Properties

Label 2205.4.a.bh
Level $2205$
Weight $4$
Character orbit 2205.a
Self dual yes
Analytic conductor $130.099$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2205 = 3^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2205.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(130.099211563\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \(x^{2} - x - 4\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 105)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 4 - \beta ) q^{2} + ( 12 - 7 \beta ) q^{4} -5 q^{5} + ( 44 - 25 \beta ) q^{8} +O(q^{10})\) \( q + ( 4 - \beta ) q^{2} + ( 12 - 7 \beta ) q^{4} -5 q^{5} + ( 44 - 25 \beta ) q^{8} + ( -20 + 5 \beta ) q^{10} + ( 18 - 10 \beta ) q^{11} + ( 4 - 22 \beta ) q^{13} + ( 180 - 63 \beta ) q^{16} + ( 22 - 28 \beta ) q^{17} + ( -74 - 26 \beta ) q^{19} + ( -60 + 35 \beta ) q^{20} + ( 112 - 48 \beta ) q^{22} + ( -120 + 56 \beta ) q^{23} + 25 q^{25} + ( 104 - 70 \beta ) q^{26} + ( 58 - 84 \beta ) q^{29} + ( -174 + 18 \beta ) q^{31} + ( 620 - 169 \beta ) q^{32} + ( 200 - 106 \beta ) q^{34} + ( -54 - 24 \beta ) q^{37} + ( -192 - 4 \beta ) q^{38} + ( -220 + 125 \beta ) q^{40} + ( 170 - 140 \beta ) q^{41} + ( 148 + 68 \beta ) q^{43} + ( 496 - 176 \beta ) q^{44} + ( -704 + 288 \beta ) q^{46} + ( 200 - 108 \beta ) q^{47} + ( 100 - 25 \beta ) q^{50} + ( 664 - 138 \beta ) q^{52} + ( -124 + 214 \beta ) q^{53} + ( -90 + 50 \beta ) q^{55} + ( 568 - 310 \beta ) q^{58} + ( 200 - 36 \beta ) q^{59} + ( -270 - 252 \beta ) q^{61} + ( -768 + 228 \beta ) q^{62} + ( 1716 - 623 \beta ) q^{64} + ( -20 + 110 \beta ) q^{65} + ( -380 - 28 \beta ) q^{67} + ( 1048 - 294 \beta ) q^{68} + ( -62 - 330 \beta ) q^{71} + ( -300 - 178 \beta ) q^{73} + ( -120 - 18 \beta ) q^{74} + ( -160 + 388 \beta ) q^{76} + ( 248 - 88 \beta ) q^{79} + ( -900 + 315 \beta ) q^{80} + ( 1240 - 590 \beta ) q^{82} + ( 436 + 264 \beta ) q^{83} + ( -110 + 140 \beta ) q^{85} + ( 320 + 56 \beta ) q^{86} + ( 1792 - 640 \beta ) q^{88} + ( -346 + 728 \beta ) q^{89} + ( -3008 + 1120 \beta ) q^{92} + ( 1232 - 524 \beta ) q^{94} + ( 370 + 130 \beta ) q^{95} + ( 176 + 146 \beta ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 7 q^{2} + 17 q^{4} - 10 q^{5} + 63 q^{8} + O(q^{10}) \) \( 2 q + 7 q^{2} + 17 q^{4} - 10 q^{5} + 63 q^{8} - 35 q^{10} + 26 q^{11} - 14 q^{13} + 297 q^{16} + 16 q^{17} - 174 q^{19} - 85 q^{20} + 176 q^{22} - 184 q^{23} + 50 q^{25} + 138 q^{26} + 32 q^{29} - 330 q^{31} + 1071 q^{32} + 294 q^{34} - 132 q^{37} - 388 q^{38} - 315 q^{40} + 200 q^{41} + 364 q^{43} + 816 q^{44} - 1120 q^{46} + 292 q^{47} + 175 q^{50} + 1190 q^{52} - 34 q^{53} - 130 q^{55} + 826 q^{58} + 364 q^{59} - 792 q^{61} - 1308 q^{62} + 2809 q^{64} + 70 q^{65} - 788 q^{67} + 1802 q^{68} - 454 q^{71} - 778 q^{73} - 258 q^{74} + 68 q^{76} + 408 q^{79} - 1485 q^{80} + 1890 q^{82} + 1136 q^{83} - 80 q^{85} + 696 q^{86} + 2944 q^{88} + 36 q^{89} - 4896 q^{92} + 1940 q^{94} + 870 q^{95} + 498 q^{97} + O(q^{100}) \)

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
2.56155
−1.56155
1.43845 0 −5.93087 −5.00000 0 0 −20.0388 0 −7.19224
1.2 5.56155 0 22.9309 −5.00000 0 0 83.0388 0 −27.8078
\(n\): 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 2205.4.a.bh 2
3.b odd 2 1 735.4.a.k 2
7.b odd 2 1 315.4.a.m 2
21.c even 2 1 105.4.a.c 2
35.c odd 2 1 1575.4.a.m 2
84.h odd 2 1 1680.4.a.bk 2
105.g even 2 1 525.4.a.p 2
105.k odd 4 2 525.4.d.i 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.4.a.c 2 21.c even 2 1
315.4.a.m 2 7.b odd 2 1
525.4.a.p 2 105.g even 2 1
525.4.d.i 4 105.k odd 4 2
735.4.a.k 2 3.b odd 2 1
1575.4.a.m 2 35.c odd 2 1
1680.4.a.bk 2 84.h odd 2 1
2205.4.a.bh 2 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2205))\):

\( T_{2}^{2} - 7 T_{2} + 8 \)
\( T_{11}^{2} - 26 T_{11} - 256 \)
\( T_{13}^{2} + 14 T_{13} - 2008 \)
\( T_{17}^{2} - 16 T_{17} - 3268 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 8 - 7 T + T^{2} \)
$3$ \( T^{2} \)
$5$ \( ( 5 + T )^{2} \)
$7$ \( T^{2} \)
$11$ \( -256 - 26 T + T^{2} \)
$13$ \( -2008 + 14 T + T^{2} \)
$17$ \( -3268 - 16 T + T^{2} \)
$19$ \( 4696 + 174 T + T^{2} \)
$23$ \( -4864 + 184 T + T^{2} \)
$29$ \( -29732 - 32 T + T^{2} \)
$31$ \( 25848 + 330 T + T^{2} \)
$37$ \( 1908 + 132 T + T^{2} \)
$41$ \( -73300 - 200 T + T^{2} \)
$43$ \( 13472 - 364 T + T^{2} \)
$47$ \( -28256 - 292 T + T^{2} \)
$53$ \( -194344 + 34 T + T^{2} \)
$59$ \( 27616 - 364 T + T^{2} \)
$61$ \( -113076 + 792 T + T^{2} \)
$67$ \( 151904 + 788 T + T^{2} \)
$71$ \( -411296 + 454 T + T^{2} \)
$73$ \( 16664 + 778 T + T^{2} \)
$79$ \( 8704 - 408 T + T^{2} \)
$83$ \( 26416 - 1136 T + T^{2} \)
$89$ \( -2252108 - 36 T + T^{2} \)
$97$ \( -28592 - 498 T + T^{2} \)
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