Properties

Label 45.10.b.b
Level 45
Weight 10
Character orbit 45.b
Analytic conductor 23.177
Analytic rank 0
Dimension 4
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 45 = 3^{2} \cdot 5 \)
Weight: \( k \) = \( 10 \)
Character orbit: \([\chi]\) = 45.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(23.1766126274\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.0.49740556.1
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{5}\cdot 3^{2}\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 5)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + ( -342 - \beta_{3} ) q^{4} + ( -285 + 19 \beta_{1} + 7 \beta_{2} - \beta_{3} ) q^{5} + ( -133 \beta_{1} + 49 \beta_{2} ) q^{7} + ( -684 \beta_{1} + 8 \beta_{2} ) q^{8} +O(q^{10})\) \( q + \beta_{1} q^{2} + ( -342 - \beta_{3} ) q^{4} + ( -285 + 19 \beta_{1} + 7 \beta_{2} - \beta_{3} ) q^{5} + ( -133 \beta_{1} + 49 \beta_{2} ) q^{7} + ( -684 \beta_{1} + 8 \beta_{2} ) q^{8} + ( -17290 - 1139 \beta_{1} + 8 \beta_{2} - 19 \beta_{3} ) q^{10} + ( -27492 + 20 \beta_{3} ) q^{11} + ( 1918 \beta_{1} + 110 \beta_{2} ) q^{13} + ( 106134 + 133 \beta_{3} ) q^{14} + ( 407816 + 172 \beta_{3} ) q^{16} + ( -7448 \beta_{1} - 1632 \beta_{2} ) q^{17} + ( -159220 + 476 \beta_{3} ) q^{19} + ( 825570 - 23788 \beta_{1} + 3736 \beta_{2} + 627 \beta_{3} ) q^{20} + ( -10412 \beta_{1} - 160 \beta_{2} ) q^{22} + ( 17005 \beta_{1} + 5199 \beta_{2} ) q^{23} + ( -334475 - 45410 \beta_{1} + 8270 \beta_{2} + 1140 \beta_{3} ) q^{25} + ( -1654692 - 1918 \beta_{3} ) q^{26} + ( 151620 \beta_{1} + 24024 \beta_{2} ) q^{28} + ( 882930 + 1976 \beta_{3} ) q^{29} + ( -2646928 - 760 \beta_{3} ) q^{31} + ( 204496 \beta_{1} + 2720 \beta_{2} ) q^{32} + ( 6608656 + 7448 \beta_{3} ) q^{34} + ( -3407460 + 144039 \beta_{1} + 26817 \beta_{2} + 9044 \beta_{3} ) q^{35} + ( 335370 \beta_{1} + 51378 \beta_{2} ) q^{37} + ( 247284 \beta_{1} - 3808 \beta_{2} ) q^{38} + ( 10894600 + 777860 \beta_{1} - 920 \beta_{2} + 14060 \beta_{3} ) q^{40} + ( 4197138 - 24890 \beta_{3} ) q^{41} + ( -719131 \beta_{1} + 132643 \beta_{2} ) q^{43} + ( -5159736 + 20652 \beta_{3} ) q^{44} + ( -15312518 - 17005 \beta_{3} ) q^{46} + ( -1012111 \beta_{1} + 214259 \beta_{2} ) q^{47} + ( -11730257 + 27930 \beta_{3} ) q^{49} + ( 37523100 + 639085 \beta_{1} - 9120 \beta_{2} + 45410 \beta_{3} ) q^{50} + ( -2310648 \beta_{1} + 71664 \beta_{2} ) q^{52} + ( -1349666 \beta_{1} - 259794 \beta_{2} ) q^{53} + ( -6726780 - 176548 \beta_{1} - 315044 \beta_{2} + 21792 \beta_{3} ) q^{55} + ( -78794520 - 83524 \beta_{3} ) q^{56} + ( 2570434 \beta_{1} - 15808 \beta_{2} ) q^{58} + ( 115207260 + 52972 \beta_{3} ) q^{59} + ( 90122642 - 43150 \beta_{3} ) q^{61} + ( -3295968 \beta_{1} + 6080 \beta_{2} ) q^{62} + ( 33748768 - 116432 \beta_{3} ) q^{64} + ( -45973920 - 2201322 \beta_{1} + 77934 \beta_{2} - 21812 \beta_{3} ) q^{65} + ( -6669647 \beta_{1} - 1448953 \beta_{2} ) q^{67} + ( 9155872 \beta_{1} - 895168 \beta_{2} ) q^{68} + ( -127085490 + 4316116 \beta_{1} - 72352 \beta_{2} - 144039 \beta_{3} ) q^{70} + ( 11902968 + 91200 \beta_{3} ) q^{71} + ( 1847940 \beta_{1} - 90564 \beta_{2} ) q^{73} + ( -294215436 - 335370 \beta_{3} ) q^{74} + ( -292122360 - 3572 \beta_{3} ) q^{76} + ( 1533756 \beta_{1} - 2162748 \beta_{2} ) q^{77} + ( -182010880 - 267976 \beta_{3} ) q^{79} + ( -241460760 + 10722384 \beta_{1} + 1800352 \beta_{2} - 456836 \beta_{3} ) q^{80} + ( -17058922 \beta_{1} + 199120 \beta_{2} ) q^{82} + ( -12737657 \beta_{1} + 1180353 \beta_{2} ) q^{83} + ( 318854960 + 8731336 \beta_{1} - 988192 \beta_{2} - 75544 \beta_{3} ) q^{85} + ( 593976138 + 719131 \beta_{3} ) q^{86} + ( 7146128 \beta_{1} - 247136 \beta_{2} ) q^{88} + ( 395675190 - 185592 \beta_{3} ) q^{89} + ( 118330632 + 357504 \beta_{3} ) q^{91} + ( -21128228 \beta_{1} + 2797928 \beta_{2} ) q^{92} + ( 831775426 + 1012111 \beta_{3} ) q^{94} + ( -301197900 + 5204860 \beta_{1} - 4032420 \beta_{2} + 23560 \beta_{3} ) q^{95} + ( -14786464 \beta_{1} + 1954216 \beta_{2} ) q^{97} + ( 12121963 \beta_{1} - 223440 \beta_{2} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 1368q^{4} - 1140q^{5} + O(q^{10}) \) \( 4q - 1368q^{4} - 1140q^{5} - 69160q^{10} - 109968q^{11} + 424536q^{14} + 1631264q^{16} - 636880q^{19} + 3302280q^{20} - 1337900q^{25} - 6618768q^{26} + 3531720q^{29} - 10587712q^{31} + 26434624q^{34} - 13629840q^{35} + 43578400q^{40} + 16788552q^{41} - 20638944q^{44} - 61250072q^{46} - 46921028q^{49} + 150092400q^{50} - 26907120q^{55} - 315178080q^{56} + 460829040q^{59} + 360490568q^{61} + 134995072q^{64} - 183895680q^{65} - 508341960q^{70} + 47611872q^{71} - 1176861744q^{74} - 1168489440q^{76} - 728043520q^{79} - 965843040q^{80} + 1275419840q^{85} + 2375904552q^{86} + 1582700760q^{89} + 473322528q^{91} + 3327101704q^{94} - 1204791600q^{95} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 45 x^{2} + 304\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} + 37 \nu \)\()/2\)
\(\beta_{2}\)\(=\)\( -\nu^{3} - 7 \nu \)
\(\beta_{3}\)\(=\)\( 60 \nu^{2} + 1350 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} + 2 \beta_{1}\)\()/30\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} - 1350\)\()/60\)
\(\nu^{3}\)\(=\)\((\)\(-37 \beta_{2} - 14 \beta_{1}\)\()/30\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/45\mathbb{Z}\right)^\times\).

\(n\) \(11\) \(37\)
\(\chi(n)\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
2.87724i
6.05982i
6.05982i
2.87724i
41.3193i 0 −1195.29 −1138.29 810.818i 0 5315.22i 28233.0i 0 −33502.5 + 47033.3i
19.2 0.843944i 0 511.288 568.288 1276.78i 0 8712.99i 863.597i 0 −1077.53 479.603i
19.3 0.843944i 0 511.288 568.288 + 1276.78i 0 8712.99i 863.597i 0 −1077.53 + 479.603i
19.4 41.3193i 0 −1195.29 −1138.29 + 810.818i 0 5315.22i 28233.0i 0 −33502.5 47033.3i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 45.10.b.b 4
3.b odd 2 1 5.10.b.a 4
5.b even 2 1 inner 45.10.b.b 4
5.c odd 4 2 225.10.a.s 4
12.b even 2 1 80.10.c.c 4
15.d odd 2 1 5.10.b.a 4
15.e even 4 2 25.10.a.e 4
60.h even 2 1 80.10.c.c 4
60.l odd 4 2 400.10.a.ba 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.10.b.a 4 3.b odd 2 1
5.10.b.a 4 15.d odd 2 1
25.10.a.e 4 15.e even 4 2
45.10.b.b 4 1.a even 1 1 trivial
45.10.b.b 4 5.b even 2 1 inner
80.10.c.c 4 12.b even 2 1
80.10.c.c 4 60.h even 2 1
225.10.a.s 4 5.c odd 4 2
400.10.a.ba 4 60.l odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 1708 T_{2}^{2} + 1216 \) acting on \(S_{10}^{\mathrm{new}}(45, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - 340 T^{2} - 174912 T^{4} - 89128960 T^{6} + 68719476736 T^{8} \)
$3$ 1
$5$ \( 1 + 1140 T + 1318750 T^{2} + 2226562500 T^{3} + 3814697265625 T^{4} \)
$7$ \( 1 - 57246700 T^{2} + 3508143545353398 T^{4} - \)\(93\!\cdots\!00\)\( T^{6} + \)\(26\!\cdots\!01\)\( T^{8} \)
$11$ \( ( 1 + 54984 T + 5180465446 T^{2} + 129649395841944 T^{3} + 5559917313492231481 T^{4} )^{2} \)
$13$ \( 1 - 35613791860 T^{2} + \)\(53\!\cdots\!58\)\( T^{4} - \)\(40\!\cdots\!40\)\( T^{6} + \)\(12\!\cdots\!41\)\( T^{8} \)
$17$ \( 1 - 285780369220 T^{2} + \)\(48\!\cdots\!18\)\( T^{4} - \)\(40\!\cdots\!80\)\( T^{6} + \)\(19\!\cdots\!81\)\( T^{8} \)
$19$ \( ( 1 + 318440 T + 505756418358 T^{2} + 102756670480744760 T^{3} + \)\(10\!\cdots\!41\)\( T^{4} )^{2} \)
$23$ \( 1 - 5779790962540 T^{2} + \)\(14\!\cdots\!38\)\( T^{4} - \)\(18\!\cdots\!60\)\( T^{6} + \)\(10\!\cdots\!61\)\( T^{8} \)
$29$ \( ( 1 - 1765860 T + 26950935551038 T^{2} - 25617588792948032340 T^{3} + \)\(21\!\cdots\!61\)\( T^{4} )^{2} \)
$31$ \( ( 1 + 5293856 T + 59464921598526 T^{2} + \)\(13\!\cdots\!76\)\( T^{3} + \)\(69\!\cdots\!41\)\( T^{4} )^{2} \)
$37$ \( 1 - 231603274936660 T^{2} + \)\(44\!\cdots\!58\)\( T^{4} - \)\(39\!\cdots\!40\)\( T^{6} + \)\(28\!\cdots\!41\)\( T^{8} \)
$41$ \( ( 1 - 8394276 T + 221313076168966 T^{2} - \)\(27\!\cdots\!36\)\( T^{3} + \)\(10\!\cdots\!21\)\( T^{4} )^{2} \)
$43$ \( 1 - 614109141147100 T^{2} + \)\(57\!\cdots\!98\)\( T^{4} - \)\(15\!\cdots\!00\)\( T^{6} + \)\(63\!\cdots\!01\)\( T^{8} \)
$47$ \( 1 - 1368976020813580 T^{2} + \)\(29\!\cdots\!78\)\( T^{4} - \)\(17\!\cdots\!20\)\( T^{6} + \)\(15\!\cdots\!21\)\( T^{8} \)
$53$ \( 1 - 7684297973864980 T^{2} + \)\(36\!\cdots\!78\)\( T^{4} - \)\(83\!\cdots\!20\)\( T^{6} + \)\(11\!\cdots\!21\)\( T^{8} \)
$59$ \( ( 1 - 230414520 T + 28555631923987078 T^{2} - \)\(19\!\cdots\!80\)\( T^{3} + \)\(75\!\cdots\!21\)\( T^{4} )^{2} \)
$61$ \( ( 1 - 180245284 T + 30154717014478446 T^{2} - \)\(21\!\cdots\!44\)\( T^{3} + \)\(13\!\cdots\!81\)\( T^{4} )^{2} \)
$67$ \( 1 + 41160407446058180 T^{2} + \)\(18\!\cdots\!18\)\( T^{4} + \)\(30\!\cdots\!20\)\( T^{6} + \)\(54\!\cdots\!81\)\( T^{8} \)
$71$ \( ( 1 - 23805936 T + 85782754020107086 T^{2} - \)\(10\!\cdots\!16\)\( T^{3} + \)\(21\!\cdots\!61\)\( T^{4} )^{2} \)
$73$ \( 1 - 229489314868712740 T^{2} + \)\(20\!\cdots\!38\)\( T^{4} - \)\(79\!\cdots\!60\)\( T^{6} + \)\(12\!\cdots\!61\)\( T^{8} \)
$79$ \( ( 1 + 364021760 T + 220545463862625438 T^{2} + \)\(43\!\cdots\!40\)\( T^{3} + \)\(14\!\cdots\!61\)\( T^{4} )^{2} \)
$83$ \( 1 - 434569632367965820 T^{2} + \)\(10\!\cdots\!18\)\( T^{4} - \)\(15\!\cdots\!80\)\( T^{6} + \)\(12\!\cdots\!81\)\( T^{8} \)
$89$ \( ( 1 - 791350380 T + 832192702699668118 T^{2} - \)\(27\!\cdots\!20\)\( T^{3} + \)\(12\!\cdots\!81\)\( T^{4} )^{2} \)
$97$ \( 1 - 2561123777205326980 T^{2} + \)\(27\!\cdots\!78\)\( T^{4} - \)\(14\!\cdots\!20\)\( T^{6} + \)\(33\!\cdots\!21\)\( T^{8} \)
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