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Properties

Label 93.2.m.b
Level $93$
Weight $2$
Character orbit 93.m
Analytic conductor $0.743$
Analytic rank $0$
Dimension $24$
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] = [93,2,Mod(7,93)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(93, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([0, 28]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("93.7");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 93 = 3 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 93.m (of order \(15\), degree \(8\), 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: \(0.742608738798\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(3\) over \(\Q(\zeta_{15})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 3 q^{3} - 4 q^{4} - 6 q^{5} - 5 q^{6} - q^{7} - 22 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 3 q^{3} - 4 q^{4} - 6 q^{5} - 5 q^{6} - q^{7} - 22 q^{8} + 3 q^{9} + 24 q^{10} - 22 q^{11} + 12 q^{12} - 8 q^{13} + 10 q^{14} - 8 q^{15} - 2 q^{16} - 17 q^{17} + 5 q^{19} - 22 q^{20} - 11 q^{21} - 37 q^{22} + 26 q^{23} + 26 q^{24} - 8 q^{25} + 4 q^{26} - 6 q^{27} - 36 q^{28} + 2 q^{29} + 42 q^{30} + 36 q^{32} + 14 q^{33} + 40 q^{34} + 9 q^{35} - 13 q^{36} - 13 q^{37} - q^{38} - 19 q^{39} - 27 q^{40} + 36 q^{41} + 10 q^{42} - 11 q^{43} - 38 q^{44} - q^{45} - 23 q^{46} - 13 q^{47} - 14 q^{48} - 22 q^{49} + 71 q^{50} - 17 q^{51} + 9 q^{52} - 20 q^{53} - 5 q^{54} + 26 q^{55} + 28 q^{56} - 15 q^{57} + 40 q^{58} - 16 q^{59} - 61 q^{60} + 70 q^{61} - 2 q^{62} + 12 q^{63} + 34 q^{64} + 94 q^{65} - 16 q^{66} + 4 q^{67} + 51 q^{68} - 13 q^{69} - 43 q^{70} - 5 q^{71} + q^{72} - 12 q^{73} + 74 q^{74} - 8 q^{75} + 71 q^{76} + 25 q^{77} + 17 q^{78} - 29 q^{79} - 113 q^{80} + 3 q^{81} - 60 q^{82} - 11 q^{83} - 41 q^{84} + 16 q^{85} - 144 q^{86} + 14 q^{87} + 26 q^{88} - 27 q^{89} + 24 q^{90} - 81 q^{91} + 28 q^{92} - 36 q^{93} - 80 q^{94} - 56 q^{95} - 43 q^{96} - 21 q^{97} - 14 q^{98} - 2 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1 −0.346673 − 1.06695i −0.978148 + 0.207912i 0.599831 − 0.435803i 0.868701 − 1.50463i 0.560929 + 0.971558i −0.532922 − 0.237272i −2.48813 − 1.80773i 0.913545 − 0.406737i −1.90653 − 0.405245i
7.2 0.362683 + 1.11622i −0.978148 + 0.207912i 0.503620 − 0.365901i −2.03997 + 3.53333i −0.586833 − 1.01642i 1.97984 + 0.881481i 2.49011 + 1.80917i 0.913545 − 0.406737i −4.68384 − 0.995581i
7.3 0.857610 + 2.63945i −0.978148 + 0.207912i −4.61318 + 3.35167i 0.980284 − 1.69790i −1.38764 − 2.40347i 2.54096 + 1.13131i −8.31238 − 6.03930i 0.913545 − 0.406737i 5.32223 + 1.13128i
10.1 −1.75230 − 1.27312i −0.104528 − 0.994522i 0.831686 + 2.55967i −1.24535 − 2.15701i −1.08298 + 1.87578i −1.22375 − 0.260117i 0.462763 − 1.42424i −0.978148 + 0.207912i −0.563906 + 5.36520i
10.2 0.620473 + 0.450800i −0.104528 − 0.994522i −0.436268 − 1.34269i 0.582742 + 1.00934i 0.383473 − 0.664195i 3.17698 + 0.675287i 0.808593 − 2.48859i −0.978148 + 0.207912i −0.0934342 + 0.888967i
10.3 1.90549 + 1.38442i −0.104528 − 0.994522i 1.09623 + 3.37386i −0.646410 − 1.11962i 1.17766 − 2.03976i −3.84035 − 0.816291i −1.12631 + 3.46643i −0.978148 + 0.207912i 0.318289 − 3.02831i
19.1 −2.08748 − 1.51664i 0.913545 + 0.406737i 1.43933 + 4.42980i −1.93212 + 3.34652i −1.29013 − 2.23457i −2.07563 + 2.30522i 2.11916 − 6.52210i 0.669131 + 0.743145i 9.10873 − 4.05547i
19.2 −0.731871 − 0.531736i 0.913545 + 0.406737i −0.365141 − 1.12379i 0.192370 − 0.333194i −0.452321 − 0.783444i 2.25360 − 2.50288i −0.889422 + 2.73736i 0.669131 + 0.743145i −0.317961 + 0.141565i
19.3 0.927656 + 0.673982i 0.913545 + 0.406737i −0.211739 − 0.651666i 0.430729 − 0.746045i 0.573323 + 0.993025i −2.45397 + 2.72541i 0.951456 − 2.92828i 0.669131 + 0.743145i 0.902389 − 0.401770i
28.1 −1.75230 + 1.27312i −0.104528 + 0.994522i 0.831686 − 2.55967i −1.24535 + 2.15701i −1.08298 − 1.87578i −1.22375 + 0.260117i 0.462763 + 1.42424i −0.978148 − 0.207912i −0.563906 − 5.36520i
28.2 0.620473 − 0.450800i −0.104528 + 0.994522i −0.436268 + 1.34269i 0.582742 − 1.00934i 0.383473 + 0.664195i 3.17698 − 0.675287i 0.808593 + 2.48859i −0.978148 − 0.207912i −0.0934342 − 0.888967i
28.3 1.90549 − 1.38442i −0.104528 + 0.994522i 1.09623 − 3.37386i −0.646410 + 1.11962i 1.17766 + 2.03976i −3.84035 + 0.816291i −1.12631 − 3.46643i −0.978148 − 0.207912i 0.318289 + 3.02831i
40.1 −0.346673 + 1.06695i −0.978148 − 0.207912i 0.599831 + 0.435803i 0.868701 + 1.50463i 0.560929 − 0.971558i −0.532922 + 0.237272i −2.48813 + 1.80773i 0.913545 + 0.406737i −1.90653 + 0.405245i
40.2 0.362683 − 1.11622i −0.978148 − 0.207912i 0.503620 + 0.365901i −2.03997 − 3.53333i −0.586833 + 1.01642i 1.97984 − 0.881481i 2.49011 − 1.80917i 0.913545 + 0.406737i −4.68384 + 0.995581i
40.3 0.857610 − 2.63945i −0.978148 − 0.207912i −4.61318 − 3.35167i 0.980284 + 1.69790i −1.38764 + 2.40347i 2.54096 − 1.13131i −8.31238 + 6.03930i 0.913545 + 0.406737i 5.32223 − 1.13128i
49.1 −2.08748 + 1.51664i 0.913545 − 0.406737i 1.43933 − 4.42980i −1.93212 − 3.34652i −1.29013 + 2.23457i −2.07563 − 2.30522i 2.11916 + 6.52210i 0.669131 − 0.743145i 9.10873 + 4.05547i
49.2 −0.731871 + 0.531736i 0.913545 − 0.406737i −0.365141 + 1.12379i 0.192370 + 0.333194i −0.452321 + 0.783444i 2.25360 + 2.50288i −0.889422 − 2.73736i 0.669131 − 0.743145i −0.317961 − 0.141565i
49.3 0.927656 − 0.673982i 0.913545 − 0.406737i −0.211739 + 0.651666i 0.430729 + 0.746045i 0.573323 − 0.993025i −2.45397 − 2.72541i 0.951456 + 2.92828i 0.669131 − 0.743145i 0.902389 + 0.401770i
76.1 −0.418031 + 1.28657i 0.669131 − 0.743145i 0.137526 + 0.0999185i 1.63328 − 2.82893i 0.676389 + 1.17154i −0.149090 + 1.41850i −2.37488 + 1.72545i −0.104528 − 0.994522i 2.95685 + 3.28391i
76.2 −0.0419030 + 0.128964i 0.669131 − 0.743145i 1.60316 + 1.16476i −1.42565 + 2.46931i 0.0678004 + 0.117434i 0.302129 − 2.87456i −0.436796 + 0.317351i −0.104528 − 0.994522i −0.258713 − 0.287329i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.g even 15 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 93.2.m.b ✓ 24
3.b odd 2 1 279.2.y.d 24
31.g even 15 1 inner 93.2.m.b ✓ 24
31.g even 15 1 2883.2.a.t 12
31.h odd 30 1 2883.2.a.s 12
93.o odd 30 1 279.2.y.d 24
93.o odd 30 1 8649.2.a.bk 12
93.p even 30 1 8649.2.a.bl 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
93.2.m.b ✓ 24 1.a even 1 1 trivial
93.2.m.b ✓ 24 31.g even 15 1 inner
279.2.y.d 24 3.b odd 2 1
279.2.y.d 24 93.o odd 30 1
2883.2.a.s 12 31.h odd 30 1
2883.2.a.t 12 31.g even 15 1
8649.2.a.bk 12 93.o odd 30 1
8649.2.a.bl 12 93.p even 30 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{24} + 8 T_{2}^{22} + 14 T_{2}^{21} + 58 T_{2}^{20} - 12 T_{2}^{19} + 483 T_{2}^{18} + 134 T_{2}^{17} + \cdots + 256 \) acting on \(S_{2}^{\mathrm{new}}(93, [\chi])\). Copy content Toggle raw display