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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 algebraic \(q\)-expansion of this newform has not been computed, 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