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Properties

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

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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: \( x^{12} + 18x^{10} + 119x^{8} + 364x^{6} + 519x^{4} + 278x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{3} q^{3} + (\beta_{2} - 1) q^{4} + (\beta_{5} + 1) q^{6} + (\beta_{10} - \beta_{8} - \beta_{7} + \cdots - \beta_1) q^{7}+ \cdots + ( - \beta_{11} + \beta_{10} + \cdots - \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + \beta_{3} q^{3} + (\beta_{2} - 1) q^{4} + (\beta_{5} + 1) q^{6} + (\beta_{10} - \beta_{8} - \beta_{7} + \cdots - \beta_1) q^{7}+ \cdots + ( - 3 \beta_{11} - \beta_{9} + \cdots + 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 2 q^{3} - 12 q^{4} + 6 q^{6}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 2 q^{3} - 12 q^{4} + 6 q^{6} - 4 q^{11} - 4 q^{12} + 12 q^{13} + 14 q^{14} + 4 q^{16} + 6 q^{17} - 4 q^{18} + 8 q^{21} + 10 q^{22} + 12 q^{23} - 8 q^{24} + 22 q^{27} - 34 q^{28} - 6 q^{29} - 6 q^{31} + 4 q^{33} + 30 q^{34} + 18 q^{37} - 12 q^{38} + 16 q^{39} + 6 q^{41} + 32 q^{44} + 30 q^{46} + 36 q^{47} - 10 q^{48} - 40 q^{51} - 64 q^{52} + 16 q^{54} - 56 q^{56} - 18 q^{57} - 6 q^{58} - 8 q^{61} - 22 q^{62} - 30 q^{63} - 44 q^{64} + 20 q^{67} + 10 q^{68} - 72 q^{69} - 20 q^{71} - 92 q^{72} - 18 q^{73} - 26 q^{74} + 38 q^{78} - 14 q^{79} - 8 q^{81} - 10 q^{82} + 36 q^{84} + 84 q^{86} + 66 q^{88} - 24 q^{89} + 12 q^{91} - 32 q^{92} + 22 q^{96} - 52 q^{97} + 76 q^{98} + 30 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 18x^{10} + 119x^{8} + 364x^{6} + 519x^{4} + 278x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2 \nu^{11} - \nu^{10} + 33 \nu^{9} - 14 \nu^{8} + 186 \nu^{7} - 58 \nu^{6} + 414 \nu^{5} - 62 \nu^{4} + \cdots + 6 ) / 20 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2 \nu^{11} + \nu^{10} + 33 \nu^{9} + 14 \nu^{8} + 186 \nu^{7} + 58 \nu^{6} + 414 \nu^{5} + 62 \nu^{4} + \cdots - 6 ) / 20 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - \nu^{11} - 3 \nu^{10} - 14 \nu^{9} - 52 \nu^{8} - 58 \nu^{7} - 314 \nu^{6} - 62 \nu^{5} - 776 \nu^{4} + \cdots - 22 ) / 20 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - \nu^{11} + 3 \nu^{10} - 14 \nu^{9} + 52 \nu^{8} - 58 \nu^{7} + 314 \nu^{6} - 62 \nu^{5} + 776 \nu^{4} + \cdots + 22 ) / 20 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2\nu^{11} + 33\nu^{9} + 191\nu^{7} + 479\nu^{5} + 507\nu^{3} + 168\nu ) / 10 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 3\nu^{10} + 47\nu^{8} + 249\nu^{6} + 531\nu^{4} + 388\nu^{2} + 2 ) / 10 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -3\nu^{11} - 52\nu^{9} - 324\nu^{7} - 906\nu^{5} - 1133\nu^{3} - 502\nu ) / 10 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 3 \nu^{11} + 8 \nu^{10} + 52 \nu^{9} + 127 \nu^{8} + 324 \nu^{7} + 684 \nu^{6} + 906 \nu^{5} + \cdots + 37 ) / 20 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 3 \nu^{11} + 8 \nu^{10} - 52 \nu^{9} + 127 \nu^{8} - 324 \nu^{7} + 684 \nu^{6} - 906 \nu^{5} + \cdots + 37 ) / 20 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{11} + \beta_{10} + \beta_{9} - 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{11} + 2\beta_{10} - 4\beta_{8} - \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} - 6\beta_{2} + 13 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 9\beta_{11} - 9\beta_{10} - 10\beta_{9} - 2\beta_{7} - \beta_{6} - \beta_{5} + 29\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -20\beta_{11} - 20\beta_{10} + 42\beta_{8} + 9\beta_{6} - 9\beta_{5} + 7\beta_{4} - 7\beta_{3} + 39\beta_{2} - 67 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 68 \beta_{11} + 68 \beta_{10} + 81 \beta_{9} + 28 \beta_{7} + 13 \beta_{6} + 13 \beta_{5} + \cdots - 185 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 162 \beta_{11} + 162 \beta_{10} - 352 \beta_{8} - 66 \beta_{6} + 66 \beta_{5} - 42 \beta_{4} + \cdots + 389 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 494 \beta_{11} - 494 \beta_{10} - 618 \beta_{9} - 276 \beta_{7} - 120 \beta_{6} - 120 \beta_{5} + \cdots + 1245 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 1232 \beta_{11} - 1232 \beta_{10} + 2740 \beta_{8} + 464 \beta_{6} - 464 \beta_{5} + 254 \beta_{4} + \cdots - 2447 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 3559 \beta_{11} + 3559 \beta_{10} + 4603 \beta_{9} + 2364 \beta_{7} + 978 \beta_{6} + \cdots - 8637 \beta_1 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(52\) \(326\)
\(\chi(n)\) \(1\) \(-\beta_{7}\)

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
− 2.08389i
− 1.44718i
− 0.0601793i
1.21647i
1.68228i
2.69251i
− 2.69251i
− 1.68228i
− 1.21647i
0.0601793i
1.44718i
2.08389i
− 2.08389i 1.31060 + 1.31060i −2.34261 0 2.73116 − 2.73116i 0.971529 − 0.971529i 0.713960i 0.435362i 0
251.2 − 1.44718i −1.14672 − 1.14672i −0.0943296 0 −1.65950 + 1.65950i −2.69805 + 2.69805i − 2.75785i − 0.370087i 0
251.3 − 0.0601793i 0.294708 + 0.294708i 1.99638 0 0.0177353 − 0.0177353i 0.900793 − 0.900793i − 0.240499i − 2.82629i 0
251.4 1.21647i −2.23861 − 2.23861i 0.520205 0 2.72320 − 2.72320i −0.679950 + 0.679950i 3.06575i 7.02277i 0
251.5 1.68228i 1.27457 + 1.27457i −0.830060 0 −2.14419 + 2.14419i −1.92759 + 1.92759i 1.96816i 0.249077i 0
251.6 2.69251i −0.494558 − 0.494558i −5.24959 0 1.33160 − 1.33160i 3.43326 − 3.43326i − 8.74953i − 2.51083i 0
276.1 − 2.69251i −0.494558 + 0.494558i −5.24959 0 1.33160 + 1.33160i 3.43326 + 3.43326i 8.74953i 2.51083i 0
276.2 − 1.68228i 1.27457 − 1.27457i −0.830060 0 −2.14419 − 2.14419i −1.92759 − 1.92759i − 1.96816i − 0.249077i 0
276.3 − 1.21647i −2.23861 + 2.23861i 0.520205 0 2.72320 + 2.72320i −0.679950 − 0.679950i − 3.06575i − 7.02277i 0
276.4 0.0601793i 0.294708 − 0.294708i 1.99638 0 0.0177353 + 0.0177353i 0.900793 + 0.900793i 0.240499i 2.82629i 0
276.5 1.44718i −1.14672 + 1.14672i −0.0943296 0 −1.65950 − 1.65950i −2.69805 − 2.69805i 2.75785i 0.370087i 0
276.6 2.08389i 1.31060 − 1.31060i −2.34261 0 2.73116 + 2.73116i 0.971529 + 0.971529i − 0.713960i − 0.435362i 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
17.c 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.c ✓ 12
5.b even 2 1 425.2.e.e yes 12
5.c odd 4 1 425.2.j.a 12
5.c odd 4 1 425.2.j.d 12
17.c even 4 1 inner 425.2.e.c ✓ 12
17.d even 8 2 7225.2.a.bm 12
85.f odd 4 1 425.2.j.d 12
85.i odd 4 1 425.2.j.a 12
85.j even 4 1 425.2.e.e yes 12
85.m even 8 2 7225.2.a.br 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
425.2.e.c ✓ 12 1.a even 1 1 trivial
425.2.e.c ✓ 12 17.c even 4 1 inner
425.2.e.e yes 12 5.b even 2 1
425.2.e.e yes 12 85.j even 4 1
425.2.j.a 12 5.c odd 4 1
425.2.j.a 12 85.i odd 4 1
425.2.j.d 12 5.c odd 4 1
425.2.j.d 12 85.f odd 4 1
7225.2.a.bm 12 17.d even 8 2
7225.2.a.br 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 \(S_{2}^{\mathrm{new}}(425, [\chi])\):

\( T_{2}^{12} + 18T_{2}^{10} + 119T_{2}^{8} + 364T_{2}^{6} + 519T_{2}^{4} + 278T_{2}^{2} + 1 \) Copy content Toggle raw display
\( T_{3}^{12} + 2 T_{3}^{11} + 2 T_{3}^{10} - 10 T_{3}^{9} + 43 T_{3}^{8} + 26 T_{3}^{7} + 16 T_{3}^{6} + \cdots + 25 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 18 T^{10} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{12} + 2 T^{11} + \cdots + 25 \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( T^{12} + 20 T^{9} + \cdots + 7225 \) Copy content Toggle raw display
$11$ \( T^{12} + 4 T^{11} + \cdots + 4 \) Copy content Toggle raw display
$13$ \( (T^{6} - 6 T^{5} + \cdots + 755)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} - 6 T^{11} + \cdots + 24137569 \) Copy content Toggle raw display
$19$ \( T^{12} + 76 T^{10} + \cdots + 16 \) Copy content Toggle raw display
$23$ \( T^{12} - 12 T^{11} + \cdots + 8584900 \) Copy content Toggle raw display
$29$ \( T^{12} + 6 T^{11} + \cdots + 35344 \) Copy content Toggle raw display
$31$ \( T^{12} + 6 T^{11} + \cdots + 10609 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 5396371600 \) Copy content Toggle raw display
$41$ \( T^{12} - 6 T^{11} + \cdots + 35344 \) Copy content Toggle raw display
$43$ \( T^{12} + 272 T^{10} + \cdots + 1157776 \) Copy content Toggle raw display
$47$ \( (T^{6} - 18 T^{5} + \cdots + 234104)^{2} \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 578354401 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 240994576 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 80947078144 \) Copy content Toggle raw display
$67$ \( (T^{6} - 10 T^{5} + \cdots - 9824)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} + 20 T^{11} + \cdots + 72982849 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 762163920400 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 1809056089 \) Copy content Toggle raw display
$83$ \( T^{12} + 376 T^{10} + \cdots + 118336 \) Copy content Toggle raw display
$89$ \( (T^{6} + 12 T^{5} + \cdots - 500)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 41697640000 \) Copy content Toggle raw display
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