LMFDB - Embedded newform 380.3.g.a.189.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [380,3,Mod(189,380)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(380, 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("380.189");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 380 = 2^{2} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	380.g (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:	$10.3542500457$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-6}, \sqrt{14})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 4x^{2} + 25 $ x^4 - 4*x^2 + 25
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			189.1
Root			$1.87083 - 1.22474i$ of defining polynomial
Character	$\chi$	$=$	380.189
Dual form			380.3.g.a.189.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.74166 q^{3} -5.00000 q^{5} -9.79796i q^{7} +5.00000 q^{9} +O(q^{10})$	$q-3.74166 q^{3} -5.00000 q^{5} -9.79796i q^{7} +5.00000 q^{9} +4.00000 q^{11} -11.2250 q^{13} +18.7083 q^{15} -19.5959i q^{17} +(-5.00000 + 18.3303i) q^{19} +36.6606i q^{21} +9.79796i q^{23} +25.0000 q^{25} +14.9666 q^{27} +36.6606i q^{29} +36.6606i q^{31} -14.9666 q^{33} +48.9898i q^{35} -33.6749 q^{37} +42.0000 q^{39} -36.6606i q^{41} +68.5857i q^{43} -25.0000 q^{45} +9.79796i q^{47} -47.0000 q^{49} +73.3212i q^{51} +56.1249 q^{53} -20.0000 q^{55} +(18.7083 - 68.5857i) q^{57} -73.3212i q^{59} +100.000 q^{61} -48.9898i q^{63} +56.1249 q^{65} -11.2250 q^{67} -36.6606i q^{69} +36.6606i q^{71} +19.5959i q^{73} -93.5414 q^{75} -39.1918i q^{77} +109.982i q^{79} -101.000 q^{81} +29.3939i q^{83} +97.9796i q^{85} -137.171i q^{87} -146.642i q^{89} +109.982i q^{91} -137.171i q^{93} +(25.0000 - 91.6515i) q^{95} -123.475 q^{97} +20.0000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 20 q^{5} + 20 q^{9}+O(q^{10}) $ 4 * q - 20 * q^5 + 20 * q^9	$ 4 q - 20 q^{5} + 20 q^{9} + 16 q^{11} - 20 q^{19} + 100 q^{25} + 168 q^{39} - 100 q^{45} - 188 q^{49} - 80 q^{55} + 400 q^{61} - 404 q^{81} + 100 q^{95} + 80 q^{99}+O(q^{100}) $ 4 * q - 20 * q^5 + 20 * q^9 + 16 * q^11 - 20 * q^19 + 100 * q^25 + 168 * q^39 - 100 * q^45 - 188 * q^49 - 80 * q^55 + 400 * q^61 - 404 * q^81 + 100 * q^95 + 80 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/380\mathbb{Z}\right)^\times$.

$n$	$21$	$77$	$191$
$\chi(n)$	$-1$	$-1$	$1$

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
$3$	−3.74166			−1.24722			−0.623610	−	0.781736i	$-0.714334\pi$
$3$	−3.74166			−1.24722			−0.623610	+	0.781736i	$0.714334\pi$
$4$		0			0
$5$	−5.00000			−1.00000
$5$	−5.00000			−1.00000
$6$		0			0
$7$		−	9.79796i		−	1.39971i	−0.714286	−	0.699854i	$-0.753248\pi$
$7$		−	9.79796i		−	1.39971i	0.714286	−	0.699854i	$-0.246752\pi$
$8$		0			0
$9$	5.00000			0.555556
$10$		0			0
$11$	4.00000			0.363636			0.181818	−	0.983332i	$-0.441802\pi$
$11$	4.00000			0.363636			0.181818	+	0.983332i	$0.441802\pi$
$12$		0			0
$13$	−11.2250			−0.863459			−0.431730	−	0.902003i	$-0.642097\pi$
$13$	−11.2250			−0.863459			−0.431730	+	0.902003i	$0.642097\pi$
$14$		0			0
$15$	18.7083			1.24722
$16$		0			0
$17$		−	19.5959i		−	1.15270i	−0.817203	−	0.576351i	$-0.804476\pi$
$17$		−	19.5959i		−	1.15270i	0.817203	−	0.576351i	$-0.195524\pi$
$18$		0			0
$19$	−5.00000	+	18.3303i	−0.263158	+	0.964753i
$19$	−5.00000	+	18.3303i	−0.263158	+	0.964753i
$20$		0			0
$21$			36.6606i			1.74574i
$22$		0			0
$23$			9.79796i			0.425998i	0.977052	+	0.212999i	$0.0683231\pi$
$23$			9.79796i			0.425998i	−0.977052	+	0.212999i	$0.931677\pi$
$24$		0			0
$25$	25.0000			1.00000
$26$		0			0
$27$	14.9666			0.554320
$28$		0			0
$29$			36.6606i			1.26416i	0.774904	+	0.632079i	$0.217798\pi$
$29$			36.6606i			1.26416i	−0.774904	+	0.632079i	$0.782202\pi$
$30$		0			0
$31$			36.6606i			1.18260i	0.806452	+	0.591300i	$0.201385\pi$
$31$			36.6606i			1.18260i	−0.806452	+	0.591300i	$0.798615\pi$
$32$		0			0
$33$	−14.9666			−0.453534
$34$		0			0
$35$			48.9898i			1.39971i
$36$		0			0
$37$	−33.6749			−0.910133			−0.455066	−	0.890457i	$-0.650384\pi$
$37$	−33.6749			−0.910133			−0.455066	+	0.890457i	$0.650384\pi$
$38$		0			0
$39$	42.0000			1.07692
$40$		0			0
$41$		−	36.6606i		−	0.894161i	−0.894494	−	0.447081i	$-0.852464\pi$
$41$		−	36.6606i		−	0.894161i	0.894494	−	0.447081i	$-0.147536\pi$
$42$		0			0
$43$			68.5857i			1.59502i	0.603308	+	0.797508i	$0.293849\pi$
$43$			68.5857i			1.59502i	−0.603308	+	0.797508i	$0.706151\pi$
$44$		0			0
$45$	−25.0000			−0.555556
$46$		0			0
$47$			9.79796i			0.208467i	0.994553	+	0.104234i	$0.0332390\pi$
$47$			9.79796i			0.208467i	−0.994553	+	0.104234i	$0.966761\pi$
$48$		0			0
$49$	−47.0000			−0.959184
$50$		0			0
$51$			73.3212i			1.43767i
$52$		0			0
$53$	56.1249			1.05896			0.529480	−	0.848323i	$-0.322387\pi$
$53$	56.1249			1.05896			0.529480	+	0.848323i	$0.322387\pi$
$54$		0			0
$55$	−20.0000			−0.363636
$56$		0			0
$57$	18.7083	−	68.5857i	0.328216	−	1.20326i
$58$		0			0
$59$		−	73.3212i		−	1.24273i	−0.783520	−	0.621366i	$-0.786578\pi$
$59$		−	73.3212i		−	1.24273i	0.783520	−	0.621366i	$-0.213422\pi$
$60$		0			0
$61$	100.000			1.63934			0.819672	−	0.572833i	$-0.194156\pi$
$61$	100.000			1.63934			0.819672	+	0.572833i	$0.194156\pi$
$62$		0			0
$63$		−	48.9898i		−	0.777616i
$64$		0			0
$65$	56.1249			0.863459
$66$		0			0
$67$	−11.2250			−0.167537			−0.0837684	−	0.996485i	$-0.526696\pi$
$67$	−11.2250			−0.167537			−0.0837684	+	0.996485i	$0.526696\pi$
$68$		0			0
$69$		−	36.6606i		−	0.531313i
$70$		0			0
$71$			36.6606i			0.516347i	0.966099	+	0.258173i	$0.0831205\pi$
$71$			36.6606i			0.516347i	−0.966099	+	0.258173i	$0.916879\pi$
$72$		0			0
$73$			19.5959i			0.268437i	0.990952	+	0.134219i	$0.0428524\pi$
$73$			19.5959i			0.268437i	−0.990952	+	0.134219i	$0.957148\pi$
$74$		0			0
$75$	−93.5414			−1.24722
$76$		0			0
$77$		−	39.1918i		−	0.508985i
$78$		0			0
$79$			109.982i			1.39217i	0.717957	+	0.696087i	$0.245077\pi$
$79$			109.982i			1.39217i	−0.717957	+	0.696087i	$0.754923\pi$
$80$		0			0
$81$	−101.000			−1.24691
$82$		0			0
$83$			29.3939i			0.354143i	0.984198	+	0.177072i	$0.0566624\pi$
$83$			29.3939i			0.354143i	−0.984198	+	0.177072i	$0.943338\pi$
$84$		0			0
$85$			97.9796i			1.15270i
$86$		0			0
$87$		−	137.171i		−	1.57668i
$88$		0			0
$89$		−	146.642i		−	1.64767i	−0.566831	−	0.823834i	$-0.691831\pi$
$89$		−	146.642i		−	1.64767i	0.566831	−	0.823834i	$-0.308169\pi$
$90$		0			0
$91$			109.982i			1.20859i
$92$		0			0
$93$		−	137.171i		−	1.47496i
$94$		0			0
$95$	25.0000	−	91.6515i	0.263158	−	0.964753i
$96$		0			0
$97$	−123.475			−1.27293			−0.636467	−	0.771304i	$-0.719605\pi$
$97$	−123.475			−1.27293			−0.636467	+	0.771304i	$0.719605\pi$
$98$		0			0
$99$	20.0000			0.202020
$100$		0			0
$101$	4.00000			0.0396040			0.0198020	−	0.999804i	$-0.493696\pi$
$101$	4.00000			0.0396040			0.0198020	+	0.999804i	$0.493696\pi$
$102$		0			0
$103$	78.5748			0.762862			0.381431	−	0.924397i	$-0.375431\pi$
$103$	78.5748			0.762862			0.381431	+	0.924397i	$0.375431\pi$
$104$		0			0
$105$		−	183.303i		−	1.74574i
$106$		0			0
$107$	−101.025			−0.944157			−0.472078	−	0.881557i	$-0.656496\pi$
$107$	−101.025			−0.944157			−0.472078	+	0.881557i	$0.656496\pi$
$108$		0			0
$109$		−	73.3212i		−	0.672672i	−0.941742	−	0.336336i	$-0.890812\pi$
$109$		−	73.3212i		−	0.672672i	0.941742	−	0.336336i	$-0.109188\pi$
$110$		0			0
$111$	126.000			1.13514
$112$		0			0
$113$	−11.2250			−0.0993360			−0.0496680	−	0.998766i	$-0.515816\pi$
$113$	−11.2250			−0.0993360			−0.0496680	+	0.998766i	$0.515816\pi$
$114$		0			0
$115$		−	48.9898i		−	0.425998i
$116$		0			0
$117$	−56.1249			−0.479700
$118$		0			0
$119$	−192.000			−1.61345
$120$		0			0
$121$	−105.000			−0.867769
$122$		0			0
$123$			137.171i			1.11521i
$124$		0			0
$125$	−125.000			−1.00000
$126$		0			0
$127$	235.724			1.85610			0.928049	−	0.372458i	$-0.121485\pi$
$127$	235.724			1.85610			0.928049	+	0.372458i	$0.121485\pi$
$128$		0			0
$129$		−	256.624i		−	1.98934i
$130$		0			0
$131$	26.0000			0.198473			0.0992366	−	0.995064i	$-0.468360\pi$
$131$	26.0000			0.198473			0.0992366	+	0.995064i	$0.468360\pi$
$132$		0			0
$133$	179.600	+	48.9898i	1.35037	+	0.368344i
$134$		0			0
$135$	−74.8331			−0.554320
$136$		0			0
$137$			254.747i			1.85947i	0.368234	+	0.929733i	$0.379963\pi$
$137$			254.747i			1.85947i	−0.368234	+	0.929733i	$0.620037\pi$
$138$		0			0
$139$	148.000			1.06475			0.532374	−	0.846509i	$-0.321300\pi$
$139$	148.000			1.06475			0.532374	+	0.846509i	$0.321300\pi$
$140$		0			0
$141$		−	36.6606i		−	0.260004i
$142$		0			0
$143$	−44.8999			−0.313985
$144$		0			0
$145$		−	183.303i		−	1.26416i
$146$		0			0
$147$	175.858			1.19631
$148$		0			0
$149$	52.0000			0.348993			0.174497	−	0.984658i	$-0.444170\pi$
$149$	52.0000			0.348993			0.174497	+	0.984658i	$0.444170\pi$
$150$		0			0
$151$			73.3212i			0.485571i	0.970080	+	0.242785i	$0.0780611\pi$
$151$			73.3212i			0.485571i	−0.970080	+	0.242785i	$0.921939\pi$
$152$		0			0
$153$		−	97.9796i		−	0.640389i
$154$		0			0
$155$		−	183.303i		−	1.18260i
$156$		0			0
$157$			117.576i			0.748889i	0.927249	+	0.374444i	$0.122167\pi$
$157$			117.576i			0.748889i	−0.927249	+	0.374444i	$0.877833\pi$
$158$		0			0
$159$	−210.000			−1.32075
$160$		0			0
$161$	96.0000			0.596273
$162$		0			0
$163$		−	9.79796i		−	0.0601102i	−0.999548	−	0.0300551i	$-0.990432\pi$
$163$		−	9.79796i		−	0.0601102i	0.999548	−	0.0300551i	$-0.00956827\pi$
$164$		0			0
$165$	74.8331			0.453534
$166$		0			0
$167$	−33.6749			−0.201646			−0.100823	−	0.994904i	$-0.532148\pi$
$167$	−33.6749			−0.201646			−0.100823	+	0.994904i	$0.532148\pi$
$168$		0			0
$169$	−43.0000			−0.254438
$170$		0			0
$171$	−25.0000	+	91.6515i	−0.146199	+	0.535974i
$172$		0			0
$173$	−280.624			−1.62211			−0.811053	−	0.584973i	$-0.801105\pi$
$173$	−280.624			−1.62211			−0.811053	+	0.584973i	$0.801105\pi$
$174$		0			0
$175$		−	244.949i		−	1.39971i
$176$		0			0
$177$			274.343i			1.54996i
$178$		0			0
$179$			146.642i			0.819231i	0.912258	+	0.409616i	$0.134337\pi$
$179$			146.642i			0.819231i	−0.912258	+	0.409616i	$0.865663\pi$
$180$		0			0
$181$			256.624i			1.41781i	0.705302	+	0.708907i	$0.250811\pi$
$181$			256.624i			1.41781i	−0.705302	+	0.708907i	$0.749189\pi$
$182$		0			0
$183$	−374.166			−2.04462
$184$		0			0
$185$	168.375			0.910133
$186$		0			0
$187$		−	78.3837i		−	0.419164i
$188$		0			0
$189$		−	146.642i		−	0.775886i
$190$		0			0
$191$	158.000			0.827225			0.413613	−	0.910453i	$-0.364267\pi$
$191$	158.000			0.827225			0.413613	+	0.910453i	$0.364267\pi$
$192$		0			0
$193$	−213.274			−1.10505			−0.552525	−	0.833497i	$-0.686335\pi$
$193$	−213.274			−1.10505			−0.552525	+	0.833497i	$0.686335\pi$
$194$		0			0
$195$	−210.000			−1.07692
$196$		0			0
$197$			176.363i			0.895245i	0.894223	+	0.447622i	$0.147729\pi$
$197$			176.363i			0.895245i	−0.894223	+	0.447622i	$0.852271\pi$
$198$		0			0
$199$	302.000			1.51759			0.758794	−	0.651331i	$-0.225789\pi$
$199$	302.000			1.51759			0.758794	+	0.651331i	$0.225789\pi$
$200$		0			0
$201$	42.0000			0.208955
$202$		0			0
$203$	359.199			1.76945
$204$		0			0
$205$			183.303i			0.894161i
$206$		0			0
$207$			48.9898i			0.236666i
$208$		0			0
$209$	−20.0000	+	73.3212i	−0.0956938	+	0.350819i
$210$		0			0
$211$			183.303i			0.868735i	0.900736	+	0.434367i	$0.143028\pi$
$211$			183.303i			0.868735i	−0.900736	+	0.434367i	$0.856972\pi$
$212$		0			0
$213$		−	137.171i		−	0.643997i
$214$		0			0
$215$		−	342.929i		−	1.59502i
$216$		0			0
$217$	359.199			1.65530
$218$		0			0
$219$		−	73.3212i		−	0.334800i
$220$		0			0
$221$			219.964i			0.995311i
$222$		0			0
$223$	−213.274			−0.956388			−0.478194	−	0.878254i	$-0.658708\pi$
$223$	−213.274			−0.956388			−0.478194	+	0.878254i	$0.658708\pi$
$224$		0			0
$225$	125.000			0.555556
$226$		0			0
$227$	−280.624			−1.23623			−0.618115	−	0.786088i	$-0.712103\pi$
$227$	−280.624			−1.23623			−0.618115	+	0.786088i	$0.712103\pi$
$228$		0			0
$229$	−236.000			−1.03057			−0.515284	−	0.857020i	$-0.672313\pi$
$229$	−236.000			−1.03057			−0.515284	+	0.857020i	$0.672313\pi$
$230$		0			0
$231$			146.642i			0.634816i
$232$		0			0
$233$		−	215.555i		−	0.925129i	−0.886586	−	0.462565i	$-0.846929\pi$
$233$		−	215.555i		−	0.925129i	0.886586	−	0.462565i	$-0.153071\pi$
$234$		0			0
$235$		−	48.9898i		−	0.208467i
$236$		0			0
$237$		−	411.514i		−	1.73635i
$238$		0			0
$239$	146.000			0.610879			0.305439	−	0.952212i	$-0.401197\pi$
$239$	146.000			0.610879			0.305439	+	0.952212i	$0.401197\pi$
$240$		0			0
$241$			183.303i			0.760593i	0.924865	+	0.380297i	$0.124178\pi$
$241$			183.303i			0.760593i	−0.924865	+	0.380297i	$0.875822\pi$
$242$		0			0
$243$	243.208			1.00085
$244$		0			0
$245$	235.000			0.959184
$246$		0			0
$247$	56.1249	−	205.757i	0.227226	−	0.833025i
$248$		0			0
$249$		−	109.982i		−	0.441694i
$250$		0			0
$251$	−250.000			−0.996016			−0.498008	−	0.867172i	$-0.665935\pi$
$251$	−250.000			−0.996016			−0.498008	+	0.867172i	$0.665935\pi$
$252$		0			0
$253$			39.1918i			0.154908i
$254$		0			0
$255$		−	366.606i		−	1.43767i
$256$		0			0
$257$	145.925			0.567800			0.283900	−	0.958854i	$-0.408372\pi$
$257$	145.925			0.567800			0.283900	+	0.958854i	$0.408372\pi$
$258$		0			0
$259$			329.945i			1.27392i
$260$		0			0
$261$			183.303i			0.702310i
$262$		0			0
$263$			88.1816i			0.335291i	0.985847	+	0.167646i	$0.0536165\pi$
$263$			88.1816i			0.335291i	−0.985847	+	0.167646i	$0.946384\pi$
$264$		0			0
$265$	−280.624			−1.05896
$266$		0			0
$267$			548.686i			2.05500i
$268$		0			0
$269$		−	73.3212i		−	0.272570i	−0.990670	−	0.136285i	$-0.956484\pi$
$269$		−	73.3212i		−	0.272570i	0.990670	−	0.136285i	$-0.0435162\pi$
$270$		0			0
$271$	−68.0000			−0.250923			−0.125461	−	0.992099i	$-0.540041\pi$
$271$	−68.0000			−0.250923			−0.125461	+	0.992099i	$0.540041\pi$
$272$		0			0
$273$		−	411.514i		−	1.50738i
$274$		0			0
$275$	100.000			0.363636
$276$		0			0
$277$			156.767i			0.565947i	0.959128	+	0.282974i	$0.0913209\pi$
$277$			156.767i			0.565947i	−0.959128	+	0.282974i	$0.908679\pi$
$278$		0			0
$279$			183.303i			0.657000i
$280$		0			0
$281$			109.982i			0.391394i	0.980664	+	0.195697i	$0.0626970\pi$
$281$			109.982i			0.391394i	−0.980664	+	0.195697i	$0.937303\pi$
$282$		0			0
$283$			480.100i			1.69647i	0.529623	+	0.848233i	$0.322333\pi$
$283$			480.100i			1.69647i	−0.529623	+	0.848233i	$0.677667\pi$
$284$		0			0
$285$	−93.5414	+	342.929i	−0.328216	+	1.20326i
$286$		0			0
$287$	−359.199			−1.25156
$288$		0			0
$289$	−95.0000			−0.328720
$290$		0			0
$291$	462.000			1.58763
$292$		0			0
$293$	505.124			1.72397			0.861986	−	0.506932i	$-0.169221\pi$
$293$	505.124			1.72397			0.861986	+	0.506932i	$0.169221\pi$
$294$		0			0
$295$			366.606i			1.24273i
$296$		0			0
$297$	59.8665			0.201571
$298$		0			0
$299$		−	109.982i		−	0.367832i
$300$		0			0
$301$	672.000			2.23256
$302$		0			0
$303$	−14.9666			−0.0493948
$304$		0			0
$305$	−500.000			−1.63934
$306$		0			0
$307$	−101.025			−0.329071			−0.164535	−	0.986371i	$-0.552612\pi$
$307$	−101.025			−0.329071			−0.164535	+	0.986371i	$0.552612\pi$
$308$		0			0
$309$	−294.000			−0.951456
$310$		0			0
$311$	−500.000			−1.60772			−0.803859	−	0.594821i	$-0.797223\pi$
$311$	−500.000			−1.60772			−0.803859	+	0.594821i	$0.797223\pi$
$312$		0			0
$313$		−	215.555i		−	0.688674i	−0.938846	−	0.344337i	$-0.888104\pi$
$313$		−	215.555i		−	0.688674i	0.938846	−	0.344337i	$-0.111896\pi$
$314$		0			0
$315$			244.949i			0.777616i
$316$		0			0
$317$	−280.624			−0.885250			−0.442625	−	0.896707i	$-0.645953\pi$
$317$	−280.624			−0.885250			−0.442625	+	0.896707i	$0.645953\pi$
$318$		0			0
$319$			146.642i			0.459694i
$320$		0			0
$321$	378.000			1.17757
$322$		0			0
$323$	359.199	+	97.9796i	1.11207	+	0.303342i
$324$		0			0
$325$	−280.624			−0.863459
$326$		0			0
$327$			274.343i			0.838969i
$328$		0			0
$329$	96.0000			0.291793
$330$		0			0
$331$			403.267i			1.21833i	0.793044	+	0.609164i	$0.208495\pi$
$331$			403.267i			1.21833i	−0.793044	+	0.609164i	$0.791505\pi$
$332$		0			0
$333$	−168.375			−0.505629
$334$		0			0
$335$	56.1249			0.167537
$336$		0			0
$337$	−101.025			−0.299777			−0.149888	−	0.988703i	$-0.547891\pi$
$337$	−101.025			−0.299777			−0.149888	+	0.988703i	$0.547891\pi$
$338$		0			0
$339$	42.0000			0.123894
$340$		0			0
$341$			146.642i			0.430036i
$342$		0			0
$343$		−	19.5959i		−	0.0571310i
$344$		0			0
$345$			183.303i			0.531313i
$346$		0			0
$347$		−	617.271i		−	1.77888i	−0.457052	−	0.889440i	$-0.651095\pi$
$347$		−	617.271i		−	1.77888i	0.457052	−	0.889440i	$-0.348905\pi$
$348$		0			0
$349$	230.000			0.659026			0.329513	−	0.944151i	$-0.393115\pi$
$349$	230.000			0.659026			0.329513	+	0.944151i	$0.393115\pi$
$350$		0			0
$351$	−168.000			−0.478632
$352$		0			0
$353$			548.686i			1.55435i	0.629284	+	0.777175i	$0.283348\pi$
$353$			548.686i			1.55435i	−0.629284	+	0.777175i	$0.716652\pi$
$354$		0			0
$355$		−	183.303i		−	0.516347i
$356$		0			0
$357$	718.398			2.01232
$358$		0			0
$359$	604.000			1.68245			0.841226	−	0.540684i	$-0.181835\pi$
$359$	604.000			1.68245			0.841226	+	0.540684i	$0.181835\pi$
$360$		0			0
$361$	−311.000	−	183.303i	−0.861496	−	0.507765i
$362$		0			0
$363$	392.874			1.08230
$364$		0			0
$365$		−	97.9796i		−	0.268437i
$366$		0			0
$367$			342.929i			0.934410i	0.884149	+	0.467205i	$0.154739\pi$
$367$			342.929i			0.934410i	−0.884149	+	0.467205i	$0.845261\pi$
$368$		0			0
$369$		−	183.303i		−	0.496756i
$370$		0			0
$371$		−	549.909i		−	1.48223i
$372$		0			0
$373$	−123.475			−0.331031			−0.165516	−	0.986207i	$-0.552929\pi$
$373$	−123.475			−0.331031			−0.165516	+	0.986207i	$0.552929\pi$
$374$		0			0
$375$	467.707			1.24722
$376$		0			0
$377$		−	411.514i		−	1.09155i
$378$		0			0
$379$			329.945i			0.870568i	0.900293	+	0.435284i	$0.143352\pi$
$379$			329.945i			0.870568i	−0.900293	+	0.435284i	$0.856648\pi$
$380$		0			0
$381$	−882.000			−2.31496
$382$		0			0
$383$	−11.2250			−0.0293080			−0.0146540	−	0.999893i	$-0.504665\pi$
$383$	−11.2250			−0.0293080			−0.0146540	+	0.999893i	$0.504665\pi$
$384$		0			0
$385$			195.959i			0.508985i
$386$		0			0
$387$			342.929i			0.886120i
$388$		0			0
$389$	122.000			0.313625			0.156812	−	0.987628i	$-0.449878\pi$
$389$	122.000			0.313625			0.156812	+	0.987628i	$0.449878\pi$
$390$		0			0
$391$	192.000			0.491049
$392$		0			0
$393$	−97.2831			−0.247540
$394$		0			0
$395$		−	549.909i		−	1.39217i
$396$		0			0
$397$		−	470.302i		−	1.18464i	−0.805703	−	0.592320i	$-0.798212\pi$
$397$		−	470.302i		−	1.18464i	0.805703	−	0.592320i	$-0.201788\pi$
$398$		0			0
$399$	−672.000	−	183.303i	−1.68421	−	0.459406i
$400$		0			0
$401$		−	219.964i		−	0.548538i	−0.961653	−	0.274269i	$-0.911564\pi$
$401$		−	219.964i		−	0.548538i	0.961653	−	0.274269i	$-0.0884358\pi$
$402$		0			0
$403$		−	411.514i		−	1.02113i
$404$		0			0
$405$	505.000			1.24691
$406$		0			0
$407$	−134.700			−0.330957
$408$		0			0
$409$		−	329.945i		−	0.806713i	−0.915043	−	0.403356i	$-0.867844\pi$
$409$		−	329.945i		−	0.806713i	0.915043	−	0.403356i	$-0.132156\pi$
$410$		0			0
$411$		−	953.176i		−	2.31916i
$412$		0			0
$413$	−718.398			−1.73946
$414$		0			0
$415$		−	146.969i		−	0.354143i
$416$		0			0
$417$	−553.765			−1.32797
$418$		0			0
$419$	−358.000			−0.854415			−0.427208	−	0.904154i	$-0.640503\pi$
$419$	−358.000			−0.854415			−0.427208	+	0.904154i	$0.640503\pi$
$420$		0			0
$421$			73.3212i			0.174160i	0.996201	+	0.0870798i	$0.0277535\pi$
$421$			73.3212i			0.174160i	−0.996201	+	0.0870798i	$0.972246\pi$
$422$		0			0
$423$			48.9898i			0.115815i
$424$		0			0
$425$		−	489.898i		−	1.15270i
$426$		0			0
$427$		−	979.796i		−	2.29460i
$428$		0			0
$429$	168.000			0.391608
$430$		0			0
$431$		−	733.212i		−	1.70119i	−0.525823	−	0.850594i	$-0.676243\pi$
$431$		−	733.212i		−	1.70119i	0.525823	−	0.850594i	$-0.323757\pi$
$432$		0			0
$433$	−190.825			−0.440703			−0.220352	−	0.975420i	$-0.570720\pi$
$433$	−190.825			−0.440703			−0.220352	+	0.975420i	$0.570720\pi$
$434$		0			0
$435$			685.857i			1.57668i
$436$		0			0
$437$	−179.600	−	48.9898i	−0.410983	−	0.112105i
$438$		0			0
$439$			183.303i			0.417547i	0.977964	+	0.208773i	$0.0669471\pi$
$439$			183.303i			0.417547i	−0.977964	+	0.208773i	$0.933053\pi$
$440$		0			0
$441$	−235.000			−0.532880
$442$		0			0
$443$		−	264.545i		−	0.597167i	−0.954384	−	0.298583i	$-0.903486\pi$
$443$		−	264.545i		−	0.597167i	0.954384	−	0.298583i	$-0.0965141\pi$
$444$		0			0
$445$			733.212i			1.64767i
$446$		0			0
$447$	−194.566			−0.435271
$448$		0			0
$449$		−	476.588i		−	1.06144i	−0.847546	−	0.530721i	$-0.821921\pi$
$449$		−	476.588i		−	1.06144i	0.847546	−	0.530721i	$-0.178079\pi$
$450$		0			0
$451$		−	146.642i		−	0.325149i
$452$		0			0
$453$		−	274.343i		−	0.605613i
$454$		0			0
$455$		−	549.909i		−	1.20859i
$456$		0			0
$457$		−	470.302i		−	1.02911i	−0.857458	−	0.514554i	$-0.827958\pi$
$457$		−	470.302i		−	1.02911i	0.857458	−	0.514554i	$-0.172042\pi$
$458$		0			0
$459$		−	293.285i		−	0.638965i
$460$		0			0
$461$	−22.0000			−0.0477223			−0.0238612	−	0.999715i	$-0.507596\pi$
$461$	−22.0000			−0.0477223			−0.0238612	+	0.999715i	$0.507596\pi$
$462$		0			0
$463$			578.080i			1.24855i	0.781204	+	0.624276i	$0.214606\pi$
$463$			578.080i			1.24855i	−0.781204	+	0.624276i	$0.785394\pi$
$464$		0			0
$465$			685.857i			1.47496i
$466$		0			0
$467$		−	421.312i		−	0.902168i	−0.892482	−	0.451084i	$-0.851038\pi$
$467$		−	421.312i		−	0.902168i	0.892482	−	0.451084i	$-0.148962\pi$
$468$		0			0
$469$			109.982i			0.234503i
$470$		0			0
$471$		−	439.927i		−	0.934028i
$472$		0			0
$473$			274.343i			0.580006i
$474$		0			0
$475$	−125.000	+	458.258i	−0.263158	+	0.964753i
$476$		0			0
$477$	280.624			0.588311
$478$		0			0
$479$	−500.000			−1.04384			−0.521921	−	0.852994i	$-0.674784\pi$
$479$	−500.000			−1.04384			−0.521921	+	0.852994i	$0.674784\pi$
$480$		0			0
$481$	378.000			0.785863
$482$		0			0
$483$	−359.199			−0.743683
$484$		0			0
$485$	617.373			1.27293
$486$		0			0
$487$	−11.2250			−0.0230492			−0.0115246	−	0.999934i	$-0.503668\pi$
$487$	−11.2250			−0.0230492			−0.0115246	+	0.999934i	$0.503668\pi$
$488$		0			0
$489$			36.6606i			0.0749706i
$490$		0			0
$491$	230.000			0.468432			0.234216	−	0.972185i	$-0.424748\pi$
$491$	230.000			0.468432			0.234216	+	0.972185i	$0.424748\pi$
$492$		0			0
$493$	718.398			1.45720
$494$		0			0
$495$	−100.000			−0.202020
$496$		0			0
$497$	359.199			0.722735
$498$		0			0
$499$	−620.000			−1.24248			−0.621242	−	0.783618i	$-0.713372\pi$
$499$	−620.000			−1.24248			−0.621242	+	0.783618i	$0.713372\pi$
$500$		0			0
$501$	126.000			0.251497
$502$		0			0
$503$			656.463i			1.30510i	0.757747	+	0.652548i	$0.226300\pi$
$503$			656.463i			1.30510i	−0.757747	+	0.652548i	$0.773700\pi$
$504$		0			0
$505$	−20.0000			−0.0396040
$506$		0			0
$507$	160.891			0.317340
$508$		0			0
$509$			769.873i			1.51252i	0.654271	+	0.756260i	$0.272976\pi$
$509$			769.873i			1.51252i	−0.654271	+	0.756260i	$0.727024\pi$
$510$		0			0
$511$	192.000			0.375734
$512$		0			0
$513$	−74.8331	+	274.343i	−0.145874	+	0.534781i
$514$		0			0
$515$	−392.874			−0.762862
$516$		0			0
$517$			39.1918i			0.0758063i
$518$		0			0
$519$	1050.00			2.02312
$520$		0			0
$521$		−	659.891i		−	1.26659i	−0.773912	−	0.633293i	$-0.781703\pi$
$521$		−	659.891i		−	1.26659i	0.773912	−	0.633293i	$-0.218297\pi$
$522$		0			0
$523$	−841.873			−1.60970			−0.804850	−	0.593479i	$-0.797754\pi$
$523$	−841.873			−1.60970			−0.804850	+	0.593479i	$0.797754\pi$
$524$		0			0
$525$			916.515i			1.74574i
$526$		0			0
$527$	718.398			1.36318
$528$		0			0
$529$	433.000			0.818526
$530$		0			0
$531$		−	366.606i		−	0.690407i
$532$		0			0
$533$			411.514i			0.772072i
$534$		0			0
$535$	505.124			0.944157
$536$		0			0
$537$		−	548.686i		−	1.02176i
$538$		0			0
$539$	−188.000			−0.348794
$540$		0			0
$541$	−620.000			−1.14603			−0.573013	−	0.819546i	$-0.694226\pi$
$541$	−620.000			−1.14603			−0.573013	+	0.819546i	$0.694226\pi$
$542$		0			0
$543$		−	960.200i		−	1.76832i
$544$		0			0
$545$			366.606i			0.672672i
$546$		0			0
$547$	−931.673			−1.70324			−0.851620	−	0.524159i	$-0.824380\pi$
$547$	−931.673			−1.70324			−0.851620	+	0.524159i	$0.824380\pi$
$548$		0			0
$549$	500.000			0.910747
$550$		0			0
$551$	−672.000	−	183.303i	−1.21960	−	0.332673i
$552$		0			0
$553$	1077.60			1.94864
$554$		0			0
$555$	−630.000			−1.13514
$556$		0			0
$557$			548.686i			0.985073i	0.870292	+	0.492537i	$0.163930\pi$
$557$			548.686i			0.985073i	−0.870292	+	0.492537i	$0.836070\pi$
$558$		0			0
$559$		−	769.873i		−	1.37723i
$560$		0			0
$561$			293.285i			0.522789i
$562$		0			0
$563$	325.524			0.578196			0.289098	−	0.957300i	$-0.406645\pi$
$563$	325.524			0.578196			0.289098	+	0.957300i	$0.406645\pi$
$564$		0			0
$565$	56.1249			0.0993360
$566$		0			0
$567$			989.594i			1.74532i
$568$		0			0
$569$		−	36.6606i		−	0.0644299i	−0.999481	−	0.0322149i	$-0.989744\pi$
$569$		−	36.6606i		−	0.0644299i	0.999481	−	0.0322149i	$-0.0102561\pi$
$570$		0			0
$571$	−332.000			−0.581436			−0.290718	−	0.956809i	$-0.593894\pi$
$571$	−332.000			−0.581436			−0.290718	+	0.956809i	$0.593894\pi$
$572$		0			0
$573$	−591.182			−1.03173
$574$		0			0
$575$			244.949i			0.425998i
$576$		0			0
$577$			979.796i			1.69809i	0.528323	+	0.849043i	$0.322821\pi$
$577$			979.796i			1.69809i	−0.528323	+	0.849043i	$0.677179\pi$
$578$		0			0
$579$	798.000			1.37824
$580$		0			0
$581$	288.000			0.495697
$582$		0			0
$583$	224.499			0.385076
$584$		0			0
$585$	280.624			0.479700
$586$		0			0
$587$		−	852.422i		−	1.45217i	−0.687606	−	0.726084i	$-0.741338\pi$
$587$		−	852.422i		−	1.45217i	0.687606	−	0.726084i	$-0.258662\pi$
$588$		0			0
$589$	−672.000	−	183.303i	−1.14092	−	0.311211i
$590$		0			0
$591$		−	659.891i		−	1.11657i
$592$		0			0
$593$			489.898i			0.826135i	0.910700	+	0.413067i	$0.135543\pi$
$593$			489.898i			0.826135i	−0.910700	+	0.413067i	$0.864457\pi$
$594$		0			0
$595$	960.000			1.61345
$596$		0			0
$597$	−1129.98			−1.89276
$598$		0			0
$599$			916.515i			1.53008i	0.643986	+	0.765038i	$0.277280\pi$
$599$			916.515i			1.53008i	−0.643986	+	0.765038i	$0.722720\pi$
$600$		0			0
$601$			843.194i			1.40298i	0.712677	+	0.701492i	$0.247483\pi$
$601$			843.194i			1.40298i	−0.712677	+	0.701492i	$0.752517\pi$
$602$		0			0
$603$	−56.1249			−0.0930761
$604$		0			0
$605$	525.000			0.867769
$606$		0			0
$607$	976.573			1.60885			0.804426	−	0.594054i	$-0.202473\pi$
$607$	976.573			1.60885			0.804426	+	0.594054i	$0.202473\pi$
$608$		0			0
$609$	−1344.00			−2.20690
$610$		0			0
$611$		−	109.982i		−	0.180003i
$612$		0			0
$613$			1077.78i			1.75820i	0.476639	+	0.879099i	$0.341855\pi$
$613$			1077.78i			1.75820i	−0.476639	+	0.879099i	$0.658145\pi$
$614$		0			0
$615$		−	685.857i		−	1.11521i
$616$		0			0
$617$			685.857i			1.11160i	0.831316	+	0.555800i	$0.187588\pi$
$617$			685.857i			1.11160i	−0.831316	+	0.555800i	$0.812412\pi$
$618$		0			0
$619$	676.000			1.09208			0.546042	−	0.837758i	$-0.316134\pi$
$619$	676.000			1.09208			0.546042	+	0.837758i	$0.316134\pi$
$620$		0			0
$621$			146.642i			0.236139i
$622$		0			0
$623$	−1436.80			−2.30625
$624$		0			0
$625$	625.000			1.00000
$626$		0			0
$627$	74.8331	−	274.343i	0.119351	−	0.437548i
$628$		0			0
$629$			659.891i			1.04911i
$630$		0			0
$631$	124.000			0.196513			0.0982567	−	0.995161i	$-0.468673\pi$
$631$	124.000			0.196513			0.0982567	+	0.995161i	$0.468673\pi$
$632$		0			0
$633$		−	685.857i		−	1.08350i
$634$		0			0
$635$	−1178.62			−1.85610
$636$		0			0
$637$	527.574			0.828216
$638$		0			0
$639$			183.303i			0.286859i
$640$		0			0
$641$		−	623.230i		−	0.972278i	−0.873881	−	0.486139i	$-0.838405\pi$
$641$		−	623.230i		−	0.972278i	0.873881	−	0.486139i	$-0.161595\pi$
$642$		0			0
$643$		−	244.949i		−	0.380947i	−0.981692	−	0.190474i	$-0.938998\pi$
$643$		−	244.949i		−	0.380947i	0.981692	−	0.190474i	$-0.0610023\pi$
$644$		0			0
$645$			1283.12i			1.98934i
$646$		0			0
$647$			342.929i			0.530029i	0.964244	+	0.265014i	$0.0853767\pi$
$647$			342.929i			0.530029i	−0.964244	+	0.265014i	$0.914623\pi$
$648$		0			0
$649$		−	293.285i		−	0.451903i
$650$		0			0
$651$	−1344.00			−2.06452
$652$		0			0
$653$		−	1038.58i		−	1.59048i	−0.606295	−	0.795240i	$-0.707345\pi$
$653$		−	1038.58i		−	1.59048i	0.606295	−	0.795240i	$-0.292655\pi$
$654$		0			0
$655$	−130.000			−0.198473
$656$		0			0
$657$			97.9796i			0.149132i
$658$		0			0
$659$			879.855i			1.33514i	0.744549	+	0.667568i	$0.232665\pi$
$659$			879.855i			1.33514i	−0.744549	+	0.667568i	$0.767335\pi$
$660$		0			0
$661$		−	586.570i		−	0.887397i	−0.896176	−	0.443699i	$-0.853666\pi$
$661$		−	586.570i		−	0.887397i	0.896176	−	0.443699i	$-0.146334\pi$
$662$		0			0
$663$		−	823.029i		−	1.24137i
$664$		0			0
$665$	−897.998	−	244.949i	−1.35037	−	0.368344i
$666$		0			0
$667$	−359.199			−0.538529
$668$		0			0
$669$	798.000			1.19283
$670$		0			0
$671$	400.000			0.596125
$672$		0			0
$673$	1156.17			1.71794			0.858969	−	0.512028i	$-0.171106\pi$
$673$	1156.17			1.71794			0.858969	+	0.512028i	$0.171106\pi$
$674$		0			0
$675$	374.166			0.554320
$676$		0			0
$677$	325.524			0.480833			0.240417	−	0.970670i	$-0.422716\pi$
$677$	325.524			0.480833			0.240417	+	0.970670i	$0.422716\pi$
$678$		0			0
$679$			1209.80i			1.78174i
$680$		0			0
$681$	1050.00			1.54185
$682$		0			0
$683$	1133.72			1.65992			0.829958	−	0.557826i	$-0.188364\pi$
$683$	1133.72			1.65992			0.829958	+	0.557826i	$0.188364\pi$
$684$		0			0
$685$		−	1273.73i		−	1.85947i
$686$		0			0
$687$	883.031			1.28534
$688$		0			0
$689$	−630.000			−0.914369
$690$		0			0
$691$	−572.000			−0.827786			−0.413893	−	0.910326i	$-0.635831\pi$
$691$	−572.000			−0.827786			−0.413893	+	0.910326i	$0.635831\pi$
$692$		0			0
$693$		−	195.959i		−	0.282769i
$694$		0			0
$695$	−740.000			−1.06475
$696$		0			0
$697$	−718.398			−1.03070
$698$		0			0
$699$			806.533i			1.15384i
$700$		0			0
$701$	820.000			1.16976			0.584879	−	0.811121i	$-0.301142\pi$
$701$	820.000			1.16976			0.584879	+	0.811121i	$0.301142\pi$
$702$		0			0
$703$	168.375	−	617.271i	0.239509	−	0.878053i
$704$		0			0
$705$			183.303i			0.260004i
$706$		0			0
$707$		−	39.1918i		−	0.0554340i
$708$		0			0
$709$	278.000			0.392102			0.196051	−	0.980594i	$-0.437188\pi$
$709$	278.000			0.392102			0.196051	+	0.980594i	$0.437188\pi$
$710$		0			0
$711$			549.909i			0.773430i
$712$		0			0
$713$	−359.199			−0.503786
$714$		0			0
$715$	224.499			0.313985
$716$		0			0
$717$	−546.282			−0.761900
$718$		0			0
$719$	604.000			0.840056			0.420028	−	0.907511i	$-0.362020\pi$
$719$	604.000			0.840056			0.420028	+	0.907511i	$0.362020\pi$
$720$		0			0
$721$		−	769.873i		−	1.06778i
$722$		0			0
$723$		−	685.857i		−	0.948627i
$724$		0			0
$725$			916.515i			1.26416i
$726$		0			0
$727$			617.271i			0.849067i	0.905412	+	0.424533i	$0.139562\pi$
$727$			617.271i			0.849067i	−0.905412	+	0.424533i	$0.860438\pi$
$728$		0			0
$729$	−1.00000			−0.00137174
$730$		0			0
$731$	1344.00			1.83858
$732$		0			0
$733$			137.171i			0.187137i	0.995613	+	0.0935685i	$0.0298274\pi$
$733$			137.171i			0.187137i	−0.995613	+	0.0935685i	$0.970173\pi$
$734$		0			0
$735$	−879.289			−1.19631
$736$		0			0
$737$	−44.8999			−0.0609225
$738$		0			0
$739$	−298.000			−0.403248			−0.201624	−	0.979463i	$-0.564622\pi$
$739$	−298.000			−0.403248			−0.201624	+	0.979463i	$0.564622\pi$
$740$		0			0
$741$	−210.000	+	769.873i	−0.283401	+	1.03896i
$742$		0			0
$743$	1313.32			1.76759			0.883797	−	0.467871i	$-0.154979\pi$
$743$	1313.32			1.76759			0.883797	+	0.467871i	$0.154979\pi$
$744$		0			0
$745$	−260.000			−0.348993
$746$		0			0
$747$			146.969i			0.196746i
$748$		0			0
$749$			989.836i			1.32154i
$750$		0			0
$751$		−	659.891i		−	0.878683i	−0.898320	−	0.439341i	$-0.855212\pi$
$751$		−	659.891i		−	0.878683i	0.898320	−	0.439341i	$-0.144788\pi$
$752$		0			0
$753$	935.414			1.24225
$754$		0			0
$755$		−	366.606i		−	0.485571i
$756$		0			0
$757$			1058.18i			1.39786i	0.715190	+	0.698930i	$0.246340\pi$
$757$			1058.18i			1.39786i	−0.715190	+	0.698930i	$0.753660\pi$
$758$		0			0
$759$		−	146.642i		−	0.193205i
$760$		0			0
$761$	−788.000			−1.03548			−0.517740	−	0.855538i	$-0.673226\pi$
$761$	−788.000			−1.03548			−0.517740	+	0.855538i	$0.673226\pi$
$762$		0			0
$763$	−718.398			−0.941544
$764$		0			0
$765$			489.898i			0.640389i
$766$		0			0
$767$			823.029i			1.07305i
$768$		0			0
$769$	−548.000			−0.712614			−0.356307	−	0.934369i	$-0.615964\pi$
$769$	−548.000			−0.712614			−0.356307	+	0.934369i	$0.615964\pi$
$770$		0			0
$771$	−546.000			−0.708171
$772$		0			0
$773$	−819.423			−1.06006			−0.530028	−	0.847980i	$-0.677819\pi$
$773$	−819.423			−1.06006			−0.530028	+	0.847980i	$0.677819\pi$
$774$		0			0
$775$			916.515i			1.18260i
$776$		0			0
$777$		−	1234.54i		−	1.58886i
$778$		0			0
$779$	672.000	+	183.303i	0.862644	+	0.235306i
$780$		0			0
$781$			146.642i			0.187762i
$782$		0			0
$783$			548.686i			0.700748i
$784$		0			0
$785$		−	587.878i		−	0.748889i
$786$		0			0
$787$	−931.673			−1.18383			−0.591914	−	0.806001i	$-0.701628\pi$
$787$	−931.673			−1.18383			−0.591914	+	0.806001i	$0.701628\pi$
$788$		0			0
$789$		−	329.945i		−	0.418182i
$790$		0			0
$791$			109.982i			0.139041i
$792$		0			0
$793$	−1122.50			−1.41551
$794$		0			0
$795$	1050.00			1.32075
$796$		0			0
$797$	774.523			0.971798			0.485899	−	0.874015i	$-0.338492\pi$
$797$	774.523			0.971798			0.485899	+	0.874015i	$0.338492\pi$
$798$		0			0
$799$	192.000			0.240300
$800$		0			0
$801$		−	733.212i		−	0.915371i
$802$		0			0
$803$			78.3837i			0.0976135i
$804$		0			0
$805$	−480.000			−0.596273
$806$		0			0
$807$			274.343i			0.339954i
$808$		0			0
$809$	638.000			0.788628			0.394314	−	0.918976i	$-0.370982\pi$
$809$	638.000			0.788628			0.394314	+	0.918976i	$0.370982\pi$
$810$		0			0
$811$		−	73.3212i		−	0.0904084i	−0.998978	−	0.0452042i	$-0.985606\pi$
$811$		−	73.3212i		−	0.0904084i	0.998978	−	0.0452042i	$-0.0143938\pi$
$812$		0			0
$813$	254.433			0.312955
$814$		0			0
$815$			48.9898i			0.0601102i
$816$		0			0
$817$	−1257.20	−	342.929i	−1.53880	−	0.419741i
$818$		0			0
$819$			549.909i			0.671440i
$820$		0			0
$821$	−1318.00			−1.60536			−0.802680	−	0.596410i	$-0.796593\pi$
$821$	−1318.00			−1.60536			−0.802680	+	0.596410i	$0.796593\pi$
$822$		0			0
$823$		−	88.1816i		−	0.107147i	−0.998564	−	0.0535733i	$-0.982939\pi$
$823$		−	88.1816i		−	0.107147i	0.998564	−	0.0535733i	$-0.0170611\pi$
$824$		0			0
$825$	−374.166			−0.453534
$826$		0			0
$827$	1133.72			1.37089			0.685443	−	0.728127i	$-0.259609\pi$
$827$	1133.72			1.37089			0.685443	+	0.728127i	$0.259609\pi$
$828$		0			0
$829$		−	1466.42i		−	1.76891i	−0.466628	−	0.884454i	$-0.654531\pi$
$829$		−	1466.42i		−	1.76891i	0.466628	−	0.884454i	$-0.345469\pi$
$830$		0			0
$831$		−	586.570i		−	0.705860i
$832$		0			0
$833$			921.008i			1.10565i
$834$		0			0
$835$	168.375			0.201646
$836$		0			0
$837$			548.686i			0.655538i
$838$		0			0
$839$			623.230i			0.742825i	0.928468	+	0.371413i	$0.121126\pi$
$839$			623.230i			0.742825i	−0.928468	+	0.371413i	$0.878874\pi$
$840$		0			0
$841$	−503.000			−0.598098
$842$		0			0
$843$		−	411.514i		−	0.488155i
$844$		0			0
$845$	215.000			0.254438
$846$		0			0
$847$			1028.79i			1.21462i
$848$		0			0
$849$		−	1796.37i		−	2.11587i
$850$		0			0
$851$		−	329.945i		−	0.387715i
$852$		0			0
$853$			921.008i			1.07973i	0.841752	+	0.539864i	$0.181524\pi$
$853$			921.008i			1.07973i	−0.841752	+	0.539864i	$0.818476\pi$
$854$		0			0
$855$	125.000	−	458.258i	0.146199	−	0.535974i
$856$		0			0
$857$	1133.72			1.32290			0.661448	−	0.749991i	$-0.269942\pi$
$857$	1133.72			1.32290			0.661448	+	0.749991i	$0.269942\pi$
$858$		0			0
$859$	746.000			0.868452			0.434226	−	0.900804i	$-0.357022\pi$
$859$	746.000			0.868452			0.434226	+	0.900804i	$0.357022\pi$
$860$		0			0
$861$	1344.00			1.56098
$862$		0			0
$863$	235.724			0.273145			0.136573	−	0.990630i	$-0.456391\pi$
$863$	235.724			0.273145			0.136573	+	0.990630i	$0.456391\pi$
$864$		0			0
$865$	1403.12			1.62211
$866$		0			0
$867$	355.457			0.409986
$868$		0			0
$869$			439.927i			0.506245i
$870$		0			0
$871$	126.000			0.144661
$872$		0			0
$873$	−617.373			−0.707186
$874$		0			0
$875$			1224.74i			1.39971i
$876$		0			0
$877$	145.925			0.166391			0.0831953	−	0.996533i	$-0.473487\pi$
$877$	145.925			0.166391			0.0831953	+	0.996533i	$0.473487\pi$
$878$		0			0
$879$	−1890.00			−2.15017
$880$		0			0
$881$	−1028.00			−1.16686			−0.583428	−	0.812165i	$-0.698289\pi$
$881$	−1028.00			−1.16686			−0.583428	+	0.812165i	$0.698289\pi$
$882$		0			0
$883$			930.806i			1.05414i	0.849822	+	0.527070i	$0.176710\pi$
$883$			930.806i			1.05414i	−0.849822	+	0.527070i	$0.823290\pi$
$884$		0			0
$885$		−	1371.71i		−	1.54996i
$886$		0			0
$887$	168.375			0.189825			0.0949124	−	0.995486i	$-0.469743\pi$
$887$	168.375			0.189825			0.0949124	+	0.995486i	$0.469743\pi$
$888$		0			0
$889$		−	2309.62i		−	2.59800i
$890$		0			0
$891$	−404.000			−0.453423
$892$		0			0
$893$	−179.600	−	48.9898i	−0.201119	−	0.0548598i
$894$		0			0
$895$		−	733.212i		−	0.819231i
$896$		0			0
$897$			411.514i			0.458767i
$898$		0			0
$899$	−1344.00			−1.49499
$900$		0			0
$901$		−	1099.82i		−	1.22066i
$902$		0			0
$903$	−2514.39			−2.78449
$904$		0			0
$905$		−	1283.12i		−	1.41781i
$906$		0			0
$907$	347.974			0.383654			0.191827	−	0.981429i	$-0.438559\pi$
$907$	347.974			0.383654			0.191827	+	0.981429i	$0.438559\pi$
$908$		0			0
$909$	20.0000			0.0220022
$910$		0			0
$911$			1246.46i			1.36823i	0.729373	+	0.684117i	$0.239812\pi$
$911$			1246.46i			1.36823i	−0.729373	+	0.684117i	$0.760188\pi$
$912$		0			0
$913$			117.576i			0.128779i
$914$		0			0
$915$	1870.83			2.04462
$916$		0			0
$917$		−	254.747i		−	0.277805i
$918$		0			0
$919$	−1150.00			−1.25136			−0.625680	−	0.780080i	$-0.715178\pi$
$919$	−1150.00			−1.25136			−0.625680	+	0.780080i	$0.715178\pi$
$920$		0			0
$921$	378.000			0.410423
$922$		0			0
$923$		−	411.514i		−	0.445844i
$924$		0			0
$925$	−841.873			−0.910133
$926$		0			0
$927$	392.874			0.423812
$928$		0			0
$929$	1454.00			1.56512			0.782562	−	0.622573i	$-0.213913\pi$
$929$	1454.00			1.56512			0.782562	+	0.622573i	$0.213913\pi$
$930$		0			0
$931$	235.000	−	861.524i	0.252417	−	0.925375i
$932$		0			0
$933$	1870.83			2.00518
$934$		0			0
$935$			391.918i			0.419164i
$936$		0			0
$937$		−	823.029i		−	0.878366i	−0.898398	−	0.439183i	$-0.855268\pi$
$937$		−	823.029i		−	0.878366i	0.898398	−	0.439183i	$-0.144732\pi$
$938$		0			0
$939$			806.533i			0.858928i
$940$		0			0
$941$			329.945i			0.350633i	0.984512	+	0.175316i	$0.0560948\pi$
$941$			329.945i			0.350633i	−0.984512	+	0.175316i	$0.943905\pi$
$942$		0			0
$943$	359.199			0.380911
$944$		0			0
$945$			733.212i			0.775886i
$946$		0			0
$947$		−	754.443i		−	0.796666i	−0.917241	−	0.398333i	$-0.869589\pi$
$947$		−	754.443i		−	0.796666i	0.917241	−	0.398333i	$-0.130411\pi$
$948$		0			0
$949$		−	219.964i		−	0.231785i
$950$		0			0
$951$	1050.00			1.10410
$952$		0			0
$953$	−931.673			−0.977621			−0.488810	−	0.872390i	$-0.662569\pi$
$953$	−931.673			−0.977621			−0.488810	+	0.872390i	$0.662569\pi$
$954$		0			0
$955$	−790.000			−0.827225
$956$		0			0
$957$		−	548.686i		−	0.573339i
$958$		0			0
$959$	2496.00			2.60271
$960$		0			0
$961$	−383.000			−0.398543
$962$		0			0
$963$	−505.124			−0.524531
$964$		0			0
$965$	1066.37			1.10505
$966$		0			0
$967$			891.614i			0.922042i	0.887389	+	0.461021i	$0.152517\pi$
$967$			891.614i			0.922042i	−0.887389	+	0.461021i	$0.847483\pi$
$968$		0			0
$969$	−1344.00	−	366.606i	−1.38700	−	0.378334i
$970$		0			0
$971$		−	73.3212i		−	0.0755110i	−0.999287	−	0.0377555i	$-0.987979\pi$
$971$		−	73.3212i		−	0.0755110i	0.999287	−	0.0377555i	$-0.0120208\pi$
$972$		0			0
$973$		−	1450.10i		−	1.49034i
$974$		0			0
$975$	1050.00			1.07692
$976$		0			0
$977$	−280.624			−0.287231			−0.143615	−	0.989634i	$-0.545873\pi$
$977$	−280.624			−0.287231			−0.143615	+	0.989634i	$0.545873\pi$
$978$		0			0
$979$		−	586.570i		−	0.599152i
$980$		0			0
$981$		−	366.606i		−	0.373706i
$982$		0			0
$983$	−460.224			−0.468183			−0.234091	−	0.972215i	$-0.575212\pi$
$983$	−460.224			−0.468183			−0.234091	+	0.972215i	$0.575212\pi$
$984$		0			0
$985$		−	881.816i		−	0.895245i
$986$		0			0
$987$	−359.199			−0.363930
$988$		0			0
$989$	−672.000			−0.679474
$990$		0			0
$991$			73.3212i			0.0739871i	0.999316	+	0.0369935i	$0.0117781\pi$
$991$			73.3212i			0.0739871i	−0.999316	+	0.0369935i	$0.988222\pi$
$992$		0			0
$993$		−	1508.89i		−	1.51952i
$994$		0			0
$995$	−1510.00			−1.51759
$996$		0			0
$997$		−	529.090i		−	0.530682i	−0.964155	−	0.265341i	$-0.914515\pi$
$997$		−	529.090i		−	0.530682i	0.964155	−	0.265341i	$-0.0854845\pi$
$998$		0			0
$999$	−504.000			−0.504505

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	380.3.g.a.189.1	✓	4
3.2	odd	2		3420.3.h.c.2089.1		4
5.2	odd	4		1900.3.e.c.1101.4		4
5.3	odd	4		1900.3.e.c.1101.1		4
5.4	even	2	inner	380.3.g.a.189.4	yes	4
15.14	odd	2		3420.3.h.c.2089.4		4
19.18	odd	2	inner	380.3.g.a.189.3	yes	4
57.56	even	2		3420.3.h.c.2089.2		4
95.18	even	4		1900.3.e.c.1101.3		4
95.37	even	4		1900.3.e.c.1101.2		4
95.94	odd	2	inner	380.3.g.a.189.2	yes	4
285.284	even	2		3420.3.h.c.2089.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
380.3.g.a.189.1	✓	4	1.1	even	1	trivial
380.3.g.a.189.2	yes	4	95.94	odd	2	inner
380.3.g.a.189.3	yes	4	19.18	odd	2	inner
380.3.g.a.189.4	yes	4	5.4	even	2	inner
1900.3.e.c.1101.1		4	5.3	odd	4
1900.3.e.c.1101.2		4	95.37	even	4
1900.3.e.c.1101.3		4	95.18	even	4
1900.3.e.c.1101.4		4	5.2	odd	4
3420.3.h.c.2089.1		4	3.2	odd	2
3420.3.h.c.2089.2		4	57.56	even	2
3420.3.h.c.2089.3		4	285.284	even	2
3420.3.h.c.2089.4		4	15.14	odd	2

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:	\( N \)	\(=\)	\( 380 = 2^{2} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	380.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.74166 q^{3} -5.00000 q^{5} -9.79796i q^{7} +5.00000 q^{9} +O(q^{10})\)	\(q-3.74166 q^{3} -5.00000 q^{5} -9.79796i q^{7} +5.00000 q^{9} +4.00000 q^{11} -11.2250 q^{13} +18.7083 q^{15} -19.5959i q^{17} +(-5.00000 + 18.3303i) q^{19} +36.6606i q^{21} +9.79796i q^{23} +25.0000 q^{25} +14.9666 q^{27} +36.6606i q^{29} +36.6606i q^{31} -14.9666 q^{33} +48.9898i q^{35} -33.6749 q^{37} +42.0000 q^{39} -36.6606i q^{41} +68.5857i q^{43} -25.0000 q^{45} +9.79796i q^{47} -47.0000 q^{49} +73.3212i q^{51} +56.1249 q^{53} -20.0000 q^{55} +(18.7083 - 68.5857i) q^{57} -73.3212i q^{59} +100.000 q^{61} -48.9898i q^{63} +56.1249 q^{65} -11.2250 q^{67} -36.6606i q^{69} +36.6606i q^{71} +19.5959i q^{73} -93.5414 q^{75} -39.1918i q^{77} +109.982i q^{79} -101.000 q^{81} +29.3939i q^{83} +97.9796i q^{85} -137.171i q^{87} -146.642i q^{89} +109.982i q^{91} -137.171i q^{93} +(25.0000 - 91.6515i) q^{95} -123.475 q^{97} +20.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 20 q^{5} + 20 q^{9}+O(q^{10}) \) 4 * q - 20 * q^5 + 20 * q^9	\( 4 q - 20 q^{5} + 20 q^{9} + 16 q^{11} - 20 q^{19} + 100 q^{25} + 168 q^{39} - 100 q^{45} - 188 q^{49} - 80 q^{55} + 400 q^{61} - 404 q^{81} + 100 q^{95} + 80 q^{99}+O(q^{100}) \) 4 * q - 20 * q^5 + 20 * q^9 + 16 * q^11 - 20 * q^19 + 100 * q^25 + 168 * q^39 - 100 * q^45 - 188 * q^49 - 80 * q^55 + 400 * q^61 - 404 * q^81 + 100 * q^95 + 80 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more