# Properties

 Label 756.2.ci.a Level 756 Weight 2 Character orbit 756.ci Analytic conductor 6.037 Analytic rank 0 Dimension 840 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$756 = 2^{2} \cdot 3^{3} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 756.ci (of order $$18$$, degree $$6$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.03669039281$$ Analytic rank: $$0$$ Dimension: $$840$$ Relative dimension: $$140$$ over $$\Q(\zeta_{18})$$ Coefficient ring index: multiple of None 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.

 $$\operatorname{Tr}(f)(q) =$$ $$840q - 3q^{2} - 3q^{4} - 6q^{5} - 12q^{6} - 18q^{8} - 6q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$840q - 3q^{2} - 3q^{4} - 6q^{5} - 12q^{6} - 18q^{8} - 6q^{9} + 3q^{10} - 3q^{12} - 24q^{13} + 24q^{14} - 3q^{16} - 18q^{17} - 3q^{18} - 12q^{20} - 12q^{21} - 12q^{22} + 48q^{24} - 6q^{25} - 12q^{28} - 12q^{29} - 21q^{30} - 63q^{32} - 6q^{33} - 6q^{34} - 42q^{36} - 12q^{37} + 45q^{38} - 33q^{40} - 24q^{41} - 21q^{42} + 6q^{45} - 6q^{46} - 78q^{48} - 12q^{49} - 27q^{50} - 3q^{52} + 39q^{54} - 27q^{56} - 6q^{57} - 3q^{58} - 63q^{60} - 6q^{61} - 117q^{62} - 6q^{64} - 54q^{65} - 3q^{66} - 12q^{68} - 48q^{69} - 21q^{70} - 9q^{72} - 12q^{73} + 15q^{74} + 12q^{77} + 6q^{78} - 54q^{81} - 6q^{82} - 39q^{84} + 6q^{85} - 9q^{86} - 27q^{88} - 18q^{89} - 39q^{90} + 48q^{92} - 6q^{93} - 3q^{94} + 213q^{96} - 24q^{97} + 162q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
95.1 −1.41419 + 0.00832720i 1.09084 1.34539i 1.99986 0.0235525i 2.13665 + 0.376749i −1.53144 + 1.91172i 2.48173 + 0.917059i −2.82799 + 0.0499609i −0.620154 2.93520i −3.02476 0.515002i
95.2 −1.41419 + 0.00841960i −1.72851 + 0.110651i 1.99986 0.0238138i 0.00617965 + 0.00108964i 2.44351 0.171035i 2.62747 + 0.310501i −2.82798 + 0.0505152i 2.97551 0.382524i −0.00874837 0.00148893i
95.3 −1.41401 0.0242190i −0.531095 + 1.64862i 1.99883 + 0.0684917i 1.58913 + 0.280207i 0.790900 2.31829i −2.09916 1.61045i −2.82469 0.145257i −2.43588 1.75115i −2.24025 0.434701i
95.4 −1.41331 + 0.0506390i −0.250798 1.71380i 1.99487 0.143137i −1.97895 0.348942i 0.441239 + 2.40942i −1.91178 + 1.82896i −2.81212 + 0.303315i −2.87420 + 0.859633i 2.81453 + 0.392950i
95.5 −1.40530 + 0.158515i 1.04908 + 1.37820i 1.94975 0.445523i 0.0136217 + 0.00240188i −1.69274 1.77049i −2.34832 1.21877i −2.66936 + 0.935159i −0.798872 + 2.89168i −0.0195234 0.00121611i
95.6 −1.40027 0.198086i 0.186948 + 1.72193i 1.92152 + 0.554748i −2.73494 0.482245i 0.0793128 2.44821i 0.265748 + 2.63237i −2.58077 1.15742i −2.93010 + 0.643822i 3.73414 + 1.21703i
95.7 −1.39717 + 0.218922i −0.617768 1.61814i 1.90415 0.611740i 3.83680 + 0.676532i 1.21737 + 2.12556i 1.64737 2.07031i −2.52649 + 1.27156i −2.23673 + 1.99926i −5.50876 0.105268i
95.8 −1.39122 0.253995i 1.51307 + 0.842975i 1.87097 + 0.706724i 3.16384 + 0.557871i −1.89090 1.55707i 0.270632 + 2.63187i −2.42343 1.45842i 1.57879 + 2.55097i −4.25990 1.57972i
95.9 −1.38746 + 0.273772i 0.796278 1.53816i 1.85010 0.759696i −0.188974 0.0333212i −0.683700 + 2.35214i −1.65353 2.06539i −2.35896 + 1.56055i −1.73188 2.44961i 0.271317 0.00550392i
95.10 −1.37324 0.337961i −1.07725 + 1.35630i 1.77156 + 0.928203i −0.713698 0.125844i 1.93769 1.49845i 2.30108 1.30576i −2.11908 1.87336i −0.679077 2.92213i 0.937547 + 0.414016i
95.11 −1.37052 0.348809i −1.72044 0.200188i 1.75666 + 0.956102i −4.20241 0.740998i 2.28808 + 0.874469i −1.31632 2.29506i −2.07405 1.92310i 2.91985 + 0.688824i 5.50103 + 2.48139i
95.12 −1.36638 + 0.364685i 1.60153 0.659617i 1.73401 0.996600i −2.53354 0.446731i −1.94776 + 1.48535i 2.18735 + 1.48846i −2.00588 + 1.99411i 2.12981 2.11280i 3.62470 0.313538i
95.13 −1.35923 0.390498i −1.12398 1.31783i 1.69502 + 1.06155i −0.140742 0.0248166i 1.01314 + 2.23015i 1.14582 + 2.38476i −1.88939 2.10480i −0.473344 + 2.96242i 0.181610 + 0.0886910i
95.14 −1.35369 + 0.409293i −1.06364 + 1.36699i 1.66496 1.10811i 4.32435 + 0.762500i 0.880342 2.28583i −0.0657043 + 2.64494i −1.80029 + 2.18150i −0.737329 2.90798i −6.16592 + 0.737740i
95.15 −1.34821 0.427004i 1.72966 0.0910470i 1.63533 + 1.15138i −3.12490 0.551004i −2.37082 0.615820i −2.64376 0.102727i −1.71313 2.25060i 2.98342 0.314960i 3.97773 + 2.07721i
95.16 −1.34200 + 0.446123i −1.17175 1.27554i 1.60195 1.19740i −2.14959 0.379031i 2.14154 + 1.18903i 1.07635 2.41691i −1.61563 + 2.32158i −0.253990 + 2.98923i 3.05386 0.450322i
95.17 −1.33299 0.472387i −1.55434 0.764214i 1.55370 + 1.25937i 3.05802 + 0.539212i 1.71091 + 1.75294i −2.62079 0.362599i −1.47615 2.41267i 1.83195 + 2.37570i −3.82158 2.16333i
95.18 −1.32216 0.501896i 1.68643 0.394900i 1.49620 + 1.32717i 0.496060 + 0.0874688i −2.42793 0.324294i 1.26084 2.32600i −1.31211 2.50567i 2.68811 1.33194i −0.611969 0.364618i
95.19 −1.32113 0.504600i 0.893683 + 1.48369i 1.49076 + 1.33328i 3.15348 + 0.556044i −0.431999 2.41109i 1.10140 2.40560i −1.29670 2.51367i −1.40266 + 2.65189i −3.88557 2.32585i
95.20 −1.30138 + 0.553549i −1.64933 0.528879i 1.38717 1.44075i 0.746991 + 0.131715i 2.43916 0.224713i −2.50378 + 0.855036i −1.00770 + 2.64283i 2.44057 + 1.74459i −1.04503 + 0.242086i
See next 80 embeddings (of 840 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 695.140 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
189.bc odd 18 1 inner
756.ci even 18 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 756.2.ci.a yes 840
4.b odd 2 1 inner 756.2.ci.a yes 840
7.c even 3 1 756.2.bs.a 840
27.f odd 18 1 756.2.bs.a 840
28.g odd 6 1 756.2.bs.a 840
108.l even 18 1 756.2.bs.a 840
189.bc odd 18 1 inner 756.2.ci.a yes 840
756.ci even 18 1 inner 756.2.ci.a yes 840

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
756.2.bs.a 840 7.c even 3 1
756.2.bs.a 840 27.f odd 18 1
756.2.bs.a 840 28.g odd 6 1
756.2.bs.a 840 108.l even 18 1
756.2.ci.a yes 840 1.a even 1 1 trivial
756.2.ci.a yes 840 4.b odd 2 1 inner
756.2.ci.a yes 840 189.bc odd 18 1 inner
756.2.ci.a yes 840 756.ci even 18 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(756, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database