# Properties

 Label 224.2.bd.a Level $224$ Weight $2$ Character orbit 224.bd Analytic conductor $1.789$ Analytic rank $0$ Dimension $240$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$224 = 2^{5} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 224.bd (of order $$24$$, degree $$8$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.78864900528$$ Analytic rank: $$0$$ Dimension: $$240$$ Relative dimension: $$30$$ over $$\Q(\zeta_{24})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{24}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$240 q - 4 q^{2} - 4 q^{3} - 4 q^{4} - 4 q^{5} - 16 q^{6} - 8 q^{7} - 16 q^{8} - 4 q^{9}+O(q^{10})$$ 240 * q - 4 * q^2 - 4 * q^3 - 4 * q^4 - 4 * q^5 - 16 * q^6 - 8 * q^7 - 16 * q^8 - 4 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$240 q - 4 q^{2} - 4 q^{3} - 4 q^{4} - 4 q^{5} - 16 q^{6} - 8 q^{7} - 16 q^{8} - 4 q^{9} - 4 q^{10} - 4 q^{11} - 4 q^{12} - 16 q^{13} + 24 q^{14} - 24 q^{16} - 24 q^{18} - 4 q^{19} - 48 q^{20} - 8 q^{21} - 24 q^{22} - 12 q^{23} + 36 q^{24} - 4 q^{25} - 4 q^{26} - 16 q^{27} + 12 q^{28} - 16 q^{29} + 44 q^{30} - 56 q^{31} - 4 q^{32} - 8 q^{33} - 32 q^{34} - 32 q^{35} - 96 q^{36} - 4 q^{37} + 36 q^{38} - 4 q^{39} - 68 q^{40} - 16 q^{41} - 28 q^{42} + 44 q^{44} + 8 q^{45} - 4 q^{46} - 16 q^{48} + 8 q^{50} - 28 q^{51} - 28 q^{52} - 20 q^{53} - 92 q^{54} - 16 q^{55} - 48 q^{56} - 16 q^{57} + 28 q^{58} - 36 q^{59} + 60 q^{60} - 4 q^{61} - 16 q^{63} + 56 q^{64} - 8 q^{65} - 36 q^{66} + 36 q^{67} - 36 q^{68} - 16 q^{69} - 44 q^{70} + 48 q^{71} - 4 q^{72} - 4 q^{73} + 68 q^{74} + 16 q^{75} - 16 q^{76} - 8 q^{77} - 120 q^{78} - 16 q^{80} - 44 q^{82} - 96 q^{83} - 28 q^{84} - 56 q^{85} - 4 q^{86} - 4 q^{87} + 40 q^{88} - 4 q^{89} + 224 q^{90} - 56 q^{91} - 80 q^{92} + 20 q^{93} + 4 q^{94} - 8 q^{95} - 32 q^{96} - 32 q^{97} + 96 q^{98} + 8 q^{99}+O(q^{100})$$ 240 * q - 4 * q^2 - 4 * q^3 - 4 * q^4 - 4 * q^5 - 16 * q^6 - 8 * q^7 - 16 * q^8 - 4 * q^9 - 4 * q^10 - 4 * q^11 - 4 * q^12 - 16 * q^13 + 24 * q^14 - 24 * q^16 - 24 * q^18 - 4 * q^19 - 48 * q^20 - 8 * q^21 - 24 * q^22 - 12 * q^23 + 36 * q^24 - 4 * q^25 - 4 * q^26 - 16 * q^27 + 12 * q^28 - 16 * q^29 + 44 * q^30 - 56 * q^31 - 4 * q^32 - 8 * q^33 - 32 * q^34 - 32 * q^35 - 96 * q^36 - 4 * q^37 + 36 * q^38 - 4 * q^39 - 68 * q^40 - 16 * q^41 - 28 * q^42 + 44 * q^44 + 8 * q^45 - 4 * q^46 - 16 * q^48 + 8 * q^50 - 28 * q^51 - 28 * q^52 - 20 * q^53 - 92 * q^54 - 16 * q^55 - 48 * q^56 - 16 * q^57 + 28 * q^58 - 36 * q^59 + 60 * q^60 - 4 * q^61 - 16 * q^63 + 56 * q^64 - 8 * q^65 - 36 * q^66 + 36 * q^67 - 36 * q^68 - 16 * q^69 - 44 * q^70 + 48 * q^71 - 4 * q^72 - 4 * q^73 + 68 * q^74 + 16 * q^75 - 16 * q^76 - 8 * q^77 - 120 * q^78 - 16 * q^80 - 44 * q^82 - 96 * q^83 - 28 * q^84 - 56 * q^85 - 4 * q^86 - 4 * q^87 + 40 * q^88 - 4 * q^89 + 224 * q^90 - 56 * q^91 - 80 * q^92 + 20 * q^93 + 4 * q^94 - 8 * q^95 - 32 * q^96 - 32 * q^97 + 96 * q^98 + 8 * 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 −1.41418 0.00960786i −1.76562 + 2.30100i 1.99982 + 0.0271745i 2.61963 2.01011i 2.51901 3.23706i −1.37753 2.25885i −2.82784 0.0576436i −1.40073 5.22759i −3.72395 + 2.81750i
37.2 −1.41287 + 0.0616517i −0.0537004 + 0.0699837i 1.99240 0.174212i −0.644996 + 0.494923i 0.0715570 0.102189i −2.22522 + 1.43123i −2.80426 + 0.368973i 0.774443 + 2.89026i 0.880782 0.739026i
37.3 −1.32342 + 0.498548i 0.615442 0.802059i 1.50290 1.31958i −1.98693 + 1.52462i −0.414625 + 1.36829i 2.32280 1.26674i −1.33110 + 2.49563i 0.511927 + 1.91054i 1.86945 3.00830i
37.4 −1.30182 0.552515i 1.55155 2.02202i 1.38945 + 1.43855i 0.189180 0.145163i −3.13703 + 1.77505i −1.52456 2.16234i −1.01400 2.64042i −0.904803 3.37677i −0.326482 + 0.0844507i
37.5 −1.29489 0.568565i −1.02607 + 1.33720i 1.35347 + 1.47246i −1.35700 + 1.04126i 2.08893 1.14813i 2.49879 0.869520i −0.915400 2.67620i 0.0411741 + 0.153664i 2.34919 0.576775i
37.6 −1.16516 + 0.801498i 1.71349 2.23306i 0.715201 1.86775i 1.90405 1.46103i −0.206695 + 3.97523i −0.262719 + 2.63268i 0.663674 + 2.74946i −1.27406 4.75485i −1.04751 + 3.22843i
37.7 −1.13266 0.846802i 0.0986144 0.128517i 0.565853 + 1.91828i 2.39507 1.83780i −0.220525 + 0.0620594i 0.134369 + 2.64234i 0.983485 2.65193i 0.769665 + 2.87243i −4.26906 + 0.0534610i
37.8 −0.940911 + 1.05579i −1.04486 + 1.36169i −0.229373 1.98680i 1.80014 1.38129i −0.454533 2.38438i 2.29881 + 1.30976i 2.31346 + 1.62724i 0.0139922 + 0.0522197i −0.235417 + 3.20024i
37.9 −0.760837 + 1.19211i −1.44087 + 1.87778i −0.842255 1.81400i −2.75892 + 2.11700i −1.14225 3.14636i −1.14468 2.38531i 2.80331 + 0.376098i −0.673491 2.51350i −0.424604 4.89963i
37.10 −0.631870 1.26520i 0.678058 0.883663i −1.20148 + 1.59889i −2.77837 + 2.13192i −1.54646 0.299522i −2.64040 0.168196i 2.78210 + 0.509831i 0.455360 + 1.69943i 4.45288 + 2.16811i
37.11 −0.480321 1.33015i −0.758838 + 0.988936i −1.53858 + 1.27780i 0.171111 0.131298i 1.67992 + 0.534359i 1.01488 2.44336i 2.43867 + 1.43279i 0.374296 + 1.39689i −0.256834 0.164537i
37.12 −0.460775 + 1.33704i 1.47397 1.92092i −1.57537 1.23215i 0.169674 0.130196i 1.88918 + 2.85588i −0.114544 2.64327i 2.37333 1.53860i −0.740878 2.76499i 0.0958956 + 0.286853i
37.13 −0.436149 + 1.34528i 0.756979 0.986515i −1.61955 1.17348i −2.52010 + 1.93374i 0.996981 + 1.44862i −0.259491 + 2.63300i 2.28503 1.66693i 0.376264 + 1.40424i −1.50228 4.23364i
37.14 −0.399779 1.35653i 1.62045 2.11181i −1.68035 + 1.08463i 1.05240 0.807534i −3.51256 1.35393i 2.64452 0.0808662i 2.14310 + 1.84584i −1.05744 3.94640i −1.51617 1.10478i
37.15 0.0713999 + 1.41241i 0.118403 0.154306i −1.98980 + 0.201692i 2.39583 1.83839i 0.226398 + 0.156217i 1.47323 2.19764i −0.426943 2.79602i 0.766666 + 2.86124i 2.76762 + 3.25263i
37.16 0.131818 1.40806i −0.751048 + 0.978785i −1.96525 0.371216i −1.09067 + 0.836901i 1.27918 + 1.18654i 0.448114 + 2.60753i −0.781748 + 2.71825i 0.382510 + 1.42755i 1.03463 + 1.64604i
37.17 0.148173 + 1.40643i −1.23048 + 1.60359i −1.95609 + 0.416790i 0.591453 0.453838i −2.43766 1.49297i −2.18592 + 1.49056i −0.876026 2.68935i −0.280969 1.04859i 0.725929 + 0.764591i
37.18 0.361256 1.36729i 0.338207 0.440760i −1.73899 0.987888i 3.17926 2.43953i −0.480469 0.621656i −2.54805 0.712336i −1.97895 + 2.02083i 0.696572 + 2.59964i −2.18703 5.22828i
37.19 0.674919 1.24277i −1.68352 + 2.19400i −1.08897 1.67754i −1.03656 + 0.795384i 1.59041 + 3.57300i −2.17678 1.50387i −2.81977 + 0.221138i −1.20296 4.48951i 0.288885 + 1.82503i
37.20 0.699836 + 1.22891i 0.666690 0.868847i −1.02046 + 1.72008i −0.367682 + 0.282133i 1.53431 + 0.211254i 1.90974 + 1.83109i −2.82798 0.0502881i 0.466037 + 1.73928i −0.604034 0.254404i
See next 80 embeddings (of 240 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 221.30 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
32.g even 8 1 inner
224.bd even 24 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 224.2.bd.a 240
4.b odd 2 1 896.2.bh.a 240
7.c even 3 1 inner 224.2.bd.a 240
28.g odd 6 1 896.2.bh.a 240
32.g even 8 1 inner 224.2.bd.a 240
32.h odd 8 1 896.2.bh.a 240
224.bd even 24 1 inner 224.2.bd.a 240
224.bf odd 24 1 896.2.bh.a 240

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.2.bd.a 240 1.a even 1 1 trivial
224.2.bd.a 240 7.c even 3 1 inner
224.2.bd.a 240 32.g even 8 1 inner
224.2.bd.a 240 224.bd even 24 1 inner
896.2.bh.a 240 4.b odd 2 1
896.2.bh.a 240 28.g odd 6 1
896.2.bh.a 240 32.h odd 8 1
896.2.bh.a 240 224.bf odd 24 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(224, [\chi])$$.