LMFDB - Embedded newform 441.7.b.a.197.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [441,7,Mod(197,441)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("441.197"); S:= CuspForms(chi, 7); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(441, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 7, names="a")

Level:	$ N $	$=$	$ 441 = 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	441.b (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,-196,0,0,0,0,0,1620] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(10)] == traces)

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

Self dual:	no
Analytic conductor:	$101.453850876$
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:	$ 3^{2} $
Twist minimal:	no (minimal twist has level 9)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			197.2
Root			$1.41421i$ of defining polynomial
Character	$\chi$	$=$	441.197
Dual form			441.7.b.a.197.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+12.7279i q^{2} -98.0000 q^{4} -63.6396i q^{5} -432.749i q^{8} +810.000 q^{10} -865.499i q^{11} -344.000 q^{13} -764.000 q^{16} +7140.36i q^{17} +2320.00 q^{19} +6236.68i q^{20} +11016.0 q^{22} +5753.02i q^{23} +11575.0 q^{25} -4378.41i q^{26} +23152.1i q^{29} +10564.0 q^{31} -37420.1i q^{32} -90882.0 q^{34} -24082.0 q^{37} +29528.8i q^{38} -27540.0 q^{40} +108836. i q^{41} -90952.0 q^{43} +84818.9i q^{44} -73224.0 q^{46} -128959. i q^{47} +147326. i q^{50} +33712.0 q^{52} -196685. i q^{53} -55080.0 q^{55} -294678. q^{58} -39812.9i q^{59} -251138. q^{61} +134458. i q^{62} +427384. q^{64} +21892.0i q^{65} -216088. q^{67} -699756. i q^{68} +53915.5i q^{71} +308176. q^{73} -306514. i q^{74} -227360. q^{76} -540124. q^{79} +48620.7i q^{80} -1.38526e6 q^{82} -932346. i q^{83} +454410. q^{85} -1.15763e6i q^{86} -374544. q^{88} -223413. i q^{89} -563796. i q^{92} +1.64138e6 q^{94} -147644. i q^{95} +37168.0 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 196 q^{4} + 1620 q^{10} - 688 q^{13} - 1528 q^{16} + 4640 q^{19} + 22032 q^{22} + 23150 q^{25} + 21128 q^{31} - 181764 q^{34} - 48164 q^{37} - 55080 q^{40} - 181904 q^{43} - 146448 q^{46} + 67424 q^{52}+ \cdots + 74336 q^{97}+O(q^{100}) $ 2 * q - 196 * q^4 + 1620 * q^10 - 688 * q^13 - 1528 * q^16 + 4640 * q^19 + 22032 * q^22 + 23150 * q^25 + 21128 * q^31 - 181764 * q^34 - 48164 * q^37 - 55080 * q^40 - 181904 * q^43 - 146448 * q^46 + 67424 * q^52 - 110160 * q^55 - 589356 * q^58 - 502276 * q^61 + 854768 * q^64 - 432176 * q^67 + 616352 * q^73 - 454720 * q^76 - 1080248 * q^79 - 2770524 * q^82 + 908820 * q^85 - 749088 * q^88 + 3282768 * q^94 + 74336 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/441\mathbb{Z}\right)^\times$.

$n$	$199$	$344$
$\chi(n)$	$1$	$-1$

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 $

$2$	12.7279i	1.59099i	0.605960	+	0.795495i	$0.292789\pi$
$2$	12.7279i	1.59099i	−0.605960	+	0.795495i	$0.707211\pi$
$3$	0	0
$3$	0	0
$4$	−98.0000	−1.53125
$5$	− 63.6396i	− 0.509117i	−0.967057	−	0.254558i	$-0.918070\pi$
$5$	− 63.6396i	− 0.509117i	0.967057	−	0.254558i	$-0.0819301\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	− 432.749i	− 0.845214i
$9$	0	0
$10$	810.000	0.810000
$11$	− 865.499i	− 0.650262i	−0.945669	−	0.325131i	$-0.894592\pi$
$11$	− 865.499i	− 0.650262i	0.945669	−	0.325131i	$-0.105408\pi$
$12$	0	0
$13$	−344.000	−0.156577	−0.0782886	−	0.996931i	$-0.524946\pi$
$13$	−344.000	−0.156577	−0.0782886	+	0.996931i	$0.524946\pi$
$14$	0	0
$15$	0	0
$16$	−764.000	−0.186523
$17$	7140.36i	1.45336i	0.686975	+	0.726681i	$0.258938\pi$
$17$	7140.36i	1.45336i	−0.686975	+	0.726681i	$0.741062\pi$
$18$	0	0
$19$	2320.00	0.338242	0.169121	−	0.985595i	$-0.445907\pi$
$19$	2320.00	0.338242	0.169121	+	0.985595i	$0.445907\pi$
$20$	6236.68i	0.779585i
$21$	0	0
$22$	11016.0	1.03456
$23$	5753.02i	0.472838i	0.971651	+	0.236419i	$0.0759738\pi$
$23$	5753.02i	0.472838i	−0.971651	+	0.236419i	$0.924026\pi$
$24$	0	0
$25$	11575.0	0.740800
$26$	− 4378.41i	− 0.249113i
$27$	0	0
$28$	0	0
$29$	23152.1i	0.949284i	0.880179	+	0.474642i	$0.157422\pi$
$29$	23152.1i	0.949284i	−0.880179	+	0.474642i	$0.842578\pi$
$30$	0	0
$31$	10564.0	0.354604	0.177302	−	0.984157i	$-0.443263\pi$
$31$	10564.0	0.354604	0.177302	+	0.984157i	$0.443263\pi$
$32$	− 37420.1i	− 1.14197i
$33$	0	0
$34$	−90882.0	−2.31228
$35$	0	0
$36$	0	0
$37$	−24082.0	−0.475431	−0.237715	−	0.971335i	$-0.576399\pi$
$37$	−24082.0	−0.475431	−0.237715	+	0.971335i	$0.576399\pi$
$38$	29528.8i	0.538139i
$39$	0	0
$40$	−27540.0	−0.430312
$41$	108836.i	1.57915i	0.613655	+	0.789574i	$0.289698\pi$
$41$	108836.i	1.57915i	−0.613655	+	0.789574i	$0.710302\pi$
$42$	0	0
$43$	−90952.0	−1.14395	−0.571975	−	0.820271i	$-0.693822\pi$
$43$	−90952.0	−1.14395	−0.571975	+	0.820271i	$0.693822\pi$
$44$	84818.9i	0.995714i
$45$	0	0
$46$	−73224.0	−0.752281
$47$	− 128959.i	− 1.24211i	−0.783768	−	0.621054i	$-0.786705\pi$
$47$	− 128959.i	− 1.24211i	0.783768	−	0.621054i	$-0.213295\pi$
$48$	0	0
$49$	0	0
$50$	147326.i	1.17861i
$51$	0	0
$52$	33712.0	0.239759
$53$	− 196685.i	− 1.32112i	−0.750773	−	0.660561i	$-0.770319\pi$
$53$	− 196685.i	− 1.32112i	0.750773	−	0.660561i	$-0.229681\pi$
$54$	0	0
$55$	−55080.0	−0.331059
$56$	0	0
$57$	0	0
$58$	−294678.	−1.51030
$59$	− 39812.9i	− 0.193851i	−0.995292	−	0.0969255i	$-0.969099\pi$
$59$	− 39812.9i	− 0.193851i	0.995292	−	0.0969255i	$-0.0309009\pi$
$60$	0	0
$61$	−251138.	−1.10643	−0.553214	−	0.833039i	$-0.686599\pi$
$61$	−251138.	−1.10643	−0.553214	+	0.833039i	$0.686599\pi$
$62$	134458.i	0.564171i
$63$	0	0
$64$	427384.	1.63034
$65$	21892.0i	0.0797161i
$66$	0	0
$67$	−216088.	−0.718466	−0.359233	−	0.933248i	$-0.616962\pi$
$67$	−216088.	−0.718466	−0.359233	+	0.933248i	$0.616962\pi$
$68$	− 699756.i	− 2.22546i
$69$	0	0
$70$	0	0
$71$	53915.5i	0.150639i	0.997159	+	0.0753197i	$0.0239977\pi$
$71$	53915.5i	0.150639i	−0.997159	+	0.0753197i	$0.976002\pi$
$72$	0	0
$73$	308176.	0.792192	0.396096	−	0.918209i	$-0.370365\pi$
$73$	308176.	0.792192	0.396096	+	0.918209i	$0.370365\pi$
$74$	− 306514.i	− 0.756406i
$75$	0	0
$76$	−227360.	−0.517933
$77$	0	0
$78$	0	0
$79$	−540124.	−1.09550	−0.547750	−	0.836642i	$-0.684515\pi$
$79$	−540124.	−1.09550	−0.547750	+	0.836642i	$0.684515\pi$
$80$	48620.7i	0.0949622i
$81$	0	0
$82$	−1.38526e6	−2.51241
$83$	− 932346.i	− 1.63058i	−0.579051	−	0.815291i	$-0.696577\pi$
$83$	− 932346.i	− 1.63058i	0.579051	−	0.815291i	$-0.303423\pi$
$84$	0	0
$85$	454410.	0.739931
$86$	− 1.15763e6i	− 1.82001i
$87$	0	0
$88$	−374544.	−0.549610
$89$	− 223413.i	− 0.316912i	−0.987366	−	0.158456i	$-0.949348\pi$
$89$	− 223413.i	− 0.316912i	0.987366	−	0.158456i	$-0.0506516\pi$
$90$	0	0
$91$	0	0
$92$	− 563796.i	− 0.724033i
$93$	0	0
$94$	1.64138e6	1.97618
$95$	− 147644.i	− 0.172205i
$96$	0	0
$97$	37168.0	0.0407243	0.0203622	−	0.999793i	$-0.493518\pi$
$97$	37168.0	0.0407243	0.0203622	+	0.999793i	$0.493518\pi$
$98$	0	0
$99$	0	0
$100$	−1.13435e6	−1.13435
$101$	− 559787.i	− 0.543323i	−0.962393	−	0.271662i	$-0.912427\pi$
$101$	− 559787.i	− 0.543323i	0.962393	−	0.271662i	$-0.0875732\pi$
$102$	0	0
$103$	−1.46018e6	−1.33627	−0.668136	−	0.744039i	$-0.732907\pi$
$103$	−1.46018e6	−1.33627	−0.668136	+	0.744039i	$0.732907\pi$
$104$	148866.i	0.132341i
$105$	0	0
$106$	2.50339e6	2.10189
$107$	− 1.29031e6i	− 1.05327i	−0.850090	−	0.526637i	$-0.823453\pi$
$107$	− 1.29031e6i	− 1.05327i	0.850090	−	0.526637i	$-0.176547\pi$
$108$	0	0
$109$	−1.43548e6	−1.10845	−0.554227	−	0.832366i	$-0.686986\pi$
$109$	−1.43548e6	−1.10845	−0.554227	+	0.832366i	$0.686986\pi$
$110$	− 701054.i	− 0.526712i
$111$	0	0
$112$	0	0
$113$	− 186426.i	− 0.129202i	−0.997911	−	0.0646012i	$-0.979422\pi$
$113$	− 186426.i	− 0.129202i	0.997911	−	0.0646012i	$-0.0205775\pi$
$114$	0	0
$115$	366120.	0.240730
$116$	− 2.26890e6i	− 1.45359i
$117$	0	0
$118$	506736.	0.308415
$119$	0	0
$120$	0	0
$121$	1.02247e6	0.577159
$122$	− 3.19646e6i	− 1.76032i
$123$	0	0
$124$	−1.03527e6	−0.542987
$125$	− 1.73100e6i	− 0.886271i
$126$	0	0
$127$	−127060.	−0.0620294	−0.0310147	−	0.999519i	$-0.509874\pi$
$127$	−127060.	−0.0620294	−0.0310147	+	0.999519i	$0.509874\pi$
$128$	3.04482e6i	1.45189i
$129$	0	0
$130$	−278640.	−0.126827
$131$	− 2.62348e6i	− 1.16698i	−0.812120	−	0.583490i	$-0.801687\pi$
$131$	− 2.62348e6i	− 1.16698i	0.812120	−	0.583490i	$-0.198313\pi$
$132$	0	0
$133$	0	0
$134$	− 2.75035e6i	− 1.14307i
$135$	0	0
$136$	3.08999e6	1.22840
$137$	202310.i	0.0786785i	0.999226	+	0.0393393i	$0.0125253\pi$
$137$	202310.i	0.0786785i	−0.999226	+	0.0393393i	$0.987475\pi$
$138$	0	0
$139$	−2.02642e6	−0.754546	−0.377273	−	0.926102i	$-0.623138\pi$
$139$	−2.02642e6	−0.754546	−0.377273	+	0.926102i	$0.623138\pi$
$140$	0	0
$141$	0	0
$142$	−686232.	−0.239666
$143$	297732.i	0.101816i
$144$	0	0
$145$	1.47339e6	0.483297
$146$	3.92244e6i	1.26037i
$147$	0	0
$148$	2.36004e6	0.728004
$149$	4.25534e6i	1.28640i	0.765699	+	0.643199i	$0.222393\pi$
$149$	4.25534e6i	1.28640i	−0.765699	+	0.643199i	$0.777607\pi$
$150$	0	0
$151$	3.74035e6	1.08638	0.543189	−	0.839610i	$-0.317217\pi$
$151$	3.74035e6	1.08638	0.543189	+	0.839610i	$0.317217\pi$
$152$	− 1.00398e6i	− 0.285886i
$153$	0	0
$154$	0	0
$155$	− 672289.i	− 0.180535i
$156$	0	0
$157$	2.38813e6	0.617105	0.308552	−	0.951207i	$-0.400155\pi$
$157$	2.38813e6	0.617105	0.308552	+	0.951207i	$0.400155\pi$
$158$	− 6.87466e6i	− 1.74293i
$159$	0	0
$160$	−2.38140e6	−0.581396
$161$	0	0
$162$	0	0
$163$	−6.74519e6	−1.55751	−0.778756	−	0.627327i	$-0.784149\pi$
$163$	−6.74519e6	−1.55751	−0.778756	+	0.627327i	$0.784149\pi$
$164$	− 1.06660e7i	− 2.41807i
$165$	0	0
$166$	1.18668e7	2.59424
$167$	6.61699e6i	1.42073i	0.703834	+	0.710364i	$0.251470\pi$
$167$	6.61699e6i	1.42073i	−0.703834	+	0.710364i	$0.748530\pi$
$168$	0	0
$169$	−4.70847e6	−0.975484
$170$	5.78370e6i	1.17722i
$171$	0	0
$172$	8.91330e6	1.75167
$173$	5.58077e6i	1.07784i	0.842356	+	0.538922i	$0.181168\pi$
$173$	5.58077e6i	1.07784i	−0.842356	+	0.538922i	$0.818832\pi$
$174$	0	0
$175$	0	0
$176$	661241.i	0.121289i
$177$	0	0
$178$	2.84359e6	0.504204
$179$	4.87103e6i	0.849301i	0.905357	+	0.424650i	$0.139603\pi$
$179$	4.87103e6i	0.849301i	−0.905357	+	0.424650i	$0.860397\pi$
$180$	0	0
$181$	−8.47546e6	−1.42931	−0.714657	−	0.699475i	$-0.753417\pi$
$181$	−8.47546e6	−1.42931	−0.714657	+	0.699475i	$0.753417\pi$
$182$	0	0
$183$	0	0
$184$	2.48962e6	0.399649
$185$	1.53257e6i	0.242050i
$186$	0	0
$187$	6.17998e6	0.945066
$188$	1.26380e7i	1.90198i
$189$	0	0
$190$	1.87920e6	0.273976
$191$	97037.7i	0.0139264i	0.999976	+	0.00696322i	$0.00221648\pi$
$191$	97037.7i	0.0139264i	−0.999976	+	0.00696322i	$0.997784\pi$
$192$	0	0
$193$	7.49473e6	1.04252	0.521260	−	0.853398i	$-0.325462\pi$
$193$	7.49473e6	1.04252	0.521260	+	0.853398i	$0.325462\pi$
$194$	473071.i	0.0647920i
$195$	0	0
$196$	0	0
$197$	− 9.84656e6i	− 1.28791i	−0.765063	−	0.643956i	$-0.777292\pi$
$197$	− 9.84656e6i	− 1.28791i	0.765063	−	0.643956i	$-0.222708\pi$
$198$	0	0
$199$	−3.54170e6	−0.449420	−0.224710	−	0.974426i	$-0.572144\pi$
$199$	−3.54170e6	−0.449420	−0.224710	+	0.974426i	$0.572144\pi$
$200$	− 5.00907e6i	− 0.626134i
$201$	0	0
$202$	7.12492e6	0.864422
$203$	0	0
$204$	0	0
$205$	6.92631e6	0.803971
$206$	− 1.85851e7i	− 2.12600i
$207$	0	0
$208$	262816.	0.0292053
$209$	− 2.00796e6i	− 0.219946i
$210$	0	0
$211$	−6.77298e6	−0.720996	−0.360498	−	0.932760i	$-0.617393\pi$
$211$	−6.77298e6	−0.720996	−0.360498	+	0.932760i	$0.617393\pi$
$212$	1.92751e7i	2.02297i
$213$	0	0
$214$	1.64229e7	1.67575
$215$	5.78815e6i	0.582404i
$216$	0	0
$217$	0	0
$218$	− 1.82707e7i	− 1.76354i
$219$	0	0
$220$	5.39784e6	0.506935
$221$	− 2.45629e6i	− 0.227563i
$222$	0	0
$223$	3.34186e6	0.301352	0.150676	−	0.988583i	$-0.451855\pi$
$223$	3.34186e6	0.301352	0.150676	+	0.988583i	$0.451855\pi$
$224$	0	0
$225$	0	0
$226$	2.37281e6	0.205560
$227$	− 1.62013e7i	− 1.38507i	−0.721385	−	0.692535i	$-0.756494\pi$
$227$	− 1.62013e7i	− 1.38507i	0.721385	−	0.692535i	$-0.243506\pi$
$228$	0	0
$229$	1.66351e7	1.38522	0.692611	−	0.721311i	$-0.256460\pi$
$229$	1.66351e7	1.38522	0.692611	+	0.721311i	$0.256460\pi$
$230$	4.65995e6i	0.382999i
$231$	0	0
$232$	1.00191e7	0.802348
$233$	− 8.85600e6i	− 0.700116i	−0.936728	−	0.350058i	$-0.886162\pi$
$233$	− 8.85600e6i	− 0.700116i	0.936728	−	0.350058i	$-0.113838\pi$
$234$	0	0
$235$	−8.20692e6	−0.632378
$236$	3.90167e6i	0.296834i
$237$	0	0
$238$	0	0
$239$	− 1.12995e7i	− 0.827689i	−0.910348	−	0.413845i	$-0.864186\pi$
$239$	− 1.12995e7i	− 0.827689i	0.910348	−	0.413845i	$-0.135814\pi$
$240$	0	0
$241$	−1.50090e7	−1.07226	−0.536129	−	0.844136i	$-0.680114\pi$
$241$	−1.50090e7	−1.07226	−0.536129	+	0.844136i	$0.680114\pi$
$242$	1.30140e7i	0.918255i
$243$	0	0
$244$	2.46115e7	1.69422
$245$	0	0
$246$	0	0
$247$	−798080.	−0.0529609
$248$	− 4.57156e6i	− 0.299716i
$249$	0	0
$250$	2.20320e7	1.41005
$251$	− 3.04076e7i	− 1.92292i	−0.274950	−	0.961458i	$-0.588661\pi$
$251$	− 3.04076e7i	− 1.92292i	0.274950	−	0.961458i	$-0.411339\pi$
$252$	0	0
$253$	4.97923e6	0.307469
$254$	− 1.61721e6i	− 0.0986882i
$255$	0	0
$256$	−1.14017e7	−0.679595
$257$	1.97422e7i	1.16305i	0.813530	+	0.581523i	$0.197543\pi$
$257$	1.97422e7i	1.16305i	−0.813530	+	0.581523i	$0.802457\pi$
$258$	0	0
$259$	0	0
$260$	− 2.14542e6i	− 0.122065i
$261$	0	0
$262$	3.33914e7	1.85666
$263$	3.48531e6i	0.191591i	0.995401	+	0.0957954i	$0.0305394\pi$
$263$	3.48531e6i	0.191591i	−0.995401	+	0.0957954i	$0.969461\pi$
$264$	0	0
$265$	−1.25169e7	−0.672605
$266$	0	0
$267$	0	0
$268$	2.11766e7	1.10015
$269$	− 5.65391e6i	− 0.290464i	−0.989398	−	0.145232i	$-0.953607\pi$
$269$	− 5.65391e6i	− 0.290464i	0.989398	−	0.145232i	$-0.0463928\pi$
$270$	0	0
$271$	−2.91893e7	−1.46662	−0.733308	−	0.679897i	$-0.762024\pi$
$271$	−2.91893e7	−1.46662	−0.733308	+	0.679897i	$0.762024\pi$
$272$	− 5.45524e6i	− 0.271086i
$273$	0	0
$274$	−2.57499e6	−0.125177
$275$	− 1.00181e7i	− 0.481714i
$276$	0	0
$277$	2.29938e7	1.08186	0.540931	−	0.841067i	$-0.318072\pi$
$277$	2.29938e7	1.08186	0.540931	+	0.841067i	$0.318072\pi$
$278$	− 2.57922e7i	− 1.20048i
$279$	0	0
$280$	0	0
$281$	− 1.25303e6i	− 0.0564730i	−0.999601	−	0.0282365i	$-0.991011\pi$
$281$	− 1.25303e6i	− 0.0564730i	0.999601	−	0.0282365i	$-0.00898916\pi$
$282$	0	0
$283$	1.45129e7	0.640317	0.320159	−	0.947364i	$-0.396264\pi$
$283$	1.45129e7	0.640317	0.320159	+	0.947364i	$0.396264\pi$
$284$	− 5.28372e6i	− 0.230666i
$285$	0	0
$286$	−3.78950e6	−0.161989
$287$	0	0
$288$	0	0
$289$	−2.68472e7	−1.11226
$290$	1.87532e7i	0.768920i
$291$	0	0
$292$	−3.02012e7	−1.21304
$293$	− 1.12729e7i	− 0.448160i	−0.974571	−	0.224080i	$-0.928062\pi$
$293$	− 1.12729e7i	− 0.448160i	0.974571	−	0.224080i	$-0.0719377\pi$
$294$	0	0
$295$	−2.53368e6	−0.0986929
$296$	1.04215e7i	0.401841i
$297$	0	0
$298$	−5.41616e7	−2.04665
$299$	− 1.97904e6i	− 0.0740356i
$300$	0	0
$301$	0	0
$302$	4.76069e7i	1.72842i
$303$	0	0
$304$	−1.77248e6	−0.0630900
$305$	1.59823e7i	0.563301i
$306$	0	0
$307$	−4.51916e7	−1.56186	−0.780930	−	0.624618i	$-0.785255\pi$
$307$	−4.51916e7	−1.56186	−0.780930	+	0.624618i	$0.785255\pi$
$308$	0	0
$309$	0	0
$310$	8.55684e6	0.287229
$311$	9.74134e6i	0.323845i	0.986803	+	0.161923i	$0.0517695\pi$
$311$	9.74134e6i	0.323845i	−0.986803	+	0.161923i	$0.948230\pi$
$312$	0	0
$313$	5.30265e6	0.172926	0.0864630	−	0.996255i	$-0.472444\pi$
$313$	5.30265e6	0.172926	0.0864630	+	0.996255i	$0.472444\pi$
$314$	3.03959e7i	0.981808i
$315$	0	0
$316$	5.29322e7	1.67748
$317$	− 1.05462e7i	− 0.331068i	−0.986204	−	0.165534i	$-0.947065\pi$
$317$	− 1.05462e7i	− 0.331068i	0.986204	−	0.165534i	$-0.0529348\pi$
$318$	0	0
$319$	2.00381e7	0.617283
$320$	− 2.71986e7i	− 0.830034i
$321$	0	0
$322$	0	0
$323$	1.65656e7i	0.491587i
$324$	0	0
$325$	−3.98180e6	−0.115992
$326$	− 8.58523e7i	− 2.47799i
$327$	0	0
$328$	4.70989e7	1.33472
$329$	0	0
$330$	0	0
$331$	−3.81242e7	−1.05128	−0.525638	−	0.850708i	$-0.676174\pi$
$331$	−3.81242e7	−1.05128	−0.525638	+	0.850708i	$0.676174\pi$
$332$	9.13699e7i	2.49683i
$333$	0	0
$334$	−8.42206e7	−2.26037
$335$	1.37518e7i	0.365783i
$336$	0	0
$337$	−1.22682e7	−0.320548	−0.160274	−	0.987073i	$-0.551238\pi$
$337$	−1.22682e7	−0.320548	−0.160274	+	0.987073i	$0.551238\pi$
$338$	− 5.99291e7i	− 1.55198i
$339$	0	0
$340$	−4.45322e7	−1.13302
$341$	− 9.14313e6i	− 0.230585i
$342$	0	0
$343$	0	0
$344$	3.93594e7i	0.966882i
$345$	0	0
$346$	−7.10317e7	−1.71484
$347$	1.65809e7i	0.396843i	0.980117	+	0.198422i	$0.0635815\pi$
$347$	1.65809e7i	0.396843i	−0.980117	+	0.198422i	$0.936418\pi$
$348$	0	0
$349$	−4.81038e7	−1.13163	−0.565813	−	0.824534i	$-0.691438\pi$
$349$	−4.81038e7	−1.13163	−0.565813	+	0.824534i	$0.691438\pi$
$350$	0	0
$351$	0	0
$352$	−3.23870e7	−0.742580
$353$	− 1.47570e7i	− 0.335485i	−0.985831	−	0.167742i	$-0.946352\pi$
$353$	− 1.47570e7i	− 0.335485i	0.985831	−	0.167742i	$-0.0536477\pi$
$354$	0	0
$355$	3.43116e6	0.0766930
$356$	2.18945e7i	0.485272i
$357$	0	0
$358$	−6.19980e7	−1.35123
$359$	1.78509e7i	0.385813i	0.981217	+	0.192907i	$0.0617914\pi$
$359$	1.78509e7i	0.385813i	−0.981217	+	0.192907i	$0.938209\pi$
$360$	0	0
$361$	−4.16635e7	−0.885593
$362$	− 1.07875e8i	− 2.27403i
$363$	0	0
$364$	0	0
$365$	− 1.96122e7i	− 0.403318i
$366$	0	0
$367$	−8.67940e6	−0.175587	−0.0877934	−	0.996139i	$-0.527982\pi$
$367$	−8.67940e6	−0.175587	−0.0877934	+	0.996139i	$0.527982\pi$
$368$	− 4.39531e6i	− 0.0881954i
$369$	0	0
$370$	−1.95064e7	−0.385099
$371$	0	0
$372$	0	0
$373$	−7.94052e7	−1.53011	−0.765055	−	0.643965i	$-0.777288\pi$
$373$	−7.94052e7	−1.53011	−0.765055	+	0.643965i	$0.777288\pi$
$374$	7.86583e7i	1.50359i
$375$	0	0
$376$	−5.58071e7	−1.04985
$377$	− 7.96432e6i	− 0.148636i
$378$	0	0
$379$	−1.46346e7	−0.268821	−0.134410	−	0.990926i	$-0.542914\pi$
$379$	−1.46346e7	−0.268821	−0.134410	+	0.990926i	$0.542914\pi$
$380$	1.44691e7i	0.263688i
$381$	0	0
$382$	−1.23509e6	−0.0221568
$383$	9.16736e6i	0.163173i	0.996666	+	0.0815865i	$0.0259987\pi$
$383$	9.16736e6i	0.163173i	−0.996666	+	0.0815865i	$0.974001\pi$
$384$	0	0
$385$	0	0
$386$	9.53924e7i	1.65864i
$387$	0	0
$388$	−3.64246e6	−0.0623591
$389$	9.26668e6i	0.157426i	0.996897	+	0.0787128i	$0.0250810\pi$
$389$	9.26668e6i	0.157426i	−0.996897	+	0.0787128i	$0.974919\pi$
$390$	0	0
$391$	−4.10787e7	−0.687205
$392$	0	0
$393$	0	0
$394$	1.25326e8	2.04905
$395$	3.43733e7i	0.557737i
$396$	0	0
$397$	−7.25544e7	−1.15956	−0.579779	−	0.814774i	$-0.696861\pi$
$397$	−7.25544e7	−1.15956	−0.579779	+	0.814774i	$0.696861\pi$
$398$	− 4.50785e7i	− 0.715023i
$399$	0	0
$400$	−8.84330e6	−0.138177
$401$	− 7.37246e7i	− 1.14335i	−0.820480	−	0.571675i	$-0.806294\pi$
$401$	− 7.37246e7i	− 1.14335i	0.820480	−	0.571675i	$-0.193706\pi$
$402$	0	0
$403$	−3.63402e6	−0.0555228
$404$	5.48591e7i	0.831964i
$405$	0	0
$406$	0	0
$407$	2.08429e7i	0.309155i
$408$	0	0
$409$	−3.43558e7	−0.502146	−0.251073	−	0.967968i	$-0.580783\pi$
$409$	−3.43558e7	−0.502146	−0.251073	+	0.967968i	$0.580783\pi$
$410$	8.81575e7i	1.27911i
$411$	0	0
$412$	1.43098e8	2.04617
$413$	0	0
$414$	0	0
$415$	−5.93341e7	−0.830157
$416$	1.28725e7i	0.178806i
$417$	0	0
$418$	2.55571e7	0.349932
$419$	− 1.31347e8i	− 1.78557i	−0.450479	−	0.892787i	$-0.648747\pi$
$419$	− 1.31347e8i	− 1.78557i	0.450479	−	0.892787i	$-0.351253\pi$
$420$	0	0
$421$	2.36756e6	0.0317289	0.0158644	−	0.999874i	$-0.494950\pi$
$421$	2.36756e6	0.0317289	0.0158644	+	0.999874i	$0.494950\pi$
$422$	− 8.62060e7i	− 1.14710i
$423$	0	0
$424$	−8.51151e7	−1.11663
$425$	8.26497e7i	1.07665i
$426$	0	0
$427$	0	0
$428$	1.26450e8i	1.61283i
$429$	0	0
$430$	−7.36711e7	−0.926599
$431$	1.13049e8i	1.41200i	0.708212	+	0.705999i	$0.249502\pi$
$431$	1.13049e8i	1.41200i	−0.708212	+	0.705999i	$0.750498\pi$
$432$	0	0
$433$	−4.50927e7	−0.555447	−0.277723	−	0.960661i	$-0.589580\pi$
$433$	−4.50927e7	−0.555447	−0.277723	+	0.960661i	$0.589580\pi$
$434$	0	0
$435$	0	0
$436$	1.40677e8	1.69732
$437$	1.33470e7i	0.159934i
$438$	0	0
$439$	1.61605e8	1.91013	0.955064	−	0.296399i	$-0.0957858\pi$
$439$	1.61605e8	1.91013	0.955064	+	0.296399i	$0.0957858\pi$
$440$	2.38358e7i	0.279816i
$441$	0	0
$442$	3.12634e7	0.362051
$443$	2.54011e7i	0.292174i	0.989272	+	0.146087i	$0.0466679\pi$
$443$	2.54011e7i	0.292174i	−0.989272	+	0.146087i	$0.953332\pi$
$444$	0	0
$445$	−1.42179e7	−0.161345
$446$	4.25349e7i	0.479448i
$447$	0	0
$448$	0	0
$449$	− 9.43898e7i	− 1.04276i	−0.853323	−	0.521382i	$-0.825417\pi$
$449$	− 9.43898e7i	− 1.04276i	0.853323	−	0.521382i	$-0.174583\pi$
$450$	0	0
$451$	9.41978e7	1.02686
$452$	1.82697e7i	0.197841i
$453$	0	0
$454$	2.06209e8	2.20363
$455$	0	0
$456$	0	0
$457$	4.68452e7	0.490813	0.245407	−	0.969420i	$-0.421078\pi$
$457$	4.68452e7	0.490813	0.245407	+	0.969420i	$0.421078\pi$
$458$	2.11730e8i	2.20387i
$459$	0	0
$460$	−3.58798e7	−0.368618
$461$	− 2.33797e7i	− 0.238636i	−0.992856	−	0.119318i	$-0.961929\pi$
$461$	− 2.33797e7i	− 0.238636i	0.992856	−	0.119318i	$-0.0380708\pi$
$462$	0	0
$463$	−4.98269e7	−0.502019	−0.251010	−	0.967985i	$-0.580763\pi$
$463$	−4.98269e7	−0.502019	−0.251010	+	0.967985i	$0.580763\pi$
$464$	− 1.76882e7i	− 0.177064i
$465$	0	0
$466$	1.12718e8	1.11388
$467$	9.52369e7i	0.935092i	0.883969	+	0.467546i	$0.154862\pi$
$467$	9.52369e7i	0.935092i	−0.883969	+	0.467546i	$0.845138\pi$
$468$	0	0
$469$	0	0
$470$	− 1.04457e8i	− 1.00611i
$471$	0	0
$472$	−1.72290e7	−0.163846
$473$	7.87188e7i	0.743867i
$474$	0	0
$475$	2.68540e7	0.250569
$476$	0	0
$477$	0	0
$478$	1.43820e8	1.31685
$479$	− 1.97141e8i	− 1.79378i	−0.442251	−	0.896891i	$-0.645820\pi$
$479$	− 1.97141e8i	− 1.79378i	0.442251	−	0.896891i	$-0.354180\pi$
$480$	0	0
$481$	8.28421e6	0.0744416
$482$	− 1.91033e8i	− 1.70595i
$483$	0	0
$484$	−1.00202e8	−0.883775
$485$	− 2.36536e6i	− 0.0207334i
$486$	0	0
$487$	1.92602e6	0.0166753	0.00833765	−	0.999965i	$-0.497346\pi$
$487$	1.92602e6	0.0166753	0.00833765	+	0.999965i	$0.497346\pi$
$488$	1.08680e8i	0.935167i
$489$	0	0
$490$	0	0
$491$	− 1.79722e8i	− 1.51830i	−0.650916	−	0.759150i	$-0.725615\pi$
$491$	− 1.79722e8i	− 1.51830i	0.650916	−	0.759150i	$-0.274385\pi$
$492$	0	0
$493$	−1.65314e8	−1.37965
$494$	− 1.01579e7i	− 0.0842603i
$495$	0	0
$496$	−8.07090e6	−0.0661419
$497$	0	0
$498$	0	0
$499$	1.54018e8	1.23956	0.619782	−	0.784774i	$-0.287221\pi$
$499$	1.54018e8	1.23956	0.619782	+	0.784774i	$0.287221\pi$
$500$	1.69638e8i	1.35710i
$501$	0	0
$502$	3.87025e8	3.05934
$503$	− 2.25142e7i	− 0.176910i	−0.996080	−	0.0884551i	$-0.971807\pi$
$503$	− 2.25142e7i	− 0.176910i	0.996080	−	0.0884551i	$-0.0281930\pi$
$504$	0	0
$505$	−3.56246e7	−0.276615
$506$	6.33753e7i	0.489180i
$507$	0	0
$508$	1.24519e7	0.0949825
$509$	2.16126e8i	1.63891i	0.573147	+	0.819453i	$0.305722\pi$
$509$	2.16126e8i	1.63891i	−0.573147	+	0.819453i	$0.694278\pi$
$510$	0	0
$511$	0	0
$512$	4.97487e7i	0.370656i
$513$	0	0
$514$	−2.51278e8	−1.85040
$515$	9.29253e7i	0.680318i
$516$	0	0
$517$	−1.11614e8	−0.807695
$518$	0	0
$519$	0	0
$520$	9.47376e6	0.0673771
$521$	− 8.35166e7i	− 0.590554i	−0.955412	−	0.295277i	$-0.904588\pi$
$521$	− 8.35166e7i	− 0.590554i	0.955412	−	0.295277i	$-0.0954120\pi$
$522$	0	0
$523$	−2.08856e8	−1.45997	−0.729983	−	0.683466i	$-0.760472\pi$
$523$	−2.08856e8	−1.45997	−0.729983	+	0.683466i	$0.760472\pi$
$524$	2.57101e8i	1.78694i
$525$	0	0
$526$	−4.43608e7	−0.304819
$527$	7.54308e7i	0.515367i
$528$	0	0
$529$	1.14939e8	0.776424
$530$	− 1.59315e8i	− 1.07011i
$531$	0	0
$532$	0	0
$533$	− 3.74397e7i	− 0.247258i
$534$	0	0
$535$	−8.21146e7	−0.536240
$536$	9.35119e7i	0.607257i
$537$	0	0
$538$	7.19625e7	0.462125
$539$	0	0
$540$	0	0
$541$	−1.21245e8	−0.765727	−0.382863	−	0.923805i	$-0.625062\pi$
$541$	−1.21245e8	−0.765727	−0.382863	+	0.923805i	$0.625062\pi$
$542$	− 3.71519e8i	− 2.33337i
$543$	0	0
$544$	2.67193e8	1.65970
$545$	9.13534e7i	0.564333i
$546$	0	0
$547$	−1.33857e8	−0.817861	−0.408931	−	0.912565i	$-0.634098\pi$
$547$	−1.33857e8	−0.817861	−0.408931	+	0.912565i	$0.634098\pi$
$548$	− 1.98264e7i	− 0.120477i
$549$	0	0
$550$	1.27510e8	0.766402
$551$	5.37128e7i	0.321087i
$552$	0	0
$553$	0	0
$554$	2.92664e8i	1.72123i
$555$	0	0
$556$	1.98590e8	1.15540
$557$	− 8.20694e7i	− 0.474915i	−0.971398	−	0.237457i	$-0.923686\pi$
$557$	− 8.20694e7i	− 0.474915i	0.971398	−	0.237457i	$-0.0763140\pi$
$558$	0	0
$559$	3.12875e7	0.179116
$560$	0	0
$561$	0	0
$562$	1.59484e7	0.0898480
$563$	2.21977e8i	1.24389i	0.783059	+	0.621947i	$0.213658\pi$
$563$	2.21977e8i	1.24389i	−0.783059	+	0.621947i	$0.786342\pi$
$564$	0	0
$565$	−1.18641e7	−0.0657792
$566$	1.84719e8i	1.01874i
$567$	0	0
$568$	2.33319e7	0.127322
$569$	2.88087e8i	1.56382i	0.623391	+	0.781911i	$0.285755\pi$
$569$	2.88087e8i	1.56382i	−0.623391	+	0.781911i	$0.714245\pi$
$570$	0	0
$571$	1.83227e8	0.984197	0.492098	−	0.870540i	$-0.336230\pi$
$571$	1.83227e8	0.984197	0.492098	+	0.870540i	$0.336230\pi$
$572$	− 2.91777e7i	− 0.155906i
$573$	0	0
$574$	0	0
$575$	6.65912e7i	0.350278i
$576$	0	0
$577$	2.07783e8	1.08164	0.540820	−	0.841139i	$-0.318114\pi$
$577$	2.07783e8	1.08164	0.540820	+	0.841139i	$0.318114\pi$
$578$	− 3.41709e8i	− 1.76959i
$579$	0	0
$580$	−1.44392e8	−0.740048
$581$	0	0
$582$	0	0
$583$	−1.70230e8	−0.859075
$584$	− 1.33363e8i	− 0.669571i
$585$	0	0
$586$	1.43481e8	0.713018
$587$	3.28908e8i	1.62615i	0.582161	+	0.813073i	$0.302207\pi$
$587$	3.28908e8i	1.62615i	−0.582161	+	0.813073i	$0.697793\pi$
$588$	0	0
$589$	2.45085e7	0.119942
$590$	− 3.22485e7i	− 0.157019i
$591$	0	0
$592$	1.83986e7	0.0886790
$593$	1.97249e7i	0.0945914i	0.998881	+	0.0472957i	$0.0150603\pi$
$593$	1.97249e7i	0.0945914i	−0.998881	+	0.0472957i	$0.984940\pi$
$594$	0	0
$595$	0	0
$596$	− 4.17023e8i	− 1.96980i
$597$	0	0
$598$	2.51891e7	0.117790
$599$	9.38486e7i	0.436664i	0.975875	+	0.218332i	$0.0700616\pi$
$599$	9.38486e7i	0.436664i	−0.975875	+	0.218332i	$0.929938\pi$
$600$	0	0
$601$	−1.53106e8	−0.705293	−0.352646	−	0.935757i	$-0.614718\pi$
$601$	−1.53106e8	−0.705293	−0.352646	+	0.935757i	$0.614718\pi$
$602$	0	0
$603$	0	0
$604$	−3.66554e8	−1.66352
$605$	− 6.50698e7i	− 0.293842i
$606$	0	0
$607$	−1.08279e8	−0.484147	−0.242074	−	0.970258i	$-0.577828\pi$
$607$	−1.08279e8	−0.484147	−0.242074	+	0.970258i	$0.577828\pi$
$608$	− 8.68146e7i	− 0.386262i
$609$	0	0
$610$	−2.03422e8	−0.896206
$611$	4.43620e7i	0.194486i
$612$	0	0
$613$	−2.60288e8	−1.12999	−0.564993	−	0.825096i	$-0.691121\pi$
$613$	−2.60288e8	−1.12999	−0.564993	+	0.825096i	$0.691121\pi$
$614$	− 5.75195e8i	− 2.48491i
$615$	0	0
$616$	0	0
$617$	1.65376e8i	0.704073i	0.935986	+	0.352036i	$0.114511\pi$
$617$	1.65376e8i	0.704073i	−0.935986	+	0.352036i	$0.885489\pi$
$618$	0	0
$619$	1.36836e8	0.576935	0.288468	−	0.957490i	$-0.406854\pi$
$619$	1.36836e8	0.576935	0.288468	+	0.957490i	$0.406854\pi$
$620$	6.58843e7i	0.276444i
$621$	0	0
$622$	−1.23987e8	−0.515235
$623$	0	0
$624$	0	0
$625$	7.06994e7	0.289585
$626$	6.74918e7i	0.275124i
$627$	0	0
$628$	−2.34037e8	−0.944942
$629$	− 1.71954e8i	− 0.690973i
$630$	0	0
$631$	2.66941e8	1.06249	0.531247	−	0.847217i	$-0.321723\pi$
$631$	2.66941e8	1.06249	0.531247	+	0.847217i	$0.321723\pi$
$632$	2.33738e8i	0.925931i
$633$	0	0
$634$	1.34231e8	0.526727
$635$	8.08605e6i	0.0315802i
$636$	0	0
$637$	0	0
$638$	2.55043e8i	0.982092i
$639$	0	0
$640$	1.93771e8	0.739179
$641$	− 3.69866e8i	− 1.40433i	−0.712013	−	0.702167i	$-0.752216\pi$
$641$	− 3.69866e8i	− 1.40433i	0.712013	−	0.702167i	$-0.247784\pi$
$642$	0	0
$643$	9.29168e7	0.349511	0.174756	−	0.984612i	$-0.444086\pi$
$643$	9.29168e7	0.349511	0.174756	+	0.984612i	$0.444086\pi$
$644$	0	0
$645$	0	0
$646$	−2.10846e8	−0.782111
$647$	− 9.21336e7i	− 0.340177i	−0.985429	−	0.170089i	$-0.945595\pi$
$647$	− 9.21336e7i	− 0.340177i	0.985429	−	0.170089i	$-0.0544054\pi$
$648$	0	0
$649$	−3.44580e7	−0.126054
$650$	− 5.06800e7i	− 0.184543i
$651$	0	0
$652$	6.61029e8	2.38494
$653$	− 2.20689e8i	− 0.792576i	−0.918126	−	0.396288i	$-0.870298\pi$
$653$	− 2.20689e8i	− 0.792576i	0.918126	−	0.396288i	$-0.129702\pi$
$654$	0	0
$655$	−1.66957e8	−0.594130
$656$	− 8.31511e7i	− 0.294548i
$657$	0	0
$658$	0	0
$659$	− 5.01619e8i	− 1.75274i	−0.481639	−	0.876370i	$-0.659958\pi$
$659$	− 5.01619e8i	− 1.75274i	0.481639	−	0.876370i	$-0.340042\pi$
$660$	0	0
$661$	2.78166e8	0.963163	0.481582	−	0.876401i	$-0.340063\pi$
$661$	2.78166e8	0.963163	0.481582	+	0.876401i	$0.340063\pi$
$662$	− 4.85242e8i	− 1.67257i
$663$	0	0
$664$	−4.03472e8	−1.37819
$665$	0	0
$666$	0	0
$667$	−1.33194e8	−0.448858
$668$	− 6.48465e8i	− 2.17549i
$669$	0	0
$670$	−1.75031e8	−0.581957
$671$	2.17360e8i	0.719468i
$672$	0	0
$673$	5.34850e8	1.75464	0.877318	−	0.479910i	$-0.159331\pi$
$673$	5.34850e8	1.75464	0.877318	+	0.479910i	$0.159331\pi$
$674$	− 1.56149e8i	− 0.509988i
$675$	0	0
$676$	4.61430e8	1.49371
$677$	− 1.41358e6i	− 0.00455568i	−0.999997	−	0.00227784i	$-0.999275\pi$
$677$	− 1.41358e6i	− 0.00455568i	0.999997	−	0.00227784i	$-0.000725059\pi$
$678$	0	0
$679$	0	0
$680$	− 1.96646e8i	− 0.625400i
$681$	0	0
$682$	1.16373e8	0.366859
$683$	2.41616e8i	0.758340i	0.925327	+	0.379170i	$0.123791\pi$
$683$	2.41616e8i	0.758340i	−0.925327	+	0.379170i	$0.876209\pi$
$684$	0	0
$685$	1.28750e7	0.0400566
$686$	0	0
$687$	0	0
$688$	6.94873e7	0.213373
$689$	6.76595e7i	0.206857i
$690$	0	0
$691$	−5.30543e8	−1.60800	−0.804001	−	0.594628i	$-0.797299\pi$
$691$	−5.30543e8	−1.60800	−0.804001	+	0.594628i	$0.797299\pi$
$692$	− 5.46916e8i	− 1.65045i
$693$	0	0
$694$	−2.11040e8	−0.631374
$695$	1.28961e8i	0.384152i
$696$	0	0
$697$	−7.77132e8	−2.29507
$698$	− 6.12261e8i	− 1.80041i
$699$	0	0
$700$	0	0
$701$	− 5.49256e7i	− 0.159449i	−0.996817	−	0.0797244i	$-0.974596\pi$
$701$	− 5.49256e7i	− 0.159449i	0.996817	−	0.0797244i	$-0.0254040\pi$
$702$	0	0
$703$	−5.58702e7	−0.160811
$704$	− 3.69900e8i	− 1.06015i
$705$	0	0
$706$	1.87826e8	0.533753
$707$	0	0
$708$	0	0
$709$	3.16706e7	0.0888622	0.0444311	−	0.999012i	$-0.485852\pi$
$709$	3.16706e7	0.0888622	0.0444311	+	0.999012i	$0.485852\pi$
$710$	4.36715e7i	0.122018i
$711$	0	0
$712$	−9.66819e7	−0.267858
$713$	6.07749e7i	0.167670i
$714$	0	0
$715$	1.89475e7	0.0518363
$716$	− 4.77361e8i	− 1.30049i
$717$	0	0
$718$	−2.27205e8	−0.613825
$719$	4.37723e8i	1.17764i	0.808264	+	0.588820i	$0.200407\pi$
$719$	4.37723e8i	1.17764i	−0.808264	+	0.588820i	$0.799593\pi$
$720$	0	0
$721$	0	0
$722$	− 5.30290e8i	− 1.40897i
$723$	0	0
$724$	8.30595e8	2.18864
$725$	2.67985e8i	0.703230i
$726$	0	0
$727$	4.64180e8	1.20805	0.604023	−	0.796967i	$-0.293564\pi$
$727$	4.64180e8	1.20805	0.604023	+	0.796967i	$0.293564\pi$
$728$	0	0
$729$	0	0
$730$	2.49623e8	0.641675
$731$	− 6.49430e8i	− 1.66257i
$732$	0	0
$733$	−7.04886e8	−1.78981	−0.894905	−	0.446256i	$-0.852757\pi$
$733$	−7.04886e8	−1.78981	−0.894905	+	0.446256i	$0.852757\pi$
$734$	− 1.10471e8i	− 0.279357i
$735$	0	0
$736$	2.15279e8	0.539967
$737$	1.87024e8i	0.467191i
$738$	0	0
$739$	−2.83900e8	−0.703448	−0.351724	−	0.936104i	$-0.614404\pi$
$739$	−2.83900e8	−0.703448	−0.351724	+	0.936104i	$0.614404\pi$
$740$	− 1.50192e8i	− 0.370639i
$741$	0	0
$742$	0	0
$743$	5.01018e8i	1.22148i	0.791830	+	0.610741i	$0.209128\pi$
$743$	5.01018e8i	1.22148i	−0.791830	+	0.610741i	$0.790872\pi$
$744$	0	0
$745$	2.70808e8	0.654927
$746$	− 1.01066e9i	− 2.43439i
$747$	0	0
$748$	−6.05638e8	−1.44713
$749$	0	0
$750$	0	0
$751$	−1.48249e8	−0.350004	−0.175002	−	0.984568i	$-0.555993\pi$
$751$	−1.48249e8	−0.350004	−0.175002	+	0.984568i	$0.555993\pi$
$752$	9.85249e7i	0.231682i
$753$	0	0
$754$	1.01369e8	0.236479
$755$	− 2.38034e8i	− 0.553094i
$756$	0	0
$757$	2.14422e8	0.494291	0.247145	−	0.968978i	$-0.420507\pi$
$757$	2.14422e8	0.494291	0.247145	+	0.968978i	$0.420507\pi$
$758$	− 1.86268e8i	− 0.427691i
$759$	0	0
$760$	−6.38928e7	−0.145550
$761$	− 2.69760e7i	− 0.0612101i	−0.999532	−	0.0306051i	$-0.990257\pi$
$761$	− 2.69760e7i	− 0.0612101i	0.999532	−	0.0306051i	$-0.00974341\pi$
$762$	0	0
$763$	0	0
$764$	− 9.50969e6i	− 0.0213249i
$765$	0	0
$766$	−1.16681e8	−0.259607
$767$	1.36957e7i	0.0303526i
$768$	0	0
$769$	4.90064e8	1.07764	0.538821	−	0.842421i	$-0.318870\pi$
$769$	4.90064e8	1.07764	0.538821	+	0.842421i	$0.318870\pi$
$770$	0	0
$771$	0	0
$772$	−7.34484e8	−1.59636
$773$	4.17032e8i	0.902882i	0.892301	+	0.451441i	$0.149090\pi$
$773$	4.17032e8i	0.902882i	−0.892301	+	0.451441i	$0.850910\pi$
$774$	0	0
$775$	1.22278e8	0.262690
$776$	− 1.60844e7i	− 0.0344208i
$777$	0	0
$778$	−1.17946e8	−0.250463
$779$	2.52501e8i	0.534134i
$780$	0	0
$781$	4.66638e7	0.0979550
$782$	− 5.22846e8i	− 1.09334i
$783$	0	0
$784$	0	0
$785$	− 1.51980e8i	− 0.314179i
$786$	0	0
$787$	3.46111e8	0.710054	0.355027	−	0.934856i	$-0.384472\pi$
$787$	3.46111e8	0.710054	0.355027	+	0.934856i	$0.384472\pi$
$788$	9.64963e8i	1.97211i
$789$	0	0
$790$	−4.37500e8	−0.887355
$791$	0	0
$792$	0	0
$793$	8.63915e7	0.173241
$794$	− 9.23467e8i	− 1.84484i
$795$	0	0
$796$	3.47087e8	0.688175
$797$	7.25306e8i	1.43267i	0.697756	+	0.716335i	$0.254182\pi$
$797$	7.25306e8i	1.43267i	−0.697756	+	0.716335i	$0.745818\pi$
$798$	0	0
$799$	9.20816e8	1.80523
$800$	− 4.33138e8i	− 0.845972i
$801$	0	0
$802$	9.38361e8	1.81906
$803$	− 2.66726e8i	− 0.515132i
$804$	0	0
$805$	0	0
$806$	− 4.62535e7i	− 0.0883363i
$807$	0	0
$808$	−2.42247e8	−0.459224
$809$	4.93161e8i	0.931415i	0.884939	+	0.465707i	$0.154200\pi$
$809$	4.93161e8i	0.931415i	−0.884939	+	0.465707i	$0.845800\pi$
$810$	0	0
$811$	5.34731e7	0.100247	0.0501237	−	0.998743i	$-0.484038\pi$
$811$	5.34731e7	0.100247	0.0501237	+	0.998743i	$0.484038\pi$
$812$	0	0
$813$	0	0
$814$	−2.65287e8	−0.491862
$815$	4.29261e8i	0.792956i
$816$	0	0
$817$	−2.11009e8	−0.386931
$818$	− 4.37278e8i	− 0.798909i
$819$	0	0
$820$	−6.78778e8	−1.23108
$821$	1.07674e9i	1.94573i	0.231377	+	0.972864i	$0.425677\pi$
$821$	1.07674e9i	1.94573i	−0.231377	+	0.972864i	$0.574323\pi$
$822$	0	0
$823$	8.88441e7	0.159378	0.0796891	−	0.996820i	$-0.474607\pi$
$823$	8.88441e7	0.159378	0.0796891	+	0.996820i	$0.474607\pi$
$824$	6.31892e8i	1.12943i
$825$	0	0
$826$	0	0
$827$	9.82047e7i	0.173626i	0.996225	+	0.0868132i	$0.0276683\pi$
$827$	9.82047e7i	0.173626i	−0.996225	+	0.0868132i	$0.972332\pi$
$828$	0	0
$829$	−2.25045e8	−0.395008	−0.197504	−	0.980302i	$-0.563283\pi$
$829$	−2.25045e8	−0.395008	−0.197504	+	0.980302i	$0.563283\pi$
$830$	− 7.55200e8i	− 1.32077i
$831$	0	0
$832$	−1.47020e8	−0.255274
$833$	0	0
$834$	0	0
$835$	4.21103e8	0.723317
$836$	1.96780e8i	0.336792i
$837$	0	0
$838$	1.67177e9	2.84083
$839$	7.84433e8i	1.32822i	0.747635	+	0.664110i	$0.231189\pi$
$839$	7.84433e8i	1.32822i	−0.747635	+	0.664110i	$0.768811\pi$
$840$	0	0
$841$	5.88040e7	0.0988597
$842$	3.01341e7i	0.0504803i
$843$	0	0
$844$	6.63752e8	1.10402
$845$	2.99645e8i	0.496635i
$846$	0	0
$847$	0	0
$848$	1.50267e8i	0.246420i
$849$	0	0
$850$	−1.05196e9	−1.71294
$851$	− 1.38544e8i	− 0.224802i
$852$	0	0
$853$	7.97906e7	0.128560	0.0642798	−	0.997932i	$-0.479525\pi$
$853$	7.97906e7	0.128560	0.0642798	+	0.997932i	$0.479525\pi$
$854$	0	0
$855$	0	0
$856$	−5.58379e8	−0.890241
$857$	− 1.06107e9i	− 1.68579i	−0.538080	−	0.842894i	$-0.680850\pi$
$857$	− 1.06107e9i	− 1.68579i	0.538080	−	0.842894i	$-0.319150\pi$
$858$	0	0
$859$	−3.34286e8	−0.527399	−0.263699	−	0.964605i	$-0.584943\pi$
$859$	−3.34286e8	−0.527399	−0.263699	+	0.964605i	$0.584943\pi$
$860$	− 5.67239e8i	− 0.891806i
$861$	0	0
$862$	−1.43888e9	−2.24648
$863$	− 2.65292e7i	− 0.0412754i	−0.999787	−	0.0206377i	$-0.993430\pi$
$863$	− 2.65292e7i	− 0.0412754i	0.999787	−	0.0206377i	$-0.00656965\pi$
$864$	0	0
$865$	3.55158e8	0.548749
$866$	− 5.73936e8i	− 0.883711i
$867$	0	0
$868$	0	0
$869$	4.67477e8i	0.712362i
$870$	0	0
$871$	7.43343e7	0.112495
$872$	6.21203e8i	0.936880i
$873$	0	0
$874$	−1.69880e8	−0.254453
$875$	0	0
$876$	0	0
$877$	−1.30791e8	−0.193901	−0.0969504	−	0.995289i	$-0.530909\pi$
$877$	−1.30791e8	−0.193901	−0.0969504	+	0.995289i	$0.530909\pi$
$878$	2.05690e9i	3.03900i
$879$	0	0
$880$	4.20811e7	0.0617503
$881$	9.18953e8i	1.34390i	0.740598	+	0.671948i	$0.234542\pi$
$881$	9.18953e8i	1.34390i	−0.740598	+	0.671948i	$0.765458\pi$
$882$	0	0
$883$	−1.09112e9	−1.58486	−0.792432	−	0.609961i	$-0.791185\pi$
$883$	−1.09112e9	−1.58486	−0.792432	+	0.609961i	$0.791185\pi$
$884$	2.40716e8i	0.348456i
$885$	0	0
$886$	−3.23303e8	−0.464846
$887$	5.36427e8i	0.768670i	0.923194	+	0.384335i	$0.125569\pi$
$887$	5.36427e8i	0.768670i	−0.923194	+	0.384335i	$0.874431\pi$
$888$	0	0
$889$	0	0
$890$	− 1.80965e8i	− 0.256699i
$891$	0	0
$892$	−3.27502e8	−0.461445
$893$	− 2.99186e8i	− 0.420133i
$894$	0	0
$895$	3.09990e8	0.432393
$896$	0	0
$897$	0	0
$898$	1.20139e9	1.65903
$899$	2.44579e8i	0.336620i
$900$	0	0
$901$	1.40440e9	1.92007
$902$	1.19894e9i	1.63372i
$903$	0	0
$904$	−8.06757e7	−0.109204
$905$	5.39375e8i	0.727688i
$906$	0	0
$907$	−4.60985e8	−0.617824	−0.308912	−	0.951091i	$-0.599965\pi$
$907$	−4.60985e8	−0.617824	−0.308912	+	0.951091i	$0.599965\pi$
$908$	1.58772e9i	2.12089i
$909$	0	0
$910$	0	0
$911$	− 8.68400e8i	− 1.14859i	−0.818649	−	0.574295i	$-0.805276\pi$
$911$	− 8.68400e8i	− 1.14859i	0.818649	−	0.574295i	$-0.194724\pi$
$912$	0	0
$913$	−8.06944e8	−1.06031
$914$	5.96242e8i	0.780879i
$915$	0	0
$916$	−1.63024e9	−2.12112
$917$	0	0
$918$	0	0
$919$	1.28061e9	1.64995	0.824976	−	0.565168i	$-0.191189\pi$
$919$	1.28061e9	1.64995	0.824976	+	0.565168i	$0.191189\pi$
$920$	− 1.58438e8i	− 0.203468i
$921$	0	0
$922$	2.97575e8	0.379668
$923$	− 1.85469e7i	− 0.0235867i
$924$	0	0
$925$	−2.78749e8	−0.352199
$926$	− 6.34192e8i	− 0.798708i
$927$	0	0
$928$	8.66353e8	1.08405
$929$	3.23960e8i	0.404058i	0.979380	+	0.202029i	$0.0647536\pi$
$929$	3.23960e8i	0.404058i	−0.979380	+	0.202029i	$0.935246\pi$
$930$	0	0
$931$	0	0
$932$	8.67888e8i	1.07205i
$933$	0	0
$934$	−1.21217e9	−1.48772
$935$	− 3.93291e8i	− 0.481149i
$936$	0	0
$937$	1.33188e9	1.61900	0.809498	−	0.587123i	$-0.199739\pi$
$937$	1.33188e9	1.61900	0.809498	+	0.587123i	$0.199739\pi$
$938$	0	0
$939$	0	0
$940$	8.04278e8	0.968329
$941$	− 8.24869e8i	− 0.989957i	−0.868905	−	0.494978i	$-0.835176\pi$
$941$	− 8.24869e8i	− 0.989957i	0.868905	−	0.494978i	$-0.164824\pi$
$942$	0	0
$943$	−6.26138e8	−0.746681
$944$	3.04171e7i	0.0361578i
$945$	0	0
$946$	−1.00193e9	−1.18349
$947$	7.33660e8i	0.863863i	0.901906	+	0.431931i	$0.142168\pi$
$947$	7.33660e8i	0.863863i	−0.901906	+	0.431931i	$0.857832\pi$
$948$	0	0
$949$	−1.06013e8	−0.124039
$950$	3.41796e8i	0.398654i
$951$	0	0
$952$	0	0
$953$	6.87744e8i	0.794600i	0.917689	+	0.397300i	$0.130053\pi$
$953$	6.87744e8i	0.794600i	−0.917689	+	0.397300i	$0.869947\pi$
$954$	0	0
$955$	6.17544e6	0.00709019
$956$	1.10736e9i	1.26740i
$957$	0	0
$958$	2.50919e9	2.85389
$959$	0	0
$960$	0	0
$961$	−7.75906e8	−0.874256
$962$	1.05441e8i	0.118436i
$963$	0	0
$964$	1.47088e9	1.64190
$965$	− 4.76962e8i	− 0.530764i
$966$	0	0
$967$	1.43001e9	1.58147	0.790733	−	0.612162i	$-0.209700\pi$
$967$	1.43001e9	1.58147	0.790733	+	0.612162i	$0.209700\pi$
$968$	− 4.42475e8i	− 0.487823i
$969$	0	0
$970$	3.01061e7	0.0329867
$971$	1.05628e9i	1.15377i	0.816824	+	0.576887i	$0.195733\pi$
$971$	1.05628e9i	1.15377i	−0.816824	+	0.576887i	$0.804267\pi$
$972$	0	0
$973$	0	0
$974$	2.45142e7i	0.0265303i
$975$	0	0
$976$	1.91869e8	0.206375
$977$	8.72371e8i	0.935443i	0.883876	+	0.467722i	$0.154925\pi$
$977$	8.72371e8i	0.935443i	−0.883876	+	0.467722i	$0.845075\pi$
$978$	0	0
$979$	−1.93364e8	−0.206076
$980$	0	0
$981$	0	0
$982$	2.28749e9	2.41560
$983$	7.90860e8i	0.832605i	0.909226	+	0.416302i	$0.136674\pi$
$983$	7.90860e8i	0.832605i	−0.909226	+	0.416302i	$0.863326\pi$
$984$	0	0
$985$	−6.26631e8	−0.655697
$986$	− 2.10411e9i	− 2.19501i
$987$	0	0
$988$	7.82118e7	0.0810964
$989$	− 5.23249e8i	− 0.540903i
$990$	0	0
$991$	1.38796e8	0.142612	0.0713062	−	0.997454i	$-0.477283\pi$
$991$	1.38796e8	0.142612	0.0713062	+	0.997454i	$0.477283\pi$
$992$	− 3.95306e8i	− 0.404947i
$993$	0	0
$994$	0	0
$995$	2.25392e8i	0.228807i
$996$	0	0
$997$	3.29036e8	0.332015	0.166008	−	0.986124i	$-0.446912\pi$
$997$	3.29036e8	0.332015	0.166008	+	0.986124i	$0.446912\pi$
$998$	1.96032e9i	1.97213i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.7.b.a.197.2		2
3.2	odd	2	inner	441.7.b.a.197.1		2
7.6	odd	2		9.7.b.a.8.2	yes	2
21.20	even	2		9.7.b.a.8.1	✓	2
28.27	even	2		144.7.e.a.17.2		2
35.13	even	4		225.7.d.a.224.3		4
35.27	even	4		225.7.d.a.224.2		4
35.34	odd	2		225.7.c.a.26.1		2
56.13	odd	2		576.7.e.l.449.1		2
56.27	even	2		576.7.e.a.449.1		2
63.13	odd	6		81.7.d.d.26.1		4
63.20	even	6		81.7.d.d.53.1		4
63.34	odd	6		81.7.d.d.53.2		4
63.41	even	6		81.7.d.d.26.2		4
84.83	odd	2		144.7.e.a.17.1		2
105.62	odd	4		225.7.d.a.224.4		4
105.83	odd	4		225.7.d.a.224.1		4
105.104	even	2		225.7.c.a.26.2		2
168.83	odd	2		576.7.e.a.449.2		2
168.125	even	2		576.7.e.l.449.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9.7.b.a.8.1	✓	2	21.20	even	2
9.7.b.a.8.2	yes	2	7.6	odd	2
81.7.d.d.26.1		4	63.13	odd	6
81.7.d.d.26.2		4	63.41	even	6
81.7.d.d.53.1		4	63.20	even	6
81.7.d.d.53.2		4	63.34	odd	6
144.7.e.a.17.1		2	84.83	odd	2
144.7.e.a.17.2		2	28.27	even	2
225.7.c.a.26.1		2	35.34	odd	2
225.7.c.a.26.2		2	105.104	even	2
225.7.d.a.224.1		4	105.83	odd	4
225.7.d.a.224.2		4	35.27	even	4
225.7.d.a.224.3		4	35.13	even	4
225.7.d.a.224.4		4	105.62	odd	4
441.7.b.a.197.1		2	3.2	odd	2	inner
441.7.b.a.197.2		2	1.1	even	1	trivial
576.7.e.a.449.1		2	56.27	even	2
576.7.e.a.449.2		2	168.83	odd	2
576.7.e.l.449.1		2	56.13	odd	2
576.7.e.l.449.2		2	168.125	even	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 441 = 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	441.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+12.7279i q^{2} -98.0000 q^{4} -63.6396i q^{5} -432.749i q^{8} +810.000 q^{10} -865.499i q^{11} -344.000 q^{13} -764.000 q^{16} +7140.36i q^{17} +2320.00 q^{19} +6236.68i q^{20} +11016.0 q^{22} +5753.02i q^{23} +11575.0 q^{25} -4378.41i q^{26} +23152.1i q^{29} +10564.0 q^{31} -37420.1i q^{32} -90882.0 q^{34} -24082.0 q^{37} +29528.8i q^{38} -27540.0 q^{40} +108836. i q^{41} -90952.0 q^{43} +84818.9i q^{44} -73224.0 q^{46} -128959. i q^{47} +147326. i q^{50} +33712.0 q^{52} -196685. i q^{53} -55080.0 q^{55} -294678. q^{58} -39812.9i q^{59} -251138. q^{61} +134458. i q^{62} +427384. q^{64} +21892.0i q^{65} -216088. q^{67} -699756. i q^{68} +53915.5i q^{71} +308176. q^{73} -306514. i q^{74} -227360. q^{76} -540124. q^{79} +48620.7i q^{80} -1.38526e6 q^{82} -932346. i q^{83} +454410. q^{85} -1.15763e6i q^{86} -374544. q^{88} -223413. i q^{89} -563796. i q^{92} +1.64138e6 q^{94} -147644. i q^{95} +37168.0 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 196 q^{4} + 1620 q^{10} - 688 q^{13} - 1528 q^{16} + 4640 q^{19} + 22032 q^{22} + 23150 q^{25} + 21128 q^{31} - 181764 q^{34} - 48164 q^{37} - 55080 q^{40} - 181904 q^{43} - 146448 q^{46} + 67424 q^{52}+ \cdots + 74336 q^{97}+O(q^{100}) \) 2 * q - 196 * q^4 + 1620 * q^10 - 688 * q^13 - 1528 * q^16 + 4640 * q^19 + 22032 * q^22 + 23150 * q^25 + 21128 * q^31 - 181764 * q^34 - 48164 * q^37 - 55080 * q^40 - 181904 * q^43 - 146448 * q^46 + 67424 * q^52 - 110160 * q^55 - 589356 * q^58 - 502276 * q^61 + 854768 * q^64 - 432176 * q^67 + 616352 * q^73 - 454720 * q^76 - 1080248 * q^79 - 2770524 * q^82 + 908820 * q^85 - 749088 * q^88 + 3282768 * q^94 + 74336 * q^97

Introduction

L-functions

Modular forms

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