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Properties

Label 190.3.g.b
Level $190$
Weight $3$
Character orbit 190.g
Analytic conductor $5.177$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [190,3,Mod(77,190)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(190, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("190.77");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 190 = 2 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 190.g (of order \(4\), 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: \(5.17712502285\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 12 x^{17} + 1656 x^{16} - 232 x^{15} + 72 x^{14} - 15472 x^{13} + 721008 x^{12} + \cdots + 10653696 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{14}\cdot 5 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{3} + 1) q^{2} - \beta_{2} q^{3} + 2 \beta_{3} q^{4} + \beta_{16} q^{5} + ( - \beta_{2} - \beta_1) q^{6} + (\beta_{15} - \beta_{12} + \beta_{3} + 1) q^{7} + (2 \beta_{3} - 2) q^{8} + ( - \beta_{17} - \beta_{15} + \cdots - \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} + 1) q^{2} - \beta_{2} q^{3} + 2 \beta_{3} q^{4} + \beta_{16} q^{5} + ( - \beta_{2} - \beta_1) q^{6} + (\beta_{15} - \beta_{12} + \beta_{3} + 1) q^{7} + (2 \beta_{3} - 2) q^{8} + ( - \beta_{17} - \beta_{15} + \cdots - \beta_{2}) q^{9}+ \cdots + (4 \beta_{19} + 6 \beta_{18} + \cdots - 5 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 20 q^{2} + 2 q^{5} + 14 q^{7} - 40 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 20 q^{2} + 2 q^{5} + 14 q^{7} - 40 q^{8} + 2 q^{10} - 8 q^{11} - 24 q^{13} + 16 q^{15} - 80 q^{16} + 22 q^{17} + 92 q^{18} - 80 q^{21} - 8 q^{22} - 24 q^{23} - 2 q^{25} - 48 q^{26} - 36 q^{27} - 28 q^{28} + 68 q^{30} + 176 q^{31} - 80 q^{32} + 40 q^{33} - 78 q^{35} + 184 q^{36} - 4 q^{40} - 136 q^{41} - 80 q^{42} + 38 q^{43} - 40 q^{45} - 48 q^{46} + 110 q^{47} - 18 q^{50} + 464 q^{51} - 48 q^{52} - 116 q^{53} - 64 q^{55} - 56 q^{56} + 104 q^{60} - 160 q^{61} + 176 q^{62} + 274 q^{63} - 76 q^{65} + 80 q^{66} - 64 q^{67} - 44 q^{68} - 254 q^{70} + 184 q^{72} + 78 q^{73} + 288 q^{75} - 294 q^{77} - 184 q^{78} - 8 q^{80} - 900 q^{81} - 136 q^{82} - 48 q^{83} - 148 q^{85} + 76 q^{86} - 496 q^{87} + 16 q^{88} - 90 q^{90} - 200 q^{91} - 48 q^{92} - 208 q^{93} + 38 q^{95} + 556 q^{97} - 148 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 12 x^{17} + 1656 x^{16} - 232 x^{15} + 72 x^{14} - 15472 x^{13} + 721008 x^{12} + \cdots + 10653696 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 83\!\cdots\!33 \nu^{19} + \cdots + 28\!\cdots\!20 ) / 45\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 55\!\cdots\!55 \nu^{19} + \cdots - 19\!\cdots\!92 ) / 92\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 10\!\cdots\!29 \nu^{19} + \cdots + 19\!\cdots\!60 ) / 16\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 29\!\cdots\!84 \nu^{19} + \cdots + 15\!\cdots\!40 ) / 26\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 22\!\cdots\!93 \nu^{19} + \cdots + 61\!\cdots\!32 ) / 16\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 54\!\cdots\!13 \nu^{19} + \cdots - 54\!\cdots\!88 ) / 32\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 30\!\cdots\!11 \nu^{19} + \cdots - 98\!\cdots\!16 ) / 16\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 15\!\cdots\!09 \nu^{19} + \cdots + 27\!\cdots\!32 ) / 80\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 64\!\cdots\!43 \nu^{19} + \cdots + 57\!\cdots\!28 ) / 32\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 10\!\cdots\!81 \nu^{19} + \cdots + 89\!\cdots\!16 ) / 32\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 54\!\cdots\!55 \nu^{19} + \cdots - 21\!\cdots\!64 ) / 10\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 10\!\cdots\!19 \nu^{19} + \cdots - 41\!\cdots\!64 ) / 17\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 29\!\cdots\!88 \nu^{19} + \cdots - 28\!\cdots\!20 ) / 40\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 26\!\cdots\!95 \nu^{19} + \cdots - 83\!\cdots\!32 ) / 32\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 92\!\cdots\!65 \nu^{19} + \cdots - 15\!\cdots\!44 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 11\!\cdots\!35 \nu^{19} + \cdots + 19\!\cdots\!28 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 98\!\cdots\!33 \nu^{19} + \cdots - 10\!\cdots\!04 ) / 80\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 43\!\cdots\!65 \nu^{19} + \cdots + 80\!\cdots\!04 ) / 16\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{17} + \beta_{15} - \beta_{12} + \beta_{11} + \beta_{7} + \beta_{5} + 14\beta_{3} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{18} - \beta_{16} + \beta_{14} - \beta_{12} + 4 \beta_{11} + 3 \beta_{9} + \beta_{8} + 3 \beta_{7} + \cdots + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 5 \beta_{19} + 34 \beta_{18} - 34 \beta_{17} - 32 \beta_{15} - 38 \beta_{14} - 12 \beta_{13} + \cdots - 346 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 56 \beta_{19} + 17 \beta_{18} - 18 \beta_{17} + 13 \beta_{16} + 14 \beta_{15} + 33 \beta_{14} + \cdots + 11 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 214 \beta_{19} - \beta_{18} - 1065 \beta_{17} - 218 \beta_{16} - 1005 \beta_{15} + 2000 \beta_{12} + \cdots - 95 \beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 1036 \beta_{18} - 84 \beta_{16} - 908 \beta_{14} - 576 \beta_{13} + 2828 \beta_{12} - 5416 \beta_{11} + \cdots + 436 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 7272 \beta_{19} - 30796 \beta_{18} + 32792 \beta_{17} + 1892 \beta_{16} + 31596 \beta_{15} + \cdots + 277180 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 87364 \beta_{19} - 45216 \beta_{18} + 65436 \beta_{17} - 23296 \beta_{16} + 24680 \beta_{15} + \cdots + 43736 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 231876 \beta_{19} - 3612 \beta_{18} + 1007364 \beta_{17} + 354864 \beta_{16} + 993372 \beta_{15} + \cdots + 168356 \beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 1092596 \beta_{18} + 675932 \beta_{16} + 578580 \beta_{14} + 873944 \beta_{13} - 3800788 \beta_{12} + \cdots - 2398396 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 7261760 \beta_{19} + 27614140 \beta_{18} - 31025836 \beta_{17} - 3165400 \beta_{16} - 31223748 \beta_{15} + \cdots - 249685040 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 105085488 \beta_{19} + 64140656 \beta_{18} - 101389984 \beta_{17} + 27732624 \beta_{16} + \cdots - 107896688 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 226981728 \beta_{19} + 2962208 \beta_{18} - 959615504 \beta_{17} - 430371136 \beta_{16} + \cdots - 260103712 \beta_1 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 1224102096 \beta_{18} - 974596400 \beta_{16} - 285914704 \beta_{14} - 985533568 \beta_{13} + \cdots + 4386835696 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 7124064752 \beta_{19} - 25767489984 \beta_{18} + 29818557440 \beta_{17} + 3955835776 \beta_{16} + \cdots + 235733278496 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 115815986752 \beta_{19} - 77260518576 \beta_{18} + 126650109984 \beta_{17} - 30209914224 \beta_{16} + \cdots + 167831132528 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 224978353376 \beta_{19} - 52664976 \beta_{18} + 930763580656 \beta_{17} + 469520714080 \beta_{16} + \cdots + 355026238608 \beta_1 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 1378193737984 \beta_{18} + 1141645436928 \beta_{16} + 36468270336 \beta_{14} + 1034604399232 \beta_{13} + \cdots - 6169679785472 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(21\) \(77\)
\(\chi(n)\) \(1\) \(\beta_{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
77.1
4.00973 + 4.00973i
3.17941 + 3.17941i
2.42472 + 2.42472i
0.792805 + 0.792805i
0.414060 + 0.414060i
0.245041 + 0.245041i
−1.28247 − 1.28247i
−2.27436 − 2.27436i
−3.57482 − 3.57482i
−3.93413 − 3.93413i
4.00973 − 4.00973i
3.17941 − 3.17941i
2.42472 − 2.42472i
0.792805 − 0.792805i
0.414060 − 0.414060i
0.245041 − 0.245041i
−1.28247 + 1.28247i
−2.27436 + 2.27436i
−3.57482 + 3.57482i
−3.93413 + 3.93413i
1.00000 + 1.00000i −4.00973 + 4.00973i 2.00000i −3.26192 + 3.78944i −8.01946 8.63917 + 8.63917i −2.00000 + 2.00000i − 23.1559i −7.05136 + 0.527527i
77.2 1.00000 + 1.00000i −3.17941 + 3.17941i 2.00000i −1.30090 − 4.82780i −6.35882 −5.76288 − 5.76288i −2.00000 + 2.00000i − 11.2173i 3.52690 − 6.12870i
77.3 1.00000 + 1.00000i −2.42472 + 2.42472i 2.00000i 4.59856 + 1.96296i −4.84945 1.10345 + 1.10345i −2.00000 + 2.00000i − 2.75857i 2.63560 + 6.56152i
77.4 1.00000 + 1.00000i −0.792805 + 0.792805i 2.00000i −0.900251 − 4.91829i −1.58561 7.77008 + 7.77008i −2.00000 + 2.00000i 7.74292i 4.01804 − 5.81854i
77.5 1.00000 + 1.00000i −0.414060 + 0.414060i 2.00000i −4.81549 + 1.34577i −0.828119 −4.76033 − 4.76033i −2.00000 + 2.00000i 8.65711i −6.16125 − 3.46972i
77.6 1.00000 + 1.00000i −0.245041 + 0.245041i 2.00000i 2.59478 + 4.27400i −0.490082 −0.856777 − 0.856777i −2.00000 + 2.00000i 8.87991i −1.67922 + 6.86878i
77.7 1.00000 + 1.00000i 1.28247 − 1.28247i 2.00000i 4.48082 − 2.21862i 2.56494 −1.86333 − 1.86333i −2.00000 + 2.00000i 5.71054i 6.69944 + 2.26219i
77.8 1.00000 + 1.00000i 2.27436 − 2.27436i 2.00000i −2.10328 + 4.53610i 4.54872 4.65675 + 4.65675i −2.00000 + 2.00000i − 1.34543i −6.63938 + 2.43283i
77.9 1.00000 + 1.00000i 3.57482 − 3.57482i 2.00000i −3.28917 − 3.76582i 7.14963 −6.53779 − 6.53779i −2.00000 + 2.00000i − 16.5586i 0.476651 − 7.05498i
77.10 1.00000 + 1.00000i 3.93413 − 3.93413i 2.00000i 4.99684 − 0.177752i 7.86825 4.61165 + 4.61165i −2.00000 + 2.00000i − 21.9547i 5.17459 + 4.81909i
153.1 1.00000 − 1.00000i −4.00973 − 4.00973i − 2.00000i −3.26192 − 3.78944i −8.01946 8.63917 − 8.63917i −2.00000 − 2.00000i 23.1559i −7.05136 − 0.527527i
153.2 1.00000 − 1.00000i −3.17941 − 3.17941i − 2.00000i −1.30090 + 4.82780i −6.35882 −5.76288 + 5.76288i −2.00000 − 2.00000i 11.2173i 3.52690 + 6.12870i
153.3 1.00000 − 1.00000i −2.42472 − 2.42472i − 2.00000i 4.59856 − 1.96296i −4.84945 1.10345 − 1.10345i −2.00000 − 2.00000i 2.75857i 2.63560 − 6.56152i
153.4 1.00000 − 1.00000i −0.792805 − 0.792805i − 2.00000i −0.900251 + 4.91829i −1.58561 7.77008 − 7.77008i −2.00000 − 2.00000i − 7.74292i 4.01804 + 5.81854i
153.5 1.00000 − 1.00000i −0.414060 − 0.414060i − 2.00000i −4.81549 − 1.34577i −0.828119 −4.76033 + 4.76033i −2.00000 − 2.00000i − 8.65711i −6.16125 + 3.46972i
153.6 1.00000 − 1.00000i −0.245041 − 0.245041i − 2.00000i 2.59478 − 4.27400i −0.490082 −0.856777 + 0.856777i −2.00000 − 2.00000i − 8.87991i −1.67922 − 6.86878i
153.7 1.00000 − 1.00000i 1.28247 + 1.28247i − 2.00000i 4.48082 + 2.21862i 2.56494 −1.86333 + 1.86333i −2.00000 − 2.00000i − 5.71054i 6.69944 − 2.26219i
153.8 1.00000 − 1.00000i 2.27436 + 2.27436i − 2.00000i −2.10328 − 4.53610i 4.54872 4.65675 − 4.65675i −2.00000 − 2.00000i 1.34543i −6.63938 − 2.43283i
153.9 1.00000 − 1.00000i 3.57482 + 3.57482i − 2.00000i −3.28917 + 3.76582i 7.14963 −6.53779 + 6.53779i −2.00000 − 2.00000i 16.5586i 0.476651 + 7.05498i
153.10 1.00000 − 1.00000i 3.93413 + 3.93413i − 2.00000i 4.99684 + 0.177752i 7.86825 4.61165 − 4.61165i −2.00000 − 2.00000i 21.9547i 5.17459 − 4.81909i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 77.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 190.3.g.b ✓ 20
5.c odd 4 1 inner 190.3.g.b ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
190.3.g.b ✓ 20 1.a even 1 1 trivial
190.3.g.b ✓ 20 5.c odd 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{20} + 12 T_{3}^{17} + 1656 T_{3}^{16} + 232 T_{3}^{15} + 72 T_{3}^{14} + 15472 T_{3}^{13} + \cdots + 10653696 \) acting on \(S_{3}^{\mathrm{new}}(190, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 2 T + 2)^{10} \) Copy content Toggle raw display
$3$ \( T^{20} + 12 T^{17} + \cdots + 10653696 \) Copy content Toggle raw display
$5$ \( T^{20} + \cdots + 95367431640625 \) Copy content Toggle raw display
$7$ \( T^{20} + \cdots + 212426194223104 \) Copy content Toggle raw display
$11$ \( (T^{10} + 4 T^{9} + \cdots - 496750720)^{2} \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots + 89\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 26\!\cdots\!36 \) Copy content Toggle raw display
$19$ \( (T^{2} + 19)^{10} \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 82\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{10} + \cdots + 2212510269440)^{2} \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 54\!\cdots\!04 \) Copy content Toggle raw display
$41$ \( (T^{10} + \cdots + 47\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 19\!\cdots\!76 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 55\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 10\!\cdots\!24 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{10} + \cdots + 29\!\cdots\!00)^{2} \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 24\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( (T^{10} + \cdots + 13\!\cdots\!40)^{2} \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 65\!\cdots\!84 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 20\!\cdots\!16 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
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