Properties

Label 100.6.c.a
Level $100$
Weight $6$
Character orbit 100.c
Analytic conductor $16.038$
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: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 100.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(16.0383819813\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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.

\(f(q)\) \(=\) \( q + 22 i q^{3} -218 i q^{7} -241 q^{9} +O(q^{10})\) \( q + 22 i q^{3} -218 i q^{7} -241 q^{9} -480 q^{11} -622 i q^{13} -186 i q^{17} + 1204 q^{19} + 4796 q^{21} -3186 i q^{23} + 44 i q^{27} -5526 q^{29} + 9356 q^{31} -10560 i q^{33} -5618 i q^{37} + 13684 q^{39} -14394 q^{41} -370 i q^{43} -16146 i q^{47} -30717 q^{49} + 4092 q^{51} -4374 i q^{53} + 26488 i q^{57} + 11748 q^{59} + 13202 q^{61} + 52538 i q^{63} + 11542 i q^{67} + 70092 q^{69} -29532 q^{71} + 33698 i q^{73} + 104640 i q^{77} -31208 q^{79} -59531 q^{81} -38466 i q^{83} -121572 i q^{87} -119514 q^{89} -135596 q^{91} + 205832 i q^{93} -94658 i q^{97} + 115680 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 482q^{9} + O(q^{10}) \) \( 2q - 482q^{9} - 960q^{11} + 2408q^{19} + 9592q^{21} - 11052q^{29} + 18712q^{31} + 27368q^{39} - 28788q^{41} - 61434q^{49} + 8184q^{51} + 23496q^{59} + 26404q^{61} + 140184q^{69} - 59064q^{71} - 62416q^{79} - 119062q^{81} - 239028q^{89} - 271192q^{91} + 231360q^{99} + O(q^{100}) \)

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 22.0000i 0 0 0 218.000i 0 −241.000 0
49.2 0 22.0000i 0 0 0 218.000i 0 −241.000 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 100.6.c.a 2
3.b odd 2 1 900.6.d.h 2
4.b odd 2 1 400.6.c.c 2
5.b even 2 1 inner 100.6.c.a 2
5.c odd 4 1 20.6.a.a 1
5.c odd 4 1 100.6.a.a 1
15.d odd 2 1 900.6.d.h 2
15.e even 4 1 180.6.a.e 1
15.e even 4 1 900.6.a.b 1
20.d odd 2 1 400.6.c.c 2
20.e even 4 1 80.6.a.b 1
20.e even 4 1 400.6.a.m 1
35.f even 4 1 980.6.a.b 1
40.i odd 4 1 320.6.a.c 1
40.k even 4 1 320.6.a.n 1
60.l odd 4 1 720.6.a.l 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.6.a.a 1 5.c odd 4 1
80.6.a.b 1 20.e even 4 1
100.6.a.a 1 5.c odd 4 1
100.6.c.a 2 1.a even 1 1 trivial
100.6.c.a 2 5.b even 2 1 inner
180.6.a.e 1 15.e even 4 1
320.6.a.c 1 40.i odd 4 1
320.6.a.n 1 40.k even 4 1
400.6.a.m 1 20.e even 4 1
400.6.c.c 2 4.b odd 2 1
400.6.c.c 2 20.d odd 2 1
720.6.a.l 1 60.l odd 4 1
900.6.a.b 1 15.e even 4 1
900.6.d.h 2 3.b odd 2 1
900.6.d.h 2 15.d odd 2 1
980.6.a.b 1 35.f even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 484 \) acting on \(S_{6}^{\mathrm{new}}(100, [\chi])\).