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Properties

Label 190.3.d.a
Level $190$
Weight $3$
Character orbit 190.d
Analytic conductor $5.177$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [190,3,Mod(189,190)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(190, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("190.189");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 190 = 2 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 190.d (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: \(5.17712502285\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 2 x^{19} + 23 x^{18} + 226 x^{17} + 1835 x^{16} + 568 x^{15} + 106900 x^{14} + \cdots + 110823862587543 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{14}\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} - \beta_{4} q^{3} + 2 q^{4} - \beta_{3} q^{5} + \beta_{6} q^{6} + \beta_{7} q^{7} + 2 \beta_{2} q^{8} + ( - \beta_{11} + 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} - \beta_{4} q^{3} + 2 q^{4} - \beta_{3} q^{5} + \beta_{6} q^{6} + \beta_{7} q^{7} + 2 \beta_{2} q^{8} + ( - \beta_{11} + 3) q^{9} - \beta_1 q^{10} + (\beta_{12} - \beta_{11} - \beta_{6} + 1) q^{11} - 2 \beta_{4} q^{12} + (\beta_{9} + \beta_{4} + \beta_{2}) q^{13} - \beta_{10} q^{14} + ( - \beta_{14} - \beta_{9} + \cdots + 2 \beta_{2}) q^{15}+ \cdots + (2 \beta_{12} - 9 \beta_{11} + \cdots + 50) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 40 q^{4} - 2 q^{5} + 60 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 40 q^{4} - 2 q^{5} + 60 q^{9} + 28 q^{11} + 80 q^{16} + 20 q^{19} - 4 q^{20} - 82 q^{25} + 32 q^{26} + 88 q^{30} + 142 q^{35} + 120 q^{36} - 288 q^{39} + 56 q^{44} - 174 q^{45} - 440 q^{49} - 144 q^{54} - 54 q^{55} - 12 q^{61} + 160 q^{64} - 544 q^{66} + 40 q^{76} - 8 q^{80} - 380 q^{81} - 266 q^{85} + 434 q^{95} + 1044 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 2 x^{19} + 23 x^{18} + 226 x^{17} + 1835 x^{16} + 568 x^{15} + 106900 x^{14} + \cdots + 110823862587543 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 38\!\cdots\!83 \nu^{19} + \cdots - 65\!\cdots\!89 ) / 84\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 13\!\cdots\!49 \nu^{19} + \cdots - 90\!\cdots\!37 ) / 25\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 13\!\cdots\!49 \nu^{19} + \cdots + 90\!\cdots\!37 ) / 25\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 37\!\cdots\!99 \nu^{19} + \cdots + 35\!\cdots\!13 ) / 66\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 37\!\cdots\!67 \nu^{19} + \cdots + 12\!\cdots\!79 ) / 66\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 31\!\cdots\!43 \nu^{19} + \cdots + 23\!\cdots\!18 ) / 25\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 84\!\cdots\!11 \nu^{19} + \cdots - 48\!\cdots\!39 ) / 66\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 48\!\cdots\!59 \nu^{19} + \cdots + 26\!\cdots\!91 ) / 19\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 26\!\cdots\!53 \nu^{19} + \cdots - 15\!\cdots\!36 ) / 99\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 14\!\cdots\!08 \nu^{19} + \cdots - 57\!\cdots\!21 ) / 50\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 39\!\cdots\!99 \nu^{19} + \cdots - 61\!\cdots\!24 ) / 84\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 13\!\cdots\!11 \nu^{19} + \cdots + 36\!\cdots\!36 ) / 25\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 10\!\cdots\!79 \nu^{19} + \cdots + 16\!\cdots\!23 ) / 99\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 63\!\cdots\!52 \nu^{19} + \cdots + 87\!\cdots\!51 ) / 50\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 13\!\cdots\!30 \nu^{19} + \cdots + 34\!\cdots\!73 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 34\!\cdots\!99 \nu^{19} + \cdots + 12\!\cdots\!01 ) / 19\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 35\!\cdots\!73 \nu^{19} + \cdots + 87\!\cdots\!27 ) / 19\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 12\!\cdots\!93 \nu^{19} + \cdots + 20\!\cdots\!57 ) / 66\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 22\!\cdots\!78 \nu^{19} + \cdots - 54\!\cdots\!31 ) / 10\!\cdots\!00 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_{3} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{17} + \beta_{16} + \beta_{11} - 2\beta_{8} - \beta_{7} + \beta_{3} + 2\beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - 3 \beta_{19} + 3 \beta_{17} - \beta_{16} - 3 \beta_{15} - 3 \beta_{14} + 3 \beta_{13} - 7 \beta_{12} + \cdots - 34 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 12 \beta_{19} - 8 \beta_{18} - 3 \beta_{17} + 21 \beta_{16} - 4 \beta_{15} + 12 \beta_{14} + 4 \beta_{13} + \cdots - 428 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 25 \beta_{19} + 72 \beta_{18} + 82 \beta_{17} - 86 \beta_{16} - 155 \beta_{15} + 105 \beta_{14} + \cdots + 381 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 252 \beta_{19} - 200 \beta_{18} + 346 \beta_{17} - 60 \beta_{16} + 620 \beta_{15} - 612 \beta_{14} + \cdots - 9611 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 1498 \beta_{19} + 2480 \beta_{18} - 994 \beta_{17} - 2794 \beta_{16} + 2534 \beta_{15} + 2058 \beta_{14} + \cdots + 46044 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 15568 \beta_{19} - 1528 \beta_{18} + 4127 \beta_{17} - 22781 \beta_{16} + 28912 \beta_{15} + \cdots + 70860 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 24969 \beta_{19} - 74616 \beta_{18} - 151207 \beta_{17} + 38013 \beta_{16} + 212691 \beta_{15} + \cdots + 1027014 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 495814 \beta_{19} + 98336 \beta_{18} - 246473 \beta_{17} - 628335 \beta_{16} - 694910 \beta_{15} + \cdots + 6086078 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 33935 \beta_{19} - 7136584 \beta_{18} - 1962078 \beta_{17} + 7485018 \beta_{16} + 3526545 \beta_{15} + \cdots + 8661313 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 1619368 \beta_{19} + 28411424 \beta_{18} + 13824552 \beta_{17} - 9691052 \beta_{16} - 71666360 \beta_{15} + \cdots + 89067513 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 120348800 \beta_{19} - 13273176 \beta_{18} + 58664512 \beta_{17} + 252509808 \beta_{16} + \cdots - 772117928 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 209442464 \beta_{19} + 1060792656 \beta_{18} + 1218454427 \beta_{17} - 282002847 \beta_{16} + \cdots - 2344290014 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 3891799669 \beta_{19} + 5202253784 \beta_{18} + 235996519 \beta_{17} + 327387139 \beta_{16} + \cdots - 106437526150 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 5961640336 \beta_{19} - 29050460040 \beta_{18} + 5910853161 \beta_{17} + 3560744381 \beta_{16} + \cdots + 266593538228 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 128997835303 \beta_{19} + 63350360016 \beta_{18} - 65740885622 \beta_{17} - 352456926414 \beta_{16} + \cdots - 3651856009819 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 16943805836 \beta_{19} - 2247066153944 \beta_{18} - 2245465302390 \beta_{17} + 1224806786504 \beta_{16} + \cdots + 20700608428213 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( - 11550490813318 \beta_{19} + 805295587016 \beta_{18} + 4427344013070 \beta_{17} + \cdots + 12887275665044 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/190\mathbb{Z}\right)^\times\).

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

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
189.1
−4.64737 + 3.81401i
−4.64737 − 3.81401i
1.40251 + 4.13111i
1.40251 − 4.13111i
−5.71586 + 2.54869i
−5.71586 − 2.54869i
2.12873 + 3.52811i
2.12873 − 3.52811i
0.260931 + 4.71104i
0.260931 − 4.71104i
3.08936 + 4.71104i
3.08936 − 4.71104i
4.95715 + 3.52811i
4.95715 − 3.52811i
−2.88744 + 2.54869i
−2.88744 − 2.54869i
4.23094 + 4.13111i
4.23094 − 4.13111i
−1.81895 + 3.81401i
−1.81895 − 3.81401i
−1.41421 −3.91491 2.00000 3.23316 − 3.81401i 5.53652 9.18796i −2.82843 6.32652 −4.57238 + 5.39383i
189.2 −1.41421 −3.91491 2.00000 3.23316 + 3.81401i 5.53652 − 9.18796i −2.82843 6.32652 −4.57238 − 5.39383i
189.3 −1.41421 −3.78357 2.00000 −2.81673 − 4.13111i 5.35077 − 4.58145i −2.82843 5.31539 3.98345 + 5.84227i
189.4 −1.41421 −3.78357 2.00000 −2.81673 + 4.13111i 5.35077 4.58145i −2.82843 5.31539 3.98345 − 5.84227i
189.5 −1.41421 0.541520 2.00000 4.30165 − 2.54869i −0.765825 − 8.53138i −2.82843 −8.70676 −6.08345 + 3.60439i
189.6 −1.41421 0.541520 2.00000 4.30165 + 2.54869i −0.765825 8.53138i −2.82843 −8.70676 −6.08345 − 3.60439i
189.7 −1.41421 2.08620 2.00000 −3.54294 − 3.52811i −2.95033 − 4.86635i −2.82843 −4.64779 5.01047 + 4.98950i
189.8 −1.41421 2.08620 2.00000 −3.54294 + 3.52811i −2.95033 4.86635i −2.82843 −4.64779 5.01047 − 4.98950i
189.9 −1.41421 5.07076 2.00000 −1.67514 − 4.71104i −7.17114 12.3744i −2.82843 16.7126 2.36901 + 6.66242i
189.10 −1.41421 5.07076 2.00000 −1.67514 + 4.71104i −7.17114 − 12.3744i −2.82843 16.7126 2.36901 − 6.66242i
189.11 1.41421 −5.07076 2.00000 −1.67514 − 4.71104i −7.17114 12.3744i 2.82843 16.7126 −2.36901 − 6.66242i
189.12 1.41421 −5.07076 2.00000 −1.67514 + 4.71104i −7.17114 − 12.3744i 2.82843 16.7126 −2.36901 + 6.66242i
189.13 1.41421 −2.08620 2.00000 −3.54294 − 3.52811i −2.95033 − 4.86635i 2.82843 −4.64779 −5.01047 − 4.98950i
189.14 1.41421 −2.08620 2.00000 −3.54294 + 3.52811i −2.95033 4.86635i 2.82843 −4.64779 −5.01047 + 4.98950i
189.15 1.41421 −0.541520 2.00000 4.30165 − 2.54869i −0.765825 − 8.53138i 2.82843 −8.70676 6.08345 − 3.60439i
189.16 1.41421 −0.541520 2.00000 4.30165 + 2.54869i −0.765825 8.53138i 2.82843 −8.70676 6.08345 + 3.60439i
189.17 1.41421 3.78357 2.00000 −2.81673 − 4.13111i 5.35077 − 4.58145i 2.82843 5.31539 −3.98345 − 5.84227i
189.18 1.41421 3.78357 2.00000 −2.81673 + 4.13111i 5.35077 4.58145i 2.82843 5.31539 −3.98345 + 5.84227i
189.19 1.41421 3.91491 2.00000 3.23316 − 3.81401i 5.53652 9.18796i 2.82843 6.32652 4.57238 − 5.39383i
189.20 1.41421 3.91491 2.00000 3.23316 + 3.81401i 5.53652 − 9.18796i 2.82843 6.32652 4.57238 + 5.39383i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 189.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
19.b odd 2 1 inner
95.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 190.3.d.a ✓ 20
3.b odd 2 1 1710.3.e.a 20
5.b even 2 1 inner 190.3.d.a ✓ 20
5.c odd 4 2 950.3.c.e 20
15.d odd 2 1 1710.3.e.a 20
19.b odd 2 1 inner 190.3.d.a ✓ 20
57.d even 2 1 1710.3.e.a 20
95.d odd 2 1 inner 190.3.d.a ✓ 20
95.g even 4 2 950.3.c.e 20
285.b even 2 1 1710.3.e.a 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
190.3.d.a ✓ 20 1.a even 1 1 trivial
190.3.d.a ✓ 20 5.b even 2 1 inner
190.3.d.a ✓ 20 19.b odd 2 1 inner
190.3.d.a ✓ 20 95.d odd 2 1 inner
950.3.c.e 20 5.c odd 4 2
950.3.c.e 20 95.g even 4 2
1710.3.e.a 20 3.b odd 2 1
1710.3.e.a 20 15.d odd 2 1
1710.3.e.a 20 57.d even 2 1
1710.3.e.a 20 285.b even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(190, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 2)^{10} \) Copy content Toggle raw display
$3$ \( (T^{10} - 60 T^{8} + \cdots - 7200)^{2} \) Copy content Toggle raw display
$5$ \( (T^{10} + T^{9} + \cdots + 9765625)^{2} \) Copy content Toggle raw display
$7$ \( (T^{10} + 355 T^{8} + \cdots + 467668080)^{2} \) Copy content Toggle raw display
$11$ \( (T^{5} - 7 T^{4} + \cdots - 178260)^{4} \) Copy content Toggle raw display
$13$ \( (T^{10} - 412 T^{8} + \cdots - 30233088)^{2} \) Copy content Toggle raw display
$17$ \( (T^{10} + 1635 T^{8} + \cdots + 1734670080)^{2} \) Copy content Toggle raw display
$19$ \( (T^{10} + \cdots + 6131066257801)^{2} \) Copy content Toggle raw display
$23$ \( (T^{10} + 2648 T^{8} + \cdots + 74742497280)^{2} \) Copy content Toggle raw display
$29$ \( (T^{10} + \cdots + 402361344000)^{2} \) Copy content Toggle raw display
$31$ \( (T^{10} + \cdots + 554027449536000)^{2} \) Copy content Toggle raw display
$37$ \( (T^{10} + \cdots - 64496357695008)^{2} \) Copy content Toggle raw display
$41$ \( (T^{10} + \cdots + 18\!\cdots\!60)^{2} \) Copy content Toggle raw display
$43$ \( (T^{10} + \cdots + 110511843750000)^{2} \) Copy content Toggle raw display
$47$ \( (T^{10} + \cdots + 363474203629680)^{2} \) Copy content Toggle raw display
$53$ \( (T^{10} + \cdots - 13\!\cdots\!92)^{2} \) Copy content Toggle raw display
$59$ \( (T^{10} + \cdots + 71\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{5} + 3 T^{4} + \cdots + 485273420)^{4} \) Copy content Toggle raw display
$67$ \( (T^{10} + \cdots - 26\!\cdots\!48)^{2} \) Copy content Toggle raw display
$71$ \( (T^{10} + \cdots + 44\!\cdots\!40)^{2} \) Copy content Toggle raw display
$73$ \( (T^{10} + \cdots + 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( (T^{10} + \cdots + 78\!\cdots\!40)^{2} \) Copy content Toggle raw display
$83$ \( (T^{10} + \cdots + 2190857518080)^{2} \) Copy content Toggle raw display
$89$ \( (T^{10} + \cdots + 55\!\cdots\!40)^{2} \) Copy content Toggle raw display
$97$ \( (T^{10} + \cdots - 81\!\cdots\!88)^{2} \) Copy content Toggle raw display
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