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Properties

Label 153.4.r.a
Level $153$
Weight $4$
Character orbit 153.r
Analytic conductor $9.027$
Analytic rank $0$
Dimension $416$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [153,4,Mod(25,153)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(153, base_ring=CyclotomicField(24))
 
chi = DirichletCharacter(H, H._module([16, 15]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("153.25");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 153 = 3^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 153.r (of order \(24\), degree \(8\), 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: \(9.02729223088\)
Analytic rank: \(0\)
Dimension: \(416\)
Relative dimension: \(52\) over \(\Q(\zeta_{24})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{24}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 416 q - 4 q^{2} - 8 q^{3} - 4 q^{5} - 8 q^{6} - 4 q^{7} - 80 q^{8} + 52 q^{9} - 16 q^{10} + 96 q^{11} - 212 q^{12} - 4 q^{14} - 256 q^{15} + 2936 q^{16} - 16 q^{17} - 240 q^{18} - 16 q^{19} - 516 q^{20}+ \cdots - 18916 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
25.1 −1.43544 − 5.35713i 4.01610 + 3.29711i −19.7101 + 11.3797i 0.208951 + 1.58714i 11.8982 − 26.2476i 1.19470 − 9.07462i 57.8814 + 57.8814i 5.25815 + 26.4830i 8.20256 − 3.39761i
25.2 −1.33497 − 4.98219i 0.502950 − 5.17175i −16.1118 + 9.30216i 0.892751 + 6.78112i −26.4381 + 4.39836i −2.46465 + 18.7209i 38.6762 + 38.6762i −26.4941 − 5.20227i 32.5930 − 13.5005i
25.3 −1.31129 − 4.89379i −4.16115 + 3.11204i −15.3015 + 8.83432i 1.23908 + 9.41172i 20.6861 + 16.2830i −0.0959621 + 0.728904i 34.6379 + 34.6379i 7.63036 − 25.8994i 44.4342 − 18.4052i
25.4 −1.30359 − 4.86508i −4.73407 − 2.14210i −15.0415 + 8.68419i −2.04204 − 15.5108i −4.25018 + 25.8241i −2.11270 + 16.0476i 33.3654 + 33.3654i 17.8228 + 20.2817i −72.7993 + 30.1545i
25.5 −1.19391 − 4.45572i 4.29216 − 2.92871i −11.4998 + 6.63941i 0.666026 + 5.05897i −18.1739 − 15.6281i 2.16384 − 16.4360i 17.2185 + 17.2185i 9.84532 − 25.1410i 21.7462 − 9.00756i
25.6 −1.17482 − 4.38447i −1.81442 + 4.86908i −10.9152 + 6.30188i −1.91562 − 14.5506i 23.4799 + 2.23499i 0.873593 − 6.63560i 14.7765 + 14.7765i −20.4158 − 17.6691i −61.5462 + 25.4933i
25.7 −1.14940 − 4.28962i −1.17168 − 5.06233i −10.1515 + 5.86099i −1.17583 − 8.93132i −20.3687 + 10.8447i 4.32711 − 32.8677i 11.6879 + 11.6879i −24.2543 + 11.8629i −36.9605 + 15.3095i
25.8 −1.05324 − 3.93073i 3.54407 + 3.79994i −7.41316 + 4.27999i −0.637464 − 4.84202i 11.2038 − 17.9330i −4.66708 + 35.4500i 1.61133 + 1.61133i −1.87908 + 26.9345i −18.3613 + 7.60550i
25.9 −1.04078 − 3.88424i 5.17689 − 0.447034i −7.07588 + 4.08526i −2.37395 − 18.0319i −7.12438 − 19.6430i 0.696851 − 5.29311i 0.484882 + 0.484882i 26.6003 − 4.62849i −67.5695 + 27.9882i
25.10 −1.03544 − 3.86430i −4.67565 − 2.26678i −6.93247 + 4.00246i 1.68998 + 12.8367i −3.91819 + 20.4152i −0.237977 + 1.80761i 0.0139318 + 0.0139318i 16.7234 + 21.1974i 47.8549 − 19.8222i
25.11 −0.962731 − 3.59296i 1.26904 + 5.03880i −5.05431 + 2.91810i 2.48660 + 18.8876i 16.8825 − 9.41061i 2.04066 − 15.5003i −5.69126 − 5.69126i −23.7791 + 12.7888i 65.4686 − 27.1180i
25.12 −0.951683 − 3.55173i 5.11799 − 0.897897i −4.78089 + 2.76025i 2.09200 + 15.8903i −8.05979 − 17.3232i −2.44821 + 18.5960i −6.44684 − 6.44684i 25.3876 − 9.19085i 54.4472 − 22.5528i
25.13 −0.797442 − 2.97609i −5.14098 + 0.755177i −1.29302 + 0.746523i −0.128250 − 0.974157i 6.34711 + 14.6978i 3.79607 − 28.8340i −14.1764 − 14.1764i 25.8594 − 7.76471i −2.79691 + 1.15852i
25.14 −0.718328 − 2.68084i 0.600327 + 5.16136i 0.257315 − 0.148561i −0.378976 − 2.87861i 13.4055 − 5.31693i 0.717706 − 5.45152i −16.2832 − 16.2832i −26.2792 + 6.19701i −7.44485 + 3.08376i
25.15 −0.694656 − 2.59249i 1.75166 − 4.89200i 0.689737 − 0.398220i −1.45111 − 11.0223i −13.8993 − 1.14292i −2.29011 + 17.3951i −16.6942 − 16.6942i −20.8633 − 17.1383i −27.5672 + 11.4187i
25.16 −0.644442 − 2.40509i −4.50009 + 2.59792i 1.55905 − 0.900121i −0.693294 − 5.26609i 9.14828 + 9.14891i −4.07107 + 30.9228i −17.2548 − 17.2548i 13.5016 − 23.3818i −12.2186 + 5.06113i
25.17 −0.599551 − 2.23756i −2.68184 − 4.45059i 2.28101 − 1.31694i 0.917615 + 6.96998i −8.35054 + 8.66912i −1.21719 + 9.24544i −17.4183 − 17.4183i −12.6155 + 23.8715i 15.0456 − 6.23208i
25.18 −0.462394 − 1.72568i 4.51777 + 2.56705i 4.16405 − 2.40411i −1.01340 − 7.69757i 2.34091 − 8.98320i 3.21991 − 24.4576i −16.1804 − 16.1804i 13.8205 + 23.1947i −12.8149 + 5.30812i
25.19 −0.398107 − 1.48575i 1.15867 − 5.06532i 4.87923 − 2.81702i 2.31421 + 17.5782i −7.98710 + 0.295044i 3.44586 − 26.1739i −14.8290 − 14.8290i −24.3150 − 11.7380i 25.1955 − 10.4363i
25.20 −0.397941 − 1.48514i −3.33481 + 3.98486i 4.88093 − 2.81801i 2.12951 + 16.1752i 7.24511 + 3.36690i −2.44027 + 18.5357i −14.8250 − 14.8250i −4.75814 − 26.5774i 23.1750 − 9.59939i
See next 80 embeddings (of 416 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 25.52
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner
17.d even 8 1 inner
153.r even 24 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 153.4.r.a ✓ 416
9.c even 3 1 inner 153.4.r.a ✓ 416
17.d even 8 1 inner 153.4.r.a ✓ 416
153.r even 24 1 inner 153.4.r.a ✓ 416
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
153.4.r.a ✓ 416 1.a even 1 1 trivial
153.4.r.a ✓ 416 9.c even 3 1 inner
153.4.r.a ✓ 416 17.d even 8 1 inner
153.4.r.a ✓ 416 153.r even 24 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(153, [\chi])\).