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Properties

Label 384.4.c.a
Level $384$
Weight $4$
Character orbit 384.c
Analytic conductor $22.657$
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] = [384,4,Mod(383,384)]
 
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([1, 0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("384.383");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 384.c (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: \(22.6567334422\)
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: \( x^{12} + 39x^{10} + 549x^{8} + 3500x^{6} + 10236x^{4} + 11952x^{2} + 4624 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{30}\cdot 3^{2} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{5} - 1) q^{3} + \beta_1 q^{5} - \beta_{2} q^{7} + ( - \beta_{10} - \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{5} - 1) q^{3} + \beta_1 q^{5} - \beta_{2} q^{7} + ( - \beta_{10} - \beta_{2}) q^{9} + ( - \beta_{10} + \beta_{8} + \beta_1 - 3) q^{11} + (\beta_{8} - 2 \beta_{5} + \cdots + \beta_1) q^{13}+ \cdots + (14 \beta_{11} - 10 \beta_{10} + \cdots + 503) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 10 q^{3}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 10 q^{3} - 36 q^{11} - 84 q^{15} + 136 q^{21} + 120 q^{23} - 300 q^{25} + 266 q^{27} - 116 q^{33} - 432 q^{35} + 528 q^{37} - 620 q^{39} + 440 q^{45} + 1248 q^{47} - 948 q^{49} + 1072 q^{51} - 172 q^{57} - 2508 q^{59} + 624 q^{61} - 2744 q^{63} - 24 q^{69} + 2040 q^{71} - 216 q^{73} + 3894 q^{75} - 1076 q^{81} - 4572 q^{83} + 480 q^{85} - 4156 q^{87} - 112 q^{93} + 5448 q^{95} - 48 q^{97} + 6044 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 39x^{10} + 549x^{8} + 3500x^{6} + 10236x^{4} + 11952x^{2} + 4624 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{11} - 29\nu^{9} - 223\nu^{7} - 82\nu^{5} + 3220\nu^{3} + 5456\nu ) / 216 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -19\nu^{11} - 401\nu^{9} + 449\nu^{7} + 42164\nu^{5} + 171832\nu^{3} + 22472\nu ) / 3672 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{11} - 22\nu^{9} + 12\nu^{7} + 2399\nu^{5} + 11762\nu^{3} + 9468\nu ) / 153 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 6 \nu^{11} - 85 \nu^{10} - 285 \nu^{9} - 3026 \nu^{8} - 4977 \nu^{7} - 36958 \nu^{6} + \cdots - 193936 ) / 918 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 121 \nu^{11} + 136 \nu^{10} + 4277 \nu^{9} + 4556 \nu^{8} + 51367 \nu^{7} + 49300 \nu^{6} + \cdots + 48688 ) / 7344 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 2 \nu^{11} + 85 \nu^{10} + 95 \nu^{9} + 2839 \nu^{8} + 1659 \nu^{7} + 30753 \nu^{6} + 12899 \nu^{5} + \cdots + 55590 ) / 306 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 13 \nu^{11} - 544 \nu^{10} + 337 \nu^{9} - 18428 \nu^{8} + 1323 \nu^{7} - 205156 \nu^{6} + \cdots - 706384 ) / 2448 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 27 \nu^{11} + 34 \nu^{10} + 985 \nu^{9} + 1394 \nu^{8} + 12511 \nu^{7} + 20638 \nu^{6} + 68388 \nu^{5} + \cdots + 242896 ) / 612 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 121 \nu^{11} + 136 \nu^{10} - 4277 \nu^{9} + 4556 \nu^{8} - 51367 \nu^{7} + 49300 \nu^{6} + \cdots + 48688 ) / 2448 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 145 \nu^{11} - 748 \nu^{10} + 5417 \nu^{9} - 25568 \nu^{8} + 71275 \nu^{7} - 289408 \nu^{6} + \cdots - 1073992 ) / 3672 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 121 \nu^{11} + 68 \nu^{10} + 4277 \nu^{9} + 2176 \nu^{8} + 51367 \nu^{7} + 21488 \nu^{6} + \cdots + 4964 ) / 1836 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 4\beta_{11} + 3\beta_{9} + \beta_{8} + 2\beta_{7} + \beta_{6} - 11\beta_{5} - \beta _1 + 1 ) / 48 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{9} + \beta_{8} + \beta_{6} - 5\beta_{5} + 2\beta_{4} + \beta _1 - 103 ) / 16 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 40 \beta_{11} - 12 \beta_{10} - 39 \beta_{9} - 13 \beta_{8} - 14 \beta_{7} - 13 \beta_{6} + \cdots - 13 ) / 48 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 4\beta_{11} + 10\beta_{9} - 9\beta_{8} - 4\beta_{7} - 12\beta_{6} + 36\beta_{5} - 17\beta_{4} - 5\beta _1 + 558 ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 502 \beta_{11} + 294 \beta_{10} + 591 \beta_{9} + 199 \beta_{8} + 104 \beta_{7} + 199 \beta_{6} + \cdots + 199 ) / 48 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 192 \beta_{11} - 12 \beta_{10} - 399 \beta_{9} + 285 \beta_{8} + 192 \beta_{7} + 447 \beta_{6} + \cdots - 14689 ) / 16 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 7186 \beta_{11} - 5454 \beta_{10} - 9318 \beta_{9} - 3160 \beta_{8} - 866 \beta_{7} - 3160 \beta_{6} + \cdots - 3160 ) / 48 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 3576 \beta_{11} + 372 \beta_{10} + 7477 \beta_{9} - 4417 \beta_{8} - 3576 \beta_{7} - 7729 \beta_{6} + \cdots + 212587 ) / 16 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 109840 \beta_{11} + 93540 \beta_{10} + 149223 \beta_{9} + 50845 \beta_{8} + 8150 \beta_{7} + \cdots + 50845 ) / 48 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 30844 \beta_{11} - 4056 \beta_{10} - 66394 \beta_{9} + 34437 \beta_{8} + 30844 \beta_{7} + \cdots - 1618422 ) / 8 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 1731022 \beta_{11} - 1559166 \beta_{10} - 2407227 \beta_{9} - 822547 \beta_{8} - 85928 \beta_{7} + \cdots - 822547 ) / 48 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
383.1
− 0.910871i
0.910871i
3.14286i
− 3.14286i
1.08600i
− 1.08600i
2.36157i
− 2.36157i
4.03251i
− 4.03251i
− 2.29679i
2.29679i
0 −4.80343 − 1.98169i 0 − 11.9846i 0 − 22.6995i 0 19.1458 + 19.0378i 0
383.2 0 −4.80343 + 1.98169i 0 11.9846i 0 22.6995i 0 19.1458 − 19.0378i 0
383.3 0 −4.12202 − 3.16369i 0 − 21.4043i 0 20.9034i 0 6.98212 + 26.0816i 0
383.4 0 −4.12202 + 3.16369i 0 21.4043i 0 − 20.9034i 0 6.98212 − 26.0816i 0
383.5 0 −3.04120 − 4.21320i 0 9.33303i 0 36.3792i 0 −8.50216 + 25.6264i 0
383.6 0 −3.04120 + 4.21320i 0 − 9.33303i 0 − 36.3792i 0 −8.50216 − 25.6264i 0
383.7 0 −0.556921 − 5.16622i 0 10.6077i 0 − 7.90379i 0 −26.3797 + 5.75435i 0
383.8 0 −0.556921 + 5.16622i 0 − 10.6077i 0 7.90379i 0 −26.3797 − 5.75435i 0
383.9 0 2.52186 − 4.54315i 0 − 8.01133i 0 − 12.6015i 0 −14.2805 − 22.9144i 0
383.10 0 2.52186 + 4.54315i 0 8.01133i 0 12.6015i 0 −14.2805 + 22.9144i 0
383.11 0 5.00172 − 1.40813i 0 − 5.86626i 0 − 5.92149i 0 23.0343 − 14.0861i 0
383.12 0 5.00172 + 1.40813i 0 5.86626i 0 5.92149i 0 23.0343 + 14.0861i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 383.12
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 384.4.c.a ✓ 12
3.b odd 2 1 384.4.c.d yes 12
4.b odd 2 1 384.4.c.d yes 12
8.b even 2 1 384.4.c.c yes 12
8.d odd 2 1 384.4.c.b yes 12
12.b even 2 1 inner 384.4.c.a ✓ 12
16.e even 4 1 768.4.f.e 12
16.e even 4 1 768.4.f.h 12
16.f odd 4 1 768.4.f.f 12
16.f odd 4 1 768.4.f.g 12
24.f even 2 1 384.4.c.c yes 12
24.h odd 2 1 384.4.c.b yes 12
48.i odd 4 1 768.4.f.f 12
48.i odd 4 1 768.4.f.g 12
48.k even 4 1 768.4.f.e 12
48.k even 4 1 768.4.f.h 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.4.c.a ✓ 12 1.a even 1 1 trivial
384.4.c.a ✓ 12 12.b even 2 1 inner
384.4.c.b yes 12 8.d odd 2 1
384.4.c.b yes 12 24.h odd 2 1
384.4.c.c yes 12 8.b even 2 1
384.4.c.c yes 12 24.f even 2 1
384.4.c.d yes 12 3.b odd 2 1
384.4.c.d yes 12 4.b odd 2 1
768.4.f.e 12 16.e even 4 1
768.4.f.e 12 48.k even 4 1
768.4.f.f 12 16.f odd 4 1
768.4.f.f 12 48.i odd 4 1
768.4.f.g 12 16.f odd 4 1
768.4.f.g 12 48.i odd 4 1
768.4.f.h 12 16.e even 4 1
768.4.f.h 12 48.k even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(384, [\chi])\):

\( T_{11}^{6} + 18T_{11}^{5} - 4152T_{11}^{4} - 106488T_{11}^{3} + 2349600T_{11}^{2} + 67009824T_{11} + 370690624 \) Copy content Toggle raw display
\( T_{13}^{6} - 7452T_{13}^{4} + 39936T_{13}^{3} + 12415920T_{13}^{2} - 231124992T_{13} + 15061696 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} + \cdots + 387420489 \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 1424528183296 \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 103646082371584 \) Copy content Toggle raw display
$11$ \( (T^{6} + 18 T^{5} + \cdots + 370690624)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} - 7452 T^{4} + \cdots + 15061696)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 50\!\cdots\!44 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 21\!\cdots\!36 \) Copy content Toggle raw display
$23$ \( (T^{6} - 60 T^{5} + \cdots - 261412163584)^{2} \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 19\!\cdots\!44 \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 12\!\cdots\!84 \) Copy content Toggle raw display
$37$ \( (T^{6} - 264 T^{5} + \cdots + 847281526208)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 19\!\cdots\!96 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 34\!\cdots\!76 \) Copy content Toggle raw display
$47$ \( (T^{6} + \cdots - 144142667350016)^{2} \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 17\!\cdots\!24 \) Copy content Toggle raw display
$59$ \( (T^{6} + \cdots - 43\!\cdots\!48)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} + \cdots - 96\!\cdots\!96)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 45\!\cdots\!36 \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots - 379600692064256)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + \cdots - 10\!\cdots\!72)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 43\!\cdots\!76 \) Copy content Toggle raw display
$83$ \( (T^{6} + \cdots + 11\!\cdots\!12)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 18\!\cdots\!36 \) Copy content Toggle raw display
$97$ \( (T^{6} + \cdots + 63\!\cdots\!68)^{2} \) Copy content Toggle raw display
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