Properties

Label 3450.2.d.v
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: \(\Q(i, \sqrt{17})\)
Defining polynomial: \(x^{4} + 9 x^{2} + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 690)
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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + \beta_{1} q^{3} - q^{4} - q^{6} + ( -\beta_{1} + \beta_{2} ) q^{7} -\beta_{1} q^{8} - q^{9} +O(q^{10})\) \( q + \beta_{1} q^{2} + \beta_{1} q^{3} - q^{4} - q^{6} + ( -\beta_{1} + \beta_{2} ) q^{7} -\beta_{1} q^{8} - q^{9} + ( 1 - \beta_{3} ) q^{11} -\beta_{1} q^{12} -2 \beta_{1} q^{13} + ( 1 - \beta_{3} ) q^{14} + q^{16} + ( 3 \beta_{1} - \beta_{2} ) q^{17} -\beta_{1} q^{18} -4 q^{19} + ( 1 - \beta_{3} ) q^{21} + ( \beta_{1} - \beta_{2} ) q^{22} -\beta_{1} q^{23} + q^{24} + 2 q^{26} -\beta_{1} q^{27} + ( \beta_{1} - \beta_{2} ) q^{28} -2 q^{29} + \beta_{1} q^{32} + ( \beta_{1} - \beta_{2} ) q^{33} + ( -3 + \beta_{3} ) q^{34} + q^{36} + ( -3 \beta_{1} + \beta_{2} ) q^{37} -4 \beta_{1} q^{38} + 2 q^{39} + 2 q^{41} + ( \beta_{1} - \beta_{2} ) q^{42} + ( -1 + \beta_{3} ) q^{44} + q^{46} -8 \beta_{1} q^{47} + \beta_{1} q^{48} + ( -11 + 2 \beta_{3} ) q^{49} + ( -3 + \beta_{3} ) q^{51} + 2 \beta_{1} q^{52} + ( -4 \beta_{1} - 2 \beta_{2} ) q^{53} + q^{54} + ( -1 + \beta_{3} ) q^{56} -4 \beta_{1} q^{57} -2 \beta_{1} q^{58} + ( 6 - 2 \beta_{3} ) q^{59} + ( 5 + \beta_{3} ) q^{61} + ( \beta_{1} - \beta_{2} ) q^{63} - q^{64} + ( -1 + \beta_{3} ) q^{66} -8 \beta_{1} q^{67} + ( -3 \beta_{1} + \beta_{2} ) q^{68} + q^{69} + ( -2 - 2 \beta_{3} ) q^{71} + \beta_{1} q^{72} + ( -4 \beta_{1} + 2 \beta_{2} ) q^{73} + ( 3 - \beta_{3} ) q^{74} + 4 q^{76} + ( -18 \beta_{1} + 2 \beta_{2} ) q^{77} + 2 \beta_{1} q^{78} + ( 1 - \beta_{3} ) q^{79} + q^{81} + 2 \beta_{1} q^{82} + ( \beta_{1} + 3 \beta_{2} ) q^{83} + ( -1 + \beta_{3} ) q^{84} -2 \beta_{1} q^{87} + ( -\beta_{1} + \beta_{2} ) q^{88} + ( 1 + \beta_{3} ) q^{89} + ( -2 + 2 \beta_{3} ) q^{91} + \beta_{1} q^{92} + 8 q^{94} - q^{96} + ( -8 \beta_{1} - 2 \beta_{2} ) q^{97} + ( -11 \beta_{1} + 2 \beta_{2} ) q^{98} + ( -1 + \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{4} - 4q^{6} - 4q^{9} + O(q^{10}) \) \( 4q - 4q^{4} - 4q^{6} - 4q^{9} + 4q^{11} + 4q^{14} + 4q^{16} - 16q^{19} + 4q^{21} + 4q^{24} + 8q^{26} - 8q^{29} - 12q^{34} + 4q^{36} + 8q^{39} + 8q^{41} - 4q^{44} + 4q^{46} - 44q^{49} - 12q^{51} + 4q^{54} - 4q^{56} + 24q^{59} + 20q^{61} - 4q^{64} - 4q^{66} + 4q^{69} - 8q^{71} + 12q^{74} + 16q^{76} + 4q^{79} + 4q^{81} - 4q^{84} + 4q^{89} - 8q^{91} + 32q^{94} - 4q^{96} - 4q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 9 x^{2} + 16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} + 5 \nu \)\()/4\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} + 13 \nu \)\()/4\)
\(\beta_{3}\)\(=\)\( 2 \nu^{2} + 9 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} - \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} - 9\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(-5 \beta_{2} + 13 \beta_{1}\)\()/2\)

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
1.56155i
2.56155i
2.56155i
1.56155i
1.00000i 1.00000i −1.00000 0 −1.00000 3.12311i 1.00000i −1.00000 0
2899.2 1.00000i 1.00000i −1.00000 0 −1.00000 5.12311i 1.00000i −1.00000 0
2899.3 1.00000i 1.00000i −1.00000 0 −1.00000 5.12311i 1.00000i −1.00000 0
2899.4 1.00000i 1.00000i −1.00000 0 −1.00000 3.12311i 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.v 4
5.b even 2 1 inner 3450.2.d.v 4
5.c odd 4 1 690.2.a.l 2
5.c odd 4 1 3450.2.a.bi 2
15.e even 4 1 2070.2.a.t 2
20.e even 4 1 5520.2.a.bs 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
690.2.a.l 2 5.c odd 4 1
2070.2.a.t 2 15.e even 4 1
3450.2.a.bi 2 5.c odd 4 1
3450.2.d.v 4 1.a even 1 1 trivial
3450.2.d.v 4 5.b even 2 1 inner
5520.2.a.bs 2 20.e even 4 1

Hecke kernels

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

\( T_{7}^{4} + 36 T_{7}^{2} + 256 \)
\( T_{11}^{2} - 2 T_{11} - 16 \)
\( T_{13}^{2} + 4 \)
\( T_{17}^{4} + 52 T_{17}^{2} + 64 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{2} )^{2} \)
$3$ \( ( 1 + T^{2} )^{2} \)
$5$ \( T^{4} \)
$7$ \( 256 + 36 T^{2} + T^{4} \)
$11$ \( ( -16 - 2 T + T^{2} )^{2} \)
$13$ \( ( 4 + T^{2} )^{2} \)
$17$ \( 64 + 52 T^{2} + T^{4} \)
$19$ \( ( 4 + T )^{4} \)
$23$ \( ( 1 + T^{2} )^{2} \)
$29$ \( ( 2 + T )^{4} \)
$31$ \( T^{4} \)
$37$ \( 64 + 52 T^{2} + T^{4} \)
$41$ \( ( -2 + T )^{4} \)
$43$ \( T^{4} \)
$47$ \( ( 64 + T^{2} )^{2} \)
$53$ \( 2704 + 168 T^{2} + T^{4} \)
$59$ \( ( -32 - 12 T + T^{2} )^{2} \)
$61$ \( ( 8 - 10 T + T^{2} )^{2} \)
$67$ \( ( 64 + T^{2} )^{2} \)
$71$ \( ( -64 + 4 T + T^{2} )^{2} \)
$73$ \( 2704 + 168 T^{2} + T^{4} \)
$79$ \( ( -16 - 2 T + T^{2} )^{2} \)
$83$ \( 23104 + 308 T^{2} + T^{4} \)
$89$ \( ( -16 - 2 T + T^{2} )^{2} \)
$97$ \( 16 + 264 T^{2} + T^{4} \)
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