Properties

Label 1110.2.e.d
Level $1110$
Weight $2$
Character orbit 1110.e
Analytic conductor $8.863$
Analytic rank $0$
Dimension $16$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1110,2,Mod(739,1110)]
 
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, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1110.739");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1110 = 2 \cdot 3 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1110.e (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: \(8.86339462436\)
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: \( x^{16} - 2 x^{15} + 2 x^{14} + 8 x^{13} + 138 x^{12} - 220 x^{11} + 196 x^{10} + 744 x^{9} + 4241 x^{8} + \cdots + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{6} \)
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.

\(f(q)\) \(=\) \( q + q^{2} + \beta_{7} q^{3} + q^{4} + \beta_{11} q^{5} + \beta_{7} q^{6} - \beta_{14} q^{7} + q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + \beta_{7} q^{3} + q^{4} + \beta_{11} q^{5} + \beta_{7} q^{6} - \beta_{14} q^{7} + q^{8} - q^{9} + \beta_{11} q^{10} + \beta_{2} q^{11} + \beta_{7} q^{12} + \beta_{8} q^{13} - \beta_{14} q^{14} - \beta_{6} q^{15} + q^{16} + ( - \beta_{9} - \beta_{6} - \beta_{5} + 3) q^{17} - q^{18} + ( - \beta_{13} + \beta_{7} + \cdots + \beta_1) q^{19}+ \cdots - \beta_{2} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 16 q^{2} + 16 q^{4} + 2 q^{5} + 16 q^{8} - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 16 q^{2} + 16 q^{4} + 2 q^{5} + 16 q^{8} - 16 q^{9} + 2 q^{10} + 2 q^{11} - 6 q^{15} + 16 q^{16} + 38 q^{17} - 16 q^{18} + 2 q^{20} + 6 q^{21} + 2 q^{22} + 20 q^{23} - 4 q^{25} - 6 q^{30} + 16 q^{32} + 38 q^{34} + 10 q^{35} - 16 q^{36} + 4 q^{37} + 2 q^{40} - 6 q^{41} + 6 q^{42} + 2 q^{43} + 2 q^{44} - 2 q^{45} + 20 q^{46} - 18 q^{49} - 4 q^{50} - 20 q^{55} - 16 q^{57} - 6 q^{60} + 16 q^{64} + 12 q^{65} + 38 q^{68} + 10 q^{70} - 24 q^{71} - 16 q^{72} + 4 q^{74} + 2 q^{80} + 16 q^{81} - 6 q^{82} + 6 q^{84} + 2 q^{86} - 2 q^{87} + 2 q^{88} - 2 q^{90} + 20 q^{92} - 22 q^{93} - 16 q^{95} - 38 q^{97} - 18 q^{98} - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 2 x^{15} + 2 x^{14} + 8 x^{13} + 138 x^{12} - 220 x^{11} + 196 x^{10} + 744 x^{9} + 4241 x^{8} + \cdots + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 35\!\cdots\!56 \nu^{15} + \cdots - 19\!\cdots\!68 ) / 87\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 50\!\cdots\!44 \nu^{15} + \cdots + 23\!\cdots\!60 ) / 87\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 95\!\cdots\!46 \nu^{15} + \cdots - 38\!\cdots\!40 ) / 87\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 10\!\cdots\!49 \nu^{15} + \cdots + 38\!\cdots\!40 ) / 87\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 23\!\cdots\!03 \nu^{15} + \cdots + 39\!\cdots\!12 ) / 17\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 27\!\cdots\!09 \nu^{15} + \cdots + 41\!\cdots\!00 ) / 17\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 18\!\cdots\!46 \nu^{15} + \cdots - 10\!\cdots\!00 ) / 87\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 21\!\cdots\!85 \nu^{15} + \cdots + 35\!\cdots\!52 ) / 87\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 24\!\cdots\!59 \nu^{15} + \cdots - 52\!\cdots\!04 ) / 87\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 12\!\cdots\!34 \nu^{15} + \cdots - 13\!\cdots\!72 ) / 43\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 83\!\cdots\!77 \nu^{15} + \cdots - 46\!\cdots\!88 ) / 17\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 87\!\cdots\!83 \nu^{15} + \cdots + 48\!\cdots\!76 ) / 17\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 48\!\cdots\!94 \nu^{15} + \cdots - 28\!\cdots\!00 ) / 87\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 58\!\cdots\!38 \nu^{15} + \cdots + 35\!\cdots\!12 ) / 87\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 59\!\cdots\!79 \nu^{15} + \cdots + 63\!\cdots\!12 ) / 87\!\cdots\!36 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{12} + \beta_{11} - \beta_{6} + \beta_{5} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{15} - \beta_{14} + \beta_{12} - \beta_{11} + 5\beta_{7} + \beta_{4} + \beta_{3} - \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 2 \beta_{14} - 2 \beta_{13} - 7 \beta_{12} - 7 \beta_{11} + 2 \beta_{10} + 2 \beta_{7} - 7 \beta_{6} + \cdots - 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 10 \beta_{15} - 2 \beta_{12} - 2 \beta_{11} + 11 \beta_{10} + 11 \beta_{9} + 11 \beta_{8} + 10 \beta_{6} + \cdots - 41 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 22 \beta_{15} + 22 \beta_{14} + 26 \beta_{13} - 61 \beta_{12} - 61 \beta_{11} + 22 \beta_{10} + \cdots - 34 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 97 \beta_{15} + 109 \beta_{14} - 26 \beta_{13} - 99 \beta_{12} + 99 \beta_{11} - 379 \beta_{7} + \cdots + 111 \beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 222 \beta_{14} + 278 \beta_{13} + 569 \beta_{12} + 577 \beta_{11} - 222 \beta_{10} - 8 \beta_{9} + \cdots + 394 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 942 \beta_{15} + 306 \beta_{12} + 306 \beta_{11} - 1073 \beta_{10} - 1141 \beta_{9} - 1097 \beta_{8} + \cdots + 3643 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 2122 \beta_{15} - 2218 \beta_{14} - 2822 \beta_{13} + 5615 \beta_{12} + 5439 \beta_{11} - 2218 \beta_{10} + \cdots + 4126 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 9163 \beta_{15} - 10567 \beta_{14} + 2630 \beta_{13} + 9657 \beta_{12} - 9657 \beta_{11} + \cdots - 10765 \beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 22122 \beta_{14} - 28178 \beta_{13} - 52459 \beta_{12} - 55187 \beta_{11} + 22122 \beta_{10} + \cdots - 41854 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 89202 \beta_{15} - 31214 \beta_{12} - 31214 \beta_{11} + 104123 \beta_{10} + 114855 \beta_{9} + \cdots - 347593 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 199358 \beta_{15} + 220574 \beta_{14} + 279810 \beta_{13} - 544365 \beta_{12} - 507629 \beta_{11} + \cdots - 420186 ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 868761 \beta_{15} + 1026381 \beta_{14} - 236434 \beta_{13} - 941739 \beta_{12} + 941739 \beta_{11} + \cdots + 1030535 \beta_1 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 2199438 \beta_{14} + 2773286 \beta_{13} + 4918065 \beta_{12} + 5377337 \beta_{11} - 2199438 \beta_{10} + \cdots + 4204746 ) / 2 \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
739.1
−2.20888 + 2.20888i
−1.32920 + 1.32920i
−1.28817 + 1.28817i
0.0603067 0.0603067i
0.623809 0.623809i
0.718347 0.718347i
2.18974 2.18974i
2.23405 2.23405i
−2.20888 2.20888i
−1.32920 1.32920i
−1.28817 1.28817i
0.0603067 + 0.0603067i
0.623809 + 0.623809i
0.718347 + 0.718347i
2.18974 + 2.18974i
2.23405 + 2.23405i
1.00000 1.00000i 1.00000 −2.20888 0.347650i 1.00000i 2.08112i 1.00000 −1.00000 −2.20888 0.347650i
739.2 1.00000 1.00000i 1.00000 −1.32920 + 1.79812i 1.00000i 1.87266i 1.00000 −1.00000 −1.32920 + 1.79812i
739.3 1.00000 1.00000i 1.00000 −1.28817 1.82773i 1.00000i 1.25548i 1.00000 −1.00000 −1.28817 1.82773i
739.4 1.00000 1.00000i 1.00000 0.0603067 2.23525i 1.00000i 3.18244i 1.00000 −1.00000 0.0603067 2.23525i
739.5 1.00000 1.00000i 1.00000 0.623809 2.14729i 1.00000i 5.25373i 1.00000 −1.00000 0.623809 2.14729i
739.6 1.00000 1.00000i 1.00000 0.718347 + 2.11754i 1.00000i 2.74990i 1.00000 −1.00000 0.718347 + 2.11754i
739.7 1.00000 1.00000i 1.00000 2.18974 0.452819i 1.00000i 0.0776078i 1.00000 −1.00000 2.18974 0.452819i
739.8 1.00000 1.00000i 1.00000 2.23405 + 0.0950871i 1.00000i 3.20752i 1.00000 −1.00000 2.23405 + 0.0950871i
739.9 1.00000 1.00000i 1.00000 −2.20888 + 0.347650i 1.00000i 2.08112i 1.00000 −1.00000 −2.20888 + 0.347650i
739.10 1.00000 1.00000i 1.00000 −1.32920 1.79812i 1.00000i 1.87266i 1.00000 −1.00000 −1.32920 1.79812i
739.11 1.00000 1.00000i 1.00000 −1.28817 + 1.82773i 1.00000i 1.25548i 1.00000 −1.00000 −1.28817 + 1.82773i
739.12 1.00000 1.00000i 1.00000 0.0603067 + 2.23525i 1.00000i 3.18244i 1.00000 −1.00000 0.0603067 + 2.23525i
739.13 1.00000 1.00000i 1.00000 0.623809 + 2.14729i 1.00000i 5.25373i 1.00000 −1.00000 0.623809 + 2.14729i
739.14 1.00000 1.00000i 1.00000 0.718347 2.11754i 1.00000i 2.74990i 1.00000 −1.00000 0.718347 2.11754i
739.15 1.00000 1.00000i 1.00000 2.18974 + 0.452819i 1.00000i 0.0776078i 1.00000 −1.00000 2.18974 + 0.452819i
739.16 1.00000 1.00000i 1.00000 2.23405 0.0950871i 1.00000i 3.20752i 1.00000 −1.00000 2.23405 0.0950871i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 739.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
185.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1110.2.e.d yes 16
3.b odd 2 1 3330.2.e.e 16
5.b even 2 1 1110.2.e.c 16
15.d odd 2 1 3330.2.e.f 16
37.b even 2 1 1110.2.e.c 16
111.d odd 2 1 3330.2.e.f 16
185.d even 2 1 inner 1110.2.e.d yes 16
555.b odd 2 1 3330.2.e.e 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1110.2.e.c 16 5.b even 2 1
1110.2.e.c 16 37.b even 2 1
1110.2.e.d yes 16 1.a even 1 1 trivial
1110.2.e.d yes 16 185.d even 2 1 inner
3330.2.e.e 16 3.b odd 2 1
3330.2.e.e 16 555.b odd 2 1
3330.2.e.f 16 15.d odd 2 1
3330.2.e.f 16 111.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1110, [\chi])\):

\( T_{7}^{16} + 65 T_{7}^{14} + 1582 T_{7}^{12} + 19194 T_{7}^{10} + 126197 T_{7}^{8} + 448469 T_{7}^{6} + \cdots + 3136 \) Copy content Toggle raw display
\( T_{13}^{8} - 34T_{13}^{6} + 285T_{13}^{4} - 16T_{13}^{3} - 600T_{13}^{2} - 80T_{13} + 64 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{16} \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{8} \) Copy content Toggle raw display
$5$ \( T^{16} - 2 T^{15} + \cdots + 390625 \) Copy content Toggle raw display
$7$ \( T^{16} + 65 T^{14} + \cdots + 3136 \) Copy content Toggle raw display
$11$ \( (T^{8} - T^{7} - 50 T^{6} + \cdots + 716)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} - 34 T^{6} + \cdots + 64)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} - 19 T^{7} + \cdots - 4504)^{2} \) Copy content Toggle raw display
$19$ \( T^{16} + 156 T^{14} + \cdots + 86415616 \) Copy content Toggle raw display
$23$ \( (T^{8} - 10 T^{7} + \cdots + 31168)^{2} \) Copy content Toggle raw display
$29$ \( T^{16} + 193 T^{14} + \cdots + 25563136 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 1349533696 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 3512479453921 \) Copy content Toggle raw display
$41$ \( (T^{8} + 3 T^{7} + \cdots - 1327648)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} - T^{7} + \cdots + 1418624)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 1124663296 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 24049806400 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 50462269542400 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 229356703744 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 505031950336 \) Copy content Toggle raw display
$71$ \( (T^{8} + 12 T^{7} + \cdots - 6032896)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 67\!\cdots\!04 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 19573129216 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 502941945856 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 60\!\cdots\!24 \) Copy content Toggle raw display
$97$ \( (T^{8} + 19 T^{7} + \cdots + 26655952)^{2} \) Copy content Toggle raw display
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