Properties

Label 308.2.n.c
Level $308$
Weight $2$
Character orbit 308.n
Analytic conductor $2.459$
Analytic rank $0$
Dimension $80$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [308,2,Mod(219,308)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(308, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 2, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("308.219");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 308 = 2^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 308.n (of order \(6\), degree \(2\), 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: \(2.45939238226\)
Analytic rank: \(0\)
Dimension: \(80\)
Relative dimension: \(40\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 80 q - 6 q^{4} - 4 q^{5} + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 80 q - 6 q^{4} - 4 q^{5} + 16 q^{9} - 18 q^{14} + 6 q^{16} + 8 q^{22} - 60 q^{25} + 24 q^{26} - 46 q^{33} - 52 q^{36} - 52 q^{37} + 16 q^{38} - 70 q^{42} + 22 q^{44} + 48 q^{45} + 148 q^{48} + 52 q^{49} - 44 q^{53} - 88 q^{56} - 36 q^{58} + 72 q^{60} - 120 q^{64} - 76 q^{66} + 40 q^{69} - 14 q^{70} + 10 q^{77} - 4 q^{78} + 62 q^{80} - 24 q^{81} + 26 q^{82} + 46 q^{86} - 32 q^{88} - 76 q^{89} - 116 q^{92} + 84 q^{93} - 152 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
219.1 −1.41411 0.0168770i 2.56217 1.47927i 1.99943 + 0.0477320i −1.86389 + 3.22836i −3.64817 + 2.04861i 2.46730 0.955225i −2.82661 0.101243i 2.87648 4.98222i 2.69024 4.53381i
219.2 −1.40099 + 0.192974i 0.581000 0.335441i 1.92552 0.540707i −0.251848 + 0.436213i −0.749242 + 0.582066i −1.57544 + 2.12556i −2.59329 + 1.12910i −1.27496 + 2.20829i 0.268657 0.659728i
219.3 −1.35934 + 0.390127i 0.349115 0.201562i 1.69560 1.06063i 1.01720 1.76184i −0.395931 + 0.410190i 2.11437 + 1.59042i −1.89112 + 2.10326i −1.41875 + 2.45734i −0.695377 + 2.79178i
219.4 −1.31677 0.515860i −0.751169 + 0.433688i 1.46778 + 1.35854i −0.818239 + 1.41723i 1.21284 0.183570i 2.64271 + 0.126788i −1.23191 2.54605i −1.12383 + 1.94653i 1.80853 1.44407i
219.5 −1.27563 + 0.610541i −0.910192 + 0.525500i 1.25448 1.55765i −1.88689 + 3.26820i 0.840232 1.22605i −1.60297 2.10487i −0.649247 + 2.75290i −0.947700 + 1.64147i 0.411618 5.32105i
219.6 −1.27184 0.618406i −1.51941 + 0.877234i 1.23515 + 1.57303i 1.94446 3.36791i 2.47494 0.176086i −1.65729 + 2.06238i −0.598141 2.76446i 0.0390796 0.0676878i −4.55578 + 3.08097i
219.7 −1.20943 + 0.732996i −2.75123 + 1.58842i 0.925433 1.77301i 0.319329 0.553095i 2.16311 3.93773i 2.28868 + 1.32739i 0.180368 + 2.82267i 3.54618 6.14217i 0.0192105 + 0.902995i
219.8 −1.17147 0.792242i 1.51941 0.877234i 0.744706 + 1.85618i 1.94446 3.36791i −2.47494 0.176086i 1.65729 2.06238i 0.598141 2.76446i 0.0390796 0.0676878i −4.94609 + 2.40493i
219.9 −1.10513 0.882428i 0.751169 0.433688i 0.442641 + 1.95040i −0.818239 + 1.41723i −1.21284 0.183570i −2.64271 0.126788i 1.23191 2.54605i −1.12383 + 1.94653i 2.15487 0.844193i
219.10 −1.09680 + 0.892770i 1.70506 0.984418i 0.405924 1.95837i 0.622247 1.07776i −0.991247 + 2.60193i 1.08267 2.41409i 1.30316 + 2.51033i 0.438159 0.758913i 0.279716 + 1.73761i
219.11 −1.06558 + 0.929809i −1.31926 + 0.761674i 0.270911 1.98157i 1.45192 2.51479i 0.697560 2.03828i −2.53420 0.760154i 1.55380 + 2.36341i −0.339706 + 0.588387i 0.791147 + 4.02971i
219.12 −0.775952 + 1.18233i 1.59924 0.923319i −0.795796 1.83486i −1.03428 + 1.79143i −0.149265 + 2.60727i −0.278241 + 2.63108i 2.78690 + 0.482873i 0.205037 0.355134i −1.31550 2.61292i
219.13 −0.721672 1.21622i −2.56217 + 1.47927i −0.958378 + 1.75542i −1.86389 + 3.22836i 3.64817 + 2.04861i −2.46730 + 0.955225i 2.82661 0.101243i 2.87648 4.98222i 5.27151 0.0629139i
219.14 −0.635949 + 1.26316i −1.59924 + 0.923319i −1.19114 1.60661i −1.03428 + 1.79143i −0.149265 2.60727i −0.278241 + 2.63108i 2.78690 0.482873i 0.205037 0.355134i −1.60511 2.44572i
219.15 −0.533373 1.30978i −0.581000 + 0.335441i −1.43103 + 1.39720i −0.251848 + 0.436213i 0.749242 + 0.582066i 1.57544 2.12556i 2.59329 + 1.12910i −1.27496 + 2.20829i 0.705670 + 0.0972001i
219.16 −0.341809 1.37229i −0.349115 + 0.201562i −1.76633 + 0.938119i 1.01720 1.76184i 0.395931 + 0.410190i −2.11437 1.59042i 1.89112 + 2.10326i −1.41875 + 2.45734i −2.76544 0.793675i
219.17 −0.272449 + 1.38772i 1.31926 0.761674i −1.85154 0.756168i 1.45192 2.51479i 0.697560 + 2.03828i −2.53420 0.760154i 1.55380 2.36341i −0.339706 + 0.588387i 3.09426 + 2.70001i
219.18 −0.224763 + 1.39624i −1.70506 + 0.984418i −1.89896 0.627646i 0.622247 1.07776i −0.991247 2.60193i 1.08267 2.41409i 1.30316 2.51033i 0.438159 0.758913i 1.36496 + 1.11105i
219.19 −0.109073 1.41000i 0.910192 0.525500i −1.97621 + 0.307586i −1.88689 + 3.26820i −0.840232 1.22605i 1.60297 + 2.10487i 0.649247 + 2.75290i −0.947700 + 1.64147i 4.81397 + 2.30405i
219.20 −0.0300795 + 1.41389i 2.75123 1.58842i −1.99819 0.0850584i 0.319329 0.553095i 2.16311 + 3.93773i 2.28868 + 1.32739i 0.180368 2.82267i 3.54618 6.14217i 0.772412 + 0.468134i
See all 80 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 219.40
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
7.c even 3 1 inner
11.b odd 2 1 inner
28.g odd 6 1 inner
44.c even 2 1 inner
77.h odd 6 1 inner
308.n even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 308.2.n.c 80
4.b odd 2 1 inner 308.2.n.c 80
7.c even 3 1 inner 308.2.n.c 80
11.b odd 2 1 inner 308.2.n.c 80
28.g odd 6 1 inner 308.2.n.c 80
44.c even 2 1 inner 308.2.n.c 80
77.h odd 6 1 inner 308.2.n.c 80
308.n even 6 1 inner 308.2.n.c 80
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
308.2.n.c 80 1.a even 1 1 trivial
308.2.n.c 80 4.b odd 2 1 inner
308.2.n.c 80 7.c even 3 1 inner
308.2.n.c 80 11.b odd 2 1 inner
308.2.n.c 80 28.g odd 6 1 inner
308.2.n.c 80 44.c even 2 1 inner
308.2.n.c 80 77.h odd 6 1 inner
308.2.n.c 80 308.n even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(308, [\chi])\):

\( T_{3}^{40} - 34 T_{3}^{38} + 689 T_{3}^{36} - 9098 T_{3}^{34} + 88474 T_{3}^{32} - 643760 T_{3}^{30} + 3651971 T_{3}^{28} - 16307078 T_{3}^{26} + 58317871 T_{3}^{24} - 166885326 T_{3}^{22} + 383836347 T_{3}^{20} + \cdots + 257049 \) Copy content Toggle raw display
\( T_{43}^{20} - 404 T_{43}^{18} + 69400 T_{43}^{16} - 6601616 T_{43}^{14} + 379126059 T_{43}^{12} - 13424006278 T_{43}^{10} + 286439448279 T_{43}^{8} - 3429281218448 T_{43}^{6} + \cdots + 1770219606016 \) Copy content Toggle raw display