Properties

Label 69.6.e.b
Level $69$
Weight $6$
Character orbit 69.e
Analytic conductor $11.066$
Analytic rank $0$
Dimension $100$
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] = [69,6,Mod(4,69)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(69, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([0, 4]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("69.4");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 69 = 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 69.e (of order \(11\), degree \(10\), 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: \(11.0664835671\)
Analytic rank: \(0\)
Dimension: \(100\)
Relative dimension: \(10\) over \(\Q(\zeta_{11})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 100 q + 12 q^{2} - 90 q^{3} - 148 q^{4} - 50 q^{5} + 9 q^{6} - 78 q^{7} - 1844 q^{8} - 810 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 100 q + 12 q^{2} - 90 q^{3} - 148 q^{4} - 50 q^{5} + 9 q^{6} - 78 q^{7} - 1844 q^{8} - 810 q^{9} - 38 q^{10} + 404 q^{11} - 1332 q^{12} - 2850 q^{13} - 724 q^{14} + 639 q^{15} - 5556 q^{16} - 2519 q^{17} + 972 q^{18} + 1117 q^{19} - 22411 q^{20} - 702 q^{21} + 32798 q^{22} + 12040 q^{23} - 756 q^{24} - 5928 q^{25} - 788 q^{26} - 7290 q^{27} - 54024 q^{28} + 13774 q^{29} - 342 q^{30} + 12501 q^{31} + 37412 q^{32} - 4185 q^{33} + 9441 q^{34} + 28894 q^{35} - 21789 q^{36} - 38902 q^{37} - 15042 q^{38} - 25650 q^{39} - 133835 q^{40} - 19456 q^{41} + 85950 q^{42} - 8126 q^{43} + 120643 q^{44} + 40500 q^{45} + 90500 q^{46} + 60846 q^{47} + 62955 q^{48} - 25762 q^{49} - 110197 q^{50} - 51084 q^{51} + 91725 q^{52} - 83896 q^{53} - 31347 q^{54} + 190331 q^{55} - 358140 q^{56} - 40437 q^{57} + 150977 q^{58} + 248368 q^{59} + 122625 q^{60} - 81516 q^{61} + 82657 q^{62} - 6318 q^{63} - 298028 q^{64} - 25475 q^{65} - 6669 q^{66} + 51337 q^{67} + 491228 q^{68} + 17082 q^{69} + 395832 q^{70} - 206653 q^{71} - 149364 q^{72} - 121007 q^{73} + 716844 q^{74} - 59490 q^{75} + 171145 q^{76} + 71982 q^{77} + 11916 q^{78} - 661118 q^{79} - 1386592 q^{80} - 65610 q^{81} - 136394 q^{82} - 368400 q^{83} + 165204 q^{84} + 45505 q^{85} - 99668 q^{86} + 50112 q^{87} - 474416 q^{88} + 328408 q^{89} - 3078 q^{90} + 1042320 q^{91} + 1554071 q^{92} + 160722 q^{93} - 299081 q^{94} - 434011 q^{95} + 412740 q^{96} - 88136 q^{97} + 515807 q^{98} - 37665 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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 −4.05453 8.87818i −8.63544 + 2.53559i −41.4274 + 47.8097i −10.1643 6.53221i 57.5241 + 66.3863i 18.4916 + 128.612i 292.757 + 85.9613i 68.1415 43.7919i −16.7826 + 116.726i
4.2 −2.89326 6.33535i −8.63544 + 2.53559i −10.8102 + 12.4756i 59.0516 + 37.9501i 41.0484 + 47.3724i −20.5053 142.618i −103.530 30.3990i 68.1415 43.7919i 69.5761 483.912i
4.3 −2.87865 6.30338i −8.63544 + 2.53559i −10.4903 + 12.1065i −49.4324 31.7683i 40.8412 + 47.1333i −20.2428 140.792i −106.255 31.1992i 68.1415 43.7919i −57.9485 + 403.041i
4.4 −0.582830 1.27622i −8.63544 + 2.53559i 19.6665 22.6963i −65.1725 41.8838i 8.26897 + 9.54290i 14.8045 + 102.968i −83.5054 24.5194i 68.1415 43.7919i −15.4685 + 107.586i
4.5 −0.310130 0.679089i −8.63544 + 2.53559i 20.5906 23.7628i 14.4659 + 9.29667i 4.40000 + 5.07787i 1.15767 + 8.05176i −45.4448 13.3438i 68.1415 43.7919i 1.82696 12.7068i
4.6 2.02749 + 4.43958i −8.63544 + 2.53559i 5.35641 6.18163i 66.9970 + 43.0564i −28.7652 33.1968i −25.3233 176.127i 188.158 + 55.2480i 68.1415 43.7919i −55.3165 + 384.735i
4.7 2.17624 + 4.76530i −8.63544 + 2.53559i 2.98345 3.44308i 20.2644 + 13.0232i −30.8757 35.6324i 8.80898 + 61.2678i 183.748 + 53.9534i 68.1415 43.7919i −17.9590 + 124.908i
4.8 2.60326 + 5.70034i −8.63544 + 2.53559i −4.76135 + 5.49489i −65.3942 42.0263i −36.9340 42.6241i 8.25660 + 57.4259i 148.692 + 43.6598i 68.1415 43.7919i 69.3261 482.174i
4.9 4.09417 + 8.96497i −8.63544 + 2.53559i −42.6530 + 49.2242i −26.2487 16.8690i −58.0864 67.0353i −28.2239 196.302i −313.318 91.9985i 68.1415 43.7919i 43.7638 304.384i
4.10 4.34821 + 9.52125i −8.63544 + 2.53559i −50.7917 + 58.6168i 82.5719 + 53.0657i −61.6907 71.1949i 29.0096 + 201.766i −457.577 134.357i 68.1415 43.7919i −146.212 + 1016.93i
13.1 −6.62196 7.64215i 7.57128 + 4.86577i −9.99801 + 69.5377i 22.3300 48.8959i −12.9518 90.0818i −79.5712 23.3642i 325.407 209.127i 33.6486 + 73.6802i −521.539 + 153.138i
13.2 −5.44390 6.28260i 7.57128 + 4.86577i −5.28090 + 36.7294i −18.8225 + 41.2154i −10.6477 74.0561i −66.5270 19.5341i 35.7163 22.9535i 33.6486 + 73.6802i 361.408 106.119i
13.3 −3.21781 3.71355i 7.57128 + 4.86577i 1.11793 7.77535i 37.1130 81.2661i −6.29366 43.7734i 196.204 + 57.6107i −164.749 + 105.878i 33.6486 + 73.6802i −421.208 + 123.678i
13.4 −2.79827 3.22937i 7.57128 + 4.86577i 1.95553 13.6010i −19.4524 + 42.5948i −5.47309 38.0662i 36.6143 + 10.7509i −164.426 + 105.670i 33.6486 + 73.6802i 191.988 56.3726i
13.5 −0.220291 0.254230i 7.57128 + 4.86577i 4.53797 31.5623i 21.3588 46.7692i −0.430865 2.99673i −219.257 64.3798i −18.0795 + 11.6190i 33.6486 + 73.6802i −16.5953 + 4.87282i
13.6 2.18855 + 2.52572i 7.57128 + 4.86577i 2.96456 20.6189i −1.08863 + 2.38376i 4.28056 + 29.7719i 60.1265 + 17.6547i 148.533 95.4564i 33.6486 + 73.6802i −8.40324 + 2.46741i
13.7 4.37852 + 5.05308i 7.57128 + 4.86577i −1.80812 + 12.5757i 41.9916 91.9487i 8.56389 + 59.5631i −7.37026 2.16411i 108.530 69.7479i 33.6486 + 73.6802i 648.485 190.412i
13.8 4.58877 + 5.29572i 7.57128 + 4.86577i −2.43379 + 16.9274i −42.3452 + 92.7231i 8.97511 + 62.4233i −216.718 63.6341i 87.8249 56.4417i 33.6486 + 73.6802i −685.348 + 201.236i
13.9 5.18515 + 5.98398i 7.57128 + 4.86577i −4.36819 + 30.3814i −11.2004 + 24.5255i 10.1416 + 70.5362i 164.984 + 48.4438i 8.70029 5.59133i 33.6486 + 73.6802i −204.836 + 60.1453i
13.10 7.28389 + 8.40605i 7.57128 + 4.86577i −13.0526 + 90.7832i 7.34378 16.0806i 14.2465 + 99.0863i −9.51310 2.79330i −558.775 + 359.103i 33.6486 + 73.6802i 188.666 55.3973i
See all 100 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.10
Significant digits:
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Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 69.6.e.b 100
23.c even 11 1 inner 69.6.e.b 100
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
69.6.e.b 100 1.a even 1 1 trivial
69.6.e.b 100 23.c even 11 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{100} - 12 T_{2}^{99} + 306 T_{2}^{98} - 1564 T_{2}^{97} + 39671 T_{2}^{96} - 127472 T_{2}^{95} + 6051408 T_{2}^{94} - 22478280 T_{2}^{93} + 833275204 T_{2}^{92} - 2513656325 T_{2}^{91} + \cdots + 59\!\cdots\!44 \) acting on \(S_{6}^{\mathrm{new}}(69, [\chi])\). Copy content Toggle raw display