# Properties

 Label 112.2.w.c Level $112$ Weight $2$ Character orbit 112.w Analytic conductor $0.894$ Analytic rank $0$ Dimension $48$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$112 = 2^{4} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 112.w (of order $$12$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.894324502638$$ Analytic rank: $$0$$ Dimension: $$48$$ Relative dimension: $$12$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$48 q - 4 q^{2} - 4 q^{4} + 4 q^{5} - 4 q^{6} - 4 q^{8}+O(q^{10})$$ 48 * q - 4 * q^2 - 4 * q^4 + 4 * q^5 - 4 * q^6 - 4 * q^8 $$\operatorname{Tr}(f)(q) =$$ $$48 q - 4 q^{2} - 4 q^{4} + 4 q^{5} - 4 q^{6} - 4 q^{8} - 2 q^{10} - 4 q^{11} + 2 q^{12} - 24 q^{13} - 34 q^{14} - 40 q^{15} + 16 q^{16} + 8 q^{17} + 18 q^{18} - 4 q^{19} - 16 q^{20} - 8 q^{21} + 18 q^{24} - 10 q^{26} - 24 q^{27} - 10 q^{28} + 24 q^{29} - 4 q^{30} + 28 q^{31} + 16 q^{32} + 16 q^{33} - 44 q^{34} + 28 q^{35} - 72 q^{36} - 24 q^{37} + 20 q^{38} + 26 q^{40} + 6 q^{42} - 40 q^{43} + 6 q^{44} - 28 q^{45} - 14 q^{46} - 20 q^{47} + 56 q^{48} + 56 q^{50} + 24 q^{51} - 16 q^{52} - 16 q^{53} + 64 q^{54} + 40 q^{56} - 6 q^{58} - 20 q^{59} + 46 q^{60} + 8 q^{61} + 24 q^{62} - 16 q^{63} + 80 q^{64} + 8 q^{65} - 20 q^{66} + 48 q^{67} - 40 q^{69} + 82 q^{70} - 32 q^{72} - 8 q^{74} - 4 q^{75} - 36 q^{76} - 20 q^{77} + 116 q^{78} - 36 q^{79} - 28 q^{80} + 2 q^{82} - 8 q^{83} + 28 q^{84} - 20 q^{86} - 42 q^{88} - 20 q^{90} + 64 q^{91} + 76 q^{92} + 8 q^{93} - 72 q^{94} - 4 q^{95} - 120 q^{96} - 48 q^{97} - 2 q^{98} - 24 q^{99}+O(q^{100})$$ 48 * q - 4 * q^2 - 4 * q^4 + 4 * q^5 - 4 * q^6 - 4 * q^8 - 2 * q^10 - 4 * q^11 + 2 * q^12 - 24 * q^13 - 34 * q^14 - 40 * q^15 + 16 * q^16 + 8 * q^17 + 18 * q^18 - 4 * q^19 - 16 * q^20 - 8 * q^21 + 18 * q^24 - 10 * q^26 - 24 * q^27 - 10 * q^28 + 24 * q^29 - 4 * q^30 + 28 * q^31 + 16 * q^32 + 16 * q^33 - 44 * q^34 + 28 * q^35 - 72 * q^36 - 24 * q^37 + 20 * q^38 + 26 * q^40 + 6 * q^42 - 40 * q^43 + 6 * q^44 - 28 * q^45 - 14 * q^46 - 20 * q^47 + 56 * q^48 + 56 * q^50 + 24 * q^51 - 16 * q^52 - 16 * q^53 + 64 * q^54 + 40 * q^56 - 6 * q^58 - 20 * q^59 + 46 * q^60 + 8 * q^61 + 24 * q^62 - 16 * q^63 + 80 * q^64 + 8 * q^65 - 20 * q^66 + 48 * q^67 - 40 * q^69 + 82 * q^70 - 32 * q^72 - 8 * q^74 - 4 * q^75 - 36 * q^76 - 20 * q^77 + 116 * q^78 - 36 * q^79 - 28 * q^80 + 2 * q^82 - 8 * q^83 + 28 * q^84 - 20 * q^86 - 42 * q^88 - 20 * q^90 + 64 * q^91 + 76 * q^92 + 8 * q^93 - 72 * q^94 - 4 * q^95 - 120 * q^96 - 48 * q^97 - 2 * q^98 - 24 * q^99

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
37.1 −1.40462 + 0.164410i −0.839165 0.224854i 1.94594 0.461868i −3.16320 + 0.847576i 1.21568 + 0.177868i 0.654939 + 2.56341i −2.65738 + 0.968682i −1.94444 1.12262i 4.30375 1.71059i
37.2 −1.36184 0.381311i −1.60845 0.430984i 1.70920 + 1.03857i 1.94900 0.522232i 2.02612 + 1.20025i −1.89075 1.85069i −1.93164 2.06610i −0.196696 0.113563i −2.85335 0.0319777i
37.3 −1.25281 + 0.656100i 1.95279 + 0.523249i 1.13907 1.64394i 0.959072 0.256983i −2.78978 + 0.625694i 0.292831 2.62950i −0.348448 + 2.80688i 0.941530 + 0.543593i −1.03293 + 0.951197i
37.4 −0.892252 + 1.09722i −2.20008 0.589510i −0.407774 1.95799i 2.32205 0.622192i 2.60985 1.88798i 2.41058 + 1.09047i 2.51218 + 1.29960i 1.89476 + 1.09394i −1.38918 + 3.10295i
37.5 −0.805226 1.16259i 1.16533 + 0.312249i −0.703221 + 1.87229i 0.980942 0.262843i −0.575337 1.60623i 2.60251 + 0.476386i 2.74296 0.690063i −1.33758 0.772254i −1.09546 0.928784i
37.6 −0.246810 + 1.39251i 2.51077 + 0.672759i −1.87817 0.687371i −2.91203 + 0.780276i −1.55651 + 3.33023i 1.41955 + 2.23268i 1.42072 2.44572i 3.25328 + 1.87828i −0.367824 4.24761i
37.7 0.236566 + 1.39429i 0.827840 + 0.221819i −1.88807 + 0.659683i 4.10878 1.10095i −0.113440 + 1.20672i −2.50325 + 0.856573i −1.36644 2.47646i −1.96196 1.13274i 2.50703 + 5.46838i
37.8 0.262311 1.38967i 2.91496 + 0.781062i −1.86239 0.729053i −0.745506 + 0.199758i 1.85005 3.84597i −2.51194 0.830765i −1.50167 + 2.39687i 5.28887 + 3.05353i 0.0820438 + 1.08841i
37.9 0.713936 + 1.22078i −3.04092 0.814813i −0.980592 + 1.74311i −1.87270 + 0.501787i −1.17632 4.29401i −1.89831 + 1.84294i −2.82803 + 0.0473857i 5.98523 + 3.45557i −1.94955 1.92790i
37.10 0.945375 1.05179i −0.0543752 0.0145698i −0.212530 1.98868i 1.25781 0.337028i −0.0667293 + 0.0434174i 0.230738 + 2.63567i −2.29259 1.65651i −2.59533 1.49842i 0.834616 1.64157i
37.11 1.39181 0.250740i −2.46036 0.659252i 1.87426 0.697963i 1.87532 0.502490i −3.58965 0.300642i 0.364450 2.62053i 2.43360 1.44138i 3.02070 + 1.74400i 2.48409 1.16959i
37.12 1.41356 + 0.0428532i 0.831674 + 0.222846i 1.99633 + 0.121151i −2.02749 + 0.543265i 1.16607 + 0.350647i −2.63544 + 0.233350i 2.81674 + 0.256804i −1.95606 1.12933i −2.88927 + 0.681055i
53.1 −1.41419 + 0.00789795i 0.814813 + 3.04092i 1.99988 0.0223384i 0.501787 1.87270i −1.17632 4.29401i 1.89831 + 1.84294i −2.82803 + 0.0473857i −5.98523 + 3.45557i −0.694833 + 2.65231i
53.2 −1.32577 0.492271i −0.221819 0.827840i 1.51534 + 1.30528i −1.10095 + 4.10878i −0.113440 + 1.20672i 2.50325 + 0.856573i −1.36644 2.47646i 1.96196 1.13274i 3.48224 4.90534i
53.3 −1.08254 0.909999i −0.672759 2.51077i 0.343804 + 1.97023i 0.780276 2.91203i −1.55651 + 3.33023i −1.41955 + 2.23268i 1.42072 2.44572i −3.25328 + 1.87828i −3.49463 + 2.44235i
53.4 −0.743894 + 1.20276i −0.222846 0.831674i −0.893243 1.78945i 0.543265 2.02749i 1.16607 + 0.350647i 2.63544 + 0.233350i 2.81674 + 0.256804i 1.95606 1.12933i 2.03445 + 2.16165i
53.5 −0.504093 1.32132i 0.589510 + 2.20008i −1.49178 + 1.33214i −0.622192 + 2.32205i 2.60985 1.88798i −2.41058 + 1.09047i 2.51218 + 1.29960i −1.89476 + 1.09394i 3.38182 0.348414i
53.6 −0.478757 + 1.33071i 0.659252 + 2.46036i −1.54158 1.27417i −0.502490 + 1.87532i −3.58965 0.300642i −0.364450 2.62053i 2.43360 1.44138i −3.02070 + 1.74400i −2.25494 1.56649i
53.7 0.0582062 1.41302i −0.523249 1.95279i −1.99322 0.164492i −0.256983 + 0.959072i −2.78978 + 0.625694i −0.292831 2.62950i −0.348448 + 2.80688i −0.941530 + 0.543593i 1.34023 + 0.418944i
53.8 0.438190 + 1.34461i 0.0145698 + 0.0543752i −1.61598 + 1.17839i −0.337028 + 1.25781i −0.0667293 + 0.0434174i −0.230738 + 2.63567i −2.29259 1.65651i 2.59533 1.49842i −1.83895 + 0.0979855i
See all 48 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 109.12 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.c even 3 1 inner
16.e even 4 1 inner
112.w even 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 112.2.w.c 48
4.b odd 2 1 448.2.ba.c 48
7.b odd 2 1 784.2.x.o 48
7.c even 3 1 inner 112.2.w.c 48
7.c even 3 1 784.2.m.j 24
7.d odd 6 1 784.2.m.k 24
7.d odd 6 1 784.2.x.o 48
8.b even 2 1 896.2.ba.f 48
8.d odd 2 1 896.2.ba.e 48
16.e even 4 1 inner 112.2.w.c 48
16.e even 4 1 896.2.ba.f 48
16.f odd 4 1 448.2.ba.c 48
16.f odd 4 1 896.2.ba.e 48
28.g odd 6 1 448.2.ba.c 48
56.k odd 6 1 896.2.ba.e 48
56.p even 6 1 896.2.ba.f 48
112.l odd 4 1 784.2.x.o 48
112.u odd 12 1 448.2.ba.c 48
112.u odd 12 1 896.2.ba.e 48
112.w even 12 1 inner 112.2.w.c 48
112.w even 12 1 784.2.m.j 24
112.w even 12 1 896.2.ba.f 48
112.x odd 12 1 784.2.m.k 24
112.x odd 12 1 784.2.x.o 48

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
112.2.w.c 48 1.a even 1 1 trivial
112.2.w.c 48 7.c even 3 1 inner
112.2.w.c 48 16.e even 4 1 inner
112.2.w.c 48 112.w even 12 1 inner
448.2.ba.c 48 4.b odd 2 1
448.2.ba.c 48 16.f odd 4 1
448.2.ba.c 48 28.g odd 6 1
448.2.ba.c 48 112.u odd 12 1
784.2.m.j 24 7.c even 3 1
784.2.m.j 24 112.w even 12 1
784.2.m.k 24 7.d odd 6 1
784.2.m.k 24 112.x odd 12 1
784.2.x.o 48 7.b odd 2 1
784.2.x.o 48 7.d odd 6 1
784.2.x.o 48 112.l odd 4 1
784.2.x.o 48 112.x odd 12 1
896.2.ba.e 48 8.d odd 2 1
896.2.ba.e 48 16.f odd 4 1
896.2.ba.e 48 56.k odd 6 1
896.2.ba.e 48 112.u odd 12 1
896.2.ba.f 48 8.b even 2 1
896.2.ba.f 48 16.e even 4 1
896.2.ba.f 48 56.p even 6 1
896.2.ba.f 48 112.w even 12 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{48} + 8 T_{3}^{45} - 162 T_{3}^{44} - 24 T_{3}^{43} + 32 T_{3}^{42} - 1116 T_{3}^{41} + 18293 T_{3}^{40} + 864 T_{3}^{39} - 3456 T_{3}^{38} + 103652 T_{3}^{37} - 1004406 T_{3}^{36} - 80664 T_{3}^{35} + 285960 T_{3}^{34} + \cdots + 194481$$ acting on $$S_{2}^{\mathrm{new}}(112, [\chi])$$.