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Properties

Label 2303.4.a.f
Level $2303$
Weight $4$
Character orbit 2303.a
Self dual yes
Analytic conductor $135.881$
Analytic rank $1$
Dimension $21$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2303,4,Mod(1,2303)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2303, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2303.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2303 = 7^{2} \cdot 47 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2303.a (trivial)

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: yes
Analytic conductor: \(135.881398743\)
Analytic rank: \(1\)
Dimension: \(21\)
Twist minimal: no (minimal twist has level 329)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 21 q + 2 q^{2} - 20 q^{3} + 102 q^{4} - 16 q^{5} - 27 q^{6} + 27 q^{8} + 247 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 21 q + 2 q^{2} - 20 q^{3} + 102 q^{4} - 16 q^{5} - 27 q^{6} + 27 q^{8} + 247 q^{9} - 96 q^{10} + 10 q^{11} - 216 q^{12} - 126 q^{13} + 8 q^{15} + 358 q^{16} - 204 q^{17} + 223 q^{18} - 340 q^{19} + 307 q^{20} + 39 q^{22} + 46 q^{23} - 708 q^{24} + 665 q^{25} - 616 q^{26} - 596 q^{27} - 54 q^{29} - 573 q^{30} - 1016 q^{31} + 93 q^{32} - 632 q^{33} - 334 q^{34} + 805 q^{36} + 490 q^{37} + 56 q^{38} + 1170 q^{39} - 1235 q^{40} - 726 q^{41} + 522 q^{43} + 126 q^{44} - 384 q^{45} + 1006 q^{46} - 987 q^{47} - 1775 q^{48} - 347 q^{50} + 1728 q^{51} - 1487 q^{52} - 450 q^{53} - 1618 q^{54} - 2848 q^{55} - 1298 q^{57} + 1408 q^{58} - 10 q^{59} - 2118 q^{60} - 1974 q^{61} - 2335 q^{62} + 2183 q^{64} + 1390 q^{65} + 97 q^{66} - 292 q^{67} - 1333 q^{68} - 1964 q^{69} + 1096 q^{71} + 3516 q^{72} - 3714 q^{73} - 1382 q^{74} - 2520 q^{75} - 4207 q^{76} + 2146 q^{78} + 5456 q^{79} + 2715 q^{80} + 701 q^{81} - 2267 q^{82} - 2908 q^{83} + 328 q^{85} + 346 q^{86} + 2564 q^{87} - 2224 q^{88} + 486 q^{89} + 4288 q^{90} - 5106 q^{92} + 1126 q^{93} - 94 q^{94} - 3676 q^{95} + 1881 q^{96} - 3242 q^{97} - 4692 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −5.54796 0.279045 22.7799 10.6122 −1.54813 0 −81.9983 −26.9221 −58.8763
1.2 −4.88427 −3.87760 15.8560 −14.7532 18.9392 0 −38.3710 −11.9642 72.0586
1.3 −4.58871 −6.60590 13.0563 18.1846 30.3126 0 −23.2018 16.6380 −83.4438
1.4 −4.25040 8.77955 10.0659 10.1389 −37.3166 0 −8.78094 50.0806 −43.0944
1.5 −3.28181 6.71722 2.77030 −12.9670 −22.0447 0 17.1629 18.1210 42.5552
1.6 −3.06883 2.21244 1.41770 0.359215 −6.78959 0 20.2000 −22.1051 −1.10237
1.7 −2.76295 −9.98657 −0.366111 −13.2869 27.5924 0 23.1151 72.7316 36.7110
1.8 −2.44586 −6.54325 −2.01775 9.20997 16.0039 0 24.5021 15.8141 −22.5263
1.9 −1.32012 3.35597 −6.25728 2.79107 −4.43029 0 18.8213 −15.7375 −3.68456
1.10 −0.751773 −6.84916 −7.43484 −14.3037 5.14901 0 11.6035 19.9110 10.7532
1.11 0.724099 0.934134 −7.47568 19.0851 0.676406 0 −11.2059 −26.1274 13.8195
1.12 0.783136 −2.20443 −7.38670 −9.28772 −1.72637 0 −12.0499 −22.1405 −7.27355
1.13 1.44132 6.44452 −5.92260 −21.6603 9.28860 0 −20.0669 14.5318 −31.2195
1.14 2.38025 −8.62552 −2.33440 6.85571 −20.5309 0 −24.5985 47.3995 16.3183
1.15 2.64032 9.01270 −1.02872 1.73341 23.7964 0 −23.8387 54.2287 4.57675
1.16 3.27257 4.70320 2.70969 14.3176 15.3915 0 −17.3129 −4.87991 46.8554
1.17 3.67588 −7.79738 5.51210 −16.9103 −28.6622 0 −9.14523 33.7991 −62.1601
1.18 4.63491 −1.67883 13.4824 −9.70825 −7.78120 0 25.4103 −24.1815 −44.9969
1.19 5.00595 5.58183 17.0595 −9.39402 27.9423 0 45.3514 4.15682 −47.0260
1.20 5.09325 −9.34545 17.9412 15.3464 −47.5987 0 50.6330 60.3374 78.1630
See all 21 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.21
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(7\) \(-1\)
\(47\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2303.4.a.f 21
7.b odd 2 1 329.4.a.d ✓ 21
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
329.4.a.d ✓ 21 7.b odd 2 1
2303.4.a.f 21 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2303))\):

\( T_{2}^{21} - 2 T_{2}^{20} - 133 T_{2}^{19} + 249 T_{2}^{18} + 7509 T_{2}^{17} - 13002 T_{2}^{16} + \cdots - 1368784896 \) Copy content Toggle raw display
\( T_{3}^{21} + 20 T_{3}^{20} - 207 T_{3}^{19} - 6248 T_{3}^{18} + 6433 T_{3}^{17} + 797218 T_{3}^{16} + \cdots + 20915520120832 \) Copy content Toggle raw display