Properties

Label 65.2.n.a
Level $65$
Weight $2$
Character orbit 65.n
Analytic conductor $0.519$
Analytic rank $0$
Dimension $12$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [65,2,Mod(9,65)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(65, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("65.9");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 65 = 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 65.n (of order \(6\), degree \(2\), 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: \(0.519027613138\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 8x^{10} + 54x^{8} - 78x^{6} + 92x^{4} - 10x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{4} q^{2} + ( - \beta_{11} + \beta_{4}) q^{3} + ( - \beta_{10} - \beta_{6} - \beta_{2}) q^{4} + (\beta_{8} - \beta_{7} + \beta_{6}) q^{5} + (\beta_{10} + 2 \beta_{6} + \beta_{2}) q^{6} + ( - \beta_{3} - \beta_1) q^{7} + (\beta_{11} - \beta_{9} - \beta_{8} + \cdots - 1) q^{8}+ \cdots + (\beta_{7} - 2 \beta_{6} - \beta_{5}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{4} q^{2} + ( - \beta_{11} + \beta_{4}) q^{3} + ( - \beta_{10} - \beta_{6} - \beta_{2}) q^{4} + (\beta_{8} - \beta_{7} + \beta_{6}) q^{5} + (\beta_{10} + 2 \beta_{6} + \beta_{2}) q^{6} + ( - \beta_{3} - \beta_1) q^{7} + (\beta_{11} - \beta_{9} - \beta_{8} + \cdots - 1) q^{8}+ \cdots + (2 \beta_{2} + 8) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 4 q^{4} - 6 q^{5} - 10 q^{6} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 4 q^{4} - 6 q^{5} - 10 q^{6} + 6 q^{9} + 7 q^{10} - 44 q^{14} - 4 q^{15} - 16 q^{16} + 12 q^{19} - q^{20} - 8 q^{21} + 32 q^{24} - 2 q^{25} + 24 q^{26} + 18 q^{29} + 4 q^{30} - 16 q^{31} + 16 q^{34} + 10 q^{35} - 2 q^{36} - 32 q^{39} + 70 q^{40} + 14 q^{41} - 4 q^{44} - 29 q^{45} + 10 q^{46} + 6 q^{49} - 31 q^{50} + 24 q^{51} - 22 q^{54} - 26 q^{55} - 16 q^{56} - 4 q^{59} - 96 q^{60} + 6 q^{61} - 12 q^{64} + 23 q^{65} + 4 q^{66} - 24 q^{69} + 20 q^{70} - 12 q^{71} + 8 q^{74} + 2 q^{75} - 10 q^{76} - 104 q^{79} + 33 q^{80} + 14 q^{81} + 90 q^{84} + 21 q^{85} - 4 q^{86} + 20 q^{89} + 62 q^{90} - 44 q^{91} + 56 q^{94} + 20 q^{95} + 12 q^{96} + 104 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 8x^{10} + 54x^{8} - 78x^{6} + 92x^{4} - 10x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 16\nu^{10} - 108\nu^{8} + 729\nu^{6} - 184\nu^{4} + 20\nu^{2} + 3531 ) / 1222 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 108\nu^{11} - 729\nu^{9} + 4768\nu^{7} - 1242\nu^{5} + 135\nu^{3} + 11156\nu ) / 1222 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 135\nu^{11} - 1064\nu^{9} + 7182\nu^{7} - 9801\nu^{5} + 12236\nu^{3} - 1330\nu ) / 1222 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 92 \nu^{11} + 293 \nu^{10} + 621 \nu^{9} - 2436 \nu^{8} - 4039 \nu^{7} + 16443 \nu^{6} + \cdots - 3045 ) / 2444 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 135\nu^{10} - 1064\nu^{8} + 7182\nu^{6} - 9801\nu^{4} + 12236\nu^{2} - 1330 ) / 1222 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 92 \nu^{11} - 293 \nu^{10} + 621 \nu^{9} + 2436 \nu^{8} - 4039 \nu^{7} - 16443 \nu^{6} + \cdots + 3045 ) / 2444 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 563 \nu^{11} - 655 \nu^{10} - 4564 \nu^{9} + 5185 \nu^{8} + 30807 \nu^{7} - 34846 \nu^{6} + \cdots + 524 ) / 2444 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 655 \nu^{11} + 92 \nu^{10} - 5185 \nu^{9} - 621 \nu^{8} + 34846 \nu^{7} + 4039 \nu^{6} - 47553 \nu^{5} + \cdots + 2737 ) / 2444 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -405\nu^{10} + 3192\nu^{8} - 21546\nu^{6} + 29403\nu^{4} - 35486\nu^{2} + 324 ) / 1222 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( \nu^{11} - 8\nu^{9} + 54\nu^{7} - 78\nu^{5} + 92\nu^{3} - 10\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{10} + 3\beta_{6} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{11} + \beta_{9} + \beta_{8} - \beta_{7} + \beta_{6} + 5\beta_{4} - \beta_{3} + 5\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 7\beta_{10} - \beta_{7} + 18\beta_{6} + \beta_{5} + 7\beta_{2} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -7\beta_{11} + 8\beta_{9} + 8\beta_{8} + 8\beta_{6} + 8\beta_{5} + 30\beta_{4} + 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -8\beta_{9} + 8\beta_{8} - 8\beta_{7} + 8\beta_{6} + 46\beta_{2} - 107 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 54\beta_{7} + 54\beta_{5} + 46\beta_{3} - 191\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -299\beta_{10} - 54\beta_{9} + 54\beta_{8} - 689\beta_{6} - 54\beta_{5} - 689 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 299 \beta_{11} - 353 \beta_{9} - 353 \beta_{8} + 353 \beta_{7} - 353 \beta_{6} - 1233 \beta_{4} + \cdots - 353 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -1939\beta_{10} + 353\beta_{7} - 4812\beta_{6} - 353\beta_{5} - 1939\beta_{2} \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 1939\beta_{11} - 2292\beta_{9} - 2292\beta_{8} - 2292\beta_{6} - 2292\beta_{5} - 7984\beta_{4} - 2292 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/65\mathbb{Z}\right)^\times\).

\(n\) \(27\) \(41\)
\(\chi(n)\) \(-1\) \(\beta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1
−2.20467 + 1.27287i
−1.02826 + 0.593667i
−0.286513 + 0.165418i
0.286513 0.165418i
1.02826 0.593667i
2.20467 1.27287i
−2.20467 1.27287i
−1.02826 0.593667i
−0.286513 0.165418i
0.286513 + 0.165418i
1.02826 + 0.593667i
2.20467 + 1.27287i
−2.20467 1.27287i 1.86449 + 1.07646i 2.24039 + 3.88048i −0.817544 + 2.08125i −2.74039 4.74650i 2.54486 1.46928i 6.31544i 0.817544 + 1.41603i 4.45158 3.54786i
9.2 −1.02826 0.593667i 0.298874 + 0.172555i −0.295120 0.511162i 1.44045 1.71029i −0.204880 0.354863i 1.75765 1.01478i 3.07548i −1.44045 2.49493i −2.49650 + 0.903481i
9.3 −0.286513 0.165418i −2.33117 1.34590i −0.945274 1.63726i −2.12291 + 0.702335i 0.445274 + 0.771236i 2.90420 1.67674i 1.28714i 2.12291 + 3.67698i 0.724419 + 0.149939i
9.4 0.286513 + 0.165418i 2.33117 + 1.34590i −0.945274 1.63726i −2.12291 0.702335i 0.445274 + 0.771236i −2.90420 + 1.67674i 1.28714i 2.12291 + 3.67698i −0.492061 0.552395i
9.5 1.02826 + 0.593667i −0.298874 0.172555i −0.295120 0.511162i 1.44045 + 1.71029i −0.204880 0.354863i −1.75765 + 1.01478i 3.07548i −1.44045 2.49493i 0.465813 + 2.61378i
9.6 2.20467 + 1.27287i −1.86449 1.07646i 2.24039 + 3.88048i −0.817544 2.08125i −2.74039 4.74650i −2.54486 + 1.46928i 6.31544i 0.817544 + 1.41603i 0.846746 5.62912i
29.1 −2.20467 + 1.27287i 1.86449 1.07646i 2.24039 3.88048i −0.817544 2.08125i −2.74039 + 4.74650i 2.54486 + 1.46928i 6.31544i 0.817544 1.41603i 4.45158 + 3.54786i
29.2 −1.02826 + 0.593667i 0.298874 0.172555i −0.295120 + 0.511162i 1.44045 + 1.71029i −0.204880 + 0.354863i 1.75765 + 1.01478i 3.07548i −1.44045 + 2.49493i −2.49650 0.903481i
29.3 −0.286513 + 0.165418i −2.33117 + 1.34590i −0.945274 + 1.63726i −2.12291 0.702335i 0.445274 0.771236i 2.90420 + 1.67674i 1.28714i 2.12291 3.67698i 0.724419 0.149939i
29.4 0.286513 0.165418i 2.33117 1.34590i −0.945274 + 1.63726i −2.12291 + 0.702335i 0.445274 0.771236i −2.90420 1.67674i 1.28714i 2.12291 3.67698i −0.492061 + 0.552395i
29.5 1.02826 0.593667i −0.298874 + 0.172555i −0.295120 + 0.511162i 1.44045 1.71029i −0.204880 + 0.354863i −1.75765 1.01478i 3.07548i −1.44045 + 2.49493i 0.465813 2.61378i
29.6 2.20467 1.27287i −1.86449 + 1.07646i 2.24039 3.88048i −0.817544 + 2.08125i −2.74039 + 4.74650i −2.54486 1.46928i 6.31544i 0.817544 1.41603i 0.846746 + 5.62912i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 9.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
13.c even 3 1 inner
65.n even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 65.2.n.a 12
3.b odd 2 1 585.2.bs.a 12
4.b odd 2 1 1040.2.dh.a 12
5.b even 2 1 inner 65.2.n.a 12
5.c odd 4 2 325.2.e.e 12
13.b even 2 1 845.2.n.e 12
13.c even 3 1 inner 65.2.n.a 12
13.c even 3 1 845.2.b.d 6
13.d odd 4 2 845.2.l.f 24
13.e even 6 1 845.2.b.e 6
13.e even 6 1 845.2.n.e 12
13.f odd 12 2 845.2.d.d 12
13.f odd 12 2 845.2.l.f 24
15.d odd 2 1 585.2.bs.a 12
20.d odd 2 1 1040.2.dh.a 12
39.i odd 6 1 585.2.bs.a 12
52.j odd 6 1 1040.2.dh.a 12
65.d even 2 1 845.2.n.e 12
65.g odd 4 2 845.2.l.f 24
65.l even 6 1 845.2.b.e 6
65.l even 6 1 845.2.n.e 12
65.n even 6 1 inner 65.2.n.a 12
65.n even 6 1 845.2.b.d 6
65.q odd 12 2 325.2.e.e 12
65.q odd 12 2 4225.2.a.br 6
65.r odd 12 2 4225.2.a.bq 6
65.s odd 12 2 845.2.d.d 12
65.s odd 12 2 845.2.l.f 24
195.x odd 6 1 585.2.bs.a 12
260.v odd 6 1 1040.2.dh.a 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.n.a 12 1.a even 1 1 trivial
65.2.n.a 12 5.b even 2 1 inner
65.2.n.a 12 13.c even 3 1 inner
65.2.n.a 12 65.n even 6 1 inner
325.2.e.e 12 5.c odd 4 2
325.2.e.e 12 65.q odd 12 2
585.2.bs.a 12 3.b odd 2 1
585.2.bs.a 12 15.d odd 2 1
585.2.bs.a 12 39.i odd 6 1
585.2.bs.a 12 195.x odd 6 1
845.2.b.d 6 13.c even 3 1
845.2.b.d 6 65.n even 6 1
845.2.b.e 6 13.e even 6 1
845.2.b.e 6 65.l even 6 1
845.2.d.d 12 13.f odd 12 2
845.2.d.d 12 65.s odd 12 2
845.2.l.f 24 13.d odd 4 2
845.2.l.f 24 13.f odd 12 2
845.2.l.f 24 65.g odd 4 2
845.2.l.f 24 65.s odd 12 2
845.2.n.e 12 13.b even 2 1
845.2.n.e 12 13.e even 6 1
845.2.n.e 12 65.d even 2 1
845.2.n.e 12 65.l even 6 1
1040.2.dh.a 12 4.b odd 2 1
1040.2.dh.a 12 20.d odd 2 1
1040.2.dh.a 12 52.j odd 6 1
1040.2.dh.a 12 260.v odd 6 1
4225.2.a.bq 6 65.r odd 12 2
4225.2.a.br 6 65.q odd 12 2

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(65, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - 8 T^{10} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{12} - 12 T^{10} + \cdots + 16 \) Copy content Toggle raw display
$5$ \( (T^{6} + 3 T^{5} + \cdots + 125)^{2} \) Copy content Toggle raw display
$7$ \( T^{12} - 24 T^{10} + \cdots + 160000 \) Copy content Toggle raw display
$11$ \( (T^{6} + 13 T^{4} + \cdots + 64)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} - 15 T^{10} + \cdots + 4826809 \) Copy content Toggle raw display
$17$ \( T^{12} - 35 T^{10} + \cdots + 28561 \) Copy content Toggle raw display
$19$ \( (T^{6} - 6 T^{5} + \cdots + 100)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} - 12 T^{10} + \cdots + 16 \) Copy content Toggle raw display
$29$ \( (T^{2} - 3 T + 9)^{6} \) Copy content Toggle raw display
$31$ \( (T^{3} + 4 T^{2} - 40 T + 40)^{4} \) Copy content Toggle raw display
$37$ \( T^{12} - 35 T^{10} + \cdots + 28561 \) Copy content Toggle raw display
$41$ \( (T^{6} - 7 T^{5} + 78 T^{4} + \cdots + 25)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} - 80 T^{10} + \cdots + 65536 \) Copy content Toggle raw display
$47$ \( (T^{6} + 236 T^{4} + \cdots + 270400)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + 171 T^{4} + \cdots + 400)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + 2 T^{5} + \cdots + 18496)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} - 3 T^{5} + \cdots + 13225)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 406586896 \) Copy content Toggle raw display
$71$ \( (T^{6} + 6 T^{5} + \cdots + 676)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + 215 T^{4} + \cdots + 250000)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} + 26 T^{2} + \cdots + 160)^{4} \) Copy content Toggle raw display
$83$ \( (T^{6} + 276 T^{4} + \cdots + 640000)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} - 10 T^{5} + \cdots + 2515396)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 41740124416 \) Copy content Toggle raw display
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