Properties

Label 261.6.a.a
Level $261$
Weight $6$
Character orbit 261.a
Self dual yes
Analytic conductor $41.860$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 261 = 3^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 261.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(41.8601769712\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.3257317.1
Defining polynomial: \( x^{4} - 34x^{2} - 27x + 10 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 29)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + ( - \beta_{3} + 5 \beta_{2} - 3 \beta_1 + 2) q^{4} + ( - 4 \beta_{3} - 6 \beta_{2} + 2 \beta_1 + 15) q^{5} + ( - 12 \beta_{3} - 4 \beta_{2} - 58) q^{7} + (5 \beta_{2} - 14 \beta_1 + 126) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} + ( - \beta_{3} + 5 \beta_{2} - 3 \beta_1 + 2) q^{4} + ( - 4 \beta_{3} - 6 \beta_{2} + 2 \beta_1 + 15) q^{5} + ( - 12 \beta_{3} - 4 \beta_{2} - 58) q^{7} + (5 \beta_{2} - 14 \beta_1 + 126) q^{8} + (26 \beta_{3} - 17 \beta_{2} + 22 \beta_1 - 184) q^{10} + (20 \beta_{3} + 29 \beta_{2} + 21 \beta_1 + 41) q^{11} + (60 \beta_{3} - 38 \beta_{2} - 10 \beta_1 - 85) q^{13} + (64 \beta_{3} - 66 \beta_{2} + 24 \beta_1 - 160) q^{14} + (27 \beta_{3} + 33 \beta_{2} + 81 \beta_1 - 90) q^{16} + (4 \beta_{3} - 50 \beta_{2} - 14 \beta_1 - 44) q^{17} + (116 \beta_{3} - 80 \beta_{2} + 168 \beta_1 - 540) q^{19} + (15 \beta_{3} - 169 \beta_{2} - 39 \beta_1 - 698) q^{20} + ( - 129 \beta_{3} + 103 \beta_{2} - 107 \beta_1 + 1320) q^{22} + (140 \beta_{3} - 14 \beta_{2} - 146 \beta_1 + 368) q^{23} + ( - 280 \beta_{3} - 220 \beta_{2} - 140 \beta_1 + 316) q^{25} + ( - 262 \beta_{3} - 305 \beta_{2} + 54 \beta_1 - 1312) q^{26} + (130 \beta_{3} - 498 \beta_{2} + 134 \beta_1 + 76) q^{28} + 841 q^{29} + ( - 92 \beta_{3} - 319 \beta_{2} + 485 \beta_1 - 4849) q^{31} + ( - 168 \beta_{3} - 355 \beta_{2} + 322 \beta_1 - 1722) q^{32} + (30 \beta_{3} - 256 \beta_{2} + 146 \beta_1 - 1888) q^{34} + ( - 220 \beta_{3} - 304 \beta_{2} - 564 \beta_1 + 5626) q^{35} + (40 \beta_{3} + 544 \beta_{2} - 712 \beta_1 - 2712) q^{37} + ( - 500 \beta_{3} - 1560 \beta_{2} + 124 \beta_1 - 136) q^{38} + ( - 738 \beta_{3} - 897 \beta_{2} - 212 \beta_1 - 374) q^{40} + (804 \beta_{3} - 1950 \beta_{2} + 366 \beta_1 + 682) q^{41} + (772 \beta_{3} + 1603 \beta_{2} - 341 \beta_1 - 4969) q^{43} + ( - 98 \beta_{3} + 1357 \beta_{2} - 852 \beta_1 + 434) q^{44} + ( - 686 \beta_{3} + 596 \beta_{2} - 98 \beta_1 - 2240) q^{46} + ( - 72 \beta_{3} - 449 \beta_{2} + 1675 \beta_1 - 5979) q^{47} + (1040 \beta_{3} - 1424 \beta_{2} - 1152 \beta_1 + 3133) q^{49} + (1620 \beta_{3} - 84 \beta_{2} + 940 \beta_1 - 10000) q^{50} + ( - 305 \beta_{3} - 1521 \beta_{2} + 1497 \beta_1 - 7418) q^{52} + ( - 628 \beta_{3} + 2258 \beta_{2} - 1898 \beta_1 - 2529) q^{53} + (1716 \beta_{3} + 545 \beta_{2} + 2181 \beta_1 - 12305) q^{55} + ( - 2200 \beta_{3} - 834 \beta_{2} + 596 \beta_1 - 9676) q^{56} + 841 \beta_{2} q^{58} + (140 \beta_{3} + 2466 \beta_{2} - 3098 \beta_1 + 2780) q^{59} + ( - 2396 \beta_{3} + 3582 \beta_{2} - 3142 \beta_1 + 11164) q^{61} + (779 \beta_{3} - 7807 \beta_{2} + 1049 \beta_1 - 4240) q^{62} + (331 \beta_{3} - 5351 \beta_{2} - 1359 \beta_1 - 5018) q^{64} + (832 \beta_{3} + 4716 \beta_{2} + 384 \beta_1 - 24043) q^{65} + ( - 1032 \beta_{3} + 4676 \beta_{2} - 4428 \beta_1 - 2476) q^{67} + ( - 22 \beta_{3} - 2036 \beta_{2} + 1186 \beta_1 - 5192) q^{68} + (1404 \beta_{3} + 6018 \beta_{2} + 1132 \beta_1 - 18672) q^{70} + (96 \beta_{3} + 3598 \beta_{2} - 3250 \beta_1 + 12234) q^{71} + ( - 1504 \beta_{3} - 2200 \beta_{2} + 3784 \beta_1 - 19500) q^{73} + ( - 744 \beta_{3} + 2104 \beta_{2} - 1672 \beta_1 + 8608) q^{74} + (348 \beta_{3} - 5248 \beta_{2} - 196 \beta_1 - 35024) q^{76} + (1876 \beta_{3} + 1122 \beta_{2} + 1742 \beta_1 - 31226) q^{77} + (2864 \beta_{3} - 5901 \beta_{2} - 1457 \beta_1 - 25087) q^{79} + (4107 \beta_{3} + 1923 \beta_{2} + 4677 \beta_1 - 12606) q^{80} + ( - 2070 \beta_{3} - 10970 \beta_{2} + 5046 \beta_1 - 59568) q^{82} + (1108 \beta_{3} - 2566 \beta_{2} + 5038 \beta_1 - 15168) q^{83} + ( - 1372 \beta_{3} + 1298 \beta_{2} - 1642 \beta_1 + 5276) q^{85} + ( - 5463 \beta_{3} + 3297 \beta_{2} - 5581 \beta_1 + 51272) q^{86} + (3261 \beta_{3} + 6577 \beta_{2} - 549 \beta_1 - 8226) q^{88} + ( - 4380 \beta_{3} + 7382 \beta_{2} + 842 \beta_1 - 29082) q^{89} + ( - 5132 \beta_{3} + 12448 \beta_{2} + 4708 \beta_1 - 69790) q^{91} + ( - 1646 \beta_{3} + 2168 \beta_{2} + 3570 \beta_1 + 5744) q^{92} + (809 \beta_{3} - 13177 \beta_{2} + 1419 \beta_1 + 8040) q^{94} + (7812 \beta_{3} + 12232 \beta_{2} + 7528 \beta_1 - 32932) q^{95} + (7180 \beta_{3} + 7150 \beta_{2} + 3962 \beta_1 - 8790) q^{97} + ( - 3776 \beta_{3} - 1571 \beta_{2} + 3232 \beta_1 - 62464) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 10 q^{4} + 68 q^{5} - 208 q^{7} + 504 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 10 q^{4} + 68 q^{5} - 208 q^{7} + 504 q^{8} - 788 q^{10} + 124 q^{11} - 460 q^{13} - 768 q^{14} - 414 q^{16} - 184 q^{17} - 2392 q^{19} - 2822 q^{20} + 5538 q^{22} + 1192 q^{23} + 1824 q^{25} - 4724 q^{26} + 44 q^{28} + 3364 q^{29} - 19212 q^{31} - 6552 q^{32} - 7612 q^{34} + 22944 q^{35} - 10928 q^{37} + 456 q^{38} - 20 q^{40} + 1120 q^{41} - 21420 q^{43} + 1932 q^{44} - 7588 q^{46} - 23772 q^{47} + 10452 q^{49} - 43240 q^{50} - 29062 q^{52} - 8860 q^{53} - 52652 q^{55} - 34304 q^{56} + 10840 q^{59} + 49448 q^{61} - 18518 q^{62} - 20734 q^{64} - 97836 q^{65} - 7840 q^{67} - 20724 q^{68} - 77496 q^{70} + 48744 q^{71} - 74992 q^{73} + 35920 q^{74} - 140792 q^{76} - 128656 q^{77} - 106076 q^{79} - 58638 q^{80} - 234132 q^{82} - 62888 q^{83} + 23848 q^{85} + 216014 q^{86} - 39426 q^{88} - 107568 q^{89} - 268896 q^{91} + 26268 q^{92} + 30542 q^{94} - 147352 q^{95} - 49520 q^{97} - 242304 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 34x^{2} - 27x + 10 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{2} + 3\nu - 17 ) / 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} - 3\nu - 17 ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2\nu^{3} - \nu^{2} - 67\nu - 25 ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3\beta_{2} + 3\beta _1 + 34 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3\beta_{3} - 32\beta_{2} + 35\beta _1 + 42 ) / 2 \) 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
0.275208
−1.10057
6.17343
−5.34807
−5.91663 0 3.00648 97.7313 0 139.558 171.544 0 −578.240
1.2 −4.16235 0 −14.6748 −32.5670 0 −220.793 194.277 0 135.555
1.3 0.863638 0 −31.2541 44.4758 0 −36.7447 −54.6287 0 38.4110
1.4 9.21534 0 52.9225 −41.6400 0 −90.0205 192.808 0 −383.727
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(29\) \(-1\)

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 261.6.a.a 4
3.b odd 2 1 29.6.a.a 4
12.b even 2 1 464.6.a.i 4
15.d odd 2 1 725.6.a.a 4
87.d odd 2 1 841.6.a.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
29.6.a.a 4 3.b odd 2 1
261.6.a.a 4 1.a even 1 1 trivial
464.6.a.i 4 12.b even 2 1
725.6.a.a 4 15.d odd 2 1
841.6.a.a 4 87.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - 69T_{2}^{2} - 168T_{2} + 196 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(261))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 69 T^{2} - 168 T + 196 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 68 T^{3} - 4850 T^{2} + \cdots + 5894489 \) Copy content Toggle raw display
$7$ \( T^{4} + 208 T^{3} + \cdots - 101924272 \) Copy content Toggle raw display
$11$ \( T^{4} - 124 T^{3} + \cdots - 3717303119 \) Copy content Toggle raw display
$13$ \( T^{4} + 460 T^{3} + \cdots - 116053863479 \) Copy content Toggle raw display
$17$ \( T^{4} + 184 T^{3} + \cdots + 11464717824 \) Copy content Toggle raw display
$19$ \( T^{4} + 2392 T^{3} + \cdots + 2620094791680 \) Copy content Toggle raw display
$23$ \( T^{4} - 1192 T^{3} + \cdots + 2033080361984 \) Copy content Toggle raw display
$29$ \( (T - 841)^{4} \) Copy content Toggle raw display
$31$ \( T^{4} + \cdots - 421127233952247 \) Copy content Toggle raw display
$37$ \( T^{4} + 10928 T^{3} + \cdots + 16079593861120 \) Copy content Toggle raw display
$41$ \( T^{4} - 1120 T^{3} + \cdots + 10\!\cdots\!56 \) Copy content Toggle raw display
$43$ \( T^{4} + 21420 T^{3} + \cdots + 47\!\cdots\!37 \) Copy content Toggle raw display
$47$ \( T^{4} + 23772 T^{3} + \cdots - 35\!\cdots\!87 \) Copy content Toggle raw display
$53$ \( T^{4} + 8860 T^{3} + \cdots + 16\!\cdots\!53 \) Copy content Toggle raw display
$59$ \( T^{4} - 10840 T^{3} + \cdots - 43\!\cdots\!80 \) Copy content Toggle raw display
$61$ \( T^{4} - 49448 T^{3} + \cdots + 90\!\cdots\!24 \) Copy content Toggle raw display
$67$ \( T^{4} + 7840 T^{3} + \cdots + 20\!\cdots\!92 \) Copy content Toggle raw display
$71$ \( T^{4} - 48744 T^{3} + \cdots - 18\!\cdots\!16 \) Copy content Toggle raw display
$73$ \( T^{4} + 74992 T^{3} + \cdots - 40\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{4} + 106076 T^{3} + \cdots - 53\!\cdots\!75 \) Copy content Toggle raw display
$83$ \( T^{4} + 62888 T^{3} + \cdots - 18\!\cdots\!88 \) Copy content Toggle raw display
$89$ \( T^{4} + 107568 T^{3} + \cdots - 18\!\cdots\!60 \) Copy content Toggle raw display
$97$ \( T^{4} + 49520 T^{3} + \cdots - 12\!\cdots\!28 \) Copy content Toggle raw display
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