# Properties

 Label 315.2.k.b Level 315 Weight 2 Character orbit 315.k Analytic conductor 2.515 Analytic rank 0 Dimension 24 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$315 = 3^{2} \cdot 5 \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 315.k (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.51528766367$$ Analytic rank: $$0$$ Dimension: $$24$$ Relative dimension: $$12$$ over $$\Q(\zeta_{3})$$ Coefficient ring index: multiple of None Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$24q - q^{2} + q^{3} - 7q^{4} - 24q^{5} + 3q^{6} + 7q^{7} + 12q^{8} + 9q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$24q - q^{2} + q^{3} - 7q^{4} - 24q^{5} + 3q^{6} + 7q^{7} + 12q^{8} + 9q^{9} + q^{10} - 2q^{11} - 16q^{12} - 4q^{13} - 13q^{14} - q^{15} - 5q^{16} - 7q^{17} - 12q^{18} - 2q^{19} + 7q^{20} - 5q^{21} + 19q^{22} - 2q^{23} - 30q^{24} + 24q^{25} + 11q^{26} - 11q^{27} - 28q^{28} - 3q^{30} + 8q^{31} + 17q^{32} + q^{33} + q^{34} - 7q^{35} - 25q^{36} + 17q^{37} + 70q^{38} - 8q^{39} - 12q^{40} + 20q^{41} - 15q^{42} + 31q^{43} - 7q^{44} - 9q^{45} - 10q^{46} - 31q^{47} - 36q^{48} - 11q^{49} - q^{50} - 37q^{51} + 8q^{52} + 8q^{53} + 111q^{54} + 2q^{55} + 45q^{56} + 5q^{57} - 90q^{58} - 21q^{59} + 16q^{60} + 5q^{61} + 14q^{62} - 9q^{63} - 56q^{64} + 4q^{65} + 46q^{66} + 43q^{67} + 96q^{68} + 26q^{69} + 13q^{70} + 24q^{71} - 69q^{72} - 18q^{73} - 18q^{74} + q^{75} - 13q^{76} + 5q^{77} + 19q^{78} + 40q^{79} + 5q^{80} - 39q^{81} + 5q^{82} - 60q^{83} + 47q^{84} + 7q^{85} - 24q^{86} - 61q^{87} - 100q^{88} - 4q^{89} + 12q^{90} - 33q^{91} - 18q^{92} - 8q^{93} - 11q^{94} + 2q^{95} + 6q^{96} + 6q^{97} - 15q^{98} - q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
16.1 −1.17004 2.02657i 1.32145 + 1.11972i −1.73798 + 3.01027i −1.00000 0.723033 3.98812i −1.00344 + 2.44808i 3.45385 0.492465 + 2.95930i 1.17004 + 2.02657i
16.2 −1.16277 2.01398i 0.434379 1.67670i −1.70408 + 2.95156i −1.00000 −3.88192 + 1.07479i 0.459529 2.60554i 3.27475 −2.62263 1.45664i 1.16277 + 2.01398i
16.3 −1.08176 1.87367i −1.55999 + 0.752618i −1.34041 + 2.32167i −1.00000 3.09769 + 2.10874i 2.14928 + 1.54292i 1.47299 1.86713 2.34815i 1.08176 + 1.87367i
16.4 −0.599034 1.03756i 1.72885 + 0.105192i 0.282317 0.488988i −1.00000 −0.926499 1.85680i 2.04342 1.68061i −3.07261 2.97787 + 0.363723i 0.599034 + 1.03756i
16.5 −0.259245 0.449026i −1.70948 + 0.278682i 0.865584 1.49923i −1.00000 0.568312 + 0.695356i −1.91816 + 1.82226i −1.93458 2.84467 0.952806i 0.259245 + 0.449026i
16.6 −0.148731 0.257610i 0.310785 1.70394i 0.955758 1.65542i −1.00000 −0.485175 + 0.173367i −2.64436 + 0.0857253i −1.16353 −2.80683 1.05912i 0.148731 + 0.257610i
16.7 0.154039 + 0.266804i 1.05264 1.37548i 0.952544 1.64985i −1.00000 0.529131 + 0.0689719i 2.20433 + 1.46319i 1.20307 −0.783880 2.89578i −0.154039 0.266804i
16.8 0.304907 + 0.528114i −1.22862 + 1.22086i 0.814064 1.41000i −1.00000 −1.01937 0.276604i 1.83653 1.90451i 2.21248 0.0190166 2.99994i −0.304907 0.528114i
16.9 0.517769 + 0.896802i −1.26722 1.18074i 0.463831 0.803378i −1.00000 0.402766 1.74780i −1.07729 2.41649i 3.03170 0.211690 + 2.99252i −0.517769 0.896802i
16.10 0.805191 + 1.39463i 1.04104 + 1.38428i −0.296664 + 0.513837i −1.00000 −1.09233 + 2.56648i 2.48389 + 0.911196i 2.26528 −0.832482 + 2.88218i −0.805191 1.39463i
16.11 0.859635 + 1.48893i −1.31121 1.13170i −0.477944 + 0.827824i −1.00000 0.557856 2.92514i 0.594106 + 2.57818i 1.79511 0.438533 + 2.96778i −0.859635 1.48893i
16.12 1.28004 + 2.21710i 1.68737 0.390872i −2.27701 + 3.94390i −1.00000 3.02651 + 3.24073i −1.62783 + 2.08571i −6.53853 2.69444 1.31909i −1.28004 2.21710i
256.1 −1.17004 + 2.02657i 1.32145 1.11972i −1.73798 3.01027i −1.00000 0.723033 + 3.98812i −1.00344 2.44808i 3.45385 0.492465 2.95930i 1.17004 2.02657i
256.2 −1.16277 + 2.01398i 0.434379 + 1.67670i −1.70408 2.95156i −1.00000 −3.88192 1.07479i 0.459529 + 2.60554i 3.27475 −2.62263 + 1.45664i 1.16277 2.01398i
256.3 −1.08176 + 1.87367i −1.55999 0.752618i −1.34041 2.32167i −1.00000 3.09769 2.10874i 2.14928 1.54292i 1.47299 1.86713 + 2.34815i 1.08176 1.87367i
256.4 −0.599034 + 1.03756i 1.72885 0.105192i 0.282317 + 0.488988i −1.00000 −0.926499 + 1.85680i 2.04342 + 1.68061i −3.07261 2.97787 0.363723i 0.599034 1.03756i
256.5 −0.259245 + 0.449026i −1.70948 0.278682i 0.865584 + 1.49923i −1.00000 0.568312 0.695356i −1.91816 1.82226i −1.93458 2.84467 + 0.952806i 0.259245 0.449026i
256.6 −0.148731 + 0.257610i 0.310785 + 1.70394i 0.955758 + 1.65542i −1.00000 −0.485175 0.173367i −2.64436 0.0857253i −1.16353 −2.80683 + 1.05912i 0.148731 0.257610i
256.7 0.154039 0.266804i 1.05264 + 1.37548i 0.952544 + 1.64985i −1.00000 0.529131 0.0689719i 2.20433 1.46319i 1.20307 −0.783880 + 2.89578i −0.154039 + 0.266804i
256.8 0.304907 0.528114i −1.22862 1.22086i 0.814064 + 1.41000i −1.00000 −1.01937 + 0.276604i 1.83653 + 1.90451i 2.21248 0.0190166 + 2.99994i −0.304907 + 0.528114i
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 256.12 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.g even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 315.2.k.b 24
3.b odd 2 1 945.2.k.b 24
7.c even 3 1 315.2.l.b yes 24
9.c even 3 1 315.2.l.b yes 24
9.d odd 6 1 945.2.l.b 24
21.h odd 6 1 945.2.l.b 24
63.g even 3 1 inner 315.2.k.b 24
63.n odd 6 1 945.2.k.b 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.2.k.b 24 1.a even 1 1 trivial
315.2.k.b 24 63.g even 3 1 inner
315.2.l.b yes 24 7.c even 3 1
315.2.l.b yes 24 9.c even 3 1
945.2.k.b 24 3.b odd 2 1
945.2.k.b 24 63.n odd 6 1
945.2.l.b 24 9.d odd 6 1
945.2.l.b 24 21.h odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{24} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(315, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database