↑

Properties

Label 190.4.e.d
Level $190$
Weight $4$
Character orbit 190.e
Analytic conductor $11.210$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [190,4,Mod(11,190)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(190, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 4]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("190.11");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 190 = 2 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 190.e (of order \(3\), degree \(2\), 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: \(11.2103629011\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{7} + 35x^{6} + 307x^{5} + 1306x^{4} + 5584x^{3} + 49561x^{2} + 188916x + 348537 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (2 \beta_1 + 2) q^{2} + (\beta_{3} + \beta_{2} + \beta_1 + 1) q^{3} + 4 \beta_1 q^{4} + (5 \beta_1 + 5) q^{5} + (2 \beta_{3} + 2 \beta_1) q^{6} + (2 \beta_{5} - 2 \beta_{4} + \beta_{2} - 10) q^{7} - 8 q^{8} + ( - 3 \beta_{5} + 3 \beta_{3} + 19 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (2 \beta_1 + 2) q^{2} + (\beta_{3} + \beta_{2} + \beta_1 + 1) q^{3} + 4 \beta_1 q^{4} + (5 \beta_1 + 5) q^{5} + (2 \beta_{3} + 2 \beta_1) q^{6} + (2 \beta_{5} - 2 \beta_{4} + \beta_{2} - 10) q^{7} - 8 q^{8} + ( - 3 \beta_{5} + 3 \beta_{3} + 19 \beta_1) q^{9} + 10 \beta_1 q^{10} + (\beta_{7} + \beta_{6} - 2 \beta_{2} + 2) q^{11} + ( - 4 \beta_{2} - 4) q^{12} + (\beta_{7} + 4 \beta_{5} + \cdots + 2 \beta_1) q^{13}+ \cdots + (2 \beta_{7} + 27 \beta_{5} + \cdots - 238 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{2} + 3 q^{3} - 16 q^{4} + 20 q^{5} - 6 q^{6} - 82 q^{7} - 64 q^{8} - 73 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{2} + 3 q^{3} - 16 q^{4} + 20 q^{5} - 6 q^{6} - 82 q^{7} - 64 q^{8} - 73 q^{9} - 40 q^{10} + 20 q^{11} - 24 q^{12} - 11 q^{13} - 82 q^{14} - 15 q^{15} - 64 q^{16} - 81 q^{17} - 292 q^{18} + 219 q^{19} - 160 q^{20} + 84 q^{21} + 20 q^{22} + 75 q^{23} - 24 q^{24} - 100 q^{25} - 44 q^{26} - 828 q^{27} + 164 q^{28} - 28 q^{29} - 60 q^{30} - 370 q^{31} + 128 q^{32} - 367 q^{33} + 162 q^{34} - 205 q^{35} - 292 q^{36} + 678 q^{37} - 198 q^{38} + 766 q^{39} - 160 q^{40} - 5 q^{41} - 168 q^{42} + 349 q^{43} - 40 q^{44} - 730 q^{45} + 300 q^{46} - 132 q^{47} + 48 q^{48} - 90 q^{49} - 400 q^{50} + 8 q^{51} - 44 q^{52} + 4 q^{53} - 828 q^{54} + 50 q^{55} + 656 q^{56} - 731 q^{57} - 112 q^{58} + 1368 q^{59} - 60 q^{60} + 1197 q^{61} - 370 q^{62} - 784 q^{63} + 512 q^{64} - 110 q^{65} + 734 q^{66} - 1816 q^{67} + 648 q^{68} + 2942 q^{69} + 410 q^{70} + 137 q^{71} + 584 q^{72} - 1592 q^{73} + 678 q^{74} - 150 q^{75} - 1272 q^{76} - 958 q^{77} + 766 q^{78} + 398 q^{79} + 320 q^{80} + 596 q^{81} + 10 q^{82} + 3222 q^{83} - 672 q^{84} + 405 q^{85} - 698 q^{86} + 4672 q^{87} - 160 q^{88} - 495 q^{89} - 730 q^{90} + 1894 q^{91} + 300 q^{92} - 462 q^{93} - 528 q^{94} - 495 q^{95} + 192 q^{96} + 2087 q^{97} - 90 q^{98} + 944 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 3x^{7} + 35x^{6} + 307x^{5} + 1306x^{4} + 5584x^{3} + 49561x^{2} + 188916x + 348537 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 33221 \nu^{7} + 37084 \nu^{6} - 71291 \nu^{5} + 22573894 \nu^{4} + 20029432 \nu^{3} + \cdots + 6602711633 ) / 3049712078 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 1822 \nu^{7} + 7269 \nu^{6} - 81080 \nu^{5} - 457982 \nu^{4} - 2481116 \nu^{3} + \cdots - 348831637 ) / 42123095 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 770449 \nu^{7} - 996704 \nu^{6} + 28708341 \nu^{5} - 641246970 \nu^{4} + \cdots - 173780786291 ) / 15248560390 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 1288079 \nu^{7} + 5358846 \nu^{6} - 33523349 \nu^{5} - 411591530 \nu^{4} + \cdots - 82873489741 ) / 15248560390 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1343299 \nu^{7} - 8801508 \nu^{6} + 91226195 \nu^{5} + 49003514 \nu^{4} + 1443871362 \nu^{3} + \cdots + 53134753819 ) / 15248560390 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 2846117 \nu^{7} + 40021042 \nu^{6} - 309608169 \nu^{5} + 466353832 \nu^{4} + \cdots + 349448958061 ) / 15248560390 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 16852113 \nu^{7} + 101479690 \nu^{6} - 803222547 \nu^{5} - 3330093116 \nu^{4} + \cdots - 1198983257969 ) / 15248560390 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} - \beta _1 + 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} - \beta_{6} + 7\beta_{5} + 4\beta_{4} - 11\beta_{3} - 10\beta_{2} - 32\beta _1 - 40 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{7} + \beta_{6} - 16\beta_{5} + 38\beta_{4} - 31\beta_{3} - 104\beta_{2} - 217\beta _1 - 584 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -43\beta_{7} - 8\beta_{6} - 585\beta_{5} + 114\beta_{4} + 343\beta_{3} - 367\beta_{2} + 1249\beta _1 - 2374 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 292 \beta_{7} + 82 \beta_{6} - 3265 \beta_{5} - 1492 \beta_{4} + 5951 \beta_{3} + 2644 \beta_{2} + \cdots + 14635 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 471 \beta_{7} + 1887 \beta_{6} + 7175 \beta_{5} - 26260 \beta_{4} + 31945 \beta_{3} + 48797 \beta_{2} + \cdots + 246983 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 19800 \beta_{7} + 11772 \beta_{6} + 262678 \beta_{5} - 177119 \beta_{4} - 72328 \beta_{3} + \cdots + 1412095 ) / 3 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(21\) \(77\)
\(\chi(n)\) \(\beta_{1}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1
−3.30180 + 2.02322i
−2.40235 − 2.44561i
2.45602 − 5.04989i
4.74814 + 6.33831i
−3.30180 − 2.02322i
−2.40235 + 2.44561i
2.45602 + 5.04989i
4.74814 − 6.33831i
1.00000 − 1.73205i −3.30180 + 5.71889i −2.00000 − 3.46410i 2.50000 − 4.33013i 6.60360 + 11.4378i −3.99135 −8.00000 −8.30378 − 14.3826i −5.00000 − 8.66025i
11.2 1.00000 − 1.73205i −2.40235 + 4.16099i −2.00000 − 3.46410i 2.50000 − 4.33013i 4.80470 + 8.32199i −19.4719 −8.00000 1.95742 + 3.39035i −5.00000 − 8.66025i
11.3 1.00000 − 1.73205i 2.45602 − 4.25394i −2.00000 − 3.46410i 2.50000 − 4.33013i −4.91203 − 8.50789i −28.4933 −8.00000 1.43598 + 2.48719i −5.00000 − 8.66025i
11.4 1.00000 − 1.73205i 4.74814 − 8.22401i −2.00000 − 3.46410i 2.50000 − 4.33013i −9.49627 − 16.4480i 10.9565 −8.00000 −31.5896 − 54.7148i −5.00000 − 8.66025i
121.1 1.00000 + 1.73205i −3.30180 − 5.71889i −2.00000 + 3.46410i 2.50000 + 4.33013i 6.60360 − 11.4378i −3.99135 −8.00000 −8.30378 + 14.3826i −5.00000 + 8.66025i
121.2 1.00000 + 1.73205i −2.40235 − 4.16099i −2.00000 + 3.46410i 2.50000 + 4.33013i 4.80470 − 8.32199i −19.4719 −8.00000 1.95742 − 3.39035i −5.00000 + 8.66025i
121.3 1.00000 + 1.73205i 2.45602 + 4.25394i −2.00000 + 3.46410i 2.50000 + 4.33013i −4.91203 + 8.50789i −28.4933 −8.00000 1.43598 − 2.48719i −5.00000 + 8.66025i
121.4 1.00000 + 1.73205i 4.74814 + 8.22401i −2.00000 + 3.46410i 2.50000 + 4.33013i −9.49627 + 16.4480i 10.9565 −8.00000 −31.5896 + 54.7148i −5.00000 + 8.66025i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 190.4.e.d ✓ 8
19.c even 3 1 inner 190.4.e.d ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
190.4.e.d ✓ 8 1.a even 1 1 trivial
190.4.e.d ✓ 8 19.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(190, [\chi])\):

\( T_{3}^{8} - 3T_{3}^{7} + 95T_{3}^{6} + 108T_{3}^{5} + 6141T_{3}^{4} + 2430T_{3}^{3} + 132905T_{3}^{2} + 111000T_{3} + 2190400 \) Copy content Toggle raw display
\( T_{7}^{4} + 41T_{7}^{3} + 177T_{7}^{2} - 5962T_{7} - 24263 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 2 T + 4)^{4} \) Copy content Toggle raw display
$3$ \( T^{8} - 3 T^{7} + \cdots + 2190400 \) Copy content Toggle raw display
$5$ \( (T^{2} - 5 T + 25)^{4} \) Copy content Toggle raw display
$7$ \( (T^{4} + 41 T^{3} + \cdots - 24263)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} - 10 T^{3} + \cdots - 200979)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 3628133514756 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 2925796566016 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 22\!\cdots\!61 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 543074505073681 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 10\!\cdots\!76 \) Copy content Toggle raw display
$31$ \( (T^{4} + 185 T^{3} + \cdots + 632774044)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 339 T^{3} + \cdots - 4709132909)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 57226153651209 \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 986392488216996 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 27\!\cdots\!44 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 11\!\cdots\!41 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 53\!\cdots\!04 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 56\!\cdots\!56 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 39\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 64\!\cdots\!56 \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 48\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 93\!\cdots\!84 \) Copy content Toggle raw display
$83$ \( (T^{4} - 1611 T^{3} + \cdots - 360734716526)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 31\!\cdots\!76 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 32\!\cdots\!04 \) Copy content Toggle raw display
show more
show less