Properties

Label 92.4.h.a
Level $92$
Weight $4$
Character orbit 92.h
Analytic conductor $5.428$
Analytic rank $0$
Dimension $340$
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] = [92,4,Mod(7,92)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(92, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([11, 19]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("92.7");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 92 = 2^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 92.h (of order \(22\), 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: \(5.42817572053\)
Analytic rank: \(0\)
Dimension: \(340\)
Relative dimension: \(34\) over \(\Q(\zeta_{22})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{22}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 340 q - 11 q^{2} + 13 q^{4} - 22 q^{5} + 2 q^{6} + 22 q^{8} + 252 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 340 q - 11 q^{2} + 13 q^{4} - 22 q^{5} + 2 q^{6} + 22 q^{8} + 252 q^{9} - 11 q^{10} - 102 q^{12} - 18 q^{13} - 11 q^{14} - 27 q^{16} - 22 q^{17} - 50 q^{18} - 11 q^{20} - 22 q^{21} - 302 q^{24} + 632 q^{25} - 418 q^{26} - 11 q^{28} + 326 q^{29} - 11 q^{30} + 69 q^{32} - 22 q^{33} + 1903 q^{34} + 1213 q^{36} - 22 q^{37} + 594 q^{38} - 1331 q^{40} - 314 q^{41} - 3641 q^{42} - 3322 q^{44} - 4907 q^{46} - 1794 q^{48} - 1200 q^{49} - 2344 q^{50} - 268 q^{52} - 22 q^{53} + 2646 q^{54} + 3564 q^{56} - 22 q^{57} + 5235 q^{58} + 3619 q^{60} - 22 q^{61} + 1522 q^{62} - 1478 q^{64} - 22 q^{65} - 308 q^{66} - 218 q^{69} + 3202 q^{70} + 3019 q^{72} - 18 q^{73} - 2376 q^{74} - 8206 q^{76} + 7514 q^{77} - 10560 q^{78} - 4070 q^{80} + 2864 q^{81} + 2804 q^{82} + 5379 q^{84} - 7586 q^{85} + 6699 q^{86} + 6941 q^{88} - 3762 q^{89} + 18678 q^{90} + 11126 q^{92} - 20060 q^{93} + 5981 q^{94} + 17245 q^{96} - 2706 q^{97} + 1957 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1 −2.82761 + 0.0677825i 1.81046 + 2.81713i 7.99081 0.383326i 12.5112 5.71369i −5.31024 7.84305i −5.24129 + 1.53898i −22.5690 + 1.62553i 6.55773 14.3594i −34.9897 + 17.0042i
7.2 −2.81852 0.236485i 1.10868 + 1.72513i 7.88815 + 1.33308i −11.1319 + 5.08378i −2.71686 5.12452i −3.75061 + 1.10128i −21.9177 5.62274i 9.46928 20.7348i 32.5778 11.6962i
7.3 −2.76439 + 0.598463i −3.90833 6.08148i 7.28368 3.30877i 2.35661 1.07623i 14.4437 + 14.4726i 30.6621 9.00321i −18.1547 + 13.5057i −10.4932 + 22.9768i −5.87051 + 4.38546i
7.4 −2.62792 1.04596i −3.88100 6.03895i 5.81193 + 5.49740i −5.21193 + 2.38021i 3.88245 + 19.9293i −18.1422 + 5.32703i −9.52324 20.5258i −10.1906 + 22.3143i 16.1861 0.803526i
7.5 −2.44815 1.41653i 5.07187 + 7.89198i 3.98688 + 6.93576i 3.38966 1.54801i −1.23745 26.5052i 22.4823 6.60139i 0.0642622 22.6273i −25.3433 + 55.4942i −10.4912 1.01181i
7.6 −2.42335 + 1.45856i 4.96798 + 7.73032i 3.74522 7.06918i −11.1313 + 5.08348i −23.3143 11.4872i −5.60740 + 1.64648i 1.23482 + 22.5937i −23.8609 + 52.2481i 19.5604 28.5546i
7.7 −2.33549 + 1.59546i −0.208207 0.323976i 2.90901 7.45236i −6.50566 + 2.97103i 1.00315 + 0.424456i 6.33191 1.85922i 5.09597 + 22.0461i 11.1546 24.4252i 10.4537 17.3183i
7.8 −2.18875 + 1.79147i −3.00503 4.67592i 1.58124 7.84217i 13.7050 6.25888i 14.9540 + 4.85098i −30.1106 + 8.84128i 10.5881 + 19.9973i −1.61781 + 3.54250i −18.7843 + 38.2513i
7.9 −1.95779 2.04134i −2.24967 3.50055i −0.334121 + 7.99302i 15.6896 7.16519i −2.74143 + 11.4457i 8.54748 2.50977i 16.9706 14.9666i 4.02336 8.80993i −45.3434 17.9998i
7.10 −1.87947 2.11367i 0.164413 + 0.255832i −0.935191 + 7.94515i −17.0252 + 7.77513i 0.231734 0.828343i 28.9597 8.50335i 18.5511 12.9560i 11.1778 24.4759i 48.4323 + 21.3724i
7.11 −1.83494 2.15244i 2.38392 + 3.70945i −1.26598 + 7.89920i −0.833192 + 0.380506i 3.61001 11.9379i −32.5869 + 9.56837i 19.3255 11.7696i 3.13926 6.87402i 2.34787 + 1.09519i
7.12 −1.46175 + 2.42142i 3.00503 + 4.67592i −3.72659 7.07902i 13.7050 6.25888i −15.7150 + 0.441437i 30.1106 8.84128i 22.5886 + 1.32411i −1.61781 + 3.54250i −4.87791 + 42.3346i
7.13 −1.24685 + 2.53877i 0.208207 + 0.323976i −4.89075 6.33092i −6.50566 + 2.97103i −1.08210 + 0.124642i −6.33191 + 1.85922i 22.1708 4.52283i 11.1546 24.4252i 0.568767 20.2208i
7.14 −1.09883 + 2.60626i −4.96798 7.73032i −5.58513 5.72768i −11.1313 + 5.08348i 25.6062 4.45349i 5.60740 1.64648i 21.0649 8.26253i −23.8609 + 52.2481i −1.01745 34.5968i
7.15 −0.722671 2.73455i 2.75621 + 4.28874i −6.95549 + 3.95235i 5.20290 2.37608i 9.73593 10.6363i −4.87145 + 1.43039i 15.8344 + 16.1639i 0.419588 0.918770i −10.2575 12.5104i
7.16 −0.437400 2.79440i −3.57074 5.55619i −7.61736 + 2.44454i −5.68745 + 2.59737i −13.9644 + 12.4084i −0.744179 + 0.218511i 10.1629 + 20.2167i −6.90479 + 15.1194i 9.74579 + 14.7569i
7.17 −0.198958 + 2.82142i 3.90833 + 6.08148i −7.92083 1.12269i 2.35661 1.07623i −17.9360 + 9.81708i −30.6621 + 9.00321i 4.74350 22.1246i −10.4932 + 22.9768i 2.56763 + 6.86313i
7.18 0.335319 + 2.80848i −1.81046 2.81713i −7.77512 + 1.88347i 12.5112 5.71369i 7.30478 6.02929i 5.24129 1.53898i −7.89684 21.2047i 6.55773 14.3594i 20.2420 + 33.2217i
7.19 0.426542 2.79608i 0.814062 + 1.26671i −7.63612 2.38529i 9.73779 4.44710i 3.88904 1.73588i 15.5856 4.57635i −9.92658 + 20.3338i 10.2744 22.4977i −8.28087 29.1245i
7.20 0.635196 + 2.75618i −1.10868 1.72513i −7.19305 + 3.50143i −11.1319 + 5.08378i 4.05055 4.15151i 3.75061 1.10128i −14.2196 17.6013i 9.46928 20.7348i −21.0827 27.4524i
See next 80 embeddings (of 340 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.34
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
23.d odd 22 1 inner
92.h even 22 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 92.4.h.a 340
4.b odd 2 1 inner 92.4.h.a 340
23.d odd 22 1 inner 92.4.h.a 340
92.h even 22 1 inner 92.4.h.a 340
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
92.4.h.a 340 1.a even 1 1 trivial
92.4.h.a 340 4.b odd 2 1 inner
92.4.h.a 340 23.d odd 22 1 inner
92.4.h.a 340 92.h even 22 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(92, [\chi])\).