Properties

Label 1045.4.a.h
Level $1045$
Weight $4$
Character orbit 1045.a
Self dual yes
Analytic conductor $61.657$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1045 = 5 \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1045.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(61.6569959560\)
Analytic rank: \(0\)
Dimension: \(24\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 4 q^{2} + 21 q^{3} + 98 q^{4} + 120 q^{5} + 15 q^{6} + 71 q^{7} + 84 q^{8} + 339 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 4 q^{2} + 21 q^{3} + 98 q^{4} + 120 q^{5} + 15 q^{6} + 71 q^{7} + 84 q^{8} + 339 q^{9} + 20 q^{10} - 264 q^{11} + 164 q^{12} - 15 q^{13} + 77 q^{14} + 105 q^{15} + 230 q^{16} + 187 q^{17} - 109 q^{18} + 456 q^{19} + 490 q^{20} + 295 q^{21} - 44 q^{22} + 451 q^{23} + 416 q^{24} + 600 q^{25} + 375 q^{26} + 1335 q^{27} + 815 q^{28} + 271 q^{29} + 75 q^{30} + 302 q^{31} + 1181 q^{32} - 231 q^{33} + 285 q^{34} + 355 q^{35} + 2445 q^{36} + 974 q^{37} + 76 q^{38} + 601 q^{39} + 420 q^{40} + 316 q^{41} + 2158 q^{42} + 686 q^{43} - 1078 q^{44} + 1695 q^{45} - 217 q^{46} + 1798 q^{47} + 353 q^{48} + 1845 q^{49} + 100 q^{50} + 383 q^{51} - 134 q^{52} + 815 q^{53} - 974 q^{54} - 1320 q^{55} + 2001 q^{56} + 399 q^{57} - 888 q^{58} + 1793 q^{59} + 820 q^{60} + 62 q^{61} + 3994 q^{62} + 366 q^{63} - 588 q^{64} - 75 q^{65} - 165 q^{66} + 2363 q^{67} - 1720 q^{68} - 287 q^{69} + 385 q^{70} + 1266 q^{71} + 3838 q^{72} + 127 q^{73} - 2861 q^{74} + 525 q^{75} + 1862 q^{76} - 781 q^{77} - 3916 q^{78} - 1922 q^{79} + 1150 q^{80} + 3688 q^{81} + 2666 q^{82} + 3666 q^{83} + 438 q^{84} + 935 q^{85} + 78 q^{86} + 2685 q^{87} - 924 q^{88} + 2344 q^{89} - 545 q^{90} + 127 q^{91} + 4800 q^{92} + 1344 q^{93} + 1756 q^{94} + 2280 q^{95} + 2874 q^{96} + 1182 q^{97} - 4328 q^{98} - 3729 q^{99}+O(q^{100}) \) 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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −5.10664 5.96143 18.0778 5.00000 −30.4429 −27.7506 −51.4637 8.53861 −25.5332
1.2 −4.91078 −7.41926 16.1158 5.00000 36.4343 29.5366 −39.8547 28.0453 −24.5539
1.3 −4.21988 −3.62548 9.80739 5.00000 15.2991 8.81364 −7.62695 −13.8559 −21.0994
1.4 −4.17313 9.97600 9.41500 5.00000 −41.6311 32.7762 −5.90498 72.5206 −20.8656
1.5 −3.69819 1.56352 5.67662 5.00000 −5.78219 −5.10978 8.59231 −24.5554 −18.4910
1.6 −3.62064 −7.41696 5.10901 5.00000 26.8541 −14.1058 10.4672 28.0113 −18.1032
1.7 −2.96820 4.96634 0.810213 5.00000 −14.7411 12.3435 21.3407 −2.33550 −14.8410
1.8 −2.74596 10.1412 −0.459676 5.00000 −27.8474 −22.6095 23.2300 75.8438 −13.7298
1.9 −1.46816 0.999287 −5.84451 5.00000 −1.46711 27.4651 20.3259 −26.0014 −7.34079
1.10 −1.39354 −7.96001 −6.05804 5.00000 11.0926 −16.3788 19.5905 36.3617 −6.96772
1.11 −0.539029 −5.36314 −7.70945 5.00000 2.89089 3.34499 8.46785 1.76332 −2.69514
1.12 −0.177794 8.83316 −7.96839 5.00000 −1.57048 5.98363 2.83908 51.0247 −0.888970
1.13 0.817278 −5.32952 −7.33206 5.00000 −4.35569 30.6226 −12.5305 1.40375 4.08639
1.14 0.998459 4.82825 −7.00308 5.00000 4.82081 −5.45452 −14.9800 −3.68800 4.99229
1.15 1.40259 2.52468 −6.03275 5.00000 3.54108 −33.4174 −19.6821 −20.6260 7.01293
1.16 2.06823 −3.71006 −3.72243 5.00000 −7.67325 −13.2300 −24.2447 −13.2355 10.3411
1.17 2.68048 7.90527 −0.815035 5.00000 21.1899 31.9517 −23.6285 35.4933 13.4024
1.18 3.34284 −2.02433 3.17460 5.00000 −6.76702 −13.1927 −16.1306 −22.9021 16.7142
1.19 3.48416 −0.598091 4.13934 5.00000 −2.08384 25.1991 −13.4511 −26.6423 17.4208
1.20 4.08044 −9.09119 8.65002 5.00000 −37.0961 −18.7255 2.65236 55.6498 20.4022
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.24
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 1045.4.a.h 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1045.4.a.h 24 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{24} - 4 T_{2}^{23} - 137 T_{2}^{22} + 520 T_{2}^{21} + 8147 T_{2}^{20} - 29129 T_{2}^{19} - 275730 T_{2}^{18} + 921785 T_{2}^{17} + 5849056 T_{2}^{16} - 18149924 T_{2}^{15} - 80686705 T_{2}^{14} + \cdots + 1815552000 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1045))\). Copy content Toggle raw display