LMFDB - Embedded newform 882.3.b.h.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:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			197.1
Root			$0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	882.197
Dual form			882.3.b.h.197.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421i q^{2} -2.00000 q^{4} -4.00000i q^{5} +2.82843i q^{8} +O(q^{10})$	$q-1.41421i q^{2} -2.00000 q^{4} -4.00000i q^{5} +2.82843i q^{8} -5.65685 q^{10} +2.82843i q^{11} +12.7279 q^{13} +4.00000 q^{16} -4.00000i q^{17} +22.6274 q^{19} +8.00000i q^{20} +4.00000 q^{22} -36.7696i q^{23} +9.00000 q^{25} -18.0000i q^{26} -32.5269i q^{29} -50.9117 q^{31} -5.65685i q^{32} -5.65685 q^{34} -32.0000 q^{37} -32.0000i q^{38} +11.3137 q^{40} +38.0000i q^{41} +20.0000 q^{43} -5.65685i q^{44} -52.0000 q^{46} +20.0000i q^{47} -12.7279i q^{50} -25.4558 q^{52} -94.7523i q^{53} +11.3137 q^{55} -46.0000 q^{58} -4.00000i q^{59} +83.4386 q^{61} +72.0000i q^{62} -8.00000 q^{64} -50.9117i q^{65} -48.0000 q^{67} +8.00000i q^{68} -76.3675i q^{71} -120.208 q^{73} +45.2548i q^{74} -45.2548 q^{76} -148.000 q^{79} -16.0000i q^{80} +53.7401 q^{82} -80.0000i q^{83} -16.0000 q^{85} -28.2843i q^{86} -8.00000 q^{88} +106.000i q^{89} +73.5391i q^{92} +28.2843 q^{94} -90.5097i q^{95} +154.149 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{4}+O(q^{10}) $ 4 * q - 8 * q^4	$ 4 q - 8 q^{4} + 16 q^{16} + 16 q^{22} + 36 q^{25} - 128 q^{37} + 80 q^{43} - 208 q^{46} - 184 q^{58} - 32 q^{64} - 192 q^{67} - 592 q^{79} - 64 q^{85} - 32 q^{88}+O(q^{100}) $ 4 * q - 8 * q^4 + 16 * q^16 + 16 * q^22 + 36 * q^25 - 128 * q^37 + 80 * q^43 - 208 * q^46 - 184 * q^58 - 32 * q^64 - 192 * q^67 - 592 * q^79 - 64 * q^85 - 32 * q^88

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
$5$	− 4.00000i	− 0.800000i	−0.916515	−	0.400000i	$-0.869010\pi$
$5$	− 4.00000i	− 0.800000i	0.916515	−	0.400000i	$-0.130990\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	2.82843i	0.353553i
$9$	0	0
$10$	−5.65685	−0.565685
$11$	2.82843i	0.257130i	0.991701	+	0.128565i	$0.0410371\pi$
$11$	2.82843i	0.257130i	−0.991701	+	0.128565i	$0.958963\pi$
$12$	0	0
$13$	12.7279	0.979071	0.489535	−	0.871983i	$-0.337166\pi$
$13$	12.7279	0.979071	0.489535	+	0.871983i	$0.337166\pi$
$14$	0	0
$15$	0	0
$16$	4.00000	0.250000
$17$	− 4.00000i	− 0.235294i	−0.993055	−	0.117647i	$-0.962465\pi$
$17$	− 4.00000i	− 0.235294i	0.993055	−	0.117647i	$-0.0375352\pi$
$18$	0	0
$19$	22.6274	1.19092	0.595458	−	0.803386i	$-0.296970\pi$
$19$	22.6274	1.19092	0.595458	+	0.803386i	$0.296970\pi$
$20$	8.00000i	0.400000i
$21$	0	0
$22$	4.00000	0.181818
$23$	− 36.7696i	− 1.59868i	−0.600882	−	0.799338i	$-0.705184\pi$
$23$	− 36.7696i	− 1.59868i	0.600882	−	0.799338i	$-0.294816\pi$
$24$	0	0
$25$	9.00000	0.360000
$26$	− 18.0000i	− 0.692308i
$27$	0	0
$28$	0	0
$29$	− 32.5269i	− 1.12162i	−0.827945	−	0.560809i	$-0.810490\pi$
$29$	− 32.5269i	− 1.12162i	0.827945	−	0.560809i	$-0.189510\pi$
$30$	0	0
$31$	−50.9117	−1.64231	−0.821156	−	0.570703i	$-0.806671\pi$
$31$	−50.9117	−1.64231	−0.821156	+	0.570703i	$0.806671\pi$
$32$	− 5.65685i	− 0.176777i
$33$	0	0
$34$	−5.65685	−0.166378
$35$	0	0
$36$	0	0
$37$	−32.0000	−0.864865	−0.432432	−	0.901666i	$-0.642345\pi$
$37$	−32.0000	−0.864865	−0.432432	+	0.901666i	$0.642345\pi$
$38$	− 32.0000i	− 0.842105i
$39$	0	0
$40$	11.3137	0.282843
$41$	38.0000i	0.926829i	0.886142	+	0.463415i	$0.153376\pi$
$41$	38.0000i	0.926829i	−0.886142	+	0.463415i	$0.846624\pi$
$42$	0	0
$43$	20.0000	0.465116	0.232558	−	0.972582i	$-0.425290\pi$
$43$	20.0000	0.465116	0.232558	+	0.972582i	$0.425290\pi$
$44$	− 5.65685i	− 0.128565i
$45$	0	0
$46$	−52.0000	−1.13043
$47$	20.0000i	0.425532i	0.977103	+	0.212766i	$0.0682472\pi$
$47$	20.0000i	0.425532i	−0.977103	+	0.212766i	$0.931753\pi$
$48$	0	0
$49$	0	0
$50$	− 12.7279i	− 0.254558i
$51$	0	0
$52$	−25.4558	−0.489535
$53$	− 94.7523i	− 1.78778i	−0.448287	−	0.893890i	$-0.647966\pi$
$53$	− 94.7523i	− 1.78778i	0.448287	−	0.893890i	$-0.352034\pi$
$54$	0	0
$55$	11.3137	0.205704
$56$	0	0
$57$	0	0
$58$	−46.0000	−0.793103
$59$	− 4.00000i	− 0.0677966i	−0.999425	−	0.0338983i	$-0.989208\pi$
$59$	− 4.00000i	− 0.0677966i	0.999425	−	0.0338983i	$-0.0107922\pi$
$60$	0	0
$61$	83.4386	1.36785	0.683923	−	0.729554i	$-0.260272\pi$
$61$	83.4386	1.36785	0.683923	+	0.729554i	$0.260272\pi$
$62$	72.0000i	1.16129i
$63$	0	0
$64$	−8.00000	−0.125000
$65$	− 50.9117i	− 0.783257i
$66$	0	0
$67$	−48.0000	−0.716418	−0.358209	−	0.933641i	$-0.616612\pi$
$67$	−48.0000	−0.716418	−0.358209	+	0.933641i	$0.616612\pi$
$68$	8.00000i	0.117647i
$69$	0	0
$70$	0	0
$71$	− 76.3675i	− 1.07560i	−0.843073	−	0.537800i	$-0.819256\pi$
$71$	− 76.3675i	− 1.07560i	0.843073	−	0.537800i	$-0.180744\pi$
$72$	0	0
$73$	−120.208	−1.64669	−0.823344	−	0.567543i	$-0.807894\pi$
$73$	−120.208	−1.64669	−0.823344	+	0.567543i	$0.807894\pi$
$74$	45.2548i	0.611552i
$75$	0	0
$76$	−45.2548	−0.595458
$77$	0	0
$78$	0	0
$79$	−148.000	−1.87342	−0.936709	−	0.350109i	$-0.886144\pi$
$79$	−148.000	−1.87342	−0.936709	+	0.350109i	$0.886144\pi$
$80$	− 16.0000i	− 0.200000i
$81$	0	0
$82$	53.7401	0.655367
$83$	− 80.0000i	− 0.963855i	−0.876211	−	0.481928i	$-0.839937\pi$
$83$	− 80.0000i	− 0.963855i	0.876211	−	0.481928i	$-0.160063\pi$
$84$	0	0
$85$	−16.0000	−0.188235
$86$	− 28.2843i	− 0.328887i
$87$	0	0
$88$	−8.00000	−0.0909091
$89$	106.000i	1.19101i	0.803351	+	0.595506i	$0.203048\pi$
$89$	106.000i	1.19101i	−0.803351	+	0.595506i	$0.796952\pi$
$90$	0	0
$91$	0	0
$92$	73.5391i	0.799338i
$93$	0	0
$94$	28.2843	0.300897
$95$	− 90.5097i	− 0.952733i
$96$	0	0
$97$	154.149	1.58917	0.794584	−	0.607154i	$-0.207689\pi$
$97$	154.149	1.58917	0.794584	+	0.607154i	$0.207689\pi$
$98$	0	0
$99$	0	0
$100$	−18.0000	−0.180000
$101$	− 126.000i	− 1.24752i	−0.781614	−	0.623762i	$-0.785603\pi$
$101$	− 126.000i	− 1.24752i	0.781614	−	0.623762i	$-0.214397\pi$
$102$	0	0
$103$	−73.5391	−0.713972	−0.356986	−	0.934110i	$-0.616196\pi$
$103$	−73.5391	−0.713972	−0.356986	+	0.934110i	$0.616196\pi$
$104$	36.0000i	0.346154i
$105$	0	0
$106$	−134.000	−1.26415
$107$	− 70.7107i	− 0.660847i	−0.943833	−	0.330424i	$-0.892808\pi$
$107$	− 70.7107i	− 0.660847i	0.943833	−	0.330424i	$-0.107192\pi$
$108$	0	0
$109$	−86.0000	−0.788991	−0.394495	−	0.918898i	$-0.629081\pi$
$109$	−86.0000	−0.788991	−0.394495	+	0.918898i	$0.629081\pi$
$110$	− 16.0000i	− 0.145455i
$111$	0	0
$112$	0	0
$113$	21.2132i	0.187727i	0.995585	+	0.0938637i	$0.0299218\pi$
$113$	21.2132i	0.187727i	−0.995585	+	0.0938637i	$0.970078\pi$
$114$	0	0
$115$	−147.078	−1.27894
$116$	65.0538i	0.560809i
$117$	0	0
$118$	−5.65685	−0.0479394
$119$	0	0
$120$	0	0
$121$	113.000	0.933884
$122$	− 118.000i	− 0.967213i
$123$	0	0
$124$	101.823	0.821156
$125$	− 136.000i	− 1.08800i
$126$	0	0
$127$	4.00000	0.0314961	0.0157480	−	0.999876i	$-0.494987\pi$
$127$	4.00000	0.0314961	0.0157480	+	0.999876i	$0.494987\pi$
$128$	11.3137i	0.0883883i
$129$	0	0
$130$	−72.0000	−0.553846
$131$	24.0000i	0.183206i	0.995796	+	0.0916031i	$0.0291991\pi$
$131$	24.0000i	0.183206i	−0.995796	+	0.0916031i	$0.970801\pi$
$132$	0	0
$133$	0	0
$134$	67.8823i	0.506584i
$135$	0	0
$136$	11.3137	0.0831890
$137$	− 72.1249i	− 0.526459i	−0.964733	−	0.263230i	$-0.915212\pi$
$137$	− 72.1249i	− 0.526459i	0.964733	−	0.263230i	$-0.0847877\pi$
$138$	0	0
$139$	141.421	1.01742	0.508710	−	0.860938i	$-0.330123\pi$
$139$	141.421	1.01742	0.508710	+	0.860938i	$0.330123\pi$
$140$	0	0
$141$	0	0
$142$	−108.000	−0.760563
$143$	36.0000i	0.251748i
$144$	0	0
$145$	−130.108	−0.897294
$146$	170.000i	1.16438i
$147$	0	0
$148$	64.0000	0.432432
$149$	− 26.8701i	− 0.180336i	−0.995927	−	0.0901680i	$-0.971260\pi$
$149$	− 26.8701i	− 0.180336i	0.995927	−	0.0901680i	$-0.0287404\pi$
$150$	0	0
$151$	152.000	1.00662	0.503311	−	0.864105i	$-0.332115\pi$
$151$	152.000	1.00662	0.503311	+	0.864105i	$0.332115\pi$
$152$	64.0000i	0.421053i
$153$	0	0
$154$	0	0
$155$	203.647i	1.31385i
$156$	0	0
$157$	94.7523	0.603518	0.301759	−	0.953384i	$-0.402426\pi$
$157$	94.7523	0.603518	0.301759	+	0.953384i	$0.402426\pi$
$158$	209.304i	1.32471i
$159$	0	0
$160$	−22.6274	−0.141421
$161$	0	0
$162$	0	0
$163$	−152.000	−0.932515	−0.466258	−	0.884649i	$-0.654398\pi$
$163$	−152.000	−0.932515	−0.466258	+	0.884649i	$0.654398\pi$
$164$	− 76.0000i	− 0.463415i
$165$	0	0
$166$	−113.137	−0.681549
$167$	− 116.000i	− 0.694611i	−0.937752	−	0.347305i	$-0.887097\pi$
$167$	− 116.000i	− 0.694611i	0.937752	−	0.347305i	$-0.112903\pi$
$168$	0	0
$169$	−7.00000	−0.0414201
$170$	22.6274i	0.133102i
$171$	0	0
$172$	−40.0000	−0.232558
$173$	222.000i	1.28324i	0.767024	+	0.641618i	$0.221737\pi$
$173$	222.000i	1.28324i	−0.767024	+	0.641618i	$0.778263\pi$
$174$	0	0
$175$	0	0
$176$	11.3137i	0.0642824i
$177$	0	0
$178$	149.907	0.842172
$179$	− 59.3970i	− 0.331827i	−0.986140	−	0.165913i	$-0.946943\pi$
$179$	− 59.3970i	− 0.331827i	0.986140	−	0.165913i	$-0.0530572\pi$
$180$	0	0
$181$	80.6102	0.445360	0.222680	−	0.974892i	$-0.428519\pi$
$181$	80.6102	0.445360	0.222680	+	0.974892i	$0.428519\pi$
$182$	0	0
$183$	0	0
$184$	104.000	0.565217
$185$	128.000i	0.691892i
$186$	0	0
$187$	11.3137	0.0605011
$188$	− 40.0000i	− 0.212766i
$189$	0	0
$190$	−128.000	−0.673684
$191$	302.642i	1.58451i	0.610189	+	0.792256i	$0.291093\pi$
$191$	302.642i	1.58451i	−0.610189	+	0.792256i	$0.708907\pi$
$192$	0	0
$193$	218.000	1.12953	0.564767	−	0.825251i	$-0.308966\pi$
$193$	218.000	1.12953	0.564767	+	0.825251i	$0.308966\pi$
$194$	− 218.000i	− 1.12371i
$195$	0	0
$196$	0	0
$197$	− 287.085i	− 1.45729i	−0.684894	−	0.728643i	$-0.740151\pi$
$197$	− 287.085i	− 1.45729i	0.684894	−	0.728643i	$-0.259849\pi$
$198$	0	0
$199$	−67.8823	−0.341117	−0.170558	−	0.985348i	$-0.554557\pi$
$199$	−67.8823	−0.341117	−0.170558	+	0.985348i	$0.554557\pi$
$200$	25.4558i	0.127279i
$201$	0	0
$202$	−178.191	−0.882133
$203$	0	0
$204$	0	0
$205$	152.000	0.741463
$206$	104.000i	0.504854i
$207$	0	0
$208$	50.9117	0.244768
$209$	64.0000i	0.306220i
$210$	0	0
$211$	132.000	0.625592	0.312796	−	0.949820i	$-0.398734\pi$
$211$	132.000	0.625592	0.312796	+	0.949820i	$0.398734\pi$
$212$	189.505i	0.893890i
$213$	0	0
$214$	−100.000	−0.467290
$215$	− 80.0000i	− 0.372093i
$216$	0	0
$217$	0	0
$218$	121.622i	0.557901i
$219$	0	0
$220$	−22.6274	−0.102852
$221$	− 50.9117i	− 0.230370i
$222$	0	0
$223$	−169.706	−0.761012	−0.380506	−	0.924778i	$-0.624250\pi$
$223$	−169.706	−0.761012	−0.380506	+	0.924778i	$0.624250\pi$
$224$	0	0
$225$	0	0
$226$	30.0000	0.132743
$227$	300.000i	1.32159i	0.750568	+	0.660793i	$0.229780\pi$
$227$	300.000i	1.32159i	−0.750568	+	0.660793i	$0.770220\pi$
$228$	0	0
$229$	−250.316	−1.09308	−0.546541	−	0.837432i	$-0.684056\pi$
$229$	−250.316	−1.09308	−0.546541	+	0.837432i	$0.684056\pi$
$230$	208.000i	0.904348i
$231$	0	0
$232$	92.0000	0.396552
$233$	− 29.6985i	− 0.127461i	−0.997967	−	0.0637307i	$-0.979700\pi$
$233$	− 29.6985i	− 0.127461i	0.997967	−	0.0637307i	$-0.0202999\pi$
$234$	0	0
$235$	80.0000	0.340426
$236$	8.00000i	0.0338983i
$237$	0	0
$238$	0	0
$239$	− 161.220i	− 0.674562i	−0.941404	−	0.337281i	$-0.890493\pi$
$239$	− 161.220i	− 0.674562i	0.941404	−	0.337281i	$-0.109507\pi$
$240$	0	0
$241$	377.595	1.56678	0.783392	−	0.621528i	$-0.213488\pi$
$241$	377.595	1.56678	0.783392	+	0.621528i	$0.213488\pi$
$242$	− 159.806i	− 0.660356i
$243$	0	0
$244$	−166.877	−0.683923
$245$	0	0
$246$	0	0
$247$	288.000	1.16599
$248$	− 144.000i	− 0.580645i
$249$	0	0
$250$	−192.333	−0.769332
$251$	− 476.000i	− 1.89641i	−0.317653	−	0.948207i	$-0.602895\pi$
$251$	− 476.000i	− 1.89641i	0.317653	−	0.948207i	$-0.397105\pi$
$252$	0	0
$253$	104.000	0.411067
$254$	− 5.65685i	− 0.0222711i
$255$	0	0
$256$	16.0000	0.0625000
$257$	− 102.000i	− 0.396887i	−0.980112	−	0.198444i	$-0.936411\pi$
$257$	− 102.000i	− 0.396887i	0.980112	−	0.198444i	$-0.0635887\pi$
$258$	0	0
$259$	0	0
$260$	101.823i	0.391628i
$261$	0	0
$262$	33.9411	0.129546
$263$	364.867i	1.38733i	0.720299	+	0.693664i	$0.244005\pi$
$263$	364.867i	1.38733i	−0.720299	+	0.693664i	$0.755995\pi$
$264$	0	0
$265$	−379.009	−1.43022
$266$	0	0
$267$	0	0
$268$	96.0000	0.358209
$269$	196.000i	0.728625i	0.931277	+	0.364312i	$0.118696\pi$
$269$	196.000i	0.728625i	−0.931277	+	0.364312i	$0.881304\pi$
$270$	0	0
$271$	175.362	0.647094	0.323547	−	0.946212i	$-0.395125\pi$
$271$	175.362	0.647094	0.323547	+	0.946212i	$0.395125\pi$
$272$	− 16.0000i	− 0.0588235i
$273$	0	0
$274$	−102.000	−0.372263
$275$	25.4558i	0.0925667i
$276$	0	0
$277$	256.000	0.924188	0.462094	−	0.886831i	$-0.347098\pi$
$277$	256.000	0.924188	0.462094	+	0.886831i	$0.347098\pi$
$278$	− 200.000i	− 0.719424i
$279$	0	0
$280$	0	0
$281$	− 26.8701i	− 0.0956230i	−0.998856	−	0.0478115i	$-0.984775\pi$
$281$	− 26.8701i	− 0.0956230i	0.998856	−	0.0478115i	$-0.0152247\pi$
$282$	0	0
$283$	503.460	1.77901	0.889505	−	0.456925i	$-0.151049\pi$
$283$	503.460	1.77901	0.889505	+	0.456925i	$0.151049\pi$
$284$	152.735i	0.537800i
$285$	0	0
$286$	50.9117	0.178013
$287$	0	0
$288$	0	0
$289$	273.000	0.944637
$290$	184.000i	0.634483i
$291$	0	0
$292$	240.416	0.823344
$293$	190.000i	0.648464i	0.945978	+	0.324232i	$0.105106\pi$
$293$	190.000i	0.648464i	−0.945978	+	0.324232i	$0.894894\pi$
$294$	0	0
$295$	−16.0000	−0.0542373
$296$	− 90.5097i	− 0.305776i
$297$	0	0
$298$	−38.0000	−0.127517
$299$	− 468.000i	− 1.56522i
$300$	0	0
$301$	0	0
$302$	− 214.960i	− 0.711790i
$303$	0	0
$304$	90.5097	0.297729
$305$	− 333.754i	− 1.09428i
$306$	0	0
$307$	265.872	0.866033	0.433017	−	0.901386i	$-0.357449\pi$
$307$	265.872	0.866033	0.433017	+	0.901386i	$0.357449\pi$
$308$	0	0
$309$	0	0
$310$	288.000	0.929032
$311$	532.000i	1.71061i	0.518124	+	0.855305i	$0.326631\pi$
$311$	532.000i	1.71061i	−0.518124	+	0.855305i	$0.673369\pi$
$312$	0	0
$313$	148.492	0.474417	0.237208	−	0.971459i	$-0.423768\pi$
$313$	148.492	0.474417	0.237208	+	0.971459i	$0.423768\pi$
$314$	− 134.000i	− 0.426752i
$315$	0	0
$316$	296.000	0.936709
$317$	− 230.517i	− 0.727182i	−0.931559	−	0.363591i	$-0.881551\pi$
$317$	− 230.517i	− 0.727182i	0.931559	−	0.363591i	$-0.118449\pi$
$318$	0	0
$319$	92.0000	0.288401
$320$	32.0000i	0.100000i
$321$	0	0
$322$	0	0
$323$	− 90.5097i	− 0.280216i
$324$	0	0
$325$	114.551	0.352466
$326$	214.960i	0.659388i
$327$	0	0
$328$	−107.480	−0.327684
$329$	0	0
$330$	0	0
$331$	268.000	0.809668	0.404834	−	0.914390i	$-0.367329\pi$
$331$	268.000	0.809668	0.404834	+	0.914390i	$0.367329\pi$
$332$	160.000i	0.481928i
$333$	0	0
$334$	−164.049	−0.491164
$335$	192.000i	0.573134i
$336$	0	0
$337$	170.000	0.504451	0.252226	−	0.967668i	$-0.418838\pi$
$337$	170.000	0.504451	0.252226	+	0.967668i	$0.418838\pi$
$338$	9.89949i	0.0292884i
$339$	0	0
$340$	32.0000	0.0941176
$341$	− 144.000i	− 0.422287i
$342$	0	0
$343$	0	0
$344$	56.5685i	0.164443i
$345$	0	0
$346$	313.955	0.907386
$347$	280.014i	0.806958i	0.914989	+	0.403479i	$0.132199\pi$
$347$	280.014i	0.806958i	−0.914989	+	0.403479i	$0.867801\pi$
$348$	0	0
$349$	−241.831	−0.692924	−0.346462	−	0.938064i	$-0.612617\pi$
$349$	−241.831	−0.692924	−0.346462	+	0.938064i	$0.612617\pi$
$350$	0	0
$351$	0	0
$352$	16.0000	0.0454545
$353$	− 532.000i	− 1.50708i	−0.657401	−	0.753541i	$-0.728344\pi$
$353$	− 532.000i	− 1.50708i	0.657401	−	0.753541i	$-0.271656\pi$
$354$	0	0
$355$	−305.470	−0.860479
$356$	− 212.000i	− 0.595506i
$357$	0	0
$358$	−84.0000	−0.234637
$359$	342.240i	0.953314i	0.879089	+	0.476657i	$0.158152\pi$
$359$	342.240i	0.953314i	−0.879089	+	0.476657i	$0.841848\pi$
$360$	0	0
$361$	151.000	0.418283
$362$	− 114.000i	− 0.314917i
$363$	0	0
$364$	0	0
$365$	480.833i	1.31735i
$366$	0	0
$367$	−684.479	−1.86507	−0.932533	−	0.361084i	$-0.882407\pi$
$367$	−684.479	−1.86507	−0.932533	+	0.361084i	$0.882407\pi$
$368$	− 147.078i	− 0.399669i
$369$	0	0
$370$	181.019	0.489241
$371$	0	0
$372$	0	0
$373$	382.000	1.02413	0.512064	−	0.858947i	$-0.328881\pi$
$373$	382.000	1.02413	0.512064	+	0.858947i	$0.328881\pi$
$374$	− 16.0000i	− 0.0427807i
$375$	0	0
$376$	−56.5685	−0.150448
$377$	− 414.000i	− 1.09814i
$378$	0	0
$379$	−140.000	−0.369393	−0.184697	−	0.982796i	$-0.559130\pi$
$379$	−140.000	−0.369393	−0.184697	+	0.982796i	$0.559130\pi$
$380$	181.019i	0.476367i
$381$	0	0
$382$	428.000	1.12042
$383$	− 248.000i	− 0.647520i	−0.946139	−	0.323760i	$-0.895053\pi$
$383$	− 248.000i	− 0.647520i	0.946139	−	0.323760i	$-0.104947\pi$
$384$	0	0
$385$	0	0
$386$	− 308.299i	− 0.798701i
$387$	0	0
$388$	−308.299	−0.794584
$389$	340.825i	0.876158i	0.898936	+	0.438079i	$0.144341\pi$
$389$	340.825i	0.876158i	−0.898936	+	0.438079i	$0.855659\pi$
$390$	0	0
$391$	−147.078	−0.376159
$392$	0	0
$393$	0	0
$394$	−406.000	−1.03046
$395$	592.000i	1.49873i
$396$	0	0
$397$	−244.659	−0.616269	−0.308135	−	0.951343i	$-0.599705\pi$
$397$	−244.659	−0.616269	−0.308135	+	0.951343i	$0.599705\pi$
$398$	96.0000i	0.241206i
$399$	0	0
$400$	36.0000	0.0900000
$401$	− 29.6985i	− 0.0740611i	−0.999314	−	0.0370305i	$-0.988210\pi$
$401$	− 29.6985i	− 0.0740611i	0.999314	−	0.0370305i	$-0.0117899\pi$
$402$	0	0
$403$	−648.000	−1.60794
$404$	252.000i	0.623762i
$405$	0	0
$406$	0	0
$407$	− 90.5097i	− 0.222382i
$408$	0	0
$409$	606.698	1.48337	0.741684	−	0.670749i	$-0.234027\pi$
$409$	606.698	1.48337	0.741684	+	0.670749i	$0.234027\pi$
$410$	− 214.960i	− 0.524294i
$411$	0	0
$412$	147.078	0.356986
$413$	0	0
$414$	0	0
$415$	−320.000	−0.771084
$416$	− 72.0000i	− 0.173077i
$417$	0	0
$418$	90.5097	0.216530
$419$	692.000i	1.65155i	0.563999	+	0.825776i	$0.309262\pi$
$419$	692.000i	1.65155i	−0.563999	+	0.825776i	$0.690738\pi$
$420$	0	0
$421$	384.000	0.912114	0.456057	−	0.889951i	$-0.349261\pi$
$421$	384.000	0.912114	0.456057	+	0.889951i	$0.349261\pi$
$422$	− 186.676i	− 0.442361i
$423$	0	0
$424$	268.000	0.632075
$425$	− 36.0000i	− 0.0847059i
$426$	0	0
$427$	0	0
$428$	141.421i	0.330424i
$429$	0	0
$430$	−113.137	−0.263109
$431$	− 110.309i	− 0.255937i	−0.991778	−	0.127968i	$-0.959154\pi$
$431$	− 110.309i	− 0.255937i	0.991778	−	0.127968i	$-0.0408456\pi$
$432$	0	0
$433$	−156.978	−0.362535	−0.181268	−	0.983434i	$-0.558020\pi$
$433$	−156.978	−0.362535	−0.181268	+	0.983434i	$0.558020\pi$
$434$	0	0
$435$	0	0
$436$	172.000	0.394495
$437$	− 832.000i	− 1.90389i
$438$	0	0
$439$	203.647	0.463888	0.231944	−	0.972729i	$-0.425491\pi$
$439$	203.647	0.463888	0.231944	+	0.972729i	$0.425491\pi$
$440$	32.0000i	0.0727273i
$441$	0	0
$442$	−72.0000	−0.162896
$443$	811.759i	1.83241i	0.400707	+	0.916206i	$0.368764\pi$
$443$	811.759i	1.83241i	−0.400707	+	0.916206i	$0.631236\pi$
$444$	0	0
$445$	424.000	0.952809
$446$	240.000i	0.538117i
$447$	0	0
$448$	0	0
$449$	− 284.257i	− 0.633089i	−0.948578	−	0.316544i	$-0.897477\pi$
$449$	− 284.257i	− 0.633089i	0.948578	−	0.316544i	$-0.102523\pi$
$450$	0	0
$451$	−107.480	−0.238315
$452$	− 42.4264i	− 0.0938637i
$453$	0	0
$454$	424.264	0.934502
$455$	0	0
$456$	0	0
$457$	−288.000	−0.630197	−0.315098	−	0.949059i	$-0.602038\pi$
$457$	−288.000	−0.630197	−0.315098	+	0.949059i	$0.602038\pi$
$458$	354.000i	0.772926i
$459$	0	0
$460$	294.156	0.639470
$461$	706.000i	1.53145i	0.643166	+	0.765727i	$0.277620\pi$
$461$	706.000i	1.53145i	−0.643166	+	0.765727i	$0.722380\pi$
$462$	0	0
$463$	−356.000	−0.768898	−0.384449	−	0.923146i	$-0.625609\pi$
$463$	−356.000	−0.768898	−0.384449	+	0.923146i	$0.625609\pi$
$464$	− 130.108i	− 0.280404i
$465$	0	0
$466$	−42.0000	−0.0901288
$467$	884.000i	1.89293i	0.322801	+	0.946467i	$0.395375\pi$
$467$	884.000i	1.89293i	−0.322801	+	0.946467i	$0.604625\pi$
$468$	0	0
$469$	0	0
$470$	− 113.137i	− 0.240717i
$471$	0	0
$472$	11.3137	0.0239697
$473$	56.5685i	0.119595i
$474$	0	0
$475$	203.647	0.428730
$476$	0	0
$477$	0	0
$478$	−228.000	−0.476987
$479$	− 508.000i	− 1.06054i	−0.847828	−	0.530271i	$-0.822090\pi$
$479$	− 508.000i	− 1.06054i	0.847828	−	0.530271i	$-0.177910\pi$
$480$	0	0
$481$	−407.294	−0.846764
$482$	− 534.000i	− 1.10788i
$483$	0	0
$484$	−226.000	−0.466942
$485$	− 616.597i	− 1.27133i
$486$	0	0
$487$	−624.000	−1.28131	−0.640657	−	0.767827i	$-0.721338\pi$
$487$	−624.000	−1.28131	−0.640657	+	0.767827i	$0.721338\pi$
$488$	236.000i	0.483607i
$489$	0	0
$490$	0	0
$491$	840.043i	1.71088i	0.517901	+	0.855441i	$0.326714\pi$
$491$	840.043i	1.71088i	−0.517901	+	0.855441i	$0.673286\pi$
$492$	0	0
$493$	−130.108	−0.263910
$494$	− 407.294i	− 0.824481i
$495$	0	0
$496$	−203.647	−0.410578
$497$	0	0
$498$	0	0
$499$	−940.000	−1.88377	−0.941884	−	0.335939i	$-0.890946\pi$
$499$	−940.000	−1.88377	−0.941884	+	0.335939i	$0.890946\pi$
$500$	272.000i	0.544000i
$501$	0	0
$502$	−673.166	−1.34097
$503$	− 632.000i	− 1.25646i	−0.778027	−	0.628231i	$-0.783779\pi$
$503$	− 632.000i	− 1.25646i	0.778027	−	0.628231i	$-0.216221\pi$
$504$	0	0
$505$	−504.000	−0.998020
$506$	− 147.078i	− 0.290668i
$507$	0	0
$508$	−8.00000	−0.0157480
$509$	220.000i	0.432220i	0.976369	+	0.216110i	$0.0693370\pi$
$509$	220.000i	0.432220i	−0.976369	+	0.216110i	$0.930663\pi$
$510$	0	0
$511$	0	0
$512$	− 22.6274i	− 0.0441942i
$513$	0	0
$514$	−144.250	−0.280642
$515$	294.156i	0.571178i
$516$	0	0
$517$	−56.5685	−0.109417
$518$	0	0
$519$	0	0
$520$	144.000	0.276923
$521$	− 508.000i	− 0.975048i	−0.873110	−	0.487524i	$-0.837900\pi$
$521$	− 508.000i	− 0.975048i	0.873110	−	0.487524i	$-0.162100\pi$
$522$	0	0
$523$	814.587	1.55753	0.778764	−	0.627317i	$-0.215847\pi$
$523$	814.587	1.55753	0.778764	+	0.627317i	$0.215847\pi$
$524$	− 48.0000i	− 0.0916031i
$525$	0	0
$526$	516.000	0.980989
$527$	203.647i	0.386426i
$528$	0	0
$529$	−823.000	−1.55577
$530$	536.000i	1.01132i
$531$	0	0
$532$	0	0
$533$	483.661i	0.907432i
$534$	0	0
$535$	−282.843	−0.528678
$536$	− 135.765i	− 0.253292i
$537$	0	0
$538$	277.186	0.515215
$539$	0	0
$540$	0	0
$541$	−464.000	−0.857671	−0.428835	−	0.903383i	$-0.641076\pi$
$541$	−464.000	−0.857671	−0.428835	+	0.903383i	$0.641076\pi$
$542$	− 248.000i	− 0.457565i
$543$	0	0
$544$	−22.6274	−0.0415945
$545$	344.000i	0.631193i
$546$	0	0
$547$	−80.0000	−0.146252	−0.0731261	−	0.997323i	$-0.523298\pi$
$547$	−80.0000	−0.146252	−0.0731261	+	0.997323i	$0.523298\pi$
$548$	144.250i	0.263230i
$549$	0	0
$550$	36.0000	0.0654545
$551$	− 736.000i	− 1.33575i
$552$	0	0
$553$	0	0
$554$	− 362.039i	− 0.653499i
$555$	0	0
$556$	−282.843	−0.508710
$557$	− 912.168i	− 1.63764i	−0.574047	−	0.818822i	$-0.694628\pi$
$557$	− 912.168i	− 1.63764i	0.574047	−	0.818822i	$-0.305372\pi$
$558$	0	0
$559$	254.558	0.455382
$560$	0	0
$561$	0	0
$562$	−38.0000	−0.0676157
$563$	− 316.000i	− 0.561279i	−0.959813	−	0.280639i	$-0.909453\pi$
$563$	− 316.000i	− 0.561279i	0.959813	−	0.280639i	$-0.0905465\pi$
$564$	0	0
$565$	84.8528	0.150182
$566$	− 712.000i	− 1.25795i
$567$	0	0
$568$	216.000	0.380282
$569$	− 1112.99i	− 1.95604i	−0.208514	−	0.978019i	$-0.566863\pi$
$569$	− 1112.99i	− 1.95604i	0.208514	−	0.978019i	$-0.433137\pi$
$570$	0	0
$571$	−440.000	−0.770578	−0.385289	−	0.922796i	$-0.625898\pi$
$571$	−440.000	−0.770578	−0.385289	+	0.922796i	$0.625898\pi$
$572$	− 72.0000i	− 0.125874i
$573$	0	0
$574$	0	0
$575$	− 330.926i	− 0.575523i
$576$	0	0
$577$	−318.198	−0.551470	−0.275735	−	0.961234i	$-0.588921\pi$
$577$	−318.198	−0.551470	−0.275735	+	0.961234i	$0.588921\pi$
$578$	− 386.080i	− 0.667959i
$579$	0	0
$580$	260.215	0.448647
$581$	0	0
$582$	0	0
$583$	268.000	0.459691
$584$	− 340.000i	− 0.582192i
$585$	0	0
$586$	268.701	0.458533
$587$	300.000i	0.511073i	0.966799	+	0.255537i	$0.0822521\pi$
$587$	300.000i	0.511073i	−0.966799	+	0.255537i	$0.917748\pi$
$588$	0	0
$589$	−1152.00	−1.95586
$590$	22.6274i	0.0383516i
$591$	0	0
$592$	−128.000	−0.216216
$593$	1036.00i	1.74705i	0.486780	+	0.873524i	$0.338171\pi$
$593$	1036.00i	1.74705i	−0.486780	+	0.873524i	$0.661829\pi$
$594$	0	0
$595$	0	0
$596$	53.7401i	0.0901680i
$597$	0	0
$598$	−661.852	−1.10678
$599$	981.464i	1.63850i	0.573433	+	0.819252i	$0.305611\pi$
$599$	981.464i	1.63850i	−0.573433	+	0.819252i	$0.694389\pi$
$600$	0	0
$601$	−352.139	−0.585922	−0.292961	−	0.956124i	$-0.594641\pi$
$601$	−352.139	−0.585922	−0.292961	+	0.956124i	$0.594641\pi$
$602$	0	0
$603$	0	0
$604$	−304.000	−0.503311
$605$	− 452.000i	− 0.747107i
$606$	0	0
$607$	865.499	1.42586	0.712931	−	0.701234i	$-0.247367\pi$
$607$	865.499	1.42586	0.712931	+	0.701234i	$0.247367\pi$
$608$	− 128.000i	− 0.210526i
$609$	0	0
$610$	−472.000	−0.773770
$611$	254.558i	0.416626i
$612$	0	0
$613$	−264.000	−0.430669	−0.215334	−	0.976540i	$-0.569084\pi$
$613$	−264.000	−0.430669	−0.215334	+	0.976540i	$0.569084\pi$
$614$	− 376.000i	− 0.612378i
$615$	0	0
$616$	0	0
$617$	15.5563i	0.0252129i	0.999921	+	0.0126064i	$0.00401286\pi$
$617$	15.5563i	0.0252129i	−0.999921	+	0.0126064i	$0.995987\pi$
$618$	0	0
$619$	−401.637	−0.648848	−0.324424	−	0.945912i	$-0.605170\pi$
$619$	−401.637	−0.648848	−0.324424	+	0.945912i	$0.605170\pi$
$620$	− 407.294i	− 0.656925i
$621$	0	0
$622$	752.362	1.20958
$623$	0	0
$624$	0	0
$625$	−319.000	−0.510400
$626$	− 210.000i	− 0.335463i
$627$	0	0
$628$	−189.505	−0.301759
$629$	128.000i	0.203498i
$630$	0	0
$631$	−676.000	−1.07132	−0.535658	−	0.844435i	$-0.679936\pi$
$631$	−676.000	−1.07132	−0.535658	+	0.844435i	$0.679936\pi$
$632$	− 418.607i	− 0.662353i
$633$	0	0
$634$	−326.000	−0.514196
$635$	− 16.0000i	− 0.0251969i
$636$	0	0
$637$	0	0
$638$	− 130.108i	− 0.203930i
$639$	0	0
$640$	45.2548	0.0707107
$641$	909.339i	1.41863i	0.704894	+	0.709313i	$0.250995\pi$
$641$	909.339i	1.41863i	−0.704894	+	0.709313i	$0.749005\pi$
$642$	0	0
$643$	480.833	0.747796	0.373898	−	0.927470i	$-0.378021\pi$
$643$	480.833	0.747796	0.373898	+	0.927470i	$0.378021\pi$
$644$	0	0
$645$	0	0
$646$	−128.000	−0.198142
$647$	588.000i	0.908810i	0.890795	+	0.454405i	$0.150148\pi$
$647$	588.000i	0.908810i	−0.890795	+	0.454405i	$0.849852\pi$
$648$	0	0
$649$	11.3137	0.0174325
$650$	− 162.000i	− 0.249231i
$651$	0	0
$652$	304.000	0.466258
$653$	− 453.963i	− 0.695195i	−0.937644	−	0.347598i	$-0.886998\pi$
$653$	− 453.963i	− 0.695195i	0.937644	−	0.347598i	$-0.113002\pi$
$654$	0	0
$655$	96.0000	0.146565
$656$	152.000i	0.231707i
$657$	0	0
$658$	0	0
$659$	755.190i	1.14596i	0.819568	+	0.572982i	$0.194213\pi$
$659$	755.190i	1.14596i	−0.819568	+	0.572982i	$0.805787\pi$
$660$	0	0
$661$	241.831	0.365856	0.182928	−	0.983126i	$-0.441443\pi$
$661$	241.831	0.365856	0.182928	+	0.983126i	$0.441443\pi$
$662$	− 379.009i	− 0.572522i
$663$	0	0
$664$	226.274	0.340774
$665$	0	0
$666$	0	0
$667$	−1196.00	−1.79310
$668$	232.000i	0.347305i
$669$	0	0
$670$	271.529	0.405267
$671$	236.000i	0.351714i
$672$	0	0
$673$	1176.00	1.74740	0.873700	−	0.486465i	$-0.161714\pi$
$673$	1176.00	1.74740	0.873700	+	0.486465i	$0.161714\pi$
$674$	− 240.416i	− 0.356701i
$675$	0	0
$676$	14.0000	0.0207101
$677$	222.000i	0.327917i	0.986467	+	0.163959i	$0.0524264\pi$
$677$	222.000i	0.327917i	−0.986467	+	0.163959i	$0.947574\pi$
$678$	0	0
$679$	0	0
$680$	− 45.2548i	− 0.0665512i
$681$	0	0
$682$	−203.647	−0.298602
$683$	− 517.602i	− 0.757836i	−0.925430	−	0.378918i	$-0.876296\pi$
$683$	− 517.602i	− 0.757836i	0.925430	−	0.378918i	$-0.123704\pi$
$684$	0	0
$685$	−288.500	−0.421167
$686$	0	0
$687$	0	0
$688$	80.0000	0.116279
$689$	− 1206.00i	− 1.75036i
$690$	0	0
$691$	644.881	0.933258	0.466629	−	0.884453i	$-0.345468\pi$
$691$	644.881	0.933258	0.466629	+	0.884453i	$0.345468\pi$
$692$	− 444.000i	− 0.641618i
$693$	0	0
$694$	396.000	0.570605
$695$	− 565.685i	− 0.813936i
$696$	0	0
$697$	152.000	0.218077
$698$	342.000i	0.489971i
$699$	0	0
$700$	0	0
$701$	− 657.609i	− 0.938102i	−0.883171	−	0.469051i	$-0.844596\pi$
$701$	− 657.609i	− 0.938102i	0.883171	−	0.469051i	$-0.155404\pi$
$702$	0	0
$703$	−724.077	−1.02998
$704$	− 22.6274i	− 0.0321412i
$705$	0	0
$706$	−752.362	−1.06567
$707$	0	0
$708$	0	0
$709$	106.000	0.149506	0.0747532	−	0.997202i	$-0.476183\pi$
$709$	106.000	0.149506	0.0747532	+	0.997202i	$0.476183\pi$
$710$	432.000i	0.608451i
$711$	0	0
$712$	−299.813	−0.421086
$713$	1872.00i	2.62553i
$714$	0	0
$715$	144.000	0.201399
$716$	118.794i	0.165913i
$717$	0	0
$718$	484.000	0.674095
$719$	− 1216.00i	− 1.69124i	−0.533787	−	0.845619i	$-0.679232\pi$
$719$	− 1216.00i	− 1.69124i	0.533787	−	0.845619i	$-0.320768\pi$
$720$	0	0
$721$	0	0
$722$	− 213.546i	− 0.295770i
$723$	0	0
$724$	−161.220	−0.222680
$725$	− 292.742i	− 0.403782i
$726$	0	0
$727$	73.5391	0.101154	0.0505771	−	0.998720i	$-0.483894\pi$
$727$	73.5391	0.101154	0.0505771	+	0.998720i	$0.483894\pi$
$728$	0	0
$729$	0	0
$730$	680.000	0.931507
$731$	− 80.0000i	− 0.109439i
$732$	0	0
$733$	420.021	0.573017	0.286508	−	0.958078i	$-0.407505\pi$
$733$	420.021	0.573017	0.286508	+	0.958078i	$0.407505\pi$
$734$	968.000i	1.31880i
$735$	0	0
$736$	−208.000	−0.282609
$737$	− 135.765i	− 0.184212i
$738$	0	0
$739$	−408.000	−0.552097	−0.276049	−	0.961144i	$-0.589025\pi$
$739$	−408.000	−0.552097	−0.276049	+	0.961144i	$0.589025\pi$
$740$	− 256.000i	− 0.345946i
$741$	0	0
$742$	0	0
$743$	− 489.318i	− 0.658571i	−0.944230	−	0.329285i	$-0.893192\pi$
$743$	− 489.318i	− 0.658571i	0.944230	−	0.329285i	$-0.106808\pi$
$744$	0	0
$745$	−107.480	−0.144269
$746$	− 540.230i	− 0.724168i
$747$	0	0
$748$	−22.6274	−0.0302506
$749$	0	0
$750$	0	0
$751$	−136.000	−0.181092	−0.0905459	−	0.995892i	$-0.528861\pi$
$751$	−136.000	−0.181092	−0.0905459	+	0.995892i	$0.528861\pi$
$752$	80.0000i	0.106383i
$753$	0	0
$754$	−585.484	−0.776505
$755$	− 608.000i	− 0.805298i
$756$	0	0
$757$	758.000	1.00132	0.500661	−	0.865644i	$-0.333091\pi$
$757$	758.000	1.00132	0.500661	+	0.865644i	$0.333091\pi$
$758$	197.990i	0.261200i
$759$	0	0
$760$	256.000	0.336842
$761$	58.0000i	0.0762155i	0.999274	+	0.0381078i	$0.0121330\pi$
$761$	58.0000i	0.0762155i	−0.999274	+	0.0381078i	$0.987867\pi$
$762$	0	0
$763$	0	0
$764$	− 605.283i	− 0.792256i
$765$	0	0
$766$	−350.725	−0.457865
$767$	− 50.9117i	− 0.0663777i
$768$	0	0
$769$	292.742	0.380679	0.190340	−	0.981718i	$-0.439041\pi$
$769$	292.742	0.380679	0.190340	+	0.981718i	$0.439041\pi$
$770$	0	0
$771$	0	0
$772$	−436.000	−0.564767
$773$	− 302.000i	− 0.390686i	−0.980735	−	0.195343i	$-0.937418\pi$
$773$	− 302.000i	− 0.390686i	0.980735	−	0.195343i	$-0.0625819\pi$
$774$	0	0
$775$	−458.205	−0.591233
$776$	436.000i	0.561856i
$777$	0	0
$778$	482.000	0.619537
$779$	859.842i	1.10378i
$780$	0	0
$781$	216.000	0.276569
$782$	208.000i	0.265985i
$783$	0	0
$784$	0	0
$785$	− 379.009i	− 0.482814i
$786$	0	0
$787$	−497.803	−0.632533	−0.316266	−	0.948670i	$-0.602429\pi$
$787$	−497.803	−0.632533	−0.316266	+	0.948670i	$0.602429\pi$
$788$	574.171i	0.728643i
$789$	0	0
$790$	837.214	1.05977
$791$	0	0
$792$	0	0
$793$	1062.00	1.33922
$794$	346.000i	0.435768i
$795$	0	0
$796$	135.765	0.170558
$797$	− 30.0000i	− 0.0376412i	−0.999823	−	0.0188206i	$-0.994009\pi$
$797$	− 30.0000i	− 0.0376412i	0.999823	−	0.0188206i	$-0.00599113\pi$
$798$	0	0
$799$	80.0000	0.100125
$800$	− 50.9117i	− 0.0636396i
$801$	0	0
$802$	−42.0000	−0.0523691
$803$	− 340.000i	− 0.423412i
$804$	0	0
$805$	0	0
$806$	916.410i	1.13699i
$807$	0	0
$808$	356.382	0.441067
$809$	− 982.878i	− 1.21493i	−0.794346	−	0.607465i	$-0.792186\pi$
$809$	− 982.878i	− 1.21493i	0.794346	−	0.607465i	$-0.207814\pi$
$810$	0	0
$811$	−458.205	−0.564988	−0.282494	−	0.959269i	$-0.591162\pi$
$811$	−458.205	−0.564988	−0.282494	+	0.959269i	$0.591162\pi$
$812$	0	0
$813$	0	0
$814$	−128.000	−0.157248
$815$	608.000i	0.746012i
$816$	0	0
$817$	452.548	0.553915
$818$	− 858.000i	− 1.04890i
$819$	0	0
$820$	−304.000	−0.370732
$821$	606.698i	0.738974i	0.929236	+	0.369487i	$0.120467\pi$
$821$	606.698i	0.738974i	−0.929236	+	0.369487i	$0.879533\pi$
$822$	0	0
$823$	−68.0000	−0.0826245	−0.0413123	−	0.999146i	$-0.513154\pi$
$823$	−68.0000	−0.0826245	−0.0413123	+	0.999146i	$0.513154\pi$
$824$	− 208.000i	− 0.252427i
$825$	0	0
$826$	0	0
$827$	− 195.161i	− 0.235987i	−0.993014	−	0.117994i	$-0.962354\pi$
$827$	− 195.161i	− 0.235987i	0.993014	−	0.117994i	$-0.0376462\pi$
$828$	0	0
$829$	1245.92	1.50292	0.751461	−	0.659778i	$-0.229350\pi$
$829$	1245.92	1.50292	0.751461	+	0.659778i	$0.229350\pi$
$830$	452.548i	0.545239i
$831$	0	0
$832$	−101.823	−0.122384
$833$	0	0
$834$	0	0
$835$	−464.000	−0.555689
$836$	− 128.000i	− 0.153110i
$837$	0	0
$838$	978.636	1.16782
$839$	1196.00i	1.42551i	0.701415	+	0.712753i	$0.252552\pi$
$839$	1196.00i	1.42551i	−0.701415	+	0.712753i	$0.747448\pi$
$840$	0	0
$841$	−217.000	−0.258026
$842$	− 543.058i	− 0.644962i
$843$	0	0
$844$	−264.000	−0.312796
$845$	28.0000i	0.0331361i
$846$	0	0
$847$	0	0
$848$	− 379.009i	− 0.446945i
$849$	0	0
$850$	−50.9117	−0.0598961
$851$	1176.63i	1.38264i
$852$	0	0
$853$	−1217.64	−1.42748	−0.713738	−	0.700412i	$-0.752999\pi$
$853$	−1217.64	−1.42748	−0.713738	+	0.700412i	$0.752999\pi$
$854$	0	0
$855$	0	0
$856$	200.000	0.233645
$857$	634.000i	0.739790i	0.929074	+	0.369895i	$0.120606\pi$
$857$	634.000i	0.739790i	−0.929074	+	0.369895i	$0.879394\pi$
$858$	0	0
$859$	−1600.89	−1.86367	−0.931833	−	0.362886i	$-0.881791\pi$
$859$	−1600.89	−1.86367	−0.931833	+	0.362886i	$0.881791\pi$
$860$	160.000i	0.186047i
$861$	0	0
$862$	−156.000	−0.180974
$863$	− 461.034i	− 0.534222i	−0.963666	−	0.267111i	$-0.913931\pi$
$863$	− 461.034i	− 0.534222i	0.963666	−	0.267111i	$-0.0860690\pi$
$864$	0	0
$865$	888.000	1.02659
$866$	222.000i	0.256351i
$867$	0	0
$868$	0	0
$869$	− 418.607i	− 0.481711i
$870$	0	0
$871$	−610.940	−0.701424
$872$	− 243.245i	− 0.278950i
$873$	0	0
$874$	−1176.63	−1.34625
$875$	0	0
$876$	0	0
$877$	1256.00	1.43216	0.716078	−	0.698021i	$-0.245936\pi$
$877$	1256.00	1.43216	0.716078	+	0.698021i	$0.245936\pi$
$878$	− 288.000i	− 0.328018i
$879$	0	0
$880$	45.2548	0.0514259
$881$	− 1092.00i	− 1.23950i	−0.784799	−	0.619750i	$-0.787234\pi$
$881$	− 1092.00i	− 1.23950i	0.784799	−	0.619750i	$-0.212766\pi$
$882$	0	0
$883$	284.000	0.321631	0.160815	−	0.986985i	$-0.448588\pi$
$883$	284.000	0.321631	0.160815	+	0.986985i	$0.448588\pi$
$884$	101.823i	0.115185i
$885$	0	0
$886$	1148.00	1.29571
$887$	− 716.000i	− 0.807215i	−0.914932	−	0.403608i	$-0.867756\pi$
$887$	− 716.000i	− 0.807215i	0.914932	−	0.403608i	$-0.132244\pi$
$888$	0	0
$889$	0	0
$890$	− 599.627i	− 0.673738i
$891$	0	0
$892$	339.411	0.380506
$893$	452.548i	0.506773i
$894$	0	0
$895$	−237.588	−0.265461
$896$	0	0
$897$	0	0
$898$	−402.000	−0.447661
$899$	1656.00i	1.84205i
$900$	0	0
$901$	−379.009	−0.420654
$902$	152.000i	0.168514i
$903$	0	0
$904$	−60.0000	−0.0663717
$905$	− 322.441i	− 0.356288i
$906$	0	0
$907$	724.000	0.798236	0.399118	−	0.916900i	$-0.369316\pi$
$907$	724.000	0.798236	0.399118	+	0.916900i	$0.369316\pi$
$908$	− 600.000i	− 0.660793i
$909$	0	0
$910$	0	0
$911$	302.642i	0.332208i	0.986108	+	0.166104i	$0.0531188\pi$
$911$	302.642i	0.332208i	−0.986108	+	0.166104i	$0.946881\pi$
$912$	0	0
$913$	226.274	0.247836
$914$	407.294i	0.445617i
$915$	0	0
$916$	500.632	0.546541
$917$	0	0
$918$	0	0
$919$	376.000	0.409140	0.204570	−	0.978852i	$-0.434420\pi$
$919$	376.000	0.409140	0.204570	+	0.978852i	$0.434420\pi$
$920$	− 416.000i	− 0.452174i
$921$	0	0
$922$	998.435	1.08290
$923$	− 972.000i	− 1.05309i
$924$	0	0
$925$	−288.000	−0.311351
$926$	503.460i	0.543693i
$927$	0	0
$928$	−184.000	−0.198276
$929$	− 1210.00i	− 1.30248i	−0.758874	−	0.651238i	$-0.774250\pi$
$929$	− 1210.00i	− 1.30248i	0.758874	−	0.651238i	$-0.225750\pi$
$930$	0	0
$931$	0	0
$932$	59.3970i	0.0637307i
$933$	0	0
$934$	1250.16	1.33851
$935$	− 45.2548i	− 0.0484009i
$936$	0	0
$937$	994.192	1.06104	0.530519	−	0.847673i	$-0.321997\pi$
$937$	994.192	1.06104	0.530519	+	0.847673i	$0.321997\pi$
$938$	0	0
$939$	0	0
$940$	−160.000	−0.170213
$941$	44.0000i	0.0467588i	0.999727	+	0.0233794i	$0.00744257\pi$
$941$	44.0000i	0.0467588i	−0.999727	+	0.0233794i	$0.992557\pi$
$942$	0	0
$943$	1397.24	1.48170
$944$	− 16.0000i	− 0.0169492i
$945$	0	0
$946$	80.0000	0.0845666
$947$	1009.75i	1.06626i	0.846033	+	0.533130i	$0.178984\pi$
$947$	1009.75i	1.06626i	−0.846033	+	0.533130i	$0.821016\pi$
$948$	0	0
$949$	−1530.00	−1.61222
$950$	− 288.000i	− 0.303158i
$951$	0	0
$952$	0	0
$953$	1322.29i	1.38750i	0.720215	+	0.693751i	$0.244043\pi$
$953$	1322.29i	1.38750i	−0.720215	+	0.693751i	$0.755957\pi$
$954$	0	0
$955$	1210.57	1.26761
$956$	322.441i	0.337281i
$957$	0	0
$958$	−718.420	−0.749917
$959$	0	0
$960$	0	0
$961$	1631.00	1.69719
$962$	576.000i	0.598753i
$963$	0	0
$964$	−755.190	−0.783392
$965$	− 872.000i	− 0.903627i
$966$	0	0
$967$	876.000	0.905895	0.452947	−	0.891537i	$-0.350373\pi$
$967$	876.000	0.905895	0.452947	+	0.891537i	$0.350373\pi$
$968$	319.612i	0.330178i
$969$	0	0
$970$	−872.000	−0.898969
$971$	1448.00i	1.49125i	0.666368	+	0.745623i	$0.267848\pi$
$971$	1448.00i	1.49125i	−0.666368	+	0.745623i	$0.732152\pi$
$972$	0	0
$973$	0	0
$974$	882.469i	0.906026i
$975$	0	0
$976$	333.754	0.341961
$977$	− 1489.17i	− 1.52422i	−0.647445	−	0.762112i	$-0.724163\pi$
$977$	− 1489.17i	− 1.52422i	0.647445	−	0.762112i	$-0.275837\pi$
$978$	0	0
$979$	−299.813	−0.306244
$980$	0	0
$981$	0	0
$982$	1188.00	1.20978
$983$	− 32.0000i	− 0.0325534i	−0.999868	−	0.0162767i	$-0.994819\pi$
$983$	− 32.0000i	− 0.0325534i	0.999868	−	0.0162767i	$-0.00518126\pi$
$984$	0	0
$985$	−1148.34	−1.16583
$986$	184.000i	0.186613i
$987$	0	0
$988$	−576.000	−0.582996
$989$	− 735.391i	− 0.743570i
$990$	0	0
$991$	−280.000	−0.282543	−0.141271	−	0.989971i	$-0.545119\pi$
$991$	−280.000	−0.282543	−0.141271	+	0.989971i	$0.545119\pi$
$992$	288.000i	0.290323i
$993$	0	0
$994$	0	0
$995$	271.529i	0.272893i
$996$	0	0
$997$	1093.19	1.09648	0.548238	−	0.836322i	$-0.315299\pi$
$997$	1093.19	1.09648	0.548238	+	0.836322i	$0.315299\pi$
$998$	1329.36i	1.33202i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	882.3.b.h.197.1	✓	4
3.2	odd	2	inner	882.3.b.h.197.4	yes	4
7.2	even	3		882.3.s.g.557.1		8
7.3	odd	6		882.3.s.g.863.3		8
7.4	even	3		882.3.s.g.863.4		8
7.5	odd	6		882.3.s.g.557.2		8
7.6	odd	2	inner	882.3.b.h.197.2	yes	4
21.2	odd	6		882.3.s.g.557.4		8
21.5	even	6		882.3.s.g.557.3		8
21.11	odd	6		882.3.s.g.863.1		8
21.17	even	6		882.3.s.g.863.2		8
21.20	even	2	inner	882.3.b.h.197.3	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
882.3.b.h.197.1	✓	4	1.1	even	1	trivial
882.3.b.h.197.2	yes	4	7.6	odd	2	inner
882.3.b.h.197.3	yes	4	21.20	even	2	inner
882.3.b.h.197.4	yes	4	3.2	odd	2	inner
882.3.s.g.557.1		8	7.2	even	3
882.3.s.g.557.2		8	7.5	odd	6
882.3.s.g.557.3		8	21.5	even	6
882.3.s.g.557.4		8	21.2	odd	6
882.3.s.g.863.1		8	21.11	odd	6
882.3.s.g.863.2		8	21.17	even	6
882.3.s.g.863.3		8	7.3	odd	6
882.3.s.g.863.4		8	7.4	even	3

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)

\(f(q)\)	\(=\)	\(q-1.41421i q^{2} -2.00000 q^{4} -4.00000i q^{5} +2.82843i q^{8} +O(q^{10})\)	\(q-1.41421i q^{2} -2.00000 q^{4} -4.00000i q^{5} +2.82843i q^{8} -5.65685 q^{10} +2.82843i q^{11} +12.7279 q^{13} +4.00000 q^{16} -4.00000i q^{17} +22.6274 q^{19} +8.00000i q^{20} +4.00000 q^{22} -36.7696i q^{23} +9.00000 q^{25} -18.0000i q^{26} -32.5269i q^{29} -50.9117 q^{31} -5.65685i q^{32} -5.65685 q^{34} -32.0000 q^{37} -32.0000i q^{38} +11.3137 q^{40} +38.0000i q^{41} +20.0000 q^{43} -5.65685i q^{44} -52.0000 q^{46} +20.0000i q^{47} -12.7279i q^{50} -25.4558 q^{52} -94.7523i q^{53} +11.3137 q^{55} -46.0000 q^{58} -4.00000i q^{59} +83.4386 q^{61} +72.0000i q^{62} -8.00000 q^{64} -50.9117i q^{65} -48.0000 q^{67} +8.00000i q^{68} -76.3675i q^{71} -120.208 q^{73} +45.2548i q^{74} -45.2548 q^{76} -148.000 q^{79} -16.0000i q^{80} +53.7401 q^{82} -80.0000i q^{83} -16.0000 q^{85} -28.2843i q^{86} -8.00000 q^{88} +106.000i q^{89} +73.5391i q^{92} +28.2843 q^{94} -90.5097i q^{95} +154.149 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{4}+O(q^{10}) \) 4 * q - 8 * q^4	\( 4 q - 8 q^{4} + 16 q^{16} + 16 q^{22} + 36 q^{25} - 128 q^{37} + 80 q^{43} - 208 q^{46} - 184 q^{58} - 32 q^{64} - 192 q^{67} - 592 q^{79} - 64 q^{85} - 32 q^{88}+O(q^{100}) \) 4 * q - 8 * q^4 + 16 * q^16 + 16 * q^22 + 36 * q^25 - 128 * q^37 + 80 * q^43 - 208 * q^46 - 184 * q^58 - 32 * q^64 - 192 * q^67 - 592 * q^79 - 64 * q^85 - 32 * q^88

Introduction

L-functions

Modular forms

Varieties

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Database

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