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Properties

Label 325.2.k.d
Level $325$
Weight $2$
Character orbit 325.k
Analytic conductor $2.595$
Analytic rank $0$
Dimension $16$
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(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: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 111x^{12} + 329x^{8} + 168x^{4} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{12} q^{2} + \beta_{3} q^{3} - \beta_1 q^{4} + ( - \beta_{9} - \beta_{4} + 1) q^{6} - \beta_{13} q^{7} + (\beta_{14} + \beta_{10} - \beta_{5}) q^{8} + (\beta_{15} + \beta_{7} + \beta_{6} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{12} q^{2} + \beta_{3} q^{3} - \beta_1 q^{4} + ( - \beta_{9} - \beta_{4} + 1) q^{6} - \beta_{13} q^{7} + (\beta_{14} + \beta_{10} - \beta_{5}) q^{8} + (\beta_{15} + \beta_{7} + \beta_{6} + 1) q^{9} + ( - \beta_{15} - \beta_{8} + \cdots + \beta_1) q^{11}+ \cdots + ( - 4 \beta_{15} + \beta_{9} + \cdots - 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 8 q^{4} + 12 q^{6}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 8 q^{4} + 12 q^{6} - 20 q^{11} - 8 q^{16} + 16 q^{19} + 12 q^{21} + 16 q^{24} - 16 q^{26} + 8 q^{31} - 44 q^{34} + 28 q^{39} + 4 q^{41} - 76 q^{44} + 12 q^{46} + 72 q^{49} - 4 q^{54} + 24 q^{59} + 24 q^{61} - 16 q^{64} - 48 q^{66} - 112 q^{69} - 20 q^{71} + 8 q^{76} + 40 q^{84} + 12 q^{86} + 36 q^{89} + 8 q^{91} - 72 q^{96} - 52 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 111x^{12} + 329x^{8} + 168x^{4} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -357\nu^{12} - 39679\nu^{8} - 123305\nu^{4} - 62220 ) / 12928 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 201\nu^{13} + 22123\nu^{9} + 45469\nu^{5} - 4676\nu ) / 6464 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 447\nu^{15} + 48813\nu^{11} + 58571\nu^{7} - 106780\nu^{3} ) / 25856 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 357\nu^{14} + 39679\nu^{10} + 123305\nu^{6} + 88076\nu^{2} ) / 25856 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -1447\nu^{13} - 160357\nu^{9} - 446803\nu^{5} - 128452\nu ) / 25856 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 1537 \nu^{14} + 1466 \nu^{12} + 169491 \nu^{10} + 161998 \nu^{8} + 382069 \nu^{6} + 400386 \nu^{4} + \cdots + 8024 ) / 51712 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 1537 \nu^{14} - 1466 \nu^{12} + 169491 \nu^{10} - 161998 \nu^{8} + 382069 \nu^{6} - 400386 \nu^{4} + \cdots - 59736 ) / 51712 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 201\nu^{14} + 201\nu^{12} + 22123\nu^{10} + 22123\nu^{8} + 45469\nu^{6} + 45469\nu^{4} - 4676\nu^{2} + 1788 ) / 6464 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -201\nu^{14} + 201\nu^{12} - 22123\nu^{10} + 22123\nu^{8} - 45469\nu^{6} + 45469\nu^{4} + 4676\nu^{2} + 1788 ) / 6464 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 1259 \nu^{15} + 357 \nu^{13} - 139697 \nu^{11} + 39679 \nu^{9} - 408359 \nu^{7} + \cdots + 62220 \nu ) / 25856 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 1259 \nu^{15} - 357 \nu^{13} - 139697 \nu^{11} - 39679 \nu^{9} - 408359 \nu^{7} + \cdots - 62220 \nu ) / 25856 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 1616 \nu^{15} + 357 \nu^{13} - 179376 \nu^{11} + 39679 \nu^{9} - 531664 \nu^{7} + \cdots + 88076 \nu ) / 25856 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 1616 \nu^{15} - 357 \nu^{13} - 179376 \nu^{11} - 39679 \nu^{9} - 531664 \nu^{7} + \cdots - 88076 \nu ) / 25856 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( -6573\nu^{15} - 728279\nu^{11} - 2015505\nu^{7} - 667244\nu^{3} ) / 51712 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 1259\nu^{14} + 139697\nu^{10} + 408359\nu^{6} + 183412\nu^{2} ) / 12928 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{13} + \beta_{12} + \beta_{11} - \beta_{10} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{15} + \beta_{9} - \beta_{8} + 2\beta_{7} + 2\beta_{6} + 10\beta_{4} + 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{14} - 2\beta_{13} - 2\beta_{12} + 5\beta_{11} + 5\beta_{10} - \beta_{3} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 13\beta_{9} + 13\beta_{8} + 24\beta_{7} - 24\beta_{6} - 20\beta _1 - 72 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 35\beta_{13} - 35\beta_{12} - 103\beta_{11} + 103\beta_{10} + 48\beta_{5} + 26\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 103\beta_{15} - 70\beta_{9} + 70\beta_{8} - 127\beta_{7} - 127\beta_{6} - 495\beta_{4} - 127 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 508\beta_{14} + 355\beta_{13} + 355\beta_{12} - 1069\beta_{11} - 1069\beta_{10} + 280\beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -1463\beta_{9} - 1463\beta_{8} - 2646\beta_{7} + 2646\beta_{6} + 2138\beta _1 + 7632 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -1838\beta_{13} + 1838\beta_{12} + 5553\beta_{11} - 5553\beta_{10} - 2646\beta_{5} - 1463\beta_{2} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 22212 \beta_{15} + 15215 \beta_{9} - 15215 \beta_{8} + 27504 \beta_{7} + 27504 \beta_{6} + \cdots + 27504 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( -55008\beta_{14} - 38177\beta_{13} - 38177\beta_{12} + 115397\beta_{11} + 115397\beta_{10} - 30430\beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 79058\beta_{9} + 79058\beta_{8} + 142901\beta_{7} - 142901\beta_{6} - 115397\beta _1 - 411872 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 396657 \beta_{13} - 396657 \beta_{12} - 1199055 \beta_{11} + 1199055 \beta_{10} + \cdots + 316232 \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 2398110 \beta_{15} - 1642973 \beta_{9} + 1642973 \beta_{8} - 2969714 \beta_{7} - 2969714 \beta_{6} + \cdots - 2969714 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 2969714 \beta_{14} + 2060754 \beta_{13} + 2060754 \beta_{12} - 6229523 \beta_{11} + \cdots + 1642973 \beta_{3} \) Copy content Toggle raw display

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

Character values

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

\(n\) \(27\) \(301\)
\(\chi(n)\) \(\beta_{4}\) \(-\beta_{4}\)

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
−0.419746 − 0.419746i
−0.592914 − 0.592914i
−0.881421 − 0.881421i
−2.27933 − 2.27933i
2.27933 + 2.27933i
0.881421 + 0.881421i
0.592914 + 0.592914i
0.419746 + 0.419746i
−0.419746 + 0.419746i
−0.592914 + 0.592914i
−0.881421 + 0.881421i
−2.27933 + 2.27933i
2.27933 − 2.27933i
0.881421 − 0.881421i
0.592914 − 0.592914i
0.419746 − 0.419746i
−2.38239 −0.648130 + 0.648130i 3.67579 0 1.54410 − 1.54410i − 2.38239i −3.99239 2.15985i 0
57.2 −1.68658 −1.99698 + 1.99698i 0.844569 0 3.36807 − 3.36807i − 1.68658i 1.94873 − 4.97584i 0
57.3 −1.13453 1.59157 − 1.59157i −0.712838 0 −1.80569 + 1.80569i − 1.13453i 3.07780 − 2.06618i 0
57.4 −0.438725 0.242722 − 0.242722i −1.80752 0 −0.106488 + 0.106488i − 0.438725i 1.67045 2.88217i 0
57.5 0.438725 −0.242722 + 0.242722i −1.80752 0 −0.106488 + 0.106488i 0.438725i −1.67045 2.88217i 0
57.6 1.13453 −1.59157 + 1.59157i −0.712838 0 −1.80569 + 1.80569i 1.13453i −3.07780 − 2.06618i 0
57.7 1.68658 1.99698 − 1.99698i 0.844569 0 3.36807 − 3.36807i 1.68658i −1.94873 − 4.97584i 0
57.8 2.38239 0.648130 − 0.648130i 3.67579 0 1.54410 − 1.54410i 2.38239i 3.99239 2.15985i 0
268.1 −2.38239 −0.648130 − 0.648130i 3.67579 0 1.54410 + 1.54410i 2.38239i −3.99239 − 2.15985i 0
268.2 −1.68658 −1.99698 − 1.99698i 0.844569 0 3.36807 + 3.36807i 1.68658i 1.94873 4.97584i 0
268.3 −1.13453 1.59157 + 1.59157i −0.712838 0 −1.80569 − 1.80569i 1.13453i 3.07780 2.06618i 0
268.4 −0.438725 0.242722 + 0.242722i −1.80752 0 −0.106488 − 0.106488i 0.438725i 1.67045 − 2.88217i 0
268.5 0.438725 −0.242722 − 0.242722i −1.80752 0 −0.106488 − 0.106488i − 0.438725i −1.67045 − 2.88217i 0
268.6 1.13453 −1.59157 − 1.59157i −0.712838 0 −1.80569 − 1.80569i − 1.13453i −3.07780 2.06618i 0
268.7 1.68658 1.99698 + 1.99698i 0.844569 0 3.36807 + 3.36807i − 1.68658i −1.94873 4.97584i 0
268.8 2.38239 0.648130 + 0.648130i 3.67579 0 1.54410 + 1.54410i − 2.38239i 3.99239 − 2.15985i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 57.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
65.f even 4 1 inner
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.d yes 16
5.b even 2 1 inner 325.2.k.d yes 16
5.c odd 4 2 325.2.f.d ✓ 16
13.d odd 4 1 325.2.f.d ✓ 16
65.f even 4 1 inner 325.2.k.d yes 16
65.g odd 4 1 325.2.f.d ✓ 16
65.k even 4 1 inner 325.2.k.d yes 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
325.2.f.d ✓ 16 5.c odd 4 2
325.2.f.d ✓ 16 13.d odd 4 1
325.2.f.d ✓ 16 65.g odd 4 1
325.2.k.d yes 16 1.a even 1 1 trivial
325.2.k.d yes 16 5.b even 2 1 inner
325.2.k.d yes 16 65.f even 4 1 inner
325.2.k.d yes 16 65.k even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} - 10T_{2}^{6} + 29T_{2}^{4} - 26T_{2}^{2} + 4 \) acting on \(S_{2}^{\mathrm{new}}(325, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} - 10 T^{6} + 29 T^{4} + \cdots + 4)^{2} \) Copy content Toggle raw display
$3$ \( T^{16} + 90 T^{12} + \cdots + 16 \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( (T^{8} + 10 T^{6} + 29 T^{4} + \cdots + 4)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} + 10 T^{7} + \cdots + 33124)^{2} \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 815730721 \) Copy content Toggle raw display
$17$ \( T^{16} + 1890 T^{12} + \cdots + 160000 \) Copy content Toggle raw display
$19$ \( (T^{8} - 8 T^{7} + \cdots + 4096)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 48796810000 \) Copy content Toggle raw display
$29$ \( (T^{8} + 120 T^{6} + \cdots + 169)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} - 4 T^{7} + \cdots + 7396)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + 294 T^{6} + \cdots + 25040016)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} - 2 T^{7} + 2 T^{6} + \cdots + 64)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} + 1530 T^{12} + \cdots + 6250000 \) Copy content Toggle raw display
$47$ \( (T^{8} + 88 T^{6} + \cdots + 64)^{2} \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 51769445841 \) Copy content Toggle raw display
$59$ \( (T^{8} - 12 T^{7} + \cdots + 11236)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 6 T^{3} + \cdots + 2259)^{4} \) Copy content Toggle raw display
$67$ \( (T^{8} - 308 T^{6} + \cdots + 7761796)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + 10 T^{7} + \cdots + 355216)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} - 382 T^{6} + \cdots + 13512976)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + 234 T^{6} + \cdots + 6091024)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + 586 T^{6} + \cdots + 145395364)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} - 18 T^{7} + \cdots + 1882384)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} - 242 T^{6} + \cdots + 4096)^{2} \) Copy content Toggle raw display
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