↑

Properties

Label 213.4.f.b
Level $213$
Weight $4$
Character orbit 213.f
Analytic conductor $12.567$
Analytic rank $0$
Dimension $108$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [213,4,Mod(37,213)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(213, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([0, 4]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("213.37");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 213 = 3 \cdot 71 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 213.f (of order \(7\), degree \(6\), 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: \(12.5674068312\)
Analytic rank: \(0\)
Dimension: \(108\)
Relative dimension: \(18\) over \(\Q(\zeta_{7})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{7}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 108 q + 2 q^{2} + 54 q^{3} - 72 q^{4} - 4 q^{5} - 6 q^{6} + 24 q^{8} - 162 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 108 q + 2 q^{2} + 54 q^{3} - 72 q^{4} - 4 q^{5} - 6 q^{6} + 24 q^{8} - 162 q^{9} + 97 q^{10} - 26 q^{11} + 153 q^{12} - 2 q^{13} + 114 q^{14} - 30 q^{15} + 316 q^{16} + 388 q^{17} - 45 q^{18} + 124 q^{19} + 16 q^{20} - 40 q^{22} - 314 q^{23} + 537 q^{24} + 2680 q^{25} - 360 q^{26} + 486 q^{27} - 428 q^{28} - 1402 q^{29} - 270 q^{30} - 510 q^{31} - 2375 q^{32} + 414 q^{33} - 514 q^{34} - 824 q^{35} - 774 q^{36} + 338 q^{37} - 194 q^{38} + 6 q^{39} + 3208 q^{40} + 96 q^{41} + 1191 q^{42} + 772 q^{43} + 1216 q^{44} - 36 q^{45} - 3292 q^{46} + 336 q^{47} + 2265 q^{48} - 2122 q^{49} - 1082 q^{50} + 600 q^{51} + 1443 q^{52} - 686 q^{53} - 54 q^{54} + 5432 q^{55} - 5829 q^{56} + 2484 q^{57} + 2715 q^{58} - 1032 q^{59} - 1623 q^{60} - 338 q^{61} + 1137 q^{62} - 6796 q^{64} + 2682 q^{65} - 2694 q^{66} - 1660 q^{67} - 5313 q^{68} + 942 q^{69} + 1088 q^{70} - 2462 q^{71} + 216 q^{72} - 2980 q^{73} + 1096 q^{74} + 2250 q^{75} - 6434 q^{76} - 864 q^{77} + 660 q^{78} + 1398 q^{79} + 1530 q^{80} - 1458 q^{81} + 2031 q^{82} + 1508 q^{83} + 843 q^{84} + 13984 q^{85} + 4867 q^{86} - 414 q^{87} + 6096 q^{88} + 2320 q^{89} - 1080 q^{90} + 6814 q^{91} - 4136 q^{92} - 2418 q^{93} - 280 q^{94} + 5416 q^{95} - 2262 q^{96} - 10372 q^{97} + 13162 q^{98} - 1242 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1 −3.10758 + 3.89678i 0.667563 + 2.92478i −3.74769 − 16.4197i 0.0616071 −13.4718 − 6.48765i 12.3080 + 15.4338i 39.7056 + 19.1212i −8.10872 + 3.90495i −0.191449 + 0.240069i
37.2 −2.85364 + 3.57835i 0.667563 + 2.92478i −2.88116 − 12.6232i 2.20373 −12.3709 − 5.95750i 8.04225 + 10.0847i 20.4029 + 9.82553i −8.10872 + 3.90495i −6.28864 + 7.88571i
37.3 −2.56024 + 3.21044i 0.667563 + 2.92478i −1.97193 − 8.63961i −20.5443 −11.0990 − 5.34498i −0.456310 − 0.572194i 3.18833 + 1.53542i −8.10872 + 3.90495i 52.5984 − 65.9563i
37.4 −2.52318 + 3.16397i 0.667563 + 2.92478i −1.86409 − 8.16711i 20.9303 −10.9383 − 5.26761i −8.62397 − 10.8141i 1.37510 + 0.662214i −8.10872 + 3.90495i −52.8110 + 66.2229i
37.5 −2.06543 + 2.58996i 0.667563 + 2.92478i −0.661757 − 2.89935i −9.05144 −8.95389 − 4.31197i −14.4945 − 18.1756i −15.0010 − 7.22411i −8.10872 + 3.90495i 18.6951 − 23.4429i
37.6 −1.34660 + 1.68858i 0.667563 + 2.92478i 0.742187 + 3.25174i 10.3174 −5.83768 − 2.81128i −6.93330 − 8.69409i −22.0574 − 10.6223i −8.10872 + 3.90495i −13.8933 + 17.4217i
37.7 −1.16648 + 1.46272i 0.667563 + 2.92478i 1.00129 + 4.38695i 4.13549 −5.05684 − 2.43525i 18.2855 + 22.9292i −21.0698 − 10.1467i −8.10872 + 3.90495i −4.82397 + 6.04907i
37.8 −1.03897 + 1.30283i 0.667563 + 2.92478i 1.16226 + 5.09221i −10.0268 −4.50408 − 2.16905i −5.24248 − 6.57386i −19.8527 − 9.56057i −8.10872 + 3.90495i 10.4175 − 13.0632i
37.9 0.104264 − 0.130743i 0.667563 + 2.92478i 1.77394 + 7.77216i −15.2787 0.451999 + 0.217671i 9.15925 + 11.4853i 2.40645 + 1.15888i −8.10872 + 3.90495i −1.59302 + 1.99758i
37.10 0.468283 − 0.587209i 0.667563 + 2.92478i 1.65464 + 7.24946i 19.1055 2.03007 + 0.977629i 15.9942 + 20.0561i 10.4453 + 5.03019i −8.10872 + 3.90495i 8.94677 − 11.2189i
37.11 0.545817 − 0.684433i 0.667563 + 2.92478i 1.60964 + 7.05227i −3.78743 2.36618 + 1.13949i −21.9001 − 27.4619i 12.0152 + 5.78621i −8.10872 + 3.90495i −2.06724 + 2.59224i
37.12 0.854804 − 1.07189i 0.667563 + 2.92478i 1.36191 + 5.96691i 6.98840 3.70568 + 1.78456i −5.35745 − 6.71803i 17.4419 + 8.39956i −8.10872 + 3.90495i 5.97371 − 7.49080i
37.13 1.29020 − 1.61785i 0.667563 + 2.92478i 0.827321 + 3.62473i 0.174921 5.59316 + 2.69352i 9.27712 + 11.6331i 21.8468 + 10.5209i −8.10872 + 3.90495i 0.225682 − 0.282996i
37.14 2.43864 − 3.05795i 0.667563 + 2.92478i −1.62396 − 7.11504i 17.9643 10.5718 + 5.09111i 1.39063 + 1.74380i 2.47375 + 1.19129i −8.10872 + 3.90495i 43.8083 − 54.9339i
37.15 2.48014 − 3.11000i 0.667563 + 2.92478i −1.74083 − 7.62708i −14.5164 10.7517 + 5.17776i −4.25394 − 5.33427i 0.633549 + 0.305101i −8.10872 + 3.90495i −36.0028 + 45.1461i
37.16 2.67100 − 3.34933i 0.667563 + 2.92478i −2.30360 − 10.0927i 6.90652 11.5791 + 5.57622i −19.7034 − 24.7073i −9.07918 − 4.37230i −8.10872 + 3.90495i 18.4473 − 23.1322i
37.17 2.72692 − 3.41945i 0.667563 + 2.92478i −2.47638 − 10.8497i −4.82267 11.8215 + 5.69296i 19.7356 + 24.7476i −12.3288 − 5.93726i −8.10872 + 3.90495i −13.1510 + 16.4909i
37.18 3.35953 − 4.21272i 0.667563 + 2.92478i −4.68039 − 20.5061i −11.6505 14.5640 + 7.01365i −7.22701 − 9.06238i −63.2733 − 30.4708i −8.10872 + 3.90495i −39.1402 + 49.0803i
91.1 −1.22258 − 5.35648i 2.70291 − 1.30165i −19.9894 + 9.62639i 17.0967 −10.2768 − 12.8867i −6.59915 + 28.9128i 48.5975 + 60.9393i 5.61141 − 7.03648i −20.9021 − 91.5783i
91.2 −1.17759 − 5.15936i 2.70291 − 1.30165i −18.0246 + 8.68018i −16.7890 −9.89861 − 12.4125i 2.13960 − 9.37420i 39.6135 + 49.6737i 5.61141 − 7.03648i 19.7705 + 86.6204i
See next 80 embeddings (of 108 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 37.18
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
71.d even 7 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 213.4.f.b ✓ 108
71.d even 7 1 inner 213.4.f.b ✓ 108
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
213.4.f.b ✓ 108 1.a even 1 1 trivial
213.4.f.b ✓ 108 71.d even 7 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{108} - 2 T_{2}^{107} + 110 T_{2}^{106} - 236 T_{2}^{105} + 6889 T_{2}^{104} + \cdots + 18\!\cdots\!36 \) acting on \(S_{4}^{\mathrm{new}}(213, [\chi])\). Copy content Toggle raw display