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Properties

Label 35.10.k.a
Level $35$
Weight $10$
Character orbit 35.k
Analytic conductor $18.026$
Analytic rank $0$
Dimension $136$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [35,10,Mod(3,35)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(35, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([9, 2]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("35.3");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 35 = 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 35.k (of order \(12\), degree \(4\), 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: \(18.0262542657\)
Analytic rank: \(0\)
Dimension: \(136\)
Relative dimension: \(34\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 136 q - 2 q^{2} - 6 q^{3} - 1710 q^{5} + 5940 q^{7} - 3484 q^{8} + 153786 q^{10} - 87752 q^{11} + 3066 q^{12} - 318872 q^{15} + 3132804 q^{16} - 1578678 q^{17} - 239036 q^{18} - 5745396 q^{21} + 2132088 q^{22}+ \cdots + 5117393178 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1 −42.4707 − 11.3800i 54.8950 + 204.871i 1230.85 + 710.632i −1352.09 + 353.508i − 9325.71i −5765.26 + 2667.46i −28269.7 − 28269.7i −21912.6 + 12651.2i 61447.3 + 373.073i
3.2 −39.0059 − 10.4516i −2.23944 − 8.35771i 968.820 + 559.349i 42.9502 − 1396.88i 349.406i 6351.67 − 99.7449i −17323.8 − 17323.8i 16981.1 − 9804.07i −16275.0 + 54037.8i
3.3 −38.4135 − 10.2929i −42.0355 − 156.879i 926.246 + 534.768i −1180.00 + 748.818i 6458.92i 2179.94 − 5966.70i −15678.3 − 15678.3i −5797.93 + 3347.44i 53035.3 − 16619.1i
3.4 −38.3290 − 10.2702i −32.9968 − 123.146i 920.231 + 531.296i 1155.42 + 786.218i 5058.95i −1359.24 + 6205.33i −15448.9 − 15448.9i 2969.87 − 1714.65i −36211.3 − 42001.4i
3.5 −33.5878 − 8.99984i 42.7497 + 159.544i 603.741 + 348.570i 1387.01 + 171.250i − 5743.49i −142.293 − 6350.86i −4552.24 − 4552.24i −6580.85 + 3799.45i −45045.5 − 18234.8i
3.6 −30.1591 − 8.08111i −1.23807 − 4.62053i 400.862 + 231.438i −158.539 − 1388.52i 149.356i −6321.02 + 631.073i 1084.56 + 1084.56i 17026.2 − 9830.06i −6439.39 + 43157.7i
3.7 −27.8720 − 7.46827i 30.2294 + 112.818i 277.667 + 160.311i 211.406 + 1381.46i − 3370.21i 3847.43 + 5054.79i 3904.82 + 3904.82i 5231.97 − 3020.68i 4424.83 − 40082.9i
3.8 −26.6836 − 7.14985i −65.2186 − 243.399i 217.489 + 125.567i 1313.38 − 477.656i 6961.07i −2565.03 − 5811.56i 5095.68 + 5095.68i −37943.7 + 21906.8i −38460.9 + 3355.11i
3.9 −23.4249 − 6.27668i 42.5378 + 158.753i 65.9235 + 38.0610i −1392.00 − 124.303i − 3985.77i 5750.30 − 2699.57i 7474.53 + 7474.53i −6347.10 + 3664.50i 31827.3 + 11649.0i
3.10 −22.3519 − 5.98917i −10.0261 − 37.4181i 20.3322 + 11.7388i −924.440 + 1048.11i 896.413i −6331.22 − 518.872i 7993.56 + 7993.56i 15746.4 − 9091.18i 26940.3 − 17890.6i
3.11 −21.6149 − 5.79170i −55.2156 − 206.067i −9.74350 − 5.62541i −1252.88 − 619.202i 4773.93i 2224.73 + 5950.14i 8279.52 + 8279.52i −22369.0 + 12914.8i 23494.7 + 20640.3i
3.12 −17.7903 − 4.76691i 63.8550 + 238.310i −149.632 − 86.3901i 439.946 − 1326.49i − 4544.01i −516.855 + 6331.39i 8918.19 + 8918.19i −35668.3 + 20593.1i −14150.1 + 21501.5i
3.13 −12.1913 − 3.26666i −24.7239 − 92.2709i −305.447 − 176.350i 348.098 + 1353.50i 1205.67i 4216.39 − 4751.39i 7717.17 + 7717.17i 9143.33 − 5278.91i 177.633 − 17638.0i
3.14 −10.2381 − 2.74328i −21.6326 − 80.7338i −346.113 − 199.828i 1190.45 − 732.094i 885.901i 5772.19 + 2652.45i 6832.66 + 6832.66i 10996.0 − 6348.54i −14196.2 + 4229.50i
3.15 −6.27232 − 1.68066i 15.1291 + 56.4624i −406.888 − 234.917i −873.884 − 1090.62i − 379.577i 259.777 − 6347.14i 4508.24 + 4508.24i 14086.9 − 8133.05i 3648.32 + 8309.42i
3.16 −6.05635 − 1.62279i 21.6603 + 80.8372i −409.359 − 236.344i 1391.85 + 125.989i − 524.728i −6352.13 + 63.6305i 4365.66 + 4365.66i 10980.5 − 6339.59i −8225.08 − 3021.72i
3.17 −0.878798 − 0.235473i 69.2202 + 258.333i −442.688 − 255.586i −379.993 + 1344.89i − 243.323i −3345.20 − 5400.30i 658.232 + 658.232i −44898.7 + 25922.3i 650.623 − 1092.41i
3.18 2.78516 + 0.746280i −58.8948 − 219.798i −436.205 − 251.843i 332.404 + 1357.44i − 656.125i −4609.57 + 4370.98i −2070.86 − 2070.86i −27796.8 + 16048.5i −87.2304 + 4028.74i
3.19 3.77794 + 1.01230i 19.1492 + 71.4659i −430.157 − 248.351i −1392.86 + 114.359i 289.378i −868.604 + 6292.78i −2789.71 − 2789.71i 12305.3 − 7104.46i −5377.89 − 977.939i
3.20 3.83230 + 1.02686i −48.4413 − 180.785i −429.773 − 248.130i −576.593 − 1273.05i − 742.566i −2972.03 − 5614.33i −2828.61 − 2828.61i −13290.8 + 7673.46i −902.428 − 5470.80i
See next 80 embeddings (of 136 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.34
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
7.d odd 6 1 inner
35.k even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 35.10.k.a ✓ 136
5.c odd 4 1 inner 35.10.k.a ✓ 136
7.d odd 6 1 inner 35.10.k.a ✓ 136
35.k even 12 1 inner 35.10.k.a ✓ 136
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.10.k.a ✓ 136 1.a even 1 1 trivial
35.10.k.a ✓ 136 5.c odd 4 1 inner
35.10.k.a ✓ 136 7.d odd 6 1 inner
35.10.k.a ✓ 136 35.k even 12 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{10}^{\mathrm{new}}(35, [\chi])\).