Properties

Label 153.10.a.f
Level $153$
Weight $10$
Character orbit 153.a
Self dual yes
Analytic conductor $78.800$
Analytic rank $0$
Dimension $7$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [153,10,Mod(1,153)]
 
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, 0]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("153.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 153 = 3^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 153.a (trivial)

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: yes
Analytic conductor: \(78.8004829331\)
Analytic rank: \(0\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - x^{6} - 2986x^{5} + 8252x^{4} + 2252056x^{3} - 10388768x^{2} - 243559296x - 675998208 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{5} \)
Twist minimal: no (minimal twist has level 17)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} - 3 \beta_1 + 342) q^{4} + ( - \beta_{4} + \beta_{3} + \beta_{2} + 26 \beta_1 - 198) q^{5} + (6 \beta_{6} - 7 \beta_{5} - 6 \beta_{4} + 3 \beta_{3} - 3 \beta_{2} - 69 \beta_1 + 1349) q^{7} + (8 \beta_{6} - 17 \beta_{5} + 10 \beta_{3} - \beta_{2} + 464 \beta_1 - 2468) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{2} - 3 \beta_1 + 342) q^{4} + ( - \beta_{4} + \beta_{3} + \beta_{2} + 26 \beta_1 - 198) q^{5} + (6 \beta_{6} - 7 \beta_{5} - 6 \beta_{4} + 3 \beta_{3} - 3 \beta_{2} - 69 \beta_1 + 1349) q^{7} + (8 \beta_{6} - 17 \beta_{5} + 10 \beta_{3} - \beta_{2} + 464 \beta_1 - 2468) q^{8} + (28 \beta_{6} - 7 \beta_{5} - 40 \beta_{4} + 26 \beta_{3} + 46 \beta_{2} + \cdots + 22030) q^{10}+ \cdots + ( - 668946 \beta_{6} + 391129 \beta_{5} + \cdots + 313296656) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q + q^{2} + 2389 q^{4} - 1362 q^{5} + 9388 q^{7} - 16821 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 7 q + q^{2} + 2389 q^{4} - 1362 q^{5} + 9388 q^{7} - 16821 q^{8} + 154226 q^{10} - 135536 q^{11} + 166122 q^{13} - 447252 q^{14} + 1463585 q^{16} - 584647 q^{17} + 777172 q^{19} + 917162 q^{20} - 1222520 q^{22} - 1357764 q^{23} + 1065785 q^{25} + 14379966 q^{26} - 3328892 q^{28} - 967002 q^{29} + 3546740 q^{31} - 4825461 q^{32} - 83521 q^{34} + 530736 q^{35} + 18296498 q^{37} + 49363020 q^{38} + 127155062 q^{40} - 10285686 q^{41} + 21913204 q^{43} - 96696624 q^{44} - 151509484 q^{46} - 56639800 q^{47} + 27010351 q^{49} + 261150303 q^{50} - 156226378 q^{52} - 121813562 q^{53} + 40793128 q^{55} + 196175436 q^{56} - 236833910 q^{58} - 29222388 q^{59} - 49915846 q^{61} + 73506556 q^{62} + 317922057 q^{64} + 122633668 q^{65} + 301863420 q^{67} - 199531669 q^{68} + 966315960 q^{70} - 652473940 q^{71} + 306656342 q^{73} - 249173874 q^{74} + 128694700 q^{76} + 102442536 q^{77} + 959147884 q^{79} + 692173602 q^{80} + 1046441254 q^{82} + 1512945268 q^{83} + 113755602 q^{85} + 164953236 q^{86} + 1132038848 q^{88} + 1971327114 q^{89} - 1061062864 q^{91} - 901186756 q^{92} + 2534831232 q^{94} + 3249631512 q^{95} + 2006526254 q^{97} + 2170640009 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{7} - x^{6} - 2986x^{5} + 8252x^{4} + 2252056x^{3} - 10388768x^{2} - 243559296x - 675998208 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 3\nu - 854 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 8711 \nu^{6} + 479085 \nu^{5} - 21966986 \nu^{4} - 962897524 \nu^{3} + 9962276152 \nu^{2} + 249666517824 \nu + 1595734267008 ) / 3478080000 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 41053 \nu^{6} - 292545 \nu^{5} - 124232878 \nu^{4} + 1085670148 \nu^{3} + 93593409896 \nu^{2} - 961758589248 \nu - 7071932313216 ) / 2086848000 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 106349 \nu^{6} - 239985 \nu^{5} - 307414574 \nu^{4} + 1241764484 \nu^{3} + 214005649768 \nu^{2} - 1407118526784 \nu - 13060049459328 ) / 5217120000 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 419317 \nu^{6} - 2816505 \nu^{5} - 1224135742 \nu^{4} + 10192644772 \nu^{3} + 873469755944 \nu^{2} - 8853358980672 \nu - 59384105783424 ) / 10434240000 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 3\beta _1 + 854 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 8\beta_{6} - 17\beta_{5} + 10\beta_{3} - \beta_{2} + 1488\beta _1 - 2468 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 336\beta_{6} - 203\beta_{5} - 488\beta_{4} + 94\beta_{3} + 1817\beta_{2} - 6760\beta _1 + 1258948 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 14912\beta_{6} - 35837\beta_{5} + 3144\beta_{4} + 27714\beta_{3} + 1331\beta_{2} + 2442620\beta _1 - 5766748 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 911488 \beta_{6} - 420111 \beta_{5} - 1403528 \beta_{4} + 217494 \beta_{3} + 3254641 \beta_{2} - 12134956 \beta _1 + 2059247660 \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−43.1213
−34.1532
−5.44491
−4.12962
16.8116
28.6400
42.3973
−43.1213 0 1347.45 −1536.21 0 3027.69 −36025.5 0 66243.5
1.2 −34.1532 0 654.438 −195.287 0 −356.628 −4864.71 0 6669.66
1.3 −5.44491 0 −482.353 −1303.94 0 9199.27 5414.17 0 7099.84
1.4 −4.12962 0 −494.946 −151.544 0 9407.97 4158.31 0 625.818
1.5 16.8116 0 −229.369 1103.40 0 −5164.29 −12463.6 0 18549.9
1.6 28.6400 0 308.250 −1776.79 0 −9598.61 −5835.40 0 −50887.2
1.7 42.3973 0 1285.53 2498.37 0 2872.61 32795.8 0 105924.
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(17\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 153.10.a.f 7
3.b odd 2 1 17.10.a.b 7
12.b even 2 1 272.10.a.g 7
51.c odd 2 1 289.10.a.b 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
17.10.a.b 7 3.b odd 2 1
153.10.a.f 7 1.a even 1 1 trivial
272.10.a.g 7 12.b even 2 1
289.10.a.b 7 51.c odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{7} - T_{2}^{6} - 2986T_{2}^{5} + 8252T_{2}^{4} + 2252056T_{2}^{3} - 10388768T_{2}^{2} - 243559296T_{2} - 675998208 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(153))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{7} - T^{6} - 2986 T^{5} + \cdots - 675998208 \) Copy content Toggle raw display
$3$ \( T^{7} \) Copy content Toggle raw display
$5$ \( T^{7} + 1362 T^{6} + \cdots + 29\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{7} - 9388 T^{6} + \cdots + 13\!\cdots\!04 \) Copy content Toggle raw display
$11$ \( T^{7} + 135536 T^{6} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{7} - 166122 T^{6} + \cdots + 51\!\cdots\!32 \) Copy content Toggle raw display
$17$ \( (T + 83521)^{7} \) Copy content Toggle raw display
$19$ \( T^{7} - 777172 T^{6} + \cdots + 35\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{7} + 1357764 T^{6} + \cdots - 31\!\cdots\!72 \) Copy content Toggle raw display
$29$ \( T^{7} + 967002 T^{6} + \cdots - 34\!\cdots\!60 \) Copy content Toggle raw display
$31$ \( T^{7} - 3546740 T^{6} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{7} - 18296498 T^{6} + \cdots + 26\!\cdots\!04 \) Copy content Toggle raw display
$41$ \( T^{7} + 10285686 T^{6} + \cdots - 11\!\cdots\!40 \) Copy content Toggle raw display
$43$ \( T^{7} - 21913204 T^{6} + \cdots + 32\!\cdots\!64 \) Copy content Toggle raw display
$47$ \( T^{7} + 56639800 T^{6} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{7} + 121813562 T^{6} + \cdots + 68\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{7} + 29222388 T^{6} + \cdots - 53\!\cdots\!60 \) Copy content Toggle raw display
$61$ \( T^{7} + 49915846 T^{6} + \cdots - 16\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{7} - 301863420 T^{6} + \cdots + 18\!\cdots\!16 \) Copy content Toggle raw display
$71$ \( T^{7} + 652473940 T^{6} + \cdots + 21\!\cdots\!16 \) Copy content Toggle raw display
$73$ \( T^{7} - 306656342 T^{6} + \cdots + 86\!\cdots\!04 \) Copy content Toggle raw display
$79$ \( T^{7} - 959147884 T^{6} + \cdots - 18\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{7} - 1512945268 T^{6} + \cdots + 61\!\cdots\!12 \) Copy content Toggle raw display
$89$ \( T^{7} - 1971327114 T^{6} + \cdots + 45\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{7} - 2006526254 T^{6} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
show more
show less