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Properties

Label 57.3.b.a
Level $57$
Weight $3$
Character orbit 57.b
Analytic conductor $1.553$
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] = [57,3,Mod(20,57)]
 
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, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("57.20");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 57 = 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 57.b (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: \(1.55313750685\)
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: \( x^{12} + 36x^{10} + 462x^{8} + 2636x^{6} + 6813x^{4} + 7296x^{2} + 2052 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2\cdot 3 \)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{6} q^{3} + (\beta_{2} - 2) q^{4} - \beta_{3} q^{5} + ( - \beta_{5} - \beta_{4} + \cdots + \beta_1) q^{6}+ \cdots + (\beta_{11} - \beta_{9} + \beta_{3} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + \beta_{6} q^{3} + (\beta_{2} - 2) q^{4} - \beta_{3} q^{5} + ( - \beta_{5} - \beta_{4} + \cdots + \beta_1) q^{6}+ \cdots + (6 \beta_{11} - 2 \beta_{10} + \cdots + 29) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 2 q^{3} - 24 q^{4} - 2 q^{6} + 8 q^{7} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 2 q^{3} - 24 q^{4} - 2 q^{6} + 8 q^{7} + 10 q^{9} - 8 q^{10} - 36 q^{12} - 32 q^{13} + 26 q^{15} + 72 q^{16} - 24 q^{18} + 10 q^{21} + 44 q^{22} + 102 q^{24} + 16 q^{25} - 70 q^{27} - 116 q^{28} + 68 q^{30} + 32 q^{31} - 34 q^{33} - 116 q^{34} - 158 q^{36} - 16 q^{37} - 138 q^{39} - 12 q^{40} + 38 q^{42} - 84 q^{43} + 176 q^{45} + 356 q^{46} + 264 q^{48} - 100 q^{49} + 58 q^{51} + 44 q^{52} - 164 q^{54} + 276 q^{55} + 220 q^{58} - 260 q^{60} - 76 q^{61} - 150 q^{63} - 600 q^{64} + 128 q^{66} - 336 q^{67} - 12 q^{69} - 132 q^{70} + 468 q^{72} + 120 q^{73} + 64 q^{75} - 248 q^{78} + 164 q^{79} + 142 q^{81} + 400 q^{82} - 828 q^{84} + 156 q^{85} + 354 q^{87} + 96 q^{88} + 344 q^{90} - 356 q^{91} - 456 q^{93} + 72 q^{94} - 574 q^{96} + 428 q^{97} + 364 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 36x^{10} + 462x^{8} + 2636x^{6} + 6813x^{4} + 7296x^{2} + 2052 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 6 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 3\nu^{11} + 101\nu^{9} + 1151\nu^{7} + 5243\nu^{5} + 8406\nu^{3} + 2952\nu ) / 24 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 15 \nu^{11} - 14 \nu^{10} - 505 \nu^{9} - 470 \nu^{8} - 5749 \nu^{7} - 5330 \nu^{6} - 26059 \nu^{5} + \cdots - 11400 ) / 96 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -13\nu^{10} - 439\nu^{8} - 5023\nu^{6} - 22965\nu^{4} - 36612\nu^{2} - 11796 ) / 48 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 52 \nu^{11} + 45 \nu^{10} - 1752 \nu^{9} + 1515 \nu^{8} - 19980 \nu^{7} + 17247 \nu^{6} + \cdots + 38484 ) / 288 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 26 \nu^{11} - 9 \nu^{10} + 876 \nu^{9} - 303 \nu^{8} + 9990 \nu^{7} - 3453 \nu^{6} + 45472 \nu^{5} + \cdots - 8712 ) / 72 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 9\nu^{10} + 303\nu^{8} + 3451\nu^{6} + 15669\nu^{4} + 24708\nu^{2} + 7828 ) / 16 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 191 \nu^{11} + 3 \nu^{10} + 6429 \nu^{9} + 105 \nu^{8} + 73197 \nu^{7} + 1257 \nu^{6} + 332191 \nu^{5} + \cdots + 4284 ) / 288 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 139 \nu^{11} + 78 \nu^{10} + 4677 \nu^{9} + 2622 \nu^{8} + 53217 \nu^{7} + 29802 \nu^{6} + \cdots + 69048 ) / 288 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 52 \nu^{11} + 279 \nu^{10} + 1752 \nu^{9} + 9393 \nu^{8} + 19980 \nu^{7} + 106989 \nu^{6} + \cdots + 247356 ) / 288 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{10} + \beta_{9} + \beta_{8} - \beta_{7} - \beta_{6} + \beta_{5} - \beta_{2} - 10\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{11} - 4\beta_{8} + 2\beta_{6} - 15\beta_{2} + 62 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 17 \beta_{10} - 19 \beta_{9} - 21 \beta_{8} + 17 \beta_{7} + 19 \beta_{6} - 21 \beta_{5} + \cdots + 126 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 48 \beta_{11} + 6 \beta_{10} - 6 \beta_{9} + 94 \beta_{8} - 6 \beta_{7} - 66 \beta_{6} - 6 \beta_{5} + \cdots - 772 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 255 \beta_{10} + 307 \beta_{9} + 367 \beta_{8} - 255 \beta_{7} - 307 \beta_{6} + 367 \beta_{5} + \cdots - 1744 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 878 \beta_{11} - 172 \beta_{10} + 172 \beta_{9} - 1716 \beta_{8} + 172 \beta_{7} + 1394 \beta_{6} + \cdots + 10478 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 36 \beta_{11} + 3733 \beta_{10} - 4779 \beta_{9} - 6037 \beta_{8} + 3805 \beta_{7} + 4743 \beta_{6} + \cdots + 25246 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 14636 \beta_{11} + 3490 \beta_{10} - 3490 \beta_{9} + 28694 \beta_{8} - 3490 \beta_{7} + \cdots - 149080 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 1212 \beta_{11} - 54751 \beta_{10} + 73511 \beta_{9} + 96339 \beta_{8} - 57175 \beta_{7} + \cdots - 374004 \beta_1 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
20.1
− 3.90587i
− 3.08351i
− 2.47948i
− 1.53106i
− 1.52359i
− 0.650284i
0.650284i
1.52359i
1.53106i
2.47948i
3.08351i
3.90587i
− 3.90587i 2.60001 − 1.49664i −11.2558 2.21947i −5.84567 − 10.1553i 7.28226 28.3402i 4.52015 − 7.78256i 8.66895
20.2 − 3.08351i −2.98600 − 0.289485i −5.50802 − 1.98277i −0.892630 + 9.20735i −5.99504 4.64999i 8.83240 + 1.72881i −6.11389
20.3 − 2.47948i 0.758838 + 2.90244i −2.14782 − 8.07989i 7.19655 − 1.88152i 6.82311 − 4.59244i −7.84833 + 4.40496i −20.0339
20.4 − 1.53106i 2.72716 + 1.25003i 1.65584 4.38402i 1.91388 − 4.17546i −7.54587 − 8.65946i 5.87483 + 6.81809i 6.71222
20.5 − 1.52359i 0.292648 − 2.98569i 1.67867 1.53790i −4.54898 − 0.445876i −3.05882 − 8.65197i −8.82871 − 1.74751i 2.34312
20.6 − 0.650284i −2.39266 + 1.80974i 3.57713 6.80244i 1.17685 + 1.55591i 6.49436 − 4.92729i 2.44967 − 8.66020i 4.42352
20.7 0.650284i −2.39266 − 1.80974i 3.57713 − 6.80244i 1.17685 − 1.55591i 6.49436 4.92729i 2.44967 + 8.66020i 4.42352
20.8 1.52359i 0.292648 + 2.98569i 1.67867 − 1.53790i −4.54898 + 0.445876i −3.05882 8.65197i −8.82871 + 1.74751i 2.34312
20.9 1.53106i 2.72716 − 1.25003i 1.65584 − 4.38402i 1.91388 + 4.17546i −7.54587 8.65946i 5.87483 − 6.81809i 6.71222
20.10 2.47948i 0.758838 − 2.90244i −2.14782 8.07989i 7.19655 + 1.88152i 6.82311 4.59244i −7.84833 − 4.40496i −20.0339
20.11 3.08351i −2.98600 + 0.289485i −5.50802 1.98277i −0.892630 − 9.20735i −5.99504 − 4.64999i 8.83240 − 1.72881i −6.11389
20.12 3.90587i 2.60001 + 1.49664i −11.2558 − 2.21947i −5.84567 + 10.1553i 7.28226 − 28.3402i 4.52015 + 7.78256i 8.66895
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 20.12
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 57.3.b.a ✓ 12
3.b odd 2 1 inner 57.3.b.a ✓ 12
4.b odd 2 1 912.3.h.a 12
12.b even 2 1 912.3.h.a 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
57.3.b.a ✓ 12 1.a even 1 1 trivial
57.3.b.a ✓ 12 3.b odd 2 1 inner
912.3.h.a 12 4.b odd 2 1
912.3.h.a 12 12.b even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(57, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 36 T^{10} + \cdots + 2052 \) Copy content Toggle raw display
$3$ \( T^{12} - 2 T^{11} + \cdots + 531441 \) Copy content Toggle raw display
$5$ \( T^{12} + 142 T^{10} + \cdots + 2659392 \) Copy content Toggle raw display
$7$ \( (T^{6} - 4 T^{5} + \cdots - 44652)^{2} \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 1002580146432 \) Copy content Toggle raw display
$13$ \( (T^{6} + 16 T^{5} + \cdots + 2159796)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 106222499020800 \) Copy content Toggle raw display
$19$ \( (T^{2} - 19)^{6} \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 703725396083712 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 639798602867712 \) Copy content Toggle raw display
$31$ \( (T^{6} - 16 T^{5} + \cdots + 3916512)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} + 8 T^{5} + \cdots - 97373376)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 13487117165568 \) Copy content Toggle raw display
$43$ \( (T^{6} + 42 T^{5} + \cdots + 100058944)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 12\!\cdots\!88 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 23\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 87\!\cdots\!72 \) Copy content Toggle raw display
$61$ \( (T^{6} + 38 T^{5} + \cdots - 4353200)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} + 168 T^{5} + \cdots + 166317551104)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 12\!\cdots\!72 \) Copy content Toggle raw display
$73$ \( (T^{6} - 60 T^{5} + \cdots - 3513904168)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} - 82 T^{5} + \cdots - 271256387328)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 32\!\cdots\!88 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 30\!\cdots\!32 \) Copy content Toggle raw display
$97$ \( (T^{6} - 214 T^{5} + \cdots - 4007585216)^{2} \) Copy content Toggle raw display
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