Properties

Label 3900.2.h.b
Level $3900$
Weight $2$
Character orbit 3900.h
Analytic conductor $31.142$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 3900 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3900.h (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(31.1416567883\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 156)
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.

\(f(q)\) \(=\) \( q -i q^{3} + 2 i q^{7} - q^{9} +O(q^{10})\) \( q -i q^{3} + 2 i q^{7} - q^{9} -4 q^{11} + i q^{13} -2 i q^{17} + 2 q^{19} + 2 q^{21} + i q^{27} + 6 q^{29} -10 q^{31} + 4 i q^{33} -10 i q^{37} + q^{39} + 8 q^{41} + 4 i q^{43} + 4 i q^{47} + 3 q^{49} -2 q^{51} -10 i q^{53} -2 i q^{57} + 8 q^{59} -14 q^{61} -2 i q^{63} -2 i q^{67} + 16 q^{71} -10 i q^{73} -8 i q^{77} + 16 q^{79} + q^{81} -6 i q^{87} + 4 q^{89} -2 q^{91} + 10 i q^{93} + 2 i q^{97} + 4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{9} + O(q^{10}) \) \( 2q - 2q^{9} - 8q^{11} + 4q^{19} + 4q^{21} + 12q^{29} - 20q^{31} + 2q^{39} + 16q^{41} + 6q^{49} - 4q^{51} + 16q^{59} - 28q^{61} + 32q^{71} + 32q^{79} + 2q^{81} + 8q^{89} - 4q^{91} + 8q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3900\mathbb{Z}\right)^\times\).

\(n\) \(301\) \(1301\) \(1951\) \(3277\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1249.1
1.00000i
1.00000i
0 1.00000i 0 0 0 2.00000i 0 −1.00000 0
1249.2 0 1.00000i 0 0 0 2.00000i 0 −1.00000 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 3900.2.h.b 2
5.b even 2 1 inner 3900.2.h.b 2
5.c odd 4 1 156.2.a.a 1
5.c odd 4 1 3900.2.a.m 1
15.e even 4 1 468.2.a.d 1
20.e even 4 1 624.2.a.e 1
35.f even 4 1 7644.2.a.k 1
40.i odd 4 1 2496.2.a.bc 1
40.k even 4 1 2496.2.a.o 1
45.k odd 12 2 4212.2.i.l 2
45.l even 12 2 4212.2.i.b 2
60.l odd 4 1 1872.2.a.s 1
65.f even 4 1 2028.2.b.a 2
65.h odd 4 1 2028.2.a.c 1
65.k even 4 1 2028.2.b.a 2
65.o even 12 2 2028.2.q.h 4
65.q odd 12 2 2028.2.i.e 2
65.r odd 12 2 2028.2.i.g 2
65.t even 12 2 2028.2.q.h 4
120.q odd 4 1 7488.2.a.d 1
120.w even 4 1 7488.2.a.c 1
195.j odd 4 1 6084.2.b.j 2
195.s even 4 1 6084.2.a.b 1
195.u odd 4 1 6084.2.b.j 2
260.p even 4 1 8112.2.a.bi 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
156.2.a.a 1 5.c odd 4 1
468.2.a.d 1 15.e even 4 1
624.2.a.e 1 20.e even 4 1
1872.2.a.s 1 60.l odd 4 1
2028.2.a.c 1 65.h odd 4 1
2028.2.b.a 2 65.f even 4 1
2028.2.b.a 2 65.k even 4 1
2028.2.i.e 2 65.q odd 12 2
2028.2.i.g 2 65.r odd 12 2
2028.2.q.h 4 65.o even 12 2
2028.2.q.h 4 65.t even 12 2
2496.2.a.o 1 40.k even 4 1
2496.2.a.bc 1 40.i odd 4 1
3900.2.a.m 1 5.c odd 4 1
3900.2.h.b 2 1.a even 1 1 trivial
3900.2.h.b 2 5.b even 2 1 inner
4212.2.i.b 2 45.l even 12 2
4212.2.i.l 2 45.k odd 12 2
6084.2.a.b 1 195.s even 4 1
6084.2.b.j 2 195.j odd 4 1
6084.2.b.j 2 195.u odd 4 1
7488.2.a.c 1 120.w even 4 1
7488.2.a.d 1 120.q odd 4 1
7644.2.a.k 1 35.f even 4 1
8112.2.a.bi 1 260.p even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3900, [\chi])\):

\( T_{7}^{2} + 4 \)
\( T_{11} + 4 \)
\( T_{19} - 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 1 + T^{2} \)
$5$ \( T^{2} \)
$7$ \( 4 + T^{2} \)
$11$ \( ( 4 + T )^{2} \)
$13$ \( 1 + T^{2} \)
$17$ \( 4 + T^{2} \)
$19$ \( ( -2 + T )^{2} \)
$23$ \( T^{2} \)
$29$ \( ( -6 + T )^{2} \)
$31$ \( ( 10 + T )^{2} \)
$37$ \( 100 + T^{2} \)
$41$ \( ( -8 + T )^{2} \)
$43$ \( 16 + T^{2} \)
$47$ \( 16 + T^{2} \)
$53$ \( 100 + T^{2} \)
$59$ \( ( -8 + T )^{2} \)
$61$ \( ( 14 + T )^{2} \)
$67$ \( 4 + T^{2} \)
$71$ \( ( -16 + T )^{2} \)
$73$ \( 100 + T^{2} \)
$79$ \( ( -16 + T )^{2} \)
$83$ \( T^{2} \)
$89$ \( ( -4 + T )^{2} \)
$97$ \( 4 + T^{2} \)
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