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Properties

Label 57.4.d.c
Level $57$
Weight $4$
Character orbit 57.d
Analytic conductor $3.363$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [57,4,Mod(56,57)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(57, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("57.56");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 57 = 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 57.d (of order \(2\), degree \(1\), 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: \(3.36310887033\)
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: \( x^{12} - 23x^{10} + 1659x^{8} + 14266x^{6} + 685507x^{4} - 16582767x^{2} + 113486409 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
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.

\(f(q)\) \(=\) \( q + \beta_{7} q^{2} + ( - \beta_{7} + \beta_1) q^{3} + ( - \beta_{5} + \beta_{4} + \beta_{2} + 2) q^{4} + \beta_{6} q^{5} + (\beta_{5} - 2 \beta_{4} - \beta_{3} - 5) q^{6} + ( - 2 \beta_{4} + 11) q^{7} + (\beta_{10} + \beta_{9} + 2 \beta_{8} + \cdots - \beta_1) q^{8}+ \cdots + ( - 3 \beta_{2} + 6) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{7} q^{2} + ( - \beta_{7} + \beta_1) q^{3} + ( - \beta_{5} + \beta_{4} + \beta_{2} + 2) q^{4} + \beta_{6} q^{5} + (\beta_{5} - 2 \beta_{4} - \beta_{3} - 5) q^{6} + ( - 2 \beta_{4} + 11) q^{7} + (\beta_{10} + \beta_{9} + 2 \beta_{8} + \cdots - \beta_1) q^{8}+ \cdots + (72 \beta_{6} + 123 \beta_{5} + \cdots + 84) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 28 q^{4} - 66 q^{6} + 124 q^{7} + 54 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 28 q^{4} - 66 q^{6} + 124 q^{7} + 54 q^{9} + 20 q^{16} - 200 q^{19} - 198 q^{24} - 156 q^{25} + 92 q^{28} - 672 q^{30} - 1146 q^{36} + 414 q^{39} + 654 q^{42} + 3376 q^{43} - 276 q^{45} - 360 q^{49} + 900 q^{54} - 1912 q^{55} + 1506 q^{57} - 2332 q^{58} + 696 q^{61} - 1002 q^{63} + 836 q^{64} + 4320 q^{66} + 404 q^{73} - 4904 q^{76} - 2106 q^{81} - 3704 q^{82} + 736 q^{85} - 4230 q^{87} - 4452 q^{93} + 426 q^{96} + 1044 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 23x^{10} + 1659x^{8} + 14266x^{6} + 685507x^{4} - 16582767x^{2} + 113486409 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 16865436 \nu^{10} - 1651226381 \nu^{8} + 28003615347 \nu^{6} - 4837127882418 \nu^{4} + \cdots - 865847993239029 ) / 236702405185380 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 40001461 \nu^{10} + 1408446416 \nu^{8} - 76307668347 \nu^{6} + 4417221835723 \nu^{4} + \cdots - 631130779956741 ) / 473404810370760 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 10688831 \nu^{10} + 388398567 \nu^{8} - 13389401808 \nu^{6} + 110113200037 \nu^{4} + \cdots + 404244260612946 ) / 47340481037076 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 154061876 \nu^{10} + 3931972159 \nu^{8} - 221141464773 \nu^{6} - 3605335928558 \nu^{4} + \cdots + 22\!\cdots\!11 ) / 236702405185380 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 102332582 \nu^{10} + 1258276607 \nu^{8} - 151117139367 \nu^{6} - 3020444354624 \nu^{4} + \cdots + 888025576319553 ) / 47340481037076 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 86860782903 \nu^{11} - 1880061132442 \nu^{9} + 113581602740829 \nu^{7} + \cdots - 12\!\cdots\!13 \nu ) / 16\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 131743338433 \nu^{11} + 8544319663902 \nu^{9} - 315350689423659 \nu^{7} + \cdots + 92\!\cdots\!63 \nu ) / 16\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 7348547620 \nu^{11} - 206972634141 \nu^{9} + 13570443812997 \nu^{7} + \cdots - 12\!\cdots\!59 \nu ) / 84\!\cdots\!38 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 11232933828 \nu^{11} + 108504287315 \nu^{9} - 15420018939643 \nu^{7} + \cdots + 89\!\cdots\!97 \nu ) / 28\!\cdots\!46 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 269097948745 \nu^{11} + 3563570281188 \nu^{9} - 414996385072989 \nu^{7} + \cdots + 18\!\cdots\!45 \nu ) / 33\!\cdots\!52 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} - 3\beta_{4} - 2\beta_{3} - 2\beta_{2} + 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 6\beta_{11} - 11\beta_{10} + 16\beta_{9} - \beta_{8} - 21\beta_{7} + 14\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 15\beta_{6} - 79\beta_{5} + 88\beta_{4} - 20\beta_{3} - 36\beta_{2} - 430 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 92\beta_{11} + 5\beta_{10} + 1023\beta_{9} + 362\beta_{8} + 282\beta_{7} - 840\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -351\beta_{6} - 1805\beta_{5} + 7615\beta_{4} + 414\beta_{3} + 3497\beta_{2} - 27706 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -10774\beta_{11} + 20833\beta_{10} + 6596\beta_{9} + 9276\beta_{8} - 2334\beta_{7} - 48791\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -40815\beta_{6} + 110810\beta_{5} + 78072\beta_{4} + 102980\beta_{3} + 100745\beta_{2} - 462885 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -515052\beta_{11} + 571424\beta_{10} - 2276758\beta_{9} - 299951\beta_{8} - 147021\beta_{7} - 657548\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -888882\beta_{6} + 5487478\beta_{5} - 10265839\beta_{4} + 2989766\beta_{3} - 1118205\beta_{2} + 51358525 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 240154 \beta_{11} - 14290844 \beta_{10} - 84662493 \beta_{9} - 27661394 \beta_{8} + \cdots + 84299736 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/57\mathbb{Z}\right)^\times\).

\(n\) \(20\) \(40\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
56.1
−2.13847 − 4.47250i
−2.13847 + 4.47250i
3.00654 − 0.806605i
3.00654 + 0.806605i
−5.62474 − 3.61886i
−5.62474 + 3.61886i
5.62474 − 3.61886i
5.62474 + 3.61886i
−3.00654 − 0.806605i
−3.00654 + 0.806605i
2.13847 − 4.47250i
2.13847 + 4.47250i
−4.78360 2.64513 − 4.47250i 14.8829 3.10139i −12.6533 + 21.3947i 8.57634 −32.9249 −13.0066 − 23.6607i − 14.8358i
56.2 −4.78360 2.64513 + 4.47250i 14.8829 − 3.10139i −12.6533 − 21.3947i 8.57634 −32.9249 −13.0066 + 23.6607i 14.8358i
56.3 −2.12662 5.13317 − 0.806605i −3.47747 17.1306i −10.9163 + 1.71535i −6.31010 24.4083 25.6988 − 8.28087i − 36.4304i
56.4 −2.12662 5.13317 + 0.806605i −3.47747 − 17.1306i −10.9163 − 1.71535i −6.31010 24.4083 25.6988 + 8.28087i 36.4304i
56.5 −1.89594 −3.72879 − 3.61886i −4.40540 10.5320i 7.06958 + 6.86115i 28.7338 23.5199 0.807773 + 26.9879i − 19.9681i
56.6 −1.89594 −3.72879 + 3.61886i −4.40540 − 10.5320i 7.06958 − 6.86115i 28.7338 23.5199 0.807773 − 26.9879i 19.9681i
56.7 1.89594 3.72879 − 3.61886i −4.40540 − 10.5320i 7.06958 − 6.86115i 28.7338 −23.5199 0.807773 − 26.9879i − 19.9681i
56.8 1.89594 3.72879 + 3.61886i −4.40540 10.5320i 7.06958 + 6.86115i 28.7338 −23.5199 0.807773 + 26.9879i 19.9681i
56.9 2.12662 −5.13317 − 0.806605i −3.47747 − 17.1306i −10.9163 − 1.71535i −6.31010 −24.4083 25.6988 + 8.28087i − 36.4304i
56.10 2.12662 −5.13317 + 0.806605i −3.47747 17.1306i −10.9163 + 1.71535i −6.31010 −24.4083 25.6988 − 8.28087i 36.4304i
56.11 4.78360 −2.64513 − 4.47250i 14.8829 − 3.10139i −12.6533 − 21.3947i 8.57634 32.9249 −13.0066 + 23.6607i − 14.8358i
56.12 4.78360 −2.64513 + 4.47250i 14.8829 3.10139i −12.6533 + 21.3947i 8.57634 32.9249 −13.0066 − 23.6607i 14.8358i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 56.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
19.b odd 2 1 inner
57.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 57.4.d.c ✓ 12
3.b odd 2 1 inner 57.4.d.c ✓ 12
19.b odd 2 1 inner 57.4.d.c ✓ 12
57.d even 2 1 inner 57.4.d.c ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
57.4.d.c ✓ 12 1.a even 1 1 trivial
57.4.d.c ✓ 12 3.b odd 2 1 inner
57.4.d.c ✓ 12 19.b odd 2 1 inner
57.4.d.c ✓ 12 57.d even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} - 31T_{2}^{4} + 202T_{2}^{2} - 372 \) acting on \(S_{4}^{\mathrm{new}}(57, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} - 31 T^{4} + \cdots - 372)^{2} \) Copy content Toggle raw display
$3$ \( T^{12} + \cdots + 387420489 \) Copy content Toggle raw display
$5$ \( (T^{6} + 414 T^{4} + \cdots + 313100)^{2} \) Copy content Toggle raw display
$7$ \( (T^{3} - 31 T^{2} + \cdots + 1555)^{4} \) Copy content Toggle raw display
$11$ \( (T^{6} + 6102 T^{4} + \cdots + 6260108876)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} + 7319 T^{4} + \cdots + 10828154652)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} + 15029 T^{4} + \cdots + 84825130275)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} + 100 T^{5} + \cdots + 322687697779)^{2} \) Copy content Toggle raw display
$23$ \( (T^{6} + 15075 T^{4} + \cdots + 38354750000)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + \cdots - 6807852364800)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + \cdots + 179322214470000)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} + \cdots + 4848988390848)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} - 95356 T^{4} + \cdots - 515896147200)^{2} \) Copy content Toggle raw display
$43$ \( (T^{3} - 844 T^{2} + \cdots - 12897290)^{4} \) Copy content Toggle raw display
$47$ \( (T^{6} + \cdots + 31\!\cdots\!00)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + \cdots - 346769744694228)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + \cdots - 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} - 174 T^{2} + \cdots + 164977960)^{4} \) Copy content Toggle raw display
$67$ \( (T^{6} + \cdots + 51654379664832)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots - 13262913172800)^{2} \) Copy content Toggle raw display
$73$ \( (T^{3} - 101 T^{2} + \cdots - 8248275)^{4} \) Copy content Toggle raw display
$79$ \( (T^{6} + \cdots + 59\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} + \cdots + 76\!\cdots\!00)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots - 23\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + \cdots + 11\!\cdots\!72)^{2} \) Copy content Toggle raw display
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