Properties

Label 100.4.a.d
Level $100$
Weight $4$
Character orbit 100.a
Self dual yes
Analytic conductor $5.900$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 100 = 2^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 100.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(5.90019100057\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{19}) \)
Defining polynomial: \( x^{2} - 19 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 20)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{19}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{3} + \beta q^{7} + 49 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{3} + \beta q^{7} + 49 q^{9} + 20 q^{11} - 6 \beta q^{13} - 8 \beta q^{17} + 84 q^{19} + 76 q^{21} + 7 \beta q^{23} + 22 \beta q^{27} - 6 q^{29} - 224 q^{31} + 20 \beta q^{33} - 14 \beta q^{37} - 456 q^{39} + 266 q^{41} - 35 \beta q^{43} - 43 \beta q^{47} - 267 q^{49} - 608 q^{51} - 42 \beta q^{53} + 84 \beta q^{57} + 28 q^{59} + 182 q^{61} + 49 \beta q^{63} - 49 \beta q^{67} + 532 q^{69} + 408 q^{71} + 124 \beta q^{73} + 20 \beta q^{77} - 48 q^{79} + 349 q^{81} - 23 \beta q^{83} - 6 \beta q^{87} + 1526 q^{89} - 456 q^{91} - 224 \beta q^{93} - 64 \beta q^{97} + 980 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 98 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 98 q^{9} + 40 q^{11} + 168 q^{19} + 152 q^{21} - 12 q^{29} - 448 q^{31} - 912 q^{39} + 532 q^{41} - 534 q^{49} - 1216 q^{51} + 56 q^{59} + 364 q^{61} + 1064 q^{69} + 816 q^{71} - 96 q^{79} + 698 q^{81} + 3052 q^{89} - 912 q^{91} + 1960 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−4.35890
4.35890
0 −8.71780 0 0 0 −8.71780 0 49.0000 0
1.2 0 8.71780 0 0 0 8.71780 0 49.0000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(-1\)

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 100.4.a.d 2
3.b odd 2 1 900.4.a.s 2
4.b odd 2 1 400.4.a.w 2
5.b even 2 1 inner 100.4.a.d 2
5.c odd 4 2 20.4.c.a 2
8.b even 2 1 1600.4.a.cj 2
8.d odd 2 1 1600.4.a.ck 2
15.d odd 2 1 900.4.a.s 2
15.e even 4 2 180.4.d.a 2
20.d odd 2 1 400.4.a.w 2
20.e even 4 2 80.4.c.b 2
35.f even 4 2 980.4.e.a 2
40.e odd 2 1 1600.4.a.ck 2
40.f even 2 1 1600.4.a.cj 2
40.i odd 4 2 320.4.c.a 2
40.k even 4 2 320.4.c.b 2
60.l odd 4 2 720.4.f.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.4.c.a 2 5.c odd 4 2
80.4.c.b 2 20.e even 4 2
100.4.a.d 2 1.a even 1 1 trivial
100.4.a.d 2 5.b even 2 1 inner
180.4.d.a 2 15.e even 4 2
320.4.c.a 2 40.i odd 4 2
320.4.c.b 2 40.k even 4 2
400.4.a.w 2 4.b odd 2 1
400.4.a.w 2 20.d odd 2 1
720.4.f.a 2 60.l odd 4 2
900.4.a.s 2 3.b odd 2 1
900.4.a.s 2 15.d odd 2 1
980.4.e.a 2 35.f even 4 2
1600.4.a.cj 2 8.b even 2 1
1600.4.a.cj 2 40.f even 2 1
1600.4.a.ck 2 8.d odd 2 1
1600.4.a.ck 2 40.e odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 76 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(100))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 76 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 76 \) Copy content Toggle raw display
$11$ \( (T - 20)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 2736 \) Copy content Toggle raw display
$17$ \( T^{2} - 4864 \) Copy content Toggle raw display
$19$ \( (T - 84)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 3724 \) Copy content Toggle raw display
$29$ \( (T + 6)^{2} \) Copy content Toggle raw display
$31$ \( (T + 224)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} - 14896 \) Copy content Toggle raw display
$41$ \( (T - 266)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} - 93100 \) Copy content Toggle raw display
$47$ \( T^{2} - 140524 \) Copy content Toggle raw display
$53$ \( T^{2} - 134064 \) Copy content Toggle raw display
$59$ \( (T - 28)^{2} \) Copy content Toggle raw display
$61$ \( (T - 182)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} - 182476 \) Copy content Toggle raw display
$71$ \( (T - 408)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 1168576 \) Copy content Toggle raw display
$79$ \( (T + 48)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} - 40204 \) Copy content Toggle raw display
$89$ \( (T - 1526)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} - 311296 \) Copy content Toggle raw display
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