.80325i	−0.432515	−	0.901627i	$-0.642374\pi$
$269$	− 485.075i	− 1.80325i	0.432515	−	0.901627i	$-0.357626\pi$
$270$	0	0
$271$	216.000	0.797048	0.398524	−	0.917158i	$-0.369523\pi$
$271$	216.000	0.797048	0.398524	+	0.917158i	$0.369523\pi$
$272$	− 45.2548i	− 0.166378i
$273$	0	0
$274$	304.000	1.10949
$275$	− 162.635i	− 0.591398i
$276$	0	0
$277$	305.000	1.10108	0.550542	−	0.834808i	$-0.314421\pi$
$277$	305.000	1.10108	0.550542	+	0.834808i	$0.314421\pi$
$278$	117.380i	0.422229i
$279$	0	0
$280$	0	0
$281$	22.6274i	0.0805246i	0.999189	+	0.0402623i	$0.0128194\pi$
$281$	22.6274i	0.0805246i	−0.999189	+	0.0402623i	$0.987181\pi$
$282$	0	0
$283$	157.000	0.554770	0.277385	−	0.960759i	$-0.410532\pi$
$283$	157.000	0.554770	0.277385	+	0.960759i	$0.410532\pi$
$284$	− 104.652i	− 0.368492i
$285$	0	0
$286$	−150.000	−0.524476
$287$	0	0
$288$	0	0
$289$	161.000	0.557093
$290$	45.2548i	0.156051i
$291$	0	0
$292$	190.000	0.650685
$293$	101.823i	0.347520i	0.984788	+	0.173760i	$0.0555917\pi$
$293$	101.823i	0.347520i	−0.984788	+	0.173760i	$0.944408\pi$
$294$	0	0
$295$	−128.000	−0.433898
$296$	48.0833i	0.162443i
$297$	0	0
$298$	144.000	0.483221
$299$	339.411i	1.13515i
$300$	0	0
$301$	0	0
$302$	− 56.5685i	− 0.187313i
$303$	0	0
$304$	−52.0000	−0.171053
$305$	101.823i	0.333847i
$306$	0	0
$307$	−11.0000	−0.0358306	−0.0179153	−	0.999840i	$-0.505703\pi$
$307$	−11.0000	−0.0358306	−0.0179153	+	0.999840i	$0.505703\pi$
$308$	0	0
$309$	0	0
$310$	−6.00000	−0.0193548
$311$	524.673i	1.68705i	0.537088	+	0.843526i	$0.319524\pi$
$311$	524.673i	1.68705i	−0.537088	+	0.843526i	$0.680476\pi$
$312$	0	0
$313$	553.000	1.76677	0.883387	−	0.468645i	$-0.155258\pi$
$313$	553.000	1.76677	0.883387	+	0.468645i	$0.155258\pi$
$314$	− 418.607i	− 1.33314i
$315$	0	0
$316$	−138.000	−0.436709
$317$	135.765i	0.428279i	0.976803	+	0.214140i	$0.0686947\pi$
$317$	135.765i	0.428279i	−0.976803	+	0.214140i	$0.931305\pi$
$318$	0	0
$319$	−160.000	−0.501567
$320$	11.3137i	0.0353553i
$321$	0	0
$322$	0	0
$323$	147.078i	0.455350i
$324$	0	0
$325$	345.000	1.06154
$326$	− 181.019i	− 0.555274i
$327$	0	0
$328$	228.000	0.695122
$329$	0	0
$330$	0	0
$331$	−61.0000	−0.184290	−0.0921450	−	0.995746i	$-0.529372\pi$
$331$	−61.0000	−0.184290	−0.0921450	+	0.995746i	$0.529372\pi$
$332$	− 121.622i	− 0.366332i
$333$	0	0
$334$	−86.0000	−0.257485
$335$	− 60.8112i	− 0.181526i
$336$	0	0
$337$	135.000	0.400593	0.200297	−	0.979735i	$-0.435809\pi$
$337$	135.000	0.400593	0.200297	+	0.979735i	$0.435809\pi$
$338$	− 79.1960i	− 0.234308i
$339$	0	0
$340$	32.0000	0.0941176
$341$	− 21.2132i	− 0.0622088i
$342$	0	0
$343$	0	0
$344$	− 240.416i	− 0.698885i
$345$	0	0
$346$	160.000	0.462428
$347$	101.823i	0.293439i	0.989178	+	0.146720i	$0.0468715\pi$
$347$	101.823i	0.293439i	−0.989178	+	0.146720i	$0.953129\pi$
$348$	0	0
$349$	−152.000	−0.435530	−0.217765	−	0.976001i	$-0.569877\pi$
$349$	−152.000	−0.435530	−0.217765	+	0.976001i	$0.569877\pi$
$350$	0	0
$351$	0	0
$352$	−40.0000	−0.113636
$353$	386.080i	1.09371i	0.837227	+	0.546856i	$0.184175\pi$
$353$	386.080i	1.09371i	−0.837227	+	0.546856i	$0.815825\pi$
$354$	0	0
$355$	74.0000	0.208451
$356$	271.529i	0.762722i
$357$	0	0
$358$	−42.0000	−0.117318
$359$	45.2548i	0.126058i	0.998012	+	0.0630290i	$0.0200761\pi$
$359$	45.2548i	0.126058i	−0.998012	+	0.0630290i	$0.979924\pi$
$360$	0	0
$361$	−192.000	−0.531856
$362$	− 114.551i	− 0.316440i
$363$	0	0
$364$	0	0
$365$	134.350i	0.368083i
$366$	0	0
$367$	101.000	0.275204	0.137602	−	0.990488i	$-0.456060\pi$
$367$	101.000	0.275204	0.137602	+	0.990488i	$0.456060\pi$
$368$	90.5097i	0.245950i
$369$	0	0
$370$	−34.0000	−0.0918919
$371$	0	0
$372$	0	0
$373$	−311.000	−0.833780	−0.416890	−	0.908957i	$-0.636880\pi$
$373$	−311.000	−0.833780	−0.416890	+	0.908957i	$0.636880\pi$
$374$	113.137i	0.302506i
$375$	0	0
$376$	204.000	0.542553
$377$	− 339.411i	− 0.900295i
$378$	0	0
$379$	91.0000	0.240106	0.120053	−	0.992768i	$-0.461694\pi$
$379$	91.0000	0.240106	0.120053	+	0.992768i	$0.461694\pi$
$380$	− 36.7696i	− 0.0967620i
$381$	0	0
$382$	−90.0000	−0.235602
$383$	328.098i	0.856652i	0.903624	+	0.428326i	$0.140896\pi$
$383$	328.098i	0.856652i	−0.903624	+	0.428326i	$0.859104\pi$
$384$	0	0
$385$	0	0
$386$	315.370i	0.817020i
$387$	0	0
$388$	−32.0000	−0.0824742
$389$	− 688.722i	− 1.77049i	−0.465122	−	0.885247i	$-0.653989\pi$
$389$	− 688.722i	− 1.77049i	0.465122	−	0.885247i	$-0.346011\pi$
$390$	0	0
$391$	256.000	0.654731
$392$	0	0
$393$	0	0
$394$	224.000	0.568528
$395$	− 97.5807i	− 0.247040i
$396$	0	0
$397$	−279.000	−0.702771	−0.351385	−	0.936231i	$-0.614289\pi$
$397$	−279.000	−0.702771	−0.351385	+	0.936231i	$0.614289\pi$
$398$	192.333i	0.483249i
$399$	0	0
$400$	92.0000	0.230000
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	0	0
$403$	45.0000	0.111663
$404$	138.593i	0.343052i
$405$	0	0
$406$	0	0
$407$	− 120.208i	− 0.295352i
$408$	0	0
$409$	−223.000	−0.545232	−0.272616	−	0.962123i	$-0.587889\pi$
$409$	−223.000	−0.545232	−0.272616	+	0.962123i	$0.587889\pi$
$410$	161.220i	0.393220i
$411$	0	0
$412$	122.000	0.296117
$413$	0	0
$414$	0	0
$415$	86.0000	0.207229
$416$	− 84.8528i	− 0.203973i
$417$	0	0
$418$	130.000	0.311005
$419$	581.242i	1.38721i	0.720355	+	0.693606i	$0.243979\pi$
$419$	581.242i	1.38721i	−0.720355	+	0.693606i	$0.756021\pi$
$420$	0	0
$421$	153.000	0.363420	0.181710	−	0.983352i	$-0.441837\pi$
$421$	153.000	0.363420	0.181710	+	0.983352i	$0.441837\pi$
$422$	− 384.666i	− 0.911531i
$423$	0	0
$424$	−96.0000	−0.226415
$425$	− 260.215i	− 0.612271i
$426$	0	0
$427$	0	0
$428$	339.411i	0.793017i
$429$	0	0
$430$	170.000	0.395349
$431$	− 140.007i	− 0.324843i	−0.986722	−	0.162421i	$-0.948070\pi$
$431$	− 140.007i	− 0.324843i	0.986722	−	0.162421i	$-0.0519304\pi$
$432$	0	0
$433$	137.000	0.316397	0.158199	−	0.987407i	$-0.449431\pi$
$433$	137.000	0.316397	0.158199	+	0.987407i	$0.449431\pi$
$434$	0	0
$435$	0	0
$436$	130.000	0.298165
$437$	− 294.156i	− 0.673127i
$438$	0	0
$439$	−600.000	−1.36674	−0.683371	−	0.730071i	$-0.739487\pi$
$439$	−600.000	−1.36674	−0.683371	+	0.730071i	$0.739487\pi$
$440$	− 28.2843i	− 0.0642824i
$441$	0	0
$442$	−240.000	−0.542986
$443$	633.568i	1.43018i	0.699035	+	0.715088i	$0.253613\pi$
$443$	633.568i	1.43018i	−0.699035	+	0.715088i	$0.746387\pi$
$444$	0	0
$445$	−192.000	−0.431461
$446$	− 350.725i	− 0.786379i
$447$	0	0
$448$	0	0
$449$	− 383.252i	− 0.853568i	−0.904354	−	0.426784i	$-0.859647\pi$
$449$	− 383.252i	− 0.853568i	0.904354	−	0.426784i	$-0.140353\pi$
$450$	0	0
$451$	−570.000	−1.26386
$452$	274.357i	0.606985i
$453$	0	0
$454$	234.000	0.515419
$455$	0	0
$456$	0	0
$457$	−239.000	−0.522976	−0.261488	−	0.965207i	$-0.584213\pi$
$457$	−239.000	−0.522976	−0.261488	+	0.965207i	$0.584213\pi$
$458$	− 612.354i	− 1.33702i
$459$	0	0
$460$	−64.0000	−0.139130
$461$	452.548i	0.981667i	0.871253	+	0.490833i	$0.163308\pi$
$461$	452.548i	0.981667i	−0.871253	+	0.490833i	$0.836692\pi$
$462$	0	0
$463$	211.000	0.455724	0.227862	−	0.973693i	$-0.426827\pi$
$463$	211.000	0.455724	0.227862	+	0.973693i	$0.426827\pi$
$464$	− 90.5097i	− 0.195064i
$465$	0	0
$466$	−322.000	−0.690987
$467$	312.541i	0.669253i	0.942351	+	0.334627i	$0.108610\pi$
$467$	312.541i	0.669253i	−0.942351	+	0.334627i	$0.891390\pi$
$468$	0	0
$469$	0	0
$470$	144.250i	0.306914i
$471$	0	0
$472$	256.000	0.542373
$473$	601.041i	1.27070i
$474$	0	0
$475$	−299.000	−0.629474
$476$	0	0
$477$	0	0
$478$	486.000	1.01674
$479$	701.450i	1.46440i	0.681087	+	0.732202i	$0.261508\pi$
$479$	701.450i	1.46440i	−0.681087	+	0.732202i	$0.738492\pi$
$480$	0	0
$481$	255.000	0.530146
$482$	− 223.446i	− 0.463580i
$483$	0	0
$484$	−142.000	−0.293388
$485$	− 22.6274i	− 0.0466545i
$486$	0	0
$487$	419.000	0.860370	0.430185	−	0.902741i	$-0.358448\pi$
$487$	419.000	0.860370	0.430185	+	0.902741i	$0.358448\pi$
$488$	− 203.647i	− 0.417309i
$489$	0	0
$490$	0	0
$491$	− 169.706i	− 0.345633i	−0.984954	−	0.172816i	$-0.944713\pi$
$491$	− 169.706i	− 0.345633i	0.984954	−	0.172816i	$-0.0552867\pi$
$492$	0	0
$493$	−256.000	−0.519270
$494$	275.772i	0.558242i
$495$	0	0
$496$	12.0000	0.0241935
$497$	0	0
$498$	0	0
$499$	187.000	0.374749	0.187375	−	0.982289i	$-0.440002\pi$
$499$	187.000	0.374749	0.187375	+	0.982289i	$0.440002\pi$
$500$	135.765i	0.271529i
$501$	0	0
$502$	−336.000	−0.669323
$503$	− 173.948i	− 0.345822i	−0.984937	−	0.172911i	$-0.944683\pi$
$503$	− 173.948i	− 0.345822i	0.984937	−	0.172911i	$-0.0553172\pi$
$504$	0	0
$505$	−98.0000	−0.194059
$506$	− 226.274i	− 0.447182i
$507$	0	0
$508$	342.000	0.673228
$509$	− 258.801i	− 0.508450i	−0.967145	−	0.254225i	$-0.918180\pi$
$509$	− 258.801i	− 0.508450i	0.967145	−	0.254225i	$-0.0818204\pi$
$510$	0	0
$511$	0	0
$512$	− 22.6274i	− 0.0441942i
$513$	0	0
$514$	−450.000	−0.875486
$515$	86.2670i	0.167509i
$516$	0	0
$517$	−510.000	−0.986460
$518$	0	0
$519$	0	0
$520$	60.0000	0.115385
$521$	− 407.294i	− 0.781753i	−0.920443	−	0.390877i	$-0.872172\pi$
$521$	− 407.294i	− 0.781753i	0.920443	−	0.390877i	$-0.127828\pi$
$522$	0	0
$523$	−811.000	−1.55067	−0.775335	−	0.631551i	$-0.782419\pi$
$523$	−811.000	−1.55067	−0.775335	+	0.631551i	$0.782419\pi$
$524$	− 234.759i	− 0.448014i
$525$	0	0
$526$	−240.000	−0.456274
$527$	− 33.9411i	− 0.0644044i
$528$	0	0
$529$	17.0000	0.0321361
$530$	− 67.8823i	− 0.128080i
$531$	0	0
$532$	0	0
$533$	− 1209.15i	− 2.26858i
$534$	0	0
$535$	−240.000	−0.448598
$536$	121.622i	0.226907i
$537$	0	0
$538$	−686.000	−1.27509
$539$	0	0
$540$	0	0
$541$	−345.000	−0.637708	−0.318854	−	0.947804i	$-0.603298\pi$
$541$	−345.000	−0.637708	−0.318854	+	0.947804i	$0.603298\pi$
$542$	− 305.470i	− 0.563598i
$543$	0	0
$544$	−64.0000	−0.117647
$545$	91.9239i	0.168668i
$546$	0	0
$547$	−864.000	−1.57952	−0.789762	−	0.613413i	$-0.789796\pi$
$547$	−864.000	−1.57952	−0.789762	+	0.613413i	$0.789796\pi$
$548$	− 429.921i	− 0.784527i
$549$	0	0
$550$	−230.000	−0.418182
$551$	294.156i	0.533859i
$552$	0	0
$553$	0	0
$554$	− 431.335i	− 0.778583i
$555$	0	0
$556$	166.000	0.298561
$557$	948.937i	1.70366i	0.523820	+	0.851829i	$0.324506\pi$
$557$	948.937i	1.70366i	−0.523820	+	0.851829i	$0.675494\pi$
$558$	0	0
$559$	−1275.00	−2.28086
$560$	0	0
$561$	0	0
$562$	32.0000	0.0569395
$563$	739.634i	1.31374i	0.754005	+	0.656868i	$0.228119\pi$
$563$	739.634i	1.31374i	−0.754005	+	0.656868i	$0.771881\pi$
$564$	0	0
$565$	−194.000	−0.343363
$566$	− 222.032i	− 0.392282i
$567$	0	0
$568$	−148.000	−0.260563
$569$	767.918i	1.34959i	0.738004	+	0.674796i	$0.235768\pi$
$569$	767.918i	1.34959i	−0.738004	+	0.674796i	$0.764232\pi$
$570$	0	0
$571$	477.000	0.835377	0.417688	−	0.908590i	$-0.362840\pi$
$571$	477.000	0.835377	0.417688	+	0.908590i	$0.362840\pi$
$572$	212.132i	0.370860i
$573$	0	0
$574$	0	0
$575$	520.431i	0.905097i
$576$	0	0
$577$	−263.000	−0.455806	−0.227903	−	0.973684i	$-0.573187\pi$
$577$	−263.000	−0.455806	−0.227903	+	0.973684i	$0.573187\pi$
$578$	− 227.688i	− 0.393925i
$579$	0	0
$580$	64.0000	0.110345
$581$	0	0
$582$	0	0
$583$	240.000	0.411664
$584$	− 268.701i	− 0.460104i
$585$	0	0
$586$	144.000	0.245734
$587$	531.744i	0.905868i	0.891544	+	0.452934i	$0.149623\pi$
$587$	531.744i	0.905868i	−0.891544	+	0.452934i	$0.850377\pi$
$588$	0	0
$589$	−39.0000	−0.0662139
$590$	181.019i	0.306812i
$591$	0	0
$592$	68.0000	0.114865
$593$	− 801.859i	− 1.35221i	−0.736806	−	0.676104i	$-0.763667\pi$
$593$	− 801.859i	− 1.35221i	0.736806	−	0.676104i	$-0.236333\pi$
$594$	0	0
$595$	0	0
$596$	− 203.647i	− 0.341689i
$597$	0	0
$598$	480.000	0.802676
$599$	− 780.646i	− 1.30325i	−0.758542	−	0.651624i	$-0.774088\pi$
$599$	− 780.646i	− 1.30325i	0.758542	−	0.651624i	$-0.225912\pi$
$600$	0	0
$601$	383.000	0.637271	0.318636	−	0.947877i	$-0.396775\pi$
$601$	383.000	0.637271	0.318636	+	0.947877i	$0.396775\pi$
$602$	0	0
$603$	0	0
$604$	−80.0000	−0.132450
$605$	− 100.409i	− 0.165966i
$606$	0	0
$607$	−765.000	−1.26030	−0.630148	−	0.776475i	$-0.717006\pi$
$607$	−765.000	−1.26030	−0.630148	+	0.776475i	$0.717006\pi$
$608$	73.5391i	0.120952i
$609$	0	0
$610$	144.000	0.236066
$611$	− 1081.87i	− 1.77066i
$612$	0	0
$613$	408.000	0.665579	0.332790	−	0.943001i	$-0.392010\pi$
$613$	408.000	0.665579	0.332790	+	0.943001i	$0.392010\pi$
$614$	15.5563i	0.0253361i
$615$	0	0
$616$	0	0
$617$	− 914.996i	− 1.48298i	−0.670966	−	0.741488i	$-0.734120\pi$
$617$	− 914.996i	− 1.48298i	0.670966	−	0.741488i	$-0.265880\pi$
$618$	0	0
$619$	−947.000	−1.52989	−0.764943	−	0.644097i	$-0.777233\pi$
$619$	−947.000	−1.52989	−0.764943	+	0.644097i	$0.777233\pi$
$620$	8.48528i	0.0136859i
$621$	0	0
$622$	742.000	1.19293
$623$	0	0
$624$	0	0
$625$	479.000	0.766400
$626$	− 782.060i	− 1.24930i
$627$	0	0
$628$	−592.000	−0.942675
$629$	− 192.333i	− 0.305776i
$630$	0	0
$631$	−760.000	−1.20444	−0.602219	−	0.798331i	$-0.705716\pi$
$631$	−760.000	−1.20444	−0.602219	+	0.798331i	$0.705716\pi$
$632$	195.161i	0.308800i
$633$	0	0
$634$	192.000	0.302839
$635$	241.831i	0.380835i
$636$	0	0
$637$	0	0
$638$	226.274i	0.354662i
$639$	0	0
$640$	16.0000	0.0250000
$641$	463.862i	0.723654i	0.932245	+	0.361827i	$0.117847\pi$
$641$	463.862i	0.723654i	−0.932245	+	0.361827i	$0.882153\pi$
$642$	0	0
$643$	−19.0000	−0.0295490	−0.0147745	−	0.999891i	$-0.504703\pi$
$643$	−19.0000	−0.0295490	−0.0147745	+	0.999891i	$0.504703\pi$
$644$	0	0
$645$	0	0
$646$	208.000	0.321981
$647$	− 1158.24i	− 1.79017i	−0.445894	−	0.895086i	$-0.647114\pi$
$647$	− 1158.24i	− 1.79017i	0.445894	−	0.895086i	$-0.352886\pi$
$648$	0	0
$649$	−640.000	−0.986133
$650$	− 487.904i	− 0.750621i
$651$	0	0
$652$	−256.000	−0.392638
$653$	1030.96i	1.57881i	0.613874	+	0.789404i	$0.289610\pi$
$653$	1030.96i	1.57881i	−0.613874	+	0.789404i	$0.710390\pi$
$654$	0	0
$655$	166.000	0.253435
$656$	− 322.441i	− 0.491525i
$657$	0	0
$658$	0	0
$659$	972.979i	1.47645i	0.674556	+	0.738224i	$0.264335\pi$
$659$	972.979i	1.47645i	−0.674556	+	0.738224i	$0.735665\pi$
$660$	0	0
$661$	−401.000	−0.606657	−0.303328	−	0.952886i	$-0.598098\pi$
$661$	−401.000	−0.606657	−0.303328	+	0.952886i	$0.598098\pi$
$662$	86.2670i	0.130313i
$663$	0	0
$664$	−172.000	−0.259036
$665$	0	0
$666$	0	0
$667$	512.000	0.767616
$668$	121.622i	0.182069i
$669$	0	0
$670$	−86.0000	−0.128358
$671$	509.117i	0.758743i
$672$	0	0
$673$	665.000	0.988113	0.494056	−	0.869430i	$-0.335514\pi$
$673$	665.000	0.988113	0.494056	+	0.869430i	$0.335514\pi$
$674$	− 190.919i	− 0.283262i
$675$	0	0
$676$	−112.000	−0.165680
$677$	− 916.410i	− 1.35363i	−0.736151	−	0.676817i	$-0.763359\pi$
$677$	− 916.410i	− 1.35363i	0.736151	−	0.676817i	$-0.236641\pi$
$678$	0	0
$679$	0	0
$680$	− 45.2548i	− 0.0665512i
$681$	0	0
$682$	−30.0000	−0.0439883
$683$	135.765i	0.198777i	0.995049	+	0.0993884i	$0.0316886\pi$
$683$	135.765i	0.198777i	−0.995049	+	0.0993884i	$0.968311\pi$
$684$	0	0
$685$	304.000	0.443796
$686$	0	0
$687$	0	0
$688$	−340.000	−0.494186
$689$	509.117i	0.738921i
$690$	0	0
$691$	221.000	0.319826	0.159913	−	0.987131i	$-0.448879\pi$
$691$	221.000	0.319826	0.159913	+	0.987131i	$0.448879\pi$
$692$	− 226.274i	− 0.326986i
$693$	0	0
$694$	144.000	0.207493
$695$	117.380i	0.168892i
$696$	0	0
$697$	−912.000	−1.30846
$698$	214.960i	0.307966i
$699$	0	0
$700$	0	0
$701$	− 192.333i	− 0.274370i	−0.990545	−	0.137185i	$-0.956195\pi$
$701$	− 192.333i	− 0.274370i	0.990545	−	0.137185i	$-0.0438054\pi$
$702$	0	0
$703$	−221.000	−0.314367
$704$	56.5685i	0.0803530i
$705$	0	0
$706$	546.000	0.773371
$707$	0	0
$708$	0	0
$709$	−328.000	−0.462623	−0.231312	−	0.972880i	$-0.574302\pi$
$709$	−328.000	−0.462623	−0.231312	+	0.972880i	$0.574302\pi$
$710$	− 104.652i	− 0.147397i
$711$	0	0
$712$	384.000	0.539326
$713$	67.8823i	0.0952065i
$714$	0	0
$715$	−150.000	−0.209790
$716$	59.3970i	0.0829567i
$717$	0	0
$718$	64.0000	0.0891365
$719$	− 1013.99i	− 1.41028i	−0.709068	−	0.705140i	$-0.750884\pi$
$719$	− 1013.99i	− 1.41028i	0.709068	−	0.705140i	$-0.249116\pi$
$720$	0	0
$721$	0	0
$722$	271.529i	0.376079i
$723$	0	0
$724$	−162.000	−0.223757
$725$	− 520.431i	− 0.717835i
$726$	0	0
$727$	1069.00	1.47043	0.735213	−	0.677836i	$-0.237082\pi$
$727$	1069.00	1.47043	0.735213	+	0.677836i	$0.237082\pi$
$728$	0	0
$729$	0	0
$730$	190.000	0.260274
$731$	961.665i	1.31555i
$732$	0	0
$733$	495.000	0.675307	0.337653	−	0.941270i	$-0.390367\pi$
$733$	495.000	0.675307	0.337653	+	0.941270i	$0.390367\pi$
$734$	− 142.836i	− 0.194599i
$735$	0	0
$736$	128.000	0.173913
$737$	− 304.056i	− 0.412559i
$738$	0	0
$739$	453.000	0.612991	0.306495	−	0.951872i	$-0.400844\pi$
$739$	453.000	0.612991	0.306495	+	0.951872i	$0.400844\pi$
$740$	48.0833i	0.0649774i
$741$	0	0
$742$	0	0
$743$	− 538.815i	− 0.725189i	−0.931947	−	0.362594i	$-0.881891\pi$
$743$	− 538.815i	− 0.725189i	0.931947	−	0.362594i	$-0.118109\pi$
$744$	0	0
$745$	144.000	0.193289
$746$	439.820i	0.589572i
$747$	0	0
$748$	160.000	0.213904
$749$	0	0
$750$	0	0
$751$	−339.000	−0.451398	−0.225699	−	0.974197i	$-0.572467\pi$
$751$	−339.000	−0.451398	−0.225699	+	0.974197i	$0.572467\pi$
$752$	− 288.500i	− 0.383643i
$753$	0	0
$754$	−480.000	−0.636605
$755$	− 56.5685i	− 0.0749252i
$756$	0	0
$757$	198.000	0.261559	0.130779	−	0.991411i	$-0.458252\pi$
$757$	198.000	0.261559	0.130779	+	0.991411i	$0.458252\pi$
$758$	− 128.693i	− 0.169780i
$759$	0	0
$760$	−52.0000	−0.0684211
$761$	362.039i	0.475741i	0.971297	+	0.237870i	$0.0764493\pi$
$761$	362.039i	0.475741i	−0.971297	+	0.237870i	$0.923551\pi$
$762$	0	0
$763$	0	0
$764$	127.279i	0.166596i
$765$	0	0
$766$	464.000	0.605744
$767$	− 1357.65i	− 1.77007i
$768$	0	0
$769$	−929.000	−1.20806	−0.604031	−	0.796961i	$-0.706440\pi$
$769$	−929.000	−1.20806	−0.604031	+	0.796961i	$0.706440\pi$
$770$	0	0
$771$	0	0
$772$	446.000	0.577720
$773$	− 171.120i	− 0.221371i	−0.993855	−	0.110686i	$-0.964695\pi$
$773$	− 171.120i	− 0.221371i	0.993855	−	0.110686i	$-0.0353046\pi$
$774$	0	0
$775$	69.0000	0.0890323
$776$	45.2548i	0.0583181i
$777$	0	0
$778$	−974.000	−1.25193
$779$	1047.93i	1.34523i
$780$	0	0
$781$	370.000	0.473752
$782$	− 362.039i	− 0.462965i
$783$	0	0
$784$	0	0
$785$	− 418.607i	− 0.533258i
$786$	0	0
$787$	90.0000	0.114358	0.0571792	−	0.998364i	$-0.481789\pi$
$787$	90.0000	0.114358	0.0571792	+	0.998364i	$0.481789\pi$
$788$	− 316.784i	− 0.402010i
$789$	0	0
$790$	−138.000	−0.174684
$791$	0	0
$792$	0	0
$793$	−1080.00	−1.36192
$794$	394.566i	0.496934i
$795$	0	0
$796$	272.000	0.341709
$797$	− 1154.00i	− 1.44793i	−0.689838	−	0.723964i	$-0.742318\pi$
$797$	− 1154.00i	− 1.44793i	0.689838	−	0.723964i	$-0.257682\pi$
$798$	0	0
$799$	−816.000	−1.02128
$800$	− 130.108i	− 0.162635i
$801$	0	0
$802$	0	0
$803$	671.751i	0.836552i
$804$	0	0
$805$	0	0
$806$	− 63.6396i	− 0.0789573i
$807$	0	0
$808$	196.000	0.242574
$809$	− 171.120i	− 0.211520i	−0.994392	−	0.105760i	$-0.966272\pi$
$809$	− 171.120i	− 0.211520i	0.994392	−	0.105760i	$-0.0337276\pi$
$810$	0	0
$811$	−554.000	−0.683107	−0.341554	−	0.939862i	$-0.610953\pi$
$811$	−554.000	−0.683107	−0.341554	+	0.939862i	$0.610953\pi$
$812$	0	0
$813$	0	0
$814$	−170.000	−0.208845
$815$	− 181.019i	− 0.222110i
$816$	0	0
$817$	1105.00	1.35251
$818$	315.370i	0.385537i
$819$	0	0
$820$	228.000	0.278049
$821$	1062.07i	1.29364i	0.762645	+	0.646818i	$0.223900\pi$
$821$	1062.07i	1.29364i	−0.762645	+	0.646818i	$0.776100\pi$
$822$	0	0
$823$	856.000	1.04010	0.520049	−	0.854137i	$-0.325914\pi$
$823$	856.000	1.04010	0.520049	+	0.854137i	$0.325914\pi$
$824$	− 172.534i	− 0.209386i
$825$	0	0
$826$	0	0
$827$	1022.48i	1.23637i	0.786033	+	0.618184i	$0.212131\pi$
$827$	1022.48i	1.23637i	−0.786033	+	0.618184i	$0.787869\pi$
$828$	0	0
$829$	1039.00	1.25332	0.626659	−	0.779294i	$-0.284422\pi$
$829$	1039.00	1.25332	0.626659	+	0.779294i	$0.284422\pi$
$830$	− 121.622i	− 0.146533i
$831$	0	0
$832$	−120.000	−0.144231
$833$	0	0
$834$	0	0
$835$	−86.0000	−0.102994
$836$	− 183.848i	− 0.219914i
$837$	0	0
$838$	822.000	0.980907
$839$	− 656.195i	− 0.782116i	−0.920366	−	0.391058i	$-0.872109\pi$
$839$	− 656.195i	− 0.782116i	0.920366	−	0.391058i	$-0.127891\pi$
$840$	0	0
$841$	329.000	0.391201
$842$	− 216.375i	− 0.256977i
$843$	0	0
$844$	−544.000	−0.644550
$845$	− 79.1960i	− 0.0937230i
$846$	0	0
$847$	0	0
$848$	135.765i	0.160100i
$849$	0	0
$850$	−368.000	−0.432941
$851$	384.666i	0.452017i
$852$	0	0
$853$	1463.00	1.71512	0.857562	−	0.514381i	$-0.171978\pi$
$853$	1463.00	1.71512	0.857562	+	0.514381i	$0.171978\pi$
$854$	0	0
$855$	0	0
$856$	480.000	0.560748
$857$	− 67.8823i	− 0.0792092i	−0.999215	−	0.0396046i	$-0.987390\pi$
$857$	− 67.8823i	− 0.0792092i	0.999215	−	0.0396046i	$-0.0126098\pi$
$858$	0	0
$859$	−1462.00	−1.70198	−0.850990	−	0.525183i	$-0.823997\pi$
$859$	−1462.00	−1.70198	−0.850990	+	0.525183i	$0.823997\pi$
$860$	− 240.416i	− 0.279554i
$861$	0	0
$862$	−198.000	−0.229698
$863$	− 807.516i	− 0.935708i	−0.883806	−	0.467854i	$-0.845027\pi$
$863$	− 807.516i	− 0.935708i	0.883806	−	0.467854i	$-0.154973\pi$
$864$	0	0
$865$	160.000	0.184971
$866$	− 193.747i	− 0.223727i
$867$	0	0
$868$	0	0
$869$	− 487.904i	− 0.561454i
$870$	0	0
$871$	645.000	0.740528
$872$	− 183.848i	− 0.210835i
$873$	0	0
$874$	−416.000	−0.475973
$875$	0	0
$876$	0	0
$877$	1480.00	1.68757	0.843786	−	0.536680i	$-0.180322\pi$
$877$	1480.00	1.68757	0.843786	+	0.536680i	$0.180322\pi$
$878$	848.528i	0.966433i
$879$	0	0
$880$	−40.0000	−0.0454545
$881$	− 712.764i	− 0.809039i	−0.914529	−	0.404520i	$-0.867439\pi$
$881$	− 712.764i	− 0.809039i	0.914529	−	0.404520i	$-0.132561\pi$
$882$	0	0
$883$	−115.000	−0.130238	−0.0651189	−	0.997878i	$-0.520743\pi$
$883$	−115.000	−0.130238	−0.0651189	+	0.997878i	$0.520743\pi$
$884$	339.411i	0.383949i
$885$	0	0
$886$	896.000	1.01129
$887$	1192.18i	1.34406i	0.740524	+	0.672030i	$0.234578\pi$
$887$	1192.18i	1.34406i	−0.740524	+	0.672030i	$0.765422\pi$
$888$	0	0
$889$	0	0
$890$	271.529i	0.305089i
$891$	0	0
$892$	−496.000	−0.556054
$893$	937.624i	1.04997i
$894$	0	0
$895$	−42.0000	−0.0469274
$896$	0	0
$897$	0	0
$898$	−542.000	−0.603563
$899$	− 67.8823i	− 0.0755086i
$900$	0	0
$901$	384.000	0.426193
$902$	806.102i	0.893683i
$903$	0	0
$904$	388.000	0.429204
$905$	− 114.551i	− 0.126576i
$906$	0	0
$907$	−501.000	−0.552370	−0.276185	−	0.961104i	$-0.589070\pi$
$907$	−501.000	−0.552370	−0.276185	+	0.961104i	$0.589070\pi$
$908$	− 330.926i	− 0.364456i
$909$	0	0
$910$	0	0
$911$	203.647i	0.223542i	0.993734	+	0.111771i	$0.0356523\pi$
$911$	203.647i	0.223542i	−0.993734	+	0.111771i	$0.964348\pi$
$912$	0	0
$913$	430.000	0.470975
$914$	337.997i	0.369800i
$915$	0	0
$916$	−866.000	−0.945415
$917$	0	0
$918$	0	0
$919$	−653.000	−0.710555	−0.355277	−	0.934761i	$-0.615614\pi$
$919$	−653.000	−0.710555	−0.355277	+	0.934761i	$0.615614\pi$
$920$	90.5097i	0.0983801i
$921$	0	0
$922$	640.000	0.694143
$923$	784.889i	0.850367i
$924$	0	0
$925$	391.000	0.422703
$926$	− 298.399i	− 0.322245i
$927$	0	0
$928$	−128.000	−0.137931
$929$	12.7279i	0.0137007i	0.999977	+	0.00685033i	$0.00218055\pi$
$929$	12.7279i	0.0137007i	−0.999977	+	0.00685033i	$0.997819\pi$
$930$	0	0
$931$	0	0
$932$	455.377i	0.488602i
$933$	0	0
$934$	442.000	0.473233
$935$	113.137i	0.121002i
$936$	0	0
$937$	761.000	0.812166	0.406083	−	0.913836i	$-0.366894\pi$
$937$	761.000	0.812166	0.406083	+	0.913836i	$0.366894\pi$
$938$	0	0
$939$	0	0
$940$	204.000	0.217021
$941$	282.843i	0.300577i	0.988642	+	0.150288i	$0.0480202\pi$
$941$	282.843i	0.300577i	−0.988642	+	0.150288i	$0.951980\pi$
$942$	0	0
$943$	1824.00	1.93425
$944$	− 362.039i	− 0.383516i
$945$	0	0
$946$	850.000	0.898520
$947$	− 1079.04i	− 1.13944i	−0.821841	−	0.569718i	$-0.807053\pi$
$947$	− 1079.04i	− 1.13944i	0.821841	−	0.569718i	$-0.192947\pi$
$948$	0	0
$949$	−1425.00	−1.50158
$950$	422.850i	0.445105i
$951$	0	0
$952$	0	0
$953$	1074.80i	1.12781i	0.825840	+	0.563905i	$0.190701\pi$
$953$	1074.80i	1.12781i	−0.825840	+	0.563905i	$0.809299\pi$
$954$	0	0
$955$	−90.0000	−0.0942408
$956$	− 687.308i	− 0.718941i
$957$	0	0
$958$	992.000	1.03549
$959$	0	0
$960$	0	0
$961$	−952.000	−0.990635
$962$	− 360.624i	− 0.374869i
$963$	0	0
$964$	−316.000	−0.327801
$965$	315.370i	0.326808i
$966$	0	0
$967$	1163.00	1.20269	0.601344	−	0.798990i	$-0.294632\pi$
$967$	1163.00	1.20269	0.601344	+	0.798990i	$0.294632\pi$
$968$	200.818i	0.207457i
$969$	0	0
$970$	−32.0000	−0.0329897
$971$	1561.29i	1.60792i	0.594682	+	0.803961i	$0.297278\pi$
$971$	1561.29i	1.60792i	−0.594682	+	0.803961i	$0.702722\pi$
$972$	0	0
$973$	0	0
$974$	− 592.555i	− 0.608373i
$975$	0	0
$976$	−288.000	−0.295082
$977$	− 1132.79i	− 1.15945i	−0.814811	−	0.579726i	$-0.803160\pi$
$977$	− 1132.79i	− 1.15945i	0.814811	−	0.579726i	$-0.196840\pi$
$978$	0	0
$979$	−960.000	−0.980592
$980$	0	0
$981$	0	0
$982$	−240.000	−0.244399
$983$	305.470i	0.310753i	0.987855	+	0.155376i	$0.0496591\pi$
$983$	305.470i	0.310753i	−0.987855	+	0.155376i	$0.950341\pi$
$984$	0	0
$985$	224.000	0.227411
$986$	362.039i	0.367179i
$987$	0	0
$988$	390.000	0.394737
$989$	− 1923.33i	− 1.94472i
$990$	0	0
$991$	−973.000	−0.981837	−0.490918	−	0.871206i	$-0.663339\pi$
$991$	−973.000	−0.981837	−0.490918	+	0.871206i	$0.663339\pi$
$992$	− 16.9706i	− 0.0171074i
$993$	0	0
$994$	0	0
$995$	192.333i	0.193300i
$996$	0	0
$997$	−79.0000	−0.0792377	−0.0396189	−	0.999215i	$-0.512614\pi$
$997$	−79.0000	−0.0792377	−0.0396189	+	0.999215i	$0.512614\pi$
$998$	− 264.458i	− 0.264988i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	882.3.b.c.197.1		2
3.2	odd	2	inner	882.3.b.c.197.2		2
7.2	even	3		126.3.s.b.53.1	✓	4
7.3	odd	6		882.3.s.c.863.2		4
7.4	even	3		126.3.s.b.107.2	yes	4
7.5	odd	6		882.3.s.c.557.1		4
7.6	odd	2		882.3.b.d.197.1		2
21.2	odd	6		126.3.s.b.53.2	yes	4
21.5	even	6		882.3.s.c.557.2		4
21.11	odd	6		126.3.s.b.107.1	yes	4
21.17	even	6		882.3.s.c.863.1		4
21.20	even	2		882.3.b.d.197.2		2
28.11	odd	6		1008.3.dc.a.737.2		4
28.23	odd	6		1008.3.dc.a.305.1		4
84.11	even	6		1008.3.dc.a.737.1		4
84.23	even	6		1008.3.dc.a.305.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
126.3.s.b.53.1	✓	4	7.2	even	3
126.3.s.b.53.2	yes	4	21.2	odd	6
126.3.s.b.107.1	yes	4	21.11	odd	6
126.3.s.b.107.2	yes	4	7.4	even	3
882.3.b.c.197.1		2	1.1	even	1	trivial
882.3.b.c.197.2		2	3.2	odd	2	inner
882.3.b.d.197.1		2	7.6	odd	2
882.3.b.d.197.2		2	21.20	even	2
882.3.s.c.557.1		4	7.5	odd	6
882.3.s.c.557.2		4	21.5	even	6
882.3.s.c.863.1		4	21.17	even	6
882.3.s.c.863.2		4	7.3	odd	6
1008.3.dc.a.305.1		4	28.23	odd	6
1008.3.dc.a.305.2		4	84.23	even	6
1008.3.dc.a.737.1		4	84.11	even	6
1008.3.dc.a.737.2		4	28.11	odd	6

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:	\( N \)	\(=\)	\( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	882.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.41421i q^{2} -2.00000 q^{4} -1.41421i q^{5} +2.82843i q^{8} +O(q^{10})\)	\(q-1.41421i q^{2} -2.00000 q^{4} -1.41421i q^{5} +2.82843i q^{8} -2.00000 q^{10} -7.07107i q^{11} +15.0000 q^{13} +4.00000 q^{16} -11.3137i q^{17} -13.0000 q^{19} +2.82843i q^{20} -10.0000 q^{22} +22.6274i q^{23} +23.0000 q^{25} -21.2132i q^{26} -22.6274i q^{29} +3.00000 q^{31} -5.65685i q^{32} -16.0000 q^{34} +17.0000 q^{37} +18.3848i q^{38} +4.00000 q^{40} -80.6102i q^{41} -85.0000 q^{43} +14.1421i q^{44} +32.0000 q^{46} -72.1249i q^{47} -32.5269i q^{50} -30.0000 q^{52} +33.9411i q^{53} -10.0000 q^{55} -32.0000 q^{58} -90.5097i q^{59} -72.0000 q^{61} -4.24264i q^{62} -8.00000 q^{64} -21.2132i q^{65} +43.0000 q^{67} +22.6274i q^{68} +52.3259i q^{71} -95.0000 q^{73} -24.0416i q^{74} +26.0000 q^{76} +69.0000 q^{79} -5.65685i q^{80} -114.000 q^{82} +60.8112i q^{83} -16.0000 q^{85} +120.208i q^{86} +20.0000 q^{88} -135.765i q^{89} -45.2548i q^{92} -102.000 q^{94} +18.3848i q^{95} +16.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{4}+O(q^{10}) \) 2 * q - 4 * q^4	\( 2 q - 4 q^{4} - 4 q^{10} + 30 q^{13} + 8 q^{16} - 26 q^{19} - 20 q^{22} + 46 q^{25} + 6 q^{31} - 32 q^{34} + 34 q^{37} + 8 q^{40} - 170 q^{43} + 64 q^{46} - 60 q^{52} - 20 q^{55} - 64 q^{58} - 144 q^{61} - 16 q^{64} + 86 q^{67} - 190 q^{73} + 52 q^{76} + 138 q^{79} - 228 q^{82} - 32 q^{85} + 40 q^{88} - 204 q^{94} + 32 q^{97}+O(q^{100}) \) 2 * q - 4 * q^4 - 4 * q^10 + 30 * q^13 + 8 * q^16 - 26 * q^19 - 20 * q^22 + 46 * q^25 + 6 * q^31 - 32 * q^34 + 34 * q^37 + 8 * q^40 - 170 * q^43 + 64 * q^46 - 60 * q^52 - 20 * q^55 - 64 * q^58 - 144 * q^61 - 16 * q^64 + 86 * q^67 - 190 * q^73 + 52 * q^76 + 138 * q^79 - 228 * q^82 - 32 * q^85 + 40 * q^88 - 204 * q^94 + 32 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more