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Properties

Label 688.5.b.d
Level $688$
Weight $5$
Character orbit 688.b
Analytic conductor $71.119$
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] = [688,5,Mod(257,688)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(688, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("688.257");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 688 = 2^{4} \cdot 43 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 688.b (of order \(2\), degree \(1\), not 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: \(71.1185346017\)
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: \( x^{12} + 142x^{10} + 7173x^{8} + 157368x^{6} + 1510016x^{4} + 5098688x^{2} + 90352 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{11} \)
Twist minimal: no (minimal twist has level 43)
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.

\(f(q)\) \(=\) \( q - \beta_{3} q^{3} + \beta_{7} q^{5} + (\beta_{10} - 2 \beta_{3} + \beta_1) q^{7} + (\beta_{8} - 2 \beta_{6} + \beta_{5} + \cdots - 39) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{3} + \beta_{7} q^{5} + (\beta_{10} - 2 \beta_{3} + \beta_1) q^{7} + (\beta_{8} - 2 \beta_{6} + \beta_{5} + \cdots - 39) q^{9}+ \cdots + (58 \beta_{8} - 380 \beta_{6} + \cdots + 2320) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 462 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 462 q^{9} + 180 q^{11} - 216 q^{13} + 92 q^{15} + 678 q^{17} - 2392 q^{21} - 1566 q^{23} - 174 q^{25} - 5710 q^{31} - 936 q^{35} + 4878 q^{41} + 1108 q^{43} + 5526 q^{47} - 8544 q^{49} + 1212 q^{53} - 7692 q^{57} - 14016 q^{59} + 1088 q^{67} - 24302 q^{79} - 23660 q^{81} + 7032 q^{83} - 17850 q^{87} - 606 q^{95} - 5842 q^{97} + 25924 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 142x^{10} + 7173x^{8} + 157368x^{6} + 1510016x^{4} + 5098688x^{2} + 90352 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 1013\nu^{10} + 51724\nu^{8} - 3751835\nu^{6} - 249791190\nu^{4} - 2971827868\nu^{2} + 390161384 ) / 218680368 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 7373 \nu^{11} + 1060068 \nu^{9} + 54814785 \nu^{7} + 1256312182 \nu^{5} + 12979904372 \nu^{3} + 48750171768 \nu ) / 437360736 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 8029 \nu^{10} - 1003616 \nu^{8} - 42059193 \nu^{6} - 669406178 \nu^{4} - 3108074228 \nu^{2} + 2072740888 ) / 218680368 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3645\nu^{10} + 491936\nu^{8} + 22519538\nu^{6} + 398171095\nu^{4} + 2226381792\nu^{2} + 109255312 ) / 54670092 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 15593 \nu^{10} - 2019468 \nu^{8} - 86326317 \nu^{6} - 1342893190 \nu^{4} - 5496338564 \nu^{2} + 9669475032 ) / 218680368 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 10113 \nu^{11} - 1379868 \nu^{9} - 65318629 \nu^{7} - 1285172518 \nu^{5} + \cdots - 28620915064 \nu ) / 145786912 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 28089 \nu^{10} - 3610960 \nu^{8} - 153145033 \nu^{6} - 2392704686 \nu^{4} - 11543951796 \nu^{2} - 967119080 ) / 218680368 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 36865 \nu^{11} - 5300340 \nu^{9} - 274073925 \nu^{7} - 6281560910 \nu^{5} + \cdots - 208761999960 \nu ) / 437360736 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 43059 \nu^{11} - 6156292 \nu^{9} - 313089127 \nu^{7} - 6867724298 \nu^{5} + \cdots - 193079423624 \nu ) / 437360736 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 130223 \nu^{11} - 18990012 \nu^{9} - 996377731 \nu^{7} - 23117282490 \nu^{5} + \cdots - 898096461064 \nu ) / 437360736 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{6} + \beta_{5} + \beta_{2} - 48 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{9} + 5\beta_{3} - 40\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -7\beta_{8} - 21\beta_{6} - 32\beta_{5} + 3\beta_{4} - 33\beta_{2} + 992 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -3\beta_{11} + 9\beta_{10} - 34\beta_{9} - 5\beta_{7} - 191\beta_{3} + 946\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 491\beta_{8} + 1051\beta_{6} + 1766\beta_{5} - 327\beta_{4} + 1783\beta_{2} - 47912 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 327\beta_{11} - 801\beta_{10} + 1938\beta_{9} + 517\beta_{7} + 12915\beta_{3} - 47576\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -28143\beta_{8} - 55849\beta_{6} - 93232\beta_{5} + 25791\beta_{4} - 93745\beta_{2} + 2454152 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -25791\beta_{11} + 53421\beta_{10} - 106384\beta_{9} - 37293\beta_{7} - 828917\beta_{3} + 2458520\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 1529399\beta_{8} + 3032785\beta_{6} + 4877264\beta_{5} - 1788243\beta_{4} + 4935709\beta_{2} - 128941408 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 1788243 \beta_{11} - 3259197 \beta_{10} + 5800608 \beta_{9} + 2370193 \beta_{7} + 51365613 \beta_{3} - 129062592 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/688\mathbb{Z}\right)^\times\).

\(n\) \(431\) \(433\) \(517\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
257.1
0.133471i
− 7.49282i
2.75662i
6.72223i
− 4.43775i
− 3.65497i
3.65497i
4.43775i
− 6.72223i
− 2.75662i
7.49282i
− 0.133471i
0 − 14.8068i 0 − 26.0324i 0 − 87.9232i 0 −138.241 0
257.2 0 − 14.5182i 0 9.40180i 0 − 53.5326i 0 −129.777 0
257.3 0 − 12.3317i 0 21.9831i 0 63.3272i 0 −71.0696 0
257.4 0 − 8.31879i 0 − 1.48242i 0 − 13.7963i 0 11.7977 0
257.5 0 − 6.93950i 0 − 22.3554i 0 51.9837i 0 32.8434 0
257.6 0 − 4.18965i 0 45.6695i 0 − 34.3337i 0 63.4469 0
257.7 0 4.18965i 0 − 45.6695i 0 34.3337i 0 63.4469 0
257.8 0 6.93950i 0 22.3554i 0 − 51.9837i 0 32.8434 0
257.9 0 8.31879i 0 1.48242i 0 13.7963i 0 11.7977 0
257.10 0 12.3317i 0 − 21.9831i 0 − 63.3272i 0 −71.0696 0
257.11 0 14.5182i 0 − 9.40180i 0 53.5326i 0 −129.777 0
257.12 0 14.8068i 0 26.0324i 0 87.9232i 0 −138.241 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 257.12
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 688.5.b.d 12
4.b odd 2 1 43.5.b.b ✓ 12
12.b even 2 1 387.5.b.c 12
43.b odd 2 1 inner 688.5.b.d 12
172.d even 2 1 43.5.b.b ✓ 12
516.h odd 2 1 387.5.b.c 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
43.5.b.b ✓ 12 4.b odd 2 1
43.5.b.b ✓ 12 172.d even 2 1
387.5.b.c 12 12.b even 2 1
387.5.b.c 12 516.h odd 2 1
688.5.b.d 12 1.a even 1 1 trivial
688.5.b.d 12 43.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{5}^{\mathrm{new}}(688, [\chi])\):

\( T_{3}^{12} + 717T_{3}^{10} + 195527T_{3}^{8} + 25281430T_{3}^{6} + 1583947512T_{3}^{4} + 44423599176T_{3}^{2} + 411073500528 \) Copy content Toggle raw display
\( T_{11}^{6} - 90T_{11}^{5} - 18220T_{11}^{4} + 2312718T_{11}^{3} - 63695785T_{11}^{2} + 475717044T_{11} - 473184052 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} + \cdots + 411073500528 \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 66311345932912 \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 53\!\cdots\!32 \) Copy content Toggle raw display
$11$ \( (T^{6} - 90 T^{5} + \cdots - 473184052)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} + \cdots - 5455125964820)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} + \cdots + 26460282371555)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 52\!\cdots\!48 \) Copy content Toggle raw display
$23$ \( (T^{6} + \cdots - 11\!\cdots\!43)^{2} \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 25\!\cdots\!88 \) Copy content Toggle raw display
$31$ \( (T^{6} + \cdots - 49\!\cdots\!55)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 39\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( (T^{6} + \cdots - 73\!\cdots\!07)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 15\!\cdots\!01 \) Copy content Toggle raw display
$47$ \( (T^{6} + \cdots - 19\!\cdots\!52)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + \cdots - 68\!\cdots\!76)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + \cdots - 33\!\cdots\!80)^{2} \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 32\!\cdots\!28 \) Copy content Toggle raw display
$67$ \( (T^{6} + \cdots - 20\!\cdots\!96)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 25\!\cdots\!32 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 31\!\cdots\!92 \) Copy content Toggle raw display
$79$ \( (T^{6} + \cdots + 19\!\cdots\!68)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} + \cdots - 33\!\cdots\!72)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 42\!\cdots\!72 \) Copy content Toggle raw display
$97$ \( (T^{6} + \cdots - 33\!\cdots\!93)^{2} \) Copy content Toggle raw display
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