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Properties

Label 275.6.b.f
Level $275$
Weight $6$
Character orbit 275.b
Analytic conductor $44.106$
Analytic rank $0$
Dimension $12$
Inner twists $2$

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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] = [275,6,Mod(199,275)] 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("275.199"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(275, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 275 = 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 275.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,0,-230] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(44.1055504486\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 307x^{10} + 32123x^{8} + 1354853x^{6} + 24286348x^{4} + 189182512x^{2} + 529736256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{8}\cdot 5^{4} \)
Twist minimal: no (minimal twist has level 55)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{9} - \beta_1) q^{3} + (\beta_{2} - 19) q^{4} + ( - \beta_{6} + \beta_{4} - 2 \beta_{2} + 54) q^{6} + ( - \beta_{11} - \beta_{10} + \cdots + 8 \beta_1) q^{7} + ( - 2 \beta_{11} - 2 \beta_{10} + \cdots - 33 \beta_1) q^{8}+ \cdots + (242 \beta_{6} + 1210 \beta_{5} + \cdots - 17061) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 230 q^{4} + 650 q^{6} - 1664 q^{9} + 1452 q^{11} - 4034 q^{14} + 13350 q^{16} - 1220 q^{19} + 5016 q^{21} - 26894 q^{24} - 15484 q^{26} - 21928 q^{29} + 17736 q^{31} + 85346 q^{34} - 60232 q^{36}+ \cdots - 201344 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 307x^{10} + 32123x^{8} + 1354853x^{6} + 24286348x^{4} + 189182512x^{2} + 529736256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 51 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 79883 \nu^{10} + 22238485 \nu^{8} + 1896728821 \nu^{6} + 45022360531 \nu^{4} + \cdots - 4070876074512 ) / 7554540864 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 115839 \nu^{10} - 34584577 \nu^{8} - 3425242657 \nu^{6} - 127549977799 \nu^{4} + \cdots - 7333580639280 ) / 7554540864 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 10493 \nu^{10} - 3131220 \nu^{8} - 310128501 \nu^{6} - 11540784510 \nu^{4} + \cdots - 643241165316 ) / 472158804 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 190595 \nu^{10} + 56487565 \nu^{8} + 5534187997 \nu^{6} + 201995167291 \nu^{4} + \cdots + 10769122162032 ) / 7554540864 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 960047 \nu^{11} - 287864153 \nu^{9} - 28759553305 \nu^{7} - 1090250778223 \nu^{5} + \cdots - 65794573616720 \nu ) / 557292668352 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 2387481 \nu^{11} - 721887889 \nu^{9} - 73341882285 \nu^{7} - 2887852196223 \nu^{5} + \cdots - 202158741225824 \nu ) / 905600586072 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 66856827 \nu^{11} - 19937305477 \nu^{9} - 1972965485365 \nu^{7} + \cdots - 42\!\cdots\!32 \nu ) / 14489609377152 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 25971305 \nu^{11} + 7685152519 \nu^{9} + 749070490519 \nu^{7} + 26881857938257 \nu^{5} + \cdots + 11\!\cdots\!60 \nu ) / 2069944196736 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 98601537 \nu^{11} - 29334269735 \nu^{9} - 2889240754519 \nu^{7} + \cdots - 55\!\cdots\!28 \nu ) / 7244804688576 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 51 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{11} - 2\beta_{10} - 2\beta_{9} + 3\beta_{8} + 2\beta_{7} - 97\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 5\beta_{6} + 19\beta_{5} - 20\beta_{4} - \beta_{3} - 121\beta_{2} + 4974 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 355\beta_{11} + 285\beta_{10} + 55\beta_{9} - 353\beta_{8} - 336\beta_{7} + 10984\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -458\beta_{6} - 3282\beta_{5} + 4050\beta_{4} + 68\beta_{3} + 14025\beta_{2} - 565405 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -51056\beta_{11} - 36472\beta_{10} + 7012\beta_{9} + 34151\beta_{8} + 66682\beta_{7} - 1289419\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -969\beta_{6} + 441557\beta_{5} - 639952\beta_{4} + 2321\beta_{3} - 1642485\beta_{2} + 66715588 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 6929577 \beta_{11} + 4568547 \beta_{10} - 1846431 \beta_{9} - 2966001 \beta_{8} - 11987760 \beta_{7} + 153310498 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 8326416\beta_{6} - 55705464\beta_{5} + 93264630\beta_{4} - 1602546\beta_{3} + 193944445\beta_{2} - 7977222927 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 919846466 \beta_{11} - 569296826 \beta_{10} + 312763606 \beta_{9} + 219716343 \beta_{8} + \cdots - 18350703001 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/275\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(177\)
\(\chi(n)\) \(1\) \(-1\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
199.1
− 11.2169i
− 10.5183i
− 6.32252i
− 3.74230i
− 3.01414i
− 2.73538i
2.73538i
3.01414i
3.74230i
6.32252i
10.5183i
11.2169i
− 11.2169i 11.4706i −93.8183 0 128.664 132.518i 693.408i 111.426 0
199.2 − 10.5183i 10.1483i −78.6353 0 106.743 − 228.119i 490.525i 140.013 0
199.3 − 6.32252i 28.4424i −7.97424 0 179.828 − 115.579i − 151.903i −565.969 0
199.4 − 3.74230i − 13.5338i 17.9952 0 −50.6477 134.766i − 187.097i 59.8351 0
199.5 − 3.01414i 13.7145i 22.9149 0 41.3374 − 131.565i − 165.521i 54.9135 0
199.6 − 2.73538i − 29.5841i 24.5177 0 −80.9238 − 175.856i − 154.598i −632.218 0
199.7 2.73538i 29.5841i 24.5177 0 −80.9238 175.856i 154.598i −632.218 0
199.8 3.01414i − 13.7145i 22.9149 0 41.3374 131.565i 165.521i 54.9135 0
199.9 3.74230i 13.5338i 17.9952 0 −50.6477 − 134.766i 187.097i 59.8351 0
199.10 6.32252i − 28.4424i −7.97424 0 179.828 115.579i 151.903i −565.969 0
199.11 10.5183i − 10.1483i −78.6353 0 106.743 228.119i − 490.525i 140.013 0
199.12 11.2169i − 11.4706i −93.8183 0 128.664 − 132.518i − 693.408i 111.426 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 199.12
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 275.6.b.f 12
5.b even 2 1 inner 275.6.b.f 12
5.c odd 4 1 55.6.a.d ✓ 6
5.c odd 4 1 275.6.a.f 6
15.e even 4 1 495.6.a.l 6
20.e even 4 1 880.6.a.u 6
55.e even 4 1 605.6.a.e 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.6.a.d ✓ 6 5.c odd 4 1
275.6.a.f 6 5.c odd 4 1
275.6.b.f 12 1.a even 1 1 trivial
275.6.b.f 12 5.b even 2 1 inner
495.6.a.l 6 15.e even 4 1
605.6.a.e 6 55.e even 4 1
880.6.a.u 6 20.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} + 307T_{2}^{10} + 32123T_{2}^{8} + 1354853T_{2}^{6} + 24286348T_{2}^{4} + 189182512T_{2}^{2} + 529736256 \) acting on \(S_{6}^{\mathrm{new}}(275, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + \cdots + 529736256 \) Copy content Toggle raw display
$3$ \( T^{12} + \cdots + 330522871762944 \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 11\!\cdots\!56 \) Copy content Toggle raw display
$11$ \( (T - 121)^{12} \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 58\!\cdots\!24 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 43\!\cdots\!64 \) Copy content Toggle raw display
$19$ \( (T^{6} + \cdots - 13\!\cdots\!00)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 78\!\cdots\!24 \) Copy content Toggle raw display
$29$ \( (T^{6} + \cdots + 94\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + \cdots + 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 48\!\cdots\!44 \) Copy content Toggle raw display
$41$ \( (T^{6} + \cdots + 80\!\cdots\!56)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 13\!\cdots\!76 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 49\!\cdots\!24 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 32\!\cdots\!36 \) Copy content Toggle raw display
$59$ \( (T^{6} + \cdots - 84\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} + \cdots - 24\!\cdots\!64)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 71\!\cdots\!56 \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots - 10\!\cdots\!28)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 56\!\cdots\!84 \) Copy content Toggle raw display
$79$ \( (T^{6} + \cdots + 29\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 91\!\cdots\!56 \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots + 30\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 17\!\cdots\!84 \) Copy content Toggle raw display
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