Properties

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

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 100 = 2^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 100.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.90019100057\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 20)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 \beta q^{3} + 8 \beta q^{7} + 11 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 \beta q^{3} + 8 \beta q^{7} + 11 q^{9} - 60 q^{11} + 43 \beta q^{13} - 9 \beta q^{17} - 44 q^{19} - 64 q^{21} + 24 \beta q^{23} + 76 \beta q^{27} + 186 q^{29} + 176 q^{31} - 120 \beta q^{33} - 127 \beta q^{37} - 344 q^{39} + 186 q^{41} - 50 \beta q^{43} - 84 \beta q^{47} + 87 q^{49} + 72 q^{51} - 249 \beta q^{53} - 88 \beta q^{57} + 252 q^{59} - 58 q^{61} + 88 \beta q^{63} + 518 \beta q^{67} - 192 q^{69} + 168 q^{71} + 253 \beta q^{73} - 480 \beta q^{77} - 272 q^{79} - 311 q^{81} + 474 \beta q^{83} + 372 \beta q^{87} + 1014 q^{89} - 1376 q^{91} + 352 \beta q^{93} + 383 \beta q^{97} - 660 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 22 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 22 q^{9} - 120 q^{11} - 88 q^{19} - 128 q^{21} + 372 q^{29} + 352 q^{31} - 688 q^{39} + 372 q^{41} + 174 q^{49} + 144 q^{51} + 504 q^{59} - 116 q^{61} - 384 q^{69} + 336 q^{71} - 544 q^{79} - 622 q^{81} + 2028 q^{89} - 2752 q^{91} - 1320 q^{99}+O(q^{100}) \) 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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
1.00000i
1.00000i
0 4.00000i 0 0 0 16.0000i 0 11.0000 0
49.2 0 4.00000i 0 0 0 16.0000i 0 11.0000 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
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.c.a 2
3.b odd 2 1 900.4.d.k 2
4.b odd 2 1 400.4.c.j 2
5.b even 2 1 inner 100.4.c.a 2
5.c odd 4 1 20.4.a.a 1
5.c odd 4 1 100.4.a.a 1
15.d odd 2 1 900.4.d.k 2
15.e even 4 1 180.4.a.a 1
15.e even 4 1 900.4.a.m 1
20.d odd 2 1 400.4.c.j 2
20.e even 4 1 80.4.a.c 1
20.e even 4 1 400.4.a.o 1
35.f even 4 1 980.4.a.c 1
35.k even 12 2 980.4.i.n 2
35.l odd 12 2 980.4.i.e 2
40.i odd 4 1 320.4.a.d 1
40.i odd 4 1 1600.4.a.bl 1
40.k even 4 1 320.4.a.k 1
40.k even 4 1 1600.4.a.p 1
45.k odd 12 2 1620.4.i.d 2
45.l even 12 2 1620.4.i.j 2
55.e even 4 1 2420.4.a.d 1
60.l odd 4 1 720.4.a.k 1
80.i odd 4 1 1280.4.d.n 2
80.j even 4 1 1280.4.d.c 2
80.s even 4 1 1280.4.d.c 2
80.t odd 4 1 1280.4.d.n 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.4.a.a 1 5.c odd 4 1
80.4.a.c 1 20.e even 4 1
100.4.a.a 1 5.c odd 4 1
100.4.c.a 2 1.a even 1 1 trivial
100.4.c.a 2 5.b even 2 1 inner
180.4.a.a 1 15.e even 4 1
320.4.a.d 1 40.i odd 4 1
320.4.a.k 1 40.k even 4 1
400.4.a.o 1 20.e even 4 1
400.4.c.j 2 4.b odd 2 1
400.4.c.j 2 20.d odd 2 1
720.4.a.k 1 60.l odd 4 1
900.4.a.m 1 15.e even 4 1
900.4.d.k 2 3.b odd 2 1
900.4.d.k 2 15.d odd 2 1
980.4.a.c 1 35.f even 4 1
980.4.i.e 2 35.l odd 12 2
980.4.i.n 2 35.k even 12 2
1280.4.d.c 2 80.j even 4 1
1280.4.d.c 2 80.s even 4 1
1280.4.d.n 2 80.i odd 4 1
1280.4.d.n 2 80.t odd 4 1
1600.4.a.p 1 40.k even 4 1
1600.4.a.bl 1 40.i odd 4 1
1620.4.i.d 2 45.k odd 12 2
1620.4.i.j 2 45.l even 12 2
2420.4.a.d 1 55.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 16 \) acting on \(S_{4}^{\mathrm{new}}(100, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 16 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 256 \) Copy content Toggle raw display
$11$ \( (T + 60)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 7396 \) Copy content Toggle raw display
$17$ \( T^{2} + 324 \) Copy content Toggle raw display
$19$ \( (T + 44)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 2304 \) Copy content Toggle raw display
$29$ \( (T - 186)^{2} \) Copy content Toggle raw display
$31$ \( (T - 176)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 64516 \) Copy content Toggle raw display
$41$ \( (T - 186)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 10000 \) Copy content Toggle raw display
$47$ \( T^{2} + 28224 \) Copy content Toggle raw display
$53$ \( T^{2} + 248004 \) Copy content Toggle raw display
$59$ \( (T - 252)^{2} \) Copy content Toggle raw display
$61$ \( (T + 58)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 1073296 \) Copy content Toggle raw display
$71$ \( (T - 168)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 256036 \) Copy content Toggle raw display
$79$ \( (T + 272)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 898704 \) Copy content Toggle raw display
$89$ \( (T - 1014)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 586756 \) Copy content Toggle raw display
show more
show less