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Properties

Label 33.8.e.a
Level $33$
Weight $8$
Character orbit 33.e
Analytic conductor $10.309$
Analytic rank $0$
Dimension $28$
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] = [33,8,Mod(4,33)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(33, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 2]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("33.4");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 33 = 3 \cdot 11 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 33.e (of order \(5\), 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: \(10.3087058410\)
Analytic rank: \(0\)
Dimension: \(28\)
Relative dimension: \(7\) over \(\Q(\zeta_{5})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 28 q - 22 q^{2} + 189 q^{3} - 692 q^{4} - 777 q^{5} + 594 q^{6} - 83 q^{7} + 3568 q^{8} - 5103 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 28 q - 22 q^{2} + 189 q^{3} - 692 q^{4} - 777 q^{5} + 594 q^{6} - 83 q^{7} + 3568 q^{8} - 5103 q^{9} - 1352 q^{10} - 18851 q^{11} - 60966 q^{12} + 14635 q^{13} + 62912 q^{14} - 22896 q^{15} - 36148 q^{16} - 10338 q^{17} + 9477 q^{18} + 90342 q^{19} - 289719 q^{20} + 28836 q^{21} - 202339 q^{22} + 138508 q^{23} + 187704 q^{24} - 174068 q^{25} + 466130 q^{26} + 137781 q^{27} - 640233 q^{28} - 487290 q^{29} - 372546 q^{30} - 388352 q^{31} + 2726322 q^{32} - 265923 q^{33} - 1543708 q^{34} + 742659 q^{35} - 504468 q^{36} + 670209 q^{37} + 2282072 q^{38} - 395145 q^{39} + 4641741 q^{40} - 2721357 q^{41} + 620676 q^{42} - 199784 q^{43} - 5887437 q^{44} - 103518 q^{45} + 3389447 q^{46} + 2112053 q^{47} + 2292111 q^{48} + 3927748 q^{49} - 3663090 q^{50} + 615681 q^{51} - 4343053 q^{52} - 1743854 q^{53} - 354294 q^{54} - 8387710 q^{55} - 6307818 q^{56} + 1223721 q^{57} + 546605 q^{58} - 218820 q^{59} + 7822413 q^{60} - 743157 q^{61} + 3996183 q^{62} - 60507 q^{63} - 21725666 q^{64} + 4895794 q^{65} - 8684037 q^{66} - 3235686 q^{67} + 961671 q^{68} + 4712499 q^{69} + 31155916 q^{70} + 9848757 q^{71} - 5068008 q^{72} - 6667768 q^{73} - 31115192 q^{74} - 2544939 q^{75} - 6535748 q^{76} + 8320585 q^{77} + 20060730 q^{78} - 10121851 q^{79} + 44592526 q^{80} - 3720087 q^{81} + 24927115 q^{82} + 13288897 q^{83} - 17477019 q^{84} - 28050005 q^{85} + 20937351 q^{86} - 22226400 q^{87} + 39687014 q^{88} - 40241460 q^{89} - 9565938 q^{90} + 8733043 q^{91} + 11607651 q^{92} + 10485504 q^{93} - 7877385 q^{94} + 14371097 q^{95} + 2790126 q^{96} + 30518151 q^{97} - 16954218 q^{98} + 11659626 q^{99}+O(q^{100}) \) 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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1 −18.2616 − 13.2678i −8.34346 + 25.6785i 117.897 + 362.849i −299.083 + 217.296i 493.063 − 358.231i 233.965 + 720.070i 1768.39 − 5442.55i −589.773 − 428.495i 8344.78
4.2 −12.1414 − 8.82122i −8.34346 + 25.6785i 30.0447 + 92.4681i 158.638 − 115.257i 327.817 − 238.173i −372.740 − 1147.18i −142.714 + 439.227i −589.773 − 428.495i −2942.80
4.3 −8.43584 − 6.12900i −8.34346 + 25.6785i −5.95538 − 18.3288i −186.461 + 135.472i 227.768 − 165.483i −43.6776 − 134.426i −474.540 + 1460.49i −589.773 − 428.495i 2403.26
4.4 −2.14212 − 1.55634i −8.34346 + 25.6785i −37.3877 − 115.067i 121.253 − 88.0954i 57.8373 − 42.0213i 231.374 + 712.095i −203.727 + 627.009i −589.773 − 428.495i −396.845
4.5 6.67280 + 4.84807i −8.34346 + 25.6785i −18.5317 − 57.0347i 269.190 − 195.578i −180.166 + 130.898i −36.4929 − 112.314i 479.095 − 1474.50i −589.773 − 428.495i 2744.43
4.6 8.90518 + 6.46999i −8.34346 + 25.6785i −2.11276 − 6.50241i −328.988 + 239.024i −240.440 + 174.690i −385.790 − 1187.34i 458.645 − 1411.56i −589.773 − 428.495i −4476.18
4.7 15.4308 + 11.2111i −8.34346 + 25.6785i 72.8662 + 224.259i 0.205524 − 0.149322i −416.632 + 302.701i 242.486 + 746.295i −635.379 + 1955.49i −589.773 − 428.495i 4.84548
16.1 −6.06001 − 18.6508i 21.8435 + 15.8702i −207.575 + 150.812i −96.8862 + 298.185i 163.620 − 503.572i 902.102 − 655.415i 2039.90 + 1482.08i 225.273 + 693.320i 6148.53
16.2 −5.89947 − 18.1567i 21.8435 + 15.8702i −191.308 + 138.993i 133.778 − 411.726i 159.286 − 490.231i −1338.24 + 972.291i 1675.32 + 1217.19i 225.273 + 693.320i −8264.81
16.3 −1.78222 − 5.48510i 21.8435 + 15.8702i 76.6441 − 55.6852i −142.765 + 439.384i 48.1199 − 148.098i −506.310 + 367.856i −1039.27 − 755.075i 225.273 + 693.320i 2664.50
16.4 −0.907839 − 2.79404i 21.8435 + 15.8702i 96.5717 − 70.1634i 61.3322 − 188.761i 24.5117 − 75.4391i −198.219 + 144.015i −587.936 − 427.160i 225.273 + 693.320i −583.086
16.5 2.98016 + 9.17200i 21.8435 + 15.8702i 28.3100 − 20.5684i 33.3401 − 102.610i −80.4644 + 247.644i 1247.17 − 906.120i 1271.70 + 923.944i 225.273 + 693.320i 1040.50
16.6 3.92793 + 12.0889i 21.8435 + 15.8702i −27.1593 + 19.7324i −17.6507 + 54.3232i −106.054 + 326.401i −962.124 + 699.024i 971.059 + 705.516i 225.273 + 693.320i −726.039
16.7 6.71359 + 20.6623i 21.8435 + 15.8702i −278.304 + 202.200i −94.4035 + 290.544i −181.267 + 557.882i 945.004 − 686.586i −3796.54 − 2758.35i 225.273 + 693.320i −6637.09
25.1 −18.2616 + 13.2678i −8.34346 − 25.6785i 117.897 − 362.849i −299.083 − 217.296i 493.063 + 358.231i 233.965 − 720.070i 1768.39 + 5442.55i −589.773 + 428.495i 8344.78
25.2 −12.1414 + 8.82122i −8.34346 − 25.6785i 30.0447 − 92.4681i 158.638 + 115.257i 327.817 + 238.173i −372.740 + 1147.18i −142.714 − 439.227i −589.773 + 428.495i −2942.80
25.3 −8.43584 + 6.12900i −8.34346 − 25.6785i −5.95538 + 18.3288i −186.461 − 135.472i 227.768 + 165.483i −43.6776 + 134.426i −474.540 − 1460.49i −589.773 + 428.495i 2403.26
25.4 −2.14212 + 1.55634i −8.34346 − 25.6785i −37.3877 + 115.067i 121.253 + 88.0954i 57.8373 + 42.0213i 231.374 − 712.095i −203.727 − 627.009i −589.773 + 428.495i −396.845
25.5 6.67280 − 4.84807i −8.34346 − 25.6785i −18.5317 + 57.0347i 269.190 + 195.578i −180.166 − 130.898i −36.4929 + 112.314i 479.095 + 1474.50i −589.773 + 428.495i 2744.43
25.6 8.90518 − 6.46999i −8.34346 − 25.6785i −2.11276 + 6.50241i −328.988 − 239.024i −240.440 − 174.690i −385.790 + 1187.34i 458.645 + 1411.56i −589.773 + 428.495i −4476.18
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.7
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 33.8.e.a ✓ 28
3.b odd 2 1 99.8.f.c 28
11.c even 5 1 inner 33.8.e.a ✓ 28
11.c even 5 1 363.8.a.q 14
11.d odd 10 1 363.8.a.p 14
33.h odd 10 1 99.8.f.c 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.8.e.a ✓ 28 1.a even 1 1 trivial
33.8.e.a ✓ 28 11.c even 5 1 inner
99.8.f.c 28 3.b odd 2 1
99.8.f.c 28 33.h odd 10 1
363.8.a.p 14 11.d odd 10 1
363.8.a.q 14 11.c even 5 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{28} + 22 T_{2}^{27} + 1036 T_{2}^{26} + 16944 T_{2}^{25} + 574889 T_{2}^{24} + 6855516 T_{2}^{23} + 234979339 T_{2}^{22} + 2089735763 T_{2}^{21} + 94011380549 T_{2}^{20} + \cdots + 74\!\cdots\!00 \) acting on \(S_{8}^{\mathrm{new}}(33, [\chi])\). Copy content Toggle raw display