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Properties

Label 115.4.g.a
Level $115$
Weight $4$
Character orbit 115.g
Analytic conductor $6.785$
Analytic rank $0$
Dimension $110$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [115,4,Mod(6,115)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(115, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([0, 18]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("115.6");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 115 = 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 115.g (of order \(11\), degree \(10\), 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: \(6.78521965066\)
Analytic rank: \(0\)
Dimension: \(110\)
Relative dimension: \(11\) over \(\Q(\zeta_{11})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 110 q - 2 q^{2} + 6 q^{3} - 40 q^{4} - 55 q^{5} - 3 q^{6} - 10 q^{7} + 243 q^{8} - 233 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 110 q - 2 q^{2} + 6 q^{3} - 40 q^{4} - 55 q^{5} - 3 q^{6} - 10 q^{7} + 243 q^{8} - 233 q^{9} - 10 q^{10} + 2 q^{11} - 57 q^{12} + 34 q^{13} - 154 q^{14} + 30 q^{15} + 16 q^{16} + 252 q^{17} - 33 q^{18} - 160 q^{19} - 200 q^{20} - 684 q^{21} - 704 q^{22} + 127 q^{23} - 3090 q^{24} - 275 q^{25} - 567 q^{26} + 384 q^{27} - 188 q^{28} + 340 q^{29} - 15 q^{30} + 582 q^{31} - 1364 q^{32} + 1760 q^{33} + 1819 q^{34} + 665 q^{35} + 2794 q^{36} + 312 q^{37} + 3123 q^{38} - 1560 q^{39} - 1205 q^{40} + 1324 q^{41} + 1454 q^{42} - 1740 q^{43} - 369 q^{44} + 3730 q^{45} - 4322 q^{46} - 2858 q^{47} - 2686 q^{48} + 2015 q^{49} - 325 q^{50} - 1426 q^{51} + 1717 q^{52} + 882 q^{53} + 2541 q^{54} + 450 q^{55} + 7755 q^{56} + 3688 q^{57} + 6622 q^{58} + 2605 q^{59} + 1530 q^{60} - 104 q^{61} - 9360 q^{62} - 7618 q^{63} + 3483 q^{64} + 170 q^{65} - 1766 q^{66} - 166 q^{67} - 5018 q^{68} - 1194 q^{69} - 770 q^{70} - 1222 q^{71} + 3303 q^{72} - 922 q^{73} + 1245 q^{74} + 150 q^{75} + 1360 q^{76} - 3093 q^{77} - 1600 q^{78} + 2448 q^{79} + 2115 q^{80} - 1371 q^{81} + 6609 q^{82} + 6704 q^{83} + 3894 q^{84} - 720 q^{85} - 5074 q^{86} - 6324 q^{87} + 1918 q^{88} - 5380 q^{89} - 165 q^{90} + 7828 q^{91} - 16225 q^{92} - 24948 q^{93} - 16490 q^{94} + 685 q^{95} + 675 q^{96} - 4400 q^{97} + 8790 q^{98} - 3626 q^{99}+O(q^{100}) \) 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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
6.1 −2.30963 + 5.05739i −3.54080 − 1.03967i −15.0039 − 17.3154i 4.20627 − 2.70320i 13.4360 − 15.5060i −0.413013 + 2.87257i 79.5474 − 23.3572i −11.2575 − 7.23474i 3.95622 + 27.5161i
6.2 −1.61506 + 3.53648i 0.822656 + 0.241553i −4.65942 − 5.37726i 4.20627 − 2.70320i −2.18289 + 2.51919i 3.70179 − 25.7465i −3.30086 + 0.969220i −22.0954 − 14.1999i 2.76647 + 19.2412i
6.3 −1.54534 + 3.38383i 4.29324 + 1.26061i −3.82332 − 4.41235i 4.20627 − 2.70320i −10.9002 + 12.5795i −1.59229 + 11.0746i −7.71551 + 2.26548i −5.87108 − 3.77311i 2.64705 + 18.4107i
6.4 −1.13426 + 2.48368i −4.84173 − 1.42166i 0.356777 + 0.411743i 4.20627 − 2.70320i 9.02270 − 10.4128i −3.26809 + 22.7301i −22.3859 + 6.57308i −1.29264 − 0.830727i 1.94290 + 13.5131i
6.5 −0.743310 + 1.62762i 9.31398 + 2.73483i 3.14224 + 3.62634i 4.20627 − 2.70320i −11.3745 + 13.1268i −1.18121 + 8.21547i −21.9727 + 6.45176i 56.5571 + 36.3470i 1.27323 + 8.85554i
6.6 −0.502951 + 1.10131i −7.38951 − 2.16976i 4.27896 + 4.93819i 4.20627 − 2.70320i 6.10614 − 7.04686i 2.63135 − 18.3014i −16.8840 + 4.95759i 27.1832 + 17.4696i 0.861517 + 5.99198i
6.7 0.274193 − 0.600398i 0.833837 + 0.244837i 4.95359 + 5.71675i 4.20627 − 2.70320i 0.375631 − 0.433502i −4.72226 + 32.8441i 9.85703 − 2.89429i −22.0785 − 14.1890i −0.469671 − 3.26663i
6.8 0.356908 − 0.781519i 5.20863 + 1.52939i 4.75550 + 5.48814i 4.20627 − 2.70320i 3.05425 − 3.52479i 3.53166 − 24.5632i 12.5812 − 3.69418i 2.07692 + 1.33475i −0.611355 − 4.25207i
6.9 0.857442 − 1.87754i −4.53505 − 1.33161i 2.44895 + 2.82624i 4.20627 − 2.70320i −6.38869 + 7.37294i 1.83914 − 12.7915i 23.2498 − 6.82676i −3.92038 − 2.51948i −1.46873 − 10.2153i
6.10 1.58073 − 3.46132i 6.46368 + 1.89791i −4.24314 − 4.89684i 4.20627 − 2.70320i 16.7866 − 19.3728i −1.14374 + 7.95486i 5.55162 − 1.63010i 15.4632 + 9.93762i −2.70767 − 18.8323i
6.11 2.10300 − 4.60492i −0.871970 − 0.256034i −11.5438 − 13.3223i 4.20627 − 2.70320i −3.01277 + 3.47692i 1.35658 − 9.43524i −46.7662 + 13.7318i −22.0191 − 14.1508i −3.60228 − 25.0544i
16.1 −3.13461 + 3.61754i −7.41459 + 4.76507i −2.12225 − 14.7606i 2.07708 + 4.54816i 6.00407 − 41.7592i 30.1381 − 8.84935i 27.8349 + 17.8884i 21.0541 − 46.1020i −22.9640 − 6.74283i
16.2 −2.78120 + 3.20967i 0.620793 − 0.398960i −1.42842 − 9.93487i 2.07708 + 4.54816i −0.446019 + 3.10213i −1.32223 + 0.388243i 7.27793 + 4.67724i −10.9900 + 24.0647i −20.3748 − 5.98259i
16.3 −2.62935 + 3.03443i 5.69405 − 3.65935i −1.15577 − 8.03858i 2.07708 + 4.54816i −3.86762 + 26.8999i −1.94859 + 0.572159i 0.409529 + 0.263188i 7.81524 − 17.1130i −19.2624 − 5.65596i
16.4 −1.69252 + 1.95327i −4.05620 + 2.60676i 0.187875 + 1.30670i 2.07708 + 4.54816i 1.77348 − 12.3348i −18.0212 + 5.29151i −20.2644 − 13.0231i −1.55865 + 3.41296i −12.3993 − 3.64075i
16.5 −0.861960 + 0.994755i −0.892278 + 0.573432i 0.891957 + 6.20369i 2.07708 + 4.54816i 0.198683 − 1.38187i 31.2385 − 9.17246i −15.7984 − 10.1530i −10.7489 + 23.5367i −6.31466 − 1.85415i
16.6 0.0310213 − 0.0358005i 1.55482 − 0.999222i 1.13820 + 7.91635i 2.07708 + 4.54816i 0.0124599 − 0.0866606i −22.1880 + 6.51498i 0.637525 + 0.409713i −9.79718 + 21.4528i 0.227260 + 0.0667296i
16.7 0.694897 − 0.801954i −7.82619 + 5.02959i 0.978270 + 6.80402i 2.07708 + 4.54816i −1.40490 + 9.77129i −10.2408 + 3.00696i 13.2778 + 8.53312i 24.7363 − 54.1649i 5.09077 + 1.49479i
16.8 0.784981 − 0.905916i 6.23621 − 4.00777i 0.934030 + 6.49632i 2.07708 + 4.54816i 1.26460 − 8.79550i 8.12768 − 2.38650i 14.6856 + 9.43786i 11.6119 − 25.4265i 5.75072 + 1.68856i
16.9 2.11818 − 2.44451i −0.807164 + 0.518733i −0.350419 − 2.43722i 2.07708 + 4.54816i −0.441670 + 3.07188i 31.6351 − 9.28890i 15.0685 + 9.68396i −10.8338 + 23.7227i 15.5176 + 4.55638i
See next 80 embeddings (of 110 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 101.11
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.c even 11 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 115.4.g.a ✓ 110
23.c even 11 1 inner 115.4.g.a ✓ 110
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
115.4.g.a ✓ 110 1.a even 1 1 trivial
115.4.g.a ✓ 110 23.c even 11 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{110} + 2 T_{2}^{109} + 66 T_{2}^{108} - 117 T_{2}^{107} + 1942 T_{2}^{106} - 12222 T_{2}^{105} + 74496 T_{2}^{104} - 439372 T_{2}^{103} + 3457423 T_{2}^{102} - 13210094 T_{2}^{101} + 120932488 T_{2}^{100} + \cdots + 13\!\cdots\!56 \) acting on \(S_{4}^{\mathrm{new}}(115, [\chi])\). Copy content Toggle raw display