Properties

Label 325.2.k.b
Level $325$
Weight $2$
Character orbit 325.k
Analytic conductor $2.595$
Analytic rank $0$
Dimension $8$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [325,2,Mod(57,325)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(325, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([1, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("325.57");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 325 = 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 325.k (of order \(4\), 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: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.619810816.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{5} + 14x^{4} - 8x^{3} + 2x^{2} + 2x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 65)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{5} q^{2} + ( - \beta_{3} + \beta_{2} + 1) q^{3} + ( - \beta_{7} - \beta_{5} + \beta_{4} + 1) q^{4} + (\beta_{6} - \beta_{5} - \beta_{3} + \cdots - 1) q^{6}+ \cdots + ( - \beta_{5} + \beta_{4} + \cdots - \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{5} q^{2} + ( - \beta_{3} + \beta_{2} + 1) q^{3} + ( - \beta_{7} - \beta_{5} + \beta_{4} + 1) q^{4} + (\beta_{6} - \beta_{5} - \beta_{3} + \cdots - 1) q^{6}+ \cdots + ( - \beta_{7} - \beta_{5} + \cdots + \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{2} + 6 q^{3} + 8 q^{4} - 6 q^{6}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{2} + 6 q^{3} + 8 q^{4} - 6 q^{6} + 6 q^{11} - 2 q^{12} + 2 q^{13} - 8 q^{16} + 16 q^{17} + 14 q^{19} - 12 q^{21} - 10 q^{22} - 14 q^{23} + 2 q^{24} + 6 q^{26} - 12 q^{27} + 2 q^{31} + 4 q^{32} + 8 q^{33} - 2 q^{38} - 6 q^{39} + 16 q^{41} - 24 q^{42} - 6 q^{43} + 10 q^{44} + 2 q^{46} + 14 q^{48} - 24 q^{49} + 22 q^{52} + 24 q^{53} - 20 q^{54} + 40 q^{57} + 22 q^{59} + 20 q^{61} - 30 q^{62} - 16 q^{63} - 48 q^{64} - 36 q^{66} + 12 q^{67} - 4 q^{68} - 4 q^{69} - 10 q^{71} + 4 q^{73} + 6 q^{76} - 24 q^{77} - 30 q^{78} - 20 q^{81} + 20 q^{82} + 16 q^{84} - 46 q^{86} - 16 q^{87} - 10 q^{88} - 28 q^{89} + 20 q^{91} - 50 q^{92} + 30 q^{96} - 12 q^{97} - 92 q^{98} - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{5} + 14x^{4} - 8x^{3} + 2x^{2} + 2x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 64\nu^{7} + 16\nu^{6} + 4\nu^{5} - 127\nu^{4} + 944\nu^{3} - 276\nu^{2} + 378\nu + 63 ) / 319 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -63\nu^{7} + 64\nu^{6} + 16\nu^{5} + 130\nu^{4} - 1009\nu^{3} + 1448\nu^{2} - 402\nu - 67 ) / 319 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -67\nu^{7} + 63\nu^{6} - 64\nu^{5} + 118\nu^{4} - 1068\nu^{3} + 1545\nu^{2} - 1263\nu + 268 ) / 319 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 83\nu^{7} - 59\nu^{6} + 65\nu^{5} - 70\nu^{4} + 1304\nu^{3} - 1614\nu^{2} + 1198\nu + 306 ) / 319 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -172\nu^{7} - 43\nu^{6} + 69\nu^{5} + 441\nu^{4} - 2218\nu^{3} + 662\nu^{2} + 619\nu - 269 ) / 319 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -196\nu^{7} - 49\nu^{6} - 92\nu^{5} + 369\nu^{4} - 2572\nu^{3} + 1244\nu^{2} - 1038\nu - 173 ) / 319 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( \nu^{7} - 2\nu^{4} + 14\nu^{3} - 8\nu^{2} + \nu + 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{7} - \beta_{5} + \beta_{4} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} + 2\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3\beta_{7} + 5\beta_{5} - 5\beta_{4} - 2\beta_{2} + 3\beta _1 + 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{7} - \beta_{5} + 5\beta_{4} + 4\beta_{3} - 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 11\beta_{7} + 2\beta_{6} + 9\beta_{5} - 11\beta_{4} - 12\beta_{3} + 12\beta_{2} - 11\beta _1 + 12 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -15\beta_{6} + 16\beta_{5} - 16\beta_{4} + 16\beta_{3} - 28\beta_{2} + 7\beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -43\beta_{7} + 16\beta_{6} - 89\beta_{5} + 105\beta_{4} + 60\beta_{2} - 43\beta _1 - 60 ) / 2 \) Copy content Toggle raw display

Character values

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

\(n\) \(27\) \(301\)
\(\chi(n)\) \(-\beta_{2}\) \(\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
57.1
0.561103 0.561103i
−1.49094 + 1.49094i
−0.252709 + 0.252709i
1.18254 1.18254i
0.561103 + 0.561103i
−1.49094 1.49094i
−0.252709 0.252709i
1.18254 + 1.18254i
−2.03032 1.33000 1.33000i 2.12221 0 −2.70032 + 2.70032i 1.61845i −0.248119 0.537789i 0
57.2 0.134632 2.15558 2.15558i −1.98187 0 0.290209 0.290209i 1.90970i −0.536087 6.29303i 0
57.3 1.57942 −0.725850 + 0.725850i 0.494582 0 −1.14643 + 1.14643i 4.24997i −2.37769 1.94628i 0
57.4 2.31627 0.240275 0.240275i 3.36509 0 0.556540 0.556540i 3.95872i 3.16190 2.88454i 0
268.1 −2.03032 1.33000 + 1.33000i 2.12221 0 −2.70032 2.70032i 1.61845i −0.248119 0.537789i 0
268.2 0.134632 2.15558 + 2.15558i −1.98187 0 0.290209 + 0.290209i 1.90970i −0.536087 6.29303i 0
268.3 1.57942 −0.725850 0.725850i 0.494582 0 −1.14643 1.14643i 4.24997i −2.37769 1.94628i 0
268.4 2.31627 0.240275 + 0.240275i 3.36509 0 0.556540 + 0.556540i 3.95872i 3.16190 2.88454i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 57.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
65.k even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 325.2.k.b 8
5.b even 2 1 65.2.k.b yes 8
5.c odd 4 1 65.2.f.b 8
5.c odd 4 1 325.2.f.b 8
13.d odd 4 1 325.2.f.b 8
15.d odd 2 1 585.2.w.e 8
15.e even 4 1 585.2.n.e 8
20.d odd 2 1 1040.2.bg.n 8
20.e even 4 1 1040.2.cd.n 8
65.d even 2 1 845.2.k.b 8
65.f even 4 1 65.2.k.b yes 8
65.g odd 4 1 65.2.f.b 8
65.g odd 4 1 845.2.f.b 8
65.h odd 4 1 845.2.f.b 8
65.k even 4 1 inner 325.2.k.b 8
65.k even 4 1 845.2.k.b 8
65.l even 6 2 845.2.o.c 16
65.n even 6 2 845.2.o.d 16
65.o even 12 2 845.2.o.c 16
65.q odd 12 2 845.2.t.c 16
65.r odd 12 2 845.2.t.d 16
65.s odd 12 2 845.2.t.c 16
65.s odd 12 2 845.2.t.d 16
65.t even 12 2 845.2.o.d 16
195.n even 4 1 585.2.n.e 8
195.u odd 4 1 585.2.w.e 8
260.l odd 4 1 1040.2.bg.n 8
260.u even 4 1 1040.2.cd.n 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.f.b 8 5.c odd 4 1
65.2.f.b 8 65.g odd 4 1
65.2.k.b yes 8 5.b even 2 1
65.2.k.b yes 8 65.f even 4 1
325.2.f.b 8 5.c odd 4 1
325.2.f.b 8 13.d odd 4 1
325.2.k.b 8 1.a even 1 1 trivial
325.2.k.b 8 65.k even 4 1 inner
585.2.n.e 8 15.e even 4 1
585.2.n.e 8 195.n even 4 1
585.2.w.e 8 15.d odd 2 1
585.2.w.e 8 195.u odd 4 1
845.2.f.b 8 65.g odd 4 1
845.2.f.b 8 65.h odd 4 1
845.2.k.b 8 65.d even 2 1
845.2.k.b 8 65.k even 4 1
845.2.o.c 16 65.l even 6 2
845.2.o.c 16 65.o even 12 2
845.2.o.d 16 65.n even 6 2
845.2.o.d 16 65.t even 12 2
845.2.t.c 16 65.q odd 12 2
845.2.t.c 16 65.s odd 12 2
845.2.t.d 16 65.r odd 12 2
845.2.t.d 16 65.s odd 12 2
1040.2.bg.n 8 20.d odd 2 1
1040.2.bg.n 8 260.l odd 4 1
1040.2.cd.n 8 20.e even 4 1
1040.2.cd.n 8 260.u even 4 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 2 T^{3} - 4 T^{2} + \cdots - 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} - 6 T^{7} + \cdots + 4 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + 40 T^{6} + \cdots + 2704 \) Copy content Toggle raw display
$11$ \( T^{8} - 6 T^{7} + \cdots + 4 \) Copy content Toggle raw display
$13$ \( T^{8} - 2 T^{7} + \cdots + 28561 \) Copy content Toggle raw display
$17$ \( T^{8} - 16 T^{7} + \cdots + 13456 \) Copy content Toggle raw display
$19$ \( T^{8} - 14 T^{7} + \cdots + 100 \) Copy content Toggle raw display
$23$ \( T^{8} + 14 T^{7} + \cdots + 40804 \) Copy content Toggle raw display
$29$ \( T^{8} + 44 T^{6} + \cdots + 10000 \) Copy content Toggle raw display
$31$ \( T^{8} - 2 T^{7} + \cdots + 16900 \) Copy content Toggle raw display
$37$ \( T^{8} + 140 T^{6} + \cdots + 336400 \) Copy content Toggle raw display
$41$ \( T^{8} - 16 T^{7} + \cdots + 13456 \) Copy content Toggle raw display
$43$ \( T^{8} + 6 T^{7} + \cdots + 8836 \) Copy content Toggle raw display
$47$ \( T^{8} + 160 T^{6} + \cdots + 26896 \) Copy content Toggle raw display
$53$ \( T^{8} - 24 T^{7} + \cdots + 19600 \) Copy content Toggle raw display
$59$ \( T^{8} - 22 T^{7} + \cdots + 119716 \) Copy content Toggle raw display
$61$ \( (T^{4} - 10 T^{3} + \cdots - 3628)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 6 T^{3} + \cdots - 148)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + 10 T^{7} + \cdots + 1223236 \) Copy content Toggle raw display
$73$ \( (T^{4} - 2 T^{3} + \cdots + 740)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} + 292 T^{6} + \cdots + 13719616 \) Copy content Toggle raw display
$83$ \( T^{8} + 416 T^{6} + \cdots + 54346384 \) Copy content Toggle raw display
$89$ \( T^{8} + 28 T^{7} + \cdots + 1795600 \) Copy content Toggle raw display
$97$ \( (T^{4} + 6 T^{3} + \cdots + 3704)^{2} \) Copy content Toggle raw display
show more
show less