LMFDB - Embedded newform 441.4.c.b.440.23

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [441,4,Mod(440,441)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(441, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("441.440");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 441 = 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	441.c (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:	$26.0198423125$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			440.23
Character	$\chi$	$=$	441.440
Dual form			441.4.c.b.440.24

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.38285i q^{2} -20.9751 q^{4} -18.3258 q^{5} +69.8432i q^{8} +O(q^{10})$	$q-5.38285i q^{2} -20.9751 q^{4} -18.3258 q^{5} +69.8432i q^{8} +98.6449i q^{10} +23.7703i q^{11} -11.7971i q^{13} +208.155 q^{16} -96.4198 q^{17} +21.0078i q^{19} +384.385 q^{20} +127.952 q^{22} -152.655i q^{23} +210.834 q^{25} -63.5019 q^{26} -77.5438i q^{29} +199.733i q^{31} -561.721i q^{32} +519.014i q^{34} +164.391 q^{37} +113.082 q^{38} -1279.93i q^{40} -435.473 q^{41} -106.148 q^{43} -498.585i q^{44} -821.722 q^{46} +11.0459 q^{47} -1134.89i q^{50} +247.445i q^{52} -110.936i q^{53} -435.609i q^{55} -417.407 q^{58} +829.036 q^{59} +22.5689i q^{61} +1075.14 q^{62} -1358.43 q^{64} +216.190i q^{65} +398.530 q^{67} +2022.42 q^{68} +228.153i q^{71} -266.960i q^{73} -884.895i q^{74} -440.641i q^{76} +920.667 q^{79} -3814.60 q^{80} +2344.09i q^{82} -642.787 q^{83} +1766.97 q^{85} +571.379i q^{86} -1660.19 q^{88} +445.254 q^{89} +3201.97i q^{92} -59.4582i q^{94} -384.984i q^{95} -1617.57i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 96 q^{4}+O(q^{10}) $ 24 * q - 96 * q^4	$ 24 q - 96 q^{4} + 144 q^{16} + 624 q^{22} + 312 q^{25} - 864 q^{37} + 1248 q^{43} - 3888 q^{46} - 7440 q^{58} - 3360 q^{64} - 2688 q^{67} + 480 q^{79} + 13248 q^{85} - 7248 q^{88}+O(q^{100}) $ 24 * q - 96 * q^4 + 144 * q^16 + 624 * q^22 + 312 * q^25 - 864 * q^37 + 1248 * q^43 - 3888 * q^46 - 7440 * q^58 - 3360 * q^64 - 2688 * q^67 + 480 * q^79 + 13248 * q^85 - 7248 * q^88

Display 100 coefficients 10 coefficients

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 $

$2$	− 5.38285i	− 1.90313i	−0.307453	−	0.951563i	$-0.599477\pi$
$2$	− 5.38285i	− 1.90313i	0.307453	−	0.951563i	$-0.400523\pi$
$3$	0	0
$3$	0	0
$4$	−20.9751	−2.62189
$5$	−18.3258	−1.63911	−0.819553	−	0.573003i	$-0.805778\pi$
$5$	−18.3258	−1.63911	−0.819553	+	0.573003i	$0.805778\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	69.8432i	3.08666i
$9$	0	0
$10$	98.6449i	3.11943i
$11$	23.7703i	0.651547i	0.945448	+	0.325773i	$0.105625\pi$
$11$	23.7703i	0.651547i	−0.945448	+	0.325773i	$0.894375\pi$
$12$	0	0
$13$	− 11.7971i	− 0.251686i	−0.992050	−	0.125843i	$-0.959836\pi$
$13$	− 11.7971i	− 0.251686i	0.992050	−	0.125843i	$-0.0401636\pi$
$14$	0	0
$15$	0	0
$16$	208.155	3.25242
$17$	−96.4198	−1.37560	−0.687801	−	0.725899i	$-0.741424\pi$
$17$	−96.4198	−1.37560	−0.687801	+	0.725899i	$0.741424\pi$
$18$	0	0
$19$	21.0078i	0.253659i	0.991925	+	0.126830i	$0.0404801\pi$
$19$	21.0078i	0.253659i	−0.991925	+	0.126830i	$0.959520\pi$
$20$	384.385	4.29756
$21$	0	0
$22$	127.952	1.23998
$23$	− 152.655i	− 1.38395i	−0.721922	−	0.691975i	$-0.756741\pi$
$23$	− 152.655i	− 1.38395i	0.721922	−	0.691975i	$-0.243259\pi$
$24$	0	0
$25$	210.834	1.68667
$26$	−63.5019	−0.478990
$27$	0	0
$28$	0	0
$29$	− 77.5438i	− 0.496535i	−0.968691	−	0.248268i	$-0.920139\pi$
$29$	− 77.5438i	− 0.496535i	0.968691	−	0.248268i	$-0.0798612\pi$
$30$	0	0
$31$	199.733i	1.15720i	0.815612	+	0.578600i	$0.196401\pi$
$31$	199.733i	1.15720i	−0.815612	+	0.578600i	$0.803599\pi$
$32$	− 561.721i	− 3.10310i
$33$	0	0
$34$	519.014i	2.61795i
$35$	0	0
$36$	0	0
$37$	164.391	0.730426	0.365213	−	0.930924i	$-0.380996\pi$
$37$	164.391	0.730426	0.365213	+	0.930924i	$0.380996\pi$
$38$	113.082	0.482745
$39$	0	0
$40$	− 1279.93i	− 5.05937i
$41$	−435.473	−1.65877	−0.829384	−	0.558679i	$-0.811308\pi$
$41$	−435.473	−1.65877	−0.829384	+	0.558679i	$0.811308\pi$
$42$	0	0
$43$	−106.148	−0.376451	−0.188226	−	0.982126i	$-0.560274\pi$
$43$	−106.148	−0.376451	−0.188226	+	0.982126i	$0.560274\pi$
$44$	− 498.585i	− 1.70828i
$45$	0	0
$46$	−821.722	−2.63383
$47$	11.0459	0.0342809	0.0171405	−	0.999853i	$-0.494544\pi$
$47$	11.0459	0.0342809	0.0171405	+	0.999853i	$0.494544\pi$
$48$	0	0
$49$	0	0
$50$	− 1134.89i	− 3.20994i
$51$	0	0
$52$	247.445i	0.659893i
$53$	− 110.936i	− 0.287514i	−0.989613	−	0.143757i	$-0.954082\pi$
$53$	− 110.936i	− 0.287514i	0.989613	−	0.143757i	$-0.0459184\pi$
$54$	0	0
$55$	− 435.609i	− 1.06795i
$56$	0	0
$57$	0	0
$58$	−417.407	−0.944969
$59$	829.036	1.82934	0.914672	−	0.404197i	$-0.132449\pi$
$59$	829.036	1.82934	0.914672	+	0.404197i	$0.132449\pi$
$60$	0	0
$61$	22.5689i	0.0473713i	0.999719	+	0.0236857i	$0.00754009\pi$
$61$	22.5689i	0.0473713i	−0.999719	+	0.0236857i	$0.992460\pi$
$62$	1075.14	2.20230
$63$	0	0
$64$	−1358.43	−2.65318
$65$	216.190i	0.412540i
$66$	0	0
$67$	398.530	0.726689	0.363345	−	0.931655i	$-0.381635\pi$
$67$	398.530	0.726689	0.363345	+	0.931655i	$0.381635\pi$
$68$	2022.42	3.60668
$69$	0	0
$70$	0	0
$71$	228.153i	0.381363i	0.981652	+	0.190682i	$0.0610698\pi$
$71$	228.153i	0.381363i	−0.981652	+	0.190682i	$0.938930\pi$
$72$	0	0
$73$	− 266.960i	− 0.428018i	−0.976832	−	0.214009i	$-0.931348\pi$
$73$	− 266.960i	− 0.428018i	0.976832	−	0.214009i	$-0.0686522\pi$
$74$	− 884.895i	− 1.39009i
$75$	0	0
$76$	− 440.641i	− 0.665066i
$77$	0	0
$78$	0	0
$79$	920.667	1.31118	0.655589	−	0.755118i	$-0.272420\pi$
$79$	920.667	1.31118	0.655589	+	0.755118i	$0.272420\pi$
$80$	−3814.60	−5.33106
$81$	0	0
$82$	2344.09i	3.15684i
$83$	−642.787	−0.850061	−0.425031	−	0.905179i	$-0.639737\pi$
$83$	−642.787	−0.850061	−0.425031	+	0.905179i	$0.639737\pi$
$84$	0	0
$85$	1766.97	2.25476
$86$	571.379i	0.716434i
$87$	0	0
$88$	−1660.19	−2.01111
$89$	445.254	0.530301	0.265150	−	0.964207i	$-0.414578\pi$
$89$	445.254	0.530301	0.265150	+	0.964207i	$0.414578\pi$
$90$	0	0
$91$	0	0
$92$	3201.97i	3.62856i
$93$	0	0
$94$	− 59.4582i	− 0.0652409i
$95$	− 384.984i	− 0.415774i
$96$	0	0
$97$	− 1617.57i	− 1.69319i	−0.532236	−	0.846596i	$-0.678648\pi$
$97$	− 1617.57i	− 1.69319i	0.532236	−	0.846596i	$-0.321352\pi$
$98$	0	0
$99$	0	0
$100$	−4422.26	−4.42226
$101$	75.0332	0.0739216	0.0369608	−	0.999317i	$-0.488232\pi$
$101$	75.0332	0.0739216	0.0369608	+	0.999317i	$0.488232\pi$
$102$	0	0
$103$	964.142i	0.922327i	0.887315	+	0.461164i	$0.152568\pi$
$103$	964.142i	0.922327i	−0.887315	+	0.461164i	$0.847432\pi$
$104$	823.945	0.776870
$105$	0	0
$106$	−597.153	−0.547176
$107$	1332.28i	1.20370i	0.798609	+	0.601851i	$0.205570\pi$
$107$	1332.28i	1.20370i	−0.798609	+	0.601851i	$0.794430\pi$
$108$	0	0
$109$	−890.160	−0.782219	−0.391110	−	0.920344i	$-0.627909\pi$
$109$	−890.160	−0.782219	−0.391110	+	0.920344i	$0.627909\pi$
$110$	−2344.82	−2.03245
$111$	0	0
$112$	0	0
$113$	− 1240.25i	− 1.03251i	−0.856436	−	0.516253i	$-0.827326\pi$
$113$	− 1240.25i	− 1.03251i	0.856436	−	0.516253i	$-0.172674\pi$
$114$	0	0
$115$	2797.53i	2.26844i
$116$	1626.49i	1.30186i
$117$	0	0
$118$	− 4462.58i	− 3.48147i
$119$	0	0
$120$	0	0
$121$	765.973	0.575487
$122$	121.485	0.0901537
$123$	0	0
$124$	− 4189.43i	− 3.03405i
$125$	−1572.97	−1.12552
$126$	0	0
$127$	1338.38	0.935137	0.467569	−	0.883957i	$-0.345130\pi$
$127$	1338.38	0.935137	0.467569	+	0.883957i	$0.345130\pi$
$128$	2818.44i	1.94623i
$129$	0	0
$130$	1163.72	0.785116
$131$	2313.81	1.54319	0.771597	−	0.636111i	$-0.219458\pi$
$131$	2313.81	1.54319	0.771597	+	0.636111i	$0.219458\pi$
$132$	0	0
$133$	0	0
$134$	− 2145.23i	− 1.38298i
$135$	0	0
$136$	− 6734.27i	− 4.24602i
$137$	2211.36i	1.37905i	0.724263	+	0.689524i	$0.242180\pi$
$137$	2211.36i	1.37905i	−0.724263	+	0.689524i	$0.757820\pi$
$138$	0	0
$139$	2718.66i	1.65895i	0.558545	+	0.829474i	$0.311360\pi$
$139$	2718.66i	1.65895i	−0.558545	+	0.829474i	$0.688640\pi$
$140$	0	0
$141$	0	0
$142$	1228.12	0.725783
$143$	280.420	0.163985
$144$	0	0
$145$	1421.05i	0.813874i
$146$	−1437.01	−0.814572
$147$	0	0
$148$	−3448.13	−1.91510
$149$	− 1835.81i	− 1.00936i	−0.863305	−	0.504682i	$-0.831610\pi$
$149$	− 1835.81i	− 1.00936i	0.863305	−	0.504682i	$-0.168390\pi$
$150$	0	0
$151$	2893.56	1.55944	0.779718	−	0.626131i	$-0.215362\pi$
$151$	2893.56	1.55944	0.779718	+	0.626131i	$0.215362\pi$
$152$	−1467.25	−0.782960
$153$	0	0
$154$	0	0
$155$	− 3660.27i	− 1.89677i
$156$	0	0
$157$	1708.65i	0.868567i	0.900776	+	0.434284i	$0.142998\pi$
$157$	1708.65i	0.868567i	−0.900776	+	0.434284i	$0.857002\pi$
$158$	− 4955.82i	− 2.49534i
$159$	0	0
$160$	10294.0i	5.08631i
$161$	0	0
$162$	0	0
$163$	−3038.08	−1.45988	−0.729941	−	0.683510i	$-0.760452\pi$
$163$	−3038.08	−1.45988	−0.729941	+	0.683510i	$0.760452\pi$
$164$	9134.10	4.34911
$165$	0	0
$166$	3460.03i	1.61777i
$167$	−702.416	−0.325477	−0.162738	−	0.986669i	$-0.552033\pi$
$167$	−702.416	−0.325477	−0.162738	+	0.986669i	$0.552033\pi$
$168$	0	0
$169$	2057.83	0.936654
$170$	− 9511.33i	− 4.29109i
$171$	0	0
$172$	2226.46	0.987014
$173$	1480.79	0.650765	0.325383	−	0.945582i	$-0.394507\pi$
$173$	1480.79	0.650765	0.325383	+	0.945582i	$0.394507\pi$
$174$	0	0
$175$	0	0
$176$	4947.90i	2.11910i
$177$	0	0
$178$	− 2396.74i	− 1.00923i
$179$	− 957.357i	− 0.399756i	−0.979821	−	0.199878i	$-0.935945\pi$
$179$	− 957.357i	− 0.399756i	0.979821	−	0.199878i	$-0.0640545\pi$
$180$	0	0
$181$	505.013i	0.207389i	0.994609	+	0.103694i	$0.0330664\pi$
$181$	505.013i	0.207389i	−0.994609	+	0.103694i	$0.966934\pi$
$182$	0	0
$183$	0	0
$184$	10661.9	4.27179
$185$	−3012.60	−1.19725
$186$	0	0
$187$	− 2291.93i	− 0.896270i
$188$	−231.688	−0.0898808
$189$	0	0
$190$	−2072.31	−0.791271
$191$	2332.40i	0.883593i	0.897115	+	0.441796i	$0.145659\pi$
$191$	2332.40i	0.883593i	−0.897115	+	0.441796i	$0.854341\pi$
$192$	0	0
$193$	−3925.85	−1.46419	−0.732095	−	0.681202i	$-0.761457\pi$
$193$	−3925.85	−1.46419	−0.732095	+	0.681202i	$0.761457\pi$
$194$	−8707.16	−3.22236
$195$	0	0
$196$	0	0
$197$	− 1429.05i	− 0.516831i	−0.966034	−	0.258416i	$-0.916800\pi$
$197$	− 1429.05i	− 0.516831i	0.966034	−	0.258416i	$-0.0832004\pi$
$198$	0	0
$199$	4937.90i	1.75899i	0.475910	+	0.879494i	$0.342119\pi$
$199$	4937.90i	1.75899i	−0.475910	+	0.879494i	$0.657881\pi$
$200$	14725.3i	5.20618i
$201$	0	0
$202$	− 403.893i	− 0.140682i
$203$	0	0
$204$	0	0
$205$	7980.38	2.71890
$206$	5189.84	1.75531
$207$	0	0
$208$	− 2455.62i	− 0.818588i
$209$	−499.362	−0.165271
$210$	0	0
$211$	1803.90	0.588557	0.294279	−	0.955720i	$-0.404921\pi$
$211$	1803.90	0.588557	0.294279	+	0.955720i	$0.404921\pi$
$212$	2326.90i	0.753831i
$213$	0	0
$214$	7171.45	2.29080
$215$	1945.24	0.617043
$216$	0	0
$217$	0	0
$218$	4791.60i	1.48866i
$219$	0	0
$220$	9136.95i	2.80006i
$221$	1137.47i	0.346220i
$222$	0	0
$223$	− 2540.18i	− 0.762795i	−0.924411	−	0.381397i	$-0.875443\pi$
$223$	− 2540.18i	− 0.762795i	0.924411	−	0.381397i	$-0.124557\pi$
$224$	0	0
$225$	0	0
$226$	−6676.10	−1.96499
$227$	3979.07	1.16344	0.581719	−	0.813390i	$-0.302380\pi$
$227$	3979.07	1.16344	0.581719	+	0.813390i	$0.302380\pi$
$228$	0	0
$229$	− 2214.46i	− 0.639019i	−0.947583	−	0.319510i	$-0.896482\pi$
$229$	− 2214.46i	− 0.639019i	0.947583	−	0.319510i	$-0.103518\pi$
$230$	15058.7	4.31713
$231$	0	0
$232$	5415.91	1.53264
$233$	4688.20i	1.31817i	0.752067	+	0.659087i	$0.229057\pi$
$233$	4688.20i	1.31817i	−0.752067	+	0.659087i	$0.770943\pi$
$234$	0	0
$235$	−202.424	−0.0561901
$236$	−17389.1	−4.79634
$237$	0	0
$238$	0	0
$239$	6575.90i	1.77975i	0.456207	+	0.889874i	$0.349208\pi$
$239$	6575.90i	1.77975i	−0.456207	+	0.889874i	$0.650792\pi$
$240$	0	0
$241$	− 3878.00i	− 1.03653i	−0.855220	−	0.518265i	$-0.826578\pi$
$241$	− 3878.00i	− 1.03653i	0.855220	−	0.518265i	$-0.173422\pi$
$242$	− 4123.12i	− 1.09522i
$243$	0	0
$244$	− 473.386i	− 0.124202i
$245$	0	0
$246$	0	0
$247$	247.831	0.0638424
$248$	−13950.0	−3.57188
$249$	0	0
$250$	8467.05i	2.14201i
$251$	−2789.25	−0.701418	−0.350709	−	0.936484i	$-0.614059\pi$
$251$	−2789.25	−0.701418	−0.350709	+	0.936484i	$0.614059\pi$
$252$	0	0
$253$	3628.66	0.901708
$254$	− 7204.33i	− 1.77968i
$255$	0	0
$256$	4303.84	1.05074
$257$	3004.64	0.729276	0.364638	−	0.931149i	$-0.381193\pi$
$257$	3004.64	0.729276	0.364638	+	0.931149i	$0.381193\pi$
$258$	0	0
$259$	0	0
$260$	− 4534.62i	− 1.08163i
$261$	0	0
$262$	− 12454.9i	− 2.93690i
$263$	− 3986.79i	− 0.934737i	−0.884063	−	0.467368i	$-0.845202\pi$
$263$	− 3986.79i	− 0.934737i	0.884063	−	0.467368i	$-0.154798\pi$
$264$	0	0
$265$	2032.99i	0.471266i
$266$	0	0
$267$	0	0
$268$	−8359.21	−1.90530
$269$	−5216.90	−1.18245	−0.591227	−	0.806505i	$-0.701356\pi$
$269$	−5216.90	−1.18245	−0.591227	+	0.806505i	$0.701356\pi$
$270$	0	0
$271$	268.201i	0.0601183i	0.999548	+	0.0300591i	$0.00956956\pi$
$271$	268.201i	0.0601183i	−0.999548	+	0.0300591i	$0.990430\pi$
$272$	−20070.3	−4.47404
$273$	0	0
$274$	11903.5	2.62450
$275$	5011.58i	1.09894i
$276$	0	0
$277$	3719.76	0.806854	0.403427	−	0.915012i	$-0.367819\pi$
$277$	3719.76	0.806854	0.403427	+	0.915012i	$0.367819\pi$
$278$	14634.2	3.15719
$279$	0	0
$280$	0	0
$281$	4360.92i	0.925802i	0.886410	+	0.462901i	$0.153191\pi$
$281$	4360.92i	0.925802i	−0.886410	+	0.462901i	$0.846809\pi$
$282$	0	0
$283$	− 2933.75i	− 0.616230i	−0.951349	−	0.308115i	$-0.900302\pi$
$283$	− 2933.75i	− 0.616230i	0.951349	−	0.308115i	$-0.0996982\pi$
$284$	− 4785.54i	− 0.999893i
$285$	0	0
$286$	− 1509.46i	− 0.312085i
$287$	0	0
$288$	0	0
$289$	4383.78	0.892283
$290$	7649.30	1.54890
$291$	0	0
$292$	5599.52i	1.12222i
$293$	−1081.19	−0.215577	−0.107789	−	0.994174i	$-0.534377\pi$
$293$	−1081.19	−0.215577	−0.107789	+	0.994174i	$0.534377\pi$
$294$	0	0
$295$	−15192.7	−2.99849
$296$	11481.6i	2.25458i
$297$	0	0
$298$	−9881.89	−1.92095
$299$	−1800.89	−0.348321
$300$	0	0
$301$	0	0
$302$	− 15575.6i	− 2.96780i
$303$	0	0
$304$	4372.88i	0.825005i
$305$	− 413.592i	− 0.0776467i
$306$	0	0
$307$	4327.83i	0.804567i	0.915515	+	0.402284i	$0.131783\pi$
$307$	4327.83i	0.804567i	−0.915515	+	0.402284i	$0.868217\pi$
$308$	0	0
$309$	0	0
$310$	−19702.7	−3.60980
$311$	1524.17	0.277902	0.138951	−	0.990299i	$-0.455627\pi$
$311$	1524.17	0.277902	0.138951	+	0.990299i	$0.455627\pi$
$312$	0	0
$313$	− 6374.33i	− 1.15111i	−0.817762	−	0.575556i	$-0.804786\pi$
$313$	− 6374.33i	− 1.15111i	0.817762	−	0.575556i	$-0.195214\pi$
$314$	9197.41	1.65299
$315$	0	0
$316$	−19311.1	−3.43777
$317$	− 934.754i	− 0.165618i	−0.996565	−	0.0828091i	$-0.973611\pi$
$317$	− 934.754i	− 0.165618i	0.996565	−	0.0828091i	$-0.0263892\pi$
$318$	0	0
$319$	1843.24	0.323516
$320$	24894.2	4.34884
$321$	0	0
$322$	0	0
$323$	− 2025.57i	− 0.348934i
$324$	0	0
$325$	− 2487.22i	− 0.424511i
$326$	16353.5i	2.77834i
$327$	0	0
$328$	− 30414.8i	− 5.12006i
$329$	0	0
$330$	0	0
$331$	−4976.08	−0.826314	−0.413157	−	0.910660i	$-0.635574\pi$
$331$	−4976.08	−0.826314	−0.413157	+	0.910660i	$0.635574\pi$
$332$	13482.5	2.22877
$333$	0	0
$334$	3781.00i	0.619423i
$335$	−7303.37	−1.19112
$336$	0	0
$337$	−5127.06	−0.828750	−0.414375	−	0.910106i	$-0.636000\pi$
$337$	−5127.06	−0.828750	−0.414375	+	0.910106i	$0.636000\pi$
$338$	− 11077.0i	− 1.78257i
$339$	0	0
$340$	−37062.4	−5.91173
$341$	−4747.73	−0.753970
$342$	0	0
$343$	0	0
$344$	− 7413.71i	− 1.16198i
$345$	0	0
$346$	− 7970.88i	− 1.23849i
$347$	5157.05i	0.797824i	0.916989	+	0.398912i	$0.130612\pi$
$347$	5157.05i	0.797824i	−0.916989	+	0.398912i	$0.869388\pi$
$348$	0	0
$349$	− 2961.44i	− 0.454218i	−0.973869	−	0.227109i	$-0.927073\pi$
$349$	− 2961.44i	− 0.454218i	0.973869	−	0.227109i	$-0.0729274\pi$
$350$	0	0
$351$	0	0
$352$	13352.3	2.02182
$353$	−11298.2	−1.70352	−0.851761	−	0.523931i	$-0.824465\pi$
$353$	−11298.2	−1.70352	−0.851761	+	0.523931i	$0.824465\pi$
$354$	0	0
$355$	− 4181.08i	− 0.625095i
$356$	−9339.25	−1.39039
$357$	0	0
$358$	−5153.32	−0.760786
$359$	2627.45i	0.386272i	0.981172	+	0.193136i	$0.0618658\pi$
$359$	2627.45i	0.386272i	−0.981172	+	0.193136i	$0.938134\pi$
$360$	0	0
$361$	6417.67	0.935657
$362$	2718.41	0.394687
$363$	0	0
$364$	0	0
$365$	4892.25i	0.701567i
$366$	0	0
$367$	− 2679.13i	− 0.381062i	−0.981681	−	0.190531i	$-0.938979\pi$
$367$	− 2679.13i	− 0.381062i	0.981681	−	0.190531i	$-0.0610209\pi$
$368$	− 31776.0i	− 4.50118i
$369$	0	0
$370$	16216.4i	2.27851i
$371$	0	0
$372$	0	0
$373$	−5187.73	−0.720135	−0.360068	−	0.932926i	$-0.617246\pi$
$373$	−5187.73	−0.720135	−0.360068	+	0.932926i	$0.617246\pi$
$374$	−12337.1	−1.70571
$375$	0	0
$376$	771.478i	0.105814i
$377$	−914.789	−0.124971
$378$	0	0
$379$	−2469.97	−0.334760	−0.167380	−	0.985892i	$-0.553531\pi$
$379$	−2469.97	−0.334760	−0.167380	+	0.985892i	$0.553531\pi$
$380$	8075.09i	1.09011i
$381$	0	0
$382$	12554.9	1.68159
$383$	11265.2	1.50294	0.751472	−	0.659765i	$-0.229344\pi$
$383$	11265.2	1.50294	0.751472	+	0.659765i	$0.229344\pi$
$384$	0	0
$385$	0	0
$386$	21132.3i	2.78654i
$387$	0	0
$388$	33928.8i	4.43936i
$389$	7688.62i	1.00213i	0.865410	+	0.501065i	$0.167058\pi$
$389$	7688.62i	1.00213i	−0.865410	+	0.501065i	$0.832942\pi$
$390$	0	0
$391$	14719.0i	1.90376i
$392$	0	0
$393$	0	0
$394$	−7692.38	−0.983596
$395$	−16871.9	−2.14916
$396$	0	0
$397$	− 809.336i	− 0.102316i	−0.998691	−	0.0511579i	$-0.983709\pi$
$397$	− 809.336i	− 0.102316i	0.998691	−	0.0511579i	$-0.0162912\pi$
$398$	26580.0	3.34758
$399$	0	0
$400$	43886.0	5.48575
$401$	− 4403.92i	− 0.548432i	−0.961668	−	0.274216i	$-0.911582\pi$
$401$	− 4403.92i	− 0.548432i	0.961668	−	0.274216i	$-0.0884183\pi$
$402$	0	0
$403$	2356.27	0.291251
$404$	−1573.83	−0.193814
$405$	0	0
$406$	0	0
$407$	3907.63i	0.475907i
$408$	0	0
$409$	58.2367i	0.00704064i	0.999994	+	0.00352032i	$0.00112055\pi$
$409$	58.2367i	0.00704064i	−0.999994	+	0.00352032i	$0.998879\pi$
$410$	− 42957.2i	− 5.17440i
$411$	0	0
$412$	− 20223.0i	− 2.41824i
$413$	0	0
$414$	0	0
$415$	11779.6	1.39334
$416$	−6626.67	−0.781007
$417$	0	0
$418$	2687.99i	0.314531i
$419$	6309.98	0.735711	0.367855	−	0.929883i	$-0.380092\pi$
$419$	6309.98	0.735711	0.367855	+	0.929883i	$0.380092\pi$
$420$	0	0
$421$	939.246	0.108732	0.0543658	−	0.998521i	$-0.482686\pi$
$421$	939.246	0.108732	0.0543658	+	0.998521i	$0.482686\pi$
$422$	− 9710.13i	− 1.12010i
$423$	0	0
$424$	7748.13	0.887459
$425$	−20328.5	−2.32019
$426$	0	0
$427$	0	0
$428$	− 27944.7i	− 3.15597i
$429$	0	0
$430$	− 10470.9i	− 1.17431i
$431$	9507.61i	1.06257i	0.847195	+	0.531283i	$0.178290\pi$
$431$	9507.61i	1.06257i	−0.847195	+	0.531283i	$0.821710\pi$
$432$	0	0
$433$	9603.12i	1.06581i	0.846175	+	0.532906i	$0.178900\pi$
$433$	9603.12i	1.06581i	−0.846175	+	0.532906i	$0.821100\pi$
$434$	0	0
$435$	0	0
$436$	18671.2	2.05089
$437$	3206.95	0.351051
$438$	0	0
$439$	873.005i	0.0949118i	0.998873	+	0.0474559i	$0.0151114\pi$
$439$	873.005i	0.0949118i	−0.998873	+	0.0474559i	$0.984889\pi$
$440$	30424.3	3.29642
$441$	0	0
$442$	6122.84	0.658900
$443$	804.059i	0.0862347i	0.999070	+	0.0431174i	$0.0137289\pi$
$443$	804.059i	0.0862347i	−0.999070	+	0.0431174i	$0.986271\pi$
$444$	0	0
$445$	−8159.61	−0.869220
$446$	−13673.4	−1.45169
$447$	0	0
$448$	0	0
$449$	12649.5i	1.32955i	0.747046	+	0.664773i	$0.231472\pi$
$449$	12649.5i	1.32955i	−0.747046	+	0.664773i	$0.768528\pi$
$450$	0	0
$451$	− 10351.3i	− 1.08077i
$452$	26014.5i	2.70712i
$453$	0	0
$454$	− 21418.8i	− 2.21417i
$455$	0	0
$456$	0	0
$457$	−7626.71	−0.780662	−0.390331	−	0.920675i	$-0.627639\pi$
$457$	−7626.71	−0.780662	−0.390331	+	0.920675i	$0.627639\pi$
$458$	−11920.1	−1.21613
$459$	0	0
$460$	− 58678.5i	− 5.94760i
$461$	15305.4	1.54630	0.773148	−	0.634226i	$-0.218681\pi$
$461$	15305.4	1.54630	0.773148	+	0.634226i	$0.218681\pi$
$462$	0	0
$463$	−9675.77	−0.971212	−0.485606	−	0.874178i	$-0.661401\pi$
$463$	−9675.77	−0.971212	−0.485606	+	0.874178i	$0.661401\pi$
$464$	− 16141.1i	− 1.61494i
$465$	0	0
$466$	25235.9	2.50865
$467$	−3487.65	−0.345587	−0.172794	−	0.984958i	$-0.555279\pi$
$467$	−3487.65	−0.345587	−0.172794	+	0.984958i	$0.555279\pi$
$468$	0	0
$469$	0	0
$470$	1089.62i	0.106937i
$471$	0	0
$472$	57902.5i	5.64657i
$473$	− 2523.17i	− 0.245276i
$474$	0	0
$475$	4429.15i	0.427839i
$476$	0	0
$477$	0	0
$478$	35397.1	3.38708
$479$	18174.4	1.73364	0.866818	−	0.498625i	$-0.166162\pi$
$479$	18174.4	1.73364	0.866818	+	0.498625i	$0.166162\pi$
$480$	0	0
$481$	− 1939.34i	− 0.183838i
$482$	−20874.7	−1.97265
$483$	0	0
$484$	−16066.4	−1.50886
$485$	29643.3i	2.77532i
$486$	0	0
$487$	−14810.7	−1.37810	−0.689052	−	0.724712i	$-0.741973\pi$
$487$	−14810.7	−1.37810	−0.689052	+	0.724712i	$0.741973\pi$
$488$	−1576.28	−0.146219
$489$	0	0
$490$	0	0
$491$	− 5187.59i	− 0.476808i	−0.971166	−	0.238404i	$-0.923376\pi$
$491$	− 5187.59i	− 0.476808i	0.971166	−	0.238404i	$-0.0766242\pi$
$492$	0	0
$493$	7476.76i	0.683035i
$494$	− 1334.04i	− 0.121500i
$495$	0	0
$496$	41575.5i	3.76370i
$497$	0	0
$498$	0	0
$499$	−4585.22	−0.411348	−0.205674	−	0.978621i	$-0.565939\pi$
$499$	−4585.22	−0.411348	−0.205674	+	0.978621i	$0.565939\pi$
$500$	32993.2	2.95100
$501$	0	0
$502$	15014.1i	1.33489i
$503$	−2600.54	−0.230521	−0.115261	−	0.993335i	$-0.536770\pi$
$503$	−2600.54	−0.230521	−0.115261	+	0.993335i	$0.536770\pi$
$504$	0	0
$505$	−1375.04	−0.121165
$506$	− 19532.6i	− 1.71606i
$507$	0	0
$508$	−28072.8	−2.45183
$509$	11519.6	1.00314	0.501569	−	0.865118i	$-0.332756\pi$
$509$	11519.6	1.00314	0.501569	+	0.865118i	$0.332756\pi$
$510$	0	0
$511$	0	0
$512$	− 619.438i	− 0.0534678i
$513$	0	0
$514$	− 16173.5i	− 1.38791i
$515$	− 17668.6i	− 1.51179i
$516$	0	0
$517$	262.563i	0.0223356i
$518$	0	0
$519$	0	0
$520$	−15099.4	−1.27337
$521$	6722.28	0.565275	0.282638	−	0.959227i	$-0.408791\pi$
$521$	6722.28	0.565275	0.282638	+	0.959227i	$0.408791\pi$
$522$	0	0
$523$	− 22231.2i	− 1.85870i	−0.369194	−	0.929352i	$-0.620366\pi$
$523$	− 22231.2i	− 1.85870i	0.369194	−	0.929352i	$-0.379634\pi$
$524$	−48532.5	−4.04609
$525$	0	0
$526$	−21460.3	−1.77892
$527$	− 19258.3i	− 1.59185i
$528$	0	0
$529$	−11136.7	−0.915317
$530$	10943.3	0.896879
$531$	0	0
$532$	0	0
$533$	5137.31i	0.417489i
$534$	0	0
$535$	− 24415.0i	− 1.97299i
$536$	27834.6i	2.24304i
$537$	0	0
$538$	28081.8i	2.25036i
$539$	0	0
$540$	0	0
$541$	−5156.71	−0.409805	−0.204902	−	0.978782i	$-0.565688\pi$
$541$	−5156.71	−0.409805	−0.204902	+	0.978782i	$0.565688\pi$
$542$	1443.69	0.114413
$543$	0	0
$544$	54161.1i	4.26864i
$545$	16312.9	1.28214
$546$	0	0
$547$	22545.5	1.76230	0.881149	−	0.472839i	$-0.156771\pi$
$547$	22545.5	1.76230	0.881149	+	0.472839i	$0.156771\pi$
$548$	− 46383.6i	− 3.61571i
$549$	0	0
$550$	26976.6	2.09143
$551$	1629.02	0.125951
$552$	0	0
$553$	0	0
$554$	− 20022.9i	− 1.53555i
$555$	0	0
$556$	− 57024.3i	− 4.34958i
$557$	− 9521.72i	− 0.724323i	−0.932115	−	0.362162i	$-0.882039\pi$
$557$	− 9521.72i	− 0.724323i	0.932115	−	0.362162i	$-0.117961\pi$
$558$	0	0
$559$	1252.23i	0.0947475i
$560$	0	0
$561$	0	0
$562$	23474.2	1.76192
$563$	5522.17	0.413378	0.206689	−	0.978407i	$-0.433731\pi$
$563$	5522.17	0.413378	0.206689	+	0.978407i	$0.433731\pi$
$564$	0	0
$565$	22728.6i	1.69239i
$566$	−15791.9	−1.17276
$567$	0	0
$568$	−15934.9	−1.17714
$569$	2930.18i	0.215887i	0.994157	+	0.107943i	$0.0344265\pi$
$569$	2930.18i	0.215887i	−0.994157	+	0.107943i	$0.965573\pi$
$570$	0	0
$571$	8718.34	0.638969	0.319484	−	0.947592i	$-0.396490\pi$
$571$	8718.34	0.638969	0.319484	+	0.947592i	$0.396490\pi$
$572$	−5881.84	−0.429951
$573$	0	0
$574$	0	0
$575$	− 32184.9i	− 2.33427i
$576$	0	0
$577$	16446.9i	1.18664i	0.804967	+	0.593320i	$0.202183\pi$
$577$	16446.9i	1.18664i	−0.804967	+	0.593320i	$0.797817\pi$
$578$	− 23597.3i	− 1.69813i
$579$	0	0
$580$	− 29806.7i	− 2.13389i
$581$	0	0
$582$	0	0
$583$	2636.99	0.187329
$584$	18645.3	1.32115
$585$	0	0
$586$	5819.92i	0.410270i
$587$	−2346.72	−0.165008	−0.0825040	−	0.996591i	$-0.526292\pi$
$587$	−2346.72	−0.165008	−0.0825040	+	0.996591i	$0.526292\pi$
$588$	0	0
$589$	−4195.96	−0.293534
$590$	81780.2i	5.70650i
$591$	0	0
$592$	34218.9	2.37565
$593$	15942.8	1.10403	0.552017	−	0.833833i	$-0.313858\pi$
$593$	15942.8	1.10403	0.552017	+	0.833833i	$0.313858\pi$
$594$	0	0
$595$	0	0
$596$	38506.3i	2.64644i
$597$	0	0
$598$	9693.91i	0.662899i
$599$	20596.0i	1.40489i	0.711738	+	0.702445i	$0.247908\pi$
$599$	20596.0i	1.40489i	−0.711738	+	0.702445i	$0.752092\pi$
$600$	0	0
$601$	662.921i	0.0449935i	0.999747	+	0.0224968i	$0.00716154\pi$
$601$	662.921i	0.0449935i	−0.999747	+	0.0224968i	$0.992838\pi$
$602$	0	0
$603$	0	0
$604$	−60692.8	−4.08867
$605$	−14037.0	−0.943284
$606$	0	0
$607$	13496.4i	0.902475i	0.892404	+	0.451238i	$0.149017\pi$
$607$	13496.4i	0.902475i	−0.892404	+	0.451238i	$0.850983\pi$
$608$	11800.5	0.787130
$609$	0	0
$610$	−2226.31	−0.147771
$611$	− 130.309i	− 0.00862803i
$612$	0	0
$613$	20074.0	1.32264	0.661322	−	0.750102i	$-0.269996\pi$
$613$	20074.0	1.32264	0.661322	+	0.750102i	$0.269996\pi$
$614$	23296.1	1.53119
$615$	0	0
$616$	0	0
$617$	21664.4i	1.41358i	0.707425	+	0.706788i	$0.249857\pi$
$617$	21664.4i	1.41358i	−0.707425	+	0.706788i	$0.750143\pi$
$618$	0	0
$619$	− 2173.02i	− 0.141100i	−0.997508	−	0.0705501i	$-0.977525\pi$
$619$	− 2173.02i	− 0.141100i	0.997508	−	0.0705501i	$-0.0224755\pi$
$620$	76774.6i	4.97313i
$621$	0	0
$622$	− 8204.38i	− 0.528884i
$623$	0	0
$624$	0	0
$625$	2471.61	0.158183
$626$	−34312.1	−2.19071
$627$	0	0
$628$	− 35839.1i	− 2.27729i
$629$	−15850.6	−1.00478
$630$	0	0
$631$	−7823.69	−0.493591	−0.246796	−	0.969068i	$-0.579378\pi$
$631$	−7823.69	−0.493591	−0.246796	+	0.969068i	$0.579378\pi$
$632$	64302.3i	4.04717i
$633$	0	0
$634$	−5031.64	−0.315193
$635$	−24526.9	−1.53279
$636$	0	0
$637$	0	0
$638$	− 9921.89i	− 0.615692i
$639$	0	0
$640$	− 51650.1i	− 3.19008i
$641$	310.308i	0.0191208i	0.999954	+	0.00956040i	$0.00304322\pi$
$641$	310.308i	0.0191208i	−0.999954	+	0.00956040i	$0.996957\pi$
$642$	0	0
$643$	− 28064.5i	− 1.72124i	−0.509250	−	0.860618i	$-0.670077\pi$
$643$	− 28064.5i	− 1.72124i	0.509250	−	0.860618i	$-0.329923\pi$
$644$	0	0
$645$	0	0
$646$	−10903.3	−0.664066
$647$	−8439.33	−0.512804	−0.256402	−	0.966570i	$-0.582537\pi$
$647$	−8439.33	−0.512804	−0.256402	+	0.966570i	$0.582537\pi$
$648$	0	0
$649$	19706.4i	1.19190i
$650$	−13388.3	−0.807898
$651$	0	0
$652$	63724.1	3.82765
$653$	− 26519.3i	− 1.58925i	−0.607101	−	0.794624i	$-0.707668\pi$
$653$	− 26519.3i	− 1.58925i	0.607101	−	0.794624i	$-0.292332\pi$
$654$	0	0
$655$	−42402.3	−2.52946
$656$	−90645.8	−5.39501
$657$	0	0
$658$	0	0
$659$	− 29625.1i	− 1.75118i	−0.483054	−	0.875591i	$-0.660473\pi$
$659$	− 29625.1i	− 1.75118i	0.483054	−	0.875591i	$-0.339527\pi$
$660$	0	0
$661$	21787.2i	1.28204i	0.767526	+	0.641018i	$0.221487\pi$
$661$	21787.2i	1.28204i	−0.767526	+	0.641018i	$0.778513\pi$
$662$	26785.5i	1.57258i
$663$	0	0
$664$	− 44894.3i	− 2.62385i
$665$	0	0
$666$	0	0
$667$	−11837.5	−0.687180
$668$	14733.3	0.853364
$669$	0	0
$670$	39313.0i	2.26685i
$671$	−536.470	−0.0308647
$672$	0	0
$673$	14551.5	0.833462	0.416731	−	0.909030i	$-0.363176\pi$
$673$	14551.5	0.833462	0.416731	+	0.909030i	$0.363176\pi$
$674$	27598.2i	1.57722i
$675$	0	0
$676$	−43163.2	−2.45580
$677$	−12316.1	−0.699181	−0.349591	−	0.936903i	$-0.613679\pi$
$677$	−12316.1	−0.699181	−0.349591	+	0.936903i	$0.613679\pi$
$678$	0	0
$679$	0	0
$680$	123411.i	6.95968i
$681$	0	0
$682$	25556.3i	1.43490i
$683$	− 10956.3i	− 0.613810i	−0.951740	−	0.306905i	$-0.900707\pi$
$683$	− 10956.3i	− 0.613810i	0.951740	−	0.306905i	$-0.0992934\pi$
$684$	0	0
$685$	− 40524.9i	− 2.26041i
$686$	0	0
$687$	0	0
$688$	−22095.2	−1.22438
$689$	−1308.72	−0.0723633
$690$	0	0
$691$	− 23371.1i	− 1.28666i	−0.765591	−	0.643328i	$-0.777553\pi$
$691$	− 23371.1i	− 1.28666i	0.765591	−	0.643328i	$-0.222447\pi$
$692$	−31059.7	−1.70623
$693$	0	0
$694$	27759.6	1.51836
$695$	− 49821.5i	− 2.71919i
$696$	0	0
$697$	41988.2	2.28181
$698$	−15941.0	−0.864434
$699$	0	0
$700$	0	0
$701$	13644.5i	0.735160i	0.929992	+	0.367580i	$0.119814\pi$
$701$	13644.5i	0.735160i	−0.929992	+	0.367580i	$0.880186\pi$
$702$	0	0
$703$	3453.50i	0.185279i
$704$	− 32290.2i	− 1.72867i
$705$	0	0
$706$	60816.6i	3.24202i
$707$	0	0
$708$	0	0
$709$	−23929.2	−1.26753	−0.633767	−	0.773524i	$-0.718492\pi$
$709$	−23929.2	−1.26753	−0.633767	+	0.773524i	$0.718492\pi$
$710$	−22506.1	−1.18963
$711$	0	0
$712$	31097.9i	1.63686i
$713$	30490.4	1.60151
$714$	0	0
$715$	−5138.91	−0.268789
$716$	20080.7i	1.04812i
$717$	0	0
$718$	14143.2	0.735124
$719$	29563.1	1.53341	0.766703	−	0.642002i	$-0.221896\pi$
$719$	29563.1	1.53341	0.766703	+	0.642002i	$0.221896\pi$
$720$	0	0
$721$	0	0
$722$	− 34545.4i	− 1.78067i
$723$	0	0
$724$	− 10592.7i	− 0.543750i
$725$	− 16348.8i	− 0.837490i
$726$	0	0
$727$	3335.51i	0.170161i	0.996374	+	0.0850805i	$0.0271148\pi$
$727$	3335.51i	0.170161i	−0.996374	+	0.0850805i	$0.972885\pi$
$728$	0	0
$729$	0	0
$730$	26334.3	1.33517
$731$	10234.8	0.517847
$732$	0	0
$733$	12219.8i	0.615754i	0.951426	+	0.307877i	$0.0996185\pi$
$733$	12219.8i	0.615754i	−0.951426	+	0.307877i	$0.900382\pi$
$734$	−14421.4	−0.725209
$735$	0	0
$736$	−85749.8	−4.29454
$737$	9473.18i	0.473472i
$738$	0	0
$739$	17759.4	0.884017	0.442009	−	0.897011i	$-0.354266\pi$
$739$	17759.4	0.884017	0.442009	+	0.897011i	$0.354266\pi$
$740$	63189.6	3.13905
$741$	0	0
$742$	0	0
$743$	− 5371.15i	− 0.265207i	−0.991169	−	0.132603i	$-0.957666\pi$
$743$	− 5371.15i	− 0.265207i	0.991169	−	0.132603i	$-0.0423336\pi$
$744$	0	0
$745$	33642.6i	1.65445i
$746$	27924.8i	1.37051i
$747$	0	0
$748$	48073.5i	2.34992i
$749$	0	0
$750$	0	0
$751$	23272.8	1.13081	0.565403	−	0.824815i	$-0.308721\pi$
$751$	23272.8	1.13081	0.565403	+	0.824815i	$0.308721\pi$
$752$	2299.25	0.111496
$753$	0	0
$754$	4924.18i	0.237835i
$755$	−53026.8	−2.55608
$756$	0	0
$757$	16172.1	0.776465	0.388233	−	0.921561i	$-0.373086\pi$
$757$	16172.1	0.776465	0.388233	+	0.921561i	$0.373086\pi$
$758$	13295.5i	0.637091i
$759$	0	0
$760$	26888.5	1.28335
$761$	−18920.6	−0.901277	−0.450639	−	0.892706i	$-0.648804\pi$
$761$	−18920.6	−0.901277	−0.450639	+	0.892706i	$0.648804\pi$
$762$	0	0
$763$	0	0
$764$	− 48922.3i	− 2.31668i
$765$	0	0
$766$	− 60639.2i	− 2.86029i
$767$	− 9780.19i	− 0.460420i
$768$	0	0
$769$	− 33747.4i	− 1.58253i	−0.611475	−	0.791264i	$-0.709423\pi$
$769$	− 33747.4i	− 1.58253i	0.611475	−	0.791264i	$-0.290577\pi$
$770$	0	0
$771$	0	0
$772$	82345.2	3.83895
$773$	21612.7	1.00563	0.502817	−	0.864393i	$-0.332297\pi$
$773$	21612.7	1.00563	0.502817	+	0.864393i	$0.332297\pi$
$774$	0	0
$775$	42110.5i	1.95181i
$776$	112976.	5.22631
$777$	0	0
$778$	41386.7	1.90718
$779$	− 9148.34i	− 0.420761i
$780$	0	0
$781$	−5423.27	−0.248476
$782$	79230.3	3.62311
$783$	0	0
$784$	0	0
$785$	− 31312.3i	− 1.42367i
$786$	0	0
$787$	− 24549.2i	− 1.11193i	−0.831207	−	0.555963i	$-0.812350\pi$
$787$	− 24549.2i	− 1.11193i	0.831207	−	0.555963i	$-0.187650\pi$
$788$	29974.6i	1.35508i
$789$	0	0
$790$	90819.1i	4.09013i
$791$	0	0
$792$	0	0
$793$	266.247	0.0119227
$794$	−4356.54	−0.194720
$795$	0	0
$796$	− 103573.i	− 4.61187i
$797$	12323.1	0.547687	0.273844	−	0.961774i	$-0.411705\pi$
$797$	12323.1	0.547687	0.273844	+	0.961774i	$0.411705\pi$
$798$	0	0
$799$	−1065.04	−0.0471569
$800$	− 118430.i	− 5.23391i
$801$	0	0
$802$	−23705.7	−1.04374
$803$	6345.72	0.278874
$804$	0	0
$805$	0	0
$806$	− 12683.5i	− 0.554287i
$807$	0	0
$808$	5240.56i	0.228171i
$809$	− 3668.61i	− 0.159433i	−0.996818	−	0.0797167i	$-0.974598\pi$
$809$	− 3668.61i	− 0.159433i	0.996818	−	0.0797167i	$-0.0254016\pi$
$810$	0	0
$811$	− 24967.7i	− 1.08105i	−0.841327	−	0.540527i	$-0.818225\pi$
$811$	− 24967.7i	− 1.08105i	0.841327	−	0.540527i	$-0.181775\pi$
$812$	0	0
$813$	0	0
$814$	21034.2	0.905711
$815$	55675.1	2.39290
$816$	0	0
$817$	− 2229.93i	− 0.0954902i
$818$	313.480	0.0133992
$819$	0	0
$820$	−167389.	−7.12865
$821$	− 6264.83i	− 0.266315i	−0.991095	−	0.133157i	$-0.957488\pi$
$821$	− 6264.83i	− 0.266315i	0.991095	−	0.133157i	$-0.0425115\pi$
$822$	0	0
$823$	3333.71	0.141198	0.0705990	−	0.997505i	$-0.477509\pi$
$823$	3333.71	0.141198	0.0705990	+	0.997505i	$0.477509\pi$
$824$	−67338.8	−2.84691
$825$	0	0
$826$	0	0
$827$	− 27785.4i	− 1.16831i	−0.811643	−	0.584154i	$-0.801426\pi$
$827$	− 27785.4i	− 1.16831i	0.811643	−	0.584154i	$-0.198574\pi$
$828$	0	0
$829$	8853.71i	0.370931i	0.982651	+	0.185466i	$0.0593794\pi$
$829$	8853.71i	0.370931i	−0.982651	+	0.185466i	$0.940621\pi$
$830$	− 63407.7i	− 2.65170i
$831$	0	0
$832$	16025.4i	0.667767i
$833$	0	0
$834$	0	0
$835$	12872.3	0.533491
$836$	10474.2	0.433322
$837$	0	0
$838$	− 33965.7i	− 1.40015i
$839$	92.8332	0.00381997	0.00190999	−	0.999998i	$-0.499392\pi$
$839$	92.8332	0.00381997	0.00190999	+	0.999998i	$0.499392\pi$
$840$	0	0
$841$	18376.0	0.753453
$842$	− 5055.82i	− 0.206930i
$843$	0	0
$844$	−37837.0	−1.54313
$845$	−37711.3	−1.53528
$846$	0	0
$847$	0	0
$848$	− 23091.9i	− 0.935116i
$849$	0	0
$850$	109426.i	4.41561i
$851$	− 25095.2i	− 1.01087i
$852$	0	0
$853$	8509.71i	0.341579i	0.985308	+	0.170790i	$0.0546318\pi$
$853$	8509.71i	0.341579i	−0.985308	+	0.170790i	$0.945368\pi$
$854$	0	0
$855$	0	0
$856$	−93050.5	−3.71542
$857$	39034.5	1.55588	0.777942	−	0.628336i	$-0.216264\pi$
$857$	39034.5	1.55588	0.777942	+	0.628336i	$0.216264\pi$
$858$	0	0
$859$	− 35149.3i	− 1.39613i	−0.716033	−	0.698067i	$-0.754044\pi$
$859$	− 35149.3i	− 1.39613i	0.716033	−	0.698067i	$-0.245956\pi$
$860$	−40801.7	−1.61782
$861$	0	0
$862$	51178.1	2.02220
$863$	16037.9i	0.632604i	0.948659	+	0.316302i	$0.102441\pi$
$863$	16037.9i	0.632604i	−0.948659	+	0.316302i	$0.897559\pi$
$864$	0	0
$865$	−27136.6	−1.06667
$866$	51692.2	2.02837
$867$	0	0
$868$	0	0
$869$	21884.5i	0.854295i
$870$	0	0
$871$	− 4701.48i	− 0.182898i
$872$	− 62171.6i	− 2.41445i
$873$	0	0
$874$	− 17262.6i	− 0.668095i
$875$	0	0
$876$	0	0
$877$	22446.0	0.864251	0.432126	−	0.901813i	$-0.357764\pi$
$877$	22446.0	0.864251	0.432126	+	0.901813i	$0.357764\pi$
$878$	4699.26	0.180629
$879$	0	0
$880$	− 90674.1i	− 3.47344i
$881$	4544.20	0.173778	0.0868888	−	0.996218i	$-0.472308\pi$
$881$	4544.20	0.173778	0.0868888	+	0.996218i	$0.472308\pi$
$882$	0	0
$883$	18749.1	0.714561	0.357280	−	0.933997i	$-0.383704\pi$
$883$	18749.1	0.714561	0.357280	+	0.933997i	$0.383704\pi$
$884$	− 23858.6i	− 0.907751i
$885$	0	0
$886$	4328.13	0.164116
$887$	25107.1	0.950410	0.475205	−	0.879875i	$-0.342374\pi$
$887$	25107.1	0.950410	0.475205	+	0.879875i	$0.342374\pi$
$888$	0	0
$889$	0	0
$890$	43922.0i	1.65423i
$891$	0	0
$892$	53280.6i	1.99996i
$893$	232.049i	0.00869567i
$894$	0	0
$895$	17544.3i	0.655242i
$896$	0	0
$897$	0	0
$898$	68090.3	2.53029
$899$	15488.1	0.574590
$900$	0	0
$901$	10696.4i	0.395505i
$902$	−55719.7	−2.05683
$903$	0	0
$904$	86623.3	3.18700
$905$	− 9254.75i	− 0.339932i
$906$	0	0
$907$	−10197.1	−0.373307	−0.186654	−	0.982426i	$-0.559764\pi$
$907$	−10197.1	−0.373307	−0.186654	+	0.982426i	$0.559764\pi$
$908$	−83461.6	−3.05041
$909$	0	0
$910$	0	0
$911$	27309.8i	0.993208i	0.867977	+	0.496604i	$0.165420\pi$
$911$	27309.8i	0.993208i	−0.867977	+	0.496604i	$0.834580\pi$
$912$	0	0
$913$	− 15279.2i	− 0.553855i
$914$	41053.5i	1.48570i
$915$	0	0
$916$	46448.5i	1.67544i
$917$	0	0
$918$	0	0
$919$	−234.486	−0.00841674	−0.00420837	−	0.999991i	$-0.501340\pi$
$919$	−234.486	−0.00841674	−0.00420837	+	0.999991i	$0.501340\pi$
$920$	−195388.	−7.00191
$921$	0	0
$922$	− 82386.6i	− 2.94280i
$923$	2691.54	0.0959838
$924$	0	0
$925$	34659.2	1.23199
$926$	52083.2i	1.84834i
$927$	0	0
$928$	−43558.0	−1.54080
$929$	−25481.4	−0.899913	−0.449956	−	0.893051i	$-0.648560\pi$
$929$	−25481.4	−0.899913	−0.449956	+	0.893051i	$0.648560\pi$
$930$	0	0
$931$	0	0
$932$	− 98335.6i	− 3.45611i
$933$	0	0
$934$	18773.5i	0.657697i
$935$	42001.3i	1.46908i
$936$	0	0
$937$	− 36561.0i	− 1.27470i	−0.770574	−	0.637350i	$-0.780031\pi$
$937$	− 36561.0i	− 1.27470i	0.770574	−	0.637350i	$-0.219969\pi$
$938$	0	0
$939$	0	0
$940$	4245.86	0.147324
$941$	−35042.2	−1.21397	−0.606984	−	0.794714i	$-0.707621\pi$
$941$	−35042.2	−1.21397	−0.606984	+	0.794714i	$0.707621\pi$
$942$	0	0
$943$	66477.3i	2.29565i
$944$	172568.	5.94979
$945$	0	0
$946$	−13581.8	−0.466790
$947$	− 35644.3i	− 1.22311i	−0.791203	−	0.611554i	$-0.790545\pi$
$947$	− 35644.3i	− 1.22311i	0.791203	−	0.611554i	$-0.209455\pi$
$948$	0	0
$949$	−3149.35	−0.107726
$950$	23841.5	0.814231
$951$	0	0
$952$	0	0
$953$	35376.7i	1.20248i	0.799068	+	0.601240i	$0.205327\pi$
$953$	35376.7i	1.20248i	−0.799068	+	0.601240i	$0.794673\pi$
$954$	0	0
$955$	− 42742.9i	− 1.44830i
$956$	− 137930.i	− 4.66630i
$957$	0	0
$958$	− 97830.3i	− 3.29933i
$959$	0	0
$960$	0	0
$961$	−10102.5	−0.339111
$962$	−10439.2	−0.349867
$963$	0	0
$964$	81341.5i	2.71767i
$965$	71944.2	2.39996
$966$	0	0
$967$	−27697.2	−0.921078	−0.460539	−	0.887640i	$-0.652344\pi$
$967$	−27697.2	−0.921078	−0.460539	+	0.887640i	$0.652344\pi$
$968$	53498.0i	1.77633i
$969$	0	0
$970$	159565.	5.28179
$971$	31760.0	1.04967	0.524834	−	0.851204i	$-0.324127\pi$
$971$	31760.0	1.04967	0.524834	+	0.851204i	$0.324127\pi$
$972$	0	0
$973$	0	0
$974$	79723.8i	2.62270i
$975$	0	0
$976$	4697.83i	0.154071i
$977$	1143.07i	0.0374308i	0.999825	+	0.0187154i	$0.00595765\pi$
$977$	1143.07i	0.0374308i	−0.999825	+	0.0187154i	$0.994042\pi$
$978$	0	0
$979$	10583.8i	0.345516i
$980$	0	0
$981$	0	0
$982$	−27924.0	−0.907425
$983$	13390.5	0.434475	0.217238	−	0.976119i	$-0.430295\pi$
$983$	13390.5	0.434475	0.217238	+	0.976119i	$0.430295\pi$
$984$	0	0
$985$	26188.5i	0.847142i
$986$	40246.3	1.29990
$987$	0	0
$988$	−5198.28	−0.167388
$989$	16204.0i	0.520989i
$990$	0	0
$991$	16015.7	0.513375	0.256688	−	0.966494i	$-0.417369\pi$
$991$	16015.7	0.513375	0.256688	+	0.966494i	$0.417369\pi$
$992$	112195.	3.59091
$993$	0	0
$994$	0	0
$995$	− 90490.8i	− 2.88317i
$996$	0	0
$997$	29547.1i	0.938582i	0.883044	+	0.469291i	$0.155490\pi$
$997$	29547.1i	0.938582i	−0.883044	+	0.469291i	$0.844510\pi$
$998$	24681.6i	0.782847i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.4.c.b.440.23	yes	24
3.2	odd	2	inner	441.4.c.b.440.2	yes	24
7.2	even	3		441.4.p.d.80.2		48
7.3	odd	6		441.4.p.d.215.23		48
7.4	even	3		441.4.p.d.215.24		48
7.5	odd	6		441.4.p.d.80.1		48
7.6	odd	2	inner	441.4.c.b.440.1	✓	24
21.2	odd	6		441.4.p.d.80.23		48
21.5	even	6		441.4.p.d.80.24		48
21.11	odd	6		441.4.p.d.215.1		48
21.17	even	6		441.4.p.d.215.2		48
21.20	even	2	inner	441.4.c.b.440.24	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.4.c.b.440.1	✓	24	7.6	odd	2	inner
441.4.c.b.440.2	yes	24	3.2	odd	2	inner
441.4.c.b.440.23	yes	24	1.1	even	1	trivial
441.4.c.b.440.24	yes	24	21.20	even	2	inner
441.4.p.d.80.1		48	7.5	odd	6
441.4.p.d.80.2		48	7.2	even	3
441.4.p.d.80.23		48	21.2	odd	6
441.4.p.d.80.24		48	21.5	even	6
441.4.p.d.215.1		48	21.11	odd	6
441.4.p.d.215.2		48	21.17	even	6
441.4.p.d.215.23		48	7.3	odd	6
441.4.p.d.215.24		48	7.4	even	3

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 441 = 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	441.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-5.38285i q^{2} -20.9751 q^{4} -18.3258 q^{5} +69.8432i q^{8} +O(q^{10})\)	\(q-5.38285i q^{2} -20.9751 q^{4} -18.3258 q^{5} +69.8432i q^{8} +98.6449i q^{10} +23.7703i q^{11} -11.7971i q^{13} +208.155 q^{16} -96.4198 q^{17} +21.0078i q^{19} +384.385 q^{20} +127.952 q^{22} -152.655i q^{23} +210.834 q^{25} -63.5019 q^{26} -77.5438i q^{29} +199.733i q^{31} -561.721i q^{32} +519.014i q^{34} +164.391 q^{37} +113.082 q^{38} -1279.93i q^{40} -435.473 q^{41} -106.148 q^{43} -498.585i q^{44} -821.722 q^{46} +11.0459 q^{47} -1134.89i q^{50} +247.445i q^{52} -110.936i q^{53} -435.609i q^{55} -417.407 q^{58} +829.036 q^{59} +22.5689i q^{61} +1075.14 q^{62} -1358.43 q^{64} +216.190i q^{65} +398.530 q^{67} +2022.42 q^{68} +228.153i q^{71} -266.960i q^{73} -884.895i q^{74} -440.641i q^{76} +920.667 q^{79} -3814.60 q^{80} +2344.09i q^{82} -642.787 q^{83} +1766.97 q^{85} +571.379i q^{86} -1660.19 q^{88} +445.254 q^{89} +3201.97i q^{92} -59.4582i q^{94} -384.984i q^{95} -1617.57i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 96 q^{4}+O(q^{10}) \) 24 * q - 96 * q^4	\( 24 q - 96 q^{4} + 144 q^{16} + 624 q^{22} + 312 q^{25} - 864 q^{37} + 1248 q^{43} - 3888 q^{46} - 7440 q^{58} - 3360 q^{64} - 2688 q^{67} + 480 q^{79} + 13248 q^{85} - 7248 q^{88}+O(q^{100}) \) 24 * q - 96 * q^4 + 144 * q^16 + 624 * q^22 + 312 * q^25 - 864 * q^37 + 1248 * q^43 - 3888 * q^46 - 7440 * q^58 - 3360 * q^64 - 2688 * q^67 + 480 * q^79 + 13248 * q^85 - 7248 * q^88

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