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Properties

Label 91.10.r.a
Level $91$
Weight $10$
Character orbit 91.r
Analytic conductor $46.868$
Analytic rank $0$
Dimension $164$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [91,10,Mod(25,91)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(91, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4, 3]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("91.25");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 91 = 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 91.r (of order \(6\), degree \(2\), 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: \(46.8682610909\)
Analytic rank: \(0\)
Dimension: \(164\)
Relative dimension: \(82\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 164 q - 326 q^{3} + 20478 q^{4} - 513456 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 164 q - 326 q^{3} + 20478 q^{4} - 513456 q^{9} + 37950 q^{10} + 328586 q^{12} + 205424 q^{13} - 347348 q^{14} - 5516922 q^{16} + 638792 q^{17} - 2021136 q^{22} - 3143742 q^{23} + 29660776 q^{25} - 4349148 q^{26} + 15971728 q^{27} + 11405516 q^{29} + 3318068 q^{30} - 13283874 q^{35} - 268857760 q^{36} + 41257144 q^{38} - 43335142 q^{39} - 64638572 q^{40} + 4428624 q^{42} + 208624280 q^{43} + 365752796 q^{48} - 8021754 q^{49} - 128665496 q^{51} + 116775564 q^{52} + 274651312 q^{53} + 333747724 q^{55} + 13700502 q^{56} + 205860820 q^{61} - 1213186480 q^{62} - 4080539840 q^{64} + 331150788 q^{65} - 1310285310 q^{66} - 598025264 q^{68} - 1943339932 q^{69} + 920056770 q^{74} + 1640230784 q^{75} + 1201037654 q^{77} + 2439213164 q^{78} - 335791710 q^{79} - 4192022058 q^{81} - 617219896 q^{82} - 3240850496 q^{87} + 1326885718 q^{88} + 14332940596 q^{90} - 3162562236 q^{91} - 7831634604 q^{92} + 1733533760 q^{94} + 319654502 q^{95}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
25.1 −38.0821 + 21.9867i 50.6497 − 87.7279i 710.832 − 1231.20i 1282.12 − 740.229i 4454.48i −3356.10 + 5393.54i 40001.0i 4710.71 + 8159.19i −32550.4 + 56379.0i
25.2 −38.0646 + 21.9766i −125.081 + 216.647i 709.942 − 1229.65i 1616.40 − 933.229i − 10995.4i 5349.40 − 3426.01i 39904.4i −21449.1 − 37150.9i −41018.4 + 71045.9i
25.3 −37.7409 + 21.7897i −89.2971 + 154.667i 693.584 − 1201.32i −2225.16 + 1284.69i − 7783.04i −1704.46 + 6119.51i 38139.4i −6106.45 − 10576.7i 55986.3 − 96971.1i
25.4 −37.1560 + 21.4520i 48.7399 − 84.4199i 664.380 − 1150.74i −1709.75 + 987.124i 4182.28i 6019.28 − 2030.23i 35042.4i 5090.35 + 8816.74i 42351.7 − 73355.2i
25.5 −36.1525 + 20.8727i 4.69329 − 8.12902i 615.337 − 1065.80i −282.457 + 163.077i 391.846i −4397.89 − 4583.90i 30001.3i 9797.45 + 16969.7i 6807.70 − 11791.3i
25.6 −34.3283 + 19.8195i 124.399 − 215.465i 529.623 − 917.335i 1553.57 − 896.956i 9862.07i 3723.94 − 5146.44i 21692.3i −21108.5 − 36561.1i −35554.4 + 61582.0i
25.7 −33.1273 + 19.1261i −56.2370 + 97.4054i 475.614 − 823.787i −61.8801 + 35.7265i − 4302.38i −884.330 − 6290.59i 16801.4i 3516.29 + 6090.40i 1366.62 − 2367.05i
25.8 −33.0528 + 19.0830i −4.22719 + 7.32170i 472.325 − 818.090i 971.918 − 561.137i − 322.670i 5457.44 + 3251.15i 16512.5i 9805.76 + 16984.1i −21416.4 + 37094.3i
25.9 −32.8449 + 18.9630i −87.6907 + 151.885i 463.190 − 802.269i 254.937 − 147.188i − 6651.51i −6062.13 + 1898.46i 15715.8i −5537.81 − 9591.77i −5582.24 + 9668.73i
25.10 −32.7319 + 18.8978i 101.067 − 175.053i 458.252 − 793.716i −1472.28 + 850.023i 7639.76i −6263.52 − 1059.22i 15288.4i −10587.5 − 18338.2i 32127.1 − 55645.7i
25.11 −31.5036 + 18.1886i 118.618 − 205.453i 405.649 − 702.605i −739.872 + 427.166i 8629.99i 3468.77 + 5321.77i 10887.6i −18299.0 − 31694.9i 15539.1 − 26914.5i
25.12 −29.1378 + 16.8227i −79.9154 + 138.417i 310.007 − 536.948i −44.8261 + 25.8804i − 5377.57i 2214.78 + 5953.85i 3634.20i −2931.43 − 5077.39i 870.757 − 1508.19i
25.13 −28.7986 + 16.6269i −13.5930 + 23.5437i 296.905 − 514.254i −1526.26 + 881.189i − 904.033i 6223.14 − 1275.18i 2720.48i 9471.96 + 16405.9i 29302.8 − 50753.9i
25.14 −28.0814 + 16.2128i 63.1589 − 109.394i 269.711 − 467.153i 925.073 − 534.091i 4095.93i −98.1045 − 6351.69i 889.152i 1863.41 + 3227.52i −17318.2 + 29996.1i
25.15 −27.1983 + 15.7029i −72.6491 + 125.832i 237.164 − 410.780i 2271.10 − 1311.22i − 4563.21i −6323.56 + 605.123i − 1183.14i −714.271 − 1237.15i −41180.0 + 71325.8i
25.16 −26.8546 + 15.5045i −119.979 + 207.810i 224.779 − 389.329i −1652.19 + 953.895i − 7440.88i 3796.17 − 5093.40i − 1936.24i −18948.6 − 32819.9i 29579.3 − 51232.9i
25.17 −25.4204 + 14.6765i 35.9915 − 62.3390i 174.796 − 302.756i −716.257 + 413.531i 2112.91i −2545.78 + 5820.02i − 4767.12i 7250.73 + 12558.6i 12138.3 − 21024.2i
25.18 −24.7556 + 14.2926i 72.8951 − 126.258i 152.560 − 264.241i 1570.80 − 906.904i 4167.46i −6170.89 + 1507.87i − 5913.75i −785.900 − 1361.22i −25924.1 + 44901.9i
25.19 −22.7814 + 13.1529i 5.21434 − 9.03151i 89.9960 − 155.878i −1248.80 + 720.993i 274.334i −2793.59 + 5705.21i − 8733.71i 9787.12 + 16951.8i 18966.3 − 32850.5i
25.20 −22.1758 + 12.8032i −20.1770 + 34.9477i 71.8445 − 124.438i 2183.34 − 1260.55i − 1033.32i 5979.71 − 2143.99i − 9431.13i 9027.27 + 15635.7i −32278.2 + 55907.4i
See next 80 embeddings (of 164 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 25.82
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
13.b even 2 1 inner
91.r even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 91.10.r.a ✓ 164
7.c even 3 1 inner 91.10.r.a ✓ 164
13.b even 2 1 inner 91.10.r.a ✓ 164
91.r even 6 1 inner 91.10.r.a ✓ 164
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.10.r.a ✓ 164 1.a even 1 1 trivial
91.10.r.a ✓ 164 7.c even 3 1 inner
91.10.r.a ✓ 164 13.b even 2 1 inner
91.10.r.a ✓ 164 91.r even 6 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{10}^{\mathrm{new}}(91, [\chi])\).