LMFDB - Embedded newform 380.3.g.a.189.3

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			189.3
Root			$-1.87083 - 1.22474i$ of defining polynomial
Character	$\chi$	$=$	380.189
Dual form			380.3.g.a.189.4

$q$-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	$-0.285666\pi$
$3$	3.74166			1.24722			0.623610	+	0.781736i	$0.285666\pi$
$4$		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	$-0.357903\pi$
$13$	11.2250			0.863459			0.431730	+	0.902003i	$0.357903\pi$
$14$		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	$-0.782202\pi$
$29$		−	36.6606i		−	1.26416i	0.774904	−	0.632079i	$-0.217798\pi$
$30$		0			0
$31$		−	36.6606i		−	1.18260i	−0.806452	−	0.591300i	$-0.798615\pi$
$31$		−	36.6606i		−	1.18260i	0.806452	−	0.591300i	$-0.201385\pi$
$32$		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	$-0.349616\pi$
$37$	33.6749			0.910133			0.455066	+	0.890457i	$0.349616\pi$
$38$		0			0
$39$	42.0000			1.07692
$40$		0			0
$41$			36.6606i			0.894161i	0.894494	+	0.447081i	$0.147536\pi$
$41$			36.6606i			0.894161i	−0.894494	+	0.447081i	$0.852464\pi$
$42$		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	$-0.677613\pi$
$53$	−56.1249			−1.05896			−0.529480	+	0.848323i	$0.677613\pi$
$54$		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	$0.213422\pi$
$59$			73.3212i			1.24273i	−0.783520	+	0.621366i	$0.786578\pi$
$60$		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	$-0.473304\pi$
$67$	11.2250			0.167537			0.0837684	+	0.996485i	$0.473304\pi$
$68$		0			0
$69$			36.6606i			0.531313i
$70$		0			0
$71$		−	36.6606i		−	0.516347i	−0.966099	−	0.258173i	$-0.916879\pi$
$71$		−	36.6606i		−	0.516347i	0.966099	−	0.258173i	$-0.0831205\pi$
$72$		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	$-0.754923\pi$
$79$		−	109.982i		−	1.39217i	0.717957	−	0.696087i	$-0.245077\pi$
$80$		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	$0.308169\pi$
$89$			146.642i			1.64767i	−0.566831	+	0.823834i	$0.691831\pi$
$90$		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	$-0.280395\pi$
$97$	123.475			1.27293			0.636467	+	0.771304i	$0.280395\pi$
$98$		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	$-0.624569\pi$
$103$	−78.5748			−0.762862			−0.381431	+	0.924397i	$0.624569\pi$
$104$		0			0
$105$			183.303i			1.74574i
$106$		0			0
$107$	101.025			0.944157			0.472078	−	0.881557i	$-0.343504\pi$
$107$	101.025			0.944157			0.472078	+	0.881557i	$0.343504\pi$
$108$		0			0
$109$			73.3212i			0.672672i	0.941742	+	0.336336i	$0.109188\pi$
$109$			73.3212i			0.672672i	−0.941742	+	0.336336i	$0.890812\pi$
$110$		0			0
$111$	126.000			1.13514
$112$		0			0
$113$	11.2250			0.0993360			0.0496680	−	0.998766i	$-0.484184\pi$
$113$	11.2250			0.0993360			0.0496680	+	0.998766i	$0.484184\pi$
$114$		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	$-0.878515\pi$
$127$	−235.724			−1.85610			−0.928049	+	0.372458i	$0.878515\pi$
$128$		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	$-0.921939\pi$
$151$		−	73.3212i		−	0.485571i	0.970080	−	0.242785i	$-0.0780611\pi$
$152$		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	$-0.467852\pi$
$167$	33.6749			0.201646			0.100823	+	0.994904i	$0.467852\pi$
$168$		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	$-0.198895\pi$
$173$	280.624			1.62211			0.811053	+	0.584973i	$0.198895\pi$
$174$		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	$-0.865663\pi$
$179$		−	146.642i		−	0.819231i	0.912258	−	0.409616i	$-0.134337\pi$
$180$		0			0
$181$		−	256.624i		−	1.41781i	−0.705302	−	0.708907i	$-0.749189\pi$
$181$		−	256.624i		−	1.41781i	0.705302	−	0.708907i	$-0.250811\pi$
$182$		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	$-0.313665\pi$
$193$	213.274			1.10505			0.552525	+	0.833497i	$0.313665\pi$
$194$		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	$-0.856972\pi$
$211$		−	183.303i		−	0.868735i	0.900736	−	0.434367i	$-0.143028\pi$
$212$		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	$-0.341292\pi$
$223$	213.274			0.956388			0.478194	+	0.878254i	$0.341292\pi$
$224$		0			0
$225$	125.000			0.555556
$226$		0			0
$227$	280.624			1.23623			0.618115	−	0.786088i	$-0.287897\pi$
$227$	280.624			1.23623			0.618115	+	0.786088i	$0.287897\pi$
$228$		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	$-0.875822\pi$
$241$		−	183.303i		−	0.760593i	0.924865	−	0.380297i	$-0.124178\pi$
$242$		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	$-0.591628\pi$
$257$	−145.925			−0.567800			−0.283900	+	0.958854i	$0.591628\pi$
$258$		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	$0.0435162\pi$
$269$			73.3212i			0.272570i	−0.990670	+	0.136285i	$0.956484\pi$
$270$		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	$-0.937303\pi$
$281$		−	109.982i		−	0.391394i	0.980664	−	0.195697i	$-0.0626970\pi$
$282$		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	$-0.830779\pi$
$293$	−505.124			−1.72397			−0.861986	+	0.506932i	$0.830779\pi$
$294$		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	$-0.447388\pi$
$307$	101.025			0.329071			0.164535	+	0.986371i	$0.447388\pi$
$308$		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	$-0.354047\pi$
$317$	280.624			0.885250			0.442625	+	0.896707i	$0.354047\pi$
$318$		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	$-0.791505\pi$
$331$		−	403.267i		−	1.21833i	0.793044	−	0.609164i	$-0.208495\pi$
$332$		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	$-0.452109\pi$
$337$	101.025			0.299777			0.149888	+	0.988703i	$0.452109\pi$
$338$		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	$-0.447071\pi$
$373$	123.475			0.331031			0.165516	+	0.986207i	$0.447071\pi$
$374$		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	$-0.856648\pi$
$379$		−	329.945i		−	0.870568i	0.900293	−	0.435284i	$-0.143352\pi$
$380$		0			0
$381$	−882.000			−2.31496
$382$		0			0
$383$	11.2250			0.0293080			0.0146540	−	0.999893i	$-0.495335\pi$
$383$	11.2250			0.0293080			0.0146540	+	0.999893i	$0.495335\pi$
$384$		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	$0.0884358\pi$
$401$			219.964i			0.548538i	−0.961653	+	0.274269i	$0.911564\pi$
$402$		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	$0.132156\pi$
$409$			329.945i			0.806713i	−0.915043	+	0.403356i	$0.867844\pi$
$410$		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	$-0.972246\pi$
$421$		−	73.3212i		−	0.174160i	0.996201	−	0.0870798i	$-0.0277535\pi$
$422$		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	$0.323757\pi$
$431$			733.212i			1.70119i	−0.525823	+	0.850594i	$0.676243\pi$
$432$		0			0
$433$	190.825			0.440703			0.220352	−	0.975420i	$-0.429280\pi$
$433$	190.825			0.440703			0.220352	+	0.975420i	$0.429280\pi$
$434$		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	$-0.933053\pi$
$439$		−	183.303i		−	0.417547i	0.977964	−	0.208773i	$-0.0669471\pi$
$440$		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	$0.178079\pi$
$449$			476.588i			1.06144i	−0.847546	+	0.530721i	$0.821921\pi$
$450$		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	$-0.496332\pi$
$487$	11.2250			0.0230492			0.0115246	+	0.999934i	$0.496332\pi$
$488$		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	$-0.727024\pi$
$509$		−	769.873i		−	1.51252i	0.654271	−	0.756260i	$-0.272976\pi$
$510$		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	$0.218297\pi$
$521$			659.891i			1.26659i	−0.773912	+	0.633293i	$0.781703\pi$
$522$		0			0
$523$	841.873			1.60970			0.804850	−	0.593479i	$-0.202246\pi$
$523$	841.873			1.60970			0.804850	+	0.593479i	$0.202246\pi$
$524$		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	$-0.175620\pi$
$547$	931.673			1.70324			0.851620	+	0.524159i	$0.175620\pi$
$548$		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	$-0.593355\pi$
$563$	−325.524			−0.578196			−0.289098	+	0.957300i	$0.593355\pi$
$564$		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	$0.0102561\pi$
$569$			36.6606i			0.0644299i	−0.999481	+	0.0322149i	$0.989744\pi$
$570$		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	$-0.722720\pi$
$599$		−	916.515i		−	1.53008i	0.643986	−	0.765038i	$-0.277280\pi$
$600$		0			0
$601$		−	843.194i		−	1.40298i	−0.712677	−	0.701492i	$-0.752517\pi$
$601$		−	843.194i		−	1.40298i	0.712677	−	0.701492i	$-0.247483\pi$
$602$		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	$-0.797527\pi$
$607$	−976.573			−1.60885			−0.804426	+	0.594054i	$0.797527\pi$
$608$		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	$0.161595\pi$
$641$			623.230i			0.972278i	−0.873881	+	0.486139i	$0.838405\pi$
$642$		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	$-0.767335\pi$
$659$		−	879.855i		−	1.33514i	0.744549	−	0.667568i	$-0.232665\pi$
$660$		0			0
$661$			586.570i			0.887397i	0.896176	+	0.443699i	$0.146334\pi$
$661$			586.570i			0.887397i	−0.896176	+	0.443699i	$0.853666\pi$
$662$		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	$-0.828894\pi$
$673$	−1156.17			−1.71794			−0.858969	+	0.512028i	$0.828894\pi$
$674$		0			0
$675$	−374.166			−0.554320
$676$		0			0
$677$	−325.524			−0.480833			−0.240417	−	0.970670i	$-0.577284\pi$
$677$	−325.524			−0.480833			−0.240417	+	0.970670i	$0.577284\pi$
$678$		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	$-0.811636\pi$
$683$	−1133.72			−1.65992			−0.829958	+	0.557826i	$0.811636\pi$
$684$		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	$-0.845021\pi$
$743$	−1313.32			−1.76759			−0.883797	+	0.467871i	$0.845021\pi$
$744$		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	$0.144788\pi$
$751$			659.891i			0.878683i	−0.898320	+	0.439341i	$0.855212\pi$
$752$		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	$-0.322181\pi$
$773$	819.423			1.06006			0.530028	+	0.847980i	$0.322181\pi$
$774$		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	$-0.298372\pi$
$787$	931.673			1.18383			0.591914	+	0.806001i	$0.298372\pi$
$788$		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	$-0.661508\pi$
$797$	−774.523			−0.971798			−0.485899	+	0.874015i	$0.661508\pi$
$798$		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	$0.0143938\pi$
$811$			73.3212i			0.0904084i	−0.998978	+	0.0452042i	$0.985606\pi$
$812$		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	$-0.740391\pi$
$827$	−1133.72			−1.37089			−0.685443	+	0.728127i	$0.740391\pi$
$828$		0			0
$829$			1466.42i			1.76891i	0.466628	+	0.884454i	$0.345469\pi$
$829$			1466.42i			1.76891i	−0.466628	+	0.884454i	$0.654531\pi$
$830$		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	$-0.878874\pi$
$839$		−	623.230i		−	0.742825i	0.928468	−	0.371413i	$-0.121126\pi$
$840$		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	$-0.730058\pi$
$857$	−1133.72			−1.32290			−0.661448	+	0.749991i	$0.730058\pi$
$858$		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	$-0.543609\pi$
$863$	−235.724			−0.273145			−0.136573	+	0.990630i	$0.543609\pi$
$864$		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	$-0.526513\pi$
$877$	−145.925			−0.166391			−0.0831953	+	0.996533i	$0.526513\pi$
$878$		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	$-0.530257\pi$
$887$	−168.375			−0.189825			−0.0949124	+	0.995486i	$0.530257\pi$
$888$		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	$-0.561441\pi$
$907$	−347.974			−0.383654			−0.191827	+	0.981429i	$0.561441\pi$
$908$		0			0
$909$	20.0000			0.0220022
$910$		0			0
$911$		−	1246.46i		−	1.36823i	−0.729373	−	0.684117i	$-0.760188\pi$
$911$		−	1246.46i		−	1.36823i	0.729373	−	0.684117i	$-0.239812\pi$
$912$		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	$-0.943905\pi$
$941$		−	329.945i		−	0.350633i	0.984512	−	0.175316i	$-0.0560948\pi$
$942$		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	$-0.337431\pi$
$953$	931.673			0.977621			0.488810	+	0.872390i	$0.337431\pi$
$954$		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	$0.0120208\pi$
$971$			73.3212i			0.0755110i	−0.999287	+	0.0377555i	$0.987979\pi$
$972$		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	$-0.454127\pi$
$977$	280.624			0.287231			0.143615	+	0.989634i	$0.454127\pi$
$978$		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	$-0.424788\pi$
$983$	460.224			0.468183			0.234091	+	0.972215i	$0.424788\pi$
$984$		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	$-0.988222\pi$
$991$		−	73.3212i		−	0.0739871i	0.999316	−	0.0369935i	$-0.0117781\pi$
$992$		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.189.3	yes	4
3.2	odd	2		3420.3.h.c.2089.2		4
5.2	odd	4		1900.3.e.c.1101.2		4
5.3	odd	4		1900.3.e.c.1101.3		4
5.4	even	2	inner	380.3.g.a.189.2	yes	4
15.14	odd	2		3420.3.h.c.2089.3		4
19.18	odd	2	inner	380.3.g.a.189.1	✓	4
57.56	even	2		3420.3.h.c.2089.1		4
95.18	even	4		1900.3.e.c.1101.1		4
95.37	even	4		1900.3.e.c.1101.4		4
95.94	odd	2	inner	380.3.g.a.189.4	yes	4
285.284	even	2		3420.3.h.c.2089.4		4

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

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