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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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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [69,7,Mod(47,69)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(69, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("69.47");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 69 = 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 69.b (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
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.

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

Display 100 coefficients 10 coefficients

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
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 \(S_{7}^{\mathrm{new}}(69, [\chi])\).