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Properties

Label 3100.3.d.e
Level $3100$
Weight $3$
Character orbit 3100.d
Analytic conductor $84.469$
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] = [3100,3,Mod(1301,3100)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3100, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3100.1301");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3100 = 2^{2} \cdot 5^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 3100.d (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: \(84.4688819517\)
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: \( x^{20} + 114 x^{18} + 5280 x^{16} + 128422 x^{14} + 1776819 x^{12} + 14249420 x^{10} + 65297060 x^{8} + \cdots + 20793600 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{31}]\)
Coefficient ring index: \( 2^{23}\cdot 3 \)
Twist minimal: no (minimal twist has level 620)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + (\beta_{6} - 1) q^{7} + (\beta_{2} - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + (\beta_{6} - 1) q^{7} + (\beta_{2} - 2) q^{9} - \beta_{11} q^{11} + \beta_{15} q^{13} + (\beta_{15} - \beta_{14} - \beta_1) q^{17} + ( - \beta_{5} + 3) q^{19} + (\beta_{17} - \beta_{15} + \cdots - 2 \beta_1) q^{21}+ \cdots + ( - 2 \beta_{19} + 2 \beta_{18} + \cdots - 2 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 12 q^{7} - 48 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 12 q^{7} - 48 q^{9} + 60 q^{19} + 8 q^{31} - 68 q^{33} + 28 q^{39} - 80 q^{41} + 48 q^{47} + 84 q^{49} + 344 q^{51} + 160 q^{59} + 232 q^{63} + 180 q^{67} - 140 q^{69} - 108 q^{71} + 336 q^{81} + 236 q^{87} + 332 q^{93} + 112 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 114 x^{18} + 5280 x^{16} + 128422 x^{14} + 1776819 x^{12} + 14249420 x^{10} + 65297060 x^{8} + \cdots + 20793600 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 11 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 403729829 \nu^{18} - 44562574416 \nu^{16} - 1973941944096 \nu^{14} + \cdots - 24\!\cdots\!00 ) / 141550143181760 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 10\!\cdots\!71 \nu^{18} + \cdots + 71\!\cdots\!80 ) / 29\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 22\!\cdots\!97 \nu^{18} + \cdots + 10\!\cdots\!20 ) / 58\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 34\!\cdots\!71 \nu^{18} + \cdots + 13\!\cdots\!40 ) / 58\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 43\!\cdots\!85 \nu^{18} + \cdots + 26\!\cdots\!00 ) / 58\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 29\!\cdots\!93 \nu^{18} + \cdots - 81\!\cdots\!60 ) / 29\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 62\!\cdots\!15 \nu^{18} + \cdots + 21\!\cdots\!60 ) / 29\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 403729829 \nu^{19} + 44562574416 \nu^{17} + 1973941944096 \nu^{15} + \cdots + 24\!\cdots\!00 \nu ) / 424650429545280 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 10\!\cdots\!89 \nu^{19} + \cdots - 25\!\cdots\!00 \nu ) / 33\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 19\!\cdots\!99 \nu^{19} + \cdots + 10\!\cdots\!40 ) / 35\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 19\!\cdots\!99 \nu^{19} + \cdots - 10\!\cdots\!40 ) / 35\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 19\!\cdots\!91 \nu^{19} + \cdots + 77\!\cdots\!00 \nu ) / 33\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 35\!\cdots\!34 \nu^{19} + \cdots + 14\!\cdots\!00 \nu ) / 41\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 45\!\cdots\!11 \nu^{19} + \cdots + 24\!\cdots\!00 \nu ) / 41\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 19\!\cdots\!27 \nu^{19} + \cdots + 70\!\cdots\!20 \nu ) / 16\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 14\!\cdots\!18 \nu^{19} + \cdots + 52\!\cdots\!00 \nu ) / 83\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 23\!\cdots\!95 \nu^{19} + \cdots + 92\!\cdots\!40 \nu ) / 67\!\cdots\!60 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 11 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{19} - \beta_{18} - \beta_{15} - 2\beta_{14} - \beta_{11} - \beta_{10} - 21\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -2\beta_{9} - \beta_{8} - \beta_{7} + 6\beta_{6} + 2\beta_{5} + \beta_{3} - 29\beta_{2} + 230 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 35 \beta_{19} + 36 \beta_{18} - 4 \beta_{16} + 29 \beta_{15} + 85 \beta_{14} - 4 \beta_{13} + \cdots + 520 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 8 \beta_{13} - 8 \beta_{12} + 101 \beta_{9} + 28 \beta_{8} + 41 \beta_{7} - 287 \beta_{6} - 93 \beta_{5} + \cdots - 5697 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 1082 \beta_{19} - 1096 \beta_{18} - 14 \beta_{17} + 209 \beta_{16} - 797 \beta_{15} - 2914 \beta_{14} + \cdots - 13735 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 452 \beta_{13} + 452 \beta_{12} - 3836 \beta_{9} - 630 \beta_{8} - 1277 \beta_{7} + 10546 \beta_{6} + \cdots + 151042 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 32586 \beta_{19} + 32273 \beta_{18} + 1242 \beta_{17} - 7710 \beta_{16} + 21759 \beta_{15} + \cdots + 375886 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 17820 \beta_{13} - 17820 \beta_{12} + 131124 \beta_{9} + 13211 \beta_{8} + 36181 \beta_{7} + \cdots - 4151963 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 973381 \beta_{19} - 942343 \beta_{18} - 70492 \beta_{17} + 249073 \beta_{16} - 590517 \beta_{15} + \cdots - 10522547 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 611984 \beta_{13} + 611984 \beta_{12} - 4262295 \beta_{9} - 259867 \beta_{8} - 978061 \beta_{7} + \cdots + 116727490 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 28984329 \beta_{19} + 27450672 \beta_{18} + 3261898 \beta_{17} - 7542076 \beta_{16} + \cdots + 299115586 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 19682748 \beta_{13} - 19682748 \beta_{12} + 134871943 \beta_{9} + 4506026 \beta_{8} + 25652559 \beta_{7} + \cdots - 3331220761 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 862068880 \beta_{19} - 799361310 \beta_{18} - 134364230 \beta_{17} + 220414273 \beta_{16} + \cdots - 8593684991 \beta_1 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 611693664 \beta_{13} + 611693664 \beta_{12} - 4202736306 \beta_{9} - 53303416 \beta_{8} + \cdots + 96055645282 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 25636364028 \beta_{19} + 23290646339 \beta_{18} + 5139929446 \beta_{17} - 6308897298 \beta_{16} + \cdots + 248777063954 \beta_1 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 18655898324 \beta_{13} - 18655898324 \beta_{12} + 129767445918 \beta_{9} - 505181431 \beta_{8} + \cdots - 2790115494851 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 762691472511 \beta_{19} - 679357159293 \beta_{18} - 186940447092 \beta_{17} + 178303540077 \beta_{16} + \cdots - 7241973291955 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3100\mathbb{Z}\right)^\times\).

\(n\) \(1551\) \(1801\) \(2977\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1301.1
− 5.50062i
− 5.24263i
− 4.37942i
− 4.11296i
− 2.65144i
− 2.53909i
− 2.04380i
− 1.19949i
− 0.767468i
− 0.693074i
0.693074i
0.767468i
1.19949i
2.04380i
2.53909i
2.65144i
4.11296i
4.37942i
5.24263i
5.50062i
0 − 5.50062i 0 0 0 6.24508 0 −21.2568 0
1301.2 0 − 5.24263i 0 0 0 −8.32678 0 −18.4851 0
1301.3 0 − 4.37942i 0 0 0 3.17309 0 −10.1793 0
1301.4 0 − 4.11296i 0 0 0 −9.84977 0 −7.91641 0
1301.5 0 − 2.65144i 0 0 0 0.895989 0 1.96986 0
1301.6 0 − 2.53909i 0 0 0 −7.38978 0 2.55304 0
1301.7 0 − 2.04380i 0 0 0 1.40634 0 4.82290 0
1301.8 0 − 1.19949i 0 0 0 8.82419 0 7.56122 0
1301.9 0 − 0.767468i 0 0 0 −9.99820 0 8.41099 0
1301.10 0 − 0.693074i 0 0 0 9.01983 0 8.51965 0
1301.11 0 0.693074i 0 0 0 9.01983 0 8.51965 0
1301.12 0 0.767468i 0 0 0 −9.99820 0 8.41099 0
1301.13 0 1.19949i 0 0 0 8.82419 0 7.56122 0
1301.14 0 2.04380i 0 0 0 1.40634 0 4.82290 0
1301.15 0 2.53909i 0 0 0 −7.38978 0 2.55304 0
1301.16 0 2.65144i 0 0 0 0.895989 0 1.96986 0
1301.17 0 4.11296i 0 0 0 −9.84977 0 −7.91641 0
1301.18 0 4.37942i 0 0 0 3.17309 0 −10.1793 0
1301.19 0 5.24263i 0 0 0 −8.32678 0 −18.4851 0
1301.20 0 5.50062i 0 0 0 6.24508 0 −21.2568 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1301.20
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3100.3.d.e 20
5.b even 2 1 620.3.d.a ✓ 20
5.c odd 4 2 3100.3.f.c 40
31.b odd 2 1 inner 3100.3.d.e 20
155.c odd 2 1 620.3.d.a ✓ 20
155.f even 4 2 3100.3.f.c 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
620.3.d.a ✓ 20 5.b even 2 1
620.3.d.a ✓ 20 155.c odd 2 1
3100.3.d.e 20 1.a even 1 1 trivial
3100.3.d.e 20 31.b odd 2 1 inner
3100.3.f.c 40 5.c odd 4 2
3100.3.f.c 40 155.f even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(3100, [\chi])\):

\( T_{3}^{20} + 114 T_{3}^{18} + 5280 T_{3}^{16} + 128422 T_{3}^{14} + 1776819 T_{3}^{12} + 14249420 T_{3}^{10} + \cdots + 20793600 \) Copy content Toggle raw display
\( T_{7}^{10} + 6 T_{7}^{9} - 248 T_{7}^{8} - 1148 T_{7}^{7} + 22740 T_{7}^{6} + 64656 T_{7}^{5} + \cdots + 12043264 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( T^{20} + 114 T^{18} + \cdots + 20793600 \) Copy content Toggle raw display
$5$ \( T^{20} \) Copy content Toggle raw display
$7$ \( (T^{10} + 6 T^{9} + \cdots + 12043264)^{2} \) Copy content Toggle raw display
$11$ \( T^{20} + \cdots + 471958592753664 \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots + 11\!\cdots\!24 \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 71\!\cdots\!04 \) Copy content Toggle raw display
$19$ \( (T^{10} + \cdots + 194035662400)^{2} \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 17\!\cdots\!84 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 53\!\cdots\!64 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 67\!\cdots\!01 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 12\!\cdots\!24 \) Copy content Toggle raw display
$41$ \( (T^{10} + \cdots + 765486843761664)^{2} \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 32\!\cdots\!64 \) Copy content Toggle raw display
$47$ \( (T^{10} + \cdots - 310988588728320)^{2} \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 68\!\cdots\!24 \) Copy content Toggle raw display
$59$ \( (T^{10} + \cdots - 41\!\cdots\!16)^{2} \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 43\!\cdots\!24 \) Copy content Toggle raw display
$67$ \( (T^{10} + \cdots + 47\!\cdots\!76)^{2} \) Copy content Toggle raw display
$71$ \( (T^{10} + \cdots - 93\!\cdots\!56)^{2} \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 48\!\cdots\!44 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 94\!\cdots\!84 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 12\!\cdots\!04 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 97\!\cdots\!84 \) Copy content Toggle raw display
$97$ \( (T^{10} + \cdots + 30\!\cdots\!36)^{2} \) Copy content Toggle raw display
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