# Properties

 Label 48.7.i.a Level 48 Weight 7 Character orbit 48.i Analytic conductor 11.043 Analytic rank 0 Dimension 92 CM no Inner twists 4

# Learn more about

## Newspace parameters

 Level: $$N$$ $$=$$ $$48 = 2^{4} \cdot 3$$ Weight: $$k$$ $$=$$ $$7$$ Character orbit: $$[\chi]$$ $$=$$ 48.i (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$11.0425960138$$ Analytic rank: $$0$$ Dimension: $$92$$ Relative dimension: $$46$$ over $$\Q(i)$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$92q - 2q^{3} - 4q^{4} + 508q^{6} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$92q - 2q^{3} - 4q^{4} + 508q^{6} + 1496q^{10} + 4288q^{12} - 4q^{13} - 4q^{15} + 10352q^{16} + 10020q^{18} - 3940q^{19} + 1456q^{21} - 42088q^{22} + 66376q^{24} - 34322q^{27} + 27000q^{28} - 127836q^{30} - 8q^{31} - 4q^{33} + 128424q^{34} + 172420q^{36} - 4q^{37} - 164392q^{40} + 221560q^{42} - 195268q^{43} - 31252q^{45} + 150712q^{46} - 188048q^{48} - 1142884q^{49} + 385056q^{51} - 122368q^{52} + 151912q^{54} - 181960q^{58} + 242304q^{60} - 326500q^{61} + 470592q^{63} - 251752q^{64} + 489172q^{66} + 1207676q^{67} - 1460q^{69} + 346392q^{70} - 903480q^{72} + 1413778q^{75} - 359992q^{76} + 1478252q^{78} + 860920q^{79} - 4q^{81} - 2412800q^{82} - 730640q^{84} + 434496q^{85} + 1259360q^{88} - 291792q^{90} + 2320992q^{91} - 1176260q^{93} - 1412496q^{94} + 1956200q^{96} - 8q^{97} - 4048228q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
5.1 −7.91573 + 1.15814i −26.8410 2.92544i 61.3174 18.3351i −131.564 + 131.564i 215.854 7.92879i 278.097i −464.137 + 216.150i 711.884 + 157.044i 889.056 1193.80i
5.2 −7.90213 + 1.24752i −6.94595 + 26.0913i 60.8874 19.7162i −0.540042 + 0.540042i 22.3384 214.842i 321.851i −456.544 + 231.758i −632.508 362.457i 3.59377 4.94120i
5.3 −7.81038 1.73145i −5.41236 26.4520i 58.0042 + 27.0465i −54.1549 + 54.1549i −3.52753 + 215.971i 639.122i −406.205 311.675i −670.413 + 286.335i 516.737 329.204i
5.4 −7.78639 + 1.83634i −19.2360 18.9466i 57.2557 28.5969i 148.490 148.490i 184.572 + 112.202i 263.428i −393.302 + 327.807i 11.0511 + 728.916i −883.522 + 1428.88i
5.5 −7.66468 2.29187i 23.2271 + 13.7659i 53.4947 + 35.1328i 133.976 133.976i −146.479 158.745i 25.6967i −329.500 391.885i 350.001 + 639.485i −1333.94 + 719.831i
5.6 −7.60798 2.47360i 14.3156 22.8925i 51.7626 + 37.6382i 4.45378 4.45378i −165.539 + 138.754i 601.125i −300.707 414.390i −319.129 655.437i −44.9011 + 22.8674i
5.7 −7.59869 + 2.50199i 26.4071 + 5.62718i 51.4801 38.0237i −81.2421 + 81.2421i −214.738 + 23.3111i 51.0073i −296.046 + 417.733i 665.670 + 297.195i 414.066 820.600i
5.8 −6.72424 4.33412i −25.2548 + 9.54954i 26.4308 + 58.2873i 70.9576 70.9576i 211.208 + 45.2441i 111.312i 74.8977 506.492i 546.612 482.344i −784.675 + 169.597i
5.9 −6.67919 4.40322i 0.0252819 + 27.0000i 25.2233 + 58.8200i −118.285 + 118.285i 118.718 180.449i 448.417i 90.5262 503.934i −728.999 + 1.36522i 1310.88 269.213i
5.10 −6.62545 + 4.48369i 17.3541 20.6842i 23.7931 59.4129i 77.8915 77.8915i −22.2368 + 214.852i 310.530i 108.749 + 500.318i −126.674 717.910i −166.825 + 865.307i
5.11 −5.62988 5.68370i 26.3911 5.70179i −0.608985 + 63.9971i −89.3741 + 89.3741i −180.986 117.899i 331.991i 367.169 356.834i 663.979 300.953i 1011.14 + 4.81082i
5.12 −5.52373 + 5.78692i −24.5429 + 11.2536i −2.97683 63.9307i 35.5126 35.5126i 70.4446 204.190i 8.76023i 386.405 + 335.909i 475.711 552.395i 9.34650 + 401.670i
5.13 −5.38773 + 5.91374i 9.03029 + 25.4451i −5.94475 63.7233i 65.5814 65.5814i −199.129 83.6886i 515.866i 408.872 + 308.168i −565.908 + 459.554i 34.4968 + 741.166i
5.14 −5.29983 + 5.99264i 2.62588 26.8720i −7.82358 63.5200i −141.081 + 141.081i 147.118 + 158.153i 308.364i 422.117 + 289.761i −715.209 141.126i −97.7428 1593.15i
5.15 −4.22435 6.79374i −18.2268 19.9194i −28.3097 + 57.3983i −14.1816 + 14.1816i −58.3307 + 207.975i 191.419i 509.539 50.1415i −64.5655 + 726.135i 156.254 + 36.4379i
5.16 −3.66778 + 7.10967i −22.4096 15.0602i −37.0947 52.1535i −13.2524 + 13.2524i 189.267 104.087i 437.375i 506.849 72.4434i 275.382 + 674.986i −45.6131 142.827i
5.17 −3.23345 7.31743i 9.91420 + 25.1139i −43.0896 + 47.3211i 41.0809 41.0809i 151.712 153.751i 200.806i 485.597 + 162.295i −532.417 + 497.969i −433.439 167.773i
5.18 −2.77352 7.50384i 17.6543 20.4285i −48.6151 + 41.6241i 132.844 132.844i −202.257 75.8159i 91.8613i 447.176 + 249.355i −105.651 721.304i −1365.28 628.393i
5.19 −2.70919 + 7.52730i 12.2138 + 24.0795i −49.3206 40.7857i −131.879 + 131.879i −214.343 + 26.7009i 543.561i 440.625 260.755i −430.647 + 588.204i −635.408 1349.98i
5.20 −1.74289 + 7.80784i 26.7985 3.29233i −57.9246 27.2165i 59.2583 59.2583i −21.0010 + 214.977i 2.09133i 313.458 404.831i 707.321 176.459i 359.398 + 565.960i
See all 92 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 29.46 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
16.e even 4 1 inner
48.i odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 48.7.i.a 92
3.b odd 2 1 inner 48.7.i.a 92
4.b odd 2 1 192.7.i.a 92
12.b even 2 1 192.7.i.a 92
16.e even 4 1 inner 48.7.i.a 92
16.f odd 4 1 192.7.i.a 92
48.i odd 4 1 inner 48.7.i.a 92
48.k even 4 1 192.7.i.a 92

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.7.i.a 92 1.a even 1 1 trivial
48.7.i.a 92 3.b odd 2 1 inner
48.7.i.a 92 16.e even 4 1 inner
48.7.i.a 92 48.i odd 4 1 inner
192.7.i.a 92 4.b odd 2 1
192.7.i.a 92 12.b even 2 1
192.7.i.a 92 16.f odd 4 1
192.7.i.a 92 48.k even 4 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{7}^{\mathrm{new}}(48, [\chi])$$.

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database