# Properties

 Label 252.2.x.a Level $252$ Weight $2$ Character orbit 252.x Analytic conductor $2.012$ Analytic rank $0$ Dimension $16$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$252 = 2^{2} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 252.x (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.01223013094$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Defining polynomial: $$x^{16} - 3 x^{14} - 9 x^{12} - 9 x^{10} + 225 x^{8} - 81 x^{6} - 729 x^{4} - 2187 x^{2} + 6561$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$3^{4}$$ 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 $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -\beta_{1} - \beta_{8} ) q^{3} + \beta_{15} q^{5} + ( -\beta_{1} - \beta_{7} - \beta_{8} - \beta_{15} ) q^{7} + \beta_{11} q^{9} +O(q^{10})$$ $$q + ( -\beta_{1} - \beta_{8} ) q^{3} + \beta_{15} q^{5} + ( -\beta_{1} - \beta_{7} - \beta_{8} - \beta_{15} ) q^{7} + \beta_{11} q^{9} + ( \beta_{2} + \beta_{5} + \beta_{9} + \beta_{11} ) q^{11} + ( 2 \beta_{1} - 2 \beta_{3} - \beta_{6} + \beta_{7} + \beta_{12} + \beta_{13} + \beta_{14} ) q^{13} + ( -\beta_{4} + \beta_{5} + \beta_{9} - \beta_{10} + \beta_{11} + \beta_{14} ) q^{15} + ( -\beta_{1} + 2 \beta_{3} + \beta_{6} - 2 \beta_{7} + \beta_{10} - 2 \beta_{12} - \beta_{13} - \beta_{14} ) q^{17} + ( \beta_{1} - \beta_{3} + \beta_{6} - \beta_{8} + \beta_{10} + \beta_{14} + \beta_{15} ) q^{19} + ( 1 - \beta_{1} + \beta_{3} - \beta_{4} - \beta_{5} - \beta_{9} - \beta_{12} - \beta_{13} - \beta_{14} - \beta_{15} ) q^{21} + ( \beta_{2} - \beta_{5} - \beta_{9} ) q^{23} + ( -\beta_{1} - 2 \beta_{2} + \beta_{5} - \beta_{7} - \beta_{8} + 2 \beta_{9} - \beta_{10} - \beta_{11} + \beta_{12} - \beta_{15} ) q^{25} + ( \beta_{1} - \beta_{3} - \beta_{6} + 2 \beta_{7} - \beta_{10} + 2 \beta_{12} + \beta_{14} ) q^{27} + ( -2 - \beta_{1} + \beta_{4} - \beta_{5} - \beta_{7} - \beta_{8} + \beta_{10} - \beta_{11} + \beta_{12} - 2 \beta_{14} - \beta_{15} ) q^{29} + ( -2 \beta_{3} - \beta_{15} ) q^{31} + ( -\beta_{1} + \beta_{3} - \beta_{6} + \beta_{7} - 2 \beta_{10} + \beta_{12} + \beta_{13} - \beta_{14} ) q^{33} + ( -1 - \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{6} + \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} - 2 \beta_{12} + \beta_{15} ) q^{35} + ( 1 - \beta_{2} - \beta_{5} - 2 \beta_{9} - 2 \beta_{11} ) q^{37} + ( 2 + \beta_{1} - 2 \beta_{4} + \beta_{5} + \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} - \beta_{12} + \beta_{15} ) q^{39} + ( 3 \beta_{1} + \beta_{6} + 3 \beta_{8} + \beta_{13} + 2 \beta_{15} ) q^{41} + ( \beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} + \beta_{7} + \beta_{8} - \beta_{9} + \beta_{10} - \beta_{11} - \beta_{12} + \beta_{15} ) q^{43} + ( -\beta_{1} + 3 \beta_{3} + \beta_{8} ) q^{45} + ( \beta_{3} - \beta_{6} + 3 \beta_{8} - \beta_{13} ) q^{47} + ( \beta_{1} - 2 \beta_{2} - 2 \beta_{5} - \beta_{6} - \beta_{8} - \beta_{9} - \beta_{11} + \beta_{12} + \beta_{13} ) q^{49} + ( -\beta_{2} - \beta_{4} - 2 \beta_{5} - 2 \beta_{9} - \beta_{10} - \beta_{11} + \beta_{14} ) q^{51} + ( -2 + 2 \beta_{1} + \beta_{2} + 4 \beta_{4} + 2 \beta_{5} + 2 \beta_{7} + 2 \beta_{8} + \beta_{9} + \beta_{10} - \beta_{11} - 2 \beta_{12} + \beta_{14} + 2 \beta_{15} ) q^{53} + ( \beta_{1} - \beta_{6} + 2 \beta_{8} + \beta_{13} ) q^{55} + ( -4 + \beta_{1} - \beta_{2} + 3 \beta_{4} + \beta_{7} + \beta_{8} + \beta_{9} + \beta_{11} - \beta_{12} + \beta_{14} + \beta_{15} ) q^{57} + ( -\beta_{1} - \beta_{6} + \beta_{7} - 2 \beta_{10} + \beta_{12} - \beta_{14} ) q^{59} + ( 3 \beta_{1} - 2 \beta_{3} + \beta_{6} + \beta_{7} + 2 \beta_{8} - \beta_{10} + \beta_{12} - \beta_{13} - 3 \beta_{15} ) q^{61} + ( -1 - \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} - 2 \beta_{9} + \beta_{10} - \beta_{11} + \beta_{12} - \beta_{13} + \beta_{15} ) q^{63} + ( -2 - \beta_{1} - \beta_{2} + \beta_{4} - \beta_{5} - \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} - \beta_{11} + \beta_{12} - 2 \beta_{14} - \beta_{15} ) q^{65} + ( -\beta_{1} + \beta_{2} + \beta_{5} - \beta_{7} - \beta_{8} + 2 \beta_{9} - \beta_{11} + \beta_{12} - \beta_{14} - \beta_{15} ) q^{67} + ( -2 \beta_{1} + 2 \beta_{3} + 2 \beta_{6} - \beta_{7} - \beta_{10} - \beta_{12} - 3 \beta_{13} - 2 \beta_{14} ) q^{69} + ( 3 - \beta_{2} - 6 \beta_{4} + \beta_{9} - \beta_{11} ) q^{71} + ( \beta_{1} + \beta_{3} + \beta_{6} - \beta_{8} + \beta_{10} + \beta_{14} - \beta_{15} ) q^{73} + ( -\beta_{3} - \beta_{6} - \beta_{7} - \beta_{8} + 2 \beta_{10} - \beta_{12} + 3 \beta_{13} + \beta_{14} - 3 \beta_{15} ) q^{75} + ( -1 + 2 \beta_{1} + \beta_{2} - \beta_{4} - \beta_{5} + \beta_{6} + \beta_{8} - \beta_{9} - 2 \beta_{10} + \beta_{12} ) q^{77} + ( \beta_{2} + \beta_{4} - 2 \beta_{5} - \beta_{9} + 2 \beta_{11} ) q^{79} + ( -2 - \beta_{1} - 2 \beta_{4} + \beta_{5} - \beta_{7} - \beta_{8} + 2 \beta_{9} + \beta_{10} + \beta_{12} - 2 \beta_{14} - \beta_{15} ) q^{81} + ( -3 \beta_{1} + 2 \beta_{3} + 2 \beta_{6} - 3 \beta_{8} - \beta_{13} ) q^{83} + ( 1 + \beta_{1} - \beta_{4} + \beta_{7} + \beta_{8} - \beta_{12} + \beta_{14} + \beta_{15} ) q^{85} + ( 2 \beta_{1} - 5 \beta_{3} - \beta_{6} + \beta_{7} - 2 \beta_{10} + \beta_{12} + \beta_{13} - \beta_{14} - 3 \beta_{15} ) q^{87} + ( 2 \beta_{1} - 2 \beta_{6} - 2 \beta_{7} - 3 \beta_{8} + \beta_{10} - 2 \beta_{12} - \beta_{13} - \beta_{14} - 2 \beta_{15} ) q^{89} + ( -1 - \beta_{3} - 2 \beta_{6} + 3 \beta_{8} - \beta_{10} + \beta_{13} + \beta_{15} ) q^{91} + ( -2 + 2 \beta_{1} + 2 \beta_{2} + \beta_{4} + \beta_{5} + 2 \beta_{7} + 2 \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} - 2 \beta_{12} + \beta_{14} + 2 \beta_{15} ) q^{93} + ( -2 - \beta_{1} - 2 \beta_{2} - 2 \beta_{4} + 2 \beta_{5} - \beta_{7} - \beta_{8} + 3 \beta_{9} - 2 \beta_{10} + \beta_{11} + \beta_{12} + \beta_{14} - \beta_{15} ) q^{95} + ( -4 \beta_{1} + \beta_{3} - 2 \beta_{6} - \beta_{7} - 4 \beta_{8} + \beta_{10} - \beta_{12} + \beta_{13} + \beta_{15} ) q^{97} + ( 5 - 2 \beta_{1} + 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{7} - 2 \beta_{8} + \beta_{9} - \beta_{10} + 2 \beta_{12} - \beta_{14} - 2 \beta_{15} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q - q^{7} + O(q^{10})$$ $$16q - q^{7} + 6q^{11} - 12q^{15} + 9q^{21} + 6q^{23} - 8q^{25} - 12q^{29} + 4q^{37} + 18q^{39} + 4q^{43} - 5q^{49} - 18q^{51} - 42q^{57} - 27q^{63} - 24q^{65} + 14q^{67} - 21q^{77} + 20q^{79} - 36q^{81} + 6q^{85} - 18q^{91} - 24q^{93} - 60q^{95} + 90q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 3 x^{14} - 9 x^{12} - 9 x^{10} + 225 x^{8} - 81 x^{6} - 729 x^{4} - 2187 x^{2} + 6561$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{15} + 15 \nu^{13} + 72 \nu^{11} + 153 \nu^{9} - 423 \nu^{7} - 891 \nu^{5} + 1944 \nu^{3} + 17496 \nu$$$$)/15309$$ $$\beta_{4}$$ $$=$$ $$($$$$5 \nu^{14} + 12 \nu^{12} - 18 \nu^{10} - 369 \nu^{8} + 153 \nu^{6} + 1782 \nu^{4} + 4617 \nu^{2} - 9477$$$$)/5103$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{14} - 3 \nu^{12} - 9 \nu^{10} - 9 \nu^{8} + 225 \nu^{6} - 81 \nu^{4} - 729 \nu^{2} - 2187$$$$)/729$$ $$\beta_{6}$$ $$=$$ $$($$$$-2 \nu^{15} - 9 \nu^{13} - 18 \nu^{11} + 72 \nu^{9} + 657 \nu^{7} - 297 \nu^{5} - 3888 \nu^{3} - 9477 \nu$$$$)/5103$$ $$\beta_{7}$$ $$=$$ $$($$$$9 \nu^{15} - 19 \nu^{14} - 33 \nu^{13} + 30 \nu^{12} - 108 \nu^{11} + 144 \nu^{10} - 135 \nu^{9} + 495 \nu^{8} + 2241 \nu^{7} - 1602 \nu^{6} + 1242 \nu^{5} - 648 \nu^{4} - 2349 \nu^{3} - 4617 \nu^{2} - 31347 \nu + 14580$$$$)/10206$$ $$\beta_{8}$$ $$=$$ $$($$$$5 \nu^{15} + 12 \nu^{13} - 18 \nu^{11} - 369 \nu^{9} + 153 \nu^{7} + 1782 \nu^{5} + 4617 \nu^{3} - 14580 \nu$$$$)/5103$$ $$\beta_{9}$$ $$=$$ $$($$$$-20 \nu^{14} + 15 \nu^{12} + 72 \nu^{10} + 342 \nu^{8} - 1179 \nu^{6} + 243 \nu^{4} - 1458 \nu^{2} + 2187$$$$)/5103$$ $$\beta_{10}$$ $$=$$ $$($$$$5 \nu^{15} - 18 \nu^{14} + 12 \nu^{13} + 27 \nu^{12} - 72 \nu^{11} + 162 \nu^{10} - 207 \nu^{9} + 648 \nu^{8} + 396 \nu^{7} - 2349 \nu^{6} + 2268 \nu^{5} - 3888 \nu^{4} - 243 \nu^{3} - 2916 \nu^{2} - 8748 \nu + 41553$$$$)/4374$$ $$\beta_{11}$$ $$=$$ $$($$$$\nu^{14} + \nu^{12} - 12 \nu^{10} - 36 \nu^{8} + 81 \nu^{6} + 306 \nu^{4} - 135 \nu^{2} - 1215$$$$)/189$$ $$\beta_{12}$$ $$=$$ $$($$$$-32 \nu^{15} - 69 \nu^{14} + 87 \nu^{13} + 99 \nu^{12} + 342 \nu^{11} + 702 \nu^{10} + 774 \nu^{9} + 3051 \nu^{8} - 5175 \nu^{7} - 11637 \nu^{6} - 5508 \nu^{5} - 25272 \nu^{4} + 12636 \nu^{3} - 6561 \nu^{2} + 67797 \nu + 247131$$$$)/30618$$ $$\beta_{13}$$ $$=$$ $$($$$$3 \nu^{15} - 4 \nu^{13} - 15 \nu^{11} - 45 \nu^{9} + 306 \nu^{7} - 90 \nu^{5} - 405 \nu^{3} - 1944 \nu$$$$)/1701$$ $$\beta_{14}$$ $$=$$ $$($$$$5 \nu^{15} + 18 \nu^{14} + 12 \nu^{13} - 27 \nu^{12} - 72 \nu^{11} - 162 \nu^{10} - 207 \nu^{9} - 648 \nu^{8} + 396 \nu^{7} + 2349 \nu^{6} + 2268 \nu^{5} + 3888 \nu^{4} - 243 \nu^{3} + 2916 \nu^{2} - 8748 \nu - 41553$$$$)/4374$$ $$\beta_{15}$$ $$=$$ $$($$$$-62 \nu^{15} + 15 \nu^{13} + 639 \nu^{11} + 2421 \nu^{9} - 7794 \nu^{7} - 17901 \nu^{5} - 3159 \nu^{3} + 139968 \nu$$$$)/15309$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2}$$ $$\nu^{3}$$ $$=$$ $$\beta_{14} + 2 \beta_{12} - \beta_{10} + 2 \beta_{7} - \beta_{6} - \beta_{3} + \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$2 \beta_{15} + \beta_{14} - 2 \beta_{12} + \beta_{10} - \beta_{9} + 2 \beta_{8} + 2 \beta_{7} - 2 \beta_{5} - 2 \beta_{4} + 2 \beta_{1} + 4$$ $$\nu^{5}$$ $$=$$ $$-3 \beta_{15} - 3 \beta_{14} - 4 \beta_{13} - 3 \beta_{12} - 3 \beta_{8} - 3 \beta_{7} - \beta_{6} + 6 \beta_{3} - 3 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$3 \beta_{15} - 3 \beta_{12} + 3 \beta_{10} + 3 \beta_{8} + 3 \beta_{7} + 6 \beta_{5} + 3 \beta_{4} + 6 \beta_{2} + 3 \beta_{1} + 15$$ $$\nu^{7}$$ $$=$$ $$3 \beta_{14} + 6 \beta_{12} - 3 \beta_{10} + 3 \beta_{8} + 6 \beta_{7} + 6 \beta_{6} + 6 \beta_{3} + 18 \beta_{1}$$ $$\nu^{8}$$ $$=$$ $$15 \beta_{15} + 3 \beta_{14} - 15 \beta_{12} + 3 \beta_{11} + 12 \beta_{10} - 12 \beta_{9} + 15 \beta_{8} + 15 \beta_{7} - 6 \beta_{5} - 24 \beta_{4} + 24 \beta_{2} + 15 \beta_{1} - 24$$ $$\nu^{9}$$ $$=$$ $$-9 \beta_{15} + 9 \beta_{14} - 9 \beta_{13} + 18 \beta_{12} - 9 \beta_{10} - 27 \beta_{8} + 18 \beta_{7} - 18 \beta_{6} + 27 \beta_{3} - 54 \beta_{1}$$ $$\nu^{10}$$ $$=$$ $$63 \beta_{15} + 18 \beta_{14} - 63 \beta_{12} - 18 \beta_{11} + 45 \beta_{10} - 45 \beta_{9} + 63 \beta_{8} + 63 \beta_{7} - 9 \beta_{5} + 18 \beta_{4} - 18 \beta_{2} + 63 \beta_{1} + 72$$ $$\nu^{11}$$ $$=$$ $$-54 \beta_{15} - 117 \beta_{14} - 63 \beta_{13} - 126 \beta_{12} + 9 \beta_{10} - 126 \beta_{7} + 54 \beta_{6} + 252 \beta_{3} - 90 \beta_{1}$$ $$\nu^{12}$$ $$=$$ $$63 \beta_{15} - 63 \beta_{14} - 63 \beta_{12} + 54 \beta_{11} + 126 \beta_{10} - 18 \beta_{9} + 63 \beta_{8} + 63 \beta_{7} + 180 \beta_{5} + 153 \beta_{4} + 216 \beta_{2} + 63 \beta_{1} - 198$$ $$\nu^{13}$$ $$=$$ $$189 \beta_{15} + 270 \beta_{14} + 144 \beta_{13} + 189 \beta_{12} + 81 \beta_{10} + 216 \beta_{8} + 189 \beta_{7} + 198 \beta_{6} + 108 \beta_{3} + 162 \beta_{1}$$ $$\nu^{14}$$ $$=$$ $$378 \beta_{15} + 81 \beta_{14} - 378 \beta_{12} + 27 \beta_{11} + 297 \beta_{10} - 648 \beta_{9} + 378 \beta_{8} + 378 \beta_{7} - 378 \beta_{5} - 432 \beta_{4} + 81 \beta_{2} + 378 \beta_{1} - 1026$$ $$\nu^{15}$$ $$=$$ $$-243 \beta_{15} - 351 \beta_{14} + 189 \beta_{13} - 540 \beta_{12} + 189 \beta_{10} - 513 \beta_{8} - 540 \beta_{7} - 513 \beta_{6} + 1242 \beta_{3} - 2187 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/252\mathbb{Z}\right)^\times$$.

 $$n$$ $$29$$ $$73$$ $$127$$ $$\chi(n)$$ $$1 - \beta_{4}$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
41.1
 0.744857 − 1.56371i 1.69483 − 0.357142i −0.604587 − 1.62311i 1.71965 + 0.206851i −1.71965 − 0.206851i 0.604587 + 1.62311i −1.69483 + 0.357142i −0.744857 + 1.56371i 0.744857 + 1.56371i 1.69483 + 0.357142i −0.604587 + 1.62311i 1.71965 − 0.206851i −1.71965 + 0.206851i 0.604587 − 1.62311i −1.69483 − 0.357142i −0.744857 − 1.56371i
0 −1.72664 + 0.136790i 0 −0.276914 + 0.479629i 0 −0.519138 2.59432i 0 2.96258 0.472374i 0
41.2 0 −1.15671 1.28920i 0 −1.21244 + 2.10001i 0 1.57244 + 2.12778i 0 −0.324049 + 2.98245i 0
41.3 0 −1.10336 + 1.33514i 0 −0.266780 + 0.462077i 0 −1.89325 + 1.84814i 0 −0.565203 2.94628i 0
41.4 0 −0.680689 1.59269i 0 2.09336 3.62580i 0 −2.64522 + 0.0532130i 0 −2.07332 + 2.16825i 0
41.5 0 0.680689 + 1.59269i 0 −2.09336 + 3.62580i 0 1.36869 2.26422i 0 −2.07332 + 2.16825i 0
41.6 0 1.10336 1.33514i 0 0.266780 0.462077i 0 2.54716 0.715531i 0 −0.565203 2.94628i 0
41.7 0 1.15671 + 1.28920i 0 1.21244 2.10001i 0 1.05649 + 2.42566i 0 −0.324049 + 2.98245i 0
41.8 0 1.72664 0.136790i 0 0.276914 0.479629i 0 −1.98718 1.74675i 0 2.96258 0.472374i 0
209.1 0 −1.72664 0.136790i 0 −0.276914 0.479629i 0 −0.519138 + 2.59432i 0 2.96258 + 0.472374i 0
209.2 0 −1.15671 + 1.28920i 0 −1.21244 2.10001i 0 1.57244 2.12778i 0 −0.324049 2.98245i 0
209.3 0 −1.10336 1.33514i 0 −0.266780 0.462077i 0 −1.89325 1.84814i 0 −0.565203 + 2.94628i 0
209.4 0 −0.680689 + 1.59269i 0 2.09336 + 3.62580i 0 −2.64522 0.0532130i 0 −2.07332 2.16825i 0
209.5 0 0.680689 1.59269i 0 −2.09336 3.62580i 0 1.36869 + 2.26422i 0 −2.07332 2.16825i 0
209.6 0 1.10336 + 1.33514i 0 0.266780 + 0.462077i 0 2.54716 + 0.715531i 0 −0.565203 + 2.94628i 0
209.7 0 1.15671 1.28920i 0 1.21244 + 2.10001i 0 1.05649 2.42566i 0 −0.324049 2.98245i 0
209.8 0 1.72664 + 0.136790i 0 0.276914 + 0.479629i 0 −1.98718 + 1.74675i 0 2.96258 + 0.472374i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 209.8 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 252.2.x.a 16
3.b odd 2 1 756.2.x.a 16
4.b odd 2 1 1008.2.cc.c 16
7.b odd 2 1 inner 252.2.x.a 16
7.c even 3 1 1764.2.w.a 16
7.c even 3 1 1764.2.bm.b 16
7.d odd 6 1 1764.2.w.a 16
7.d odd 6 1 1764.2.bm.b 16
9.c even 3 1 756.2.x.a 16
9.c even 3 1 2268.2.f.b 16
9.d odd 6 1 inner 252.2.x.a 16
9.d odd 6 1 2268.2.f.b 16
12.b even 2 1 3024.2.cc.c 16
21.c even 2 1 756.2.x.a 16
21.g even 6 1 5292.2.w.a 16
21.g even 6 1 5292.2.bm.b 16
21.h odd 6 1 5292.2.w.a 16
21.h odd 6 1 5292.2.bm.b 16
28.d even 2 1 1008.2.cc.c 16
36.f odd 6 1 3024.2.cc.c 16
36.h even 6 1 1008.2.cc.c 16
63.g even 3 1 5292.2.w.a 16
63.h even 3 1 5292.2.bm.b 16
63.i even 6 1 1764.2.bm.b 16
63.j odd 6 1 1764.2.bm.b 16
63.k odd 6 1 5292.2.w.a 16
63.l odd 6 1 756.2.x.a 16
63.l odd 6 1 2268.2.f.b 16
63.n odd 6 1 1764.2.w.a 16
63.o even 6 1 inner 252.2.x.a 16
63.o even 6 1 2268.2.f.b 16
63.s even 6 1 1764.2.w.a 16
63.t odd 6 1 5292.2.bm.b 16
84.h odd 2 1 3024.2.cc.c 16
252.s odd 6 1 1008.2.cc.c 16
252.bi even 6 1 3024.2.cc.c 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.2.x.a 16 1.a even 1 1 trivial
252.2.x.a 16 7.b odd 2 1 inner
252.2.x.a 16 9.d odd 6 1 inner
252.2.x.a 16 63.o even 6 1 inner
756.2.x.a 16 3.b odd 2 1
756.2.x.a 16 9.c even 3 1
756.2.x.a 16 21.c even 2 1
756.2.x.a 16 63.l odd 6 1
1008.2.cc.c 16 4.b odd 2 1
1008.2.cc.c 16 28.d even 2 1
1008.2.cc.c 16 36.h even 6 1
1008.2.cc.c 16 252.s odd 6 1
1764.2.w.a 16 7.c even 3 1
1764.2.w.a 16 7.d odd 6 1
1764.2.w.a 16 63.n odd 6 1
1764.2.w.a 16 63.s even 6 1
1764.2.bm.b 16 7.c even 3 1
1764.2.bm.b 16 7.d odd 6 1
1764.2.bm.b 16 63.i even 6 1
1764.2.bm.b 16 63.j odd 6 1
2268.2.f.b 16 9.c even 3 1
2268.2.f.b 16 9.d odd 6 1
2268.2.f.b 16 63.l odd 6 1
2268.2.f.b 16 63.o even 6 1
3024.2.cc.c 16 12.b even 2 1
3024.2.cc.c 16 36.f odd 6 1
3024.2.cc.c 16 84.h odd 2 1
3024.2.cc.c 16 252.bi even 6 1
5292.2.w.a 16 21.g even 6 1
5292.2.w.a 16 21.h odd 6 1
5292.2.w.a 16 63.g even 3 1
5292.2.w.a 16 63.k odd 6 1
5292.2.bm.b 16 21.g even 6 1
5292.2.bm.b 16 21.h odd 6 1
5292.2.bm.b 16 63.h even 3 1
5292.2.bm.b 16 63.t odd 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(252, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16}$$
$3$ $$6561 + 729 T^{4} - 405 T^{6} - 18 T^{8} - 45 T^{10} + 9 T^{12} + T^{16}$$
$5$ $$81 + 567 T^{2} + 2916 T^{4} + 6939 T^{6} + 12168 T^{8} + 2682 T^{10} + 459 T^{12} + 24 T^{14} + T^{16}$$
$7$ $$5764801 + 823543 T + 352947 T^{2} + 336140 T^{3} + 127253 T^{4} + 7203 T^{5} - 3626 T^{6} + 2485 T^{7} - 1314 T^{8} + 355 T^{9} - 74 T^{10} + 21 T^{11} + 53 T^{12} + 20 T^{13} + 3 T^{14} + T^{15} + T^{16}$$
$11$ $$( 3969 + 2268 T - 891 T^{2} - 756 T^{3} + 342 T^{4} + 63 T^{5} - 18 T^{6} - 3 T^{7} + T^{8} )^{2}$$
$13$ $$81 - 4536 T^{2} + 248994 T^{4} - 280368 T^{6} + 287163 T^{8} - 25776 T^{10} + 1746 T^{12} - 48 T^{14} + T^{16}$$
$17$ $$( 900 - 6741 T^{2} + 1467 T^{4} - 78 T^{6} + T^{8} )^{2}$$
$19$ $$( 900 + 5787 T^{2} + 1476 T^{4} + 75 T^{6} + T^{8} )^{2}$$
$23$ $$( 50625 + 20250 T - 6075 T^{2} - 3510 T^{3} + 1206 T^{4} + 117 T^{5} - 36 T^{6} - 3 T^{7} + T^{8} )^{2}$$
$29$ $$( 245025 - 236115 T + 38718 T^{2} + 35775 T^{3} + 4176 T^{4} - 450 T^{5} - 63 T^{6} + 6 T^{7} + T^{8} )^{2}$$
$31$ $$531441 - 1240029 T^{2} + 2125764 T^{4} - 1686177 T^{6} + 985608 T^{8} - 72414 T^{10} + 4131 T^{12} - 72 T^{14} + T^{16}$$
$37$ $$( 610 + 23 T - 66 T^{2} - T^{3} + T^{4} )^{4}$$
$41$ $$331869318561 + 96021181080 T^{2} + 22187899809 T^{4} + 1414696806 T^{6} + 64225080 T^{8} + 1385487 T^{10} + 21618 T^{12} + 177 T^{14} + T^{16}$$
$43$ $$( 461041 + 131047 T + 88174 T^{2} - 11759 T^{3} + 5332 T^{4} - 236 T^{5} + 79 T^{6} - 2 T^{7} + T^{8} )^{2}$$
$47$ $$24685405970481 + 2494659194541 T^{2} + 168710132016 T^{4} + 6221777481 T^{6} + 165301362 T^{8} + 2722068 T^{10} + 32499 T^{12} + 222 T^{14} + T^{16}$$
$53$ $$( 41990400 + 2848203 T^{2} + 56133 T^{4} + 414 T^{6} + T^{8} )^{2}$$
$59$ $$194481 + 8037225 T^{2} + 331075026 T^{4} + 44366103 T^{6} + 4198680 T^{8} + 197694 T^{10} + 6777 T^{12} + 96 T^{14} + T^{16}$$
$61$ $$1247449349924001 - 74160462871782 T^{2} + 2899235658015 T^{4} - 64949934240 T^{6} + 1054472814 T^{8} - 10802655 T^{10} + 80460 T^{12} - 351 T^{14} + T^{16}$$
$67$ $$( 3940225 - 1992940 T + 787681 T^{2} - 139234 T^{3} + 21334 T^{4} - 1231 T^{5} + 160 T^{6} - 7 T^{7} + T^{8} )^{2}$$
$71$ $$( 15876 + 97443 T^{2} + 11250 T^{4} + 207 T^{6} + T^{8} )^{2}$$
$73$ $$( 76176 + 527499 T^{2} + 19620 T^{4} + 243 T^{6} + T^{8} )^{2}$$
$79$ $$( 319225 + 470645 T + 746434 T^{2} - 66169 T^{3} + 16414 T^{4} - 736 T^{5} + 193 T^{6} - 10 T^{7} + T^{8} )^{2}$$
$83$ $$30237384321 + 13715668764 T^{2} + 4535917299 T^{4} + 671688342 T^{6} + 72720468 T^{8} + 2430279 T^{10} + 61596 T^{12} + 267 T^{14} + T^{16}$$
$89$ $$( 211004676 - 9432045 T^{2} + 133731 T^{4} - 648 T^{6} + T^{8} )^{2}$$
$97$ $$83955602727441 - 9642663582291 T^{2} + 789930535230 T^{4} - 29657333385 T^{6} + 800598564 T^{8} - 10788390 T^{10} + 103725 T^{12} - 372 T^{14} + T^{16}$$