Properties

Label 1110.4.h.e
Level $1110$
Weight $4$
Character orbit 1110.h
Analytic conductor $65.492$
Analytic rank $0$
Dimension $20$
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: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 4232 x^{18} + 7165838 x^{16} + 6242308300 x^{14} + 3021678610305 x^{12} + 829821558651956 x^{10} + \cdots + 52\!\cdots\!16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{37}]\)
Coefficient ring index: \( 2^{7}\cdot 5^{2} \)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 \beta_{10} q^{2} + 3 q^{3} - 4 q^{4} - 5 \beta_{10} q^{5} + 6 \beta_{10} q^{6} + ( - \beta_{4} - 1) q^{7} - 8 \beta_{10} q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 \beta_{10} q^{2} + 3 q^{3} - 4 q^{4} - 5 \beta_{10} q^{5} + 6 \beta_{10} q^{6} + ( - \beta_{4} - 1) q^{7} - 8 \beta_{10} q^{8} + 9 q^{9} + 10 q^{10} + ( - \beta_{7} - 3) q^{11} - 12 q^{12} + ( - \beta_{15} + 2 \beta_{10} + \beta_1) q^{13} + ( - 2 \beta_{10} + 2 \beta_1) q^{14} - 15 \beta_{10} q^{15} + 16 q^{16} + (\beta_{11} - 6 \beta_{10}) q^{17} + 18 \beta_{10} q^{18} + (\beta_{19} - \beta_{16} - \beta_{15} + 2 \beta_{10} + \beta_1) q^{19} + 20 \beta_{10} q^{20} + ( - 3 \beta_{4} - 3) q^{21} + (2 \beta_{13} - 6 \beta_{10}) q^{22} + (\beta_{15} - \beta_{13} - \beta_{12} - \beta_{11} + 11 \beta_{10} - \beta_1) q^{23} - 24 \beta_{10} q^{24} - 25 q^{25} + (2 \beta_{4} + 2 \beta_{3} - 4) q^{26} + 27 q^{27} + (4 \beta_{4} + 4) q^{28} + (\beta_{18} + \beta_{17} + \beta_{16} - \beta_{11} - 16 \beta_{10} - 2 \beta_1) q^{29} + 30 q^{30} + ( - \beta_{16} + 2 \beta_{15} - \beta_{13} + \beta_{12} - \beta_{11} - 6 \beta_{10} - 3 \beta_1) q^{31} + 32 \beta_{10} q^{32} + ( - 3 \beta_{7} - 9) q^{33} + ( - 2 \beta_{5} + 12) q^{34} + (5 \beta_{10} - 5 \beta_1) q^{35} - 36 q^{36} + (\beta_{17} - 2 \beta_{16} - \beta_{15} - \beta_{12} - 3 \beta_{10} + \beta_{8} + \beta_{7} - \beta_{5} + 2 \beta_{4} + \cdots - 6) q^{37}+ \cdots + ( - 9 \beta_{7} - 27) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 60 q^{3} - 80 q^{4} - 28 q^{7} + 180 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 60 q^{3} - 80 q^{4} - 28 q^{7} + 180 q^{9} + 200 q^{10} - 56 q^{11} - 240 q^{12} + 320 q^{16} - 84 q^{21} - 500 q^{25} - 64 q^{26} + 540 q^{27} + 112 q^{28} + 600 q^{30} - 168 q^{33} + 240 q^{34} - 720 q^{36} - 110 q^{37} - 52 q^{38} - 800 q^{40} + 406 q^{41} + 224 q^{44} - 444 q^{46} - 1458 q^{47} + 960 q^{48} + 1640 q^{49} + 410 q^{53} + 592 q^{58} + 196 q^{62} - 252 q^{63} - 1280 q^{64} + 160 q^{65} - 720 q^{67} - 280 q^{70} - 1320 q^{71} - 1142 q^{73} + 100 q^{74} - 1500 q^{75} - 64 q^{77} - 192 q^{78} + 1620 q^{81} + 764 q^{83} + 336 q^{84} - 600 q^{85} - 2320 q^{86} + 1800 q^{90} + 130 q^{95} - 504 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 4232 x^{18} + 7165838 x^{16} + 6242308300 x^{14} + 3021678610305 x^{12} + 829821558651956 x^{10} + \cdots + 52\!\cdots\!16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 53\!\cdots\!13 \nu^{18} + \cdots - 25\!\cdots\!88 ) / 54\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 98\!\cdots\!57 \nu^{18} + \cdots - 58\!\cdots\!48 ) / 54\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 20\!\cdots\!63 \nu^{18} + \cdots + 37\!\cdots\!92 ) / 49\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 14\!\cdots\!81 \nu^{18} + \cdots - 13\!\cdots\!04 ) / 91\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 16\!\cdots\!63 \nu^{18} + \cdots - 32\!\cdots\!32 ) / 85\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 56\!\cdots\!71 \nu^{18} + \cdots - 95\!\cdots\!64 ) / 27\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 99\!\cdots\!23 \nu^{18} + \cdots - 40\!\cdots\!12 ) / 45\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 91\!\cdots\!41 \nu^{18} + \cdots - 17\!\cdots\!80 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 16\!\cdots\!11 \nu^{19} + \cdots - 39\!\cdots\!24 \nu ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 14\!\cdots\!21 \nu^{19} + \cdots - 26\!\cdots\!64 \nu ) / 20\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 95\!\cdots\!03 \nu^{19} + \cdots + 21\!\cdots\!00 \nu ) / 82\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 18\!\cdots\!23 \nu^{19} + \cdots - 12\!\cdots\!32 \nu ) / 12\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 25\!\cdots\!89 \nu^{19} + \cdots - 12\!\cdots\!04 \nu ) / 68\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 42\!\cdots\!49 \nu^{19} + \cdots - 88\!\cdots\!92 \nu ) / 82\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 25\!\cdots\!67 \nu^{19} + \cdots - 71\!\cdots\!88 \nu ) / 41\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 13\!\cdots\!41 \nu^{19} + \cdots - 18\!\cdots\!48 ) / 20\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 13\!\cdots\!41 \nu^{19} + \cdots + 18\!\cdots\!48 ) / 20\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 60\!\cdots\!31 \nu^{19} + \cdots - 12\!\cdots\!44 \nu ) / 32\!\cdots\!00 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{18} - 2\beta_{17} + \beta_{9} + \beta_{8} + 3\beta_{7} + \beta_{5} - 4\beta_{4} + 5\beta_{3} - 2\beta_{2} - 420 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - 60 \beta_{19} - 21 \beta_{18} - 21 \beta_{17} + 38 \beta_{16} + 81 \beta_{15} - 19 \beta_{14} - 107 \beta_{13} + 27 \beta_{12} + 27 \beta_{11} + 1576 \beta_{10} - 874 \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 2584 \beta_{18} + 2584 \beta_{17} - 1379 \beta_{9} - 1117 \beta_{8} - 3575 \beta_{7} - 870 \beta_{6} - 907 \beta_{5} + 3830 \beta_{4} - 5425 \beta_{3} + 2316 \beta_{2} + 354598 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 84408 \beta_{19} + 34591 \beta_{18} + 34591 \beta_{17} - 57594 \beta_{16} - 116983 \beta_{15} + 25925 \beta_{14} + 132977 \beta_{13} - 30845 \beta_{12} - 47053 \beta_{11} - 1361384 \beta_{10} + \cdots + 877932 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2899208 \beta_{18} - 2899208 \beta_{17} + 1610683 \beta_{9} + 1225281 \beta_{8} + 3947771 \beta_{7} + 1369266 \beta_{6} + 847035 \beta_{5} - 2351966 \beta_{4} + 6111029 \beta_{3} + \cdots - 351696322 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 98595600 \beta_{19} - 46442435 \beta_{18} - 46442435 \beta_{17} + 71155202 \beta_{16} + 139851755 \beta_{15} - 28864385 \beta_{14} - 151852221 \beta_{13} + \cdots - 915593880 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 3178131280 \beta_{18} + 3178131280 \beta_{17} - 1805586971 \beta_{9} - 1318317537 \beta_{8} - 4452812563 \beta_{7} - 1766755074 \beta_{6} + \cdots + 366763128490 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 110716192008 \beta_{19} + 58811303115 \beta_{18} + 58811303115 \beta_{17} - 83515755194 \beta_{16} - 160600780803 \beta_{15} + 30472321081 \beta_{14} + \cdots + 969069654392 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 3481112564896 \beta_{18} - 3481112564896 \beta_{17} + 2004869196035 \beta_{9} + 1406738038481 \beta_{8} + 5089856663883 \beta_{7} + \cdots - 389686692218330 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 123363039098832 \beta_{19} - 72403731830123 \beta_{18} - 72403731830123 \beta_{17} + 96651330038594 \beta_{16} + 183180552515939 \beta_{15} + \cdots - 10\!\cdots\!88 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 38\!\cdots\!04 \beta_{18} + \cdots + 41\!\cdots\!14 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 13\!\cdots\!84 \beta_{19} + \cdots + 11\!\cdots\!60 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 42\!\cdots\!76 \beta_{18} + \cdots - 45\!\cdots\!34 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 15\!\cdots\!84 \beta_{19} + \cdots - 12\!\cdots\!44 \beta_1 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 47\!\cdots\!24 \beta_{18} + \cdots + 48\!\cdots\!02 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 17\!\cdots\!16 \beta_{19} + \cdots + 13\!\cdots\!20 \beta_1 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 52\!\cdots\!60 \beta_{18} + \cdots - 53\!\cdots\!38 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( - 19\!\cdots\!76 \beta_{19} + \cdots - 14\!\cdots\!08 \beta_1 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
961.1
32.1509i
30.7816i
18.7154i
5.41516i
1.22012i
5.37596i
11.8939i
15.0522i
18.1445i
33.8167i
32.1509i
30.7816i
18.7154i
5.41516i
1.22012i
5.37596i
11.8939i
15.0522i
18.1445i
33.8167i
2.00000i 3.00000 −4.00000 5.00000i 6.00000i −33.1509 8.00000i 9.00000 10.0000
961.2 2.00000i 3.00000 −4.00000 5.00000i 6.00000i −31.7816 8.00000i 9.00000 10.0000
961.3 2.00000i 3.00000 −4.00000 5.00000i 6.00000i −19.7154 8.00000i 9.00000 10.0000
961.4 2.00000i 3.00000 −4.00000 5.00000i 6.00000i −6.41516 8.00000i 9.00000 10.0000
961.5 2.00000i 3.00000 −4.00000 5.00000i 6.00000i −2.22012 8.00000i 9.00000 10.0000
961.6 2.00000i 3.00000 −4.00000 5.00000i 6.00000i 4.37596 8.00000i 9.00000 10.0000
961.7 2.00000i 3.00000 −4.00000 5.00000i 6.00000i 10.8939 8.00000i 9.00000 10.0000
961.8 2.00000i 3.00000 −4.00000 5.00000i 6.00000i 14.0522 8.00000i 9.00000 10.0000
961.9 2.00000i 3.00000 −4.00000 5.00000i 6.00000i 17.1445 8.00000i 9.00000 10.0000
961.10 2.00000i 3.00000 −4.00000 5.00000i 6.00000i 32.8167 8.00000i 9.00000 10.0000
961.11 2.00000i 3.00000 −4.00000 5.00000i 6.00000i −33.1509 8.00000i 9.00000 10.0000
961.12 2.00000i 3.00000 −4.00000 5.00000i 6.00000i −31.7816 8.00000i 9.00000 10.0000
961.13 2.00000i 3.00000 −4.00000 5.00000i 6.00000i −19.7154 8.00000i 9.00000 10.0000
961.14 2.00000i 3.00000 −4.00000 5.00000i 6.00000i −6.41516 8.00000i 9.00000 10.0000
961.15 2.00000i 3.00000 −4.00000 5.00000i 6.00000i −2.22012 8.00000i 9.00000 10.0000
961.16 2.00000i 3.00000 −4.00000 5.00000i 6.00000i 4.37596 8.00000i 9.00000 10.0000
961.17 2.00000i 3.00000 −4.00000 5.00000i 6.00000i 10.8939 8.00000i 9.00000 10.0000
961.18 2.00000i 3.00000 −4.00000 5.00000i 6.00000i 14.0522 8.00000i 9.00000 10.0000
961.19 2.00000i 3.00000 −4.00000 5.00000i 6.00000i 17.1445 8.00000i 9.00000 10.0000
961.20 2.00000i 3.00000 −4.00000 5.00000i 6.00000i 32.8167 8.00000i 9.00000 10.0000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 961.20
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.e 20
37.b even 2 1 inner 1110.4.h.e 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1110.4.h.e 20 1.a even 1 1 trivial
1110.4.h.e 20 37.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{10} + 14 T_{7}^{9} - 2027 T_{7}^{8} - 18646 T_{7}^{7} + 1280103 T_{7}^{6} + 3221322 T_{7}^{5} - 289689021 T_{7}^{4} + 707607982 T_{7}^{3} + 13651938768 T_{7}^{2} + \cdots - 111502057472 \) acting on \(S_{4}^{\mathrm{new}}(1110, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 4)^{10} \) Copy content Toggle raw display
$3$ \( (T - 3)^{20} \) Copy content Toggle raw display
$5$ \( (T^{2} + 25)^{10} \) Copy content Toggle raw display
$7$ \( (T^{10} + 14 T^{9} + \cdots - 111502057472)^{2} \) Copy content Toggle raw display
$11$ \( (T^{10} + 28 T^{9} + \cdots - 60762453201024)^{2} \) Copy content Toggle raw display
$13$ \( T^{20} + 21550 T^{18} + \cdots + 19\!\cdots\!44 \) Copy content Toggle raw display
$17$ \( T^{20} + 53420 T^{18} + \cdots + 54\!\cdots\!64 \) Copy content Toggle raw display
$19$ \( T^{20} + 85855 T^{18} + \cdots + 32\!\cdots\!04 \) Copy content Toggle raw display
$23$ \( T^{20} + 116785 T^{18} + \cdots + 15\!\cdots\!76 \) Copy content Toggle raw display
$29$ \( T^{20} + 295572 T^{18} + \cdots + 28\!\cdots\!44 \) Copy content Toggle raw display
$31$ \( T^{20} + 332569 T^{18} + \cdots + 52\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{20} + 110 T^{19} + \cdots + 11\!\cdots\!49 \) Copy content Toggle raw display
$41$ \( (T^{10} - 203 T^{9} + \cdots + 49\!\cdots\!64)^{2} \) Copy content Toggle raw display
$43$ \( T^{20} + 1055788 T^{18} + \cdots + 83\!\cdots\!24 \) Copy content Toggle raw display
$47$ \( (T^{10} + 729 T^{9} + \cdots + 14\!\cdots\!60)^{2} \) Copy content Toggle raw display
$53$ \( (T^{10} - 205 T^{9} + \cdots - 10\!\cdots\!40)^{2} \) Copy content Toggle raw display
$59$ \( T^{20} + 1671679 T^{18} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{20} + 2514369 T^{18} + \cdots + 56\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( (T^{10} + 360 T^{9} + \cdots - 65\!\cdots\!52)^{2} \) Copy content Toggle raw display
$71$ \( (T^{10} + 660 T^{9} + \cdots + 63\!\cdots\!64)^{2} \) Copy content Toggle raw display
$73$ \( (T^{10} + 571 T^{9} + \cdots - 47\!\cdots\!76)^{2} \) Copy content Toggle raw display
$79$ \( T^{20} + 4129290 T^{18} + \cdots + 54\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( (T^{10} - 382 T^{9} + \cdots + 53\!\cdots\!00)^{2} \) Copy content Toggle raw display
$89$ \( T^{20} + 6979610 T^{18} + \cdots + 15\!\cdots\!56 \) Copy content Toggle raw display
$97$ \( T^{20} + 7855113 T^{18} + \cdots + 18\!\cdots\!76 \) Copy content Toggle raw display
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