LMFDB - Embedded newform 882.5.b.f.197.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [882,5,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, 5, names="a")

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

chi := DirichletCharacter("882.197");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 882 = 2 \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 5 $
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:	$91.1723074400$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 32x^{4} - 7932x^{3} + 185683x^{2} + 6753864x + 64548144 $ x^6 - 32x^4 - 7932x^3 + 185683x^2 + 6753864x + 64548144
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 126)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			197.6
Root			$-10.6519 - 21.4865i$ of defining polynomial
Character	$\chi$	$=$	882.197
Dual form			882.5.b.f.197.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.82843i q^{2} -8.00000 q^{4} +44.3872i q^{5} -22.6274i q^{8} +O(q^{10})$	$q+2.82843i q^{2} -8.00000 q^{4} +44.3872i q^{5} -22.6274i q^{8} -125.546 q^{10} +72.1549i q^{11} +119.188 q^{13} +64.0000 q^{16} -166.812i q^{17} +146.082 q^{19} -355.097i q^{20} -204.085 q^{22} -586.572i q^{23} -1345.22 q^{25} +337.114i q^{26} -1117.41i q^{29} -1716.30 q^{31} +181.019i q^{32} +471.814 q^{34} -2099.13 q^{37} +413.182i q^{38} +1004.37 q^{40} +2393.22i q^{41} +324.169 q^{43} -577.239i q^{44} +1659.08 q^{46} -1735.01i q^{47} -3804.86i q^{50} -953.501 q^{52} -2807.02i q^{53} -3202.75 q^{55} +3160.50 q^{58} -5423.64i q^{59} +1582.26 q^{61} -4854.44i q^{62} -512.000 q^{64} +5290.40i q^{65} -1058.38 q^{67} +1334.49i q^{68} +4249.50i q^{71} +5148.13 q^{73} -5937.25i q^{74} -1168.65 q^{76} -8619.78 q^{79} +2840.78i q^{80} -6769.05 q^{82} -4627.10i q^{83} +7404.30 q^{85} +916.887i q^{86} +1632.68 q^{88} -12128.4i q^{89} +4692.58i q^{92} +4907.35 q^{94} +6484.16i q^{95} -9813.46 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 48 q^{4}+O(q^{10}) $ 6 * q - 48 * q^4	$ 6 q - 48 q^{4} - 40 q^{10} + 278 q^{13} + 384 q^{16} + 654 q^{19} + 808 q^{22} - 1786 q^{25} - 3790 q^{31} - 2768 q^{34} - 4542 q^{37} + 320 q^{40} + 8142 q^{43} - 1136 q^{46} - 2224 q^{52} - 11432 q^{55} - 832 q^{58} - 4024 q^{61} - 3072 q^{64} - 7466 q^{67} + 5506 q^{73} - 5232 q^{76} - 3778 q^{79} + 5496 q^{82} + 37312 q^{85} - 6464 q^{88} + 31224 q^{94} - 30692 q^{97}+O(q^{100}) $ 6 * q - 48 * q^4 - 40 * q^10 + 278 * q^13 + 384 * q^16 + 654 * q^19 + 808 * q^22 - 1786 * q^25 - 3790 * q^31 - 2768 * q^34 - 4542 * q^37 + 320 * q^40 + 8142 * q^43 - 1136 * q^46 - 2224 * q^52 - 11432 * q^55 - 832 * q^58 - 4024 * q^61 - 3072 * q^64 - 7466 * q^67 + 5506 * q^73 - 5232 * q^76 - 3778 * q^79 + 5496 * q^82 + 37312 * q^85 - 6464 * q^88 + 31224 * q^94 - 30692 * 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 $

$2$	2.82843i	0.707107i
$2$	2.82843i	0.707107i
$3$	0	0
$3$	0	0
$4$	−8.00000	−0.500000
$5$	44.3872i	1.77549i	0.460338	+	0.887744i	$0.347728\pi$
$5$	44.3872i	1.77549i	−0.460338	+	0.887744i	$0.652272\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	− 22.6274i	− 0.353553i
$9$	0	0
$10$	−125.546	−1.25546
$11$	72.1549i	0.596321i	0.954516	+	0.298161i	$0.0963731\pi$
$11$	72.1549i	0.596321i	−0.954516	+	0.298161i	$0.903627\pi$
$12$	0	0
$13$	119.188	0.705252	0.352626	−	0.935764i	$-0.385289\pi$
$13$	119.188	0.705252	0.352626	+	0.935764i	$0.385289\pi$
$14$	0	0
$15$	0	0
$16$	64.0000	0.250000
$17$	− 166.812i	− 0.577203i	−0.957449	−	0.288601i	$-0.906810\pi$
$17$	− 166.812i	− 0.577203i	0.957449	−	0.288601i	$-0.0931902\pi$
$18$	0	0
$19$	146.082	0.404659	0.202329	−	0.979318i	$-0.435149\pi$
$19$	146.082	0.404659	0.202329	+	0.979318i	$0.435149\pi$
$20$	− 355.097i	− 0.887744i
$21$	0	0
$22$	−204.085	−0.421663
$23$	− 586.572i	− 1.10883i	−0.832240	−	0.554416i	$-0.812942\pi$
$23$	− 586.572i	− 1.10883i	0.832240	−	0.554416i	$-0.187058\pi$
$24$	0	0
$25$	−1345.22	−2.15236
$26$	337.114i	0.498689i
$27$	0	0
$28$	0	0
$29$	− 1117.41i	− 1.32866i	−0.747437	−	0.664332i	$-0.768716\pi$
$29$	− 1117.41i	− 1.32866i	0.747437	−	0.664332i	$-0.231284\pi$
$30$	0	0
$31$	−1716.30	−1.78596	−0.892978	−	0.450100i	$-0.851388\pi$
$31$	−1716.30	−1.78596	−0.892978	+	0.450100i	$0.851388\pi$
$32$	181.019i	0.176777i
$33$	0	0
$34$	471.814	0.408144
$35$	0	0
$36$	0	0
$37$	−2099.13	−1.53333	−0.766667	−	0.642045i	$-0.778086\pi$
$37$	−2099.13	−1.53333	−0.766667	+	0.642045i	$0.778086\pi$
$38$	413.182i	0.286137i
$39$	0	0
$40$	1004.37	0.627730
$41$	2393.22i	1.42369i	0.702337	+	0.711845i	$0.252140\pi$
$41$	2393.22i	1.42369i	−0.702337	+	0.711845i	$0.747860\pi$
$42$	0	0
$43$	324.169	0.175321	0.0876605	−	0.996150i	$-0.472061\pi$
$43$	324.169	0.175321	0.0876605	+	0.996150i	$0.472061\pi$
$44$	− 577.239i	− 0.298161i
$45$	0	0
$46$	1659.08	0.784062
$47$	− 1735.01i	− 0.785427i	−0.919661	−	0.392714i	$-0.871536\pi$
$47$	− 1735.01i	− 0.785427i	0.919661	−	0.392714i	$-0.128464\pi$
$48$	0	0
$49$	0	0
$50$	− 3804.86i	− 1.52195i
$51$	0	0
$52$	−953.501	−0.352626
$53$	− 2807.02i	− 0.999296i	−0.866228	−	0.499648i	$-0.833463\pi$
$53$	− 2807.02i	− 0.999296i	0.866228	−	0.499648i	$-0.166537\pi$
$54$	0	0
$55$	−3202.75	−1.05876
$56$	0	0
$57$	0	0
$58$	3160.50	0.939508
$59$	− 5423.64i	− 1.55807i	−0.626981	−	0.779035i	$-0.715709\pi$
$59$	− 5423.64i	− 1.55807i	0.626981	−	0.779035i	$-0.284291\pi$
$60$	0	0
$61$	1582.26	0.425224	0.212612	−	0.977137i	$-0.431803\pi$
$61$	1582.26	0.425224	0.212612	+	0.977137i	$0.431803\pi$
$62$	− 4854.44i	− 1.26286i
$63$	0	0
$64$	−512.000	−0.125000
$65$	5290.40i	1.25217i
$66$	0	0
$67$	−1058.38	−0.235772	−0.117886	−	0.993027i	$-0.537612\pi$
$67$	−1058.38	−0.235772	−0.117886	+	0.993027i	$0.537612\pi$
$68$	1334.49i	0.288601i
$69$	0	0
$70$	0	0
$71$	4249.50i	0.842988i	0.906831	+	0.421494i	$0.138494\pi$
$71$	4249.50i	0.842988i	−0.906831	+	0.421494i	$0.861506\pi$
$72$	0	0
$73$	5148.13	0.966060	0.483030	−	0.875604i	$-0.339536\pi$
$73$	5148.13	0.966060	0.483030	+	0.875604i	$0.339536\pi$
$74$	− 5937.25i	− 1.08423i
$75$	0	0
$76$	−1168.65	−0.202329
$77$	0	0
$78$	0	0
$79$	−8619.78	−1.38115	−0.690576	−	0.723259i	$-0.742643\pi$
$79$	−8619.78	−1.38115	−0.690576	+	0.723259i	$0.742643\pi$
$80$	2840.78i	0.443872i
$81$	0	0
$82$	−6769.05	−1.00670
$83$	− 4627.10i	− 0.671665i	−0.941922	−	0.335832i	$-0.890982\pi$
$83$	− 4627.10i	− 0.671665i	0.941922	−	0.335832i	$-0.109018\pi$
$84$	0	0
$85$	7404.30	1.02482
$86$	916.887i	0.123971i
$87$	0	0
$88$	1632.68	0.210831
$89$	− 12128.4i	− 1.53117i	−0.643336	−	0.765584i	$-0.722450\pi$
$89$	− 12128.4i	− 1.53117i	0.643336	−	0.765584i	$-0.277550\pi$
$90$	0	0
$91$	0	0
$92$	4692.58i	0.554416i
$93$	0	0
$94$	4907.35	0.555381
$95$	6484.16i	0.718467i
$96$	0	0
$97$	−9813.46	−1.04299	−0.521493	−	0.853255i	$-0.674625\pi$
$97$	−9813.46	−1.04299	−0.521493	+	0.853255i	$0.674625\pi$
$98$	0	0
$99$	0	0
$100$	10761.8	1.07618
$101$	− 3246.95i	− 0.318297i	−0.987255	−	0.159149i	$-0.949125\pi$
$101$	− 3246.95i	− 0.318297i	0.987255	−	0.159149i	$-0.0508749\pi$
$102$	0	0
$103$	103.524	0.00975812	0.00487906	−	0.999988i	$-0.498447\pi$
$103$	103.524	0.00975812	0.00487906	+	0.999988i	$0.498447\pi$
$104$	− 2696.91i	− 0.249344i
$105$	0	0
$106$	7939.46	0.706609
$107$	15295.0i	1.33592i	0.744196	+	0.667961i	$0.232833\pi$
$107$	15295.0i	1.33592i	−0.744196	+	0.667961i	$0.767167\pi$
$108$	0	0
$109$	−20255.0	−1.70482	−0.852410	−	0.522874i	$-0.824860\pi$
$109$	−20255.0	−1.70482	−0.852410	+	0.522874i	$0.824860\pi$
$110$	− 9058.75i	− 0.748657i
$111$	0	0
$112$	0	0
$113$	17178.7i	1.34535i	0.739939	+	0.672674i	$0.234854\pi$
$113$	17178.7i	1.34535i	−0.739939	+	0.672674i	$0.765146\pi$
$114$	0	0
$115$	26036.3	1.96872
$116$	8939.26i	0.664332i
$117$	0	0
$118$	15340.4	1.10172
$119$	0	0
$120$	0	0
$121$	9434.67	0.644401
$122$	4475.30i	0.300679i
$123$	0	0
$124$	13730.4	0.892978
$125$	− 31968.6i	− 2.04599i
$126$	0	0
$127$	29037.9	1.80036	0.900178	−	0.435522i	$-0.143436\pi$
$127$	29037.9	1.80036	0.900178	+	0.435522i	$0.143436\pi$
$128$	− 1448.15i	− 0.0883883i
$129$	0	0
$130$	−14963.5	−0.885415
$131$	− 26483.8i	− 1.54325i	−0.636075	−	0.771627i	$-0.719443\pi$
$131$	− 26483.8i	− 1.54325i	0.636075	−	0.771627i	$-0.280557\pi$
$132$	0	0
$133$	0	0
$134$	− 2993.55i	− 0.166716i
$135$	0	0
$136$	−3774.52	−0.204072
$137$	615.502i	0.0327935i	0.999866	+	0.0163968i	$0.00521949\pi$
$137$	615.502i	0.0327935i	−0.999866	+	0.0163968i	$0.994781\pi$
$138$	0	0
$139$	−3126.82	−0.161835	−0.0809177	−	0.996721i	$-0.525785\pi$
$139$	−3126.82	−0.161835	−0.0809177	+	0.996721i	$0.525785\pi$
$140$	0	0
$141$	0	0
$142$	−12019.4	−0.596083
$143$	8599.97i	0.420557i
$144$	0	0
$145$	49598.5	2.35903
$146$	14561.1i	0.683107i
$147$	0	0
$148$	16793.1	0.766667
$149$	− 7166.03i	− 0.322779i	−0.986891	−	0.161390i	$-0.948402\pi$
$149$	− 7166.03i	− 0.322779i	0.986891	−	0.161390i	$-0.0515976\pi$
$150$	0	0
$151$	−4493.18	−0.197061	−0.0985303	−	0.995134i	$-0.531414\pi$
$151$	−4493.18	−0.197061	−0.0985303	+	0.995134i	$0.531414\pi$
$152$	− 3305.45i	− 0.143069i
$153$	0	0
$154$	0	0
$155$	− 76181.9i	− 3.17094i
$156$	0	0
$157$	−32103.9	−1.30244	−0.651222	−	0.758887i	$-0.725743\pi$
$157$	−32103.9	−1.30244	−0.651222	+	0.758887i	$0.725743\pi$
$158$	− 24380.4i	− 0.976623i
$159$	0	0
$160$	−8034.94	−0.313865
$161$	0	0
$162$	0	0
$163$	14291.3	0.537892	0.268946	−	0.963155i	$-0.413325\pi$
$163$	14291.3	0.537892	0.268946	+	0.963155i	$0.413325\pi$
$164$	− 19145.8i	− 0.711845i
$165$	0	0
$166$	13087.4	0.474939
$167$	− 3107.12i	− 0.111410i	−0.998447	−	0.0557051i	$-0.982259\pi$
$167$	− 3107.12i	− 0.111410i	0.998447	−	0.0557051i	$-0.0177407\pi$
$168$	0	0
$169$	−14355.3	−0.502619
$170$	20942.5i	0.724654i
$171$	0	0
$172$	−2593.35	−0.0876605
$173$	1017.44i	0.0339952i	0.999856	+	0.0169976i	$0.00541076\pi$
$173$	1017.44i	0.0339952i	−0.999856	+	0.0169976i	$0.994589\pi$
$174$	0	0
$175$	0	0
$176$	4617.91i	0.149080i
$177$	0	0
$178$	34304.2	1.08270
$179$	31602.7i	0.986321i	0.869938	+	0.493160i	$0.164158\pi$
$179$	31602.7i	0.986321i	−0.869938	+	0.493160i	$0.835842\pi$
$180$	0	0
$181$	46558.3	1.42115	0.710575	−	0.703622i	$-0.248435\pi$
$181$	46558.3	1.42115	0.710575	+	0.703622i	$0.248435\pi$
$182$	0	0
$183$	0	0
$184$	−13272.6	−0.392031
$185$	− 93174.7i	− 2.72242i
$186$	0	0
$187$	12036.3	0.344198
$188$	13880.1i	0.392714i
$189$	0	0
$190$	−18340.0	−0.508033
$191$	− 64242.9i	− 1.76100i	−0.474049	−	0.880499i	$-0.657208\pi$
$191$	− 64242.9i	− 1.76100i	0.474049	−	0.880499i	$-0.342792\pi$
$192$	0	0
$193$	4506.44	0.120981	0.0604907	−	0.998169i	$-0.480733\pi$
$193$	4506.44	0.120981	0.0604907	+	0.998169i	$0.480733\pi$
$194$	− 27756.7i	− 0.737503i
$195$	0	0
$196$	0	0
$197$	− 19771.9i	− 0.509466i	−0.967011	−	0.254733i	$-0.918012\pi$
$197$	− 19771.9i	− 0.509466i	0.967011	−	0.254733i	$-0.0819876\pi$
$198$	0	0
$199$	39284.2	0.992001	0.496000	−	0.868322i	$-0.334801\pi$
$199$	39284.2	0.992001	0.496000	+	0.868322i	$0.334801\pi$
$200$	30438.9i	0.760973i
$201$	0	0
$202$	9183.76	0.225070
$203$	0	0
$204$	0	0
$205$	−106228.	−2.52774
$206$	292.810i	0.00690003i
$207$	0	0
$208$	7628.01	0.176313
$209$	10540.5i	0.241307i
$210$	0	0
$211$	44168.1	0.992073	0.496037	−	0.868302i	$-0.334788\pi$
$211$	44168.1	0.992073	0.496037	+	0.868302i	$0.334788\pi$
$212$	22456.2i	0.499648i
$213$	0	0
$214$	−43260.7	−0.944640
$215$	14388.9i	0.311280i
$216$	0	0
$217$	0	0
$218$	− 57289.7i	− 1.20549i
$219$	0	0
$220$	25622.0	0.529381
$221$	− 19881.9i	− 0.407074i
$222$	0	0
$223$	−40459.4	−0.813598	−0.406799	−	0.913518i	$-0.633355\pi$
$223$	−40459.4	−0.813598	−0.406799	+	0.913518i	$0.633355\pi$
$224$	0	0
$225$	0	0
$226$	−48588.8	−0.951304
$227$	− 41758.4i	− 0.810386i	−0.914231	−	0.405193i	$-0.867204\pi$
$227$	− 41758.4i	− 0.810386i	0.914231	−	0.405193i	$-0.132796\pi$
$228$	0	0
$229$	25883.7	0.493577	0.246788	−	0.969069i	$-0.420625\pi$
$229$	25883.7	0.493577	0.246788	+	0.969069i	$0.420625\pi$
$230$	73641.7i	1.39209i
$231$	0	0
$232$	−25284.0	−0.469754
$233$	30532.2i	0.562402i	0.959649	+	0.281201i	$0.0907327\pi$
$233$	30532.2i	0.562402i	−0.959649	+	0.281201i	$0.909267\pi$
$234$	0	0
$235$	77012.1	1.39452
$236$	43389.1i	0.779035i
$237$	0	0
$238$	0	0
$239$	5944.69i	0.104072i	0.998645	+	0.0520359i	$0.0165710\pi$
$239$	5944.69i	0.104072i	−0.998645	+	0.0520359i	$0.983429\pi$
$240$	0	0
$241$	−33280.9	−0.573009	−0.286504	−	0.958079i	$-0.592493\pi$
$241$	−33280.9	−0.573009	−0.286504	+	0.958079i	$0.592493\pi$
$242$	26685.3i	0.455660i
$243$	0	0
$244$	−12658.1	−0.212612
$245$	0	0
$246$	0	0
$247$	17411.1	0.285387
$248$	38835.5i	0.631431i
$249$	0	0
$250$	90420.9	1.44674
$251$	− 20452.8i	− 0.324643i	−0.986738	−	0.162322i	$-0.948102\pi$
$251$	− 20452.8i	− 0.324643i	0.986738	−	0.162322i	$-0.0518982\pi$
$252$	0	0
$253$	42324.0	0.661220
$254$	82131.7i	1.27304i
$255$	0	0
$256$	4096.00	0.0625000
$257$	− 44952.1i	− 0.680588i	−0.940319	−	0.340294i	$-0.889473\pi$
$257$	− 44952.1i	− 0.680588i	0.940319	−	0.340294i	$-0.110527\pi$
$258$	0	0
$259$	0	0
$260$	− 42323.2i	− 0.626083i
$261$	0	0
$262$	74907.5	1.09125
$263$	56919.6i	0.822906i	0.911431	+	0.411453i	$0.134979\pi$
$263$	56919.6i	0.822906i	−0.911431	+	0.411453i	$0.865021\pi$
$264$	0	0
$265$	124596.	1.77424
$266$	0	0
$267$	0	0
$268$	8467.05	0.117886
$269$	55811.4i	0.771291i	0.922647	+	0.385646i	$0.126021\pi$
$269$	55811.4i	0.771291i	−0.922647	+	0.385646i	$0.873979\pi$
$270$	0	0
$271$	48364.1	0.658544	0.329272	−	0.944235i	$-0.393197\pi$
$271$	48364.1	0.658544	0.329272	+	0.944235i	$0.393197\pi$
$272$	− 10675.9i	− 0.144301i
$273$	0	0
$274$	−1740.90	−0.0231885
$275$	− 97064.4i	− 1.28350i
$276$	0	0
$277$	−81028.3	−1.05603	−0.528016	−	0.849234i	$-0.677064\pi$
$277$	−81028.3	−1.05603	−0.528016	+	0.849234i	$0.677064\pi$
$278$	− 8843.99i	− 0.114435i
$279$	0	0
$280$	0	0
$281$	83937.9i	1.06303i	0.847049	+	0.531515i	$0.178377\pi$
$281$	83937.9i	1.06303i	−0.847049	+	0.531515i	$0.821623\pi$
$282$	0	0
$283$	37718.9	0.470962	0.235481	−	0.971879i	$-0.424333\pi$
$283$	37718.9	0.470962	0.235481	+	0.971879i	$0.424333\pi$
$284$	− 33996.0i	− 0.421494i
$285$	0	0
$286$	−24324.4	−0.297379
$287$	0	0
$288$	0	0
$289$	55694.9	0.666837
$290$	140286.i	1.66808i
$291$	0	0
$292$	−41185.1	−0.483030
$293$	61792.6i	0.719783i	0.932994	+	0.359891i	$0.117186\pi$
$293$	61792.6i	0.719783i	−0.932994	+	0.359891i	$0.882814\pi$
$294$	0	0
$295$	240740.	2.76633
$296$	47498.0i	0.542116i
$297$	0	0
$298$	20268.6	0.228240
$299$	− 69912.1i	− 0.782006i
$300$	0	0
$301$	0	0
$302$	− 12708.6i	− 0.139343i
$303$	0	0
$304$	9349.24	0.101165
$305$	70232.0i	0.754980i
$306$	0	0
$307$	−10959.4	−0.116282	−0.0581409	−	0.998308i	$-0.518517\pi$
$307$	−10959.4	−0.116282	−0.0581409	+	0.998308i	$0.518517\pi$
$308$	0	0
$309$	0	0
$310$	215475.	2.24220
$311$	− 22478.8i	− 0.232409i	−0.993225	−	0.116204i	$-0.962927\pi$
$311$	− 22478.8i	− 0.232409i	0.993225	−	0.116204i	$-0.0370728\pi$
$312$	0	0
$313$	114159.	1.16525	0.582626	−	0.812740i	$-0.302025\pi$
$313$	114159.	1.16525	0.582626	+	0.812740i	$0.302025\pi$
$314$	− 90803.7i	− 0.920967i
$315$	0	0
$316$	68958.2	0.690576
$317$	67234.6i	0.669074i	0.942383	+	0.334537i	$0.108580\pi$
$317$	67234.6i	0.669074i	−0.942383	+	0.334537i	$0.891420\pi$
$318$	0	0
$319$	80626.4	0.792311
$320$	− 22726.2i	− 0.221936i
$321$	0	0
$322$	0	0
$323$	− 24368.1i	− 0.233570i
$324$	0	0
$325$	−160334.	−1.51795
$326$	40421.8i	0.380347i
$327$	0	0
$328$	54152.4	0.503350
$329$	0	0
$330$	0	0
$331$	−201257.	−1.83694	−0.918471	−	0.395487i	$-0.870576\pi$
$331$	−201257.	−1.83694	−0.918471	+	0.395487i	$0.870576\pi$
$332$	37016.8i	0.335832i
$333$	0	0
$334$	8788.26	0.0787789
$335$	− 46978.5i	− 0.418610i
$336$	0	0
$337$	−99656.5	−0.877497	−0.438748	−	0.898610i	$-0.644578\pi$
$337$	−99656.5	−0.877497	−0.438748	+	0.898610i	$0.644578\pi$
$338$	− 40602.9i	− 0.355406i
$339$	0	0
$340$	−59234.4	−0.512408
$341$	− 123840.i	− 1.06500i
$342$	0	0
$343$	0	0
$344$	− 7335.10i	− 0.0619853i
$345$	0	0
$346$	−2877.76	−0.0240382
$347$	96676.1i	0.802898i	0.915881	+	0.401449i	$0.131493\pi$
$347$	96676.1i	0.802898i	−0.915881	+	0.401449i	$0.868507\pi$
$348$	0	0
$349$	−172234.	−1.41406	−0.707028	−	0.707185i	$-0.749965\pi$
$349$	−172234.	−1.41406	−0.707028	+	0.707185i	$0.749965\pi$
$350$	0	0
$351$	0	0
$352$	−13061.4	−0.105416
$353$	− 54895.5i	− 0.440542i	−0.975439	−	0.220271i	$-0.929306\pi$
$353$	− 54895.5i	− 0.440542i	0.975439	−	0.220271i	$-0.0706942\pi$
$354$	0	0
$355$	−188623.	−1.49671
$356$	97027.0i	0.765584i
$357$	0	0
$358$	−89386.0	−0.697434
$359$	− 145908.i	− 1.13211i	−0.824366	−	0.566057i	$-0.808468\pi$
$359$	− 145908.i	− 1.13211i	0.824366	−	0.566057i	$-0.191532\pi$
$360$	0	0
$361$	−108981.	−0.836251
$362$	131687.i	1.00490i
$363$	0	0
$364$	0	0
$365$	228511.i	1.71523i
$366$	0	0
$367$	245671.	1.82399	0.911994	−	0.410204i	$-0.134543\pi$
$367$	245671.	1.82399	0.911994	+	0.410204i	$0.134543\pi$
$368$	− 37540.6i	− 0.277208i
$369$	0	0
$370$	263538.	1.92504
$371$	0	0
$372$	0	0
$373$	−92648.8	−0.665920	−0.332960	−	0.942941i	$-0.608047\pi$
$373$	−92648.8	−0.665920	−0.332960	+	0.942941i	$0.608047\pi$
$374$	34043.7i	0.243385i
$375$	0	0
$376$	−39258.8	−0.277690
$377$	− 133181.i	− 0.937044i
$378$	0	0
$379$	−178725.	−1.24425	−0.622123	−	0.782920i	$-0.713730\pi$
$379$	−178725.	−1.24425	−0.622123	+	0.782920i	$0.713730\pi$
$380$	− 51873.3i	− 0.359233i
$381$	0	0
$382$	181706.	1.24521
$383$	139145.i	0.948570i	0.880371	+	0.474285i	$0.157293\pi$
$383$	139145.i	0.948570i	−0.880371	+	0.474285i	$0.842707\pi$
$384$	0	0
$385$	0	0
$386$	12746.1i	0.0855468i
$387$	0	0
$388$	78507.7	0.521493
$389$	80633.7i	0.532866i	0.963854	+	0.266433i	$0.0858450\pi$
$389$	80633.7i	0.532866i	−0.963854	+	0.266433i	$0.914155\pi$
$390$	0	0
$391$	−97847.0	−0.640021
$392$	0	0
$393$	0	0
$394$	55923.3	0.360247
$395$	− 382608.i	− 2.45222i
$396$	0	0
$397$	−109698.	−0.696016	−0.348008	−	0.937492i	$-0.613142\pi$
$397$	−109698.	−0.696016	−0.348008	+	0.937492i	$0.613142\pi$
$398$	111113.i	0.701450i
$399$	0	0
$400$	−86094.2	−0.538089
$401$	115774.i	0.719984i	0.932955	+	0.359992i	$0.117221\pi$
$401$	115774.i	0.719984i	−0.932955	+	0.359992i	$0.882779\pi$
$402$	0	0
$403$	−204562.	−1.25955
$404$	25975.6i	0.159149i
$405$	0	0
$406$	0	0
$407$	− 151463.i	− 0.914360i
$408$	0	0
$409$	−244131.	−1.45941	−0.729704	−	0.683763i	$-0.760342\pi$
$409$	−244131.	−1.45941	−0.729704	+	0.683763i	$0.760342\pi$
$410$	− 300459.i	− 1.78738i
$411$	0	0
$412$	−828.191	−0.00487906
$413$	0	0
$414$	0	0
$415$	205384.	1.19253
$416$	21575.3i	0.124672i
$417$	0	0
$418$	−29813.1	−0.170630
$419$	− 56932.9i	− 0.324291i	−0.986767	−	0.162146i	$-0.948159\pi$
$419$	− 56932.9i	− 0.324291i	0.986767	−	0.162146i	$-0.0518414\pi$
$420$	0	0
$421$	−345720.	−1.95057	−0.975283	−	0.220961i	$-0.929081\pi$
$421$	−345720.	−1.95057	−0.975283	+	0.220961i	$0.929081\pi$
$422$	124926.i	0.701502i
$423$	0	0
$424$	−63515.7	−0.353305
$425$	224399.i	1.24235i
$426$	0	0
$427$	0	0
$428$	− 122360.i	− 0.667961i
$429$	0	0
$430$	−40698.0	−0.220108
$431$	− 366489.i	− 1.97291i	−0.164040	−	0.986454i	$-0.552453\pi$
$431$	− 366489.i	− 1.97291i	0.164040	−	0.986454i	$-0.447547\pi$
$432$	0	0
$433$	−258911.	−1.38094	−0.690470	−	0.723361i	$-0.742596\pi$
$433$	−258911.	−1.38094	−0.690470	+	0.723361i	$0.742596\pi$
$434$	0	0
$435$	0	0
$436$	162040.	0.852410
$437$	− 85687.5i	− 0.448699i
$438$	0	0
$439$	16877.4	0.0875744	0.0437872	−	0.999041i	$-0.486058\pi$
$439$	16877.4	0.0875744	0.0437872	+	0.999041i	$0.486058\pi$
$440$	72470.0i	0.374329i
$441$	0	0
$442$	56234.4	0.287844
$443$	− 104055.i	− 0.530219i	−0.964218	−	0.265110i	$-0.914592\pi$
$443$	− 104055.i	− 0.530219i	0.964218	−	0.265110i	$-0.0854082\pi$
$444$	0	0
$445$	538345.	2.71857
$446$	− 114437.i	− 0.575301i
$447$	0	0
$448$	0	0
$449$	− 176582.i	− 0.875897i	−0.899000	−	0.437949i	$-0.855705\pi$
$449$	− 176582.i	− 0.875897i	0.899000	−	0.437949i	$-0.144295\pi$
$450$	0	0
$451$	−172683.	−0.848977
$452$	− 137430.i	− 0.672674i
$453$	0	0
$454$	118111.	0.573030
$455$	0	0
$456$	0	0
$457$	−405421.	−1.94121	−0.970607	−	0.240671i	$-0.922633\pi$
$457$	−405421.	−1.94121	−0.970607	+	0.240671i	$0.922633\pi$
$458$	73210.0i	0.349011i
$459$	0	0
$460$	−208290.	−0.984358
$461$	20288.5i	0.0954658i	0.998860	+	0.0477329i	$0.0151996\pi$
$461$	20288.5i	0.0954658i	−0.998860	+	0.0477329i	$0.984800\pi$
$462$	0	0
$463$	1177.05	0.00549075	0.00274538	−	0.999996i	$-0.499126\pi$
$463$	1177.05	0.00549075	0.00274538	+	0.999996i	$0.499126\pi$
$464$	− 71514.0i	− 0.332166i
$465$	0	0
$466$	−86358.2	−0.397678
$467$	246423.i	1.12992i	0.825119	+	0.564959i	$0.191108\pi$
$467$	246423.i	1.12992i	−0.825119	+	0.564959i	$0.808892\pi$
$468$	0	0
$469$	0	0
$470$	217823.i	0.986072i
$471$	0	0
$472$	−122723.	−0.550861
$473$	23390.4i	0.104548i
$474$	0	0
$475$	−196513.	−0.870970
$476$	0	0
$477$	0	0
$478$	−16814.1	−0.0735899
$479$	173471.i	0.756060i	0.925793	+	0.378030i	$0.123398\pi$
$479$	173471.i	0.756060i	−0.925793	+	0.378030i	$0.876602\pi$
$480$	0	0
$481$	−250191.	−1.08139
$482$	− 94132.6i	− 0.405178i
$483$	0	0
$484$	−75477.4	−0.322200
$485$	− 435592.i	− 1.85181i
$486$	0	0
$487$	61861.9	0.260835	0.130417	−	0.991459i	$-0.458368\pi$
$487$	61861.9	0.260835	0.130417	+	0.991459i	$0.458368\pi$
$488$	− 35802.4i	− 0.150339i
$489$	0	0
$490$	0	0
$491$	348540.i	1.44574i	0.690985	+	0.722869i	$0.257177\pi$
$491$	348540.i	1.44574i	−0.690985	+	0.722869i	$0.742823\pi$
$492$	0	0
$493$	−186396.	−0.766909
$494$	49246.2i	0.201799i
$495$	0	0
$496$	−109843.	−0.446489
$497$	0	0
$498$	0	0
$499$	293614.	1.17917	0.589583	−	0.807708i	$-0.299292\pi$
$499$	293614.	1.17917	0.589583	+	0.807708i	$0.299292\pi$
$500$	255749.i	1.02300i
$501$	0	0
$502$	57849.4	0.229557
$503$	14496.7i	0.0572970i	0.999590	+	0.0286485i	$0.00912035\pi$
$503$	14496.7i	0.0572970i	−0.999590	+	0.0286485i	$0.990880\pi$
$504$	0	0
$505$	144123.	0.565132
$506$	119710.i	0.467553i
$507$	0	0
$508$	−232304.	−0.900178
$509$	− 295019.i	− 1.13871i	−0.822091	−	0.569356i	$-0.807193\pi$
$509$	− 295019.i	− 1.13871i	0.822091	−	0.569356i	$-0.192807\pi$
$510$	0	0
$511$	0	0
$512$	11585.2i	0.0441942i
$513$	0	0
$514$	127144.	0.481248
$515$	4595.13i	0.0173254i
$516$	0	0
$517$	125189.	0.468367
$518$	0	0
$519$	0	0
$520$	119708.	0.442708
$521$	147051.i	0.541741i	0.962616	+	0.270871i	$0.0873115\pi$
$521$	147051.i	0.541741i	−0.962616	+	0.270871i	$0.912688\pi$
$522$	0	0
$523$	−36960.8	−0.135126	−0.0675629	−	0.997715i	$-0.521522\pi$
$523$	−36960.8	−0.135126	−0.0675629	+	0.997715i	$0.521522\pi$
$524$	211870.i	0.771627i
$525$	0	0
$526$	−160993.	−0.581882
$527$	286299.i	1.03086i
$528$	0	0
$529$	−64225.6	−0.229508
$530$	352410.i	1.25458i
$531$	0	0
$532$	0	0
$533$	285242.i	1.00406i
$534$	0	0
$535$	−678901.	−2.37191
$536$	23948.4i	0.0833580i
$537$	0	0
$538$	−157858.	−0.545385
$539$	0	0
$540$	0	0
$541$	226849.	0.775073	0.387537	−	0.921854i	$-0.373326\pi$
$541$	226849.	0.775073	0.387537	+	0.921854i	$0.373326\pi$
$542$	136794.i	0.465661i
$543$	0	0
$544$	30196.1	0.102036
$545$	− 899061.i	− 3.02689i
$546$	0	0
$547$	202374.	0.676364	0.338182	−	0.941081i	$-0.390188\pi$
$547$	202374.	0.676364	0.338182	+	0.941081i	$0.390188\pi$
$548$	− 4924.02i	− 0.0163968i
$549$	0	0
$550$	274539.	0.907568
$551$	− 163233.i	− 0.537656i
$552$	0	0
$553$	0	0
$554$	− 229183.i	− 0.746728i
$555$	0	0
$556$	25014.6	0.0809177
$557$	− 103690.i	− 0.334216i	−0.985939	−	0.167108i	$-0.946557\pi$
$557$	− 103690.i	− 0.334216i	0.985939	−	0.167108i	$-0.0534429\pi$
$558$	0	0
$559$	38636.9	0.123646
$560$	0	0
$561$	0	0
$562$	−237412.	−0.751676
$563$	271213.i	0.855645i	0.903863	+	0.427823i	$0.140719\pi$
$563$	271213.i	0.855645i	−0.903863	+	0.427823i	$0.859281\pi$
$564$	0	0
$565$	−762516.	−2.38865
$566$	106685.i	0.333021i
$567$	0	0
$568$	96155.3	0.298041
$569$	8436.46i	0.0260577i	0.999915	+	0.0130288i	$0.00414732\pi$
$569$	8436.46i	0.0260577i	−0.999915	+	0.0130288i	$0.995853\pi$
$570$	0	0
$571$	−306335.	−0.939560	−0.469780	−	0.882784i	$-0.655667\pi$
$571$	−306335.	−0.939560	−0.469780	+	0.882784i	$0.655667\pi$
$572$	− 68799.8i	− 0.210279i
$573$	0	0
$574$	0	0
$575$	789070.i	2.38660i
$576$	0	0
$577$	−50556.6	−0.151854	−0.0759270	−	0.997113i	$-0.524192\pi$
$577$	−50556.6	−0.151854	−0.0759270	+	0.997113i	$0.524192\pi$
$578$	157529.i	0.471525i
$579$	0	0
$580$	−396788.	−1.17951
$581$	0	0
$582$	0	0
$583$	202540.	0.595902
$584$	− 116489.i	− 0.341554i
$585$	0	0
$586$	−174776.	−0.508963
$587$	293006.i	0.850355i	0.905110	+	0.425178i	$0.139788\pi$
$587$	293006.i	0.850355i	−0.905110	+	0.425178i	$0.860212\pi$
$588$	0	0
$589$	−250721.	−0.722703
$590$	680916.i	1.95609i
$591$	0	0
$592$	−134345.	−0.383334
$593$	− 233531.i	− 0.664103i	−0.943261	−	0.332052i	$-0.892259\pi$
$593$	− 233531.i	− 0.664103i	0.943261	−	0.332052i	$-0.107741\pi$
$594$	0	0
$595$	0	0
$596$	57328.2i	0.161390i
$597$	0	0
$598$	197741.	0.552962
$599$	− 79356.2i	− 0.221171i	−0.993867	−	0.110585i	$-0.964727\pi$
$599$	− 79356.2i	− 0.221171i	0.993867	−	0.110585i	$-0.0352725\pi$
$600$	0	0
$601$	−6957.97	−0.0192634	−0.00963172	−	0.999954i	$-0.503066\pi$
$601$	−6957.97	−0.0192634	−0.00963172	+	0.999954i	$0.503066\pi$
$602$	0	0
$603$	0	0
$604$	35945.4	0.0985303
$605$	418778.i	1.14413i
$606$	0	0
$607$	−630178.	−1.71035	−0.855176	−	0.518338i	$-0.826551\pi$
$607$	−630178.	−1.71035	−0.855176	+	0.518338i	$0.826551\pi$
$608$	26443.6i	0.0715343i
$609$	0	0
$610$	−198646.	−0.533851
$611$	− 206792.i	− 0.553924i
$612$	0	0
$613$	117222.	0.311953	0.155976	−	0.987761i	$-0.450148\pi$
$613$	117222.	0.311953	0.155976	+	0.987761i	$0.450148\pi$
$614$	− 30998.0i	− 0.0822237i
$615$	0	0
$616$	0	0
$617$	− 271651.i	− 0.713578i	−0.934185	−	0.356789i	$-0.883872\pi$
$617$	− 271651.i	− 0.713578i	0.934185	−	0.356789i	$-0.116128\pi$
$618$	0	0
$619$	−61643.3	−0.160881	−0.0804405	−	0.996759i	$-0.525633\pi$
$619$	−61643.3	−0.160881	−0.0804405	+	0.996759i	$0.525633\pi$
$620$	609455.i	1.58547i
$621$	0	0
$622$	63579.7	0.164338
$623$	0	0
$624$	0	0
$625$	578234.	1.48028
$626$	322889.i	0.823958i
$627$	0	0
$628$	256832.	0.651222
$629$	350160.i	0.885045i
$630$	0	0
$631$	70660.7	0.177468	0.0887338	−	0.996055i	$-0.471718\pi$
$631$	70660.7	0.177468	0.0887338	+	0.996055i	$0.471718\pi$
$632$	195043.i	0.488311i
$633$	0	0
$634$	−190168.	−0.473107
$635$	1.28891e6i	3.19651i
$636$	0	0
$637$	0	0
$638$	228046.i	0.560249i
$639$	0	0
$640$	64279.5	0.156932
$641$	− 335937.i	− 0.817601i	−0.912624	−	0.408801i	$-0.865947\pi$
$641$	− 335937.i	− 0.817601i	0.912624	−	0.408801i	$-0.134053\pi$
$642$	0	0
$643$	397762.	0.962057	0.481029	−	0.876705i	$-0.340263\pi$
$643$	397762.	0.962057	0.481029	+	0.876705i	$0.340263\pi$
$644$	0	0
$645$	0	0
$646$	68923.5	0.165159
$647$	− 322094.i	− 0.769439i	−0.923033	−	0.384720i	$-0.874298\pi$
$647$	− 322094.i	− 0.769439i	0.923033	−	0.384720i	$-0.125702\pi$
$648$	0	0
$649$	391342.	0.929110
$650$	− 453493.i	− 1.07336i
$651$	0	0
$652$	−114330.	−0.268946
$653$	− 484796.i	− 1.13693i	−0.822708	−	0.568464i	$-0.807538\pi$
$653$	− 484796.i	− 1.13693i	0.822708	−	0.568464i	$-0.192462\pi$
$654$	0	0
$655$	1.17554e6	2.74003
$656$	153166.i	0.355922i
$657$	0	0
$658$	0	0
$659$	335125.i	0.771677i	0.922566	+	0.385838i	$0.126088\pi$
$659$	335125.i	0.771677i	−0.922566	+	0.385838i	$0.873912\pi$
$660$	0	0
$661$	596207.	1.36457	0.682283	−	0.731088i	$-0.260987\pi$
$661$	596207.	1.36457	0.682283	+	0.731088i	$0.260987\pi$
$662$	− 569242.i	− 1.29891i
$663$	0	0
$664$	−104699.	−0.237469
$665$	0	0
$666$	0	0
$667$	−655440.	−1.47327
$668$	24857.0i	0.0557051i
$669$	0	0
$670$	132875.	0.296002
$671$	114168.i	0.253570i
$672$	0	0
$673$	−119153.	−0.263071	−0.131536	−	0.991311i	$-0.541991\pi$
$673$	−119153.	−0.263071	−0.131536	+	0.991311i	$0.541991\pi$
$674$	− 281871.i	− 0.620484i
$675$	0	0
$676$	114842.	0.251310
$677$	− 93308.3i	− 0.203584i	−0.994806	−	0.101792i	$-0.967542\pi$
$677$	− 93308.3i	− 0.203584i	0.994806	−	0.101792i	$-0.0324576\pi$
$678$	0	0
$679$	0	0
$680$	− 167540.i	− 0.362327i
$681$	0	0
$682$	350272.	0.753072
$683$	− 409719.i	− 0.878303i	−0.898413	−	0.439151i	$-0.855279\pi$
$683$	− 409719.i	− 0.878303i	0.898413	−	0.439151i	$-0.144721\pi$
$684$	0	0
$685$	−27320.4	−0.0582245
$686$	0	0
$687$	0	0
$688$	20746.8	0.0438303
$689$	− 334562.i	− 0.704756i
$690$	0	0
$691$	208229.	0.436099	0.218049	−	0.975938i	$-0.430031\pi$
$691$	208229.	0.436099	0.218049	+	0.975938i	$0.430031\pi$
$692$	− 8139.53i	− 0.0169976i
$693$	0	0
$694$	−273441.	−0.567734
$695$	− 138791.i	− 0.287337i
$696$	0	0
$697$	399217.	0.821758
$698$	− 487150.i	− 0.999889i
$699$	0	0
$700$	0	0
$701$	− 390386.i	− 0.794436i	−0.917724	−	0.397218i	$-0.869976\pi$
$701$	− 390386.i	− 0.794436i	0.917724	−	0.397218i	$-0.130024\pi$
$702$	0	0
$703$	−306645.	−0.620477
$704$	− 36943.3i	− 0.0745402i
$705$	0	0
$706$	155268.	0.311510
$707$	0	0
$708$	0	0
$709$	174300.	0.346741	0.173371	−	0.984857i	$-0.444534\pi$
$709$	174300.	0.346741	0.173371	+	0.984857i	$0.444534\pi$
$710$	− 533508.i	− 1.05834i
$711$	0	0
$712$	−274434.	−0.541349
$713$	1.00674e6i	1.98032i
$714$	0	0
$715$	−381728.	−0.746694
$716$	− 252822.i	− 0.493160i
$717$	0	0
$718$	412690.	0.800525
$719$	− 903515.i	− 1.74774i	−0.486158	−	0.873871i	$-0.661602\pi$
$719$	− 903515.i	− 1.74774i	0.486158	−	0.873871i	$-0.338398\pi$
$720$	0	0
$721$	0	0
$722$	− 308245.i	− 0.591319i
$723$	0	0
$724$	−372466.	−0.710575
$725$	1.50316e6i	2.85976i
$726$	0	0
$727$	441387.	0.835124	0.417562	−	0.908648i	$-0.362885\pi$
$727$	441387.	0.835124	0.417562	+	0.908648i	$0.362885\pi$
$728$	0	0
$729$	0	0
$730$	−646327.	−1.21285
$731$	− 54075.1i	− 0.101196i
$732$	0	0
$733$	−405762.	−0.755202	−0.377601	−	0.925968i	$-0.623251\pi$
$733$	−405762.	−0.755202	−0.377601	+	0.925968i	$0.623251\pi$
$734$	694863.i	1.28975i
$735$	0	0
$736$	106181.	0.196016
$737$	− 76367.4i	− 0.140596i
$738$	0	0
$739$	−926305.	−1.69615	−0.848077	−	0.529873i	$-0.822239\pi$
$739$	−926305.	−1.69615	−0.848077	+	0.529873i	$0.822239\pi$
$740$	745397.i	1.36121i
$741$	0	0
$742$	0	0
$743$	− 506683.i	− 0.917822i	−0.888482	−	0.458911i	$-0.848240\pi$
$743$	− 506683.i	− 0.917822i	0.888482	−	0.458911i	$-0.151760\pi$
$744$	0	0
$745$	318080.	0.573091
$746$	− 262051.i	− 0.470877i
$747$	0	0
$748$	−96290.2	−0.172099
$749$	0	0
$750$	0	0
$751$	−684679.	−1.21397	−0.606984	−	0.794714i	$-0.707621\pi$
$751$	−684679.	−1.21397	−0.606984	+	0.794714i	$0.707621\pi$
$752$	− 111041.i	− 0.196357i
$753$	0	0
$754$	376693.	0.662590
$755$	− 199440.i	− 0.349879i
$756$	0	0
$757$	225679.	0.393822	0.196911	−	0.980421i	$-0.436909\pi$
$757$	225679.	0.393822	0.196911	+	0.980421i	$0.436909\pi$
$758$	− 505510.i	− 0.879815i
$759$	0	0
$760$	146720.	0.254016
$761$	1.06664e6i	1.84183i	0.389767	+	0.920914i	$0.372556\pi$
$761$	1.06664e6i	1.84183i	−0.389767	+	0.920914i	$0.627444\pi$
$762$	0	0
$763$	0	0
$764$	513943.i	0.880499i
$765$	0	0
$766$	−393561.	−0.670740
$767$	− 646431.i	− 1.09883i
$768$	0	0
$769$	−688607.	−1.16444	−0.582222	−	0.813030i	$-0.697817\pi$
$769$	−688607.	−1.16444	−0.582222	+	0.813030i	$0.697817\pi$
$770$	0	0
$771$	0	0
$772$	−36051.5	−0.0604907
$773$	− 724599.i	− 1.21266i	−0.795213	−	0.606330i	$-0.792641\pi$
$773$	− 724599.i	− 1.21266i	0.795213	−	0.606330i	$-0.207359\pi$
$774$	0	0
$775$	2.30881e6	3.84401
$776$	222053.i	0.368751i
$777$	0	0
$778$	−228067.	−0.376793
$779$	349606.i	0.576109i
$780$	0	0
$781$	−306622.	−0.502692
$782$	− 276753.i	− 0.452563i
$783$	0	0
$784$	0	0
$785$	− 1.42500e6i	− 2.31247i
$786$	0	0
$787$	75971.1	0.122659	0.0613294	−	0.998118i	$-0.480466\pi$
$787$	75971.1	0.122659	0.0613294	+	0.998118i	$0.480466\pi$
$788$	158175.i	0.254733i
$789$	0	0
$790$	1.08218e6	1.73398
$791$	0	0
$792$	0	0
$793$	188586.	0.299890
$794$	− 310274.i	− 0.492158i
$795$	0	0
$796$	−314274.	−0.496000
$797$	− 374918.i	− 0.590227i	−0.955462	−	0.295114i	$-0.904643\pi$
$797$	− 374918.i	− 0.590227i	0.955462	−	0.295114i	$-0.0953575\pi$
$798$	0	0
$799$	−289420.	−0.453351
$800$	− 243511.i	− 0.380486i
$801$	0	0
$802$	−327459.	−0.509106
$803$	371463.i	0.576082i
$804$	0	0
$805$	0	0
$806$	− 578589.i	− 0.890636i
$807$	0	0
$808$	−73470.0	−0.112535
$809$	465964.i	0.711960i	0.934494	+	0.355980i	$0.115853\pi$
$809$	465964.i	0.711960i	−0.934494	+	0.355980i	$0.884147\pi$
$810$	0	0
$811$	630677.	0.958883	0.479442	−	0.877574i	$-0.340839\pi$
$811$	630677.	0.958883	0.479442	+	0.877574i	$0.340839\pi$
$812$	0	0
$813$	0	0
$814$	428402.	0.646550
$815$	634349.i	0.955021i
$816$	0	0
$817$	47355.1	0.0709452
$818$	− 690508.i	− 1.03196i
$819$	0	0
$820$	849827.	1.26387
$821$	− 28266.6i	− 0.0419361i	−0.999780	−	0.0209680i	$-0.993325\pi$
$821$	− 28266.6i	− 0.0419361i	0.999780	−	0.0209680i	$-0.00667482\pi$
$822$	0	0
$823$	−922971.	−1.36266	−0.681331	−	0.731975i	$-0.738599\pi$
$823$	−922971.	−1.36266	−0.681331	+	0.731975i	$0.738599\pi$
$824$	− 2342.48i	− 0.00345002i
$825$	0	0
$826$	0	0
$827$	293811.i	0.429593i	0.976659	+	0.214796i	$0.0689088\pi$
$827$	293811.i	0.429593i	−0.976659	+	0.214796i	$0.931091\pi$
$828$	0	0
$829$	770289.	1.12084	0.560421	−	0.828208i	$-0.310639\pi$
$829$	770289.	1.12084	0.560421	+	0.828208i	$0.310639\pi$
$830$	580913.i	0.843248i
$831$	0	0
$832$	−61024.1	−0.0881565
$833$	0	0
$834$	0	0
$835$	137916.	0.197807
$836$	− 84324.2i	− 0.120653i
$837$	0	0
$838$	161031.	0.229309
$839$	218921.i	0.311002i	0.987836	+	0.155501i	$0.0496991\pi$
$839$	218921.i	0.311002i	−0.987836	+	0.155501i	$0.950301\pi$
$840$	0	0
$841$	−541317.	−0.765350
$842$	− 977844.i	− 1.37926i
$843$	0	0
$844$	−353345.	−0.496037
$845$	− 637192.i	− 0.892394i
$846$	0	0
$847$	0	0
$848$	− 179649.i	− 0.249824i
$849$	0	0
$850$	−634695.	−0.878471
$851$	1.23129e6i	1.70021i
$852$	0	0
$853$	1.04357e6	1.43425	0.717123	−	0.696947i	$-0.245459\pi$
$853$	1.04357e6	1.43425	0.717123	+	0.696947i	$0.245459\pi$
$854$	0	0
$855$	0	0
$856$	346086.	0.472320
$857$	− 385674.i	− 0.525120i	−0.964916	−	0.262560i	$-0.915433\pi$
$857$	− 385674.i	− 0.525120i	0.964916	−	0.262560i	$-0.0845668\pi$
$858$	0	0
$859$	371370.	0.503292	0.251646	−	0.967819i	$-0.419028\pi$
$859$	371370.	0.503292	0.251646	+	0.967819i	$0.419028\pi$
$860$	− 115111.i	− 0.155640i
$861$	0	0
$862$	1.03659e6	1.39506
$863$	533992.i	0.716990i	0.933532	+	0.358495i	$0.116710\pi$
$863$	533992.i	0.716990i	−0.933532	+	0.358495i	$0.883290\pi$
$864$	0	0
$865$	−45161.4	−0.0603580
$866$	− 732311.i	− 0.976472i
$867$	0	0
$868$	0	0
$869$	− 621959.i	− 0.823611i
$870$	0	0
$871$	−126146.	−0.166279
$872$	458318.i	0.602745i
$873$	0	0
$874$	242361.	0.317278
$875$	0	0
$876$	0	0
$877$	−851427.	−1.10700	−0.553501	−	0.832848i	$-0.686709\pi$
$877$	−851427.	−1.10700	−0.553501	+	0.832848i	$0.686709\pi$
$878$	47736.6i	0.0619244i
$879$	0	0
$880$	−204976.	−0.264690
$881$	− 22124.7i	− 0.0285053i	−0.999898	−	0.0142526i	$-0.995463\pi$
$881$	− 22124.7i	− 0.0285053i	0.999898	−	0.0142526i	$-0.00453691\pi$
$882$	0	0
$883$	−561314.	−0.719920	−0.359960	−	0.932968i	$-0.617210\pi$
$883$	−561314.	−0.719920	−0.359960	+	0.932968i	$0.617210\pi$
$884$	159055.i	0.203537i
$885$	0	0
$886$	294312.	0.374922
$887$	− 536497.i	− 0.681900i	−0.940082	−	0.340950i	$-0.889251\pi$
$887$	− 536497.i	− 0.681900i	0.940082	−	0.340950i	$-0.110749\pi$
$888$	0	0
$889$	0	0
$890$	1.52267e6i	1.92232i
$891$	0	0
$892$	323675.	0.406799
$893$	− 253453.i	− 0.317830i
$894$	0	0
$895$	−1.40276e6	−1.75120
$896$	0	0
$897$	0	0
$898$	499449.	0.619353
$899$	1.91781e6i	2.37294i
$900$	0	0
$901$	−468244.	−0.576797
$902$	− 488420.i	− 0.600317i
$903$	0	0
$904$	388711.	0.475652
$905$	2.06659e6i	2.52323i
$906$	0	0
$907$	−744570.	−0.905089	−0.452544	−	0.891742i	$-0.649484\pi$
$907$	−744570.	−0.905089	−0.452544	+	0.891742i	$0.649484\pi$
$908$	334067.i	0.405193i
$909$	0	0
$910$	0	0
$911$	− 267278.i	− 0.322053i	−0.986950	−	0.161026i	$-0.948520\pi$
$911$	− 267278.i	− 0.322053i	0.986950	−	0.161026i	$-0.0514804\pi$
$912$	0	0
$913$	333868.	0.400528
$914$	− 1.14670e6i	− 1.37265i
$915$	0	0
$916$	−207069.	−0.246788
$917$	0	0
$918$	0	0
$919$	−834263.	−0.987807	−0.493904	−	0.869517i	$-0.664430\pi$
$919$	−834263.	−0.987807	−0.493904	+	0.869517i	$0.664430\pi$
$920$	− 589134.i	− 0.696046i
$921$	0	0
$922$	−57384.5	−0.0675045
$923$	506488.i	0.594519i
$924$	0	0
$925$	2.82380e6	3.30028
$926$	3329.19i	0.00388255i
$927$	0	0
$928$	202272.	0.234877
$929$	− 1.63779e6i	− 1.89769i	−0.315735	−	0.948847i	$-0.602251\pi$
$929$	− 1.63779e6i	− 1.89769i	0.315735	−	0.948847i	$-0.397749\pi$
$930$	0	0
$931$	0	0
$932$	− 244258.i	− 0.281201i
$933$	0	0
$934$	−696988.	−0.798972
$935$	534256.i	0.611120i
$936$	0	0
$937$	−335014.	−0.381578	−0.190789	−	0.981631i	$-0.561105\pi$
$937$	−335014.	−0.381578	−0.190789	+	0.981631i	$0.561105\pi$
$938$	0	0
$939$	0	0
$940$	−616097.	−0.697258
$941$	97662.9i	0.110294i	0.998478	+	0.0551468i	$0.0175627\pi$
$941$	97662.9i	0.110294i	−0.998478	+	0.0551468i	$0.982437\pi$
$942$	0	0
$943$	1.40380e6	1.57863
$944$	− 347113.i	− 0.389517i
$945$	0	0
$946$	−66157.9	−0.0739264
$947$	1.18968e6i	1.32657i	0.748367	+	0.663285i	$0.230838\pi$
$947$	1.18968e6i	1.32657i	−0.748367	+	0.663285i	$0.769162\pi$
$948$	0	0
$949$	613594.	0.681316
$950$	− 555821.i	− 0.615869i
$951$	0	0
$952$	0	0
$953$	− 277245.i	− 0.305266i	−0.988283	−	0.152633i	$-0.951225\pi$
$953$	− 277245.i	− 0.305266i	0.988283	−	0.152633i	$-0.0487752\pi$
$954$	0	0
$955$	2.85156e6	3.12663
$956$	− 47557.5i	− 0.0520359i
$957$	0	0
$958$	−490651.	−0.534615
$959$	0	0
$960$	0	0
$961$	2.02218e6	2.18964
$962$	− 707647.i	− 0.764656i
$963$	0	0
$964$	266247.	0.286504
$965$	200028.i	0.214801i
$966$	0	0
$967$	1.26308e6	1.35076	0.675380	−	0.737470i	$-0.263980\pi$
$967$	1.26308e6	1.35076	0.675380	+	0.737470i	$0.263980\pi$
$968$	− 213482.i	− 0.227830i
$969$	0	0
$970$	1.23204e6	1.30943
$971$	− 318176.i	− 0.337465i	−0.985662	−	0.168732i	$-0.946033\pi$
$971$	− 318176.i	− 0.337465i	0.985662	−	0.168732i	$-0.0539674\pi$
$972$	0	0
$973$	0	0
$974$	174972.i	0.184438i
$975$	0	0
$976$	101265.	0.106306
$977$	− 1.72474e6i	− 1.80690i	−0.428692	−	0.903451i	$-0.641026\pi$
$977$	− 1.72474e6i	− 1.80690i	0.428692	−	0.903451i	$-0.358974\pi$
$978$	0	0
$979$	875122.	0.913068
$980$	0	0
$981$	0	0
$982$	−985821.	−1.02229
$983$	1.56285e6i	1.61737i	0.588241	+	0.808686i	$0.299821\pi$
$983$	1.56285e6i	1.61737i	−0.588241	+	0.808686i	$0.700179\pi$
$984$	0	0
$985$	877618.	0.904551
$986$	− 527209.i	− 0.542286i
$987$	0	0
$988$	−139289.	−0.142693
$989$	− 190148.i	− 0.194402i
$990$	0	0
$991$	353628.	0.360080	0.180040	−	0.983659i	$-0.442377\pi$
$991$	353628.	0.360080	0.180040	+	0.983659i	$0.442377\pi$
$992$	− 310684.i	− 0.315715i
$993$	0	0
$994$	0	0
$995$	1.74372e6i	1.76128i
$996$	0	0
$997$	267909.	0.269524	0.134762	−	0.990878i	$-0.456973\pi$
$997$	267909.	0.269524	0.134762	+	0.990878i	$0.456973\pi$
$998$	830465.i	0.833797i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	882.5.b.f.197.6		6
3.2	odd	2	inner	882.5.b.f.197.1		6
7.2	even	3		126.5.s.a.53.6	yes	12
7.4	even	3		126.5.s.a.107.1	yes	12
7.6	odd	2		882.5.b.g.197.4		6
21.2	odd	6		126.5.s.a.53.1	✓	12
21.11	odd	6		126.5.s.a.107.6	yes	12
21.20	even	2		882.5.b.g.197.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
126.5.s.a.53.1	✓	12	21.2	odd	6
126.5.s.a.53.6	yes	12	7.2	even	3
126.5.s.a.107.1	yes	12	7.4	even	3
126.5.s.a.107.6	yes	12	21.11	odd	6
882.5.b.f.197.1		6	3.2	odd	2	inner
882.5.b.f.197.6		6	1.1	even	1	trivial
882.5.b.g.197.3		6	21.20	even	2
882.5.b.g.197.4		6	7.6	odd	2

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:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	882.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.82843i q^{2} -8.00000 q^{4} +44.3872i q^{5} -22.6274i q^{8} +O(q^{10})\)	\(q+2.82843i q^{2} -8.00000 q^{4} +44.3872i q^{5} -22.6274i q^{8} -125.546 q^{10} +72.1549i q^{11} +119.188 q^{13} +64.0000 q^{16} -166.812i q^{17} +146.082 q^{19} -355.097i q^{20} -204.085 q^{22} -586.572i q^{23} -1345.22 q^{25} +337.114i q^{26} -1117.41i q^{29} -1716.30 q^{31} +181.019i q^{32} +471.814 q^{34} -2099.13 q^{37} +413.182i q^{38} +1004.37 q^{40} +2393.22i q^{41} +324.169 q^{43} -577.239i q^{44} +1659.08 q^{46} -1735.01i q^{47} -3804.86i q^{50} -953.501 q^{52} -2807.02i q^{53} -3202.75 q^{55} +3160.50 q^{58} -5423.64i q^{59} +1582.26 q^{61} -4854.44i q^{62} -512.000 q^{64} +5290.40i q^{65} -1058.38 q^{67} +1334.49i q^{68} +4249.50i q^{71} +5148.13 q^{73} -5937.25i q^{74} -1168.65 q^{76} -8619.78 q^{79} +2840.78i q^{80} -6769.05 q^{82} -4627.10i q^{83} +7404.30 q^{85} +916.887i q^{86} +1632.68 q^{88} -12128.4i q^{89} +4692.58i q^{92} +4907.35 q^{94} +6484.16i q^{95} -9813.46 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 48 q^{4}+O(q^{10}) \) 6 * q - 48 * q^4	\( 6 q - 48 q^{4} - 40 q^{10} + 278 q^{13} + 384 q^{16} + 654 q^{19} + 808 q^{22} - 1786 q^{25} - 3790 q^{31} - 2768 q^{34} - 4542 q^{37} + 320 q^{40} + 8142 q^{43} - 1136 q^{46} - 2224 q^{52} - 11432 q^{55} - 832 q^{58} - 4024 q^{61} - 3072 q^{64} - 7466 q^{67} + 5506 q^{73} - 5232 q^{76} - 3778 q^{79} + 5496 q^{82} + 37312 q^{85} - 6464 q^{88} + 31224 q^{94} - 30692 q^{97}+O(q^{100}) \) 6 * q - 48 * q^4 - 40 * q^10 + 278 * q^13 + 384 * q^16 + 654 * q^19 + 808 * q^22 - 1786 * q^25 - 3790 * q^31 - 2768 * q^34 - 4542 * q^37 + 320 * q^40 + 8142 * q^43 - 1136 * q^46 - 2224 * q^52 - 11432 * q^55 - 832 * q^58 - 4024 * q^61 - 3072 * q^64 - 7466 * q^67 + 5506 * q^73 - 5232 * q^76 - 3778 * q^79 + 5496 * q^82 + 37312 * q^85 - 6464 * q^88 + 31224 * q^94 - 30692 * q^97

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