↑

Properties

Label 308.2.j.c
Level $308$
Weight $2$
Character orbit 308.j
Analytic conductor $2.459$
Analytic rank $0$
Dimension $12$
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] = [308,2,Mod(113,308)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(308, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 0, 8]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("308.113");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 308 = 2^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 308.j (of order \(5\), degree \(4\), 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: \(2.45939238226\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(3\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 4x^{11} + 11x^{10} - 18x^{9} + 48x^{8} - 22x^{7} + 80x^{6} + 68x^{5} + 26x^{4} - 24x^{3} + 9x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{11} - \beta_{9} + \beta_{8} + \cdots + \beta_1) q^{3}+ \cdots + (2 \beta_{11} + 2 \beta_{10} + \cdots + \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{11} - \beta_{9} + \beta_{8} + \cdots + \beta_1) q^{3}+ \cdots + ( - 3 \beta_{11} - \beta_{10} + \cdots + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 6 q^{3} + q^{5} - 3 q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 6 q^{3} + q^{5} - 3 q^{7} + q^{9} + 2 q^{11} - 7 q^{13} - 4 q^{15} + 3 q^{17} - 23 q^{19} - 4 q^{21} - 38 q^{23} + 2 q^{25} - 18 q^{27} + 29 q^{29} + 9 q^{31} - 4 q^{33} + q^{35} + 3 q^{37} + 25 q^{39} - 18 q^{41} + 34 q^{43} + 14 q^{45} + 9 q^{47} - 3 q^{49} + 35 q^{51} + 13 q^{53} + 16 q^{55} + 9 q^{57} - 17 q^{59} - 19 q^{61} - 9 q^{63} + 8 q^{65} - 20 q^{67} - 14 q^{69} + 15 q^{71} + 9 q^{73} - 47 q^{75} - 8 q^{77} - 14 q^{79} - 49 q^{81} + 41 q^{83} - 66 q^{85} + 40 q^{87} + 34 q^{89} + 13 q^{91} - 40 q^{93} + 42 q^{95} - 10 q^{97} - 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 4x^{11} + 11x^{10} - 18x^{9} + 48x^{8} - 22x^{7} + 80x^{6} + 68x^{5} + 26x^{4} - 24x^{3} + 9x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 580196 \nu^{11} + 11620973 \nu^{10} - 42771822 \nu^{9} + 108655768 \nu^{8} - 181822650 \nu^{7} + \cdots + 75452715 ) / 43682836 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 575253 \nu^{11} - 2507728 \nu^{10} + 7436560 \nu^{9} - 13870429 \nu^{8} + 34817756 \nu^{7} + \cdots - 6886398 ) / 21841418 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3458527 \nu^{11} - 7947697 \nu^{10} + 14308392 \nu^{9} + 4419258 \nu^{8} + 51551422 \nu^{7} + \cdots + 46024261 ) / 87365672 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2303078 \nu^{11} - 4739705 \nu^{10} + 7782846 \nu^{9} + 6817656 \nu^{8} + 31483778 \nu^{7} + \cdots + 30076993 ) / 43682836 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 8168487 \nu^{11} - 23864237 \nu^{10} + 49117940 \nu^{9} - 26586546 \nu^{8} + 167969998 \nu^{7} + \cdots + 202660601 ) / 87365672 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 8809711 \nu^{11} - 40735417 \nu^{10} + 120446220 \nu^{9} - 224117378 \nu^{8} + 534728398 \nu^{7} + \cdots - 8168487 ) / 87365672 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 4472607 \nu^{11} + 17551012 \nu^{10} - 48273060 \nu^{9} + 79063966 \nu^{8} - 215565558 \nu^{7} + \cdots + 2303078 ) / 43682836 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 14129725 \nu^{11} + 57666529 \nu^{10} - 156958688 \nu^{9} + 255592350 \nu^{8} + \cdots - 31934409 ) / 87365672 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 11562349 \nu^{11} - 50565972 \nu^{10} + 144249404 \nu^{9} - 253830336 \nu^{8} + 626857792 \nu^{7} + \cdots - 27666684 ) / 43682836 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 46024261 \nu^{11} - 187555571 \nu^{10} + 514214568 \nu^{9} - 842745090 \nu^{8} + 2204745270 \nu^{7} + \cdots + 110534303 ) / 87365672 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{11} + 2\beta_{9} + \beta_{8} - \beta_{7} + \beta_{5} - 2\beta_{4} + \beta_{3} + 2\beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{10} + \beta_{6} + 7\beta_{5} - 7\beta_{4} + 2\beta_{3} - 2\beta_{2} + 2\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 5 \beta_{11} - 5 \beta_{10} - 7 \beta_{9} - 8 \beta_{8} + 7 \beta_{7} + 7 \beta_{6} + 13 \beta_{5} + \cdots - 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -10\beta_{11} - 26\beta_{8} + 18\beta_{6} + 30\beta_{3} - 10\beta_{2} - 26\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 26 \beta_{11} + 30 \beta_{10} + 14 \beta_{9} - 92 \beta_{8} - 26 \beta_{7} + 26 \beta_{6} - 92 \beta_{5} + \cdots + 14 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 65 \beta_{11} + 92 \beta_{10} - 8 \beta_{9} - 213 \beta_{8} - \beta_{7} - 475 \beta_{5} + \cdots - 474 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 261 \beta_{10} + 44 \beta_{7} - 261 \beta_{6} - 1431 \beta_{5} + 1375 \beta_{4} - 828 \beta_{3} + \cdots - 44 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 547 \beta_{11} + 547 \beta_{10} + 53 \beta_{9} + 1688 \beta_{8} - 53 \beta_{7} - 1375 \beta_{6} + \cdots + 73 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 2420 \beta_{11} - 96 \beta_{9} + 7376 \beta_{8} - 4108 \beta_{6} + 96 \beta_{5} - 4708 \beta_{3} + \cdots + 313 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 7376 \beta_{11} - 4708 \beta_{10} + 248 \beta_{9} + 21784 \beta_{8} + 352 \beta_{7} - 7376 \beta_{6} + \cdots + 248 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/308\mathbb{Z}\right)^\times\).

\(n\) \(45\) \(57\) \(155\)
\(\chi(n)\) \(1\) \(-\beta_{5}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
113.1
0.348189 + 0.252974i
−0.632917 − 0.459841i
2.40276 + 1.74571i
−0.607074 − 1.86838i
0.573758 + 1.76585i
−0.0847181 − 0.260736i
0.348189 − 0.252974i
−0.632917 + 0.459841i
2.40276 − 1.74571i
−0.607074 + 1.86838i
0.573758 − 1.76585i
−0.0847181 + 0.260736i
0 −0.585004 − 1.80046i 0 −1.81637 − 1.31967i 0 0.309017 − 0.951057i 0 −0.472365 + 0.343194i 0
113.2 0 0.153244 + 0.471635i 0 −0.489968 − 0.355982i 0 0.309017 − 0.951057i 0 2.22809 − 1.61881i 0
113.3 0 0.813726 + 2.50439i 0 3.11535 + 2.26344i 0 0.309017 − 0.951057i 0 −3.18278 + 2.31243i 0
141.1 0 −1.17753 + 0.855526i 0 0.0324904 + 0.0999953i 0 −0.809017 − 0.587785i 0 −0.272398 + 0.838356i 0
141.2 0 1.06640 − 0.774782i 0 −1.02103 − 3.14242i 0 −0.809017 − 0.587785i 0 −0.390138 + 1.20072i 0
141.3 0 2.72917 − 1.98286i 0 0.679526 + 2.09137i 0 −0.809017 − 0.587785i 0 2.58959 − 7.96993i 0
169.1 0 −0.585004 + 1.80046i 0 −1.81637 + 1.31967i 0 0.309017 + 0.951057i 0 −0.472365 − 0.343194i 0
169.2 0 0.153244 − 0.471635i 0 −0.489968 + 0.355982i 0 0.309017 + 0.951057i 0 2.22809 + 1.61881i 0
169.3 0 0.813726 − 2.50439i 0 3.11535 − 2.26344i 0 0.309017 + 0.951057i 0 −3.18278 − 2.31243i 0
225.1 0 −1.17753 − 0.855526i 0 0.0324904 − 0.0999953i 0 −0.809017 + 0.587785i 0 −0.272398 − 0.838356i 0
225.2 0 1.06640 + 0.774782i 0 −1.02103 + 3.14242i 0 −0.809017 + 0.587785i 0 −0.390138 − 1.20072i 0
225.3 0 2.72917 + 1.98286i 0 0.679526 − 2.09137i 0 −0.809017 + 0.587785i 0 2.58959 + 7.96993i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 113.3
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 308.2.j.c ✓ 12
11.c even 5 1 inner 308.2.j.c ✓ 12
11.c even 5 1 3388.2.a.u 6
11.d odd 10 1 3388.2.a.t 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
308.2.j.c ✓ 12 1.a even 1 1 trivial
308.2.j.c ✓ 12 11.c even 5 1 inner
3388.2.a.t 6 11.d odd 10 1
3388.2.a.u 6 11.c even 5 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} - 6 T_{3}^{11} + 22 T_{3}^{10} - 46 T_{3}^{9} + 96 T_{3}^{8} - 92 T_{3}^{7} + 271 T_{3}^{6} + \cdots + 256 \) acting on \(S_{2}^{\mathrm{new}}(308, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} - 6 T^{11} + \cdots + 256 \) Copy content Toggle raw display
$5$ \( T^{12} - T^{11} + \cdots + 16 \) Copy content Toggle raw display
$7$ \( (T^{4} + T^{3} + T^{2} + \cdots + 1)^{3} \) Copy content Toggle raw display
$11$ \( T^{12} - 2 T^{11} + \cdots + 1771561 \) Copy content Toggle raw display
$13$ \( T^{12} + 7 T^{11} + \cdots + 1936 \) Copy content Toggle raw display
$17$ \( T^{12} - 3 T^{11} + \cdots + 698896 \) Copy content Toggle raw display
$19$ \( T^{12} + 23 T^{11} + \cdots + 26873856 \) Copy content Toggle raw display
$23$ \( (T^{6} + 19 T^{5} + \cdots - 11741)^{2} \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 4071588481 \) Copy content Toggle raw display
$31$ \( T^{12} - 9 T^{11} + \cdots + 51322896 \) Copy content Toggle raw display
$37$ \( T^{12} - 3 T^{11} + \cdots + 81 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 1317980416 \) Copy content Toggle raw display
$43$ \( (T^{6} - 17 T^{5} + \cdots + 10919)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} - 9 T^{11} + \cdots + 33223696 \) Copy content Toggle raw display
$53$ \( T^{12} - 13 T^{11} + \cdots + 8874441 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 11873153296 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 12008653056 \) Copy content Toggle raw display
$67$ \( (T^{6} + 10 T^{5} + \cdots + 23819)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 455352921 \) Copy content Toggle raw display
$73$ \( T^{12} - 9 T^{11} + \cdots + 68624656 \) Copy content Toggle raw display
$79$ \( T^{12} + 14 T^{11} + \cdots + 421201 \) Copy content Toggle raw display
$83$ \( T^{12} - 41 T^{11} + \cdots + 18011536 \) Copy content Toggle raw display
$89$ \( (T^{6} - 17 T^{5} + \cdots + 20596)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + 10 T^{11} + \cdots + 99856 \) Copy content Toggle raw display
show more
show less