Properties

Label 405.4.a.k
Level $405$
Weight $4$
Character orbit 405.a
Self dual yes
Analytic conductor $23.896$
Analytic rank $1$
Dimension $6$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 405 = 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 405.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(23.8957735523\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Defining polynomial: \(x^{6} - 2 x^{5} - 38 x^{4} + 42 x^{3} + 393 x^{2} - 72 x - 432\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2\cdot 3^{3} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 + \beta_{1} ) q^{2} + ( 6 - \beta_{1} + \beta_{3} ) q^{4} -5 q^{5} + ( 7 - 2 \beta_{1} - \beta_{2} - \beta_{4} ) q^{7} + ( -12 + 4 \beta_{1} - 2 \beta_{3} + \beta_{4} ) q^{8} +O(q^{10})\) \( q + ( -1 + \beta_{1} ) q^{2} + ( 6 - \beta_{1} + \beta_{3} ) q^{4} -5 q^{5} + ( 7 - 2 \beta_{1} - \beta_{2} - \beta_{4} ) q^{7} + ( -12 + 4 \beta_{1} - 2 \beta_{3} + \beta_{4} ) q^{8} + ( 5 - 5 \beta_{1} ) q^{10} + ( -14 - 2 \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{5} ) q^{11} + ( 4 + 3 \beta_{2} - \beta_{3} + 2 \beta_{4} - \beta_{5} ) q^{13} + ( -32 + 7 \beta_{1} + \beta_{2} - 4 \beta_{3} + \beta_{4} - 3 \beta_{5} ) q^{14} + ( 14 - 16 \beta_{1} - 3 \beta_{4} + 2 \beta_{5} ) q^{16} + ( -19 - 6 \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{4} + 3 \beta_{5} ) q^{17} + ( -7 + 2 \beta_{1} - 4 \beta_{2} - 4 \beta_{3} + 4 \beta_{4} - 2 \beta_{5} ) q^{19} + ( -30 + 5 \beta_{1} - 5 \beta_{3} ) q^{20} + ( -5 - 20 \beta_{1} + 10 \beta_{2} - \beta_{3} + 2 \beta_{4} + 2 \beta_{5} ) q^{22} + ( -34 - 6 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 3 \beta_{4} - 6 \beta_{5} ) q^{23} + 25 q^{25} + ( -2 - 2 \beta_{1} - 15 \beta_{2} + 5 \beta_{3} - 6 \beta_{4} + 7 \beta_{5} ) q^{26} + ( 70 - 40 \beta_{1} - 29 \beta_{2} + 13 \beta_{3} - 6 \beta_{4} + 3 \beta_{5} ) q^{28} + ( -52 + 12 \beta_{1} - 8 \beta_{2} - \beta_{3} + 4 \beta_{4} + 3 \beta_{5} ) q^{29} + ( -13 - 16 \beta_{1} + 6 \beta_{2} - 10 \beta_{3} - 3 \beta_{4} - 4 \beta_{5} ) q^{31} + ( -114 - 18 \beta_{1} + 24 \beta_{2} - 6 \beta_{3} + \beta_{4} - 6 \beta_{5} ) q^{32} + ( -60 - 25 \beta_{1} + 38 \beta_{2} - 7 \beta_{3} + 9 \beta_{4} - 4 \beta_{5} ) q^{34} + ( -35 + 10 \beta_{1} + 5 \beta_{2} + 5 \beta_{4} ) q^{35} + ( -29 - 10 \beta_{1} - 28 \beta_{2} - 5 \beta_{3} + 5 \beta_{4} + 3 \beta_{5} ) q^{37} + ( 9 - 31 \beta_{1} - 20 \beta_{2} + 14 \beta_{3} - 14 \beta_{4} + 4 \beta_{5} ) q^{38} + ( 60 - 20 \beta_{1} + 10 \beta_{3} - 5 \beta_{4} ) q^{40} + ( -62 + 16 \beta_{1} + 7 \beta_{2} + 26 \beta_{3} + 2 \beta_{4} - 10 \beta_{5} ) q^{41} + ( 99 - 42 \beta_{1} + 38 \beta_{2} + 25 \beta_{3} - \beta_{4} + \beta_{5} ) q^{43} + ( -120 + 5 \beta_{1} - 2 \beta_{2} - 7 \beta_{3} + 3 \beta_{4} + 6 \beta_{5} ) q^{44} + ( -28 - 22 \beta_{1} - 74 \beta_{2} - 14 \beta_{3} - 13 \beta_{4} - 4 \beta_{5} ) q^{46} + ( -23 - 44 \beta_{1} - 14 \beta_{2} + 29 \beta_{3} + 6 \beta_{4} + 9 \beta_{5} ) q^{47} + ( 15 - 26 \beta_{1} - 12 \beta_{2} + 16 \beta_{3} - 15 \beta_{4} + 12 \beta_{5} ) q^{49} + ( -25 + 25 \beta_{1} ) q^{50} + ( -82 + 28 \beta_{1} + 75 \beta_{2} - 11 \beta_{3} + 16 \beta_{4} - 19 \beta_{5} ) q^{52} + ( -127 - 20 \beta_{1} + 58 \beta_{2} + 27 \beta_{3} + 24 \beta_{4} - 5 \beta_{5} ) q^{53} + ( 70 + 10 \beta_{1} - 10 \beta_{2} + 5 \beta_{3} - 5 \beta_{5} ) q^{55} + ( -410 + 92 \beta_{1} + 57 \beta_{2} - 33 \beta_{3} + 20 \beta_{4} - 17 \beta_{5} ) q^{56} + ( 169 - 58 \beta_{1} + 44 \beta_{2} + 21 \beta_{3} + 4 \beta_{4} ) q^{58} + ( -294 - 30 \beta_{1} - 15 \beta_{2} - 6 \beta_{3} + 8 \beta_{4} ) q^{59} + ( -84 + 72 \beta_{1} + 30 \beta_{2} - 38 \beta_{3} + 2 \beta_{4} + 22 \beta_{5} ) q^{61} + ( -155 - 73 \beta_{1} - 54 \beta_{2} - 12 \beta_{3} - 19 \beta_{4} ) q^{62} + ( -158 - 22 \beta_{1} - 96 \beta_{2} - 10 \beta_{3} - \beta_{4} + 10 \beta_{5} ) q^{64} + ( -20 - 15 \beta_{2} + 5 \beta_{3} - 10 \beta_{4} + 5 \beta_{5} ) q^{65} + ( -36 + 164 \beta_{1} - 8 \beta_{2} - 30 \beta_{3} + 12 \beta_{4} + 10 \beta_{5} ) q^{67} + ( -28 - 54 \beta_{1} - 70 \beta_{2} + 8 \beta_{3} - 20 \beta_{4} + 32 \beta_{5} ) q^{68} + ( 160 - 35 \beta_{1} - 5 \beta_{2} + 20 \beta_{3} - 5 \beta_{4} + 15 \beta_{5} ) q^{70} + ( -268 + 82 \beta_{1} - 52 \beta_{2} + 9 \beta_{3} - 40 \beta_{4} - 9 \beta_{5} ) q^{71} + ( 101 + 138 \beta_{1} - 50 \beta_{2} + 33 \beta_{3} + 23 \beta_{4} - 7 \beta_{5} ) q^{73} + ( -200 - 59 \beta_{1} + 64 \beta_{2} + 5 \beta_{3} - \beta_{4} - 18 \beta_{5} ) q^{74} + ( -374 + 77 \beta_{1} + 100 \beta_{2} - 41 \beta_{3} + 8 \beta_{4} - 32 \beta_{5} ) q^{76} + ( -123 - 46 \beta_{1} + 40 \beta_{2} - 19 \beta_{3} + 5 \beta_{4} + 9 \beta_{5} ) q^{77} + ( -172 + 144 \beta_{1} + 10 \beta_{2} + 12 \beta_{3} - 32 \beta_{4} - 12 \beta_{5} ) q^{79} + ( -70 + 80 \beta_{1} + 15 \beta_{4} - 10 \beta_{5} ) q^{80} + ( 257 + 94 \beta_{1} - 127 \beta_{2} - 6 \beta_{3} - 6 \beta_{4} + 11 \beta_{5} ) q^{82} + ( -369 + 6 \beta_{1} - 34 \beta_{2} - 45 \beta_{3} - 27 \beta_{4} + 19 \beta_{5} ) q^{83} + ( 95 + 30 \beta_{1} + 10 \beta_{2} + 5 \beta_{3} + 5 \beta_{4} - 15 \beta_{5} ) q^{85} + ( -552 + 249 \beta_{1} - 26 \beta_{2} - 69 \beta_{3} + 29 \beta_{4} + 36 \beta_{5} ) q^{86} + ( 214 - 2 \beta_{1} - 6 \beta_{2} + 26 \beta_{3} - 8 \beta_{4} - 12 \beta_{5} ) q^{88} + ( -474 - 96 \beta_{1} + 76 \beta_{2} - 33 \beta_{3} - 18 \beta_{4} - 21 \beta_{5} ) q^{89} + ( -538 + 106 \beta_{1} + 44 \beta_{2} - 34 \beta_{3} + 21 \beta_{4} - 18 \beta_{5} ) q^{91} + ( -142 - 64 \beta_{1} + 10 \beta_{2} - 50 \beta_{3} + 11 \beta_{4} - 52 \beta_{5} ) q^{92} + ( -644 + 151 \beta_{1} + 122 \beta_{2} - 61 \beta_{3} + 50 \beta_{4} - 2 \beta_{5} ) q^{94} + ( 35 - 10 \beta_{1} + 20 \beta_{2} + 20 \beta_{3} - 20 \beta_{4} + 10 \beta_{5} ) q^{95} + ( -125 + 30 \beta_{1} + 72 \beta_{2} - 13 \beta_{3} - 39 \beta_{4} - 41 \beta_{5} ) q^{97} + ( -345 + 111 \beta_{1} + 156 \beta_{2} - 72 \beta_{3} + 67 \beta_{4} - 42 \beta_{5} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 4 q^{2} + 34 q^{4} - 30 q^{5} + 40 q^{7} - 66 q^{8} + O(q^{10}) \) \( 6 q - 4 q^{2} + 34 q^{4} - 30 q^{5} + 40 q^{7} - 66 q^{8} + 20 q^{10} - 88 q^{11} + 20 q^{13} - 180 q^{14} + 58 q^{16} - 124 q^{17} - 46 q^{19} - 170 q^{20} - 74 q^{22} - 210 q^{23} + 150 q^{25} - 4 q^{26} + 352 q^{28} - 296 q^{29} - 104 q^{31} - 722 q^{32} - 428 q^{34} - 200 q^{35} - 204 q^{37} + 20 q^{38} + 330 q^{40} - 344 q^{41} + 512 q^{43} - 716 q^{44} - 186 q^{46} - 238 q^{47} + 68 q^{49} - 100 q^{50} - 468 q^{52} - 850 q^{53} + 440 q^{55} - 2316 q^{56} + 890 q^{58} - 1840 q^{59} - 364 q^{61} - 1038 q^{62} - 990 q^{64} - 100 q^{65} + 88 q^{67} - 236 q^{68} + 900 q^{70} - 1364 q^{71} + 836 q^{73} - 1316 q^{74} - 2106 q^{76} - 840 q^{77} - 680 q^{79} - 290 q^{80} + 1742 q^{82} - 2148 q^{83} + 620 q^{85} - 2872 q^{86} + 1296 q^{88} - 3000 q^{89} - 3058 q^{91} - 1002 q^{92} - 3662 q^{94} + 230 q^{95} - 612 q^{97} - 1982 q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 2 x^{5} - 38 x^{4} + 42 x^{3} + 393 x^{2} - 72 x - 432\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{5} - 2 \nu^{4} - 26 \nu^{3} + 30 \nu^{2} + 141 \nu - 36 \)\()/24\)
\(\beta_{3}\)\(=\)\( \nu^{2} - \nu - 13 \)
\(\beta_{4}\)\(=\)\( \nu^{3} - \nu^{2} - 19 \nu + 1 \)
\(\beta_{5}\)\(=\)\((\)\( \nu^{4} - \nu^{3} - 21 \nu^{2} + 3 \nu + 30 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + \beta_{1} + 13\)
\(\nu^{3}\)\(=\)\(\beta_{4} + \beta_{3} + 20 \beta_{1} + 12\)
\(\nu^{4}\)\(=\)\(2 \beta_{5} + \beta_{4} + 22 \beta_{3} + 38 \beta_{1} + 255\)
\(\nu^{5}\)\(=\)\(4 \beta_{5} + 28 \beta_{4} + 40 \beta_{3} + 24 \beta_{2} + 425 \beta_{1} + 468\)

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
−4.23336
−3.53444
−1.07326
1.14915
4.57457
5.11734
−5.23336 0 19.3881 −5.00000 0 33.0180 −59.5981 0 26.1668
1.2 −4.53444 0 12.5612 −5.00000 0 −2.63618 −20.6823 0 22.6722
1.3 −2.07326 0 −3.70159 −5.00000 0 −4.66112 24.2604 0 10.3663
1.4 0.149150 0 −7.97775 −5.00000 0 20.1424 −2.38308 0 −0.745751
1.5 3.57457 0 4.77759 −5.00000 0 14.1597 −11.5188 0 −17.8729
1.6 4.11734 0 8.95250 −5.00000 0 −20.0229 3.92177 0 −20.5867
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Atkin-Lehner signs

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

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 405.4.a.k 6
3.b odd 2 1 405.4.a.l yes 6
5.b even 2 1 2025.4.a.z 6
9.c even 3 2 405.4.e.x 12
9.d odd 6 2 405.4.e.w 12
15.d odd 2 1 2025.4.a.y 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
405.4.a.k 6 1.a even 1 1 trivial
405.4.a.l yes 6 3.b odd 2 1
405.4.e.w 12 9.d odd 6 2
405.4.e.x 12 9.c even 3 2
2025.4.a.y 6 15.d odd 2 1
2025.4.a.z 6 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} + 4 T_{2}^{5} - 33 T_{2}^{4} - 110 T_{2}^{3} + 286 T_{2}^{2} + 684 T_{2} - 108 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(405))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -108 + 684 T + 286 T^{2} - 110 T^{3} - 33 T^{4} + 4 T^{5} + T^{6} \)
$3$ \( T^{6} \)
$5$ \( ( 5 + T )^{6} \)
$7$ \( -2316924 - 1142856 T - 49260 T^{2} + 18900 T^{3} - 263 T^{4} - 40 T^{5} + T^{6} \)
$11$ \( 1197108 - 426912 T - 167759 T^{2} - 4520 T^{3} + 1518 T^{4} + 88 T^{5} + T^{6} \)
$13$ \( -185793728 - 11327360 T + 2055488 T^{2} + 93472 T^{3} - 4519 T^{4} - 20 T^{5} + T^{6} \)
$17$ \( 105966288 + 34954656 T - 3790268 T^{2} - 762824 T^{3} - 5004 T^{4} + 124 T^{5} + T^{6} \)
$19$ \( -36821611175 + 1527158350 T + 46339391 T^{2} - 872156 T^{3} - 18169 T^{4} + 46 T^{5} + T^{6} \)
$23$ \( 332569842768 + 29896763040 T + 135198072 T^{2} - 5308560 T^{3} - 25779 T^{4} + 210 T^{5} + T^{6} \)
$29$ \( 635447099088 + 9092541024 T - 206203583 T^{2} - 3275896 T^{3} + 10626 T^{4} + 296 T^{5} + T^{6} \)
$31$ \( -69859854216 - 1251193716 T + 509538609 T^{2} - 1378812 T^{3} - 58274 T^{4} + 104 T^{5} + T^{6} \)
$37$ \( 12008297128192 + 211123326144 T - 305431068 T^{2} - 20257048 T^{3} - 74400 T^{4} + 204 T^{5} + T^{6} \)
$41$ \( -86213802377403 + 1228171376184 T + 10310796571 T^{2} - 48204400 T^{3} - 185193 T^{4} + 344 T^{5} + T^{6} \)
$43$ \( -180465347194400 - 3519864391760 T + 8186569412 T^{2} + 89502136 T^{3} - 172552 T^{4} - 512 T^{5} + T^{6} \)
$47$ \( -49451750433900 + 1521993107760 T + 23432551972 T^{2} - 86562956 T^{3} - 427923 T^{4} + 238 T^{5} + T^{6} \)
$53$ \( 15741889277692500 + 87258267127200 T - 7800157964 T^{2} - 601620140 T^{3} - 550155 T^{4} + 850 T^{5} + T^{6} \)
$59$ \( -168320389359483 + 2504531324400 T + 64312689211 T^{2} + 431982880 T^{3} + 1299447 T^{4} + 1840 T^{5} + T^{6} \)
$61$ \( -20318301594331136 + 69945317346304 T + 298491534848 T^{2} - 318570368 T^{3} - 1014088 T^{4} + 364 T^{5} + T^{6} \)
$67$ \( 19496902162867200 - 171798967607040 T + 382729270992 T^{2} + 289302624 T^{3} - 1313672 T^{4} - 88 T^{5} + T^{6} \)
$71$ \( -11833359413893884 + 215551314974244 T - 225597721799 T^{2} - 1811192200 T^{3} - 995994 T^{4} + 1364 T^{5} + T^{6} \)
$73$ \( -10768933998500336 - 63694541190656 T + 254161845668 T^{2} + 781045192 T^{3} - 1130908 T^{4} - 836 T^{5} + T^{6} \)
$79$ \( -15947821725492672 + 246823997051520 T + 413753779248 T^{2} - 1090263744 T^{3} - 1627796 T^{4} + 680 T^{5} + T^{6} \)
$83$ \( 4163598986306832 - 30732426001728 T - 464661188604 T^{2} - 848431368 T^{3} + 655092 T^{4} + 2148 T^{5} + T^{6} \)
$89$ \( 167521946305912848 + 24233669446680 T - 1254009337479 T^{2} - 1004808240 T^{3} + 2101338 T^{4} + 3000 T^{5} + T^{6} \)
$97$ \( -979999359726233024 + 975674052130752 T + 3181554001188 T^{2} - 1545360136 T^{3} - 3167712 T^{4} + 612 T^{5} + T^{6} \)
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