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Properties

Label 2303.4.a.n
Level $2303$
Weight $4$
Character orbit 2303.a
Self dual yes
Analytic conductor $135.881$
Analytic rank $0$
Dimension $68$
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: \(0\)
Dimension: \(68\)
Twist minimal: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 68 q - 2 q^{2} + 24 q^{3} + 254 q^{4} + 40 q^{5} + 48 q^{6} - 66 q^{8} + 576 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 68 q - 2 q^{2} + 24 q^{3} + 254 q^{4} + 40 q^{5} + 48 q^{6} - 66 q^{8} + 576 q^{9} + 200 q^{10} - 20 q^{11} + 288 q^{12} + 520 q^{13} + 88 q^{15} + 1062 q^{16} + 784 q^{17} - 2 q^{18} + 532 q^{19} + 400 q^{20} - 4 q^{22} - 268 q^{23} + 576 q^{24} + 1864 q^{25} + 312 q^{26} + 864 q^{27} + 200 q^{29} + 792 q^{30} + 936 q^{31} + 30 q^{32} + 2112 q^{33} + 1088 q^{34} + 2130 q^{36} - 356 q^{37} + 1192 q^{38} - 488 q^{39} + 2400 q^{40} + 1476 q^{41} - 92 q^{43} + 192 q^{44} + 1848 q^{45} - 424 q^{46} - 3196 q^{47} + 2688 q^{48} - 1338 q^{50} - 148 q^{51} + 4980 q^{52} - 80 q^{53} + 4944 q^{54} + 2200 q^{55} + 2244 q^{57} - 356 q^{58} + 560 q^{59} - 736 q^{60} + 3944 q^{61} + 1488 q^{62} + 3778 q^{64} + 2004 q^{65} - 1000 q^{66} + 2768 q^{67} + 8192 q^{68} + 2208 q^{69} - 2448 q^{71} - 5234 q^{72} + 9532 q^{73} - 2000 q^{74} + 11136 q^{75} + 6384 q^{76} - 3460 q^{78} - 1520 q^{79} + 616 q^{80} + 6976 q^{81} + 4976 q^{82} + 3320 q^{83} + 3244 q^{85} - 2892 q^{86} + 2360 q^{87} - 2868 q^{88} + 8152 q^{89} + 5400 q^{90} - 4684 q^{92} + 2840 q^{93} + 94 q^{94} - 4256 q^{95} + 5376 q^{96} + 13968 q^{97} + 2380 q^{99}+O(q^{100}) \) 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 −5.54672 −4.12273 22.7661 −1.19438 22.8676 0 −81.9032 −10.0031 6.62489
1.2 −5.42513 7.75006 21.4320 −20.7842 −42.0451 0 −72.8703 33.0634 112.757
1.3 −5.23818 −8.02981 19.4385 −12.1642 42.0616 0 −59.9169 37.4779 63.7181
1.4 −5.07646 5.02841 17.7704 1.46556 −25.5265 0 −49.5992 −1.71512 −7.43986
1.5 −4.98878 −0.891652 16.8879 −13.0126 4.44826 0 −44.3397 −26.2050 64.9169
1.6 −4.92650 2.78611 16.2704 14.4979 −13.7257 0 −40.7439 −19.2376 −71.4239
1.7 −4.92287 −8.87891 16.2346 12.9365 43.7097 0 −40.5380 51.8350 −63.6847
1.8 −4.80817 9.29565 15.1185 20.1348 −44.6951 0 −34.2269 59.4092 −96.8114
1.9 −4.44134 7.09644 11.7255 −11.0798 −31.5177 0 −16.5462 23.3594 49.2091
1.10 −4.41806 −1.05773 11.5193 10.7523 4.67309 0 −15.5483 −25.8812 −47.5044
1.11 −4.29295 5.15174 10.4294 −11.0725 −22.1161 0 −10.4294 −0.459613 47.5339
1.12 −4.23805 1.59282 9.96111 11.2954 −6.75046 0 −8.31128 −24.4629 −47.8707
1.13 −3.80432 −7.12022 6.47285 −5.10593 27.0876 0 5.80976 23.6975 19.4246
1.14 −3.74103 −10.2661 5.99534 7.67916 38.4059 0 7.49951 78.3932 −28.7280
1.15 −3.71434 3.20753 5.79632 1.58594 −11.9139 0 8.18522 −16.7118 −5.89072
1.16 −3.62360 1.56421 5.13048 −17.7675 −5.66806 0 10.3980 −24.5533 64.3823
1.17 −3.26875 −5.67634 2.68473 −2.63434 18.5546 0 17.3743 5.22089 8.61100
1.18 −3.07952 7.08333 1.48344 18.8521 −21.8133 0 20.0679 23.1736 −58.0553
1.19 −2.73998 0.0952364 −0.492507 9.66207 −0.260946 0 23.2693 −26.9909 −26.4739
1.20 −2.56494 −4.05159 −1.42107 −18.6425 10.3921 0 24.1645 −10.5846 47.8169
See all 68 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.68
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.n yes 68
7.b odd 2 1 2303.4.a.m ✓ 68
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2303.4.a.m ✓ 68 7.b odd 2 1
2303.4.a.n yes 68 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2303))\):

\( T_{2}^{68} + 2 T_{2}^{67} - 397 T_{2}^{66} - 764 T_{2}^{65} + 74903 T_{2}^{64} + 138314 T_{2}^{63} + \cdots - 26\!\cdots\!76 \) Copy content Toggle raw display
\( T_{3}^{68} - 24 T_{3}^{67} - 918 T_{3}^{66} + 25920 T_{3}^{65} + 372800 T_{3}^{64} + \cdots - 82\!\cdots\!16 \) Copy content Toggle raw display