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Properties

Label 1110.2.x.e
Level $1110$
Weight $2$
Character orbit 1110.x
Analytic conductor $8.863$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1110,2,Mod(751,1110)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1110, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1110.751");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1110 = 2 \cdot 3 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1110.x (of order \(6\), degree \(2\), 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: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 44x^{14} + 724x^{12} + 5750x^{10} + 23344x^{8} + 47024x^{6} + 43297x^{4} + 13976x^{2} + 676 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{6} q^{2} - \beta_1 q^{3} - \beta_1 q^{4} - \beta_{7} q^{5} + ( - \beta_{7} + \beta_{6}) q^{6} + ( - \beta_{12} + \beta_{11} - \beta_{8} + \cdots - 1) q^{7}+ \cdots + ( - \beta_1 - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{6} q^{2} - \beta_1 q^{3} - \beta_1 q^{4} - \beta_{7} q^{5} + ( - \beta_{7} + \beta_{6}) q^{6} + ( - \beta_{12} + \beta_{11} - \beta_{8} + \cdots - 1) q^{7}+ \cdots + (\beta_{14} + \beta_{13} - \beta_{8} + \cdots - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 8 q^{3} + 8 q^{4} - 4 q^{7} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 8 q^{3} + 8 q^{4} - 4 q^{7} - 8 q^{9} - 16 q^{10} - 4 q^{11} - 8 q^{12} - 8 q^{16} - 6 q^{17} - 6 q^{19} + 4 q^{21} + 12 q^{22} + 8 q^{25} - 8 q^{26} - 16 q^{27} + 4 q^{28} - 8 q^{30} - 2 q^{33} + 2 q^{34} + 18 q^{35} - 16 q^{36} - 12 q^{37} + 12 q^{38} - 8 q^{40} + 4 q^{41} + 18 q^{42} - 2 q^{44} + 12 q^{47} - 16 q^{48} - 42 q^{49} - 16 q^{53} - 12 q^{55} + 18 q^{56} - 6 q^{57} + 2 q^{58} + 48 q^{59} - 12 q^{61} - 4 q^{62} + 8 q^{63} - 16 q^{64} + 4 q^{65} + 6 q^{67} - 18 q^{69} + 4 q^{70} - 6 q^{71} - 52 q^{73} - 16 q^{74} + 16 q^{75} - 6 q^{76} + 10 q^{77} - 4 q^{78} + 78 q^{79} - 8 q^{81} + 36 q^{83} + 8 q^{84} - 4 q^{85} - 6 q^{86} + 12 q^{87} + 18 q^{89} + 8 q^{90} - 66 q^{91} - 18 q^{92} - 36 q^{93} + 12 q^{94} - 6 q^{95} - 12 q^{98} + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 44x^{14} + 724x^{12} + 5750x^{10} + 23344x^{8} + 47024x^{6} + 43297x^{4} + 13976x^{2} + 676 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 13206 \nu^{15} - 582741 \nu^{13} - 9594515 \nu^{11} - 75642377 \nu^{9} - 298884555 \nu^{7} + \cdots - 9055852 ) / 18111704 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 10324114 \nu^{15} - 231600785 \nu^{14} - 431418313 \nu^{13} - 9788447929 \nu^{12} + \cdots - 77890999576 ) / 13348325848 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 54424395 \nu^{15} - 147256174 \nu^{14} + 2303097077 \nu^{13} - 6219207696 \nu^{12} + \cdots - 122757597300 ) / 13348325848 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2649 \nu^{14} + 111621 \nu^{12} + 1710274 \nu^{10} + 12057906 \nu^{8} + 39518805 \nu^{6} + \cdots + 980070 ) / 46766 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 71079673 \nu^{15} + 53593657 \nu^{14} - 2995690422 \nu^{13} + 2247594011 \nu^{12} + \cdots - 2258341020 ) / 6674162924 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 18809805 \nu^{15} - 44185843 \nu^{14} - 793263294 \nu^{13} - 1863894877 \nu^{12} + \cdots - 17830174076 ) / 1213484168 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 18809805 \nu^{15} - 44185843 \nu^{14} + 793263294 \nu^{13} - 1863894877 \nu^{12} + \cdots - 17830174076 ) / 1213484168 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 109394411 \nu^{15} - 394487353 \nu^{14} - 4631516922 \nu^{13} - 16655667834 \nu^{12} + \cdots - 152403200300 ) / 6674162924 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 152483460 \nu^{15} - 6422799157 \nu^{13} - 98320814245 \nu^{11} - 691602201221 \nu^{9} + \cdots + 36856818194 \nu ) / 6674162924 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 180474084 \nu^{15} - 826130396 \nu^{14} - 7627207344 \nu^{13} - 34833141239 \nu^{12} + \cdots - 283340406336 ) / 6674162924 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 180474084 \nu^{15} - 826130396 \nu^{14} + 7627207344 \nu^{13} - 34833141239 \nu^{12} + \cdots - 283340406336 ) / 6674162924 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 664665858 \nu^{15} - 716837485 \nu^{14} - 28023362596 \nu^{13} - 30213515345 \nu^{12} + \cdots - 246614842136 ) / 13348325848 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 817149318 \nu^{15} - 841250956 \nu^{14} + 34446161753 \nu^{13} - 35506775252 \nu^{12} + \cdots - 315674197904 ) / 13348325848 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 1240593227 \nu^{15} - 431724501 \nu^{14} - 52322147503 \nu^{13} - 18205033405 \nu^{12} + \cdots - 111042129536 ) / 13348325848 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 1240593227 \nu^{15} - 431724501 \nu^{14} + 52322147503 \nu^{13} - 18205033405 \nu^{12} + \cdots - 111042129536 ) / 13348325848 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{10} - \beta_{9} - \beta_{8} - \beta_{7} + \beta_{4} + \beta_{3} + \beta_{2} - 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 2 \beta_{13} - 2 \beta_{12} + \beta_{10} - \beta_{9} - \beta_{8} + 3 \beta_{7} + 2 \beta_{5} + \cdots - 9 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - 2 \beta_{15} + 2 \beta_{14} + 4 \beta_{13} - 4 \beta_{12} + 3 \beta_{11} - 5 \beta_{10} + 6 \beta_{9} + \cdots + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 8 \beta_{15} - 8 \beta_{14} + 14 \beta_{13} + 14 \beta_{12} + 20 \beta_{11} - 11 \beta_{10} + \cdots + 91 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 82 \beta_{15} - 82 \beta_{14} - 178 \beta_{13} + 178 \beta_{12} - 152 \beta_{11} + 137 \beta_{10} + \cdots + 25 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 119 \beta_{15} + 119 \beta_{14} - 31 \beta_{13} - 31 \beta_{12} - 256 \beta_{11} + 70 \beta_{10} + \cdots - 615 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 1456 \beta_{15} + 1456 \beta_{14} + 3334 \beta_{13} - 3334 \beta_{12} + 2998 \beta_{11} - 2137 \beta_{10} + \cdots - 721 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 5016 \beta_{15} - 5016 \beta_{14} - 406 \beta_{13} - 406 \beta_{12} + 10228 \beta_{11} - 2165 \beta_{10} + \cdots + 19109 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 12624 \beta_{15} - 12624 \beta_{14} - 29928 \beta_{13} + 29928 \beta_{12} - 27525 \beta_{11} + \cdots + 7056 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 95208 \beta_{15} + 95208 \beta_{14} + 19626 \beta_{13} + 19626 \beta_{12} - 189672 \beta_{11} + \cdots - 317701 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 438354 \beta_{15} + 438354 \beta_{14} + 1061474 \beta_{13} - 1061474 \beta_{12} + 986636 \beta_{11} + \cdots - 255349 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 868373 \beta_{15} - 868373 \beta_{14} - 226657 \beta_{13} - 226657 \beta_{12} + 1710670 \beta_{11} + \cdots + 2726774 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 7652012 \beta_{15} - 7652012 \beta_{14} - 18758706 \beta_{13} + 18758706 \beta_{12} - 17527426 \beta_{11} + \cdots + 4525447 ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 31149580 \beta_{15} + 31149580 \beta_{14} + 8945262 \beta_{13} + 8945262 \beta_{12} - 61014980 \beta_{11} + \cdots - 95017697 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 67095230 \beta_{15} + 67095230 \beta_{14} + 165621522 \beta_{13} - 165621522 \beta_{12} + \cdots - 39904483 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(371\) \(631\) \(667\)
\(\chi(n)\) \(1\) \(-\beta_{1}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
751.1
3.06293i
− 4.20281i
− 1.35087i
− 0.241301i
2.47236i
− 1.35400i
− 2.57049i
0.720074i
− 3.06293i
4.20281i
1.35087i
0.241301i
− 2.47236i
1.35400i
2.57049i
− 0.720074i
−0.866025 + 0.500000i 0.500000 − 0.866025i 0.500000 − 0.866025i 0.866025 + 0.500000i 1.00000i −1.55211 + 2.68834i 1.00000i −0.500000 − 0.866025i −1.00000
751.2 −0.866025 + 0.500000i 0.500000 − 0.866025i 0.500000 − 0.866025i 0.866025 + 0.500000i 1.00000i −0.839478 + 1.45402i 1.00000i −0.500000 − 0.866025i −1.00000
751.3 −0.866025 + 0.500000i 0.500000 − 0.866025i 0.500000 − 0.866025i 0.866025 + 0.500000i 1.00000i 1.60020 − 2.77163i 1.00000i −0.500000 − 0.866025i −1.00000
751.4 −0.866025 + 0.500000i 0.500000 − 0.866025i 0.500000 − 0.866025i 0.866025 + 0.500000i 1.00000i 2.38947 − 4.13868i 1.00000i −0.500000 − 0.866025i −1.00000
751.5 0.866025 − 0.500000i 0.500000 − 0.866025i 0.500000 − 0.866025i −0.866025 − 0.500000i − 1.00000i −2.53766 + 4.39536i − 1.00000i −0.500000 − 0.866025i −1.00000
751.6 0.866025 − 0.500000i 0.500000 − 0.866025i 0.500000 − 0.866025i −0.866025 − 0.500000i − 1.00000i −2.34998 + 4.07029i − 1.00000i −0.500000 − 0.866025i −1.00000
751.7 0.866025 − 0.500000i 0.500000 − 0.866025i 0.500000 − 0.866025i −0.866025 − 0.500000i − 1.00000i 0.243282 − 0.421377i − 1.00000i −0.500000 − 0.866025i −1.00000
751.8 0.866025 − 0.500000i 0.500000 − 0.866025i 0.500000 − 0.866025i −0.866025 − 0.500000i − 1.00000i 1.04628 − 1.81222i − 1.00000i −0.500000 − 0.866025i −1.00000
841.1 −0.866025 − 0.500000i 0.500000 + 0.866025i 0.500000 + 0.866025i 0.866025 − 0.500000i − 1.00000i −1.55211 − 2.68834i − 1.00000i −0.500000 + 0.866025i −1.00000
841.2 −0.866025 − 0.500000i 0.500000 + 0.866025i 0.500000 + 0.866025i 0.866025 − 0.500000i − 1.00000i −0.839478 − 1.45402i − 1.00000i −0.500000 + 0.866025i −1.00000
841.3 −0.866025 − 0.500000i 0.500000 + 0.866025i 0.500000 + 0.866025i 0.866025 − 0.500000i − 1.00000i 1.60020 + 2.77163i − 1.00000i −0.500000 + 0.866025i −1.00000
841.4 −0.866025 − 0.500000i 0.500000 + 0.866025i 0.500000 + 0.866025i 0.866025 − 0.500000i − 1.00000i 2.38947 + 4.13868i − 1.00000i −0.500000 + 0.866025i −1.00000
841.5 0.866025 + 0.500000i 0.500000 + 0.866025i 0.500000 + 0.866025i −0.866025 + 0.500000i 1.00000i −2.53766 − 4.39536i 1.00000i −0.500000 + 0.866025i −1.00000
841.6 0.866025 + 0.500000i 0.500000 + 0.866025i 0.500000 + 0.866025i −0.866025 + 0.500000i 1.00000i −2.34998 − 4.07029i 1.00000i −0.500000 + 0.866025i −1.00000
841.7 0.866025 + 0.500000i 0.500000 + 0.866025i 0.500000 + 0.866025i −0.866025 + 0.500000i 1.00000i 0.243282 + 0.421377i 1.00000i −0.500000 + 0.866025i −1.00000
841.8 0.866025 + 0.500000i 0.500000 + 0.866025i 0.500000 + 0.866025i −0.866025 + 0.500000i 1.00000i 1.04628 + 1.81222i 1.00000i −0.500000 + 0.866025i −1.00000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 751.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1110.2.x.e ✓ 16
37.e even 6 1 inner 1110.2.x.e ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1110.2.x.e ✓ 16 1.a even 1 1 trivial
1110.2.x.e ✓ 16 37.e even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{16} + 4 T_{7}^{15} + 57 T_{7}^{14} + 128 T_{7}^{13} + 1763 T_{7}^{12} + 3366 T_{7}^{11} + \cdots + 3748096 \) acting on \(S_{2}^{\mathrm{new}}(1110, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - T^{2} + 1)^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} - T + 1)^{8} \) Copy content Toggle raw display
$5$ \( (T^{4} - T^{2} + 1)^{4} \) Copy content Toggle raw display
$7$ \( T^{16} + 4 T^{15} + \cdots + 3748096 \) Copy content Toggle raw display
$11$ \( (T^{8} + 2 T^{7} + \cdots - 16316)^{2} \) Copy content Toggle raw display
$13$ \( T^{16} - 64 T^{14} + \cdots + 33721249 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 33962066944 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 1234872117504 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 301925376 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 290225296 \) Copy content Toggle raw display
$31$ \( T^{16} + 224 T^{14} + \cdots + 31719424 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 3512479453921 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 33398293504 \) Copy content Toggle raw display
$43$ \( T^{16} + 216 T^{14} + \cdots + 99680256 \) Copy content Toggle raw display
$47$ \( (T^{8} - 6 T^{7} + \cdots - 924282)^{2} \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 2390818816 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 8625839772484 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 219503494144 \) Copy content Toggle raw display
$67$ \( T^{16} - 6 T^{15} + \cdots + 389376 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 39359261500416 \) Copy content Toggle raw display
$73$ \( (T^{8} + 26 T^{7} + \cdots + 467968)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 815702845702144 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 54711081216 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 300268929024 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 14\!\cdots\!36 \) Copy content Toggle raw display
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