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Properties

Label 308.2.j.b
Level $308$
Weight $2$
Character orbit 308.j
Analytic conductor $2.459$
Analytic rank $0$
Dimension $8$
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(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: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + 13x^{6} - 25x^{5} + 126x^{4} + 135x^{3} + 717x^{2} + 1068x + 7921 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{6} - \beta_{3} - 2 \beta_{2} - 2) q^{3} + \beta_{7} q^{5} - \beta_{3} q^{7} + (3 \beta_{3} + 2 \beta_{2} + 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{6} - \beta_{3} - 2 \beta_{2} - 2) q^{3} + \beta_{7} q^{5} - \beta_{3} q^{7} + (3 \beta_{3} + 2 \beta_{2} + 3) q^{9} + ( - \beta_{7} + \beta_{6} - \beta_{5} + \cdots - 1) q^{11}+ \cdots + ( - \beta_{5} - 3 \beta_{4} - \beta_{3} + \cdots - 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{3} - q^{5} + 2 q^{7} + 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{3} - q^{5} + 2 q^{7} + 14 q^{9} - 5 q^{11} + 11 q^{13} - 4 q^{15} - 7 q^{17} - 5 q^{19} - 12 q^{21} + 28 q^{23} - 15 q^{25} + 10 q^{27} + 7 q^{29} + 3 q^{31} + q^{35} + 10 q^{37} + 9 q^{39} + 8 q^{41} - 44 q^{43} + 2 q^{45} + q^{47} - 2 q^{49} - 13 q^{51} - 6 q^{53} - 12 q^{55} + 5 q^{57} + 17 q^{59} - 23 q^{61} + 16 q^{63} - 80 q^{65} + 38 q^{67} - 38 q^{69} + 10 q^{71} - 5 q^{73} + 15 q^{75} + 5 q^{77} - 27 q^{79} + 28 q^{81} + 21 q^{83} + 38 q^{85} - 32 q^{87} + 26 q^{89} + 9 q^{91} + 12 q^{93} - 66 q^{95} - 28 q^{97} - 20 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} + 13x^{6} - 25x^{5} + 126x^{4} + 135x^{3} + 717x^{2} + 1068x + 7921 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 15315436 \nu^{7} + 461671965 \nu^{6} + 297691842 \nu^{5} - 216651878 \nu^{4} + \cdots - 121034107755 ) / 616964103482 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 38804845 \nu^{7} + 32912779 \nu^{6} - 152484018 \nu^{5} + 809961111 \nu^{4} + \cdots + 39857169307 ) / 616964103482 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 785169 \nu^{7} - 1371676 \nu^{6} - 62450258 \nu^{5} - 12105993 \nu^{4} - 83689073 \nu^{3} + \cdots - 18301378782 ) / 6932180938 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 805816 \nu^{7} + 7381427 \nu^{6} - 20000924 \nu^{5} + 93045740 \nu^{4} - 882499627 \nu^{3} + \cdots + 3453631205 ) / 6932180938 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 127743796 \nu^{7} + 197623837 \nu^{6} - 1538590184 \nu^{5} + 8751667862 \nu^{4} + \cdots - 136936011163 ) / 616964103482 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 6950394 \nu^{7} - 4901985 \nu^{6} + 84318980 \nu^{5} - 97366914 \nu^{4} + 791956581 \nu^{3} + \cdots + 6552492835 ) / 6932180938 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} + 9\beta_{3} + \beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 9\beta_{6} - 10\beta_{5} + \beta_{4} + 9\beta_{3} + 10\beta_{2} - \beta _1 + 11 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 21\beta_{7} + 91\beta_{6} + 2\beta_{5} + 2\beta_{4} - 20\beta_{3} - 20\beta_{2} + 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -22\beta_{7} - 39\beta_{6} - 132\beta_{4} - 39\beta_{2} + 22\beta _1 - 344 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -61\beta_{7} - 61\beta_{5} + 330\beta_{3} + 1342\beta_{2} - 283\beta _1 + 330 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 1686\beta_{7} + 893\beta_{6} + 1295\beta_{5} + 1686\beta_{4} - 2608\beta_{3} - 1295\beta _1 + 2579 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(45\) \(57\) \(155\)
\(\chi(n)\) \(1\) \(\beta_{3}\) \(1\)

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
2.78142 − 2.02082i
−1.97240 + 1.43303i
0.864219 − 2.65979i
−1.17324 + 3.61085i
2.78142 + 2.02082i
−1.97240 − 1.43303i
0.864219 + 2.65979i
−1.17324 − 3.61085i
0 0.118034 + 0.363271i 0 −2.78142 − 2.02082i 0 −0.309017 + 0.951057i 0 2.30902 − 1.67760i 0
113.2 0 0.118034 + 0.363271i 0 1.97240 + 1.43303i 0 −0.309017 + 0.951057i 0 2.30902 − 1.67760i 0
141.1 0 −2.11803 + 1.53884i 0 −0.864219 − 2.65979i 0 0.809017 + 0.587785i 0 1.19098 − 3.66547i 0
141.2 0 −2.11803 + 1.53884i 0 1.17324 + 3.61085i 0 0.809017 + 0.587785i 0 1.19098 − 3.66547i 0
169.1 0 0.118034 − 0.363271i 0 −2.78142 + 2.02082i 0 −0.309017 − 0.951057i 0 2.30902 + 1.67760i 0
169.2 0 0.118034 − 0.363271i 0 1.97240 − 1.43303i 0 −0.309017 − 0.951057i 0 2.30902 + 1.67760i 0
225.1 0 −2.11803 − 1.53884i 0 −0.864219 + 2.65979i 0 0.809017 − 0.587785i 0 1.19098 + 3.66547i 0
225.2 0 −2.11803 − 1.53884i 0 1.17324 − 3.61085i 0 0.809017 − 0.587785i 0 1.19098 + 3.66547i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 113.2
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.b ✓ 8
11.c even 5 1 inner 308.2.j.b ✓ 8
11.c even 5 1 3388.2.a.r 4
11.d odd 10 1 3388.2.a.s 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
308.2.j.b ✓ 8 1.a even 1 1 trivial
308.2.j.b ✓ 8 11.c even 5 1 inner
3388.2.a.r 4 11.c even 5 1
3388.2.a.s 4 11.d odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 4T_{3}^{3} + 6T_{3}^{2} - T_{3} + 1 \) acting on \(S_{2}^{\mathrm{new}}(308, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} + 4 T^{3} + 6 T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} + T^{7} + \cdots + 7921 \) Copy content Toggle raw display
$7$ \( (T^{4} - T^{3} + T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} + 5 T^{7} + \cdots + 14641 \) Copy content Toggle raw display
$13$ \( T^{8} - 11 T^{7} + \cdots + 81 \) Copy content Toggle raw display
$17$ \( T^{8} + 7 T^{7} + \cdots + 361 \) Copy content Toggle raw display
$19$ \( T^{8} + 5 T^{7} + \cdots + 5041 \) Copy content Toggle raw display
$23$ \( (T^{2} - 7 T + 11)^{4} \) Copy content Toggle raw display
$29$ \( T^{8} - 7 T^{7} + \cdots + 81 \) Copy content Toggle raw display
$31$ \( T^{8} - 3 T^{7} + \cdots + 63001 \) Copy content Toggle raw display
$37$ \( T^{8} - 10 T^{7} + \cdots + 1185921 \) Copy content Toggle raw display
$41$ \( (T^{4} - 4 T^{3} + 6 T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 22 T^{3} + \cdots - 5301)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} - T^{7} + \cdots + 3207681 \) Copy content Toggle raw display
$53$ \( T^{8} + 6 T^{7} + \cdots + 912025 \) Copy content Toggle raw display
$59$ \( T^{8} - 17 T^{7} + \cdots + 12257001 \) Copy content Toggle raw display
$61$ \( T^{8} + 23 T^{7} + \cdots + 18879025 \) Copy content Toggle raw display
$67$ \( (T^{4} - 19 T^{3} + \cdots - 709)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} - 10 T^{7} + \cdots + 52548001 \) Copy content Toggle raw display
$73$ \( T^{8} + 5 T^{7} + \cdots + 9801 \) Copy content Toggle raw display
$79$ \( T^{8} + 27 T^{7} + \cdots + 81 \) Copy content Toggle raw display
$83$ \( T^{8} - 21 T^{7} + \cdots + 11256025 \) Copy content Toggle raw display
$89$ \( (T^{4} - 13 T^{3} + \cdots - 6381)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + 28 T^{7} + \cdots + 21986721 \) Copy content Toggle raw display
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