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Properties

Label 145.4.k.b
Level $145$
Weight $4$
Character orbit 145.k
Analytic conductor $8.555$
Analytic rank $0$
Dimension $96$
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] = [145,4,Mod(16,145)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(145, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([0, 2]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("145.16");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 145 = 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 145.k (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: \(8.55527695083\)
Analytic rank: \(0\)
Dimension: \(96\)
Relative dimension: \(16\) over \(\Q(\zeta_{7})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{7}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 96 q + 2 q^{2} + 4 q^{3} - 82 q^{4} + 80 q^{5} - 62 q^{6} - 18 q^{7} - 3 q^{8} - 94 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 96 q + 2 q^{2} + 4 q^{3} - 82 q^{4} + 80 q^{5} - 62 q^{6} - 18 q^{7} - 3 q^{8} - 94 q^{9} + 25 q^{10} + 90 q^{11} - 14 q^{12} + 38 q^{13} + 232 q^{14} - 20 q^{15} - 90 q^{16} + 524 q^{17} - 314 q^{18} - 36 q^{19} + 235 q^{20} + 730 q^{21} + 273 q^{22} - 278 q^{23} - 1784 q^{24} - 400 q^{25} + 328 q^{26} + 364 q^{27} + 800 q^{28} - 88 q^{29} - 670 q^{30} + 98 q^{31} - 1858 q^{32} + 1450 q^{33} + 182 q^{34} + 90 q^{35} - 3432 q^{36} + 52 q^{37} + 512 q^{38} + 2532 q^{39} + 15 q^{40} - 840 q^{41} - 3209 q^{42} - 378 q^{43} + 2565 q^{44} + 470 q^{45} + 10176 q^{46} - 234 q^{47} + 3143 q^{48} - 2974 q^{49} + 50 q^{50} - 1024 q^{51} - 2065 q^{52} - 150 q^{53} - 7101 q^{54} + 530 q^{55} - 2230 q^{56} + 172 q^{57} + 1236 q^{58} + 556 q^{59} - 1295 q^{60} + 680 q^{61} + 373 q^{62} + 208 q^{63} - 4383 q^{64} + 930 q^{65} - 4014 q^{66} + 1078 q^{67} + 252 q^{68} + 372 q^{69} + 730 q^{70} - 1676 q^{71} + 14141 q^{72} + 3812 q^{73} + 3121 q^{74} + 100 q^{75} - 8272 q^{76} + 486 q^{77} + 2180 q^{78} - 1030 q^{79} + 1850 q^{80} - 1352 q^{81} + 7812 q^{82} - 4842 q^{83} - 13962 q^{84} - 100 q^{85} + 3286 q^{86} - 9608 q^{87} - 3484 q^{88} + 3734 q^{89} - 950 q^{90} + 296 q^{91} - 10353 q^{92} + 586 q^{93} - 8780 q^{94} + 180 q^{95} + 1197 q^{96} - 5624 q^{97} + 10230 q^{98} + 12088 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
16.1 −4.78694 − 2.30527i 2.00818 − 2.51818i 12.6126 + 15.8157i 4.50484 + 2.16942i −15.4181 + 7.42497i 14.8705 − 18.6470i −14.4582 − 63.3453i 3.69963 + 16.2092i −16.5633 − 20.7698i
16.2 −4.40546 − 2.12156i −0.444863 + 0.557841i 9.91916 + 12.4382i 4.50484 + 2.16942i 3.14332 − 1.51374i −19.2311 + 24.1150i −8.60556 − 37.7034i 5.89478 + 25.8267i −15.2434 − 19.1146i
16.3 −3.65182 − 1.75863i −5.75592 + 7.21770i 5.25513 + 6.58973i 4.50484 + 2.16942i 33.7128 − 16.2352i 7.20933 − 9.04022i −0.386548 − 1.69358i −12.9565 − 56.7660i −12.6357 − 15.8447i
16.4 −3.12175 − 1.50335i 5.64173 − 7.07450i 2.49731 + 3.13153i 4.50484 + 2.16942i −28.2475 + 13.6033i 10.2096 − 12.8024i 3.07988 + 13.4939i −12.2114 − 53.5018i −10.8016 − 13.5448i
16.5 −2.22972 − 1.07378i 3.80345 − 4.76937i −1.16926 − 1.46620i 4.50484 + 2.16942i −13.6019 + 6.55032i −17.8934 + 22.4376i 5.43831 + 23.8268i −2.27264 − 9.95707i −7.71508 − 9.67440i
16.6 −2.19468 − 1.05690i 0.272359 − 0.341527i −1.28834 − 1.61553i 4.50484 + 2.16942i −0.958700 + 0.461686i 1.57629 − 1.97660i 5.45637 + 23.9059i 5.96560 + 26.1370i −7.59382 − 9.52236i
16.7 −1.28666 − 0.619624i −3.44035 + 4.31406i −3.71635 − 4.66016i 4.50484 + 2.16942i 7.09966 − 3.41902i 14.1752 − 17.7752i 4.43638 + 19.4370i −0.767057 − 3.36070i −4.45199 − 5.58262i
16.8 0.122277 + 0.0588855i −5.54417 + 6.95216i −4.97643 − 6.24025i 4.50484 + 2.16942i −1.08731 + 0.523619i −18.2121 + 22.8373i −0.482642 − 2.11459i −11.5867 − 50.7649i 0.423091 + 0.530540i
16.9 0.662845 + 0.319209i 2.88467 − 3.61727i −4.65045 − 5.83148i 4.50484 + 2.16942i 3.06675 − 1.47687i 3.02817 − 3.79721i −2.53074 − 11.0879i 1.24479 + 5.45379i 2.29351 + 2.87598i
16.10 0.702663 + 0.338385i −0.832230 + 1.04358i −4.60869 − 5.77911i 4.50484 + 2.16942i −0.937910 + 0.451673i −9.11604 + 11.4312i −2.67114 − 11.7030i 5.61161 + 24.5861i 2.43129 + 3.04874i
16.11 2.14520 + 1.03307i −1.22511 + 1.53624i −1.45329 − 1.82237i 4.50484 + 2.16942i −4.21516 + 2.02991i 20.9111 − 26.2217i −5.47351 − 23.9810i 5.14892 + 22.5589i 7.42261 + 9.30766i
16.12 2.18074 + 1.05019i 6.04078 − 7.57490i −1.33520 − 1.67428i 4.50484 + 2.16942i 21.1284 − 10.1749i 2.63079 − 3.29890i −5.46218 − 23.9314i −14.8800 − 65.1937i 7.54559 + 9.46186i
16.13 3.51657 + 1.69349i −2.92714 + 3.67052i 4.51042 + 5.65589i 4.50484 + 2.16942i −16.5095 + 7.95055i −9.43295 + 11.8286i −0.665153 − 2.91423i 1.10350 + 4.83477i 12.1677 + 15.2578i
16.14 4.11743 + 1.98285i 3.46788 − 4.34858i 8.03362 + 10.0738i 4.50484 + 2.16942i 22.9013 − 11.0287i 2.34209 − 2.93688i 4.96758 + 21.7644i −0.875911 − 3.83762i 14.2467 + 17.8649i
16.15 4.26877 + 2.05573i 1.78467 − 2.23791i 9.00843 + 11.2962i 4.50484 + 2.16942i 12.2189 − 5.88430i −2.94482 + 3.69268i 6.79854 + 29.7863i 4.18489 + 18.3352i 14.7704 + 18.5215i
16.16 5.08404 + 2.44835i −5.17897 + 6.49423i 14.8652 + 18.6404i 4.50484 + 2.16942i −42.2302 + 20.3370i 8.05118 − 10.0959i 19.8920 + 87.1524i −9.34515 − 40.9438i 17.5913 + 22.0588i
36.1 −1.25099 − 5.48093i 6.23471 + 3.00248i −21.2679 + 10.2421i 1.11260 + 4.87464i 8.65683 − 37.9281i 5.74525 + 2.76677i 54.7005 + 68.5922i 13.0225 + 16.3297i 25.3257 − 12.1962i
36.2 −1.13604 − 4.97730i −6.88653 − 3.31638i −16.2752 + 7.83772i 1.11260 + 4.87464i −8.68326 + 38.0438i −30.0208 − 14.4573i 32.0350 + 40.1707i 19.5917 + 24.5672i 22.9986 − 11.0755i
36.3 −0.942150 − 4.12783i 1.08757 + 0.523744i −8.94356 + 4.30699i 1.11260 + 4.87464i 1.13728 − 4.98273i −4.45637 − 2.14608i 5.08594 + 6.37757i −15.9257 − 19.9702i 19.0734 − 9.18528i
36.4 −0.786113 − 3.44419i −0.0947235 − 0.0456164i −4.03669 + 1.94397i 1.11260 + 4.87464i −0.0826481 + 0.362105i 22.3503 + 10.7634i −7.75243 − 9.72124i −16.8273 − 21.1008i 15.9145 − 7.66403i
See all 96 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 16.16
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 145.4.k.b ✓ 96
29.d even 7 1 inner 145.4.k.b ✓ 96
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
145.4.k.b ✓ 96 1.a even 1 1 trivial
145.4.k.b ✓ 96 29.d even 7 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{96} - 2 T_{2}^{95} + 107 T_{2}^{94} - 221 T_{2}^{93} + 6559 T_{2}^{92} - 12629 T_{2}^{91} + \cdots + 35\!\cdots\!44 \) acting on \(S_{4}^{\mathrm{new}}(145, [\chi])\). Copy content Toggle raw display