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Properties

Label 3380.2.f.i
Level $3380$
Weight $2$
Character orbit 3380.f
Analytic conductor $26.989$
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] = [3380,2,Mod(3041,3380)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3380, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("3380.3041");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3380 = 2^{2} \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3380.f (of order \(2\), degree \(1\), not 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: \(26.9894358832\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.22581504.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} + 5x^{6} + 2x^{5} - 11x^{4} + 4x^{3} + 20x^{2} - 32x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 260)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{4} + \beta_{2}) q^{3} - \beta_{5} q^{5} + (\beta_{5} - \beta_{3}) q^{7} + ( - \beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{4} + \beta_{2}) q^{3} - \beta_{5} q^{5} + (\beta_{5} - \beta_{3}) q^{7} + ( - \beta_1 + 1) q^{9} + \beta_{6} q^{11} + ( - \beta_{6} + \beta_{3}) q^{15} + ( - \beta_{4} - \beta_{2} - \beta_1 - 1) q^{17} + (\beta_{7} + \beta_{6} - \beta_{5} - 2 \beta_{3}) q^{19} + (\beta_{6} + 2 \beta_{5} - 2 \beta_{3}) q^{21} + ( - \beta_{4} + \beta_{2} - \beta_1 + 1) q^{23} - q^{25} + ( - \beta_{4} - \beta_{2} - \beta_1 + 1) q^{27} + ( - 2 \beta_{4} + \beta_1) q^{29} + ( - 2 \beta_{6} + 2 \beta_{5} + 4 \beta_{3}) q^{31} + ( - \beta_{7} + 2 \beta_{5} + \beta_{3}) q^{33} + (\beta_{2} + 1) q^{35} + (2 \beta_{6} + 3 \beta_{5} - 3 \beta_{3}) q^{37} + (\beta_{6} - 6 \beta_{5}) q^{41} + ( - \beta_{4} - \beta_{2} + 2 \beta_1 - 2) q^{43} + (\beta_{7} - \beta_{5}) q^{45} + 2 \beta_{6} q^{47} + ( - 4 \beta_{2} - \beta_1 + 3) q^{49} + ( - 2 \beta_{4} - 2 \beta_{2} - 2 \beta_1 + 1) q^{51} + ( - 4 \beta_{2} - 3 \beta_1 + 5) q^{53} - \beta_{4} q^{55} + ( - 4 \beta_{6} + 5 \beta_{5} + \beta_{3}) q^{57} + (2 \beta_{7} + 3 \beta_{6} + 2 \beta_{5} - 2 \beta_{3}) q^{59} + (4 \beta_{4} - 2 \beta_{2} + \beta_1 + 2) q^{61} + ( - \beta_{7} + 2 \beta_{6} + 3 \beta_{5}) q^{63} + ( - 4 \beta_{6} - 5 \beta_{5} - \beta_{3}) q^{67} + ( - 4 \beta_{4} + 2 \beta_{2} - 2 \beta_1 + 5) q^{69} + ( - 3 \beta_{7} + 3 \beta_{6} - 3 \beta_{5}) q^{71} + (\beta_{7} - 2 \beta_{6} + 7 \beta_{5} + 2 \beta_{3}) q^{73} + (\beta_{4} - \beta_{2}) q^{75} + (\beta_{4} + \beta_{2} + \beta_1 + 1) q^{77} + ( - 6 \beta_{4} + 2 \beta_{2} - \beta_1 - 3) q^{79} + ( - 4 \beta_{4} + \beta_1 - 2) q^{81} + ( - 2 \beta_{7} + 10 \beta_{5} + 2 \beta_{3}) q^{83} + (\beta_{7} - \beta_{6} + \beta_{5} - \beta_{3}) q^{85} + (3 \beta_{4} - 3 \beta_{2} - \beta_1 + 3) q^{87} + (2 \beta_{7} - \beta_{6} + 2 \beta_{5} - 2 \beta_{3}) q^{89} + (2 \beta_{7} + 2 \beta_{6} - 12 \beta_{5}) q^{93} + ( - \beta_{4} + 2 \beta_{2} + \beta_1 - 1) q^{95} + ( - 2 \beta_{7} + 4 \beta_{6} - 3 \beta_{5} - 3 \beta_{3}) q^{97} + ( - \beta_{7} + 2 \beta_{6} - \beta_{5} - 2 \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{3} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{3} + 8 q^{9} - 12 q^{17} + 12 q^{23} - 8 q^{25} + 4 q^{27} + 12 q^{35} - 20 q^{43} + 8 q^{49} + 24 q^{53} + 8 q^{61} + 48 q^{69} - 4 q^{75} + 12 q^{77} - 16 q^{79} - 16 q^{81} + 12 q^{87}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} + 5x^{6} + 2x^{5} - 11x^{4} + 4x^{3} + 20x^{2} - 32x + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{7} + 2\nu^{6} + \nu^{5} - 4\nu^{4} + \nu^{3} + 6\nu^{2} - 10\nu + 2 ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 3\nu^{7} - 8\nu^{6} + 3\nu^{5} + 10\nu^{4} - 13\nu^{3} - 8\nu^{2} + 32\nu - 24 ) / 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -5\nu^{7} + 12\nu^{6} - 5\nu^{5} - 18\nu^{4} + 19\nu^{3} + 28\nu^{2} - 64\nu + 40 ) / 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -3\nu^{7} + 9\nu^{6} - 5\nu^{5} - 13\nu^{4} + 21\nu^{3} + 13\nu^{2} - 54\nu + 40 ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -4\nu^{7} + 11\nu^{6} - 6\nu^{5} - 17\nu^{4} + 24\nu^{3} + 15\nu^{2} - 62\nu + 48 ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 3\nu^{7} - 7\nu^{6} + 3\nu^{5} + 11\nu^{4} - 15\nu^{3} - 11\nu^{2} + 40\nu - 30 ) / 2 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 5\nu^{7} - 15\nu^{6} + 11\nu^{5} + 19\nu^{4} - 35\nu^{3} - 11\nu^{2} + 90\nu - 80 ) / 4 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} + \beta_{5} - 2\beta_{2} - \beta _1 + 3 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} - \beta_{5} + 2\beta_{4} + 2\beta_{3} - \beta _1 + 3 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{7} - \beta_{6} - \beta_{5} + 2\beta_{4} - \beta_{3} - \beta_{2} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -\beta_{7} - 2\beta_{6} - 11\beta_{5} + 6\beta_{4} + 2\beta_{3} - 4\beta_{2} - \beta _1 + 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( \beta_{7} - 2\beta_{6} - 7\beta_{5} + 8\beta_{4} - 6\beta_{3} - 4\beta_{2} + 3\beta _1 + 9 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 3\beta_{6} - 2\beta_{5} + 4\beta_{4} - 8\beta_{2} + \beta _1 + 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 3\beta_{7} + 16\beta_{6} + 11\beta_{5} + 16\beta_{4} - 4\beta_{3} - 2\beta_{2} + 5\beta _1 + 17 ) / 4 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3380\mathbb{Z}\right)^\times\).

\(n\) \(677\) \(1691\) \(1861\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3041.1
1.40994 − 0.109843i
1.40994 + 0.109843i
0.665665 + 1.24775i
0.665665 − 1.24775i
−1.27597 + 0.609843i
−1.27597 − 0.609843i
1.20036 − 0.747754i
1.20036 + 0.747754i
0 −2.33225 0 − 1.00000i 0 0.399804i 0 2.43937 0
3041.2 0 −2.33225 0 1.00000i 0 − 0.399804i 0 2.43937 0
3041.3 0 −0.0947876 0 − 1.00000i 0 − 0.826838i 0 −2.99102 0
3041.4 0 −0.0947876 0 1.00000i 0 0.826838i 0 −2.99102 0
3041.5 0 1.60020 0 − 1.00000i 0 4.33225i 0 −0.439374 0
3041.6 0 1.60020 0 1.00000i 0 − 4.33225i 0 −0.439374 0
3041.7 0 2.82684 0 − 1.00000i 0 2.09479i 0 4.99102 0
3041.8 0 2.82684 0 1.00000i 0 − 2.09479i 0 4.99102 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3041.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3380.2.f.i 8
13.b even 2 1 inner 3380.2.f.i 8
13.c even 3 1 260.2.x.a ✓ 8
13.d odd 4 1 3380.2.a.p 4
13.d odd 4 1 3380.2.a.q 4
13.e even 6 1 260.2.x.a ✓ 8
39.h odd 6 1 2340.2.dj.d 8
39.i odd 6 1 2340.2.dj.d 8
52.i odd 6 1 1040.2.da.c 8
52.j odd 6 1 1040.2.da.c 8
65.l even 6 1 1300.2.y.b 8
65.n even 6 1 1300.2.y.b 8
65.q odd 12 1 1300.2.ba.b 8
65.q odd 12 1 1300.2.ba.c 8
65.r odd 12 1 1300.2.ba.b 8
65.r odd 12 1 1300.2.ba.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
260.2.x.a ✓ 8 13.c even 3 1
260.2.x.a ✓ 8 13.e even 6 1
1040.2.da.c 8 52.i odd 6 1
1040.2.da.c 8 52.j odd 6 1
1300.2.y.b 8 65.l even 6 1
1300.2.y.b 8 65.n even 6 1
1300.2.ba.b 8 65.q odd 12 1
1300.2.ba.b 8 65.r odd 12 1
1300.2.ba.c 8 65.q odd 12 1
1300.2.ba.c 8 65.r odd 12 1
2340.2.dj.d 8 39.h odd 6 1
2340.2.dj.d 8 39.i odd 6 1
3380.2.a.p 4 13.d odd 4 1
3380.2.a.q 4 13.d odd 4 1
3380.2.f.i 8 1.a even 1 1 trivial
3380.2.f.i 8 13.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3380, [\chi])\):

\( T_{3}^{4} - 2T_{3}^{3} - 6T_{3}^{2} + 10T_{3} + 1 \) Copy content Toggle raw display
\( T_{19}^{4} + 30T_{19}^{2} + 33 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} - 2 T^{3} - 6 T^{2} + 10 T + 1)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$7$ \( T^{8} + 24 T^{6} + 102 T^{4} + 72 T^{2} + \cdots + 9 \) Copy content Toggle raw display
$11$ \( (T^{2} + 3)^{4} \) Copy content Toggle raw display
$13$ \( T^{8} \) Copy content Toggle raw display
$17$ \( (T^{4} + 6 T^{3} - 18 T^{2} - 54 T - 27)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 30 T^{2} + 33)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} - 6 T^{3} - 18 T^{2} + 30 T - 3)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 42 T^{2} - 96 T - 39)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 96 T^{2} + 2112)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + 192 T^{6} + 9774 T^{4} + \cdots + 42849 \) Copy content Toggle raw display
$41$ \( (T^{4} + 78 T^{2} + 1089)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 10 T^{3} - 66 T^{2} - 914 T - 2243)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 12)^{4} \) Copy content Toggle raw display
$53$ \( (T^{4} - 12 T^{3} - 156 T^{2} + 1920 T - 624)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + 228 T^{6} + 14958 T^{4} + \cdots + 558009 \) Copy content Toggle raw display
$61$ \( (T^{4} - 4 T^{3} - 102 T^{2} + 644 T - 971)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + 336 T^{6} + 31278 T^{4} + \cdots + 1083681 \) Copy content Toggle raw display
$71$ \( T^{8} + 468 T^{6} + \cdots + 45198729 \) Copy content Toggle raw display
$73$ \( T^{8} + 264 T^{6} + 21168 T^{4} + \cdots + 2509056 \) Copy content Toggle raw display
$79$ \( (T^{4} + 8 T^{3} - 180 T^{2} - 1504 T - 368)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 480 T^{6} + 63936 T^{4} + \cdots + 331776 \) Copy content Toggle raw display
$89$ \( T^{8} + 228 T^{6} + 9774 T^{4} + \cdots + 13689 \) Copy content Toggle raw display
$97$ \( T^{8} + 408 T^{6} + \cdots + 12981609 \) Copy content Toggle raw display
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