LMFDB - Embedded newform 882.3.b.e.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.e.197.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421i q^{2} -2.00000 q^{4} +4.24264i q^{5} +2.82843i q^{8} +O(q^{10})$	$q-1.41421i q^{2} -2.00000 q^{4} +4.24264i q^{5} +2.82843i q^{8} +6.00000 q^{10} +12.7279i q^{11} +1.00000 q^{13} +4.00000 q^{16} -16.9706i q^{17} -23.0000 q^{19} -8.48528i q^{20} +18.0000 q^{22} +16.9706i q^{23} +7.00000 q^{25} -1.41421i q^{26} -33.9411i q^{29} -47.0000 q^{31} -5.65685i q^{32} -24.0000 q^{34} -55.0000 q^{37} +32.5269i q^{38} -12.0000 q^{40} -46.6690i q^{41} +23.0000 q^{43} -25.4558i q^{44} +24.0000 q^{46} -4.24264i q^{47} -9.89949i q^{50} -2.00000 q^{52} +50.9117i q^{53} -54.0000 q^{55} -48.0000 q^{58} +84.8528i q^{59} -104.000 q^{61} +66.4680i q^{62} -8.00000 q^{64} +4.24264i q^{65} -97.0000 q^{67} +33.9411i q^{68} -97.5807i q^{71} -65.0000 q^{73} +77.7817i q^{74} +46.0000 q^{76} +113.000 q^{79} +16.9706i q^{80} -66.0000 q^{82} -29.6985i q^{83} +72.0000 q^{85} -32.5269i q^{86} -36.0000 q^{88} +135.765i q^{89} -33.9411i q^{92} -6.00000 q^{94} -97.5807i q^{95} -104.000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{4}+O(q^{10}) $ 2 * q - 4 * q^4	$ 2 q - 4 q^{4} + 12 q^{10} + 2 q^{13} + 8 q^{16} - 46 q^{19} + 36 q^{22} + 14 q^{25} - 94 q^{31} - 48 q^{34} - 110 q^{37} - 24 q^{40} + 46 q^{43} + 48 q^{46} - 4 q^{52} - 108 q^{55} - 96 q^{58} - 208 q^{61} - 16 q^{64} - 194 q^{67} - 130 q^{73} + 92 q^{76} + 226 q^{79} - 132 q^{82} + 144 q^{85} - 72 q^{88} - 12 q^{94} - 208 q^{97}+O(q^{100}) $ 2 * q - 4 * q^4 + 12 * q^10 + 2 * q^13 + 8 * q^16 - 46 * q^19 + 36 * q^22 + 14 * q^25 - 94 * q^31 - 48 * q^34 - 110 * q^37 - 24 * q^40 + 46 * q^43 + 48 * q^46 - 4 * q^52 - 108 * q^55 - 96 * q^58 - 208 * q^61 - 16 * q^64 - 194 * q^67 - 130 * q^73 + 92 * q^76 + 226 * q^79 - 132 * q^82 + 144 * q^85 - 72 * q^88 - 12 * q^94 - 208 * 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
$5$	4.24264i	0.848528i	0.905539	+	0.424264i	$0.139467\pi$
$5$	4.24264i	0.848528i	−0.905539	+	0.424264i	$0.860533\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	2.82843i	0.353553i
$9$	0	0
$10$	6.00000	0.600000
$11$	12.7279i	1.15708i	0.815653	+	0.578542i	$0.196378\pi$
$11$	12.7279i	1.15708i	−0.815653	+	0.578542i	$0.803622\pi$
$12$	0	0
$13$	1.00000	0.0769231	0.0384615	−	0.999260i	$-0.487754\pi$
$13$	1.00000	0.0769231	0.0384615	+	0.999260i	$0.487754\pi$
$14$	0	0
$15$	0	0
$16$	4.00000	0.250000
$17$	− 16.9706i	− 0.998268i	−0.866525	−	0.499134i	$-0.833651\pi$
$17$	− 16.9706i	− 0.998268i	0.866525	−	0.499134i	$-0.166349\pi$
$18$	0	0
$19$	−23.0000	−1.21053	−0.605263	−	0.796025i	$-0.706932\pi$
$19$	−23.0000	−1.21053	−0.605263	+	0.796025i	$0.706932\pi$
$20$	− 8.48528i	− 0.424264i
$21$	0	0
$22$	18.0000	0.818182
$23$	16.9706i	0.737851i	0.929459	+	0.368925i	$0.120274\pi$
$23$	16.9706i	0.737851i	−0.929459	+	0.368925i	$0.879726\pi$
$24$	0	0
$25$	7.00000	0.280000
$26$	− 1.41421i	− 0.0543928i
$27$	0	0
$28$	0	0
$29$	− 33.9411i	− 1.17038i	−0.810895	−	0.585192i	$-0.801019\pi$
$29$	− 33.9411i	− 1.17038i	0.810895	−	0.585192i	$-0.198981\pi$
$30$	0	0
$31$	−47.0000	−1.51613	−0.758065	−	0.652180i	$-0.773855\pi$
$31$	−47.0000	−1.51613	−0.758065	+	0.652180i	$0.773855\pi$
$32$	− 5.65685i	− 0.176777i
$33$	0	0
$34$	−24.0000	−0.705882
$35$	0	0
$36$	0	0
$37$	−55.0000	−1.48649	−0.743243	−	0.669021i	$-0.766713\pi$
$37$	−55.0000	−1.48649	−0.743243	+	0.669021i	$0.766713\pi$
$38$	32.5269i	0.855971i
$39$	0	0
$40$	−12.0000	−0.300000
$41$	− 46.6690i	− 1.13827i	−0.822244	−	0.569135i	$-0.807278\pi$
$41$	− 46.6690i	− 1.13827i	0.822244	−	0.569135i	$-0.192722\pi$
$42$	0	0
$43$	23.0000	0.534884	0.267442	−	0.963574i	$-0.413822\pi$
$43$	23.0000	0.534884	0.267442	+	0.963574i	$0.413822\pi$
$44$	− 25.4558i	− 0.578542i
$45$	0	0
$46$	24.0000	0.521739
$47$	− 4.24264i	− 0.0902690i	−0.998981	−	0.0451345i	$-0.985628\pi$
$47$	− 4.24264i	− 0.0902690i	0.998981	−	0.0451345i	$-0.0143716\pi$
$48$	0	0
$49$	0	0
$50$	− 9.89949i	− 0.197990i
$51$	0	0
$52$	−2.00000	−0.0384615
$53$	50.9117i	0.960598i	0.877105	+	0.480299i	$0.159472\pi$
$53$	50.9117i	0.960598i	−0.877105	+	0.480299i	$0.840528\pi$
$54$	0	0
$55$	−54.0000	−0.981818
$56$	0	0
$57$	0	0
$58$	−48.0000	−0.827586
$59$	84.8528i	1.43818i	0.694915	+	0.719092i	$0.255442\pi$
$59$	84.8528i	1.43818i	−0.694915	+	0.719092i	$0.744558\pi$
$60$	0	0
$61$	−104.000	−1.70492	−0.852459	−	0.522794i	$-0.824890\pi$
$61$	−104.000	−1.70492	−0.852459	+	0.522794i	$0.824890\pi$
$62$	66.4680i	1.07207i
$63$	0	0
$64$	−8.00000	−0.125000
$65$	4.24264i	0.0652714i
$66$	0	0
$67$	−97.0000	−1.44776	−0.723881	−	0.689925i	$-0.757643\pi$
$67$	−97.0000	−1.44776	−0.723881	+	0.689925i	$0.757643\pi$
$68$	33.9411i	0.499134i
$69$	0	0
$70$	0	0
$71$	− 97.5807i	− 1.37438i	−0.726479	−	0.687188i	$-0.758845\pi$
$71$	− 97.5807i	− 1.37438i	0.726479	−	0.687188i	$-0.241155\pi$
$72$	0	0
$73$	−65.0000	−0.890411	−0.445205	−	0.895428i	$-0.646869\pi$
$73$	−65.0000	−0.890411	−0.445205	+	0.895428i	$0.646869\pi$
$74$	77.7817i	1.05110i
$75$	0	0
$76$	46.0000	0.605263
$77$	0	0
$78$	0	0
$79$	113.000	1.43038	0.715190	−	0.698930i	$-0.246340\pi$
$79$	113.000	1.43038	0.715190	+	0.698930i	$0.246340\pi$
$80$	16.9706i	0.212132i
$81$	0	0
$82$	−66.0000	−0.804878
$83$	− 29.6985i	− 0.357813i	−0.983866	−	0.178907i	$-0.942744\pi$
$83$	− 29.6985i	− 0.357813i	0.983866	−	0.178907i	$-0.0572560\pi$
$84$	0	0
$85$	72.0000	0.847059
$86$	− 32.5269i	− 0.378220i
$87$	0	0
$88$	−36.0000	−0.409091
$89$	135.765i	1.52544i	0.646727	+	0.762722i	$0.276137\pi$
$89$	135.765i	1.52544i	−0.646727	+	0.762722i	$0.723863\pi$
$90$	0	0
$91$	0	0
$92$	− 33.9411i	− 0.368925i
$93$	0	0
$94$	−6.00000	−0.0638298
$95$	− 97.5807i	− 1.02717i
$96$	0	0
$97$	−104.000	−1.07216	−0.536082	−	0.844166i	$-0.680096\pi$
$97$	−104.000	−1.07216	−0.536082	+	0.844166i	$0.680096\pi$
$98$	0	0
$99$	0	0
$100$	−14.0000	−0.140000
$101$	− 148.492i	− 1.47022i	−0.677947	−	0.735111i	$-0.737130\pi$
$101$	− 148.492i	− 1.47022i	0.677947	−	0.735111i	$-0.262870\pi$
$102$	0	0
$103$	−119.000	−1.15534	−0.577670	−	0.816270i	$-0.696038\pi$
$103$	−119.000	−1.15534	−0.577670	+	0.816270i	$0.696038\pi$
$104$	2.82843i	0.0271964i
$105$	0	0
$106$	72.0000	0.679245
$107$	118.794i	1.11022i	0.831776	+	0.555112i	$0.187325\pi$
$107$	118.794i	1.11022i	−0.831776	+	0.555112i	$0.812675\pi$
$108$	0	0
$109$	−49.0000	−0.449541	−0.224771	−	0.974412i	$-0.572163\pi$
$109$	−49.0000	−0.449541	−0.224771	+	0.974412i	$0.572163\pi$
$110$	76.3675i	0.694250i
$111$	0	0
$112$	0	0
$113$	97.5807i	0.863546i	0.901982	+	0.431773i	$0.142112\pi$
$113$	97.5807i	0.863546i	−0.901982	+	0.431773i	$0.857888\pi$
$114$	0	0
$115$	−72.0000	−0.626087
$116$	67.8823i	0.585192i
$117$	0	0
$118$	120.000	1.01695
$119$	0	0
$120$	0	0
$121$	−41.0000	−0.338843
$122$	147.078i	1.20556i
$123$	0	0
$124$	94.0000	0.758065
$125$	135.765i	1.08612i
$126$	0	0
$127$	113.000	0.889764	0.444882	−	0.895589i	$-0.353246\pi$
$127$	113.000	0.889764	0.444882	+	0.895589i	$0.353246\pi$
$128$	11.3137i	0.0883883i
$129$	0	0
$130$	6.00000	0.0461538
$131$	21.2132i	0.161933i	0.996717	+	0.0809664i	$0.0258007\pi$
$131$	21.2132i	0.161933i	−0.996717	+	0.0809664i	$0.974199\pi$
$132$	0	0
$133$	0	0
$134$	137.179i	1.02372i
$135$	0	0
$136$	48.0000	0.352941
$137$	− 67.8823i	− 0.495491i	−0.968825	−	0.247745i	$-0.920310\pi$
$137$	− 67.8823i	− 0.495491i	0.968825	−	0.247745i	$-0.0796897\pi$
$138$	0	0
$139$	103.000	0.741007	0.370504	−	0.928831i	$-0.379185\pi$
$139$	103.000	0.741007	0.370504	+	0.928831i	$0.379185\pi$
$140$	0	0
$141$	0	0
$142$	−138.000	−0.971831
$143$	12.7279i	0.0890064i
$144$	0	0
$145$	144.000	0.993103
$146$	91.9239i	0.629616i
$147$	0	0
$148$	110.000	0.743243
$149$	169.706i	1.13896i	0.822004	+	0.569482i	$0.192856\pi$
$149$	169.706i	1.13896i	−0.822004	+	0.569482i	$0.807144\pi$
$150$	0	0
$151$	104.000	0.688742	0.344371	−	0.938834i	$-0.388092\pi$
$151$	104.000	0.688742	0.344371	+	0.938834i	$0.388092\pi$
$152$	− 65.0538i	− 0.427986i
$153$	0	0
$154$	0	0
$155$	− 199.404i	− 1.28648i
$156$	0	0
$157$	−152.000	−0.968153	−0.484076	−	0.875026i	$-0.660844\pi$
$157$	−152.000	−0.968153	−0.484076	+	0.875026i	$0.660844\pi$
$158$	− 159.806i	− 1.01143i
$159$	0	0
$160$	24.0000	0.150000
$161$	0	0
$162$	0	0
$163$	56.0000	0.343558	0.171779	−	0.985135i	$-0.445048\pi$
$163$	56.0000	0.343558	0.171779	+	0.985135i	$0.445048\pi$
$164$	93.3381i	0.569135i
$165$	0	0
$166$	−42.0000	−0.253012
$167$	− 4.24264i	− 0.0254050i	−0.999919	−	0.0127025i	$-0.995957\pi$
$167$	− 4.24264i	− 0.0254050i	0.999919	−	0.0127025i	$-0.00404345\pi$
$168$	0	0
$169$	−168.000	−0.994083
$170$	− 101.823i	− 0.598961i
$171$	0	0
$172$	−46.0000	−0.267442
$173$	− 152.735i	− 0.882862i	−0.897295	−	0.441431i	$-0.854471\pi$
$173$	− 152.735i	− 0.882862i	0.897295	−	0.441431i	$-0.145529\pi$
$174$	0	0
$175$	0	0
$176$	50.9117i	0.289271i
$177$	0	0
$178$	192.000	1.07865
$179$	− 4.24264i	− 0.0237019i	−0.999930	−	0.0118510i	$-0.996228\pi$
$179$	− 4.24264i	− 0.0237019i	0.999930	−	0.0118510i	$-0.00377236\pi$
$180$	0	0
$181$	55.0000	0.303867	0.151934	−	0.988391i	$-0.451450\pi$
$181$	55.0000	0.303867	0.151934	+	0.988391i	$0.451450\pi$
$182$	0	0
$183$	0	0
$184$	−48.0000	−0.260870
$185$	− 233.345i	− 1.26133i
$186$	0	0
$187$	216.000	1.15508
$188$	8.48528i	0.0451345i
$189$	0	0
$190$	−138.000	−0.726316
$191$	63.6396i	0.333192i	0.986025	+	0.166596i	$0.0532775\pi$
$191$	63.6396i	0.333192i	−0.986025	+	0.166596i	$0.946722\pi$
$192$	0	0
$193$	−151.000	−0.782383	−0.391192	−	0.920309i	$-0.627937\pi$
$193$	−151.000	−0.782383	−0.391192	+	0.920309i	$0.627937\pi$
$194$	147.078i	0.758135i
$195$	0	0
$196$	0	0
$197$	− 16.9706i	− 0.0861450i	−0.999072	−	0.0430725i	$-0.986285\pi$
$197$	− 16.9706i	− 0.0861450i	0.999072	−	0.0430725i	$-0.0137146\pi$
$198$	0	0
$199$	160.000	0.804020	0.402010	−	0.915635i	$-0.368312\pi$
$199$	160.000	0.804020	0.402010	+	0.915635i	$0.368312\pi$
$200$	19.7990i	0.0989949i
$201$	0	0
$202$	−210.000	−1.03960
$203$	0	0
$204$	0	0
$205$	198.000	0.965854
$206$	168.291i	0.816949i
$207$	0	0
$208$	4.00000	0.0192308
$209$	− 292.742i	− 1.40068i
$210$	0	0
$211$	−208.000	−0.985782	−0.492891	−	0.870091i	$-0.664060\pi$
$211$	−208.000	−0.985782	−0.492891	+	0.870091i	$0.664060\pi$
$212$	− 101.823i	− 0.480299i
$213$	0	0
$214$	168.000	0.785047
$215$	97.5807i	0.453864i
$216$	0	0
$217$	0	0
$218$	69.2965i	0.317874i
$219$	0	0
$220$	108.000	0.490909
$221$	− 16.9706i	− 0.0767899i
$222$	0	0
$223$	40.0000	0.179372	0.0896861	−	0.995970i	$-0.471414\pi$
$223$	40.0000	0.179372	0.0896861	+	0.995970i	$0.471414\pi$
$224$	0	0
$225$	0	0
$226$	138.000	0.610619
$227$	− 224.860i	− 0.990572i	−0.868730	−	0.495286i	$-0.835063\pi$
$227$	− 224.860i	− 0.990572i	0.868730	−	0.495286i	$-0.164937\pi$
$228$	0	0
$229$	−377.000	−1.64629	−0.823144	−	0.567833i	$-0.807782\pi$
$229$	−377.000	−1.64629	−0.823144	+	0.567833i	$0.807782\pi$
$230$	101.823i	0.442710i
$231$	0	0
$232$	96.0000	0.413793
$233$	− 343.654i	− 1.47491i	−0.675397	−	0.737455i	$-0.736028\pi$
$233$	− 343.654i	− 1.47491i	0.675397	−	0.737455i	$-0.263972\pi$
$234$	0	0
$235$	18.0000	0.0765957
$236$	− 169.706i	− 0.719092i
$237$	0	0
$238$	0	0
$239$	12.7279i	0.0532549i	0.999645	+	0.0266275i	$0.00847678\pi$
$239$	12.7279i	0.0532549i	−0.999645	+	0.0266275i	$0.991523\pi$
$240$	0	0
$241$	130.000	0.539419	0.269710	−	0.962942i	$-0.413072\pi$
$241$	130.000	0.539419	0.269710	+	0.962942i	$0.413072\pi$
$242$	57.9828i	0.239598i
$243$	0	0
$244$	208.000	0.852459
$245$	0	0
$246$	0	0
$247$	−23.0000	−0.0931174
$248$	− 132.936i	− 0.536033i
$249$	0	0
$250$	192.000	0.768000
$251$	− 50.9117i	− 0.202835i	−0.994844	−	0.101418i	$-0.967662\pi$
$251$	− 50.9117i	− 0.202835i	0.994844	−	0.101418i	$-0.0323379\pi$
$252$	0	0
$253$	−216.000	−0.853755
$254$	− 159.806i	− 0.629158i
$255$	0	0
$256$	16.0000	0.0625000
$257$	− 470.933i	− 1.83242i	−0.400693	−	0.916212i	$-0.631231\pi$
$257$	− 470.933i	− 1.83242i	0.400693	−	0.916212i	$-0.368769\pi$
$258$	0	0
$259$	0	0
$260$	− 8.48528i	− 0.0326357i
$261$	0	0
$262$	30.0000	0.114504
$263$	101.823i	0.387161i	0.981084	+	0.193581i	$0.0620101\pi$
$263$	101.823i	0.387161i	−0.981084	+	0.193581i	$0.937990\pi$
$264$	0	0
$265$	−216.000	−0.815094
$266$	0	0
$267$	0	0
$268$	194.000	0.723881
$269$	284.257i	1.05672i	0.849021	+	0.528359i	$0.177192\pi$
$269$	284.257i	1.05672i	−0.849021	+	0.528359i	$0.822808\pi$
$270$	0	0
$271$	520.000	1.91882	0.959410	−	0.282016i	$-0.0910032\pi$
$271$	520.000	1.91882	0.959410	+	0.282016i	$0.0910032\pi$
$272$	− 67.8823i	− 0.249567i
$273$	0	0
$274$	−96.0000	−0.350365
$275$	89.0955i	0.323983i
$276$	0	0
$277$	113.000	0.407942	0.203971	−	0.978977i	$-0.434615\pi$
$277$	113.000	0.407942	0.203971	+	0.978977i	$0.434615\pi$
$278$	− 145.664i	− 0.523971i
$279$	0	0
$280$	0	0
$281$	458.205i	1.63062i	0.579022	+	0.815312i	$0.303434\pi$
$281$	458.205i	1.63062i	−0.579022	+	0.815312i	$0.696566\pi$
$282$	0	0
$283$	−89.0000	−0.314488	−0.157244	−	0.987560i	$-0.550261\pi$
$283$	−89.0000	−0.314488	−0.157244	+	0.987560i	$0.550261\pi$
$284$	195.161i	0.687188i
$285$	0	0
$286$	18.0000	0.0629371
$287$	0	0
$288$	0	0
$289$	1.00000	0.00346021
$290$	− 203.647i	− 0.702230i
$291$	0	0
$292$	130.000	0.445205
$293$	− 67.8823i	− 0.231680i	−0.993268	−	0.115840i	$-0.963044\pi$
$293$	− 67.8823i	− 0.231680i	0.993268	−	0.115840i	$-0.0369560\pi$
$294$	0	0
$295$	−360.000	−1.22034
$296$	− 155.563i	− 0.525552i
$297$	0	0
$298$	240.000	0.805369
$299$	16.9706i	0.0567577i
$300$	0	0
$301$	0	0
$302$	− 147.078i	− 0.487014i
$303$	0	0
$304$	−92.0000	−0.302632
$305$	− 441.235i	− 1.44667i
$306$	0	0
$307$	−233.000	−0.758958	−0.379479	−	0.925200i	$-0.623897\pi$
$307$	−233.000	−0.758958	−0.379479	+	0.925200i	$0.623897\pi$
$308$	0	0
$309$	0	0
$310$	−282.000	−0.909677
$311$	− 250.316i	− 0.804874i	−0.915448	−	0.402437i	$-0.868163\pi$
$311$	− 250.316i	− 0.804874i	0.915448	−	0.402437i	$-0.131837\pi$
$312$	0	0
$313$	151.000	0.482428	0.241214	−	0.970472i	$-0.422454\pi$
$313$	151.000	0.482428	0.241214	+	0.970472i	$0.422454\pi$
$314$	214.960i	0.684587i
$315$	0	0
$316$	−226.000	−0.715190
$317$	407.294i	1.28484i	0.766354	+	0.642419i	$0.222069\pi$
$317$	407.294i	1.28484i	−0.766354	+	0.642419i	$0.777931\pi$
$318$	0	0
$319$	432.000	1.35423
$320$	− 33.9411i	− 0.106066i
$321$	0	0
$322$	0	0
$323$	390.323i	1.20843i
$324$	0	0
$325$	7.00000	0.0215385
$326$	− 79.1960i	− 0.242932i
$327$	0	0
$328$	132.000	0.402439
$329$	0	0
$330$	0	0
$331$	−73.0000	−0.220544	−0.110272	−	0.993901i	$-0.535172\pi$
$331$	−73.0000	−0.220544	−0.110272	+	0.993901i	$0.535172\pi$
$332$	59.3970i	0.178907i
$333$	0	0
$334$	−6.00000	−0.0179641
$335$	− 411.536i	− 1.22847i
$336$	0	0
$337$	527.000	1.56380	0.781899	−	0.623405i	$-0.214251\pi$
$337$	527.000	1.56380	0.781899	+	0.623405i	$0.214251\pi$
$338$	237.588i	0.702923i
$339$	0	0
$340$	−144.000	−0.423529
$341$	− 598.212i	− 1.75429i
$342$	0	0
$343$	0	0
$344$	65.0538i	0.189110i
$345$	0	0
$346$	−216.000	−0.624277
$347$	424.264i	1.22266i	0.791375	+	0.611332i	$0.209366\pi$
$347$	424.264i	1.22266i	−0.791375	+	0.611332i	$0.790634\pi$
$348$	0	0
$349$	112.000	0.320917	0.160458	−	0.987043i	$-0.448703\pi$
$349$	112.000	0.320917	0.160458	+	0.987043i	$0.448703\pi$
$350$	0	0
$351$	0	0
$352$	72.0000	0.204545
$353$	− 309.713i	− 0.877373i	−0.898640	−	0.438687i	$-0.855444\pi$
$353$	− 309.713i	− 0.877373i	0.898640	−	0.438687i	$-0.144556\pi$
$354$	0	0
$355$	414.000	1.16620
$356$	− 271.529i	− 0.762722i
$357$	0	0
$358$	−6.00000	−0.0167598
$359$	− 509.117i	− 1.41815i	−0.705132	−	0.709076i	$-0.749112\pi$
$359$	− 509.117i	− 1.41815i	0.705132	−	0.709076i	$-0.250888\pi$
$360$	0	0
$361$	168.000	0.465374
$362$	− 77.7817i	− 0.214867i
$363$	0	0
$364$	0	0
$365$	− 275.772i	− 0.755539i
$366$	0	0
$367$	319.000	0.869210	0.434605	−	0.900621i	$-0.356888\pi$
$367$	319.000	0.869210	0.434605	+	0.900621i	$0.356888\pi$
$368$	67.8823i	0.184463i
$369$	0	0
$370$	−330.000	−0.891892
$371$	0	0
$372$	0	0
$373$	209.000	0.560322	0.280161	−	0.959953i	$-0.409612\pi$
$373$	209.000	0.560322	0.280161	+	0.959953i	$0.409612\pi$
$374$	− 305.470i	− 0.816765i
$375$	0	0
$376$	12.0000	0.0319149
$377$	− 33.9411i	− 0.0900295i
$378$	0	0
$379$	−433.000	−1.14248	−0.571240	−	0.820783i	$-0.693537\pi$
$379$	−433.000	−1.14248	−0.571240	+	0.820783i	$0.693537\pi$
$380$	195.161i	0.513583i
$381$	0	0
$382$	90.0000	0.235602
$383$	390.323i	1.01912i	0.860435	+	0.509560i	$0.170192\pi$
$383$	390.323i	1.01912i	−0.860435	+	0.509560i	$0.829808\pi$
$384$	0	0
$385$	0	0
$386$	213.546i	0.553229i
$387$	0	0
$388$	208.000	0.536082
$389$	− 369.110i	− 0.948868i	−0.880291	−	0.474434i	$-0.842653\pi$
$389$	− 369.110i	− 0.948868i	0.880291	−	0.474434i	$-0.157347\pi$
$390$	0	0
$391$	288.000	0.736573
$392$	0	0
$393$	0	0
$394$	−24.0000	−0.0609137
$395$	479.418i	1.21372i
$396$	0	0
$397$	−113.000	−0.284635	−0.142317	−	0.989821i	$-0.545455\pi$
$397$	−113.000	−0.284635	−0.142317	+	0.989821i	$0.545455\pi$
$398$	− 226.274i	− 0.568528i
$399$	0	0
$400$	28.0000	0.0700000
$401$	− 356.382i	− 0.888733i	−0.895845	−	0.444366i	$-0.853429\pi$
$401$	− 356.382i	− 0.888733i	0.895845	−	0.444366i	$-0.146571\pi$
$402$	0	0
$403$	−47.0000	−0.116625
$404$	296.985i	0.735111i
$405$	0	0
$406$	0	0
$407$	− 700.036i	− 1.71999i
$408$	0	0
$409$	319.000	0.779951	0.389976	−	0.920825i	$-0.372483\pi$
$409$	319.000	0.779951	0.389976	+	0.920825i	$0.372483\pi$
$410$	− 280.014i	− 0.682962i
$411$	0	0
$412$	238.000	0.577670
$413$	0	0
$414$	0	0
$415$	126.000	0.303614
$416$	− 5.65685i	− 0.0135982i
$417$	0	0
$418$	−414.000	−0.990431
$419$	767.918i	1.83274i	0.400333	+	0.916370i	$0.368895\pi$
$419$	767.918i	1.83274i	−0.400333	+	0.916370i	$0.631105\pi$
$420$	0	0
$421$	65.0000	0.154394	0.0771971	−	0.997016i	$-0.475403\pi$
$421$	65.0000	0.154394	0.0771971	+	0.997016i	$0.475403\pi$
$422$	294.156i	0.697053i
$423$	0	0
$424$	−144.000	−0.339623
$425$	− 118.794i	− 0.279515i
$426$	0	0
$427$	0	0
$428$	− 237.588i	− 0.555112i
$429$	0	0
$430$	138.000	0.320930
$431$	479.418i	1.11234i	0.831069	+	0.556170i	$0.187730\pi$
$431$	479.418i	1.11234i	−0.831069	+	0.556170i	$0.812270\pi$
$432$	0	0
$433$	367.000	0.847575	0.423788	−	0.905762i	$-0.360700\pi$
$433$	367.000	0.847575	0.423788	+	0.905762i	$0.360700\pi$
$434$	0	0
$435$	0	0
$436$	98.0000	0.224771
$437$	− 390.323i	− 0.893188i
$438$	0	0
$439$	−536.000	−1.22096	−0.610478	−	0.792033i	$-0.709023\pi$
$439$	−536.000	−1.22096	−0.610478	+	0.792033i	$0.709023\pi$
$440$	− 152.735i	− 0.347125i
$441$	0	0
$442$	−24.0000	−0.0542986
$443$	− 84.8528i	− 0.191541i	−0.995403	−	0.0957707i	$-0.969468\pi$
$443$	− 84.8528i	− 0.191541i	0.995403	−	0.0957707i	$-0.0305315\pi$
$444$	0	0
$445$	−576.000	−1.29438
$446$	− 56.5685i	− 0.126835i
$447$	0	0
$448$	0	0
$449$	615.183i	1.37012i	0.728488	+	0.685059i	$0.240224\pi$
$449$	615.183i	1.37012i	−0.728488	+	0.685059i	$0.759776\pi$
$450$	0	0
$451$	594.000	1.31707
$452$	− 195.161i	− 0.431773i
$453$	0	0
$454$	−318.000	−0.700441
$455$	0	0
$456$	0	0
$457$	−463.000	−1.01313	−0.506565	−	0.862202i	$-0.669085\pi$
$457$	−463.000	−1.01313	−0.506565	+	0.862202i	$0.669085\pi$
$458$	533.159i	1.16410i
$459$	0	0
$460$	144.000	0.313043
$461$	271.529i	0.589000i	0.955651	+	0.294500i	$0.0951531\pi$
$461$	271.529i	0.589000i	−0.955651	+	0.294500i	$0.904847\pi$
$462$	0	0
$463$	47.0000	0.101512	0.0507559	−	0.998711i	$-0.483837\pi$
$463$	47.0000	0.101512	0.0507559	+	0.998711i	$0.483837\pi$
$464$	− 135.765i	− 0.292596i
$465$	0	0
$466$	−486.000	−1.04292
$467$	742.462i	1.58985i	0.606705	+	0.794927i	$0.292491\pi$
$467$	742.462i	1.58985i	−0.606705	+	0.794927i	$0.707509\pi$
$468$	0	0
$469$	0	0
$470$	− 25.4558i	− 0.0541614i
$471$	0	0
$472$	−240.000	−0.508475
$473$	292.742i	0.618905i
$474$	0	0
$475$	−161.000	−0.338947
$476$	0	0
$477$	0	0
$478$	18.0000	0.0376569
$479$	424.264i	0.885729i	0.896589	+	0.442864i	$0.146038\pi$
$479$	424.264i	0.885729i	−0.896589	+	0.442864i	$0.853962\pi$
$480$	0	0
$481$	−55.0000	−0.114345
$482$	− 183.848i	− 0.381427i
$483$	0	0
$484$	82.0000	0.169421
$485$	− 441.235i	− 0.909762i
$486$	0	0
$487$	239.000	0.490760	0.245380	−	0.969427i	$-0.421087\pi$
$487$	239.000	0.490760	0.245380	+	0.969427i	$0.421087\pi$
$488$	− 294.156i	− 0.602780i
$489$	0	0
$490$	0	0
$491$	− 203.647i	− 0.414759i	−0.978261	−	0.207380i	$-0.933506\pi$
$491$	− 203.647i	− 0.414759i	0.978261	−	0.207380i	$-0.0664935\pi$
$492$	0	0
$493$	−576.000	−1.16836
$494$	32.5269i	0.0658440i
$495$	0	0
$496$	−188.000	−0.379032
$497$	0	0
$498$	0	0
$499$	−49.0000	−0.0981964	−0.0490982	−	0.998794i	$-0.515635\pi$
$499$	−49.0000	−0.0981964	−0.0490982	+	0.998794i	$0.515635\pi$
$500$	− 271.529i	− 0.543058i
$501$	0	0
$502$	−72.0000	−0.143426
$503$	589.727i	1.17242i	0.810159	+	0.586210i	$0.199381\pi$
$503$	589.727i	1.17242i	−0.810159	+	0.586210i	$0.800619\pi$
$504$	0	0
$505$	630.000	1.24752
$506$	305.470i	0.603696i
$507$	0	0
$508$	−226.000	−0.444882
$509$	− 241.831i	− 0.475109i	−0.971374	−	0.237555i	$-0.923654\pi$
$509$	− 241.831i	− 0.475109i	0.971374	−	0.237555i	$-0.0763458\pi$
$510$	0	0
$511$	0	0
$512$	− 22.6274i	− 0.0441942i
$513$	0	0
$514$	−666.000	−1.29572
$515$	− 504.874i	− 0.980338i
$516$	0	0
$517$	54.0000	0.104449
$518$	0	0
$519$	0	0
$520$	−12.0000	−0.0230769
$521$	848.528i	1.62865i	0.580407	+	0.814326i	$0.302893\pi$
$521$	848.528i	1.62865i	−0.580407	+	0.814326i	$0.697107\pi$
$522$	0	0
$523$	391.000	0.747610	0.373805	−	0.927507i	$-0.378053\pi$
$523$	391.000	0.747610	0.373805	+	0.927507i	$0.378053\pi$
$524$	− 42.4264i	− 0.0809664i
$525$	0	0
$526$	144.000	0.273764
$527$	797.616i	1.51350i
$528$	0	0
$529$	241.000	0.455577
$530$	305.470i	0.576359i
$531$	0	0
$532$	0	0
$533$	− 46.6690i	− 0.0875592i
$534$	0	0
$535$	−504.000	−0.942056
$536$	− 274.357i	− 0.511861i
$537$	0	0
$538$	402.000	0.747212
$539$	0	0
$540$	0	0
$541$	−73.0000	−0.134935	−0.0674677	−	0.997721i	$-0.521492\pi$
$541$	−73.0000	−0.134935	−0.0674677	+	0.997721i	$0.521492\pi$
$542$	− 735.391i	− 1.35681i
$543$	0	0
$544$	−96.0000	−0.176471
$545$	− 207.889i	− 0.381448i
$546$	0	0
$547$	32.0000	0.0585009	0.0292505	−	0.999572i	$-0.490688\pi$
$547$	32.0000	0.0585009	0.0292505	+	0.999572i	$0.490688\pi$
$548$	135.765i	0.247745i
$549$	0	0
$550$	126.000	0.229091
$551$	780.646i	1.41678i
$552$	0	0
$553$	0	0
$554$	− 159.806i	− 0.288459i
$555$	0	0
$556$	−206.000	−0.370504
$557$	46.6690i	0.0837864i	0.999122	+	0.0418932i	$0.0133389\pi$
$557$	46.6690i	0.0837864i	−0.999122	+	0.0418932i	$0.986661\pi$
$558$	0	0
$559$	23.0000	0.0411449
$560$	0	0
$561$	0	0
$562$	648.000	1.15302
$563$	886.712i	1.57498i	0.616330	+	0.787488i	$0.288619\pi$
$563$	886.712i	1.57498i	−0.616330	+	0.787488i	$0.711381\pi$
$564$	0	0
$565$	−414.000	−0.732743
$566$	125.865i	0.222376i
$567$	0	0
$568$	276.000	0.485915
$569$	318.198i	0.559223i	0.960113	+	0.279612i	$0.0902057\pi$
$569$	318.198i	0.559223i	−0.960113	+	0.279612i	$0.909794\pi$
$570$	0	0
$571$	−535.000	−0.936953	−0.468476	−	0.883476i	$-0.655197\pi$
$571$	−535.000	−0.936953	−0.468476	+	0.883476i	$0.655197\pi$
$572$	− 25.4558i	− 0.0445032i
$573$	0	0
$574$	0	0
$575$	118.794i	0.206598i
$576$	0	0
$577$	−905.000	−1.56846	−0.784229	−	0.620472i	$-0.786941\pi$
$577$	−905.000	−1.56846	−0.784229	+	0.620472i	$0.786941\pi$
$578$	− 1.41421i	− 0.00244674i
$579$	0	0
$580$	−288.000	−0.496552
$581$	0	0
$582$	0	0
$583$	−648.000	−1.11149
$584$	− 183.848i	− 0.314808i
$585$	0	0
$586$	−96.0000	−0.163823
$587$	576.999i	0.982963i	0.870888	+	0.491481i	$0.163544\pi$
$587$	576.999i	0.982963i	−0.870888	+	0.491481i	$0.836456\pi$
$588$	0	0
$589$	1081.00	1.83531
$590$	509.117i	0.862910i
$591$	0	0
$592$	−220.000	−0.371622
$593$	555.786i	0.937244i	0.883399	+	0.468622i	$0.155249\pi$
$593$	555.786i	0.937244i	−0.883399	+	0.468622i	$0.844751\pi$
$594$	0	0
$595$	0	0
$596$	− 339.411i	− 0.569482i
$597$	0	0
$598$	24.0000	0.0401338
$599$	− 203.647i	− 0.339978i	−0.985446	−	0.169989i	$-0.945627\pi$
$599$	− 203.647i	− 0.339978i	0.985446	−	0.169989i	$-0.0543732\pi$
$600$	0	0
$601$	−143.000	−0.237937	−0.118968	−	0.992898i	$-0.537959\pi$
$601$	−143.000	−0.237937	−0.118968	+	0.992898i	$0.537959\pi$
$602$	0	0
$603$	0	0
$604$	−208.000	−0.344371
$605$	− 173.948i	− 0.287518i
$606$	0	0
$607$	−599.000	−0.986820	−0.493410	−	0.869797i	$-0.664250\pi$
$607$	−599.000	−0.986820	−0.493410	+	0.869797i	$0.664250\pi$
$608$	130.108i	0.213993i
$609$	0	0
$610$	−624.000	−1.02295
$611$	− 4.24264i	− 0.00694377i
$612$	0	0
$613$	−304.000	−0.495922	−0.247961	−	0.968770i	$-0.579760\pi$
$613$	−304.000	−0.495922	−0.247961	+	0.968770i	$0.579760\pi$
$614$	329.512i	0.536664i
$615$	0	0
$616$	0	0
$617$	292.742i	0.474461i	0.971453	+	0.237230i	$0.0762396\pi$
$617$	292.742i	0.474461i	−0.971453	+	0.237230i	$0.923760\pi$
$618$	0	0
$619$	343.000	0.554120	0.277060	−	0.960853i	$-0.410640\pi$
$619$	343.000	0.554120	0.277060	+	0.960853i	$0.410640\pi$
$620$	398.808i	0.643239i
$621$	0	0
$622$	−354.000	−0.569132
$623$	0	0
$624$	0	0
$625$	−401.000	−0.641600
$626$	− 213.546i	− 0.341128i
$627$	0	0
$628$	304.000	0.484076
$629$	933.381i	1.48391i
$630$	0	0
$631$	272.000	0.431062	0.215531	−	0.976497i	$-0.430852\pi$
$631$	272.000	0.431062	0.215531	+	0.976497i	$0.430852\pi$
$632$	319.612i	0.505716i
$633$	0	0
$634$	576.000	0.908517
$635$	479.418i	0.754990i
$636$	0	0
$637$	0	0
$638$	− 610.940i	− 0.957587i
$639$	0	0
$640$	−48.0000	−0.0750000
$641$	424.264i	0.661878i	0.943652	+	0.330939i	$0.107366\pi$
$641$	424.264i	0.661878i	−0.943652	+	0.330939i	$0.892634\pi$
$642$	0	0
$643$	679.000	1.05599	0.527994	−	0.849248i	$-0.322944\pi$
$643$	679.000	1.05599	0.527994	+	0.849248i	$0.322944\pi$
$644$	0	0
$645$	0	0
$646$	552.000	0.854489
$647$	963.079i	1.48853i	0.667884	+	0.744265i	$0.267200\pi$
$647$	963.079i	1.48853i	−0.667884	+	0.744265i	$0.732800\pi$
$648$	0	0
$649$	−1080.00	−1.66410
$650$	− 9.89949i	− 0.0152300i
$651$	0	0
$652$	−112.000	−0.171779
$653$	− 895.197i	− 1.37090i	−0.728120	−	0.685450i	$-0.759606\pi$
$653$	− 895.197i	− 1.37090i	0.728120	−	0.685450i	$-0.240394\pi$
$654$	0	0
$655$	−90.0000	−0.137405
$656$	− 186.676i	− 0.284567i
$657$	0	0
$658$	0	0
$659$	− 16.9706i	− 0.0257520i	−0.999917	−	0.0128760i	$-0.995901\pi$
$659$	− 16.9706i	− 0.0257520i	0.999917	−	0.0128760i	$-0.00409867\pi$
$660$	0	0
$661$	433.000	0.655068	0.327534	−	0.944839i	$-0.393782\pi$
$661$	433.000	0.655068	0.327534	+	0.944839i	$0.393782\pi$
$662$	103.238i	0.155948i
$663$	0	0
$664$	84.0000	0.126506
$665$	0	0
$666$	0	0
$667$	576.000	0.863568
$668$	8.48528i	0.0127025i
$669$	0	0
$670$	−582.000	−0.868657
$671$	− 1323.70i	− 1.97273i
$672$	0	0
$673$	737.000	1.09510	0.547548	−	0.836774i	$-0.315561\pi$
$673$	737.000	1.09510	0.547548	+	0.836774i	$0.315561\pi$
$674$	− 745.291i	− 1.10577i
$675$	0	0
$676$	336.000	0.497041
$677$	135.765i	0.200538i	0.994960	+	0.100269i	$0.0319704\pi$
$677$	135.765i	0.200538i	−0.994960	+	0.100269i	$0.968030\pi$
$678$	0	0
$679$	0	0
$680$	203.647i	0.299481i
$681$	0	0
$682$	−846.000	−1.24047
$683$	1272.79i	1.86353i	0.363060	+	0.931766i	$0.381732\pi$
$683$	1272.79i	1.86353i	−0.363060	+	0.931766i	$0.618268\pi$
$684$	0	0
$685$	288.000	0.420438
$686$	0	0
$687$	0	0
$688$	92.0000	0.133721
$689$	50.9117i	0.0738921i
$690$	0	0
$691$	247.000	0.357453	0.178726	−	0.983899i	$-0.442802\pi$
$691$	247.000	0.357453	0.178726	+	0.983899i	$0.442802\pi$
$692$	305.470i	0.441431i
$693$	0	0
$694$	600.000	0.864553
$695$	436.992i	0.628765i
$696$	0	0
$697$	−792.000	−1.13630
$698$	− 158.392i	− 0.226923i
$699$	0	0
$700$	0	0
$701$	322.441i	0.459972i	0.973194	+	0.229986i	$0.0738681\pi$
$701$	322.441i	0.459972i	−0.973194	+	0.229986i	$0.926132\pi$
$702$	0	0
$703$	1265.00	1.79943
$704$	− 101.823i	− 0.144635i
$705$	0	0
$706$	−438.000	−0.620397
$707$	0	0
$708$	0	0
$709$	−1048.00	−1.47814	−0.739069	−	0.673630i	$-0.764734\pi$
$709$	−1048.00	−1.47814	−0.739069	+	0.673630i	$0.764734\pi$
$710$	− 585.484i	− 0.824626i
$711$	0	0
$712$	−384.000	−0.539326
$713$	− 797.616i	− 1.11868i
$714$	0	0
$715$	−54.0000	−0.0755245
$716$	8.48528i	0.0118510i
$717$	0	0
$718$	−720.000	−1.00279
$719$	− 250.316i	− 0.348144i	−0.984733	−	0.174072i	$-0.944307\pi$
$719$	− 250.316i	− 0.348144i	0.984733	−	0.174072i	$-0.0556926\pi$
$720$	0	0
$721$	0	0
$722$	− 237.588i	− 0.329069i
$723$	0	0
$724$	−110.000	−0.151934
$725$	− 237.588i	− 0.327707i
$726$	0	0
$727$	−641.000	−0.881706	−0.440853	−	0.897579i	$-0.645324\pi$
$727$	−641.000	−0.881706	−0.440853	+	0.897579i	$0.645324\pi$
$728$	0	0
$729$	0	0
$730$	−390.000	−0.534247
$731$	− 390.323i	− 0.533958i
$732$	0	0
$733$	25.0000	0.0341064	0.0170532	−	0.999855i	$-0.494572\pi$
$733$	25.0000	0.0341064	0.0170532	+	0.999855i	$0.494572\pi$
$734$	− 451.134i	− 0.614624i
$735$	0	0
$736$	96.0000	0.130435
$737$	− 1234.61i	− 1.67518i
$738$	0	0
$739$	−535.000	−0.723951	−0.361976	−	0.932188i	$-0.617898\pi$
$739$	−535.000	−0.723951	−0.361976	+	0.932188i	$0.617898\pi$
$740$	466.690i	0.630663i
$741$	0	0
$742$	0	0
$743$	− 937.624i	− 1.26194i	−0.775806	−	0.630971i	$-0.782656\pi$
$743$	− 937.624i	− 1.26194i	0.775806	−	0.630971i	$-0.217344\pi$
$744$	0	0
$745$	−720.000	−0.966443
$746$	− 295.571i	− 0.396207i
$747$	0	0
$748$	−432.000	−0.577540
$749$	0	0
$750$	0	0
$751$	41.0000	0.0545939	0.0272969	−	0.999627i	$-0.491310\pi$
$751$	41.0000	0.0545939	0.0272969	+	0.999627i	$0.491310\pi$
$752$	− 16.9706i	− 0.0225672i
$753$	0	0
$754$	−48.0000	−0.0636605
$755$	441.235i	0.584417i
$756$	0	0
$757$	−1114.00	−1.47160	−0.735799	−	0.677200i	$-0.763193\pi$
$757$	−1114.00	−1.47160	−0.735799	+	0.677200i	$0.763193\pi$
$758$	612.354i	0.807856i
$759$	0	0
$760$	276.000	0.363158
$761$	− 288.500i	− 0.379106i	−0.981871	−	0.189553i	$-0.939296\pi$
$761$	− 288.500i	− 0.379106i	0.981871	−	0.189553i	$-0.0607039\pi$
$762$	0	0
$763$	0	0
$764$	− 127.279i	− 0.166596i
$765$	0	0
$766$	552.000	0.720627
$767$	84.8528i	0.110629i
$768$	0	0
$769$	−1127.00	−1.46554	−0.732770	−	0.680477i	$-0.761773\pi$
$769$	−1127.00	−1.46554	−0.732770	+	0.680477i	$0.761773\pi$
$770$	0	0
$771$	0	0
$772$	302.000	0.391192
$773$	− 216.375i	− 0.279915i	−0.990157	−	0.139958i	$-0.955303\pi$
$773$	− 216.375i	− 0.279915i	0.990157	−	0.139958i	$-0.0446967\pi$
$774$	0	0
$775$	−329.000	−0.424516
$776$	− 294.156i	− 0.379068i
$777$	0	0
$778$	−522.000	−0.670951
$779$	1073.39i	1.37791i
$780$	0	0
$781$	1242.00	1.59027
$782$	− 407.294i	− 0.520836i
$783$	0	0
$784$	0	0
$785$	− 644.881i	− 0.821505i
$786$	0	0
$787$	−170.000	−0.216010	−0.108005	−	0.994150i	$-0.534446\pi$
$787$	−170.000	−0.216010	−0.108005	+	0.994150i	$0.534446\pi$
$788$	33.9411i	0.0430725i
$789$	0	0
$790$	678.000	0.858228
$791$	0	0
$792$	0	0
$793$	−104.000	−0.131148
$794$	159.806i	0.201267i
$795$	0	0
$796$	−320.000	−0.402010
$797$	− 1170.97i	− 1.46922i	−0.678489	−	0.734610i	$-0.737365\pi$
$797$	− 1170.97i	− 1.46922i	0.678489	−	0.734610i	$-0.262635\pi$
$798$	0	0
$799$	−72.0000	−0.0901126
$800$	− 39.5980i	− 0.0494975i
$801$	0	0
$802$	−504.000	−0.628429
$803$	− 827.315i	− 1.03028i
$804$	0	0
$805$	0	0
$806$	66.4680i	0.0824665i
$807$	0	0
$808$	420.000	0.519802
$809$	335.169i	0.414300i	0.978309	+	0.207150i	$0.0664188\pi$
$809$	335.169i	0.414300i	−0.978309	+	0.207150i	$0.933581\pi$
$810$	0	0
$811$	1114.00	1.37361	0.686806	−	0.726840i	$-0.259012\pi$
$811$	1114.00	1.37361	0.686806	+	0.726840i	$0.259012\pi$
$812$	0	0
$813$	0	0
$814$	−990.000	−1.21622
$815$	237.588i	0.291519i
$816$	0	0
$817$	−529.000	−0.647491
$818$	− 451.134i	− 0.551509i
$819$	0	0
$820$	−396.000	−0.482927
$821$	97.5807i	0.118856i	0.998233	+	0.0594280i	$0.0189277\pi$
$821$	97.5807i	0.118856i	−0.998233	+	0.0594280i	$0.981072\pi$
$822$	0	0
$823$	1592.00	1.93439	0.967193	−	0.254042i	$-0.0817601\pi$
$823$	1592.00	1.93439	0.967193	+	0.254042i	$0.0817601\pi$
$824$	− 336.583i	− 0.408474i
$825$	0	0
$826$	0	0
$827$	− 1022.48i	− 1.23637i	−0.786033	−	0.618184i	$-0.787869\pi$
$827$	− 1022.48i	− 1.23637i	0.786033	−	0.618184i	$-0.212131\pi$
$828$	0	0
$829$	−383.000	−0.462002	−0.231001	−	0.972953i	$-0.574200\pi$
$829$	−383.000	−0.462002	−0.231001	+	0.972953i	$0.574200\pi$
$830$	− 178.191i	− 0.214688i
$831$	0	0
$832$	−8.00000	−0.00961538
$833$	0	0
$834$	0	0
$835$	18.0000	0.0215569
$836$	585.484i	0.700340i
$837$	0	0
$838$	1086.00	1.29594
$839$	576.999i	0.687722i	0.939020	+	0.343861i	$0.111735\pi$
$839$	576.999i	0.687722i	−0.939020	+	0.343861i	$0.888265\pi$
$840$	0	0
$841$	−311.000	−0.369798
$842$	− 91.9239i	− 0.109173i
$843$	0	0
$844$	416.000	0.492891
$845$	− 712.764i	− 0.843507i
$846$	0	0
$847$	0	0
$848$	203.647i	0.240149i
$849$	0	0
$850$	−168.000	−0.197647
$851$	− 933.381i	− 1.09680i
$852$	0	0
$853$	−527.000	−0.617819	−0.308910	−	0.951091i	$-0.599964\pi$
$853$	−527.000	−0.617819	−0.308910	+	0.951091i	$0.599964\pi$
$854$	0	0
$855$	0	0
$856$	−336.000	−0.392523
$857$	− 1323.70i	− 1.54458i	−0.635271	−	0.772289i	$-0.719112\pi$
$857$	− 1323.70i	− 1.54458i	0.635271	−	0.772289i	$-0.280888\pi$
$858$	0	0
$859$	1414.00	1.64610	0.823050	−	0.567969i	$-0.192271\pi$
$859$	1414.00	1.64610	0.823050	+	0.567969i	$0.192271\pi$
$860$	− 195.161i	− 0.226932i
$861$	0	0
$862$	678.000	0.786543
$863$	− 1370.37i	− 1.58792i	−0.607972	−	0.793959i	$-0.708017\pi$
$863$	− 1370.37i	− 1.58792i	0.607972	−	0.793959i	$-0.291983\pi$
$864$	0	0
$865$	648.000	0.749133
$866$	− 519.016i	− 0.599326i
$867$	0	0
$868$	0	0
$869$	1438.26i	1.65507i
$870$	0	0
$871$	−97.0000	−0.111366
$872$	− 138.593i	− 0.158937i
$873$	0	0
$874$	−552.000	−0.631579
$875$	0	0
$876$	0	0
$877$	224.000	0.255416	0.127708	−	0.991812i	$-0.459238\pi$
$877$	224.000	0.255416	0.127708	+	0.991812i	$0.459238\pi$
$878$	758.018i	0.863347i
$879$	0	0
$880$	−216.000	−0.245455
$881$	1612.20i	1.82997i	0.403488	+	0.914985i	$0.367798\pi$
$881$	1612.20i	1.82997i	−0.403488	+	0.914985i	$0.632202\pi$
$882$	0	0
$883$	329.000	0.372593	0.186297	−	0.982494i	$-0.440351\pi$
$883$	329.000	0.372593	0.186297	+	0.982494i	$0.440351\pi$
$884$	33.9411i	0.0383949i
$885$	0	0
$886$	−120.000	−0.135440
$887$	55.1543i	0.0621808i	0.999517	+	0.0310904i	$0.00989797\pi$
$887$	55.1543i	0.0621808i	−0.999517	+	0.0310904i	$0.990102\pi$
$888$	0	0
$889$	0	0
$890$	814.587i	0.915266i
$891$	0	0
$892$	−80.0000	−0.0896861
$893$	97.5807i	0.109273i
$894$	0	0
$895$	18.0000	0.0201117
$896$	0	0
$897$	0	0
$898$	870.000	0.968820
$899$	1595.23i	1.77445i
$900$	0	0
$901$	864.000	0.958935
$902$	− 840.043i	− 0.931311i
$903$	0	0
$904$	−276.000	−0.305310
$905$	233.345i	0.257840i
$906$	0	0
$907$	311.000	0.342889	0.171444	−	0.985194i	$-0.445157\pi$
$907$	311.000	0.342889	0.171444	+	0.985194i	$0.445157\pi$
$908$	449.720i	0.495286i
$909$	0	0
$910$	0	0
$911$	− 356.382i	− 0.391198i	−0.980684	−	0.195599i	$-0.937335\pi$
$911$	− 356.382i	− 0.391198i	0.980684	−	0.195599i	$-0.0626652\pi$
$912$	0	0
$913$	378.000	0.414020
$914$	654.781i	0.716390i
$915$	0	0
$916$	754.000	0.823144
$917$	0	0
$918$	0	0
$919$	−145.000	−0.157780	−0.0788901	−	0.996883i	$-0.525138\pi$
$919$	−145.000	−0.157780	−0.0788901	+	0.996883i	$0.525138\pi$
$920$	− 203.647i	− 0.221355i
$921$	0	0
$922$	384.000	0.416486
$923$	− 97.5807i	− 0.105721i
$924$	0	0
$925$	−385.000	−0.416216
$926$	− 66.4680i	− 0.0717797i
$927$	0	0
$928$	−192.000	−0.206897
$929$	− 309.713i	− 0.333383i	−0.986009	−	0.166691i	$-0.946692\pi$
$929$	− 309.713i	− 0.333383i	0.986009	−	0.166691i	$-0.0533084\pi$
$930$	0	0
$931$	0	0
$932$	687.308i	0.737455i
$933$	0	0
$934$	1050.00	1.12420
$935$	916.410i	0.980118i
$936$	0	0
$937$	487.000	0.519744	0.259872	−	0.965643i	$-0.416320\pi$
$937$	487.000	0.519744	0.259872	+	0.965643i	$0.416320\pi$
$938$	0	0
$939$	0	0
$940$	−36.0000	−0.0382979
$941$	− 1374.62i	− 1.46080i	−0.683018	−	0.730401i	$-0.739333\pi$
$941$	− 1374.62i	− 1.46080i	0.683018	−	0.730401i	$-0.260667\pi$
$942$	0	0
$943$	792.000	0.839873
$944$	339.411i	0.359546i
$945$	0	0
$946$	414.000	0.437632
$947$	190.919i	0.201604i	0.994907	+	0.100802i	$0.0321408\pi$
$947$	190.919i	0.201604i	−0.994907	+	0.100802i	$0.967859\pi$
$948$	0	0
$949$	−65.0000	−0.0684932
$950$	227.688i	0.239672i
$951$	0	0
$952$	0	0
$953$	1035.20i	1.08626i	0.839649	+	0.543129i	$0.182761\pi$
$953$	1035.20i	1.08626i	−0.839649	+	0.543129i	$0.817239\pi$
$954$	0	0
$955$	−270.000	−0.282723
$956$	− 25.4558i	− 0.0266275i
$957$	0	0
$958$	600.000	0.626305
$959$	0	0
$960$	0	0
$961$	1248.00	1.29865
$962$	77.7817i	0.0808542i
$963$	0	0
$964$	−260.000	−0.269710
$965$	− 640.639i	− 0.663874i
$966$	0	0
$967$	−1681.00	−1.73837	−0.869183	−	0.494490i	$-0.835355\pi$
$967$	−1681.00	−1.73837	−0.869183	+	0.494490i	$0.835355\pi$
$968$	− 115.966i	− 0.119799i
$969$	0	0
$970$	−624.000	−0.643299
$971$	1204.91i	1.24090i	0.784248	+	0.620448i	$0.213049\pi$
$971$	1204.91i	1.24090i	−0.784248	+	0.620448i	$0.786951\pi$
$972$	0	0
$973$	0	0
$974$	− 337.997i	− 0.347020i
$975$	0	0
$976$	−416.000	−0.426230
$977$	− 1175.21i	− 1.20288i	−0.798919	−	0.601439i	$-0.794594\pi$
$977$	− 1175.21i	− 1.20288i	0.798919	−	0.601439i	$-0.205406\pi$
$978$	0	0
$979$	−1728.00	−1.76507
$980$	0	0
$981$	0	0
$982$	−288.000	−0.293279
$983$	− 1917.67i	− 1.95084i	−0.220359	−	0.975419i	$-0.570723\pi$
$983$	− 1917.67i	− 1.95084i	0.220359	−	0.975419i	$-0.429277\pi$
$984$	0	0
$985$	72.0000	0.0730964
$986$	814.587i	0.826153i
$987$	0	0
$988$	46.0000	0.0465587
$989$	390.323i	0.394664i
$990$	0	0
$991$	−625.000	−0.630676	−0.315338	−	0.948979i	$-0.602118\pi$
$991$	−625.000	−0.630676	−0.315338	+	0.948979i	$0.602118\pi$
$992$	265.872i	0.268016i
$993$	0	0
$994$	0	0
$995$	678.823i	0.682234i
$996$	0	0
$997$	943.000	0.945838	0.472919	−	0.881106i	$-0.343200\pi$
$997$	943.000	0.945838	0.472919	+	0.881106i	$0.343200\pi$
$998$	69.2965i	0.0694353i
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
126.3.s.a.53.1	✓	4	7.5	odd	6
126.3.s.a.53.2	yes	4	21.5	even	6
126.3.s.a.107.1	yes	4	21.17	even	6
126.3.s.a.107.2	yes	4	7.3	odd	6
882.3.b.b.197.1		2	7.6	odd	2
882.3.b.b.197.2		2	21.20	even	2
882.3.b.e.197.1		2	1.1	even	1	trivial
882.3.b.e.197.2		2	3.2	odd	2	inner
882.3.s.a.557.1		4	7.2	even	3
882.3.s.a.557.2		4	21.2	odd	6
882.3.s.a.863.1		4	21.11	odd	6
882.3.s.a.863.2		4	7.4	even	3
1008.3.dc.b.305.1		4	28.19	even	6
1008.3.dc.b.305.2		4	84.47	odd	6
1008.3.dc.b.737.1		4	84.59	odd	6
1008.3.dc.b.737.2		4	28.3	even	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)

\(f(q)\)	\(=\)	\(q-1.41421i q^{2} -2.00000 q^{4} +4.24264i q^{5} +2.82843i q^{8} +O(q^{10})\)	\(q-1.41421i q^{2} -2.00000 q^{4} +4.24264i q^{5} +2.82843i q^{8} +6.00000 q^{10} +12.7279i q^{11} +1.00000 q^{13} +4.00000 q^{16} -16.9706i q^{17} -23.0000 q^{19} -8.48528i q^{20} +18.0000 q^{22} +16.9706i q^{23} +7.00000 q^{25} -1.41421i q^{26} -33.9411i q^{29} -47.0000 q^{31} -5.65685i q^{32} -24.0000 q^{34} -55.0000 q^{37} +32.5269i q^{38} -12.0000 q^{40} -46.6690i q^{41} +23.0000 q^{43} -25.4558i q^{44} +24.0000 q^{46} -4.24264i q^{47} -9.89949i q^{50} -2.00000 q^{52} +50.9117i q^{53} -54.0000 q^{55} -48.0000 q^{58} +84.8528i q^{59} -104.000 q^{61} +66.4680i q^{62} -8.00000 q^{64} +4.24264i q^{65} -97.0000 q^{67} +33.9411i q^{68} -97.5807i q^{71} -65.0000 q^{73} +77.7817i q^{74} +46.0000 q^{76} +113.000 q^{79} +16.9706i q^{80} -66.0000 q^{82} -29.6985i q^{83} +72.0000 q^{85} -32.5269i q^{86} -36.0000 q^{88} +135.765i q^{89} -33.9411i q^{92} -6.00000 q^{94} -97.5807i q^{95} -104.000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{4}+O(q^{10}) \) 2 * q - 4 * q^4	\( 2 q - 4 q^{4} + 12 q^{10} + 2 q^{13} + 8 q^{16} - 46 q^{19} + 36 q^{22} + 14 q^{25} - 94 q^{31} - 48 q^{34} - 110 q^{37} - 24 q^{40} + 46 q^{43} + 48 q^{46} - 4 q^{52} - 108 q^{55} - 96 q^{58} - 208 q^{61} - 16 q^{64} - 194 q^{67} - 130 q^{73} + 92 q^{76} + 226 q^{79} - 132 q^{82} + 144 q^{85} - 72 q^{88} - 12 q^{94} - 208 q^{97}+O(q^{100}) \) 2 * q - 4 * q^4 + 12 * q^10 + 2 * q^13 + 8 * q^16 - 46 * q^19 + 36 * q^22 + 14 * q^25 - 94 * q^31 - 48 * q^34 - 110 * q^37 - 24 * q^40 + 46 * q^43 + 48 * q^46 - 4 * q^52 - 108 * q^55 - 96 * q^58 - 208 * q^61 - 16 * q^64 - 194 * q^67 - 130 * q^73 + 92 * q^76 + 226 * q^79 - 132 * q^82 + 144 * q^85 - 72 * q^88 - 12 * q^94 - 208 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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Database

Properties

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