Properties

Label 2205.4.a.bb
Level $2205$
Weight $4$
Character orbit 2205.a
Self dual yes
Analytic conductor $130.099$
Analytic rank $1$
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
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 \(\beta = 2\sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta + 1) q^{2} + (2 \beta + 1) q^{4} + 5 q^{5} + ( - 5 \beta + 9) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta + 1) q^{2} + (2 \beta + 1) q^{4} + 5 q^{5} + ( - 5 \beta + 9) q^{8} + (5 \beta + 5) q^{10} + ( - 20 \beta + 8) q^{11} + (2 \beta + 38) q^{13} + ( - 12 \beta - 39) q^{16} + ( - 2 \beta - 62) q^{17} + (16 \beta + 48) q^{19} + (10 \beta + 5) q^{20} + ( - 12 \beta - 152) q^{22} + (34 \beta + 8) q^{23} + 25 q^{25} + (40 \beta + 54) q^{26} + (54 \beta - 94) q^{29} + ( - 18 \beta + 60) q^{31} + ( - 11 \beta - 207) q^{32} + ( - 64 \beta - 78) q^{34} + ( - 66 \beta - 66) q^{37} + (64 \beta + 176) q^{38} + ( - 25 \beta + 45) q^{40} + (80 \beta + 50) q^{41} + (62 \beta - 268) q^{43} + ( - 4 \beta - 312) q^{44} + (42 \beta + 280) q^{46} + ( - 42 \beta - 464) q^{47} + (25 \beta + 25) q^{50} + (78 \beta + 70) q^{52} + ( - 64 \beta - 442) q^{53} + ( - 100 \beta + 40) q^{55} + ( - 40 \beta + 338) q^{58} + ( - 204 \beta + 52) q^{59} + (42 \beta + 234) q^{61} + (42 \beta - 84) q^{62} + ( - 122 \beta + 17) q^{64} + (10 \beta + 190) q^{65} + (38 \beta - 844) q^{67} + ( - 126 \beta - 94) q^{68} + (150 \beta + 68) q^{71} + ( - 322 \beta - 254) q^{73} + ( - 132 \beta - 594) q^{74} + (112 \beta + 304) q^{76} + ( - 232 \beta - 216) q^{79} + ( - 60 \beta - 195) q^{80} + (130 \beta + 690) q^{82} + ( - 84 \beta - 292) q^{83} + ( - 10 \beta - 310) q^{85} + ( - 206 \beta + 228) q^{86} + ( - 220 \beta + 872) q^{88} + (112 \beta - 702) q^{89} + (50 \beta + 552) q^{92} + ( - 506 \beta - 800) q^{94} + (80 \beta + 240) q^{95} + ( - 46 \beta + 594) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} + 10 q^{5} + 18 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 2 q^{4} + 10 q^{5} + 18 q^{8} + 10 q^{10} + 16 q^{11} + 76 q^{13} - 78 q^{16} - 124 q^{17} + 96 q^{19} + 10 q^{20} - 304 q^{22} + 16 q^{23} + 50 q^{25} + 108 q^{26} - 188 q^{29} + 120 q^{31} - 414 q^{32} - 156 q^{34} - 132 q^{37} + 352 q^{38} + 90 q^{40} + 100 q^{41} - 536 q^{43} - 624 q^{44} + 560 q^{46} - 928 q^{47} + 50 q^{50} + 140 q^{52} - 884 q^{53} + 80 q^{55} + 676 q^{58} + 104 q^{59} + 468 q^{61} - 168 q^{62} + 34 q^{64} + 380 q^{65} - 1688 q^{67} - 188 q^{68} + 136 q^{71} - 508 q^{73} - 1188 q^{74} + 608 q^{76} - 432 q^{79} - 390 q^{80} + 1380 q^{82} - 584 q^{83} - 620 q^{85} + 456 q^{86} + 1744 q^{88} - 1404 q^{89} + 1104 q^{92} - 1600 q^{94} + 480 q^{95} + 1188 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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
−1.82843 0 −4.65685 5.00000 0 0 23.1421 0 −9.14214
1.2 3.82843 0 6.65685 5.00000 0 0 −5.14214 0 19.1421
\(n\): 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.bb 2
3.b odd 2 1 735.4.a.o 2
7.b odd 2 1 315.4.a.k 2
21.c even 2 1 105.4.a.e 2
35.c odd 2 1 1575.4.a.q 2
84.h odd 2 1 1680.4.a.bo 2
105.g even 2 1 525.4.a.l 2
105.k odd 4 2 525.4.d.l 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.4.a.e 2 21.c even 2 1
315.4.a.k 2 7.b odd 2 1
525.4.a.l 2 105.g even 2 1
525.4.d.l 4 105.k odd 4 2
735.4.a.o 2 3.b odd 2 1
1575.4.a.q 2 35.c odd 2 1
1680.4.a.bo 2 84.h odd 2 1
2205.4.a.bb 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2205))\):

\( T_{2}^{2} - 2T_{2} - 7 \) Copy content Toggle raw display
\( T_{11}^{2} - 16T_{11} - 3136 \) Copy content Toggle raw display
\( T_{13}^{2} - 76T_{13} + 1412 \) Copy content Toggle raw display
\( T_{17}^{2} + 124T_{17} + 3812 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 2T - 7 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( (T - 5)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 16T - 3136 \) Copy content Toggle raw display
$13$ \( T^{2} - 76T + 1412 \) Copy content Toggle raw display
$17$ \( T^{2} + 124T + 3812 \) Copy content Toggle raw display
$19$ \( T^{2} - 96T + 256 \) Copy content Toggle raw display
$23$ \( T^{2} - 16T - 9184 \) Copy content Toggle raw display
$29$ \( T^{2} + 188T - 14492 \) Copy content Toggle raw display
$31$ \( T^{2} - 120T + 1008 \) Copy content Toggle raw display
$37$ \( T^{2} + 132T - 30492 \) Copy content Toggle raw display
$41$ \( T^{2} - 100T - 48700 \) Copy content Toggle raw display
$43$ \( T^{2} + 536T + 41072 \) Copy content Toggle raw display
$47$ \( T^{2} + 928T + 201184 \) Copy content Toggle raw display
$53$ \( T^{2} + 884T + 162596 \) Copy content Toggle raw display
$59$ \( T^{2} - 104T - 330224 \) Copy content Toggle raw display
$61$ \( T^{2} - 468T + 40644 \) Copy content Toggle raw display
$67$ \( T^{2} + 1688 T + 700784 \) Copy content Toggle raw display
$71$ \( T^{2} - 136T - 175376 \) Copy content Toggle raw display
$73$ \( T^{2} + 508T - 764956 \) Copy content Toggle raw display
$79$ \( T^{2} + 432T - 383936 \) Copy content Toggle raw display
$83$ \( T^{2} + 584T + 28816 \) Copy content Toggle raw display
$89$ \( T^{2} + 1404 T + 392452 \) Copy content Toggle raw display
$97$ \( T^{2} - 1188 T + 335908 \) Copy content Toggle raw display
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