Properties

Label 1045.6.a.b
Level $1045$
Weight $6$
Character orbit 1045.a
Self dual yes
Analytic conductor $167.601$
Analytic rank $1$
Dimension $36$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1045 = 5 \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 1045.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(167.601091705\)
Analytic rank: \(1\)
Dimension: \(36\)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 36 q - 8 q^{2} - 63 q^{3} + 520 q^{4} + 900 q^{5} + 5 q^{6} - 509 q^{7} - 690 q^{8} + 1935 q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 36 q - 8 q^{2} - 63 q^{3} + 520 q^{4} + 900 q^{5} + 5 q^{6} - 509 q^{7} - 690 q^{8} + 1935 q^{9} - 200 q^{10} - 4356 q^{11} - 2008 q^{12} - 43 q^{13} - 1937 q^{14} - 1575 q^{15} + 3612 q^{16} - 2431 q^{17} - 6225 q^{18} + 12996 q^{19} + 13000 q^{20} + 2863 q^{21} + 968 q^{22} - 11444 q^{23} - 6210 q^{24} + 22500 q^{25} - 6339 q^{26} - 12960 q^{27} - 1083 q^{28} - 873 q^{29} + 125 q^{30} - 1405 q^{31} - 14283 q^{32} + 7623 q^{33} + 19937 q^{34} - 12725 q^{35} - 1169 q^{36} - 22729 q^{37} - 2888 q^{38} + 3710 q^{39} - 17250 q^{40} - 17043 q^{41} - 39996 q^{42} - 42231 q^{43} - 62920 q^{44} + 48375 q^{45} + 50947 q^{46} - 72440 q^{47} + 42475 q^{48} + 54119 q^{49} - 5000 q^{50} - 114970 q^{51} + 16786 q^{52} - 67603 q^{53} - 26080 q^{54} - 108900 q^{55} - 216071 q^{56} - 22743 q^{57} - 115746 q^{58} - 247439 q^{59} - 50200 q^{60} - 66627 q^{61} - 262438 q^{62} - 226118 q^{63} + 1078 q^{64} - 1075 q^{65} - 605 q^{66} - 189550 q^{67} - 140936 q^{68} - 65684 q^{69} - 48425 q^{70} - 320146 q^{71} - 509978 q^{72} - 55266 q^{73} - 63309 q^{74} - 39375 q^{75} + 187720 q^{76} + 61589 q^{77} - 284264 q^{78} - 1033 q^{79} + 90300 q^{80} - 58588 q^{81} - 328242 q^{82} - 451983 q^{83} + 43932 q^{84} - 60775 q^{85} - 44142 q^{86} - 457510 q^{87} + 83490 q^{88} + 13940 q^{89} - 155625 q^{90} - 211732 q^{91} - 735304 q^{92} + 4486 q^{93} + 152164 q^{94} + 324900 q^{95} + 195996 q^{96} - 234346 q^{97} - 58328 q^{98} - 234135 q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −10.8010 −21.4786 84.6626 25.0000 231.991 40.5180 −568.812 218.330 −270.026
1.2 −10.6840 22.9747 82.1485 25.0000 −245.462 48.0772 −535.788 284.836 −267.101
1.3 −9.59224 −21.5591 60.0111 25.0000 206.800 −136.266 −268.689 221.796 −239.806
1.4 −9.55197 −7.71824 59.2400 25.0000 73.7243 175.466 −260.196 −183.429 −238.799
1.5 −9.26988 10.9289 53.9306 25.0000 −101.309 145.096 −203.294 −123.560 −231.747
1.6 −9.10660 19.2419 50.9302 25.0000 −175.228 −172.410 −172.390 127.251 −227.665
1.7 −7.64458 −6.22870 26.4397 25.0000 47.6158 101.320 42.5064 −204.203 −191.115
1.8 −7.00111 −3.95422 17.0156 25.0000 27.6840 −243.106 104.908 −227.364 −175.028
1.9 −6.95850 −10.9341 16.4207 25.0000 76.0849 −81.2421 108.408 −123.446 −173.963
1.10 −6.79783 −25.2972 14.2104 25.0000 171.966 60.0170 120.930 396.947 −169.946
1.11 −5.96881 24.3951 3.62665 25.0000 −145.610 53.9217 169.355 352.122 −149.220
1.12 −4.65971 14.3366 −10.2871 25.0000 −66.8044 127.161 197.046 −37.4616 −116.493
1.13 −3.77729 −30.8759 −17.7321 25.0000 116.627 −222.797 187.852 710.323 −94.4321
1.14 −3.50600 −5.81498 −19.7080 25.0000 20.3873 29.7384 181.288 −209.186 −87.6500
1.15 −3.31390 −19.0151 −21.0181 25.0000 63.0141 82.9347 175.697 118.573 −82.8475
1.16 −2.55950 0.0390510 −25.4489 25.0000 −0.0999513 −217.307 147.041 −242.998 −63.9876
1.17 −1.33765 19.4352 −30.2107 25.0000 −25.9975 −167.518 83.2162 134.726 −33.4413
1.18 −0.114186 8.15329 −31.9870 25.0000 −0.930987 179.478 7.30638 −176.524 −2.85464
1.19 0.0328525 −8.60052 −31.9989 25.0000 −0.282548 −22.0929 −2.10252 −169.031 0.821311
1.20 1.07599 26.0994 −30.8422 25.0000 28.0828 −166.075 −67.6179 438.177 26.8998
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.36
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(-1\)
\(11\) \(1\)
\(19\) \(-1\)

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 1045.6.a.b 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1045.6.a.b 36 1.a even 1 1 trivial