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Properties

Label 325.2.e.e
Level $325$
Weight $2$
Character orbit 325.e
Analytic conductor $2.595$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [325,2,Mod(126,325)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(325, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("325.126");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 325 = 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 325.e (of order \(3\), 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: \(2.59513806569\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 65)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 8x^{10} + 54x^{8} + 78x^{6} + 92x^{4} + 10x^{2} + 1 \) : 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
\(\beta_{3}\)\(=\) \( ( 92\nu^{10} + 621\nu^{8} + 4039\nu^{6} + 1058\nu^{4} + 115\nu^{2} - 3959 ) / 1222 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -92\nu^{11} - 621\nu^{9} - 4039\nu^{7} - 1058\nu^{5} - 115\nu^{3} + 3959\nu ) / 1222 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 108\nu^{11} + 729\nu^{9} + 4768\nu^{7} + 1242\nu^{5} + 135\nu^{3} - 11156\nu ) / 1222 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -135\nu^{11} - 1064\nu^{9} - 7182\nu^{7} - 9801\nu^{5} - 12236\nu^{3} - 1330\nu ) / 1222 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -135\nu^{10} - 1064\nu^{8} - 7182\nu^{6} - 9801\nu^{4} - 12236\nu^{2} - 108 ) / 1222 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -405\nu^{10} - 3192\nu^{8} - 21546\nu^{6} - 29403\nu^{4} - 35486\nu^{2} - 324 ) / 1222 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 428\nu^{10} + 3500\nu^{8} + 23625\nu^{6} + 36694\nu^{4} + 40250\nu^{2} + 4375 ) / 1222 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 428\nu^{11} + 3500\nu^{9} + 23625\nu^{7} + 36694\nu^{5} + 40250\nu^{3} + 4375\nu ) / 1222 \) 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

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{8} - 3\beta_{7} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{11} + \beta_{10} - 6\beta_{6} - \beta_{5} - \beta_{4} - 6\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{9} - 7\beta_{8} + 17\beta_{7} + 7\beta_{2} - 17 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -7\beta_{11} - 8\beta_{10} + 38\beta_{6} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -8\beta_{3} - 46\beta_{2} + 107 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 46\beta_{5} + 54\beta_{4} + 245\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 54\beta_{9} + 299\beta_{8} - 689\beta_{7} + 54\beta_{3} \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 299\beta_{11} + 353\beta_{10} - 1586\beta_{6} - 299\beta_{5} - 353\beta_{4} - 1586\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -353\beta_{9} - 1939\beta_{8} + 4459\beta_{7} + 1939\beta_{2} - 4459 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( -1939\beta_{11} - 2292\beta_{10} + 10276\beta_{6} \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/325\mathbb{Z}\right)^\times\).

\(n\) \(27\) \(301\)
\(\chi(n)\) \(1\) \(-1 + \beta_{7}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
126.1
−1.27287 − 2.20467i
−0.593667 − 1.02826i
−0.165418 − 0.286513i
0.165418 + 0.286513i
0.593667 + 1.02826i
1.27287 + 2.20467i
−1.27287 + 2.20467i
−0.593667 + 1.02826i
−0.165418 + 0.286513i
0.165418 − 0.286513i
0.593667 − 1.02826i
1.27287 − 2.20467i
−1.27287 + 2.20467i −1.07646 + 1.86449i −2.24039 − 3.88048i 0 −2.74039 − 4.74650i −1.46928 − 2.54486i 6.31544 −0.817544 − 1.41603i 0
126.2 −0.593667 + 1.02826i −0.172555 + 0.298874i 0.295120 + 0.511162i 0 −0.204880 − 0.354863i −1.01478 − 1.75765i −3.07548 1.44045 + 2.49493i 0
126.3 −0.165418 + 0.286513i 1.34590 − 2.33117i 0.945274 + 1.63726i 0 0.445274 + 0.771236i −1.67674 − 2.90420i −1.28714 −2.12291 − 3.67698i 0
126.4 0.165418 − 0.286513i −1.34590 + 2.33117i 0.945274 + 1.63726i 0 0.445274 + 0.771236i 1.67674 + 2.90420i 1.28714 −2.12291 − 3.67698i 0
126.5 0.593667 − 1.02826i 0.172555 − 0.298874i 0.295120 + 0.511162i 0 −0.204880 − 0.354863i 1.01478 + 1.75765i 3.07548 1.44045 + 2.49493i 0
126.6 1.27287 − 2.20467i 1.07646 − 1.86449i −2.24039 − 3.88048i 0 −2.74039 − 4.74650i 1.46928 + 2.54486i −6.31544 −0.817544 − 1.41603i 0
276.1 −1.27287 − 2.20467i −1.07646 − 1.86449i −2.24039 + 3.88048i 0 −2.74039 + 4.74650i −1.46928 + 2.54486i 6.31544 −0.817544 + 1.41603i 0
276.2 −0.593667 − 1.02826i −0.172555 − 0.298874i 0.295120 − 0.511162i 0 −0.204880 + 0.354863i −1.01478 + 1.75765i −3.07548 1.44045 − 2.49493i 0
276.3 −0.165418 − 0.286513i 1.34590 + 2.33117i 0.945274 − 1.63726i 0 0.445274 − 0.771236i −1.67674 + 2.90420i −1.28714 −2.12291 + 3.67698i 0
276.4 0.165418 + 0.286513i −1.34590 − 2.33117i 0.945274 − 1.63726i 0 0.445274 − 0.771236i 1.67674 − 2.90420i 1.28714 −2.12291 + 3.67698i 0
276.5 0.593667 + 1.02826i 0.172555 + 0.298874i 0.295120 − 0.511162i 0 −0.204880 + 0.354863i 1.01478 − 1.75765i 3.07548 1.44045 − 2.49493i 0
276.6 1.27287 + 2.20467i 1.07646 + 1.86449i −2.24039 + 3.88048i 0 −2.74039 + 4.74650i 1.46928 − 2.54486i −6.31544 −0.817544 + 1.41603i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 126.6
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 325.2.e.e 12
5.b even 2 1 inner 325.2.e.e 12
5.c odd 4 2 65.2.n.a ✓ 12
13.c even 3 1 inner 325.2.e.e 12
13.c even 3 1 4225.2.a.br 6
13.e even 6 1 4225.2.a.bq 6
15.e even 4 2 585.2.bs.a 12
20.e even 4 2 1040.2.dh.a 12
65.f even 4 2 845.2.l.f 24
65.h odd 4 2 845.2.n.e 12
65.k even 4 2 845.2.l.f 24
65.l even 6 1 4225.2.a.bq 6
65.n even 6 1 inner 325.2.e.e 12
65.n even 6 1 4225.2.a.br 6
65.o even 12 2 845.2.d.d 12
65.o even 12 2 845.2.l.f 24
65.q odd 12 2 65.2.n.a ✓ 12
65.q odd 12 2 845.2.b.d 6
65.r odd 12 2 845.2.b.e 6
65.r odd 12 2 845.2.n.e 12
65.t even 12 2 845.2.d.d 12
65.t even 12 2 845.2.l.f 24
195.bl even 12 2 585.2.bs.a 12
260.bj even 12 2 1040.2.dh.a 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.n.a ✓ 12 5.c odd 4 2
65.2.n.a ✓ 12 65.q odd 12 2
325.2.e.e 12 1.a even 1 1 trivial
325.2.e.e 12 5.b even 2 1 inner
325.2.e.e 12 13.c even 3 1 inner
325.2.e.e 12 65.n even 6 1 inner
585.2.bs.a 12 15.e even 4 2
585.2.bs.a 12 195.bl even 12 2
845.2.b.d 6 65.q odd 12 2
845.2.b.e 6 65.r odd 12 2
845.2.d.d 12 65.o even 12 2
845.2.d.d 12 65.t even 12 2
845.2.l.f 24 65.f even 4 2
845.2.l.f 24 65.k even 4 2
845.2.l.f 24 65.o even 12 2
845.2.l.f 24 65.t even 12 2
845.2.n.e 12 65.h odd 4 2
845.2.n.e 12 65.r odd 12 2
1040.2.dh.a 12 20.e even 4 2
1040.2.dh.a 12 260.bj even 12 2
4225.2.a.bq 6 13.e even 6 1
4225.2.a.bq 6 65.l even 6 1
4225.2.a.br 6 13.c even 3 1
4225.2.a.br 6 65.n even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} + 8T_{2}^{10} + 54T_{2}^{8} + 78T_{2}^{6} + 92T_{2}^{4} + 10T_{2}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(325, [\chi])\). Copy content Toggle raw display

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
$5$ \( T^{12} \) 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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