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Properties

Label 455.2.c.b
Level $455$
Weight $2$
Character orbit 455.c
Analytic conductor $3.633$
Analytic rank $0$
Dimension $14$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [455,2,Mod(274,455)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("455.274"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(455, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 455 = 5 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 455.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [14] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.63319329197\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} + 16x^{12} + 96x^{10} + 272x^{8} + 372x^{6} + 225x^{4} + 56x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{13} q^{3} + ( - \beta_{12} + \beta_{8} - \beta_{5} + \cdots - 1) q^{4} + (\beta_{13} - \beta_{12} + \cdots + \beta_{7}) q^{5} + ( - \beta_{12} + \beta_{8} + \beta_{6} + \cdots - 2) q^{6}+ \cdots + ( - \beta_{11} + \beta_{10} + \cdots + 2 \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 4 q^{4} + 2 q^{5} - 16 q^{6} - 2 q^{9} - 4 q^{10} - 12 q^{11} - 6 q^{15} - 8 q^{16} + 20 q^{19} + 6 q^{20} - 4 q^{21} + 10 q^{24} - 4 q^{25} + 52 q^{29} + 28 q^{30} - 12 q^{31} + 2 q^{34} + 2 q^{35}+ \cdots + 36 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} + 16x^{12} + 96x^{10} + 272x^{8} + 372x^{6} + 225x^{4} + 56x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -3\nu^{13} - 46\nu^{11} - 256\nu^{9} - 628\nu^{7} - 620\nu^{5} - 115\nu^{3} + 26\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{12} + 16\nu^{10} + 95\nu^{8} + 259\nu^{6} + 316\nu^{4} + 134\nu^{2} + 14 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -5\nu^{12} - 78\nu^{10} - 448\nu^{8} - 1170\nu^{6} - 1344\nu^{4} - 509\nu^{2} - 42 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -7\nu^{12} - 108\nu^{10} - 612\nu^{8} - 1576\nu^{6} - 1794\nu^{4} - 693\nu^{2} - 64 ) / 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -4\nu^{12} - 62\nu^{10} - 353\nu^{8} - 912\nu^{6} - 1037\nu^{4} - 395\nu^{2} - 35 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -17\nu^{13} - 262\nu^{11} - 1480\nu^{9} - 3780\nu^{7} - 4208\nu^{5} - 1501\nu^{3} - 102\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 23 \nu^{13} + 16 \nu^{12} + 358 \nu^{11} + 252 \nu^{10} + 2052 \nu^{9} + 1464 \nu^{8} + 5360 \nu^{7} + \cdots + 164 ) / 8 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -21\nu^{13} - 326\nu^{11} - 1860\nu^{9} - 4816\nu^{7} - 5472\nu^{5} - 2037\nu^{3} - 158\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - \nu^{13} - 21 \nu^{12} - 16 \nu^{11} - 326 \nu^{10} - 96 \nu^{9} - 1860 \nu^{8} - 272 \nu^{7} + \cdots - 158 ) / 4 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - \nu^{13} + 21 \nu^{12} - 16 \nu^{11} + 326 \nu^{10} - 96 \nu^{9} + 1860 \nu^{8} - 272 \nu^{7} + \cdots + 158 ) / 4 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 23 \nu^{13} + 24 \nu^{12} + 358 \nu^{11} + 372 \nu^{10} + 2052 \nu^{9} + 2120 \nu^{8} + 5360 \nu^{7} + \cdots + 228 ) / 8 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 17\nu^{13} + 264\nu^{11} + 1508\nu^{9} + 3918\nu^{7} + 4500\nu^{5} + 1751\nu^{3} + 164\nu ) / 2 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{12} + \beta_{8} - \beta_{5} + \beta_{4} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - 2 \beta_{13} + \beta_{12} - 2 \beta_{11} - 2 \beta_{10} - \beta_{9} + \beta_{8} - \beta_{7} + \cdots - 4 \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 6\beta_{12} - \beta_{11} + \beta_{10} - 6\beta_{8} - \beta_{6} + 6\beta_{5} - 9\beta_{4} - \beta_{3} + 13 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 16 \beta_{13} - 9 \beta_{12} + 15 \beta_{11} + 15 \beta_{10} + 7 \beta_{9} - 9 \beta_{8} + \cdots + 18 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -34\beta_{12} + 9\beta_{11} - 9\beta_{10} + 34\beta_{8} + 8\beta_{6} - 34\beta_{5} + 63\beta_{4} + 10\beta_{3} - 64 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 105 \beta_{13} + 63 \beta_{12} - 97 \beta_{11} - 97 \beta_{10} - 41 \beta_{9} + 63 \beta_{8} + \cdots - 90 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 195 \beta_{12} - 63 \beta_{11} + 63 \beta_{10} - 195 \beta_{8} - 50 \beta_{6} + 193 \beta_{5} + \cdots + 339 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 651 \beta_{13} - 406 \beta_{12} + 601 \beta_{11} + 601 \beta_{10} + 233 \beta_{9} - 406 \beta_{8} + \cdots + 484 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 1135 \beta_{12} + 406 \beta_{11} - 406 \beta_{10} + 1135 \beta_{8} + 293 \beta_{6} - 1108 \beta_{5} + \cdots - 1883 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 3954 \beta_{13} + 2526 \beta_{12} - 3661 \beta_{11} - 3661 \beta_{10} - 1330 \beta_{9} + \cdots - 2725 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 6679 \beta_{12} - 2526 \beta_{11} + 2526 \beta_{10} - 6679 \beta_{8} - 1694 \beta_{6} + 6437 \beta_{5} + \cdots + 10779 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 23826 \beta_{13} - 15453 \beta_{12} + 22132 \beta_{11} + 22132 \beta_{10} + 7685 \beta_{9} + \cdots + 15764 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/455\mathbb{Z}\right)^\times\).

\(n\) \(66\) \(92\) \(106\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
274.1
− 2.44914i
− 1.92480i
− 1.71085i
− 1.51372i
− 0.700339i
− 0.687115i
− 0.340436i
0.340436i
0.687115i
0.700339i
1.51372i
1.71085i
1.92480i
2.44914i
− 2.44914i − 1.71242i −3.99829 −0.124172 − 2.23262i −4.19396 1.00000i 4.89410i 0.0676101 −5.46800 + 0.304116i
274.2 − 1.92480i − 2.17582i −1.70485 −1.38003 − 1.75941i −4.18801 − 1.00000i − 0.568104i −1.73417 −3.38651 + 2.65628i
274.3 − 1.71085i − 0.260291i −0.927009 −0.395001 + 2.20090i −0.445320 − 1.00000i − 1.83573i 2.93225 3.76542 + 0.675787i
274.4 − 1.51372i 1.55652i −0.291349 2.12532 + 0.694983i 2.35614 1.00000i − 2.58642i 0.577244 1.05201 − 3.21714i
274.5 − 0.700339i − 1.90743i 1.50953 0.743143 + 2.10897i −1.33585 1.00000i − 2.45786i −0.638282 1.47699 − 0.520452i
274.6 − 0.687115i 1.04121i 1.52787 2.17805 + 0.506082i 0.715431 − 1.00000i − 2.42406i 1.91588 0.347737 − 1.49657i
274.7 − 0.340436i − 2.66843i 1.88410 −2.14731 + 0.623754i −0.908430 − 1.00000i − 1.32229i −4.12053 0.212348 + 0.731020i
274.8 0.340436i 2.66843i 1.88410 −2.14731 − 0.623754i −0.908430 1.00000i 1.32229i −4.12053 0.212348 − 0.731020i
274.9 0.687115i − 1.04121i 1.52787 2.17805 − 0.506082i 0.715431 1.00000i 2.42406i 1.91588 0.347737 + 1.49657i
274.10 0.700339i 1.90743i 1.50953 0.743143 − 2.10897i −1.33585 − 1.00000i 2.45786i −0.638282 1.47699 + 0.520452i
274.11 1.51372i − 1.55652i −0.291349 2.12532 − 0.694983i 2.35614 − 1.00000i 2.58642i 0.577244 1.05201 + 3.21714i
274.12 1.71085i 0.260291i −0.927009 −0.395001 − 2.20090i −0.445320 1.00000i 1.83573i 2.93225 3.76542 − 0.675787i
274.13 1.92480i 2.17582i −1.70485 −1.38003 + 1.75941i −4.18801 1.00000i 0.568104i −1.73417 −3.38651 − 2.65628i
274.14 2.44914i 1.71242i −3.99829 −0.124172 + 2.23262i −4.19396 − 1.00000i − 4.89410i 0.0676101 −5.46800 − 0.304116i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 274.14
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 455.2.c.b ✓ 14
5.b even 2 1 inner 455.2.c.b ✓ 14
5.c odd 4 1 2275.2.a.w 7
5.c odd 4 1 2275.2.a.y 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
455.2.c.b ✓ 14 1.a even 1 1 trivial
455.2.c.b ✓ 14 5.b even 2 1 inner
2275.2.a.w 7 5.c odd 4 1
2275.2.a.y 7 5.c odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{14} + 16T_{2}^{12} + 96T_{2}^{10} + 272T_{2}^{8} + 372T_{2}^{6} + 225T_{2}^{4} + 56T_{2}^{2} + 4 \) acting on \(S_{2}^{\mathrm{new}}(455, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} + 16 T^{12} + \cdots + 4 \) Copy content Toggle raw display
$3$ \( T^{14} + 22 T^{12} + \cdots + 64 \) Copy content Toggle raw display
$5$ \( T^{14} - 2 T^{13} + \cdots + 78125 \) Copy content Toggle raw display
$7$ \( (T^{2} + 1)^{7} \) Copy content Toggle raw display
$11$ \( (T^{7} + 6 T^{6} + \cdots - 512)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 1)^{7} \) Copy content Toggle raw display
$17$ \( T^{14} + 98 T^{12} + \cdots + 16 \) Copy content Toggle raw display
$19$ \( (T^{7} - 10 T^{6} + \cdots - 15782)^{2} \) Copy content Toggle raw display
$23$ \( T^{14} + \cdots + 164660224 \) Copy content Toggle raw display
$29$ \( (T^{7} - 26 T^{6} + \cdots - 302)^{2} \) Copy content Toggle raw display
$31$ \( (T^{7} + 6 T^{6} + \cdots - 43898)^{2} \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots + 880190224 \) Copy content Toggle raw display
$41$ \( (T^{7} + 28 T^{6} + \cdots - 46208)^{2} \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots + 503194624 \) Copy content Toggle raw display
$47$ \( T^{14} + 382 T^{12} + \cdots + 4194304 \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots + 9130184704 \) Copy content Toggle raw display
$59$ \( (T^{7} - 28 T^{6} + \cdots - 54848)^{2} \) Copy content Toggle raw display
$61$ \( (T^{7} + 26 T^{6} + \cdots - 1967744)^{2} \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 386306999296 \) Copy content Toggle raw display
$71$ \( (T^{7} + 14 T^{6} + \cdots + 794624)^{2} \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots + 219513856 \) Copy content Toggle raw display
$79$ \( (T^{7} + 14 T^{6} + \cdots + 436112)^{2} \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots + 872023734882304 \) Copy content Toggle raw display
$89$ \( (T^{7} - 30 T^{6} + \cdots - 32188)^{2} \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots + 1297995375616 \) Copy content Toggle raw display
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