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Properties

Label 425.2.e.d
Level $425$
Weight $2$
Character orbit 425.e
Analytic conductor $3.394$
Analytic rank $0$
Dimension $12$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [425,2,Mod(251,425)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(425, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("425.251");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 425 = 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 425.e (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: \(3.39364208590\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 24x^{10} + 188x^{8} + 572x^{6} + 776x^{4} + 464x^{2} + 100 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 85)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{4} q^{2} + ( - \beta_{7} + \beta_{4} + \beta_1) q^{3} + (\beta_{6} - 1) q^{4} + ( - \beta_{11} + \beta_{6} + \beta_{5} + \cdots - 2) q^{6}+ \cdots + ( - \beta_{11} - \beta_{10} + \cdots + 3 \beta_{3}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{4} q^{2} + ( - \beta_{7} + \beta_{4} + \beta_1) q^{3} + (\beta_{6} - 1) q^{4} + ( - \beta_{11} + \beta_{6} + \beta_{5} + \cdots - 2) q^{6}+ \cdots + ( - 3 \beta_{10} - \beta_{6} + \cdots - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{4} - 20 q^{6}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 12 q^{4} - 20 q^{6} + 16 q^{11} + 16 q^{14} + 4 q^{16} - 24 q^{21} + 32 q^{24} - 4 q^{29} + 4 q^{31} - 12 q^{39} + 16 q^{41} - 28 q^{44} - 4 q^{46} + 44 q^{51} - 100 q^{54} + 4 q^{56} - 48 q^{61} + 76 q^{64} + 88 q^{69} - 40 q^{71} - 28 q^{74} + 4 q^{79} - 60 q^{81} - 40 q^{84} - 16 q^{86} + 16 q^{89} + 36 q^{91} - 4 q^{96} - 36 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 24x^{10} + 188x^{8} + 572x^{6} + 776x^{4} + 464x^{2} + 100 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 8\nu^{10} + 185\nu^{8} + 1344\nu^{6} + 3440\nu^{4} + 3438\nu^{2} + 1080 ) / 20 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 9\nu^{10} + 210\nu^{8} + 1552\nu^{6} + 4110\nu^{4} + 4204\nu^{2} + 1320 ) / 20 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 11\nu^{11} + 256\nu^{9} + 1883\nu^{7} + 4948\nu^{5} + 5096\nu^{3} + 1666\nu ) / 20 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -11\nu^{11} - 256\nu^{9} - 1883\nu^{7} - 4948\nu^{5} - 5096\nu^{3} - 1646\nu ) / 20 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -14\nu^{11} - 327\nu^{9} - 2422\nu^{7} - 6456\nu^{5} - 6754\nu^{3} - 2272\nu ) / 20 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 8\nu^{10} + 185\nu^{8} + 1344\nu^{6} + 3440\nu^{4} + 3448\nu^{2} + 1120 ) / 10 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 11 \nu^{11} + 26 \nu^{10} - 254 \nu^{9} + 610 \nu^{8} - 1838 \nu^{7} + 4558 \nu^{6} - 4642 \nu^{5} + \cdots + 4420 ) / 40 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 11 \nu^{11} + 26 \nu^{10} + 254 \nu^{9} + 610 \nu^{8} + 1838 \nu^{7} + 4558 \nu^{6} + 4642 \nu^{5} + \cdots + 4420 ) / 40 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -20\nu^{11} - 469\nu^{9} - 3500\nu^{7} - 9472\nu^{5} - 10050\nu^{3} - 3324\nu ) / 20 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 15 \nu^{11} - 36 \nu^{10} + 352 \nu^{9} - 840 \nu^{8} + 2630 \nu^{7} - 6208 \nu^{6} + 7126 \nu^{5} + \cdots - 5560 ) / 40 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 15 \nu^{11} + 36 \nu^{10} + 352 \nu^{9} + 840 \nu^{8} + 2630 \nu^{7} + 6208 \nu^{6} + 7126 \nu^{5} + \cdots + 5560 ) / 40 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_{4} + \beta_{3} \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} - 2\beta _1 - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{9} - 3\beta_{5} - 8\beta_{4} - 10\beta_{3} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{11} - \beta_{10} - 12\beta_{6} - 4\beta_{2} + 24\beta _1 + 34 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -5\beta_{11} - 5\beta_{10} - 18\beta_{9} + \beta_{8} - \beta_{7} + 40\beta_{5} + 82\beta_{4} + 106\beta_{3} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -24\beta_{11} + 24\beta_{10} + 6\beta_{8} + 6\beta_{7} + 140\beta_{6} + 68\beta_{2} - 268\beta _1 - 350 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 98\beta_{11} + 98\beta_{10} + 256\beta_{9} - 30\beta_{8} + 30\beta_{7} - 476\beta_{5} - 898\beta_{4} - 1142\beta_{3} \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 384\beta_{11} - 384\beta_{10} - 128\beta_{8} - 128\beta_{7} - 1630\beta_{6} - 928\beta_{2} + 2992\beta _1 + 3776 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 1440 \beta_{11} - 1440 \beta_{10} - 3326 \beta_{9} + 512 \beta_{8} - 512 \beta_{7} + \cdots + 12496 \beta_{3} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 5278 \beta_{11} + 5278 \beta_{10} + 1952 \beta_{8} + 1952 \beta_{7} + 18904 \beta_{6} + 11756 \beta_{2} + \cdots - 41556 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 18986 \beta_{11} + 18986 \beta_{10} + 41216 \beta_{9} - 7230 \beta_{8} + 7230 \beta_{7} + \cdots - 138524 \beta_{3} \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/425\mathbb{Z}\right)^\times\).

\(n\) \(52\) \(326\)
\(\chi(n)\) \(1\) \(\beta_{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
251.1
− 1.38621i
− 0.803330i
0.767611i
1.23239i
2.80333i
3.38621i
− 3.38621i
− 2.80333i
− 1.23239i
− 0.767611i
0.803330i
1.38621i
− 2.38621i −2.23065 − 2.23065i −3.69399 0 −5.32280 + 5.32280i −0.155559 + 0.155559i 4.04223i 6.95160i 0
251.2 − 1.80333i 0.397180 + 0.397180i −1.25200 0 0.716248 − 0.716248i −2.20051 + 2.20051i − 1.34889i − 2.68450i 0
251.3 − 0.232389i −1.69306 − 1.69306i 1.94600 0 −0.393449 + 0.393449i 1.46067 − 1.46067i − 0.917007i 2.73289i 0
251.4 0.232389i 1.69306 + 1.69306i 1.94600 0 −0.393449 + 0.393449i −1.46067 + 1.46067i 0.917007i 2.73289i 0
251.5 1.80333i −0.397180 − 0.397180i −1.25200 0 0.716248 − 0.716248i 2.20051 − 2.20051i 1.34889i − 2.68450i 0
251.6 2.38621i 2.23065 + 2.23065i −3.69399 0 −5.32280 + 5.32280i 0.155559 − 0.155559i − 4.04223i 6.95160i 0
276.1 − 2.38621i 2.23065 − 2.23065i −3.69399 0 −5.32280 − 5.32280i 0.155559 + 0.155559i 4.04223i − 6.95160i 0
276.2 − 1.80333i −0.397180 + 0.397180i −1.25200 0 0.716248 + 0.716248i 2.20051 + 2.20051i − 1.34889i 2.68450i 0
276.3 − 0.232389i 1.69306 − 1.69306i 1.94600 0 −0.393449 − 0.393449i −1.46067 − 1.46067i − 0.917007i − 2.73289i 0
276.4 0.232389i −1.69306 + 1.69306i 1.94600 0 −0.393449 − 0.393449i 1.46067 + 1.46067i 0.917007i − 2.73289i 0
276.5 1.80333i 0.397180 − 0.397180i −1.25200 0 0.716248 + 0.716248i −2.20051 − 2.20051i 1.34889i 2.68450i 0
276.6 2.38621i −2.23065 + 2.23065i −3.69399 0 −5.32280 − 5.32280i −0.155559 − 0.155559i − 4.04223i − 6.95160i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 251.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
17.c even 4 1 inner
85.j even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 425.2.e.d 12
5.b even 2 1 inner 425.2.e.d 12
5.c odd 4 2 85.2.j.c ✓ 12
15.e even 4 2 765.2.t.e 12
17.c even 4 1 inner 425.2.e.d 12
17.d even 8 2 7225.2.a.bp 12
85.f odd 4 1 85.2.j.c ✓ 12
85.i odd 4 1 85.2.j.c ✓ 12
85.j even 4 1 inner 425.2.e.d 12
85.k odd 8 2 1445.2.b.f 12
85.m even 8 2 7225.2.a.bp 12
85.n odd 8 2 1445.2.b.f 12
255.k even 4 1 765.2.t.e 12
255.r even 4 1 765.2.t.e 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
85.2.j.c ✓ 12 5.c odd 4 2
85.2.j.c ✓ 12 85.f odd 4 1
85.2.j.c ✓ 12 85.i odd 4 1
425.2.e.d 12 1.a even 1 1 trivial
425.2.e.d 12 5.b even 2 1 inner
425.2.e.d 12 17.c even 4 1 inner
425.2.e.d 12 85.j even 4 1 inner
765.2.t.e 12 15.e even 4 2
765.2.t.e 12 255.k even 4 1
765.2.t.e 12 255.r even 4 1
1445.2.b.f 12 85.k odd 8 2
1445.2.b.f 12 85.n odd 8 2
7225.2.a.bp 12 17.d even 8 2
7225.2.a.bp 12 85.m even 8 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(425, [\chi])\):

\( T_{2}^{6} + 9T_{2}^{4} + 19T_{2}^{2} + 1 \) Copy content Toggle raw display
\( T_{3}^{12} + 132T_{3}^{8} + 3268T_{3}^{4} + 324 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} + 9 T^{4} + 19 T^{2} + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{12} + 132 T^{8} + \cdots + 324 \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( T^{12} + 112 T^{8} + \cdots + 4 \) Copy content Toggle raw display
$11$ \( (T^{6} - 8 T^{5} + 32 T^{4} + \cdots + 2)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} - 22 T^{4} + \cdots - 100)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} + 70 T^{10} + \cdots + 24137569 \) Copy content Toggle raw display
$19$ \( (T^{2} + 16)^{6} \) Copy content Toggle raw display
$23$ \( T^{12} + 4444 T^{8} + \cdots + 2500 \) Copy content Toggle raw display
$29$ \( (T^{6} + 2 T^{5} + \cdots + 2312)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} - 2 T^{5} + \cdots + 6498)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} + 1084 T^{8} + \cdots + 40000 \) Copy content Toggle raw display
$41$ \( (T^{6} - 8 T^{5} + \cdots + 2592)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + 66 T^{4} + \cdots + 1444)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} - 82 T^{4} + \cdots - 13924)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + 156 T^{4} + \cdots + 110224)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 36)^{6} \) Copy content Toggle raw display
$61$ \( (T^{2} + 8 T + 32)^{6} \) Copy content Toggle raw display
$67$ \( (T^{6} - 214 T^{4} + \cdots - 93636)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + 20 T^{5} + \cdots + 13122)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + 21952 T^{8} + \cdots + 262144 \) Copy content Toggle raw display
$79$ \( (T^{6} - 2 T^{5} + \cdots + 114242)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} + 58 T^{4} + \cdots + 2916)^{2} \) Copy content Toggle raw display
$89$ \( (T^{3} - 4 T^{2} + \cdots + 1590)^{4} \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 13286025000000 \) Copy content Toggle raw display
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