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Properties

Label 387.5.b.c
Level $387$
Weight $5$
Character orbit 387.b
Analytic conductor $40.004$
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] = [387,5,Mod(343,387)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(387, 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("387.343");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 387 = 3^{2} \cdot 43 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 387.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: \(40.0041757134\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{8} \)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} - 8) q^{4} + (\beta_{9} - \beta_1) q^{5} + (\beta_{10} - 2 \beta_{4} + 2 \beta_1) q^{7} + (\beta_{6} + 2 \beta_{4} - 8 \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{2} - 8) q^{4} + (\beta_{9} - \beta_1) q^{5} + (\beta_{10} - 2 \beta_{4} + 2 \beta_1) q^{7} + (\beta_{6} + 2 \beta_{4} - 8 \beta_1) q^{8} + (3 \beta_{8} - 3 \beta_{7} - \beta_{2} + 17) q^{10} + (\beta_{8} - \beta_{7} - \beta_{5} + \cdots + 14) q^{11}+ \cdots + (24 \beta_{11} - 12 \beta_{10} + \cdots + 606 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 92 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 92 q^{4} + 182 q^{10} + 180 q^{11} - 216 q^{13} - 732 q^{14} + 1076 q^{16} - 678 q^{17} - 1566 q^{23} - 174 q^{25} + 5710 q^{31} - 936 q^{35} - 1242 q^{38} - 2618 q^{40} - 4878 q^{41} - 1108 q^{43} + 15168 q^{44} + 5526 q^{47} - 8544 q^{49} + 24084 q^{52} - 1212 q^{53} + 10152 q^{56} - 4666 q^{58} - 14016 q^{59} - 15580 q^{64} - 1088 q^{67} - 15186 q^{68} + 7674 q^{74} + 24302 q^{79} + 7032 q^{83} + 14412 q^{86} - 48354 q^{92} - 606 q^{95} - 5842 q^{97}+O(q^{100}) \) 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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 24 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -1013\nu^{10} - 51724\nu^{8} + 3751835\nu^{6} + 249791190\nu^{4} + 3190508236\nu^{2} + 4858167448 ) / 218680368 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 7373 \nu^{11} + 1060068 \nu^{9} + 54814785 \nu^{7} + 1256312182 \nu^{5} + 12979904372 \nu^{3} + 48750171768 \nu ) / 437360736 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 8029 \nu^{10} - 1003616 \nu^{8} - 42059193 \nu^{6} - 669406178 \nu^{4} - 3108074228 \nu^{2} + 2072740888 ) / 218680368 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 7373 \nu^{11} - 1060068 \nu^{9} - 54814785 \nu^{7} - 1256312182 \nu^{5} + \cdots - 40002957048 \nu ) / 218680368 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -3124\nu^{10} - 397873\nu^{8} - 16704679\nu^{6} - 262452874\nu^{4} - 1457233216\nu^{2} - 1347066320 ) / 54670092 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 15593 \nu^{10} + 2019468 \nu^{8} + 86326317 \nu^{6} + 1342893190 \nu^{4} + 5715018932 \nu^{2} - 4421146200 ) / 218680368 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 10113 \nu^{11} - 1379868 \nu^{9} - 65318629 \nu^{7} - 1285172518 \nu^{5} + \cdots - 28475128152 \nu ) / 145786912 \) 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
\(\beta_{11}\)\(=\) \( ( - 130223 \nu^{11} - 18990012 \nu^{9} - 996377731 \nu^{7} - 23117282490 \nu^{5} + \cdots - 897659100328 \nu ) / 437360736 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 24 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{6} + 2\beta_{4} - 40\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -4\beta_{8} - 7\beta_{7} + 3\beta_{5} + \beta_{3} - 54\beta_{2} + 992 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -3\beta_{11} + 9\beta_{10} - 5\beta_{9} - 68\beta_{6} - 157\beta_{4} + 1900\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 224\beta_{8} + 491\beta_{7} - 327\beta_{5} - 17\beta_{3} + 2834\beta_{2} - 47912 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 327\beta_{11} - 801\beta_{10} + 517\beta_{9} + 3876\beta_{6} + 10977\beta_{4} - 95996\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -9240\beta_{8} - 28143\beta_{7} + 25791\beta_{5} + 513\beta_{3} - 149594\beta_{2} + 2454152 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -25791\beta_{11} + 53421\beta_{10} - 37293\beta_{9} - 212768\beta_{6} - 722533\beta_{4} + 4980124\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 315080\beta_{8} + 1529399\beta_{7} - 1788243\beta_{5} - 58445\beta_{3} + 7968494\beta_{2} - 128941408 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 1788243 \beta_{11} - 3259197 \beta_{10} + 2370193 \beta_{9} + 11601216 \beta_{6} + \cdots - 262283620 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/387\mathbb{Z}\right)^\times\).

\(n\) \(46\) \(173\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
343.1
− 7.49282i
− 6.72223i
− 4.43775i
− 3.65497i
− 2.75662i
− 0.133471i
0.133471i
2.75662i
3.65497i
4.43775i
6.72223i
7.49282i
− 7.49282i 0 −40.1424 9.40180i 0 − 53.5326i 180.895i 0 70.4460
343.2 − 6.72223i 0 −29.1884 1.48242i 0 13.7963i 88.6554i 0 9.96518
343.3 − 4.43775i 0 −3.69363 − 22.3554i 0 51.9837i − 54.6126i 0 −99.2077
343.4 − 3.65497i 0 2.64117 45.6695i 0 − 34.3337i − 68.1330i 0 166.921
343.5 − 2.75662i 0 8.40105 − 21.9831i 0 − 63.3272i − 67.2644i 0 −60.5989
343.6 − 0.133471i 0 15.9822 26.0324i 0 87.9232i − 4.26869i 0 3.47456
343.7 0.133471i 0 15.9822 − 26.0324i 0 − 87.9232i 4.26869i 0 3.47456
343.8 2.75662i 0 8.40105 21.9831i 0 63.3272i 67.2644i 0 −60.5989
343.9 3.65497i 0 2.64117 − 45.6695i 0 34.3337i 68.1330i 0 166.921
343.10 4.43775i 0 −3.69363 22.3554i 0 − 51.9837i 54.6126i 0 −99.2077
343.11 6.72223i 0 −29.1884 − 1.48242i 0 − 13.7963i − 88.6554i 0 9.96518
343.12 7.49282i 0 −40.1424 − 9.40180i 0 53.5326i − 180.895i 0 70.4460
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 343.12
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 387.5.b.c 12
3.b odd 2 1 43.5.b.b ✓ 12
12.b even 2 1 688.5.b.d 12
43.b odd 2 1 inner 387.5.b.c 12
129.d even 2 1 43.5.b.b ✓ 12
516.h odd 2 1 688.5.b.d 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
43.5.b.b ✓ 12 3.b odd 2 1
43.5.b.b ✓ 12 129.d even 2 1
387.5.b.c 12 1.a even 1 1 trivial
387.5.b.c 12 43.b odd 2 1 inner
688.5.b.d 12 12.b even 2 1
688.5.b.d 12 516.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{5}^{\mathrm{new}}(387, [\chi])\):

\( T_{2}^{12} + 142T_{2}^{10} + 7173T_{2}^{8} + 157368T_{2}^{6} + 1510016T_{2}^{4} + 5098688T_{2}^{2} + 90352 \) 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

$p$ $F_p(T)$
$2$ \( T^{12} + 142 T^{10} + \cdots + 90352 \) Copy content Toggle raw display
$3$ \( T^{12} \) 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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