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Properties

Label 153.10.d.b
Level $153$
Weight $10$
Character orbit 153.d
Analytic conductor $78.800$
Analytic rank $0$
Dimension $12$
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] = [153,10,Mod(118,153)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(153, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("153.118");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 153 = 3^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 153.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: \(78.8004829331\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 122690 x^{10} + 5157152560 x^{8} + 87983684680032 x^{6} + \cdots + 20\!\cdots\!28 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{17}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 17)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{2} - 2) q^{2} + ( - \beta_{3} + 4 \beta_{2} + 154) q^{4} + (\beta_{8} + \beta_1) q^{5} + ( - \beta_{9} - \beta_1) q^{7} + (\beta_{5} + 3 \beta_{3} - 117 \beta_{2} - 1904) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} - 2) q^{2} + ( - \beta_{3} + 4 \beta_{2} + 154) q^{4} + (\beta_{8} + \beta_1) q^{5} + ( - \beta_{9} - \beta_1) q^{7} + (\beta_{5} + 3 \beta_{3} - 117 \beta_{2} - 1904) q^{8} + (\beta_{11} + \beta_{9} + \cdots - 2 \beta_1) q^{10}+ \cdots + ( - 2770 \beta_{6} + 17133 \beta_{5} + \cdots + 26424984) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 30 q^{2} + 1874 q^{4} - 23550 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 30 q^{2} + 1874 q^{4} - 23550 q^{8} - 63204 q^{13} + 38978 q^{16} + 105960 q^{17} + 1110672 q^{19} - 4441796 q^{25} - 1336332 q^{26} + 1934850 q^{32} - 15085546 q^{34} - 3519864 q^{35} - 28748136 q^{38} + 10004616 q^{43} + 112552440 q^{47} + 121354720 q^{49} + 164889018 q^{50} - 59093180 q^{52} - 76804272 q^{53} + 300732568 q^{55} - 11618904 q^{59} - 260062974 q^{64} - 304208752 q^{67} + 444301206 q^{68} + 460311456 q^{70} + 416024248 q^{76} - 138357828 q^{77} + 845042136 q^{83} - 388949632 q^{85} - 127952904 q^{86} + 938223804 q^{89} - 238629952 q^{94} + 152046078 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 122690 x^{10} + 5157152560 x^{8} + 87983684680032 x^{6} + \cdots + 20\!\cdots\!28 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 17\!\cdots\!43 \nu^{10} + \cdots + 13\!\cdots\!24 ) / 33\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 15\!\cdots\!99 \nu^{10} + \cdots - 88\!\cdots\!08 ) / 11\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 96\!\cdots\!67 \nu^{10} + \cdots + 13\!\cdots\!36 ) / 12\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 54\!\cdots\!05 \nu^{10} + \cdots - 99\!\cdots\!48 ) / 66\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 28\!\cdots\!77 \nu^{10} + \cdots + 39\!\cdots\!16 ) / 33\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 17\!\cdots\!43 \nu^{11} + \cdots - 13\!\cdots\!84 \nu ) / 33\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 68\!\cdots\!89 \nu^{11} + \cdots + 91\!\cdots\!72 \nu ) / 44\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 19\!\cdots\!13 \nu^{11} + \cdots - 76\!\cdots\!56 \nu ) / 44\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 42\!\cdots\!39 \nu^{11} + \cdots + 39\!\cdots\!52 \nu ) / 82\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 28\!\cdots\!79 \nu^{11} + \cdots - 30\!\cdots\!52 \nu ) / 49\!\cdots\!60 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} - 8\beta_{3} + 69\beta_{2} - 20484 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -16\beta_{11} + 17\beta_{10} - 117\beta_{9} - 721\beta_{8} + 392\beta_{7} - 38585\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -255\beta_{6} + 1818\beta_{5} - 52846\beta_{4} + 398660\beta_{3} + 4044453\beta_{2} + 787789536 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 844672 \beta_{11} - 957386 \beta_{10} + 5636790 \beta_{9} + 38978278 \beta_{8} - 28327880 \beta_{7} + 1729287230 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 669498 \beta_{6} - 317420460 \beta_{5} + 2581679884 \beta_{4} - 19478780120 \beta_{3} + \cdots - 35253701400576 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 33315004576 \beta_{11} + 52033077140 \beta_{10} - 257894873532 \beta_{9} + \cdots - 81193337780012 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 828626755356 \beta_{6} + 27163756037592 \beta_{5} - 125092695479896 \beta_{4} + 917219353093424 \beta_{3} + \cdots + 16\!\cdots\!88 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 11\!\cdots\!80 \beta_{11} + \cdots + 38\!\cdots\!04 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 82\!\cdots\!32 \beta_{6} + \cdots - 78\!\cdots\!76 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 29\!\cdots\!40 \beta_{11} + \cdots - 18\!\cdots\!12 \beta_1 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(37\) \(137\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
118.1
− 12.8394i
12.8394i
− 225.146i
225.146i
59.5904i
− 59.5904i
105.759i
− 105.759i
− 206.667i
206.667i
119.947i
− 119.947i
−39.2436 0 1028.06 − 2413.73i 0 − 2302.99i −20252.0 0 94723.3i
118.2 −39.2436 0 1028.06 2413.73i 0 2302.99i −20252.0 0 − 94723.3i
118.3 −25.8215 0 154.751 − 96.4328i 0 1385.77i 9224.72 0 2490.04i
118.4 −25.8215 0 154.751 96.4328i 0 − 1385.77i 9224.72 0 − 2490.04i
118.5 −11.8575 0 −371.399 − 633.821i 0 10932.1i 10474.9 0 7515.55i
118.6 −11.8575 0 −371.399 633.821i 0 − 10932.1i 10474.9 0 − 7515.55i
118.7 9.15386 0 −428.207 − 1752.99i 0 − 5869.78i −8606.52 0 − 16046.7i
118.8 9.15386 0 −428.207 1752.99i 0 5869.78i −8606.52 0 16046.7i
118.9 16.7531 0 −231.335 − 1946.29i 0 1633.30i −12453.1 0 − 32606.3i
118.10 16.7531 0 −231.335 1946.29i 0 − 1633.30i −12453.1 0 32606.3i
118.11 36.0157 0 785.131 − 917.335i 0 4193.73i 9837.00 0 − 33038.5i
118.12 36.0157 0 785.131 917.335i 0 − 4193.73i 9837.00 0 33038.5i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 118.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 153.10.d.b 12
3.b odd 2 1 17.10.b.a ✓ 12
12.b even 2 1 272.10.b.c 12
17.b even 2 1 inner 153.10.d.b 12
51.c odd 2 1 17.10.b.a ✓ 12
51.f odd 4 2 289.10.a.c 12
204.h even 2 1 272.10.b.c 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
17.10.b.a ✓ 12 3.b odd 2 1
17.10.b.a ✓ 12 51.c odd 2 1
153.10.d.b 12 1.a even 1 1 trivial
153.10.d.b 12 17.b even 2 1 inner
272.10.b.c 12 12.b even 2 1
272.10.b.c 12 204.h even 2 1
289.10.a.c 12 51.f odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} + 15T_{2}^{5} - 1892T_{2}^{4} - 20460T_{2}^{3} + 770176T_{2}^{2} + 3195840T_{2} - 66364416 \) acting on \(S_{10}^{\mathrm{new}}(153, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} + 15 T^{5} + \cdots - 66364416)^{2} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 72\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{6} + \cdots - 63\!\cdots\!00)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 27\!\cdots\!29 \) Copy content Toggle raw display
$19$ \( (T^{6} + \cdots + 83\!\cdots\!00)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 16\!\cdots\!88 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 89\!\cdots\!52 \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 53\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 81\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 55\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{6} + \cdots + 77\!\cdots\!24)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} + \cdots + 48\!\cdots\!36)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + \cdots + 11\!\cdots\!28)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + \cdots + 19\!\cdots\!80)^{2} \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( (T^{6} + \cdots - 20\!\cdots\!40)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 27\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 15\!\cdots\!32 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 23\!\cdots\!92 \) Copy content Toggle raw display
$83$ \( (T^{6} + \cdots - 80\!\cdots\!24)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots + 17\!\cdots\!80)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 48\!\cdots\!52 \) Copy content Toggle raw display
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