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Properties

Label 57.5.c.a
Level $57$
Weight $5$
Character orbit 57.c
Analytic conductor $5.892$
Analytic rank $0$
Dimension $14$
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] = [57,5,Mod(37,57)]
 
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([0, 1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("57.37");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 57 = 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 57.c (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: \(5.89208789578\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} + 190x^{12} + 14073x^{10} + 516256x^{8} + 9846472x^{6} + 92351712x^{4} + 334182672x^{2} + 45349632 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{11}\cdot 3^{4} \)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - \beta_{5} q^{3} + (\beta_{2} - 11) q^{4} + ( - \beta_{7} + 1) q^{5} + ( - \beta_{4} + 3) q^{6} + ( - \beta_{8} - \beta_{2} - 4) q^{7} + (\beta_{9} - 4 \beta_{5} - 10 \beta_1) q^{8} - 27 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - \beta_{5} q^{3} + (\beta_{2} - 11) q^{4} + ( - \beta_{7} + 1) q^{5} + ( - \beta_{4} + 3) q^{6} + ( - \beta_{8} - \beta_{2} - 4) q^{7} + (\beta_{9} - 4 \beta_{5} - 10 \beta_1) q^{8} - 27 q^{9} + ( - \beta_{13} - \beta_{11} + \cdots - 2 \beta_1) q^{10}+ \cdots + ( - 27 \beta_{8} + 27 \beta_{7} + \cdots + 594) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 156 q^{4} + 18 q^{5} + 36 q^{6} - 54 q^{7} - 378 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 156 q^{4} + 18 q^{5} + 36 q^{6} - 54 q^{7} - 378 q^{9} - 318 q^{11} + 1252 q^{16} - 654 q^{17} - 694 q^{19} + 960 q^{20} - 492 q^{23} - 1764 q^{24} + 3736 q^{25} + 1296 q^{26} - 3536 q^{28} - 1152 q^{30} + 4566 q^{35} + 4212 q^{36} - 4440 q^{38} - 2952 q^{39} + 2304 q^{42} - 8422 q^{43} + 11232 q^{44} - 486 q^{45} + 12258 q^{47} + 2376 q^{49} - 972 q^{54} + 12782 q^{55} - 4860 q^{57} - 24856 q^{58} + 5378 q^{61} - 41088 q^{62} + 1458 q^{63} - 26012 q^{64} + 792 q^{66} + 26832 q^{68} + 12594 q^{73} + 23856 q^{74} - 12652 q^{76} - 20610 q^{77} - 64056 q^{80} + 10206 q^{81} + 59704 q^{82} - 29436 q^{83} + 26182 q^{85} - 12384 q^{87} + 103704 q^{92} + 38016 q^{93} + 22818 q^{95} - 4932 q^{96} + 8586 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} + 190x^{12} + 14073x^{10} + 516256x^{8} + 9846472x^{6} + 92351712x^{4} + 334182672x^{2} + 45349632 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 27 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 38437 \nu^{12} + 10189924 \nu^{10} + 974951925 \nu^{8} + 42199145518 \nu^{6} + \cdots + 6859489027968 ) / 22506564096 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1441 \nu^{12} + 253028 \nu^{10} + 16356369 \nu^{8} + 473625574 \nu^{6} + 5917953548 \nu^{4} + \cdots + 7465580928 ) / 833576448 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 53207 \nu^{13} - 9409004 \nu^{11} - 625810503 \nu^{9} - 19519237658 \nu^{7} + \cdots - 5885298393984 \nu ) / 405118153728 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 33871 \nu^{12} - 5153584 \nu^{10} - 271667031 \nu^{8} - 5772478654 \nu^{6} + \cdots - 1576791722880 ) / 11253282048 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 27131 \nu^{12} - 4676612 \nu^{10} - 300435915 \nu^{8} - 8882638298 \nu^{6} + \cdots - 162457757568 ) / 5626641024 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 109579 \nu^{12} - 18650668 \nu^{10} - 1176804987 \nu^{8} - 33776592130 \nu^{6} + \cdots - 389743996032 ) / 11253282048 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 53207 \nu^{13} - 9409004 \nu^{11} - 625810503 \nu^{9} - 19519237658 \nu^{7} + \cdots - 1631557779840 \nu ) / 101279538432 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 156325 \nu^{13} + 22845748 \nu^{11} + 1022459541 \nu^{9} + 5442764734 \nu^{7} + \cdots - 69351148321920 \nu ) / 101279538432 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 27269 \nu^{13} + 4854500 \nu^{11} + 331395861 \nu^{9} + 10993977326 \nu^{7} + \cdots + 5720909185152 \nu ) / 15004376064 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 1187429 \nu^{13} + 221105804 \nu^{11} + 15718919781 \nu^{9} + 530202576950 \nu^{7} + \cdots + 80619008462208 \nu ) / 405118153728 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 1371451 \nu^{13} - 238598284 \nu^{11} - 15602442315 \nu^{9} - 477953291458 \nu^{7} + \cdots - 78123598471296 \nu ) / 202559076864 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 27 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{9} - 4\beta_{5} - 42\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -2\beta_{8} + 6\beta_{7} - 2\beta_{6} + \beta_{4} + \beta_{3} - 60\beta_{2} + 1130 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 4\beta_{13} + 8\beta_{11} + 2\beta_{10} - 77\beta_{9} + 236\beta_{5} + 2050\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 232\beta_{8} - 612\beta_{7} + 142\beta_{6} - 71\beta_{4} - 83\beta_{3} + 3374\beta_{2} - 54904 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -404\beta_{13} - 48\beta_{12} - 856\beta_{11} - 232\beta_{10} + 4851\beta_{9} - 14220\beta_{5} - 107180\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -18654\beta_{8} + 45282\beta_{7} - 8190\beta_{6} + 1455\beta_{4} + 5583\beta_{3} - 187690\beta_{2} + 2866824 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 29604 \beta_{13} + 5952 \beta_{12} + 68064 \beta_{11} + 18654 \beta_{10} - 288433 \beta_{9} + \cdots + 5795586 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 1315436 \beta_{8} - 2996088 \beta_{7} + 450590 \beta_{6} + 181553 \beta_{4} - 348595 \beta_{3} + \cdots - 155247092 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 1927252 \beta_{13} - 493200 \beta_{12} - 4841672 \beta_{11} - 1315436 \beta_{10} + \cdots - 319229848 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 87283594 \beta_{8} + 188616366 \beta_{7} - 24616342 \beta_{6} - 28586725 \beta_{4} + \cdots + 8578126168 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 118742324 \beta_{13} + 34819104 \beta_{12} + 325499632 \beta_{11} + 87283594 \beta_{10} + \cdots + 17785988606 \beta_1 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1
− 7.66280i
− 7.14518i
− 5.72212i
− 4.67191i
− 4.04168i
− 3.03019i
− 0.375666i
0.375666i
3.03019i
4.04168i
4.67191i
5.72212i
7.14518i
7.66280i
− 7.66280i 5.19615i −42.7185 −31.0118 39.8171 66.5198 204.739i −27.0000 237.637i
37.2 − 7.14518i − 5.19615i −35.0537 4.37368 −37.1275 −34.1721 136.142i −27.0000 − 31.2507i
37.3 − 5.72212i 5.19615i −16.7427 44.3284 29.7330 9.09531 4.24968i −27.0000 − 253.653i
37.4 − 4.67191i 5.19615i −5.82672 −16.7066 24.2759 −87.8991 − 47.5286i −27.0000 78.0516i
37.5 − 4.04168i − 5.19615i −0.335204 33.4922 −21.0012 61.0484 − 63.3121i −27.0000 − 135.365i
37.6 − 3.03019i − 5.19615i 6.81792 −40.7717 −15.7454 −21.6742 − 69.1427i −27.0000 123.546i
37.7 − 0.375666i − 5.19615i 15.8589 15.2958 −1.95202 −19.9181 − 11.9683i −27.0000 − 5.74610i
37.8 0.375666i 5.19615i 15.8589 15.2958 −1.95202 −19.9181 11.9683i −27.0000 5.74610i
37.9 3.03019i 5.19615i 6.81792 −40.7717 −15.7454 −21.6742 69.1427i −27.0000 − 123.546i
37.10 4.04168i 5.19615i −0.335204 33.4922 −21.0012 61.0484 63.3121i −27.0000 135.365i
37.11 4.67191i − 5.19615i −5.82672 −16.7066 24.2759 −87.8991 47.5286i −27.0000 − 78.0516i
37.12 5.72212i − 5.19615i −16.7427 44.3284 29.7330 9.09531 − 4.24968i −27.0000 253.653i
37.13 7.14518i 5.19615i −35.0537 4.37368 −37.1275 −34.1721 − 136.142i −27.0000 31.2507i
37.14 7.66280i − 5.19615i −42.7185 −31.0118 39.8171 66.5198 − 204.739i −27.0000 − 237.637i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 37.14
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 57.5.c.a ✓ 14
3.b odd 2 1 171.5.c.e 14
4.b odd 2 1 912.5.o.b 14
19.b odd 2 1 inner 57.5.c.a ✓ 14
57.d even 2 1 171.5.c.e 14
76.d even 2 1 912.5.o.b 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
57.5.c.a ✓ 14 1.a even 1 1 trivial
57.5.c.a ✓ 14 19.b odd 2 1 inner
171.5.c.e 14 3.b odd 2 1
171.5.c.e 14 57.d even 2 1
912.5.o.b 14 4.b odd 2 1
912.5.o.b 14 76.d even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(57, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} + 190 T^{12} + \cdots + 45349632 \) Copy content Toggle raw display
$3$ \( (T^{2} + 27)^{7} \) Copy content Toggle raw display
$5$ \( (T^{7} - 9 T^{6} + \cdots + 2098057824)^{2} \) Copy content Toggle raw display
$7$ \( (T^{7} + 27 T^{6} + \cdots - 47894867872)^{2} \) Copy content Toggle raw display
$11$ \( (T^{7} + \cdots - 1564857510168)^{2} \) Copy content Toggle raw display
$13$ \( T^{14} + \cdots + 18\!\cdots\!48 \) Copy content Toggle raw display
$17$ \( (T^{7} + \cdots + 31\!\cdots\!68)^{2} \) Copy content Toggle raw display
$19$ \( T^{14} + \cdots + 63\!\cdots\!41 \) Copy content Toggle raw display
$23$ \( (T^{7} + \cdots - 37\!\cdots\!48)^{2} \) Copy content Toggle raw display
$29$ \( T^{14} + \cdots + 67\!\cdots\!68 \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots + 58\!\cdots\!52 \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots + 43\!\cdots\!28 \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots + 12\!\cdots\!92 \) Copy content Toggle raw display
$43$ \( (T^{7} + \cdots + 31\!\cdots\!44)^{2} \) Copy content Toggle raw display
$47$ \( (T^{7} + \cdots - 98\!\cdots\!52)^{2} \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots + 82\!\cdots\!68 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 68\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{7} + \cdots - 11\!\cdots\!68)^{2} \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 10\!\cdots\!72 \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots + 35\!\cdots\!72 \) Copy content Toggle raw display
$73$ \( (T^{7} + \cdots - 19\!\cdots\!40)^{2} \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots + 61\!\cdots\!28 \) Copy content Toggle raw display
$83$ \( (T^{7} + \cdots + 32\!\cdots\!80)^{2} \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots + 81\!\cdots\!92 \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots + 94\!\cdots\!00 \) Copy content Toggle raw display
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