Properties

Label 735.4.a.q
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: \(\Q(\sqrt{41}) \)
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 \(\beta = \frac{1}{2}(1 + \sqrt{41})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 + \beta ) q^{2} -3 q^{3} + ( 3 + 3 \beta ) q^{4} + 5 q^{5} + ( -3 - 3 \beta ) q^{6} + ( 25 + \beta ) q^{8} + 9 q^{9} +O(q^{10})\) \( q + ( 1 + \beta ) q^{2} -3 q^{3} + ( 3 + 3 \beta ) q^{4} + 5 q^{5} + ( -3 - 3 \beta ) q^{6} + ( 25 + \beta ) q^{8} + 9 q^{9} + ( 5 + 5 \beta ) q^{10} + ( 32 - 2 \beta ) q^{11} + ( -9 - 9 \beta ) q^{12} + ( -2 + 10 \beta ) q^{13} -15 q^{15} + ( 11 + 3 \beta ) q^{16} + ( -26 + 12 \beta ) q^{17} + ( 9 + 9 \beta ) q^{18} + ( 60 + 2 \beta ) q^{19} + ( 15 + 15 \beta ) q^{20} + ( 12 + 28 \beta ) q^{22} + ( 32 - 48 \beta ) q^{23} + ( -75 - 3 \beta ) q^{24} + 25 q^{25} + ( 98 + 18 \beta ) q^{26} -27 q^{27} + ( 170 + 12 \beta ) q^{29} + ( -15 - 15 \beta ) q^{30} + ( -52 + 38 \beta ) q^{31} + ( -159 + 9 \beta ) q^{32} + ( -96 + 6 \beta ) q^{33} + ( 94 - 2 \beta ) q^{34} + ( 27 + 27 \beta ) q^{36} + ( -134 + 80 \beta ) q^{37} + ( 80 + 64 \beta ) q^{38} + ( 6 - 30 \beta ) q^{39} + ( 125 + 5 \beta ) q^{40} + ( -62 + 108 \beta ) q^{41} + ( -248 + 100 \beta ) q^{43} + ( 36 + 84 \beta ) q^{44} + 45 q^{45} + ( -448 - 64 \beta ) q^{46} + ( 164 - 140 \beta ) q^{47} + ( -33 - 9 \beta ) q^{48} + ( 25 + 25 \beta ) q^{50} + ( 78 - 36 \beta ) q^{51} + ( 294 + 54 \beta ) q^{52} + ( 462 + 58 \beta ) q^{53} + ( -27 - 27 \beta ) q^{54} + ( 160 - 10 \beta ) q^{55} + ( -180 - 6 \beta ) q^{57} + ( 290 + 194 \beta ) q^{58} + ( -220 - 76 \beta ) q^{59} + ( -45 - 45 \beta ) q^{60} + ( 398 + 84 \beta ) q^{61} + ( 328 + 24 \beta ) q^{62} + ( -157 - 165 \beta ) q^{64} + ( -10 + 50 \beta ) q^{65} + ( -36 - 84 \beta ) q^{66} + ( -64 - 228 \beta ) q^{67} + ( 282 - 6 \beta ) q^{68} + ( -96 + 144 \beta ) q^{69} + ( 112 + 86 \beta ) q^{71} + ( 225 + 9 \beta ) q^{72} + ( -182 + 38 \beta ) q^{73} + ( 666 + 26 \beta ) q^{74} -75 q^{75} + ( 240 + 192 \beta ) q^{76} + ( -294 - 54 \beta ) q^{78} + ( 920 - 8 \beta ) q^{79} + ( 55 + 15 \beta ) q^{80} + 81 q^{81} + ( 1018 + 154 \beta ) q^{82} + ( 228 + 224 \beta ) q^{83} + ( -130 + 60 \beta ) q^{85} + ( 752 - 48 \beta ) q^{86} + ( -510 - 36 \beta ) q^{87} + ( 780 - 20 \beta ) q^{88} + ( -230 - 336 \beta ) q^{89} + ( 45 + 45 \beta ) q^{90} + ( -1344 - 192 \beta ) q^{92} + ( 156 - 114 \beta ) q^{93} + ( -1236 - 116 \beta ) q^{94} + ( 300 + 10 \beta ) q^{95} + ( 477 - 27 \beta ) q^{96} + ( 474 - 278 \beta ) q^{97} + ( 288 - 18 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 3q^{2} - 6q^{3} + 9q^{4} + 10q^{5} - 9q^{6} + 51q^{8} + 18q^{9} + O(q^{10}) \) \( 2q + 3q^{2} - 6q^{3} + 9q^{4} + 10q^{5} - 9q^{6} + 51q^{8} + 18q^{9} + 15q^{10} + 62q^{11} - 27q^{12} + 6q^{13} - 30q^{15} + 25q^{16} - 40q^{17} + 27q^{18} + 122q^{19} + 45q^{20} + 52q^{22} + 16q^{23} - 153q^{24} + 50q^{25} + 214q^{26} - 54q^{27} + 352q^{29} - 45q^{30} - 66q^{31} - 309q^{32} - 186q^{33} + 186q^{34} + 81q^{36} - 188q^{37} + 224q^{38} - 18q^{39} + 255q^{40} - 16q^{41} - 396q^{43} + 156q^{44} + 90q^{45} - 960q^{46} + 188q^{47} - 75q^{48} + 75q^{50} + 120q^{51} + 642q^{52} + 982q^{53} - 81q^{54} + 310q^{55} - 366q^{57} + 774q^{58} - 516q^{59} - 135q^{60} + 880q^{61} + 680q^{62} - 479q^{64} + 30q^{65} - 156q^{66} - 356q^{67} + 558q^{68} - 48q^{69} + 310q^{71} + 459q^{72} - 326q^{73} + 1358q^{74} - 150q^{75} + 672q^{76} - 642q^{78} + 1832q^{79} + 125q^{80} + 162q^{81} + 2190q^{82} + 680q^{83} - 200q^{85} + 1456q^{86} - 1056q^{87} + 1540q^{88} - 796q^{89} + 135q^{90} - 2880q^{92} + 198q^{93} - 2588q^{94} + 610q^{95} + 927q^{96} + 670q^{97} + 558q^{99} + O(q^{100}) \)

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
−2.70156
3.70156
−1.70156 −3.00000 −5.10469 5.00000 5.10469 0 22.2984 9.00000 −8.50781
1.2 4.70156 −3.00000 14.1047 5.00000 −14.1047 0 28.7016 9.00000 23.5078
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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.q 2
3.b odd 2 1 2205.4.a.v 2
7.b odd 2 1 105.4.a.g 2
21.c even 2 1 315.4.a.g 2
28.d even 2 1 1680.4.a.y 2
35.c odd 2 1 525.4.a.i 2
35.f even 4 2 525.4.d.j 4
105.g even 2 1 1575.4.a.y 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.4.a.g 2 7.b odd 2 1
315.4.a.g 2 21.c even 2 1
525.4.a.i 2 35.c odd 2 1
525.4.d.j 4 35.f even 4 2
735.4.a.q 2 1.a even 1 1 trivial
1575.4.a.y 2 105.g even 2 1
1680.4.a.y 2 28.d even 2 1
2205.4.a.v 2 3.b odd 2 1

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(5\) \(-1\)
\(7\) \(-1\)

Hecke kernels

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

\( T_{2}^{2} - 3 T_{2} - 8 \)
\( T_{11}^{2} - 62 T_{11} + 920 \)
\( T_{13}^{2} - 6 T_{13} - 1016 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 3 T + 8 T^{2} - 24 T^{3} + 64 T^{4} \)
$3$ \( ( 1 + 3 T )^{2} \)
$5$ \( ( 1 - 5 T )^{2} \)
$7$ 1
$11$ \( 1 - 62 T + 3582 T^{2} - 82522 T^{3} + 1771561 T^{4} \)
$13$ \( 1 - 6 T + 3378 T^{2} - 13182 T^{3} + 4826809 T^{4} \)
$17$ \( 1 + 40 T + 8750 T^{2} + 196520 T^{3} + 24137569 T^{4} \)
$19$ \( 1 - 122 T + 17398 T^{2} - 836798 T^{3} + 47045881 T^{4} \)
$23$ \( 1 - 16 T + 782 T^{2} - 194672 T^{3} + 148035889 T^{4} \)
$29$ \( 1 - 352 T + 78278 T^{2} - 8584928 T^{3} + 594823321 T^{4} \)
$31$ \( 1 + 66 T + 45870 T^{2} + 1966206 T^{3} + 887503681 T^{4} \)
$37$ \( 1 + 188 T + 44542 T^{2} + 9522764 T^{3} + 2565726409 T^{4} \)
$41$ \( 1 + 16 T + 18350 T^{2} + 1102736 T^{3} + 4750104241 T^{4} \)
$43$ \( 1 + 396 T + 95718 T^{2} + 31484772 T^{3} + 6321363049 T^{4} \)
$47$ \( 1 - 188 T + 15582 T^{2} - 19518724 T^{3} + 10779215329 T^{4} \)
$53$ \( 1 - 982 T + 504354 T^{2} - 146197214 T^{3} + 22164361129 T^{4} \)
$59$ \( 1 + 516 T + 418118 T^{2} + 105975564 T^{3} + 42180533641 T^{4} \)
$61$ \( 1 - 880 T + 575238 T^{2} - 199743280 T^{3} + 51520374361 T^{4} \)
$67$ \( 1 + 356 T + 100374 T^{2} + 107071628 T^{3} + 90458382169 T^{4} \)
$71$ \( 1 - 310 T + 664038 T^{2} - 110952410 T^{3} + 128100283921 T^{4} \)
$73$ \( 1 + 326 T + 789802 T^{2} + 126819542 T^{3} + 151334226289 T^{4} \)
$79$ \( 1 - 1832 T + 1824478 T^{2} - 903247448 T^{3} + 243087455521 T^{4} \)
$83$ \( 1 - 680 T + 744870 T^{2} - 388815160 T^{3} + 326940373369 T^{4} \)
$89$ \( 1 + 796 T + 411158 T^{2} + 561155324 T^{3} + 496981290961 T^{4} \)
$97$ \( 1 - 670 T + 1145410 T^{2} - 611490910 T^{3} + 832972004929 T^{4} \)
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