# Properties

 Label 1815.2.c.i Level $1815$ Weight $2$ Character orbit 1815.c Analytic conductor $14.493$ Analytic rank $0$ Dimension $12$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1815 = 3 \cdot 5 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1815.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$14.4928479669$$ Analytic rank: $$0$$ Dimension: $$12$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} + \cdots)$$ Defining polynomial: $$x^{12} + 21 x^{10} + 164 x^{8} + 589 x^{6} + 965 x^{4} + 576 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} -\beta_{4} q^{3} + ( -2 + \beta_{2} ) q^{4} -\beta_{5} q^{5} + \beta_{3} q^{6} + ( -\beta_{1} + \beta_{5} - \beta_{6} - \beta_{8} ) q^{7} + ( -\beta_{1} + \beta_{4} - \beta_{7} - \beta_{8} - \beta_{9} ) q^{8} - q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} -\beta_{4} q^{3} + ( -2 + \beta_{2} ) q^{4} -\beta_{5} q^{5} + \beta_{3} q^{6} + ( -\beta_{1} + \beta_{5} - \beta_{6} - \beta_{8} ) q^{7} + ( -\beta_{1} + \beta_{4} - \beta_{7} - \beta_{8} - \beta_{9} ) q^{8} - q^{9} + ( 1 + \beta_{6} - \beta_{8} - \beta_{10} ) q^{10} + ( \beta_{4} + \beta_{8} ) q^{12} + ( -\beta_{5} + \beta_{6} - \beta_{10} - \beta_{11} ) q^{13} + ( 2 - 2 \beta_{2} + 2 \beta_{3} + \beta_{10} - \beta_{11} ) q^{14} + \beta_{7} q^{15} + ( 2 - 2 \beta_{2} + \beta_{3} + \beta_{5} + \beta_{6} - \beta_{7} + \beta_{9} + \beta_{10} - \beta_{11} ) q^{16} + ( -\beta_{1} + \beta_{4} + \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} - \beta_{11} ) q^{17} -\beta_{1} q^{18} + ( 1 - \beta_{2} - \beta_{3} + \beta_{7} - \beta_{9} ) q^{19} + ( \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{5} + \beta_{7} + \beta_{8} - \beta_{9} + \beta_{11} ) q^{20} + ( -1 + \beta_{2} - \beta_{3} - \beta_{7} + \beta_{9} ) q^{21} + ( \beta_{1} - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} - \beta_{10} - \beta_{11} ) q^{23} + ( \beta_{2} - \beta_{3} - \beta_{5} - \beta_{6} ) q^{24} + ( 2 \beta_{1} - \beta_{3} - 2 \beta_{4} + \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} ) q^{25} + ( \beta_{5} + \beta_{6} + 2 \beta_{7} - 2 \beta_{9} - \beta_{10} + \beta_{11} ) q^{26} + \beta_{4} q^{27} + ( 2 \beta_{1} + 2 \beta_{4} + 2 \beta_{7} + 2 \beta_{8} + 2 \beta_{9} ) q^{28} + ( -2 + 2 \beta_{3} - \beta_{5} - \beta_{6} + 2 \beta_{7} - 2 \beta_{9} ) q^{29} + ( -1 + \beta_{2} - \beta_{4} - \beta_{9} + \beta_{11} ) q^{30} + ( 3 - \beta_{5} - \beta_{6} + \beta_{10} - \beta_{11} ) q^{31} + ( 2 \beta_{1} + 3 \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} + 3 \beta_{8} + \beta_{9} ) q^{32} + ( \beta_{3} + \beta_{5} + \beta_{6} + 3 \beta_{7} - 3 \beta_{9} - \beta_{10} + \beta_{11} ) q^{34} + ( 2 + \beta_{2} + \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} ) q^{35} + ( 2 - \beta_{2} ) q^{36} + ( -\beta_{1} + 4 \beta_{4} + \beta_{8} - \beta_{10} - \beta_{11} ) q^{37} + ( 2 \beta_{1} - 6 \beta_{4} + \beta_{10} + \beta_{11} ) q^{38} + ( \beta_{7} - \beta_{9} - \beta_{10} + \beta_{11} ) q^{39} + ( -2 + \beta_{1} + 2 \beta_{2} - 3 \beta_{3} + 4 \beta_{4} - \beta_{6} - \beta_{7} + 2 \beta_{8} + \beta_{9} + \beta_{11} ) q^{40} + ( -2 + 2 \beta_{3} + \beta_{5} + \beta_{6} + \beta_{10} - \beta_{11} ) q^{41} + ( -2 \beta_{1} - 2 \beta_{8} - \beta_{10} - \beta_{11} ) q^{42} + ( -2 \beta_{1} - 2 \beta_{4} + 2 \beta_{7} + 2 \beta_{9} ) q^{43} + \beta_{5} q^{45} + ( -2 - 2 \beta_{2} + 3 \beta_{3} + 2 \beta_{5} + 2 \beta_{6} + \beta_{7} - \beta_{9} ) q^{46} + ( \beta_{1} + \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} ) q^{47} + ( -\beta_{1} - \beta_{5} + \beta_{6} - \beta_{7} - 2 \beta_{8} - \beta_{9} - \beta_{10} - \beta_{11} ) q^{48} + ( -2 + \beta_{2} - 3 \beta_{3} - \beta_{7} + \beta_{9} ) q^{49} + ( -6 + 3 \beta_{2} - 2 \beta_{4} - \beta_{5} + 2 \beta_{9} - \beta_{10} ) q^{50} + ( \beta_{2} - \beta_{3} + \beta_{5} + \beta_{6} - \beta_{10} + \beta_{11} ) q^{51} + ( -2 \beta_{4} + \beta_{5} - \beta_{6} - 2 \beta_{7} + 2 \beta_{8} - 2 \beta_{9} + \beta_{10} + \beta_{11} ) q^{52} + ( -\beta_{1} - 3 \beta_{4} - \beta_{7} + \beta_{8} - \beta_{9} ) q^{53} -\beta_{3} q^{54} + ( -8 + 4 \beta_{2} - 2 \beta_{3} - 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} - 2 \beta_{9} ) q^{56} + ( \beta_{1} + \beta_{5} - \beta_{6} - \beta_{8} ) q^{57} + ( -2 \beta_{1} + 2 \beta_{4} - \beta_{5} + \beta_{6} - 2 \beta_{7} - 2 \beta_{9} + \beta_{10} + \beta_{11} ) q^{58} + ( 2 + 2 \beta_{2} + 2 \beta_{3} - \beta_{10} + \beta_{11} ) q^{59} + ( 2 - 2 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} + \beta_{10} ) q^{60} + ( -1 + 4 \beta_{2} - 2 \beta_{3} + \beta_{5} + \beta_{6} - 2 \beta_{7} + 2 \beta_{9} - \beta_{10} + \beta_{11} ) q^{61} + ( 3 \beta_{1} - 2 \beta_{4} - 3 \beta_{5} + 3 \beta_{6} - 2 \beta_{8} - \beta_{10} - \beta_{11} ) q^{62} + ( \beta_{1} - \beta_{5} + \beta_{6} + \beta_{8} ) q^{63} + ( -4 + 3 \beta_{2} - 7 \beta_{3} - \beta_{7} + \beta_{9} ) q^{64} + ( -2 + \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{5} - 2 \beta_{6} - 2 \beta_{8} + 2 \beta_{9} + \beta_{10} ) q^{65} + ( \beta_{1} + 2 \beta_{4} - \beta_{5} + \beta_{6} - 2 \beta_{7} + \beta_{8} - 2 \beta_{9} ) q^{67} + ( -2 \beta_{1} + \beta_{4} + 3 \beta_{5} - 3 \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} + 2 \beta_{10} + 2 \beta_{11} ) q^{68} + ( -2 + \beta_{2} + \beta_{3} - \beta_{5} - \beta_{6} + \beta_{7} - \beta_{9} - \beta_{10} + \beta_{11} ) q^{69} + ( -1 + \beta_{1} + \beta_{2} - 3 \beta_{3} - \beta_{5} - \beta_{6} - \beta_{7} - 3 \beta_{8} - 3 \beta_{9} ) q^{70} + ( 2 \beta_{3} - \beta_{10} + \beta_{11} ) q^{71} + ( \beta_{1} - \beta_{4} + \beta_{7} + \beta_{8} + \beta_{9} ) q^{72} + ( \beta_{1} - 4 \beta_{4} + \beta_{5} - \beta_{6} + 3 \beta_{8} ) q^{73} + ( 2 - 6 \beta_{3} - \beta_{5} - \beta_{6} + 2 \beta_{7} - 2 \beta_{9} ) q^{74} + ( -1 + \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{5} + \beta_{6} - \beta_{9} - \beta_{11} ) q^{75} + ( -4 + 4 \beta_{3} ) q^{76} + ( 2 \beta_{5} - 2 \beta_{6} - \beta_{7} - \beta_{9} + \beta_{10} + \beta_{11} ) q^{78} + ( -1 + 2 \beta_{2} - 2 \beta_{3} + \beta_{5} + \beta_{6} + 2 \beta_{7} - 2 \beta_{9} - \beta_{10} + \beta_{11} ) q^{79} + ( -4 - 2 \beta_{1} + \beta_{2} - 4 \beta_{3} - 2 \beta_{4} - 3 \beta_{6} - 4 \beta_{8} - 2 \beta_{9} - \beta_{10} ) q^{80} + q^{81} + ( -2 \beta_{1} + 4 \beta_{4} - \beta_{5} + \beta_{6} + 4 \beta_{8} + \beta_{10} + \beta_{11} ) q^{82} + ( -2 \beta_{1} + 6 \beta_{4} - 4 \beta_{8} - \beta_{10} - \beta_{11} ) q^{83} + ( 4 - 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{5} + 2 \beta_{6} ) q^{84} + ( -1 + 2 \beta_{1} + \beta_{2} + 4 \beta_{3} - 4 \beta_{4} - \beta_{5} - 2 \beta_{6} - \beta_{8} - \beta_{11} ) q^{85} + ( 4 + 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{7} - 2 \beta_{9} - 2 \beta_{10} + 2 \beta_{11} ) q^{86} + ( -2 \beta_{1} + 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} + \beta_{7} + \beta_{9} ) q^{87} + ( 4 - 4 \beta_{2} + 2 \beta_{3} - \beta_{5} - \beta_{6} + 2 \beta_{7} - 2 \beta_{9} + 2 \beta_{10} - 2 \beta_{11} ) q^{89} + ( -1 - \beta_{6} + \beta_{8} + \beta_{10} ) q^{90} + ( 4 - 4 \beta_{3} + 2 \beta_{5} + 2 \beta_{6} ) q^{91} + ( 2 \beta_{1} + 3 \beta_{4} - \beta_{7} + 7 \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} ) q^{92} + ( -3 \beta_{4} + \beta_{7} + \beta_{9} - \beta_{10} - \beta_{11} ) q^{93} + ( -3 \beta_{3} - \beta_{7} + \beta_{9} ) q^{94} + ( -\beta_{2} + 5 \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} - 3 \beta_{11} ) q^{95} + ( 6 - 3 \beta_{2} + 2 \beta_{3} + \beta_{5} + \beta_{6} + \beta_{7} - \beta_{9} ) q^{96} + ( 5 \beta_{1} + 2 \beta_{4} - 2 \beta_{7} - \beta_{8} - 2 \beta_{9} + \beta_{10} + \beta_{11} ) q^{97} + ( -3 \beta_{1} - 6 \beta_{4} - 4 \beta_{8} - \beta_{10} - \beta_{11} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12q - 18q^{4} - 2q^{5} + 2q^{6} - 12q^{9} + O(q^{10})$$ $$12q - 18q^{4} - 2q^{5} + 2q^{6} - 12q^{9} + 12q^{10} + 20q^{14} + 22q^{16} + 4q^{19} - 2q^{20} - 8q^{21} + 2q^{25} - 24q^{29} - 8q^{30} + 36q^{31} + 2q^{34} + 24q^{35} + 18q^{36} - 4q^{39} - 22q^{40} - 12q^{41} + 2q^{45} - 22q^{46} - 24q^{49} - 58q^{50} + 4q^{51} - 2q^{54} - 84q^{56} + 36q^{59} + 22q^{60} + 8q^{61} - 44q^{64} - 14q^{65} - 24q^{69} - 16q^{70} + 8q^{74} - 12q^{75} - 40q^{76} - 4q^{79} - 58q^{80} + 12q^{81} + 48q^{84} - 2q^{85} + 56q^{86} + 32q^{89} - 12q^{90} + 48q^{91} - 6q^{94} + 62q^{96} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} + 21 x^{10} + 164 x^{8} + 589 x^{6} + 965 x^{4} + 576 x^{2} + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} + 4$$ $$\beta_{3}$$ $$=$$ $$($$$$2 \nu^{10} + 37 \nu^{8} + 224 \nu^{6} + 480 \nu^{4} + 247 \nu^{2} - 6$$$$)/92$$ $$\beta_{4}$$ $$=$$ $$($$$$-3 \nu^{11} - 67 \nu^{9} - 566 \nu^{7} - 2215 \nu^{5} - 3855 \nu^{3} - 2222 \nu$$$$)/184$$ $$\beta_{5}$$ $$=$$ $$($$$$7 \nu^{11} - 5 \nu^{10} + 141 \nu^{9} - 81 \nu^{8} + 1014 \nu^{7} - 422 \nu^{6} + 3083 \nu^{5} - 717 \nu^{4} + 3521 \nu^{3} + 15 \nu^{2} + 830 \nu + 406$$$$)/184$$ $$\beta_{6}$$ $$=$$ $$($$$$-7 \nu^{11} - 5 \nu^{10} - 141 \nu^{9} - 81 \nu^{8} - 1014 \nu^{7} - 422 \nu^{6} - 3083 \nu^{5} - 717 \nu^{4} - 3521 \nu^{3} + 15 \nu^{2} - 830 \nu + 406$$$$)/184$$ $$\beta_{7}$$ $$=$$ $$($$$$-8 \nu^{11} - 14 \nu^{10} - 171 \nu^{9} - 259 \nu^{8} - 1356 \nu^{7} - 1660 \nu^{6} - 4910 \nu^{5} - 4280 \nu^{4} - 8049 \nu^{3} - 3661 \nu^{2} - 4898 \nu + 42$$$$)/184$$ $$\beta_{8}$$ $$=$$ $$($$$$13 \nu^{11} + 275 \nu^{9} + 2146 \nu^{7} + 7605 \nu^{5} + 12059 \nu^{3} + 6654 \nu$$$$)/184$$ $$\beta_{9}$$ $$=$$ $$($$$$-8 \nu^{11} + 14 \nu^{10} - 171 \nu^{9} + 259 \nu^{8} - 1356 \nu^{7} + 1660 \nu^{6} - 4910 \nu^{5} + 4280 \nu^{4} - 8049 \nu^{3} + 3661 \nu^{2} - 4898 \nu - 42$$$$)/184$$ $$\beta_{10}$$ $$=$$ $$($$$$-25 \nu^{11} - 11 \nu^{10} - 497 \nu^{9} - 215 \nu^{8} - 3582 \nu^{7} - 1462 \nu^{6} - 11405 \nu^{5} - 3951 \nu^{4} - 15565 \nu^{3} - 3187 \nu^{2} - 7078 \nu + 562$$$$)/184$$ $$\beta_{11}$$ $$=$$ $$($$$$-25 \nu^{11} + 11 \nu^{10} - 497 \nu^{9} + 215 \nu^{8} - 3582 \nu^{7} + 1462 \nu^{6} - 11405 \nu^{5} + 3951 \nu^{4} - 15565 \nu^{3} + 3187 \nu^{2} - 7078 \nu - 562$$$$)/184$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} - 4$$ $$\nu^{3}$$ $$=$$ $$-\beta_{9} - \beta_{8} - \beta_{7} + \beta_{4} - 5 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$-\beta_{11} + \beta_{10} + \beta_{9} - \beta_{7} + \beta_{6} + \beta_{5} + \beta_{3} - 8 \beta_{2} + 22$$ $$\nu^{5}$$ $$=$$ $$9 \beta_{9} + 11 \beta_{8} + 9 \beta_{7} + \beta_{6} - \beta_{5} - 5 \beta_{4} + 30 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$10 \beta_{11} - 10 \beta_{10} - 9 \beta_{9} + 9 \beta_{7} - 10 \beta_{6} - 10 \beta_{5} - 17 \beta_{3} + 59 \beta_{2} - 136$$ $$\nu^{7}$$ $$=$$ $$-\beta_{11} - \beta_{10} - 68 \beta_{9} - 96 \beta_{8} - 68 \beta_{7} - 10 \beta_{6} + 10 \beta_{5} + 10 \beta_{4} - 195 \beta_{1}$$ $$\nu^{8}$$ $$=$$ $$-78 \beta_{11} + 78 \beta_{10} + 66 \beta_{9} - 66 \beta_{7} + 86 \beta_{6} + 86 \beta_{5} + 182 \beta_{3} - 427 \beta_{2} + 894$$ $$\nu^{9}$$ $$=$$ $$20 \beta_{11} + 20 \beta_{10} + 493 \beta_{9} + 781 \beta_{8} + 493 \beta_{7} + 70 \beta_{6} - 70 \beta_{5} + 95 \beta_{4} + 1321 \beta_{1}$$ $$\nu^{10}$$ $$=$$ $$563 \beta_{11} - 563 \beta_{10} - 453 \beta_{9} + 453 \beta_{7} - 711 \beta_{6} - 711 \beta_{5} - 1657 \beta_{3} + 3088 \beta_{2} - 6090$$ $$\nu^{11}$$ $$=$$ $$-258 \beta_{11} - 258 \beta_{10} - 3541 \beta_{9} - 6167 \beta_{8} - 3541 \beta_{7} - 415 \beta_{6} + 415 \beta_{5} - 1663 \beta_{4} - 9178 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1815\mathbb{Z}\right)^\times$$.

 $$n$$ $$727$$ $$1211$$ $$1696$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
364.1
 − 2.73519i − 2.42911i − 1.95455i − 1.53672i − 1.19557i − 0.0838261i 0.0838261i 1.19557i 1.53672i 1.95455i 2.42911i 2.73519i
2.73519i 1.00000i −5.48125 −1.65271 + 1.50617i 2.73519 4.20410i 9.52186i −1.00000 4.11965 + 4.52048i
364.2 2.42911i 1.00000i −3.90056 2.04405 0.906565i −2.42911 1.34168i 4.61665i −1.00000 −2.20214 4.96521i
364.3 1.95455i 1.00000i −1.82026 −1.24280 + 1.85889i −1.95455 2.58348i 0.351308i −1.00000 3.63328 + 2.42911i
364.4 1.53672i 1.00000i −0.361523 1.20807 + 1.88164i 1.53672 2.86503i 2.51789i −1.00000 2.89156 1.85647i
364.5 1.19557i 1.00000i 0.570614 0.849151 2.06856i 1.19557 3.76208i 3.07335i −1.00000 −2.47311 1.01522i
364.6 0.0838261i 1.00000i 1.99297 −2.20576 + 0.366926i −0.0838261 2.34295i 0.334715i −1.00000 0.0307580 + 0.184900i
364.7 0.0838261i 1.00000i 1.99297 −2.20576 0.366926i −0.0838261 2.34295i 0.334715i −1.00000 0.0307580 0.184900i
364.8 1.19557i 1.00000i 0.570614 0.849151 + 2.06856i 1.19557 3.76208i 3.07335i −1.00000 −2.47311 + 1.01522i
364.9 1.53672i 1.00000i −0.361523 1.20807 1.88164i 1.53672 2.86503i 2.51789i −1.00000 2.89156 + 1.85647i
364.10 1.95455i 1.00000i −1.82026 −1.24280 1.85889i −1.95455 2.58348i 0.351308i −1.00000 3.63328 2.42911i
364.11 2.42911i 1.00000i −3.90056 2.04405 + 0.906565i −2.42911 1.34168i 4.61665i −1.00000 −2.20214 + 4.96521i
364.12 2.73519i 1.00000i −5.48125 −1.65271 1.50617i 2.73519 4.20410i 9.52186i −1.00000 4.11965 4.52048i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 364.12 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1815.2.c.i yes 12
5.b even 2 1 inner 1815.2.c.i yes 12
5.c odd 4 1 9075.2.a.do 6
5.c odd 4 1 9075.2.a.ds 6
11.b odd 2 1 1815.2.c.h 12
55.d odd 2 1 1815.2.c.h 12
55.e even 4 1 9075.2.a.dp 6
55.e even 4 1 9075.2.a.dr 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1815.2.c.h 12 11.b odd 2 1
1815.2.c.h 12 55.d odd 2 1
1815.2.c.i yes 12 1.a even 1 1 trivial
1815.2.c.i yes 12 5.b even 2 1 inner
9075.2.a.do 6 5.c odd 4 1
9075.2.a.dp 6 55.e even 4 1
9075.2.a.dr 6 55.e even 4 1
9075.2.a.ds 6 5.c odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1815, [\chi])$$:

 $$T_{2}^{12} + 21 T_{2}^{10} + 164 T_{2}^{8} + 589 T_{2}^{6} + 965 T_{2}^{4} + 576 T_{2}^{2} + 4$$ $$T_{19}^{6} - 2 T_{19}^{5} - 47 T_{19}^{4} + 64 T_{19}^{3} + 368 T_{19}^{2} - 128 T_{19} - 512$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$4 + 576 T^{2} + 965 T^{4} + 589 T^{6} + 164 T^{8} + 21 T^{10} + T^{12}$$
$3$ $$( 1 + T^{2} )^{6}$$
$5$ $$15625 + 6250 T + 625 T^{2} + 750 T^{3} + 825 T^{4} + 160 T^{5} - 78 T^{6} + 32 T^{7} + 33 T^{8} + 6 T^{9} + T^{10} + 2 T^{11} + T^{12}$$
$7$ $$135424 + 153920 T^{2} + 61248 T^{4} + 11604 T^{6} + 1129 T^{8} + 54 T^{10} + T^{12}$$
$11$ $$T^{12}$$
$13$ $$1024 + 167424 T^{2} + 182032 T^{4} + 41964 T^{6} + 3249 T^{8} + 98 T^{10} + T^{12}$$
$17$ $$2116 + 4050996 T^{2} + 1476001 T^{4} + 158688 T^{6} + 7098 T^{8} + 140 T^{10} + T^{12}$$
$19$ $$( -512 - 128 T + 368 T^{2} + 64 T^{3} - 47 T^{4} - 2 T^{5} + T^{6} )^{2}$$
$23$ $$1149184 + 20546304 T^{2} + 5575776 T^{4} + 409808 T^{6} + 12921 T^{8} + 186 T^{10} + T^{12}$$
$29$ $$( -7688 - 13196 T - 6522 T^{2} - 1218 T^{3} - 41 T^{4} + 12 T^{5} + T^{6} )^{2}$$
$31$ $$( -2656 - 4296 T - 841 T^{2} + 450 T^{3} + 48 T^{4} - 18 T^{5} + T^{6} )^{2}$$
$37$ $$9710131600 + 1537138060 T^{2} + 92004409 T^{4} + 2702288 T^{6} + 41686 T^{8} + 324 T^{10} + T^{12}$$
$41$ $$( -52544 + 11600 T + 4044 T^{2} - 600 T^{3} - 119 T^{4} + 6 T^{5} + T^{6} )^{2}$$
$43$ $$287641600 + 419159040 T^{2} + 41533184 T^{4} + 1628288 T^{6} + 30960 T^{8} + 284 T^{10} + T^{12}$$
$47$ $$25600 + 115840 T^{2} + 113444 T^{4} + 32356 T^{6} + 2893 T^{8} + 94 T^{10} + T^{12}$$
$53$ $$84419344 + 34582616 T^{2} + 5292929 T^{4} + 378240 T^{6} + 13070 T^{8} + 200 T^{10} + T^{12}$$
$59$ $$( -8224 + 13008 T - 6592 T^{2} + 1212 T^{3} - 6 T^{4} - 18 T^{5} + T^{6} )^{2}$$
$61$ $$( -104284 + 46588 T + 17505 T^{2} + 56 T^{3} - 246 T^{4} - 4 T^{5} + T^{6} )^{2}$$
$67$ $$215296 + 2755136 T^{2} + 8931392 T^{4} + 874788 T^{6} + 24105 T^{8} + 262 T^{10} + T^{12}$$
$71$ $$( 11744 + 15296 T + 4224 T^{2} - 240 T^{3} - 130 T^{4} + T^{6} )^{2}$$
$73$ $$1961426944 + 5367189504 T^{2} + 502670336 T^{4} + 13448528 T^{6} + 140481 T^{8} + 626 T^{10} + T^{12}$$
$79$ $$( -196624 + 27712 T + 17331 T^{2} - 938 T^{3} - 300 T^{4} + 2 T^{5} + T^{6} )^{2}$$
$83$ $$70883737600 + 13025935360 T^{2} + 707416064 T^{4} + 14553088 T^{6} + 137380 T^{8} + 604 T^{10} + T^{12}$$
$89$ $$( -80776 - 82236 T - 7310 T^{2} + 4926 T^{3} - 261 T^{4} - 16 T^{5} + T^{6} )^{2}$$
$97$ $$555639184 + 425887292 T^{2} + 104886809 T^{4} + 8993904 T^{6} + 145718 T^{8} + 692 T^{10} + T^{12}$$