# Properties

 Label 225.4.k.c Level $225$ Weight $4$ Character orbit 225.k Analytic conductor $13.275$ Analytic rank $0$ Dimension $12$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$225 = 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 225.k (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$13.2754297513$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$6$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ Defining polynomial: $$x^{12} - 23 x^{10} + 198 x^{8} - 719 x^{6} + 886 x^{4} + 585 x^{2} + 81$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{6}\cdot 3^{2}$$ Twist minimal: no (minimal twist has level 45) 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.

 $$f(q)$$ $$=$$ $$q + ( -\beta_{7} + \beta_{11} ) q^{2} + ( \beta_{5} + \beta_{8} + \beta_{11} ) q^{3} + ( -\beta_{1} - 4 \beta_{3} - 2 \beta_{4} - \beta_{6} ) q^{4} + ( 17 - \beta_{1} - 3 \beta_{2} + 5 \beta_{3} + \beta_{4} ) q^{6} + ( -5 \beta_{5} - \beta_{7} - 5 \beta_{9} + 6 \beta_{10} + \beta_{11} ) q^{7} + ( -3 \beta_{5} - 3 \beta_{7} + 3 \beta_{8} ) q^{8} + ( 2 - \beta_{1} - 5 \beta_{2} + 22 \beta_{3} + 4 \beta_{4} + \beta_{6} ) q^{9} +O(q^{10})$$ $$q + ( -\beta_{7} + \beta_{11} ) q^{2} + ( \beta_{5} + \beta_{8} + \beta_{11} ) q^{3} + ( -\beta_{1} - 4 \beta_{3} - 2 \beta_{4} - \beta_{6} ) q^{4} + ( 17 - \beta_{1} - 3 \beta_{2} + 5 \beta_{3} + \beta_{4} ) q^{6} + ( -5 \beta_{5} - \beta_{7} - 5 \beta_{9} + 6 \beta_{10} + \beta_{11} ) q^{7} + ( -3 \beta_{5} - 3 \beta_{7} + 3 \beta_{8} ) q^{8} + ( 2 - \beta_{1} - 5 \beta_{2} + 22 \beta_{3} + 4 \beta_{4} + \beta_{6} ) q^{9} + ( -3 + 2 \beta_{2} - 3 \beta_{3} - 2 \beta_{4} - 7 \beta_{6} ) q^{11} + ( -3 \beta_{5} - 11 \beta_{7} - 7 \beta_{8} + 7 \beta_{9} + 10 \beta_{10} + 17 \beta_{11} ) q^{12} + ( -3 \beta_{8} + 10 \beta_{9} + 3 \beta_{10} + 14 \beta_{11} ) q^{13} + ( -12 \beta_{1} + 18 \beta_{3} + 15 \beta_{4} - 12 \beta_{6} ) q^{14} + ( 11 - 19 \beta_{2} + 11 \beta_{3} + 19 \beta_{4} - \beta_{6} ) q^{16} + ( 27 \beta_{5} - 2 \beta_{7} - 3 \beta_{8} ) q^{17} + ( -5 \beta_{5} + 14 \beta_{7} - \beta_{8} - 4 \beta_{9} + 5 \beta_{10} + 17 \beta_{11} ) q^{18} + ( 55 + 5 \beta_{1} + 4 \beta_{2} ) q^{19} + ( -51 - 12 \beta_{1} + 18 \beta_{2} - 33 \beta_{3} - 3 \beta_{4} + 6 \beta_{6} ) q^{21} + ( 33 \beta_{8} + 45 \beta_{9} - 33 \beta_{10} - 40 \beta_{11} ) q^{22} + ( -36 \beta_{8} + 42 \beta_{9} + 36 \beta_{10} + 9 \beta_{11} ) q^{23} + ( -3 - 12 \beta_{1} - 21 \beta_{3} + 3 \beta_{4} - 9 \beta_{6} ) q^{24} + ( 153 - 7 \beta_{1} - 32 \beta_{2} ) q^{26} + ( -33 \beta_{5} + 45 \beta_{7} - 33 \beta_{8} - 9 \beta_{9} + 9 \beta_{10} - 24 \beta_{11} ) q^{27} + ( -74 \beta_{5} - 19 \beta_{7} + 27 \beta_{8} ) q^{28} + ( -117 + 23 \beta_{2} - 117 \beta_{3} - 23 \beta_{4} - 7 \beta_{6} ) q^{29} + ( 31 \beta_{1} - 127 \beta_{3} + 2 \beta_{4} + 31 \beta_{6} ) q^{31} + ( 21 \beta_{9} + 49 \beta_{11} ) q^{32} + ( -22 \beta_{5} - 58 \beta_{7} + 33 \beta_{8} + 23 \beta_{9} + 38 \beta_{10} + 18 \beta_{11} ) q^{33} + ( -39 + 37 \beta_{2} - 39 \beta_{3} - 37 \beta_{4} + 28 \beta_{6} ) q^{34} + ( 191 - 10 \beta_{1} - 29 \beta_{2} + 28 \beta_{3} + 49 \beta_{4} + 13 \beta_{6} ) q^{36} + ( -46 \beta_{5} - 24 \beta_{7} + 51 \beta_{8} ) q^{37} + ( 27 \beta_{5} - 26 \beta_{7} + 27 \beta_{9} - 21 \beta_{10} + 26 \beta_{11} ) q^{38} + ( 153 + 3 \beta_{1} - 32 \beta_{2} + 179 \beta_{3} + 69 \beta_{4} + 10 \beta_{6} ) q^{39} + ( 20 \beta_{1} + 72 \beta_{3} + 49 \beta_{4} + 20 \beta_{6} ) q^{41} + ( -114 \beta_{5} - 27 \beta_{7} - 33 \beta_{8} - 162 \beta_{9} + 108 \beta_{10} + 21 \beta_{11} ) q^{42} + ( 128 \beta_{5} + 8 \beta_{7} + 128 \beta_{9} + 87 \beta_{10} - 8 \beta_{11} ) q^{43} + ( -291 + 62 \beta_{1} - 47 \beta_{2} ) q^{44} + ( -72 - 3 \beta_{1} + 12 \beta_{2} ) q^{46} + ( 96 \beta_{5} + 59 \beta_{7} + 96 \beta_{9} - 15 \beta_{10} - 59 \beta_{11} ) q^{47} + ( -146 \beta_{5} + 61 \beta_{7} + 24 \beta_{8} - 101 \beta_{9} - 41 \beta_{10} - 36 \beta_{11} ) q^{48} + ( 13 \beta_{1} - 189 \beta_{3} - 124 \beta_{4} + 13 \beta_{6} ) q^{49} + ( -20 + 31 \beta_{1} - 53 \beta_{2} - 39 \beta_{3} - 37 \beta_{4} + 28 \beta_{6} ) q^{51} + ( -65 \beta_{5} - 108 \beta_{7} - 65 \beta_{9} - 21 \beta_{10} + 108 \beta_{11} ) q^{52} + ( -102 \beta_{5} - 70 \beta_{7} + 24 \beta_{8} ) q^{53} + ( 87 + 6 \beta_{1} + 420 \beta_{3} + 75 \beta_{4} + 36 \beta_{6} ) q^{54} + ( -237 + 30 \beta_{2} - 237 \beta_{3} - 30 \beta_{4} - 24 \beta_{6} ) q^{56} + ( 89 \beta_{5} - 18 \beta_{7} + 113 \beta_{8} + 27 \beta_{9} - 21 \beta_{10} + 26 \beta_{11} ) q^{57} + ( 12 \beta_{8} + 3 \beta_{9} - 12 \beta_{10} - 175 \beta_{11} ) q^{58} + ( 41 \beta_{1} + 36 \beta_{3} - 149 \beta_{4} + 41 \beta_{6} ) q^{59} + ( 448 + 85 \beta_{2} + 448 \beta_{3} - 85 \beta_{4} + 10 \beta_{6} ) q^{61} + ( 213 \beta_{5} + 30 \beta_{7} - 153 \beta_{8} ) q^{62} + ( 69 \beta_{5} - 6 \beta_{7} - 141 \beta_{8} - 141 \beta_{9} + 21 \beta_{10} + 21 \beta_{11} ) q^{63} + ( 500 - 36 \beta_{1} + 33 \beta_{2} ) q^{64} + ( -315 - 33 \beta_{1} - 31 \beta_{2} - 341 \beta_{3} + 48 \beta_{4} - 151 \beta_{6} ) q^{66} + ( 21 \beta_{8} + 308 \beta_{9} - 21 \beta_{10} + 41 \beta_{11} ) q^{67} + ( -153 \beta_{8} - 54 \beta_{9} + 153 \beta_{10} + 80 \beta_{11} ) q^{68} + ( -72 + 36 \beta_{1} + 12 \beta_{2} - 399 \beta_{3} + 117 \beta_{4} + 39 \beta_{6} ) q^{69} + ( 81 - 38 \beta_{1} - 211 \beta_{2} ) q^{71} + ( -78 \beta_{5} - 57 \beta_{7} - 48 \beta_{8} - 111 \beta_{9} + 78 \beta_{10} + 87 \beta_{11} ) q^{72} + ( 10 \beta_{5} - 44 \beta_{7} + 192 \beta_{8} ) q^{73} + ( -33 - 100 \beta_{2} - 33 \beta_{3} + 100 \beta_{4} - 121 \beta_{6} ) q^{74} + ( -18 \beta_{1} + 23 \beta_{3} - 153 \beta_{4} - 18 \beta_{6} ) q^{76} + ( 3 \beta_{8} + 117 \beta_{9} - 3 \beta_{10} - 240 \beta_{11} ) q^{77} + ( -53 \beta_{5} + 78 \beta_{7} + 19 \beta_{8} + 15 \beta_{9} + 3 \beta_{10} + 220 \beta_{11} ) q^{78} + ( -154 + 136 \beta_{2} - 154 \beta_{3} - 136 \beta_{4} - 56 \beta_{6} ) q^{79} + ( -300 + 69 \beta_{1} + 210 \beta_{2} - 87 \beta_{3} - 33 \beta_{4} + 39 \beta_{6} ) q^{81} + ( 42 \beta_{5} + 221 \beta_{7} - 51 \beta_{8} ) q^{82} + ( 3 \beta_{5} + 105 \beta_{7} + 3 \beta_{9} - 354 \beta_{10} - 105 \beta_{11} ) q^{83} + ( 27 - 147 \beta_{1} + 102 \beta_{2} - 93 \beta_{3} + 90 \beta_{4} - 120 \beta_{6} ) q^{84} + ( 49 \beta_{1} + 531 \beta_{3} + 62 \beta_{4} + 49 \beta_{6} ) q^{86} + ( 32 \beta_{5} - 235 \beta_{7} + 12 \beta_{8} + 35 \beta_{9} - 13 \beta_{10} + 60 \beta_{11} ) q^{87} + ( 168 \beta_{5} + 234 \beta_{7} + 168 \beta_{9} - 93 \beta_{10} - 234 \beta_{11} ) q^{88} + ( -549 - 84 \beta_{1} - 156 \beta_{2} ) q^{89} + ( -377 - 143 \beta_{1} + 278 \beta_{2} ) q^{91} + ( -381 \beta_{5} + 141 \beta_{7} - 381 \beta_{9} - 261 \beta_{10} - 141 \beta_{11} ) q^{92} + ( 213 \beta_{5} + 30 \beta_{7} + 288 \beta_{8} + 12 \beta_{9} - 441 \beta_{10} - 96 \beta_{11} ) q^{93} + ( 170 \beta_{1} + 633 \beta_{3} - 8 \beta_{4} + 170 \beta_{6} ) q^{94} + ( 588 - 119 \beta_{2} + 791 \beta_{3} + 210 \beta_{4} + 28 \beta_{6} ) q^{96} + ( -80 \beta_{5} + 78 \beta_{7} - 80 \beta_{9} - 60 \beta_{10} - 78 \beta_{11} ) q^{97} + ( 339 \beta_{5} - 248 \beta_{7} - 189 \beta_{8} ) q^{98} + ( -208 - 184 \beta_{1} - 68 \beta_{2} - 575 \beta_{3} + 133 \beta_{4} - 80 \beta_{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q + 22 q^{4} + 168 q^{6} - 114 q^{9} + O(q^{10})$$ $$12 q + 22 q^{4} + 168 q^{6} - 114 q^{9} - 28 q^{11} - 54 q^{14} + 26 q^{16} + 656 q^{19} - 288 q^{21} + 126 q^{24} + 1736 q^{26} - 670 q^{29} + 704 q^{31} - 104 q^{34} + 2172 q^{36} + 780 q^{39} - 374 q^{41} - 3928 q^{44} - 804 q^{46} + 860 q^{49} - 360 q^{51} - 1278 q^{54} - 1410 q^{56} - 596 q^{59} + 2878 q^{61} + 6276 q^{64} - 1932 q^{66} + 1746 q^{69} + 280 q^{71} - 640 q^{74} - 408 q^{76} - 764 q^{79} - 2502 q^{81} + 1818 q^{84} - 3160 q^{86} - 6876 q^{89} - 2840 q^{91} - 4154 q^{94} + 2310 q^{96} + 1524 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 23 x^{10} + 198 x^{8} - 719 x^{6} + 886 x^{4} + 585 x^{2} + 81$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$17 \nu^{10} - 260 \nu^{8} + 1311 \nu^{6} - 2275 \nu^{4} - 1388 \nu^{2} + 33786$$$$)/7875$$ $$\beta_{2}$$ $$=$$ $$($$$$-19 \nu^{10} + 445 \nu^{8} - 3627 \nu^{6} + 9800 \nu^{4} + 5566 \nu^{2} - 24327$$$$)/7875$$ $$\beta_{3}$$ $$=$$ $$($$$$-18 \nu^{10} + 415 \nu^{8} - 3594 \nu^{6} + 13350 \nu^{4} - 18398 \nu^{2} - 5994$$$$)/1125$$ $$\beta_{4}$$ $$=$$ $$($$$$-368 \nu^{10} + 8665 \nu^{8} - 77019 \nu^{6} + 295225 \nu^{4} - 411623 \nu^{2} - 133794$$$$)/7875$$ $$\beta_{5}$$ $$=$$ $$($$$$181 \nu^{11} - 4055 \nu^{9} + 33723 \nu^{7} - 115325 \nu^{5} + 124141 \nu^{3} + 132273 \nu$$$$)/23625$$ $$\beta_{6}$$ $$=$$ $$($$$$106 \nu^{10} - 2455 \nu^{8} + 21423 \nu^{6} - 80150 \nu^{4} + 111416 \nu^{2} + 29448$$$$)/1575$$ $$\beta_{7}$$ $$=$$ $$($$$$-88 \nu^{11} + 2015 \nu^{9} - 17154 \nu^{7} + 60725 \nu^{5} - 67168 \nu^{3} - 70929 \nu$$$$)/6750$$ $$\beta_{8}$$ $$=$$ $$($$$$-316 \nu^{11} + 7355 \nu^{9} - 64053 \nu^{7} + 233450 \nu^{5} - 264751 \nu^{3} - 277353 \nu$$$$)/23625$$ $$\beta_{9}$$ $$=$$ $$($$$$-1528 \nu^{11} + 35465 \nu^{9} - 309924 \nu^{7} + 1163225 \nu^{5} - 1604758 \nu^{3} - 514449 \nu$$$$)/23625$$ $$\beta_{10}$$ $$=$$ $$($$$$2818 \nu^{11} - 65540 \nu^{9} + 575244 \nu^{7} - 2181725 \nu^{5} + 3102073 \nu^{3} + 800019 \nu$$$$)/23625$$ $$\beta_{11}$$ $$=$$ $$($$$$-811 \nu^{11} + 18830 \nu^{9} - 164763 \nu^{7} + 620450 \nu^{5} - 860821 \nu^{3} - 275688 \nu$$$$)/6750$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$4 \beta_{11} + 4 \beta_{10} + \beta_{9} - 2 \beta_{8} - 2 \beta_{7} + 2 \beta_{5}$$$$)/6$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{6} + \beta_{4} + \beta_{3} + \beta_{2} - \beta_{1} + 11$$$$)/3$$ $$\nu^{3}$$ $$=$$ $$($$$$26 \beta_{11} + 20 \beta_{10} - 7 \beta_{9} - 10 \beta_{8} - 4 \beta_{7} + 13 \beta_{5}$$$$)/6$$ $$\nu^{4}$$ $$=$$ $$($$$$14 \beta_{6} + 8 \beta_{4} + 35 \beta_{3} - \beta_{2} - 5 \beta_{1} + 79$$$$)/3$$ $$\nu^{5}$$ $$=$$ $$($$$$154 \beta_{11} + 94 \beta_{10} - 92 \beta_{9} - 92 \beta_{8} + 58 \beta_{7} + 113 \beta_{5}$$$$)/6$$ $$\nu^{6}$$ $$=$$ $$($$$$151 \beta_{6} + 46 \beta_{4} + 505 \beta_{3} - 26 \beta_{2} + 2 \beta_{1} + 560$$$$)/3$$ $$\nu^{7}$$ $$=$$ $$($$$$692 \beta_{11} + 350 \beta_{10} - 565 \beta_{9} - 868 \beta_{8} + 1166 \beta_{7} + 1102 \beta_{5}$$$$)/6$$ $$\nu^{8}$$ $$=$$ $$($$$$1430 \beta_{6} + 242 \beta_{4} + 5399 \beta_{3} - 190 \beta_{2} + 460 \beta_{1} + 3580$$$$)/3$$ $$\nu^{9}$$ $$=$$ $$($$$$262 \beta_{11} - 218 \beta_{10} - 947 \beta_{9} - 7154 \beta_{8} + 14116 \beta_{7} + 11039 \beta_{5}$$$$)/6$$ $$\nu^{10}$$ $$=$$ $$($$$$12181 \beta_{6} + 1306 \beta_{4} + 48394 \beta_{3} - 953 \beta_{2} + 7520 \beta_{1} + 17075$$$$)/3$$ $$\nu^{11}$$ $$=$$ $$($$$$-45694 \beta_{11} - 26842 \beta_{10} + 29504 \beta_{9} - 48850 \beta_{8} + 140162 \beta_{7} + 104395 \beta_{5}$$$$)/6$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/225\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$127$$ $$\chi(n)$$ $$\beta_{3}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1
 −2.88506 − 0.500000i −0.0378788 + 0.500000i 1.98116 − 0.500000i −1.98116 + 0.500000i 0.0378788 − 0.500000i 2.88506 + 0.500000i −2.88506 + 0.500000i −0.0378788 − 0.500000i 1.98116 + 0.500000i −1.98116 − 0.500000i 0.0378788 + 0.500000i 2.88506 − 0.500000i
−3.96084 + 2.28679i −3.96084 3.36330i 6.45882 11.1870i 0 23.3794 + 4.26387i −17.4197 + 10.0573i 22.4912i 4.37646 + 26.6429i 0
49.2 −3.24635 + 1.87428i −3.24635 4.05724i 3.02587 5.24096i 0 18.1432 + 7.08665i 27.1492 15.6746i 7.30318i −5.92239 + 26.3425i 0
49.3 −0.151541 + 0.0874923i −0.151541 5.19394i −3.98469 + 6.90169i 0 0.477395 + 0.773837i 7.32979 4.23186i 2.79440i −26.9541 + 1.57419i 0
49.4 0.151541 0.0874923i 0.151541 + 5.19394i −3.98469 + 6.90169i 0 0.477395 + 0.773837i −7.32979 + 4.23186i 2.79440i −26.9541 + 1.57419i 0
49.5 3.24635 1.87428i 3.24635 + 4.05724i 3.02587 5.24096i 0 18.1432 + 7.08665i −27.1492 + 15.6746i 7.30318i −5.92239 + 26.3425i 0
49.6 3.96084 2.28679i 3.96084 + 3.36330i 6.45882 11.1870i 0 23.3794 + 4.26387i 17.4197 10.0573i 22.4912i 4.37646 + 26.6429i 0
124.1 −3.96084 2.28679i −3.96084 + 3.36330i 6.45882 + 11.1870i 0 23.3794 4.26387i −17.4197 10.0573i 22.4912i 4.37646 26.6429i 0
124.2 −3.24635 1.87428i −3.24635 + 4.05724i 3.02587 + 5.24096i 0 18.1432 7.08665i 27.1492 + 15.6746i 7.30318i −5.92239 26.3425i 0
124.3 −0.151541 0.0874923i −0.151541 + 5.19394i −3.98469 6.90169i 0 0.477395 0.773837i 7.32979 + 4.23186i 2.79440i −26.9541 1.57419i 0
124.4 0.151541 + 0.0874923i 0.151541 5.19394i −3.98469 6.90169i 0 0.477395 0.773837i −7.32979 4.23186i 2.79440i −26.9541 1.57419i 0
124.5 3.24635 + 1.87428i 3.24635 4.05724i 3.02587 + 5.24096i 0 18.1432 7.08665i −27.1492 15.6746i 7.30318i −5.92239 26.3425i 0
124.6 3.96084 + 2.28679i 3.96084 3.36330i 6.45882 + 11.1870i 0 23.3794 4.26387i 17.4197 + 10.0573i 22.4912i 4.37646 26.6429i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 124.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
9.c even 3 1 inner
45.j even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 225.4.k.c 12
5.b even 2 1 inner 225.4.k.c 12
5.c odd 4 1 45.4.e.b 6
5.c odd 4 1 225.4.e.c 6
9.c even 3 1 inner 225.4.k.c 12
15.e even 4 1 135.4.e.b 6
45.j even 6 1 inner 225.4.k.c 12
45.k odd 12 1 45.4.e.b 6
45.k odd 12 1 225.4.e.c 6
45.k odd 12 1 405.4.a.h 3
45.k odd 12 1 2025.4.a.s 3
45.l even 12 1 135.4.e.b 6
45.l even 12 1 405.4.a.j 3
45.l even 12 1 2025.4.a.q 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.4.e.b 6 5.c odd 4 1
45.4.e.b 6 45.k odd 12 1
135.4.e.b 6 15.e even 4 1
135.4.e.b 6 45.l even 12 1
225.4.e.c 6 5.c odd 4 1
225.4.e.c 6 45.k odd 12 1
225.4.k.c 12 1.a even 1 1 trivial
225.4.k.c 12 5.b even 2 1 inner
225.4.k.c 12 9.c even 3 1 inner
225.4.k.c 12 45.j even 6 1 inner
405.4.a.h 3 45.k odd 12 1
405.4.a.j 3 45.l even 12 1
2025.4.a.q 3 45.l even 12 1
2025.4.a.s 3 45.k odd 12 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{12} - 35 T_{2}^{10} + 930 T_{2}^{8} - 10307 T_{2}^{6} + 86710 T_{2}^{4} - 2655 T_{2}^{2} + 81$$ acting on $$S_{4}^{\mathrm{new}}(225, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$81 - 2655 T^{2} + 86710 T^{4} - 10307 T^{6} + 930 T^{8} - 35 T^{10} + T^{12}$$
$3$ $$387420489 + 30292137 T^{2} + 1640250 T^{4} + 77517 T^{6} + 2250 T^{8} + 57 T^{10} + T^{12}$$
$5$ $$T^{12}$$
$7$ $$811313702977761 - 14156533177983 T^{2} + 205458430878 T^{4} - 668166075 T^{6} + 1631674 T^{8} - 1459 T^{10} + T^{12}$$
$11$ $$( 1896428304 - 122631168 T + 7320184 T^{2} - 126520 T^{3} + 3012 T^{4} + 14 T^{5} + T^{6} )^{2}$$
$13$ $$32259361226118390016 - 68392067190914304 T^{2} + 108054911793792 T^{4} - 66957956704 T^{6} + 30260592 T^{8} - 6504 T^{10} + T^{12}$$
$17$ $$( 24437192976 + 27666832 T^{2} + 9716 T^{4} + T^{6} )^{2}$$
$19$ $$( -57316 + 7292 T - 164 T^{2} + T^{3} )^{4}$$
$23$ $$14036574567470305761 - 165939452573967927 T^{2} + 1815958493546022 T^{4} - 1715751132507 T^{6} + 1469463282 T^{8} - 38907 T^{10} + T^{12}$$
$29$ $$( 11463342489 - 2926248177 T + 782851006 T^{2} + 9370019 T^{3} + 84894 T^{4} + 335 T^{5} + T^{6} )^{2}$$
$31$ $$( 97238137683600 - 137382616080 T + 3665151504 T^{2} - 14817816 T^{3} + 137836 T^{4} - 352 T^{5} + T^{6} )^{2}$$
$37$ $$( 11124119360656 + 3292053600 T^{2} + 112188 T^{4} + T^{6} )^{2}$$
$41$ $$( 52247959475625 + 322359380175 T + 3340579834 T^{2} + 6116911 T^{3} + 79566 T^{4} + 187 T^{5} + T^{6} )^{2}$$
$43$ $$56\!\cdots\!76$$$$-$$$$44\!\cdots\!32$$$$T^{2} +$$$$26\!\cdots\!32$$$$T^{4} - 5925591427156192 T^{6} + 98606441232 T^{8} - 342492 T^{10} + T^{12}$$
$47$ $$38\!\cdots\!21$$$$-$$$$35\!\cdots\!19$$$$T^{2} + 31737062392410542434 T^{4} - 1439749703256995 T^{6} + 57564148998 T^{8} - 251663 T^{10} + T^{12}$$
$53$ $$( 10565984287296 + 6553907440 T^{2} + 320300 T^{4} + T^{6} )^{2}$$
$59$ $$( 16224618881802816 + 45173097261024 T + 163730383744 T^{2} + 149067880 T^{3} + 443448 T^{4} + 298 T^{5} + T^{6} )^{2}$$
$61$ $$( 3092861048569009 - 32773423076579 T + 267254918066 T^{2} - 736785779 T^{3} + 1481414 T^{4} - 1439 T^{5} + T^{6} )^{2}$$
$67$ $$18\!\cdots\!61$$$$-$$$$18\!\cdots\!07$$$$T^{2} +$$$$12\!\cdots\!02$$$$T^{4} - 435051095147093547 T^{6} + 1092386569462 T^{8} - 1231447 T^{10} + T^{12}$$
$71$ $$( 223775052 - 685460 T - 70 T^{2} + T^{3} )^{4}$$
$73$ $$( 18205496100929536 + 238382112768 T^{2} + 881280 T^{4} + T^{6} )^{2}$$
$79$ $$( 19133137244437056 + 53658650075616 T + 203324256864 T^{2} + 128458200 T^{3} + 533848 T^{4} + 382 T^{5} + T^{6} )^{2}$$
$83$ $$83\!\cdots\!81$$$$-$$$$26\!\cdots\!71$$$$T^{2} +$$$$58\!\cdots\!74$$$$T^{4} - 7018863830915788035 T^{6} + 6138727687818 T^{8} - 3012507 T^{10} + T^{12}$$
$89$ $$( -125506395 + 238491 T + 1719 T^{2} + T^{3} )^{4}$$
$97$ $$81\!\cdots\!16$$$$-$$$$40\!\cdots\!20$$$$T^{2} +$$$$13\!\cdots\!40$$$$T^{4} - 2460481517295808 T^{6} + 32055240480 T^{8} - 214860 T^{10} + T^{12}$$