# Properties

 Label 912.2.ci.g Level $912$ Weight $2$ Character orbit 912.ci Analytic conductor $7.282$ Analytic rank $0$ Dimension $24$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$912 = 2^{4} \cdot 3 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 912.ci (of order $$18$$, degree $$6$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.28235666434$$ Analytic rank: $$0$$ Dimension: $$24$$ Relative dimension: $$4$$ over $$\Q(\zeta_{18})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$24 q - 9 q^{7}+O(q^{10})$$ 24 * q - 9 * q^7 $$\operatorname{Tr}(f)(q) =$$ $$24 q - 9 q^{7} + 9 q^{11} - 9 q^{13} - 6 q^{17} - 3 q^{19} - 6 q^{21} + 15 q^{23} + 6 q^{25} - 12 q^{27} - 6 q^{29} + 12 q^{31} - 3 q^{33} + 30 q^{41} - 9 q^{43} + 3 q^{45} - 15 q^{47} + 27 q^{49} + 3 q^{51} + 6 q^{53} + 21 q^{55} - 9 q^{57} - 36 q^{59} - 21 q^{61} - 3 q^{63} - 9 q^{65} + 45 q^{67} - 36 q^{71} + 42 q^{75} + 108 q^{77} + 36 q^{79} - 27 q^{83} - 9 q^{85} - 9 q^{87} - 27 q^{89} - 36 q^{91} - 18 q^{93} + 30 q^{95} - 51 q^{97} + 3 q^{99}+O(q^{100})$$ 24 * q - 9 * q^7 + 9 * q^11 - 9 * q^13 - 6 * q^17 - 3 * q^19 - 6 * q^21 + 15 * q^23 + 6 * q^25 - 12 * q^27 - 6 * q^29 + 12 * q^31 - 3 * q^33 + 30 * q^41 - 9 * q^43 + 3 * q^45 - 15 * q^47 + 27 * q^49 + 3 * q^51 + 6 * q^53 + 21 * q^55 - 9 * q^57 - 36 * q^59 - 21 * q^61 - 3 * q^63 - 9 * q^65 + 45 * q^67 - 36 * q^71 + 42 * q^75 + 108 * q^77 + 36 * q^79 - 27 * q^83 - 9 * q^85 - 9 * q^87 - 27 * q^89 - 36 * q^91 - 18 * q^93 + 30 * q^95 - 51 * q^97 + 3 * 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
79.1 0 −0.939693 + 0.342020i 0 −0.467043 + 2.64873i 0 −0.480437 + 0.277380i 0 0.766044 0.642788i 0
79.2 0 −0.939693 + 0.342020i 0 −0.134718 + 0.764021i 0 −0.911489 + 0.526248i 0 0.766044 0.642788i 0
79.3 0 −0.939693 + 0.342020i 0 0.494739 2.80581i 0 4.05139 2.33907i 0 0.766044 0.642788i 0
79.4 0 −0.939693 + 0.342020i 0 0.525770 2.98179i 0 −3.04612 + 1.75868i 0 0.766044 0.642788i 0
127.1 0 −0.939693 0.342020i 0 −0.467043 2.64873i 0 −0.480437 0.277380i 0 0.766044 + 0.642788i 0
127.2 0 −0.939693 0.342020i 0 −0.134718 0.764021i 0 −0.911489 0.526248i 0 0.766044 + 0.642788i 0
127.3 0 −0.939693 0.342020i 0 0.494739 + 2.80581i 0 4.05139 + 2.33907i 0 0.766044 + 0.642788i 0
127.4 0 −0.939693 0.342020i 0 0.525770 + 2.98179i 0 −3.04612 1.75868i 0 0.766044 + 0.642788i 0
223.1 0 0.173648 0.984808i 0 −3.37195 2.82940i 0 1.97482 1.14016i 0 −0.939693 0.342020i 0
223.2 0 0.173648 0.984808i 0 −1.40574 1.17956i 0 −4.46660 + 2.57879i 0 −0.939693 0.342020i 0
223.3 0 0.173648 0.984808i 0 1.00588 + 0.844034i 0 3.15949 1.82413i 0 −0.939693 0.342020i 0
223.4 0 0.173648 0.984808i 0 1.30003 + 1.09086i 0 −1.57531 + 0.909508i 0 −0.939693 0.342020i 0
319.1 0 0.173648 + 0.984808i 0 −3.37195 + 2.82940i 0 1.97482 + 1.14016i 0 −0.939693 + 0.342020i 0
319.2 0 0.173648 + 0.984808i 0 −1.40574 + 1.17956i 0 −4.46660 2.57879i 0 −0.939693 + 0.342020i 0
319.3 0 0.173648 + 0.984808i 0 1.00588 0.844034i 0 3.15949 + 1.82413i 0 −0.939693 + 0.342020i 0
319.4 0 0.173648 + 0.984808i 0 1.30003 1.09086i 0 −1.57531 0.909508i 0 −0.939693 + 0.342020i 0
751.1 0 0.766044 0.642788i 0 −2.67232 0.972646i 0 −2.07863 1.20010i 0 0.173648 0.984808i 0
751.2 0 0.766044 0.642788i 0 −0.800469 0.291347i 0 1.70402 + 0.983816i 0 0.173648 0.984808i 0
751.3 0 0.766044 0.642788i 0 1.72729 + 0.628683i 0 −3.53090 2.03857i 0 0.173648 0.984808i 0
751.4 0 0.766044 0.642788i 0 3.79853 + 1.38255i 0 0.699779 + 0.404018i 0 0.173648 0.984808i 0
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 895.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
76.k even 18 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 912.2.ci.g 24
4.b odd 2 1 912.2.ci.h yes 24
19.f odd 18 1 912.2.ci.h yes 24
76.k even 18 1 inner 912.2.ci.g 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
912.2.ci.g 24 1.a even 1 1 trivial
912.2.ci.g 24 76.k even 18 1 inner
912.2.ci.h yes 24 4.b odd 2 1
912.2.ci.h yes 24 19.f odd 18 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(912, [\chi])$$:

 $$T_{5}^{24} - 3 T_{5}^{22} - 34 T_{5}^{21} + 84 T_{5}^{20} - 549 T_{5}^{19} + 2661 T_{5}^{18} + 4995 T_{5}^{17} + 7083 T_{5}^{16} + 43658 T_{5}^{15} + 11538 T_{5}^{14} - 281328 T_{5}^{13} + 774727 T_{5}^{12} - 1715466 T_{5}^{11} + \cdots + 34012224$$ T5^24 - 3*T5^22 - 34*T5^21 + 84*T5^20 - 549*T5^19 + 2661*T5^18 + 4995*T5^17 + 7083*T5^16 + 43658*T5^15 + 11538*T5^14 - 281328*T5^13 + 774727*T5^12 - 1715466*T5^11 + 3319938*T5^10 + 1264836*T5^9 + 1162818*T5^8 - 28829601*T5^7 + 51651441*T5^6 + 8232840*T5^5 - 30221424*T5^4 + 8992944*T5^3 + 20785248*T5^2 + 28343520*T5 + 34012224 $$T_{7}^{24} + 9 T_{7}^{23} - 15 T_{7}^{22} - 378 T_{7}^{21} + 36 T_{7}^{20} + 10998 T_{7}^{19} + 17656 T_{7}^{18} - 161091 T_{7}^{17} - 381534 T_{7}^{16} + 1697328 T_{7}^{15} + 5641794 T_{7}^{14} - 8508843 T_{7}^{13} + \cdots + 136118889$$ T7^24 + 9*T7^23 - 15*T7^22 - 378*T7^21 + 36*T7^20 + 10998*T7^19 + 17656*T7^18 - 161091*T7^17 - 381534*T7^16 + 1697328*T7^15 + 5641794*T7^14 - 8508843*T7^13 - 40858616*T7^12 + 27661365*T7^11 + 213772962*T7^10 + 21652596*T7^9 - 609061311*T7^8 - 276611085*T7^7 + 1210282413*T7^6 + 1197947151*T7^5 - 421902612*T7^4 - 787872825*T7^3 + 169674534*T7^2 + 417386925*T7 + 136118889