Properties

Label 105.4.x.a
Level $105$
Weight $4$
Character orbit 105.x
Analytic conductor $6.195$
Analytic rank $0$
Dimension $176$
CM no
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 105 = 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 105.x (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.19520055060\)
Analytic rank: \(0\)
Dimension: \(176\)
Relative dimension: \(44\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 176q - 2q^{3} + 4q^{7} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 176q - 2q^{3} + 4q^{7} - 28q^{10} - 94q^{12} - 16q^{13} - 176q^{15} + 1016q^{16} + 14q^{18} - 148q^{21} - 32q^{22} - 76q^{25} - 32q^{27} + 776q^{28} - 500q^{30} - 296q^{31} - 430q^{33} - 880q^{36} + 212q^{37} - 1296q^{40} - 910q^{42} - 1360q^{43} + 490q^{45} + 1000q^{46} + 1604q^{48} - 268q^{51} + 68q^{52} + 1240q^{55} + 3548q^{57} - 1572q^{58} - 470q^{60} - 608q^{61} - 2974q^{63} - 92q^{66} - 1012q^{67} - 6720q^{70} + 866q^{72} + 2804q^{73} + 4086q^{75} - 8256q^{76} + 6888q^{78} - 2992q^{81} - 1112q^{82} + 4736q^{85} - 52q^{87} + 1284q^{88} - 5724q^{90} - 4192q^{91} + 3034q^{93} - 1824q^{96} + 7496q^{97} + O(q^{100}) \)

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.

Label \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1 −1.39762 5.21600i 5.16396 0.577475i −18.3251 + 10.5800i −10.0542 + 4.89013i −10.2294 26.1281i −17.8038 + 5.10128i 50.2499 + 50.2499i 26.3330 5.96412i 39.5589 + 45.6081i
2.2 −1.35648 5.06246i −2.89313 + 4.31623i −16.8602 + 9.73427i 4.47697 + 10.2448i 25.7752 + 8.79147i −1.97131 18.4150i 42.5021 + 42.5021i −10.2596 24.9748i 45.7911 36.5614i
2.3 −1.29573 4.83572i −2.41255 4.60213i −14.7771 + 8.53155i 7.35527 + 8.42022i −19.1286 + 17.6295i 7.27945 + 17.0297i 32.0833 + 32.0833i −15.3592 + 22.2058i 31.1874 46.4783i
2.4 −1.26251 4.71175i −4.27909 2.94778i −13.6784 + 7.89724i −3.51839 10.6123i −8.48683 + 23.8836i −15.2757 10.4716i 26.8850 + 26.8850i 9.62114 + 25.2276i −45.5605 + 29.9759i
2.5 −1.24693 4.65360i 3.48035 + 3.85839i −13.1730 + 7.60541i 9.76680 5.44147i 13.6157 21.0073i 7.78888 + 16.8028i 24.5649 + 24.5649i −2.77439 + 26.8571i −37.5009 38.6657i
2.6 −1.19670 4.46614i 3.39774 3.93133i −11.5861 + 6.68922i 2.95094 10.7839i −21.6239 10.4701i 16.8977 7.58076i 17.5845 + 17.5845i −3.91074 26.7153i −51.6936 0.274265i
2.7 −1.07388 4.00779i 1.26476 + 5.03988i −7.98097 + 4.60782i −9.67232 5.60770i 18.8406 10.4811i 11.5335 14.4906i 3.56656 + 3.56656i −23.8008 + 12.7485i −12.0875 + 44.7867i
2.8 −1.04291 3.89220i −5.08196 + 1.08337i −7.13339 + 4.11846i −10.8347 + 2.75860i 9.51675 + 18.6502i 11.6318 + 14.4118i 0.675065 + 0.675065i 24.6526 11.0113i 22.0367 + 39.2938i
2.9 −0.906743 3.38401i −3.54568 + 3.79844i −3.70115 + 2.13686i 7.59616 8.20356i 16.0690 + 8.55440i −17.0889 + 7.13925i −9.23100 9.23100i −1.85636 26.9361i −34.6487 18.2670i
2.10 −0.904480 3.37557i 2.72714 4.42297i −3.64816 + 2.10627i 10.0776 + 4.84170i −17.3967 5.20516i −16.2054 8.96573i −9.35916 9.35916i −12.1254 24.1242i 7.22850 38.3968i
2.11 −0.791452 2.95374i 5.19588 0.0534027i −1.16997 + 0.675482i −1.94897 + 11.0092i −4.27002 15.3050i 17.5402 5.94478i −14.3771 14.3771i 26.9943 0.554948i 34.0607 2.95647i
2.12 −0.760875 2.83962i 1.74277 + 4.89518i −0.556327 + 0.321196i 0.186008 + 11.1788i 12.5744 8.67341i −10.7614 + 15.0729i −15.2946 15.2946i −20.9255 + 17.0623i 31.6020 9.03385i
2.13 −0.718246 2.68053i 1.46722 4.98470i 0.258841 0.149442i −10.9729 2.14346i −14.4155 0.352673i −5.48584 + 17.6891i −16.2848 16.2848i −22.6946 14.6273i 2.13568 + 30.9528i
2.14 −0.701027 2.61627i −5.19561 0.0748371i 0.574786 0.331853i 11.1637 0.610079i 3.44647 + 13.6456i 15.4775 10.1709i −16.5931 16.5931i 26.9888 + 0.777649i −9.42217 28.7795i
2.15 −0.620524 2.31583i −3.07792 4.18646i 1.95020 1.12595i −5.94896 + 9.46625i −7.78518 + 9.72572i −4.27644 18.0198i −17.3801 17.3801i −8.05285 + 25.7711i 25.6137 + 7.90273i
2.16 −0.557385 2.08019i 4.92343 + 1.66126i 2.91169 1.68107i −2.90737 10.7957i 0.711494 11.1676i −17.2324 6.78566i −17.3023 17.3023i 21.4804 + 16.3582i −20.8366 + 12.0652i
2.17 −0.363732 1.35747i −2.21697 4.69947i 5.21779 3.01249i 0.729495 11.1565i −5.57299 + 4.71882i 18.5013 0.838755i −13.9371 13.9371i −17.1701 + 20.8372i −15.4099 + 3.06772i
2.18 −0.229611 0.856918i −2.99971 + 4.24285i 6.24662 3.60649i −9.50945 + 5.87965i 4.32454 + 1.59630i −14.5995 11.3954i −9.54321 9.54321i −9.00352 25.4546i 7.22185 + 6.79879i
2.19 −0.212644 0.793597i 5.02395 1.32663i 6.34362 3.66249i 10.8037 2.87737i −2.12112 3.70489i 4.51416 + 17.9617i −8.90311 8.90311i 23.4801 13.3299i −4.58082 7.96196i
2.20 −0.189195 0.706085i −1.86875 + 4.84848i 6.46544 3.73282i −2.65237 10.8612i 3.77700 + 0.402189i 15.8305 + 9.61223i −7.99404 7.99404i −20.0155 18.1212i −7.16709 + 3.92768i
See next 80 embeddings (of 176 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 53.44
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
7.c even 3 1 inner
15.e even 4 1 inner
21.h odd 6 1 inner
35.l odd 12 1 inner
105.x even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 105.4.x.a 176
3.b odd 2 1 inner 105.4.x.a 176
5.c odd 4 1 inner 105.4.x.a 176
7.c even 3 1 inner 105.4.x.a 176
15.e even 4 1 inner 105.4.x.a 176
21.h odd 6 1 inner 105.4.x.a 176
35.l odd 12 1 inner 105.4.x.a 176
105.x even 12 1 inner 105.4.x.a 176
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.4.x.a 176 1.a even 1 1 trivial
105.4.x.a 176 3.b odd 2 1 inner
105.4.x.a 176 5.c odd 4 1 inner
105.4.x.a 176 7.c even 3 1 inner
105.4.x.a 176 15.e even 4 1 inner
105.4.x.a 176 21.h odd 6 1 inner
105.4.x.a 176 35.l odd 12 1 inner
105.4.x.a 176 105.x even 12 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(105, [\chi])\).