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Properties

Label 308.2.d.a
Level $308$
Weight $2$
Character orbit 308.d
Analytic conductor $2.459$
Analytic rank $0$
Dimension $18$
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] = [308,2,Mod(43,308)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(308, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("308.43");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 308 = 2^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 308.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: \(2.45939238226\)
Analytic rank: \(0\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - x^{17} - 2 x^{15} + 3 x^{14} - 3 x^{13} + 4 x^{12} - 14 x^{11} + 12 x^{10} + 16 x^{9} + \cdots + 512 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{7} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{12} q^{2} - \beta_{4} q^{3} - \beta_{8} q^{4} - \beta_{10} q^{5} - \beta_{3} q^{6} + q^{7} + \beta_{11} q^{8} + ( - \beta_{17} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{12} q^{2} - \beta_{4} q^{3} - \beta_{8} q^{4} - \beta_{10} q^{5} - \beta_{3} q^{6} + q^{7} + \beta_{11} q^{8} + ( - \beta_{17} - 1) q^{9} - \beta_{13} q^{10} + \beta_{9} q^{11} + ( - \beta_{13} - \beta_{10} + \beta_{9} + \cdots - 1) q^{12}+ \cdots + ( - \beta_{17} + \beta_{16} + \cdots + 2 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - q^{2} + q^{4} + 18 q^{7} - 7 q^{8} - 22 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q - q^{2} + q^{4} + 18 q^{7} - 7 q^{8} - 22 q^{9} + 6 q^{10} + 4 q^{11} - 8 q^{12} - q^{14} - 3 q^{16} + 5 q^{18} + 8 q^{19} - 10 q^{20} - 7 q^{22} + 28 q^{24} + 18 q^{25} - 26 q^{26} + q^{28} + 32 q^{30} - 11 q^{32} - 12 q^{33} + 10 q^{34} + 11 q^{36} - 8 q^{37} - 4 q^{38} - 18 q^{40} - 40 q^{43} + 15 q^{44} - 12 q^{45} - 38 q^{46} + 16 q^{48} + 18 q^{49} - 43 q^{50} - 8 q^{51} - 30 q^{52} + 4 q^{53} + 16 q^{54} - 20 q^{55} - 7 q^{56} + 36 q^{60} + 36 q^{62} - 22 q^{63} + q^{64} - 48 q^{66} + 26 q^{68} - 12 q^{69} + 6 q^{70} + 67 q^{72} + 30 q^{74} - 24 q^{76} + 4 q^{77} + 52 q^{78} - 16 q^{79} - 42 q^{80} + 50 q^{81} + 2 q^{82} - 24 q^{83} - 8 q^{84} - 8 q^{86} + 72 q^{87} + 39 q^{88} + 8 q^{89} - 50 q^{90} + 18 q^{92} - 4 q^{93} - 48 q^{94} + 8 q^{95} - 80 q^{96} + 8 q^{97} - q^{98} - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - x^{17} - 2 x^{15} + 3 x^{14} - 3 x^{13} + 4 x^{12} - 14 x^{11} + 12 x^{10} + 16 x^{9} + \cdots + 512 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -\nu^{3} + \nu^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - \nu^{16} + \nu^{15} - 4 \nu^{14} - 2 \nu^{13} + 5 \nu^{12} - 5 \nu^{11} - 16 \nu^{10} + 2 \nu^{9} + \cdots - 128 ) / 256 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{17} - \nu^{16} + 4 \nu^{15} + 2 \nu^{14} - 5 \nu^{13} + 5 \nu^{12} + 16 \nu^{11} - 2 \nu^{10} + \cdots + 128 \nu ) / 512 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{17} + \nu^{16} - 6 \nu^{15} - 6 \nu^{14} + 7 \nu^{13} + 11 \nu^{12} - 30 \nu^{11} - 50 \nu^{10} + \cdots - 768 ) / 512 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{17} - \nu^{16} + 2 \nu^{14} + 7 \nu^{13} - 3 \nu^{12} + 4 \nu^{11} - 18 \nu^{10} + 24 \nu^{9} + \cdots - 128 \nu ) / 256 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - \nu^{16} + \nu^{15} - 2 \nu^{14} - 3 \nu^{12} + 3 \nu^{11} - 2 \nu^{10} + 8 \nu^{9} - 4 \nu^{8} + \cdots + 64 \nu ) / 128 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( \nu^{17} + \nu^{16} - 2 \nu^{15} - 2 \nu^{14} - \nu^{13} + 3 \nu^{12} - 2 \nu^{11} - 6 \nu^{10} + \cdots - 256 ) / 256 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( \nu^{17} + 3 \nu^{16} - 6 \nu^{14} - 5 \nu^{13} + \nu^{12} + 36 \nu^{11} - 42 \nu^{10} - 92 \nu^{9} + \cdots - 1536 ) / 512 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 3 \nu^{17} - \nu^{16} - 6 \nu^{14} - \nu^{13} + 21 \nu^{12} + 20 \nu^{11} + 22 \nu^{10} + \cdots + 512 ) / 512 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( \nu^{17} + \nu^{16} + 2 \nu^{15} - 6 \nu^{14} - \nu^{13} - 5 \nu^{12} + 10 \nu^{11} - 18 \nu^{10} + \cdots - 768 ) / 256 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( \nu^{17} - \nu^{16} - 2 \nu^{14} + 3 \nu^{13} - 3 \nu^{12} + 4 \nu^{11} - 14 \nu^{10} + 12 \nu^{9} + \cdots - 256 ) / 256 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( \nu^{17} + 2 \nu^{16} + \nu^{15} - 2 \nu^{14} + 9 \nu^{13} - 2 \nu^{12} - 17 \nu^{11} - 34 \nu^{10} + \cdots - 640 ) / 256 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( \nu^{17} + 2 \nu^{16} + \nu^{15} - 6 \nu^{14} - 3 \nu^{13} - 10 \nu^{12} + 15 \nu^{11} + 2 \nu^{10} + \cdots - 640 ) / 256 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 2 \nu^{17} + 3 \nu^{16} - \nu^{15} - 4 \nu^{14} + 9 \nu^{12} - 3 \nu^{11} - 40 \nu^{10} - 18 \nu^{9} + \cdots - 384 ) / 256 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 2 \nu^{17} - \nu^{16} + 3 \nu^{15} + 4 \nu^{13} + 5 \nu^{12} - 7 \nu^{11} - 28 \nu^{10} - 6 \nu^{9} + \cdots - 384 ) / 256 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 7 \nu^{17} + 5 \nu^{16} - 2 \nu^{15} + 10 \nu^{14} - 25 \nu^{13} + 39 \nu^{12} - 18 \nu^{11} + \cdots + 1792 ) / 512 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{14} - \beta_{11} - \beta_{8} + \beta_{5} + \beta_{4} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{14} - \beta_{11} - \beta_{8} + \beta_{5} + \beta_{4} - \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( \beta_{14} - 2 \beta_{12} - \beta_{11} + \beta_{8} + 2 \beta_{6} - \beta_{5} - \beta_{4} + 2 \beta_{3} + \cdots - 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 2\beta_{16} + \beta_{14} - 2\beta_{13} + \beta_{11} - 3\beta_{8} - 2\beta_{7} + \beta_{5} - 3\beta_{4} + \beta_{2} + 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 2 \beta_{16} - \beta_{14} - 2 \beta_{13} - 6 \beta_{12} + 3 \beta_{11} + \beta_{8} + 2 \beta_{7} + \cdots - \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 4 \beta_{15} - \beta_{14} - 4 \beta_{12} + 5 \beta_{11} + 5 \beta_{8} - 4 \beta_{7} + 4 \beta_{6} + \cdots + 8 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 4 \beta_{17} + 4 \beta_{16} - 4 \beta_{15} + \beta_{14} + 6 \beta_{12} + 3 \beta_{11} + 5 \beta_{8} + \cdots + 6 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 4 \beta_{17} - 6 \beta_{16} + 4 \beta_{15} + 5 \beta_{14} + 2 \beta_{13} + 12 \beta_{12} - 3 \beta_{11} + \cdots - 22 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 4 \beta_{17} + 2 \beta_{16} - 4 \beta_{15} + 11 \beta_{14} + 2 \beta_{13} - 10 \beta_{12} - 9 \beta_{11} + \cdots - 16 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 4 \beta_{17} - 5 \beta_{14} - 4 \beta_{13} + 8 \beta_{12} + 9 \beta_{11} + 8 \beta_{10} + 8 \beta_{9} + \cdots - 2 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 12 \beta_{16} + 8 \beta_{15} - 19 \beta_{14} - 4 \beta_{13} + 2 \beta_{12} + 15 \beta_{11} + 24 \beta_{10} + \cdots - 34 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 8 \beta_{17} + 18 \beta_{16} - 8 \beta_{15} - 15 \beta_{14} + 22 \beta_{13} - 8 \beta_{12} + \cdots + 26 ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 8 \beta_{17} - 14 \beta_{16} - 24 \beta_{15} - 65 \beta_{14} + 6 \beta_{13} - 14 \beta_{12} + 51 \beta_{11} + \cdots - 16 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 24 \beta_{17} + 16 \beta_{16} - 44 \beta_{15} + 15 \beta_{14} + 88 \beta_{13} + 4 \beta_{12} + 5 \beta_{11} + \cdots + 40 ) / 2 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( 44 \beta_{17} - 60 \beta_{16} + 100 \beta_{15} + 57 \beta_{14} + 88 \beta_{13} + 158 \beta_{12} + \cdots - 74 ) / 2 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( - 100 \beta_{17} - 22 \beta_{16} + 156 \beta_{15} + 29 \beta_{14} - 6 \beta_{13} - 76 \beta_{12} + \cdots - 182 ) / 2 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(45\) \(57\) \(155\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
43.1
1.39867 − 0.209120i
1.39867 + 0.209120i
1.35077 − 0.418821i
1.35077 + 0.418821i
0.916371 − 1.07716i
0.916371 + 1.07716i
0.730056 − 1.21121i
0.730056 + 1.21121i
−0.166850 − 1.40434i
−0.166850 + 1.40434i
−0.308387 − 1.38018i
−0.308387 + 1.38018i
−0.963054 − 1.03563i
−0.963054 + 1.03563i
−1.11440 − 0.870694i
−1.11440 + 0.870694i
−1.34317 − 0.442587i
−1.34317 + 0.442587i
−1.39867 − 0.209120i 2.69604i 1.91254 + 0.584978i −3.01989 0.563795 − 3.77086i 1.00000 −2.55267 − 1.21814i −4.26863 4.22382 + 0.631519i
43.2 −1.39867 + 0.209120i − 2.69604i 1.91254 − 0.584978i −3.01989 0.563795 + 3.77086i 1.00000 −2.55267 + 1.21814i −4.26863 4.22382 − 0.631519i
43.3 −1.35077 − 0.418821i − 1.57463i 1.64918 + 1.13146i 1.58188 −0.659487 + 2.12696i 1.00000 −1.75378 − 2.21906i 0.520549 −2.13676 − 0.662524i
43.4 −1.35077 + 0.418821i 1.57463i 1.64918 − 1.13146i 1.58188 −0.659487 − 2.12696i 1.00000 −1.75378 + 2.21906i 0.520549 −2.13676 + 0.662524i
43.5 −0.916371 − 1.07716i 2.19521i −0.320527 + 1.97415i 3.16783 2.36458 − 2.01163i 1.00000 2.42019 − 1.46380i −1.81894 −2.90291 − 3.41224i
43.6 −0.916371 + 1.07716i − 2.19521i −0.320527 − 1.97415i 3.16783 2.36458 + 2.01163i 1.00000 2.42019 + 1.46380i −1.81894 −2.90291 + 3.41224i
43.7 −0.730056 − 1.21121i 0.700517i −0.934037 + 1.76850i −3.85785 0.848470 − 0.511417i 1.00000 2.82391 − 0.159790i 2.50928 2.81644 + 4.67264i
43.8 −0.730056 + 1.21121i − 0.700517i −0.934037 − 1.76850i −3.85785 0.848470 + 0.511417i 1.00000 2.82391 + 0.159790i 2.50928 2.81644 − 4.67264i
43.9 0.166850 − 1.40434i − 3.22189i −1.94432 − 0.468627i −1.36931 −4.52462 − 0.537572i 1.00000 −0.982520 + 2.65229i −7.38059 −0.228469 + 1.92297i
43.10 0.166850 + 1.40434i 3.22189i −1.94432 + 0.468627i −1.36931 −4.52462 + 0.537572i 1.00000 −0.982520 − 2.65229i −7.38059 −0.228469 − 1.92297i
43.11 0.308387 − 1.38018i − 0.280638i −1.80980 − 0.851259i 2.41173 −0.387331 − 0.0865450i 1.00000 −1.73301 + 2.23533i 2.92124 0.743745 − 3.32862i
43.12 0.308387 + 1.38018i 0.280638i −1.80980 + 0.851259i 2.41173 −0.387331 + 0.0865450i 1.00000 −1.73301 − 2.23533i 2.92124 0.743745 + 3.32862i
43.13 0.963054 − 1.03563i 3.04909i −0.145053 − 1.99473i 2.80597 3.15773 + 2.93644i 1.00000 −2.20550 − 1.77082i −6.29696 2.70230 − 2.90594i
43.14 0.963054 + 1.03563i − 3.04909i −0.145053 + 1.99473i 2.80597 3.15773 − 2.93644i 1.00000 −2.20550 + 1.77082i −6.29696 2.70230 + 2.90594i
43.15 1.11440 − 0.870694i − 1.74833i 0.483785 − 1.94061i −0.404609 −1.52226 − 1.94835i 1.00000 −1.15054 − 2.58384i −0.0566695 −0.450897 + 0.352290i
43.16 1.11440 + 0.870694i 1.74833i 0.483785 + 1.94061i −0.404609 −1.52226 + 1.94835i 1.00000 −1.15054 + 2.58384i −0.0566695 −0.450897 − 0.352290i
43.17 1.34317 − 0.442587i 0.359546i 1.60823 − 1.18894i −1.31575 0.159130 + 0.482933i 1.00000 1.63393 − 2.30874i 2.87073 −1.76728 + 0.582334i
43.18 1.34317 + 0.442587i − 0.359546i 1.60823 + 1.18894i −1.31575 0.159130 − 0.482933i 1.00000 1.63393 + 2.30874i 2.87073 −1.76728 − 0.582334i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 43.18
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
44.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 308.2.d.a ✓ 18
4.b odd 2 1 308.2.d.b yes 18
11.b odd 2 1 308.2.d.b yes 18
44.c even 2 1 inner 308.2.d.a ✓ 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
308.2.d.a ✓ 18 1.a even 1 1 trivial
308.2.d.a ✓ 18 44.c even 2 1 inner
308.2.d.b yes 18 4.b odd 2 1
308.2.d.b yes 18 11.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{19}^{9} - 4 T_{19}^{8} - 84 T_{19}^{7} + 264 T_{19}^{6} + 2204 T_{19}^{5} - 6160 T_{19}^{4} + \cdots - 12288 \) acting on \(S_{2}^{\mathrm{new}}(308, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} + T^{17} + \cdots + 512 \) Copy content Toggle raw display
$3$ \( T^{18} + 38 T^{16} + \cdots + 128 \) Copy content Toggle raw display
$5$ \( (T^{9} - 27 T^{7} + \cdots + 288)^{2} \) Copy content Toggle raw display
$7$ \( (T - 1)^{18} \) Copy content Toggle raw display
$11$ \( T^{18} + \cdots + 2357947691 \) Copy content Toggle raw display
$13$ \( T^{18} + \cdots + 214990848 \) Copy content Toggle raw display
$17$ \( T^{18} + 124 T^{16} + \cdots + 131072 \) Copy content Toggle raw display
$19$ \( (T^{9} - 4 T^{8} + \cdots - 12288)^{2} \) Copy content Toggle raw display
$23$ \( T^{18} + \cdots + 70904741888 \) Copy content Toggle raw display
$29$ \( T^{18} + \cdots + 3699376128 \) Copy content Toggle raw display
$31$ \( T^{18} + \cdots + 1831819392 \) Copy content Toggle raw display
$37$ \( (T^{9} + 4 T^{8} + \cdots - 277504)^{2} \) Copy content Toggle raw display
$41$ \( T^{18} + \cdots + 2717908992 \) Copy content Toggle raw display
$43$ \( (T^{9} + 20 T^{8} + \cdots - 68608)^{2} \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots + 492095490048 \) Copy content Toggle raw display
$53$ \( (T^{9} - 2 T^{8} + \cdots + 8051712)^{2} \) Copy content Toggle raw display
$59$ \( T^{18} + \cdots + 68765298647168 \) Copy content Toggle raw display
$61$ \( T^{18} + \cdots + 35044924096512 \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots + 125753131008 \) Copy content Toggle raw display
$71$ \( T^{18} + \cdots + 185585370693632 \) Copy content Toggle raw display
$73$ \( T^{18} + \cdots + 8321633353728 \) Copy content Toggle raw display
$79$ \( (T^{9} + 8 T^{8} + \cdots + 9453568)^{2} \) Copy content Toggle raw display
$83$ \( (T^{9} + 12 T^{8} + \cdots - 1161216)^{2} \) Copy content Toggle raw display
$89$ \( (T^{9} - 4 T^{8} + \cdots - 624000)^{2} \) Copy content Toggle raw display
$97$ \( (T^{9} - 4 T^{8} + \cdots - 29749888)^{2} \) Copy content Toggle raw display
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