# Properties

 Label 378.2.m.a Level 378 Weight 2 Character orbit 378.m Analytic conductor 3.018 Analytic rank 0 Dimension 16 CM no Inner twists 4

# Learn more about

## Newspace parameters

 Level: $$N$$ = $$378 = 2 \cdot 3^{3} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 378.m (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.01834519640$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$3^{4}$$ Twist minimal: no (minimal twist has level 126) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{7} q^{2} + ( 1 + \beta_{5} ) q^{4} + ( \beta_{6} + \beta_{8} - \beta_{15} ) q^{5} + ( -1 + \beta_{2} - \beta_{10} + \beta_{13} ) q^{7} + \beta_{3} q^{8} +O(q^{10})$$ $$q + \beta_{7} q^{2} + ( 1 + \beta_{5} ) q^{4} + ( \beta_{6} + \beta_{8} - \beta_{15} ) q^{5} + ( -1 + \beta_{2} - \beta_{10} + \beta_{13} ) q^{7} + \beta_{3} q^{8} + ( \beta_{8} - \beta_{15} ) q^{10} + ( 2 - \beta_{1} + \beta_{3} + \beta_{5} - \beta_{7} + \beta_{10} - \beta_{12} ) q^{11} + ( \beta_{4} - \beta_{15} ) q^{13} + ( 1 - \beta_{2} + \beta_{3} - \beta_{9} + \beta_{11} ) q^{14} + \beta_{5} q^{16} + ( -1 + \beta_{1} + \beta_{2} - \beta_{3} - \beta_{11} + \beta_{13} - \beta_{14} ) q^{17} + ( -1 - \beta_{1} + \beta_{2} + \beta_{4} + \beta_{6} + 2 \beta_{9} - \beta_{10} - \beta_{11} - \beta_{12} + \beta_{13} + \beta_{14} ) q^{19} + ( \beta_{4} + \beta_{8} - \beta_{15} ) q^{20} + ( 2 - 2 \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{5} + \beta_{10} - \beta_{12} ) q^{22} + ( 1 + \beta_{2} - 2 \beta_{3} - 2 \beta_{5} + 2 \beta_{7} - \beta_{10} + \beta_{12} ) q^{23} + ( -1 + 2 \beta_{1} + \beta_{2} + 3 \beta_{3} - 2 \beta_{7} + \beta_{12} - \beta_{13} - \beta_{14} ) q^{25} + ( \beta_{4} - \beta_{6} + \beta_{8} + \beta_{15} ) q^{26} + ( \beta_{7} - \beta_{10} ) q^{28} + ( -1 + \beta_{1} + \beta_{2} - 2 \beta_{7} - \beta_{10} + \beta_{13} + \beta_{14} ) q^{29} + ( 1 + \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{9} + 2 \beta_{10} - 2 \beta_{11} - \beta_{12} - \beta_{13} - \beta_{14} ) q^{31} + ( \beta_{3} - \beta_{7} ) q^{32} + ( 1 - \beta_{2} + \beta_{3} - \beta_{9} + 2 \beta_{11} + \beta_{12} - \beta_{13} ) q^{34} + ( -2 + 3 \beta_{1} + \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + \beta_{7} - \beta_{8} - \beta_{10} + \beta_{11} + 2 \beta_{12} - 2 \beta_{14} - \beta_{15} ) q^{35} + ( -2 + 3 \beta_{3} - 4 \beta_{7} - \beta_{10} - \beta_{12} + 2 \beta_{13} + 2 \beta_{14} ) q^{37} + ( \beta_{1} - \beta_{3} - \beta_{4} - \beta_{6} - \beta_{9} + \beta_{10} + \beta_{12} - \beta_{14} + \beta_{15} ) q^{38} + ( \beta_{4} - \beta_{6} + \beta_{8} ) q^{40} + ( -1 - 2 \beta_{1} + \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 3 \beta_{9} + \beta_{10} - 3 \beta_{11} - \beta_{12} + \beta_{13} + 2 \beta_{14} + 2 \beta_{15} ) q^{41} + ( -2 + \beta_{1} + 2 \beta_{2} - 3 \beta_{3} - \beta_{5} + \beta_{7} - \beta_{10} + \beta_{12} ) q^{43} + ( 1 - 2 \beta_{1} + 2 \beta_{3} + 2 \beta_{5} - \beta_{7} - \beta_{12} + \beta_{13} + \beta_{14} ) q^{44} + ( 1 - \beta_{3} + 3 \beta_{7} - \beta_{12} + \beta_{13} + \beta_{14} ) q^{46} + ( 1 - \beta_{1} - \beta_{2} + 2 \beta_{3} + 2 \beta_{6} - 2 \beta_{8} - \beta_{10} - 2 \beta_{12} - \beta_{13} + \beta_{14} ) q^{47} + ( 1 - \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{4} - \beta_{9} + 2 \beta_{10} - \beta_{11} - \beta_{12} - \beta_{14} + 2 \beta_{15} ) q^{49} + ( -2 + \beta_{1} + 2 \beta_{5} - \beta_{7} + \beta_{10} - \beta_{13} - \beta_{14} ) q^{50} + ( -\beta_{6} - \beta_{8} ) q^{52} + ( -2 - 2 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} - 2 \beta_{8} + 4 \beta_{9} + \beta_{10} - 2 \beta_{11} + \beta_{12} + 2 \beta_{13} + 2 \beta_{14} + 2 \beta_{15} ) q^{55} + ( \beta_{1} - \beta_{9} ) q^{56} + ( -1 + \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{5} + 2 \beta_{7} + \beta_{12} - \beta_{13} - \beta_{14} ) q^{58} + ( 1 - \beta_{2} - 3 \beta_{4} - 2 \beta_{6} - 2 \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} + \beta_{12} - \beta_{13} - \beta_{15} ) q^{59} + ( 1 + \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{8} - 2 \beta_{9} - \beta_{10} + 4 \beta_{11} + 2 \beta_{12} - \beta_{13} - \beta_{14} + \beta_{15} ) q^{61} + ( 2 - 2 \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{10} - \beta_{12} - 2 \beta_{13} + 2 \beta_{14} ) q^{62} - q^{64} + ( -5 - \beta_{1} - \beta_{2} - 3 \beta_{5} - \beta_{7} + \beta_{10} - \beta_{13} - \beta_{14} ) q^{65} + ( -1 - \beta_{1} - 2 \beta_{2} - 5 \beta_{3} - 3 \beta_{5} - 4 \beta_{7} + \beta_{10} - \beta_{13} - \beta_{14} ) q^{67} + ( -1 + \beta_{2} + \beta_{9} - \beta_{10} - \beta_{11} - \beta_{12} + \beta_{13} ) q^{68} + ( -1 + 3 \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{5} + 2 \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} + 2 \beta_{11} + 2 \beta_{12} - \beta_{13} - 2 \beta_{14} - \beta_{15} ) q^{70} + ( 1 - 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{5} + \beta_{7} + \beta_{10} - \beta_{13} - \beta_{14} ) q^{71} + ( 1 + \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{4} - 2 \beta_{6} - 2 \beta_{9} + \beta_{11} - \beta_{13} - \beta_{14} ) q^{73} + ( -3 + \beta_{1} - \beta_{2} - 2 \beta_{3} - 2 \beta_{5} + 2 \beta_{7} - \beta_{10} + 2 \beta_{12} - \beta_{13} - \beta_{14} ) q^{74} + ( -\beta_{1} + \beta_{3} + \beta_{6} - \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} + \beta_{14} - \beta_{15} ) q^{76} + ( -3 + 2 \beta_{1} + 3 \beta_{3} + 3 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} - 2 \beta_{8} - 2 \beta_{9} - \beta_{10} + 2 \beta_{11} + \beta_{12} - \beta_{13} - 2 \beta_{14} + 2 \beta_{15} ) q^{77} + ( 3 \beta_{1} - 9 \beta_{3} - \beta_{5} + 4 \beta_{7} + \beta_{10} + \beta_{12} - 2 \beta_{13} - 2 \beta_{14} ) q^{79} + ( \beta_{4} - \beta_{6} ) q^{80} + ( -1 - \beta_{1} + \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{6} - 2 \beta_{8} + 2 \beta_{9} + \beta_{10} - \beta_{11} + \beta_{12} + \beta_{13} + \beta_{14} + 2 \beta_{15} ) q^{82} + ( -2 - 4 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - \beta_{6} + 3 \beta_{8} + 6 \beta_{9} - 4 \beta_{10} - 2 \beta_{12} + 2 \beta_{13} + 4 \beta_{14} - 2 \beta_{15} ) q^{83} + ( -1 + \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{5} + 2 \beta_{7} + \beta_{12} - \beta_{13} - \beta_{14} ) q^{85} + ( 2 - \beta_{2} - \beta_{5} + \beta_{10} - \beta_{12} ) q^{86} + ( 1 - 2 \beta_{1} - \beta_{2} + 3 \beta_{3} + \beta_{5} - \beta_{7} - \beta_{12} + \beta_{13} + \beta_{14} ) q^{88} + ( 2 - 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{6} + 2 \beta_{11} - 2 \beta_{13} + 2 \beta_{14} ) q^{89} + ( 2 + 2 \beta_{3} - \beta_{4} - \beta_{6} - 2 \beta_{7} + \beta_{8} - 2 \beta_{12} - \beta_{15} ) q^{91} + ( 5 - \beta_{2} - \beta_{3} + 2 \beta_{5} + 3 \beta_{7} + \beta_{12} - \beta_{13} - \beta_{14} ) q^{92} + ( 1 + \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} - 2 \beta_{9} + 2 \beta_{10} - 2 \beta_{11} - \beta_{12} - \beta_{13} - \beta_{14} + 2 \beta_{15} ) q^{94} + ( -1 - \beta_{1} - 2 \beta_{3} + \beta_{5} + 3 \beta_{7} - \beta_{10} + \beta_{13} + \beta_{14} ) q^{95} + ( 4 - 3 \beta_{1} - 4 \beta_{2} + \beta_{3} + 2 \beta_{4} - 4 \beta_{6} - 2 \beta_{8} - \beta_{9} + 3 \beta_{10} + 2 \beta_{11} + \beta_{12} - 4 \beta_{13} + 3 \beta_{14} + 2 \beta_{15} ) q^{97} + ( 2 - 3 \beta_{1} - \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} - \beta_{7} - 2 \beta_{8} + \beta_{11} - \beta_{12} + 2 \beta_{14} - 2 \beta_{15} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + 8q^{4} + 2q^{7} + O(q^{10})$$ $$16q + 8q^{4} + 2q^{7} + 12q^{11} + 6q^{14} - 8q^{16} + 48q^{23} - 8q^{25} + 4q^{28} + 12q^{29} - 8q^{37} + 4q^{43} + 24q^{46} - 8q^{49} - 60q^{50} + 6q^{56} - 12q^{58} - 16q^{64} - 84q^{65} - 28q^{67} - 36q^{74} - 78q^{77} - 4q^{79} - 12q^{85} + 24q^{86} + 24q^{91} + 48q^{92} - 12q^{95} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 6 x^{14} + 9 x^{12} + 54 x^{10} - 288 x^{8} + 486 x^{6} + 729 x^{4} - 4374 x^{2} + 6561$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{14} - \nu^{12} + 6 \nu^{10} - 36 \nu^{8} + 72 \nu^{6} + 234 \nu^{4} + 729 \nu^{2} - 243$$$$)/1944$$ $$\beta_{2}$$ $$=$$ $$($$$$-2 \nu^{14} + \nu^{12} - 6 \nu^{10} + 36 \nu^{8} - 180 \nu^{6} - 396 \nu^{4} + 972 \nu^{2} - 4131$$$$)/1944$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{14} - 3 \nu^{12} - 9 \nu^{10} + 81 \nu^{8} - 126 \nu^{6} - 135 \nu^{4} + 1458 \nu^{2} - 2187$$$$)/1458$$ $$\beta_{4}$$ $$=$$ $$($$$$-2 \nu^{15} - 9 \nu^{13} + 18 \nu^{11} - 396 \nu^{7} + 216 \nu^{5} - 324 \nu^{3} - 9477 \nu$$$$)/5832$$ $$\beta_{5}$$ $$=$$ $$($$$$2 \nu^{14} - 21 \nu^{12} + 18 \nu^{10} + 108 \nu^{8} - 576 \nu^{6} + 648 \nu^{4} + 972 \nu^{2} - 9477$$$$)/5832$$ $$\beta_{6}$$ $$=$$ $$($$$$-7 \nu^{15} + 24 \nu^{13} - 36 \nu^{11} - 540 \nu^{9} + 1044 \nu^{7} - 1134 \nu^{5} - 8019 \nu^{3} + 13122 \nu$$$$)/17496$$ $$\beta_{7}$$ $$=$$ $$($$$$3 \nu^{14} - 10 \nu^{12} - 12 \nu^{10} + 180 \nu^{8} - 432 \nu^{6} - 198 \nu^{4} + 3483 \nu^{2} - 5832$$$$)/1944$$ $$\beta_{8}$$ $$=$$ $$($$$$-4 \nu^{15} + 9 \nu^{13} + 18 \nu^{11} - 216 \nu^{9} + 504 \nu^{7} + 432 \nu^{5} - 2754 \nu^{3} + 9477 \nu$$$$)/5832$$ $$\beta_{9}$$ $$=$$ $$($$$$-8 \nu^{15} - 3 \nu^{14} + 48 \nu^{13} - 36 \nu^{12} - 18 \nu^{11} + 216 \nu^{10} - 270 \nu^{9} - 162 \nu^{8} + 1332 \nu^{7} - 594 \nu^{6} - 2430 \nu^{5} + 5346 \nu^{4} + 486 \nu^{3} - 729 \nu^{2} + 17496 \nu - 8748$$$$)/17496$$ $$\beta_{10}$$ $$=$$ $$($$$$-8 \nu^{15} - 27 \nu^{14} + 12 \nu^{13} + 72 \nu^{12} + 90 \nu^{11} + 54 \nu^{10} - 432 \nu^{9} - 810 \nu^{8} - 126 \nu^{7} + 2916 \nu^{6} + 2106 \nu^{5} - 3240 \nu^{4} - 7776 \nu^{3} - 6561 \nu^{2} + 4374 \nu + 21870$$$$)/17496$$ $$\beta_{11}$$ $$=$$ $$($$$$-4 \nu^{15} - 39 \nu^{14} + 24 \nu^{13} + 108 \nu^{12} - 90 \nu^{11} + 162 \nu^{10} + 108 \nu^{9} - 1782 \nu^{8} + 666 \nu^{7} + 4428 \nu^{6} - 3402 \nu^{5} - 1620 \nu^{4} + 3888 \nu^{3} - 24057 \nu^{2} + 13122 \nu + 48114$$$$)/17496$$ $$\beta_{12}$$ $$=$$ $$($$$$-8 \nu^{15} + 39 \nu^{14} + 12 \nu^{13} - 108 \nu^{12} + 90 \nu^{11} - 162 \nu^{10} - 432 \nu^{9} + 1782 \nu^{8} - 126 \nu^{7} - 4428 \nu^{6} + 2106 \nu^{5} + 1620 \nu^{4} - 7776 \nu^{3} + 24057 \nu^{2} + 4374 \nu - 48114$$$$)/17496$$ $$\beta_{13}$$ $$=$$ $$($$$$14 \nu^{15} - 6 \nu^{14} - 66 \nu^{13} + 99 \nu^{12} + 72 \nu^{11} - 108 \nu^{10} + 594 \nu^{9} - 972 \nu^{8} - 2574 \nu^{7} + 5130 \nu^{6} + 1620 \nu^{5} - 5022 \nu^{4} + 7776 \nu^{3} - 14580 \nu^{2} - 17496 \nu + 85293$$$$)/17496$$ $$\beta_{14}$$ $$=$$ $$($$$$-22 \nu^{15} - 6 \nu^{14} + 78 \nu^{13} + 27 \nu^{12} + 18 \nu^{11} - 162 \nu^{10} - 1026 \nu^{9} - 162 \nu^{8} + 2448 \nu^{7} + 1242 \nu^{6} + 486 \nu^{5} - 3240 \nu^{4} - 15552 \nu^{3} + 7290 \nu^{2} + 21870 \nu + 6561$$$$)/17496$$ $$\beta_{15}$$ $$=$$ $$($$$$-35 \nu^{15} + 138 \nu^{13} + 36 \nu^{11} - 2052 \nu^{9} + 6192 \nu^{7} - 2106 \nu^{5} - 37179 \nu^{3} + 87480 \nu$$$$)/17496$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{14} + 2 \beta_{12} + 2 \beta_{11} - \beta_{9} + \beta_{8} - \beta_{4} + \beta_{1}$$$$)/3$$ $$\nu^{2}$$ $$=$$ $$\beta_{7} - \beta_{5} - \beta_{3} + \beta_{1}$$ $$\nu^{3}$$ $$=$$ $$-\beta_{15} + \beta_{14} + \beta_{13} - \beta_{12} - \beta_{11} - \beta_{10} + 2 \beta_{9} + 2 \beta_{8} + \beta_{6} + \beta_{2} - \beta_{1} - 1$$ $$\nu^{4}$$ $$=$$ $$-\beta_{14} - \beta_{13} + 2 \beta_{12} - \beta_{10} - 4 \beta_{7} - 3 \beta_{5} + 3 \beta_{3} + 3 \beta_{1}$$ $$\nu^{5}$$ $$=$$ $$-2 \beta_{15} - 5 \beta_{13} + \beta_{11} + 4 \beta_{10} - 5 \beta_{9} + 3 \beta_{8} - 2 \beta_{6} - 3 \beta_{4} + \beta_{3} - 5 \beta_{2} + 5$$ $$\nu^{6}$$ $$=$$ $$-3 \beta_{12} + 3 \beta_{10} - 6 \beta_{7} + 6 \beta_{5} + 18 \beta_{3} - 6 \beta_{2} - 6$$ $$\nu^{7}$$ $$=$$ $$3 \beta_{15} + 3 \beta_{14} - 18 \beta_{12} - 12 \beta_{11} - 6 \beta_{10} + 3 \beta_{9} + 6 \beta_{8} - 9 \beta_{6} + 6 \beta_{4} + 6 \beta_{3} - 3 \beta_{1}$$ $$\nu^{8}$$ $$=$$ $$-12 \beta_{14} - 12 \beta_{13} - 3 \beta_{12} + 15 \beta_{10} - 33 \beta_{7} + 3 \beta_{5} + 75 \beta_{3} - 12 \beta_{2} - 15 \beta_{1} + 27$$ $$\nu^{9}$$ $$=$$ $$18 \beta_{15} - 15 \beta_{14} - 27 \beta_{13} + 21 \beta_{12} + 39 \beta_{11} + 9 \beta_{10} - 42 \beta_{9} - 21 \beta_{8} - 72 \beta_{6} + 3 \beta_{4} + 18 \beta_{3} - 27 \beta_{2} + 15 \beta_{1} + 27$$ $$\nu^{10}$$ $$=$$ $$-45 \beta_{14} - 45 \beta_{13} - 45 \beta_{12} + 90 \beta_{10} + 90 \beta_{7} - 27 \beta_{5} - 27 \beta_{3} - 72 \beta_{2} - 45 \beta_{1} + 27$$ $$\nu^{11}$$ $$=$$ $$63 \beta_{15} + 36 \beta_{14} + 153 \beta_{13} - 54 \beta_{12} - 135 \beta_{11} - 72 \beta_{10} + 189 \beta_{9} - 36 \beta_{8} - 117 \beta_{6} + 72 \beta_{4} - 81 \beta_{3} + 153 \beta_{2} - 36 \beta_{1} - 153$$ $$\nu^{12}$$ $$=$$ $$-126 \beta_{14} - 126 \beta_{13} + 90 \beta_{12} + 36 \beta_{10} + 90 \beta_{7} - 648 \beta_{5} - 216 \beta_{3} - 243$$ $$\nu^{13}$$ $$=$$ $$-36 \beta_{15} + 189 \beta_{14} - 36 \beta_{13} + 216 \beta_{12} + 18 \beta_{11} + 234 \beta_{10} + 153 \beta_{9} - 459 \beta_{8} - 144 \beta_{6} - 297 \beta_{4} - 198 \beta_{3} - 36 \beta_{2} - 189 \beta_{1} + 36$$ $$\nu^{14}$$ $$=$$ $$54 \beta_{14} + 54 \beta_{13} - 54 \beta_{10} + 999 \beta_{7} - 621 \beta_{5} - 1377 \beta_{3} - 432 \beta_{2} - 243 \beta_{1} - 1242$$ $$\nu^{15}$$ $$=$$ $$81 \beta_{15} + 297 \beta_{14} + 837 \beta_{13} - 891 \beta_{12} - 1809 \beta_{11} + 81 \beta_{10} + 1134 \beta_{9} - 1026 \beta_{8} + 999 \beta_{6} - 864 \beta_{4} - 918 \beta_{3} + 837 \beta_{2} - 297 \beta_{1} - 837$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/378\mathbb{Z}\right)^\times$$.

 $$n$$ $$29$$ $$325$$ $$\chi(n)$$ $$1 + \beta_{5}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
125.1
 −1.69547 − 0.354107i 0.0967785 − 1.72934i −0.0967785 + 1.72934i 1.69547 + 0.354107i −1.62181 + 0.608059i 1.40917 + 1.00709i −1.40917 − 1.00709i 1.62181 − 0.608059i −1.69547 + 0.354107i 0.0967785 + 1.72934i −0.0967785 − 1.72934i 1.69547 − 0.354107i −1.62181 − 0.608059i 1.40917 − 1.00709i −1.40917 + 1.00709i 1.62181 + 0.608059i
−0.866025 + 0.500000i 0 0.500000 0.866025i −0.895175 + 1.55049i 0 0.0213944 2.64566i 1.00000i 0 1.79035i
125.2 −0.866025 + 0.500000i 0 0.500000 0.866025i −0.183299 + 0.317483i 0 −0.624224 + 2.57106i 1.00000i 0 0.366598i
125.3 −0.866025 + 0.500000i 0 0.500000 0.866025i 0.183299 0.317483i 0 2.53871 + 0.744936i 1.00000i 0 0.366598i
125.4 −0.866025 + 0.500000i 0 0.500000 0.866025i 0.895175 1.55049i 0 −2.30191 1.30430i 1.00000i 0 1.79035i
125.5 0.866025 0.500000i 0 0.500000 0.866025i −1.94556 + 3.36980i 0 0.343982 + 2.62329i 1.00000i 0 3.89111i
125.6 0.866025 0.500000i 0 0.500000 0.866025i −1.17468 + 2.03460i 0 1.55364 2.14154i 1.00000i 0 2.34936i
125.7 0.866025 0.500000i 0 0.500000 0.866025i 1.17468 2.03460i 0 −2.63145 + 0.274725i 1.00000i 0 2.34936i
125.8 0.866025 0.500000i 0 0.500000 0.866025i 1.94556 3.36980i 0 2.09985 + 1.60954i 1.00000i 0 3.89111i
251.1 −0.866025 0.500000i 0 0.500000 + 0.866025i −0.895175 1.55049i 0 0.0213944 + 2.64566i 1.00000i 0 1.79035i
251.2 −0.866025 0.500000i 0 0.500000 + 0.866025i −0.183299 0.317483i 0 −0.624224 2.57106i 1.00000i 0 0.366598i
251.3 −0.866025 0.500000i 0 0.500000 + 0.866025i 0.183299 + 0.317483i 0 2.53871 0.744936i 1.00000i 0 0.366598i
251.4 −0.866025 0.500000i 0 0.500000 + 0.866025i 0.895175 + 1.55049i 0 −2.30191 + 1.30430i 1.00000i 0 1.79035i
251.5 0.866025 + 0.500000i 0 0.500000 + 0.866025i −1.94556 3.36980i 0 0.343982 2.62329i 1.00000i 0 3.89111i
251.6 0.866025 + 0.500000i 0 0.500000 + 0.866025i −1.17468 2.03460i 0 1.55364 + 2.14154i 1.00000i 0 2.34936i
251.7 0.866025 + 0.500000i 0 0.500000 + 0.866025i 1.17468 + 2.03460i 0 −2.63145 0.274725i 1.00000i 0 2.34936i
251.8 0.866025 + 0.500000i 0 0.500000 + 0.866025i 1.94556 + 3.36980i 0 2.09985 1.60954i 1.00000i 0 3.89111i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 251.8 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.b odd 2 1 inner
9.d odd 6 1 inner
63.o even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 378.2.m.a 16
3.b odd 2 1 126.2.m.a 16
4.b odd 2 1 3024.2.cc.b 16
7.b odd 2 1 inner 378.2.m.a 16
7.c even 3 1 2646.2.l.b 16
7.c even 3 1 2646.2.t.a 16
7.d odd 6 1 2646.2.l.b 16
7.d odd 6 1 2646.2.t.a 16
9.c even 3 1 126.2.m.a 16
9.c even 3 1 1134.2.d.a 16
9.d odd 6 1 inner 378.2.m.a 16
9.d odd 6 1 1134.2.d.a 16
12.b even 2 1 1008.2.cc.b 16
21.c even 2 1 126.2.m.a 16
21.g even 6 1 882.2.l.a 16
21.g even 6 1 882.2.t.b 16
21.h odd 6 1 882.2.l.a 16
21.h odd 6 1 882.2.t.b 16
28.d even 2 1 3024.2.cc.b 16
36.f odd 6 1 1008.2.cc.b 16
36.h even 6 1 3024.2.cc.b 16
63.g even 3 1 882.2.l.a 16
63.h even 3 1 882.2.t.b 16
63.i even 6 1 2646.2.t.a 16
63.j odd 6 1 2646.2.t.a 16
63.k odd 6 1 882.2.l.a 16
63.l odd 6 1 126.2.m.a 16
63.l odd 6 1 1134.2.d.a 16
63.n odd 6 1 2646.2.l.b 16
63.o even 6 1 inner 378.2.m.a 16
63.o even 6 1 1134.2.d.a 16
63.s even 6 1 2646.2.l.b 16
63.t odd 6 1 882.2.t.b 16
84.h odd 2 1 1008.2.cc.b 16
252.s odd 6 1 3024.2.cc.b 16
252.bi even 6 1 1008.2.cc.b 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.m.a 16 3.b odd 2 1
126.2.m.a 16 9.c even 3 1
126.2.m.a 16 21.c even 2 1
126.2.m.a 16 63.l odd 6 1
378.2.m.a 16 1.a even 1 1 trivial
378.2.m.a 16 7.b odd 2 1 inner
378.2.m.a 16 9.d odd 6 1 inner
378.2.m.a 16 63.o even 6 1 inner
882.2.l.a 16 21.g even 6 1
882.2.l.a 16 21.h odd 6 1
882.2.l.a 16 63.g even 3 1
882.2.l.a 16 63.k odd 6 1
882.2.t.b 16 21.g even 6 1
882.2.t.b 16 21.h odd 6 1
882.2.t.b 16 63.h even 3 1
882.2.t.b 16 63.t odd 6 1
1008.2.cc.b 16 12.b even 2 1
1008.2.cc.b 16 36.f odd 6 1
1008.2.cc.b 16 84.h odd 2 1
1008.2.cc.b 16 252.bi even 6 1
1134.2.d.a 16 9.c even 3 1
1134.2.d.a 16 9.d odd 6 1
1134.2.d.a 16 63.l odd 6 1
1134.2.d.a 16 63.o even 6 1
2646.2.l.b 16 7.c even 3 1
2646.2.l.b 16 7.d odd 6 1
2646.2.l.b 16 63.n odd 6 1
2646.2.l.b 16 63.s even 6 1
2646.2.t.a 16 7.c even 3 1
2646.2.t.a 16 7.d odd 6 1
2646.2.t.a 16 63.i even 6 1
2646.2.t.a 16 63.j odd 6 1
3024.2.cc.b 16 4.b odd 2 1
3024.2.cc.b 16 28.d even 2 1
3024.2.cc.b 16 36.h even 6 1
3024.2.cc.b 16 252.s odd 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(378, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T^{2} + T^{4} )^{4}$$
$3$ 
$5$ $$1 - 16 T^{2} + 123 T^{4} - 584 T^{6} + 1481 T^{8} + 1416 T^{10} - 59414 T^{12} + 576968 T^{14} - 3477114 T^{16} + 14424200 T^{18} - 37133750 T^{20} + 22125000 T^{22} + 578515625 T^{24} - 5703125000 T^{26} + 30029296875 T^{28} - 97656250000 T^{30} + 152587890625 T^{32}$$
$7$ $$1 - 2 T + 6 T^{2} + 8 T^{3} - 58 T^{4} + 222 T^{5} - 104 T^{6} - 662 T^{7} + 3483 T^{8} - 4634 T^{9} - 5096 T^{10} + 76146 T^{11} - 139258 T^{12} + 134456 T^{13} + 705894 T^{14} - 1647086 T^{15} + 5764801 T^{16}$$
$11$ $$( 1 - 6 T + 35 T^{2} - 138 T^{3} + 481 T^{4} - 1512 T^{5} + 3854 T^{6} - 13116 T^{7} + 37618 T^{8} - 144276 T^{9} + 466334 T^{10} - 2012472 T^{11} + 7042321 T^{12} - 22225038 T^{13} + 62004635 T^{14} - 116923026 T^{15} + 214358881 T^{16} )^{2}$$
$13$ $$1 + 68 T^{2} + 2376 T^{4} + 57352 T^{6} + 1082018 T^{8} + 16951644 T^{10} + 232506496 T^{12} + 2987085740 T^{14} + 38351015667 T^{16} + 504817490060 T^{18} + 6640618032256 T^{20} + 81822347823996 T^{22} + 882635323274978 T^{24} + 7906460224523848 T^{26} + 55356250251014856 T^{28} + 267741594227551652 T^{30} + 665416609183179841 T^{32}$$
$17$ $$( 1 + 94 T^{2} + 4285 T^{4} + 124198 T^{6} + 2503180 T^{8} + 35893222 T^{10} + 357887485 T^{12} + 2268931486 T^{14} + 6975757441 T^{16} )^{2}$$
$19$ $$( 1 - 98 T^{2} + 4546 T^{4} - 134984 T^{6} + 2932423 T^{8} - 48729224 T^{10} + 592439266 T^{12} - 4610496338 T^{14} + 16983563041 T^{16} )^{2}$$
$23$ $$( 1 - 24 T + 317 T^{2} - 3000 T^{3} + 22111 T^{4} - 134028 T^{5} + 704756 T^{6} - 3411156 T^{7} + 16228318 T^{8} - 78456588 T^{9} + 372815924 T^{10} - 1630718676 T^{11} + 6187564351 T^{12} - 19309029000 T^{13} + 46927376813 T^{14} - 81715810728 T^{15} + 78310985281 T^{16} )^{2}$$
$29$ $$( 1 - 6 T + 98 T^{2} - 516 T^{3} + 4846 T^{4} - 25650 T^{5} + 193448 T^{6} - 972210 T^{7} + 6347347 T^{8} - 28194090 T^{9} + 162689768 T^{10} - 625577850 T^{11} + 3427483726 T^{12} - 10583752884 T^{13} + 58292685458 T^{14} - 103499257854 T^{15} + 500246412961 T^{16} )^{2}$$
$31$ $$1 + 104 T^{2} + 3888 T^{4} + 96880 T^{6} + 4455362 T^{8} + 148421160 T^{10} + 1870813504 T^{12} + 70884338648 T^{14} + 4079738375235 T^{16} + 68119849440728 T^{18} + 1727735558027584 T^{20} + 131724325838289960 T^{22} + 3799938318355208642 T^{24} + 79405588442700000880 T^{26} +$$$$30\!\cdots\!68$$$$T^{28} +$$$$78\!\cdots\!84$$$$T^{30} +$$$$72\!\cdots\!81$$$$T^{32}$$
$37$ $$( 1 + 2 T + 46 T^{2} + 38 T^{3} + 2002 T^{4} + 1406 T^{5} + 62974 T^{6} + 101306 T^{7} + 1874161 T^{8} )^{4}$$
$41$ $$1 - 70 T^{2} - 1569 T^{4} + 150526 T^{6} + 5171081 T^{8} - 280356900 T^{10} - 8583026738 T^{12} + 255096492224 T^{14} + 9422146980954 T^{16} + 428817203428544 T^{18} - 24253582218197618 T^{20} - 1331724499683612900 T^{22} + 41290695138728249801 T^{24} +$$$$20\!\cdots\!26$$$$T^{26} -$$$$35\!\cdots\!89$$$$T^{28} -$$$$26\!\cdots\!70$$$$T^{30} +$$$$63\!\cdots\!41$$$$T^{32}$$
$43$ $$( 1 - 2 T - 129 T^{2} - 46 T^{3} + 9833 T^{4} + 11184 T^{5} - 521114 T^{6} - 232628 T^{7} + 22298490 T^{8} - 10003004 T^{9} - 963539786 T^{10} + 889206288 T^{11} + 33617070233 T^{12} - 6762388378 T^{13} - 815455833321 T^{14} - 543637222214 T^{15} + 11688200277601 T^{16} )^{2}$$
$47$ $$1 - 136 T^{2} + 8016 T^{4} - 225584 T^{6} - 1533310 T^{8} + 489880632 T^{10} - 30253322048 T^{12} + 1486359308360 T^{14} - 68731587628605 T^{16} + 3283367712167240 T^{18} - 147626560784506688 T^{20} + 5280528817834607928 T^{22} - 36510083951344758910 T^{24} -$$$$11\!\cdots\!16$$$$T^{26} +$$$$93\!\cdots\!56$$$$T^{28} -$$$$34\!\cdots\!84$$$$T^{30} +$$$$56\!\cdots\!21$$$$T^{32}$$
$53$ $$( 1 - 53 T^{2} )^{16}$$
$59$ $$1 - 178 T^{2} + 20643 T^{4} - 1496150 T^{6} + 80456981 T^{8} - 3515943660 T^{10} + 180649052698 T^{12} - 13219132050040 T^{14} + 857467356385554 T^{16} - 46015798666189240 T^{18} + 2188989785849689978 T^{20} -$$$$14\!\cdots\!60$$$$T^{22} +$$$$11\!\cdots\!01$$$$T^{24} -$$$$76\!\cdots\!50$$$$T^{26} +$$$$36\!\cdots\!83$$$$T^{28} -$$$$11\!\cdots\!58$$$$T^{30} +$$$$21\!\cdots\!41$$$$T^{32}$$
$61$ $$1 + 248 T^{2} + 28191 T^{4} + 2052448 T^{6} + 122525357 T^{8} + 7198089144 T^{10} + 457346362462 T^{12} + 32095759051208 T^{14} + 2131627513941198 T^{16} + 119428319429544968 T^{18} + 6332345016577220542 T^{20} +$$$$37\!\cdots\!84$$$$T^{22} +$$$$23\!\cdots\!17$$$$T^{24} +$$$$14\!\cdots\!48$$$$T^{26} +$$$$74\!\cdots\!11$$$$T^{28} +$$$$24\!\cdots\!68$$$$T^{30} +$$$$36\!\cdots\!61$$$$T^{32}$$
$67$ $$( 1 + 14 T + 39 T^{2} - 110 T^{3} - 2659 T^{4} - 53760 T^{5} - 92054 T^{6} + 2669060 T^{7} + 22240746 T^{8} + 178827020 T^{9} - 413230406 T^{10} - 16169018880 T^{11} - 53581830739 T^{12} - 148513761770 T^{13} + 3527876904591 T^{14} + 84849962474522 T^{15} + 406067677556641 T^{16} )^{2}$$
$71$ $$( 1 - 478 T^{2} + 105553 T^{4} - 13986142 T^{6} + 1213269316 T^{8} - 70504141822 T^{10} + 2682279164593 T^{12} - 61231935714238 T^{14} + 645753531245761 T^{16} )^{2}$$
$73$ $$( 1 - 362 T^{2} + 64045 T^{4} - 7316714 T^{6} + 612211324 T^{8} - 38990768906 T^{10} + 1818765344845 T^{12} - 54782989916618 T^{14} + 806460091894081 T^{16} )^{2}$$
$79$ $$( 1 + 2 T - 183 T^{2} + 982 T^{3} + 19715 T^{4} - 144312 T^{5} - 491612 T^{6} + 7480148 T^{7} - 8945118 T^{8} + 590931692 T^{9} - 3068150492 T^{10} - 71151444168 T^{11} + 767900846915 T^{12} + 3021669383818 T^{13} - 44485004360343 T^{14} + 38407817972318 T^{15} + 1517108809906561 T^{16} )^{2}$$
$83$ $$1 + 44 T^{2} - 13368 T^{4} - 595880 T^{6} + 87112226 T^{8} + 3596762100 T^{10} - 160886956928 T^{12} - 11841264313180 T^{14} - 673218489607821 T^{16} - 81574469853497020 T^{18} - 7635424846602197888 T^{20} +$$$$11\!\cdots\!00$$$$T^{22} +$$$$19\!\cdots\!66$$$$T^{24} -$$$$92\!\cdots\!20$$$$T^{26} -$$$$14\!\cdots\!48$$$$T^{28} +$$$$32\!\cdots\!76$$$$T^{30} +$$$$50\!\cdots\!81$$$$T^{32}$$
$89$ $$( 1 + 496 T^{2} + 119404 T^{4} + 18272464 T^{6} + 1934931814 T^{8} + 144736187344 T^{10} + 7491674544364 T^{12} + 246502720316656 T^{14} + 3936588805702081 T^{16} )^{2}$$
$97$ $$1 + 74 T^{2} + 11511 T^{4} - 803858 T^{6} - 150210775 T^{8} - 25062425316 T^{10} - 114467134418 T^{12} + 73367970993632 T^{14} + 27832786456667274 T^{16} + 690319239079083488 T^{18} - 10133693108155893458 T^{20} -$$$$20\!\cdots\!64$$$$T^{22} -$$$$11\!\cdots\!75$$$$T^{24} -$$$$59\!\cdots\!42$$$$T^{26} +$$$$79\!\cdots\!51$$$$T^{28} +$$$$48\!\cdots\!06$$$$T^{30} +$$$$61\!\cdots\!21$$$$T^{32}$$
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