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Properties

Label 1104.3.g.b
Level $1104$
Weight $3$
Character orbit 1104.g
Analytic conductor $30.082$
Analytic rank $0$
Dimension $14$
CM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1104,3,Mod(737,1104)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1104, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1104.737");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1104 = 2^{4} \cdot 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1104.g (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: \(30.0818211854\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} + 40x^{12} + 598x^{10} + 4207x^{8} + 14465x^{6} + 23786x^{4} + 17144x^{2} + 3887 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 69)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{6} q^{3} + \beta_{4} q^{5} - \beta_{5} q^{7} + (\beta_{13} - \beta_{7} + \beta_{3}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{6} q^{3} + \beta_{4} q^{5} - \beta_{5} q^{7} + (\beta_{13} - \beta_{7} + \beta_{3}) q^{9} + (\beta_{12} + \beta_{11} - \beta_{9} + \cdots - 1) q^{11}+ \cdots + ( - 3 \beta_{13} + 3 \beta_{12} + \cdots - 16) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + 4 q^{3} + 4 q^{7} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q + 4 q^{3} + 4 q^{7} - 8 q^{9} + 14 q^{15} - 8 q^{19} - 2 q^{21} + 38 q^{25} - 62 q^{27} + 144 q^{31} + 90 q^{33} + 48 q^{37} + 78 q^{39} + 48 q^{43} - 18 q^{45} - 30 q^{49} - 18 q^{51} - 232 q^{55} + 76 q^{57} - 140 q^{61} + 206 q^{63} - 204 q^{67} - 224 q^{73} + 80 q^{75} + 344 q^{79} - 232 q^{81} + 480 q^{85} - 86 q^{87} + 172 q^{91} + 62 q^{93} - 24 q^{97} - 234 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} + 40x^{12} + 598x^{10} + 4207x^{8} + 14465x^{6} + 23786x^{4} + 17144x^{2} + 3887 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu + 6 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 43\nu^{12} + 1536\nu^{10} + 19201\nu^{8} + 99693\nu^{6} + 189536\nu^{4} + 82695\nu^{2} - 16408 ) / 6840 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -32\nu^{13} - 1839\nu^{11} - 39104\nu^{9} - 384237\nu^{7} - 1758889\nu^{5} - 3225120\nu^{3} - 1623643\nu ) / 88920 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 409\nu^{12} + 15783\nu^{10} + 221923\nu^{8} + 1396044\nu^{6} + 3838193\nu^{4} + 3932805\nu^{2} + 990011 ) / 13680 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 295 \nu^{13} + 117 \nu^{12} - 11280 \nu^{11} + 4524 \nu^{10} - 156715 \nu^{9} + 65169 \nu^{8} + \cdots + 799578 ) / 88920 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -131\nu^{13} - 5097\nu^{11} - 73385\nu^{9} - 489744\nu^{7} - 1548127\nu^{5} - 2150079\nu^{3} - 913741\nu ) / 35568 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 131\nu^{13} + 5097\nu^{11} + 73385\nu^{9} + 489744\nu^{7} + 1548127\nu^{5} + 2185647\nu^{3} + 1269421\nu ) / 35568 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 59 \nu^{13} + 468 \nu^{12} + 2256 \nu^{11} + 18096 \nu^{10} + 31343 \nu^{9} + 256230 \nu^{8} + \cdots + 1513278 ) / 17784 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 295 \nu^{13} - 117 \nu^{12} - 11280 \nu^{11} - 4524 \nu^{10} - 156715 \nu^{9} - 65169 \nu^{8} + \cdots - 829218 ) / 29640 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 2986 \nu^{13} + 1209 \nu^{12} + 115332 \nu^{11} + 45513 \nu^{10} + 1625962 \nu^{9} + 621543 \nu^{8} + \cdots - 159159 ) / 177840 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 1192 \nu^{13} + 1157 \nu^{12} + 45964 \nu^{11} + 45149 \nu^{10} + 646464 \nu^{9} + 646919 \nu^{8} + \cdots + 4800913 ) / 59280 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 3763 \nu^{13} - 1508 \nu^{12} + 145281 \nu^{11} - 57486 \nu^{10} + 2045641 \nu^{9} - 790556 \nu^{8} + \cdots - 2305342 ) / 177840 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{2} + \beta _1 - 12 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{8} + \beta_{7} - 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 2 \beta_{12} - 2 \beta_{11} + 4 \beta_{10} + 4 \beta_{9} - 6 \beta_{6} - 4 \beta_{5} + 2 \beta_{3} + \cdots + 124 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 12 \beta_{13} - 6 \beta_{12} - 10 \beta_{11} + 2 \beta_{10} + 6 \beta_{9} - 40 \beta_{8} - 44 \beta_{7} + \cdots + 12 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 20 \beta_{12} + 20 \beta_{11} - 49 \beta_{10} - 51 \beta_{9} + 76 \beta_{6} + 48 \beta_{5} + \cdots - 753 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 294 \beta_{13} + 154 \beta_{12} + 252 \beta_{11} - 36 \beta_{10} - 154 \beta_{9} + 672 \beta_{8} + \cdots - 288 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 652 \beta_{12} - 652 \beta_{11} + 1798 \beta_{10} + 1934 \beta_{9} - 2808 \beta_{6} - 1768 \beta_{5} + \cdots + 19708 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 2697 \beta_{13} - 1457 \beta_{12} - 2356 \beta_{11} + 258 \beta_{10} + 1457 \beta_{9} - 5319 \beta_{8} + \cdots + 2614 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 10038 \beta_{12} + 10038 \beta_{11} - 29750 \beta_{10} - 32778 \beta_{9} + 46434 \beta_{6} + \cdots - 270104 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 89250 \beta_{13} + 49244 \beta_{12} + 78994 \beta_{11} - 7010 \beta_{10} - 49244 \beta_{9} + \cdots - 86004 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 75674 \beta_{12} - 75674 \beta_{11} + 234702 \beta_{10} + 263056 \beta_{9} - 365376 \beta_{6} + \cdots + 1908451 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 1408212 \beta_{13} - 788432 \beta_{12} - 1257836 \beta_{11} + 94640 \beta_{10} + 788432 \beta_{9} + \cdots + 1352476 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1104\mathbb{Z}\right)^\times\).

\(n\) \(97\) \(277\) \(415\) \(737\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
737.1
− 2.92693i
2.92693i
0.634638i
− 0.634638i
− 1.90763i
1.90763i
3.07980i
− 3.07980i
− 1.29748i
1.29748i
− 3.86310i
3.86310i
1.13976i
− 1.13976i
0 −2.91955 − 0.690107i 0 − 2.77451i 0 1.17249 0 8.04750 + 4.02960i 0
737.2 0 −2.91955 + 0.690107i 0 2.77451i 0 1.17249 0 8.04750 − 4.02960i 0
737.3 0 −1.99717 − 2.23859i 0 − 4.18173i 0 4.15888 0 −1.02261 + 8.94172i 0
737.4 0 −1.99717 + 2.23859i 0 4.18173i 0 4.15888 0 −1.02261 − 8.94172i 0
737.5 0 −0.570904 − 2.94518i 0 9.03892i 0 −1.87935 0 −8.34814 + 3.36283i 0
737.6 0 −0.570904 + 2.94518i 0 − 9.03892i 0 −1.87935 0 −8.34814 − 3.36283i 0
737.7 0 0.479940 − 2.96136i 0 − 0.529218i 0 −5.42850 0 −8.53932 − 2.84255i 0
737.8 0 0.479940 + 2.96136i 0 0.529218i 0 −5.42850 0 −8.53932 + 2.84255i 0
737.9 0 2.20436 − 2.03489i 0 − 5.31444i 0 −12.0394 0 0.718425 − 8.97128i 0
737.10 0 2.20436 + 2.03489i 0 5.31444i 0 −12.0394 0 0.718425 + 8.97128i 0
737.11 0 2.26727 − 1.96456i 0 − 1.45241i 0 6.75781 0 1.28104 − 8.90836i 0
737.12 0 2.26727 + 1.96456i 0 1.45241i 0 6.75781 0 1.28104 + 8.90836i 0
737.13 0 2.53605 − 1.60264i 0 4.29888i 0 9.25811 0 3.86309 − 8.12875i 0
737.14 0 2.53605 + 1.60264i 0 − 4.29888i 0 9.25811 0 3.86309 + 8.12875i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 737.14
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 1104.3.g.b 14
3.b odd 2 1 inner 1104.3.g.b 14
4.b odd 2 1 69.3.b.a ✓ 14
12.b even 2 1 69.3.b.a ✓ 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
69.3.b.a ✓ 14 4.b odd 2 1
69.3.b.a ✓ 14 12.b even 2 1
1104.3.g.b 14 1.a even 1 1 trivial
1104.3.g.b 14 3.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{14} + 156T_{5}^{12} + 8076T_{5}^{10} + 187728T_{5}^{8} + 2067024T_{5}^{6} + 9802560T_{5}^{4} + 14696704T_{5}^{2} + 3391488 \) acting on \(S_{3}^{\mathrm{new}}(1104, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} \) Copy content Toggle raw display
$3$ \( T^{14} - 4 T^{13} + \cdots + 4782969 \) Copy content Toggle raw display
$5$ \( T^{14} + 156 T^{12} + \cdots + 3391488 \) Copy content Toggle raw display
$7$ \( (T^{7} - 2 T^{6} + \cdots + 37472)^{2} \) Copy content Toggle raw display
$11$ \( T^{14} + \cdots + 44251280863232 \) Copy content Toggle raw display
$13$ \( (T^{7} - 263 T^{5} + \cdots - 839808)^{2} \) Copy content Toggle raw display
$17$ \( T^{14} + \cdots + 76\!\cdots\!52 \) Copy content Toggle raw display
$19$ \( (T^{7} + 4 T^{6} + \cdots + 369654688)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 23)^{7} \) Copy content Toggle raw display
$29$ \( T^{14} + \cdots + 19\!\cdots\!92 \) Copy content Toggle raw display
$31$ \( (T^{7} - 72 T^{6} + \cdots + 1909922976)^{2} \) Copy content Toggle raw display
$37$ \( (T^{7} - 24 T^{6} + \cdots + 31114889632)^{2} \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots + 24\!\cdots\!92 \) Copy content Toggle raw display
$43$ \( (T^{7} - 24 T^{6} + \cdots - 8823945600)^{2} \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots + 13\!\cdots\!08 \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots + 85\!\cdots\!48 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 32\!\cdots\!12 \) Copy content Toggle raw display
$61$ \( (T^{7} + \cdots + 1148734281696)^{2} \) Copy content Toggle raw display
$67$ \( (T^{7} + 102 T^{6} + \cdots + 173963774816)^{2} \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots + 41\!\cdots\!68 \) Copy content Toggle raw display
$73$ \( (T^{7} + 112 T^{6} + \cdots + 921220683136)^{2} \) Copy content Toggle raw display
$79$ \( (T^{7} - 172 T^{6} + \cdots - 607693691232)^{2} \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots + 16\!\cdots\!32 \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T^{7} + 12 T^{6} + \cdots + 144273643904)^{2} \) Copy content Toggle raw display
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