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Properties

Label 368.3.d.c
Level $368$
Weight $3$
Character orbit 368.d
Analytic conductor $10.027$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [368,3,Mod(47,368)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(368, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("368.47");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 368 = 2^{4} \cdot 23 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 368.d (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: \(10.0272737285\)
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: \( x^{16} + 106 x^{14} + 4335 x^{12} + 87548 x^{10} + 943407 x^{8} + 5489962 x^{6} + 16555009 x^{4} + \cdots + 10969344 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2^{24} \)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + (\beta_{4} + 1) q^{5} - \beta_{9} q^{7} + ( - \beta_{4} + \beta_{3} - 5) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + (\beta_{4} + 1) q^{5} - \beta_{9} q^{7} + ( - \beta_{4} + \beta_{3} - 5) q^{9} + ( - \beta_{12} - \beta_{7}) q^{11} - \beta_{10} q^{13} + (2 \beta_{7} - \beta_{6} + \cdots + 2 \beta_1) q^{15}+ \cdots + (\beta_{15} + 3 \beta_{14} + \cdots - 3 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 8 q^{5} - 68 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 8 q^{5} - 68 q^{9} + 4 q^{13} + 4 q^{17} - 44 q^{21} + 112 q^{25} + 116 q^{29} - 4 q^{33} + 48 q^{37} + 44 q^{41} - 372 q^{45} - 360 q^{49} + 180 q^{53} + 96 q^{57} + 196 q^{61} - 92 q^{65} + 260 q^{73} + 80 q^{77} + 704 q^{81} - 216 q^{85} - 464 q^{89} - 76 q^{93} - 28 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 106 x^{14} + 4335 x^{12} + 87548 x^{10} + 943407 x^{8} + 5489962 x^{6} + 16555009 x^{4} + \cdots + 10969344 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 134 \nu^{15} + 12134 \nu^{13} + 381963 \nu^{11} + 4701712 \nu^{9} + 13247568 \nu^{7} + \cdots - 2625786720 \nu ) / 377528256 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 8017 \nu^{14} + 861394 \nu^{12} + 35842446 \nu^{10} + 738226208 \nu^{8} + 8055133233 \nu^{6} + \cdots + 115864000704 ) / 1635955776 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 8017 \nu^{14} + 861394 \nu^{12} + 35842446 \nu^{10} + 738226208 \nu^{8} + 8055133233 \nu^{6} + \cdots + 92960619840 ) / 1635955776 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 6224 \nu^{14} + 653787 \nu^{12} + 26245868 \nu^{10} + 508914010 \nu^{8} + 4993404276 \nu^{6} + \cdots + 15767924160 ) / 545318592 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 4697 \nu^{15} + 421628 \nu^{13} + 12753153 \nu^{11} + 128464864 \nu^{9} + \cdots - 68064406368 \nu ) / 1635955776 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 13 \nu^{15} + 1306 \nu^{13} + 49137 \nu^{11} + 867404 \nu^{9} + 7501743 \nu^{7} + \cdots + 11618496 \nu ) / 3199392 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 28799 \nu^{14} + 2829410 \nu^{12} + 102585714 \nu^{10} + 1697023336 \nu^{8} + \cdots - 18407778048 ) / 1635955776 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 31676 \nu^{15} - 3303935 \nu^{13} - 131452536 \nu^{11} - 2528000530 \nu^{9} + \cdots - 173736033984 \nu ) / 4907867328 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 7437 \nu^{14} + 745358 \nu^{12} + 27964270 \nu^{10} + 491917272 \nu^{8} + 4239113085 \nu^{6} + \cdots + 15816473664 ) / 272659296 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 1174 \nu^{14} - 116739 \nu^{12} - 4330520 \nu^{10} - 75016626 \nu^{8} - 636321098 \nu^{6} + \cdots - 2074932672 ) / 41947584 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 50014 \nu^{15} - 4950565 \nu^{13} - 181741236 \nu^{11} - 3071060726 \nu^{9} + \cdots - 72832741344 \nu ) / 4907867328 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 71405 \nu^{14} - 7291688 \nu^{12} - 281168358 \nu^{10} - 5155902172 \nu^{8} + \cdots - 235942362432 ) / 1635955776 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 100849 \nu^{15} - 10316629 \nu^{13} - 398858766 \nu^{11} - 7340644730 \nu^{9} + \cdots - 282614410656 \nu ) / 4907867328 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 114599 \nu^{15} + 11887043 \nu^{13} + 470429346 \nu^{11} + 9027902710 \nu^{9} + \cdots + 960279213312 \nu ) / 4907867328 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{4} + \beta_{3} - 14 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{15} + 2\beta_{12} + \beta_{9} - \beta_{7} + 3\beta_{6} - 3\beta_{2} - 23\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{13} + 6\beta_{11} + 7\beta_{10} + 3\beta_{8} - 2\beta_{5} + 41\beta_{4} - 34\beta_{3} + 349 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -40\beta_{15} - 2\beta_{14} - 84\beta_{12} - 26\beta_{9} + 50\beta_{7} - 126\beta_{6} + 78\beta_{2} + 663\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 112 \beta_{13} - 284 \beta_{11} - 330 \beta_{10} - 164 \beta_{8} + 68 \beta_{5} - 1459 \beta_{4} + \cdots - 10366 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 1391 \beta_{15} + 84 \beta_{14} + 2898 \beta_{12} + 663 \beta_{9} - 2229 \beta_{7} + 4453 \beta_{6} + \cdots - 20627 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 4514 \beta_{13} + 10578 \beta_{11} + 12069 \beta_{10} + 6795 \beta_{8} - 2086 \beta_{5} + 49931 \beta_{4} + \cdots + 326161 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 46618 \beta_{15} - 1862 \beta_{14} - 95876 \beta_{12} - 19740 \beta_{9} + 88506 \beta_{7} + \cdots + 660051 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 160784 \beta_{13} - 368480 \beta_{11} - 407380 \beta_{10} - 256712 \beta_{8} + 67136 \beta_{5} + \cdots - 10485454 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 1544685 \beta_{15} + 6280 \beta_{14} + 3140658 \beta_{12} + 676909 \beta_{9} - 3292273 \beta_{7} + \cdots - 21363375 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 5409714 \beta_{13} + 12539518 \beta_{11} + 13305843 \beta_{10} + 9316171 \beta_{8} - 2266602 \beta_{5} + \cdots + 340114349 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 50999612 \beta_{15} + 1963574 \beta_{14} - 102655364 \beta_{12} - 24928478 \beta_{9} + \cdots + 694955855 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 176947680 \beta_{13} - 422822020 \beta_{11} - 428023438 \beta_{10} - 331299124 \beta_{8} + \cdots - 11077436022 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 1682005243 \beta_{15} - 140594660 \beta_{14} + 3354561250 \beta_{12} + 937079811 \beta_{9} + \cdots - 22662653355 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/368\mathbb{Z}\right)^\times\).

\(n\) \(47\) \(97\) \(277\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
47.1
− 5.76329i
− 5.65527i
− 4.02937i
− 3.21690i
− 2.53261i
− 2.18930i
− 1.42213i
− 0.994205i
0.994205i
1.42213i
2.18930i
2.53261i
3.21690i
4.02937i
5.65527i
5.76329i
0 − 5.76329i 0 7.33520 0 − 11.3358i 0 −24.2155 0
47.2 0 − 5.65527i 0 2.36583 0 9.00850i 0 −22.9821 0
47.3 0 − 4.02937i 0 −5.54141 0 − 12.4504i 0 −7.23584 0
47.4 0 − 3.21690i 0 −0.0537932 0 9.68698i 0 −1.34846 0
47.5 0 − 2.53261i 0 1.96600 0 0.999609i 0 2.58587 0
47.6 0 − 2.18930i 0 −2.15962 0 − 2.09808i 0 4.20696 0
47.7 0 − 1.42213i 0 −8.82613 0 10.3233i 0 6.97753 0
47.8 0 − 0.994205i 0 8.91392 0 − 1.23559i 0 8.01156 0
47.9 0 0.994205i 0 8.91392 0 1.23559i 0 8.01156 0
47.10 0 1.42213i 0 −8.82613 0 − 10.3233i 0 6.97753 0
47.11 0 2.18930i 0 −2.15962 0 2.09808i 0 4.20696 0
47.12 0 2.53261i 0 1.96600 0 − 0.999609i 0 2.58587 0
47.13 0 3.21690i 0 −0.0537932 0 − 9.68698i 0 −1.34846 0
47.14 0 4.02937i 0 −5.54141 0 12.4504i 0 −7.23584 0
47.15 0 5.65527i 0 2.36583 0 − 9.00850i 0 −22.9821 0
47.16 0 5.76329i 0 7.33520 0 11.3358i 0 −24.2155 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 47.16
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 368.3.d.c ✓ 16
4.b odd 2 1 inner 368.3.d.c ✓ 16
8.b even 2 1 1472.3.d.c 16
8.d odd 2 1 1472.3.d.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
368.3.d.c ✓ 16 1.a even 1 1 trivial
368.3.d.c ✓ 16 4.b odd 2 1 inner
1472.3.d.c 16 8.b even 2 1
1472.3.d.c 16 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{16} + 106 T_{3}^{14} + 4335 T_{3}^{12} + 87548 T_{3}^{10} + 943407 T_{3}^{8} + 5489962 T_{3}^{6} + \cdots + 10969344 \) acting on \(S_{3}^{\mathrm{new}}(368, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} + 106 T^{14} + \cdots + 10969344 \) Copy content Toggle raw display
$5$ \( (T^{8} - 4 T^{7} + \cdots + 1728)^{2} \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 108551798784 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 38662529126400 \) Copy content Toggle raw display
$13$ \( (T^{8} - 2 T^{7} + \cdots + 572212)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} - 2 T^{7} + \cdots - 148640832)^{2} \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 13\!\cdots\!56 \) Copy content Toggle raw display
$23$ \( (T^{2} + 23)^{8} \) Copy content Toggle raw display
$29$ \( (T^{8} - 58 T^{7} + \cdots - 33224282964)^{2} \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 69\!\cdots\!04 \) Copy content Toggle raw display
$37$ \( (T^{8} - 24 T^{7} + \cdots - 269788502592)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} - 22 T^{7} + \cdots + 98950661004)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 64\!\cdots\!76 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 381398401771776 \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots - 8206605871872)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 23\!\cdots\!36 \) Copy content Toggle raw display
$61$ \( (T^{8} - 98 T^{7} + \cdots - 89980220160)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 13\!\cdots\!56 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 85\!\cdots\!84 \) Copy content Toggle raw display
$73$ \( (T^{8} - 130 T^{7} + \cdots + 797179218924)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 17\!\cdots\!84 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 10\!\cdots\!44 \) Copy content Toggle raw display
$89$ \( (T^{8} + 232 T^{7} + \cdots - 929942310912)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 10\!\cdots\!92)^{2} \) Copy content Toggle raw display
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