LMFDB - Embedded newform 1444.3.c.d.721.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1444,3,Mod(721,1444)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1444, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1444.721");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1444 = 2^{2} \cdot 19^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1444.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:	$39.3461501736$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			721.11
Character	$\chi$	$=$	1444.721
Dual form			1444.3.c.d.721.14

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.174362i q^{3} -0.292748 q^{5} -5.22349 q^{7} +8.96960 q^{9} +O(q^{10})$	$q-0.174362i q^{3} -0.292748 q^{5} -5.22349 q^{7} +8.96960 q^{9} +3.31534 q^{11} -12.1425i q^{13} +0.0510442i q^{15} -7.62097 q^{17} +0.910778i q^{21} +8.91299 q^{23} -24.9143 q^{25} -3.13322i q^{27} +3.24116i q^{29} -15.5816i q^{31} -0.578069i q^{33} +1.52917 q^{35} +35.5500i q^{37} -2.11719 q^{39} -35.0238i q^{41} +45.0456 q^{43} -2.62583 q^{45} -56.0380 q^{47} -21.7152 q^{49} +1.32881i q^{51} +53.1203i q^{53} -0.970560 q^{55} -95.4339i q^{59} +64.9620 q^{61} -46.8526 q^{63} +3.55470i q^{65} -37.3961i q^{67} -1.55409i q^{69} -86.4658i q^{71} -23.5945 q^{73} +4.34411i q^{75} -17.3176 q^{77} -120.503i q^{79} +80.1801 q^{81} -106.176 q^{83} +2.23103 q^{85} +0.565135 q^{87} -112.628i q^{89} +63.4262i q^{91} -2.71684 q^{93} -43.1255i q^{97} +29.7373 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 12 q^{5} + 40 q^{7} - 104 q^{9}+O(q^{10}) $ 24 * q - 12 * q^5 + 40 * q^7 - 104 * q^9	$ 24 q - 12 q^{5} + 40 q^{7} - 104 q^{9} - 64 q^{11} - 60 q^{17} + 100 q^{23} + 24 q^{25} - 144 q^{35} + 68 q^{39} - 16 q^{43} + 248 q^{45} + 180 q^{47} - 208 q^{49} - 532 q^{55} + 104 q^{61} - 1216 q^{63} - 484 q^{73} + 136 q^{77} + 460 q^{81} + 744 q^{83} + 32 q^{85} - 1296 q^{87} - 256 q^{93} + 804 q^{99}+O(q^{100}) $ 24 * q - 12 * q^5 + 40 * q^7 - 104 * q^9 - 64 * q^11 - 60 * q^17 + 100 * q^23 + 24 * q^25 - 144 * q^35 + 68 * q^39 - 16 * q^43 + 248 * q^45 + 180 * q^47 - 208 * q^49 - 532 * q^55 + 104 * q^61 - 1216 * q^63 - 484 * q^73 + 136 * q^77 + 460 * q^81 + 744 * q^83 + 32 * q^85 - 1296 * q^87 - 256 * q^93 + 804 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1444\mathbb{Z}\right)^\times$.

$n$	$723$	$1085$
$\chi(n)$	$1$	$-1$

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	0
$2$	0	0
$3$	− 0.174362i	− 0.0581207i	−0.999578	−	0.0290603i	$-0.990749\pi$
$3$	− 0.174362i	− 0.0581207i	0.999578	−	0.0290603i	$-0.00925150\pi$
$4$	0	0
$5$	−0.292748	−0.0585496	−0.0292748	−	0.999571i	$-0.509320\pi$
$5$	−0.292748	−0.0585496	−0.0292748	+	0.999571i	$0.509320\pi$
$6$	0	0
$7$	−5.22349	−0.746212	−0.373106	−	0.927789i	$-0.621707\pi$
$7$	−5.22349	−0.746212	−0.373106	+	0.927789i	$0.621707\pi$
$8$	0	0
$9$	8.96960	0.996622
$10$	0	0
$11$	3.31534	0.301395	0.150697	−	0.988580i	$-0.451848\pi$
$11$	3.31534	0.301395	0.150697	+	0.988580i	$0.451848\pi$
$12$	0	0
$13$	− 12.1425i	− 0.934039i	−0.884247	−	0.467020i	$-0.845328\pi$
$13$	− 12.1425i	− 0.934039i	0.884247	−	0.467020i	$-0.154672\pi$
$14$	0	0
$15$	0.0510442i	0.00340295i
$16$	0	0
$17$	−7.62097	−0.448293	−0.224146	−	0.974556i	$-0.571959\pi$
$17$	−7.62097	−0.448293	−0.224146	+	0.974556i	$0.571959\pi$
$18$	0	0
$19$	0	0
$19$	0	0
$20$	0	0
$21$	0.910778i	0.0433704i
$22$	0	0
$23$	8.91299	0.387521	0.193761	−	0.981049i	$-0.437931\pi$
$23$	8.91299	0.387521	0.193761	+	0.981049i	$0.437931\pi$
$24$	0	0
$25$	−24.9143	−0.996572
$26$	0	0
$27$	− 3.13322i	− 0.116045i
$28$	0	0
$29$	3.24116i	0.111764i	0.998437	+	0.0558820i	$0.0177971\pi$
$29$	3.24116i	0.111764i	−0.998437	+	0.0558820i	$0.982203\pi$
$30$	0	0
$31$	− 15.5816i	− 0.502632i	−0.967905	−	0.251316i	$-0.919137\pi$
$31$	− 15.5816i	− 0.502632i	0.967905	−	0.251316i	$-0.0808633\pi$
$32$	0	0
$33$	− 0.578069i	− 0.0175173i
$34$	0	0
$35$	1.52917	0.0436905
$36$	0	0
$37$	35.5500i	0.960810i	0.877047	+	0.480405i	$0.159510\pi$
$37$	35.5500i	0.960810i	−0.877047	+	0.480405i	$0.840490\pi$
$38$	0	0
$39$	−2.11719	−0.0542870
$40$	0	0
$41$	− 35.0238i	− 0.854240i	−0.904195	−	0.427120i	$-0.859528\pi$
$41$	− 35.0238i	− 0.854240i	0.904195	−	0.427120i	$-0.140472\pi$
$42$	0	0
$43$	45.0456	1.04757	0.523786	−	0.851850i	$-0.324519\pi$
$43$	45.0456	1.04757	0.523786	+	0.851850i	$0.324519\pi$
$44$	0	0
$45$	−2.62583	−0.0583519
$46$	0	0
$47$	−56.0380	−1.19230	−0.596149	−	0.802874i	$-0.703303\pi$
$47$	−56.0380	−1.19230	−0.596149	+	0.802874i	$0.703303\pi$
$48$	0	0
$49$	−21.7152	−0.443167
$50$	0	0
$51$	1.32881i	0.0260551i
$52$	0	0
$53$	53.1203i	1.00227i	0.865369	+	0.501135i	$0.167084\pi$
$53$	53.1203i	1.00227i	−0.865369	+	0.501135i	$0.832916\pi$
$54$	0	0
$55$	−0.970560	−0.0176465
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 95.4339i	− 1.61752i	−0.588136	−	0.808762i	$-0.700138\pi$
$59$	− 95.4339i	− 1.61752i	0.588136	−	0.808762i	$-0.299862\pi$
$60$	0	0
$61$	64.9620	1.06495	0.532475	−	0.846445i	$-0.321262\pi$
$61$	64.9620	1.06495	0.532475	+	0.846445i	$0.321262\pi$
$62$	0	0
$63$	−46.8526	−0.743692
$64$	0	0
$65$	3.55470i	0.0546877i
$66$	0	0
$67$	− 37.3961i	− 0.558150i	−0.960269	−	0.279075i	$-0.909972\pi$
$67$	− 37.3961i	− 0.558150i	0.960269	−	0.279075i	$-0.0900279\pi$
$68$	0	0
$69$	− 1.55409i	− 0.0225230i
$70$	0	0
$71$	− 86.4658i	− 1.21783i	−0.793236	−	0.608914i	$-0.791605\pi$
$71$	− 86.4658i	− 1.21783i	0.793236	−	0.608914i	$-0.208395\pi$
$72$	0	0
$73$	−23.5945	−0.323212	−0.161606	−	0.986855i	$-0.551667\pi$
$73$	−23.5945	−0.323212	−0.161606	+	0.986855i	$0.551667\pi$
$74$	0	0
$75$	4.34411i	0.0579214i
$76$	0	0
$77$	−17.3176	−0.224904
$78$	0	0
$79$	− 120.503i	− 1.52535i	−0.646781	−	0.762675i	$-0.723885\pi$
$79$	− 120.503i	− 1.52535i	0.646781	−	0.762675i	$-0.276115\pi$
$80$	0	0
$81$	80.1801	0.989877
$82$	0	0
$83$	−106.176	−1.27922	−0.639612	−	0.768698i	$-0.720905\pi$
$83$	−106.176	−1.27922	−0.639612	+	0.768698i	$0.720905\pi$
$84$	0	0
$85$	2.23103	0.0262474
$86$	0	0
$87$	0.565135	0.00649580
$88$	0	0
$89$	− 112.628i	− 1.26548i	−0.774365	−	0.632739i	$-0.781931\pi$
$89$	− 112.628i	− 1.26548i	0.774365	−	0.632739i	$-0.218069\pi$
$90$	0	0
$91$	63.4262i	0.696992i
$92$	0	0
$93$	−2.71684	−0.0292133
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 43.1255i	− 0.444593i	−0.974979	−	0.222296i	$-0.928645\pi$
$97$	− 43.1255i	− 0.444593i	0.974979	−	0.222296i	$-0.0713552\pi$
$98$	0	0
$99$	29.7373	0.300376
$100$	0	0
$101$	−140.809	−1.39414	−0.697072	−	0.717001i	$-0.745514\pi$
$101$	−140.809	−1.39414	−0.697072	+	0.717001i	$0.745514\pi$
$102$	0	0
$103$	− 16.2617i	− 0.157880i	−0.996879	−	0.0789401i	$-0.974846\pi$
$103$	− 16.2617i	− 0.157880i	0.996879	−	0.0789401i	$-0.0251536\pi$
$104$	0	0
$105$	− 0.266629i	− 0.00253932i
$106$	0	0
$107$	− 154.304i	− 1.44209i	−0.692889	−	0.721044i	$-0.743662\pi$
$107$	− 154.304i	− 1.44209i	0.692889	−	0.721044i	$-0.256338\pi$
$108$	0	0
$109$	− 62.5925i	− 0.574243i	−0.957894	−	0.287121i	$-0.907302\pi$
$109$	− 62.5925i	− 0.574243i	0.957894	−	0.287121i	$-0.0926983\pi$
$110$	0	0
$111$	6.19856	0.0558429
$112$	0	0
$113$	158.802i	1.40533i	0.711522	+	0.702664i	$0.248006\pi$
$113$	158.802i	1.40533i	−0.711522	+	0.702664i	$0.751994\pi$
$114$	0	0
$115$	−2.60926	−0.0226892
$116$	0	0
$117$	− 108.913i	− 0.930884i
$118$	0	0
$119$	39.8081	0.334521
$120$	0	0
$121$	−110.009	−0.909161
$122$	0	0
$123$	−6.10683	−0.0496490
$124$	0	0
$125$	14.6123	0.116899
$126$	0	0
$127$	− 156.747i	− 1.23423i	−0.786874	−	0.617114i	$-0.788302\pi$
$127$	− 156.747i	− 1.23423i	0.786874	−	0.617114i	$-0.211698\pi$
$128$	0	0
$129$	− 7.85424i	− 0.0608856i
$130$	0	0
$131$	−182.901	−1.39619	−0.698095	−	0.716005i	$-0.745969\pi$
$131$	−182.901	−1.39619	−0.698095	+	0.716005i	$0.745969\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0.917243i	0.00679440i
$136$	0	0
$137$	139.500	1.01825	0.509125	−	0.860693i	$-0.329969\pi$
$137$	139.500	1.01825	0.509125	+	0.860693i	$0.329969\pi$
$138$	0	0
$139$	−232.172	−1.67030	−0.835150	−	0.550023i	$-0.814619\pi$
$139$	−232.172	−1.67030	−0.835150	+	0.550023i	$0.814619\pi$
$140$	0	0
$141$	9.77089i	0.0692971i
$142$	0	0
$143$	− 40.2566i	− 0.281514i
$144$	0	0
$145$	− 0.948843i	− 0.00654374i
$146$	0	0
$147$	3.78631i	0.0257572i
$148$	0	0
$149$	30.3358	0.203596	0.101798	−	0.994805i	$-0.467540\pi$
$149$	30.3358	0.203596	0.101798	+	0.994805i	$0.467540\pi$
$150$	0	0
$151$	116.639i	0.772446i	0.922405	+	0.386223i	$0.126220\pi$
$151$	116.639i	0.772446i	−0.922405	+	0.386223i	$0.873780\pi$
$152$	0	0
$153$	−68.3571	−0.446778
$154$	0	0
$155$	4.56148i	0.0294289i
$156$	0	0
$157$	256.980	1.63681	0.818407	−	0.574640i	$-0.194858\pi$
$157$	256.980	1.63681	0.818407	+	0.574640i	$0.194858\pi$
$158$	0	0
$159$	9.26217	0.0582526
$160$	0	0
$161$	−46.5569	−0.289173
$162$	0	0
$163$	−66.6571	−0.408939	−0.204470	−	0.978873i	$-0.565547\pi$
$163$	−66.6571	−0.408939	−0.204470	+	0.978873i	$0.565547\pi$
$164$	0	0
$165$	0.169229i	0.00102563i
$166$	0	0
$167$	273.919i	1.64024i	0.572195	+	0.820118i	$0.306092\pi$
$167$	273.919i	1.64024i	−0.572195	+	0.820118i	$0.693908\pi$
$168$	0	0
$169$	21.5594	0.127570
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 182.388i	− 1.05426i	−0.849783	−	0.527132i	$-0.823267\pi$
$173$	− 182.388i	− 1.05426i	0.849783	−	0.527132i	$-0.176733\pi$
$174$	0	0
$175$	130.139	0.743654
$176$	0	0
$177$	−16.6400	−0.0940116
$178$	0	0
$179$	− 148.715i	− 0.830811i	−0.909636	−	0.415405i	$-0.863640\pi$
$179$	− 148.715i	− 0.830811i	0.909636	−	0.415405i	$-0.136360\pi$
$180$	0	0
$181$	75.2066i	0.415506i	0.978181	+	0.207753i	$0.0666151\pi$
$181$	75.2066i	0.415506i	−0.978181	+	0.207753i	$0.933385\pi$
$182$	0	0
$183$	− 11.3269i	− 0.0618957i
$184$	0	0
$185$	− 10.4072i	− 0.0562551i
$186$	0	0
$187$	−25.2661	−0.135113
$188$	0	0
$189$	16.3663i	0.0865942i
$190$	0	0
$191$	214.541	1.12325	0.561627	−	0.827391i	$-0.310176\pi$
$191$	214.541	1.12325	0.561627	+	0.827391i	$0.310176\pi$
$192$	0	0
$193$	− 255.791i	− 1.32534i	−0.748910	−	0.662672i	$-0.769422\pi$
$193$	− 255.791i	− 1.32534i	0.748910	−	0.662672i	$-0.230578\pi$
$194$	0	0
$195$	0.619805	0.00317848
$196$	0	0
$197$	−84.8620	−0.430772	−0.215386	−	0.976529i	$-0.569101\pi$
$197$	−84.8620	−0.430772	−0.215386	+	0.976529i	$0.569101\pi$
$198$	0	0
$199$	−81.5447	−0.409772	−0.204886	−	0.978786i	$-0.565682\pi$
$199$	−81.5447	−0.409772	−0.204886	+	0.978786i	$0.565682\pi$
$200$	0	0
$201$	−6.52046	−0.0324401
$202$	0	0
$203$	− 16.9301i	− 0.0833997i
$204$	0	0
$205$	10.2532i	0.0500154i
$206$	0	0
$207$	79.9459	0.386212
$208$	0	0
$209$	0	0
$210$	0	0
$211$	− 121.687i	− 0.576715i	−0.957523	−	0.288357i	$-0.906891\pi$
$211$	− 121.687i	− 0.576715i	0.957523	−	0.288357i	$-0.0931091\pi$
$212$	0	0
$213$	−15.0764	−0.0707810
$214$	0	0
$215$	−13.1870	−0.0613350
$216$	0	0
$217$	81.3902i	0.375070i
$218$	0	0
$219$	4.11399i	0.0187853i
$220$	0	0
$221$	92.5378i	0.418723i
$222$	0	0
$223$	− 190.354i	− 0.853604i	−0.904345	−	0.426802i	$-0.859640\pi$
$223$	− 190.354i	− 0.853604i	0.904345	−	0.426802i	$-0.140360\pi$
$224$	0	0
$225$	−223.471	−0.993206
$226$	0	0
$227$	− 54.6157i	− 0.240598i	−0.992738	−	0.120299i	$-0.961615\pi$
$227$	− 54.6157i	− 0.240598i	0.992738	−	0.120299i	$-0.0383853\pi$
$228$	0	0
$229$	−276.771	−1.20861	−0.604303	−	0.796755i	$-0.706548\pi$
$229$	−276.771	−1.20861	−0.604303	+	0.796755i	$0.706548\pi$
$230$	0	0
$231$	3.01954i	0.0130716i
$232$	0	0
$233$	141.042	0.605330	0.302665	−	0.953097i	$-0.402124\pi$
$233$	141.042	0.605330	0.302665	+	0.953097i	$0.402124\pi$
$234$	0	0
$235$	16.4050	0.0698086
$236$	0	0
$237$	−21.0111	−0.0886544
$238$	0	0
$239$	−131.214	−0.549012	−0.274506	−	0.961585i	$-0.588514\pi$
$239$	−131.214	−0.549012	−0.274506	+	0.961585i	$0.588514\pi$
$240$	0	0
$241$	56.8804i	0.236018i	0.993013	+	0.118009i	$0.0376512\pi$
$241$	56.8804i	0.236018i	−0.993013	+	0.118009i	$0.962349\pi$
$242$	0	0
$243$	− 42.1793i	− 0.173577i
$244$	0	0
$245$	6.35708	0.0259473
$246$	0	0
$247$	0	0
$248$	0	0
$249$	18.5130i	0.0743494i
$250$	0	0
$251$	−28.6887	−0.114297	−0.0571487	−	0.998366i	$-0.518201\pi$
$251$	−28.6887	−0.114297	−0.0571487	+	0.998366i	$0.518201\pi$
$252$	0	0
$253$	29.5496	0.116797
$254$	0	0
$255$	− 0.389006i	− 0.00152552i
$256$	0	0
$257$	− 240.125i	− 0.934339i	−0.884168	−	0.467169i	$-0.845274\pi$
$257$	− 240.125i	− 0.934339i	0.884168	−	0.467169i	$-0.154726\pi$
$258$	0	0
$259$	− 185.695i	− 0.716968i
$260$	0	0
$261$	29.0719i	0.111387i
$262$	0	0
$263$	399.493	1.51899	0.759493	−	0.650516i	$-0.225447\pi$
$263$	399.493	1.51899	0.759493	+	0.650516i	$0.225447\pi$
$264$	0	0
$265$	− 15.5509i	− 0.0586826i
$266$	0	0
$267$	−19.6380	−0.0735504
$268$	0	0
$269$	− 42.9822i	− 0.159785i	−0.996803	−	0.0798925i	$-0.974542\pi$
$269$	− 42.9822i	− 0.159785i	0.996803	−	0.0798925i	$-0.0254577\pi$
$270$	0	0
$271$	−347.842	−1.28355	−0.641776	−	0.766893i	$-0.721802\pi$
$271$	−347.842	−1.28355	−0.641776	+	0.766893i	$0.721802\pi$
$272$	0	0
$273$	11.0591	0.0405096
$274$	0	0
$275$	−82.5994	−0.300361
$276$	0	0
$277$	274.783	0.991997	0.495998	−	0.868323i	$-0.334802\pi$
$277$	274.783	0.991997	0.495998	+	0.868323i	$0.334802\pi$
$278$	0	0
$279$	− 139.761i	− 0.500934i
$280$	0	0
$281$	− 320.931i	− 1.14210i	−0.820914	−	0.571052i	$-0.806535\pi$
$281$	− 320.931i	− 1.14210i	0.820914	−	0.571052i	$-0.193465\pi$
$282$	0	0
$283$	3.55529	0.0125629	0.00628144	−	0.999980i	$-0.498001\pi$
$283$	3.55529	0.0125629	0.00628144	+	0.999980i	$0.498001\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	182.946i	0.637444i
$288$	0	0
$289$	−230.921	−0.799034
$290$	0	0
$291$	−7.51945	−0.0258400
$292$	0	0
$293$	401.937i	1.37180i	0.727696	+	0.685900i	$0.240591\pi$
$293$	401.937i	1.37180i	−0.727696	+	0.685900i	$0.759409\pi$
$294$	0	0
$295$	27.9381i	0.0947054i
$296$	0	0
$297$	− 10.3877i	− 0.0349753i
$298$	0	0
$299$	− 108.226i	− 0.361960i
$300$	0	0
$301$	−235.295	−0.781711
$302$	0	0
$303$	24.5517i	0.0810286i
$304$	0	0
$305$	−19.0175	−0.0623525
$306$	0	0
$307$	361.693i	1.17815i	0.808077	+	0.589077i	$0.200509\pi$
$307$	361.693i	1.17815i	−0.808077	+	0.589077i	$0.799491\pi$
$308$	0	0
$309$	−2.83542	−0.00917611
$310$	0	0
$311$	−165.716	−0.532849	−0.266425	−	0.963856i	$-0.585842\pi$
$311$	−165.716	−0.532849	−0.266425	+	0.963856i	$0.585842\pi$
$312$	0	0
$313$	287.796	0.919478	0.459739	−	0.888054i	$-0.347943\pi$
$313$	287.796	0.919478	0.459739	+	0.888054i	$0.347943\pi$
$314$	0	0
$315$	13.7160	0.0435429
$316$	0	0
$317$	250.354i	0.789760i	0.918733	+	0.394880i	$0.129214\pi$
$317$	250.354i	0.789760i	−0.918733	+	0.394880i	$0.870786\pi$
$318$	0	0
$319$	10.7455i	0.0336851i
$320$	0	0
$321$	−26.9047	−0.0838152
$322$	0	0
$323$	0	0
$324$	0	0
$325$	302.522i	0.930837i
$326$	0	0
$327$	−10.9138	−0.0333754
$328$	0	0
$329$	292.714	0.889707
$330$	0	0
$331$	− 272.358i	− 0.822833i	−0.911448	−	0.411416i	$-0.865034\pi$
$331$	− 272.358i	− 0.822833i	0.911448	−	0.411416i	$-0.134966\pi$
$332$	0	0
$333$	318.869i	0.957564i
$334$	0	0
$335$	10.9476i	0.0326795i
$336$	0	0
$337$	556.457i	1.65121i	0.564249	+	0.825604i	$0.309166\pi$
$337$	556.457i	1.65121i	−0.564249	+	0.825604i	$0.690834\pi$
$338$	0	0
$339$	27.6890	0.0816786
$340$	0	0
$341$	− 51.6583i	− 0.151491i
$342$	0	0
$343$	369.380	1.07691
$344$	0	0
$345$	0.454956i	0.00131871i
$346$	0	0
$347$	−289.875	−0.835375	−0.417688	−	0.908591i	$-0.637159\pi$
$347$	−289.875	−0.835375	−0.417688	+	0.908591i	$0.637159\pi$
$348$	0	0
$349$	400.968	1.14891	0.574453	−	0.818538i	$-0.305215\pi$
$349$	400.968	1.14891	0.574453	+	0.818538i	$0.305215\pi$
$350$	0	0
$351$	−38.0451	−0.108391
$352$	0	0
$353$	−34.0370	−0.0964222	−0.0482111	−	0.998837i	$-0.515352\pi$
$353$	−34.0370	−0.0964222	−0.0482111	+	0.998837i	$0.515352\pi$
$354$	0	0
$355$	25.3127i	0.0713034i
$356$	0	0
$357$	− 6.94101i	− 0.0194426i
$358$	0	0
$359$	78.3786	0.218325	0.109162	−	0.994024i	$-0.465183\pi$
$359$	78.3786	0.218325	0.109162	+	0.994024i	$0.465183\pi$
$360$	0	0
$361$	0	0
$362$	0	0
$363$	19.1813i	0.0528411i
$364$	0	0
$365$	6.90725	0.0189240
$366$	0	0
$367$	315.805	0.860505	0.430253	−	0.902709i	$-0.358425\pi$
$367$	315.805	0.860505	0.430253	+	0.902709i	$0.358425\pi$
$368$	0	0
$369$	− 314.150i	− 0.851354i
$370$	0	0
$371$	− 277.473i	− 0.747906i
$372$	0	0
$373$	515.730i	1.38265i	0.722542	+	0.691327i	$0.242974\pi$
$373$	515.730i	1.38265i	−0.722542	+	0.691327i	$0.757026\pi$
$374$	0	0
$375$	− 2.54783i	− 0.00679422i
$376$	0	0
$377$	39.3558	0.104392
$378$	0	0
$379$	− 666.800i	− 1.75937i	−0.475561	−	0.879683i	$-0.657755\pi$
$379$	− 666.800i	− 1.75937i	0.475561	−	0.879683i	$-0.342245\pi$
$380$	0	0
$381$	−27.3307	−0.0717342
$382$	0	0
$383$	657.540i	1.71681i	0.512969	+	0.858407i	$0.328546\pi$
$383$	657.540i	1.71681i	−0.512969	+	0.858407i	$0.671454\pi$
$384$	0	0
$385$	5.06971	0.0131681
$386$	0	0
$387$	404.041	1.04403
$388$	0	0
$389$	524.051	1.34718	0.673588	−	0.739107i	$-0.264752\pi$
$389$	524.051	1.34718	0.673588	+	0.739107i	$0.264752\pi$
$390$	0	0
$391$	−67.9257	−0.173723
$392$	0	0
$393$	31.8910i	0.0811475i
$394$	0	0
$395$	35.2770i	0.0893087i
$396$	0	0
$397$	134.549	0.338914	0.169457	−	0.985538i	$-0.445799\pi$
$397$	134.549	0.338914	0.169457	+	0.985538i	$0.445799\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	37.6119i	0.0937953i	0.998900	+	0.0468977i	$0.0149335\pi$
$401$	37.6119i	0.0937953i	−0.998900	+	0.0468977i	$0.985067\pi$
$402$	0	0
$403$	−189.200	−0.469478
$404$	0	0
$405$	−23.4726	−0.0579570
$406$	0	0
$407$	117.860i	0.289583i
$408$	0	0
$409$	548.225i	1.34040i	0.742179	+	0.670202i	$0.233793\pi$
$409$	548.225i	1.34040i	−0.742179	+	0.670202i	$0.766207\pi$
$410$	0	0
$411$	− 24.3235i	− 0.0591814i
$412$	0	0
$413$	498.498i	1.20702i
$414$	0	0
$415$	31.0827	0.0748981
$416$	0	0
$417$	40.4819i	0.0970789i
$418$	0	0
$419$	167.251	0.399167	0.199584	−	0.979881i	$-0.436041\pi$
$419$	167.251	0.399167	0.199584	+	0.979881i	$0.436041\pi$
$420$	0	0
$421$	254.971i	0.605632i	0.953049	+	0.302816i	$0.0979267\pi$
$421$	254.971i	0.605632i	−0.953049	+	0.302816i	$0.902073\pi$
$422$	0	0
$423$	−502.638	−1.18827
$424$	0	0
$425$	189.871	0.446756
$426$	0	0
$427$	−339.328	−0.794679
$428$	0	0
$429$	−7.01921	−0.0163618
$430$	0	0
$431$	806.053i	1.87019i	0.354395	+	0.935096i	$0.384687\pi$
$431$	806.053i	1.87019i	−0.354395	+	0.935096i	$0.615313\pi$
$432$	0	0
$433$	− 442.305i	− 1.02149i	−0.859732	−	0.510745i	$-0.829370\pi$
$433$	− 442.305i	− 1.02149i	0.859732	−	0.510745i	$-0.170630\pi$
$434$	0	0
$435$	−0.165442	−0.000380327 0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	190.927i	0.434913i	0.976070	+	0.217456i	$0.0697759\pi$
$439$	190.927i	0.434913i	−0.976070	+	0.217456i	$0.930224\pi$
$440$	0	0
$441$	−194.777	−0.441670
$442$	0	0
$443$	703.141	1.58722	0.793612	−	0.608424i	$-0.208198\pi$
$443$	703.141	1.58722	0.793612	+	0.608424i	$0.208198\pi$
$444$	0	0
$445$	32.9715i	0.0740933i
$446$	0	0
$447$	− 5.28942i	− 0.0118331i
$448$	0	0
$449$	741.132i	1.65063i	0.564674	+	0.825314i	$0.309002\pi$
$449$	741.132i	1.65063i	−0.564674	+	0.825314i	$0.690998\pi$
$450$	0	0
$451$	− 116.116i	− 0.257463i
$452$	0	0
$453$	20.3375	0.0448951
$454$	0	0
$455$	− 18.5679i	− 0.0408086i
$456$	0	0
$457$	−163.274	−0.357274	−0.178637	−	0.983915i	$-0.557169\pi$
$457$	−163.274	−0.357274	−0.178637	+	0.983915i	$0.557169\pi$
$458$	0	0
$459$	23.8782i	0.0520221i
$460$	0	0
$461$	176.160	0.382125	0.191062	−	0.981578i	$-0.438807\pi$
$461$	176.160	0.382125	0.191062	+	0.981578i	$0.438807\pi$
$462$	0	0
$463$	214.968	0.464293	0.232146	−	0.972681i	$-0.425425\pi$
$463$	214.968	0.464293	0.232146	+	0.972681i	$0.425425\pi$
$464$	0	0
$465$	0.795350	0.00171043
$466$	0	0
$467$	79.9859	0.171276	0.0856380	−	0.996326i	$-0.472707\pi$
$467$	79.9859	0.171276	0.0856380	+	0.996326i	$0.472707\pi$
$468$	0	0
$469$	195.338i	0.416499i
$470$	0	0
$471$	− 44.8075i	− 0.0951327i
$472$	0	0
$473$	149.341	0.315733
$474$	0	0
$475$	0	0
$476$	0	0
$477$	476.468i	0.998884i
$478$	0	0
$479$	−752.664	−1.57132	−0.785662	−	0.618656i	$-0.787678\pi$
$479$	−752.664	−1.57132	−0.785662	+	0.618656i	$0.787678\pi$
$480$	0	0
$481$	431.666	0.897434
$482$	0	0
$483$	8.11775i	0.0168069i
$484$	0	0
$485$	12.6249i	0.0260307i
$486$	0	0
$487$	− 688.105i	− 1.41295i	−0.707740	−	0.706473i	$-0.750285\pi$
$487$	− 688.105i	− 1.41295i	0.707740	−	0.706473i	$-0.249715\pi$
$488$	0	0
$489$	11.6225i	0.0237678i
$490$	0	0
$491$	499.410	1.01713	0.508564	−	0.861024i	$-0.330177\pi$
$491$	499.410	1.01713	0.508564	+	0.861024i	$0.330177\pi$
$492$	0	0
$493$	− 24.7008i	− 0.0501030i
$494$	0	0
$495$	−8.70553	−0.0175869
$496$	0	0
$497$	451.653i	0.908758i
$498$	0	0
$499$	−901.718	−1.80705	−0.903525	−	0.428535i	$-0.859030\pi$
$499$	−901.718	−1.80705	−0.903525	+	0.428535i	$0.859030\pi$
$500$	0	0
$501$	47.7611	0.0953316
$502$	0	0
$503$	−5.24574	−0.0104289	−0.00521445	−	0.999986i	$-0.501660\pi$
$503$	−5.24574	−0.0104289	−0.00521445	+	0.999986i	$0.501660\pi$
$504$	0	0
$505$	41.2215	0.0816267
$506$	0	0
$507$	− 3.75914i	− 0.00741448i
$508$	0	0
$509$	− 772.423i	− 1.51753i	−0.651364	−	0.758765i	$-0.725803\pi$
$509$	− 772.423i	− 1.51753i	0.651364	−	0.758765i	$-0.274197\pi$
$510$	0	0
$511$	123.246	0.241185
$512$	0	0
$513$	0	0
$514$	0	0
$515$	4.76057i	0.00924383i
$516$	0	0
$517$	−185.785	−0.359352
$518$	0	0
$519$	−31.8015	−0.0612746
$520$	0	0
$521$	924.538i	1.77454i	0.461246	+	0.887272i	$0.347403\pi$
$521$	924.538i	1.77454i	−0.461246	+	0.887272i	$0.652597\pi$
$522$	0	0
$523$	− 599.716i	− 1.14668i	−0.819316	−	0.573342i	$-0.805647\pi$
$523$	− 599.716i	− 1.14668i	0.819316	−	0.573342i	$-0.194353\pi$
$524$	0	0
$525$	− 22.6914i	− 0.0432217i
$526$	0	0
$527$	118.747i	0.225326i
$528$	0	0
$529$	−449.559	−0.849827
$530$	0	0
$531$	− 856.004i	− 1.61206i
$532$	0	0
$533$	−425.277	−0.797894
$534$	0	0
$535$	45.1721i	0.0844338i
$536$	0	0
$537$	−25.9303	−0.0482873
$538$	0	0
$539$	−71.9932	−0.133568
$540$	0	0
$541$	248.930	0.460129	0.230065	−	0.973175i	$-0.426106\pi$
$541$	248.930	0.460129	0.230065	+	0.973175i	$0.426106\pi$
$542$	0	0
$543$	13.1132	0.0241495
$544$	0	0
$545$	18.3238i	0.0336217i
$546$	0	0
$547$	− 210.495i	− 0.384816i	−0.981315	−	0.192408i	$-0.938370\pi$
$547$	− 210.495i	− 0.384816i	0.981315	−	0.192408i	$-0.0616298\pi$
$548$	0	0
$549$	582.683	1.06135
$550$	0	0
$551$	0	0
$552$	0	0
$553$	629.444i	1.13824i
$554$	0	0
$555$	−1.81462	−0.00326958
$556$	0	0
$557$	775.558	1.39238	0.696192	−	0.717856i	$-0.254876\pi$
$557$	775.558	1.39238	0.696192	+	0.717856i	$0.254876\pi$
$558$	0	0
$559$	− 546.967i	− 0.978474i
$560$	0	0
$561$	4.40545i	0.00785286i
$562$	0	0
$563$	525.266i	0.932977i	0.884527	+	0.466489i	$0.154481\pi$
$563$	525.266i	0.932977i	−0.884527	+	0.466489i	$0.845519\pi$
$564$	0	0
$565$	− 46.4890i	− 0.0822814i
$566$	0	0
$567$	−418.819	−0.738659
$568$	0	0
$569$	238.642i	0.419407i	0.977765	+	0.209703i	$0.0672498\pi$
$569$	238.642i	0.419407i	−0.977765	+	0.209703i	$0.932750\pi$
$570$	0	0
$571$	37.4826	0.0656437	0.0328219	−	0.999461i	$-0.489551\pi$
$571$	37.4826	0.0656437	0.0328219	+	0.999461i	$0.489551\pi$
$572$	0	0
$573$	− 37.4079i	− 0.0652843i
$574$	0	0
$575$	−222.061	−0.386193
$576$	0	0
$577$	142.149	0.246358	0.123179	−	0.992384i	$-0.460691\pi$
$577$	142.149	0.246358	0.123179	+	0.992384i	$0.460691\pi$
$578$	0	0
$579$	−44.6003	−0.0770299
$580$	0	0
$581$	554.607	0.954573
$582$	0	0
$583$	176.112i	0.302079i
$584$	0	0
$585$	31.8842i	0.0545029i
$586$	0	0
$587$	−674.791	−1.14956	−0.574779	−	0.818309i	$-0.694912\pi$
$587$	−674.791	−1.14956	−0.574779	+	0.818309i	$0.694912\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	14.7967i	0.0250367i
$592$	0	0
$593$	297.336	0.501410	0.250705	−	0.968064i	$-0.419338\pi$
$593$	297.336	0.501410	0.250705	+	0.968064i	$0.419338\pi$
$594$	0	0
$595$	−11.6537	−0.0195861
$596$	0	0
$597$	14.2183i	0.0238162i
$598$	0	0
$599$	305.773i	0.510473i	0.966879	+	0.255237i	$0.0821534\pi$
$599$	305.773i	0.510473i	−0.966879	+	0.255237i	$0.917847\pi$
$600$	0	0
$601$	247.113i	0.411170i	0.978639	+	0.205585i	$0.0659097\pi$
$601$	247.113i	0.411170i	−0.978639	+	0.205585i	$0.934090\pi$
$602$	0	0
$603$	− 335.428i	− 0.556265i
$604$	0	0
$605$	32.2048	0.0532311
$606$	0	0
$607$	− 672.112i	− 1.10727i	−0.832760	−	0.553634i	$-0.813241\pi$
$607$	− 672.112i	− 1.10727i	0.832760	−	0.553634i	$-0.186759\pi$
$608$	0	0
$609$	−2.95197	−0.00484725
$610$	0	0
$611$	680.442i	1.11365i
$612$	0	0
$613$	−106.368	−0.173521	−0.0867603	−	0.996229i	$-0.527651\pi$
$613$	−106.368	−0.173521	−0.0867603	+	0.996229i	$0.527651\pi$
$614$	0	0
$615$	1.78776	0.00290693
$616$	0	0
$617$	436.946	0.708178	0.354089	−	0.935212i	$-0.384791\pi$
$617$	436.946	0.708178	0.354089	+	0.935212i	$0.384791\pi$
$618$	0	0
$619$	−244.400	−0.394831	−0.197415	−	0.980320i	$-0.563255\pi$
$619$	−244.400	−0.394831	−0.197415	+	0.980320i	$0.563255\pi$
$620$	0	0
$621$	− 27.9263i	− 0.0449699i
$622$	0	0
$623$	588.308i	0.944315i
$624$	0	0
$625$	618.580	0.989728
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 270.925i	− 0.430724i
$630$	0	0
$631$	−447.601	−0.709351	−0.354676	−	0.934989i	$-0.615409\pi$
$631$	−447.601	−0.709351	−0.354676	+	0.934989i	$0.615409\pi$
$632$	0	0
$633$	−21.2176	−0.0335191
$634$	0	0
$635$	45.8874i	0.0722636i
$636$	0	0
$637$	263.677i	0.413936i
$638$	0	0
$639$	− 775.564i	− 1.21371i
$640$	0	0
$641$	728.377i	1.13631i	0.822920	+	0.568157i	$0.192343\pi$
$641$	728.377i	1.13631i	−0.822920	+	0.568157i	$0.807657\pi$
$642$	0	0
$643$	353.282	0.549427	0.274714	−	0.961526i	$-0.411417\pi$
$643$	353.282	0.549427	0.274714	+	0.961526i	$0.411417\pi$
$644$	0	0
$645$	2.29932i	0.00356483i
$646$	0	0
$647$	839.172	1.29702	0.648510	−	0.761206i	$-0.275393\pi$
$647$	839.172	1.29702	0.648510	+	0.761206i	$0.275393\pi$
$648$	0	0
$649$	− 316.396i	− 0.487513i
$650$	0	0
$651$	14.1914	0.0217993
$652$	0	0
$653$	−399.499	−0.611790	−0.305895	−	0.952065i	$-0.598956\pi$
$653$	−399.499	−0.611790	−0.305895	+	0.952065i	$0.598956\pi$
$654$	0	0
$655$	53.5439	0.0817464
$656$	0	0
$657$	−211.633	−0.322120
$658$	0	0
$659$	− 66.9789i	− 0.101637i	−0.998708	−	0.0508186i	$-0.983817\pi$
$659$	− 66.9789i	− 0.101637i	0.998708	−	0.0508186i	$-0.0161830\pi$
$660$	0	0
$661$	6.00669i	0.00908727i	0.999990	+	0.00454364i	$0.00144629\pi$
$661$	6.00669i	0.00908727i	−0.999990	+	0.00454364i	$0.998554\pi$
$662$	0	0
$663$	16.1351	0.0243365
$664$	0	0
$665$	0	0
$666$	0	0
$667$	28.8884i	0.0433109i
$668$	0	0
$669$	−33.1905	−0.0496121
$670$	0	0
$671$	215.371	0.320970
$672$	0	0
$673$	− 90.5848i	− 0.134599i	−0.997733	−	0.0672993i	$-0.978562\pi$
$673$	− 90.5848i	− 0.134599i	0.997733	−	0.0672993i	$-0.0214382\pi$
$674$	0	0
$675$	78.0619i	0.115647i
$676$	0	0
$677$	885.013i	1.30726i	0.756816	+	0.653628i	$0.226754\pi$
$677$	885.013i	1.30726i	−0.756816	+	0.653628i	$0.773246\pi$
$678$	0	0
$679$	225.265i	0.331760i
$680$	0	0
$681$	−9.52290	−0.0139837
$682$	0	0
$683$	1033.98i	1.51387i	0.653488	+	0.756937i	$0.273305\pi$
$683$	1033.98i	1.51387i	−0.653488	+	0.756937i	$0.726695\pi$
$684$	0	0
$685$	−40.8384	−0.0596182
$686$	0	0
$687$	48.2583i	0.0702450i
$688$	0	0
$689$	645.014	0.936160
$690$	0	0
$691$	736.320	1.06559	0.532793	−	0.846245i	$-0.321142\pi$
$691$	736.320	1.06559	0.532793	+	0.846245i	$0.321142\pi$
$692$	0	0
$693$	−155.332	−0.224145
$694$	0	0
$695$	67.9678	0.0977954
$696$	0	0
$697$	266.916i	0.382949i
$698$	0	0
$699$	− 24.5923i	− 0.0351822i
$700$	0	0
$701$	−916.635	−1.30761	−0.653806	−	0.756663i	$-0.726829\pi$
$701$	−916.635	−1.30761	−0.653806	+	0.756663i	$0.726829\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	− 2.86041i	− 0.00405732i
$706$	0	0
$707$	735.512	1.04033
$708$	0	0
$709$	−603.418	−0.851083	−0.425541	−	0.904939i	$-0.639916\pi$
$709$	−603.418	−0.851083	−0.425541	+	0.904939i	$0.639916\pi$
$710$	0	0
$711$	− 1080.86i	− 1.52020i
$712$	0	0
$713$	− 138.879i	− 0.194781i
$714$	0	0
$715$	11.7850i	0.0164826i
$716$	0	0
$717$	22.8787i	0.0319089i
$718$	0	0
$719$	688.997	0.958271	0.479135	−	0.877741i	$-0.340950\pi$
$719$	688.997	0.958271	0.479135	+	0.877741i	$0.340950\pi$
$720$	0	0
$721$	84.9426i	0.117812i
$722$	0	0
$723$	9.91779	0.0137175
$724$	0	0
$725$	− 80.7512i	− 0.111381i
$726$	0	0
$727$	163.234	0.224530	0.112265	−	0.993678i	$-0.464189\pi$
$727$	163.234	0.224530	0.112265	+	0.993678i	$0.464189\pi$
$728$	0	0
$729$	714.266	0.979789
$730$	0	0
$731$	−343.291	−0.469619
$732$	0	0
$733$	−735.387	−1.00326	−0.501628	−	0.865083i	$-0.667266\pi$
$733$	−735.387	−1.00326	−0.501628	+	0.865083i	$0.667266\pi$
$734$	0	0
$735$	− 1.10843i	− 0.00150807i
$736$	0	0
$737$	− 123.981i	− 0.168223i
$738$	0	0
$739$	−194.299	−0.262922	−0.131461	−	0.991321i	$-0.541967\pi$
$739$	−194.299	−0.262922	−0.131461	+	0.991321i	$0.541967\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 507.324i	− 0.682805i	−0.939917	−	0.341403i	$-0.889098\pi$
$743$	− 507.324i	− 0.682805i	0.939917	−	0.341403i	$-0.110902\pi$
$744$	0	0
$745$	−8.88076	−0.0119205
$746$	0	0
$747$	−952.352	−1.27490
$748$	0	0
$749$	806.002i	1.07610i
$750$	0	0
$751$	509.368i	0.678253i	0.940741	+	0.339126i	$0.110131\pi$
$751$	509.368i	0.678253i	−0.940741	+	0.339126i	$0.889869\pi$
$752$	0	0
$753$	5.00221i	0.00664305i
$754$	0	0
$755$	− 34.1460i	− 0.0452264i
$756$	0	0
$757$	−460.829	−0.608757	−0.304378	−	0.952551i	$-0.598449\pi$
$757$	−460.829	−0.608757	−0.304378	+	0.952551i	$0.598449\pi$
$758$	0	0
$759$	− 5.15233i	− 0.00678831i
$760$	0	0
$761$	−805.958	−1.05908	−0.529539	−	0.848286i	$-0.677635\pi$
$761$	−805.958	−1.05908	−0.529539	+	0.848286i	$0.677635\pi$
$762$	0	0
$763$	326.951i	0.428507i
$764$	0	0
$765$	20.0114	0.0261587
$766$	0	0
$767$	−1158.81	−1.51083
$768$	0	0
$769$	−292.257	−0.380048	−0.190024	−	0.981779i	$-0.560857\pi$
$769$	−292.257	−0.380048	−0.190024	+	0.981779i	$0.560857\pi$
$770$	0	0
$771$	−41.8687	−0.0543044
$772$	0	0
$773$	− 178.228i	− 0.230566i	−0.993333	−	0.115283i	$-0.963222\pi$
$773$	− 178.228i	− 0.230566i	0.993333	−	0.115283i	$-0.0367776\pi$
$774$	0	0
$775$	388.204i	0.500909i
$776$	0	0
$777$	−32.3781	−0.0416707
$778$	0	0
$779$	0	0
$780$	0	0
$781$	− 286.664i	− 0.367047i
$782$	0	0
$783$	10.1552	0.0129697
$784$	0	0
$785$	−75.2303	−0.0958348
$786$	0	0
$787$	− 443.009i	− 0.562908i	−0.959575	−	0.281454i	$-0.909183\pi$
$787$	− 443.009i	− 0.562908i	0.959575	−	0.281454i	$-0.0908167\pi$
$788$	0	0
$789$	− 69.6564i	− 0.0882845i
$790$	0	0
$791$	− 829.500i	− 1.04867i
$792$	0	0
$793$	− 788.802i	− 0.994706i
$794$	0	0
$795$	−2.71148	−0.00341067
$796$	0	0
$797$	666.229i	0.835921i	0.908465	+	0.417961i	$0.137255\pi$
$797$	666.229i	0.835921i	−0.908465	+	0.417961i	$0.862745\pi$
$798$	0	0
$799$	427.064	0.534498
$800$	0	0
$801$	− 1010.22i	− 1.26120i
$802$	0	0
$803$	−78.2238	−0.0974144
$804$	0	0
$805$	13.6294	0.0169310
$806$	0	0
$807$	−7.49446	−0.00928681
$808$	0	0
$809$	507.989	0.627922	0.313961	−	0.949436i	$-0.398344\pi$
$809$	507.989	0.627922	0.313961	+	0.949436i	$0.398344\pi$
$810$	0	0
$811$	− 1367.04i	− 1.68563i	−0.538207	−	0.842813i	$-0.680898\pi$
$811$	− 1367.04i	− 1.68563i	0.538207	−	0.842813i	$-0.319102\pi$
$812$	0	0
$813$	60.6505i	0.0746009i
$814$	0	0
$815$	19.5137	0.0239433
$816$	0	0
$817$	0	0
$818$	0	0
$819$	568.908i	0.694637i
$820$	0	0
$821$	−143.846	−0.175208	−0.0876041	−	0.996155i	$-0.527921\pi$
$821$	−143.846	−0.175208	−0.0876041	+	0.996155i	$0.527921\pi$
$822$	0	0
$823$	−647.892	−0.787232	−0.393616	−	0.919275i	$-0.628776\pi$
$823$	−647.892	−0.787232	−0.393616	+	0.919275i	$0.628776\pi$
$824$	0	0
$825$	14.4022i	0.0174572i
$826$	0	0
$827$	− 842.574i	− 1.01883i	−0.860520	−	0.509416i	$-0.829861\pi$
$827$	− 842.574i	− 1.01883i	0.860520	−	0.509416i	$-0.170139\pi$
$828$	0	0
$829$	− 761.850i	− 0.918999i	−0.888178	−	0.459499i	$-0.848029\pi$
$829$	− 761.850i	− 0.918999i	0.888178	−	0.459499i	$-0.151971\pi$
$830$	0	0
$831$	− 47.9117i	− 0.0576555i
$832$	0	0
$833$	165.491	0.198669
$834$	0	0
$835$	− 80.1894i	− 0.0960352i
$836$	0	0
$837$	−48.8205	−0.0583280
$838$	0	0
$839$	− 537.935i	− 0.641162i	−0.947221	−	0.320581i	$-0.896122\pi$
$839$	− 537.935i	− 0.641162i	0.947221	−	0.320581i	$-0.103878\pi$
$840$	0	0
$841$	830.495	0.987509
$842$	0	0
$843$	−55.9583	−0.0663799
$844$	0	0
$845$	−6.31148	−0.00746921
$846$	0	0
$847$	574.628	0.678427
$848$	0	0
$849$	− 0.619908i	0 0.000730163i
$850$	0	0
$851$	316.856i	0.372334i
$852$	0	0
$853$	637.161	0.746965	0.373483	−	0.927637i	$-0.378164\pi$
$853$	637.161	0.746965	0.373483	+	0.927637i	$0.378164\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 1278.94i	− 1.49234i	−0.665755	−	0.746171i	$-0.731890\pi$
$857$	− 1278.94i	− 1.49234i	0.665755	−	0.746171i	$-0.268110\pi$
$858$	0	0
$859$	1285.58	1.49660	0.748298	−	0.663363i	$-0.230871\pi$
$859$	1285.58	1.49660	0.748298	+	0.663363i	$0.230871\pi$
$860$	0	0
$861$	31.8989	0.0370487
$862$	0	0
$863$	1231.16i	1.42660i	0.700857	+	0.713302i	$0.252801\pi$
$863$	1231.16i	1.42660i	−0.700857	+	0.713302i	$0.747199\pi$
$864$	0	0
$865$	53.3937i	0.0617268i
$866$	0	0
$867$	40.2638i	0.0464404i
$868$	0	0
$869$	− 399.507i	− 0.459732i
$870$	0	0
$871$	−454.082	−0.521334
$872$	0	0
$873$	− 386.818i	− 0.443091i
$874$	0	0
$875$	−76.3273	−0.0872312
$876$	0	0
$877$	− 1576.56i	− 1.79768i	−0.438279	−	0.898839i	$-0.644412\pi$
$877$	− 1576.56i	− 1.79768i	0.438279	−	0.898839i	$-0.355588\pi$
$878$	0	0
$879$	70.0826	0.0797299
$880$	0	0
$881$	−1607.31	−1.82441	−0.912207	−	0.409730i	$-0.865623\pi$
$881$	−1607.31	−1.82441	−0.912207	+	0.409730i	$0.865623\pi$
$882$	0	0
$883$	100.440	0.113748	0.0568741	−	0.998381i	$-0.481887\pi$
$883$	100.440	0.113748	0.0568741	+	0.998381i	$0.481887\pi$
$884$	0	0
$885$	4.87134	0.00550434
$886$	0	0
$887$	396.592i	0.447116i	0.974691	+	0.223558i	$0.0717673\pi$
$887$	396.592i	0.447116i	−0.974691	+	0.223558i	$0.928233\pi$
$888$	0	0
$889$	818.766i	0.920996i
$890$	0	0
$891$	265.824	0.298344
$892$	0	0
$893$	0	0
$894$	0	0
$895$	43.5361i	0.0486437i
$896$	0	0
$897$	−18.8705	−0.0210374
$898$	0	0
$899$	50.5024	0.0561762
$900$	0	0
$901$	− 404.829i	− 0.449310i
$902$	0	0
$903$	41.0265i	0.0454336i
$904$	0	0
$905$	− 22.0166i	− 0.0243277i
$906$	0	0
$907$	− 1179.36i	− 1.30029i	−0.759810	−	0.650145i	$-0.774708\pi$
$907$	− 1179.36i	− 1.30029i	0.759810	−	0.650145i	$-0.225292\pi$
$908$	0	0
$909$	−1263.00	−1.38943
$910$	0	0
$911$	722.809i	0.793424i	0.917943	+	0.396712i	$0.129849\pi$
$911$	722.809i	0.793424i	−0.917943	+	0.396712i	$0.870151\pi$
$912$	0	0
$913$	−352.008	−0.385551
$914$	0	0
$915$	3.31593i	0.00362397i
$916$	0	0
$917$	955.380	1.04185
$918$	0	0
$919$	−1055.81	−1.14887	−0.574436	−	0.818550i	$-0.694778\pi$
$919$	−1055.81	−1.14887	−0.574436	+	0.818550i	$0.694778\pi$
$920$	0	0
$921$	63.0656	0.0684751
$922$	0	0
$923$	−1049.91	−1.13750
$924$	0	0
$925$	− 885.702i	− 0.957516i
$926$	0	0
$927$	− 145.861i	− 0.157347i
$928$	0	0
$929$	−912.804	−0.982566	−0.491283	−	0.871000i	$-0.663472\pi$
$929$	−912.804	−0.982566	−0.491283	+	0.871000i	$0.663472\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	28.8946i	0.0309696i
$934$	0	0
$935$	7.39661	0.00791081
$936$	0	0
$937$	−1272.97	−1.35856	−0.679281	−	0.733879i	$-0.737708\pi$
$937$	−1272.97	−1.35856	−0.679281	+	0.733879i	$0.737708\pi$
$938$	0	0
$939$	− 50.1808i	− 0.0534407i
$940$	0	0
$941$	− 1484.81i	− 1.57790i	−0.614457	−	0.788951i	$-0.710625\pi$
$941$	− 1484.81i	− 1.57790i	0.614457	−	0.788951i	$-0.289375\pi$
$942$	0	0
$943$	− 312.167i	− 0.331036i
$944$	0	0
$945$	− 4.79121i	− 0.00507006i
$946$	0	0
$947$	825.557	0.871760	0.435880	−	0.900005i	$-0.356437\pi$
$947$	825.557	0.871760	0.435880	+	0.900005i	$0.356437\pi$
$948$	0	0
$949$	286.496i	0.301893i
$950$	0	0
$951$	43.6522	0.0459014
$952$	0	0
$953$	194.271i	0.203852i	0.994792	+	0.101926i	$0.0325005\pi$
$953$	194.271i	0.203852i	−0.994792	+	0.101926i	$0.967500\pi$
$954$	0	0
$955$	−62.8066	−0.0657661
$956$	0	0
$957$	1.87361	0.00195780
$958$	0	0
$959$	−728.677	−0.759830
$960$	0	0
$961$	718.214	0.747361
$962$	0	0
$963$	− 1384.04i	− 1.43722i
$964$	0	0
$965$	74.8824i	0.0775984i
$966$	0	0
$967$	−146.703	−0.151709	−0.0758545	−	0.997119i	$-0.524168\pi$
$967$	−146.703	−0.151709	−0.0758545	+	0.997119i	$0.524168\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 1856.82i	− 1.91227i	−0.292924	−	0.956136i	$-0.594628\pi$
$971$	− 1856.82i	− 1.91227i	0.292924	−	0.956136i	$-0.405372\pi$
$972$	0	0
$973$	1212.75	1.24640
$974$	0	0
$975$	52.7484	0.0541009
$976$	0	0
$977$	274.398i	0.280858i	0.990091	+	0.140429i	$0.0448482\pi$
$977$	274.398i	0.280858i	−0.990091	+	0.140429i	$0.955152\pi$
$978$	0	0
$979$	− 373.398i	− 0.381408i
$980$	0	0
$981$	− 561.429i	− 0.572303i
$982$	0	0
$983$	− 920.324i	− 0.936240i	−0.883665	−	0.468120i	$-0.844932\pi$
$983$	− 920.324i	− 0.936240i	0.883665	−	0.468120i	$-0.155068\pi$
$984$	0	0
$985$	24.8432	0.0252215
$986$	0	0
$987$	− 51.0381i	− 0.0517104i
$988$	0	0
$989$	401.491	0.405957
$990$	0	0
$991$	− 1639.39i	− 1.65428i	−0.561998	−	0.827138i	$-0.689967\pi$
$991$	− 1639.39i	− 1.65428i	0.561998	−	0.827138i	$-0.310033\pi$
$992$	0	0
$993$	−47.4888	−0.0478236
$994$	0	0
$995$	23.8721	0.0239920
$996$	0	0
$997$	−601.411	−0.603221	−0.301611	−	0.953431i	$-0.597524\pi$
$997$	−601.411	−0.603221	−0.301611	+	0.953431i	$0.597524\pi$
$998$	0	0
$999$	111.386	0.111497

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1444.3.c.d.721.11	✓	24
19.18	odd	2	inner	1444.3.c.d.721.14	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1444.3.c.d.721.11	✓	24	1.1	even	1	trivial
1444.3.c.d.721.14	yes	24	19.18	odd	2	inner

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:	\( N \)	\(=\)	\( 1444 = 2^{2} \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1444.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-0.174362i q^{3} -0.292748 q^{5} -5.22349 q^{7} +8.96960 q^{9} +O(q^{10})\)	\(q-0.174362i q^{3} -0.292748 q^{5} -5.22349 q^{7} +8.96960 q^{9} +3.31534 q^{11} -12.1425i q^{13} +0.0510442i q^{15} -7.62097 q^{17} +0.910778i q^{21} +8.91299 q^{23} -24.9143 q^{25} -3.13322i q^{27} +3.24116i q^{29} -15.5816i q^{31} -0.578069i q^{33} +1.52917 q^{35} +35.5500i q^{37} -2.11719 q^{39} -35.0238i q^{41} +45.0456 q^{43} -2.62583 q^{45} -56.0380 q^{47} -21.7152 q^{49} +1.32881i q^{51} +53.1203i q^{53} -0.970560 q^{55} -95.4339i q^{59} +64.9620 q^{61} -46.8526 q^{63} +3.55470i q^{65} -37.3961i q^{67} -1.55409i q^{69} -86.4658i q^{71} -23.5945 q^{73} +4.34411i q^{75} -17.3176 q^{77} -120.503i q^{79} +80.1801 q^{81} -106.176 q^{83} +2.23103 q^{85} +0.565135 q^{87} -112.628i q^{89} +63.4262i q^{91} -2.71684 q^{93} -43.1255i q^{97} +29.7373 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 12 q^{5} + 40 q^{7} - 104 q^{9}+O(q^{10}) \) 24 * q - 12 * q^5 + 40 * q^7 - 104 * q^9	\( 24 q - 12 q^{5} + 40 q^{7} - 104 q^{9} - 64 q^{11} - 60 q^{17} + 100 q^{23} + 24 q^{25} - 144 q^{35} + 68 q^{39} - 16 q^{43} + 248 q^{45} + 180 q^{47} - 208 q^{49} - 532 q^{55} + 104 q^{61} - 1216 q^{63} - 484 q^{73} + 136 q^{77} + 460 q^{81} + 744 q^{83} + 32 q^{85} - 1296 q^{87} - 256 q^{93} + 804 q^{99}+O(q^{100}) \) 24 * q - 12 * q^5 + 40 * q^7 - 104 * q^9 - 64 * q^11 - 60 * q^17 + 100 * q^23 + 24 * q^25 - 144 * q^35 + 68 * q^39 - 16 * q^43 + 248 * q^45 + 180 * q^47 - 208 * q^49 - 532 * q^55 + 104 * q^61 - 1216 * q^63 - 484 * q^73 + 136 * q^77 + 460 * q^81 + 744 * q^83 + 32 * q^85 - 1296 * q^87 - 256 * q^93 + 804 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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