# Properties

 Label 91.10.i.a Level $91$ Weight $10$ Character orbit 91.i Analytic conductor $46.868$ Analytic rank $0$ Dimension $164$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$91 = 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$10$$ Character orbit: $$[\chi]$$ $$=$$ 91.i (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$46.8682610909$$ Analytic rank: $$0$$ Dimension: $$164$$ Relative dimension: $$82$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$164 q - 4 q^{2} - 1142 q^{7} + 2044 q^{8} - 1023524 q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$164 q - 4 q^{2} - 1142 q^{7} + 2044 q^{8} - 1023524 q^{9} + 83020 q^{11} + 342380 q^{14} + 108440 q^{15} - 9961480 q^{16} + 733960 q^{18} - 2430528 q^{21} - 2436232 q^{22} - 3224028 q^{28} - 12730912 q^{29} + 14102504 q^{32} + 1383128 q^{35} - 22260172 q^{37} + 7370608 q^{39} + 142705660 q^{42} - 29990964 q^{44} - 340921064 q^{46} - 250003828 q^{50} + 408277152 q^{53} + 172857368 q^{57} - 25520428 q^{58} + 541182148 q^{60} - 173245586 q^{63} + 743607836 q^{65} + 460418836 q^{67} - 1611790784 q^{70} - 937506828 q^{71} - 1730965108 q^{72} + 1230722160 q^{74} + 2491131268 q^{78} + 324907200 q^{79} + 7601803508 q^{81} - 4681602688 q^{84} - 700010456 q^{85} - 108608508 q^{86} - 750548542 q^{91} - 1994311080 q^{92} + 491721312 q^{93} - 6546627460 q^{98} - 6566823268 q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
34.1 −30.6699 30.6699i 210.836i 1369.29i 1324.26 1324.26i −6466.33 + 6466.33i −5803.54 + 2583.12i 26293.0 26293.0i −24768.9 −81229.8
34.2 −30.6699 30.6699i 210.836i 1369.29i −1324.26 + 1324.26i 6466.33 6466.33i −2583.12 + 5803.54i 26293.0 26293.0i −24768.9 81229.8
34.3 −30.1481 30.1481i 41.9129i 1305.82i 47.1446 47.1446i −1263.60 + 1263.60i 6068.45 + 1878.17i 23932.2 23932.2i 17926.3 −2842.65
34.4 −30.1481 30.1481i 41.9129i 1305.82i −47.1446 + 47.1446i 1263.60 1263.60i −1878.17 6068.45i 23932.2 23932.2i 17926.3 2842.65
34.5 −29.6596 29.6596i 221.983i 1247.39i −1582.09 + 1582.09i −6583.93 + 6583.93i −4505.15 4478.53i 21811.2 21811.2i −29593.5 93848.6
34.6 −29.6596 29.6596i 221.983i 1247.39i 1582.09 1582.09i 6583.93 6583.93i 4478.53 + 4505.15i 21811.2 21811.2i −29593.5 −93848.6
34.7 −26.0602 26.0602i 88.5050i 846.263i 553.439 553.439i −2306.45 + 2306.45i −5157.40 + 3708.75i 8710.95 8710.95i 11849.9 −28845.4
34.8 −26.0602 26.0602i 88.5050i 846.263i −553.439 + 553.439i 2306.45 2306.45i −3708.75 + 5157.40i 8710.95 8710.95i 11849.9 28845.4
34.9 −26.0147 26.0147i 172.872i 841.530i −924.961 + 924.961i −4497.22 + 4497.22i 3450.17 + 5333.85i 8572.61 8572.61i −10201.8 48125.2
34.10 −26.0147 26.0147i 172.872i 841.530i 924.961 924.961i 4497.22 4497.22i −5333.85 3450.17i 8572.61 8572.61i −10201.8 −48125.2
34.11 −25.7334 25.7334i 103.954i 812.416i 1902.22 1902.22i −2675.09 + 2675.09i 4687.23 4287.59i 7730.72 7730.72i 8876.58 −97901.0
34.12 −25.7334 25.7334i 103.954i 812.416i −1902.22 + 1902.22i 2675.09 2675.09i 4287.59 4687.23i 7730.72 7730.72i 8876.58 97901.0
34.13 −23.9539 23.9539i 256.019i 635.577i 408.587 408.587i −6132.64 + 6132.64i 4798.95 4162.18i 2960.16 2960.16i −45862.6 −19574.5
34.14 −23.9539 23.9539i 256.019i 635.577i −408.587 + 408.587i 6132.64 6132.64i 4162.18 4798.95i 2960.16 2960.16i −45862.6 19574.5
34.15 −21.7082 21.7082i 61.2207i 430.495i −1460.95 + 1460.95i −1328.99 + 1328.99i −5891.36 + 2376.03i −1769.32 + 1769.32i 15935.0 63429.3
34.16 −21.7082 21.7082i 61.2207i 430.495i 1460.95 1460.95i 1328.99 1328.99i −2376.03 + 5891.36i −1769.32 + 1769.32i 15935.0 −63429.3
34.17 −21.2010 21.2010i 146.854i 386.961i −617.378 + 617.378i −3113.45 + 3113.45i 737.960 6309.44i −2650.95 + 2650.95i −1883.14 26178.0
34.18 −21.2010 21.2010i 146.854i 386.961i 617.378 617.378i 3113.45 3113.45i 6309.44 737.960i −2650.95 + 2650.95i −1883.14 −26178.0
34.19 −20.1032 20.1032i 68.1439i 296.276i 476.883 476.883i −1369.91 + 1369.91i −2725.61 5738.00i −4336.75 + 4336.75i 15039.4 −19173.7
34.20 −20.1032 20.1032i 68.1439i 296.276i −476.883 + 476.883i 1369.91 1369.91i 5738.00 + 2725.61i −4336.75 + 4336.75i 15039.4 19173.7
See next 80 embeddings (of 164 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 83.82 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
13.d odd 4 1 inner
91.i even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 91.10.i.a 164
7.b odd 2 1 inner 91.10.i.a 164
13.d odd 4 1 inner 91.10.i.a 164
91.i even 4 1 inner 91.10.i.a 164

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.10.i.a 164 1.a even 1 1 trivial
91.10.i.a 164 7.b odd 2 1 inner
91.10.i.a 164 13.d odd 4 1 inner
91.10.i.a 164 91.i even 4 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{10}^{\mathrm{new}}(91, [\chi])$$.