# Properties

 Label 315.2.bj.a Level 315 Weight 2 Character orbit 315.bj Analytic conductor 2.515 Analytic rank 0 Dimension 12 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.bj (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.51528766367$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$6$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} + \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$3^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{2} + ( -\beta_{2} + \beta_{4} + \beta_{5} ) q^{4} + ( -1 + \beta_{4} ) q^{5} + ( \beta_{1} - \beta_{7} ) q^{7} + ( 2 \beta_{1} - \beta_{5} - \beta_{7} - \beta_{9} ) q^{8} +O(q^{10})$$ $$q + \beta_{3} q^{2} + ( -\beta_{2} + \beta_{4} + \beta_{5} ) q^{4} + ( -1 + \beta_{4} ) q^{5} + ( \beta_{1} - \beta_{7} ) q^{7} + ( 2 \beta_{1} - \beta_{5} - \beta_{7} - \beta_{9} ) q^{8} + ( \beta_{1} - \beta_{3} ) q^{10} + ( 2 - \beta_{4} - \beta_{7} + \beta_{8} + \beta_{11} ) q^{11} + ( 1 - 2 \beta_{4} + \beta_{5} - \beta_{6} + \beta_{8} + \beta_{9} - \beta_{11} ) q^{13} + ( -2 - \beta_{2} - \beta_{3} + 2 \beta_{4} + \beta_{5} + \beta_{10} ) q^{14} + ( -2 - 2 \beta_{1} + \beta_{3} + 2 \beta_{4} + \beta_{5} + \beta_{6} - \beta_{8} + \beta_{11} ) q^{16} + ( -\beta_{2} - \beta_{4} + \beta_{5} - \beta_{6} - \beta_{8} ) q^{17} + ( -\beta_{3} - \beta_{6} - \beta_{8} - \beta_{11} ) q^{19} + ( -1 + \beta_{2} ) q^{20} + ( 2 - \beta_{1} + 2 \beta_{3} - \beta_{6} + \beta_{7} - \beta_{8} - \beta_{11} ) q^{22} + ( \beta_{2} + \beta_{3} - 2 \beta_{6} + 2 \beta_{7} - 2 \beta_{8} - \beta_{10} - \beta_{11} ) q^{23} -\beta_{4} q^{25} + ( -1 - 2 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} - 2 \beta_{6} + 2 \beta_{7} + 2 \beta_{8} + 2 \beta_{9} - \beta_{10} - 3 \beta_{11} ) q^{26} + ( -1 + 3 \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{4} - 2 \beta_{5} + \beta_{10} ) q^{28} + ( -\beta_{1} + \beta_{2} - \beta_{5} + \beta_{7} - \beta_{8} + \beta_{9} + \beta_{11} ) q^{29} + ( 2 + \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} + 2 \beta_{6} - 3 \beta_{7} + \beta_{8} + 3 \beta_{11} ) q^{31} + ( 6 + 3 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} - 3 \beta_{4} - 2 \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} ) q^{32} + ( 1 + 3 \beta_{1} - 2 \beta_{4} - 2 \beta_{5} + \beta_{6} - \beta_{7} - 2 \beta_{9} ) q^{34} + ( -\beta_{3} + \beta_{8} ) q^{35} + ( -1 - 2 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - 2 \beta_{7} - 2 \beta_{9} + \beta_{10} + \beta_{11} ) q^{37} + ( \beta_{2} - 5 \beta_{4} - \beta_{5} + \beta_{6} + \beta_{8} ) q^{38} + ( -\beta_{2} - 2 \beta_{3} + \beta_{10} + \beta_{11} ) q^{40} + ( 3 + 2 \beta_{1} - 4 \beta_{3} + \beta_{6} - \beta_{7} + 2 \beta_{8} + 2 \beta_{11} ) q^{41} + ( -1 + \beta_{1} - 2 \beta_{3} - \beta_{5} - 2 \beta_{6} + \beta_{7} - 2 \beta_{8} - \beta_{9} + 2 \beta_{10} ) q^{43} + ( -\beta_{2} + 3 \beta_{3} + \beta_{5} - \beta_{6} - \beta_{8} - \beta_{10} - 2 \beta_{11} ) q^{44} + ( \beta_{1} + \beta_{3} - \beta_{4} + 2 \beta_{6} + \beta_{7} - \beta_{8} - 2 \beta_{9} - 2 \beta_{10} + \beta_{11} ) q^{46} + ( -1 - 2 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + 3 \beta_{7} + 2 \beta_{9} - \beta_{10} - 2 \beta_{11} ) q^{47} + ( \beta_{2} - \beta_{3} + 2 \beta_{4} - \beta_{6} + \beta_{8} + 2 \beta_{9} + \beta_{11} ) q^{49} -\beta_{1} q^{50} + ( -2 - 3 \beta_{1} + \beta_{2} + 3 \beta_{3} + \beta_{4} + \beta_{5} - 2 \beta_{6} + 4 \beta_{7} + \beta_{8} + 3 \beta_{9} - 3 \beta_{10} - 4 \beta_{11} ) q^{52} + ( 2 + \beta_{1} - \beta_{3} - \beta_{4} + \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} ) q^{53} + ( -1 + 2 \beta_{4} + \beta_{6} + \beta_{7} ) q^{55} + ( -6 - 5 \beta_{1} + \beta_{2} + 5 \beta_{4} + \beta_{5} + 2 \beta_{6} - \beta_{7} + 2 \beta_{8} + \beta_{9} + \beta_{11} ) q^{56} + ( 2 - 2 \beta_{1} + \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + \beta_{7} - \beta_{11} ) q^{58} + ( \beta_{1} + \beta_{3} - 4 \beta_{4} + \beta_{6} - \beta_{9} - \beta_{10} ) q^{59} + ( 1 + 2 \beta_{2} - 4 \beta_{3} + \beta_{4} - \beta_{5} + 2 \beta_{6} + 2 \beta_{8} + 2 \beta_{11} ) q^{61} + ( 1 - 2 \beta_{1} - \beta_{2} + 4 \beta_{3} - \beta_{5} - \beta_{6} - 3 \beta_{8} - \beta_{9} + 2 \beta_{10} - \beta_{11} ) q^{62} + ( -4 - 4 \beta_{1} + 2 \beta_{2} + 8 \beta_{3} ) q^{64} + ( 1 + \beta_{2} + \beta_{4} + \beta_{7} - \beta_{10} ) q^{65} + ( -2 \beta_{2} + 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} - \beta_{7} + 3 \beta_{8} - \beta_{11} ) q^{67} + ( -9 - 6 \beta_{1} + 3 \beta_{3} + 9 \beta_{4} + 3 \beta_{5} + 2 \beta_{6} + \beta_{7} - 2 \beta_{8} + \beta_{11} ) q^{68} + ( -\beta_{1} + 2 \beta_{2} + \beta_{3} - 2 \beta_{4} - \beta_{6} + \beta_{7} + \beta_{9} - \beta_{10} - \beta_{11} ) q^{70} + ( -4 + 8 \beta_{4} - \beta_{6} - \beta_{7} + \beta_{8} - \beta_{11} ) q^{71} + ( -4 + \beta_{1} + 2 \beta_{2} - \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - \beta_{6} + \beta_{7} + 2 \beta_{8} + 2 \beta_{9} - 2 \beta_{10} - \beta_{11} ) q^{73} + ( 10 + 4 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} - 5 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - 3 \beta_{7} - \beta_{8} - 2 \beta_{9} + 2 \beta_{10} + 3 \beta_{11} ) q^{74} + ( -1 - 7 \beta_{1} + 2 \beta_{4} + 2 \beta_{5} + \beta_{6} + 3 \beta_{7} + 2 \beta_{9} ) q^{76} + ( -2 + 2 \beta_{1} - 3 \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} + 3 \beta_{6} - 2 \beta_{7} + \beta_{8} + 2 \beta_{11} ) q^{77} + ( 2 - 4 \beta_{1} + 2 \beta_{3} - 2 \beta_{4} + \beta_{5} + 2 \beta_{7} - 2 \beta_{11} ) q^{79} + ( \beta_{1} + \beta_{2} + \beta_{3} - 2 \beta_{4} - \beta_{5} - \beta_{7} + \beta_{8} - \beta_{11} ) q^{80} + ( -4 + 4 \beta_{2} + 3 \beta_{3} - 4 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + \beta_{7} - 2 \beta_{8} - \beta_{11} ) q^{82} + ( -1 + 2 \beta_{1} - 3 \beta_{2} - 4 \beta_{3} - \beta_{5} + 3 \beta_{6} - 4 \beta_{7} + \beta_{8} - \beta_{9} + 2 \beta_{10} + 3 \beta_{11} ) q^{83} + ( 1 + \beta_{2} + \beta_{6} - \beta_{7} + \beta_{8} + \beta_{11} ) q^{85} + ( -3 + 4 \beta_{2} - 2 \beta_{3} - 3 \beta_{4} - 2 \beta_{5} + 3 \beta_{6} - 2 \beta_{7} + 3 \beta_{8} + \beta_{11} ) q^{86} + ( -3 \beta_{2} + \beta_{4} + 3 \beta_{5} + 3 \beta_{6} + 2 \beta_{8} - \beta_{9} - \beta_{10} ) q^{88} + ( 1 - 2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - 2 \beta_{9} + \beta_{10} - \beta_{11} ) q^{89} + ( -5 + 2 \beta_{1} + \beta_{2} + 2 \beta_{3} + 4 \beta_{4} + \beta_{5} - 2 \beta_{6} + 4 \beta_{7} - 2 \beta_{8} + 2 \beta_{9} - 2 \beta_{10} - 2 \beta_{11} ) q^{91} + ( -1 + \beta_{1} - 3 \beta_{2} + 2 \beta_{4} + 2 \beta_{5} + 3 \beta_{6} - \beta_{7} - 3 \beta_{8} - 4 \beta_{9} + 3 \beta_{11} ) q^{92} + ( -2 \beta_{6} + 2 \beta_{7} + 2 \beta_{8} + 2 \beta_{9} - 2 \beta_{10} - 2 \beta_{11} ) q^{94} + ( -\beta_{1} + \beta_{3} - \beta_{7} + \beta_{8} + \beta_{11} ) q^{95} + ( -2 + 5 \beta_{1} + 4 \beta_{4} - 2 \beta_{5} - \beta_{6} - 3 \beta_{7} - \beta_{8} - 2 \beta_{9} + \beta_{11} ) q^{97} + ( 2 \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - 3 \beta_{6} + 4 \beta_{7} + \beta_{8} + 2 \beta_{9} - \beta_{10} - 5 \beta_{11} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12q + 8q^{4} - 6q^{5} - 2q^{7} + O(q^{10})$$ $$12q + 8q^{4} - 6q^{5} - 2q^{7} + 12q^{11} - 12q^{14} - 16q^{16} + 6q^{19} - 16q^{20} + 32q^{22} + 12q^{23} - 6q^{25} - 20q^{28} + 6q^{31} + 60q^{32} - 2q^{35} - 10q^{37} - 36q^{38} + 24q^{41} - 4q^{43} + 12q^{44} - 4q^{46} + 6q^{49} + 12q^{53} - 60q^{56} + 20q^{58} - 24q^{59} + 24q^{62} - 56q^{64} + 18q^{65} + 6q^{67} - 60q^{68} - 12q^{70} - 42q^{73} + 84q^{74} - 36q^{77} + 18q^{79} - 16q^{80} - 72q^{82} - 24q^{83} - 84q^{86} + 4q^{88} + 12q^{89} - 18q^{91} + 12q^{94} - 6q^{95} + 12q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} + 20 x^{10} + 144 x^{8} + 452 x^{6} + 604 x^{4} + 312 x^{2} + 36$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} + 3$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{6} + 10 \nu^{4} + 22 \nu^{2} + 4 \nu + 6$$$$)/8$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{11} + 20 \nu^{9} + 144 \nu^{7} + 446 \nu^{5} + 544 \nu^{3} + 180 \nu + 24$$$$)/48$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{11} - 20 \nu^{9} - 142 \nu^{7} - 426 \nu^{5} - 500 \nu^{3} + 8 \nu^{2} - 168 \nu + 24$$$$)/16$$ $$\beta_{6}$$ $$=$$ $$($$$$-5 \nu^{11} - 94 \nu^{9} - 606 \nu^{7} - 6 \nu^{6} - 1510 \nu^{5} - 84 \nu^{4} - 1064 \nu^{3} - 300 \nu^{2} + 72 \nu - 156$$$$)/48$$ $$\beta_{7}$$ $$=$$ $$($$$$-5 \nu^{11} - 94 \nu^{9} - 606 \nu^{7} + 6 \nu^{6} - 1510 \nu^{5} + 84 \nu^{4} - 1064 \nu^{3} + 300 \nu^{2} + 72 \nu + 156$$$$)/48$$ $$\beta_{8}$$ $$=$$ $$($$$$-5 \nu^{11} + 6 \nu^{10} - 94 \nu^{9} + 114 \nu^{8} - 612 \nu^{7} + 756 \nu^{6} - 1594 \nu^{5} + 2040 \nu^{4} - 1412 \nu^{3} + 1932 \nu^{2} - 324 \nu + 384$$$$)/48$$ $$\beta_{9}$$ $$=$$ $$($$$$4 \nu^{11} + 77 \nu^{9} + 516 \nu^{7} - 3 \nu^{6} + 1394 \nu^{5} - 42 \nu^{4} + 1306 \nu^{3} - 162 \nu^{2} + 360 \nu - 114$$$$)/24$$ $$\beta_{10}$$ $$=$$ $$($$$$-5 \nu^{11} - 6 \nu^{10} - 94 \nu^{9} - 108 \nu^{8} - 612 \nu^{7} - 660 \nu^{6} - 1594 \nu^{5} - 1548 \nu^{4} - 1388 \nu^{3} - 1056 \nu^{2} - 180 \nu - 24$$$$)/48$$ $$\beta_{11}$$ $$=$$ $$($$$$5 \nu^{11} + 6 \nu^{10} + 94 \nu^{9} + 114 \nu^{8} + 612 \nu^{7} + 756 \nu^{6} + 1594 \nu^{5} + 2040 \nu^{4} + 1412 \nu^{3} + 1932 \nu^{2} + 324 \nu + 384$$$$)/48$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} - 3$$ $$\nu^{3}$$ $$=$$ $$\beta_{9} + \beta_{7} + \beta_{5} - 6 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$\beta_{7} - \beta_{6} - 2 \beta_{3} - 7 \beta_{2} + \beta_{1} + 16$$ $$\nu^{5}$$ $$=$$ $$\beta_{11} - 9 \beta_{9} - \beta_{8} - 8 \beta_{7} + \beta_{6} - 11 \beta_{5} - 6 \beta_{4} + \beta_{2} + 39 \beta_{1} + 3$$ $$\nu^{6}$$ $$=$$ $$-10 \beta_{7} + 10 \beta_{6} + 28 \beta_{3} + 48 \beta_{2} - 14 \beta_{1} - 100$$ $$\nu^{7}$$ $$=$$ $$-10 \beta_{11} + 68 \beta_{9} + 10 \beta_{8} + 58 \beta_{7} - 10 \beta_{6} + 96 \beta_{5} + 84 \beta_{4} - 14 \beta_{2} - 264 \beta_{1} - 42$$ $$\nu^{8}$$ $$=$$ $$8 \beta_{11} + 8 \beta_{10} - 4 \beta_{9} + 74 \beta_{7} - 78 \beta_{6} - 4 \beta_{5} - 284 \beta_{3} - 340 \beta_{2} + 142 \beta_{1} + 666$$ $$\nu^{9}$$ $$=$$ $$70 \beta_{11} - 488 \beta_{9} - 70 \beta_{8} - 414 \beta_{7} + 74 \beta_{6} - 780 \beta_{5} - 836 \beta_{4} + 146 \beta_{2} + 1830 \beta_{1} + 418$$ $$\nu^{10}$$ $$=$$ $$-148 \beta_{11} - 152 \beta_{10} + 76 \beta_{9} + 4 \beta_{8} - 486 \beta_{7} + 562 \beta_{6} + 76 \beta_{5} + 2548 \beta_{3} + 2470 \beta_{2} - 1274 \beta_{1} - 4592$$ $$\nu^{11}$$ $$=$$ $$-406 \beta_{11} + 3438 \beta_{9} + 406 \beta_{8} + 2952 \beta_{7} - 486 \beta_{6} + 6138 \beta_{5} + 7348 \beta_{4} - 1350 \beta_{2} - 12894 \beta_{1} - 3674$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/315\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$136$$ $$281$$ $$\chi(n)$$ $$1$$ $$1 - \beta_{4}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
26.1
 − 2.74137i − 1.13965i − 0.979668i 0.398211i 1.99567i 2.46680i 2.74137i 1.13965i 0.979668i − 0.398211i − 1.99567i − 2.46680i
−2.37409 1.37068i 0 2.75754 + 4.77621i −0.500000 + 0.866025i 0 −2.06138 1.65853i 9.63615i 0 2.37409 1.37068i
26.2 −0.986962 0.569823i 0 −0.350603 0.607263i −0.500000 + 0.866025i 0 2.18931 1.48558i 3.07842i 0 0.986962 0.569823i
26.3 −0.848417 0.489834i 0 −0.520126 0.900884i −0.500000 + 0.866025i 0 1.24698 + 2.33346i 2.97844i 0 0.848417 0.489834i
26.4 0.344861 + 0.199105i 0 −0.920714 1.59472i −0.500000 + 0.866025i 0 −2.30243 1.30338i 1.52970i 0 −0.344861 + 0.199105i
26.5 1.72830 + 0.997835i 0 0.991350 + 1.71707i −0.500000 + 0.866025i 0 1.78020 + 1.95727i 0.0345244i 0 −1.72830 + 0.997835i
26.6 2.13631 + 1.23340i 0 2.04255 + 3.53780i −0.500000 + 0.866025i 0 −1.85267 + 1.88881i 5.14351i 0 −2.13631 + 1.23340i
206.1 −2.37409 + 1.37068i 0 2.75754 4.77621i −0.500000 0.866025i 0 −2.06138 + 1.65853i 9.63615i 0 2.37409 + 1.37068i
206.2 −0.986962 + 0.569823i 0 −0.350603 + 0.607263i −0.500000 0.866025i 0 2.18931 + 1.48558i 3.07842i 0 0.986962 + 0.569823i
206.3 −0.848417 + 0.489834i 0 −0.520126 + 0.900884i −0.500000 0.866025i 0 1.24698 2.33346i 2.97844i 0 0.848417 + 0.489834i
206.4 0.344861 0.199105i 0 −0.920714 + 1.59472i −0.500000 0.866025i 0 −2.30243 + 1.30338i 1.52970i 0 −0.344861 0.199105i
206.5 1.72830 0.997835i 0 0.991350 1.71707i −0.500000 0.866025i 0 1.78020 1.95727i 0.0345244i 0 −1.72830 0.997835i
206.6 2.13631 1.23340i 0 2.04255 3.53780i −0.500000 0.866025i 0 −1.85267 1.88881i 5.14351i 0 −2.13631 1.23340i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 206.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
21.g even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 315.2.bj.a 12
3.b odd 2 1 315.2.bj.b yes 12
5.b even 2 1 1575.2.bk.f 12
5.c odd 4 2 1575.2.bc.d 24
7.c even 3 1 2205.2.b.b 12
7.d odd 6 1 315.2.bj.b yes 12
7.d odd 6 1 2205.2.b.a 12
15.d odd 2 1 1575.2.bk.e 12
15.e even 4 2 1575.2.bc.c 24
21.g even 6 1 inner 315.2.bj.a 12
21.g even 6 1 2205.2.b.b 12
21.h odd 6 1 2205.2.b.a 12
35.i odd 6 1 1575.2.bk.e 12
35.k even 12 2 1575.2.bc.c 24
105.p even 6 1 1575.2.bk.f 12
105.w odd 12 2 1575.2.bc.d 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.2.bj.a 12 1.a even 1 1 trivial
315.2.bj.a 12 21.g even 6 1 inner
315.2.bj.b yes 12 3.b odd 2 1
315.2.bj.b yes 12 7.d odd 6 1
1575.2.bc.c 24 15.e even 4 2
1575.2.bc.c 24 35.k even 12 2
1575.2.bc.d 24 5.c odd 4 2
1575.2.bc.d 24 105.w odd 12 2
1575.2.bk.e 12 15.d odd 2 1
1575.2.bk.e 12 35.i odd 6 1
1575.2.bk.f 12 5.b even 2 1
1575.2.bk.f 12 105.p even 6 1
2205.2.b.a 12 7.d odd 6 1
2205.2.b.a 12 21.h odd 6 1
2205.2.b.b 12 7.c even 3 1
2205.2.b.b 12 21.g even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{12} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(315, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 + 2 T^{2} + 2 T^{4} - 12 T^{5} - 24 T^{7} - 8 T^{8} - 24 T^{9} + 52 T^{10} + 24 T^{11} + 148 T^{12} + 48 T^{13} + 208 T^{14} - 192 T^{15} - 128 T^{16} - 768 T^{17} - 1536 T^{19} + 512 T^{20} + 2048 T^{22} + 4096 T^{24}$$
$3$ 1
$5$ $$( 1 + T + T^{2} )^{6}$$
$7$ $$1 + 2 T - T^{2} + 6 T^{3} + 130 T^{4} + 130 T^{5} - 131 T^{6} + 910 T^{7} + 6370 T^{8} + 2058 T^{9} - 2401 T^{10} + 33614 T^{11} + 117649 T^{12}$$
$11$ $$1 - 12 T + 104 T^{2} - 672 T^{3} + 3641 T^{4} - 16332 T^{5} + 62340 T^{6} - 195852 T^{7} + 477010 T^{8} - 649368 T^{9} - 1262450 T^{10} + 12891636 T^{11} - 53694803 T^{12} + 141807996 T^{13} - 152756450 T^{14} - 864308808 T^{15} + 6983903410 T^{16} - 31542160452 T^{17} + 110439112740 T^{18} - 318264476772 T^{19} + 780480685721 T^{20} - 1584540848352 T^{21} + 2697492158504 T^{22} - 3423740047332 T^{23} + 3138428376721 T^{24}$$
$13$ $$1 - 54 T^{2} + 1725 T^{4} - 42910 T^{6} + 855078 T^{8} - 14180646 T^{10} + 200139573 T^{12} - 2396529174 T^{14} + 24421882758 T^{16} - 207118374190 T^{18} + 1407135493725 T^{20} - 7444358559846 T^{22} + 23298085122481 T^{24}$$
$17$ $$1 - 42 T^{2} + 120 T^{3} + 831 T^{4} - 5256 T^{5} - 3958 T^{6} + 127332 T^{7} - 227496 T^{8} - 2018952 T^{9} + 9082140 T^{10} + 14965308 T^{11} - 192427863 T^{12} + 254410236 T^{13} + 2624738460 T^{14} - 9919111176 T^{15} - 19000693416 T^{16} + 180793231524 T^{17} - 95536498102 T^{18} - 2156740065288 T^{19} + 5796854433471 T^{20} + 14230545179640 T^{21} - 84671743818858 T^{22} + 582622237229761 T^{24}$$
$19$ $$1 - 6 T + 75 T^{2} - 378 T^{3} + 2703 T^{4} - 15096 T^{5} + 85196 T^{6} - 494928 T^{7} + 2381289 T^{8} - 12436578 T^{9} + 56967225 T^{10} - 267312150 T^{11} + 1188448446 T^{12} - 5078930850 T^{13} + 20565168225 T^{14} - 85302488502 T^{15} + 310331963769 T^{16} - 1225490725872 T^{17} + 4008120877676 T^{18} - 13493887771944 T^{19} + 45906570899823 T^{20} - 121975949760462 T^{21} + 459829969335075 T^{22} - 698941553389314 T^{23} + 2213314919066161 T^{24}$$
$23$ $$1 - 12 T + 110 T^{2} - 744 T^{3} + 3947 T^{4} - 16104 T^{5} + 39450 T^{6} + 32328 T^{7} - 1232708 T^{8} + 9040500 T^{9} - 50848868 T^{10} + 256937364 T^{11} - 1217123567 T^{12} + 5909559372 T^{13} - 26899051172 T^{14} + 109995763500 T^{15} - 344962239428 T^{16} + 208074096504 T^{17} + 5840015821050 T^{18} - 54831308998488 T^{19} + 309093458904107 T^{20} - 1340057580128472 T^{21} + 4556916233501390 T^{22} - 11433717094967124 T^{23} + 21914624432020321 T^{24}$$
$29$ $$1 - 184 T^{2} + 16934 T^{4} - 1029132 T^{6} + 46624399 T^{8} - 1703051756 T^{10} + 52962115192 T^{12} - 1432266526796 T^{14} + 32976551549119 T^{16} - 612151713987372 T^{18} + 8471172757081574 T^{20} - 77410130927236984 T^{22} + 353814783205469041 T^{24}$$
$31$ $$1 - 6 T + 75 T^{2} - 378 T^{3} + 2451 T^{4} - 14268 T^{5} + 15464 T^{6} - 69696 T^{7} - 2002827 T^{8} + 11519550 T^{9} - 72958179 T^{10} + 685842102 T^{11} - 2463303858 T^{12} + 21261105162 T^{13} - 70112810019 T^{14} + 343178914050 T^{15} - 1849652793867 T^{16} - 1995337308096 T^{17} + 13724356922984 T^{18} - 392549978135748 T^{19} + 2090435932767891 T^{20} - 9994177176733638 T^{21} + 61472121523560075 T^{22} - 152450861378428986 T^{23} + 787662783788549761 T^{24}$$
$37$ $$1 + 10 T - T^{2} + 206 T^{3} + 1893 T^{4} - 19356 T^{5} - 25070 T^{6} - 11384 T^{7} - 4236169 T^{8} + 22256290 T^{9} - 6662617 T^{10} - 355484390 T^{11} + 9259385786 T^{12} - 13152922430 T^{13} - 9121122673 T^{14} + 1127347857370 T^{15} - 7939262729209 T^{16} - 789411606488 T^{17} - 64322761073630 T^{18} - 1837501413786348 T^{19} + 6649123606272453 T^{20} + 26772118397785862 T^{21} - 4808584372417849 T^{22} + 1779176217794604130 T^{23} + 6582952005840035281 T^{24}$$
$41$ $$( 1 - 12 T + 192 T^{2} - 1704 T^{3} + 18087 T^{4} - 122340 T^{5} + 939022 T^{6} - 5015940 T^{7} + 30404247 T^{8} - 117441384 T^{9} + 542546112 T^{10} - 1390274412 T^{11} + 4750104241 T^{12} )^{2}$$
$43$ $$( 1 + 2 T + 65 T^{2} + 318 T^{3} + 5080 T^{4} + 14554 T^{5} + 241357 T^{6} + 625822 T^{7} + 9392920 T^{8} + 25283226 T^{9} + 222222065 T^{10} + 294016886 T^{11} + 6321363049 T^{12} )^{2}$$
$47$ $$1 - 162 T^{2} + 192 T^{3} + 11733 T^{4} - 23328 T^{5} - 672838 T^{6} + 760896 T^{7} + 42600858 T^{8} + 8087904 T^{9} - 2486828586 T^{10} - 573202080 T^{11} + 123583933965 T^{12} - 26940497760 T^{13} - 5493404346474 T^{14} + 839710456992 T^{15} + 207878597366298 T^{16} + 174507698446272 T^{17} - 7252665683533702 T^{18} - 11818504154160864 T^{19} + 279377826402441813 T^{20} + 214873050835731264 T^{21} - 8521059422204467938 T^{22} +$$$$11\!\cdots\!41$$$$T^{24}$$
$53$ $$1 - 12 T + 290 T^{2} - 2904 T^{3} + 41501 T^{4} - 314064 T^{5} + 3427926 T^{6} - 18869580 T^{7} + 172940890 T^{8} - 534860628 T^{9} + 5470600330 T^{10} + 1694225472 T^{11} + 171503876389 T^{12} + 89793950016 T^{13} + 15366916326970 T^{14} - 79628445714756 T^{15} + 1364586806668090 T^{16} - 7891173310802940 T^{17} + 75977789787488454 T^{18} - 368934479421767568 T^{19} + 2583839411761892861 T^{20} - 9582513470593394232 T^{21} + 50717366405998784210 T^{22} -$$$$11\!\cdots\!64$$$$T^{23} +$$$$49\!\cdots\!41$$$$T^{24}$$
$59$ $$1 + 24 T + 84 T^{2} - 1128 T^{3} + 9885 T^{4} + 197484 T^{5} - 900376 T^{6} - 12268224 T^{7} + 92002038 T^{8} + 411544404 T^{9} - 9205294494 T^{10} - 19136016684 T^{11} + 471190351413 T^{12} - 1129024984356 T^{13} - 32043630133614 T^{14} + 84522578149116 T^{15} + 1114821907181718 T^{16} - 8770851443174976 T^{17} - 37978340157549016 T^{18} + 491468849827995396 T^{19} + 1451418875718713085 T^{20} - 9771859283442771192 T^{21} + 42933807277253877684 T^{22} +$$$$72\!\cdots\!16$$$$T^{23} +$$$$17\!\cdots\!81$$$$T^{24}$$
$61$ $$1 + 138 T^{2} + 5889 T^{4} - 41004 T^{5} - 102418 T^{6} - 5059224 T^{7} - 8898402 T^{8} - 158765292 T^{9} + 1660743870 T^{10} + 9579987756 T^{11} + 189206615181 T^{12} + 584379253116 T^{13} + 6179627940270 T^{14} - 36036704743452 T^{15} - 123205859246082 T^{16} - 4273001876330424 T^{17} - 5276613701304898 T^{18} - 128865027248205084 T^{19} + 1128964366240987809 T^{20} + 98441321809477798938 T^{22} +$$$$26\!\cdots\!21$$$$T^{24}$$
$67$ $$1 - 6 T - 219 T^{2} + 1306 T^{3} + 22971 T^{4} - 120540 T^{5} - 1928636 T^{6} + 7225512 T^{7} + 163638939 T^{8} - 309627182 T^{9} - 13816766313 T^{10} + 6348931902 T^{11} + 1027244050774 T^{12} + 425378437434 T^{13} - 62023463979057 T^{14} - 93124400139866 T^{15} + 3297508060100619 T^{16} + 9755345162129784 T^{17} - 174461292352891484 T^{18} - 730558176905634420 T^{19} + 9327780621153600411 T^{20} + 35531733921561200782 T^{21} -$$$$39\!\cdots\!31$$$$T^{22} -$$$$73\!\cdots\!98$$$$T^{23} +$$$$81\!\cdots\!61$$$$T^{24}$$
$71$ $$1 - 460 T^{2} + 108410 T^{4} - 17445312 T^{6} + 2127087511 T^{8} - 205744741412 T^{10} + 16152616390624 T^{12} - 1037159241457892 T^{14} + 54052869288615991 T^{16} - 2234749420290428352 T^{18} + 70006140322352950010 T^{20} -$$$$14\!\cdots\!60$$$$T^{22} +$$$$16\!\cdots\!41$$$$T^{24}$$
$73$ $$1 + 42 T + 1119 T^{2} + 22302 T^{3} + 373665 T^{4} + 5463480 T^{5} + 72046694 T^{6} + 868793688 T^{7} + 9715685451 T^{8} + 101446836642 T^{9} + 996453337299 T^{10} + 9237792089286 T^{11} + 81107402998458 T^{12} + 674358822517878 T^{13} + 5310099834466371 T^{14} + 39464544049960914 T^{15} + 275908376917691691 T^{16} + 1801071514770504984 T^{17} + 10903130693170338566 T^{18} + 60357240861116077560 T^{19} +$$$$30\!\cdots\!65$$$$T^{20} +$$$$13\!\cdots\!26$$$$T^{21} +$$$$48\!\cdots\!31$$$$T^{22} +$$$$13\!\cdots\!34$$$$T^{23} +$$$$22\!\cdots\!21$$$$T^{24}$$
$79$ $$1 - 18 T - 177 T^{2} + 3346 T^{3} + 40923 T^{4} - 454344 T^{5} - 6959816 T^{6} + 45880152 T^{7} + 889910193 T^{8} - 3054243566 T^{9} - 93786004551 T^{10} + 82100387598 T^{11} + 8378036930038 T^{12} + 6485930620242 T^{13} - 585318454402791 T^{14} - 1505861193537074 T^{15} + 34662074100075633 T^{16} + 141175815298692648 T^{17} - 1691843962334344136 T^{18} - 8725180824407424696 T^{19} + 62084643827806195803 T^{20} +$$$$40\!\cdots\!74$$$$T^{21} -$$$$16\!\cdots\!77$$$$T^{22} -$$$$13\!\cdots\!22$$$$T^{23} +$$$$59\!\cdots\!41$$$$T^{24}$$
$83$ $$( 1 + 12 T + 300 T^{2} + 3204 T^{3} + 50769 T^{4} + 448284 T^{5} + 5149954 T^{6} + 37207572 T^{7} + 349747641 T^{8} + 1832005548 T^{9} + 14237496300 T^{10} + 47268487716 T^{11} + 326940373369 T^{12} )^{2}$$
$89$ $$1 - 12 T - 282 T^{2} + 648 T^{3} + 81261 T^{4} + 90852 T^{5} - 10133890 T^{6} - 68061132 T^{7} + 906987750 T^{8} + 7493819688 T^{9} - 22404081252 T^{10} - 403242866028 T^{11} + 351385075713 T^{12} - 35888615076492 T^{13} - 177462727597092 T^{14} + 5282910571629672 T^{15} + 56906443994547750 T^{16} - 380057407254236268 T^{17} - 5036353734656768290 T^{18} + 4018505237928600708 T^{19} +$$$$31\!\cdots\!41$$$$T^{20} +$$$$22\!\cdots\!32$$$$T^{21} -$$$$87\!\cdots\!82$$$$T^{22} -$$$$33\!\cdots\!68$$$$T^{23} +$$$$24\!\cdots\!21$$$$T^{24}$$
$97$ $$1 - 432 T^{2} + 85686 T^{4} - 12118408 T^{6} + 1564041327 T^{8} - 184182428136 T^{10} + 19001698536324 T^{12} - 1732972466331624 T^{14} + 138463454133595887 T^{16} - 10094294608307633032 T^{18} +$$$$67\!\cdots\!46$$$$T^{20} -$$$$31\!\cdots\!68$$$$T^{22} +$$$$69\!\cdots\!41$$$$T^{24}$$