LMFDB - Embedded newform 1900.3.g.c.949.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1900,3,Mod(949,1900)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1900, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1900.949");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1900 = 2^{2} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1900.g (of order $2$, degree $1$, not 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:	$51.7712502285$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	no (minimal twist has level 380)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			949.15
Character	$\chi$	$=$	1900.949
Dual form			1900.3.g.c.949.16

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.03369 q^{3} -4.27843i q^{7} -4.86411 q^{9} +O(q^{10})$	$q+2.03369 q^{3} -4.27843i q^{7} -4.86411 q^{9} +5.47217 q^{11} +0.634081 q^{13} +1.69271i q^{17} +(6.46218 + 17.8673i) q^{19} -8.70099i q^{21} +25.9867i q^{23} -28.1953 q^{27} +48.8696i q^{29} -4.47053i q^{31} +11.1287 q^{33} +6.12833 q^{37} +1.28952 q^{39} -16.9988i q^{41} +0.690441i q^{43} -23.0801i q^{47} +30.6950 q^{49} +3.44245i q^{51} +54.3934 q^{53} +(13.1420 + 36.3365i) q^{57} -0.251179i q^{59} +39.5570 q^{61} +20.8108i q^{63} +96.3354 q^{67} +52.8489i q^{69} +16.0568i q^{71} +70.0291i q^{73} -23.4123i q^{77} -74.6211i q^{79} -13.5634 q^{81} +1.29672i q^{83} +99.3856i q^{87} +141.348i q^{89} -2.71287i q^{91} -9.09167i q^{93} +125.110 q^{97} -26.6173 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 32 q^{9}+O(q^{10}) $ 24 * q + 32 * q^9	$ 24 q + 32 q^{9} - 64 q^{11} - 48 q^{19} - 248 q^{39} + 48 q^{49} + 304 q^{61} - 936 q^{81} - 784 q^{99}+O(q^{100}) $ 24 * q + 32 * q^9 - 64 * q^11 - 48 * q^19 - 248 * q^39 + 48 * q^49 + 304 * q^61 - 936 * q^81 - 784 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1900\mathbb{Z}\right)^\times$.

$n$	$77$	$401$	$951$
$\chi(n)$	$-1$	$-1$	$1$

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
$3$	2.03369			0.677896			0.338948	−	0.940805i	$-0.389929\pi$
$3$	2.03369			0.677896			0.338948	+	0.940805i	$0.389929\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$		−	4.27843i		−	0.611204i	−0.952159	−	0.305602i	$-0.901142\pi$
$7$		−	4.27843i		−	0.611204i	0.952159	−	0.305602i	$-0.0988577\pi$
$8$		0			0
$9$	−4.86411			−0.540457
$10$		0			0
$11$	5.47217			0.497470			0.248735	−	0.968572i	$-0.419985\pi$
$11$	5.47217			0.497470			0.248735	+	0.968572i	$0.419985\pi$
$12$		0			0
$13$	0.634081			0.0487754			0.0243877	−	0.999703i	$-0.492236\pi$
$13$	0.634081			0.0487754			0.0243877	+	0.999703i	$0.492236\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			1.69271i			0.0995712i	0.998760	+	0.0497856i	$0.0158538\pi$
$17$			1.69271i			0.0995712i	−0.998760	+	0.0497856i	$0.984146\pi$
$18$		0			0
$19$	6.46218	+	17.8673i	0.340115	+	0.940384i
$19$	6.46218	+	17.8673i	0.340115	+	0.940384i
$20$		0			0
$21$		−	8.70099i		−	0.414333i
$22$		0			0
$23$			25.9867i			1.12986i	0.825139	+	0.564929i	$0.191096\pi$
$23$			25.9867i			1.12986i	−0.825139	+	0.564929i	$0.808904\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	−28.1953			−1.04427
$28$		0			0
$29$			48.8696i			1.68516i	0.538571	+	0.842580i	$0.318964\pi$
$29$			48.8696i			1.68516i	−0.538571	+	0.842580i	$0.681036\pi$
$30$		0			0
$31$		−	4.47053i		−	0.144211i	−0.997397	−	0.0721054i	$-0.977028\pi$
$31$		−	4.47053i		−	0.144211i	0.997397	−	0.0721054i	$-0.0229718\pi$
$32$		0			0
$33$	11.1287			0.337233
$34$		0			0
$35$		0			0
$36$		0			0
$37$	6.12833			0.165630			0.0828152	−	0.996565i	$-0.473609\pi$
$37$	6.12833			0.165630			0.0828152	+	0.996565i	$0.473609\pi$
$38$		0			0
$39$	1.28952			0.0330647
$40$		0			0
$41$		−	16.9988i		−	0.414605i	−0.978277	−	0.207303i	$-0.933532\pi$
$41$		−	16.9988i		−	0.414605i	0.978277	−	0.207303i	$-0.0664685\pi$
$42$		0			0
$43$			0.690441i			0.0160568i	0.999968	+	0.00802838i	$0.00255554\pi$
$43$			0.690441i			0.0160568i	−0.999968	+	0.00802838i	$0.997444\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	23.0801i		−	0.491066i	−0.969388	−	0.245533i	$-0.921037\pi$
$47$		−	23.0801i		−	0.491066i	0.969388	−	0.245533i	$-0.0789630\pi$
$48$		0			0
$49$	30.6950			0.626429
$50$		0			0
$51$			3.44245i			0.0674989i
$52$		0			0
$53$	54.3934			1.02629			0.513145	−	0.858302i	$-0.328480\pi$
$53$	54.3934			1.02629			0.513145	+	0.858302i	$0.328480\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	13.1420	+	36.3365i	0.230562	+	0.637482i
$58$		0			0
$59$		−	0.251179i		−	0.00425727i	−0.999998	−	0.00212863i	$-0.999322\pi$
$59$		−	0.251179i		−	0.00425727i	0.999998	−	0.00212863i	$-0.000677566\pi$
$60$		0			0
$61$	39.5570			0.648475			0.324237	−	0.945976i	$-0.394892\pi$
$61$	39.5570			0.648475			0.324237	+	0.945976i	$0.394892\pi$
$62$		0			0
$63$			20.8108i			0.330330i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	96.3354			1.43784			0.718921	−	0.695092i	$-0.244637\pi$
$67$	96.3354			1.43784			0.718921	+	0.695092i	$0.244637\pi$
$68$		0			0
$69$			52.8489i			0.765926i
$70$		0			0
$71$			16.0568i			0.226152i	0.993586	+	0.113076i	$0.0360704\pi$
$71$			16.0568i			0.226152i	−0.993586	+	0.113076i	$0.963930\pi$
$72$		0			0
$73$			70.0291i			0.959302i	0.877459	+	0.479651i	$0.159237\pi$
$73$			70.0291i			0.959302i	−0.877459	+	0.479651i	$0.840763\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	23.4123i		−	0.304056i
$78$		0			0
$79$		−	74.6211i		−	0.944570i	−0.881446	−	0.472285i	$-0.843429\pi$
$79$		−	74.6211i		−	0.944570i	0.881446	−	0.472285i	$-0.156571\pi$
$80$		0			0
$81$	−13.5634			−0.167449
$82$		0			0
$83$			1.29672i			0.0156232i	0.999969	+	0.00781159i	$0.00248653\pi$
$83$			1.29672i			0.0156232i	−0.999969	+	0.00781159i	$0.997513\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$			99.3856i			1.14236i
$88$		0			0
$89$			141.348i			1.58818i	0.607800	+	0.794090i	$0.292052\pi$
$89$			141.348i			1.58818i	−0.607800	+	0.794090i	$0.707948\pi$
$90$		0			0
$91$		−	2.71287i		−	0.0298117i
$92$		0			0
$93$		−	9.09167i		−	0.0977598i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	125.110			1.28980			0.644898	−	0.764268i	$-0.276900\pi$
$97$	125.110			1.28980			0.644898	+	0.764268i	$0.276900\pi$
$98$		0			0
$99$	−26.6173			−0.268861
$100$		0			0
$101$	102.871			1.01853			0.509263	−	0.860611i	$-0.329918\pi$
$101$	102.871			1.01853			0.509263	+	0.860611i	$0.329918\pi$
$102$		0			0
$103$	−11.6950			−0.113544			−0.0567720	−	0.998387i	$-0.518081\pi$
$103$	−11.6950			−0.113544			−0.0567720	+	0.998387i	$0.518081\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	189.935			1.77509			0.887545	−	0.460722i	$-0.152409\pi$
$107$	189.935			1.77509			0.887545	+	0.460722i	$0.152409\pi$
$108$		0			0
$109$			43.6838i			0.400769i	0.979717	+	0.200384i	$0.0642191\pi$
$109$			43.6838i			0.400769i	−0.979717	+	0.200384i	$0.935781\pi$
$110$		0			0
$111$	12.4631			0.112280
$112$		0			0
$113$	−100.896			−0.892884			−0.446442	−	0.894813i	$-0.647309\pi$
$113$	−100.896			−0.892884			−0.446442	+	0.894813i	$0.647309\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−3.08424			−0.0263610
$118$		0			0
$119$	7.24214			0.0608584
$120$		0			0
$121$	−91.0554			−0.752524
$122$		0			0
$123$		−	34.5703i		−	0.281059i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	140.225			1.10414			0.552068	−	0.833799i	$-0.313839\pi$
$127$	140.225			1.10414			0.552068	+	0.833799i	$0.313839\pi$
$128$		0			0
$129$			1.40414i			0.0108848i
$130$		0			0
$131$	−158.825			−1.21241			−0.606203	−	0.795310i	$-0.707308\pi$
$131$	−158.825			−1.21241			−0.606203	+	0.795310i	$0.707308\pi$
$132$		0			0
$133$	76.4440	−	27.6480i	0.574767	−	0.207879i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			74.8377i			0.546261i	0.961977	+	0.273130i	$0.0880590\pi$
$137$			74.8377i			0.546261i	−0.961977	+	0.273130i	$0.911941\pi$
$138$		0			0
$139$	17.4925			0.125846			0.0629228	−	0.998018i	$-0.479958\pi$
$139$	17.4925			0.125846			0.0629228	+	0.998018i	$0.479958\pi$
$140$		0			0
$141$		−	46.9377i		−	0.332892i
$142$		0			0
$143$	3.46980			0.0242643
$144$		0			0
$145$		0			0
$146$		0			0
$147$	62.4241			0.424654
$148$		0			0
$149$	−163.077			−1.09448			−0.547239	−	0.836976i	$-0.684321\pi$
$149$	−163.077			−1.09448			−0.547239	+	0.836976i	$0.684321\pi$
$150$		0			0
$151$			147.367i			0.975940i	0.872861	+	0.487970i	$0.162262\pi$
$151$			147.367i			0.975940i	−0.872861	+	0.487970i	$0.837738\pi$
$152$		0			0
$153$		−	8.23354i		−	0.0538140i
$154$		0			0
$155$		0			0
$156$		0			0
$157$			208.993i			1.33117i	0.746324	+	0.665583i	$0.231817\pi$
$157$			208.993i			1.33117i	−0.746324	+	0.665583i	$0.768183\pi$
$158$		0			0
$159$	110.619			0.695718
$160$		0			0
$161$	111.182			0.690574
$162$		0			0
$163$		−	87.2475i		−	0.535261i	−0.963522	−	0.267630i	$-0.913759\pi$
$163$		−	87.2475i		−	0.535261i	0.963522	−	0.267630i	$-0.0862406\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	97.0793			0.581313			0.290657	−	0.956827i	$-0.406126\pi$
$167$	97.0793			0.581313			0.290657	+	0.956827i	$0.406126\pi$
$168$		0			0
$169$	−168.598			−0.997621
$170$		0			0
$171$	−31.4328	−	86.9086i	−0.183817	−	0.508237i
$172$		0			0
$173$	70.7451			0.408931			0.204466	−	0.978874i	$-0.434454\pi$
$173$	70.7451			0.408931			0.204466	+	0.978874i	$0.434454\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		−	0.510819i		−	0.00288598i
$178$		0			0
$179$			176.104i			0.983822i	0.870646	+	0.491911i	$0.163701\pi$
$179$			176.104i			0.983822i	−0.870646	+	0.491911i	$0.836299\pi$
$180$		0			0
$181$		−	57.5577i		−	0.317998i	−0.987279	−	0.158999i	$-0.949173\pi$
$181$		−	57.5577i		−	0.317998i	0.987279	−	0.158999i	$-0.0508267\pi$
$182$		0			0
$183$	80.4465			0.439598
$184$		0			0
$185$		0			0
$186$		0			0
$187$			9.26280i			0.0495337i
$188$		0			0
$189$			120.632i			0.638262i
$190$		0			0
$191$	−145.682			−0.762731			−0.381365	−	0.924424i	$-0.624546\pi$
$191$	−145.682			−0.762731			−0.381365	+	0.924424i	$0.624546\pi$
$192$		0			0
$193$	−64.1316			−0.332288			−0.166144	−	0.986101i	$-0.553132\pi$
$193$	−64.1316			−0.332288			−0.166144	+	0.986101i	$0.553132\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	17.9219i		−	0.0909739i	−0.998965	−	0.0454870i	$-0.985516\pi$
$197$		−	17.9219i		−	0.0909739i	0.998965	−	0.0454870i	$-0.0144839\pi$
$198$		0			0
$199$	−209.349			−1.05200			−0.526002	−	0.850483i	$-0.676310\pi$
$199$	−209.349			−1.05200			−0.526002	+	0.850483i	$0.676310\pi$
$200$		0			0
$201$	195.916			0.974707
$202$		0			0
$203$	209.085			1.02998
$204$		0			0
$205$		0			0
$206$		0			0
$207$		−	126.402i		−	0.610640i
$208$		0			0
$209$	35.3621	+	97.7729i	0.169197	+	0.467813i
$210$		0			0
$211$			232.975i			1.10415i	0.833795	+	0.552074i	$0.186163\pi$
$211$			232.975i			1.10415i	−0.833795	+	0.552074i	$0.813837\pi$
$212$		0			0
$213$			32.6546i			0.153308i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−19.1269			−0.0881422
$218$		0			0
$219$			142.417i			0.650307i
$220$		0			0
$221$			1.07332i			0.00485663i
$222$		0			0
$223$	68.2129			0.305887			0.152944	−	0.988235i	$-0.451125\pi$
$223$	68.2129			0.305887			0.152944	+	0.988235i	$0.451125\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	143.128			0.630520			0.315260	−	0.949005i	$-0.397908\pi$
$227$	143.128			0.630520			0.315260	+	0.949005i	$0.397908\pi$
$228$		0			0
$229$	−135.889			−0.593403			−0.296701	−	0.954970i	$-0.595887\pi$
$229$	−135.889			−0.593403			−0.296701	+	0.954970i	$0.595887\pi$
$230$		0			0
$231$		−	47.6133i		−	0.206118i
$232$		0			0
$233$		−	270.764i		−	1.16208i	−0.813876	−	0.581039i	$-0.802646\pi$
$233$		−	270.764i		−	1.16208i	0.813876	−	0.581039i	$-0.197354\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		−	151.756i		−	0.640320i
$238$		0			0
$239$	−300.071			−1.25553			−0.627764	−	0.778404i	$-0.716030\pi$
$239$	−300.071			−1.25553			−0.627764	+	0.778404i	$0.716030\pi$
$240$		0			0
$241$		−	177.442i		−	0.736275i	−0.929771	−	0.368138i	$-0.879996\pi$
$241$		−	177.442i		−	0.736275i	0.929771	−	0.368138i	$-0.120004\pi$
$242$		0			0
$243$	226.174			0.930757
$244$		0			0
$245$		0			0
$246$		0			0
$247$	4.09754	+	11.3293i	0.0165892	+	0.0458676i
$248$		0			0
$249$			2.63713i			0.0105909i
$250$		0			0
$251$	−80.0537			−0.318939			−0.159469	−	0.987203i	$-0.550978\pi$
$251$	−80.0537			−0.318939			−0.159469	+	0.987203i	$0.550978\pi$
$252$		0			0
$253$			142.204i			0.562070i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	147.621			0.574402			0.287201	−	0.957870i	$-0.407275\pi$
$257$	147.621			0.574402			0.287201	+	0.957870i	$0.407275\pi$
$258$		0			0
$259$		−	26.2196i		−	0.101234i
$260$		0			0
$261$		−	237.707i		−	0.910757i
$262$		0			0
$263$			344.458i			1.30973i	0.755747	+	0.654864i	$0.227274\pi$
$263$			344.458i			1.30973i	−0.755747	+	0.654864i	$0.772726\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$			287.458i			1.07662i
$268$		0			0
$269$			190.965i			0.709908i	0.934884	+	0.354954i	$0.115503\pi$
$269$			190.965i			0.709908i	−0.934884	+	0.354954i	$0.884497\pi$
$270$		0			0
$271$	−265.131			−0.978342			−0.489171	−	0.872188i	$-0.662701\pi$
$271$	−265.131			−0.978342			−0.489171	+	0.872188i	$0.662701\pi$
$272$		0			0
$273$		−	5.51713i		−	0.0202093i
$274$		0			0
$275$		0			0
$276$		0			0
$277$			87.0682i			0.314326i	0.987573	+	0.157163i	$0.0502348\pi$
$277$			87.0682i			0.314326i	−0.987573	+	0.157163i	$0.949765\pi$
$278$		0			0
$279$			21.7452i			0.0779397i
$280$		0			0
$281$		−	233.988i		−	0.832697i	−0.909205	−	0.416348i	$-0.863310\pi$
$281$		−	233.988i		−	0.832697i	0.909205	−	0.416348i	$-0.136690\pi$
$282$		0			0
$283$			190.662i			0.673718i	0.941555	+	0.336859i	$0.109365\pi$
$283$			190.662i			0.673718i	−0.941555	+	0.336859i	$0.890635\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−72.7283			−0.253409
$288$		0			0
$289$	286.135			0.990086
$290$		0			0
$291$	254.435			0.874348
$292$		0			0
$293$	230.400			0.786348			0.393174	−	0.919464i	$-0.371377\pi$
$293$	230.400			0.786348			0.393174	+	0.919464i	$0.371377\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−154.289			−0.519493
$298$		0			0
$299$			16.4777i			0.0551093i
$300$		0			0
$301$	2.95400			0.00981396
$302$		0			0
$303$	209.208			0.690454
$304$		0			0
$305$		0			0
$306$		0			0
$307$	370.642			1.20730			0.603652	−	0.797248i	$-0.293712\pi$
$307$	370.642			1.20730			0.603652	+	0.797248i	$0.293712\pi$
$308$		0			0
$309$	−23.7840			−0.0769710
$310$		0			0
$311$	−12.2047			−0.0392433			−0.0196216	−	0.999807i	$-0.506246\pi$
$311$	−12.2047			−0.0392433			−0.0196216	+	0.999807i	$0.506246\pi$
$312$		0			0
$313$		−	263.418i		−	0.841591i	−0.907156	−	0.420795i	$-0.861751\pi$
$313$		−	263.418i		−	0.841591i	0.907156	−	0.420795i	$-0.138249\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−328.524			−1.03635			−0.518177	−	0.855274i	$-0.673389\pi$
$317$	−328.524			−1.03635			−0.518177	+	0.855274i	$0.673389\pi$
$318$		0			0
$319$			267.423i			0.838316i
$320$		0			0
$321$	386.268			1.20333
$322$		0			0
$323$	−30.2442	+	10.9386i	−0.0936352	+	0.0338656i
$324$		0			0
$325$		0			0
$326$		0			0
$327$			88.8392i			0.271679i
$328$		0			0
$329$	−98.7466			−0.300142
$330$		0			0
$331$		−	29.2287i		−	0.0883043i	−0.999025	−	0.0441522i	$-0.985941\pi$
$331$		−	29.2287i		−	0.0883043i	0.999025	−	0.0441522i	$-0.0140586\pi$
$332$		0			0
$333$	−29.8089			−0.0895162
$334$		0			0
$335$		0			0
$336$		0			0
$337$	283.044			0.839894			0.419947	−	0.907549i	$-0.362049\pi$
$337$	283.044			0.839894			0.419947	+	0.907549i	$0.362049\pi$
$338$		0			0
$339$	−205.191			−0.605282
$340$		0			0
$341$		−	24.4635i		−	0.0717405i
$342$		0			0
$343$		−	340.970i		−	0.994080i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			214.566i			0.618347i	0.951006	+	0.309174i	$0.100052\pi$
$347$			214.566i			0.618347i	−0.951006	+	0.309174i	$0.899948\pi$
$348$		0			0
$349$	627.491			1.79797			0.898984	−	0.437981i	$-0.144306\pi$
$349$	627.491			1.79797			0.898984	+	0.437981i	$0.144306\pi$
$350$		0			0
$351$	−17.8781			−0.0509347
$352$		0			0
$353$			61.3501i			0.173796i	0.996217	+	0.0868982i	$0.0276955\pi$
$353$			61.3501i			0.173796i	−0.996217	+	0.0868982i	$0.972305\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	14.7283			0.0412556
$358$		0			0
$359$	−303.982			−0.846748			−0.423374	−	0.905955i	$-0.639154\pi$
$359$	−303.982			−0.846748			−0.423374	+	0.905955i	$0.639154\pi$
$360$		0			0
$361$	−277.481	+	230.923i	−0.768644	+	0.639677i
$362$		0			0
$363$	−185.178			−0.510133
$364$		0			0
$365$		0			0
$366$		0			0
$367$		−	542.618i		−	1.47852i	−0.673418	−	0.739262i	$-0.735175\pi$
$367$		−	542.618i		−	1.47852i	0.673418	−	0.739262i	$-0.264825\pi$
$368$		0			0
$369$			82.6842i			0.224076i
$370$		0			0
$371$		−	232.718i		−	0.627273i
$372$		0			0
$373$	−568.106			−1.52307			−0.761536	−	0.648123i	$-0.775554\pi$
$373$	−568.106			−1.52307			−0.761536	+	0.648123i	$0.775554\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			30.9873i			0.0821944i
$378$		0			0
$379$		−	322.531i		−	0.851005i	−0.904957	−	0.425502i	$-0.860097\pi$
$379$		−	322.531i		−	0.851005i	0.904957	−	0.425502i	$-0.139903\pi$
$380$		0			0
$381$	285.175			0.748490
$382$		0			0
$383$	134.256			0.350539			0.175269	−	0.984521i	$-0.443920\pi$
$383$	134.256			0.350539			0.175269	+	0.984521i	$0.443920\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		−	3.35838i		−	0.00867800i
$388$		0			0
$389$	47.9667			0.123308			0.0616539	−	0.998098i	$-0.480362\pi$
$389$	47.9667			0.123308			0.0616539	+	0.998098i	$0.480362\pi$
$390$		0			0
$391$	−43.9880			−0.112501
$392$		0			0
$393$	−323.001			−0.821885
$394$		0			0
$395$		0			0
$396$		0			0
$397$		−	161.948i		−	0.407928i	−0.978978	−	0.203964i	$-0.934617\pi$
$397$		−	161.948i		−	0.407928i	0.978978	−	0.203964i	$-0.0653826\pi$
$398$		0			0
$399$	155.463	−	56.2273i	0.389632	−	0.140921i
$400$		0			0
$401$			203.228i			0.506802i	0.967361	+	0.253401i	$0.0815492\pi$
$401$			203.228i			0.506802i	−0.967361	+	0.253401i	$0.918451\pi$
$402$		0			0
$403$		−	2.83468i		−	0.00703394i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	33.5352			0.0823962
$408$		0			0
$409$		−	651.844i		−	1.59375i	−0.604144	−	0.796875i	$-0.706485\pi$
$409$		−	651.844i		−	1.59375i	0.604144	−	0.796875i	$-0.293515\pi$
$410$		0			0
$411$			152.197i			0.370308i
$412$		0			0
$413$	−1.07465			−0.00260206
$414$		0			0
$415$		0			0
$416$		0			0
$417$	35.5744			0.0853102
$418$		0			0
$419$	787.718			1.88000			0.939998	−	0.341180i	$-0.110827\pi$
$419$	787.718			1.88000			0.939998	+	0.341180i	$0.110827\pi$
$420$		0			0
$421$		−	93.6475i		−	0.222441i	−0.993796	−	0.111220i	$-0.964524\pi$
$421$		−	93.6475i		−	0.222441i	0.993796	−	0.111220i	$-0.0354759\pi$
$422$		0			0
$423$			112.264i			0.265400i
$424$		0			0
$425$		0			0
$426$		0			0
$427$		−	169.242i		−	0.396351i
$428$		0			0
$429$	7.05648			0.0164487
$430$		0			0
$431$			546.360i			1.26766i	0.773474	+	0.633828i	$0.218517\pi$
$431$			546.360i			1.26766i	−0.773474	+	0.633828i	$0.781483\pi$
$432$		0			0
$433$	−843.569			−1.94820			−0.974098	−	0.226128i	$-0.927393\pi$
$433$	−843.569			−1.94820			−0.974098	+	0.226128i	$0.927393\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−464.313	+	167.931i	−1.06250	+	0.384281i
$438$		0			0
$439$			118.320i			0.269521i	0.990878	+	0.134761i	$0.0430266\pi$
$439$			118.320i			0.269521i	−0.990878	+	0.134761i	$0.956973\pi$
$440$		0			0
$441$	−149.304			−0.338558
$442$		0			0
$443$			883.490i			1.99433i	0.0752227	+	0.997167i	$0.476033\pi$
$443$			883.490i			1.99433i	−0.0752227	+	0.997167i	$0.523967\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−331.648			−0.741943
$448$		0			0
$449$		−	358.093i		−	0.797534i	−0.917052	−	0.398767i	$-0.869438\pi$
$449$		−	358.093i		−	0.797534i	0.917052	−	0.398767i	$-0.130562\pi$
$450$		0			0
$451$		−	93.0204i		−	0.206254i
$452$		0			0
$453$			299.698i			0.661585i
$454$		0			0
$455$		0			0
$456$		0			0
$457$			329.340i			0.720657i	0.932825	+	0.360328i	$0.117335\pi$
$457$			329.340i			0.720657i	−0.932825	+	0.360328i	$0.882665\pi$
$458$		0			0
$459$		−	47.7265i		−	0.103979i
$460$		0			0
$461$	−613.352			−1.33048			−0.665240	−	0.746629i	$-0.731671\pi$
$461$	−613.352			−1.33048			−0.665240	+	0.746629i	$0.731671\pi$
$462$		0			0
$463$		−	142.260i		−	0.307258i	−0.988129	−	0.153629i	$-0.950904\pi$
$463$		−	142.260i		−	0.307258i	0.988129	−	0.153629i	$-0.0490960\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	435.333i		−	0.932191i	−0.884734	−	0.466096i	$-0.845660\pi$
$467$		−	435.333i		−	0.932191i	0.884734	−	0.466096i	$-0.154340\pi$
$468$		0			0
$469$		−	412.164i		−	0.878814i
$470$		0			0
$471$			425.027i			0.902392i
$472$		0			0
$473$			3.77821i			0.00798776i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−264.575			−0.554666
$478$		0			0
$479$	80.5389			0.168140			0.0840698	−	0.996460i	$-0.473208\pi$
$479$	80.5389			0.168140			0.0840698	+	0.996460i	$0.473208\pi$
$480$		0			0
$481$	3.88585			0.00807870
$482$		0			0
$483$	226.110			0.468137
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−168.483			−0.345960			−0.172980	−	0.984925i	$-0.555340\pi$
$487$	−168.483			−0.345960			−0.172980	+	0.984925i	$0.555340\pi$
$488$		0			0
$489$		−	177.434i		−	0.362851i
$490$		0			0
$491$	467.906			0.952965			0.476482	−	0.879184i	$-0.341912\pi$
$491$	467.906			0.952965			0.476482	+	0.879184i	$0.341912\pi$
$492$		0			0
$493$	−82.7222			−0.167793
$494$		0			0
$495$		0			0
$496$		0			0
$497$	68.6980			0.138225
$498$		0			0
$499$	487.521			0.976997			0.488498	−	0.872565i	$-0.337545\pi$
$499$	487.521			0.976997			0.488498	+	0.872565i	$0.337545\pi$
$500$		0			0
$501$	197.429			0.394070
$502$		0			0
$503$		−	500.167i		−	0.994369i	−0.867645	−	0.497184i	$-0.834367\pi$
$503$		−	500.167i		−	0.994369i	0.867645	−	0.497184i	$-0.165633\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−342.876			−0.676283
$508$		0			0
$509$		−	254.504i		−	0.500007i	−0.968245	−	0.250004i	$-0.919568\pi$
$509$		−	254.504i		−	0.500007i	0.968245	−	0.250004i	$-0.0804318\pi$
$510$		0			0
$511$	299.614			0.586329
$512$		0			0
$513$	−182.203	−	503.773i	−0.355171	−	0.982014i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		−	126.298i		−	0.244291i
$518$		0			0
$519$	143.873			0.277213
$520$		0			0
$521$			676.089i			1.29767i	0.760927	+	0.648837i	$0.224744\pi$
$521$			676.089i			1.29767i	−0.760927	+	0.648837i	$0.775256\pi$
$522$		0			0
$523$	−34.9971			−0.0669162			−0.0334581	−	0.999440i	$-0.510652\pi$
$523$	−34.9971			−0.0669162			−0.0334581	+	0.999440i	$0.510652\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	7.56732			0.0143592
$528$		0			0
$529$	−146.311			−0.276580
$530$		0			0
$531$			1.22176i			0.00230087i
$532$		0			0
$533$		−	10.7786i		−	0.0202226i
$534$		0			0
$535$		0			0
$536$		0			0
$537$			358.141i			0.666929i
$538$		0			0
$539$	167.968			0.311630
$540$		0			0
$541$	411.260			0.760184			0.380092	−	0.924949i	$-0.375892\pi$
$541$	411.260			0.760184			0.380092	+	0.924949i	$0.375892\pi$
$542$		0			0
$543$		−	117.054i		−	0.215570i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−355.828			−0.650508			−0.325254	−	0.945627i	$-0.605450\pi$
$547$	−355.828			−0.650508			−0.325254	+	0.945627i	$0.605450\pi$
$548$		0			0
$549$	−192.410			−0.350473
$550$		0			0
$551$	−873.168	+	315.804i	−1.58470	+	0.573147i
$552$		0			0
$553$	−319.261			−0.577325
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	823.586i		−	1.47861i	−0.673370	−	0.739305i	$-0.735154\pi$
$557$		−	823.586i		−	1.47861i	0.673370	−	0.739305i	$-0.264846\pi$
$558$		0			0
$559$			0.437795i			0.000783176i
$560$		0			0
$561$			18.8376i			0.0335787i
$562$		0			0
$563$	753.891			1.33906			0.669531	−	0.742785i	$-0.266495\pi$
$563$	753.891			1.33906			0.669531	+	0.742785i	$0.266495\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$			58.0299i			0.102345i
$568$		0			0
$569$			183.744i			0.322924i	0.986879	+	0.161462i	$0.0516209\pi$
$569$			183.744i			0.322924i	−0.986879	+	0.161462i	$0.948379\pi$
$570$		0			0
$571$	−544.748			−0.954024			−0.477012	−	0.878897i	$-0.658280\pi$
$571$	−544.748			−0.954024			−0.477012	+	0.878897i	$0.658280\pi$
$572$		0			0
$573$	−296.271			−0.517052
$574$		0			0
$575$		0			0
$576$		0			0
$577$			68.9024i			0.119415i	0.998216	+	0.0597074i	$0.0190168\pi$
$577$			68.9024i			0.119415i	−0.998216	+	0.0597074i	$0.980983\pi$
$578$		0			0
$579$	−130.424			−0.225257
$580$		0			0
$581$	5.54794			0.00954895
$582$		0			0
$583$	297.650			0.510548
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	110.222i		−	0.187771i	−0.995583	−	0.0938857i	$-0.970071\pi$
$587$		−	110.222i		−	0.187771i	0.995583	−	0.0938857i	$-0.0299288\pi$
$588$		0			0
$589$	79.8763	−	28.8894i	0.135613	−	0.0490482i
$590$		0			0
$591$		−	36.4475i		−	0.0616708i
$592$		0			0
$593$		−	15.0895i		−	0.0254460i	−0.999919	−	0.0127230i	$-0.995950\pi$
$593$		−	15.0895i		−	0.0254460i	0.999919	−	0.0127230i	$-0.00404996\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−425.750			−0.713150
$598$		0			0
$599$		−	557.545i		−	0.930793i	−0.885102	−	0.465396i	$-0.845912\pi$
$599$		−	557.545i		−	0.930793i	0.885102	−	0.465396i	$-0.154088\pi$
$600$		0			0
$601$		−	409.791i		−	0.681849i	−0.940091	−	0.340924i	$-0.889260\pi$
$601$		−	409.791i		−	0.681849i	0.940091	−	0.340924i	$-0.110740\pi$
$602$		0			0
$603$	−468.586			−0.777092
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−720.249			−1.18657			−0.593286	−	0.804992i	$-0.702170\pi$
$607$	−720.249			−1.18657			−0.593286	+	0.804992i	$0.702170\pi$
$608$		0			0
$609$	425.214			0.698217
$610$		0			0
$611$		−	14.6346i		−	0.0239520i
$612$		0			0
$613$		−	662.266i		−	1.08037i	−0.841547	−	0.540184i	$-0.818354\pi$
$613$		−	662.266i		−	1.08037i	0.841547	−	0.540184i	$-0.181646\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	656.881i		−	1.06464i	−0.846544	−	0.532319i	$-0.821321\pi$
$617$		−	656.881i		−	1.06464i	0.846544	−	0.532319i	$-0.178679\pi$
$618$		0			0
$619$	−454.752			−0.734656			−0.367328	−	0.930092i	$-0.619727\pi$
$619$	−454.752			−0.734656			−0.367328	+	0.930092i	$0.619727\pi$
$620$		0			0
$621$		−	732.703i		−	1.17988i
$622$		0			0
$623$	604.748			0.970703
$624$		0			0
$625$		0			0
$626$		0			0
$627$	71.9155	+	198.839i	0.114698	+	0.317128i
$628$		0			0
$629$			10.3735i			0.0164920i
$630$		0			0
$631$	283.098			0.448650			0.224325	−	0.974514i	$-0.427982\pi$
$631$	283.098			0.448650			0.224325	+	0.974514i	$0.427982\pi$
$632$		0			0
$633$			473.799i			0.748497i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	19.4631			0.0305544
$638$		0			0
$639$		−	78.1022i		−	0.122226i
$640$		0			0
$641$		−	470.172i		−	0.733497i	−0.930320	−	0.366749i	$-0.880471\pi$
$641$		−	470.172i		−	0.733497i	0.930320	−	0.366749i	$-0.119529\pi$
$642$		0			0
$643$			916.438i			1.42525i	0.701544	+	0.712627i	$0.252495\pi$
$643$			916.438i			1.42525i	−0.701544	+	0.712627i	$0.747505\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			582.366i			0.900101i	0.893003	+	0.450051i	$0.148594\pi$
$647$			582.366i			0.900101i	−0.893003	+	0.450051i	$0.851406\pi$
$648$		0			0
$649$		−	1.37449i		−	0.00211786i
$650$		0			0
$651$	−38.8980			−0.0597512
$652$		0			0
$653$		−	228.094i		−	0.349301i	−0.984630	−	0.174651i	$-0.944120\pi$
$653$		−	228.094i		−	0.349301i	0.984630	−	0.174651i	$-0.0558796\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		−	340.629i		−	0.518462i
$658$		0			0
$659$		−	250.379i		−	0.379938i	−0.981790	−	0.189969i	$-0.939161\pi$
$659$		−	250.379i		−	0.379938i	0.981790	−	0.189969i	$-0.0608387\pi$
$660$		0			0
$661$		−	830.133i		−	1.25587i	−0.778264	−	0.627937i	$-0.783900\pi$
$661$		−	830.133i		−	1.25587i	0.778264	−	0.627937i	$-0.216100\pi$
$662$		0			0
$663$			2.18279i			0.00329229i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−1269.96			−1.90399
$668$		0			0
$669$	138.724			0.207360
$670$		0			0
$671$	216.462			0.322597
$672$		0			0
$673$	−1120.67			−1.66519			−0.832595	−	0.553882i	$-0.813146\pi$
$673$	−1120.67			−1.66519			−0.832595	+	0.553882i	$0.813146\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−500.474			−0.739253			−0.369626	−	0.929181i	$-0.620514\pi$
$677$	−500.474			−0.739253			−0.369626	+	0.929181i	$0.620514\pi$
$678$		0			0
$679$		−	535.275i		−	0.788329i
$680$		0			0
$681$	291.078			0.427427
$682$		0			0
$683$	253.863			0.371688			0.185844	−	0.982579i	$-0.440498\pi$
$683$	253.863			0.371688			0.185844	+	0.982579i	$0.440498\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−276.356			−0.402265
$688$		0			0
$689$	34.4898			0.0500577
$690$		0			0
$691$	210.478			0.304599			0.152300	−	0.988334i	$-0.451332\pi$
$691$	210.478			0.304599			0.152300	+	0.988334i	$0.451332\pi$
$692$		0			0
$693$			113.880i			0.164329i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	28.7741			0.0412828
$698$		0			0
$699$		−	550.650i		−	0.787768i
$700$		0			0
$701$	146.797			0.209410			0.104705	−	0.994503i	$-0.466610\pi$
$701$	146.797			0.209410			0.104705	+	0.994503i	$0.466610\pi$
$702$		0			0
$703$	39.6023	+	109.497i	0.0563333	+	0.155756i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	440.127i		−	0.622527i
$708$		0			0
$709$	−803.083			−1.13270			−0.566349	−	0.824166i	$-0.691644\pi$
$709$	−803.083			−1.13270			−0.566349	+	0.824166i	$0.691644\pi$
$710$		0			0
$711$			362.965i			0.510500i
$712$		0			0
$713$	116.175			0.162938
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−610.251			−0.851117
$718$		0			0
$719$	−637.968			−0.887299			−0.443650	−	0.896200i	$-0.646317\pi$
$719$	−637.968			−0.887299			−0.443650	+	0.896200i	$0.646317\pi$
$720$		0			0
$721$			50.0364i			0.0693986i
$722$		0			0
$723$		−	360.862i		−	0.499118i
$724$		0			0
$725$		0			0
$726$		0			0
$727$			319.305i			0.439210i	0.975589	+	0.219605i	$0.0704768\pi$
$727$			319.305i			0.439210i	−0.975589	+	0.219605i	$0.929523\pi$
$728$		0			0
$729$	582.037			0.798405
$730$		0			0
$731$	−1.16872			−0.00159879
$732$		0			0
$733$		−	1185.22i		−	1.61694i	−0.588537	−	0.808470i	$-0.700296\pi$
$733$		−	1185.22i		−	1.61694i	0.588537	−	0.808470i	$-0.299704\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	527.163			0.715283
$738$		0			0
$739$	−182.023			−0.246311			−0.123155	−	0.992387i	$-0.539301\pi$
$739$	−182.023			−0.246311			−0.123155	+	0.992387i	$0.539301\pi$
$740$		0			0
$741$	8.33312	+	23.0403i	0.0112458	+	0.0310935i
$742$		0			0
$743$	827.162			1.11327			0.556637	−	0.830756i	$-0.312092\pi$
$743$	827.162			1.11327			0.556637	+	0.830756i	$0.312092\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		−	6.30741i		−	0.00844366i
$748$		0			0
$749$		−	812.622i		−	1.08494i
$750$		0			0
$751$		−	1154.06i		−	1.53669i	−0.640034	−	0.768346i	$-0.721080\pi$
$751$		−	1154.06i		−	1.53669i	0.640034	−	0.768346i	$-0.278920\pi$
$752$		0			0
$753$	−162.804			−0.216207
$754$		0			0
$755$		0			0
$756$		0			0
$757$		−	1361.46i		−	1.79850i	−0.437435	−	0.899250i	$-0.644113\pi$
$757$		−	1361.46i		−	1.79850i	0.437435	−	0.899250i	$-0.355887\pi$
$758$		0			0
$759$			289.198i			0.381025i
$760$		0			0
$761$	−646.107			−0.849023			−0.424512	−	0.905422i	$-0.639554\pi$
$761$	−646.107			−0.849023			−0.424512	+	0.905422i	$0.639554\pi$
$762$		0			0
$763$	186.898			0.244952
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	0.159268i		−	0.000207650i
$768$		0			0
$769$	−1226.28			−1.59464			−0.797318	−	0.603559i	$-0.793749\pi$
$769$	−1226.28			−1.59464			−0.797318	+	0.603559i	$0.793749\pi$
$770$		0			0
$771$	300.216			0.389385
$772$		0			0
$773$	−544.100			−0.703881			−0.351940	−	0.936022i	$-0.614478\pi$
$773$	−544.100			−0.703881			−0.351940	+	0.936022i	$0.614478\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		−	53.3225i		−	0.0686261i
$778$		0			0
$779$	303.723	−	109.849i	0.389888	−	0.141013i
$780$		0			0
$781$			87.8656i			0.112504i
$782$		0			0
$783$		−	1377.89i		−	1.75976i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−1035.63			−1.31592			−0.657958	−	0.753054i	$-0.728580\pi$
$787$	−1035.63			−1.31592			−0.657958	+	0.753054i	$0.728580\pi$
$788$		0			0
$789$			700.521i			0.887859i
$790$		0			0
$791$			431.676i			0.545734i
$792$		0			0
$793$	25.0823			0.0316296
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−1237.45			−1.55264			−0.776318	−	0.630341i	$-0.782915\pi$
$797$	−1237.45			−1.55264			−0.776318	+	0.630341i	$0.782915\pi$
$798$		0			0
$799$	39.0680			0.0488961
$800$		0			0
$801$		−	687.533i		−	0.858344i
$802$		0			0
$803$			383.211i			0.477224i
$804$		0			0
$805$		0			0
$806$		0			0
$807$			388.363i			0.481243i
$808$		0			0
$809$	456.534			0.564319			0.282160	−	0.959367i	$-0.408949\pi$
$809$	456.534			0.564319			0.282160	+	0.959367i	$0.408949\pi$
$810$		0			0
$811$			1238.50i			1.52712i	0.645735	+	0.763562i	$0.276551\pi$
$811$			1238.50i			1.52712i	−0.645735	+	0.763562i	$0.723449\pi$
$812$		0			0
$813$	−539.193			−0.663214
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−12.3363	+	4.46175i	−0.0150995	+	0.00546114i
$818$		0			0
$819$			13.1957i			0.0161120i
$820$		0			0
$821$	503.379			0.613130			0.306565	−	0.951850i	$-0.400820\pi$
$821$	503.379			0.613130			0.306565	+	0.951850i	$0.400820\pi$
$822$		0			0
$823$			864.599i			1.05055i	0.850934	+	0.525273i	$0.176037\pi$
$823$			864.599i			1.05055i	−0.850934	+	0.525273i	$0.823963\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	1528.37			1.84809			0.924047	−	0.382280i	$-0.124861\pi$
$827$	1528.37			1.84809			0.924047	+	0.382280i	$0.124861\pi$
$828$		0			0
$829$			696.807i			0.840539i	0.907399	+	0.420270i	$0.138065\pi$
$829$			696.807i			0.840539i	−0.907399	+	0.420270i	$0.861935\pi$
$830$		0			0
$831$			177.070i			0.213080i
$832$		0			0
$833$			51.9578i			0.0623744i
$834$		0			0
$835$		0			0
$836$		0			0
$837$			126.048i			0.150595i
$838$		0			0
$839$		−	1372.41i		−	1.63577i	−0.575380	−	0.817886i	$-0.695146\pi$
$839$		−	1372.41i		−	1.63577i	0.575380	−	0.817886i	$-0.304854\pi$
$840$		0			0
$841$	−1547.24			−1.83976
$842$		0			0
$843$		−	475.858i		−	0.564482i
$844$		0			0
$845$		0			0
$846$		0			0
$847$			389.574i			0.459946i
$848$		0			0
$849$			387.748i			0.456711i
$850$		0			0
$851$			159.255i			0.187139i
$852$		0			0
$853$		−	152.915i		−	0.179268i	−0.995975	−	0.0896339i	$-0.971430\pi$
$853$		−	152.915i		−	0.179268i	0.995975	−	0.0896339i	$-0.0285697\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−621.971			−0.725754			−0.362877	−	0.931837i	$-0.618205\pi$
$857$	−621.971			−0.725754			−0.362877	+	0.931837i	$0.618205\pi$
$858$		0			0
$859$	710.294			0.826885			0.413442	−	0.910530i	$-0.364326\pi$
$859$	710.294			0.826885			0.413442	+	0.910530i	$0.364326\pi$
$860$		0			0
$861$	−147.907			−0.171785
$862$		0			0
$863$	450.250			0.521726			0.260863	−	0.965376i	$-0.415993\pi$
$863$	450.250			0.521726			0.260863	+	0.965376i	$0.415993\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	581.909			0.671175
$868$		0			0
$869$		−	408.339i		−	0.469895i
$870$		0			0
$871$	61.0844			0.0701313
$872$		0			0
$873$	−608.551			−0.697080
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−687.264			−0.783654			−0.391827	−	0.920039i	$-0.628157\pi$
$877$	−687.264			−0.783654			−0.391827	+	0.920039i	$0.628157\pi$
$878$		0			0
$879$	468.561			0.533062
$880$		0			0
$881$	−848.052			−0.962602			−0.481301	−	0.876555i	$-0.659836\pi$
$881$	−848.052			−0.962602			−0.481301	+	0.876555i	$0.659836\pi$
$882$		0			0
$883$			267.540i			0.302990i	0.988458	+	0.151495i	$0.0484088\pi$
$883$			267.540i			0.302990i	−0.988458	+	0.151495i	$0.951591\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	1189.74			1.34130			0.670651	−	0.741773i	$-0.266015\pi$
$887$	1189.74			1.34130			0.670651	+	0.741773i	$0.266015\pi$
$888$		0			0
$889$		−	599.944i		−	0.674853i
$890$		0			0
$891$	−74.2210			−0.0833008
$892$		0			0
$893$	412.379	−	149.148i	0.461791	−	0.167019i
$894$		0			0
$895$		0			0
$896$		0			0
$897$			33.5105i			0.0373584i
$898$		0			0
$899$	218.473			0.243018
$900$		0			0
$901$			92.0722i			0.102189i
$902$		0			0
$903$	6.00752			0.00665285
$904$		0			0
$905$		0			0
$906$		0			0
$907$	1069.54			1.17921			0.589605	−	0.807692i	$-0.299283\pi$
$907$	1069.54			1.17921			0.589605	+	0.807692i	$0.299283\pi$
$908$		0			0
$909$	−500.377			−0.550469
$910$		0			0
$911$			1455.41i			1.59760i	0.601597	+	0.798800i	$0.294531\pi$
$911$			1455.41i			1.59760i	−0.601597	+	0.798800i	$0.705469\pi$
$912$		0			0
$913$			7.09589i			0.00777206i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			679.522i			0.741027i
$918$		0			0
$919$	−104.189			−0.113372			−0.0566862	−	0.998392i	$-0.518053\pi$
$919$	−104.189			−0.113372			−0.0566862	+	0.998392i	$0.518053\pi$
$920$		0			0
$921$	753.771			0.818426
$922$		0			0
$923$			10.1813i			0.0110307i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	56.8860			0.0613657
$928$		0			0
$929$	−59.7100			−0.0642734			−0.0321367	−	0.999483i	$-0.510231\pi$
$929$	−59.7100			−0.0642734			−0.0321367	+	0.999483i	$0.510231\pi$
$930$		0			0
$931$	198.357	+	548.437i	0.213058	+	0.589084i
$932$		0			0
$933$	−24.8205			−0.0266028
$934$		0			0
$935$		0			0
$936$		0			0
$937$			907.749i			0.968783i	0.874851	+	0.484391i	$0.160959\pi$
$937$			907.749i			0.968783i	−0.874851	+	0.484391i	$0.839041\pi$
$938$		0			0
$939$		−	535.710i		−	0.570511i
$940$		0			0
$941$		−	60.4652i		−	0.0642563i	−0.999484	−	0.0321282i	$-0.989772\pi$
$941$		−	60.4652i		−	0.0642563i	0.999484	−	0.0321282i	$-0.0102285\pi$
$942$		0			0
$943$	441.744			0.468445
$944$		0			0
$945$		0			0
$946$		0			0
$947$			96.4321i			0.101829i	0.998703	+	0.0509145i	$0.0162136\pi$
$947$			96.4321i			0.101829i	−0.998703	+	0.0509145i	$0.983786\pi$
$948$		0			0
$949$			44.4041i			0.0467904i
$950$		0			0
$951$	−668.115			−0.702539
$952$		0			0
$953$	334.185			0.350667			0.175333	−	0.984509i	$-0.443900\pi$
$953$	334.185			0.350667			0.175333	+	0.984509i	$0.443900\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$			543.855i			0.568291i
$958$		0			0
$959$	320.188			0.333877
$960$		0			0
$961$	941.014			0.979203
$962$		0			0
$963$	−923.863			−0.959360
$964$		0			0
$965$		0			0
$966$		0			0
$967$		−	1230.05i		−	1.27203i	−0.771676	−	0.636016i	$-0.780581\pi$
$967$		−	1230.05i		−	1.27203i	0.771676	−	0.636016i	$-0.219419\pi$
$968$		0			0
$969$	−61.5072	+	22.2457i	−0.0634749	+	0.0229574i
$970$		0			0
$971$			1215.04i			1.25133i	0.780091	+	0.625667i	$0.215173\pi$
$971$			1215.04i			1.25133i	−0.780091	+	0.625667i	$0.784827\pi$
$972$		0			0
$973$		−	74.8406i		−	0.0769174i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	337.236			0.345175			0.172588	−	0.984994i	$-0.444787\pi$
$977$	337.236			0.345175			0.172588	+	0.984994i	$0.444787\pi$
$978$		0			0
$979$			773.481i			0.790072i
$980$		0			0
$981$		−	212.483i		−	0.216598i
$982$		0			0
$983$	−73.3754			−0.0746443			−0.0373222	−	0.999303i	$-0.511883\pi$
$983$	−73.3754			−0.0746443			−0.0373222	+	0.999303i	$0.511883\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−200.820			−0.203465
$988$		0			0
$989$	−17.9423			−0.0181419
$990$		0			0
$991$		−	635.844i		−	0.641618i	−0.947144	−	0.320809i	$-0.896045\pi$
$991$		−	635.844i		−	0.641618i	0.947144	−	0.320809i	$-0.103955\pi$
$992$		0			0
$993$		−	59.4421i		−	0.0598611i
$994$		0			0
$995$		0			0
$996$		0			0
$997$		−	1405.38i		−	1.40960i	−0.709404	−	0.704802i	$-0.751036\pi$
$997$		−	1405.38i		−	1.40960i	0.709404	−	0.704802i	$-0.248964\pi$
$998$		0			0
$999$	−172.790			−0.172963

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1900.3.g.c.949.15		24
5.2	odd	4		380.3.e.a.341.5	✓	12
5.3	odd	4		1900.3.e.f.1101.8		12
5.4	even	2	inner	1900.3.g.c.949.10		24
15.2	even	4		3420.3.o.a.721.11		12
19.18	odd	2	inner	1900.3.g.c.949.9		24
20.7	even	4		1520.3.h.b.721.8		12
95.18	even	4		1900.3.e.f.1101.5		12
95.37	even	4		380.3.e.a.341.8	yes	12
95.94	odd	2	inner	1900.3.g.c.949.16		24
285.227	odd	4		3420.3.o.a.721.12		12
380.227	odd	4		1520.3.h.b.721.5		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
380.3.e.a.341.5	✓	12	5.2	odd	4
380.3.e.a.341.8	yes	12	95.37	even	4
1520.3.h.b.721.5		12	380.227	odd	4
1520.3.h.b.721.8		12	20.7	even	4
1900.3.e.f.1101.5		12	95.18	even	4
1900.3.e.f.1101.8		12	5.3	odd	4
1900.3.g.c.949.9		24	19.18	odd	2	inner
1900.3.g.c.949.10		24	5.4	even	2	inner
1900.3.g.c.949.15		24	1.1	even	1	trivial
1900.3.g.c.949.16		24	95.94	odd	2	inner
3420.3.o.a.721.11		12	15.2	even	4
3420.3.o.a.721.12		12	285.227	odd	4

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:	\( N \)	\(=\)	\( 1900 = 2^{2} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1900.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.03369 q^{3} -4.27843i q^{7} -4.86411 q^{9} +O(q^{10})\)	\(q+2.03369 q^{3} -4.27843i q^{7} -4.86411 q^{9} +5.47217 q^{11} +0.634081 q^{13} +1.69271i q^{17} +(6.46218 + 17.8673i) q^{19} -8.70099i q^{21} +25.9867i q^{23} -28.1953 q^{27} +48.8696i q^{29} -4.47053i q^{31} +11.1287 q^{33} +6.12833 q^{37} +1.28952 q^{39} -16.9988i q^{41} +0.690441i q^{43} -23.0801i q^{47} +30.6950 q^{49} +3.44245i q^{51} +54.3934 q^{53} +(13.1420 + 36.3365i) q^{57} -0.251179i q^{59} +39.5570 q^{61} +20.8108i q^{63} +96.3354 q^{67} +52.8489i q^{69} +16.0568i q^{71} +70.0291i q^{73} -23.4123i q^{77} -74.6211i q^{79} -13.5634 q^{81} +1.29672i q^{83} +99.3856i q^{87} +141.348i q^{89} -2.71287i q^{91} -9.09167i q^{93} +125.110 q^{97} -26.6173 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 32 q^{9}+O(q^{10}) \) 24 * q + 32 * q^9	\( 24 q + 32 q^{9} - 64 q^{11} - 48 q^{19} - 248 q^{39} + 48 q^{49} + 304 q^{61} - 936 q^{81} - 784 q^{99}+O(q^{100}) \) 24 * q + 32 * q^9 - 64 * q^11 - 48 * q^19 - 248 * q^39 + 48 * q^49 + 304 * q^61 - 936 * q^81 - 784 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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