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Properties

Label 325.2.m.b
Level $325$
Weight $2$
Character orbit 325.m
Analytic conductor $2.595$
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] = [325,2,Mod(49,325)]
 
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([3, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("325.49");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 325 = 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 325.m (of order \(6\), degree \(2\), 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: \(2.59513806569\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 65)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{6} - \beta_{3} - \beta_1 + 1) q^{2} + ( - \beta_{6} + \beta_{5} - \beta_{4} + \cdots + 2) q^{3}+ \cdots + ( - 2 \beta_{6} + 2 \beta_{5} + \cdots + 4) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{6} - \beta_{3} - \beta_1 + 1) q^{2} + ( - \beta_{6} + \beta_{5} - \beta_{4} + \cdots + 2) q^{3}+ \cdots + ( - 8 \beta_{7} + 4 \beta_{6} + \cdots - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{2} + 6 q^{3} - 2 q^{4} - 18 q^{6} + 10 q^{7} + 12 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 2 q^{2} + 6 q^{3} - 2 q^{4} - 18 q^{6} + 10 q^{7} + 12 q^{8} + 4 q^{9} + 8 q^{13} - 4 q^{14} - 2 q^{16} + 18 q^{17} - 40 q^{18} - 12 q^{19} + 6 q^{22} + 6 q^{23} + 12 q^{24} + 10 q^{26} + 8 q^{28} + 8 q^{29} - 4 q^{32} - 18 q^{33} + 20 q^{36} - 2 q^{37} + 12 q^{41} - 42 q^{42} - 18 q^{43} - 42 q^{46} - 16 q^{47} - 6 q^{48} - 12 q^{49} - 8 q^{51} + 16 q^{52} - 18 q^{54} + 12 q^{56} - 28 q^{57} + 22 q^{58} + 12 q^{59} - 28 q^{61} + 12 q^{62} - 4 q^{63} + 8 q^{64} + 12 q^{66} - 30 q^{67} + 12 q^{68} + 16 q^{69} + 12 q^{72} - 16 q^{73} - 10 q^{74} + 54 q^{76} + 18 q^{78} + 16 q^{79} + 8 q^{81} - 6 q^{82} + 24 q^{83} + 30 q^{84} + 54 q^{87} + 42 q^{88} - 24 q^{89} + 28 q^{91} + 8 q^{93} - 32 q^{94} - 2 q^{97} + 24 q^{98}+O(q^{100}) \) 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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} - 2\nu^{6} + \nu^{5} + 4\nu^{4} - 3\nu^{3} - 2\nu^{2} + 8\nu - 8 ) / 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{7} + 2\nu^{6} - \nu^{5} - 4\nu^{4} + 3\nu^{3} + 10\nu^{2} - 16\nu + 8 ) / 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{7} - 3\nu^{6} + 3\nu^{5} + 3\nu^{4} - 7\nu^{3} - 3\nu^{2} + 18\nu - 16 ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2\nu^{7} - 5\nu^{6} + 2\nu^{5} + 7\nu^{4} - 8\nu^{3} - 9\nu^{2} + 28\nu - 20 ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -3\nu^{7} + 7\nu^{6} - 3\nu^{5} - 11\nu^{4} + 15\nu^{3} + 11\nu^{2} - 40\nu + 32 ) / 4 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 7\nu^{7} - 20\nu^{6} + 11\nu^{5} + 30\nu^{4} - 45\nu^{3} - 28\nu^{2} + 116\nu - 88 ) / 8 \) Copy content Toggle raw display

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

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

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

Character values

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

\(n\) \(27\) \(301\)
\(\chi(n)\) \(-1\) \(1 - \beta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
1.20036 + 0.747754i
1.40994 − 0.109843i
−1.27597 + 0.609843i
0.665665 − 1.24775i
1.20036 − 0.747754i
1.40994 + 0.109843i
−1.27597 − 0.609843i
0.665665 + 1.24775i
−1.24775 − 2.16117i 2.44811 − 1.41342i −2.11378 + 3.66117i 0 −6.10929 − 3.52720i 0.952606 − 1.64996i 5.55889 2.49551 − 4.32235i 0
49.2 −0.609843 − 1.05628i 2.01978 − 1.16612i 0.256182 − 0.443720i 0 −2.46350 − 1.42231i 1.80010 − 3.11786i −3.06430 1.21969 − 2.11256i 0
49.3 0.109843 + 0.190254i −1.38581 + 0.800098i 0.975869 − 1.69025i 0 −0.304444 − 0.175771i −0.166123 + 0.287734i 0.868145 −0.219687 + 0.380509i 0
49.4 0.747754 + 1.29515i −0.0820885 + 0.0473938i −0.118272 + 0.204852i 0 −0.122764 − 0.0708778i 2.41342 − 4.18016i 2.63726 −1.49551 + 2.59030i 0
199.1 −1.24775 + 2.16117i 2.44811 + 1.41342i −2.11378 − 3.66117i 0 −6.10929 + 3.52720i 0.952606 + 1.64996i 5.55889 2.49551 + 4.32235i 0
199.2 −0.609843 + 1.05628i 2.01978 + 1.16612i 0.256182 + 0.443720i 0 −2.46350 + 1.42231i 1.80010 + 3.11786i −3.06430 1.21969 + 2.11256i 0
199.3 0.109843 − 0.190254i −1.38581 − 0.800098i 0.975869 + 1.69025i 0 −0.304444 + 0.175771i −0.166123 − 0.287734i 0.868145 −0.219687 − 0.380509i 0
199.4 0.747754 − 1.29515i −0.0820885 − 0.0473938i −0.118272 − 0.204852i 0 −0.122764 + 0.0708778i 2.41342 + 4.18016i 2.63726 −1.49551 − 2.59030i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
65.l 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.m.b 8
5.b even 2 1 325.2.m.c 8
5.c odd 4 1 65.2.m.a ✓ 8
5.c odd 4 1 325.2.n.d 8
13.e even 6 1 325.2.m.c 8
15.e even 4 1 585.2.bu.c 8
20.e even 4 1 1040.2.da.b 8
65.f even 4 1 845.2.e.n 8
65.h odd 4 1 845.2.m.g 8
65.k even 4 1 845.2.e.m 8
65.l even 6 1 inner 325.2.m.b 8
65.o even 12 1 845.2.a.m 4
65.o even 12 1 845.2.e.m 8
65.o even 12 1 4225.2.a.bl 4
65.q odd 12 1 845.2.c.g 8
65.q odd 12 1 845.2.m.g 8
65.r odd 12 1 65.2.m.a ✓ 8
65.r odd 12 1 325.2.n.d 8
65.r odd 12 1 845.2.c.g 8
65.t even 12 1 845.2.a.l 4
65.t even 12 1 845.2.e.n 8
65.t even 12 1 4225.2.a.bi 4
195.bc odd 12 1 7605.2.a.cj 4
195.bf even 12 1 585.2.bu.c 8
195.bn odd 12 1 7605.2.a.cf 4
260.bg even 12 1 1040.2.da.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.m.a ✓ 8 5.c odd 4 1
65.2.m.a ✓ 8 65.r odd 12 1
325.2.m.b 8 1.a even 1 1 trivial
325.2.m.b 8 65.l even 6 1 inner
325.2.m.c 8 5.b even 2 1
325.2.m.c 8 13.e even 6 1
325.2.n.d 8 5.c odd 4 1
325.2.n.d 8 65.r odd 12 1
585.2.bu.c 8 15.e even 4 1
585.2.bu.c 8 195.bf even 12 1
845.2.a.l 4 65.t even 12 1
845.2.a.m 4 65.o even 12 1
845.2.c.g 8 65.q odd 12 1
845.2.c.g 8 65.r odd 12 1
845.2.e.m 8 65.k even 4 1
845.2.e.m 8 65.o even 12 1
845.2.e.n 8 65.f even 4 1
845.2.e.n 8 65.t even 12 1
845.2.m.g 8 65.h odd 4 1
845.2.m.g 8 65.q odd 12 1
1040.2.da.b 8 20.e even 4 1
1040.2.da.b 8 260.bg even 12 1
4225.2.a.bi 4 65.t even 12 1
4225.2.a.bl 4 65.o even 12 1
7605.2.a.cf 4 195.bn odd 12 1
7605.2.a.cj 4 195.bc odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 2T_{2}^{7} + 7T_{2}^{6} + 2T_{2}^{5} + 16T_{2}^{4} + 8T_{2}^{3} + 19T_{2}^{2} - 4T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(325, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 2 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{8} - 6 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} - 10 T^{7} + \cdots + 121 \) Copy content Toggle raw display
$11$ \( T^{8} - 30 T^{6} + \cdots + 1089 \) Copy content Toggle raw display
$13$ \( T^{8} - 8 T^{7} + \cdots + 28561 \) Copy content Toggle raw display
$17$ \( T^{8} - 18 T^{7} + \cdots + 169 \) Copy content Toggle raw display
$19$ \( (T^{4} + 6 T^{3} + \cdots + 169)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} - 6 T^{7} + \cdots + 89401 \) Copy content Toggle raw display
$29$ \( T^{8} - 8 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$31$ \( (T^{4} + 32 T^{2} + 64)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + 2 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$41$ \( (T^{4} - 6 T^{3} + 11 T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + 18 T^{7} + \cdots + 169 \) Copy content Toggle raw display
$47$ \( (T^{4} + 8 T^{3} + \cdots - 1328)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} + 72 T^{6} + \cdots + 2304 \) Copy content Toggle raw display
$59$ \( T^{8} - 12 T^{7} + \cdots + 9 \) Copy content Toggle raw display
$61$ \( T^{8} + 28 T^{7} + \cdots + 1590121 \) Copy content Toggle raw display
$67$ \( T^{8} + 30 T^{7} + \cdots + 7667361 \) Copy content Toggle raw display
$71$ \( T^{8} - 218 T^{6} + \cdots + 109767529 \) Copy content Toggle raw display
$73$ \( (T^{4} + 8 T^{3} + \cdots - 1712)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 8 T^{3} + \cdots + 4432)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} - 12 T^{3} + \cdots - 192)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + 24 T^{7} + \cdots + 78375609 \) Copy content Toggle raw display
$97$ \( T^{8} + 2 T^{7} + \cdots + 196249 \) Copy content Toggle raw display
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