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Properties

Label 1110.4.h.b
Level $1110$
Weight $4$
Character orbit 1110.h
Analytic conductor $65.492$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1110,4,Mod(961,1110)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1110, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("1110.961");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1110 = 2 \cdot 3 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1110.h (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: \(65.4921201064\)
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: \( x^{12} - 4 x^{11} + 8 x^{10} + 2504 x^{9} + 121919 x^{8} + 287122 x^{7} + 1011168 x^{6} + \cdots + 983198736 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4} \)
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.

\(f(q)\) \(=\) \( q - 2 \beta_{2} q^{2} - 3 q^{3} - 4 q^{4} - 5 \beta_{2} q^{5} + 6 \beta_{2} q^{6} + (\beta_{6} - 8) q^{7} + 8 \beta_{2} q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 \beta_{2} q^{2} - 3 q^{3} - 4 q^{4} - 5 \beta_{2} q^{5} + 6 \beta_{2} q^{6} + (\beta_{6} - 8) q^{7} + 8 \beta_{2} q^{8} + 9 q^{9} - 10 q^{10} + (\beta_{11} + \beta_{6} + 19) q^{11} + 12 q^{12} + (\beta_{9} + 2 \beta_{5} + \cdots + 8 \beta_{2}) q^{13}+ \cdots + (9 \beta_{11} + 9 \beta_{6} + 171) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 36 q^{3} - 48 q^{4} - 94 q^{7} + 108 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 36 q^{3} - 48 q^{4} - 94 q^{7} + 108 q^{9} - 120 q^{10} + 234 q^{11} + 144 q^{12} + 192 q^{16} + 282 q^{21} - 300 q^{25} + 184 q^{26} - 324 q^{27} + 376 q^{28} + 360 q^{30} - 702 q^{33} - 348 q^{34} - 432 q^{36} - 204 q^{37} - 536 q^{38} + 480 q^{40} - 934 q^{41} - 936 q^{44} + 336 q^{46} + 4 q^{47} - 576 q^{48} - 266 q^{49} + 630 q^{53} + 124 q^{58} + 412 q^{62} - 846 q^{63} - 768 q^{64} + 460 q^{65} - 2092 q^{67} + 940 q^{70} - 368 q^{71} - 944 q^{73} + 1264 q^{74} + 900 q^{75} - 82 q^{77} - 552 q^{78} + 972 q^{81} + 748 q^{83} - 1128 q^{84} - 870 q^{85} - 492 q^{86} - 1080 q^{90} - 1340 q^{95} + 2106 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 4 x^{11} + 8 x^{10} + 2504 x^{9} + 121919 x^{8} + 287122 x^{7} + 1011168 x^{6} + \cdots + 983198736 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 41\!\cdots\!11 \nu^{11} + \cdots + 34\!\cdots\!30 ) / 24\!\cdots\!82 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 14\!\cdots\!87 \nu^{11} + \cdots + 37\!\cdots\!04 ) / 32\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 19\!\cdots\!61 \nu^{11} + \cdots - 51\!\cdots\!36 ) / 32\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 31\!\cdots\!93 \nu^{11} + \cdots - 44\!\cdots\!08 ) / 61\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 22\!\cdots\!02 \nu^{11} + \cdots + 11\!\cdots\!76 ) / 34\!\cdots\!66 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 31\!\cdots\!78 \nu^{11} + \cdots - 35\!\cdots\!46 ) / 45\!\cdots\!18 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 18\!\cdots\!93 \nu^{11} + \cdots - 10\!\cdots\!48 ) / 20\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 97\!\cdots\!81 \nu^{11} + \cdots - 12\!\cdots\!32 ) / 50\!\cdots\!98 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 39\!\cdots\!21 \nu^{11} + \cdots - 46\!\cdots\!96 ) / 20\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 49\!\cdots\!42 \nu^{11} + \cdots + 41\!\cdots\!10 ) / 15\!\cdots\!94 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 49\!\cdots\!74 \nu^{11} + \cdots - 49\!\cdots\!66 ) / 15\!\cdots\!94 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{3} - \beta_{2} - \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 4\beta_{9} + 7\beta_{7} + 7\beta_{5} - 5\beta_{4} - 5\beta_{3} - 139\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 42 \beta_{11} - 51 \beta_{10} + 51 \beta_{9} - 39 \beta_{8} + 42 \beta_{7} - 143 \beta_{6} + \cdots - 1296 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -2755\beta_{11} - 1891\beta_{10} - 2415\beta_{8} - 2537\beta_{6} + 2307\beta _1 - 42384 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 33490 \beta_{11} - 33097 \beta_{10} - 33097 \beta_{9} - 28179 \beta_{8} - 33490 \beta_{7} + \cdots - 659717 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -772959\beta_{9} - 1049268\beta_{7} - 892940\beta_{5} + 1135412\beta_{4} + 1086797\beta_{3} + 15656208\beta_{2} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 18208586 \beta_{11} + 16493462 \beta_{10} - 16493462 \beta_{9} + 15140798 \beta_{8} - 18208586 \beta_{7} + \cdots + 315101839 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 411120255 \beta_{11} + 314781801 \beta_{10} + 346648262 \beta_{8} + 490363124 \beta_{6} + \cdots + 6189392756 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 8768560170 \beta_{11} + 7591979103 \beta_{10} + 7591979103 \beta_{9} + 7295639037 \beta_{8} + \cdots + 144568509194 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 130002459117 \beta_{9} + 165817558353 \beta_{7} + 139293944555 \beta_{5} + \cdots - 2528681849423 \beta_{2} \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 4004458650064 \beta_{11} - 3381573925219 \beta_{10} + 3381573925219 \beta_{9} - 3337639396575 \beta_{8} + \cdots - 64570034000627 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1110\mathbb{Z}\right)^\times\).

\(n\) \(371\) \(631\) \(667\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
961.1
7.15139 − 7.15139i
−11.3954 + 11.3954i
−0.195401 + 0.195401i
14.6402 − 14.6402i
−4.07613 + 4.07613i
−4.12466 + 4.12466i
7.15139 + 7.15139i
−11.3954 − 11.3954i
−0.195401 − 0.195401i
14.6402 + 14.6402i
−4.07613 − 4.07613i
−4.12466 − 4.12466i
− 2.00000i −3.00000 −4.00000 − 5.00000i 6.00000i −31.4390 8.00000i 9.00000 −10.0000
961.2 − 2.00000i −3.00000 −4.00000 − 5.00000i 6.00000i −21.6647 8.00000i 9.00000 −10.0000
961.3 − 2.00000i −3.00000 −4.00000 − 5.00000i 6.00000i −14.7727 8.00000i 9.00000 −10.0000
961.4 − 2.00000i −3.00000 −4.00000 − 5.00000i 6.00000i 1.59738 8.00000i 9.00000 −10.0000
961.5 − 2.00000i −3.00000 −4.00000 − 5.00000i 6.00000i 4.13473 8.00000i 9.00000 −10.0000
961.6 − 2.00000i −3.00000 −4.00000 − 5.00000i 6.00000i 15.1443 8.00000i 9.00000 −10.0000
961.7 2.00000i −3.00000 −4.00000 5.00000i − 6.00000i −31.4390 − 8.00000i 9.00000 −10.0000
961.8 2.00000i −3.00000 −4.00000 5.00000i − 6.00000i −21.6647 − 8.00000i 9.00000 −10.0000
961.9 2.00000i −3.00000 −4.00000 5.00000i − 6.00000i −14.7727 − 8.00000i 9.00000 −10.0000
961.10 2.00000i −3.00000 −4.00000 5.00000i − 6.00000i 1.59738 − 8.00000i 9.00000 −10.0000
961.11 2.00000i −3.00000 −4.00000 5.00000i − 6.00000i 4.13473 − 8.00000i 9.00000 −10.0000
961.12 2.00000i −3.00000 −4.00000 5.00000i − 6.00000i 15.1443 − 8.00000i 9.00000 −10.0000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 961.12
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1110.4.h.b ✓ 12
37.b even 2 1 inner 1110.4.h.b ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1110.4.h.b ✓ 12 1.a even 1 1 trivial
1110.4.h.b ✓ 12 37.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{6} + 47T_{7}^{5} + 142T_{7}^{4} - 14294T_{7}^{3} - 79939T_{7}^{2} + 793323T_{7} - 1006432 \) acting on \(S_{4}^{\mathrm{new}}(1110, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 4)^{6} \) Copy content Toggle raw display
$3$ \( (T + 3)^{12} \) Copy content Toggle raw display
$5$ \( (T^{2} + 25)^{6} \) Copy content Toggle raw display
$7$ \( (T^{6} + 47 T^{5} + \cdots - 1006432)^{2} \) Copy content Toggle raw display
$11$ \( (T^{6} - 117 T^{5} + \cdots - 10035276)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 671048715633664 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 45\!\cdots\!36 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 74\!\cdots\!76 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 14\!\cdots\!96 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 51\!\cdots\!56 \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 11\!\cdots\!96 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 16\!\cdots\!29 \) Copy content Toggle raw display
$41$ \( (T^{6} + 467 T^{5} + \cdots + 191939136214)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 10\!\cdots\!64 \) Copy content Toggle raw display
$47$ \( (T^{6} + \cdots + 1440534556800)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + \cdots + 90764132897952)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 17\!\cdots\!36 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 18\!\cdots\!64 \) Copy content Toggle raw display
$67$ \( (T^{6} + \cdots + 44557878795656)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + 184 T^{5} + \cdots - 697189596408)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + \cdots - 39\!\cdots\!76)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 22\!\cdots\!84 \) Copy content Toggle raw display
$83$ \( (T^{6} + \cdots - 27\!\cdots\!08)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 14\!\cdots\!04 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 67\!\cdots\!04 \) Copy content Toggle raw display
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