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Properties

Label 2303.4.a.e
Level $2303$
Weight $4$
Character orbit 2303.a
Self dual yes
Analytic conductor $135.881$
Analytic rank $0$
Dimension $20$
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: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 3 x^{19} - 130 x^{18} + 379 x^{17} + 6970 x^{16} - 19652 x^{15} - 199330 x^{14} + \cdots - 19267584 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 329)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{6} q^{3} + (\beta_{2} + 5) q^{4} + ( - \beta_{8} + \beta_{6}) q^{5} - \beta_{3} q^{6} + ( - \beta_{15} + \beta_{14} - \beta_{7} + \cdots - 1) q^{8}+ \cdots + ( - \beta_{19} + \beta_{17} + \cdots + 13) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + \beta_{6} q^{3} + (\beta_{2} + 5) q^{4} + ( - \beta_{8} + \beta_{6}) q^{5} - \beta_{3} q^{6} + ( - \beta_{15} + \beta_{14} - \beta_{7} + \cdots - 1) q^{8}+ \cdots + (8 \beta_{19} + 12 \beta_{18} + \cdots + 321) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 3 q^{2} - 2 q^{3} + 109 q^{4} + 4 q^{5} - q^{6} + 12 q^{8} + 278 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 3 q^{2} - 2 q^{3} + 109 q^{4} + 4 q^{5} - q^{6} + 12 q^{8} + 278 q^{9} - 68 q^{10} + 62 q^{11} + 6 q^{13} + 420 q^{15} + 957 q^{16} + 138 q^{17} + 120 q^{18} - 182 q^{19} - 787 q^{20} + 521 q^{22} + 326 q^{23} + 18 q^{24} + 678 q^{25} - 118 q^{26} - 68 q^{27} + 528 q^{29} + 877 q^{30} - 1092 q^{31} + 254 q^{32} + 424 q^{33} - 234 q^{34} + 2608 q^{36} + 904 q^{37} + 166 q^{38} + 606 q^{39} - 323 q^{40} + 8 q^{41} - 18 q^{43} + 98 q^{44} - 316 q^{45} + 1382 q^{46} + 940 q^{47} - 937 q^{48} + 892 q^{50} + 756 q^{51} - 761 q^{52} + 992 q^{53} - 384 q^{54} - 1060 q^{55} + 1554 q^{57} - 938 q^{58} - 2140 q^{59} + 5830 q^{60} + 186 q^{61} - 69 q^{62} + 7086 q^{64} + 662 q^{65} - 3617 q^{66} + 2852 q^{67} - 1163 q^{68} - 628 q^{69} + 3448 q^{71} - 355 q^{72} + 304 q^{73} + 1512 q^{74} - 4930 q^{75} - 339 q^{76} + 1054 q^{78} + 4248 q^{79} - 5945 q^{80} + 5696 q^{81} - 3665 q^{82} + 274 q^{83} + 232 q^{85} - 5730 q^{86} - 3756 q^{87} + 1976 q^{88} + 3168 q^{89} + 13306 q^{90} - 4214 q^{92} - 374 q^{93} + 141 q^{94} + 6100 q^{95} + 11433 q^{96} + 4112 q^{97} + 7252 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 3 x^{19} - 130 x^{18} + 379 x^{17} + 6970 x^{16} - 19652 x^{15} - 199330 x^{14} + \cdots - 19267584 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 13 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 10\!\cdots\!41 \nu^{19} + \cdots + 96\!\cdots\!68 ) / 26\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 76\!\cdots\!36 \nu^{19} + \cdots - 14\!\cdots\!92 ) / 67\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 88\!\cdots\!87 \nu^{19} + \cdots - 13\!\cdots\!84 ) / 53\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 99\!\cdots\!79 \nu^{19} + \cdots - 61\!\cdots\!08 ) / 53\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 86\!\cdots\!33 \nu^{19} + \cdots + 70\!\cdots\!36 ) / 26\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 14\!\cdots\!32 \nu^{19} + \cdots + 10\!\cdots\!44 ) / 29\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 98\!\cdots\!91 \nu^{19} + \cdots + 83\!\cdots\!32 ) / 11\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 18\!\cdots\!87 \nu^{19} + \cdots - 57\!\cdots\!24 ) / 17\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 17\!\cdots\!73 \nu^{19} + \cdots + 66\!\cdots\!36 ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 45\!\cdots\!43 \nu^{19} + \cdots + 91\!\cdots\!56 ) / 26\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 10\!\cdots\!67 \nu^{19} + \cdots + 52\!\cdots\!44 ) / 53\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 62\!\cdots\!89 \nu^{19} + \cdots - 29\!\cdots\!68 ) / 26\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 35\!\cdots\!11 \nu^{19} + \cdots - 20\!\cdots\!52 ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 55\!\cdots\!29 \nu^{19} + \cdots + 19\!\cdots\!08 ) / 17\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 17\!\cdots\!09 \nu^{19} + \cdots - 49\!\cdots\!68 ) / 53\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 69\!\cdots\!89 \nu^{19} + \cdots - 29\!\cdots\!48 ) / 17\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 24\!\cdots\!21 \nu^{19} + \cdots - 97\!\cdots\!92 ) / 53\!\cdots\!00 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 13 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{15} + \beta_{14} - \beta_{7} + \beta_{2} + 22\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3 \beta_{19} - \beta_{17} + \beta_{16} - \beta_{15} + 3 \beta_{13} - 2 \beta_{12} - \beta_{11} + \cdots + 292 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 3 \beta_{19} - 4 \beta_{18} - 3 \beta_{17} - 38 \beta_{15} + 41 \beta_{14} - \beta_{12} - 2 \beta_{11} + \cdots - 26 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 130 \beta_{19} - 6 \beta_{18} - 42 \beta_{17} + 37 \beta_{16} - 49 \beta_{15} + 9 \beta_{14} + \cdots + 7534 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 187 \beta_{19} - 198 \beta_{18} - 175 \beta_{17} - 23 \beta_{16} - 1259 \beta_{15} + 1340 \beta_{14} + \cdots - 685 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 4517 \beta_{19} - 352 \beta_{18} - 1433 \beta_{17} + 1092 \beta_{16} - 1897 \beta_{15} + 614 \beta_{14} + \cdots + 207528 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 8223 \beta_{19} - 7322 \beta_{18} - 7305 \beta_{17} - 1594 \beta_{16} - 40235 \beta_{15} + 41312 \beta_{14} + \cdots - 12429 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 146401 \beta_{19} - 15382 \beta_{18} - 45859 \beta_{17} + 30216 \beta_{16} - 66846 \beta_{15} + \cdots + 5910613 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 316214 \beta_{19} - 245622 \beta_{18} - 269526 \beta_{17} - 75199 \beta_{16} - 1265950 \beta_{15} + \cdots + 46856 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 4610702 \beta_{19} - 599126 \beta_{18} - 1426560 \beta_{17} + 826178 \beta_{16} - 2241283 \beta_{15} + \cdots + 171473240 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 11372941 \beta_{19} - 7908948 \beta_{18} - 9378771 \beta_{17} - 3019253 \beta_{16} - 39476126 \beta_{15} + \cdots + 18290058 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 143258798 \beta_{19} - 21982892 \beta_{18} - 43691584 \beta_{17} + 22819315 \beta_{16} + \cdots + 5030911083 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 393567770 \beta_{19} - 249859254 \beta_{18} - 315945432 \beta_{17} - 111368148 \beta_{16} + \cdots + 1149790754 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 4420870136 \beta_{19} - 778465760 \beta_{18} - 1325275832 \beta_{17} + 642204580 \beta_{16} + \cdots + 148725185621 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 13290977168 \beta_{19} - 7820846016 \beta_{18} - 10431186776 \beta_{17} - 3908841820 \beta_{16} + \cdots + 54520497339 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 135943616935 \beta_{19} - 26940052240 \beta_{18} - 39944979269 \beta_{17} + 18447270797 \beta_{16} + \cdots + 4420970154204 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 441475876563 \beta_{19} - 243719273644 \beta_{18} - 339729478019 \beta_{17} - 132985289740 \beta_{16} + \cdots + 2277552758406 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−5.46932
−5.36219
−5.19626
−3.72716
−3.29728
−3.14322
−2.66476
−1.08798
−0.261595
0.336166
0.555944
0.748140
1.36355
2.69602
3.49887
3.59392
4.56788
4.72446
5.54126
5.58355
−5.46932 −8.72896 21.9135 −16.5083 47.7415 0 −76.0972 49.1948 90.2894
1.2 −5.36219 −0.407439 20.7531 6.41758 2.18477 0 −68.3843 −26.8340 −34.4123
1.3 −5.19626 9.33960 19.0011 −0.446094 −48.5309 0 −57.1644 60.2280 2.31802
1.4 −3.72716 2.16417 5.89176 2.35387 −8.06620 0 7.85777 −22.3164 −8.77327
1.5 −3.29728 −4.34822 2.87207 −10.4624 14.3373 0 16.9082 −8.09301 34.4975
1.6 −3.14322 −1.23602 1.87985 20.8634 3.88510 0 19.2370 −25.4722 −65.5783
1.7 −2.66476 8.97221 −0.899055 −6.43898 −23.9088 0 23.7138 53.5005 17.1583
1.8 −1.08798 −9.47147 −6.81631 17.6627 10.3047 0 16.1198 62.7087 −19.2166
1.9 −0.261595 4.65342 −7.93157 −5.71533 −1.21731 0 4.16762 −5.34571 1.49510
1.10 0.336166 −8.97492 −7.88699 −16.7104 −3.01706 0 −5.34067 53.5491 −5.61748
1.11 0.555944 −2.26906 −7.69093 10.3131 −1.26147 0 −8.72328 −21.8514 5.73353
1.12 0.748140 0.325864 −7.44029 13.4351 0.243792 0 −11.5515 −26.8938 10.0514
1.13 1.36355 6.05426 −6.14074 14.3754 8.25525 0 −19.2815 9.65404 19.6015
1.14 2.69602 5.19857 −0.731458 −5.54179 14.0155 0 −23.5402 0.0251370 −14.9408
1.15 3.49887 −2.39451 4.24210 −20.8760 −8.37807 0 −13.1484 −21.2663 −73.0424
1.16 3.59392 8.69140 4.91626 20.1949 31.2362 0 −11.0827 48.5405 72.5789
1.17 4.56788 −7.10606 12.8656 −4.53739 −32.4597 0 22.2254 23.4961 −20.7263
1.18 4.72446 −9.01401 14.3205 −1.14435 −42.5863 0 29.8609 54.2523 −5.40644
1.19 5.54126 8.44592 22.7056 2.63242 46.8010 0 81.4873 44.3336 14.5869
1.20 5.58355 −1.89474 23.1761 −15.8675 −10.5794 0 84.7365 −23.4100 −88.5968
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.20
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.e 20
7.b odd 2 1 329.4.a.c ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
329.4.a.c ✓ 20 7.b odd 2 1
2303.4.a.e 20 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2303))\):

\( T_{2}^{20} - 3 T_{2}^{19} - 130 T_{2}^{18} + 379 T_{2}^{17} + 6970 T_{2}^{16} - 19652 T_{2}^{15} + \cdots - 19267584 \) Copy content Toggle raw display
\( T_{3}^{20} + 2 T_{3}^{19} - 407 T_{3}^{18} - 758 T_{3}^{17} + 68597 T_{3}^{16} + 119880 T_{3}^{15} + \cdots - 680736039136 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} - 3 T^{19} + \cdots - 19267584 \) Copy content Toggle raw display
$3$ \( T^{20} + \cdots - 680736039136 \) Copy content Toggle raw display
$5$ \( T^{20} + \cdots - 26\!\cdots\!92 \) Copy content Toggle raw display
$7$ \( T^{20} \) Copy content Toggle raw display
$11$ \( T^{20} + \cdots + 46\!\cdots\!48 \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots + 13\!\cdots\!56 \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 16\!\cdots\!08 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots - 22\!\cdots\!40 \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots - 90\!\cdots\!48 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots - 23\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots - 59\!\cdots\!32 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 37\!\cdots\!68 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots - 80\!\cdots\!56 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 22\!\cdots\!12 \) Copy content Toggle raw display
$47$ \( (T - 47)^{20} \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots - 61\!\cdots\!48 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots - 16\!\cdots\!60 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 39\!\cdots\!52 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots - 46\!\cdots\!48 \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots - 27\!\cdots\!72 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots - 10\!\cdots\!08 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 23\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots - 19\!\cdots\!16 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 22\!\cdots\!40 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots - 11\!\cdots\!96 \) Copy content Toggle raw display
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