Properties

Label 100.9.b.g
Level $100$
Weight $9$
Character orbit 100.b
Analytic conductor $40.738$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [100,9,Mod(51,100)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(100, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("100.51");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 100 = 2^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 100.b (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: \(40.7378610061\)
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: \( x^{20} - 94 x^{18} + 5343 x^{16} - 172772 x^{14} + 36131456 x^{12} - 3044563968 x^{10} + \cdots + 11\!\cdots\!76 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{75}\cdot 3^{4}\cdot 5^{14} \)
Twist minimal: no (minimal twist has level 20)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{8} - \beta_1) q^{3} + (\beta_{2} + 38) q^{4} + (\beta_{7} - \beta_{2} + 170) q^{6} + ( - \beta_{14} + 3 \beta_{8} + 3 \beta_1) q^{7} + (\beta_{3} + 38 \beta_1) q^{8} + (\beta_{10} - 2 \beta_{7} + \beta_{5} + \cdots - 130) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{8} - \beta_1) q^{3} + (\beta_{2} + 38) q^{4} + (\beta_{7} - \beta_{2} + 170) q^{6} + ( - \beta_{14} + 3 \beta_{8} + 3 \beta_1) q^{7} + (\beta_{3} + 38 \beta_1) q^{8} + (\beta_{10} - 2 \beta_{7} + \beta_{5} + \cdots - 130) q^{9}+ \cdots + (342 \beta_{15} - 4299 \beta_{12} + \cdots - 25347) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 752 q^{4} + 3408 q^{6} - 2556 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 752 q^{4} + 3408 q^{6} - 2556 q^{9} - 8848 q^{14} - 59200 q^{16} + 410256 q^{21} + 156672 q^{24} - 440448 q^{26} - 660136 q^{29} - 4342528 q^{34} - 7191312 q^{36} + 7068520 q^{41} - 2666880 q^{44} + 561168 q^{46} - 11719036 q^{49} - 37110816 q^{54} - 35044352 q^{56} - 17660440 q^{61} - 20201728 q^{64} + 31902720 q^{66} - 111747216 q^{69} - 19114368 q^{74} - 54998400 q^{76} - 154212444 q^{81} - 101289216 q^{84} + 94429648 q^{86} - 105006376 q^{89} + 192757872 q^{94} + 28850688 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 94 x^{18} + 5343 x^{16} - 172772 x^{14} + 36131456 x^{12} - 3044563968 x^{10} + \cdots + 11\!\cdots\!76 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 4\nu^{2} - 38 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 8\nu^{3} - 76\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3302597 \nu^{18} - 1391343702 \nu^{16} + 29876009883 \nu^{14} + 1448679820940 \nu^{12} + \cdots - 37\!\cdots\!28 ) / 12\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 4117787 \nu^{18} - 336684054 \nu^{16} + 5062818171 \nu^{14} + 677234017804 \nu^{12} + \cdots - 10\!\cdots\!56 ) / 31\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 10120793 \nu^{19} + 1538239314 \nu^{17} - 72882621561 \nu^{15} - 834264549700 \nu^{13} + \cdots + 47\!\cdots\!48 \nu ) / 16\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 607811 \nu^{18} + 30995610 \nu^{16} - 223223901 \nu^{14} + 58863964588 \nu^{12} + \cdots + 60\!\cdots\!92 ) / 39\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 21623225 \nu^{19} + 457010706 \nu^{17} - 11425127385 \nu^{15} - 2821562731204 \nu^{13} + \cdots + 20\!\cdots\!28 \nu ) / 32\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 5555591 \nu^{18} + 201530478 \nu^{16} + 2619368601 \nu^{14} - 925646290492 \nu^{12} + \cdots + 68\!\cdots\!24 ) / 31\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 3819187 \nu^{18} - 514637242 \nu^{16} + 8512799213 \nu^{14} - 487547106668 \nu^{12} + \cdots - 12\!\cdots\!60 ) / 14\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 1687693 \nu^{18} - 208464954 \nu^{16} + 10100867373 \nu^{14} + 136486344596 \nu^{12} + \cdots - 44\!\cdots\!40 ) / 52\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 11250533 \nu^{18} - 356503830 \nu^{16} + 6095187195 \nu^{14} - 1380868386292 \nu^{12} + \cdots - 16\!\cdots\!96 ) / 15\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 3849039 \nu^{19} - 564836098 \nu^{17} + 12303037905 \nu^{15} - 1038892195932 \nu^{13} + \cdots - 10\!\cdots\!76 \nu ) / 59\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 32459431 \nu^{19} - 1527461550 \nu^{17} + 42771655815 \nu^{15} + \cdots - 61\!\cdots\!48 \nu ) / 26\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 92592211 \nu^{18} + 1719020154 \nu^{16} + 10180103091 \nu^{14} + 9232416044012 \nu^{12} + \cdots + 68\!\cdots\!68 ) / 31\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 383591717 \nu^{19} - 52518900630 \nu^{17} + 661582634811 \nu^{15} + \cdots - 13\!\cdots\!44 \nu ) / 26\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 534030799 \nu^{19} + 38122392066 \nu^{17} - 484129093457 \nu^{15} + \cdots + 78\!\cdots\!20 \nu ) / 35\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 4125908705 \nu^{19} - 268226021022 \nu^{17} + 2193308998143 \nu^{15} + \cdots - 55\!\cdots\!12 \nu ) / 16\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 1651996615 \nu^{19} - 148477643538 \nu^{17} + 2203492598361 \nu^{15} + \cdots - 33\!\cdots\!40 \nu ) / 53\!\cdots\!32 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{2} + 38 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{3} + 38\beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -\beta_{15} - 3\beta_{12} + \beta_{11} + \beta_{9} + \beta_{7} + 3\beta_{5} - 4\beta_{4} + 38\beta_{2} - 2942 ) / 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - \beta_{18} + 6 \beta_{17} + 4 \beta_{16} - 33 \beta_{14} + 22 \beta_{13} + 70 \beta_{8} + \cdots - 708 \beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 13 \beta_{15} - 13 \beta_{12} - 9 \beta_{11} - 42 \beta_{10} + 199 \beta_{9} + 21 \beta_{7} + \cdots - 126284 ) / 8 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 302 \beta_{19} + 198 \beta_{18} - 76 \beta_{17} + 184 \beta_{16} - 2308 \beta_{14} + \cdots - 74600 \beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 2173 \beta_{15} + 5761 \beta_{12} - 139 \beta_{11} + 2008 \beta_{10} - 6091 \beta_{9} + \cdots - 213160862 ) / 16 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 6732 \beta_{19} - 3821 \beta_{18} - 7170 \beta_{17} - 9260 \beta_{16} - 169897 \beta_{14} + \cdots - 54757548 \beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 43657 \beta_{15} - 33706 \beta_{12} - 10014 \beta_{11} + 190175 \beta_{10} - 100502 \beta_{9} + \cdots - 75256117 ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 580346 \beta_{19} - 1258084 \beta_{18} - 118064 \beta_{17} + 615200 \beta_{16} - 9935094 \beta_{14} + \cdots - 103249184 \beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 6059779 \beta_{15} + 28400713 \beta_{12} - 14017923 \beta_{11} - 10424432 \beta_{10} + \cdots + 213163934826 ) / 16 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 41117344 \beta_{19} + 22827667 \beta_{18} - 43848850 \beta_{17} - 81976844 \beta_{16} + \cdots + 50040075636 \beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 81463825 \beta_{15} + 243032197 \beta_{12} + 344100561 \beta_{11} + 922044958 \beta_{10} + \cdots - 2477399821968 ) / 8 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 9617817002 \beta_{19} - 6285977854 \beta_{18} + 3398721388 \beta_{17} - 5988895864 \beta_{16} + \cdots - 1009722348984 \beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( 31396928847 \beta_{15} - 178835848715 \beta_{12} + 3613266745 \beta_{11} - 19365091832 \beta_{10} + \cdots + 14\!\cdots\!66 ) / 16 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( - 103165004876 \beta_{19} - 41335429089 \beta_{18} + 222830891798 \beta_{17} + \cdots + 401708096627684 \beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( ( 942674446596 \beta_{15} + 765578160849 \beta_{12} + 206826061833 \beta_{11} - 5603630468293 \beta_{10} + \cdots - 21\!\cdots\!59 ) / 4 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( - 9760000253310 \beta_{19} + 35266025760656 \beta_{18} + 6286559214840 \beta_{17} + \cdots - 20\!\cdots\!60 \beta_1 ) / 8 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/100\mathbb{Z}\right)^\times\).

\(n\) \(51\) \(77\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
51.1
−7.73824 2.02968i
−7.73824 + 2.02968i
−7.17826 3.53165i
−7.17826 + 3.53165i
−7.13604 3.61621i
−7.13604 + 3.61621i
−3.58697 7.15078i
−3.58697 + 7.15078i
−2.88145 7.46306i
−2.88145 + 7.46306i
2.88145 7.46306i
2.88145 + 7.46306i
3.58697 7.15078i
3.58697 + 7.15078i
7.13604 3.61621i
7.13604 + 3.61621i
7.17826 3.53165i
7.17826 + 3.53165i
7.73824 2.02968i
7.73824 + 2.02968i
−15.4765 4.05935i 75.2390i 223.043 + 125.649i 0 −305.422 + 1164.44i 4411.35i −2941.87 2850.02i 900.091 0
51.2 −15.4765 + 4.05935i 75.2390i 223.043 125.649i 0 −305.422 1164.44i 4411.35i −2941.87 + 2850.02i 900.091 0
51.3 −14.3565 7.06329i 134.970i 156.220 + 202.809i 0 953.329 1937.69i 1863.96i −810.276 4015.06i −11655.8 0
51.4 −14.3565 + 7.06329i 134.970i 156.220 202.809i 0 953.329 + 1937.69i 1863.96i −810.276 + 4015.06i −11655.8 0
51.5 −14.2721 7.23243i 44.3270i 151.384 + 206.443i 0 −320.592 + 632.638i 2826.75i −667.476 4041.25i 4596.11 0
51.6 −14.2721 + 7.23243i 44.3270i 151.384 206.443i 0 −320.592 632.638i 2826.75i −667.476 + 4041.25i 4596.11 0
51.7 −7.17394 14.3016i 42.6663i −153.069 + 205.197i 0 −610.194 + 306.085i 869.685i 4032.75 + 717.054i 4740.59 0
51.8 −7.17394 + 14.3016i 42.6663i −153.069 205.197i 0 −610.194 306.085i 869.685i 4032.75 717.054i 4740.59 0
51.9 −5.76290 14.9261i 76.0331i −189.578 + 172.035i 0 1134.88 438.171i 269.408i 3660.34 + 1838.24i 779.974 0
51.10 −5.76290 + 14.9261i 76.0331i −189.578 172.035i 0 1134.88 + 438.171i 269.408i 3660.34 1838.24i 779.974 0
51.11 5.76290 14.9261i 76.0331i −189.578 172.035i 0 1134.88 + 438.171i 269.408i −3660.34 + 1838.24i 779.974 0
51.12 5.76290 + 14.9261i 76.0331i −189.578 + 172.035i 0 1134.88 438.171i 269.408i −3660.34 1838.24i 779.974 0
51.13 7.17394 14.3016i 42.6663i −153.069 205.197i 0 −610.194 306.085i 869.685i −4032.75 + 717.054i 4740.59 0
51.14 7.17394 + 14.3016i 42.6663i −153.069 + 205.197i 0 −610.194 + 306.085i 869.685i −4032.75 717.054i 4740.59 0
51.15 14.2721 7.23243i 44.3270i 151.384 206.443i 0 −320.592 632.638i 2826.75i 667.476 4041.25i 4596.11 0
51.16 14.2721 + 7.23243i 44.3270i 151.384 + 206.443i 0 −320.592 + 632.638i 2826.75i 667.476 + 4041.25i 4596.11 0
51.17 14.3565 7.06329i 134.970i 156.220 202.809i 0 953.329 + 1937.69i 1863.96i 810.276 4015.06i −11655.8 0
51.18 14.3565 + 7.06329i 134.970i 156.220 + 202.809i 0 953.329 1937.69i 1863.96i 810.276 + 4015.06i −11655.8 0
51.19 15.4765 4.05935i 75.2390i 223.043 125.649i 0 −305.422 1164.44i 4411.35i 2941.87 2850.02i 900.091 0
51.20 15.4765 + 4.05935i 75.2390i 223.043 + 125.649i 0 −305.422 + 1164.44i 4411.35i 2941.87 + 2850.02i 900.091 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 51.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.b even 2 1 inner
20.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 100.9.b.g 20
4.b odd 2 1 inner 100.9.b.g 20
5.b even 2 1 inner 100.9.b.g 20
5.c odd 4 2 20.9.d.c 20
20.d odd 2 1 inner 100.9.b.g 20
20.e even 4 2 20.9.d.c 20
40.i odd 4 2 320.9.h.g 20
40.k even 4 2 320.9.h.g 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.9.d.c 20 5.c odd 4 2
20.9.d.c 20 20.e even 4 2
100.9.b.g 20 1.a even 1 1 trivial
100.9.b.g 20 4.b odd 2 1 inner
100.9.b.g 20 5.b even 2 1 inner
100.9.b.g 20 20.d odd 2 1 inner
320.9.h.g 20 40.i odd 4 2
320.9.h.g 20 40.k even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{9}^{\mathrm{new}}(100, [\chi])\):

\( T_{3}^{10} + 33444 T_{3}^{8} + 357004800 T_{3}^{6} + 1615110935040 T_{3}^{4} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
\( T_{13}^{10} - 4307527296 T_{13}^{8} + \cdots - 15\!\cdots\!00 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} + \cdots + 12\!\cdots\!76 \) Copy content Toggle raw display
$3$ \( (T^{10} + \cdots + 21\!\cdots\!00)^{2} \) Copy content Toggle raw display
$5$ \( T^{20} \) Copy content Toggle raw display
$7$ \( (T^{10} + \cdots + 29\!\cdots\!00)^{2} \) Copy content Toggle raw display
$11$ \( (T^{10} + \cdots + 15\!\cdots\!00)^{2} \) Copy content Toggle raw display
$13$ \( (T^{10} + \cdots - 15\!\cdots\!00)^{2} \) Copy content Toggle raw display
$17$ \( (T^{10} + \cdots - 57\!\cdots\!00)^{2} \) Copy content Toggle raw display
$19$ \( (T^{10} + \cdots + 54\!\cdots\!00)^{2} \) Copy content Toggle raw display
$23$ \( (T^{10} + \cdots + 72\!\cdots\!00)^{2} \) Copy content Toggle raw display
$29$ \( (T^{5} + \cdots + 30\!\cdots\!52)^{4} \) Copy content Toggle raw display
$31$ \( (T^{10} + \cdots + 51\!\cdots\!00)^{2} \) Copy content Toggle raw display
$37$ \( (T^{10} + \cdots - 30\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( (T^{5} + \cdots + 51\!\cdots\!28)^{4} \) Copy content Toggle raw display
$43$ \( (T^{10} + \cdots + 13\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( (T^{10} + \cdots + 25\!\cdots\!00)^{2} \) Copy content Toggle raw display
$53$ \( (T^{10} + \cdots - 74\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( (T^{10} + \cdots + 53\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{5} + \cdots + 50\!\cdots\!68)^{4} \) Copy content Toggle raw display
$67$ \( (T^{10} + \cdots + 19\!\cdots\!00)^{2} \) Copy content Toggle raw display
$71$ \( (T^{10} + \cdots + 92\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( (T^{10} + \cdots - 86\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( (T^{10} + \cdots + 74\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( (T^{10} + \cdots + 78\!\cdots\!00)^{2} \) Copy content Toggle raw display
$89$ \( (T^{5} + \cdots - 38\!\cdots\!28)^{4} \) Copy content Toggle raw display
$97$ \( (T^{10} + \cdots - 34\!\cdots\!00)^{2} \) Copy content Toggle raw display
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