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Properties

Label 32.11.h.a
Level $32$
Weight $11$
Character orbit 32.h
Analytic conductor $20.331$
Analytic rank $0$
Dimension $156$
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] = [32,11,Mod(3,32)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(32, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([4, 3]))
 
N = Newforms(chi, 11, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("32.3");
 
S:= CuspForms(chi, 11);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 32 = 2^{5} \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 32.h (of order \(8\), degree \(4\), 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: \(20.3314320856\)
Analytic rank: \(0\)
Dimension: \(156\)
Relative dimension: \(39\) over \(\Q(\zeta_{8})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 156 q - 4 q^{2} - 4 q^{3} - 4 q^{4} - 4 q^{5} - 4 q^{6} - 4 q^{7} - 4 q^{8} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 156 q - 4 q^{2} - 4 q^{3} - 4 q^{4} - 4 q^{5} - 4 q^{6} - 4 q^{7} - 4 q^{8} - 4 q^{9} - 125004 q^{10} - 4 q^{11} + 1240268 q^{12} - 4 q^{13} - 2259284 q^{14} - 8 q^{15} + 5020176 q^{16} - 4330264 q^{18} - 4 q^{19} - 4750004 q^{20} - 4 q^{21} + 26583712 q^{22} + 8279740 q^{23} + 19992016 q^{24} - 4 q^{25} - 48999504 q^{26} + 51979292 q^{27} + 17926136 q^{28} - 4 q^{29} + 141142340 q^{30} - 92175384 q^{32} - 8 q^{33} - 82891248 q^{34} - 68411428 q^{35} + 655427568 q^{36} - 4 q^{37} - 422517036 q^{38} + 279841724 q^{39} - 206977688 q^{40} - 4 q^{41} + 1135435016 q^{42} - 376781860 q^{43} - 1303262780 q^{44} - 236200 q^{45} - 655508132 q^{46} - 8 q^{47} + 1267668528 q^{48} + 924624788 q^{50} - 746182048 q^{51} + 634955812 q^{52} - 1214144804 q^{53} - 2262787232 q^{54} - 1427102468 q^{55} - 1685472984 q^{56} - 4 q^{57} + 3232345888 q^{58} + 1499085436 q^{59} + 1937656760 q^{60} - 3346560324 q^{61} + 1876477368 q^{62} - 1922944744 q^{64} - 8 q^{65} - 8453274100 q^{66} - 4242302244 q^{67} - 5908113008 q^{68} - 3396479044 q^{69} + 14667010136 q^{70} + 7572888316 q^{71} + 19641672980 q^{72} - 4 q^{73} - 7984418644 q^{74} - 11956280168 q^{75} - 21159784964 q^{76} + 1543017052 q^{77} + 1963003636 q^{78} + 24897943032 q^{79} - 2589388488 q^{80} + 39782080636 q^{82} - 18091079524 q^{83} + 18202566176 q^{84} + 39062496 q^{85} - 39965941952 q^{86} + 31832612668 q^{87} - 42312991328 q^{88} - 4 q^{89} + 16471040552 q^{90} - 27520202692 q^{91} + 64169680168 q^{92} + 236192 q^{93} + 72301821928 q^{94} - 108446589056 q^{96} - 8 q^{97} - 77681549032 q^{98} + 56144149080 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1 −31.9553 + 1.69169i 169.416 − 409.006i 1018.28 − 108.117i 459.056 + 1108.26i −4721.82 + 13356.5i −5316.74 − 5316.74i −32356.4 + 5177.51i −96830.5 − 96830.5i −16544.1 − 34638.1i
3.2 −31.9340 + 2.05438i 36.8133 − 88.8751i 1015.56 − 131.209i −2122.77 − 5124.82i −993.011 + 2913.76i −17012.5 − 17012.5i −32161.3 + 6276.39i 35210.4 + 35210.4i 78316.8 + 159295.i
3.3 −31.8859 − 2.69969i −8.87547 + 21.4273i 1009.42 + 172.164i 1570.98 + 3792.68i 340.849 − 659.267i 12151.8 + 12151.8i −31721.6 − 8214.73i 41373.6 + 41373.6i −39853.1 − 125174.i
3.4 −30.9289 + 8.20986i −126.155 + 304.566i 889.197 − 507.844i −1691.72 − 4084.18i 1401.41 − 10455.6i 16469.5 + 16469.5i −23332.6 + 23007.2i −35091.2 − 35091.2i 85853.7 + 112430.i
3.5 −29.7444 + 11.8013i −64.2857 + 155.199i 745.458 − 702.045i 898.143 + 2168.31i 80.5838 − 5374.97i −13836.6 − 13836.6i −13888.2 + 29679.3i 21799.8 + 21799.8i −52303.6 − 53895.8i
3.6 −28.9574 − 13.6187i −133.301 + 321.817i 653.063 + 788.723i −55.0849 − 132.987i 8242.78 − 7503.61i −6472.31 − 6472.31i −8169.67 − 31733.2i −44043.3 − 44043.3i −215.986 + 4601.13i
3.7 −28.3765 − 14.7909i 77.6669 − 187.505i 586.456 + 839.431i −1231.05 − 2972.02i −4977.29 + 4171.96i 15547.6 + 15547.6i −4225.63 − 32494.4i 12628.1 + 12628.1i −9025.99 + 102544.i
3.8 −25.5952 − 19.2064i 56.0693 − 135.363i 286.229 + 983.183i 391.704 + 945.657i −4034.94 + 2387.76i −16594.6 − 16594.6i 11557.3 − 30662.2i 26574.5 + 26574.5i 8136.92 − 31727.5i
3.9 −24.8641 + 20.1438i 35.5173 − 85.7463i 212.451 − 1001.72i 189.838 + 458.309i 844.154 + 2847.46i 924.532 + 924.532i 14896.1 + 29186.4i 35663.0 + 35663.0i −13952.3 − 7571.39i
3.10 −24.4670 + 20.6244i 106.830 − 257.909i 173.265 − 1009.23i −554.472 − 1338.61i 2705.44 + 8513.56i 12411.7 + 12411.7i 16575.6 + 28266.4i −13350.7 − 13350.7i 41174.4 + 21316.1i
3.11 −21.4613 + 23.7363i −173.119 + 417.946i −102.823 − 1018.82i 1312.23 + 3168.01i −6205.12 − 13078.9i 5581.44 + 5581.44i 26389.8 + 19424.7i −102955. − 102955.i −103359. − 36842.2i
3.12 −18.3968 − 26.1832i 124.918 − 301.579i −347.117 + 963.372i 1520.20 + 3670.09i −10194.4 + 2277.33i 3963.58 + 3963.58i 31610.0 − 8634.34i −33591.6 − 33591.6i 68127.8 − 107322.i
3.13 −15.8196 − 27.8162i −54.5691 + 131.741i −523.484 + 880.080i −971.141 − 2344.54i 4527.80 − 566.184i 5775.92 + 5775.92i 32761.8 + 638.862i 27375.9 + 27375.9i −49853.2 + 64103.0i
3.14 −12.9691 + 29.2541i 113.202 − 273.294i −687.604 − 758.799i 2344.76 + 5660.76i 6526.83 + 6856.00i −16654.5 − 16654.5i 31115.6 − 10274.3i −20120.9 − 20120.9i −196010. − 4821.08i
3.15 −12.9008 + 29.2843i −73.4708 + 177.374i −691.137 − 755.583i −1142.25 − 2757.64i −4246.44 − 4439.81i −8149.47 − 8149.47i 31042.9 − 10491.8i 15690.3 + 15690.3i 95491.3 + 2125.82i
3.16 −11.5027 − 29.8612i −117.381 + 283.384i −759.377 + 686.966i 2156.79 + 5206.96i 9812.36 + 245.475i 5098.42 + 5098.42i 29248.5 + 14773.9i −24774.0 − 24774.0i 130677. − 124298.i
3.17 −6.03667 − 31.4254i 161.981 − 391.057i −951.117 + 379.410i −1993.68 − 4813.17i −13267.0 − 2729.65i −140.048 − 140.048i 17664.7 + 27598.9i −84934.0 − 84934.0i −139221. + 91707.7i
3.18 −3.83980 + 31.7688i 137.729 − 332.507i −994.512 − 243.972i −1554.76 − 3753.53i 10034.5 + 5652.23i −1477.73 − 1477.73i 11569.4 − 30657.6i −49837.4 − 49837.4i 125215. − 34980.1i
3.19 −3.02541 + 31.8567i −25.0291 + 60.4255i −1005.69 − 192.759i 1027.09 + 2479.61i −1849.23 − 980.154i 19688.7 + 19688.7i 9183.30 − 31454.9i 38729.2 + 38729.2i −82099.6 + 25217.8i
3.20 −1.49202 − 31.9652i 25.1327 − 60.6756i −1019.55 + 95.3855i −118.675 − 286.507i −1977.01 − 712.841i −20103.5 − 20103.5i 4570.21 + 32447.7i 38704.1 + 38704.1i −8981.17 + 4220.94i
See next 80 embeddings (of 156 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.39
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
32.h odd 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 32.11.h.a ✓ 156
32.h odd 8 1 inner 32.11.h.a ✓ 156
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
32.11.h.a ✓ 156 1.a even 1 1 trivial
32.11.h.a ✓ 156 32.h odd 8 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{11}^{\mathrm{new}}(32, [\chi])\).