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Properties

Label 123.4.e.a
Level $123$
Weight $4$
Character orbit 123.e
Analytic conductor $7.257$
Analytic rank $0$
Dimension $44$
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] = [123,4,Mod(73,123)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(123, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("123.73");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 123 = 3 \cdot 41 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 123.e (of order \(4\), degree \(2\), 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: \(7.25723493071\)
Analytic rank: \(0\)
Dimension: \(44\)
Relative dimension: \(22\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 44 q - 192 q^{4}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 44 q - 192 q^{4} + 224 q^{10} + 96 q^{11} - 24 q^{12} - 196 q^{13} + 148 q^{14} + 24 q^{15} + 896 q^{16} - 308 q^{17} + 112 q^{19} - 612 q^{22} - 208 q^{23} - 180 q^{24} - 1452 q^{25} + 1172 q^{26} - 220 q^{28} + 300 q^{29} + 456 q^{30} + 128 q^{31} - 552 q^{34} + 504 q^{35} - 88 q^{37} + 1204 q^{38} - 2112 q^{40} + 1316 q^{41} + 1560 q^{42} - 1412 q^{44} - 504 q^{45} + 520 q^{47} + 192 q^{48} + 936 q^{51} + 2652 q^{52} + 1116 q^{53} - 560 q^{55} - 4088 q^{56} - 936 q^{57} - 2860 q^{58} - 992 q^{59} - 804 q^{60} - 8472 q^{64} - 1480 q^{65} + 1968 q^{66} + 2592 q^{67} + 3836 q^{68} + 1368 q^{69} + 2468 q^{70} + 1888 q^{71} + 1512 q^{72} - 48 q^{75} + 996 q^{76} + 1392 q^{78} - 3000 q^{79} - 3564 q^{81} - 236 q^{82} - 2080 q^{83} - 1160 q^{85} - 5520 q^{86} + 7940 q^{88} - 2748 q^{89} + 14712 q^{92} + 1680 q^{93} - 6400 q^{94} - 7456 q^{95} + 4320 q^{96} + 5516 q^{97} + 11008 q^{98} - 864 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
73.1 − 5.61902i −2.12132 + 2.12132i −23.5733 − 13.3921i 11.9197 + 11.9197i −10.6525 + 10.6525i 87.5068i − 9.00000i −75.2505
73.2 − 5.25757i 2.12132 − 2.12132i −19.6420 21.1469i −11.1530 − 11.1530i −8.38626 + 8.38626i 61.2088i − 9.00000i 111.181
73.3 − 4.46476i 2.12132 − 2.12132i −11.9341 − 8.04569i −9.47119 − 9.47119i 16.4598 − 16.4598i 17.5647i − 9.00000i −35.9221
73.4 − 4.24235i −2.12132 + 2.12132i −9.99752 16.7806i 8.99938 + 8.99938i 8.84825 − 8.84825i 8.47416i − 9.00000i 71.1889
73.5 − 4.04342i 2.12132 − 2.12132i −8.34925 − 13.7007i −8.57739 − 8.57739i −23.0314 + 23.0314i 1.41217i − 9.00000i −55.3977
73.6 − 3.81117i −2.12132 + 2.12132i −6.52505 − 7.20336i 8.08472 + 8.08472i 18.1690 − 18.1690i − 5.62130i − 9.00000i −27.4532
73.7 − 2.63096i 2.12132 − 2.12132i 1.07806 9.61057i −5.58111 − 5.58111i 6.96584 − 6.96584i − 23.8840i − 9.00000i 25.2850
73.8 − 2.16950i −2.12132 + 2.12132i 3.29326 − 7.68303i 4.60221 + 4.60221i −9.55652 + 9.55652i − 24.5008i − 9.00000i −16.6683
73.9 − 1.23251i −2.12132 + 2.12132i 6.48091 10.4273i 2.61456 + 2.61456i −9.33905 + 9.33905i − 17.8479i − 9.00000i 12.8517
73.10 − 0.702755i 2.12132 − 2.12132i 7.50614 − 13.0908i −1.49077 − 1.49077i 4.57630 − 4.57630i − 10.8970i − 9.00000i −9.19959
73.11 0.335020i −2.12132 + 2.12132i 7.88776 − 3.72745i −0.710685 − 0.710685i 11.7913 − 11.7913i 5.32272i − 9.00000i 1.24877
73.12 0.406748i 2.12132 − 2.12132i 7.83456 − 5.41007i 0.862843 + 0.862843i 4.20418 − 4.20418i 6.44068i − 9.00000i 2.20054
73.13 0.926424i 2.12132 − 2.12132i 7.14174 15.2132i 1.96524 + 1.96524i −14.5651 + 14.5651i 14.0277i − 9.00000i −14.0939
73.14 1.95804i −2.12132 + 2.12132i 4.16607 − 19.9285i −4.15363 − 4.15363i −25.5597 + 25.5597i 23.8217i − 9.00000i 39.0208
73.15 2.33516i −2.12132 + 2.12132i 2.54705 14.1999i −4.95361 − 4.95361i −0.414133 + 0.414133i 24.6290i − 9.00000i −33.1590
73.16 2.68829i 2.12132 − 2.12132i 0.773103 10.5013i 5.70272 + 5.70272i 23.9191 − 23.9191i 23.5846i − 9.00000i −28.2306
73.17 3.35016i 2.12132 − 2.12132i −3.22354 − 17.1459i 7.10675 + 7.10675i −8.67318 + 8.67318i 16.0019i − 9.00000i 57.4414
73.18 3.47014i −2.12132 + 2.12132i −4.04185 12.6171i −7.36127 − 7.36127i 1.00923 − 1.00923i 13.7353i − 9.00000i −43.7829
73.19 3.73883i −2.12132 + 2.12132i −5.97887 − 16.5635i −7.93126 − 7.93126i 18.0717 − 18.0717i 7.55667i − 9.00000i 61.9282
73.20 4.27333i 2.12132 − 2.12132i −10.2613 0.477644i 9.06509 + 9.06509i −14.6412 + 14.6412i − 9.66333i − 9.00000i −2.04113
See all 44 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 73.22
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
41.c even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 123.4.e.a ✓ 44
41.c even 4 1 inner 123.4.e.a ✓ 44
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
123.4.e.a ✓ 44 1.a even 1 1 trivial
123.4.e.a ✓ 44 41.c even 4 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(123, [\chi])\).