Properties

Label 100.3.b.f
Level $100$
Weight $3$
Character orbit 100.b
Analytic conductor $2.725$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [100,3,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, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("100.51");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 100 = 2^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
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: \(2.72480264360\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{4} \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{2} q^{3} + (\beta_{3} + \beta_{2} + \beta_1 - 1) q^{4} + ( - 3 \beta_{3} + \beta_{2} - \beta_1 - 1) q^{6} + (2 \beta_{3} - \beta_{2} - 2 \beta_1 + 2) q^{7} + ( - 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 2) q^{8} + ( - 2 \beta_{3} - 2 \beta_1 - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + \beta_{2} q^{3} + (\beta_{3} + \beta_{2} + \beta_1 - 1) q^{4} + ( - 3 \beta_{3} + \beta_{2} - \beta_1 - 1) q^{6} + (2 \beta_{3} - \beta_{2} - 2 \beta_1 + 2) q^{7} + ( - 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 2) q^{8} + ( - 2 \beta_{3} - 2 \beta_1 - 1) q^{9} + (4 \beta_{3} + 2 \beta_{2} - 4 \beta_1 + 4) q^{11} + ( - 4 \beta_{3} - 12) q^{12} + ( - 2 \beta_{3} - 2 \beta_1 + 4) q^{13} + (\beta_{3} - 3 \beta_{2} - \beta_1 + 11) q^{14} + ( - 8 \beta_{3} - 8) q^{16} + (8 \beta_{3} + 8 \beta_1 + 6) q^{17} + ( - 2 \beta_{3} - 2 \beta_{2} - \beta_1 - 6) q^{18} + ( - 4 \beta_{3} + 4 \beta_1 - 4) q^{19} + (10 \beta_{3} + 10 \beta_1 + 10) q^{21} + ( - 10 \beta_{3} - 2 \beta_{2} - 6 \beta_1 + 18) q^{22} + (2 \beta_{3} - 3 \beta_{2} - 2 \beta_1 + 2) q^{23} + ( - 8 \beta_1 - 16) q^{24} + ( - 2 \beta_{3} - 2 \beta_{2} + 4 \beta_1 - 6) q^{26} + (4 \beta_{3} + 6 \beta_{2} - 4 \beta_1 + 4) q^{27} + (8 \beta_{3} - 4 \beta_{2} + 12 \beta_1 + 8) q^{28} + (4 \beta_{3} + 4 \beta_1 - 2) q^{29} + (4 \beta_{3} - 10 \beta_{2} - 4 \beta_1 + 4) q^{31} - 32 q^{32} + (12 \beta_{3} + 12 \beta_1 - 20) q^{33} + (8 \beta_{3} + 8 \beta_{2} + 6 \beta_1 + 24) q^{34} + (5 \beta_{3} - 3 \beta_{2} - 3 \beta_1 - 5) q^{36} + ( - 10 \beta_{3} - 10 \beta_1 - 4) q^{37} + (4 \beta_{3} + 4 \beta_{2} + 4 \beta_1 - 20) q^{38} + (4 \beta_{3} + 2 \beta_{2} - 4 \beta_1 + 4) q^{39} + ( - 6 \beta_{3} - 6 \beta_1 - 28) q^{41} + (10 \beta_{3} + 10 \beta_{2} + 10 \beta_1 + 30) q^{42} + ( - 4 \beta_{3} - 3 \beta_{2} + 4 \beta_1 - 4) q^{43} + ( - 8 \beta_{2} + 24 \beta_1 - 32) q^{44} + (7 \beta_{3} - 5 \beta_{2} + \beta_1 + 13) q^{46} + ( - 6 \beta_{3} - 13 \beta_{2} + 6 \beta_1 - 6) q^{47} + ( - 8 \beta_{3} - 8 \beta_{2} - 24 \beta_1 + 8) q^{48} + ( - 10 \beta_{3} - 10 \beta_1 - 1) q^{49} + ( - 16 \beta_{3} + 14 \beta_{2} + 16 \beta_1 - 16) q^{51} + (10 \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 10) q^{52} + ( - 10 \beta_{3} - 10 \beta_1 + 44) q^{53} + ( - 22 \beta_{3} + 2 \beta_{2} - 10 \beta_1 + 14) q^{54} + (24 \beta_{3} + 8 \beta_{2} + 16 \beta_1 + 24) q^{56} + ( - 16 \beta_{3} - 16 \beta_1) q^{57} + (4 \beta_{3} + 4 \beta_{2} - 2 \beta_1 + 12) q^{58} + (12 \beta_{3} - 12 \beta_{2} - 12 \beta_1 + 12) q^{59} + ( - 26 \beta_{3} - 26 \beta_1 + 32) q^{61} + (26 \beta_{3} - 14 \beta_{2} + 6 \beta_1 + 30) q^{62} + ( - 2 \beta_{3} + 11 \beta_{2} + 2 \beta_1 - 2) q^{63} - 32 \beta_1 q^{64} + (12 \beta_{3} + 12 \beta_{2} - 20 \beta_1 + 36) q^{66} + ( - 20 \beta_{3} + 11 \beta_{2} + 20 \beta_1 - 20) q^{67} + ( - 18 \beta_{3} + 14 \beta_{2} + 14 \beta_1 + 18) q^{68} + (14 \beta_{3} + 14 \beta_1 + 30) q^{69} + ( - 20 \beta_{3} - 2 \beta_{2} + 20 \beta_1 - 20) q^{71} + (6 \beta_{3} - 6 \beta_{2} - 10 \beta_1 + 26) q^{72} + ( - 32 \beta_{3} - 32 \beta_1 - 66) q^{73} + ( - 10 \beta_{3} - 10 \beta_{2} - 4 \beta_1 - 30) q^{74} + ( - 8 \beta_{3} + 8 \beta_{2} - 24 \beta_1 + 8) q^{76} + (20 \beta_{3} + 20 \beta_1 - 60) q^{77} + ( - 10 \beta_{3} - 2 \beta_{2} - 6 \beta_1 + 18) q^{78} + ( - 16 \beta_{3} + 20 \beta_{2} + 16 \beta_1 - 16) q^{79} + ( - 14 \beta_{3} - 14 \beta_1 - 69) q^{81} + ( - 6 \beta_{3} - 6 \beta_{2} - 28 \beta_1 - 18) q^{82} + (12 \beta_{3} + 13 \beta_{2} - 12 \beta_1 + 12) q^{83} + ( - 20 \beta_{3} + 20 \beta_{2} + 20 \beta_1 + 20) q^{84} + (13 \beta_{3} + \beta_{2} + 7 \beta_1 - 17) q^{86} + ( - 8 \beta_{3} + 2 \beta_{2} + 8 \beta_1 - 8) q^{87} + (48 \beta_{3} + 16 \beta_{2} - 16) q^{88} + (40 \beta_{3} + 40 \beta_1 - 22) q^{89} + (8 \beta_{3} + 6 \beta_{2} - 8 \beta_1 + 8) q^{91} + (16 \beta_{3} - 4 \beta_{2} + 12 \beta_1 + 32) q^{92} + (36 \beta_{3} + 36 \beta_1 + 100) q^{93} + (45 \beta_{3} - 7 \beta_{2} + 19 \beta_1 - 17) q^{94} - 32 \beta_{2} q^{96} + ( - 12 \beta_{3} - 12 \beta_1 + 66) q^{97} + ( - 10 \beta_{3} - 10 \beta_{2} - \beta_1 - 30) q^{98} + (12 \beta_{3} + 10 \beta_{2} - 12 \beta_1 + 12) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} - 4 q^{4} + 8 q^{8} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{2} - 4 q^{4} + 8 q^{8} - 4 q^{9} - 40 q^{12} + 16 q^{13} + 40 q^{14} - 16 q^{16} + 24 q^{17} - 22 q^{18} + 40 q^{21} + 80 q^{22} - 80 q^{24} - 12 q^{26} + 40 q^{28} - 8 q^{29} - 128 q^{32} - 80 q^{33} + 92 q^{34} - 36 q^{36} - 16 q^{37} - 80 q^{38} - 112 q^{41} + 120 q^{42} - 80 q^{44} + 40 q^{46} - 4 q^{49} - 56 q^{52} + 176 q^{53} + 80 q^{54} + 80 q^{56} + 36 q^{58} + 128 q^{61} + 80 q^{62} - 64 q^{64} + 80 q^{66} + 136 q^{68} + 120 q^{69} + 72 q^{72} - 264 q^{73} - 108 q^{74} - 240 q^{77} + 80 q^{78} - 276 q^{81} - 116 q^{82} + 160 q^{84} - 80 q^{86} - 160 q^{88} - 88 q^{89} + 120 q^{92} + 400 q^{93} - 120 q^{94} + 264 q^{97} - 102 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( 2\zeta_{10} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\zeta_{10}^{3} + 2\zeta_{10}^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -2\zeta_{10}^{3} + 2\zeta_{10}^{2} - 2\zeta_{10} + 1 \) Copy content Toggle raw display
\(\zeta_{10}\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{10}^{2}\)\(=\) \( ( \beta_{3} + \beta_{2} + \beta _1 - 1 ) / 4 \) Copy content Toggle raw display
\(\zeta_{10}^{3}\)\(=\) \( ( -\beta_{3} + \beta_{2} - \beta _1 + 1 ) / 4 \) 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
−0.309017 0.951057i
−0.309017 + 0.951057i
0.809017 0.587785i
0.809017 + 0.587785i
−0.618034 1.90211i 2.35114i −3.23607 + 2.35114i 0 4.47214 1.45309i 5.25731i 6.47214 + 4.70228i 3.47214 0
51.2 −0.618034 + 1.90211i 2.35114i −3.23607 2.35114i 0 4.47214 + 1.45309i 5.25731i 6.47214 4.70228i 3.47214 0
51.3 1.61803 1.17557i 3.80423i 1.23607 3.80423i 0 −4.47214 6.15537i 8.50651i −2.47214 7.60845i −5.47214 0
51.4 1.61803 + 1.17557i 3.80423i 1.23607 + 3.80423i 0 −4.47214 + 6.15537i 8.50651i −2.47214 + 7.60845i −5.47214 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 100.3.b.f 4
3.b odd 2 1 900.3.c.k 4
4.b odd 2 1 inner 100.3.b.f 4
5.b even 2 1 20.3.b.a 4
5.c odd 4 2 100.3.d.b 8
8.b even 2 1 1600.3.b.s 4
8.d odd 2 1 1600.3.b.s 4
12.b even 2 1 900.3.c.k 4
15.d odd 2 1 180.3.c.a 4
15.e even 4 2 900.3.f.e 8
20.d odd 2 1 20.3.b.a 4
20.e even 4 2 100.3.d.b 8
40.e odd 2 1 320.3.b.c 4
40.f even 2 1 320.3.b.c 4
40.i odd 4 2 1600.3.h.n 8
40.k even 4 2 1600.3.h.n 8
60.h even 2 1 180.3.c.a 4
60.l odd 4 2 900.3.f.e 8
80.k odd 4 2 1280.3.g.e 8
80.q even 4 2 1280.3.g.e 8
120.i odd 2 1 2880.3.e.e 4
120.m even 2 1 2880.3.e.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.3.b.a 4 5.b even 2 1
20.3.b.a 4 20.d odd 2 1
100.3.b.f 4 1.a even 1 1 trivial
100.3.b.f 4 4.b odd 2 1 inner
100.3.d.b 8 5.c odd 4 2
100.3.d.b 8 20.e even 4 2
180.3.c.a 4 15.d odd 2 1
180.3.c.a 4 60.h even 2 1
320.3.b.c 4 40.e odd 2 1
320.3.b.c 4 40.f even 2 1
900.3.c.k 4 3.b odd 2 1
900.3.c.k 4 12.b even 2 1
900.3.f.e 8 15.e even 4 2
900.3.f.e 8 60.l odd 4 2
1280.3.g.e 8 80.k odd 4 2
1280.3.g.e 8 80.q even 4 2
1600.3.b.s 4 8.b even 2 1
1600.3.b.s 4 8.d odd 2 1
1600.3.h.n 8 40.i odd 4 2
1600.3.h.n 8 40.k even 4 2
2880.3.e.e 4 120.i odd 2 1
2880.3.e.e 4 120.m even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(100, [\chi])\):

\( T_{3}^{4} + 20T_{3}^{2} + 80 \) Copy content Toggle raw display
\( T_{13}^{2} - 8T_{13} - 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 2 T^{3} + 4 T^{2} - 8 T + 16 \) Copy content Toggle raw display
$3$ \( T^{4} + 20T^{2} + 80 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 100T^{2} + 2000 \) Copy content Toggle raw display
$11$ \( T^{4} + 400T^{2} + 1280 \) Copy content Toggle raw display
$13$ \( (T^{2} - 8 T - 4)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 12 T - 284)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} + 320 T^{2} + 20480 \) Copy content Toggle raw display
$23$ \( T^{4} + 260T^{2} + 80 \) Copy content Toggle raw display
$29$ \( (T^{2} + 4 T - 76)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + 2320 T^{2} + 154880 \) Copy content Toggle raw display
$37$ \( (T^{2} + 8 T - 484)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 56 T + 604)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 500T^{2} + 2000 \) Copy content Toggle raw display
$47$ \( T^{4} + 4100 T^{2} + \cdots + 3561680 \) Copy content Toggle raw display
$53$ \( (T^{2} - 88 T + 1436)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 5760 T^{2} + \cdots + 1658880 \) Copy content Toggle raw display
$61$ \( (T^{2} - 64 T - 2356)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 10420 T^{2} + \cdots + 19920080 \) Copy content Toggle raw display
$71$ \( T^{4} + 8080 T^{2} + \cdots + 10138880 \) Copy content Toggle raw display
$73$ \( (T^{2} + 132 T - 764)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} + 13120 T^{2} + \cdots + 2478080 \) Copy content Toggle raw display
$83$ \( T^{4} + 6260 T^{2} + \cdots + 2620880 \) Copy content Toggle raw display
$89$ \( (T^{2} + 44 T - 7516)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 132 T + 3636)^{2} \) Copy content Toggle raw display
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