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Properties

Label 425.2.b.f
Level $425$
Weight $2$
Character orbit 425.b
Analytic conductor $3.394$
Analytic rank $0$
Dimension $10$
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] = [425,2,Mod(324,425)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(425, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("425.324");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 425 = 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 425.b (of order \(2\), degree \(1\), 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: \(3.39364208590\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 2x^{7} + 64x^{6} - 30x^{5} + 2x^{4} + 136x^{3} + 324x^{2} + 180x + 50 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{8} q^{2} + \beta_{6} q^{3} + (\beta_{7} - \beta_{2} + \beta_1 - 2) q^{4} + ( - \beta_{4} - \beta_{2} + \beta_1 + 1) q^{6} + ( - \beta_{9} - \beta_{3}) q^{7} + ( - \beta_{9} + 2 \beta_{8} + \cdots - \beta_{5}) q^{8}+ \cdots + ( - \beta_{2} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{8} q^{2} + \beta_{6} q^{3} + (\beta_{7} - \beta_{2} + \beta_1 - 2) q^{4} + ( - \beta_{4} - \beta_{2} + \beta_1 + 1) q^{6} + ( - \beta_{9} - \beta_{3}) q^{7} + ( - \beta_{9} + 2 \beta_{8} + \cdots - \beta_{5}) q^{8}+ \cdots + ( - \beta_{7} - 2 \beta_{4} + \beta_{2} + \cdots + 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 22 q^{4} + 6 q^{6} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 22 q^{4} + 6 q^{6} - 12 q^{9} + 8 q^{11} + 14 q^{14} + 54 q^{16} - 12 q^{19} - 10 q^{21} + 38 q^{24} - 10 q^{26} - 4 q^{29} + 42 q^{31} + 2 q^{34} + 44 q^{36} - 46 q^{39} - 16 q^{41} + 8 q^{44} - 12 q^{46} - 20 q^{49} + 2 q^{51} + 50 q^{54} - 58 q^{56} - 24 q^{59} - 4 q^{61} - 86 q^{64} - 56 q^{66} + 42 q^{71} + 100 q^{74} - 28 q^{76} - 82 q^{79} - 70 q^{81} + 142 q^{84} - 72 q^{86} + 20 q^{89} + 26 q^{91} + 20 q^{94} - 170 q^{96} + 28 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 2x^{7} + 64x^{6} - 30x^{5} + 2x^{4} + 136x^{3} + 324x^{2} + 180x + 50 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 1431 \nu^{9} + 48841 \nu^{8} + 33935 \nu^{7} - 184 \nu^{6} - 222008 \nu^{5} + 2017679 \nu^{4} + \cdots - 6682775 ) / 5752719 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 7142 \nu^{9} - 83651 \nu^{8} - 27170 \nu^{7} - 20572 \nu^{6} + 857344 \nu^{5} - 5093189 \nu^{4} + \cdots - 15303501 ) / 5752719 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 20492 \nu^{9} - 8573 \nu^{8} + 34810 \nu^{7} - 47749 \nu^{6} + 1332244 \nu^{5} - 1250096 \nu^{4} + \cdots + 3566730 ) / 5752719 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 20492 \nu^{9} + 8573 \nu^{8} - 34810 \nu^{7} + 47749 \nu^{6} - 1332244 \nu^{5} + 1250096 \nu^{4} + \cdots - 3566730 ) / 5752719 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 356673 \nu^{9} - 102460 \nu^{8} + 42865 \nu^{7} - 887396 \nu^{6} + 23065817 \nu^{5} + \cdots + 34074695 ) / 28763595 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 82220 \nu^{9} - 21671 \nu^{8} - 27647 \nu^{7} - 400072 \nu^{6} + 5100937 \nu^{5} - 3804875 \nu^{4} + \cdots + 7039875 ) / 5752719 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 15857 \nu^{9} - 35255 \nu^{8} + 22525 \nu^{7} - 37390 \nu^{6} + 1075024 \nu^{5} - 2761982 \nu^{4} + \cdots - 3695547 ) / 821817 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 1015592 \nu^{9} - 301485 \nu^{8} + 309695 \nu^{7} - 1549224 \nu^{6} + 66758583 \nu^{5} + \cdots + 101099405 ) / 28763595 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 2312393 \nu^{9} + 619335 \nu^{8} + 314365 \nu^{7} + 3894846 \nu^{6} - 148655697 \nu^{5} + \cdots - 195277745 ) / 28763595 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{4} + \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{8} + \beta_{6} - 4\beta_{5} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{9} - 3\beta_{8} - \beta_{7} - \beta_{6} + \beta_{5} - 6\beta_{4} + 6\beta_{3} + 3\beta_{2} + \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{7} + 2\beta_{4} - 8\beta_{2} - 9\beta _1 - 26 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 5 \beta_{9} + 16 \beta_{8} - 5 \beta_{7} + 6 \beta_{6} - 14 \beta_{5} - 21 \beta_{4} - 21 \beta_{3} + \cdots + 14 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -11\beta_{9} - 65\beta_{8} - 73\beta_{6} + 189\beta_{5} + 31\beta_{3} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 42 \beta_{9} + 146 \beta_{8} + 42 \beta_{7} + 65 \beta_{6} - 177 \beta_{5} + 158 \beta_{4} - 158 \beta_{3} + \cdots - 177 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -107\beta_{7} - 346\beta_{4} + 553\beta_{2} + 585\beta _1 + 1455 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 346 \beta_{9} - 1277 \beta_{8} + 346 \beta_{7} - 660 \beta_{6} + 1837 \beta_{5} + 1243 \beta_{4} + \cdots - 1837 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(52\) \(326\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
324.1
1.44796 + 1.44796i
−0.328166 + 0.328166i
−2.08367 − 2.08367i
−0.740669 + 0.740669i
1.70454 − 1.70454i
1.70454 + 1.70454i
−0.740669 − 0.740669i
−2.08367 + 2.08367i
−0.328166 − 0.328166i
1.44796 − 1.44796i
− 2.80107i − 2.60789i −5.84602 0 −7.30489 − 1.33298i 10.7730i −3.80107 0
324.2 − 2.60242i 1.18219i −4.77260 0 3.07656 3.53650i 7.21549i 1.60242 0
324.3 − 2.19447i 2.48887i −2.81568 0 5.46174 3.05725i 1.78998i −3.19447 0
324.4 − 1.24214i 1.66068i 0.457096 0 2.06279 − 4.35698i − 3.05205i 0.242137 0
324.5 − 0.150980i − 1.96189i 1.97720 0 −0.296207 1.54475i − 0.600480i −0.849020 0
324.6 0.150980i 1.96189i 1.97720 0 −0.296207 − 1.54475i 0.600480i −0.849020 0
324.7 1.24214i − 1.66068i 0.457096 0 2.06279 4.35698i 3.05205i 0.242137 0
324.8 2.19447i − 2.48887i −2.81568 0 5.46174 − 3.05725i − 1.78998i −3.19447 0
324.9 2.60242i − 1.18219i −4.77260 0 3.07656 − 3.53650i − 7.21549i 1.60242 0
324.10 2.80107i 2.60789i −5.84602 0 −7.30489 1.33298i − 10.7730i −3.80107 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 324.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 425.2.b.f 10
5.b even 2 1 inner 425.2.b.f 10
5.c odd 4 1 425.2.a.i ✓ 5
5.c odd 4 1 425.2.a.j yes 5
15.e even 4 1 3825.2.a.bl 5
15.e even 4 1 3825.2.a.bq 5
20.e even 4 1 6800.2.a.bz 5
20.e even 4 1 6800.2.a.cd 5
85.g odd 4 1 7225.2.a.x 5
85.g odd 4 1 7225.2.a.y 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
425.2.a.i ✓ 5 5.c odd 4 1
425.2.a.j yes 5 5.c odd 4 1
425.2.b.f 10 1.a even 1 1 trivial
425.2.b.f 10 5.b even 2 1 inner
3825.2.a.bl 5 15.e even 4 1
3825.2.a.bq 5 15.e even 4 1
6800.2.a.bz 5 20.e even 4 1
6800.2.a.cd 5 20.e even 4 1
7225.2.a.x 5 85.g odd 4 1
7225.2.a.y 5 85.g odd 4 1

Hecke kernels

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

\( T_{2}^{10} + 21T_{2}^{8} + 154T_{2}^{6} + 450T_{2}^{4} + 405T_{2}^{2} + 9 \) Copy content Toggle raw display
\( T_{3}^{10} + 21T_{3}^{8} + 166T_{3}^{6} + 610T_{3}^{4} + 1029T_{3}^{2} + 625 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + 21 T^{8} + \cdots + 9 \) Copy content Toggle raw display
$3$ \( T^{10} + 21 T^{8} + \cdots + 625 \) Copy content Toggle raw display
$5$ \( T^{10} \) Copy content Toggle raw display
$7$ \( T^{10} + 45 T^{8} + \cdots + 9409 \) Copy content Toggle raw display
$11$ \( (T^{5} - 4 T^{4} - 22 T^{3} + \cdots + 60)^{2} \) Copy content Toggle raw display
$13$ \( T^{10} + 61 T^{8} + \cdots + 51529 \) Copy content Toggle raw display
$17$ \( (T^{2} + 1)^{5} \) Copy content Toggle raw display
$19$ \( (T^{5} + 6 T^{4} + \cdots - 400)^{2} \) Copy content Toggle raw display
$23$ \( T^{10} + 108 T^{8} + \cdots + 11664 \) Copy content Toggle raw display
$29$ \( (T^{5} + 2 T^{4} + \cdots - 240)^{2} \) Copy content Toggle raw display
$31$ \( (T^{5} - 21 T^{4} + \cdots + 2151)^{2} \) Copy content Toggle raw display
$37$ \( T^{10} + 164 T^{8} + \cdots + 14622976 \) Copy content Toggle raw display
$41$ \( (T^{5} + 8 T^{4} - 20 T^{3} + \cdots + 48)^{2} \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots + 216207616 \) Copy content Toggle raw display
$47$ \( T^{10} + 180 T^{8} + \cdots + 230400 \) Copy content Toggle raw display
$53$ \( T^{10} + 133 T^{8} + \cdots + 81 \) Copy content Toggle raw display
$59$ \( (T^{5} + 12 T^{4} + \cdots + 11760)^{2} \) Copy content Toggle raw display
$61$ \( (T^{5} + 2 T^{4} + \cdots + 800)^{2} \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 729432064 \) Copy content Toggle raw display
$71$ \( (T^{5} - 21 T^{4} + \cdots + 11853)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 1742561536 \) Copy content Toggle raw display
$79$ \( (T^{5} + 41 T^{4} + \cdots - 42575)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + 192 T^{8} + \cdots + 35426304 \) Copy content Toggle raw display
$89$ \( (T^{5} - 10 T^{4} + \cdots + 18000)^{2} \) Copy content Toggle raw display
$97$ \( T^{10} + 288 T^{8} + \cdots + 64000000 \) Copy content Toggle raw display
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