Properties

Label 3675.2.a.bf
Level $3675$
Weight $2$
Character orbit 3675.a
Self dual yes
Analytic conductor $29.345$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3675 = 3 \cdot 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3675.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(29.3450227428\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 147)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(f(q)\) \(=\) \( q + ( 1 + \beta ) q^{2} + q^{3} + ( 1 + 2 \beta ) q^{4} + ( 1 + \beta ) q^{6} + ( 3 + \beta ) q^{8} + q^{9} +O(q^{10})\) \( q + ( 1 + \beta ) q^{2} + q^{3} + ( 1 + 2 \beta ) q^{4} + ( 1 + \beta ) q^{6} + ( 3 + \beta ) q^{8} + q^{9} -2 q^{11} + ( 1 + 2 \beta ) q^{12} + ( 4 + \beta ) q^{13} + 3 q^{16} + ( 2 + 3 \beta ) q^{17} + ( 1 + \beta ) q^{18} -2 \beta q^{19} + ( -2 - 2 \beta ) q^{22} + ( 2 - 4 \beta ) q^{23} + ( 3 + \beta ) q^{24} + ( 6 + 5 \beta ) q^{26} + q^{27} + ( -4 + 2 \beta ) q^{29} + ( 4 + 2 \beta ) q^{31} + ( -3 + \beta ) q^{32} -2 q^{33} + ( 8 + 5 \beta ) q^{34} + ( 1 + 2 \beta ) q^{36} + 4 q^{37} + ( -4 - 2 \beta ) q^{38} + ( 4 + \beta ) q^{39} + ( -2 + 3 \beta ) q^{41} + 4 \beta q^{43} + ( -2 - 4 \beta ) q^{44} + ( -6 - 2 \beta ) q^{46} + 2 \beta q^{47} + 3 q^{48} + ( 2 + 3 \beta ) q^{51} + ( 8 + 9 \beta ) q^{52} + 2 q^{53} + ( 1 + \beta ) q^{54} -2 \beta q^{57} -2 \beta q^{58} + ( 4 + 2 \beta ) q^{59} + ( -8 + 3 \beta ) q^{61} + ( 8 + 6 \beta ) q^{62} + ( -7 - 2 \beta ) q^{64} + ( -2 - 2 \beta ) q^{66} -4 \beta q^{67} + ( 14 + 7 \beta ) q^{68} + ( 2 - 4 \beta ) q^{69} + ( -2 - 8 \beta ) q^{71} + ( 3 + \beta ) q^{72} + ( 4 - 7 \beta ) q^{73} + ( 4 + 4 \beta ) q^{74} + ( -8 - 2 \beta ) q^{76} + ( 6 + 5 \beta ) q^{78} + ( 8 - 4 \beta ) q^{79} + q^{81} + ( 4 + \beta ) q^{82} + ( -4 - 8 \beta ) q^{83} + ( 8 + 4 \beta ) q^{86} + ( -4 + 2 \beta ) q^{87} + ( -6 - 2 \beta ) q^{88} + ( 10 - 3 \beta ) q^{89} -14 q^{92} + ( 4 + 2 \beta ) q^{93} + ( 4 + 2 \beta ) q^{94} + ( -3 + \beta ) q^{96} + ( 4 + \beta ) q^{97} -2 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} + 2q^{3} + 2q^{4} + 2q^{6} + 6q^{8} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{2} + 2q^{3} + 2q^{4} + 2q^{6} + 6q^{8} + 2q^{9} - 4q^{11} + 2q^{12} + 8q^{13} + 6q^{16} + 4q^{17} + 2q^{18} - 4q^{22} + 4q^{23} + 6q^{24} + 12q^{26} + 2q^{27} - 8q^{29} + 8q^{31} - 6q^{32} - 4q^{33} + 16q^{34} + 2q^{36} + 8q^{37} - 8q^{38} + 8q^{39} - 4q^{41} - 4q^{44} - 12q^{46} + 6q^{48} + 4q^{51} + 16q^{52} + 4q^{53} + 2q^{54} + 8q^{59} - 16q^{61} + 16q^{62} - 14q^{64} - 4q^{66} + 28q^{68} + 4q^{69} - 4q^{71} + 6q^{72} + 8q^{73} + 8q^{74} - 16q^{76} + 12q^{78} + 16q^{79} + 2q^{81} + 8q^{82} - 8q^{83} + 16q^{86} - 8q^{87} - 12q^{88} + 20q^{89} - 28q^{92} + 8q^{93} + 8q^{94} - 6q^{96} + 8q^{97} - 4q^{99} + O(q^{100}) \)

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
−1.41421
1.41421
−0.414214 1.00000 −1.82843 0 −0.414214 0 1.58579 1.00000 0
1.2 2.41421 1.00000 3.82843 0 2.41421 0 4.41421 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(1\)
\(7\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3675.2.a.bf 2
5.b even 2 1 147.2.a.d 2
7.b odd 2 1 3675.2.a.bd 2
15.d odd 2 1 441.2.a.j 2
20.d odd 2 1 2352.2.a.be 2
35.c odd 2 1 147.2.a.e yes 2
35.i odd 6 2 147.2.e.d 4
35.j even 6 2 147.2.e.e 4
40.e odd 2 1 9408.2.a.dq 2
40.f even 2 1 9408.2.a.ef 2
60.h even 2 1 7056.2.a.cv 2
105.g even 2 1 441.2.a.i 2
105.o odd 6 2 441.2.e.f 4
105.p even 6 2 441.2.e.g 4
140.c even 2 1 2352.2.a.bc 2
140.p odd 6 2 2352.2.q.bb 4
140.s even 6 2 2352.2.q.bd 4
280.c odd 2 1 9408.2.a.di 2
280.n even 2 1 9408.2.a.dt 2
420.o odd 2 1 7056.2.a.cf 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
147.2.a.d 2 5.b even 2 1
147.2.a.e yes 2 35.c odd 2 1
147.2.e.d 4 35.i odd 6 2
147.2.e.e 4 35.j even 6 2
441.2.a.i 2 105.g even 2 1
441.2.a.j 2 15.d odd 2 1
441.2.e.f 4 105.o odd 6 2
441.2.e.g 4 105.p even 6 2
2352.2.a.bc 2 140.c even 2 1
2352.2.a.be 2 20.d odd 2 1
2352.2.q.bb 4 140.p odd 6 2
2352.2.q.bd 4 140.s even 6 2
3675.2.a.bd 2 7.b odd 2 1
3675.2.a.bf 2 1.a even 1 1 trivial
7056.2.a.cf 2 420.o odd 2 1
7056.2.a.cv 2 60.h even 2 1
9408.2.a.di 2 280.c odd 2 1
9408.2.a.dq 2 40.e odd 2 1
9408.2.a.dt 2 280.n even 2 1
9408.2.a.ef 2 40.f even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3675))\):

\( T_{2}^{2} - 2 T_{2} - 1 \)
\( T_{11} + 2 \)
\( T_{13}^{2} - 8 T_{13} + 14 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 2 T + 3 T^{2} - 4 T^{3} + 4 T^{4} \)
$3$ \( ( 1 - T )^{2} \)
$5$ 1
$7$ 1
$11$ \( ( 1 + 2 T + 11 T^{2} )^{2} \)
$13$ \( 1 - 8 T + 40 T^{2} - 104 T^{3} + 169 T^{4} \)
$17$ \( 1 - 4 T + 20 T^{2} - 68 T^{3} + 289 T^{4} \)
$19$ \( 1 + 30 T^{2} + 361 T^{4} \)
$23$ \( 1 - 4 T + 18 T^{2} - 92 T^{3} + 529 T^{4} \)
$29$ \( 1 + 8 T + 66 T^{2} + 232 T^{3} + 841 T^{4} \)
$31$ \( 1 - 8 T + 70 T^{2} - 248 T^{3} + 961 T^{4} \)
$37$ \( ( 1 - 4 T + 37 T^{2} )^{2} \)
$41$ \( 1 + 4 T + 68 T^{2} + 164 T^{3} + 1681 T^{4} \)
$43$ \( 1 + 54 T^{2} + 1849 T^{4} \)
$47$ \( 1 + 86 T^{2} + 2209 T^{4} \)
$53$ \( ( 1 - 2 T + 53 T^{2} )^{2} \)
$59$ \( 1 - 8 T + 126 T^{2} - 472 T^{3} + 3481 T^{4} \)
$61$ \( 1 + 16 T + 168 T^{2} + 976 T^{3} + 3721 T^{4} \)
$67$ \( 1 + 102 T^{2} + 4489 T^{4} \)
$71$ \( 1 + 4 T + 18 T^{2} + 284 T^{3} + 5041 T^{4} \)
$73$ \( 1 - 8 T + 64 T^{2} - 584 T^{3} + 5329 T^{4} \)
$79$ \( 1 - 16 T + 190 T^{2} - 1264 T^{3} + 6241 T^{4} \)
$83$ \( 1 + 8 T + 54 T^{2} + 664 T^{3} + 6889 T^{4} \)
$89$ \( 1 - 20 T + 260 T^{2} - 1780 T^{3} + 7921 T^{4} \)
$97$ \( 1 - 8 T + 208 T^{2} - 776 T^{3} + 9409 T^{4} \)
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