Properties

Label 69.7.b.a
Level $69$
Weight $7$
Character orbit 69.b
Analytic conductor $15.874$
Analytic rank $0$
Dimension $44$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 69 = 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 69.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(15.8737317698\)
Analytic rank: \(0\)
Dimension: \(44\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 44q + 20q^{3} - 1408q^{4} + 95q^{6} + 568q^{7} - 548q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 44q + 20q^{3} - 1408q^{4} + 95q^{6} + 568q^{7} - 548q^{9} + 1752q^{10} + 4075q^{12} + 808q^{13} + 7696q^{15} + 36776q^{16} + 12149q^{18} + 28936q^{19} - 6416q^{21} - 7764q^{22} - 11792q^{24} - 129172q^{25} - 27172q^{27} - 25988q^{28} - 54658q^{30} - 72248q^{31} + 25968q^{33} - 32100q^{34} - 217125q^{36} + 260968q^{37} + 133440q^{39} - 227880q^{40} + 63332q^{42} - 187304q^{43} + 455472q^{45} - 164849q^{48} + 959652q^{49} - 218832q^{51} - 410102q^{52} + 882504q^{54} + 517392q^{55} - 572600q^{57} - 197334q^{58} - 854196q^{60} + 914248q^{61} + 885136q^{63} - 312634q^{64} - 816874q^{66} - 310856q^{67} - 395040q^{70} + 205764q^{72} - 227912q^{73} + 1167580q^{75} - 1438412q^{76} - 6065q^{78} + 841384q^{79} + 1019636q^{81} - 291126q^{82} - 2787738q^{84} - 2823120q^{85} - 2899120q^{87} - 2657340q^{88} + 1478966q^{90} - 2848288q^{91} - 1992952q^{93} + 6985482q^{94} + 1309665q^{96} + 1079608q^{97} + 3251880q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
47.1 15.2045i −26.4076 + 5.62468i −167.177 4.50276i 85.5205 + 401.515i 443.657 1568.76i 665.726 297.069i −68.4623
47.2 14.6275i 2.75073 + 26.8595i −149.963 100.828i 392.887 40.2362i −263.111 1257.42i −713.867 + 147.767i −1474.86
47.3 14.4029i 25.1960 + 9.70372i −143.443 103.914i 139.762 362.895i 470.752 1144.21i 540.676 + 488.990i 1496.66
47.4 14.0722i 4.93651 26.5449i −134.028 199.248i −373.546 69.4677i 69.1270 985.449i −680.262 262.078i 2803.87
47.5 14.0000i −13.7042 23.2636i −132.001 135.214i −325.691 + 191.860i −169.403 952.014i −353.387 + 637.619i −1893.00
47.6 13.1440i 26.0223 7.20015i −108.766 47.7180i −94.6390 342.037i −438.649 588.401i 625.316 374.728i −627.206
47.7 12.3777i −26.6682 + 4.21999i −89.2071 195.553i 52.2337 + 330.090i −657.659 312.006i 693.383 225.079i 2420.50
47.8 11.0694i −6.47589 + 26.2119i −58.5309 94.9815i 290.149 + 71.6840i 96.5626 60.5391i −645.126 339.491i 1051.39
47.9 10.2941i 12.6170 23.8707i −41.9690 93.4381i −245.728 129.881i 330.565 226.790i −410.623 602.354i −961.863
47.10 9.87745i 22.8275 + 14.4190i −33.5640 220.404i 142.423 225.477i 473.108 300.630i 313.187 + 658.297i −2177.03
47.11 9.52809i −24.3830 + 11.5961i −26.7846 179.250i 110.488 + 232.324i −22.6651 354.592i 460.063 565.494i −1707.91
47.12 8.50016i −23.8542 12.6482i −8.25265 65.5840i −107.512 + 202.765i 552.822 473.861i 409.047 + 603.425i 557.474
47.13 8.07729i 18.6168 + 19.5554i −1.24263 135.340i 157.955 150.374i −41.9690 506.910i −35.8265 + 728.119i 1093.18
47.14 7.89562i −18.6193 19.5530i 1.65917 10.5412i −154.383 + 147.011i −269.224 518.420i −35.6409 + 728.128i 83.2291
47.15 6.32520i 24.9999 10.1983i 23.9919 138.173i −64.5060 158.129i −112.869 556.566i 520.991 509.911i 873.969
47.16 5.20732i 17.8105 + 20.2925i 36.8838 125.797i 105.670 92.7449i −679.000 525.334i −94.5737 + 722.839i −655.063
47.17 4.53332i −0.406897 26.9969i 43.4490 169.748i −122.386 + 1.84460i −134.238 487.101i −728.669 + 21.9700i 769.522
47.18 4.32001i −14.6931 + 22.6520i 45.3375 48.7369i 97.8569 + 63.4743i −61.9430 472.339i −297.226 665.656i −210.544
47.19 2.95357i 2.97092 26.8361i 55.2764 175.802i −79.2621 8.77482i −435.351 352.291i −711.347 159.456i −519.242
47.20 1.98217i 25.5387 8.76199i 60.0710 91.3581i −17.3677 50.6220i 169.692 245.929i 575.455 447.541i −181.087
See all 44 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 47.44
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 69.7.b.a 44
3.b odd 2 1 inner 69.7.b.a 44
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
69.7.b.a 44 1.a even 1 1 trivial
69.7.b.a 44 3.b odd 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{7}^{\mathrm{new}}(69, [\chi])\).