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Properties

Label 2432.2.c.j
Level $2432$
Weight $2$
Character orbit 2432.c
Analytic conductor $19.420$
Analytic rank $0$
Dimension $20$
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2432,2,Mod(1217,2432)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2432, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2432.1217"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 2432 = 2^{7} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2432.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [20,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(19.4196177716\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 170x^{16} + 6593x^{12} + 64168x^{8} + 95760x^{4} + 4096 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{19} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{5} q^{3} - \beta_{15} q^{5} + ( - \beta_{19} - \beta_{17}) q^{7} + ( - \beta_{3} + \beta_{2} - 1) q^{9} + ( - \beta_{11} + 2 \beta_{8}) q^{11} + \beta_{18} q^{13} + ( - \beta_{9} - \beta_{7}) q^{15}+ \cdots + (4 \beta_{12} - \beta_{11} - 6 \beta_{8}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 28 q^{9} + 8 q^{17} - 20 q^{25} - 16 q^{33} + 24 q^{41} + 52 q^{49} - 8 q^{57} - 48 q^{65} - 24 q^{73} + 68 q^{81} + 40 q^{89} - 56 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 170x^{16} + 6593x^{12} + 64168x^{8} + 95760x^{4} + 4096 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -29963\nu^{16} - 4251522\nu^{12} - 60648551\nu^{8} + 2552436256\nu^{4} + 13215776896 ) / 2201812976 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -242999\nu^{16} - 40138010\nu^{12} - 1424870879\nu^{8} - 10707695292\nu^{4} - 4504397056 ) / 8807251904 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -309571\nu^{16} - 51641634\nu^{12} - 1898897707\nu^{8} - 17258379212\nu^{4} - 38172293120 ) / 8807251904 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -98195\nu^{16} - 16753088\nu^{12} - 653920303\nu^{8} - 6173773922\nu^{4} - 3599744560 ) / 1100906488 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 669065\nu^{18} + 112902174\nu^{14} + 4277565505\nu^{10} + 38678132940\nu^{6} + 40140949280\nu^{2} ) / 17614503808 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 615785 \nu^{19} - 1457982 \nu^{17} + 107399710 \nu^{15} - 242810436 \nu^{13} + \cdots + 34231972352 \nu ) / 35229007616 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 615785 \nu^{19} - 1457982 \nu^{17} - 107399710 \nu^{15} - 242810436 \nu^{13} + \cdots + 34231972352 \nu ) / 35229007616 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -138593\nu^{18} - 23585018\nu^{14} - 917926785\nu^{10} - 9065609016\nu^{6} - 15653732560\nu^{2} ) / 3202637056 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 541737 \nu^{19} + 104855 \nu^{17} + 92396535 \nu^{15} + 17440896 \nu^{13} + \cdots - 13822069824 \nu ) / 4403625952 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 541737 \nu^{19} - 104855 \nu^{17} + 92396535 \nu^{15} - 17440896 \nu^{13} + \cdots + 13822069824 \nu ) / 4403625952 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 3648213 \nu^{18} + 619264250 \nu^{14} + 23883688069 \nu^{10} + 226468156432 \nu^{6} + 263656409392 \nu^{2} ) / 17614503808 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 3718111 \nu^{18} - 631772570 \nu^{14} - 24471954775 \nu^{10} - 238121531292 \nu^{6} - 366908561792 \nu^{2} ) / 17614503808 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 2346695 \nu^{18} + 400216410 \nu^{14} + 15679810351 \nu^{10} + 157489923452 \nu^{6} + 256581837344 \nu^{2} ) / 8807251904 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 17595301 \nu^{19} + 5352520 \nu^{17} + 2987313186 \nu^{15} + 903217392 \nu^{13} + \cdots + 321127594240 \nu ) / 70458015232 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 8890847 \nu^{19} + 2190262 \nu^{17} + 1510472018 \nu^{15} + 371332676 \nu^{13} + \cdots + 169169506816 \nu ) / 35229007616 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 17595301 \nu^{19} + 5352520 \nu^{17} - 2987313186 \nu^{15} + 903217392 \nu^{13} + \cdots + 321127594240 \nu ) / 70458015232 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 8890847 \nu^{19} + 2190262 \nu^{17} - 1510472018 \nu^{15} + 371332676 \nu^{13} + \cdots + 169169506816 \nu ) / 35229007616 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 30510473 \nu^{19} - 6817080 \nu^{17} - 5185315850 \nu^{15} - 1160261872 \nu^{13} + \cdots - 587014522112 \nu ) / 70458015232 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 30510473 \nu^{19} - 6817080 \nu^{17} + 5185315850 \nu^{15} - 1160261872 \nu^{13} + \cdots - 587014522112 \nu ) / 70458015232 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{19} + \beta_{18} + \beta_{17} + \beta_{16} + \beta_{15} + \beta_{14} + \beta_{7} + \beta_{6} ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{12} - 4\beta_{8} + \beta_{5} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 5 \beta_{19} + 5 \beta_{18} - \beta_{17} - 9 \beta_{16} + \beta_{15} + 9 \beta_{14} + \cdots + 9 \beta_{6} ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -11\beta_{3} + 15\beta_{2} - 2\beta _1 - 28 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 29 \beta_{19} - 29 \beta_{18} + 31 \beta_{17} - 97 \beta_{16} + 31 \beta_{15} - 97 \beta_{14} + \cdots - 89 \beta_{6} ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -32\beta_{13} - 117\beta_{12} + 6\beta_{11} + 240\beta_{8} - 185\beta_{5} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 213 \beta_{19} - 213 \beta_{18} - 551 \beta_{17} + 1081 \beta_{16} + 551 \beta_{15} + \cdots - 913 \beta_{6} ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 112\beta_{4} + 1247\beta_{3} - 2147\beta_{2} + 398\beta _1 + 2284 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 1869 \beta_{19} + 1869 \beta_{18} - 7167 \beta_{17} + 12049 \beta_{16} - 7167 \beta_{15} + \cdots + 9593 \beta_{6} ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 4588\beta_{13} + 13385\beta_{12} - 1522\beta_{11} - 23064\beta_{8} + 24221\beta_{5} \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 18213 \beta_{19} + 18213 \beta_{18} + 84759 \beta_{17} - 133449 \beta_{16} - 84759 \beta_{15} + \cdots + 102369 \beta_{6} ) / 4 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( -18468\beta_{4} - 144563\beta_{3} + 269207\beta_{2} - 51370\beta _1 - 240932 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 187485 \beta_{19} - 187485 \beta_{18} + 963215 \beta_{17} - 1469793 \beta_{16} + \cdots - 1102825 \beta_{6} ) / 4 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( -567880\beta_{13} - 1567965\beta_{12} + 212950\beta_{11} + 2565680\beta_{8} - 2967953\beta_{5} \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 1984373 \beta_{19} - 1984373 \beta_{18} - 10739239 \beta_{17} + 16127705 \beta_{16} + \cdots - 11947473 \beta_{6} ) / 4 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 2393768\beta_{4} + 17051271\beta_{3} - 32574859\beta_{2} + 6239558\beta _1 + 27619196 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( 21304685 \beta_{19} + 21304685 \beta_{18} - 118569823 \beta_{17} + 176562353 \beta_{16} + \cdots + 129851161 \beta_{6} ) / 4 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 68344804\beta_{13} + 185716849\beta_{12} - 26550682\beta_{11} - 299126792\beta_{8} + 356638533\beta_{5} \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( - 230453061 \beta_{19} + 230453061 \beta_{18} + 1302303799 \beta_{17} - 1930386409 \beta_{16} + \cdots + 1413876161 \beta_{6} ) / 4 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2432\mathbb{Z}\right)^\times\).

\(n\) \(1407\) \(1921\) \(2053\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1217.1
2.33603 − 2.33603i
−2.33603 + 2.33603i
1.74565 + 1.74565i
−1.74565 − 1.74565i
−1.31430 + 1.31430i
1.31430 − 1.31430i
−0.814454 − 0.814454i
0.814454 + 0.814454i
−0.323980 + 0.323980i
0.323980 − 0.323980i
0.323980 + 0.323980i
−0.323980 − 0.323980i
0.814454 − 0.814454i
−0.814454 + 0.814454i
1.31430 + 1.31430i
−1.31430 − 1.31430i
−1.74565 + 1.74565i
1.74565 − 1.74565i
−2.33603 − 2.33603i
2.33603 + 2.33603i
0 − 3.30364i 0 − 2.55295i 0 1.32515 0 −7.91406 0
1217.2 0 − 3.30364i 0 2.55295i 0 −1.32515 0 −7.91406 0
1217.3 0 − 2.46871i 0 − 3.22672i 0 −4.04213 0 −3.09455 0
1217.4 0 − 2.46871i 0 3.22672i 0 4.04213 0 −3.09455 0
1217.5 0 − 1.85871i 0 − 1.69983i 0 4.99365 0 −0.454788 0
1217.6 0 − 1.85871i 0 1.69983i 0 −4.99365 0 −0.454788 0
1217.7 0 − 1.15181i 0 − 1.07587i 0 0.238565 0 1.67333 0
1217.8 0 − 1.15181i 0 1.07587i 0 −0.238565 0 1.67333 0
1217.9 0 − 0.458177i 0 − 3.00397i 0 −2.21624 0 2.79007 0
1217.10 0 − 0.458177i 0 3.00397i 0 2.21624 0 2.79007 0
1217.11 0 0.458177i 0 − 3.00397i 0 2.21624 0 2.79007 0
1217.12 0 0.458177i 0 3.00397i 0 −2.21624 0 2.79007 0
1217.13 0 1.15181i 0 − 1.07587i 0 −0.238565 0 1.67333 0
1217.14 0 1.15181i 0 1.07587i 0 0.238565 0 1.67333 0
1217.15 0 1.85871i 0 − 1.69983i 0 −4.99365 0 −0.454788 0
1217.16 0 1.85871i 0 1.69983i 0 4.99365 0 −0.454788 0
1217.17 0 2.46871i 0 − 3.22672i 0 4.04213 0 −3.09455 0
1217.18 0 2.46871i 0 3.22672i 0 −4.04213 0 −3.09455 0
1217.19 0 3.30364i 0 − 2.55295i 0 −1.32515 0 −7.91406 0
1217.20 0 3.30364i 0 2.55295i 0 1.32515 0 −7.91406 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1217.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2432.2.c.j ✓ 20
4.b odd 2 1 inner 2432.2.c.j ✓ 20
8.b even 2 1 inner 2432.2.c.j ✓ 20
8.d odd 2 1 inner 2432.2.c.j ✓ 20
16.e even 4 1 4864.2.a.bs 10
16.e even 4 1 4864.2.a.bt 10
16.f odd 4 1 4864.2.a.bs 10
16.f odd 4 1 4864.2.a.bt 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2432.2.c.j ✓ 20 1.a even 1 1 trivial
2432.2.c.j ✓ 20 4.b odd 2 1 inner
2432.2.c.j ✓ 20 8.b even 2 1 inner
2432.2.c.j ✓ 20 8.d odd 2 1 inner
4864.2.a.bs 10 16.e even 4 1
4864.2.a.bs 10 16.f odd 4 1
4864.2.a.bt 10 16.e even 4 1
4864.2.a.bt 10 16.f odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2432, [\chi])\):

\( T_{3}^{10} + 22T_{3}^{8} + 157T_{3}^{6} + 428T_{3}^{4} + 388T_{3}^{2} + 64 \) Copy content Toggle raw display
\( T_{7}^{10} - 48T_{7}^{8} + 694T_{7}^{6} - 3112T_{7}^{4} + 3689T_{7}^{2} - 200 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( (T^{10} + 22 T^{8} + \cdots + 64)^{2} \) Copy content Toggle raw display
$5$ \( (T^{10} + 30 T^{8} + \cdots + 2048)^{2} \) Copy content Toggle raw display
$7$ \( (T^{10} - 48 T^{8} + \cdots - 200)^{2} \) Copy content Toggle raw display
$11$ \( (T^{10} + 82 T^{8} + \cdots + 65536)^{2} \) Copy content Toggle raw display
$13$ \( (T^{10} + 42 T^{8} + \cdots + 512)^{2} \) Copy content Toggle raw display
$17$ \( (T^{5} - 2 T^{4} - 42 T^{3} + \cdots - 22)^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} + 1)^{10} \) Copy content Toggle raw display
$23$ \( (T^{10} - 154 T^{8} + \cdots - 1548800)^{2} \) Copy content Toggle raw display
$29$ \( (T^{10} + 134 T^{8} + \cdots + 59168)^{2} \) Copy content Toggle raw display
$31$ \( (T^{10} - 172 T^{8} + \cdots - 24780800)^{2} \) Copy content Toggle raw display
$37$ \( (T^{10} + 172 T^{8} + \cdots + 12800)^{2} \) Copy content Toggle raw display
$41$ \( (T^{5} - 6 T^{4} + \cdots + 416)^{4} \) Copy content Toggle raw display
$43$ \( (T^{10} + 154 T^{8} + \cdots + 256)^{2} \) Copy content Toggle raw display
$47$ \( (T^{10} - 254 T^{8} + \cdots - 4524032)^{2} \) Copy content Toggle raw display
$53$ \( (T^{10} + 254 T^{8} + \cdots + 67931168)^{2} \) Copy content Toggle raw display
$59$ \( (T^{10} + 378 T^{8} + \cdots + 79423744)^{2} \) Copy content Toggle raw display
$61$ \( (T^{10} + 382 T^{8} + \cdots + 633110528)^{2} \) Copy content Toggle raw display
$67$ \( (T^{10} + 606 T^{8} + \cdots + 8441934400)^{2} \) Copy content Toggle raw display
$71$ \( (T^{10} - 208 T^{8} + \cdots - 991232)^{2} \) Copy content Toggle raw display
$73$ \( (T^{5} + 6 T^{4} + \cdots - 470)^{4} \) Copy content Toggle raw display
$79$ \( (T^{10} - 764 T^{8} + \cdots - 5939628032)^{2} \) Copy content Toggle raw display
$83$ \( (T^{10} + 328 T^{8} + \cdots + 112869376)^{2} \) Copy content Toggle raw display
$89$ \( (T^{5} - 10 T^{4} + \cdots - 23696)^{4} \) Copy content Toggle raw display
$97$ \( (T^{5} + 14 T^{4} + \cdots - 832)^{4} \) Copy content Toggle raw display
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