LMFDB - Embedded newform 1444.3.c.d.721.19

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			721.19
Character	$\chi$	$=$	1444.721
Dual form			1444.3.c.d.721.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.56718i q^{3} -2.91277 q^{5} -2.95359 q^{7} -3.72474 q^{9} +O(q^{10})$	$q+3.56718i q^{3} -2.91277 q^{5} -2.95359 q^{7} -3.72474 q^{9} -21.8845 q^{11} -10.0774i q^{13} -10.3903i q^{15} +8.21959 q^{17} -10.5360i q^{21} +12.5196 q^{23} -16.5158 q^{25} +18.8178i q^{27} +21.5020i q^{29} -41.3632i q^{31} -78.0660i q^{33} +8.60312 q^{35} +32.6932i q^{37} +35.9477 q^{39} -48.4085i q^{41} +49.2505 q^{43} +10.8493 q^{45} +74.2654 q^{47} -40.2763 q^{49} +29.3207i q^{51} -31.2444i q^{53} +63.7445 q^{55} -54.0781i q^{59} +105.977 q^{61} +11.0014 q^{63} +29.3530i q^{65} +101.564i q^{67} +44.6596i q^{69} -61.8458i q^{71} +26.3088 q^{73} -58.9147i q^{75} +64.6379 q^{77} +144.433i q^{79} -100.649 q^{81} +2.17164 q^{83} -23.9417 q^{85} -76.7015 q^{87} -19.7114i q^{89} +29.7644i q^{91} +147.550 q^{93} -110.583i q^{97} +81.5142 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	3.56718i	1.18906i	0.804074	+	0.594529i	$0.202662\pi$
$3$	3.56718i	1.18906i	−0.804074	+	0.594529i	$0.797338\pi$
$4$	0	0
$5$	−2.91277	−0.582553	−0.291277	−	0.956639i	$-0.594080\pi$
$5$	−2.91277	−0.582553	−0.291277	+	0.956639i	$0.594080\pi$
$6$	0	0
$7$	−2.95359	−0.421941	−0.210971	−	0.977492i	$-0.567662\pi$
$7$	−2.95359	−0.421941	−0.210971	+	0.977492i	$0.567662\pi$
$8$	0	0
$9$	−3.72474	−0.413860
$10$	0	0
$11$	−21.8845	−1.98950	−0.994751	−	0.102323i	$-0.967373\pi$
$11$	−21.8845	−1.98950	−0.994751	+	0.102323i	$0.967373\pi$
$12$	0	0
$13$	− 10.0774i	− 0.775182i	−0.921831	−	0.387591i	$-0.873307\pi$
$13$	− 10.0774i	− 0.775182i	0.921831	−	0.387591i	$-0.126693\pi$
$14$	0	0
$15$	− 10.3903i	− 0.692690i
$16$	0	0
$17$	8.21959	0.483505	0.241753	−	0.970338i	$-0.422278\pi$
$17$	8.21959	0.483505	0.241753	+	0.970338i	$0.422278\pi$
$18$	0	0
$19$	0	0
$19$	0	0
$20$	0	0
$21$	− 10.5360i	− 0.501713i
$22$	0	0
$23$	12.5196	0.544330	0.272165	−	0.962251i	$-0.412260\pi$
$23$	12.5196	0.544330	0.272165	+	0.962251i	$0.412260\pi$
$24$	0	0
$25$	−16.5158	−0.660632
$26$	0	0
$27$	18.8178i	0.696955i
$28$	0	0
$29$	21.5020i	0.741449i	0.928743	+	0.370725i	$0.120891\pi$
$29$	21.5020i	0.741449i	−0.928743	+	0.370725i	$0.879109\pi$
$30$	0	0
$31$	− 41.3632i	− 1.33430i	−0.744926	−	0.667148i	$-0.767515\pi$
$31$	− 41.3632i	− 1.33430i	0.744926	−	0.667148i	$-0.232485\pi$
$32$	0	0
$33$	− 78.0660i	− 2.36563i
$34$	0	0
$35$	8.60312	0.245803
$36$	0	0
$37$	32.6932i	0.883601i	0.897113	+	0.441800i	$0.145660\pi$
$37$	32.6932i	0.883601i	−0.897113	+	0.441800i	$0.854340\pi$
$38$	0	0
$39$	35.9477	0.921737
$40$	0	0
$41$	− 48.4085i	− 1.18070i	−0.807149	−	0.590348i	$-0.798991\pi$
$41$	− 48.4085i	− 1.18070i	0.807149	−	0.590348i	$-0.201009\pi$
$42$	0	0
$43$	49.2505	1.14536	0.572680	−	0.819779i	$-0.305904\pi$
$43$	49.2505	1.14536	0.572680	+	0.819779i	$0.305904\pi$
$44$	0	0
$45$	10.8493	0.241096
$46$	0	0
$47$	74.2654	1.58012	0.790058	−	0.613033i	$-0.210051\pi$
$47$	74.2654	1.58012	0.790058	+	0.613033i	$0.210051\pi$
$48$	0	0
$49$	−40.2763	−0.821965
$50$	0	0
$51$	29.3207i	0.574916i
$52$	0	0
$53$	− 31.2444i	− 0.589517i	−0.955572	−	0.294759i	$-0.904761\pi$
$53$	− 31.2444i	− 0.589517i	0.955572	−	0.294759i	$-0.0952393\pi$
$54$	0	0
$55$	63.7445	1.15899
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 54.0781i	− 0.916578i	−0.888803	−	0.458289i	$-0.848463\pi$
$59$	− 54.0781i	− 0.916578i	0.888803	−	0.458289i	$-0.151537\pi$
$60$	0	0
$61$	105.977	1.73733	0.868666	−	0.495398i	$-0.164978\pi$
$61$	105.977	1.73733	0.868666	+	0.495398i	$0.164978\pi$
$62$	0	0
$63$	11.0014	0.174625
$64$	0	0
$65$	29.3530i	0.451585i
$66$	0	0
$67$	101.564i	1.51589i	0.652320	+	0.757944i	$0.273796\pi$
$67$	101.564i	1.51589i	−0.652320	+	0.757944i	$0.726204\pi$
$68$	0	0
$69$	44.6596i	0.647241i
$70$	0	0
$71$	− 61.8458i	− 0.871067i	−0.900172	−	0.435534i	$-0.856560\pi$
$71$	− 61.8458i	− 0.871067i	0.900172	−	0.435534i	$-0.143440\pi$
$72$	0	0
$73$	26.3088	0.360395	0.180198	−	0.983630i	$-0.442326\pi$
$73$	26.3088	0.360395	0.180198	+	0.983630i	$0.442326\pi$
$74$	0	0
$75$	− 58.9147i	− 0.785530i
$76$	0	0
$77$	64.6379	0.839453
$78$	0	0
$79$	144.433i	1.82827i	0.405413	+	0.914134i	$0.367128\pi$
$79$	144.433i	1.82827i	−0.405413	+	0.914134i	$0.632872\pi$
$80$	0	0
$81$	−100.649	−1.24258
$82$	0	0
$83$	2.17164	0.0261643	0.0130821	−	0.999914i	$-0.495836\pi$
$83$	2.17164	0.0261643	0.0130821	+	0.999914i	$0.495836\pi$
$84$	0	0
$85$	−23.9417	−0.281668
$86$	0	0
$87$	−76.7015	−0.881626
$88$	0	0
$89$	− 19.7114i	− 0.221477i	−0.993850	−	0.110738i	$-0.964678\pi$
$89$	− 19.7114i	− 0.221477i	0.993850	−	0.110738i	$-0.0353216\pi$
$90$	0	0
$91$	29.7644i	0.327081i
$92$	0	0
$93$	147.550	1.58656
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 110.583i	− 1.14003i	−0.821635	−	0.570014i	$-0.806938\pi$
$97$	− 110.583i	− 1.14003i	0.821635	−	0.570014i	$-0.193062\pi$
$98$	0	0
$99$	81.5142	0.823376
$100$	0	0
$101$	−93.9678	−0.930374	−0.465187	−	0.885212i	$-0.654013\pi$
$101$	−93.9678	−0.930374	−0.465187	+	0.885212i	$0.654013\pi$
$102$	0	0
$103$	− 103.900i	− 1.00874i	−0.863487	−	0.504371i	$-0.831724\pi$
$103$	− 103.900i	− 1.00874i	0.863487	−	0.504371i	$-0.168276\pi$
$104$	0	0
$105$	30.6888i	0.292275i
$106$	0	0
$107$	183.795i	1.71771i	0.512216	+	0.858856i	$0.328825\pi$
$107$	183.795i	1.71771i	−0.512216	+	0.858856i	$0.671175\pi$
$108$	0	0
$109$	− 46.6611i	− 0.428083i	−0.976825	−	0.214042i	$-0.931337\pi$
$109$	− 46.6611i	− 0.428083i	0.976825	−	0.214042i	$-0.0686628\pi$
$110$	0	0
$111$	−116.623	−1.05065
$112$	0	0
$113$	− 186.762i	− 1.65276i	−0.563110	−	0.826382i	$-0.690395\pi$
$113$	− 186.762i	− 1.65276i	0.563110	−	0.826382i	$-0.309605\pi$
$114$	0	0
$115$	−36.4667	−0.317101
$116$	0	0
$117$	37.5356i	0.320817i
$118$	0	0
$119$	−24.2773	−0.204011
$120$	0	0
$121$	357.933	2.95812
$122$	0	0
$123$	172.682	1.40392
$124$	0	0
$125$	120.926	0.967406
$126$	0	0
$127$	84.8853i	0.668388i	0.942504	+	0.334194i	$0.108464\pi$
$127$	84.8853i	0.668388i	−0.942504	+	0.334194i	$0.891536\pi$
$128$	0	0
$129$	175.685i	1.36190i
$130$	0	0
$131$	34.2072	0.261124	0.130562	−	0.991440i	$-0.458322\pi$
$131$	34.2072	0.261124	0.130562	+	0.991440i	$0.458322\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	− 54.8118i	− 0.406013i
$136$	0	0
$137$	−42.7457	−0.312012	−0.156006	−	0.987756i	$-0.549862\pi$
$137$	−42.7457	−0.312012	−0.156006	+	0.987756i	$0.549862\pi$
$138$	0	0
$139$	69.4802	0.499858	0.249929	−	0.968264i	$-0.419593\pi$
$139$	69.4802	0.499858	0.249929	+	0.968264i	$0.419593\pi$
$140$	0	0
$141$	264.918i	1.87885i
$142$	0	0
$143$	220.538i	1.54223i
$144$	0	0
$145$	− 62.6304i	− 0.431934i
$146$	0	0
$147$	− 143.673i	− 0.977365i
$148$	0	0
$149$	−244.901	−1.64363	−0.821814	−	0.569755i	$-0.807038\pi$
$149$	−244.901	−1.64363	−0.821814	+	0.569755i	$0.807038\pi$
$150$	0	0
$151$	41.9634i	0.277904i	0.990299	+	0.138952i	$0.0443733\pi$
$151$	41.9634i	0.277904i	−0.990299	+	0.138952i	$0.955627\pi$
$152$	0	0
$153$	−30.6158	−0.200104
$154$	0	0
$155$	120.481i	0.777298i
$156$	0	0
$157$	179.430	1.14286	0.571432	−	0.820649i	$-0.306388\pi$
$157$	179.430	1.14286	0.571432	+	0.820649i	$0.306388\pi$
$158$	0	0
$159$	111.454	0.700971
$160$	0	0
$161$	−36.9778	−0.229676
$162$	0	0
$163$	242.887	1.49011	0.745053	−	0.667005i	$-0.232424\pi$
$163$	242.887	1.49011	0.745053	+	0.667005i	$0.232424\pi$
$164$	0	0
$165$	227.388i	1.37811i
$166$	0	0
$167$	80.3087i	0.480891i	0.970663	+	0.240445i	$0.0772935\pi$
$167$	80.3087i	0.480891i	−0.970663	+	0.240445i	$0.922707\pi$
$168$	0	0
$169$	67.4467	0.399093
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 138.007i	− 0.797730i	−0.917010	−	0.398865i	$-0.869404\pi$
$173$	− 138.007i	− 0.797730i	0.917010	−	0.398865i	$-0.130596\pi$
$174$	0	0
$175$	48.7809	0.278748
$176$	0	0
$177$	192.906	1.08986
$178$	0	0
$179$	− 66.7312i	− 0.372800i	−0.982474	−	0.186400i	$-0.940318\pi$
$179$	− 66.7312i	− 0.372800i	0.982474	−	0.186400i	$-0.0596821\pi$
$180$	0	0
$181$	− 189.905i	− 1.04920i	−0.851349	−	0.524600i	$-0.824215\pi$
$181$	− 189.905i	− 1.04920i	0.851349	−	0.524600i	$-0.175785\pi$
$182$	0	0
$183$	378.040i	2.06579i
$184$	0	0
$185$	− 95.2277i	− 0.514745i
$186$	0	0
$187$	−179.882	−0.961935
$188$	0	0
$189$	− 55.5800i	− 0.294074i
$190$	0	0
$191$	−96.8587	−0.507113	−0.253557	−	0.967321i	$-0.581600\pi$
$191$	−96.8587	−0.507113	−0.253557	+	0.967321i	$0.581600\pi$
$192$	0	0
$193$	41.6748i	0.215931i	0.994155	+	0.107966i	$0.0344337\pi$
$193$	41.6748i	0.215931i	−0.994155	+	0.107966i	$0.965566\pi$
$194$	0	0
$195$	−104.707	−0.536961
$196$	0	0
$197$	269.420	1.36761	0.683807	−	0.729663i	$-0.260323\pi$
$197$	269.420	1.36761	0.683807	+	0.729663i	$0.260323\pi$
$198$	0	0
$199$	95.7815	0.481314	0.240657	−	0.970610i	$-0.422637\pi$
$199$	95.7815	0.481314	0.240657	+	0.970610i	$0.422637\pi$
$200$	0	0
$201$	−362.298	−1.80248
$202$	0	0
$203$	− 63.5082i	− 0.312848i
$204$	0	0
$205$	141.003i	0.687818i
$206$	0	0
$207$	−46.6323	−0.225277
$208$	0	0
$209$	0	0
$210$	0	0
$211$	− 229.049i	− 1.08554i	−0.839881	−	0.542770i	$-0.817376\pi$
$211$	− 229.049i	− 1.08554i	0.839881	−	0.542770i	$-0.182624\pi$
$212$	0	0
$213$	220.615	1.03575
$214$	0	0
$215$	−143.455	−0.667233
$216$	0	0
$217$	122.170i	0.562994i
$218$	0	0
$219$	93.8482i	0.428531i
$220$	0	0
$221$	− 82.8318i	− 0.374805i
$222$	0	0
$223$	217.766i	0.976529i	0.872696	+	0.488264i	$0.162370\pi$
$223$	217.766i	0.976529i	−0.872696	+	0.488264i	$0.837630\pi$
$224$	0	0
$225$	61.5171	0.273409
$226$	0	0
$227$	48.9342i	0.215569i	0.994174	+	0.107785i	$0.0343757\pi$
$227$	48.9342i	0.215569i	−0.994174	+	0.107785i	$0.965624\pi$
$228$	0	0
$229$	314.156	1.37186	0.685930	−	0.727667i	$-0.259395\pi$
$229$	314.156	1.37186	0.685930	+	0.727667i	$0.259395\pi$
$230$	0	0
$231$	230.575i	0.998159i
$232$	0	0
$233$	−279.711	−1.20048	−0.600238	−	0.799822i	$-0.704927\pi$
$233$	−279.711	−1.20048	−0.600238	+	0.799822i	$0.704927\pi$
$234$	0	0
$235$	−216.318	−0.920501
$236$	0	0
$237$	−515.218	−2.17392
$238$	0	0
$239$	−0.573252	−0.00239854	−0.00119927	−	0.999999i	$-0.500382\pi$
$239$	−0.573252	−0.00239854	−0.00119927	+	0.999999i	$0.500382\pi$
$240$	0	0
$241$	− 288.744i	− 1.19811i	−0.800709	−	0.599054i	$-0.795543\pi$
$241$	− 288.744i	− 1.19811i	0.800709	−	0.599054i	$-0.204457\pi$
$242$	0	0
$243$	− 189.673i	− 0.780546i
$244$	0	0
$245$	117.315	0.478839
$246$	0	0
$247$	0	0
$248$	0	0
$249$	7.74661i	0.0311109i
$250$	0	0
$251$	−58.3244	−0.232368	−0.116184	−	0.993228i	$-0.537066\pi$
$251$	−58.3244	−0.232368	−0.116184	+	0.993228i	$0.537066\pi$
$252$	0	0
$253$	−273.986	−1.08295
$254$	0	0
$255$	− 85.4044i	− 0.334919i
$256$	0	0
$257$	6.31597i	0.0245757i	0.999925	+	0.0122879i	$0.00391145\pi$
$257$	6.31597i	0.0245757i	−0.999925	+	0.0122879i	$0.996089\pi$
$258$	0	0
$259$	− 96.5624i	− 0.372828i
$260$	0	0
$261$	− 80.0895i	− 0.306856i
$262$	0	0
$263$	375.469	1.42764	0.713820	−	0.700329i	$-0.246963\pi$
$263$	375.469	1.42764	0.713820	+	0.700329i	$0.246963\pi$
$264$	0	0
$265$	91.0077i	0.343425i
$266$	0	0
$267$	70.3141	0.263349
$268$	0	0
$269$	− 288.521i	− 1.07257i	−0.844038	−	0.536284i	$-0.819828\pi$
$269$	− 288.521i	− 1.07257i	0.844038	−	0.536284i	$-0.180172\pi$
$270$	0	0
$271$	63.0550	0.232675	0.116338	−	0.993210i	$-0.462885\pi$
$271$	63.0550	0.232675	0.116338	+	0.993210i	$0.462885\pi$
$272$	0	0
$273$	−106.175	−0.388919
$274$	0	0
$275$	361.440	1.31433
$276$	0	0
$277$	−117.696	−0.424896	−0.212448	−	0.977172i	$-0.568144\pi$
$277$	−117.696	−0.424896	−0.212448	+	0.977172i	$0.568144\pi$
$278$	0	0
$279$	154.067i	0.552212i
$280$	0	0
$281$	34.4275i	0.122518i	0.998122	+	0.0612589i	$0.0195115\pi$
$281$	34.4275i	0.122518i	−0.998122	+	0.0612589i	$0.980488\pi$
$282$	0	0
$283$	281.782	0.995696	0.497848	−	0.867264i	$-0.334124\pi$
$283$	281.782	0.995696	0.497848	+	0.867264i	$0.334124\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	142.979i	0.498184i
$288$	0	0
$289$	−221.438	−0.766223
$290$	0	0
$291$	394.468	1.35556
$292$	0	0
$293$	− 309.955i	− 1.05787i	−0.848663	−	0.528934i	$-0.822592\pi$
$293$	− 309.955i	− 1.05787i	0.848663	−	0.528934i	$-0.177408\pi$
$294$	0	0
$295$	157.517i	0.533955i
$296$	0	0
$297$	− 411.818i	− 1.38659i
$298$	0	0
$299$	− 126.165i	− 0.421955i
$300$	0	0
$301$	−145.466	−0.483275
$302$	0	0
$303$	− 335.200i	− 1.10627i
$304$	0	0
$305$	−308.687	−1.01209
$306$	0	0
$307$	− 414.620i	− 1.35055i	−0.737564	−	0.675277i	$-0.764024\pi$
$307$	− 414.620i	− 1.35055i	0.737564	−	0.675277i	$-0.235976\pi$
$308$	0	0
$309$	370.631	1.19945
$310$	0	0
$311$	−387.400	−1.24566	−0.622830	−	0.782357i	$-0.714017\pi$
$311$	−387.400	−1.24566	−0.622830	+	0.782357i	$0.714017\pi$
$312$	0	0
$313$	368.585	1.17759	0.588794	−	0.808283i	$-0.299603\pi$
$313$	368.585	1.17759	0.588794	+	0.808283i	$0.299603\pi$
$314$	0	0
$315$	−32.0444	−0.101728
$316$	0	0
$317$	− 213.580i	− 0.673755i	−0.941549	−	0.336877i	$-0.890629\pi$
$317$	− 213.580i	− 0.673755i	0.941549	−	0.336877i	$-0.109371\pi$
$318$	0	0
$319$	− 470.562i	− 1.47511i
$320$	0	0
$321$	−655.630	−2.04246
$322$	0	0
$323$	0	0
$324$	0	0
$325$	166.436i	0.512110i
$326$	0	0
$327$	166.448	0.509016
$328$	0	0
$329$	−219.350	−0.666716
$330$	0	0
$331$	231.565i	0.699592i	0.936826	+	0.349796i	$0.113749\pi$
$331$	231.565i	0.699592i	−0.936826	+	0.349796i	$0.886251\pi$
$332$	0	0
$333$	− 121.774i	− 0.365687i
$334$	0	0
$335$	− 295.833i	− 0.883085i
$336$	0	0
$337$	− 308.471i	− 0.915344i	−0.889121	−	0.457672i	$-0.848683\pi$
$337$	− 308.471i	− 0.915344i	0.889121	−	0.457672i	$-0.151317\pi$
$338$	0	0
$339$	666.214	1.96523
$340$	0	0
$341$	905.213i	2.65458i
$342$	0	0
$343$	263.686	0.768763
$344$	0	0
$345$	− 130.083i	− 0.377052i
$346$	0	0
$347$	270.251	0.778821	0.389411	−	0.921064i	$-0.372679\pi$
$347$	270.251	0.778821	0.389411	+	0.921064i	$0.372679\pi$
$348$	0	0
$349$	−515.867	−1.47813	−0.739065	−	0.673634i	$-0.764732\pi$
$349$	−515.867	−1.47813	−0.739065	+	0.673634i	$0.764732\pi$
$350$	0	0
$351$	189.634	0.540267
$352$	0	0
$353$	196.162	0.555699	0.277850	−	0.960625i	$-0.410378\pi$
$353$	196.162	0.555699	0.277850	+	0.960625i	$0.410378\pi$
$354$	0	0
$355$	180.142i	0.507443i
$356$	0	0
$357$	− 86.6014i	− 0.242581i
$358$	0	0
$359$	−25.8616	−0.0720378	−0.0360189	−	0.999351i	$-0.511468\pi$
$359$	−25.8616	−0.0720378	−0.0360189	+	0.999351i	$0.511468\pi$
$360$	0	0
$361$	0	0
$362$	0	0
$363$	1276.81i	3.51738i
$364$	0	0
$365$	−76.6315	−0.209949
$366$	0	0
$367$	−198.474	−0.540801	−0.270401	−	0.962748i	$-0.587156\pi$
$367$	−198.474	−0.540801	−0.270401	+	0.962748i	$0.587156\pi$
$368$	0	0
$369$	180.309i	0.488643i
$370$	0	0
$371$	92.2832i	0.248742i
$372$	0	0
$373$	41.2092i	0.110480i	0.998473	+	0.0552402i	$0.0175925\pi$
$373$	41.2092i	0.110480i	−0.998473	+	0.0552402i	$0.982408\pi$
$374$	0	0
$375$	431.364i	1.15030i
$376$	0	0
$377$	216.684	0.574758
$378$	0	0
$379$	− 419.992i	− 1.10816i	−0.832464	−	0.554079i	$-0.813071\pi$
$379$	− 419.992i	− 1.10816i	0.832464	−	0.554079i	$-0.186929\pi$
$380$	0	0
$381$	−302.801	−0.794753
$382$	0	0
$383$	7.84945i	0.0204947i	0.999947	+	0.0102473i	$0.00326188\pi$
$383$	7.84945i	0.0204947i	−0.999947	+	0.0102473i	$0.996738\pi$
$384$	0	0
$385$	−188.275	−0.489026
$386$	0	0
$387$	−183.445	−0.474019
$388$	0	0
$389$	−645.273	−1.65880	−0.829400	−	0.558656i	$-0.811317\pi$
$389$	−645.273	−1.65880	−0.829400	+	0.558656i	$0.811317\pi$
$390$	0	0
$391$	102.906	0.263187
$392$	0	0
$393$	122.023i	0.310492i
$394$	0	0
$395$	− 420.700i	− 1.06506i
$396$	0	0
$397$	−149.939	−0.377681	−0.188840	−	0.982008i	$-0.560473\pi$
$397$	−149.939	−0.377681	−0.188840	+	0.982008i	$0.560473\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 472.855i	− 1.17919i	−0.807699	−	0.589595i	$-0.799287\pi$
$401$	− 472.855i	− 1.17919i	0.807699	−	0.589595i	$-0.200713\pi$
$402$	0	0
$403$	−416.832	−1.03432
$404$	0	0
$405$	293.167	0.723869
$406$	0	0
$407$	− 715.476i	− 1.75793i
$408$	0	0
$409$	− 700.328i	− 1.71229i	−0.516734	−	0.856146i	$-0.672852\pi$
$409$	− 700.328i	− 1.71229i	0.516734	−	0.856146i	$-0.327148\pi$
$410$	0	0
$411$	− 152.481i	− 0.371001i
$412$	0	0
$413$	159.724i	0.386742i
$414$	0	0
$415$	−6.32547	−0.0152421
$416$	0	0
$417$	247.848i	0.594360i
$418$	0	0
$419$	−161.641	−0.385779	−0.192889	−	0.981220i	$-0.561786\pi$
$419$	−161.641	−0.385779	−0.192889	+	0.981220i	$0.561786\pi$
$420$	0	0
$421$	− 179.229i	− 0.425723i	−0.977082	−	0.212861i	$-0.931722\pi$
$421$	− 179.229i	− 0.425723i	0.977082	−	0.212861i	$-0.0682782\pi$
$422$	0	0
$423$	−276.620	−0.653947
$424$	0	0
$425$	−135.753	−0.319419
$426$	0	0
$427$	−313.013	−0.733053
$428$	0	0
$429$	−786.699	−1.83380
$430$	0	0
$431$	− 749.177i	− 1.73823i	−0.494610	−	0.869115i	$-0.664689\pi$
$431$	− 749.177i	− 1.73823i	0.494610	−	0.869115i	$-0.335311\pi$
$432$	0	0
$433$	− 327.825i	− 0.757101i	−0.925581	−	0.378550i	$-0.876423\pi$
$433$	− 327.825i	− 0.757101i	0.925581	−	0.378550i	$-0.123577\pi$
$434$	0	0
$435$	223.414	0.513594
$436$	0	0
$437$	0	0
$438$	0	0
$439$	40.8286i	0.0930036i	0.998918	+	0.0465018i	$0.0148073\pi$
$439$	40.8286i	0.0930036i	−0.998918	+	0.0465018i	$0.985193\pi$
$440$	0	0
$441$	150.019	0.340179
$442$	0	0
$443$	−560.481	−1.26519	−0.632597	−	0.774481i	$-0.718011\pi$
$443$	−560.481	−1.26519	−0.632597	+	0.774481i	$0.718011\pi$
$444$	0	0
$445$	57.4148i	0.129022i
$446$	0	0
$447$	− 873.604i	− 1.95437i
$448$	0	0
$449$	− 138.943i	− 0.309450i	−0.987958	−	0.154725i	$-0.950551\pi$
$449$	− 138.943i	− 0.309450i	0.987958	−	0.154725i	$-0.0494492\pi$
$450$	0	0
$451$	1059.40i	2.34900i
$452$	0	0
$453$	−149.691	−0.330444
$454$	0	0
$455$	− 86.6968i	− 0.190542i
$456$	0	0
$457$	357.592	0.782477	0.391239	−	0.920289i	$-0.372047\pi$
$457$	357.592	0.782477	0.391239	+	0.920289i	$0.372047\pi$
$458$	0	0
$459$	154.674i	0.336981i
$460$	0	0
$461$	7.35160	0.0159471	0.00797354	−	0.999968i	$-0.497462\pi$
$461$	7.35160	0.0159471	0.00797354	+	0.999968i	$0.497462\pi$
$462$	0	0
$463$	−485.832	−1.04931	−0.524656	−	0.851314i	$-0.675806\pi$
$463$	−485.832	−1.04931	−0.524656	+	0.851314i	$0.675806\pi$
$464$	0	0
$465$	−429.778	−0.924253
$466$	0	0
$467$	790.719	1.69319	0.846594	−	0.532239i	$-0.178649\pi$
$467$	790.719	1.69319	0.846594	+	0.532239i	$0.178649\pi$
$468$	0	0
$469$	− 299.980i	− 0.639615i
$470$	0	0
$471$	640.057i	1.35893i
$472$	0	0
$473$	−1077.82	−2.27870
$474$	0	0
$475$	0	0
$476$	0	0
$477$	116.377i	0.243978i
$478$	0	0
$479$	−343.994	−0.718150	−0.359075	−	0.933309i	$-0.616908\pi$
$479$	−343.994	−0.718150	−0.359075	+	0.933309i	$0.616908\pi$
$480$	0	0
$481$	329.462	0.684952
$482$	0	0
$483$	− 131.906i	− 0.273098i
$484$	0	0
$485$	322.101i	0.664126i
$486$	0	0
$487$	− 150.026i	− 0.308062i	−0.988066	−	0.154031i	$-0.950774\pi$
$487$	− 150.026i	− 0.308062i	0.988066	−	0.154031i	$-0.0492255\pi$
$488$	0	0
$489$	866.422i	1.77182i
$490$	0	0
$491$	99.6980	0.203051	0.101525	−	0.994833i	$-0.467628\pi$
$491$	99.6980	0.203051	0.101525	+	0.994833i	$0.467628\pi$
$492$	0	0
$493$	176.738i	0.358495i
$494$	0	0
$495$	−237.432	−0.479660
$496$	0	0
$497$	182.667i	0.367539i
$498$	0	0
$499$	−104.779	−0.209978	−0.104989	−	0.994473i	$-0.533481\pi$
$499$	−104.779	−0.209978	−0.104989	+	0.994473i	$0.533481\pi$
$500$	0	0
$501$	−286.475	−0.571807
$502$	0	0
$503$	−337.201	−0.670379	−0.335190	−	0.942151i	$-0.608800\pi$
$503$	−337.201	−0.670379	−0.335190	+	0.942151i	$0.608800\pi$
$504$	0	0
$505$	273.706	0.541992
$506$	0	0
$507$	240.594i	0.474545i
$508$	0	0
$509$	596.834i	1.17256i	0.810108	+	0.586281i	$0.199409\pi$
$509$	596.834i	1.17256i	−0.810108	+	0.586281i	$0.800591\pi$
$510$	0	0
$511$	−77.7055	−0.152066
$512$	0	0
$513$	0	0
$514$	0	0
$515$	302.637i	0.587646i
$516$	0	0
$517$	−1625.26	−3.14364
$518$	0	0
$519$	492.296	0.948548
$520$	0	0
$521$	407.775i	0.782677i	0.920247	+	0.391339i	$0.127988\pi$
$521$	407.775i	0.782677i	−0.920247	+	0.391339i	$0.872012\pi$
$522$	0	0
$523$	− 2.72800i	− 0.00521606i	−0.999997	−	0.00260803i	$-0.999170\pi$
$523$	− 2.72800i	− 0.00521606i	0.999997	−	0.00260803i	$-0.000830163\pi$
$524$	0	0
$525$	174.010i	0.331448i
$526$	0	0
$527$	− 339.988i	− 0.645139i
$528$	0	0
$529$	−372.260	−0.703704
$530$	0	0
$531$	201.427i	0.379335i
$532$	0	0
$533$	−487.831	−0.915254
$534$	0	0
$535$	− 535.353i	− 1.00066i
$536$	0	0
$537$	238.042	0.443281
$538$	0	0
$539$	881.428	1.63530
$540$	0	0
$541$	199.085	0.367994	0.183997	−	0.982927i	$-0.441096\pi$
$541$	199.085	0.367994	0.183997	+	0.982927i	$0.441096\pi$
$542$	0	0
$543$	677.425	1.24756
$544$	0	0
$545$	135.913i	0.249381i
$546$	0	0
$547$	181.698i	0.332172i	0.986111	+	0.166086i	$0.0531129\pi$
$547$	181.698i	0.332172i	−0.986111	+	0.166086i	$0.946887\pi$
$548$	0	0
$549$	−394.738	−0.719013
$550$	0	0
$551$	0	0
$552$	0	0
$553$	− 426.596i	− 0.771422i
$554$	0	0
$555$	339.694	0.612061
$556$	0	0
$557$	665.560	1.19490	0.597451	−	0.801906i	$-0.296180\pi$
$557$	665.560	1.19490	0.597451	+	0.801906i	$0.296180\pi$
$558$	0	0
$559$	− 496.315i	− 0.887863i
$560$	0	0
$561$	− 641.670i	− 1.14380i
$562$	0	0
$563$	813.921i	1.44569i	0.691012	+	0.722843i	$0.257165\pi$
$563$	813.921i	1.44569i	−0.691012	+	0.722843i	$0.742835\pi$
$564$	0	0
$565$	543.995i	0.962823i
$566$	0	0
$567$	297.276	0.524296
$568$	0	0
$569$	787.495i	1.38400i	0.721898	+	0.691999i	$0.243270\pi$
$569$	787.495i	1.38400i	−0.721898	+	0.691999i	$0.756730\pi$
$570$	0	0
$571$	−750.796	−1.31488	−0.657439	−	0.753508i	$-0.728360\pi$
$571$	−750.796	−1.31488	−0.657439	+	0.753508i	$0.728360\pi$
$572$	0	0
$573$	− 345.512i	− 0.602988i
$574$	0	0
$575$	−206.771	−0.359602
$576$	0	0
$577$	433.999	0.752165	0.376082	−	0.926586i	$-0.377271\pi$
$577$	433.999	0.752165	0.376082	+	0.926586i	$0.377271\pi$
$578$	0	0
$579$	−148.661	−0.256755
$580$	0	0
$581$	−6.41412	−0.0110398
$582$	0	0
$583$	683.770i	1.17285i
$584$	0	0
$585$	− 109.332i	− 0.186893i
$586$	0	0
$587$	−984.663	−1.67745	−0.838724	−	0.544556i	$-0.816698\pi$
$587$	−984.663	−1.67745	−0.838724	+	0.544556i	$0.816698\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	961.068i	1.62617i
$592$	0	0
$593$	−327.699	−0.552613	−0.276306	−	0.961070i	$-0.589110\pi$
$593$	−327.699	−0.552613	−0.276306	+	0.961070i	$0.589110\pi$
$594$	0	0
$595$	70.7141	0.118847
$596$	0	0
$597$	341.669i	0.572310i
$598$	0	0
$599$	− 366.105i	− 0.611193i	−0.952161	−	0.305597i	$-0.901144\pi$
$599$	− 366.105i	− 0.611193i	0.952161	−	0.305597i	$-0.0988559\pi$
$600$	0	0
$601$	− 827.746i	− 1.37728i	−0.725103	−	0.688641i	$-0.758208\pi$
$601$	− 827.746i	− 1.37728i	0.725103	−	0.688641i	$-0.241792\pi$
$602$	0	0
$603$	− 378.301i	− 0.627365i
$604$	0	0
$605$	−1042.57	−1.72326
$606$	0	0
$607$	− 863.473i	− 1.42253i	−0.702926	−	0.711263i	$-0.748124\pi$
$607$	− 863.473i	− 1.42253i	0.702926	−	0.711263i	$-0.251876\pi$
$608$	0	0
$609$	226.545	0.371995
$610$	0	0
$611$	− 748.400i	− 1.22488i
$612$	0	0
$613$	512.515	0.836076	0.418038	−	0.908430i	$-0.362718\pi$
$613$	512.515	0.836076	0.418038	+	0.908430i	$0.362718\pi$
$614$	0	0
$615$	−502.981	−0.817856
$616$	0	0
$617$	−167.197	−0.270984	−0.135492	−	0.990778i	$-0.543261\pi$
$617$	−167.197	−0.270984	−0.135492	+	0.990778i	$0.543261\pi$
$618$	0	0
$619$	948.222	1.53186	0.765930	−	0.642923i	$-0.222279\pi$
$619$	948.222	1.53186	0.765930	+	0.642923i	$0.222279\pi$
$620$	0	0
$621$	235.591i	0.379374i
$622$	0	0
$623$	58.2195i	0.0934502i
$624$	0	0
$625$	60.6663	0.0970661
$626$	0	0
$627$	0	0
$628$	0	0
$629$	268.725i	0.427226i
$630$	0	0
$631$	856.658	1.35762	0.678810	−	0.734314i	$-0.262496\pi$
$631$	856.658	1.35762	0.678810	+	0.734314i	$0.262496\pi$
$632$	0	0
$633$	817.058	1.29077
$634$	0	0
$635$	− 247.251i	− 0.389372i
$636$	0	0
$637$	405.879i	0.637173i
$638$	0	0
$639$	230.360i	0.360500i
$640$	0	0
$641$	521.034i	0.812845i	0.913685	+	0.406423i	$0.133224\pi$
$641$	521.034i	0.812845i	−0.913685	+	0.406423i	$0.866776\pi$
$642$	0	0
$643$	654.879	1.01847	0.509237	−	0.860626i	$-0.329928\pi$
$643$	654.879	1.01847	0.509237	+	0.860626i	$0.329928\pi$
$644$	0	0
$645$	− 511.730i	− 0.793380i
$646$	0	0
$647$	−262.226	−0.405295	−0.202648	−	0.979252i	$-0.564955\pi$
$647$	−262.226	−0.405295	−0.202648	+	0.979252i	$0.564955\pi$
$648$	0	0
$649$	1183.47i	1.82353i
$650$	0	0
$651$	−435.801	−0.669433
$652$	0	0
$653$	−371.470	−0.568866	−0.284433	−	0.958696i	$-0.591805\pi$
$653$	−371.470	−0.568866	−0.284433	+	0.958696i	$0.591805\pi$
$654$	0	0
$655$	−99.6377	−0.152119
$656$	0	0
$657$	−97.9936	−0.149153
$658$	0	0
$659$	− 515.532i	− 0.782294i	−0.920328	−	0.391147i	$-0.872078\pi$
$659$	− 515.532i	− 0.782294i	0.920328	−	0.391147i	$-0.127922\pi$
$660$	0	0
$661$	1265.78i	1.91495i	0.288518	+	0.957474i	$0.406837\pi$
$661$	1265.78i	1.91495i	−0.288518	+	0.957474i	$0.593163\pi$
$662$	0	0
$663$	295.476	0.445665
$664$	0	0
$665$	0	0
$666$	0	0
$667$	269.197i	0.403593i
$668$	0	0
$669$	−776.809	−1.16115
$670$	0	0
$671$	−2319.26	−3.45643
$672$	0	0
$673$	516.203i	0.767019i	0.923537	+	0.383509i	$0.125285\pi$
$673$	516.203i	0.767019i	−0.923537	+	0.383509i	$0.874715\pi$
$674$	0	0
$675$	− 310.790i	− 0.460430i
$676$	0	0
$677$	985.790i	1.45611i	0.685516	+	0.728057i	$0.259577\pi$
$677$	985.790i	1.45611i	−0.685516	+	0.728057i	$0.740423\pi$
$678$	0	0
$679$	326.616i	0.481025i
$680$	0	0
$681$	−174.557	−0.256324
$682$	0	0
$683$	− 74.0653i	− 0.108441i	−0.998529	−	0.0542206i	$-0.982733\pi$
$683$	− 74.0653i	− 0.108441i	0.998529	−	0.0542206i	$-0.0172674\pi$
$684$	0	0
$685$	124.508	0.181764
$686$	0	0
$687$	1120.65i	1.63122i
$688$	0	0
$689$	−314.862	−0.456983
$690$	0	0
$691$	691.086	1.00013	0.500063	−	0.865989i	$-0.333310\pi$
$691$	691.086	1.00013	0.500063	+	0.865989i	$0.333310\pi$
$692$	0	0
$693$	−240.760	−0.347416
$694$	0	0
$695$	−202.380	−0.291194
$696$	0	0
$697$	− 397.898i	− 0.570873i
$698$	0	0
$699$	− 997.777i	− 1.42744i
$700$	0	0
$701$	875.548	1.24900	0.624499	−	0.781025i	$-0.285303\pi$
$701$	875.548	1.24900	0.624499	+	0.781025i	$0.285303\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	− 771.643i	− 1.09453i
$706$	0	0
$707$	277.542	0.392563
$708$	0	0
$709$	481.812	0.679565	0.339783	−	0.940504i	$-0.389647\pi$
$709$	481.812	0.679565	0.339783	+	0.940504i	$0.389647\pi$
$710$	0	0
$711$	− 537.976i	− 0.756647i
$712$	0	0
$713$	− 517.850i	− 0.726298i
$714$	0	0
$715$	− 642.377i	− 0.898429i
$716$	0	0
$717$	− 2.04489i	− 0.00285201i
$718$	0	0
$719$	357.773	0.497598	0.248799	−	0.968555i	$-0.419964\pi$
$719$	357.773	0.497598	0.248799	+	0.968555i	$0.419964\pi$
$720$	0	0
$721$	306.879i	0.425630i
$722$	0	0
$723$	1030.00	1.42462
$724$	0	0
$725$	− 355.123i	− 0.489825i
$726$	0	0
$727$	249.823	0.343635	0.171818	−	0.985129i	$-0.445036\pi$
$727$	249.823	0.343635	0.171818	+	0.985129i	$0.445036\pi$
$728$	0	0
$729$	−229.245	−0.314465
$730$	0	0
$731$	404.819	0.553788
$732$	0	0
$733$	−1409.10	−1.92238	−0.961189	−	0.275890i	$-0.911028\pi$
$733$	−1409.10	−1.92238	−0.961189	+	0.275890i	$0.911028\pi$
$734$	0	0
$735$	418.485i	0.569367i
$736$	0	0
$737$	− 2222.69i	− 3.01586i
$738$	0	0
$739$	975.430	1.31993	0.659966	−	0.751295i	$-0.270571\pi$
$739$	975.430	1.31993	0.659966	+	0.751295i	$0.270571\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 1289.68i	− 1.73577i	−0.496763	−	0.867886i	$-0.665478\pi$
$743$	− 1289.68i	− 1.73577i	0.496763	−	0.867886i	$-0.334522\pi$
$744$	0	0
$745$	713.338	0.957501
$746$	0	0
$747$	−8.08878	−0.0108284
$748$	0	0
$749$	− 542.856i	− 0.724774i
$750$	0	0
$751$	639.407i	0.851408i	0.904863	+	0.425704i	$0.139973\pi$
$751$	639.407i	0.851408i	−0.904863	+	0.425704i	$0.860027\pi$
$752$	0	0
$753$	− 208.053i	− 0.276299i
$754$	0	0
$755$	− 122.230i	− 0.161894i
$756$	0	0
$757$	300.388	0.396813	0.198407	−	0.980120i	$-0.436423\pi$
$757$	300.388	0.396813	0.198407	+	0.980120i	$0.436423\pi$
$758$	0	0
$759$	− 977.355i	− 1.28769i
$760$	0	0
$761$	−560.059	−0.735952	−0.367976	−	0.929835i	$-0.619949\pi$
$761$	−560.059	−0.735952	−0.367976	+	0.929835i	$0.619949\pi$
$762$	0	0
$763$	137.818i	0.180626i
$764$	0	0
$765$	89.1768	0.116571
$766$	0	0
$767$	−544.965	−0.710515
$768$	0	0
$769$	596.650	0.775878	0.387939	−	0.921685i	$-0.373187\pi$
$769$	596.650	0.775878	0.387939	+	0.921685i	$0.373187\pi$
$770$	0	0
$771$	−22.5302	−0.0292220
$772$	0	0
$773$	− 1008.16i	− 1.30421i	−0.758128	−	0.652106i	$-0.773886\pi$
$773$	− 1008.16i	− 1.30421i	0.758128	−	0.652106i	$-0.226114\pi$
$774$	0	0
$775$	683.145i	0.881478i
$776$	0	0
$777$	344.455	0.443314
$778$	0	0
$779$	0	0
$780$	0	0
$781$	1353.47i	1.73299i
$782$	0	0
$783$	−404.620	−0.516756
$784$	0	0
$785$	−522.637	−0.665779
$786$	0	0
$787$	− 738.630i	− 0.938539i	−0.883055	−	0.469270i	$-0.844517\pi$
$787$	− 738.630i	− 0.938539i	0.883055	−	0.469270i	$-0.155483\pi$
$788$	0	0
$789$	1339.37i	1.69755i
$790$	0	0
$791$	551.619i	0.697370i
$792$	0	0
$793$	− 1067.97i	− 1.34675i
$794$	0	0
$795$	−324.640	−0.408353
$796$	0	0
$797$	1039.69i	1.30451i	0.758001	+	0.652254i	$0.226176\pi$
$797$	1039.69i	1.30451i	−0.758001	+	0.652254i	$0.773824\pi$
$798$	0	0
$799$	610.431	0.763994
$800$	0	0
$801$	73.4200i	0.0916604i
$802$	0	0
$803$	−575.756	−0.717007
$804$	0	0
$805$	107.708	0.133798
$806$	0	0
$807$	1029.20	1.27535
$808$	0	0
$809$	567.506	0.701491	0.350745	−	0.936471i	$-0.385928\pi$
$809$	567.506	0.701491	0.350745	+	0.936471i	$0.385928\pi$
$810$	0	0
$811$	− 233.648i	− 0.288099i	−0.989571	−	0.144049i	$-0.953988\pi$
$811$	− 233.648i	− 0.288099i	0.989571	−	0.144049i	$-0.0460124\pi$
$812$	0	0
$813$	224.928i	0.276664i
$814$	0	0
$815$	−707.474	−0.868066
$816$	0	0
$817$	0	0
$818$	0	0
$819$	− 110.865i	− 0.135366i
$820$	0	0
$821$	1515.76	1.84624	0.923120	−	0.384513i	$-0.125631\pi$
$821$	1515.76	1.84624	0.923120	+	0.384513i	$0.125631\pi$
$822$	0	0
$823$	132.814	0.161378	0.0806892	−	0.996739i	$-0.474288\pi$
$823$	132.814	0.161378	0.0806892	+	0.996739i	$0.474288\pi$
$824$	0	0
$825$	1289.32i	1.56281i
$826$	0	0
$827$	1130.49i	1.36698i	0.729959	+	0.683491i	$0.239539\pi$
$827$	1130.49i	1.36698i	−0.729959	+	0.683491i	$0.760461\pi$
$828$	0	0
$829$	− 681.015i	− 0.821490i	−0.911750	−	0.410745i	$-0.865269\pi$
$829$	− 681.015i	− 0.821490i	0.911750	−	0.410745i	$-0.134731\pi$
$830$	0	0
$831$	− 419.843i	− 0.505226i
$832$	0	0
$833$	−331.055	−0.397425
$834$	0	0
$835$	− 233.920i	− 0.280144i
$836$	0	0
$837$	778.362	0.929943
$838$	0	0
$839$	− 803.332i	− 0.957487i	−0.877955	−	0.478744i	$-0.841092\pi$
$839$	− 803.332i	− 0.957487i	0.877955	−	0.478744i	$-0.158908\pi$
$840$	0	0
$841$	378.663	0.450253
$842$	0	0
$843$	−122.809	−0.145681
$844$	0	0
$845$	−196.456	−0.232493
$846$	0	0
$847$	−1057.19	−1.24815
$848$	0	0
$849$	1005.17i	1.18394i
$850$	0	0
$851$	409.306i	0.480971i
$852$	0	0
$853$	43.0736	0.0504966	0.0252483	−	0.999681i	$-0.491962\pi$
$853$	43.0736	0.0504966	0.0252483	+	0.999681i	$0.491962\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 1302.82i	− 1.52021i	−0.649799	−	0.760106i	$-0.725147\pi$
$857$	− 1302.82i	− 1.52021i	0.649799	−	0.760106i	$-0.274853\pi$
$858$	0	0
$859$	588.536	0.685140	0.342570	−	0.939492i	$-0.388703\pi$
$859$	588.536	0.685140	0.342570	+	0.939492i	$0.388703\pi$
$860$	0	0
$861$	−510.031	−0.592370
$862$	0	0
$863$	1149.51i	1.33199i	0.745955	+	0.665996i	$0.231993\pi$
$863$	1149.51i	1.33199i	−0.745955	+	0.665996i	$0.768007\pi$
$864$	0	0
$865$	401.983i	0.464720i
$866$	0	0
$867$	− 789.909i	− 0.911084i
$868$	0	0
$869$	− 3160.85i	− 3.63734i
$870$	0	0
$871$	1023.50	1.17509
$872$	0	0
$873$	411.892i	0.471812i
$874$	0	0
$875$	−357.165	−0.408189
$876$	0	0
$877$	− 10.8876i	− 0.0124146i	−0.999981	−	0.00620731i	$-0.998024\pi$
$877$	− 10.8876i	− 0.0124146i	0.999981	−	0.00620731i	$-0.00197586\pi$
$878$	0	0
$879$	1105.67	1.25787
$880$	0	0
$881$	458.646	0.520597	0.260298	−	0.965528i	$-0.416179\pi$
$881$	458.646	0.520597	0.260298	+	0.965528i	$0.416179\pi$
$882$	0	0
$883$	204.491	0.231586	0.115793	−	0.993273i	$-0.463059\pi$
$883$	204.491	0.231586	0.115793	+	0.993273i	$0.463059\pi$
$884$	0	0
$885$	−561.890	−0.634904
$886$	0	0
$887$	1727.40i	1.94746i	0.227706	+	0.973730i	$0.426878\pi$
$887$	1727.40i	1.94746i	−0.227706	+	0.973730i	$0.573122\pi$
$888$	0	0
$889$	− 250.716i	− 0.282021i
$890$	0	0
$891$	2202.66	2.47212
$892$	0	0
$893$	0	0
$894$	0	0
$895$	194.373i	0.217176i
$896$	0	0
$897$	450.051	0.501729
$898$	0	0
$899$	889.392	0.989312
$900$	0	0
$901$	− 256.816i	− 0.285035i
$902$	0	0
$903$	− 518.902i	− 0.574642i
$904$	0	0
$905$	553.149i	0.611214i
$906$	0	0
$907$	− 855.204i	− 0.942893i	−0.881895	−	0.471447i	$-0.843732\pi$
$907$	− 855.204i	− 0.942893i	0.881895	−	0.471447i	$-0.156268\pi$
$908$	0	0
$909$	350.006	0.385045
$910$	0	0
$911$	22.7639i	0.0249878i	0.999922	+	0.0124939i	$0.00397703\pi$
$911$	22.7639i	0.0249878i	−0.999922	+	0.0124939i	$0.996023\pi$
$912$	0	0
$913$	−47.5252	−0.0520539
$914$	0	0
$915$	− 1101.14i	− 1.20343i
$916$	0	0
$917$	−101.034	−0.110179
$918$	0	0
$919$	369.613	0.402191	0.201095	−	0.979572i	$-0.435550\pi$
$919$	369.613	0.402191	0.201095	+	0.979572i	$0.435550\pi$
$920$	0	0
$921$	1479.02	1.60589
$922$	0	0
$923$	−623.243	−0.675236
$924$	0	0
$925$	− 539.955i	− 0.583735i
$926$	0	0
$927$	387.002i	0.417478i
$928$	0	0
$929$	−762.812	−0.821111	−0.410555	−	0.911836i	$-0.634665\pi$
$929$	−762.812	−0.821111	−0.410555	+	0.911836i	$0.634665\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	− 1381.93i	− 1.48116i
$934$	0	0
$935$	523.954	0.560378
$936$	0	0
$937$	978.242	1.04402	0.522008	−	0.852941i	$-0.325183\pi$
$937$	978.242	1.04402	0.522008	+	0.852941i	$0.325183\pi$
$938$	0	0
$939$	1314.81i	1.40022i
$940$	0	0
$941$	702.289i	0.746322i	0.927767	+	0.373161i	$0.121726\pi$
$941$	702.289i	0.746322i	−0.927767	+	0.373161i	$0.878274\pi$
$942$	0	0
$943$	− 606.055i	− 0.642689i
$944$	0	0
$945$	161.891i	0.171314i
$946$	0	0
$947$	660.149	0.697095	0.348547	−	0.937291i	$-0.386675\pi$
$947$	660.149	0.697095	0.348547	+	0.937291i	$0.386675\pi$
$948$	0	0
$949$	− 265.124i	− 0.279372i
$950$	0	0
$951$	761.878	0.801134
$952$	0	0
$953$	− 615.697i	− 0.646062i	−0.946388	−	0.323031i	$-0.895298\pi$
$953$	− 615.697i	− 0.646062i	0.946388	−	0.323031i	$-0.104702\pi$
$954$	0	0
$955$	282.127	0.295421
$956$	0	0
$957$	1678.58	1.75400
$958$	0	0
$959$	126.253	0.131651
$960$	0	0
$961$	−749.911	−0.780344
$962$	0	0
$963$	− 684.590i	− 0.710893i
$964$	0	0
$965$	− 121.389i	− 0.125792i
$966$	0	0
$967$	214.810	0.222141	0.111070	−	0.993813i	$-0.464572\pi$
$967$	214.810	0.222141	0.111070	+	0.993813i	$0.464572\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	679.022i	0.699301i	0.936880	+	0.349651i	$0.113700\pi$
$971$	679.022i	0.699301i	−0.936880	+	0.349651i	$0.886300\pi$
$972$	0	0
$973$	−205.216	−0.210911
$974$	0	0
$975$	−593.705	−0.608929
$976$	0	0
$977$	− 1088.94i	− 1.11457i	−0.830320	−	0.557287i	$-0.811842\pi$
$977$	− 1088.94i	− 1.11457i	0.830320	−	0.557287i	$-0.188158\pi$
$978$	0	0
$979$	431.375i	0.440629i
$980$	0	0
$981$	173.801i	0.177167i
$982$	0	0
$983$	− 1330.59i	− 1.35360i	−0.736166	−	0.676800i	$-0.763366\pi$
$983$	− 1330.59i	− 1.35360i	0.736166	−	0.676800i	$-0.236634\pi$
$984$	0	0
$985$	−784.757	−0.796708
$986$	0	0
$987$	− 782.458i	− 0.792764i
$988$	0	0
$989$	616.597	0.623455
$990$	0	0
$991$	1579.98i	1.59433i	0.603761	+	0.797166i	$0.293668\pi$
$991$	1579.98i	1.59433i	−0.603761	+	0.797166i	$0.706332\pi$
$992$	0	0
$993$	−826.033	−0.831856
$994$	0	0
$995$	−278.989	−0.280391
$996$	0	0
$997$	−1456.40	−1.46078	−0.730391	−	0.683029i	$-0.760662\pi$
$997$	−1456.40	−1.46078	−0.730391	+	0.683029i	$0.760662\pi$
$998$	0	0
$999$	−615.214	−0.615830

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1444.3.c.d.721.19	yes	24
19.18	odd	2	inner	1444.3.c.d.721.6	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1444.3.c.d.721.6	✓	24	19.18	odd	2	inner
1444.3.c.d.721.19	yes	24	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+3.56718i q^{3} -2.91277 q^{5} -2.95359 q^{7} -3.72474 q^{9} +O(q^{10})\)	\(q+3.56718i q^{3} -2.91277 q^{5} -2.95359 q^{7} -3.72474 q^{9} -21.8845 q^{11} -10.0774i q^{13} -10.3903i q^{15} +8.21959 q^{17} -10.5360i q^{21} +12.5196 q^{23} -16.5158 q^{25} +18.8178i q^{27} +21.5020i q^{29} -41.3632i q^{31} -78.0660i q^{33} +8.60312 q^{35} +32.6932i q^{37} +35.9477 q^{39} -48.4085i q^{41} +49.2505 q^{43} +10.8493 q^{45} +74.2654 q^{47} -40.2763 q^{49} +29.3207i q^{51} -31.2444i q^{53} +63.7445 q^{55} -54.0781i q^{59} +105.977 q^{61} +11.0014 q^{63} +29.3530i q^{65} +101.564i q^{67} +44.6596i q^{69} -61.8458i q^{71} +26.3088 q^{73} -58.9147i q^{75} +64.6379 q^{77} +144.433i q^{79} -100.649 q^{81} +2.17164 q^{83} -23.9417 q^{85} -76.7015 q^{87} -19.7114i q^{89} +29.7644i q^{91} +147.550 q^{93} -110.583i q^{97} +81.5142 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 12 q^{5} + 40 q^{7} - 104 q^{9}+O(q^{10}) \) 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

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Representations

Groups

Database

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