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Properties

Label 387.6.a.e
Level $387$
Weight $6$
Character orbit 387.a
Self dual yes
Analytic conductor $62.069$
Analytic rank $1$
Dimension $10$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [387,6,Mod(1,387)]
 
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, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("387.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 387 = 3^{2} \cdot 43 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 387.a (trivial)

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: yes
Analytic conductor: \(62.0685382676\)
Analytic rank: \(1\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 2 x^{9} - 256 x^{8} + 266 x^{7} + 21986 x^{6} - 10450 x^{5} - 719484 x^{4} + 384582 x^{3} + \cdots - 22734604 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 43)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 1) q^{2} + (\beta_{2} - \beta_1 + 20) q^{4} + ( - \beta_{7} - 14) q^{5} + ( - \beta_{9} - \beta_{4} - \beta_{3} + \cdots + 7) q^{7}+ \cdots + ( - \beta_{9} - 2 \beta_{8} + \cdots - 32) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 1) q^{2} + (\beta_{2} - \beta_1 + 20) q^{4} + ( - \beta_{7} - 14) q^{5} + ( - \beta_{9} - \beta_{4} - \beta_{3} + \cdots + 7) q^{7}+ \cdots + (22 \beta_{9} + 336 \beta_{8} + \cdots - 35611) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 8 q^{2} + 202 q^{4} - 138 q^{5} + 60 q^{7} - 294 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 8 q^{2} + 202 q^{4} - 138 q^{5} + 60 q^{7} - 294 q^{8} - 17 q^{10} - 745 q^{11} + 1917 q^{13} - 1936 q^{14} + 5354 q^{16} - 4017 q^{17} - 2404 q^{19} - 1311 q^{20} - 5836 q^{22} - 1733 q^{23} + 7120 q^{25} + 1484 q^{26} - 15028 q^{28} - 6996 q^{29} - 4899 q^{31} + 7554 q^{32} - 27033 q^{34} - 7084 q^{35} + 1466 q^{37} - 13905 q^{38} - 93211 q^{40} - 10297 q^{41} + 18490 q^{43} + 36140 q^{44} + 17991 q^{46} - 48592 q^{47} + 29458 q^{49} - 983 q^{50} + 14232 q^{52} - 127165 q^{53} + 106672 q^{55} + 7780 q^{56} - 10305 q^{58} - 99372 q^{59} + 17408 q^{61} - 28265 q^{62} + 47202 q^{64} - 54484 q^{65} - 2021 q^{67} - 192151 q^{68} - 33194 q^{70} - 11286 q^{71} + 49892 q^{73} + 125431 q^{74} - 249803 q^{76} - 98144 q^{77} - 91524 q^{79} - 12251 q^{80} - 158909 q^{82} + 105203 q^{83} - 87212 q^{85} - 14792 q^{86} - 461824 q^{88} + 62682 q^{89} - 295304 q^{91} - 183783 q^{92} + 7259 q^{94} + 305340 q^{95} + 108383 q^{97} - 354656 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 2 x^{9} - 256 x^{8} + 266 x^{7} + 21986 x^{6} - 10450 x^{5} - 719484 x^{4} + 384582 x^{3} + \cdots - 22734604 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 51 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 192531 \nu^{9} + 19231001 \nu^{8} - 40404153 \nu^{7} - 4078413669 \nu^{6} + \cdots + 33312422351204 ) / 197066179584 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 24107 \nu^{9} + 594245 \nu^{8} - 7832247 \nu^{7} - 116173337 \nu^{6} + 397671097 \nu^{5} + \cdots + 167478232916 ) / 24633272448 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 144543 \nu^{9} - 820999 \nu^{8} + 35600547 \nu^{7} + 213763899 \nu^{6} + \cdots - 912070606876 ) / 24633272448 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 474917 \nu^{9} + 1144977 \nu^{8} - 116809809 \nu^{7} - 421392509 \nu^{6} + \cdots + 4700999183556 ) / 21896242176 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 6235841 \nu^{9} - 47565 \nu^{8} + 1552893885 \nu^{7} + 1540088777 \nu^{6} + \cdots - 19301122595508 ) / 197066179584 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 1603183 \nu^{9} - 3158223 \nu^{8} + 411977523 \nu^{7} + 1148308531 \nu^{6} + \cdots - 12026382560220 ) / 49266544896 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 8421725 \nu^{9} + 11663207 \nu^{8} + 2044433577 \nu^{7} - 264140251 \nu^{6} + \cdots - 56109167945572 ) / 197066179584 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 51 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{9} - 2\beta_{8} + 3\beta_{7} - \beta_{6} - \beta_{5} + 2\beta_{4} - \beta_{3} + 4\beta_{2} + 84\beta _1 + 58 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 9 \beta_{9} + 2 \beta_{8} + 9 \beta_{7} - 3 \beta_{6} - 7 \beta_{5} - 4 \beta_{4} + 7 \beta_{3} + \cdots + 4411 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 152 \beta_{9} - 286 \beta_{8} + 430 \beta_{7} - 176 \beta_{6} - 226 \beta_{5} + 180 \beta_{4} + \cdots + 13438 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 1722 \beta_{9} - 22 \beta_{8} + 1924 \beta_{7} - 1166 \beta_{6} - 2292 \beta_{5} - 596 \beta_{4} + \cdots + 450581 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 21607 \beta_{9} - 34192 \beta_{8} + 55963 \beta_{7} - 24587 \beta_{6} - 41129 \beta_{5} + \cdots + 2192128 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 261787 \beta_{9} - 53612 \beta_{8} + 330037 \beta_{7} - 229441 \beta_{6} - 456771 \beta_{5} + \cdots + 49716725 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 3049570 \beta_{9} - 3990164 \beta_{8} + 7188442 \beta_{7} - 3369222 \beta_{6} - 6655062 \beta_{5} + \cdots + 318546212 \) Copy content Toggle raw display

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−9.70631
−8.57770
−6.91219
−4.38824
−1.48720
2.86024
3.50018
5.31531
9.86547
11.5305
−10.7063 0 82.6251 72.1865 0 −96.4803 −542.008 0 −772.851
1.2 −9.57770 0 59.7324 −28.1028 0 195.604 −265.613 0 269.160
1.3 −7.91219 0 30.6028 −79.5677 0 −172.354 11.0549 0 629.555
1.4 −5.38824 0 −2.96684 0.456695 0 166.517 188.410 0 −2.46078
1.5 −2.48720 0 −25.8138 −101.308 0 15.7005 143.795 0 251.974
1.6 1.86024 0 −28.5395 42.2365 0 202.971 −112.618 0 78.5700
1.7 2.50018 0 −25.7491 −47.4635 0 67.4603 −144.383 0 −118.667
1.8 4.31531 0 −13.3781 52.5837 0 −174.859 −195.821 0 226.915
1.9 8.86547 0 46.5966 37.8251 0 −124.747 129.406 0 335.337
1.10 10.5305 0 78.8905 −86.8464 0 −19.8137 493.778 0 −914.532
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(43\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 387.6.a.e 10
3.b odd 2 1 43.6.a.b ✓ 10
12.b even 2 1 688.6.a.h 10
15.d odd 2 1 1075.6.a.b 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
43.6.a.b ✓ 10 3.b odd 2 1
387.6.a.e 10 1.a even 1 1 trivial
688.6.a.h 10 12.b even 2 1
1075.6.a.b 10 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{10} + 8 T_{2}^{9} - 229 T_{2}^{8} - 1734 T_{2}^{7} + 16722 T_{2}^{6} + 112716 T_{2}^{5} + \cdots - 20373120 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(387))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + 8 T^{9} + \cdots - 20373120 \) Copy content Toggle raw display
$3$ \( T^{10} \) Copy content Toggle raw display
$5$ \( T^{10} + \cdots - 25\!\cdots\!96 \) Copy content Toggle raw display
$7$ \( T^{10} + \cdots - 50\!\cdots\!40 \) Copy content Toggle raw display
$11$ \( T^{10} + \cdots + 46\!\cdots\!72 \) Copy content Toggle raw display
$13$ \( T^{10} + \cdots + 47\!\cdots\!40 \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots - 49\!\cdots\!66 \) Copy content Toggle raw display
$19$ \( T^{10} + \cdots - 10\!\cdots\!16 \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots - 24\!\cdots\!52 \) Copy content Toggle raw display
$29$ \( T^{10} + \cdots + 29\!\cdots\!36 \) Copy content Toggle raw display
$31$ \( T^{10} + \cdots - 47\!\cdots\!36 \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots - 25\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( T^{10} + \cdots - 22\!\cdots\!50 \) Copy content Toggle raw display
$43$ \( (T - 1849)^{10} \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots - 56\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots - 13\!\cdots\!24 \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots - 14\!\cdots\!68 \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 89\!\cdots\!92 \) Copy content Toggle raw display
$71$ \( T^{10} + \cdots + 15\!\cdots\!32 \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots - 95\!\cdots\!32 \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots - 15\!\cdots\!20 \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 13\!\cdots\!40 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 29\!\cdots\!88 \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots - 18\!\cdots\!58 \) Copy content Toggle raw display
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