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Properties

Label 2304.4.c.m
Level $2304$
Weight $4$
Character orbit 2304.c
Analytic conductor $135.940$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2304,4,Mod(2303,2304)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2304, 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("2304.2303");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2304 = 2^{8} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2304.c (of order \(2\), degree \(1\), not 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: \(135.940400653\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: 16.0.196571825135013064605696.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 49x^{12} + 2145x^{8} - 12544x^{4} + 65536 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{48}\cdot 3^{12} \)
Twist minimal: no (minimal twist has level 576)
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.

\(f(q)\) \(=\) \( q + \beta_{8} q^{5} + ( - \beta_{13} - \beta_{11}) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{8} q^{5} + ( - \beta_{13} - \beta_{11}) q^{7} - \beta_{15} q^{11} + ( - \beta_{12} + 4 \beta_{6}) q^{13} + ( - \beta_{7} - 4 \beta_{4}) q^{17} - 5 \beta_1 q^{19} + \beta_{10} q^{23} + (\beta_{2} - 1) q^{25} + ( - 5 \beta_{9} + 4 \beta_{8}) q^{29} + ( - \beta_{13} - 4 \beta_{11}) q^{31} + ( - 5 \beta_{15} - 9 \beta_{14}) q^{35} + ( - 4 \beta_{12} - 17 \beta_{6}) q^{37} + ( - 5 \beta_{7} + 2 \beta_{4}) q^{41} + (\beta_{5} + 42 \beta_1) q^{43} + ( - \beta_{10} - 2 \beta_{3}) q^{47} + (4 \beta_{2} - 333) q^{49} + (17 \beta_{9} + 30 \beta_{8}) q^{53} + (10 \beta_{13} + 32 \beta_{11}) q^{55} + ( - 6 \beta_{15} + 19 \beta_{14}) q^{59} + 61 \beta_{6} q^{61} + ( - 7 \beta_{7} - 47 \beta_{4}) q^{65} + (5 \beta_{5} - 25 \beta_1) q^{67} + (2 \beta_{10} + 13 \beta_{3}) q^{71} + (2 \beta_{2} + 332) q^{73} + ( - 28 \beta_{9} - 72 \beta_{8}) q^{77} + (9 \beta_{13} + 4 \beta_{11}) q^{79} + (5 \beta_{15} - 31 \beta_{14}) q^{83} + (19 \beta_{12} - 135 \beta_{6}) q^{85} + (2 \beta_{7} + 49 \beta_{4}) q^{89} + (6 \beta_{5} + 185 \beta_1) q^{91} + ( - 5 \beta_{10} + 5 \beta_{3}) q^{95} + ( - 7 \beta_{2} + 8) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 16 q^{25} - 5328 q^{49} + 5312 q^{73} + 128 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 49x^{12} + 2145x^{8} - 12544x^{4} + 65536 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -49\nu^{12} + 2145\nu^{8} - 105105\nu^{4} + 340096 ) / 68640 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -16\nu^{12} - 640136 ) / 6435 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -112\nu^{15} - 33\nu^{13} - 32175\nu^{9} - 70785\nu^{5} + 615568\nu^{3} + 413952\nu ) / 3294720 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -16\nu^{15} - 233\nu^{13} + 12441\nu^{9} - 499785\nu^{5} - 1041680\nu^{3} + 2922752\nu ) / 439296 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{12} + 49\nu^{8} - 1889\nu^{4} + 6272 ) / 48 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 81\nu^{14} - 4225\nu^{10} + 190385\nu^{6} - 2097664\nu^{2} ) / 599040 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -752\nu^{15} - 17607\nu^{13} + 804375\nu^{9} - 37767015\nu^{5} - 80588272\nu^{3} + 220862208\nu ) / 6589440 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 16381 \nu^{15} - 22824 \nu^{13} + 760045 \nu^{11} + 909480 \nu^{9} - 33032285 \nu^{7} + \cdots - 179349504 \nu ) / 52715520 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 3307 \nu^{15} - 6168 \nu^{13} + 158587 \nu^{11} + 216216 \nu^{9} - 6416267 \nu^{7} + \cdots - 33331200 \nu ) / 10543104 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -848\nu^{15} + 14013\nu^{13} - 714285\nu^{9} + 30057885\nu^{5} - 63116368\nu^{3} - 175779072\nu ) / 3294720 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -49\nu^{14} + 2145\nu^{10} - 96657\nu^{6} + 65536\nu^{2} ) / 76032 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( -17\nu^{14} + 1089\nu^{10} - 44913\nu^{6} + 492032\nu^{2} ) / 22528 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( -30919\nu^{14} + 1353495\nu^{10} - 53691495\nu^{6} + 41353216\nu^{2} ) / 39536640 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 211 \nu^{15} - 344 \nu^{13} - 9955 \nu^{11} + 12760 \nu^{9} + 417395 \nu^{7} - 512600 \nu^{5} + \cdots - 2217984 \nu ) / 337920 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 24533 \nu^{15} - 32232 \nu^{13} - 1131845 \nu^{11} + 1321320 \nu^{9} + 49786165 \nu^{7} + \cdots - 272197632 \nu ) / 26357760 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -6\beta_{15} + 3\beta_{14} - \beta_{10} + 6\beta_{9} - 18\beta_{8} + 2\beta_{7} - 2\beta_{4} + 3\beta_{3} ) / 192 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 8\beta_{13} - 8\beta_{12} - 23\beta_{11} - 108\beta_{6} ) / 192 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -7\beta_{10} - 30\beta_{7} + 94\beta_{4} + 53\beta_{3} ) / 288 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 3\beta_{5} + 3\beta_{2} - 98\beta _1 + 392 ) / 32 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 174 \beta_{15} + 231 \beta_{14} + 5 \beta_{10} + 318 \beta_{9} - 378 \beta_{8} + \cdots - 111 \beta_{3} ) / 192 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 520\beta_{13} - 631\beta_{11} ) / 96 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 3258 \beta_{15} - 4749 \beta_{14} - 23 \beta_{10} + 6378 \beta_{9} - 6654 \beta_{8} + \cdots + 2149 \beta_{3} ) / 576 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 147\beta_{5} - 147\beta_{2} - 3778\beta _1 - 15112 ) / 32 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( -11\beta_{10} - 2330\beta_{7} + 11738\beta_{4} - 4671\beta_{3} ) / 96 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 23432\beta_{13} + 23432\beta_{12} - 25031\beta_{11} + 146988\beta_{6} ) / 192 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 136602 \beta_{15} - 208893 \beta_{14} - 665 \beta_{10} + 277194 \beta_{9} - 269214 \beta_{8} + \cdots - 91733 \beta_{3} ) / 576 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( -6435\beta_{2} - 640136 ) / 16 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 297966 \beta_{15} - 457863 \beta_{14} - 1819 \beta_{10} - 606846 \beta_{9} + 585018 \beta_{8} + \cdots - 200463 \beta_{3} ) / 192 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( -1015048\beta_{13} + 1015048\beta_{12} + 1064983\beta_{11} + 6290028\beta_{6} ) / 192 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( -38473\beta_{10} + 1953150\beta_{7} - 10073534\beta_{4} - 3944773\beta_{3} ) / 288 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2304\mathbb{Z}\right)^\times\).

\(n\) \(1279\) \(1793\) \(2053\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2303.1
−1.50834 + 0.404160i
−0.404160 + 1.50834i
1.50834 + 0.404160i
0.404160 + 1.50834i
0.662979 + 2.47427i
2.47427 + 0.662979i
−0.662979 + 2.47427i
−2.47427 + 0.662979i
−0.662979 − 2.47427i
−2.47427 − 0.662979i
0.662979 − 2.47427i
2.47427 − 0.662979i
1.50834 − 0.404160i
0.404160 − 1.50834i
−1.50834 − 0.404160i
−0.404160 − 1.50834i
0 0 0 − 14.9985i 0 − 32.7386i 0 0 0
2303.2 0 0 0 − 14.9985i 0 − 32.7386i 0 0 0
2303.3 0 0 0 − 14.9985i 0 32.7386i 0 0 0
2303.4 0 0 0 − 14.9985i 0 32.7386i 0 0 0
2303.5 0 0 0 − 5.20053i 0 − 16.7386i 0 0 0
2303.6 0 0 0 − 5.20053i 0 − 16.7386i 0 0 0
2303.7 0 0 0 − 5.20053i 0 16.7386i 0 0 0
2303.8 0 0 0 − 5.20053i 0 16.7386i 0 0 0
2303.9 0 0 0 5.20053i 0 − 16.7386i 0 0 0
2303.10 0 0 0 5.20053i 0 − 16.7386i 0 0 0
2303.11 0 0 0 5.20053i 0 16.7386i 0 0 0
2303.12 0 0 0 5.20053i 0 16.7386i 0 0 0
2303.13 0 0 0 14.9985i 0 − 32.7386i 0 0 0
2303.14 0 0 0 14.9985i 0 − 32.7386i 0 0 0
2303.15 0 0 0 14.9985i 0 32.7386i 0 0 0
2303.16 0 0 0 14.9985i 0 32.7386i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2303.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner
12.b even 2 1 inner
24.f even 2 1 inner
24.h odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2304.4.c.m 16
3.b odd 2 1 inner 2304.4.c.m 16
4.b odd 2 1 inner 2304.4.c.m 16
8.b even 2 1 inner 2304.4.c.m 16
8.d odd 2 1 inner 2304.4.c.m 16
12.b even 2 1 inner 2304.4.c.m 16
16.e even 4 2 576.4.f.b ✓ 16
16.f odd 4 2 576.4.f.b ✓ 16
24.f even 2 1 inner 2304.4.c.m 16
24.h odd 2 1 inner 2304.4.c.m 16
48.i odd 4 2 576.4.f.b ✓ 16
48.k even 4 2 576.4.f.b ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
576.4.f.b ✓ 16 16.e even 4 2
576.4.f.b ✓ 16 16.f odd 4 2
576.4.f.b ✓ 16 48.i odd 4 2
576.4.f.b ✓ 16 48.k even 4 2
2304.4.c.m 16 1.a even 1 1 trivial
2304.4.c.m 16 3.b odd 2 1 inner
2304.4.c.m 16 4.b odd 2 1 inner
2304.4.c.m 16 8.b even 2 1 inner
2304.4.c.m 16 8.d odd 2 1 inner
2304.4.c.m 16 12.b even 2 1 inner
2304.4.c.m 16 24.f even 2 1 inner
2304.4.c.m 16 24.h odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(2304, [\chi])\):

\( T_{5}^{4} + 252T_{5}^{2} + 6084 \) Copy content Toggle raw display
\( T_{11}^{4} - 3312T_{11}^{2} + 2585664 \) Copy content Toggle raw display
\( T_{13}^{4} - 5208T_{13}^{2} + 1140624 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{4} + 252 T^{2} + 6084)^{4} \) Copy content Toggle raw display
$7$ \( (T^{4} + 1352 T^{2} + 300304)^{4} \) Copy content Toggle raw display
$11$ \( (T^{4} - 3312 T^{2} + 2585664)^{4} \) Copy content Toggle raw display
$13$ \( (T^{4} - 5208 T^{2} + 1140624)^{4} \) Copy content Toggle raw display
$17$ \( (T^{4} + 14148 T^{2} + 7387524)^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} + 1200)^{8} \) Copy content Toggle raw display
$23$ \( (T^{4} - 11664 T^{2} + 11451456)^{4} \) Copy content Toggle raw display
$29$ \( (T^{4} + 37692 T^{2} + 5391684)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} + 7496 T^{2} + 6370576)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} - 86496 T^{2} + 240374016)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} + 249156 T^{2} + 14453329284)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} + 228096 T^{2} + 3057647616)^{4} \) Copy content Toggle raw display
$47$ \( (T^{4} - 39312 T^{2} + 296115264)^{4} \) Copy content Toggle raw display
$53$ \( (T^{4} + 669276 T^{2} + 90097226244)^{4} \) Copy content Toggle raw display
$59$ \( (T^{4} - 677376 T^{2} + 48922361856)^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} - 178608)^{8} \) Copy content Toggle raw display
$67$ \( (T^{4} + 1528800 T^{2} + 496179360000)^{4} \) Copy content Toggle raw display
$71$ \( (T^{4} - 962064 T^{2} + 34159910976)^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} - 664 T + 71056)^{8} \) Copy content Toggle raw display
$79$ \( (T^{4} + 99272 T^{2} + 2451042064)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} - 1654128 T^{2} + 555716575296)^{4} \) Copy content Toggle raw display
$89$ \( (T^{4} + 796068 T^{2} + 128784805956)^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} - 16 T - 479744)^{8} \) Copy content Toggle raw display
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