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Properties

Label 153.4.f.b
Level $153$
Weight $4$
Character orbit 153.f
Analytic conductor $9.027$
Analytic rank $0$
Dimension $16$
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] = [153,4,Mod(55,153)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(153, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("153.55");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 153 = 3^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 153.f (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: \(9.02729223088\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 88 x^{14} + 3086 x^{12} + 54880 x^{10} + 516641 x^{8} + 2403800 x^{6} + 4378064 x^{4} + \cdots + 295936 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{4} \)
Twist minimal: no (minimal twist has level 51)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{5} + \beta_1) q^{2} + (\beta_{4} + \beta_{3} - 4) q^{4} + (\beta_{9} + 2 \beta_{5} + 2) q^{5} + (\beta_{14} + \beta_{10} - \beta_{6} + \cdots - 2) q^{7}+ \cdots + ( - \beta_{15} + 2 \beta_{14} + \cdots - \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{5} + \beta_1) q^{2} + (\beta_{4} + \beta_{3} - 4) q^{4} + (\beta_{9} + 2 \beta_{5} + 2) q^{5} + (\beta_{14} + \beta_{10} - \beta_{6} + \cdots - 2) q^{7}+ \cdots + (19 \beta_{13} + 19 \beta_{12} + \cdots + 121) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 48 q^{4} + 32 q^{5} - 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 48 q^{4} + 32 q^{5} - 8 q^{7} - 20 q^{10} - 96 q^{11} + 120 q^{13} + 288 q^{14} - 200 q^{16} - 16 q^{17} - 384 q^{20} + 4 q^{22} - 208 q^{23} + 904 q^{28} - 320 q^{29} - 624 q^{31} - 788 q^{34} + 1184 q^{35} + 256 q^{37} + 168 q^{38} - 664 q^{40} - 800 q^{41} - 128 q^{44} + 572 q^{46} - 368 q^{47} + 720 q^{50} - 3160 q^{52} + 4296 q^{55} - 576 q^{56} + 1368 q^{58} - 1296 q^{61} - 536 q^{62} + 1752 q^{64} + 4288 q^{65} - 1248 q^{67} + 240 q^{68} - 1488 q^{71} - 1208 q^{73} - 2272 q^{74} - 1256 q^{79} - 384 q^{80} + 3564 q^{82} + 3216 q^{85} - 3224 q^{86} - 1296 q^{88} + 2608 q^{89} - 56 q^{91} + 5072 q^{92} - 160 q^{95} - 4456 q^{97} + 1208 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 88 x^{14} + 3086 x^{12} + 54880 x^{10} + 516641 x^{8} + 2403800 x^{6} + 4378064 x^{4} + \cdots + 295936 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 666831 \nu^{14} - 53499640 \nu^{12} - 1665415346 \nu^{10} - 25354867584 \nu^{8} + \cdots - 167038788224 ) / 8801563136 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 7957 \nu^{14} - 607120 \nu^{12} - 17482150 \nu^{10} - 234485168 \nu^{8} - 1435946037 \nu^{6} + \cdots - 404729472 ) / 73653248 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 7957 \nu^{14} - 607120 \nu^{12} - 17482150 \nu^{10} - 234485168 \nu^{8} - 1435946037 \nu^{6} + \cdots + 405456256 ) / 73653248 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 10941 \nu^{15} - 899152 \nu^{13} - 28906966 \nu^{11} - 460584880 \nu^{9} + \cdots - 5760768640 \nu ) / 589225984 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 1925903 \nu^{15} - 48451320 \nu^{14} - 156417744 \nu^{13} - 3624189376 \nu^{12} + \cdots + 1456215112704 ) / 140825010176 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 5364767 \nu^{15} - 2458608 \nu^{14} - 370808464 \nu^{13} - 182547904 \nu^{12} + \cdots + 134062280704 ) / 140825010176 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 5364767 \nu^{15} - 2458608 \nu^{14} + 370808464 \nu^{13} - 182547904 \nu^{12} + \cdots + 134062280704 ) / 140825010176 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 1925903 \nu^{15} + 48451320 \nu^{14} - 156417744 \nu^{13} + 3624189376 \nu^{12} + \cdots - 1456215112704 ) / 140825010176 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 1527211 \nu^{15} - 7458614 \nu^{14} - 112979168 \nu^{13} - 569781248 \nu^{12} + \cdots - 634907182848 ) / 35206252544 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 1527211 \nu^{15} + 7458614 \nu^{14} - 112979168 \nu^{13} + 569781248 \nu^{12} + \cdots + 634907182848 ) / 35206252544 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 8632335 \nu^{15} + 13129576 \nu^{14} + 668604528 \nu^{13} + 1097115264 \nu^{12} + \cdots + 8557361779712 ) / 140825010176 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 8632335 \nu^{15} + 13129576 \nu^{14} - 668604528 \nu^{13} + 1097115264 \nu^{12} + \cdots + 8557361779712 ) / 140825010176 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 56695 \nu^{15} + 5033712 \nu^{13} + 178119426 \nu^{11} + 3190552336 \nu^{9} + \cdots + 59541393280 \nu ) / 589225984 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 14087039 \nu^{15} + 1108132528 \nu^{13} + 33516210802 \nu^{11} + 489555293200 \nu^{9} + \cdots + 3022126939008 \nu ) / 140825010176 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} - \beta_{3} - 11 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{15} - \beta_{14} - \beta_{11} - \beta_{10} - \beta_{8} + \beta_{7} - 10\beta_{5} - 17\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2 \beta_{11} - 2 \beta_{10} + \beta_{9} + 5 \beta_{8} + 5 \beta_{7} - \beta_{6} - 19 \beta_{4} + \cdots + 187 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 25 \beta_{15} + 37 \beta_{14} - 3 \beta_{13} + 3 \beta_{12} + 32 \beta_{11} + 32 \beta_{10} + \cdots + 323 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 11 \beta_{13} + 11 \beta_{12} - 99 \beta_{11} + 99 \beta_{10} - 24 \beta_{9} - 192 \beta_{8} + \cdots - 3563 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 557 \beta_{15} - 1097 \beta_{14} + 118 \beta_{13} - 118 \beta_{12} - 895 \beta_{11} + \cdots - 6665 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 520 \beta_{13} - 520 \beta_{12} + 3430 \beta_{11} - 3430 \beta_{10} + 421 \beta_{9} + 5749 \beta_{8} + \cdots + 73667 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 12129 \beta_{15} + 30089 \beta_{14} - 3759 \beta_{13} + 3759 \beta_{12} + 24140 \beta_{11} + \cdots + 146567 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 17907 \beta_{13} + 17907 \beta_{12} - 104103 \beta_{11} + 104103 \beta_{10} - 5368 \beta_{9} + \cdots - 1624787 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 264501 \beta_{15} - 797133 \beta_{14} + 110502 \beta_{13} - 110502 \beta_{12} - 641655 \beta_{11} + \cdots - 3380181 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 545852 \beta_{13} - 545852 \beta_{12} + 2962854 \beta_{11} - 2962854 \beta_{10} + 13385 \beta_{9} + \cdots + 37612891 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 5840809 \beta_{15} + 20771885 \beta_{14} - 3111635 \beta_{13} + 3111635 \beta_{12} + \cdots + 80658763 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 15644299 \beta_{13} + 15644299 \beta_{12} - 81371195 \beta_{11} + 81371195 \beta_{10} + 2290008 \beta_{9} + \cdots - 901149547 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 131341581 \beta_{15} - 537060769 \beta_{14} + 85337894 \beta_{13} - 85337894 \beta_{12} + \cdots - 1971047633 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/153\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(137\)
\(\chi(n)\) \(\beta_{5}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
55.1
− 5.08109i
− 4.35402i
− 3.48829i
− 0.635039i
0.496675i
1.61897i
3.51137i
3.93142i
− 3.93142i
− 3.51137i
− 1.61897i
− 0.496675i
0.635039i
3.48829i
4.35402i
5.08109i
− 4.08109i 0 −8.65531 5.03574 − 5.03574i 0 −6.78004 − 6.78004i 2.67440i 0 −20.5513 − 20.5513i
55.2 − 3.35402i 0 −3.24945 11.9082 − 11.9082i 0 17.4318 + 17.4318i − 15.9334i 0 −39.9404 − 39.9404i
55.3 − 2.48829i 0 1.80841 −9.24941 + 9.24941i 0 −3.11013 − 3.11013i − 24.4062i 0 23.0152 + 23.0152i
55.4 0.364961i 0 7.86680 7.07138 − 7.07138i 0 19.0146 + 19.0146i 5.79077i 0 2.58078 + 2.58078i
55.5 1.49668i 0 5.75996 −0.574777 + 0.574777i 0 2.73692 + 2.73692i 20.5942i 0 −0.860255 − 0.860255i
55.6 2.61897i 0 1.14098 −9.86835 + 9.86835i 0 −11.6609 − 11.6609i 23.9400i 0 −25.8450 − 25.8450i
55.7 4.51137i 0 −12.3525 14.2465 − 14.2465i 0 −12.4732 − 12.4732i − 19.6356i 0 64.2713 + 64.2713i
55.8 4.93142i 0 −16.3189 −2.56932 + 2.56932i 0 −9.15912 − 9.15912i − 41.0241i 0 −12.6704 − 12.6704i
64.1 − 4.93142i 0 −16.3189 −2.56932 − 2.56932i 0 −9.15912 + 9.15912i 41.0241i 0 −12.6704 + 12.6704i
64.2 − 4.51137i 0 −12.3525 14.2465 + 14.2465i 0 −12.4732 + 12.4732i 19.6356i 0 64.2713 − 64.2713i
64.3 − 2.61897i 0 1.14098 −9.86835 − 9.86835i 0 −11.6609 + 11.6609i − 23.9400i 0 −25.8450 + 25.8450i
64.4 − 1.49668i 0 5.75996 −0.574777 − 0.574777i 0 2.73692 − 2.73692i − 20.5942i 0 −0.860255 + 0.860255i
64.5 − 0.364961i 0 7.86680 7.07138 + 7.07138i 0 19.0146 − 19.0146i − 5.79077i 0 2.58078 − 2.58078i
64.6 2.48829i 0 1.80841 −9.24941 − 9.24941i 0 −3.11013 + 3.11013i 24.4062i 0 23.0152 − 23.0152i
64.7 3.35402i 0 −3.24945 11.9082 + 11.9082i 0 17.4318 − 17.4318i 15.9334i 0 −39.9404 + 39.9404i
64.8 4.08109i 0 −8.65531 5.03574 + 5.03574i 0 −6.78004 + 6.78004i − 2.67440i 0 −20.5513 + 20.5513i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 55.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 153.4.f.b 16
3.b odd 2 1 51.4.e.a ✓ 16
17.c even 4 1 inner 153.4.f.b 16
51.f odd 4 1 51.4.e.a ✓ 16
51.g odd 8 1 867.4.a.p 8
51.g odd 8 1 867.4.a.q 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
51.4.e.a ✓ 16 3.b odd 2 1
51.4.e.a ✓ 16 51.f odd 4 1
153.4.f.b 16 1.a even 1 1 trivial
153.4.f.b 16 17.c even 4 1 inner
867.4.a.p 8 51.g odd 8 1
867.4.a.q 8 51.g odd 8 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{16} + 88 T_{2}^{14} + 3122 T_{2}^{12} + 57388 T_{2}^{10} + 584697 T_{2}^{8} + 3268788 T_{2}^{6} + \cdots + 1175056 \) acting on \(S_{4}^{\mathrm{new}}(153, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + 88 T^{14} + \cdots + 1175056 \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 169761199211536 \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 16\!\cdots\!36 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 38\!\cdots\!64 \) Copy content Toggle raw display
$13$ \( (T^{8} - 60 T^{7} + \cdots + 193312031744)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 33\!\cdots\!21 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 39\!\cdots\!56 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 12\!\cdots\!64 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 45\!\cdots\!84 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 14\!\cdots\!36 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 19\!\cdots\!36 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 12\!\cdots\!04 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 12\!\cdots\!16 \) Copy content Toggle raw display
$47$ \( (T^{8} + \cdots - 87\!\cdots\!96)^{2} \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 41\!\cdots\!96 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 10\!\cdots\!44 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 40\!\cdots\!84 \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 11\!\cdots\!76)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 36\!\cdots\!36 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 17\!\cdots\!76 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 16\!\cdots\!64 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 32\!\cdots\!04 \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 34\!\cdots\!92)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 41\!\cdots\!84 \) Copy content Toggle raw display
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