Properties

Label 735.4.a.o
Level $735$
Weight $4$
Character orbit 735.a
Self dual yes
Analytic conductor $43.366$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 735 = 3 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 735.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(43.3664038542\)
Analytic rank: \(0\)
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.

\(f(q)\) \(=\) \( q + (\beta - 1) q^{2} + 3 q^{3} + ( - 2 \beta + 1) q^{4} - 5 q^{5} + (3 \beta - 3) q^{6} + ( - 5 \beta - 9) q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta - 1) q^{2} + 3 q^{3} + ( - 2 \beta + 1) q^{4} - 5 q^{5} + (3 \beta - 3) q^{6} + ( - 5 \beta - 9) q^{8} + 9 q^{9} + ( - 5 \beta + 5) q^{10} + ( - 20 \beta - 8) q^{11} + ( - 6 \beta + 3) q^{12} + ( - 2 \beta + 38) q^{13} - 15 q^{15} + (12 \beta - 39) q^{16} + ( - 2 \beta + 62) q^{17} + (9 \beta - 9) q^{18} + ( - 16 \beta + 48) q^{19} + (10 \beta - 5) q^{20} + (12 \beta - 152) q^{22} + (34 \beta - 8) q^{23} + ( - 15 \beta - 27) q^{24} + 25 q^{25} + (40 \beta - 54) q^{26} + 27 q^{27} + (54 \beta + 94) q^{29} + ( - 15 \beta + 15) q^{30} + (18 \beta + 60) q^{31} + ( - 11 \beta + 207) q^{32} + ( - 60 \beta - 24) q^{33} + (64 \beta - 78) q^{34} + ( - 18 \beta + 9) q^{36} + (66 \beta - 66) q^{37} + (64 \beta - 176) q^{38} + ( - 6 \beta + 114) q^{39} + (25 \beta + 45) q^{40} + (80 \beta - 50) q^{41} + ( - 62 \beta - 268) q^{43} + ( - 4 \beta + 312) q^{44} - 45 q^{45} + ( - 42 \beta + 280) q^{46} + ( - 42 \beta + 464) q^{47} + (36 \beta - 117) q^{48} + (25 \beta - 25) q^{50} + ( - 6 \beta + 186) q^{51} + ( - 78 \beta + 70) q^{52} + ( - 64 \beta + 442) q^{53} + (27 \beta - 27) q^{54} + (100 \beta + 40) q^{55} + ( - 48 \beta + 144) q^{57} + (40 \beta + 338) q^{58} + ( - 204 \beta - 52) q^{59} + (30 \beta - 15) q^{60} + ( - 42 \beta + 234) q^{61} + (42 \beta + 84) q^{62} + (122 \beta + 17) q^{64} + (10 \beta - 190) q^{65} + (36 \beta - 456) q^{66} + ( - 38 \beta - 844) q^{67} + ( - 126 \beta + 94) q^{68} + (102 \beta - 24) q^{69} + (150 \beta - 68) q^{71} + ( - 45 \beta - 81) q^{72} + (322 \beta - 254) q^{73} + ( - 132 \beta + 594) q^{74} + 75 q^{75} + ( - 112 \beta + 304) q^{76} + (120 \beta - 162) q^{78} + (232 \beta - 216) q^{79} + ( - 60 \beta + 195) q^{80} + 81 q^{81} + ( - 130 \beta + 690) q^{82} + ( - 84 \beta + 292) q^{83} + (10 \beta - 310) q^{85} + ( - 206 \beta - 228) q^{86} + (162 \beta + 282) q^{87} + (220 \beta + 872) q^{88} + (112 \beta + 702) q^{89} + ( - 45 \beta + 45) q^{90} + (50 \beta - 552) q^{92} + (54 \beta + 180) q^{93} + (506 \beta - 800) q^{94} + (80 \beta - 240) q^{95} + ( - 33 \beta + 621) q^{96} + (46 \beta + 594) q^{97} + ( - 180 \beta - 72) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 6 q^{3} + 2 q^{4} - 10 q^{5} - 6 q^{6} - 18 q^{8} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 6 q^{3} + 2 q^{4} - 10 q^{5} - 6 q^{6} - 18 q^{8} + 18 q^{9} + 10 q^{10} - 16 q^{11} + 6 q^{12} + 76 q^{13} - 30 q^{15} - 78 q^{16} + 124 q^{17} - 18 q^{18} + 96 q^{19} - 10 q^{20} - 304 q^{22} - 16 q^{23} - 54 q^{24} + 50 q^{25} - 108 q^{26} + 54 q^{27} + 188 q^{29} + 30 q^{30} + 120 q^{31} + 414 q^{32} - 48 q^{33} - 156 q^{34} + 18 q^{36} - 132 q^{37} - 352 q^{38} + 228 q^{39} + 90 q^{40} - 100 q^{41} - 536 q^{43} + 624 q^{44} - 90 q^{45} + 560 q^{46} + 928 q^{47} - 234 q^{48} - 50 q^{50} + 372 q^{51} + 140 q^{52} + 884 q^{53} - 54 q^{54} + 80 q^{55} + 288 q^{57} + 676 q^{58} - 104 q^{59} - 30 q^{60} + 468 q^{61} + 168 q^{62} + 34 q^{64} - 380 q^{65} - 912 q^{66} - 1688 q^{67} + 188 q^{68} - 48 q^{69} - 136 q^{71} - 162 q^{72} - 508 q^{73} + 1188 q^{74} + 150 q^{75} + 608 q^{76} - 324 q^{78} - 432 q^{79} + 390 q^{80} + 162 q^{81} + 1380 q^{82} + 584 q^{83} - 620 q^{85} - 456 q^{86} + 564 q^{87} + 1744 q^{88} + 1404 q^{89} + 90 q^{90} - 1104 q^{92} + 360 q^{93} - 1600 q^{94} - 480 q^{95} + 1242 q^{96} + 1188 q^{97} - 144 q^{99}+O(q^{100}) \) 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
−3.82843 3.00000 6.65685 −5.00000 −11.4853 0 5.14214 9.00000 19.1421
1.2 1.82843 3.00000 −4.65685 −5.00000 5.48528 0 −23.1421 9.00000 −9.14214
\(n\): 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 735.4.a.o 2
3.b odd 2 1 2205.4.a.bb 2
7.b odd 2 1 105.4.a.e 2
21.c even 2 1 315.4.a.k 2
28.d even 2 1 1680.4.a.bo 2
35.c odd 2 1 525.4.a.l 2
35.f even 4 2 525.4.d.l 4
105.g even 2 1 1575.4.a.q 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.4.a.e 2 7.b odd 2 1
315.4.a.k 2 21.c even 2 1
525.4.a.l 2 35.c odd 2 1
525.4.d.l 4 35.f even 4 2
735.4.a.o 2 1.a even 1 1 trivial
1575.4.a.q 2 105.g even 2 1
1680.4.a.bo 2 28.d even 2 1
2205.4.a.bb 2 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(735))\):

\( T_{2}^{2} + 2T_{2} - 7 \) 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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 2T - 7 \) Copy content Toggle raw display
$3$ \( (T - 3)^{2} \) 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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