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Properties

Label 425.4.b.j
Level $425$
Weight $4$
Character orbit 425.b
Analytic conductor $25.076$
Analytic rank $0$
Dimension $12$
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] = [425,4,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, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("425.324");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 425 = 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
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: \(25.0758117524\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 61x^{10} + 1386x^{8} + 14450x^{6} + 68469x^{4} + 129457x^{2} + 76176 \) 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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{8} - \beta_{7}) q^{3} + (\beta_{2} - 2) q^{4} + (\beta_{5} - \beta_{3} - 4) q^{6} + ( - \beta_{10} + \beta_{9} + \cdots - 2 \beta_1) q^{7}+ \cdots + ( - \beta_{6} + \beta_{5} + \beta_{4} + \cdots + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{8} - \beta_{7}) q^{3} + (\beta_{2} - 2) q^{4} + (\beta_{5} - \beta_{3} - 4) q^{6} + ( - \beta_{10} + \beta_{9} + \cdots - 2 \beta_1) q^{7}+ \cdots + (17 \beta_{6} - 47 \beta_{5} + \cdots + 170) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 26 q^{4} - 46 q^{6} + 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 26 q^{4} - 46 q^{6} + 24 q^{9} - 40 q^{11} + 186 q^{14} - 262 q^{16} + 140 q^{19} - 564 q^{21} - 174 q^{24} - 262 q^{26} + 632 q^{29} - 1516 q^{31} - 34 q^{34} - 1516 q^{36} + 584 q^{39} - 1024 q^{41} + 568 q^{44} - 2324 q^{46} + 168 q^{49} - 204 q^{51} + 22 q^{54} - 2222 q^{56} + 476 q^{59} - 2376 q^{61} + 1830 q^{64} - 1768 q^{66} + 3472 q^{69} - 2552 q^{71} - 1892 q^{74} - 4356 q^{76} + 6132 q^{79} + 348 q^{81} - 2510 q^{84} - 1480 q^{86} + 3520 q^{89} - 3848 q^{91} + 2044 q^{94} - 3294 q^{96} + 1900 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 61x^{10} + 1386x^{8} + 14450x^{6} + 68469x^{4} + 129457x^{2} + 76176 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 10 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -29\nu^{10} - 3018\nu^{8} - 95452\nu^{6} - 1187862\nu^{4} - 5437623\nu^{2} - 5718016 ) / 197000 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 26\nu^{10} + 1517\nu^{8} + 31063\nu^{6} + 263903\nu^{4} + 849687\nu^{2} + 746304 ) / 98500 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -68\nu^{10} - 2831\nu^{8} - 36159\nu^{6} - 177629\nu^{4} - 730741\nu^{2} - 1282072 ) / 98500 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -749\nu^{10} - 38208\nu^{8} - 685162\nu^{6} - 5223472\nu^{4} - 16078313\nu^{2} - 13698096 ) / 197000 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 791\nu^{11} + 59222\nu^{9} + 1488108\nu^{7} + 14120398\nu^{5} + 33127917\nu^{3} - 54696436\nu ) / 13593000 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 676\nu^{11} + 39442\nu^{9} + 832263\nu^{7} + 7624853\nu^{5} + 28075737\nu^{3} + 28884529\nu ) / 6796500 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 2099\nu^{11} + 83258\nu^{9} + 780012\nu^{7} - 3421628\nu^{5} - 58731087\nu^{3} - 102285154\nu ) / 6796500 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -4966\nu^{11} - 289747\nu^{9} - 6179283\nu^{7} - 58039223\nu^{5} - 222128967\nu^{3} - 237350314\nu ) / 6796500 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -709\nu^{11} - 35268\nu^{9} - 606762\nu^{7} - 4307692\nu^{5} - 11184793\nu^{3} - 359576\nu ) / 906200 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 10 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{11} - \beta_{10} + \beta_{9} - 2\beta_{8} - \beta_{7} - 15\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{6} - 3\beta_{5} + 6\beta_{4} - \beta_{3} - 20\beta_{2} + 156 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -24\beta_{11} + 32\beta_{10} - 26\beta_{9} + 120\beta_{8} + 12\beta_{7} + 255\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -36\beta_{6} + 94\beta_{5} - 266\beta_{4} + 12\beta_{3} + 385\beta_{2} - 2766 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 501\beta_{11} - 853\beta_{10} + 563\beta_{9} - 3998\beta_{8} - 129\beta_{7} - 4645\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 977\beta_{6} - 2289\beta_{5} + 8072\beta_{4} - 25\beta_{3} - 7564\beta_{2} + 51896 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -10188\beta_{11} + 20856\beta_{10} - 11832\beta_{9} + 109012\beta_{8} + 1364\beta_{7} + 88429\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -24144\beta_{6} + 51700\beta_{5} - 210284\beta_{4} - 2728\beta_{3} + 151681\beta_{2} - 1008626 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 206793\beta_{11} - 486097\beta_{10} + 248941\beta_{9} - 2698666\beta_{8} - 14585\beta_{7} - 1736747\beta_1 \) Copy content Toggle raw display

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

Character values

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

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

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
− 4.62705i
− 3.95575i
− 3.74408i
− 2.55730i
− 1.52037i
− 1.03585i
1.03585i
1.52037i
2.55730i
3.74408i
3.95575i
4.62705i
− 4.62705i − 2.57794i −13.4096 0 −11.9282 22.4360i 25.0303i 20.3542 0
324.2 − 3.95575i − 3.53947i −7.64793 0 −14.0012 − 13.8166i − 1.39269i 14.4722 0
324.3 − 3.74408i 3.16721i −6.01816 0 11.8583 26.6627i − 7.42016i 16.9688 0
324.4 − 2.55730i 3.86147i 1.46020 0 9.87494 − 5.49760i − 24.1926i 12.0891 0
324.5 − 1.52037i − 9.11646i 5.68846 0 −13.8604 − 16.3642i − 20.8116i −56.1099 0
324.6 − 1.03585i − 4.77225i 6.92701 0 −4.94335 − 16.4564i − 15.4622i 4.22566 0
324.7 1.03585i 4.77225i 6.92701 0 −4.94335 16.4564i 15.4622i 4.22566 0
324.8 1.52037i 9.11646i 5.68846 0 −13.8604 16.3642i 20.8116i −56.1099 0
324.9 2.55730i − 3.86147i 1.46020 0 9.87494 5.49760i 24.1926i 12.0891 0
324.10 3.74408i − 3.16721i −6.01816 0 11.8583 − 26.6627i 7.42016i 16.9688 0
324.11 3.95575i 3.53947i −7.64793 0 −14.0012 13.8166i 1.39269i 14.4722 0
324.12 4.62705i 2.57794i −13.4096 0 −11.9282 − 22.4360i − 25.0303i 20.3542 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 324.12
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 425.4.b.j 12
5.b even 2 1 inner 425.4.b.j 12
5.c odd 4 1 425.4.a.j ✓ 6
5.c odd 4 1 425.4.a.k yes 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
425.4.a.j ✓ 6 5.c odd 4 1
425.4.a.k yes 6 5.c odd 4 1
425.4.b.j 12 1.a even 1 1 trivial
425.4.b.j 12 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(425, [\chi])\):

\( T_{2}^{12} + 61T_{2}^{10} + 1386T_{2}^{8} + 14450T_{2}^{6} + 68469T_{2}^{4} + 129457T_{2}^{2} + 76176 \) Copy content Toggle raw display
\( T_{3}^{12} + 150T_{3}^{10} + 7275T_{3}^{8} + 163736T_{3}^{6} + 1881879T_{3}^{4} + 10677414T_{3}^{2} + 23571025 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 61 T^{10} + \cdots + 76176 \) Copy content Toggle raw display
$3$ \( T^{12} + 150 T^{10} + \cdots + 23571025 \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 149730537292249 \) Copy content Toggle raw display
$11$ \( (T^{6} + 20 T^{5} + \cdots - 98280800)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 67\!\cdots\!21 \) Copy content Toggle raw display
$17$ \( (T^{2} + 289)^{6} \) Copy content Toggle raw display
$19$ \( (T^{6} - 70 T^{5} + \cdots - 977808960)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( (T^{6} + \cdots + 2867744451360)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + \cdots - 4319982391195)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 17\!\cdots\!76 \) Copy content Toggle raw display
$41$ \( (T^{6} + \cdots - 15523734022400)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 59\!\cdots\!16 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 60\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 28\!\cdots\!25 \) Copy content Toggle raw display
$59$ \( (T^{6} + \cdots - 132455594307936)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} + \cdots + 5378930561600)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 33\!\cdots\!16 \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots + 61\!\cdots\!61)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 20\!\cdots\!16 \) Copy content Toggle raw display
$79$ \( (T^{6} + \cdots - 61\!\cdots\!07)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 14\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots - 13\!\cdots\!20)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
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