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Properties

Label 175.4.h.a
Level $175$
Weight $4$
Character orbit 175.h
Analytic conductor $10.325$
Analytic rank $0$
Dimension $88$
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] = [175,4,Mod(36,175)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(175, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([8, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("175.36");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 175 = 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 175.h (of order \(5\), 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: \(10.3253342510\)
Analytic rank: \(0\)
Dimension: \(88\)
Relative dimension: \(22\) over \(\Q(\zeta_{5})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 88 q - 6 q^{2} - 2 q^{3} - 84 q^{4} + 14 q^{5} - 15 q^{6} - 616 q^{7} - 72 q^{8} - 326 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 88 q - 6 q^{2} - 2 q^{3} - 84 q^{4} + 14 q^{5} - 15 q^{6} - 616 q^{7} - 72 q^{8} - 326 q^{9} + 56 q^{10} + 68 q^{11} - 144 q^{12} - 88 q^{13} + 42 q^{14} - 68 q^{15} - 340 q^{16} - 126 q^{17} + 1038 q^{18} + 232 q^{19} + 472 q^{20} + 14 q^{21} - 607 q^{22} - 418 q^{23} + 534 q^{24} + 184 q^{25} - 296 q^{26} - 674 q^{27} + 588 q^{28} - 382 q^{29} + 68 q^{30} + 720 q^{31} + 2748 q^{32} - 210 q^{33} + 1060 q^{34} - 98 q^{35} - 1019 q^{36} - 1058 q^{37} + 1065 q^{38} + 82 q^{39} - 2575 q^{40} - 164 q^{41} + 105 q^{42} + 732 q^{43} + 393 q^{44} + 998 q^{45} - 1928 q^{46} - 2090 q^{47} - 3025 q^{48} + 4312 q^{49} + 2351 q^{50} - 1100 q^{51} - 3726 q^{52} - 136 q^{53} - 1152 q^{54} + 878 q^{55} + 504 q^{56} + 4764 q^{57} - 4483 q^{58} + 1688 q^{59} + 7056 q^{60} - 2620 q^{61} + 62 q^{62} + 2282 q^{63} - 744 q^{64} - 3488 q^{65} - 4403 q^{66} + 1310 q^{67} + 6502 q^{68} - 3320 q^{69} - 392 q^{70} + 3314 q^{71} - 2120 q^{72} - 1274 q^{73} - 6930 q^{74} + 3022 q^{75} - 9810 q^{76} - 476 q^{77} - 4301 q^{78} + 2796 q^{79} + 744 q^{80} + 6532 q^{81} + 13006 q^{82} - 1436 q^{83} + 1008 q^{84} + 1016 q^{85} + 2990 q^{86} - 1792 q^{87} + 9602 q^{88} + 928 q^{89} - 1608 q^{90} + 616 q^{91} + 4102 q^{92} + 3488 q^{93} - 10684 q^{94} - 220 q^{95} + 824 q^{96} - 3848 q^{97} - 294 q^{98} - 6576 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
36.1 −4.45785 + 3.23882i 2.73943 − 8.43111i 6.91037 − 21.2679i −0.0985657 + 11.1799i 15.0949 + 46.4572i −7.00000 24.4556 + 75.2666i −41.7356 − 30.3227i −35.7703 − 50.1576i
36.2 −4.19557 + 3.04826i 0.165045 − 0.507956i 5.83877 − 17.9699i 9.87125 − 5.24962i 0.855923 + 2.63426i −7.00000 17.4594 + 53.7344i 21.6127 + 15.7025i −25.4133 + 52.1152i
36.3 −4.10456 + 2.98213i −2.49848 + 7.68953i 5.48212 − 16.8722i −3.83647 + 10.5015i −12.6761 − 39.0129i −7.00000 15.2712 + 46.9999i −31.0430 − 22.5541i −15.5699 − 54.5449i
36.4 −3.30242 + 2.39935i −0.664737 + 2.04585i 2.67696 − 8.23884i 8.83680 + 6.84916i −2.71347 − 8.35119i −7.00000 0.836099 + 2.57325i 18.0998 + 13.1503i −45.6163 − 1.41622i
36.5 −3.14033 + 2.28158i 2.34361 − 7.21289i 2.18391 − 6.72138i −7.57731 − 8.22097i 9.09710 + 27.9980i −7.00000 −1.11879 − 3.44330i −24.6898 − 17.9382i 42.5521 + 8.52828i
36.6 −2.64384 + 1.92086i 1.24468 − 3.83074i 0.828039 − 2.54844i −9.38342 + 6.07876i 4.06758 + 12.5187i −7.00000 −5.37285 − 16.5359i 8.71811 + 6.33408i 13.1318 − 34.0955i
36.7 −2.64125 + 1.91898i −0.768302 + 2.36459i 0.821580 − 2.52856i −1.38816 − 11.0938i −2.50832 − 7.71983i −7.00000 −5.38868 − 16.5846i 16.8425 + 12.2368i 24.9553 + 26.6377i
36.8 −1.75299 + 1.27362i −2.52987 + 7.78615i −1.02128 + 3.14318i 10.4419 − 3.99577i −5.48175 − 16.8711i −7.00000 −7.56957 − 23.2967i −32.3804 − 23.5257i −13.2155 + 20.3036i
36.9 −1.41192 + 1.02582i −2.67584 + 8.23537i −1.53092 + 4.71170i −10.3810 + 4.15154i −4.66995 − 14.3726i −7.00000 −6.98626 − 21.5015i −38.8178 − 28.2028i 10.3984 − 16.5107i
36.10 −0.879137 + 0.638731i 1.68319 − 5.18032i −2.10723 + 6.48539i 9.95233 − 5.09422i 1.82908 + 5.62932i −7.00000 −4.97628 − 15.3154i −2.15913 − 1.56870i −5.49563 + 10.8354i
36.11 −0.687396 + 0.499422i 1.56806 − 4.82598i −2.24905 + 6.92185i 5.53757 + 9.71264i 1.33233 + 4.10048i −7.00000 −4.01144 − 12.3459i 1.01218 + 0.735393i −8.65721 − 3.91084i
36.12 0.0999315 − 0.0726045i −0.640406 + 1.97097i −2.46742 + 7.59394i −11.1018 − 1.32268i 0.0791043 + 0.243458i −7.00000 0.610145 + 1.87783i 18.3689 + 13.3458i −1.20545 + 0.673865i
36.13 0.615492 − 0.447181i 3.09463 − 9.52430i −2.29328 + 7.05798i −6.76573 − 8.90083i −2.35437 − 7.24600i −7.00000 3.62548 + 11.1581i −59.2921 − 43.0782i −8.14454 − 2.45289i
36.14 0.758424 − 0.551028i −0.0954005 + 0.293613i −2.20056 + 6.77263i −0.888074 − 11.1450i 0.0894345 + 0.275251i −7.00000 4.38049 + 13.4817i 21.7664 + 15.8142i −6.81475 − 7.96330i
36.15 1.09105 − 0.792696i −2.07319 + 6.38062i −1.91011 + 5.87871i 7.02303 + 8.69925i 2.79593 + 8.60500i −7.00000 5.90996 + 18.1890i −14.5708 − 10.5863i 14.5584 + 3.92421i
36.16 1.47830 − 1.07405i 1.58881 − 4.88986i −1.44034 + 4.43291i −5.95918 + 9.45982i −2.90320 − 8.93514i −7.00000 7.14920 + 22.0030i 0.457093 + 0.332097i 1.35084 + 20.3849i
36.17 2.85282 − 2.07269i 3.09367 − 9.52134i 1.37038 − 4.21759i 10.5983 + 3.56020i −10.9091 − 33.5749i −7.00000 3.88511 + 11.9571i −59.2416 − 43.0416i 37.6144 − 11.8105i
36.18 2.99963 − 2.17936i −2.51060 + 7.72685i 1.77604 − 5.46608i −7.10120 − 8.63556i 9.30868 + 28.6492i −7.00000 2.58096 + 7.94339i −31.5575 − 22.9279i −40.1209 − 10.4274i
36.19 3.05660 − 2.22075i −0.544361 + 1.67537i 1.93894 − 5.96746i −2.96767 + 10.7793i 2.05669 + 6.32984i −7.00000 2.01449 + 6.19995i 19.3329 + 14.0462i 14.8671 + 39.5384i
36.20 3.28583 − 2.38729i −2.00988 + 6.18578i 2.62535 − 8.08001i 8.19071 − 7.61002i 8.16314 + 25.1236i −7.00000 −0.622293 − 1.91522i −12.3808 − 8.99517i 8.74591 − 44.5588i
See all 88 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 36.22
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.d even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 175.4.h.a ✓ 88
25.d even 5 1 inner 175.4.h.a ✓ 88
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
175.4.h.a ✓ 88 1.a even 1 1 trivial
175.4.h.a ✓ 88 25.d even 5 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{88} + 6 T_{2}^{87} + 148 T_{2}^{86} + 872 T_{2}^{85} + 12473 T_{2}^{84} + 67562 T_{2}^{83} + \cdots + 30\!\cdots\!00 \) acting on \(S_{4}^{\mathrm{new}}(175, [\chi])\). Copy content Toggle raw display