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Properties

Label 2303.4.a.g
Level $2303$
Weight $4$
Character orbit 2303.a
Self dual yes
Analytic conductor $135.881$
Analytic rank $1$
Dimension $35$
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: \(35\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 35 q + 5 q^{2} - 12 q^{3} + 139 q^{4} - 20 q^{5} - 24 q^{6} + 39 q^{8} + 303 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 35 q + 5 q^{2} - 12 q^{3} + 139 q^{4} - 20 q^{5} - 24 q^{6} + 39 q^{8} + 303 q^{9} - 100 q^{10} + 40 q^{11} - 144 q^{12} - 328 q^{13} + 20 q^{15} + 643 q^{16} - 152 q^{17} + 51 q^{18} - 266 q^{19} - 1064 q^{20} - 168 q^{22} - 134 q^{23} - 288 q^{24} + 1137 q^{25} - 156 q^{26} - 672 q^{27} + 248 q^{29} - 216 q^{30} - 276 q^{31} + 347 q^{32} - 1056 q^{33} - 908 q^{34} + 909 q^{36} - 418 q^{37} - 164 q^{38} - 548 q^{39} - 1200 q^{40} - 918 q^{41} + 608 q^{43} + 1288 q^{44} - 876 q^{45} - 972 q^{46} - 1645 q^{47} - 1252 q^{48} - 367 q^{50} - 464 q^{51} - 3798 q^{52} - 218 q^{53} + 744 q^{54} - 1004 q^{55} - 436 q^{57} - 1270 q^{58} - 3760 q^{59} - 424 q^{60} - 956 q^{61} - 84 q^{62} + 2189 q^{64} - 596 q^{65} - 5500 q^{66} - 476 q^{67} - 256 q^{68} - 444 q^{69} + 852 q^{71} - 883 q^{72} - 6250 q^{73} + 1366 q^{74} + 2568 q^{75} - 1742 q^{76} - 1460 q^{78} + 632 q^{79} - 10124 q^{80} + 1267 q^{81} - 792 q^{82} - 796 q^{83} - 1228 q^{85} - 2864 q^{86} - 8360 q^{87} - 50 q^{88} - 908 q^{89} + 1858 q^{90} + 1696 q^{92} + 644 q^{93} - 235 q^{94} + 1320 q^{95} - 2688 q^{96} - 6184 q^{97} - 1812 q^{99}+O(q^{100}) \) 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 −5.48152 2.93802 22.0470 −19.1722 −16.1048 0 −76.9989 −18.3680 105.093
1.2 −5.33525 −8.84301 20.4649 −3.71171 47.1797 0 −66.5034 51.1989 19.8029
1.3 −4.92815 4.29114 16.2866 −2.55035 −21.1474 0 −40.8377 −8.58611 12.5685
1.4 −4.60891 5.06004 13.2421 19.1036 −23.3213 0 −24.1603 −1.39599 −88.0469
1.5 −4.14945 −8.07223 9.21794 −8.04659 33.4953 0 −5.05376 38.1609 33.3889
1.6 −3.90337 −4.20489 7.23626 15.3320 16.4132 0 2.98114 −9.31892 −59.8464
1.7 −3.81980 8.81241 6.59091 −14.5625 −33.6617 0 5.38246 50.6586 55.6259
1.8 −3.77029 −0.853932 6.21511 1.45940 3.21958 0 6.72955 −26.2708 −5.50238
1.9 −3.31842 4.09865 3.01190 16.7525 −13.6010 0 16.5526 −10.2011 −55.5919
1.10 −2.62935 −5.20657 −1.08651 5.47417 13.6899 0 23.8916 0.108399 −14.3935
1.11 −2.23222 −1.75933 −3.01718 −8.98350 3.92721 0 24.5928 −23.9048 20.0532
1.12 −2.07002 −9.29307 −3.71503 −12.3025 19.2368 0 24.2503 59.3612 25.4663
1.13 −1.67304 8.29020 −5.20094 −3.53349 −13.8698 0 22.0857 41.7274 5.91167
1.14 −1.17274 −3.53925 −6.62469 19.8836 4.15061 0 17.1509 −14.4737 −23.3182
1.15 −0.936762 −3.03943 −7.12248 −14.9851 2.84722 0 14.1662 −17.7619 14.0375
1.16 −0.479509 5.23227 −7.77007 −17.2964 −2.50892 0 7.56190 0.376695 8.29377
1.17 −0.230045 5.72475 −7.94708 2.70751 −1.31695 0 3.66854 5.77272 −0.622850
1.18 0.482933 −0.143091 −7.76678 10.0508 −0.0691036 0 −7.61430 −26.9795 4.85389
1.19 0.529428 9.49502 −7.71971 20.1615 5.02693 0 −8.32246 63.1555 10.6741
1.20 1.11403 −0.352262 −6.75894 −4.08963 −0.392430 0 −16.4419 −26.8759 −4.55597
See all 35 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.35
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.g ✓ 35
7.b odd 2 1 2303.4.a.h yes 35
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2303.4.a.g ✓ 35 1.a even 1 1 trivial
2303.4.a.h yes 35 7.b odd 2 1

Hecke kernels

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

\( T_{2}^{35} - 5 T_{2}^{34} - 197 T_{2}^{33} + 987 T_{2}^{32} + 17436 T_{2}^{31} - 87638 T_{2}^{30} + \cdots + 18089615294464 \) Copy content Toggle raw display
\( T_{3}^{35} + 12 T_{3}^{34} - 552 T_{3}^{33} - 6760 T_{3}^{32} + 136694 T_{3}^{31} + \cdots + 63\!\cdots\!56 \) Copy content Toggle raw display