Properties

Label 3450.2.d.x
Level $3450$
Weight $2$
Character orbit 3450.d
Analytic conductor $27.548$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 3450 = 2 \cdot 3 \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3450.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(27.5483886973\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
Defining polynomial: \(x^{4} + 3 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 138)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} -\beta_{1} q^{3} - q^{4} + q^{6} + 2 \beta_{2} q^{7} -\beta_{1} q^{8} - q^{9} +O(q^{10})\) \( q + \beta_{1} q^{2} -\beta_{1} q^{3} - q^{4} + q^{6} + 2 \beta_{2} q^{7} -\beta_{1} q^{8} - q^{9} + ( -3 + \beta_{3} ) q^{11} + \beta_{1} q^{12} + 2 \beta_{2} q^{13} -2 \beta_{3} q^{14} + q^{16} -4 \beta_{1} q^{17} -\beta_{1} q^{18} + ( 1 + 3 \beta_{3} ) q^{19} + 2 \beta_{3} q^{21} + ( -3 \beta_{1} + \beta_{2} ) q^{22} -\beta_{1} q^{23} - q^{24} -2 \beta_{3} q^{26} + \beta_{1} q^{27} -2 \beta_{2} q^{28} -2 \beta_{3} q^{29} + ( 2 + 2 \beta_{3} ) q^{31} + \beta_{1} q^{32} + ( 3 \beta_{1} - \beta_{2} ) q^{33} + 4 q^{34} + q^{36} + ( 9 \beta_{1} - \beta_{2} ) q^{37} + ( \beta_{1} + 3 \beta_{2} ) q^{38} + 2 \beta_{3} q^{39} -2 q^{41} + 2 \beta_{2} q^{42} + ( 7 \beta_{1} + \beta_{2} ) q^{43} + ( 3 - \beta_{3} ) q^{44} + q^{46} + 4 \beta_{1} q^{47} -\beta_{1} q^{48} -13 q^{49} -4 q^{51} -2 \beta_{2} q^{52} + ( -3 \beta_{1} + \beta_{2} ) q^{53} - q^{54} + 2 \beta_{3} q^{56} + ( -\beta_{1} - 3 \beta_{2} ) q^{57} -2 \beta_{2} q^{58} -4 \beta_{3} q^{59} + ( 3 + \beta_{3} ) q^{61} + ( 2 \beta_{1} + 2 \beta_{2} ) q^{62} -2 \beta_{2} q^{63} - q^{64} + ( -3 + \beta_{3} ) q^{66} + ( 3 \beta_{1} - 3 \beta_{2} ) q^{67} + 4 \beta_{1} q^{68} - q^{69} -4 \beta_{3} q^{71} + \beta_{1} q^{72} -2 \beta_{2} q^{73} + ( -9 + \beta_{3} ) q^{74} + ( -1 - 3 \beta_{3} ) q^{76} + ( 10 \beta_{1} - 6 \beta_{2} ) q^{77} + 2 \beta_{2} q^{78} + 2 \beta_{3} q^{79} + q^{81} -2 \beta_{1} q^{82} + ( -11 \beta_{1} + \beta_{2} ) q^{83} -2 \beta_{3} q^{84} + ( -7 - \beta_{3} ) q^{86} + 2 \beta_{2} q^{87} + ( 3 \beta_{1} - \beta_{2} ) q^{88} + ( 6 - 2 \beta_{3} ) q^{89} -20 q^{91} + \beta_{1} q^{92} + ( -2 \beta_{1} - 2 \beta_{2} ) q^{93} -4 q^{94} + q^{96} + ( -4 \beta_{1} - 2 \beta_{2} ) q^{97} -13 \beta_{1} q^{98} + ( 3 - \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{4} + 4q^{6} - 4q^{9} + O(q^{10}) \) \( 4q - 4q^{4} + 4q^{6} - 4q^{9} - 12q^{11} + 4q^{16} + 4q^{19} - 4q^{24} + 8q^{31} + 16q^{34} + 4q^{36} - 8q^{41} + 12q^{44} + 4q^{46} - 52q^{49} - 16q^{51} - 4q^{54} + 12q^{61} - 4q^{64} - 12q^{66} - 4q^{69} - 36q^{74} - 4q^{76} + 4q^{81} - 28q^{86} + 24q^{89} - 80q^{91} - 16q^{94} + 4q^{96} + 12q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 3 x^{2} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{3} + 2 \nu \)
\(\beta_{2}\)\(=\)\( \nu^{3} + 4 \nu \)
\(\beta_{3}\)\(=\)\( 2 \nu^{2} + 3 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} - \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} - 3\)\()/2\)
\(\nu^{3}\)\(=\)\(-\beta_{2} + 2 \beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3450\mathbb{Z}\right)^\times\).

\(n\) \(277\) \(1151\) \(1201\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2899.1
0.618034i
1.61803i
1.61803i
0.618034i
1.00000i 1.00000i −1.00000 0 1.00000 4.47214i 1.00000i −1.00000 0
2899.2 1.00000i 1.00000i −1.00000 0 1.00000 4.47214i 1.00000i −1.00000 0
2899.3 1.00000i 1.00000i −1.00000 0 1.00000 4.47214i 1.00000i −1.00000 0
2899.4 1.00000i 1.00000i −1.00000 0 1.00000 4.47214i 1.00000i −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 3450.2.d.x 4
5.b even 2 1 inner 3450.2.d.x 4
5.c odd 4 1 138.2.a.d 2
5.c odd 4 1 3450.2.a.be 2
15.e even 4 1 414.2.a.f 2
20.e even 4 1 1104.2.a.j 2
35.f even 4 1 6762.2.a.cb 2
40.i odd 4 1 4416.2.a.bh 2
40.k even 4 1 4416.2.a.bl 2
60.l odd 4 1 3312.2.a.bc 2
115.e even 4 1 3174.2.a.s 2
345.l odd 4 1 9522.2.a.q 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
138.2.a.d 2 5.c odd 4 1
414.2.a.f 2 15.e even 4 1
1104.2.a.j 2 20.e even 4 1
3174.2.a.s 2 115.e even 4 1
3312.2.a.bc 2 60.l odd 4 1
3450.2.a.be 2 5.c odd 4 1
3450.2.d.x 4 1.a even 1 1 trivial
3450.2.d.x 4 5.b even 2 1 inner
4416.2.a.bh 2 40.i odd 4 1
4416.2.a.bl 2 40.k even 4 1
6762.2.a.cb 2 35.f even 4 1
9522.2.a.q 2 345.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3450, [\chi])\):

\( T_{7}^{2} + 20 \)
\( T_{11}^{2} + 6 T_{11} + 4 \)
\( T_{13}^{2} + 20 \)
\( T_{17}^{2} + 16 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{2} )^{2} \)
$3$ \( ( 1 + T^{2} )^{2} \)
$5$ \( T^{4} \)
$7$ \( ( 20 + T^{2} )^{2} \)
$11$ \( ( 4 + 6 T + T^{2} )^{2} \)
$13$ \( ( 20 + T^{2} )^{2} \)
$17$ \( ( 16 + T^{2} )^{2} \)
$19$ \( ( -44 - 2 T + T^{2} )^{2} \)
$23$ \( ( 1 + T^{2} )^{2} \)
$29$ \( ( -20 + T^{2} )^{2} \)
$31$ \( ( -16 - 4 T + T^{2} )^{2} \)
$37$ \( 5776 + 172 T^{2} + T^{4} \)
$41$ \( ( 2 + T )^{4} \)
$43$ \( 1936 + 108 T^{2} + T^{4} \)
$47$ \( ( 16 + T^{2} )^{2} \)
$53$ \( 16 + 28 T^{2} + T^{4} \)
$59$ \( ( -80 + T^{2} )^{2} \)
$61$ \( ( 4 - 6 T + T^{2} )^{2} \)
$67$ \( 1296 + 108 T^{2} + T^{4} \)
$71$ \( ( -80 + T^{2} )^{2} \)
$73$ \( ( 20 + T^{2} )^{2} \)
$79$ \( ( -20 + T^{2} )^{2} \)
$83$ \( 13456 + 252 T^{2} + T^{4} \)
$89$ \( ( 16 - 12 T + T^{2} )^{2} \)
$97$ \( 16 + 72 T^{2} + T^{4} \)
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