LMFDB - Embedded newform 1444.3.c.d.721.3

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			721.3
Character	$\chi$	$=$	1444.721
Dual form			1444.3.c.d.721.22

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.89235i q^{3} -9.07521 q^{5} +11.2253 q^{7} -14.9351 q^{9} +O(q^{10})$	$q-4.89235i q^{3} -9.07521 q^{5} +11.2253 q^{7} -14.9351 q^{9} -5.07525 q^{11} -2.03857i q^{13} +44.3991i q^{15} -15.6174 q^{17} -54.9180i q^{21} -19.4729 q^{23} +57.3594 q^{25} +29.0364i q^{27} +1.45784i q^{29} +44.7102i q^{31} +24.8299i q^{33} -101.872 q^{35} +34.8762i q^{37} -9.97338 q^{39} +60.8205i q^{41} +11.1884 q^{43} +135.539 q^{45} -14.7027 q^{47} +77.0071 q^{49} +76.4058i q^{51} +16.7970i q^{53} +46.0589 q^{55} +6.20357i q^{59} +71.3685 q^{61} -167.650 q^{63} +18.5004i q^{65} -65.0771i q^{67} +95.2683i q^{69} -108.429i q^{71} -35.3311 q^{73} -280.622i q^{75} -56.9711 q^{77} +113.483i q^{79} +7.64040 q^{81} +28.8906 q^{83} +141.731 q^{85} +7.13225 q^{87} -60.0053i q^{89} -22.8835i q^{91} +218.738 q^{93} +29.7750i q^{97} +75.7991 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	− 4.89235i	− 1.63078i	−0.578910	−	0.815391i	$-0.696522\pi$
$3$	− 4.89235i	− 1.63078i	0.578910	−	0.815391i	$-0.303478\pi$
$4$	0	0
$5$	−9.07521	−1.81504	−0.907521	−	0.420007i	$-0.862028\pi$
$5$	−9.07521	−1.81504	−0.907521	+	0.420007i	$0.862028\pi$
$6$	0	0
$7$	11.2253	1.60361	0.801806	−	0.597584i	$-0.203873\pi$
$7$	11.2253	1.60361	0.801806	+	0.597584i	$0.203873\pi$
$8$	0	0
$9$	−14.9351	−1.65945
$10$	0	0
$11$	−5.07525	−0.461386	−0.230693	−	0.973027i	$-0.574099\pi$
$11$	−5.07525	−0.461386	−0.230693	+	0.973027i	$0.574099\pi$
$12$	0	0
$13$	− 2.03857i	− 0.156813i	−0.996921	−	0.0784064i	$-0.975017\pi$
$13$	− 2.03857i	− 0.156813i	0.996921	−	0.0784064i	$-0.0249832\pi$
$14$	0	0
$15$	44.3991i	2.95994i
$16$	0	0
$17$	−15.6174	−0.918672	−0.459336	−	0.888263i	$-0.651913\pi$
$17$	−15.6174	−0.918672	−0.459336	+	0.888263i	$0.651913\pi$
$18$	0	0
$19$	0	0
$19$	0	0
$20$	0	0
$21$	− 54.9180i	− 2.61514i
$22$	0	0
$23$	−19.4729	−0.846649	−0.423324	−	0.905978i	$-0.639137\pi$
$23$	−19.4729	−0.846649	−0.423324	+	0.905978i	$0.639137\pi$
$24$	0	0
$25$	57.3594	2.29438
$26$	0	0
$27$	29.0364i	1.07542i
$28$	0	0
$29$	1.45784i	0.0502703i	0.999684	+	0.0251351i	$0.00800160\pi$
$29$	1.45784i	0.0502703i	−0.999684	+	0.0251351i	$0.991998\pi$
$30$	0	0
$31$	44.7102i	1.44226i	0.692798	+	0.721132i	$0.256378\pi$
$31$	44.7102i	1.44226i	−0.692798	+	0.721132i	$0.743622\pi$
$32$	0	0
$33$	24.8299i	0.752420i
$34$	0	0
$35$	−101.872	−2.91062
$36$	0	0
$37$	34.8762i	0.942600i	0.881973	+	0.471300i	$0.156215\pi$
$37$	34.8762i	0.942600i	−0.881973	+	0.471300i	$0.843785\pi$
$38$	0	0
$39$	−9.97338	−0.255728
$40$	0	0
$41$	60.8205i	1.48343i	0.670717	+	0.741713i	$0.265987\pi$
$41$	60.8205i	1.48343i	−0.670717	+	0.741713i	$0.734013\pi$
$42$	0	0
$43$	11.1884	0.260194	0.130097	−	0.991501i	$-0.458471\pi$
$43$	11.1884	0.260194	0.130097	+	0.991501i	$0.458471\pi$
$44$	0	0
$45$	135.539	3.01197
$46$	0	0
$47$	−14.7027	−0.312823	−0.156412	−	0.987692i	$-0.549993\pi$
$47$	−14.7027	−0.312823	−0.156412	+	0.987692i	$0.549993\pi$
$48$	0	0
$49$	77.0071	1.57157
$50$	0	0
$51$	76.4058i	1.49815i
$52$	0	0
$53$	16.7970i	0.316925i	0.987365	+	0.158463i	$0.0506538\pi$
$53$	16.7970i	0.316925i	−0.987365	+	0.158463i	$0.949346\pi$
$54$	0	0
$55$	46.0589	0.837435
$56$	0	0
$57$	0	0
$58$	0	0
$59$	6.20357i	0.105145i	0.998617	+	0.0525727i	$0.0167421\pi$
$59$	6.20357i	0.105145i	−0.998617	+	0.0525727i	$0.983258\pi$
$60$	0	0
$61$	71.3685	1.16998	0.584988	−	0.811042i	$-0.301099\pi$
$61$	71.3685	1.16998	0.584988	+	0.811042i	$0.301099\pi$
$62$	0	0
$63$	−167.650	−2.66112
$64$	0	0
$65$	18.5004i	0.284622i
$66$	0	0
$67$	− 65.0771i	− 0.971300i	−0.874153	−	0.485650i	$-0.838583\pi$
$67$	− 65.0771i	− 0.971300i	0.874153	−	0.485650i	$-0.161417\pi$
$68$	0	0
$69$	95.2683i	1.38070i
$70$	0	0
$71$	− 108.429i	− 1.52716i	−0.645711	−	0.763582i	$-0.723439\pi$
$71$	− 108.429i	− 1.52716i	0.645711	−	0.763582i	$-0.276561\pi$
$72$	0	0
$73$	−35.3311	−0.483987	−0.241994	−	0.970278i	$-0.577801\pi$
$73$	−35.3311	−0.483987	−0.241994	+	0.970278i	$0.577801\pi$
$74$	0	0
$75$	− 280.622i	− 3.74163i
$76$	0	0
$77$	−56.9711	−0.739885
$78$	0	0
$79$	113.483i	1.43649i	0.695788	+	0.718247i	$0.255055\pi$
$79$	113.483i	1.43649i	−0.695788	+	0.718247i	$0.744945\pi$
$80$	0	0
$81$	7.64040	0.0943260
$82$	0	0
$83$	28.8906	0.348079	0.174040	−	0.984739i	$-0.444318\pi$
$83$	28.8906	0.348079	0.174040	+	0.984739i	$0.444318\pi$
$84$	0	0
$85$	141.731	1.66743
$86$	0	0
$87$	7.13225	0.0819799
$88$	0	0
$89$	− 60.0053i	− 0.674217i	−0.941466	−	0.337109i	$-0.890551\pi$
$89$	− 60.0053i	− 0.674217i	0.941466	−	0.337109i	$-0.109449\pi$
$90$	0	0
$91$	− 22.8835i	− 0.251467i
$92$	0	0
$93$	218.738	2.35202
$94$	0	0
$95$	0	0
$96$	0	0
$97$	29.7750i	0.306959i	0.988152	+	0.153479i	$0.0490479\pi$
$97$	29.7750i	0.306959i	−0.988152	+	0.153479i	$0.950952\pi$
$98$	0	0
$99$	75.7991	0.765648
$100$	0	0
$101$	194.467	1.92541	0.962707	−	0.270548i	$-0.0872047\pi$
$101$	194.467	1.92541	0.962707	+	0.270548i	$0.0872047\pi$
$102$	0	0
$103$	− 122.285i	− 1.18723i	−0.804748	−	0.593617i	$-0.797699\pi$
$103$	− 122.285i	− 1.18723i	0.804748	−	0.593617i	$-0.202301\pi$
$104$	0	0
$105$	498.392i	4.74659i
$106$	0	0
$107$	1.47576i	0.0137922i	0.999976	+	0.00689609i	$0.00219511\pi$
$107$	1.47576i	0.0137922i	−0.999976	+	0.00689609i	$0.997805\pi$
$108$	0	0
$109$	− 64.3521i	− 0.590386i	−0.955438	−	0.295193i	$-0.904616\pi$
$109$	− 64.3521i	− 0.590386i	0.955438	−	0.295193i	$-0.0953840\pi$
$110$	0	0
$111$	170.626	1.53717
$112$	0	0
$113$	105.595i	0.934468i	0.884134	+	0.467234i	$0.154749\pi$
$113$	105.595i	0.934468i	−0.884134	+	0.467234i	$0.845251\pi$
$114$	0	0
$115$	176.721	1.53670
$116$	0	0
$117$	30.4461i	0.260223i
$118$	0	0
$119$	−175.310	−1.47319
$120$	0	0
$121$	−95.2419	−0.787123
$122$	0	0
$123$	297.555	2.41915
$124$	0	0
$125$	−293.669	−2.34935
$126$	0	0
$127$	167.771i	1.32103i	0.750811	+	0.660517i	$0.229663\pi$
$127$	167.771i	1.32103i	−0.750811	+	0.660517i	$0.770337\pi$
$128$	0	0
$129$	− 54.7373i	− 0.424320i
$130$	0	0
$131$	14.7733	0.112773	0.0563865	−	0.998409i	$-0.482042\pi$
$131$	14.7733	0.112773	0.0563865	+	0.998409i	$0.482042\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	− 263.511i	− 1.95193i
$136$	0	0
$137$	111.662	0.815052	0.407526	−	0.913194i	$-0.366392\pi$
$137$	111.662	0.815052	0.407526	+	0.913194i	$0.366392\pi$
$138$	0	0
$139$	−213.252	−1.53418	−0.767092	−	0.641537i	$-0.778297\pi$
$139$	−213.252	−1.53418	−0.767092	+	0.641537i	$0.778297\pi$
$140$	0	0
$141$	71.9307i	0.510147i
$142$	0	0
$143$	10.3462i	0.0723513i
$144$	0	0
$145$	− 13.2302i	− 0.0912426i
$146$	0	0
$147$	− 376.745i	− 2.56289i
$148$	0	0
$149$	63.5403	0.426445	0.213223	−	0.977004i	$-0.431604\pi$
$149$	63.5403	0.426445	0.213223	+	0.977004i	$0.431604\pi$
$150$	0	0
$151$	27.9116i	0.184845i	0.995720	+	0.0924225i	$0.0294610\pi$
$151$	27.9116i	0.184845i	−0.995720	+	0.0924225i	$0.970539\pi$
$152$	0	0
$153$	233.247	1.52449
$154$	0	0
$155$	− 405.754i	− 2.61777i
$156$	0	0
$157$	229.427	1.46132	0.730660	−	0.682742i	$-0.239213\pi$
$157$	229.427	1.46132	0.730660	+	0.682742i	$0.239213\pi$
$158$	0	0
$159$	82.1770	0.516836
$160$	0	0
$161$	−218.589	−1.35770
$162$	0	0
$163$	−13.9039	−0.0853000	−0.0426500	−	0.999090i	$-0.513580\pi$
$163$	−13.9039	−0.0853000	−0.0426500	+	0.999090i	$0.513580\pi$
$164$	0	0
$165$	− 225.336i	− 1.36567i
$166$	0	0
$167$	174.780i	1.04659i	0.852153	+	0.523293i	$0.175297\pi$
$167$	174.780i	1.04659i	−0.852153	+	0.523293i	$0.824703\pi$
$168$	0	0
$169$	164.844	0.975410
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 326.690i	− 1.88838i	−0.329400	−	0.944190i	$-0.606847\pi$
$173$	− 326.690i	− 1.88838i	0.329400	−	0.944190i	$-0.393153\pi$
$174$	0	0
$175$	643.876	3.67929
$176$	0	0
$177$	30.3500	0.171469
$178$	0	0
$179$	244.659i	1.36681i	0.730039	+	0.683406i	$0.239502\pi$
$179$	244.659i	1.36681i	−0.730039	+	0.683406i	$0.760498\pi$
$180$	0	0
$181$	− 288.840i	− 1.59580i	−0.602789	−	0.797900i	$-0.705944\pi$
$181$	− 288.840i	− 1.59580i	0.602789	−	0.797900i	$-0.294056\pi$
$182$	0	0
$183$	− 349.160i	− 1.90798i
$184$	0	0
$185$	− 316.509i	− 1.71086i
$186$	0	0
$187$	79.2623	0.423862
$188$	0	0
$189$	325.941i	1.72456i
$190$	0	0
$191$	105.373	0.551691	0.275846	−	0.961202i	$-0.411042\pi$
$191$	105.373	0.551691	0.275846	+	0.961202i	$0.411042\pi$
$192$	0	0
$193$	200.893i	1.04090i	0.853893	+	0.520449i	$0.174235\pi$
$193$	200.893i	1.04090i	−0.853893	+	0.520449i	$0.825765\pi$
$194$	0	0
$195$	90.5105	0.464156
$196$	0	0
$197$	112.050	0.568782	0.284391	−	0.958708i	$-0.408209\pi$
$197$	112.050	0.568782	0.284391	+	0.958708i	$0.408209\pi$
$198$	0	0
$199$	−278.028	−1.39713	−0.698564	−	0.715548i	$-0.746177\pi$
$199$	−278.028	−1.39713	−0.698564	+	0.715548i	$0.746177\pi$
$200$	0	0
$201$	−318.380	−1.58398
$202$	0	0
$203$	16.3646i	0.0806140i
$204$	0	0
$205$	− 551.959i	− 2.69248i
$206$	0	0
$207$	290.829	1.40497
$208$	0	0
$209$	0	0
$210$	0	0
$211$	114.046i	0.540503i	0.962790	+	0.270252i	$0.0871069\pi$
$211$	114.046i	0.540503i	−0.962790	+	0.270252i	$0.912893\pi$
$212$	0	0
$213$	−530.470	−2.49047
$214$	0	0
$215$	−101.537	−0.472264
$216$	0	0
$217$	501.885i	2.31283i
$218$	0	0
$219$	172.852i	0.789278i
$220$	0	0
$221$	31.8372i	0.144060i
$222$	0	0
$223$	247.579i	1.11022i	0.831777	+	0.555109i	$0.187324\pi$
$223$	247.579i	1.11022i	−0.831777	+	0.555109i	$0.812676\pi$
$224$	0	0
$225$	−856.666	−3.80741
$226$	0	0
$227$	86.1442i	0.379490i	0.981833	+	0.189745i	$0.0607661\pi$
$227$	86.1442i	0.379490i	−0.981833	+	0.189745i	$0.939234\pi$
$228$	0	0
$229$	−125.448	−0.547808	−0.273904	−	0.961757i	$-0.588315\pi$
$229$	−125.448	−0.547808	−0.273904	+	0.961757i	$0.588315\pi$
$230$	0	0
$231$	278.722i	1.20659i
$232$	0	0
$233$	−34.0856	−0.146290	−0.0731451	−	0.997321i	$-0.523304\pi$
$233$	−34.0856	−0.146290	−0.0731451	+	0.997321i	$0.523304\pi$
$234$	0	0
$235$	133.430	0.567787
$236$	0	0
$237$	555.198	2.34261
$238$	0	0
$239$	−325.140	−1.36042	−0.680210	−	0.733017i	$-0.738111\pi$
$239$	−325.140	−1.36042	−0.680210	+	0.733017i	$0.738111\pi$
$240$	0	0
$241$	319.265i	1.32475i	0.749171	+	0.662376i	$0.230452\pi$
$241$	319.265i	1.32475i	−0.749171	+	0.662376i	$0.769548\pi$
$242$	0	0
$243$	223.948i	0.921595i
$244$	0	0
$245$	−698.856	−2.85247
$246$	0	0
$247$	0	0
$248$	0	0
$249$	− 141.343i	− 0.567642i
$250$	0	0
$251$	−219.505	−0.874521	−0.437260	−	0.899335i	$-0.644051\pi$
$251$	−219.505	−0.874521	−0.437260	+	0.899335i	$0.644051\pi$
$252$	0	0
$253$	98.8299	0.390632
$254$	0	0
$255$	− 693.399i	− 2.71921i
$256$	0	0
$257$	148.555i	0.578035i	0.957324	+	0.289018i	$0.0933286\pi$
$257$	148.555i	0.578035i	−0.957324	+	0.289018i	$0.906671\pi$
$258$	0	0
$259$	391.495i	1.51156i
$260$	0	0
$261$	− 21.7729i	− 0.0834210i
$262$	0	0
$263$	223.824	0.851043	0.425522	−	0.904948i	$-0.360091\pi$
$263$	223.824	0.851043	0.425522	+	0.904948i	$0.360091\pi$
$264$	0	0
$265$	− 152.437i	− 0.575233i
$266$	0	0
$267$	−293.567	−1.09950
$268$	0	0
$269$	351.687i	1.30739i	0.756760	+	0.653693i	$0.226781\pi$
$269$	351.687i	1.30739i	−0.756760	+	0.653693i	$0.773219\pi$
$270$	0	0
$271$	98.1840	0.362303	0.181151	−	0.983455i	$-0.442018\pi$
$271$	98.1840	0.362303	0.181151	+	0.983455i	$0.442018\pi$
$272$	0	0
$273$	−111.954	−0.410088
$274$	0	0
$275$	−291.113	−1.05859
$276$	0	0
$277$	222.229	0.802271	0.401136	−	0.916019i	$-0.368616\pi$
$277$	222.229	0.802271	0.401136	+	0.916019i	$0.368616\pi$
$278$	0	0
$279$	− 667.749i	− 2.39337i
$280$	0	0
$281$	445.385i	1.58500i	0.609872	+	0.792500i	$0.291221\pi$
$281$	445.385i	1.58500i	−0.609872	+	0.792500i	$0.708779\pi$
$282$	0	0
$283$	269.677	0.952924	0.476462	−	0.879195i	$-0.341919\pi$
$283$	269.677	0.952924	0.476462	+	0.879195i	$0.341919\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	682.728i	2.37884i
$288$	0	0
$289$	−45.0963	−0.156043
$290$	0	0
$291$	145.670	0.500583
$292$	0	0
$293$	128.310i	0.437917i	0.975734	+	0.218959i	$0.0702660\pi$
$293$	128.310i	0.437917i	−0.975734	+	0.218959i	$0.929734\pi$
$294$	0	0
$295$	− 56.2987i	− 0.190843i
$296$	0	0
$297$	− 147.367i	− 0.496184i
$298$	0	0
$299$	39.6969i	0.132765i
$300$	0	0
$301$	125.593	0.417251
$302$	0	0
$303$	− 951.399i	− 3.13993i
$304$	0	0
$305$	−647.684	−2.12356
$306$	0	0
$307$	− 246.087i	− 0.801586i	−0.916169	−	0.400793i	$-0.868735\pi$
$307$	− 246.087i	− 0.801586i	0.916169	−	0.400793i	$-0.131265\pi$
$308$	0	0
$309$	−598.261	−1.93612
$310$	0	0
$311$	383.981	1.23467	0.617333	−	0.786702i	$-0.288213\pi$
$311$	383.981	1.23467	0.617333	+	0.786702i	$0.288213\pi$
$312$	0	0
$313$	−212.369	−0.678495	−0.339247	−	0.940697i	$-0.610172\pi$
$313$	−212.369	−0.678495	−0.339247	+	0.940697i	$0.610172\pi$
$314$	0	0
$315$	1521.46	4.83004
$316$	0	0
$317$	250.664i	0.790738i	0.918522	+	0.395369i	$0.129383\pi$
$317$	250.664i	0.790738i	−0.918522	+	0.395369i	$0.870617\pi$
$318$	0	0
$319$	− 7.39889i	− 0.0231940i
$320$	0	0
$321$	7.21995	0.0224921
$322$	0	0
$323$	0	0
$324$	0	0
$325$	− 116.931i	− 0.359788i
$326$	0	0
$327$	−314.833	−0.962791
$328$	0	0
$329$	−165.042	−0.501647
$330$	0	0
$331$	− 99.5337i	− 0.300706i	−0.988632	−	0.150353i	$-0.951959\pi$
$331$	− 99.5337i	− 0.300706i	0.988632	−	0.150353i	$-0.0480410\pi$
$332$	0	0
$333$	− 520.878i	− 1.56420i
$334$	0	0
$335$	590.588i	1.76295i
$336$	0	0
$337$	654.811i	1.94306i	0.236915	+	0.971530i	$0.423864\pi$
$337$	654.811i	1.94306i	−0.236915	+	0.971530i	$0.576136\pi$
$338$	0	0
$339$	516.607	1.52391
$340$	0	0
$341$	− 226.915i	− 0.665441i
$342$	0	0
$343$	314.388	0.916583
$344$	0	0
$345$	− 864.580i	− 2.50603i
$346$	0	0
$347$	−220.584	−0.635690	−0.317845	−	0.948143i	$-0.602959\pi$
$347$	−220.584	−0.635690	−0.317845	+	0.948143i	$0.602959\pi$
$348$	0	0
$349$	−196.974	−0.564396	−0.282198	−	0.959356i	$-0.591064\pi$
$349$	−196.974	−0.564396	−0.282198	+	0.959356i	$0.591064\pi$
$350$	0	0
$351$	59.1926	0.168640
$352$	0	0
$353$	105.146	0.297864	0.148932	−	0.988847i	$-0.452416\pi$
$353$	105.146	0.297864	0.148932	+	0.988847i	$0.452416\pi$
$354$	0	0
$355$	984.012i	2.77186i
$356$	0	0
$357$	857.677i	2.40246i
$358$	0	0
$359$	−107.507	−0.299463	−0.149731	−	0.988727i	$-0.547841\pi$
$359$	−107.507	−0.299463	−0.149731	+	0.988727i	$0.547841\pi$
$360$	0	0
$361$	0	0
$362$	0	0
$363$	465.956i	1.28363i
$364$	0	0
$365$	320.637	0.878458
$366$	0	0
$367$	413.447	1.12656	0.563280	−	0.826266i	$-0.309539\pi$
$367$	413.447	1.12656	0.563280	+	0.826266i	$0.309539\pi$
$368$	0	0
$369$	− 908.358i	− 2.46167i
$370$	0	0
$371$	188.552i	0.508226i
$372$	0	0
$373$	309.835i	0.830656i	0.909672	+	0.415328i	$0.136333\pi$
$373$	309.835i	0.830656i	−0.909672	+	0.415328i	$0.863667\pi$
$374$	0	0
$375$	1436.73i	3.83128i
$376$	0	0
$377$	2.97190	0.00788302
$378$	0	0
$379$	381.191i	1.00578i	0.864350	+	0.502890i	$0.167730\pi$
$379$	381.191i	1.00578i	−0.864350	+	0.502890i	$0.832270\pi$
$380$	0	0
$381$	820.795	2.15432
$382$	0	0
$383$	674.771i	1.76180i	0.473300	+	0.880902i	$0.343063\pi$
$383$	674.771i	1.76180i	−0.473300	+	0.880902i	$0.656937\pi$
$384$	0	0
$385$	517.025	1.34292
$386$	0	0
$387$	−167.099	−0.431780
$388$	0	0
$389$	−72.7428	−0.186999	−0.0934997	−	0.995619i	$-0.529805\pi$
$389$	−72.7428	−0.186999	−0.0934997	+	0.995619i	$0.529805\pi$
$390$	0	0
$391$	304.117	0.777792
$392$	0	0
$393$	− 72.2759i	− 0.183908i
$394$	0	0
$395$	− 1029.88i	− 2.60730i
$396$	0	0
$397$	−764.241	−1.92504	−0.962520	−	0.271210i	$-0.912576\pi$
$397$	−764.241	−1.92504	−0.962520	+	0.271210i	$0.912576\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 119.673i	− 0.298437i	−0.988804	−	0.149218i	$-0.952324\pi$
$401$	− 119.673i	− 0.298437i	0.988804	−	0.149218i	$-0.0476757\pi$
$402$	0	0
$403$	91.1447	0.226166
$404$	0	0
$405$	−69.3383	−0.171206
$406$	0	0
$407$	− 177.005i	− 0.434902i
$408$	0	0
$409$	− 623.899i	− 1.52542i	−0.646738	−	0.762712i	$-0.723867\pi$
$409$	− 623.899i	− 1.52542i	0.646738	−	0.762712i	$-0.276133\pi$
$410$	0	0
$411$	− 546.290i	− 1.32917i
$412$	0	0
$413$	69.6369i	0.168612i
$414$	0	0
$415$	−262.188	−0.631779
$416$	0	0
$417$	1043.30i	2.50192i
$418$	0	0
$419$	−805.018	−1.92128	−0.960642	−	0.277791i	$-0.910398\pi$
$419$	−805.018	−1.92128	−0.960642	+	0.277791i	$0.910398\pi$
$420$	0	0
$421$	449.818i	1.06845i	0.845342	+	0.534225i	$0.179397\pi$
$421$	449.818i	1.06845i	−0.845342	+	0.534225i	$0.820603\pi$
$422$	0	0
$423$	219.586	0.519115
$424$	0	0
$425$	−895.806	−2.10778
$426$	0	0
$427$	801.133	1.87619
$428$	0	0
$429$	50.6174	0.117989
$430$	0	0
$431$	− 505.775i	− 1.17349i	−0.809771	−	0.586746i	$-0.800409\pi$
$431$	− 505.775i	− 1.17349i	0.809771	−	0.586746i	$-0.199591\pi$
$432$	0	0
$433$	394.512i	0.911113i	0.890207	+	0.455556i	$0.150560\pi$
$433$	394.512i	0.911113i	−0.890207	+	0.455556i	$0.849440\pi$
$434$	0	0
$435$	−64.7266	−0.148797
$436$	0	0
$437$	0	0
$438$	0	0
$439$	10.1333i	0.0230827i	0.999933	+	0.0115414i	$0.00367381\pi$
$439$	10.1333i	0.0230827i	−0.999933	+	0.0115414i	$0.996326\pi$
$440$	0	0
$441$	−1150.11	−2.60795
$442$	0	0
$443$	554.992	1.25280	0.626402	−	0.779500i	$-0.284527\pi$
$443$	554.992	1.25280	0.626402	+	0.779500i	$0.284527\pi$
$444$	0	0
$445$	544.561i	1.22373i
$446$	0	0
$447$	− 310.861i	− 0.695439i
$448$	0	0
$449$	533.512i	1.18822i	0.804383	+	0.594112i	$0.202496\pi$
$449$	533.512i	1.18822i	−0.804383	+	0.594112i	$0.797504\pi$
$450$	0	0
$451$	− 308.679i	− 0.684433i
$452$	0	0
$453$	136.553	0.301442
$454$	0	0
$455$	207.673i	0.456423i
$456$	0	0
$457$	−29.4713	−0.0644886	−0.0322443	−	0.999480i	$-0.510265\pi$
$457$	−29.4713	−0.0644886	−0.0322443	+	0.999480i	$0.510265\pi$
$458$	0	0
$459$	− 453.473i	− 0.987958i
$460$	0	0
$461$	−20.6073	−0.0447012	−0.0223506	−	0.999750i	$-0.507115\pi$
$461$	−20.6073	−0.0447012	−0.0223506	+	0.999750i	$0.507115\pi$
$462$	0	0
$463$	−833.016	−1.79917	−0.899586	−	0.436744i	$-0.856131\pi$
$463$	−833.016	−1.79917	−0.899586	+	0.436744i	$0.856131\pi$
$464$	0	0
$465$	−1985.09	−4.26901
$466$	0	0
$467$	123.523	0.264503	0.132251	−	0.991216i	$-0.457779\pi$
$467$	123.523	0.264503	0.132251	+	0.991216i	$0.457779\pi$
$468$	0	0
$469$	− 730.509i	− 1.55759i
$470$	0	0
$471$	− 1122.44i	− 2.38309i
$472$	0	0
$473$	−56.7837	−0.120050
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 250.865i	− 0.525922i
$478$	0	0
$479$	107.182	0.223762	0.111881	−	0.993722i	$-0.464313\pi$
$479$	107.182	0.223762	0.111881	+	0.993722i	$0.464313\pi$
$480$	0	0
$481$	71.0975	0.147812
$482$	0	0
$483$	1069.41i	2.21411i
$484$	0	0
$485$	− 270.214i	− 0.557143i
$486$	0	0
$487$	− 573.209i	− 1.17702i	−0.808490	−	0.588510i	$-0.799715\pi$
$487$	− 573.209i	− 1.17702i	0.808490	−	0.588510i	$-0.200285\pi$
$488$	0	0
$489$	68.0227i	0.139106i
$490$	0	0
$491$	247.257	0.503579	0.251789	−	0.967782i	$-0.418981\pi$
$491$	247.257	0.503579	0.251789	+	0.967782i	$0.418981\pi$
$492$	0	0
$493$	− 22.7677i	− 0.0461819i
$494$	0	0
$495$	−687.893	−1.38968
$496$	0	0
$497$	− 1217.14i	− 2.44898i
$498$	0	0
$499$	−321.230	−0.643747	−0.321874	−	0.946783i	$-0.604313\pi$
$499$	−321.230	−0.643747	−0.321874	+	0.946783i	$0.604313\pi$
$500$	0	0
$501$	855.084	1.70675
$502$	0	0
$503$	160.877	0.319835	0.159917	−	0.987130i	$-0.448877\pi$
$503$	160.877	0.319835	0.159917	+	0.987130i	$0.448877\pi$
$504$	0	0
$505$	−1764.83	−3.49471
$506$	0	0
$507$	− 806.475i	− 1.59068i
$508$	0	0
$509$	95.9149i	0.188438i	0.995552	+	0.0942189i	$0.0300354\pi$
$509$	95.9149i	0.188438i	−0.995552	+	0.0942189i	$0.969965\pi$
$510$	0	0
$511$	−396.602	−0.776128
$512$	0	0
$513$	0	0
$514$	0	0
$515$	1109.76i	2.15488i
$516$	0	0
$517$	74.6198	0.144332
$518$	0	0
$519$	−1598.28	−3.07954
$520$	0	0
$521$	− 818.234i	− 1.57051i	−0.619175	−	0.785253i	$-0.712533\pi$
$521$	− 818.234i	− 1.57051i	0.619175	−	0.785253i	$-0.287467\pi$
$522$	0	0
$523$	− 198.802i	− 0.380119i	−0.981773	−	0.190059i	$-0.939132\pi$
$523$	− 198.802i	− 0.380119i	0.981773	−	0.190059i	$-0.0608681\pi$
$524$	0	0
$525$	− 3150.06i	− 6.00012i
$526$	0	0
$527$	− 698.258i	− 1.32497i
$528$	0	0
$529$	−149.805	−0.283186
$530$	0	0
$531$	− 92.6507i	− 0.174483i
$532$	0	0
$533$	123.987	0.232620
$534$	0	0
$535$	− 13.3929i	− 0.0250334i
$536$	0	0
$537$	1196.96	2.22897
$538$	0	0
$539$	−390.830	−0.725102
$540$	0	0
$541$	−908.860	−1.67996	−0.839982	−	0.542615i	$-0.817434\pi$
$541$	−908.860	−1.67996	−0.839982	+	0.542615i	$0.817434\pi$
$542$	0	0
$543$	−1413.11	−2.60240
$544$	0	0
$545$	584.009i	1.07158i
$546$	0	0
$547$	252.201i	0.461062i	0.973065	+	0.230531i	$0.0740463\pi$
$547$	252.201i	0.461062i	−0.973065	+	0.230531i	$0.925954\pi$
$548$	0	0
$549$	−1065.89	−1.94152
$550$	0	0
$551$	0	0
$552$	0	0
$553$	1273.88i	2.30358i
$554$	0	0
$555$	−1548.47	−2.79004
$556$	0	0
$557$	−576.073	−1.03424	−0.517121	−	0.855912i	$-0.672996\pi$
$557$	−576.073	−1.03424	−0.517121	+	0.855912i	$0.672996\pi$
$558$	0	0
$559$	− 22.8082i	− 0.0408018i
$560$	0	0
$561$	− 387.778i	− 0.691227i
$562$	0	0
$563$	407.673i	0.724109i	0.932157	+	0.362054i	$0.117925\pi$
$563$	407.673i	0.724109i	−0.932157	+	0.362054i	$0.882075\pi$
$564$	0	0
$565$	− 958.296i	− 1.69610i
$566$	0	0
$567$	85.7657	0.151262
$568$	0	0
$569$	773.395i	1.35922i	0.733575	+	0.679609i	$0.237850\pi$
$569$	773.395i	1.35922i	−0.733575	+	0.679609i	$0.762150\pi$
$570$	0	0
$571$	678.192	1.18773	0.593863	−	0.804566i	$-0.297602\pi$
$571$	678.192	1.18773	0.593863	+	0.804566i	$0.297602\pi$
$572$	0	0
$573$	− 515.521i	− 0.899688i
$574$	0	0
$575$	−1116.96	−1.94253
$576$	0	0
$577$	634.074	1.09891	0.549457	−	0.835522i	$-0.314835\pi$
$577$	634.074	1.09891	0.549457	+	0.835522i	$0.314835\pi$
$578$	0	0
$579$	982.840	1.69748
$580$	0	0
$581$	324.305	0.558185
$582$	0	0
$583$	− 85.2492i	− 0.146225i
$584$	0	0
$585$	− 276.305i	− 0.472316i
$586$	0	0
$587$	−86.6998	−0.147700	−0.0738499	−	0.997269i	$-0.523529\pi$
$587$	−86.6998	−0.147700	−0.0738499	+	0.997269i	$0.523529\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	− 548.188i	− 0.927560i
$592$	0	0
$593$	942.791	1.58987	0.794933	−	0.606697i	$-0.207506\pi$
$593$	942.791	1.58987	0.794933	+	0.606697i	$0.207506\pi$
$594$	0	0
$595$	1590.98	2.67391
$596$	0	0
$597$	1360.21i	2.27841i
$598$	0	0
$599$	510.991i	0.853074i	0.904470	+	0.426537i	$0.140267\pi$
$599$	510.991i	0.853074i	−0.904470	+	0.426537i	$0.859733\pi$
$600$	0	0
$601$	− 684.427i	− 1.13881i	−0.822056	−	0.569407i	$-0.807173\pi$
$601$	− 684.427i	− 1.13881i	0.822056	−	0.569407i	$-0.192827\pi$
$602$	0	0
$603$	971.930i	1.61182i
$604$	0	0
$605$	864.340	1.42866
$606$	0	0
$607$	82.6292i	0.136127i	0.997681	+	0.0680636i	$0.0216821\pi$
$607$	82.6292i	0.136127i	−0.997681	+	0.0680636i	$0.978318\pi$
$608$	0	0
$609$	80.0615	0.131464
$610$	0	0
$611$	29.9724i	0.0490547i
$612$	0	0
$613$	−784.108	−1.27913	−0.639566	−	0.768736i	$-0.720886\pi$
$613$	−784.108	−1.27913	−0.639566	+	0.768736i	$0.720886\pi$
$614$	0	0
$615$	−2700.37	−4.39085
$616$	0	0
$617$	470.095	0.761905	0.380952	−	0.924595i	$-0.375596\pi$
$617$	470.095	0.761905	0.380952	+	0.924595i	$0.375596\pi$
$618$	0	0
$619$	−267.534	−0.432204	−0.216102	−	0.976371i	$-0.569334\pi$
$619$	−267.534	−0.432204	−0.216102	+	0.976371i	$0.569334\pi$
$620$	0	0
$621$	− 565.423i	− 0.910504i
$622$	0	0
$623$	− 673.577i	− 1.08118i
$624$	0	0
$625$	1231.12	1.96979
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 544.676i	− 0.865940i
$630$	0	0
$631$	−757.403	−1.20032	−0.600161	−	0.799879i	$-0.704897\pi$
$631$	−757.403	−1.20032	−0.600161	+	0.799879i	$0.704897\pi$
$632$	0	0
$633$	557.954	0.881443
$634$	0	0
$635$	− 1522.56i	− 2.39773i
$636$	0	0
$637$	− 156.984i	− 0.246443i
$638$	0	0
$639$	1619.39i	2.53425i
$640$	0	0
$641$	− 328.219i	− 0.512042i	−0.966671	−	0.256021i	$-0.917588\pi$
$641$	− 328.219i	− 0.512042i	0.966671	−	0.256021i	$-0.0824116\pi$
$642$	0	0
$643$	430.818	0.670012	0.335006	−	0.942216i	$-0.391262\pi$
$643$	430.818	0.670012	0.335006	+	0.942216i	$0.391262\pi$
$644$	0	0
$645$	496.753i	0.770159i
$646$	0	0
$647$	102.322	0.158148	0.0790740	−	0.996869i	$-0.474804\pi$
$647$	102.322	0.158148	0.0790740	+	0.996869i	$0.474804\pi$
$648$	0	0
$649$	− 31.4847i	− 0.0485126i
$650$	0	0
$651$	2455.39	3.77173
$652$	0	0
$653$	−269.405	−0.412565	−0.206283	−	0.978492i	$-0.566137\pi$
$653$	−269.405	−0.412565	−0.206283	+	0.978492i	$0.566137\pi$
$654$	0	0
$655$	−134.070	−0.204688
$656$	0	0
$657$	527.672	0.803153
$658$	0	0
$659$	38.7277i	0.0587674i	0.999568	+	0.0293837i	$0.00935446\pi$
$659$	38.7277i	0.0587674i	−0.999568	+	0.0293837i	$0.990646\pi$
$660$	0	0
$661$	− 967.272i	− 1.46335i	−0.681655	−	0.731674i	$-0.738740\pi$
$661$	− 967.272i	− 1.46335i	0.681655	−	0.731674i	$-0.261260\pi$
$662$	0	0
$663$	155.758	0.234930
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 28.3884i	− 0.0425613i
$668$	0	0
$669$	1211.24	1.81052
$670$	0	0
$671$	−362.213	−0.539811
$672$	0	0
$673$	478.524i	0.711031i	0.934670	+	0.355515i	$0.115695\pi$
$673$	478.524i	0.711031i	−0.934670	+	0.355515i	$0.884305\pi$
$674$	0	0
$675$	1665.51i	2.46742i
$676$	0	0
$677$	409.627i	0.605062i	0.953140	+	0.302531i	$0.0978316\pi$
$677$	409.627i	0.605062i	−0.953140	+	0.302531i	$0.902168\pi$
$678$	0	0
$679$	334.233i	0.492243i
$680$	0	0
$681$	421.447	0.618865
$682$	0	0
$683$	327.446i	0.479423i	0.970844	+	0.239711i	$0.0770528\pi$
$683$	327.446i	0.479423i	−0.970844	+	0.239711i	$0.922947\pi$
$684$	0	0
$685$	−1013.36	−1.47935
$686$	0	0
$687$	613.735i	0.893356i
$688$	0	0
$689$	34.2419	0.0496980
$690$	0	0
$691$	−280.087	−0.405336	−0.202668	−	0.979248i	$-0.564961\pi$
$691$	−280.087	−0.405336	−0.202668	+	0.979248i	$0.564961\pi$
$692$	0	0
$693$	850.867	1.22780
$694$	0	0
$695$	1935.30	2.78461
$696$	0	0
$697$	− 949.859i	− 1.36278i
$698$	0	0
$699$	166.759i	0.238568i
$700$	0	0
$701$	−448.669	−0.640041	−0.320020	−	0.947411i	$-0.603690\pi$
$701$	−448.669	−0.640041	−0.320020	+	0.947411i	$0.603690\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	− 652.786i	− 0.925937i
$706$	0	0
$707$	2182.95	3.08762
$708$	0	0
$709$	−91.6498	−0.129266	−0.0646332	−	0.997909i	$-0.520588\pi$
$709$	−91.6498	−0.129266	−0.0646332	+	0.997909i	$0.520588\pi$
$710$	0	0
$711$	− 1694.88i	− 2.38379i
$712$	0	0
$713$	− 870.638i	− 1.22109i
$714$	0	0
$715$	− 93.8942i	− 0.131321i
$716$	0	0
$717$	1590.70i	2.21855i
$718$	0	0
$719$	420.498	0.584837	0.292418	−	0.956290i	$-0.405540\pi$
$719$	420.498	0.584837	0.292418	+	0.956290i	$0.405540\pi$
$720$	0	0
$721$	− 1372.69i	− 1.90386i
$722$	0	0
$723$	1561.96	2.16038
$724$	0	0
$725$	83.6207i	0.115339i
$726$	0	0
$727$	−908.205	−1.24925	−0.624625	−	0.780925i	$-0.714748\pi$
$727$	−908.205	−1.24925	−0.624625	+	0.780925i	$0.714748\pi$
$728$	0	0
$729$	1164.39	1.59725
$730$	0	0
$731$	−174.733	−0.239033
$732$	0	0
$733$	138.411	0.188828	0.0944139	−	0.995533i	$-0.469902\pi$
$733$	138.411	0.188828	0.0944139	+	0.995533i	$0.469902\pi$
$734$	0	0
$735$	3419.04i	4.65176i
$736$	0	0
$737$	330.282i	0.448144i
$738$	0	0
$739$	40.6514	0.0550087	0.0275044	−	0.999622i	$-0.491244\pi$
$739$	40.6514	0.0550087	0.0275044	+	0.999622i	$0.491244\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 119.656i	− 0.161045i	−0.996753	−	0.0805223i	$-0.974341\pi$
$743$	− 119.656i	− 0.161045i	0.996753	−	0.0805223i	$-0.0256588\pi$
$744$	0	0
$745$	−576.642	−0.774016
$746$	0	0
$747$	−431.483	−0.577621
$748$	0	0
$749$	16.5659i	0.0221173i
$750$	0	0
$751$	− 114.859i	− 0.152942i	−0.997072	−	0.0764709i	$-0.975635\pi$
$751$	− 114.859i	− 0.152942i	0.997072	−	0.0764709i	$-0.0243652\pi$
$752$	0	0
$753$	1073.89i	1.42615i
$754$	0	0
$755$	− 253.304i	− 0.335501i
$756$	0	0
$757$	−3.91689	−0.00517423	−0.00258711	−	0.999997i	$-0.500824\pi$
$757$	−3.91689	−0.00517423	−0.00258711	+	0.999997i	$0.500824\pi$
$758$	0	0
$759$	− 483.510i	− 0.637036i
$760$	0	0
$761$	1013.16	1.33135	0.665677	−	0.746240i	$-0.268143\pi$
$761$	1013.16	1.33135	0.665677	+	0.746240i	$0.268143\pi$
$762$	0	0
$763$	− 722.371i	− 0.946751i
$764$	0	0
$765$	−2116.77	−2.76701
$766$	0	0
$767$	12.6464	0.0164881
$768$	0	0
$769$	−504.992	−0.656687	−0.328343	−	0.944558i	$-0.606490\pi$
$769$	−504.992	−0.656687	−0.328343	+	0.944558i	$0.606490\pi$
$770$	0	0
$771$	726.783	0.942650
$772$	0	0
$773$	291.252i	0.376782i	0.982094	+	0.188391i	$0.0603272\pi$
$773$	291.252i	0.376782i	−0.982094	+	0.188391i	$0.939673\pi$
$774$	0	0
$775$	2564.55i	3.30910i
$776$	0	0
$777$	1915.33	2.46503
$778$	0	0
$779$	0	0
$780$	0	0
$781$	550.302i	0.704612i
$782$	0	0
$783$	−42.3303	−0.0540617
$784$	0	0
$785$	−2082.10	−2.65236
$786$	0	0
$787$	− 804.654i	− 1.02243i	−0.859452	−	0.511216i	$-0.829195\pi$
$787$	− 804.654i	− 1.02243i	0.859452	−	0.511216i	$-0.170805\pi$
$788$	0	0
$789$	− 1095.03i	− 1.38787i
$790$	0	0
$791$	1185.33i	1.49852i
$792$	0	0
$793$	− 145.490i	− 0.183467i
$794$	0	0
$795$	−745.773	−0.938080
$796$	0	0
$797$	− 860.780i	− 1.08002i	−0.841657	−	0.540012i	$-0.818420\pi$
$797$	− 860.780i	− 1.08002i	0.841657	−	0.540012i	$-0.181580\pi$
$798$	0	0
$799$	229.618	0.287382
$800$	0	0
$801$	896.183i	1.11883i
$802$	0	0
$803$	179.314	0.223305
$804$	0	0
$805$	1983.74	2.46428
$806$	0	0
$807$	1720.57	2.13206
$808$	0	0
$809$	996.810	1.23215	0.616075	−	0.787687i	$-0.288722\pi$
$809$	996.810	1.23215	0.616075	+	0.787687i	$0.288722\pi$
$810$	0	0
$811$	− 165.611i	− 0.204205i	−0.994774	−	0.102103i	$-0.967443\pi$
$811$	− 165.611i	− 0.204205i	0.994774	−	0.102103i	$-0.0325570\pi$
$812$	0	0
$813$	− 480.350i	− 0.590837i
$814$	0	0
$815$	126.181	0.154823
$816$	0	0
$817$	0	0
$818$	0	0
$819$	341.766i	0.417297i
$820$	0	0
$821$	−1109.72	−1.35167	−0.675835	−	0.737053i	$-0.736217\pi$
$821$	−1109.72	−1.35167	−0.675835	+	0.737053i	$0.736217\pi$
$822$	0	0
$823$	542.480	0.659150	0.329575	−	0.944129i	$-0.393095\pi$
$823$	542.480	0.659150	0.329575	+	0.944129i	$0.393095\pi$
$824$	0	0
$825$	1424.23i	1.72634i
$826$	0	0
$827$	192.451i	0.232710i	0.993208	+	0.116355i	$0.0371211\pi$
$827$	192.451i	0.232710i	−0.993208	+	0.116355i	$0.962879\pi$
$828$	0	0
$829$	− 973.999i	− 1.17491i	−0.809257	−	0.587454i	$-0.800130\pi$
$829$	− 973.999i	− 1.17491i	0.809257	−	0.587454i	$-0.199870\pi$
$830$	0	0
$831$	− 1087.22i	− 1.30833i
$832$	0	0
$833$	−1202.65	−1.44376
$834$	0	0
$835$	− 1586.16i	− 1.89960i
$836$	0	0
$837$	−1298.22	−1.55104
$838$	0	0
$839$	772.253i	0.920445i	0.887804	+	0.460223i	$0.152230\pi$
$839$	772.253i	0.920445i	−0.887804	+	0.460223i	$0.847770\pi$
$840$	0	0
$841$	838.875	0.997473
$842$	0	0
$843$	2178.98	2.58479
$844$	0	0
$845$	−1496.00	−1.77041
$846$	0	0
$847$	−1069.12	−1.26224
$848$	0	0
$849$	− 1319.36i	− 1.55401i
$850$	0	0
$851$	− 679.141i	− 0.798051i
$852$	0	0
$853$	−184.691	−0.216520	−0.108260	−	0.994123i	$-0.534528\pi$
$853$	−184.691	−0.216520	−0.108260	+	0.994123i	$0.534528\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	948.525i	1.10680i	0.832916	+	0.553399i	$0.186669\pi$
$857$	948.525i	1.10680i	−0.832916	+	0.553399i	$0.813331\pi$
$858$	0	0
$859$	32.0045	0.0372579	0.0186289	−	0.999826i	$-0.494070\pi$
$859$	32.0045	0.0372579	0.0186289	+	0.999826i	$0.494070\pi$
$860$	0	0
$861$	3340.14	3.87937
$862$	0	0
$863$	1146.52i	1.32852i	0.747500	+	0.664262i	$0.231254\pi$
$863$	1146.52i	1.32852i	−0.747500	+	0.664262i	$0.768746\pi$
$864$	0	0
$865$	2964.78i	3.42749i
$866$	0	0
$867$	220.627i	0.254471i
$868$	0	0
$869$	− 575.955i	− 0.662779i
$870$	0	0
$871$	−132.664	−0.152312
$872$	0	0
$873$	− 444.691i	− 0.509383i
$874$	0	0
$875$	−3296.51	−3.76744
$876$	0	0
$877$	187.762i	0.214096i	0.994254	+	0.107048i	$0.0341399\pi$
$877$	187.762i	0.214096i	−0.994254	+	0.107048i	$0.965860\pi$
$878$	0	0
$879$	627.736	0.714148
$880$	0	0
$881$	635.725	0.721594	0.360797	−	0.932644i	$-0.382505\pi$
$881$	635.725	0.721594	0.360797	+	0.932644i	$0.382505\pi$
$882$	0	0
$883$	807.361	0.914339	0.457170	−	0.889380i	$-0.348863\pi$
$883$	807.361	0.914339	0.457170	+	0.889380i	$0.348863\pi$
$884$	0	0
$885$	−275.433	−0.311224
$886$	0	0
$887$	− 1208.85i	− 1.36286i	−0.731885	−	0.681429i	$-0.761359\pi$
$887$	− 1208.85i	− 1.36286i	0.731885	−	0.681429i	$-0.238641\pi$
$888$	0	0
$889$	1883.28i	2.11843i
$890$	0	0
$891$	−38.7769	−0.0435207
$892$	0	0
$893$	0	0
$894$	0	0
$895$	− 2220.33i	− 2.48082i
$896$	0	0
$897$	194.211	0.216511
$898$	0	0
$899$	−65.1802	−0.0725030
$900$	0	0
$901$	− 262.326i	− 0.291150i
$902$	0	0
$903$	− 614.442i	− 0.680445i
$904$	0	0
$905$	2621.28i	2.89645i
$906$	0	0
$907$	1364.17i	1.50404i	0.659138	+	0.752022i	$0.270921\pi$
$907$	1364.17i	1.50404i	−0.659138	+	0.752022i	$0.729079\pi$
$908$	0	0
$909$	−2904.37	−3.19513
$910$	0	0
$911$	492.916i	0.541071i	0.962710	+	0.270536i	$0.0872008\pi$
$911$	492.916i	0.541071i	−0.962710	+	0.270536i	$0.912799\pi$
$912$	0	0
$913$	−146.627	−0.160599
$914$	0	0
$915$	3168.70i	3.46306i
$916$	0	0
$917$	165.834	0.180844
$918$	0	0
$919$	−516.342	−0.561852	−0.280926	−	0.959729i	$-0.590641\pi$
$919$	−516.342	−0.561852	−0.280926	+	0.959729i	$0.590641\pi$
$920$	0	0
$921$	−1203.94	−1.30721
$922$	0	0
$923$	−221.039	−0.239479
$924$	0	0
$925$	2000.48i	2.16268i
$926$	0	0
$927$	1826.33i	1.97016i
$928$	0	0
$929$	−1347.03	−1.44998	−0.724988	−	0.688761i	$-0.758155\pi$
$929$	−1347.03	−1.44998	−0.724988	+	0.688761i	$0.758155\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	− 1878.57i	− 2.01347i
$934$	0	0
$935$	−719.322	−0.769328
$936$	0	0
$937$	−179.032	−0.191069	−0.0955346	−	0.995426i	$-0.530456\pi$
$937$	−179.032	−0.191069	−0.0955346	+	0.995426i	$0.530456\pi$
$938$	0	0
$939$	1038.98i	1.10648i
$940$	0	0
$941$	1000.60i	1.06333i	0.846953	+	0.531667i	$0.178434\pi$
$941$	1000.60i	1.06333i	−0.846953	+	0.531667i	$0.821566\pi$
$942$	0	0
$943$	− 1184.35i	− 1.25594i
$944$	0	0
$945$	− 2957.99i	− 3.13014i
$946$	0	0
$947$	−1345.81	−1.42113	−0.710564	−	0.703633i	$-0.751560\pi$
$947$	−1345.81	−1.42113	−0.710564	+	0.703633i	$0.751560\pi$
$948$	0	0
$949$	72.0248i	0.0758955i
$950$	0	0
$951$	1226.33	1.28952
$952$	0	0
$953$	− 466.625i	− 0.489638i	−0.969569	−	0.244819i	$-0.921271\pi$
$953$	− 466.625i	− 0.489638i	0.969569	−	0.244819i	$-0.0787286\pi$
$954$	0	0
$955$	−956.282	−1.00134
$956$	0	0
$957$	−36.1979	−0.0378244
$958$	0	0
$959$	1253.44	1.30703
$960$	0	0
$961$	−1038.00	−1.08013
$962$	0	0
$963$	− 22.0406i	− 0.0228875i
$964$	0	0
$965$	− 1823.15i	− 1.88927i
$966$	0	0
$967$	−1033.54	−1.06881	−0.534406	−	0.845228i	$-0.679465\pi$
$967$	−1033.54	−1.06881	−0.534406	+	0.845228i	$0.679465\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 1360.16i	− 1.40078i	−0.713761	−	0.700390i	$-0.753010\pi$
$971$	− 1360.16i	− 1.40078i	0.713761	−	0.700390i	$-0.246990\pi$
$972$	0	0
$973$	−2393.81	−2.46024
$974$	0	0
$975$	−572.067	−0.586735
$976$	0	0
$977$	1085.93i	1.11150i	0.831350	+	0.555749i	$0.187569\pi$
$977$	1085.93i	1.11150i	−0.831350	+	0.555749i	$0.812431\pi$
$978$	0	0
$979$	304.542i	0.311075i
$980$	0	0
$981$	961.102i	0.979717i
$982$	0	0
$983$	− 716.409i	− 0.728799i	−0.931243	−	0.364399i	$-0.881274\pi$
$983$	− 716.409i	− 0.728799i	0.931243	−	0.364399i	$-0.118726\pi$
$984$	0	0
$985$	−1016.88	−1.03236
$986$	0	0
$987$	807.443i	0.818078i
$988$	0	0
$989$	−217.870	−0.220293
$990$	0	0
$991$	514.771i	0.519446i	0.965683	+	0.259723i	$0.0836312\pi$
$991$	514.771i	0.519446i	−0.965683	+	0.259723i	$0.916369\pi$
$992$	0	0
$993$	−486.953	−0.490386
$994$	0	0
$995$	2523.17	2.53584
$996$	0	0
$997$	1037.25	1.04037	0.520187	−	0.854053i	$-0.325862\pi$
$997$	1037.25	1.04037	0.520187	+	0.854053i	$0.325862\pi$
$998$	0	0
$999$	−1012.68	−1.01369

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1444.3.c.d.721.3	✓	24
19.18	odd	2	inner	1444.3.c.d.721.22	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1444.3.c.d.721.3	✓	24	1.1	even	1	trivial
1444.3.c.d.721.22	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)

\(f(q)\)	\(=\)	\(q-4.89235i q^{3} -9.07521 q^{5} +11.2253 q^{7} -14.9351 q^{9} +O(q^{10})\)	\(q-4.89235i q^{3} -9.07521 q^{5} +11.2253 q^{7} -14.9351 q^{9} -5.07525 q^{11} -2.03857i q^{13} +44.3991i q^{15} -15.6174 q^{17} -54.9180i q^{21} -19.4729 q^{23} +57.3594 q^{25} +29.0364i q^{27} +1.45784i q^{29} +44.7102i q^{31} +24.8299i q^{33} -101.872 q^{35} +34.8762i q^{37} -9.97338 q^{39} +60.8205i q^{41} +11.1884 q^{43} +135.539 q^{45} -14.7027 q^{47} +77.0071 q^{49} +76.4058i q^{51} +16.7970i q^{53} +46.0589 q^{55} +6.20357i q^{59} +71.3685 q^{61} -167.650 q^{63} +18.5004i q^{65} -65.0771i q^{67} +95.2683i q^{69} -108.429i q^{71} -35.3311 q^{73} -280.622i q^{75} -56.9711 q^{77} +113.483i q^{79} +7.64040 q^{81} +28.8906 q^{83} +141.731 q^{85} +7.13225 q^{87} -60.0053i q^{89} -22.8835i q^{91} +218.738 q^{93} +29.7750i q^{97} +75.7991 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 12 q^{5} + 40 q^{7} - 104 q^{9}+O(q^{10}) \) 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

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Database

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