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Properties

Label 115.3.k.a
Level $115$
Weight $3$
Character orbit 115.k
Analytic conductor $3.134$
Analytic rank $0$
Dimension $440$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [115,3,Mod(2,115)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(115, base_ring=CyclotomicField(44))
 
chi = DirichletCharacter(H, H._module([11, 4]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("115.2");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 115 = 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 115.k (of order \(44\), degree \(20\), minimal)

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: no
Analytic conductor: \(3.13352304014\)
Analytic rank: \(0\)
Dimension: \(440\)
Relative dimension: \(22\) over \(\Q(\zeta_{44})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{44}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 440 q - 18 q^{2} - 22 q^{3} - 14 q^{5} - 44 q^{6} - 6 q^{7} - 34 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 440 q - 18 q^{2} - 22 q^{3} - 14 q^{5} - 44 q^{6} - 6 q^{7} - 34 q^{8} - 18 q^{10} - 20 q^{11} - 70 q^{12} - 26 q^{13} - 82 q^{15} + 180 q^{16} - 46 q^{17} - 286 q^{18} + 34 q^{20} - 52 q^{21} + 4 q^{22} + 22 q^{23} - 106 q^{25} - 100 q^{26} + 110 q^{27} + 302 q^{28} + 178 q^{30} - 144 q^{31} - 206 q^{32} + 288 q^{33} + 26 q^{35} - 596 q^{36} - 678 q^{37} + 30 q^{38} - 222 q^{40} + 88 q^{41} - 434 q^{42} - 150 q^{43} - 248 q^{45} - 44 q^{46} - 20 q^{47} + 566 q^{48} + 206 q^{50} + 220 q^{51} + 1238 q^{52} - 346 q^{53} + 134 q^{55} - 1316 q^{56} + 714 q^{57} - 386 q^{58} - 522 q^{60} - 692 q^{61} + 22 q^{62} + 198 q^{63} + 174 q^{65} + 156 q^{66} + 330 q^{67} + 572 q^{68} - 232 q^{70} + 508 q^{71} + 368 q^{72} - 190 q^{73} - 962 q^{75} + 900 q^{76} + 270 q^{77} - 618 q^{78} - 1884 q^{80} + 1432 q^{81} + 798 q^{82} + 122 q^{83} - 818 q^{85} + 212 q^{86} - 558 q^{87} + 462 q^{88} - 700 q^{90} - 400 q^{91} - 22 q^{92} + 372 q^{93} + 796 q^{95} + 1796 q^{96} + 480 q^{97} + 606 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1 −0.819631 − 3.76778i 1.97881 + 1.48132i −9.88588 + 4.51473i −2.38747 − 4.39318i 3.95940 − 8.66987i −9.02964 + 0.645813i 15.8703 + 21.2002i −0.814207 − 2.77293i −14.5957 + 12.5962i
2.2 −0.742529 − 3.41335i −2.46541 − 1.84558i −7.46109 + 3.40736i −1.45677 + 4.78308i −4.46898 + 9.78570i 3.45679 − 0.247234i 8.79706 + 11.7515i 0.136472 + 0.464782i 17.4080 + 1.42089i
2.3 −0.682712 − 3.13838i 3.04355 + 2.27838i −5.74478 + 2.62355i 2.33030 + 4.42376i 5.07253 − 11.1073i 3.77117 − 0.269719i 4.45675 + 5.95351i 1.53663 + 5.23328i 12.2925 − 10.3335i
2.4 −0.648158 − 2.97953i −3.88803 − 2.91054i −4.81899 + 2.20076i 3.80000 − 3.24962i −6.15200 + 13.4710i −10.1374 + 0.725043i 2.37138 + 3.16779i 4.10993 + 13.9971i −12.1453 − 9.21596i
2.5 −0.576224 − 2.64886i −0.0419222 − 0.0313826i −3.04589 + 1.39101i 3.82499 − 3.22017i −0.0589714 + 0.129129i 7.20637 − 0.515410i −1.05840 − 1.41386i −2.53482 − 8.63281i −10.7338 − 8.27631i
2.6 −0.427943 − 1.96722i 1.59934 + 1.19725i −0.0482935 + 0.0220549i −4.28609 − 2.57477i 1.67083 − 3.65861i 5.64972 − 0.404076i −4.76188 − 6.36113i −1.41111 − 4.80581i −3.23095 + 9.53353i
2.7 −0.405946 − 1.86610i −0.720853 − 0.539623i 0.320985 − 0.146589i −4.22376 + 2.67579i −0.714365 + 1.56424i −9.98418 + 0.714083i −4.98172 − 6.65480i −2.30716 − 7.85746i 6.70791 + 6.79574i
2.8 −0.302416 − 1.39018i 4.37415 + 3.27445i 1.79737 − 0.820833i 3.27585 − 3.77741i 3.22927 − 7.07112i −6.79057 + 0.485671i −5.09502 − 6.80615i 5.87560 + 20.0104i −6.24196 − 3.41169i
2.9 −0.231342 − 1.06346i −4.27240 − 3.19828i 2.56109 − 1.16961i −3.94764 − 3.06857i −2.41287 + 5.28345i 5.39412 − 0.385795i −4.44519 − 5.93808i 5.48883 + 18.6933i −2.35006 + 4.90806i
2.10 −0.224529 − 1.03214i −2.18717 − 1.63729i 2.62363 − 1.19817i 4.40443 + 2.36664i −1.19884 + 2.62508i 4.30051 − 0.307579i −4.35778 − 5.82131i −0.432621 − 1.47337i 1.45379 − 5.07737i
2.11 −0.0834986 − 0.383837i 3.27523 + 2.45181i 3.49817 − 1.59756i −2.36674 + 4.40438i 0.667617 − 1.46188i 2.67017 − 0.190974i −1.84691 − 2.46719i 2.18019 + 7.42503i 1.88818 + 0.540684i
2.12 0.0316510 + 0.145497i 0.999515 + 0.748228i 3.61836 − 1.65245i 3.77999 + 3.27288i −0.0772294 + 0.169109i −2.63890 + 0.188738i 0.711881 + 0.950962i −2.09641 − 7.13971i −0.356554 + 0.653567i
2.13 0.0705134 + 0.324145i −1.07020 − 0.801145i 3.53843 − 1.61595i −0.169980 − 4.99711i 0.184223 − 0.403393i −7.52080 + 0.537898i 1.56849 + 2.09526i −2.03209 − 6.92066i 1.60780 − 0.407461i
2.14 0.231926 + 1.06614i −2.38575 − 1.78595i 2.55565 − 1.16713i −3.57126 + 3.49944i 1.35077 − 2.95777i 9.74279 − 0.696818i 4.45249 + 5.94782i −0.0334007 − 0.113752i −4.55918 − 2.99587i
2.15 0.285049 + 1.31035i 2.22594 + 1.66632i 2.00277 − 0.914634i −2.50420 − 4.32770i −1.54896 + 3.39174i 11.1198 − 0.795306i 4.98389 + 6.65769i −0.357395 − 1.21718i 4.95698 − 4.51497i
2.16 0.364005 + 1.67330i −4.37570 − 3.27560i 0.971082 − 0.443478i 1.50484 + 4.76817i 3.88831 − 8.51420i −13.1563 + 0.940954i 5.20046 + 6.94700i 5.88153 + 20.0307i −7.43083 + 4.25370i
2.17 0.443586 + 2.03913i 1.92885 + 1.44392i −0.322766 + 0.147402i 4.93588 − 0.798182i −2.08873 + 4.57368i −4.03529 + 0.288610i 4.55860 + 6.08958i −0.900034 − 3.06523i 3.81709 + 9.71085i
2.18 0.465349 + 2.13918i 3.67180 + 2.74867i −0.720994 + 0.329267i −4.99563 − 0.209036i −4.17123 + 9.13371i −9.71128 + 0.694564i 4.20789 + 5.62109i 3.39130 + 11.5497i −1.87755 − 10.7838i
2.19 0.539471 + 2.47991i −2.86320 − 2.14337i −2.22039 + 1.01402i 4.01494 − 2.97998i 3.77074 − 8.25676i 7.43218 − 0.531560i 2.37114 + 3.16748i 1.06832 + 3.63835i 9.55602 + 8.34906i
2.20 0.666101 + 3.06202i 0.854227 + 0.639466i −5.29372 + 2.41756i −0.0448752 + 4.99980i −1.38905 + 3.04160i 2.51293 − 0.179728i −3.41710 − 4.56472i −2.21481 − 7.54294i −15.3394 + 3.19296i
See next 80 embeddings (of 440 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2.22
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
23.c even 11 1 inner
115.k odd 44 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 115.3.k.a ✓ 440
5.c odd 4 1 inner 115.3.k.a ✓ 440
23.c even 11 1 inner 115.3.k.a ✓ 440
115.k odd 44 1 inner 115.3.k.a ✓ 440
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
115.3.k.a ✓ 440 1.a even 1 1 trivial
115.3.k.a ✓ 440 5.c odd 4 1 inner
115.3.k.a ✓ 440 23.c even 11 1 inner
115.3.k.a ✓ 440 115.k odd 44 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(115, [\chi])\).