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Properties

Label 387.3.b.e
Level $387$
Weight $3$
Character orbit 387.b
Analytic conductor $10.545$
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] = [387,3,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, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("387.343");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 387 = 3^{2} \cdot 43 \)
Weight: \( k \) \(=\) \( 3 \)
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: \(10.5449862307\)
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: \( x^{14} + 46x^{12} + 795x^{10} + 6372x^{8} + 23531x^{6} + 34550x^{4} + 16513x^{2} + 48 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{11} \)
Twist minimal: no (minimal twist has level 129)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{2} - 3) q^{4} + \beta_{11} q^{5} + ( - \beta_{11} - \beta_{8}) q^{7} + ( - \beta_{8} - \beta_{7} + 3 \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_{2} - 3) q^{4} + \beta_{11} q^{5} + ( - \beta_{11} - \beta_{8}) q^{7} + ( - \beta_{8} - \beta_{7} + 3 \beta_1) q^{8} + (\beta_{6} - \beta_{4} + \beta_{3} + \cdots + 1) q^{10}+ \cdots + ( - 3 \beta_{12} + 5 \beta_{11} + \cdots - 11 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 36 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 36 q^{4} + 8 q^{10} + 14 q^{11} - 2 q^{13} + 12 q^{14} + 172 q^{16} + 18 q^{17} - 150 q^{23} - 34 q^{25} + 62 q^{31} + 236 q^{35} + 172 q^{38} - 224 q^{40} + 50 q^{41} + 46 q^{43} - 44 q^{44} - 152 q^{47} - 10 q^{49} + 112 q^{52} - 270 q^{53} - 300 q^{56} - 496 q^{58} + 244 q^{59} - 380 q^{64} + 134 q^{67} - 768 q^{68} + 80 q^{74} - 60 q^{79} + 294 q^{83} - 468 q^{86} + 880 q^{92} - 140 q^{95} + 534 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} + 46x^{12} + 795x^{10} + 6372x^{8} + 23531x^{6} + 34550x^{4} + 16513x^{2} + 48 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 7 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 1181 \nu^{12} - 46417 \nu^{10} - 657764 \nu^{8} - 4138752 \nu^{6} - 12185751 \nu^{4} + \cdots - 7877048 ) / 2095304 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 16\nu^{12} + 732\nu^{10} + 12537\nu^{8} + 97295\nu^{6} + 317149\nu^{4} + 270987\nu^{2} + 3072 ) / 24364 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 477\nu^{12} + 20300\nu^{10} + 307139\nu^{8} + 1886075\nu^{6} + 3268452\nu^{4} - 4217787\nu^{2} - 2673730 ) / 523826 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 2135 \nu^{12} - 87017 \nu^{10} - 1272042 \nu^{8} - 7910902 \nu^{6} - 17675003 \nu^{4} + \cdots + 21566408 ) / 2095304 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 5027 \nu^{13} + 231508 \nu^{11} + 4005589 \nu^{9} + 32107231 \nu^{7} + 117878802 \nu^{5} + \cdots + 75494660 \nu ) / 1047652 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 5027 \nu^{13} - 231508 \nu^{11} - 4005589 \nu^{9} - 32107231 \nu^{7} - 117878802 \nu^{5} + \cdots - 63970488 \nu ) / 1047652 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 3706 \nu^{13} - 166504 \nu^{11} - 2770642 \nu^{9} - 20768803 \nu^{7} - 66848618 \nu^{5} + \cdots - 14038660 \nu ) / 523826 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 8577 \nu^{12} + 384784 \nu^{10} + 6387514 \nu^{8} + 47607366 \nu^{6} + 150079269 \nu^{4} + \cdots + 3011168 ) / 1047652 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 8889 \nu^{13} + 411240 \nu^{11} + 7164948 \nu^{9} + 58144702 \nu^{7} + 219375571 \nu^{5} + \cdots + 166626604 \nu ) / 1047652 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 64\nu^{13} + 2928\nu^{11} + 50148\nu^{9} + 395271\nu^{7} + 1408689\nu^{5} + 1894051\nu^{3} + 785845\nu ) / 6091 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 57377 \nu^{13} - 2635657 \nu^{11} - 45490286 \nu^{9} - 364323414 \nu^{7} - 1347002237 \nu^{5} + \cdots - 965402992 \nu ) / 2095304 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 7 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{8} + \beta_{7} - 11\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{6} + \beta_{5} - 2\beta_{3} - 15\beta_{2} + 82 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 5\beta_{12} + 2\beta_{9} - 14\beta_{8} - 22\beta_{7} + 139\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2\beta_{10} - 42\beta_{6} - 26\beta_{5} - 18\beta_{4} + 42\beta_{3} + 217\beta_{2} - 1065 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -146\beta_{12} + 16\beta_{11} - 62\beta_{9} + 189\beta_{8} + 389\beta_{7} - 1861\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -62\beta_{10} + 700\beta_{6} + 465\beta_{5} + 586\beta_{4} - 732\beta_{3} - 3139\beta_{2} + 14494 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -32\beta_{13} + 3069\beta_{12} - 620\beta_{11} + 1380\beta_{9} - 2644\beta_{8} - 6416\beta_{7} + 25793\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 1380\beta_{10} - 10768\beta_{6} - 7428\beta_{5} - 13168\beta_{4} + 12168\beta_{3} + 45621\beta_{2} - 202991 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 1400 \beta_{13} - 56700 \beta_{12} + 15988 \beta_{11} - 26716 \beta_{9} + 38213 \beta_{8} + \cdots - 365919 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 26716 \beta_{10} + 159898 \beta_{6} + 113757 \beta_{5} + 254248 \beta_{4} - 198874 \beta_{3} + \cdots + 2899058 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 38976 \beta_{13} + 981025 \beta_{12} - 344460 \beta_{11} + 479838 \beta_{9} - 565234 \beta_{8} + \cdots + 5277871 \beta_1 \) 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
3.91274i
3.57254i
3.08959i
2.41852i
1.25449i
0.977698i
0.0540801i
− 0.0540801i
− 0.977698i
− 1.25449i
− 2.41852i
− 3.08959i
− 3.57254i
− 3.91274i
− 3.91274i 0 −11.3095 − 1.41080i 0 4.94044i 28.6003i 0 −5.52011
343.2 − 3.57254i 0 −8.76308 8.00852i 0 − 7.09524i 17.0163i 0 28.6108
343.3 − 3.08959i 0 −5.54557 − 4.25487i 0 5.84811i 4.77519i 0 −13.1458
343.4 − 2.41852i 0 −1.84924 − 0.0463223i 0 5.27660i − 5.20165i 0 −0.112031
343.5 − 1.25449i 0 2.42625 − 5.80040i 0 − 4.94237i − 8.06168i 0 −7.27657
343.6 − 0.977698i 0 3.04411 1.00365i 0 − 12.6083i − 6.88701i 0 0.981270
343.7 − 0.0540801i 0 3.99708 8.55074i 0 − 5.27366i − 0.432483i 0 0.462425
343.8 0.0540801i 0 3.99708 − 8.55074i 0 5.27366i 0.432483i 0 0.462425
343.9 0.977698i 0 3.04411 − 1.00365i 0 12.6083i 6.88701i 0 0.981270
343.10 1.25449i 0 2.42625 5.80040i 0 4.94237i 8.06168i 0 −7.27657
343.11 2.41852i 0 −1.84924 0.0463223i 0 − 5.27660i 5.20165i 0 −0.112031
343.12 3.08959i 0 −5.54557 4.25487i 0 − 5.84811i − 4.77519i 0 −13.1458
343.13 3.57254i 0 −8.76308 − 8.00852i 0 7.09524i − 17.0163i 0 28.6108
343.14 3.91274i 0 −11.3095 1.41080i 0 − 4.94044i − 28.6003i 0 −5.52011
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 343.14
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 387.3.b.e 14
3.b odd 2 1 129.3.b.a ✓ 14
12.b even 2 1 2064.3.e.b 14
43.b odd 2 1 inner 387.3.b.e 14
129.d even 2 1 129.3.b.a ✓ 14
516.h odd 2 1 2064.3.e.b 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
129.3.b.a ✓ 14 3.b odd 2 1
129.3.b.a ✓ 14 129.d even 2 1
387.3.b.e 14 1.a even 1 1 trivial
387.3.b.e 14 43.b odd 2 1 inner
2064.3.e.b 14 12.b even 2 1
2064.3.e.b 14 516.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 \(S_{3}^{\mathrm{new}}(387, [\chi])\):

\( T_{2}^{14} + 46T_{2}^{12} + 795T_{2}^{10} + 6372T_{2}^{8} + 23531T_{2}^{6} + 34550T_{2}^{4} + 16513T_{2}^{2} + 48 \) Copy content Toggle raw display
\( T_{11}^{7} - 7T_{11}^{6} - 304T_{11}^{5} + 1140T_{11}^{4} + 23445T_{11}^{3} - 22319T_{11}^{2} - 193472T_{11} + 261652 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} + 46 T^{12} + \cdots + 48 \) Copy content Toggle raw display
$3$ \( T^{14} \) Copy content Toggle raw display
$5$ \( T^{14} + 192 T^{12} + \cdots + 12288 \) Copy content Toggle raw display
$7$ \( T^{14} + \cdots + 126360521472 \) Copy content Toggle raw display
$11$ \( (T^{7} - 7 T^{6} + \cdots + 261652)^{2} \) Copy content Toggle raw display
$13$ \( (T^{7} + T^{6} + \cdots - 6590252)^{2} \) Copy content Toggle raw display
$17$ \( (T^{7} - 9 T^{6} + \cdots + 205285264)^{2} \) Copy content Toggle raw display
$19$ \( T^{14} + \cdots + 50042657046528 \) Copy content Toggle raw display
$23$ \( (T^{7} + 75 T^{6} + \cdots - 30524384)^{2} \) Copy content Toggle raw display
$29$ \( T^{14} + \cdots + 21\!\cdots\!88 \) Copy content Toggle raw display
$31$ \( (T^{7} - 31 T^{6} + \cdots + 3026004112)^{2} \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots + 40\!\cdots\!12 \) Copy content Toggle raw display
$41$ \( (T^{7} - 25 T^{6} + \cdots - 10435765184)^{2} \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots + 73\!\cdots\!49 \) Copy content Toggle raw display
$47$ \( (T^{7} + 76 T^{6} + \cdots + 124103127024)^{2} \) Copy content Toggle raw display
$53$ \( (T^{7} + 135 T^{6} + \cdots - 16376021504)^{2} \) Copy content Toggle raw display
$59$ \( (T^{7} - 122 T^{6} + \cdots - 597777982496)^{2} \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots + 54\!\cdots\!68 \) Copy content Toggle raw display
$67$ \( (T^{7} - 67 T^{6} + \cdots - 88463223872)^{2} \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots + 12\!\cdots\!52 \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots + 11\!\cdots\!32 \) Copy content Toggle raw display
$79$ \( (T^{7} + \cdots + 12285047883232)^{2} \) Copy content Toggle raw display
$83$ \( (T^{7} + \cdots + 2871616580116)^{2} \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots + 87\!\cdots\!88 \) Copy content Toggle raw display
$97$ \( (T^{7} - 267 T^{6} + \cdots - 282513599612)^{2} \) Copy content Toggle raw display
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