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Properties

Label 93.4.m.b
Level $93$
Weight $4$
Character orbit 93.m
Analytic conductor $5.487$
Analytic rank $0$
Dimension $72$
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] = [93,4,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, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("93.7");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 93 = 3 \cdot 31 \)
Weight: \( k \) \(=\) \( 4 \)
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: \(5.48717763053\)
Analytic rank: \(0\)
Dimension: \(72\)
Relative dimension: \(9\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 72 q - 6 q^{2} - 27 q^{3} - 94 q^{4} - 54 q^{5} + 21 q^{6} + 24 q^{7} + 94 q^{8} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 72 q - 6 q^{2} - 27 q^{3} - 94 q^{4} - 54 q^{5} + 21 q^{6} + 24 q^{7} + 94 q^{8} + 81 q^{9} - 22 q^{10} + 94 q^{11} - 51 q^{12} - 28 q^{13} - 229 q^{14} - 24 q^{15} - 398 q^{16} + 110 q^{17} + 27 q^{18} - 482 q^{19} + 739 q^{20} + 183 q^{21} + 1024 q^{22} + 484 q^{23} + 321 q^{24} - 1584 q^{25} + 38 q^{26} + 486 q^{27} - 1107 q^{28} - 149 q^{29} - 402 q^{30} + 157 q^{31} + 996 q^{32} - 66 q^{33} - 1294 q^{34} - 924 q^{35} - 1557 q^{36} + 720 q^{37} + 1525 q^{38} - 108 q^{39} + 118 q^{40} + 1914 q^{41} + 687 q^{42} + 2036 q^{43} - 2311 q^{44} + 279 q^{45} - 379 q^{46} - 1568 q^{47} - 2187 q^{48} - 385 q^{49} + 5899 q^{50} - 330 q^{51} + 8573 q^{52} - 913 q^{53} + 27 q^{54} - 2424 q^{55} - 3451 q^{56} + 366 q^{57} - 1212 q^{58} - 6887 q^{59} - 1461 q^{60} - 4092 q^{61} + 3035 q^{62} + 1098 q^{63} - 2504 q^{64} - 1690 q^{65} - 1011 q^{66} - 2404 q^{67} + 297 q^{68} + 726 q^{69} + 2933 q^{70} - 916 q^{71} + 612 q^{72} + 2020 q^{73} + 14872 q^{74} - 1158 q^{75} + 14555 q^{76} + 980 q^{77} - 2667 q^{78} - 3356 q^{79} - 12727 q^{80} + 729 q^{81} + 1068 q^{82} + 2861 q^{83} + 1311 q^{84} + 2070 q^{85} - 3720 q^{86} - 201 q^{87} - 5983 q^{88} + 348 q^{89} - 198 q^{90} - 7978 q^{91} - 6940 q^{92} + 2685 q^{93} - 7752 q^{94} + 2606 q^{95} - 966 q^{96} - 6486 q^{97} - 973 q^{98} + 261 q^{99}+O(q^{100}) \) 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 −1.64456 − 5.06144i 2.93444 − 0.623735i −16.4414 + 11.9454i −7.29891 + 12.6421i −7.98286 − 13.8267i 4.58132 + 2.03974i 53.0556 + 38.5471i 8.22191 − 3.66063i 75.9906 + 16.1523i
7.2 −1.24503 − 3.83180i 2.93444 − 0.623735i −6.66046 + 4.83911i 8.47191 − 14.6738i −6.04349 − 10.4676i 25.5123 + 11.3588i 0.758755 + 0.551268i 8.22191 − 3.66063i −66.7747 − 14.1934i
7.3 −1.04301 − 3.21005i 2.93444 − 0.623735i −2.74440 + 1.99392i −2.06909 + 3.58378i −5.06286 − 8.76914i −21.2407 − 9.45699i −12.5820 − 9.14137i 8.22191 − 3.66063i 13.6622 + 2.90399i
7.4 −0.214861 − 0.661275i 2.93444 − 0.623735i 6.08102 − 4.41812i −6.96841 + 12.0696i −1.04296 − 1.80646i 29.0422 + 12.9304i −8.72827 − 6.34146i 8.22191 − 3.66063i 9.47859 + 2.01474i
7.5 −0.175553 − 0.540295i 2.93444 − 0.623735i 6.21104 − 4.51258i 4.90087 − 8.48855i −0.852150 − 1.47597i −14.1535 − 6.30153i −7.20531 − 5.23496i 8.22191 − 3.66063i −5.44668 − 1.15773i
7.6 0.678611 + 2.08855i 2.93444 − 0.623735i 2.57061 − 1.86766i 3.06406 − 5.30711i 3.29405 + 5.70546i −0.560205 − 0.249419i 19.8581 + 14.4278i 8.22191 − 3.66063i 13.1635 + 2.79798i
7.7 0.961191 + 2.95824i 2.93444 − 0.623735i −1.35517 + 0.984585i −10.8927 + 18.8668i 4.66572 + 8.08126i −25.1556 − 11.2000i 15.9162 + 11.5638i 8.22191 − 3.66063i −66.2824 − 14.0888i
7.8 1.09895 + 3.38221i 2.93444 − 0.623735i −3.75954 + 2.73147i 4.90363 − 8.49334i 5.33440 + 9.23946i 10.5713 + 4.70662i 9.64672 + 7.00875i 8.22191 − 3.66063i 34.1151 + 7.25139i
7.9 1.63769 + 5.04029i 2.93444 − 0.623735i −16.2504 + 11.8066i −4.77444 + 8.26958i 7.94952 + 13.7690i 20.5654 + 9.15631i −51.8217 − 37.6507i 8.22191 − 3.66063i −49.5002 − 10.5216i
10.1 −4.23885 − 3.07970i 0.313585 + 2.98357i 6.01111 + 18.5003i 9.32364 + 16.1490i 7.85925 − 13.6126i −32.6352 − 6.93682i 18.5425 − 57.0679i −8.80333 + 1.87121i 10.2127 − 97.1672i
10.2 −4.03214 − 2.92952i 0.313585 + 2.98357i 5.20391 + 16.0160i −10.7797 − 18.6709i 7.47599 − 12.9488i 1.88910 + 0.401541i 13.6151 − 41.9031i −8.80333 + 1.87121i −11.2317 + 106.863i
10.3 −2.57741 − 1.87260i 0.313585 + 2.98357i 0.664282 + 2.04445i −0.0632694 − 0.109586i 4.77878 − 8.27709i 4.69373 + 0.997683i −5.75956 + 17.7261i −8.80333 + 1.87121i −0.0421390 + 0.400926i
10.4 −1.40193 − 1.01857i 0.313585 + 2.98357i −1.54419 − 4.75253i −1.61807 − 2.80258i 2.59933 − 4.50217i 0.211610 + 0.0449791i −6.95984 + 21.4202i −8.80333 + 1.87121i −0.586182 + 5.57715i
10.5 −0.532239 − 0.386694i 0.313585 + 2.98357i −2.33839 − 7.19682i 9.03382 + 15.6470i 0.986826 − 1.70923i −7.15113 − 1.52002i −3.16477 + 9.74015i −8.80333 + 1.87121i 1.24247 − 11.8213i
10.6 1.09721 + 0.797173i 0.313585 + 2.98357i −1.90374 − 5.85911i −5.53885 − 9.59357i −2.03435 + 3.52359i 34.6699 + 7.36932i 5.93470 − 18.2651i −8.80333 + 1.87121i 1.57043 − 14.9416i
10.7 1.32013 + 0.959129i 0.313585 + 2.98357i −1.64933 − 5.07611i −6.92578 − 11.9958i −2.44765 + 4.23946i −30.6846 − 6.52221i 6.72527 − 20.6983i −8.80333 + 1.87121i 2.36261 − 22.4787i
10.8 3.03179 + 2.20272i 0.313585 + 2.98357i 1.86762 + 5.74794i 4.47092 + 7.74385i −5.62125 + 9.73628i 12.5507 + 2.66773i 2.26543 − 6.97228i −8.80333 + 1.87121i −3.50270 + 33.3259i
10.9 4.13272 + 3.00260i 0.313585 + 2.98357i 5.59167 + 17.2094i −0.739641 − 1.28110i −7.66249 + 13.2718i −11.5918 − 2.46391i −15.9356 + 49.0447i −8.80333 + 1.87121i 0.789885 − 7.51526i
19.1 −4.38131 − 3.18321i −2.74064 − 1.22021i 6.59093 + 20.2848i −7.05449 + 12.2187i 8.12339 + 14.0701i 15.8239 − 17.5742i 22.3057 − 68.6499i 6.02218 + 6.68830i 69.8027 − 31.0782i
19.2 −3.43608 − 2.49646i −2.74064 − 1.22021i 3.10222 + 9.54765i 3.31641 − 5.74419i 6.37085 + 11.0346i −18.6810 + 20.7474i 2.67611 − 8.23623i 6.02218 + 6.68830i −25.7356 + 11.4582i
See all 72 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.9
Significant digits:
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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 93.4.m.b ✓ 72
31.g even 15 1 inner 93.4.m.b ✓ 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
93.4.m.b ✓ 72 1.a even 1 1 trivial
93.4.m.b ✓ 72 31.g even 15 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{72} + 6 T_{2}^{71} + 137 T_{2}^{70} + 724 T_{2}^{69} + 10270 T_{2}^{68} + 48534 T_{2}^{67} + \cdots + 21\!\cdots\!00 \) acting on \(S_{4}^{\mathrm{new}}(93, [\chi])\). Copy content Toggle raw display