LMFDB - Embedded newform 1900.3.g.c.949.7

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			949.7
Character	$\chi$	$=$	1900.949
Dual form			1900.3.g.c.949.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.14288 q^{3} +12.2192i q^{7} +0.877687 q^{9} +O(q^{10})$	$q-3.14288 q^{3} +12.2192i q^{7} +0.877687 q^{9} +0.636018 q^{11} +19.9122 q^{13} -20.7245i q^{17} +(4.37892 + 18.4885i) q^{19} -38.4034i q^{21} -6.16098i q^{23} +25.5274 q^{27} -11.9924i q^{29} -47.7875i q^{31} -1.99893 q^{33} +19.1997 q^{37} -62.5817 q^{39} -22.7335i q^{41} -34.1742i q^{43} +75.4198i q^{47} -100.308 q^{49} +65.1347i q^{51} +87.2484 q^{53} +(-13.7624 - 58.1071i) q^{57} +40.7398i q^{59} -10.2300 q^{61} +10.7246i q^{63} -16.8833 q^{67} +19.3632i q^{69} -23.2877i q^{71} -103.642i q^{73} +7.77162i q^{77} +123.657i q^{79} -88.1288 q^{81} -82.9966i q^{83} +37.6908i q^{87} -34.0374i q^{89} +243.311i q^{91} +150.190i q^{93} +66.0861 q^{97} +0.558225 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−3.14288			−1.04763			−0.523813	−	0.851833i	$-0.675491\pi$
$3$	−3.14288			−1.04763			−0.523813	+	0.851833i	$0.675491\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$			12.2192i			1.74560i	0.488081	+	0.872798i	$0.337697\pi$
$7$			12.2192i			1.74560i	−0.488081	+	0.872798i	$0.662303\pi$
$8$		0			0
$9$	0.877687			0.0975207
$10$		0			0
$11$	0.636018			0.0578199			0.0289099	−	0.999582i	$-0.490796\pi$
$11$	0.636018			0.0578199			0.0289099	+	0.999582i	$0.490796\pi$
$12$		0			0
$13$	19.9122			1.53171			0.765855	−	0.643014i	$-0.222316\pi$
$13$	19.9122			1.53171			0.765855	+	0.643014i	$0.222316\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	20.7245i		−	1.21909i	−0.792751	−	0.609546i	$-0.791352\pi$
$17$		−	20.7245i		−	1.21909i	0.792751	−	0.609546i	$-0.208648\pi$
$18$		0			0
$19$	4.37892	+	18.4885i	0.230470	+	0.973080i
$19$	4.37892	+	18.4885i	0.230470	+	0.973080i
$20$		0			0
$21$		−	38.4034i		−	1.82873i
$22$		0			0
$23$		−	6.16098i		−	0.267869i	−0.990990	−	0.133934i	$-0.957239\pi$
$23$		−	6.16098i		−	0.267869i	0.990990	−	0.133934i	$-0.0427611\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	25.5274			0.945461
$28$		0			0
$29$		−	11.9924i		−	0.413532i	−0.978390	−	0.206766i	$-0.933706\pi$
$29$		−	11.9924i		−	0.413532i	0.978390	−	0.206766i	$-0.0662939\pi$
$30$		0			0
$31$		−	47.7875i		−	1.54153i	−0.637118	−	0.770767i	$-0.719873\pi$
$31$		−	47.7875i		−	1.54153i	0.637118	−	0.770767i	$-0.280127\pi$
$32$		0			0
$33$	−1.99893			−0.0605736
$34$		0			0
$35$		0			0
$36$		0			0
$37$	19.1997			0.518910			0.259455	−	0.965755i	$-0.416457\pi$
$37$	19.1997			0.518910			0.259455	+	0.965755i	$0.416457\pi$
$38$		0			0
$39$	−62.5817			−1.60466
$40$		0			0
$41$		−	22.7335i		−	0.554475i	−0.960801	−	0.277238i	$-0.910581\pi$
$41$		−	22.7335i		−	0.554475i	0.960801	−	0.277238i	$-0.0894190\pi$
$42$		0			0
$43$		−	34.1742i		−	0.794750i	−0.917656	−	0.397375i	$-0.869921\pi$
$43$		−	34.1742i		−	0.794750i	0.917656	−	0.397375i	$-0.130079\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			75.4198i			1.60468i	0.596870	+	0.802338i	$0.296411\pi$
$47$			75.4198i			1.60468i	−0.596870	+	0.802338i	$0.703589\pi$
$48$		0			0
$49$	−100.308			−2.04711
$50$		0			0
$51$			65.1347i			1.27715i
$52$		0			0
$53$	87.2484			1.64620			0.823098	−	0.567900i	$-0.192244\pi$
$53$	87.2484			1.64620			0.823098	+	0.567900i	$0.192244\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−13.7624	−	58.1071i	−0.241446	−	1.01942i
$58$		0			0
$59$			40.7398i			0.690505i	0.938510	+	0.345252i	$0.112207\pi$
$59$			40.7398i			0.690505i	−0.938510	+	0.345252i	$0.887793\pi$
$60$		0			0
$61$	−10.2300			−0.167705			−0.0838525	−	0.996478i	$-0.526722\pi$
$61$	−10.2300			−0.167705			−0.0838525	+	0.996478i	$0.526722\pi$
$62$		0			0
$63$			10.7246i			0.170232i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−16.8833			−0.251989			−0.125995	−	0.992031i	$-0.540212\pi$
$67$	−16.8833			−0.251989			−0.125995	+	0.992031i	$0.540212\pi$
$68$		0			0
$69$			19.3632i			0.280626i
$70$		0			0
$71$		−	23.2877i		−	0.327996i	−0.986461	−	0.163998i	$-0.947561\pi$
$71$		−	23.2877i		−	0.327996i	0.986461	−	0.163998i	$-0.0524390\pi$
$72$		0			0
$73$		−	103.642i		−	1.41975i	−0.704328	−	0.709875i	$-0.748751\pi$
$73$		−	103.642i		−	1.41975i	0.704328	−	0.709875i	$-0.251249\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			7.77162i			0.100930i
$78$		0			0
$79$			123.657i			1.56528i	0.622473	+	0.782641i	$0.286128\pi$
$79$			123.657i			1.56528i	−0.622473	+	0.782641i	$0.713872\pi$
$80$		0			0
$81$	−88.1288			−1.08801
$82$		0			0
$83$		−	82.9966i		−	0.999959i	−0.866037	−	0.499979i	$-0.833341\pi$
$83$		−	82.9966i		−	0.999959i	0.866037	−	0.499979i	$-0.166659\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$			37.6908i			0.433227i
$88$		0			0
$89$		−	34.0374i		−	0.382443i	−0.981547	−	0.191221i	$-0.938755\pi$
$89$		−	34.0374i		−	0.382443i	0.981547	−	0.191221i	$-0.0612448\pi$
$90$		0			0
$91$			243.311i			2.67375i
$92$		0			0
$93$			150.190i			1.61495i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	66.0861			0.681300			0.340650	−	0.940190i	$-0.389353\pi$
$97$	66.0861			0.681300			0.340650	+	0.940190i	$0.389353\pi$
$98$		0			0
$99$	0.558225			0.00563864
$100$		0			0
$101$	39.1734			0.387855			0.193928	−	0.981016i	$-0.437877\pi$
$101$	39.1734			0.387855			0.193928	+	0.981016i	$0.437877\pi$
$102$		0			0
$103$	−113.940			−1.10621			−0.553105	−	0.833112i	$-0.686557\pi$
$103$	−113.940			−1.10621			−0.553105	+	0.833112i	$0.686557\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	86.3080			0.806617			0.403308	−	0.915064i	$-0.367860\pi$
$107$	86.3080			0.806617			0.403308	+	0.915064i	$0.367860\pi$
$108$		0			0
$109$			24.4905i			0.224683i	0.993670	+	0.112342i	$0.0358351\pi$
$109$			24.4905i			0.224683i	−0.993670	+	0.112342i	$0.964165\pi$
$110$		0			0
$111$	−60.3423			−0.543624
$112$		0			0
$113$	211.181			1.86886			0.934428	−	0.356153i	$-0.115912\pi$
$113$	211.181			1.86886			0.934428	+	0.356153i	$0.115912\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	17.4767			0.149373
$118$		0			0
$119$	253.237			2.12804
$120$		0			0
$121$	−120.595			−0.996657
$122$		0			0
$123$			71.4486i			0.580883i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	195.613			1.54026			0.770132	−	0.637885i	$-0.220190\pi$
$127$	195.613			1.54026			0.770132	+	0.637885i	$0.220190\pi$
$128$		0			0
$129$			107.405i			0.832600i
$130$		0			0
$131$	247.494			1.88927			0.944635	−	0.328123i	$-0.106416\pi$
$131$	247.494			1.88927			0.944635	+	0.328123i	$0.106416\pi$
$132$		0			0
$133$	−225.914	+	53.5068i	−1.69860	+	0.402307i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			51.5720i			0.376438i	0.982127	+	0.188219i	$0.0602715\pi$
$137$			51.5720i			0.376438i	−0.982127	+	0.188219i	$0.939728\pi$
$138$		0			0
$139$	−114.447			−0.823356			−0.411678	−	0.911329i	$-0.635057\pi$
$139$	−114.447			−0.823356			−0.411678	+	0.911329i	$0.635057\pi$
$140$		0			0
$141$		−	237.035i		−	1.68110i
$142$		0			0
$143$	12.6645			0.0885632
$144$		0			0
$145$		0			0
$146$		0			0
$147$	315.257			2.14460
$148$		0			0
$149$	−141.259			−0.948044			−0.474022	−	0.880513i	$-0.657198\pi$
$149$	−141.259			−0.948044			−0.474022	+	0.880513i	$0.657198\pi$
$150$		0			0
$151$			53.3624i			0.353394i	0.984265	+	0.176697i	$0.0565412\pi$
$151$			53.3624i			0.353394i	−0.984265	+	0.176697i	$0.943459\pi$
$152$		0			0
$153$		−	18.1897i		−	0.118887i
$154$		0			0
$155$		0			0
$156$		0			0
$157$			124.881i			0.795421i	0.917511	+	0.397710i	$0.130195\pi$
$157$			124.881i			0.795421i	−0.917511	+	0.397710i	$0.869805\pi$
$158$		0			0
$159$	−274.211			−1.72460
$160$		0			0
$161$	75.2821			0.467590
$162$		0			0
$163$			9.62929i			0.0590754i	0.999564	+	0.0295377i	$0.00940351\pi$
$163$			9.62929i			0.0590754i	−0.999564	+	0.0295377i	$0.990596\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	225.797			1.35208			0.676038	−	0.736867i	$-0.263696\pi$
$167$	225.797			1.35208			0.676038	+	0.736867i	$0.263696\pi$
$168$		0			0
$169$	227.497			1.34613
$170$		0			0
$171$	3.84332	+	16.2271i	0.0224756	+	0.0948954i
$172$		0			0
$173$	227.218			1.31340			0.656698	−	0.754153i	$-0.271952\pi$
$173$	227.218			1.31340			0.656698	+	0.754153i	$0.271952\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		−	128.040i		−	0.723391i
$178$		0			0
$179$		−	146.908i		−	0.820714i	−0.911925	−	0.410357i	$-0.865404\pi$
$179$		−	146.908i		−	0.820714i	0.911925	−	0.410357i	$-0.134596\pi$
$180$		0			0
$181$		−	13.1154i		−	0.0724605i	−0.999343	−	0.0362303i	$-0.988465\pi$
$181$		−	13.1154i		−	0.0724605i	0.999343	−	0.0362303i	$-0.0115350\pi$
$182$		0			0
$183$	32.1517			0.175692
$184$		0			0
$185$		0			0
$186$		0			0
$187$		−	13.1812i		−	0.0704877i
$188$		0			0
$189$			311.924i			1.65039i
$190$		0			0
$191$	−1.77858			−0.00931196			−0.00465598	−	0.999989i	$-0.501482\pi$
$191$	−1.77858			−0.00931196			−0.00465598	+	0.999989i	$0.501482\pi$
$192$		0			0
$193$	−214.837			−1.11314			−0.556571	−	0.830800i	$-0.687883\pi$
$193$	−214.837			−1.11314			−0.556571	+	0.830800i	$0.687883\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			229.807i			1.16653i	0.812281	+	0.583266i	$0.198225\pi$
$197$			229.807i			1.16653i	−0.812281	+	0.583266i	$0.801775\pi$
$198$		0			0
$199$	192.242			0.966040			0.483020	−	0.875609i	$-0.339540\pi$
$199$	192.242			0.966040			0.483020	+	0.875609i	$0.339540\pi$
$200$		0			0
$201$	53.0621			0.263991
$202$		0			0
$203$	146.538			0.721861
$204$		0			0
$205$		0			0
$206$		0			0
$207$		−	5.40741i		−	0.0261227i
$208$		0			0
$209$	2.78508	+	11.7590i	0.0133257	+	0.0562633i
$210$		0			0
$211$		−	108.302i		−	0.513280i	−0.966507	−	0.256640i	$-0.917385\pi$
$211$		−	108.302i		−	0.513280i	0.966507	−	0.256640i	$-0.0826154\pi$
$212$		0			0
$213$			73.1904i			0.343617i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	583.924			2.69089
$218$		0			0
$219$			325.733i			1.48737i
$220$		0			0
$221$		−	412.672i		−	1.86729i
$222$		0			0
$223$	248.566			1.11465			0.557323	−	0.830296i	$-0.311828\pi$
$223$	248.566			1.11465			0.557323	+	0.830296i	$0.311828\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−362.900			−1.59868			−0.799338	−	0.600881i	$-0.794817\pi$
$227$	−362.900			−1.59868			−0.799338	+	0.600881i	$0.794817\pi$
$228$		0			0
$229$	225.882			0.986385			0.493193	−	0.869920i	$-0.335830\pi$
$229$	225.882			0.986385			0.493193	+	0.869920i	$0.335830\pi$
$230$		0			0
$231$		−	24.4253i		−	0.105737i
$232$		0			0
$233$			327.528i			1.40570i	0.711338	+	0.702850i	$0.248089\pi$
$233$			327.528i			1.40570i	−0.711338	+	0.702850i	$0.751911\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		−	388.640i		−	1.63983i
$238$		0			0
$239$	341.632			1.42942			0.714711	−	0.699420i	$-0.246558\pi$
$239$	341.632			1.42942			0.714711	+	0.699420i	$0.246558\pi$
$240$		0			0
$241$		−	2.08810i		−	0.00866433i	−0.999991	−	0.00433217i	$-0.998621\pi$
$241$		−	2.08810i		−	0.00866433i	0.999991	−	0.00433217i	$-0.00137898\pi$
$242$		0			0
$243$	47.2313			0.194367
$244$		0			0
$245$		0			0
$246$		0			0
$247$	87.1941	+	368.147i	0.353012	+	1.49048i
$248$		0			0
$249$			260.848i			1.04758i
$250$		0			0
$251$	20.1361			0.0802234			0.0401117	−	0.999195i	$-0.487229\pi$
$251$	20.1361			0.0802234			0.0401117	+	0.999195i	$0.487229\pi$
$252$		0			0
$253$		−	3.91850i		−	0.0154881i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−347.814			−1.35336			−0.676682	−	0.736276i	$-0.736583\pi$
$257$	−347.814			−1.35336			−0.676682	+	0.736276i	$0.736583\pi$
$258$		0			0
$259$			234.604i			0.905808i
$260$		0			0
$261$		−	10.5256i		−	0.0403280i
$262$		0			0
$263$			486.729i			1.85068i	0.379139	+	0.925340i	$0.376220\pi$
$263$			486.729i			1.85068i	−0.379139	+	0.925340i	$0.623780\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$			106.975i			0.400657i
$268$		0			0
$269$			201.083i			0.747521i	0.927525	+	0.373760i	$0.121932\pi$
$269$			201.083i			0.747521i	−0.927525	+	0.373760i	$0.878068\pi$
$270$		0			0
$271$	207.523			0.765767			0.382884	−	0.923797i	$-0.374931\pi$
$271$	207.523			0.765767			0.382884	+	0.923797i	$0.374931\pi$
$272$		0			0
$273$		−	764.697i		−	2.80109i
$274$		0			0
$275$		0			0
$276$		0			0
$277$		−	284.708i		−	1.02783i	−0.857842	−	0.513914i	$-0.828195\pi$
$277$		−	284.708i		−	1.02783i	0.857842	−	0.513914i	$-0.171805\pi$
$278$		0			0
$279$		−	41.9425i		−	0.150331i
$280$		0			0
$281$			301.030i			1.07128i	0.844446	+	0.535640i	$0.179930\pi$
$281$			301.030i			1.07128i	−0.844446	+	0.535640i	$0.820070\pi$
$282$		0			0
$283$			520.536i			1.83935i	0.392683	+	0.919674i	$0.371547\pi$
$283$			520.536i			1.83935i	−0.392683	+	0.919674i	$0.628453\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	277.785			0.967890
$288$		0			0
$289$	−140.507			−0.486183
$290$		0			0
$291$	−207.701			−0.713748
$292$		0			0
$293$	−42.8491			−0.146243			−0.0731214	−	0.997323i	$-0.523296\pi$
$293$	−42.8491			−0.146243			−0.0731214	+	0.997323i	$0.523296\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	16.2359			0.0546664
$298$		0			0
$299$		−	122.679i		−	0.410297i
$300$		0			0
$301$	417.581			1.38731
$302$		0			0
$303$	−123.117			−0.406327
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−483.543			−1.57506			−0.787530	−	0.616277i	$-0.788640\pi$
$307$	−483.543			−1.57506			−0.787530	+	0.616277i	$0.788640\pi$
$308$		0			0
$309$	358.098			1.15889
$310$		0			0
$311$	−163.546			−0.525870			−0.262935	−	0.964814i	$-0.584690\pi$
$311$	−163.546			−0.525870			−0.262935	+	0.964814i	$0.584690\pi$
$312$		0			0
$313$			394.650i			1.26086i	0.776244	+	0.630432i	$0.217122\pi$
$313$			394.650i			1.26086i	−0.776244	+	0.630432i	$0.782878\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	245.190			0.773471			0.386736	−	0.922191i	$-0.373603\pi$
$317$	245.190			0.773471			0.386736	+	0.922191i	$0.373603\pi$
$318$		0			0
$319$		−	7.62741i		−	0.0239104i
$320$		0			0
$321$	−271.255			−0.845033
$322$		0			0
$323$	383.166	−	90.7512i	1.18627	−	0.280963i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		−	76.9706i		−	0.235384i
$328$		0			0
$329$	−921.568			−2.80112
$330$		0			0
$331$			105.067i			0.317423i	0.987325	+	0.158712i	$0.0507340\pi$
$331$			105.067i			0.317423i	−0.987325	+	0.158712i	$0.949266\pi$
$332$		0			0
$333$	16.8513			0.0506045
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−69.6657			−0.206723			−0.103362	−	0.994644i	$-0.532960\pi$
$337$	−69.6657			−0.206723			−0.103362	+	0.994644i	$0.532960\pi$
$338$		0			0
$339$	−663.715			−1.95786
$340$		0			0
$341$		−	30.3937i		−	0.0891312i
$342$		0			0
$343$		−	626.945i		−	1.82783i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			103.244i			0.297532i	0.988872	+	0.148766i	$0.0475302\pi$
$347$			103.244i			0.297532i	−0.988872	+	0.148766i	$0.952470\pi$
$348$		0			0
$349$	−542.273			−1.55379			−0.776896	−	0.629629i	$-0.783207\pi$
$349$	−542.273			−1.55379			−0.776896	+	0.629629i	$0.783207\pi$
$350$		0			0
$351$	508.308			1.44817
$352$		0			0
$353$		−	250.724i		−	0.710266i	−0.934816	−	0.355133i	$-0.884436\pi$
$353$		−	250.724i		−	0.710266i	0.934816	−	0.355133i	$-0.115564\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−795.893			−2.22939
$358$		0			0
$359$	356.137			0.992025			0.496013	−	0.868315i	$-0.334797\pi$
$359$	356.137			0.992025			0.496013	+	0.868315i	$0.334797\pi$
$360$		0			0
$361$	−322.650	+	161.920i	−0.893768	+	0.448530i
$362$		0			0
$363$	379.017			1.04412
$364$		0			0
$365$		0			0
$366$		0			0
$367$			247.991i			0.675725i	0.941195	+	0.337863i	$0.109704\pi$
$367$			247.991i			0.675725i	−0.941195	+	0.337863i	$0.890296\pi$
$368$		0			0
$369$		−	19.9529i		−	0.0540729i
$370$		0			0
$371$			1066.10i			2.87359i
$372$		0			0
$373$	−230.389			−0.617664			−0.308832	−	0.951117i	$-0.599938\pi$
$373$	−230.389			−0.617664			−0.308832	+	0.951117i	$0.599938\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	238.796i		−	0.633411i
$378$		0			0
$379$		−	15.3995i		−	0.0406320i	−0.999794	−	0.0203160i	$-0.993533\pi$
$379$		−	15.3995i		−	0.0406320i	0.999794	−	0.0203160i	$-0.00646722\pi$
$380$		0			0
$381$	−614.789			−1.61362
$382$		0			0
$383$	697.645			1.82153			0.910764	−	0.412928i	$-0.135494\pi$
$383$	697.645			1.82153			0.910764	+	0.412928i	$0.135494\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		−	29.9943i		−	0.0775046i
$388$		0			0
$389$	190.689			0.490203			0.245101	−	0.969497i	$-0.421179\pi$
$389$	190.689			0.490203			0.245101	+	0.969497i	$0.421179\pi$
$390$		0			0
$391$	−127.683			−0.326556
$392$		0			0
$393$	−777.845			−1.97925
$394$		0			0
$395$		0			0
$396$		0			0
$397$		−	251.632i		−	0.633835i	−0.948453	−	0.316917i	$-0.897352\pi$
$397$		−	251.632i		−	0.633835i	0.948453	−	0.316917i	$-0.102648\pi$
$398$		0			0
$399$	710.021	−	168.165i	1.77950	−	0.421467i
$400$		0			0
$401$		−	644.306i		−	1.60675i	−0.595475	−	0.803374i	$-0.703036\pi$
$401$		−	644.306i		−	1.60675i	0.595475	−	0.803374i	$-0.296964\pi$
$402$		0			0
$403$		−	951.556i		−	2.36118i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	12.2114			0.0300033
$408$		0			0
$409$		−	216.145i		−	0.528472i	−0.964458	−	0.264236i	$-0.914880\pi$
$409$		−	216.145i		−	0.528472i	0.964458	−	0.264236i	$-0.0851198\pi$
$410$		0			0
$411$		−	162.085i		−	0.394367i
$412$		0			0
$413$	−497.807			−1.20534
$414$		0			0
$415$		0			0
$416$		0			0
$417$	359.692			0.862570
$418$		0			0
$419$	−165.246			−0.394382			−0.197191	−	0.980365i	$-0.563182\pi$
$419$	−165.246			−0.394382			−0.197191	+	0.980365i	$0.563182\pi$
$420$		0			0
$421$			271.021i			0.643755i	0.946781	+	0.321877i	$0.104314\pi$
$421$			271.021i			0.643755i	−0.946781	+	0.321877i	$0.895686\pi$
$422$		0			0
$423$			66.1949i			0.156489i
$424$		0			0
$425$		0			0
$426$		0			0
$427$		−	125.002i		−	0.292745i
$428$		0			0
$429$	−39.8031			−0.0927812
$430$		0			0
$431$		−	290.548i		−	0.674126i	−0.941482	−	0.337063i	$-0.890566\pi$
$431$		−	290.548i		−	0.674126i	0.941482	−	0.337063i	$-0.109434\pi$
$432$		0			0
$433$	−546.395			−1.26188			−0.630941	−	0.775831i	$-0.717331\pi$
$433$	−546.395			−1.26188			−0.630941	+	0.775831i	$0.717331\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	113.907	−	26.9784i	0.260657	−	0.0617356i
$438$		0			0
$439$		−	822.087i		−	1.87264i	−0.351153	−	0.936318i	$-0.614210\pi$
$439$		−	822.087i		−	1.87264i	0.351153	−	0.936318i	$-0.385790\pi$
$440$		0			0
$441$	−88.0392			−0.199635
$442$		0			0
$443$		−	85.1709i		−	0.192259i	−0.995369	−	0.0961297i	$-0.969354\pi$
$443$		−	85.1709i		−	0.192259i	0.995369	−	0.0961297i	$-0.0306463\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	443.958			0.993196
$448$		0			0
$449$		−	535.082i		−	1.19172i	−0.803089	−	0.595860i	$-0.796811\pi$
$449$		−	535.082i		−	1.19172i	0.803089	−	0.595860i	$-0.203189\pi$
$450$		0			0
$451$		−	14.4589i		−	0.0320597i
$452$		0			0
$453$		−	167.712i		−	0.370224i
$454$		0			0
$455$		0			0
$456$		0			0
$457$			792.130i			1.73333i	0.498894	+	0.866663i	$0.333740\pi$
$457$			792.130i			1.73333i	−0.498894	+	0.866663i	$0.666260\pi$
$458$		0			0
$459$		−	529.045i		−	1.15260i
$460$		0			0
$461$	−290.229			−0.629565			−0.314782	−	0.949164i	$-0.601932\pi$
$461$	−290.229			−0.629565			−0.314782	+	0.949164i	$0.601932\pi$
$462$		0			0
$463$		−	139.806i		−	0.301957i	−0.988537	−	0.150978i	$-0.951758\pi$
$463$		−	139.806i		−	0.301957i	0.988537	−	0.150978i	$-0.0482423\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			252.356i			0.540377i	0.962807	+	0.270189i	$0.0870860\pi$
$467$			252.356i			0.540377i	−0.962807	+	0.270189i	$0.912914\pi$
$468$		0			0
$469$		−	206.300i		−	0.439872i
$470$		0			0
$471$		−	392.486i		−	0.833304i
$472$		0			0
$473$		−	21.7354i		−	0.0459523i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	76.5767			0.160538
$478$		0			0
$479$	−425.333			−0.887960			−0.443980	−	0.896037i	$-0.646434\pi$
$479$	−425.333			−0.887960			−0.443980	+	0.896037i	$0.646434\pi$
$480$		0			0
$481$	382.308			0.794820
$482$		0			0
$483$	−236.602			−0.489860
$484$		0			0
$485$		0			0
$486$		0			0
$487$	359.736			0.738678			0.369339	−	0.929295i	$-0.379584\pi$
$487$	359.736			0.738678			0.369339	+	0.929295i	$0.379584\pi$
$488$		0			0
$489$		−	30.2637i		−	0.0618889i
$490$		0			0
$491$	−466.219			−0.949529			−0.474765	−	0.880113i	$-0.657467\pi$
$491$	−466.219			−0.949529			−0.474765	+	0.880113i	$0.657467\pi$
$492$		0			0
$493$	−248.538			−0.504134
$494$		0			0
$495$		0			0
$496$		0			0
$497$	284.557			0.572549
$498$		0			0
$499$	123.063			0.246620			0.123310	−	0.992368i	$-0.460649\pi$
$499$	123.063			0.246620			0.123310	+	0.992368i	$0.460649\pi$
$500$		0			0
$501$	−709.651			−1.41647
$502$		0			0
$503$			794.478i			1.57948i	0.613442	+	0.789740i	$0.289785\pi$
$503$			794.478i			1.57948i	−0.613442	+	0.789740i	$0.710215\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−714.994			−1.41025
$508$		0			0
$509$			39.8648i			0.0783199i	0.999233	+	0.0391599i	$0.0124682\pi$
$509$			39.8648i			0.0783199i	−0.999233	+	0.0391599i	$0.987532\pi$
$510$		0			0
$511$	1266.42			2.47831
$512$		0			0
$513$	111.783	+	471.964i	0.217900	+	0.920009i
$514$		0			0
$515$		0			0
$516$		0			0
$517$			47.9684i			0.0927822i
$518$		0			0
$519$	−714.118			−1.37595
$520$		0			0
$521$		−	603.806i		−	1.15894i	−0.814995	−	0.579468i	$-0.803260\pi$
$521$		−	603.806i		−	1.15894i	0.814995	−	0.579468i	$-0.196740\pi$
$522$		0			0
$523$	461.654			0.882703			0.441352	−	0.897334i	$-0.354499\pi$
$523$	461.654			0.882703			0.441352	+	0.897334i	$0.354499\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−990.375			−1.87927
$528$		0			0
$529$	491.042			0.928246
$530$		0			0
$531$			35.7568i			0.0673385i
$532$		0			0
$533$		−	452.674i		−	0.849295i
$534$		0			0
$535$		0			0
$536$		0			0
$537$			461.713i			0.859801i
$538$		0			0
$539$	−63.7979			−0.118363
$540$		0			0
$541$	747.259			1.38125			0.690627	−	0.723211i	$-0.257334\pi$
$541$	747.259			1.38125			0.690627	+	0.723211i	$0.257334\pi$
$542$		0			0
$543$			41.2200i			0.0759115i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−210.917			−0.385588			−0.192794	−	0.981239i	$-0.561755\pi$
$547$	−210.917			−0.385588			−0.192794	+	0.981239i	$0.561755\pi$
$548$		0			0
$549$	−8.97874			−0.0163547
$550$		0			0
$551$	221.722	−	52.5139i	0.402400	−	0.0953066i
$552$		0			0
$553$	−1510.99			−2.73235
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	770.359i		−	1.38305i	−0.722353	−	0.691525i	$-0.756939\pi$
$557$		−	770.359i		−	1.38305i	0.722353	−	0.691525i	$-0.243061\pi$
$558$		0			0
$559$		−	680.485i		−	1.21733i
$560$		0			0
$561$			41.4269i			0.0738447i
$562$		0			0
$563$	567.202			1.00746			0.503731	−	0.863860i	$-0.331960\pi$
$563$	567.202			1.00746			0.503731	+	0.863860i	$0.331960\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		−	1076.86i		−	1.89923i
$568$		0			0
$569$		−	796.712i		−	1.40020i	−0.714046	−	0.700099i	$-0.753139\pi$
$569$		−	796.712i		−	1.40020i	0.714046	−	0.700099i	$-0.246861\pi$
$570$		0			0
$571$	−808.589			−1.41609			−0.708046	−	0.706166i	$-0.750423\pi$
$571$	−808.589			−1.41609			−0.708046	+	0.706166i	$0.750423\pi$
$572$		0			0
$573$	5.58987			0.00975545
$574$		0			0
$575$		0			0
$576$		0			0
$577$		−	15.4977i		−	0.0268591i	−0.999910	−	0.0134295i	$-0.995725\pi$
$577$		−	15.4977i		−	0.0268591i	0.999910	−	0.0134295i	$-0.00427488\pi$
$578$		0			0
$579$	675.205			1.16616
$580$		0			0
$581$	1014.15			1.74552
$582$		0			0
$583$	55.4916			0.0951828
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	645.782i		−	1.10014i	−0.835119	−	0.550070i	$-0.814601\pi$
$587$		−	645.782i		−	1.10014i	0.835119	−	0.550070i	$-0.185399\pi$
$588$		0			0
$589$	883.520	−	209.258i	1.50003	−	0.355276i
$590$		0			0
$591$		−	722.255i		−	1.22209i
$592$		0			0
$593$		−	974.473i		−	1.64329i	−0.569997	−	0.821647i	$-0.693056\pi$
$593$		−	974.473i		−	1.64329i	0.569997	−	0.821647i	$-0.306944\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−604.193			−1.01205
$598$		0			0
$599$			853.712i			1.42523i	0.701556	+	0.712614i	$0.252489\pi$
$599$			853.712i			1.42523i	−0.701556	+	0.712614i	$0.747511\pi$
$600$		0			0
$601$			655.593i			1.09084i	0.838164	+	0.545419i	$0.183629\pi$
$601$			655.593i			1.09084i	−0.838164	+	0.545419i	$0.816371\pi$
$602$		0			0
$603$	−14.8182			−0.0245742
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−17.3522			−0.0285868			−0.0142934	−	0.999898i	$-0.504550\pi$
$607$	−17.3522			−0.0285868			−0.0142934	+	0.999898i	$0.504550\pi$
$608$		0			0
$609$	−460.550			−0.756240
$610$		0			0
$611$			1501.78i			2.45790i
$612$		0			0
$613$		−	562.093i		−	0.916955i	−0.888706	−	0.458477i	$-0.848395\pi$
$613$		−	562.093i		−	0.916955i	0.888706	−	0.458477i	$-0.151605\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			376.298i			0.609883i	0.952371	+	0.304941i	$0.0986369\pi$
$617$			376.298i			0.609883i	−0.952371	+	0.304941i	$0.901363\pi$
$618$		0			0
$619$	−402.209			−0.649772			−0.324886	−	0.945753i	$-0.605326\pi$
$619$	−402.209			−0.649772			−0.324886	+	0.945753i	$0.605326\pi$
$620$		0			0
$621$		−	157.274i		−	0.253259i
$622$		0			0
$623$	415.909			0.667591
$624$		0			0
$625$		0			0
$626$		0			0
$627$	−8.75315	−	36.9572i	−0.0139604	−	0.0589429i
$628$		0			0
$629$		−	397.905i		−	0.632599i
$630$		0			0
$631$	−1050.52			−1.66485			−0.832427	−	0.554135i	$-0.813049\pi$
$631$	−1050.52			−1.66485			−0.832427	+	0.554135i	$0.813049\pi$
$632$		0			0
$633$			340.380i			0.537725i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−1997.36			−3.13557
$638$		0			0
$639$		−	20.4393i		−	0.0319864i
$640$		0			0
$641$			1010.76i			1.57685i	0.615132	+	0.788424i	$0.289103\pi$
$641$			1010.76i			1.57685i	−0.615132	+	0.788424i	$0.710897\pi$
$642$		0			0
$643$			546.019i			0.849174i	0.905387	+	0.424587i	$0.139581\pi$
$643$			546.019i			0.849174i	−0.905387	+	0.424587i	$0.860419\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	120.783i		−	0.186682i	−0.995634	−	0.0933408i	$-0.970245\pi$
$647$		−	120.783i		−	0.186682i	0.995634	−	0.0933408i	$-0.0297546\pi$
$648$		0			0
$649$			25.9112i			0.0399249i
$650$		0			0
$651$	−1835.20			−2.81905
$652$		0			0
$653$		−	558.984i		−	0.856024i	−0.903773	−	0.428012i	$-0.859214\pi$
$653$		−	558.984i		−	0.856024i	0.903773	−	0.428012i	$-0.140786\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		−	90.9650i		−	0.138455i
$658$		0			0
$659$			932.855i			1.41556i	0.706432	+	0.707781i	$0.250304\pi$
$659$			932.855i			1.41556i	−0.706432	+	0.707781i	$0.749696\pi$
$660$		0			0
$661$		−	188.164i		−	0.284665i	−0.989819	−	0.142332i	$-0.954540\pi$
$661$		−	188.164i		−	0.284665i	0.989819	−	0.142332i	$-0.0454602\pi$
$662$		0			0
$663$			1296.98i			1.95623i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−73.8851			−0.110772
$668$		0			0
$669$	−781.214			−1.16773
$670$		0			0
$671$	−6.50647			−0.00969668
$672$		0			0
$673$	953.791			1.41722			0.708612	−	0.705599i	$-0.249322\pi$
$673$	953.791			1.41722			0.708612	+	0.705599i	$0.249322\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−1189.20			−1.75657			−0.878284	−	0.478140i	$-0.841311\pi$
$677$	−1189.20			−1.75657			−0.878284	+	0.478140i	$0.841311\pi$
$678$		0			0
$679$			807.518i			1.18928i
$680$		0			0
$681$	1140.55			1.67482
$682$		0			0
$683$	−470.293			−0.688570			−0.344285	−	0.938865i	$-0.611879\pi$
$683$	−470.293			−0.688570			−0.344285	+	0.938865i	$0.611879\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−709.920			−1.03336
$688$		0			0
$689$	1737.31			2.52149
$690$		0			0
$691$	707.606			1.02403			0.512016	−	0.858976i	$-0.328899\pi$
$691$	707.606			1.02403			0.512016	+	0.858976i	$0.328899\pi$
$692$		0			0
$693$			6.82105i			0.00984278i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−471.141			−0.675956
$698$		0			0
$699$		−	1029.38i		−	1.47265i
$700$		0			0
$701$	−10.3658			−0.0147871			−0.00739356	−	0.999973i	$-0.502353\pi$
$701$	−10.3658			−0.0147871			−0.00739356	+	0.999973i	$0.502353\pi$
$702$		0			0
$703$	84.0739	+	354.974i	0.119593	+	0.504941i
$704$		0			0
$705$		0			0
$706$		0			0
$707$			478.666i			0.677039i
$708$		0			0
$709$	−419.639			−0.591875			−0.295937	−	0.955207i	$-0.595632\pi$
$709$	−419.639			−0.591875			−0.295937	+	0.955207i	$0.595632\pi$
$710$		0			0
$711$			108.532i			0.152648i
$712$		0			0
$713$	−294.418			−0.412928
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−1073.71			−1.49750
$718$		0			0
$719$	249.933			0.347612			0.173806	−	0.984780i	$-0.444393\pi$
$719$	249.933			0.347612			0.173806	+	0.984780i	$0.444393\pi$
$720$		0			0
$721$		−	1392.25i		−	1.93100i
$722$		0			0
$723$			6.56266i			0.00907698i
$724$		0			0
$725$		0			0
$726$		0			0
$727$		−	521.188i		−	0.716902i	−0.933549	−	0.358451i	$-0.883305\pi$
$727$		−	521.188i		−	0.716902i	0.933549	−	0.358451i	$-0.116695\pi$
$728$		0			0
$729$	644.717			0.884386
$730$		0			0
$731$	−708.245			−0.968872
$732$		0			0
$733$			806.985i			1.10093i	0.834857	+	0.550467i	$0.185551\pi$
$733$			806.985i			1.10093i	−0.834857	+	0.550467i	$0.814449\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−10.7381			−0.0145700
$738$		0			0
$739$	−852.210			−1.15319			−0.576597	−	0.817029i	$-0.695620\pi$
$739$	−852.210			−1.15319			−0.576597	+	0.817029i	$0.695620\pi$
$740$		0			0
$741$	−274.040	−	1157.04i	−0.369825	−	1.56146i
$742$		0			0
$743$	−835.888			−1.12502			−0.562508	−	0.826792i	$-0.690164\pi$
$743$	−835.888			−1.12502			−0.562508	+	0.826792i	$0.690164\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		−	72.8450i		−	0.0975167i
$748$		0			0
$749$			1054.61i			1.40803i
$750$		0			0
$751$			219.641i			0.292465i	0.989250	+	0.146232i	$0.0467147\pi$
$751$			219.641i			0.292465i	−0.989250	+	0.146232i	$0.953285\pi$
$752$		0			0
$753$	−63.2852			−0.0840441
$754$		0			0
$755$		0			0
$756$		0			0
$757$			1097.01i			1.44915i	0.689195	+	0.724576i	$0.257964\pi$
$757$			1097.01i			1.44915i	−0.689195	+	0.724576i	$0.742036\pi$
$758$		0			0
$759$			12.3154i			0.0162258i
$760$		0			0
$761$	637.660			0.837924			0.418962	−	0.908004i	$-0.362394\pi$
$761$	637.660			0.837924			0.418962	+	0.908004i	$0.362394\pi$
$762$		0			0
$763$	−299.254			−0.392206
$764$		0			0
$765$		0			0
$766$		0			0
$767$			811.220i			1.05765i
$768$		0			0
$769$	799.723			1.03995			0.519976	−	0.854181i	$-0.325941\pi$
$769$	799.723			1.03995			0.519976	+	0.854181i	$0.325941\pi$
$770$		0			0
$771$	1093.14			1.41782
$772$		0			0
$773$	1229.63			1.59072			0.795359	−	0.606138i	$-0.207282\pi$
$773$	1229.63			1.59072			0.795359	+	0.606138i	$0.207282\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		−	737.333i		−	0.948948i
$778$		0			0
$779$	420.308	−	99.5482i	0.539549	−	0.127790i
$780$		0			0
$781$		−	14.8114i		−	0.0189647i
$782$		0			0
$783$		−	306.136i		−	0.390979i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	211.715			0.269016			0.134508	−	0.990913i	$-0.457055\pi$
$787$	211.715			0.269016			0.134508	+	0.990913i	$0.457055\pi$
$788$		0			0
$789$		−	1529.73i		−	1.93882i
$790$		0			0
$791$			2580.45i			3.26227i
$792$		0			0
$793$	−203.702			−0.256875
$794$		0			0
$795$		0			0
$796$		0			0
$797$	708.469			0.888920			0.444460	−	0.895799i	$-0.353396\pi$
$797$	708.469			0.888920			0.444460	+	0.895799i	$0.353396\pi$
$798$		0			0
$799$	1563.04			1.95625
$800$		0			0
$801$		−	29.8742i		−	0.0372961i
$802$		0			0
$803$		−	65.9181i		−	0.0820898i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		−	631.980i		−	0.783122i
$808$		0			0
$809$	784.699			0.969961			0.484981	−	0.874525i	$-0.338827\pi$
$809$	784.699			0.969961			0.484981	+	0.874525i	$0.338827\pi$
$810$		0			0
$811$		−	586.588i		−	0.723290i	−0.932316	−	0.361645i	$-0.882215\pi$
$811$		−	586.588i		−	0.723290i	0.932316	−	0.361645i	$-0.117785\pi$
$812$		0			0
$813$	−652.219			−0.802238
$814$		0			0
$815$		0			0
$816$		0			0
$817$	631.831	−	149.646i	0.773354	−	0.183166i
$818$		0			0
$819$			213.551i			0.260746i
$820$		0			0
$821$	1501.33			1.82866			0.914329	−	0.404972i	$-0.132719\pi$
$821$	1501.33			1.82866			0.914329	+	0.404972i	$0.132719\pi$
$822$		0			0
$823$			649.300i			0.788943i	0.918908	+	0.394472i	$0.129072\pi$
$823$			649.300i			0.788943i	−0.918908	+	0.394472i	$0.870928\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	167.715			0.202799			0.101400	−	0.994846i	$-0.467668\pi$
$827$	167.715			0.202799			0.101400	+	0.994846i	$0.467668\pi$
$828$		0			0
$829$		−	77.3323i		−	0.0932838i	−0.998912	−	0.0466419i	$-0.985148\pi$
$829$		−	77.3323i		−	0.0932838i	0.998912	−	0.0466419i	$-0.0148520\pi$
$830$		0			0
$831$			894.804i			1.07678i
$832$		0			0
$833$			2078.84i			2.49561i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		−	1219.89i		−	1.45746i
$838$		0			0
$839$		−	61.7289i		−	0.0735743i	−0.999323	−	0.0367872i	$-0.988288\pi$
$839$		−	61.7289i		−	0.0735743i	0.999323	−	0.0367872i	$-0.0117124\pi$
$840$		0			0
$841$	697.181			0.828991
$842$		0			0
$843$		−	946.100i		−	1.12230i
$844$		0			0
$845$		0			0
$846$		0			0
$847$		−	1473.58i		−	1.73976i
$848$		0			0
$849$		−	1635.98i		−	1.92695i
$850$		0			0
$851$		−	118.289i		−	0.139000i
$852$		0			0
$853$			103.960i			0.121876i	0.998142	+	0.0609381i	$0.0194092\pi$
$853$			103.960i			0.121876i	−0.998142	+	0.0609381i	$0.980591\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	733.977			0.856449			0.428225	−	0.903672i	$-0.359139\pi$
$857$	733.977			0.856449			0.428225	+	0.903672i	$0.359139\pi$
$858$		0			0
$859$	−158.143			−0.184102			−0.0920508	−	0.995754i	$-0.529342\pi$
$859$	−158.143			−0.184102			−0.0920508	+	0.995754i	$0.529342\pi$
$860$		0			0
$861$	−873.043			−1.01399
$862$		0			0
$863$	521.014			0.603725			0.301862	−	0.953352i	$-0.402392\pi$
$863$	521.014			0.603725			0.301862	+	0.953352i	$0.402392\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	441.596			0.509338
$868$		0			0
$869$			78.6483i			0.0905044i
$870$		0			0
$871$	−336.184			−0.385975
$872$		0			0
$873$	58.0029			0.0664409
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−323.145			−0.368466			−0.184233	−	0.982883i	$-0.558980\pi$
$877$	−323.145			−0.368466			−0.184233	+	0.982883i	$0.558980\pi$
$878$		0			0
$879$	134.670			0.153208
$880$		0			0
$881$	−827.030			−0.938740			−0.469370	−	0.883002i	$-0.655519\pi$
$881$	−827.030			−0.938740			−0.469370	+	0.883002i	$0.655519\pi$
$882$		0			0
$883$		−	1065.03i		−	1.20615i	−0.797685	−	0.603075i	$-0.793942\pi$
$883$		−	1065.03i		−	1.20615i	0.797685	−	0.603075i	$-0.206058\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	1471.28			1.65871			0.829356	−	0.558721i	$-0.188708\pi$
$887$	1471.28			1.65871			0.829356	+	0.558721i	$0.188708\pi$
$888$		0			0
$889$			2390.24i			2.68868i
$890$		0			0
$891$	−56.0516			−0.0629086
$892$		0			0
$893$	−1394.40	+	330.257i	−1.56148	+	0.369829i
$894$		0			0
$895$		0			0
$896$		0			0
$897$			385.564i			0.429838i
$898$		0			0
$899$	−573.089			−0.637474
$900$		0			0
$901$		−	1808.18i		−	2.00686i
$902$		0			0
$903$	−1312.41			−1.45338
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−1307.42			−1.44148			−0.720738	−	0.693207i	$-0.756197\pi$
$907$	−1307.42			−1.44148			−0.720738	+	0.693207i	$0.756197\pi$
$908$		0			0
$909$	34.3820			0.0378239
$910$		0			0
$911$			64.9642i			0.0713108i	0.999364	+	0.0356554i	$0.0113519\pi$
$911$			64.9642i			0.0713108i	−0.999364	+	0.0356554i	$0.988648\pi$
$912$		0			0
$913$		−	52.7873i		−	0.0578175i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			3024.18i			3.29790i
$918$		0			0
$919$	−315.153			−0.342930			−0.171465	−	0.985190i	$-0.554850\pi$
$919$	−315.153			−0.342930			−0.171465	+	0.985190i	$0.554850\pi$
$920$		0			0
$921$	1519.72			1.65007
$922$		0			0
$923$		−	463.710i		−	0.502394i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−100.003			−0.107878
$928$		0			0
$929$	−787.663			−0.847861			−0.423930	−	0.905695i	$-0.639350\pi$
$929$	−787.663			−0.847861			−0.423930	+	0.905695i	$0.639350\pi$
$930$		0			0
$931$	−439.242	−	1854.55i	−0.471796	−	1.99200i
$932$		0			0
$933$	514.004			0.550915
$934$		0			0
$935$		0			0
$936$		0			0
$937$		−	761.331i		−	0.812520i	−0.913758	−	0.406260i	$-0.866833\pi$
$937$		−	761.331i		−	0.812520i	0.913758	−	0.406260i	$-0.133167\pi$
$938$		0			0
$939$		−	1240.34i		−	1.32091i
$940$		0			0
$941$		−	38.1755i		−	0.0405691i	−0.999794	−	0.0202845i	$-0.993543\pi$
$941$		−	38.1755i		−	0.0405691i	0.999794	−	0.0202845i	$-0.00645721\pi$
$942$		0			0
$943$	−140.061			−0.148527
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	802.970i		−	0.847909i	−0.905684	−	0.423954i	$-0.860642\pi$
$947$		−	802.970i		−	0.847909i	0.905684	−	0.423954i	$-0.139358\pi$
$948$		0			0
$949$		−	2063.74i		−	2.17464i
$950$		0			0
$951$	−770.604			−0.810309
$952$		0			0
$953$	−606.643			−0.636562			−0.318281	−	0.947996i	$-0.603105\pi$
$953$	−606.643			−0.636562			−0.318281	+	0.947996i	$0.603105\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$			23.9720i			0.0250491i
$958$		0			0
$959$	−630.168			−0.657109
$960$		0			0
$961$	−1322.65			−1.37632
$962$		0			0
$963$	75.7514			0.0786618
$964$		0			0
$965$		0			0
$966$		0			0
$967$			1086.46i			1.12354i	0.827293	+	0.561770i	$0.189879\pi$
$967$			1086.46i			1.12354i	−0.827293	+	0.561770i	$0.810121\pi$
$968$		0			0
$969$	−1204.24	+	285.220i	−1.24277	+	0.294345i
$970$		0			0
$971$			1004.12i			1.03411i	0.855952	+	0.517055i	$0.172972\pi$
$971$			1004.12i			1.03411i	−0.855952	+	0.517055i	$0.827028\pi$
$972$		0			0
$973$		−	1398.44i		−	1.43725i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	908.123			0.929501			0.464751	−	0.885442i	$-0.346144\pi$
$977$	908.123			0.929501			0.464751	+	0.885442i	$0.346144\pi$
$978$		0			0
$979$		−	21.6484i		−	0.0221128i
$980$		0			0
$981$			21.4950i			0.0219113i
$982$		0			0
$983$	1162.31			1.18242			0.591208	−	0.806519i	$-0.298651\pi$
$983$	1162.31			1.18242			0.591208	+	0.806519i	$0.298651\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	2896.38			2.93452
$988$		0			0
$989$	−210.547			−0.212888
$990$		0			0
$991$		−	157.641i		−	0.159072i	−0.996832	−	0.0795361i	$-0.974656\pi$
$991$		−	157.641i		−	0.159072i	0.996832	−	0.0795361i	$-0.0253439\pi$
$992$		0			0
$993$		−	330.213i		−	0.332541i
$994$		0			0
$995$		0			0
$996$		0			0
$997$		−	987.181i		−	0.990152i	−0.868850	−	0.495076i	$-0.835140\pi$
$997$		−	987.181i		−	0.990152i	0.868850	−	0.495076i	$-0.164860\pi$
$998$		0			0
$999$	490.119			0.490609

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1900.3.g.c.949.7		24
5.2	odd	4		380.3.e.a.341.9	yes	12
5.3	odd	4		1900.3.e.f.1101.4		12
5.4	even	2	inner	1900.3.g.c.949.18		24
15.2	even	4		3420.3.o.a.721.7		12
19.18	odd	2	inner	1900.3.g.c.949.17		24
20.7	even	4		1520.3.h.b.721.4		12
95.18	even	4		1900.3.e.f.1101.9		12
95.37	even	4		380.3.e.a.341.4	✓	12
95.94	odd	2	inner	1900.3.g.c.949.8		24
285.227	odd	4		3420.3.o.a.721.8		12
380.227	odd	4		1520.3.h.b.721.9		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
380.3.e.a.341.4	✓	12	95.37	even	4
380.3.e.a.341.9	yes	12	5.2	odd	4
1520.3.h.b.721.4		12	20.7	even	4
1520.3.h.b.721.9		12	380.227	odd	4
1900.3.e.f.1101.4		12	5.3	odd	4
1900.3.e.f.1101.9		12	95.18	even	4
1900.3.g.c.949.7		24	1.1	even	1	trivial
1900.3.g.c.949.8		24	95.94	odd	2	inner
1900.3.g.c.949.17		24	19.18	odd	2	inner
1900.3.g.c.949.18		24	5.4	even	2	inner
3420.3.o.a.721.7		12	15.2	even	4
3420.3.o.a.721.8		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)

\(f(q)\)	\(=\)	\(q-3.14288 q^{3} +12.2192i q^{7} +0.877687 q^{9} +O(q^{10})\)	\(q-3.14288 q^{3} +12.2192i q^{7} +0.877687 q^{9} +0.636018 q^{11} +19.9122 q^{13} -20.7245i q^{17} +(4.37892 + 18.4885i) q^{19} -38.4034i q^{21} -6.16098i q^{23} +25.5274 q^{27} -11.9924i q^{29} -47.7875i q^{31} -1.99893 q^{33} +19.1997 q^{37} -62.5817 q^{39} -22.7335i q^{41} -34.1742i q^{43} +75.4198i q^{47} -100.308 q^{49} +65.1347i q^{51} +87.2484 q^{53} +(-13.7624 - 58.1071i) q^{57} +40.7398i q^{59} -10.2300 q^{61} +10.7246i q^{63} -16.8833 q^{67} +19.3632i q^{69} -23.2877i q^{71} -103.642i q^{73} +7.77162i q^{77} +123.657i q^{79} -88.1288 q^{81} -82.9966i q^{83} +37.6908i q^{87} -34.0374i q^{89} +243.311i q^{91} +150.190i q^{93} +66.0861 q^{97} +0.558225 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 32 q^{9}+O(q^{10}) \) 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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