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Properties

Label 384.5.e.b
Level $384$
Weight $5$
Character orbit 384.e
Analytic conductor $39.694$
Analytic rank $0$
Dimension $16$
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] = [384,5,Mod(257,384)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(384, 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("384.257");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 384.e (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: \(39.6940658242\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 4 x^{15} - 32 x^{14} + 356 x^{13} + 1348 x^{12} - 8992 x^{11} + 22064 x^{10} + \cdots + 21479188203 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{54}\cdot 3^{10} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} - \beta_{3} q^{5} + (\beta_{2} - \beta_1 + 5) q^{7} + (\beta_{5} + \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} - \beta_{3} q^{5} + (\beta_{2} - \beta_1 + 5) q^{7} + (\beta_{5} + \beta_1) q^{9} - \beta_{9} q^{11} + ( - \beta_{7} - \beta_{2} + \beta_1) q^{13} + (\beta_{15} + \beta_{7} + \beta_{3} + \cdots + 26) q^{15}+ \cdots + ( - \beta_{15} + 5 \beta_{14} + \cdots - 1769) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 8 q^{3} + 80 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 8 q^{3} + 80 q^{7} + 416 q^{15} + 816 q^{19} + 608 q^{21} - 2000 q^{25} + 280 q^{27} + 592 q^{31} - 496 q^{33} + 2240 q^{37} - 16 q^{39} + 368 q^{43} + 800 q^{45} + 3984 q^{49} - 352 q^{51} + 1920 q^{55} + 560 q^{57} - 3520 q^{61} - 816 q^{63} - 3536 q^{67} - 10784 q^{69} + 3680 q^{73} + 5112 q^{75} - 14448 q^{79} - 624 q^{81} - 11136 q^{85} - 14944 q^{87} - 22944 q^{91} + 13760 q^{93} + 3264 q^{97} - 26976 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 4 x^{15} - 32 x^{14} + 356 x^{13} + 1348 x^{12} - 8992 x^{11} + 22064 x^{10} + \cdots + 21479188203 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 22\!\cdots\!28 \nu^{15} + \cdots - 87\!\cdots\!42 ) / 65\!\cdots\!05 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 35\!\cdots\!08 \nu^{15} + \cdots - 10\!\cdots\!37 ) / 65\!\cdots\!05 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 68\!\cdots\!44 \nu^{15} + \cdots - 94\!\cdots\!56 ) / 21\!\cdots\!35 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 26\!\cdots\!28 \nu^{15} + \cdots + 25\!\cdots\!53 ) / 72\!\cdots\!45 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 95\!\cdots\!12 \nu^{15} + \cdots + 31\!\cdots\!83 ) / 21\!\cdots\!35 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 30\!\cdots\!56 \nu^{15} + \cdots - 54\!\cdots\!16 ) / 65\!\cdots\!05 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 75\!\cdots\!04 \nu^{15} + \cdots - 35\!\cdots\!53 ) / 14\!\cdots\!67 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 37\!\cdots\!28 \nu^{15} + \cdots - 68\!\cdots\!67 ) / 65\!\cdots\!05 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 43\!\cdots\!52 \nu^{15} + \cdots + 16\!\cdots\!58 ) / 65\!\cdots\!05 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 46\!\cdots\!68 \nu^{15} + \cdots - 44\!\cdots\!32 ) / 67\!\cdots\!65 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 16\!\cdots\!48 \nu^{15} + \cdots + 30\!\cdots\!03 ) / 21\!\cdots\!35 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 22\!\cdots\!72 \nu^{15} + \cdots - 32\!\cdots\!08 ) / 21\!\cdots\!35 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 69\!\cdots\!28 \nu^{15} + \cdots - 93\!\cdots\!92 ) / 34\!\cdots\!95 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 13\!\cdots\!96 \nu^{15} + \cdots + 73\!\cdots\!94 ) / 65\!\cdots\!05 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 20\!\cdots\!12 \nu^{15} + \cdots + 40\!\cdots\!38 ) / 72\!\cdots\!45 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 4 \beta_{15} - 2 \beta_{14} - 4 \beta_{13} + 2 \beta_{12} + 2 \beta_{11} + 3 \beta_{10} - 5 \beta_{9} + \cdots + 163 ) / 576 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 13 \beta_{15} + 4 \beta_{14} - 10 \beta_{13} - \beta_{12} + 17 \beta_{11} + 75 \beta_{10} + \cdots + 1427 ) / 288 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 633 \beta_{15} - 9 \beta_{14} + 108 \beta_{13} - 249 \beta_{12} + 321 \beta_{11} + 1317 \beta_{10} + \cdots - 43922 ) / 1152 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 68 \beta_{15} + 194 \beta_{14} - 314 \beta_{13} - 116 \beta_{12} + 220 \beta_{11} + 3123 \beta_{10} + \cdots - 115958 ) / 288 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( \beta_{15} + 12247 \beta_{14} - 1498 \beta_{13} - 9037 \beta_{12} + 3317 \beta_{11} + 12429 \beta_{10} + \cdots - 2467896 ) / 1152 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 22141 \beta_{15} + 13751 \beta_{14} - 7790 \beta_{13} - 6947 \beta_{12} + 1475 \beta_{11} + \cdots - 3965456 ) / 192 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 463685 \beta_{15} + 312775 \beta_{14} + 165605 \beta_{13} + 46829 \beta_{12} - 51775 \beta_{11} + \cdots - 50172173 ) / 576 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 461717 \beta_{15} + 108514 \beta_{14} + 203780 \beta_{13} - 6865 \beta_{12} - 196765 \beta_{11} + \cdots - 43506712 ) / 72 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 42905517 \beta_{15} - 3543777 \beta_{14} + 35369424 \beta_{13} + 19427673 \beta_{12} + \cdots + 1531224196 ) / 1152 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 69804791 \beta_{15} - 92501603 \beta_{14} + 96246506 \beta_{13} + 34876091 \beta_{12} + \cdots + 10487785712 ) / 576 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 33516937 \beta_{15} - 1873237115 \beta_{14} + 458804858 \beta_{13} + 844244783 \beta_{12} + \cdots + 402080090394 ) / 1152 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 517056830 \beta_{15} - 640445545 \beta_{14} - 57810902 \beta_{13} + 117356734 \beta_{12} + \cdots + 118581350800 ) / 48 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 124158997358 \beta_{15} - 87249591688 \beta_{14} - 64217369096 \beta_{13} + 12515078182 \beta_{12} + \cdots + 20568054435284 ) / 1152 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 285682510210 \beta_{15} - 76313039837 \beta_{14} - 130256422525 \beta_{13} - 10557779896 \beta_{12} + \cdots + 24292276985813 ) / 288 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 6610073552139 \beta_{15} + 780869951325 \beta_{14} - 4518467449902 \beta_{13} + \cdots + 126934006033042 ) / 1152 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/384\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(133\) \(257\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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
−5.66644 − 2.02063i
−5.66644 + 2.02063i
−3.05642 − 3.40211i
−3.05642 + 3.40211i
6.34189 − 3.61720i
6.34189 + 3.61720i
2.27178 − 3.46189i
2.27178 + 3.46189i
−3.40000 + 2.59434i
−3.40000 − 2.59434i
4.43019 + 3.93201i
4.43019 − 3.93201i
2.18273 + 4.51404i
2.18273 − 4.51404i
−1.10373 − 0.840249i
−1.10373 + 0.840249i
0 −8.99926 − 0.115244i 0 25.0292i 0 −59.0069 0 80.9734 + 2.07423i 0
257.2 0 −8.99926 + 0.115244i 0 − 25.0292i 0 −59.0069 0 80.9734 − 2.07423i 0
257.3 0 −7.95805 − 4.20350i 0 − 19.3338i 0 67.6413 0 45.6611 + 66.9034i 0
257.4 0 −7.95805 + 4.20350i 0 19.3338i 0 67.6413 0 45.6611 − 66.9034i 0
257.5 0 −4.08506 − 8.01949i 0 47.0883i 0 −13.4570 0 −47.6246 + 65.5202i 0
257.6 0 −4.08506 + 8.01949i 0 − 47.0883i 0 −13.4570 0 −47.6246 − 65.5202i 0
257.7 0 −3.32132 − 8.36474i 0 − 25.4639i 0 36.1251 0 −58.9376 + 55.5640i 0
257.8 0 −3.32132 + 8.36474i 0 25.4639i 0 36.1251 0 −58.9376 − 55.5640i 0
257.9 0 −0.622561 − 8.97844i 0 0.562050i 0 −55.8203 0 −80.2248 + 11.1793i 0
257.10 0 −0.622561 + 8.97844i 0 − 0.562050i 0 −55.8203 0 −80.2248 − 11.1793i 0
257.11 0 5.70733 − 6.95891i 0 34.7105i 0 89.5593 0 −15.8529 − 79.4335i 0
257.12 0 5.70733 + 6.95891i 0 − 34.7105i 0 89.5593 0 −15.8529 + 79.4335i 0
257.13 0 6.57180 − 6.14911i 0 − 9.58700i 0 −22.6799 0 5.37699 − 80.8213i 0
257.14 0 6.57180 + 6.14911i 0 9.58700i 0 −22.6799 0 5.37699 + 80.8213i 0
257.15 0 8.70713 − 2.27724i 0 − 28.9306i 0 −2.36161 0 70.6284 − 39.6564i 0
257.16 0 8.70713 + 2.27724i 0 28.9306i 0 −2.36161 0 70.6284 + 39.6564i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 257.16
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 384.5.e.b yes 16
3.b odd 2 1 inner 384.5.e.b yes 16
4.b odd 2 1 384.5.e.c yes 16
8.b even 2 1 384.5.e.d yes 16
8.d odd 2 1 384.5.e.a ✓ 16
12.b even 2 1 384.5.e.c yes 16
24.f even 2 1 384.5.e.a ✓ 16
24.h odd 2 1 384.5.e.d yes 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.5.e.a ✓ 16 8.d odd 2 1
384.5.e.a ✓ 16 24.f even 2 1
384.5.e.b yes 16 1.a even 1 1 trivial
384.5.e.b yes 16 3.b odd 2 1 inner
384.5.e.c yes 16 4.b odd 2 1
384.5.e.c yes 16 12.b even 2 1
384.5.e.d yes 16 8.b even 2 1
384.5.e.d yes 16 24.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 \(S_{5}^{\mathrm{new}}(384, [\chi])\):

\( T_{7}^{8} - 40 T_{7}^{7} - 9800 T_{7}^{6} + 186176 T_{7}^{5} + 29617840 T_{7}^{4} - 12942592 T_{7}^{3} + \cdots - 519546316544 \) Copy content Toggle raw display
\( T_{13}^{8} - 107088 T_{13}^{6} - 4006400 T_{13}^{5} + 3129196896 T_{13}^{4} + 218176770048 T_{13}^{3} + \cdots + 115283511800064 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} + \cdots + 18\!\cdots\!41 \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 98\!\cdots\!04 \) Copy content Toggle raw display
$7$ \( (T^{8} - 40 T^{7} + \cdots - 519546316544)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 83\!\cdots\!56 \) Copy content Toggle raw display
$13$ \( (T^{8} + \cdots + 115283511800064)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 67\!\cdots\!56 \) Copy content Toggle raw display
$19$ \( (T^{8} + \cdots - 30\!\cdots\!48)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 34\!\cdots\!44 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 37\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{8} + \cdots - 45\!\cdots\!64)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + \cdots - 37\!\cdots\!20)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 71\!\cdots\!24 \) Copy content Toggle raw display
$43$ \( (T^{8} + \cdots + 12\!\cdots\!48)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 14\!\cdots\!84 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 20\!\cdots\!24 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 23\!\cdots\!04 \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots - 33\!\cdots\!40)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots - 24\!\cdots\!80)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 79\!\cdots\!84 \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 11\!\cdots\!56)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots - 56\!\cdots\!72)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 97\!\cdots\!44 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 14\!\cdots\!24 \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots - 33\!\cdots\!84)^{2} \) Copy content Toggle raw display
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