Properties

Label 325.2.m.a
Level $325$
Weight $2$
Character orbit 325.m
Analytic conductor $2.595$
Analytic rank $0$
Dimension $4$
Inner twists $4$

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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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) 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 13)
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 primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \zeta_{12}^{3} - \zeta_{12}) q^{2} + (2 \zeta_{12}^{3} - 2 \zeta_{12}) q^{3} + (\zeta_{12}^{2} - 1) q^{4} + (2 \zeta_{12}^{2} + 2) q^{6} + (\zeta_{12}^{3} - 2 \zeta_{12}) q^{8} + ( - \zeta_{12}^{2} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{12}^{3} - \zeta_{12}) q^{2} + (2 \zeta_{12}^{3} - 2 \zeta_{12}) q^{3} + (\zeta_{12}^{2} - 1) q^{4} + (2 \zeta_{12}^{2} + 2) q^{6} + (\zeta_{12}^{3} - 2 \zeta_{12}) q^{8} + ( - \zeta_{12}^{2} + 1) q^{9} - 2 \zeta_{12}^{3} q^{12} + (\zeta_{12}^{3} + 3 \zeta_{12}) q^{13} + 5 \zeta_{12}^{2} q^{16} + 3 \zeta_{12} q^{17} + (\zeta_{12}^{3} - 2 \zeta_{12}) q^{18} + (2 \zeta_{12}^{2} + 2) q^{19} + ( - 6 \zeta_{12}^{3} + 6 \zeta_{12}) q^{23} + ( - 2 \zeta_{12}^{2} + 4) q^{24} + ( - 7 \zeta_{12}^{2} + 5) q^{26} - 4 \zeta_{12}^{3} q^{27} + 3 \zeta_{12}^{2} q^{29} + ( - 4 \zeta_{12}^{2} + 2) q^{31} + ( - 6 \zeta_{12}^{3} + 3 \zeta_{12}) q^{32} + ( - 6 \zeta_{12}^{2} + 3) q^{34} + \zeta_{12}^{2} q^{36} + (5 \zeta_{12}^{3} + 5 \zeta_{12}) q^{37} - 6 \zeta_{12}^{3} q^{38} + ( - 2 \zeta_{12}^{2} - 6) q^{39} + (3 \zeta_{12}^{2} - 6) q^{41} + 8 \zeta_{12} q^{43} + ( - 6 \zeta_{12}^{2} - 6) q^{46} + (2 \zeta_{12}^{3} - 4 \zeta_{12}) q^{47} - 10 \zeta_{12} q^{48} + 7 \zeta_{12}^{2} q^{49} - 6 q^{51} + (3 \zeta_{12}^{3} - 4 \zeta_{12}) q^{52} + 3 \zeta_{12}^{3} q^{53} + (4 \zeta_{12}^{2} - 8) q^{54} + (4 \zeta_{12}^{3} - 8 \zeta_{12}) q^{57} + ( - 6 \zeta_{12}^{3} + 3 \zeta_{12}) q^{58} + ( - 4 \zeta_{12}^{2} - 4) q^{59} + (\zeta_{12}^{2} - 1) q^{61} + (6 \zeta_{12}^{3} - 6 \zeta_{12}) q^{62} + q^{64} + (2 \zeta_{12}^{3} + 2 \zeta_{12}) q^{67} + (3 \zeta_{12}^{3} - 3 \zeta_{12}) q^{68} + (12 \zeta_{12}^{2} - 12) q^{69} + (2 \zeta_{12}^{2} + 2) q^{71} + (2 \zeta_{12}^{3} - \zeta_{12}) q^{72} + (\zeta_{12}^{3} - 2 \zeta_{12}) q^{73} + ( - 15 \zeta_{12}^{2} + 15) q^{74} + (2 \zeta_{12}^{2} - 4) q^{76} + (10 \zeta_{12}^{3} + 4 \zeta_{12}) q^{78} - 4 q^{79} + 11 \zeta_{12}^{2} q^{81} + 9 \zeta_{12} q^{82} + (8 \zeta_{12}^{3} - 16 \zeta_{12}) q^{83} + ( - 16 \zeta_{12}^{2} + 8) q^{86} - 6 \zeta_{12} q^{87} + ( - 4 \zeta_{12}^{2} + 8) q^{89} + 6 \zeta_{12}^{3} q^{92} + (4 \zeta_{12}^{3} + 4 \zeta_{12}) q^{93} + 6 \zeta_{12}^{2} q^{94} + (12 \zeta_{12}^{2} - 6) q^{96} + (8 \zeta_{12}^{3} - 4 \zeta_{12}) q^{97} + ( - 14 \zeta_{12}^{3} + 7 \zeta_{12}) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{4} + 12 q^{6} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{4} + 12 q^{6} + 2 q^{9} + 10 q^{16} + 12 q^{19} + 12 q^{24} + 6 q^{26} + 6 q^{29} + 2 q^{36} - 28 q^{39} - 18 q^{41} - 36 q^{46} + 14 q^{49} - 24 q^{51} - 24 q^{54} - 24 q^{59} - 2 q^{61} + 4 q^{64} - 24 q^{69} + 12 q^{71} + 30 q^{74} - 12 q^{76} - 16 q^{79} + 22 q^{81} + 24 q^{89} + 12 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(27\) \(301\)
\(\chi(n)\) \(-1\) \(1 - \zeta_{12}^{2}\)

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
0.866025 + 0.500000i
−0.866025 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i
−0.866025 1.50000i −1.73205 + 1.00000i −0.500000 + 0.866025i 0 3.00000 + 1.73205i 0 −1.73205 0.500000 0.866025i 0
49.2 0.866025 + 1.50000i 1.73205 1.00000i −0.500000 + 0.866025i 0 3.00000 + 1.73205i 0 1.73205 0.500000 0.866025i 0
199.1 −0.866025 + 1.50000i −1.73205 1.00000i −0.500000 0.866025i 0 3.00000 1.73205i 0 −1.73205 0.500000 + 0.866025i 0
199.2 0.866025 1.50000i 1.73205 + 1.00000i −0.500000 0.866025i 0 3.00000 1.73205i 0 1.73205 0.500000 + 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
13.e even 6 1 inner
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.a 4
5.b even 2 1 inner 325.2.m.a 4
5.c odd 4 1 13.2.e.a 2
5.c odd 4 1 325.2.n.a 2
13.e even 6 1 inner 325.2.m.a 4
15.e even 4 1 117.2.q.c 2
20.e even 4 1 208.2.w.b 2
35.f even 4 1 637.2.q.a 2
35.k even 12 1 637.2.k.c 2
35.k even 12 1 637.2.u.b 2
35.l odd 12 1 637.2.k.a 2
35.l odd 12 1 637.2.u.c 2
40.i odd 4 1 832.2.w.d 2
40.k even 4 1 832.2.w.a 2
60.l odd 4 1 1872.2.by.d 2
65.f even 4 1 169.2.c.a 4
65.h odd 4 1 169.2.e.a 2
65.k even 4 1 169.2.c.a 4
65.l even 6 1 inner 325.2.m.a 4
65.o even 12 1 169.2.a.a 2
65.o even 12 1 169.2.c.a 4
65.o even 12 1 4225.2.a.v 2
65.q odd 12 1 169.2.b.a 2
65.q odd 12 1 169.2.e.a 2
65.r odd 12 1 13.2.e.a 2
65.r odd 12 1 169.2.b.a 2
65.r odd 12 1 325.2.n.a 2
65.t even 12 1 169.2.a.a 2
65.t even 12 1 169.2.c.a 4
65.t even 12 1 4225.2.a.v 2
195.bc odd 12 1 1521.2.a.k 2
195.bf even 12 1 117.2.q.c 2
195.bf even 12 1 1521.2.b.a 2
195.bl even 12 1 1521.2.b.a 2
195.bn odd 12 1 1521.2.a.k 2
260.be odd 12 1 2704.2.a.o 2
260.bg even 12 1 208.2.w.b 2
260.bg even 12 1 2704.2.f.b 2
260.bj even 12 1 2704.2.f.b 2
260.bl odd 12 1 2704.2.a.o 2
455.cf odd 12 1 8281.2.a.q 2
455.cr even 12 1 637.2.u.b 2
455.ct odd 12 1 637.2.u.c 2
455.cw even 12 1 637.2.k.c 2
455.cz even 12 1 637.2.q.a 2
455.da odd 12 1 637.2.k.a 2
455.ds odd 12 1 8281.2.a.q 2
520.co odd 12 1 832.2.w.d 2
520.cs even 12 1 832.2.w.a 2
780.cw odd 12 1 1872.2.by.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.2.e.a 2 5.c odd 4 1
13.2.e.a 2 65.r odd 12 1
117.2.q.c 2 15.e even 4 1
117.2.q.c 2 195.bf even 12 1
169.2.a.a 2 65.o even 12 1
169.2.a.a 2 65.t even 12 1
169.2.b.a 2 65.q odd 12 1
169.2.b.a 2 65.r odd 12 1
169.2.c.a 4 65.f even 4 1
169.2.c.a 4 65.k even 4 1
169.2.c.a 4 65.o even 12 1
169.2.c.a 4 65.t even 12 1
169.2.e.a 2 65.h odd 4 1
169.2.e.a 2 65.q odd 12 1
208.2.w.b 2 20.e even 4 1
208.2.w.b 2 260.bg even 12 1
325.2.m.a 4 1.a even 1 1 trivial
325.2.m.a 4 5.b even 2 1 inner
325.2.m.a 4 13.e even 6 1 inner
325.2.m.a 4 65.l even 6 1 inner
325.2.n.a 2 5.c odd 4 1
325.2.n.a 2 65.r odd 12 1
637.2.k.a 2 35.l odd 12 1
637.2.k.a 2 455.da odd 12 1
637.2.k.c 2 35.k even 12 1
637.2.k.c 2 455.cw even 12 1
637.2.q.a 2 35.f even 4 1
637.2.q.a 2 455.cz even 12 1
637.2.u.b 2 35.k even 12 1
637.2.u.b 2 455.cr even 12 1
637.2.u.c 2 35.l odd 12 1
637.2.u.c 2 455.ct odd 12 1
832.2.w.a 2 40.k even 4 1
832.2.w.a 2 520.cs even 12 1
832.2.w.d 2 40.i odd 4 1
832.2.w.d 2 520.co odd 12 1
1521.2.a.k 2 195.bc odd 12 1
1521.2.a.k 2 195.bn odd 12 1
1521.2.b.a 2 195.bf even 12 1
1521.2.b.a 2 195.bl even 12 1
1872.2.by.d 2 60.l odd 4 1
1872.2.by.d 2 780.cw odd 12 1
2704.2.a.o 2 260.be odd 12 1
2704.2.a.o 2 260.bl odd 12 1
2704.2.f.b 2 260.bg even 12 1
2704.2.f.b 2 260.bj even 12 1
4225.2.a.v 2 65.o even 12 1
4225.2.a.v 2 65.t even 12 1
8281.2.a.q 2 455.cf odd 12 1
8281.2.a.q 2 455.ds odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 3T_{2}^{2} + 9 \) acting on \(S_{2}^{\mathrm{new}}(325, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$3$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} - T^{2} + 169 \) Copy content Toggle raw display
$17$ \( T^{4} - 9T^{2} + 81 \) Copy content Toggle raw display
$19$ \( (T^{2} - 6 T + 12)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} - 36T^{2} + 1296 \) Copy content Toggle raw display
$29$ \( (T^{2} - 3 T + 9)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 75T^{2} + 5625 \) Copy content Toggle raw display
$41$ \( (T^{2} + 9 T + 27)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} - 64T^{2} + 4096 \) Copy content Toggle raw display
$47$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 9)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 12 T + 48)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 12T^{2} + 144 \) Copy content Toggle raw display
$71$ \( (T^{2} - 6 T + 12)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 3)^{2} \) Copy content Toggle raw display
$79$ \( (T + 4)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} - 192)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 12 T + 48)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 48T^{2} + 2304 \) Copy content Toggle raw display
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