LMFDB - Embedded newform 441.4.c.a.440.9

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:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 48 x^{14} + 1647 x^{12} - 27620 x^{10} + 336765 x^{8} - 1200006 x^{6} + 3242464 x^{4} + \cdots + 810000 $ x^16 - 48x^14 + 1647x^12 - 27620x^10 + 336765x^8 - 1200006x^6 + 3242464x^4 - 1762200*x^2 + 810000
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{12}\cdot 3^{8}\cdot 7^{4} $
Twist minimal:	no (minimal twist has level 63)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			440.9
Root			$0.648633 - 0.374489i$ of defining polynomial
Character	$\chi$	$=$	441.440
Dual form			441.4.c.a.440.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.748977i q^{2} +7.43903 q^{4} -10.8554 q^{5} +11.5635i q^{8} +O(q^{10})$	$q+0.748977i q^{2} +7.43903 q^{4} -10.8554 q^{5} +11.5635i q^{8} -8.13042i q^{10} -51.8611i q^{11} +32.1880i q^{13} +50.8515 q^{16} -81.4649 q^{17} -0.0485737i q^{19} -80.7534 q^{20} +38.8428 q^{22} -89.2956i q^{23} -7.16119 q^{25} -24.1081 q^{26} -175.246i q^{29} -215.049i q^{31} +130.594i q^{32} -61.0153i q^{34} +64.5458 q^{37} +0.0363806 q^{38} -125.526i q^{40} +411.485 q^{41} -234.771 q^{43} -385.797i q^{44} +66.8803 q^{46} -632.152 q^{47} -5.36357i q^{50} +239.448i q^{52} -265.638i q^{53} +562.971i q^{55} +131.255 q^{58} -351.059 q^{59} -778.069i q^{61} +161.067 q^{62} +309.000 q^{64} -349.412i q^{65} -196.009 q^{67} -606.020 q^{68} +142.632i q^{71} -780.879i q^{73} +48.3433i q^{74} -0.361341i q^{76} +1289.05 q^{79} -552.011 q^{80} +308.193i q^{82} -235.123 q^{83} +884.330 q^{85} -175.838i q^{86} +599.695 q^{88} +670.780 q^{89} -664.273i q^{92} -473.468i q^{94} +0.527285i q^{95} -655.891i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 64 q^{4}+O(q^{10}) $ 16 * q - 64 * q^4	$ 16 q - 64 q^{4} + 376 q^{16} + 528 q^{22} + 40 q^{25} + 2392 q^{37} + 328 q^{43} + 2784 q^{46} + 6744 q^{58} + 5432 q^{64} - 616 q^{67} + 4352 q^{79} - 4608 q^{85} - 1416 q^{88}+O(q^{100}) $ 16 * q - 64 * q^4 + 376 * q^16 + 528 * q^22 + 40 * q^25 + 2392 * q^37 + 328 * q^43 + 2784 * q^46 + 6744 * q^58 + 5432 * q^64 - 616 * q^67 + 4352 * q^79 - 4608 * q^85 - 1416 * 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 $

$2$	0.748977i	0.264803i	0.991196	+	0.132402i	$0.0422689\pi$
$2$	0.748977i	0.264803i	−0.991196	+	0.132402i	$0.957731\pi$
$3$	0	0
$3$	0	0
$4$	7.43903	0.929879
$5$	−10.8554	−0.970933	−0.485466	−	0.874255i	$-0.661350\pi$
$5$	−10.8554	−0.970933	−0.485466	+	0.874255i	$0.661350\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	11.5635i	0.511039i
$9$	0	0
$10$	− 8.13042i	− 0.257106i
$11$	− 51.8611i	− 1.42152i	−0.703435	−	0.710760i	$-0.748351\pi$
$11$	− 51.8611i	− 1.42152i	0.703435	−	0.710760i	$-0.251649\pi$
$12$	0	0
$13$	32.1880i	0.686719i	0.939204	+	0.343360i	$0.111565\pi$
$13$	32.1880i	0.686719i	−0.939204	+	0.343360i	$0.888435\pi$
$14$	0	0
$15$	0	0
$16$	50.8515	0.794554
$17$	−81.4649	−1.16224	−0.581121	−	0.813817i	$-0.697386\pi$
$17$	−81.4649	−1.16224	−0.581121	+	0.813817i	$0.697386\pi$
$18$	0	0
$19$	− 0.0485737i	0 0.000586504i	−1.00000		0.000293252i	$-0.999907\pi$
$19$	− 0.0485737i	0 0.000586504i	1.00000		0.000293252i	$-9.33450e-5\pi$
$20$	−80.7534	−0.902850
$21$	0	0
$22$	38.8428	0.376423
$23$	− 89.2956i	− 0.809540i	−0.914419	−	0.404770i	$-0.867352\pi$
$23$	− 89.2956i	− 0.809540i	0.914419	−	0.404770i	$-0.132648\pi$
$24$	0	0
$25$	−7.16119	−0.0572896
$26$	−24.1081	−0.181846
$27$	0	0
$28$	0	0
$29$	− 175.246i	− 1.12215i	−0.827766	−	0.561074i	$-0.810388\pi$
$29$	− 175.246i	− 1.12215i	0.827766	−	0.561074i	$-0.189612\pi$
$30$	0	0
$31$	− 215.049i	− 1.24593i	−0.782249	−	0.622966i	$-0.785927\pi$
$31$	− 215.049i	− 1.24593i	0.782249	−	0.622966i	$-0.214073\pi$
$32$	130.594i	0.721439i
$33$	0	0
$34$	− 61.0153i	− 0.307766i
$35$	0	0
$36$	0	0
$37$	64.5458	0.286791	0.143395	−	0.989665i	$-0.454198\pi$
$37$	64.5458	0.286791	0.143395	+	0.989665i	$0.454198\pi$
$38$	0.0363806	0.000155308 0
$39$	0	0
$40$	− 125.526i	− 0.496184i
$41$	411.485	1.56740	0.783698	−	0.621142i	$-0.213331\pi$
$41$	411.485	1.56740	0.783698	+	0.621142i	$0.213331\pi$
$42$	0	0
$43$	−234.771	−0.832611	−0.416305	−	0.909225i	$-0.636675\pi$
$43$	−234.771	−0.832611	−0.416305	+	0.909225i	$0.636675\pi$
$44$	− 385.797i	− 1.32184i
$45$	0	0
$46$	66.8803	0.214369
$47$	−632.152	−1.96189	−0.980946	−	0.194283i	$-0.937762\pi$
$47$	−632.152	−1.96189	−0.980946	+	0.194283i	$0.937762\pi$
$48$	0	0
$49$	0	0
$50$	− 5.36357i	− 0.0151705i
$51$	0	0
$52$	239.448i	0.638566i
$53$	− 265.638i	− 0.688456i	−0.938886	−	0.344228i	$-0.888141\pi$
$53$	− 265.638i	− 0.688456i	0.938886	−	0.344228i	$-0.111859\pi$
$54$	0	0
$55$	562.971i	1.38020i
$56$	0	0
$57$	0	0
$58$	131.255	0.297149
$59$	−351.059	−0.774645	−0.387322	−	0.921944i	$-0.626600\pi$
$59$	−351.059	−0.774645	−0.387322	+	0.921944i	$0.626600\pi$
$60$	0	0
$61$	− 778.069i	− 1.63314i	−0.577247	−	0.816569i	$-0.695873\pi$
$61$	− 778.069i	− 1.63314i	0.577247	−	0.816569i	$-0.304127\pi$
$62$	161.067	0.329927
$63$	0	0
$64$	309.000	0.603515
$65$	− 349.412i	− 0.666758i
$66$	0	0
$67$	−196.009	−0.357407	−0.178703	−	0.983903i	$-0.557190\pi$
$67$	−196.009	−0.357407	−0.178703	+	0.983903i	$0.557190\pi$
$68$	−606.020	−1.08075
$69$	0	0
$70$	0	0
$71$	142.632i	0.238412i	0.992870	+	0.119206i	$0.0380349\pi$
$71$	142.632i	0.238412i	−0.992870	+	0.119206i	$0.961965\pi$
$72$	0	0
$73$	− 780.879i	− 1.25199i	−0.779829	−	0.625993i	$-0.784694\pi$
$73$	− 780.879i	− 1.25199i	0.779829	−	0.625993i	$-0.215306\pi$
$74$	48.3433i	0.0759432i
$75$	0	0
$76$	− 0.361341i	0 0.000545378i
$77$	0	0
$78$	0	0
$79$	1289.05	1.83582	0.917908	−	0.396794i	$-0.129877\pi$
$79$	1289.05	1.83582	0.917908	+	0.396794i	$0.129877\pi$
$80$	−552.011	−0.771459
$81$	0	0
$82$	308.193i	0.415052i
$83$	−235.123	−0.310940	−0.155470	−	0.987841i	$-0.549689\pi$
$83$	−235.123	−0.310940	−0.155470	+	0.987841i	$0.549689\pi$
$84$	0	0
$85$	884.330	1.12846
$86$	− 175.838i	− 0.220478i
$87$	0	0
$88$	599.695	0.726452
$89$	670.780	0.798905	0.399453	−	0.916754i	$-0.369200\pi$
$89$	670.780	0.798905	0.399453	+	0.916754i	$0.369200\pi$
$90$	0	0
$91$	0	0
$92$	− 664.273i	− 0.752774i
$93$	0	0
$94$	− 473.468i	− 0.519515i
$95$	0.527285i	0 0.000569456i
$96$	0	0
$97$	− 655.891i	− 0.686553i	−0.939234	−	0.343276i	$-0.888463\pi$
$97$	− 655.891i	− 0.686553i	0.939234	−	0.343276i	$-0.111537\pi$
$98$	0	0
$99$	0	0
$100$	−53.2724	−0.0532724
$101$	−1163.24	−1.14600	−0.573002	−	0.819554i	$-0.694221\pi$
$101$	−1163.24	−1.14600	−0.573002	+	0.819554i	$0.694221\pi$
$102$	0	0
$103$	26.4198i	0.0252740i	0.999920	+	0.0126370i	$0.00402258\pi$
$103$	26.4198i	0.0252740i	−0.999920	+	0.0126370i	$0.995977\pi$
$104$	−372.206	−0.350940
$105$	0	0
$106$	198.957	0.182305
$107$	− 1467.08i	− 1.32549i	−0.748844	−	0.662746i	$-0.769391\pi$
$107$	− 1467.08i	− 1.32549i	0.748844	−	0.662746i	$-0.230609\pi$
$108$	0	0
$109$	−135.069	−0.118690	−0.0593450	−	0.998238i	$-0.518901\pi$
$109$	−135.069	−0.118690	−0.0593450	+	0.998238i	$0.518901\pi$
$110$	−421.653	−0.365482
$111$	0	0
$112$	0	0
$113$	288.471i	0.240151i	0.992765	+	0.120076i	$0.0383137\pi$
$113$	288.471i	0.240151i	−0.992765	+	0.120076i	$0.961686\pi$
$114$	0	0
$115$	969.335i	0.786009i
$116$	− 1303.66i	− 1.04346i
$117$	0	0
$118$	− 262.935i	− 0.205129i
$119$	0	0
$120$	0	0
$121$	−1358.58	−1.02072
$122$	582.756	0.432461
$123$	0	0
$124$	− 1599.75i	− 1.15857i
$125$	1434.66	1.02656
$126$	0	0
$127$	−2269.80	−1.58592	−0.792961	−	0.609273i	$-0.791461\pi$
$127$	−2269.80	−1.58592	−0.792961	+	0.609273i	$0.791461\pi$
$128$	1276.19i	0.881252i
$129$	0	0
$130$	261.702	0.176560
$131$	−389.692	−0.259905	−0.129952	−	0.991520i	$-0.541482\pi$
$131$	−389.692	−0.259905	−0.129952	+	0.991520i	$0.541482\pi$
$132$	0	0
$133$	0	0
$134$	− 146.806i	− 0.0946425i
$135$	0	0
$136$	− 942.017i	− 0.593951i
$137$	1468.69i	0.915905i	0.888977	+	0.457953i	$0.151417\pi$
$137$	1468.69i	0.915905i	−0.888977	+	0.457953i	$0.848583\pi$
$138$	0	0
$139$	624.712i	0.381204i	0.981667	+	0.190602i	$0.0610440\pi$
$139$	624.712i	0.381204i	−0.981667	+	0.190602i	$0.938956\pi$
$140$	0	0
$141$	0	0
$142$	−106.828	−0.0631323
$143$	1669.31	0.976185
$144$	0	0
$145$	1902.35i	1.08953i
$146$	584.860	0.331530
$147$	0	0
$148$	480.158	0.266681
$149$	1601.86i	0.880735i	0.897818	+	0.440367i	$0.145152\pi$
$149$	1601.86i	0.880735i	−0.897818	+	0.440367i	$0.854848\pi$
$150$	0	0
$151$	404.376	0.217931	0.108966	−	0.994046i	$-0.465246\pi$
$151$	404.376	0.217931	0.108966	+	0.994046i	$0.465246\pi$
$152$	0.561681	0.000299726 0
$153$	0	0
$154$	0	0
$155$	2334.43i	1.20972i
$156$	0	0
$157$	2412.07i	1.22614i	0.790028	+	0.613071i	$0.210066\pi$
$157$	2412.07i	1.22614i	−0.790028	+	0.613071i	$0.789934\pi$
$158$	965.469i	0.486130i
$159$	0	0
$160$	− 1417.65i	− 0.700469i
$161$	0	0
$162$	0	0
$163$	−945.367	−0.454275	−0.227138	−	0.973863i	$-0.572937\pi$
$163$	−945.367	−0.454275	−0.227138	+	0.973863i	$0.572937\pi$
$164$	3061.05	1.45749
$165$	0	0
$166$	− 176.101i	− 0.0823381i
$167$	−1271.18	−0.589022	−0.294511	−	0.955648i	$-0.595157\pi$
$167$	−1271.18	−0.589022	−0.294511	+	0.955648i	$0.595157\pi$
$168$	0	0
$169$	1160.93	0.528417
$170$	662.343i	0.298820i
$171$	0	0
$172$	−1746.47	−0.774228
$173$	4434.99	1.94905	0.974525	−	0.224278i	$-0.0720023\pi$
$173$	4434.99	1.94905	0.974525	+	0.224278i	$0.0720023\pi$
$174$	0	0
$175$	0	0
$176$	− 2637.22i	− 1.12947i
$177$	0	0
$178$	502.399i	0.211553i
$179$	1087.54i	0.454114i	0.973881	+	0.227057i	$0.0729103\pi$
$179$	1087.54i	0.454114i	−0.973881	+	0.227057i	$0.927090\pi$
$180$	0	0
$181$	2916.08i	1.19752i	0.800930	+	0.598758i	$0.204339\pi$
$181$	2916.08i	1.19752i	−0.800930	+	0.598758i	$0.795661\pi$
$182$	0	0
$183$	0	0
$184$	1032.57	0.413706
$185$	−700.667	−0.278455
$186$	0	0
$187$	4224.86i	1.65215i
$188$	−4702.60	−1.82432
$189$	0	0
$190$	−0.394924	−0.000150794 0
$191$	4559.87i	1.72744i	0.503973	+	0.863719i	$0.331871\pi$
$191$	4559.87i	1.72744i	−0.503973	+	0.863719i	$0.668129\pi$
$192$	0	0
$193$	−3757.20	−1.40129	−0.700645	−	0.713510i	$-0.747104\pi$
$193$	−3757.20	−1.40129	−0.700645	+	0.713510i	$0.747104\pi$
$194$	491.247	0.181802
$195$	0	0
$196$	0	0
$197$	2014.34i	0.728507i	0.931300	+	0.364253i	$0.118676\pi$
$197$	2014.34i	0.728507i	−0.931300	+	0.364253i	$0.881324\pi$
$198$	0	0
$199$	− 12.5118i	− 0.00445698i	−0.999998	−	0.00222849i	$-0.999291\pi$
$199$	− 12.5118i	− 0.00445698i	0.999998	−	0.00222849i	$-0.000709351\pi$
$200$	− 82.8083i	− 0.0292772i
$201$	0	0
$202$	− 871.238i	− 0.303466i
$203$	0	0
$204$	0	0
$205$	−4466.82	−1.52184
$206$	−19.7878	−0.00669263
$207$	0	0
$208$	1636.81i	0.545636i
$209$	−2.51909	−0.000833727 0
$210$	0	0
$211$	−2915.84	−0.951349	−0.475675	−	0.879621i	$-0.657796\pi$
$211$	−2915.84	−0.951349	−0.475675	+	0.879621i	$0.657796\pi$
$212$	− 1976.09i	− 0.640181i
$213$	0	0
$214$	1098.81	0.350995
$215$	2548.53	0.808409
$216$	0	0
$217$	0	0
$218$	− 101.163i	− 0.0314295i
$219$	0	0
$220$	4187.96i	1.28342i
$221$	− 2622.19i	− 0.798135i
$222$	0	0
$223$	− 1097.87i	− 0.329681i	−0.986320	−	0.164841i	$-0.947289\pi$
$223$	− 1097.87i	− 0.329681i	0.986320	−	0.164841i	$-0.0527110\pi$
$224$	0	0
$225$	0	0
$226$	−216.058	−0.0635929
$227$	500.594	0.146368	0.0731841	−	0.997318i	$-0.476684\pi$
$227$	500.594	0.146368	0.0731841	+	0.997318i	$0.476684\pi$
$228$	0	0
$229$	− 1133.53i	− 0.327099i	−0.986535	−	0.163549i	$-0.947706\pi$
$229$	− 1133.53i	− 0.327099i	0.986535	−	0.163549i	$-0.0522943\pi$
$230$	−726.010	−0.208138
$231$	0	0
$232$	2026.45	0.573461
$233$	3435.37i	0.965916i	0.875643	+	0.482958i	$0.160438\pi$
$233$	3435.37i	0.965916i	−0.875643	+	0.482958i	$0.839562\pi$
$234$	0	0
$235$	6862.24	1.90486
$236$	−2611.54	−0.720326
$237$	0	0
$238$	0	0
$239$	2213.97i	0.599203i	0.954064	+	0.299602i	$0.0968537\pi$
$239$	2213.97i	0.599203i	−0.954064	+	0.299602i	$0.903146\pi$
$240$	0	0
$241$	− 5951.96i	− 1.59087i	−0.606040	−	0.795435i	$-0.707243\pi$
$241$	− 5951.96i	− 1.59087i	0.606040	−	0.795435i	$-0.292757\pi$
$242$	− 1017.54i	− 0.270290i
$243$	0	0
$244$	− 5788.08i	− 1.51862i
$245$	0	0
$246$	0	0
$247$	1.56349	0.000402763 0
$248$	2486.71	0.636719
$249$	0	0
$250$	1074.53i	0.271836i
$251$	4889.86	1.22966	0.614831	−	0.788659i	$-0.289224\pi$
$251$	4889.86	1.22966	0.614831	+	0.788659i	$0.289224\pi$
$252$	0	0
$253$	−4630.97	−1.15078
$254$	− 1700.03i	− 0.419957i
$255$	0	0
$256$	1516.16	0.370156
$257$	3196.42	0.775825	0.387913	−	0.921696i	$-0.373196\pi$
$257$	3196.42	0.775825	0.387913	+	0.921696i	$0.373196\pi$
$258$	0	0
$259$	0	0
$260$	− 2599.29i	− 0.620005i
$261$	0	0
$262$	− 291.870i	− 0.0688237i
$263$	− 748.464i	− 0.175484i	−0.996143	−	0.0877419i	$-0.972035\pi$
$263$	− 748.464i	− 0.175484i	0.996143	−	0.0877419i	$-0.0279651\pi$
$264$	0	0
$265$	2883.59i	0.668444i
$266$	0	0
$267$	0	0
$268$	−1458.11	−0.332345
$269$	1299.26	0.294487	0.147244	−	0.989100i	$-0.452960\pi$
$269$	1299.26	0.294487	0.147244	+	0.989100i	$0.452960\pi$
$270$	0	0
$271$	83.5611i	0.0187305i	0.999956	+	0.00936527i	$0.00298110\pi$
$271$	83.5611i	0.0187305i	−0.999956	+	0.00936527i	$0.997019\pi$
$272$	−4142.61	−0.923465
$273$	0	0
$274$	−1100.02	−0.242535
$275$	371.388i	0.0814382i
$276$	0	0
$277$	−4641.86	−1.00687	−0.503434	−	0.864034i	$-0.667930\pi$
$277$	−4641.86	−1.00687	−0.503434	+	0.864034i	$0.667930\pi$
$278$	−467.895	−0.100944
$279$	0	0
$280$	0	0
$281$	− 179.289i	− 0.0380622i	−0.999819	−	0.0190311i	$-0.993942\pi$
$281$	− 179.289i	− 0.0380622i	0.999819	−	0.0190311i	$-0.00605816\pi$
$282$	0	0
$283$	4048.54i	0.850391i	0.905101	+	0.425196i	$0.139795\pi$
$283$	4048.54i	0.850391i	−0.905101	+	0.425196i	$0.860205\pi$
$284$	1061.04i	0.221694i
$285$	0	0
$286$	1250.27i	0.258497i
$287$	0	0
$288$	0	0
$289$	1723.52	0.350808
$290$	−1424.82	−0.288511
$291$	0	0
$292$	− 5808.98i	− 1.16420i
$293$	3389.52	0.675828	0.337914	−	0.941177i	$-0.390279\pi$
$293$	3389.52	0.675828	0.337914	+	0.941177i	$0.390279\pi$
$294$	0	0
$295$	3810.88	0.752128
$296$	746.374i	0.146561i
$297$	0	0
$298$	−1199.76	−0.233222
$299$	2874.25	0.555927
$300$	0	0
$301$	0	0
$302$	302.868i	0.0577089i
$303$	0	0
$304$	− 2.47004i	0 0.000466009i
$305$	8446.21i	1.58567i
$306$	0	0
$307$	− 2014.64i	− 0.374534i	−0.982309	−	0.187267i	$-0.940037\pi$
$307$	− 2014.64i	− 0.374534i	0.982309	−	0.187267i	$-0.0599629\pi$
$308$	0	0
$309$	0	0
$310$	−1748.44	−0.320337
$311$	−5844.02	−1.06554	−0.532772	−	0.846259i	$-0.678850\pi$
$311$	−5844.02	−1.06554	−0.532772	+	0.846259i	$0.678850\pi$
$312$	0	0
$313$	2449.45i	0.442336i	0.975236	+	0.221168i	$0.0709870\pi$
$313$	2449.45i	0.442336i	−0.975236	+	0.221168i	$0.929013\pi$
$314$	−1806.59	−0.324686
$315$	0	0
$316$	9589.28	1.70709
$317$	− 6123.66i	− 1.08498i	−0.840062	−	0.542490i	$-0.817481\pi$
$317$	− 6123.66i	− 1.08498i	0.840062	−	0.542490i	$-0.182519\pi$
$318$	0	0
$319$	−9088.44	−1.59516
$320$	−3354.30	−0.585972
$321$	0	0
$322$	0	0
$323$	3.95705i	0 0.000681660i
$324$	0	0
$325$	− 230.505i	− 0.0393418i
$326$	− 708.059i	− 0.120294i
$327$	0	0
$328$	4758.20i	0.801000i
$329$	0	0
$330$	0	0
$331$	7597.04	1.26154	0.630772	−	0.775968i	$-0.282738\pi$
$331$	7597.04	1.26154	0.630772	+	0.775968i	$0.282738\pi$
$332$	−1749.08	−0.289137
$333$	0	0
$334$	− 952.083i	− 0.155975i
$335$	2127.74	0.347018
$336$	0	0
$337$	−3863.22	−0.624460	−0.312230	−	0.950007i	$-0.601076\pi$
$337$	−3863.22	−0.624460	−0.312230	+	0.950007i	$0.601076\pi$
$338$	869.511i	0.139927i
$339$	0	0
$340$	6578.56	1.04933
$341$	−11152.7	−1.77112
$342$	0	0
$343$	0	0
$344$	− 2714.77i	− 0.425496i
$345$	0	0
$346$	3321.70i	0.516115i
$347$	− 1824.41i	− 0.282246i	−0.989992	−	0.141123i	$-0.954929\pi$
$347$	− 1824.41i	− 0.282246i	0.989992	−	0.141123i	$-0.0450713\pi$
$348$	0	0
$349$	1537.52i	0.235822i	0.993024	+	0.117911i	$0.0376197\pi$
$349$	1537.52i	0.235822i	−0.993024	+	0.117911i	$0.962380\pi$
$350$	0	0
$351$	0	0
$352$	6772.78	1.02554
$353$	5927.33	0.893711	0.446855	−	0.894606i	$-0.352544\pi$
$353$	5927.33	0.893711	0.446855	+	0.894606i	$0.352544\pi$
$354$	0	0
$355$	− 1548.32i	− 0.231482i
$356$	4989.96	0.742885
$357$	0	0
$358$	−814.541	−0.120251
$359$	− 4839.49i	− 0.711472i	−0.934587	−	0.355736i	$-0.884230\pi$
$359$	− 4839.49i	− 0.711472i	0.934587	−	0.355736i	$-0.115770\pi$
$360$	0	0
$361$	6859.00	1.00000
$362$	−2184.08	−0.317106
$363$	0	0
$364$	0	0
$365$	8476.72i	1.21559i
$366$	0	0
$367$	− 11509.1i	− 1.63698i	−0.574519	−	0.818491i	$-0.694811\pi$
$367$	− 11509.1i	− 1.63698i	0.574519	−	0.818491i	$-0.305189\pi$
$368$	− 4540.81i	− 0.643223i
$369$	0	0
$370$	− 524.784i	− 0.0737357i
$371$	0	0
$372$	0	0
$373$	187.497	0.0260275	0.0130137	−	0.999915i	$-0.495857\pi$
$373$	187.497	0.0260275	0.0130137	+	0.999915i	$0.495857\pi$
$374$	−3164.32	−0.437495
$375$	0	0
$376$	− 7309.88i	− 1.00260i
$377$	5640.81	0.770601
$378$	0	0
$379$	3515.82	0.476506	0.238253	−	0.971203i	$-0.423425\pi$
$379$	3515.82	0.476506	0.238253	+	0.971203i	$0.423425\pi$
$380$	3.92249i	0 0.000529525i
$381$	0	0
$382$	−3415.24	−0.457432
$383$	−2029.38	−0.270748	−0.135374	−	0.990795i	$-0.543224\pi$
$383$	−2029.38	−0.270748	−0.135374	+	0.990795i	$0.543224\pi$
$384$	0	0
$385$	0	0
$386$	− 2814.06i	− 0.371067i
$387$	0	0
$388$	− 4879.19i	− 0.638411i
$389$	− 6439.94i	− 0.839378i	−0.907668	−	0.419689i	$-0.862139\pi$
$389$	− 6439.94i	− 0.839378i	0.907668	−	0.419689i	$-0.137861\pi$
$390$	0	0
$391$	7274.45i	0.940882i
$392$	0	0
$393$	0	0
$394$	−1508.70	−0.192911
$395$	−13993.1	−1.78245
$396$	0	0
$397$	− 9664.61i	− 1.22180i	−0.791709	−	0.610898i	$-0.790809\pi$
$397$	− 9664.61i	− 1.22180i	0.791709	−	0.610898i	$-0.209191\pi$
$398$	9.37106	0.00118022
$399$	0	0
$400$	−364.157	−0.0455197
$401$	− 11762.8i	− 1.46485i	−0.680846	−	0.732426i	$-0.738388\pi$
$401$	− 11762.8i	− 1.46485i	0.680846	−	0.732426i	$-0.261612\pi$
$402$	0	0
$403$	6921.99	0.855606
$404$	−8653.36	−1.06565
$405$	0	0
$406$	0	0
$407$	− 3347.42i	− 0.407679i
$408$	0	0
$409$	5272.03i	0.637371i	0.947860	+	0.318686i	$0.103241\pi$
$409$	5272.03i	0.637371i	−0.947860	+	0.318686i	$0.896759\pi$
$410$	− 3345.55i	− 0.402987i
$411$	0	0
$412$	196.538i	0.0235017i
$413$	0	0
$414$	0	0
$415$	2552.34	0.301902
$416$	−4203.58	−0.495426
$417$	0	0
$418$	− 1.88674i	0 0.000220774i
$419$	−5103.18	−0.595003	−0.297502	−	0.954721i	$-0.596153\pi$
$419$	−5103.18	−0.595003	−0.297502	+	0.954721i	$0.596153\pi$
$420$	0	0
$421$	−8395.31	−0.971882	−0.485941	−	0.873992i	$-0.661523\pi$
$421$	−8395.31	−0.971882	−0.485941	+	0.873992i	$0.661523\pi$
$422$	− 2183.90i	− 0.251921i
$423$	0	0
$424$	3071.70	0.351827
$425$	583.386	0.0665844
$426$	0	0
$427$	0	0
$428$	− 10913.6i	− 1.23255i
$429$	0	0
$430$	1908.79i	0.214070i
$431$	− 2088.48i	− 0.233408i	−0.993167	−	0.116704i	$-0.962767\pi$
$431$	− 2088.48i	− 0.233408i	0.993167	−	0.116704i	$-0.0372329\pi$
$432$	0	0
$433$	− 11495.3i	− 1.27582i	−0.770111	−	0.637910i	$-0.779799\pi$
$433$	− 11495.3i	− 1.27582i	0.770111	−	0.637910i	$-0.220201\pi$
$434$	0	0
$435$	0	0
$436$	−1004.78	−0.110367
$437$	−4.33742	−0.000474798 0
$438$	0	0
$439$	4252.70i	0.462347i	0.972913	+	0.231173i	$0.0742565\pi$
$439$	4252.70i	0.462347i	−0.972913	+	0.231173i	$0.925744\pi$
$440$	−6509.91	−0.705336
$441$	0	0
$442$	1963.96	0.211349
$443$	− 3304.79i	− 0.354437i	−0.984172	−	0.177218i	$-0.943290\pi$
$443$	− 3304.79i	− 0.354437i	0.984172	−	0.177218i	$-0.0567098\pi$
$444$	0	0
$445$	−7281.56	−0.775683
$446$	822.280	0.0873007
$447$	0	0
$448$	0	0
$449$	− 6952.63i	− 0.730768i	−0.930857	−	0.365384i	$-0.880938\pi$
$449$	− 6952.63i	− 0.730768i	0.930857	−	0.365384i	$-0.119062\pi$
$450$	0	0
$451$	− 21340.1i	− 2.22808i
$452$	2145.95i	0.223312i
$453$	0	0
$454$	374.933i	0.0387588i
$455$	0	0
$456$	0	0
$457$	9741.14	0.997093	0.498546	−	0.866863i	$-0.333867\pi$
$457$	9741.14	0.997093	0.498546	+	0.866863i	$0.333867\pi$
$458$	848.987	0.0866169
$459$	0	0
$460$	7210.92i	0.730893i
$461$	5563.15	0.562043	0.281021	−	0.959702i	$-0.409327\pi$
$461$	5563.15	0.562043	0.281021	+	0.959702i	$0.409327\pi$
$462$	0	0
$463$	4114.02	0.412948	0.206474	−	0.978452i	$-0.433801\pi$
$463$	4114.02	0.412948	0.206474	+	0.978452i	$0.433801\pi$
$464$	− 8911.50i	− 0.891608i
$465$	0	0
$466$	−2573.01	−0.255778
$467$	6061.59	0.600636	0.300318	−	0.953839i	$-0.402907\pi$
$467$	6061.59	0.600636	0.300318	+	0.953839i	$0.402907\pi$
$468$	0	0
$469$	0	0
$470$	5139.66i	0.504415i
$471$	0	0
$472$	− 4059.47i	− 0.395873i
$473$	12175.5i	1.18357i
$474$	0	0
$475$	0.347846i	0 3.36005e-5i
$476$	0	0
$477$	0	0
$478$	−1658.21	−0.158671
$479$	−8246.66	−0.786638	−0.393319	−	0.919402i	$-0.628673\pi$
$479$	−8246.66	−0.786638	−0.393319	+	0.919402i	$0.628673\pi$
$480$	0	0
$481$	2077.60i	0.196945i
$482$	4457.88	0.421268
$483$	0	0
$484$	−10106.5	−0.949145
$485$	7119.93i	0.666597i
$486$	0	0
$487$	11744.2	1.09277	0.546385	−	0.837534i	$-0.316004\pi$
$487$	11744.2	1.09277	0.546385	+	0.837534i	$0.316004\pi$
$488$	8997.18	0.834597
$489$	0	0
$490$	0	0
$491$	− 6008.34i	− 0.552246i	−0.961122	−	0.276123i	$-0.910950\pi$
$491$	− 6008.34i	− 0.552246i	0.961122	−	0.276123i	$-0.0890497\pi$
$492$	0	0
$493$	14276.4i	1.30421i
$494$	1.17102i	0 0.000106653i
$495$	0	0
$496$	− 10935.5i	− 0.989961i
$497$	0	0
$498$	0	0
$499$	−13649.9	−1.22455	−0.612276	−	0.790644i	$-0.709746\pi$
$499$	−13649.9	−1.22455	−0.612276	+	0.790644i	$0.709746\pi$
$500$	10672.5	0.954574
$501$	0	0
$502$	3662.39i	0.325619i
$503$	−4862.69	−0.431047	−0.215524	−	0.976499i	$-0.569146\pi$
$503$	−4862.69	−0.431047	−0.215524	+	0.976499i	$0.569146\pi$
$504$	0	0
$505$	12627.4	1.11269
$506$	− 3468.49i	− 0.304730i
$507$	0	0
$508$	−16885.1	−1.47472
$509$	−17722.7	−1.54331	−0.771653	−	0.636043i	$-0.780570\pi$
$509$	−17722.7	−1.54331	−0.771653	+	0.636043i	$0.780570\pi$
$510$	0	0
$511$	0	0
$512$	11345.1i	0.979271i
$513$	0	0
$514$	2394.04i	0.205441i
$515$	− 286.796i	− 0.0245393i
$516$	0	0
$517$	32784.1i	2.78887i
$518$	0	0
$519$	0	0
$520$	4040.43	0.340739
$521$	13755.9	1.15673	0.578366	−	0.815777i	$-0.303691\pi$
$521$	13755.9	1.15673	0.578366	+	0.815777i	$0.303691\pi$
$522$	0	0
$523$	1311.94i	0.109689i	0.998495	+	0.0548443i	$0.0174663\pi$
$523$	1311.94i	0.109689i	−0.998495	+	0.0548443i	$0.982534\pi$
$524$	−2898.93	−0.241680
$525$	0	0
$526$	560.582	0.0464687
$527$	17518.9i	1.44808i
$528$	0	0
$529$	4193.30	0.344645
$530$	−2159.75	−0.177006
$531$	0	0
$532$	0	0
$533$	13244.9i	1.07636i
$534$	0	0
$535$	15925.6i	1.28696i
$536$	− 2266.54i	− 0.182649i
$537$	0	0
$538$	973.114i	0.0779812i
$539$	0	0
$540$	0	0
$541$	1195.91	0.0950389	0.0475195	−	0.998870i	$-0.484868\pi$
$541$	1195.91	0.0950389	0.0475195	+	0.998870i	$0.484868\pi$
$542$	−62.5854	−0.00495991
$543$	0	0
$544$	− 10638.9i	− 0.838488i
$545$	1466.22	0.115240
$546$	0	0
$547$	−6178.59	−0.482957	−0.241478	−	0.970406i	$-0.577632\pi$
$547$	−6178.59	−0.482957	−0.241478	+	0.970406i	$0.577632\pi$
$548$	10925.7i	0.851681i
$549$	0	0
$550$	−278.161	−0.0215651
$551$	−8.51233	−0.000658144 0
$552$	0	0
$553$	0	0
$554$	− 3476.65i	− 0.266622i
$555$	0	0
$556$	4647.26i	0.354474i
$557$	− 3356.20i	− 0.255309i	−0.991819	−	0.127654i	$-0.959255\pi$
$557$	− 3356.20i	− 0.255309i	0.991819	−	0.127654i	$-0.0407448\pi$
$558$	0	0
$559$	− 7556.82i	− 0.571770i
$560$	0	0
$561$	0	0
$562$	134.283	0.0100790
$563$	3585.28	0.268386	0.134193	−	0.990955i	$-0.457156\pi$
$563$	3585.28	0.268386	0.134193	+	0.990955i	$0.457156\pi$
$564$	0	0
$565$	− 3131.46i	− 0.233171i
$566$	−3032.26	−0.225187
$567$	0	0
$568$	−1649.32	−0.121838
$569$	18285.7i	1.34723i	0.739081	+	0.673617i	$0.235260\pi$
$569$	18285.7i	1.34723i	−0.739081	+	0.673617i	$0.764740\pi$
$570$	0	0
$571$	−16363.0	−1.19925	−0.599624	−	0.800282i	$-0.704683\pi$
$571$	−16363.0	−1.19925	−0.599624	+	0.800282i	$0.704683\pi$
$572$	12418.0	0.907734
$573$	0	0
$574$	0	0
$575$	639.463i	0.0463782i
$576$	0	0
$577$	7711.39i	0.556376i	0.960527	+	0.278188i	$0.0897339\pi$
$577$	7711.39i	0.556376i	−0.960527	+	0.278188i	$0.910266\pi$
$578$	1290.88i	0.0928953i
$579$	0	0
$580$	14151.7i	1.01313i
$581$	0	0
$582$	0	0
$583$	−13776.3	−0.978653
$584$	9029.68	0.639813
$585$	0	0
$586$	2538.67i	0.178962i
$587$	4182.21	0.294069	0.147034	−	0.989131i	$-0.453027\pi$
$587$	4182.21	0.294069	0.147034	+	0.989131i	$0.453027\pi$
$588$	0	0
$589$	−10.4457	−0.000730744 0
$590$	2854.26i	0.199166i
$591$	0	0
$592$	3282.25	0.227871
$593$	16189.3	1.12110	0.560552	−	0.828119i	$-0.310589\pi$
$593$	16189.3	1.12110	0.560552	+	0.828119i	$0.310589\pi$
$594$	0	0
$595$	0	0
$596$	11916.3i	0.818977i
$597$	0	0
$598$	2152.75i	0.147211i
$599$	− 12777.3i	− 0.871566i	−0.900052	−	0.435783i	$-0.856471\pi$
$599$	− 12777.3i	− 0.871566i	0.900052	−	0.435783i	$-0.143529\pi$
$600$	0	0
$601$	− 24022.7i	− 1.63046i	−0.579137	−	0.815231i	$-0.696610\pi$
$601$	− 24022.7i	− 1.63046i	0.579137	−	0.815231i	$-0.303390\pi$
$602$	0	0
$603$	0	0
$604$	3008.16	0.202650
$605$	14747.8	0.991050
$606$	0	0
$607$	9666.83i	0.646400i	0.946331	+	0.323200i	$0.104759\pi$
$607$	9666.83i	0.646400i	−0.946331	+	0.323200i	$0.895241\pi$
$608$	6.34346	0.000423127 0
$609$	0	0
$610$	−6326.02	−0.419890
$611$	− 20347.7i	− 1.34727i
$612$	0	0
$613$	14385.5	0.947836	0.473918	−	0.880569i	$-0.342839\pi$
$613$	14385.5	0.947836	0.473918	+	0.880569i	$0.342839\pi$
$614$	1508.92	0.0991778
$615$	0	0
$616$	0	0
$617$	7712.69i	0.503244i	0.967826	+	0.251622i	$0.0809639\pi$
$617$	7712.69i	0.503244i	−0.967826	+	0.251622i	$0.919036\pi$
$618$	0	0
$619$	− 14317.0i	− 0.929645i	−0.885404	−	0.464822i	$-0.846118\pi$
$619$	− 14317.0i	− 0.929645i	0.885404	−	0.464822i	$-0.153882\pi$
$620$	17365.9i	1.12489i
$621$	0	0
$622$	− 4377.04i	− 0.282160i
$623$	0	0
$624$	0	0
$625$	−14678.6	−0.939428
$626$	−1834.58	−0.117132
$627$	0	0
$628$	17943.5i	1.14016i
$629$	−5258.21	−0.333320
$630$	0	0
$631$	4971.96	0.313678	0.156839	−	0.987624i	$-0.449870\pi$
$631$	4971.96	0.313678	0.156839	+	0.987624i	$0.449870\pi$
$632$	14905.9i	0.938172i
$633$	0	0
$634$	4586.48	0.287307
$635$	24639.5	1.53982
$636$	0	0
$637$	0	0
$638$	− 6807.03i	− 0.422403i
$639$	0	0
$640$	− 13853.5i	− 0.855637i
$641$	29423.8i	1.81306i	0.422145	+	0.906529i	$0.361277\pi$
$641$	29423.8i	1.81306i	−0.422145	+	0.906529i	$0.638723\pi$
$642$	0	0
$643$	− 31273.9i	− 1.91807i	−0.283283	−	0.959036i	$-0.591424\pi$
$643$	− 31273.9i	− 1.91807i	0.283283	−	0.959036i	$-0.408576\pi$
$644$	0	0
$645$	0	0
$646$	−2.96374	−0.000180506 0
$647$	−8011.64	−0.486816	−0.243408	−	0.969924i	$-0.578265\pi$
$647$	−8011.64	−0.486816	−0.243408	+	0.969924i	$0.578265\pi$
$648$	0	0
$649$	18206.3i	1.10117i
$650$	172.643	0.0104179
$651$	0	0
$652$	−7032.62	−0.422421
$653$	15440.6i	0.925327i	0.886534	+	0.462664i	$0.153106\pi$
$653$	15440.6i	0.925327i	−0.886534	+	0.462664i	$0.846894\pi$
$654$	0	0
$655$	4230.24	0.252350
$656$	20924.6	1.24538
$657$	0	0
$658$	0	0
$659$	− 31288.9i	− 1.84953i	−0.380537	−	0.924766i	$-0.624261\pi$
$659$	− 31288.9i	− 1.84953i	0.380537	−	0.924766i	$-0.375739\pi$
$660$	0	0
$661$	30326.1i	1.78449i	0.451550	+	0.892246i	$0.350871\pi$
$661$	30326.1i	1.78449i	−0.451550	+	0.892246i	$0.649129\pi$
$662$	5690.01i	0.334061i
$663$	0	0
$664$	− 2718.84i	− 0.158903i
$665$	0	0
$666$	0	0
$667$	−15648.7	−0.908424
$668$	−9456.33	−0.547719
$669$	0	0
$670$	1593.63i	0.0918915i
$671$	−40351.5	−2.32154
$672$	0	0
$673$	12067.9	0.691207	0.345604	−	0.938381i	$-0.387674\pi$
$673$	12067.9	0.691207	0.345604	+	0.938381i	$0.387674\pi$
$674$	− 2893.46i	− 0.165359i
$675$	0	0
$676$	8636.21	0.491364
$677$	−6544.82	−0.371548	−0.185774	−	0.982593i	$-0.559479\pi$
$677$	−6544.82	−0.371548	−0.185774	+	0.982593i	$0.559479\pi$
$678$	0	0
$679$	0	0
$680$	10225.9i	0.576686i
$681$	0	0
$682$	− 8353.10i	− 0.468998i
$683$	− 31485.3i	− 1.76391i	−0.471332	−	0.881956i	$-0.656227\pi$
$683$	− 31485.3i	− 1.76391i	0.471332	−	0.881956i	$-0.343773\pi$
$684$	0	0
$685$	− 15943.2i	− 0.889282i
$686$	0	0
$687$	0	0
$688$	−11938.5	−0.661555
$689$	8550.35	0.472776
$690$	0	0
$691$	− 10035.3i	− 0.552476i	−0.961089	−	0.276238i	$-0.910912\pi$
$691$	− 10035.3i	− 0.552476i	0.961089	−	0.276238i	$-0.0890879\pi$
$692$	32992.0	1.81238
$693$	0	0
$694$	1366.44	0.0747397
$695$	− 6781.48i	− 0.370124i
$696$	0	0
$697$	−33521.6	−1.82169
$698$	−1151.57	−0.0624464
$699$	0	0
$700$	0	0
$701$	− 768.196i	− 0.0413900i	−0.999786	−	0.0206950i	$-0.993412\pi$
$701$	− 768.196i	− 0.0413900i	0.999786	−	0.0206950i	$-0.00658789\pi$
$702$	0	0
$703$	− 3.13523i	0 0.000168204i
$704$	− 16025.1i	− 0.857908i
$705$	0	0
$706$	4439.43i	0.236658i
$707$	0	0
$708$	0	0
$709$	13968.6	0.739918	0.369959	−	0.929048i	$-0.379372\pi$
$709$	13968.6	0.739918	0.369959	+	0.929048i	$0.379372\pi$
$710$	1159.65	0.0612973
$711$	0	0
$712$	7756.56i	0.408271i
$713$	−19202.9	−1.00863
$714$	0	0
$715$	−18120.9	−0.947810
$716$	8090.23i	0.422271i
$717$	0	0
$718$	3624.67	0.188400
$719$	11969.6	0.620847	0.310424	−	0.950598i	$-0.399529\pi$
$719$	11969.6	0.620847	0.310424	+	0.950598i	$0.399529\pi$
$720$	0	0
$721$	0	0
$722$	5137.23i	0.264803i
$723$	0	0
$724$	21692.8i	1.11354i
$725$	1254.97i	0.0642874i
$726$	0	0
$727$	12223.3i	0.623575i	0.950152	+	0.311787i	$0.100928\pi$
$727$	12223.3i	0.623575i	−0.950152	+	0.311787i	$0.899072\pi$
$728$	0	0
$729$	0	0
$730$	−6348.87	−0.321893
$731$	19125.6	0.967696
$732$	0	0
$733$	− 23785.0i	− 1.19853i	−0.800552	−	0.599264i	$-0.795460\pi$
$733$	− 23785.0i	− 1.19853i	0.800552	−	0.599264i	$-0.204540\pi$
$734$	8620.09	0.433478
$735$	0	0
$736$	11661.5	0.584034
$737$	10165.2i	0.508061i
$738$	0	0
$739$	11478.1	0.571351	0.285675	−	0.958326i	$-0.407782\pi$
$739$	11478.1	0.571351	0.285675	+	0.958326i	$0.407782\pi$
$740$	−5212.29	−0.258929
$741$	0	0
$742$	0	0
$743$	18604.0i	0.918593i	0.888283	+	0.459297i	$0.151899\pi$
$743$	18604.0i	0.918593i	−0.888283	+	0.459297i	$0.848101\pi$
$744$	0	0
$745$	− 17388.8i	− 0.855134i
$746$	140.431i	0.00689217i
$747$	0	0
$748$	31428.9i	1.53630i
$749$	0	0
$750$	0	0
$751$	31012.5	1.50687	0.753436	−	0.657522i	$-0.228395\pi$
$751$	31012.5	1.50687	0.753436	+	0.657522i	$0.228395\pi$
$752$	−32145.9	−1.55883
$753$	0	0
$754$	4224.84i	0.204058i
$755$	−4389.64	−0.211597
$756$	0	0
$757$	−19065.9	−0.915407	−0.457703	−	0.889105i	$-0.651328\pi$
$757$	−19065.9	−0.915407	−0.457703	+	0.889105i	$0.651328\pi$
$758$	2633.27i	0.126180i
$759$	0	0
$760$	−6.09725	−0.000291014 0
$761$	−14679.2	−0.699236	−0.349618	−	0.936892i	$-0.613689\pi$
$761$	−14679.2	−0.699236	−0.349618	+	0.936892i	$0.613689\pi$
$762$	0	0
$763$	0	0
$764$	33921.0i	1.60631i
$765$	0	0
$766$	− 1519.96i	− 0.0716949i
$767$	− 11299.9i	− 0.531963i
$768$	0	0
$769$	29972.5i	1.40551i	0.711434	+	0.702753i	$0.248046\pi$
$769$	29972.5i	1.40551i	−0.711434	+	0.702753i	$0.751954\pi$
$770$	0	0
$771$	0	0
$772$	−27949.9	−1.30303
$773$	22687.8	1.05566	0.527828	−	0.849351i	$-0.323007\pi$
$773$	22687.8	1.05566	0.527828	+	0.849351i	$0.323007\pi$
$774$	0	0
$775$	1540.01i	0.0713789i
$776$	7584.38	0.350855
$777$	0	0
$778$	4823.37	0.222270
$779$	− 19.9874i	0 0.000919284i
$780$	0	0
$781$	7397.04	0.338908
$782$	−5448.40	−0.249149
$783$	0	0
$784$	0	0
$785$	− 26183.9i	− 1.19050i
$786$	0	0
$787$	20937.9i	0.948357i	0.880429	+	0.474178i	$0.157255\pi$
$787$	20937.9i	0.948357i	−0.880429	+	0.474178i	$0.842745\pi$
$788$	14984.7i	0.677423i
$789$	0	0
$790$	− 10480.5i	− 0.472000i
$791$	0	0
$792$	0	0
$793$	25044.5	1.12151
$794$	7238.57	0.323536
$795$	0	0
$796$	− 93.0758i	− 0.00414445i
$797$	27393.4	1.21747	0.608735	−	0.793373i	$-0.291677\pi$
$797$	27393.4	1.21747	0.608735	+	0.793373i	$0.291677\pi$
$798$	0	0
$799$	51498.2	2.28019
$800$	− 935.212i	− 0.0413309i
$801$	0	0
$802$	8810.06	0.387898
$803$	−40497.3	−1.77972
$804$	0	0
$805$	0	0
$806$	5184.41i	0.226567i
$807$	0	0
$808$	− 13451.1i	− 0.585652i
$809$	26805.4i	1.16493i	0.812856	+	0.582465i	$0.197912\pi$
$809$	26805.4i	1.16493i	−0.812856	+	0.582465i	$0.802088\pi$
$810$	0	0
$811$	− 17139.6i	− 0.742113i	−0.928610	−	0.371056i	$-0.878996\pi$
$811$	− 17139.6i	− 0.742113i	0.928610	−	0.371056i	$-0.121004\pi$
$812$	0	0
$813$	0	0
$814$	2507.14	0.107955
$815$	10262.3	0.441071
$816$	0	0
$817$	11.4037i	0 0.000488329i
$818$	−3948.63	−0.168778
$819$	0	0
$820$	−33228.8	−1.41512
$821$	− 8629.28i	− 0.366826i	−0.983036	−	0.183413i	$-0.941285\pi$
$821$	− 8629.28i	− 0.366826i	0.983036	−	0.183413i	$-0.0587145\pi$
$822$	0	0
$823$	−5284.96	−0.223842	−0.111921	−	0.993717i	$-0.535700\pi$
$823$	−5284.96	−0.223842	−0.111921	+	0.993717i	$0.535700\pi$
$824$	−305.505	−0.0129160
$825$	0	0
$826$	0	0
$827$	1845.35i	0.0775926i	0.999247	+	0.0387963i	$0.0123523\pi$
$827$	1845.35i	0.0775926i	−0.999247	+	0.0387963i	$0.987648\pi$
$828$	0	0
$829$	− 19493.2i	− 0.816677i	−0.912831	−	0.408339i	$-0.866108\pi$
$829$	− 19493.2i	− 0.816677i	0.912831	−	0.408339i	$-0.133892\pi$
$830$	1911.64i	0.0799447i
$831$	0	0
$832$	9946.08i	0.414445i
$833$	0	0
$834$	0	0
$835$	13799.1	0.571901
$836$	−18.7396	−0.000775265 0
$837$	0	0
$838$	− 3822.16i	− 0.157559i
$839$	−27816.7	−1.14462	−0.572312	−	0.820036i	$-0.693953\pi$
$839$	−27816.7	−1.14462	−0.572312	+	0.820036i	$0.693953\pi$
$840$	0	0
$841$	−6322.03	−0.259217
$842$	− 6287.90i	− 0.257358i
$843$	0	0
$844$	−21691.0	−0.884640
$845$	−12602.3	−0.513057
$846$	0	0
$847$	0	0
$848$	− 13508.1i	− 0.547015i
$849$	0	0
$850$	436.942i	0.0176318i
$851$	− 5763.65i	− 0.232168i
$852$	0	0
$853$	9210.41i	0.369705i	0.982766	+	0.184852i	$0.0591807\pi$
$853$	9210.41i	0.369705i	−0.982766	+	0.184852i	$0.940819\pi$
$854$	0	0
$855$	0	0
$856$	16964.5	0.677377
$857$	−34500.9	−1.37518	−0.687589	−	0.726100i	$-0.741331\pi$
$857$	−34500.9	−1.37518	−0.687589	+	0.726100i	$0.741331\pi$
$858$	0	0
$859$	− 43698.0i	− 1.73569i	−0.496836	−	0.867845i	$-0.665505\pi$
$859$	− 43698.0i	− 1.73569i	0.496836	−	0.867845i	$-0.334495\pi$
$860$	18958.6	0.751723
$861$	0	0
$862$	1564.23	0.0618072
$863$	− 28333.6i	− 1.11760i	−0.829303	−	0.558800i	$-0.811262\pi$
$863$	− 28333.6i	− 1.11760i	0.829303	−	0.558800i	$-0.188738\pi$
$864$	0	0
$865$	−48143.4	−1.89240
$866$	8609.73	0.337841
$867$	0	0
$868$	0	0
$869$	− 66851.6i	− 2.60965i
$870$	0	0
$871$	− 6309.13i	− 0.245438i
$872$	− 1561.86i	− 0.0606552i
$873$	0	0
$874$	− 3.24863i	0 0.000125728i
$875$	0	0
$876$	0	0
$877$	−45125.8	−1.73750	−0.868751	−	0.495249i	$-0.835077\pi$
$877$	−45125.8	−1.73750	−0.868751	+	0.495249i	$0.835077\pi$
$878$	−3185.17	−0.122431
$879$	0	0
$880$	28627.9i	1.09664i
$881$	4244.09	0.162301	0.0811504	−	0.996702i	$-0.474141\pi$
$881$	4244.09	0.162301	0.0811504	+	0.996702i	$0.474141\pi$
$882$	0	0
$883$	−21.9077	−0.000834939 0	−0.000417470	−	1.00000i	$-0.500133\pi$
$883$	−21.9077	−0.000834939 0	−0.000417470		1.00000i	$0.500133\pi$
$884$	− 19506.6i	− 0.742169i
$885$	0	0
$886$	2475.21	0.0938560
$887$	7501.63	0.283969	0.141984	−	0.989869i	$-0.454652\pi$
$887$	7501.63	0.283969	0.141984	+	0.989869i	$0.454652\pi$
$888$	0	0
$889$	0	0
$890$	− 5453.72i	− 0.205404i
$891$	0	0
$892$	− 8167.10i	− 0.306564i
$893$	30.7060i	0.00115066i
$894$	0	0
$895$	− 11805.6i	− 0.440914i
$896$	0	0
$897$	0	0
$898$	5207.36	0.193510
$899$	−37686.4	−1.39812
$900$	0	0
$901$	21640.1i	0.800153i
$902$	15983.2	0.590004
$903$	0	0
$904$	−3335.73	−0.122727
$905$	− 31655.1i	− 1.16271i
$906$	0	0
$907$	41137.3	1.50600	0.753001	−	0.658020i	$-0.228606\pi$
$907$	41137.3	1.50600	0.753001	+	0.658020i	$0.228606\pi$
$908$	3723.94	0.136105
$909$	0	0
$910$	0	0
$911$	14080.6i	0.512086i	0.966665	+	0.256043i	$0.0824189\pi$
$911$	14080.6i	0.512086i	−0.966665	+	0.256043i	$0.917581\pi$
$912$	0	0
$913$	12193.7i	0.442008i
$914$	7295.89i	0.264034i
$915$	0	0
$916$	− 8432.35i	− 0.304162i
$917$	0	0
$918$	0	0
$919$	12623.2	0.453101	0.226550	−	0.973999i	$-0.427255\pi$
$919$	12623.2	0.453101	0.226550	+	0.973999i	$0.427255\pi$
$920$	−11208.9	−0.401681
$921$	0	0
$922$	4166.67i	0.148831i
$923$	−4591.03	−0.163722
$924$	0	0
$925$	−462.225	−0.0164301
$926$	3081.31i	0.109350i
$927$	0	0
$928$	22886.1	0.809562
$929$	−33473.7	−1.18217	−0.591085	−	0.806609i	$-0.701300\pi$
$929$	−33473.7	−1.18217	−0.591085	+	0.806609i	$0.701300\pi$
$930$	0	0
$931$	0	0
$932$	25555.8i	0.898186i
$933$	0	0
$934$	4539.99i	0.159050i
$935$	− 45862.4i	− 1.60413i
$936$	0	0
$937$	25652.3i	0.894371i	0.894441	+	0.447185i	$0.147574\pi$
$937$	25652.3i	0.894371i	−0.894441	+	0.447185i	$0.852426\pi$
$938$	0	0
$939$	0	0
$940$	51048.4	1.77129
$941$	2602.96	0.0901744	0.0450872	−	0.998983i	$-0.485643\pi$
$941$	2602.96	0.0901744	0.0450872	+	0.998983i	$0.485643\pi$
$942$	0	0
$943$	− 36743.8i	− 1.26887i
$944$	−17851.9	−0.615497
$945$	0	0
$946$	−9119.17	−0.313414
$947$	− 37616.8i	− 1.29079i	−0.763847	−	0.645397i	$-0.776692\pi$
$947$	− 37616.8i	− 1.29079i	0.763847	−	0.645397i	$-0.223308\pi$
$948$	0	0
$949$	25134.9	0.859763
$950$	−0.260528	−8.89754e−6 0
$951$	0	0
$952$	0	0
$953$	− 23444.5i	− 0.796896i	−0.917191	−	0.398448i	$-0.869549\pi$
$953$	− 23444.5i	− 0.796896i	0.917191	−	0.398448i	$-0.130451\pi$
$954$	0	0
$955$	− 49499.1i	− 1.67723i
$956$	16469.8i	0.557187i
$957$	0	0
$958$	− 6176.56i	− 0.208304i
$959$	0	0
$960$	0	0
$961$	−16455.0	−0.552347
$962$	−1556.07	−0.0521516
$963$	0	0
$964$	− 44276.8i	− 1.47932i
$965$	40785.7	1.36056
$966$	0	0
$967$	4312.24	0.143405	0.0717023	−	0.997426i	$-0.477157\pi$
$967$	4312.24	0.143405	0.0717023	+	0.997426i	$0.477157\pi$
$968$	− 15709.9i	− 0.521627i
$969$	0	0
$970$	−5332.66	−0.176517
$971$	−47052.7	−1.55509	−0.777545	−	0.628827i	$-0.783535\pi$
$971$	−47052.7	−1.55509	−0.777545	+	0.628827i	$0.783535\pi$
$972$	0	0
$973$	0	0
$974$	8796.11i	0.289369i
$975$	0	0
$976$	− 39565.9i	− 1.29762i
$977$	− 20463.0i	− 0.670082i	−0.942204	−	0.335041i	$-0.891250\pi$
$977$	− 20463.0i	− 0.670082i	0.942204	−	0.335041i	$-0.108750\pi$
$978$	0	0
$979$	− 34787.4i	− 1.13566i
$980$	0	0
$981$	0	0
$982$	4500.11	0.146237
$983$	−44629.0	−1.44806	−0.724031	−	0.689767i	$-0.757713\pi$
$983$	−44629.0	−1.44806	−0.724031	+	0.689767i	$0.757713\pi$
$984$	0	0
$985$	− 21866.4i	− 0.707331i
$986$	−10692.7	−0.345359
$987$	0	0
$988$	11.6309	0.000374521 0
$989$	20964.0i	0.674032i
$990$	0	0
$991$	43131.0	1.38254	0.691272	−	0.722594i	$-0.257051\pi$
$991$	43131.0	1.38254	0.691272	+	0.722594i	$0.257051\pi$
$992$	28084.2	0.898864
$993$	0	0
$994$	0	0
$995$	135.820i	0.00432743i
$996$	0	0
$997$	− 3622.25i	− 0.115063i	−0.998344	−	0.0575315i	$-0.981677\pi$
$997$	− 3622.25i	− 0.115063i	0.998344	−	0.0575315i	$-0.0183230\pi$
$998$	− 10223.4i	− 0.324266i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.4.c.a.440.9		16
3.2	odd	2	inner	441.4.c.a.440.8		16
7.2	even	3		63.4.p.a.17.5	yes	16
7.3	odd	6		63.4.p.a.26.4	yes	16
7.4	even	3		441.4.p.c.215.4		16
7.5	odd	6		441.4.p.c.80.5		16
7.6	odd	2	inner	441.4.c.a.440.10		16
21.2	odd	6		63.4.p.a.17.4	✓	16
21.5	even	6		441.4.p.c.80.4		16
21.11	odd	6		441.4.p.c.215.5		16
21.17	even	6		63.4.p.a.26.5	yes	16
21.20	even	2	inner	441.4.c.a.440.7		16
28.3	even	6		1008.4.bt.a.593.3		16
28.23	odd	6		1008.4.bt.a.17.6		16
84.23	even	6		1008.4.bt.a.17.3		16
84.59	odd	6		1008.4.bt.a.593.6		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.4.p.a.17.4	✓	16	21.2	odd	6
63.4.p.a.17.5	yes	16	7.2	even	3
63.4.p.a.26.4	yes	16	7.3	odd	6
63.4.p.a.26.5	yes	16	21.17	even	6
441.4.c.a.440.7		16	21.20	even	2	inner
441.4.c.a.440.8		16	3.2	odd	2	inner
441.4.c.a.440.9		16	1.1	even	1	trivial
441.4.c.a.440.10		16	7.6	odd	2	inner
441.4.p.c.80.4		16	21.5	even	6
441.4.p.c.80.5		16	7.5	odd	6
441.4.p.c.215.4		16	7.4	even	3
441.4.p.c.215.5		16	21.11	odd	6
1008.4.bt.a.17.3		16	84.23	even	6
1008.4.bt.a.17.6		16	28.23	odd	6
1008.4.bt.a.593.3		16	28.3	even	6
1008.4.bt.a.593.6		16	84.59	odd	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+0.748977i q^{2} +7.43903 q^{4} -10.8554 q^{5} +11.5635i q^{8} +O(q^{10})\)	\(q+0.748977i q^{2} +7.43903 q^{4} -10.8554 q^{5} +11.5635i q^{8} -8.13042i q^{10} -51.8611i q^{11} +32.1880i q^{13} +50.8515 q^{16} -81.4649 q^{17} -0.0485737i q^{19} -80.7534 q^{20} +38.8428 q^{22} -89.2956i q^{23} -7.16119 q^{25} -24.1081 q^{26} -175.246i q^{29} -215.049i q^{31} +130.594i q^{32} -61.0153i q^{34} +64.5458 q^{37} +0.0363806 q^{38} -125.526i q^{40} +411.485 q^{41} -234.771 q^{43} -385.797i q^{44} +66.8803 q^{46} -632.152 q^{47} -5.36357i q^{50} +239.448i q^{52} -265.638i q^{53} +562.971i q^{55} +131.255 q^{58} -351.059 q^{59} -778.069i q^{61} +161.067 q^{62} +309.000 q^{64} -349.412i q^{65} -196.009 q^{67} -606.020 q^{68} +142.632i q^{71} -780.879i q^{73} +48.3433i q^{74} -0.361341i q^{76} +1289.05 q^{79} -552.011 q^{80} +308.193i q^{82} -235.123 q^{83} +884.330 q^{85} -175.838i q^{86} +599.695 q^{88} +670.780 q^{89} -664.273i q^{92} -473.468i q^{94} +0.527285i q^{95} -655.891i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 64 q^{4}+O(q^{10}) \) 16 * q - 64 * q^4	\( 16 q - 64 q^{4} + 376 q^{16} + 528 q^{22} + 40 q^{25} + 2392 q^{37} + 328 q^{43} + 2784 q^{46} + 6744 q^{58} + 5432 q^{64} - 616 q^{67} + 4352 q^{79} - 4608 q^{85} - 1416 q^{88}+O(q^{100}) \) 16 * q - 64 * q^4 + 376 * q^16 + 528 * q^22 + 40 * q^25 + 2392 * q^37 + 328 * q^43 + 2784 * q^46 + 6744 * q^58 + 5432 * q^64 - 616 * q^67 + 4352 * q^79 - 4608 * q^85 - 1416 * q^88

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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