# Properties

 Label 900.3.f.f Level $900$ Weight $3$ Character orbit 900.f Analytic conductor $24.523$ Analytic rank $0$ Dimension $16$ CM no Inner twists $4$

# Learn more

## Newspace parameters

 Level: $$N$$ $$=$$ $$900 = 2^{2} \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 900.f (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$24.5232237924$$ Analytic rank: $$0$$ Dimension: $$16$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ Defining polynomial: $$x^{16} + 5 x^{14} + 12 x^{12} + 25 x^{10} + 53 x^{8} + 100 x^{6} + 192 x^{4} + 320 x^{2} + 256$$ Coefficient ring: $$\Z[a_1, \ldots, a_{29}]$$ Coefficient ring index: $$2^{28}$$ Twist minimal: no (minimal twist has level 60) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q -\beta_{6} q^{2} + ( -1 - \beta_{8} ) q^{4} + ( -\beta_{11} - \beta_{13} ) q^{7} + ( 2 \beta_{6} + 2 \beta_{9} + \beta_{13} + \beta_{15} ) q^{8} +O(q^{10})$$ $$q -\beta_{6} q^{2} + ( -1 - \beta_{8} ) q^{4} + ( -\beta_{11} - \beta_{13} ) q^{7} + ( 2 \beta_{6} + 2 \beta_{9} + \beta_{13} + \beta_{15} ) q^{8} + ( -1 + 2 \beta_{1} + \beta_{4} + \beta_{5} + \beta_{8} ) q^{11} + ( \beta_{6} + 2 \beta_{7} + 2 \beta_{9} - \beta_{11} - 2 \beta_{12} - 2 \beta_{15} ) q^{13} + ( -2 + \beta_{1} - \beta_{3} + 2 \beta_{4} - \beta_{5} ) q^{14} + ( 1 - 3 \beta_{2} - \beta_{3} + \beta_{4} + 3 \beta_{5} + 2 \beta_{8} - \beta_{10} ) q^{16} + ( 3 \beta_{6} + \beta_{7} + 2 \beta_{9} + \beta_{12} + \beta_{13} + 3 \beta_{14} + 3 \beta_{15} ) q^{17} + ( -1 - 2 \beta_{1} - 3 \beta_{4} + 2 \beta_{5} + \beta_{8} + \beta_{10} ) q^{19} + ( \beta_{6} + 2 \beta_{7} - 4 \beta_{9} + 2 \beta_{11} + 3 \beta_{12} + \beta_{13} - \beta_{14} - \beta_{15} ) q^{22} + ( -4 \beta_{6} - 2 \beta_{7} - \beta_{11} - 2 \beta_{12} + 2 \beta_{13} + 3 \beta_{14} - 3 \beta_{15} ) q^{23} + ( 1 + 4 \beta_{1} - 5 \beta_{2} - 3 \beta_{3} - \beta_{5} - 2 \beta_{10} ) q^{26} + ( 4 \beta_{6} - 2 \beta_{7} + \beta_{9} + 4 \beta_{11} - \beta_{12} - \beta_{14} - 3 \beta_{15} ) q^{28} + ( 10 - 5 \beta_{1} - 6 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - 4 \beta_{5} + 3 \beta_{8} + 2 \beta_{10} ) q^{29} + ( -7 + 4 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - \beta_{4} + 2 \beta_{5} + 5 \beta_{8} - 5 \beta_{10} ) q^{31} + ( -6 \beta_{6} + 6 \beta_{7} - 4 \beta_{9} + \beta_{13} + 2 \beta_{14} + \beta_{15} ) q^{32} + ( 9 - 2 \beta_{1} + \beta_{2} - 3 \beta_{3} + 3 \beta_{5} + 4 \beta_{8} ) q^{34} + ( 5 \beta_{6} + 4 \beta_{7} + 12 \beta_{9} - \beta_{11} + 2 \beta_{13} + 8 \beta_{14} + 6 \beta_{15} ) q^{37} + ( -2 \beta_{6} + 4 \beta_{7} + 4 \beta_{9} - 4 \beta_{11} + 2 \beta_{12} - 2 \beta_{13} + 2 \beta_{14} + 6 \beta_{15} ) q^{38} + ( 2 \beta_{4} + 4 \beta_{5} - 2 \beta_{8} - 2 \beta_{10} ) q^{41} + ( -4 \beta_{6} + 7 \beta_{7} - \beta_{11} + 7 \beta_{12} + 3 \beta_{13} - 7 \beta_{14} + 7 \beta_{15} ) q^{43} + ( 28 - 3 \beta_{1} + 7 \beta_{2} + 4 \beta_{3} - 4 \beta_{4} - 6 \beta_{5} - 2 \beta_{8} + \beta_{10} ) q^{44} + ( 20 - 2 \beta_{2} + 2 \beta_{3} - 4 \beta_{4} + 6 \beta_{5} - 8 \beta_{8} + 6 \beta_{10} ) q^{46} + ( -4 \beta_{6} + 2 \beta_{7} - \beta_{11} + 2 \beta_{12} - 4 \beta_{13} + 5 \beta_{14} - 5 \beta_{15} ) q^{47} + ( 7 - 16 \beta_{1} - 12 \beta_{2} + 4 \beta_{3} + 8 \beta_{8} ) q^{49} + ( -8 \beta_{6} - 2 \beta_{7} + 10 \beta_{9} - 4 \beta_{11} - 4 \beta_{12} - 4 \beta_{13} + 18 \beta_{14} ) q^{52} + ( -\beta_{6} - 4 \beta_{7} - 24 \beta_{9} + 2 \beta_{11} + 4 \beta_{12} - \beta_{14} + 3 \beta_{15} ) q^{53} + ( -10 - 11 \beta_{1} + \beta_{2} + 8 \beta_{3} - 10 \beta_{4} - 4 \beta_{5} + 6 \beta_{8} + 3 \beta_{10} ) q^{56} + ( -19 \beta_{6} - 8 \beta_{7} - 18 \beta_{9} - 3 \beta_{12} - 7 \beta_{13} + 13 \beta_{14} + 5 \beta_{15} ) q^{58} + ( 5 - 6 \beta_{1} - \beta_{4} - 5 \beta_{5} - 5 \beta_{8} ) q^{59} + ( -20 - 10 \beta_{1} - 12 \beta_{2} + 4 \beta_{3} - 2 \beta_{4} - 4 \beta_{5} + 4 \beta_{8} + 2 \beta_{10} ) q^{61} + ( -6 \beta_{6} + 4 \beta_{7} - 12 \beta_{11} - 6 \beta_{12} - 6 \beta_{13} + 10 \beta_{14} - 10 \beta_{15} ) q^{62} + ( 11 + 10 \beta_{1} + 5 \beta_{2} - 7 \beta_{3} - \beta_{4} - 3 \beta_{5} - 6 \beta_{8} - 5 \beta_{10} ) q^{64} + ( -8 \beta_{6} + 9 \beta_{7} + \beta_{11} + 9 \beta_{12} - \beta_{13} + \beta_{14} - \beta_{15} ) q^{67} + ( -12 \beta_{6} + 6 \beta_{7} + 8 \beta_{9} - 2 \beta_{12} - 4 \beta_{13} - 4 \beta_{14} - 2 \beta_{15} ) q^{68} + ( 2 + 2 \beta_{4} + 6 \beta_{5} - 2 \beta_{8} + 8 \beta_{10} ) q^{71} + ( 18 \beta_{6} + 8 \beta_{7} - 5 \beta_{9} - 2 \beta_{11} + 4 \beta_{13} + 8 \beta_{14} + 4 \beta_{15} ) q^{73} + ( 7 - 6 \beta_{1} - 7 \beta_{2} - 11 \beta_{3} + 7 \beta_{5} + 8 \beta_{8} - 2 \beta_{10} ) q^{74} + ( -8 + 6 \beta_{1} - 2 \beta_{2} - 12 \beta_{3} + 12 \beta_{4} - 6 \beta_{10} ) q^{76} + ( -24 \beta_{6} + 2 \beta_{7} - 20 \beta_{9} - 2 \beta_{11} - 6 \beta_{12} - 2 \beta_{13} + 14 \beta_{14} + 10 \beta_{15} ) q^{77} + ( 3 + 14 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} + 13 \beta_{4} - 6 \beta_{5} + \beta_{8} - 3 \beta_{10} ) q^{79} + ( 2 \beta_{6} + 8 \beta_{7} + 8 \beta_{13} - 8 \beta_{14} ) q^{82} + ( -4 \beta_{6} + 5 \beta_{7} + 5 \beta_{11} + 5 \beta_{12} - 5 \beta_{13} + 9 \beta_{14} - 9 \beta_{15} ) q^{83} + ( 18 + 12 \beta_{1} - 4 \beta_{2} - 8 \beta_{3} + 16 \beta_{4} - 8 \beta_{8} - 14 \beta_{10} ) q^{86} + ( -22 \beta_{6} - 12 \beta_{7} - \beta_{9} - 6 \beta_{11} - 3 \beta_{12} - 4 \beta_{13} - 7 \beta_{14} - 5 \beta_{15} ) q^{88} + ( 8 - 22 \beta_{1} + 2 \beta_{4} + 4 \beta_{5} + 20 \beta_{8} - 2 \beta_{10} ) q^{89} + ( 10 - 2 \beta_{1} + 10 \beta_{2} + 10 \beta_{3} - 2 \beta_{4} - 10 \beta_{5} ) q^{91} + ( -12 \beta_{6} + 12 \beta_{7} + 12 \beta_{9} + 4 \beta_{11} + 20 \beta_{12} + 12 \beta_{13} + 4 \beta_{14} + 32 \beta_{15} ) q^{92} + ( 32 - 14 \beta_{1} + 10 \beta_{2} + 4 \beta_{3} - 8 \beta_{4} - 16 \beta_{5} - 8 \beta_{8} + 10 \beta_{10} ) q^{94} + ( -24 \beta_{6} - 2 \beta_{7} - 37 \beta_{9} - 2 \beta_{11} - 10 \beta_{12} - 6 \beta_{13} - 10 \beta_{14} - 14 \beta_{15} ) q^{97} + ( -27 \beta_{6} - 48 \beta_{9} - 12 \beta_{12} - 4 \beta_{13} + 4 \beta_{14} + 4 \beta_{15} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q - 20q^{4} + O(q^{10})$$ $$16q - 20q^{4} - 40q^{14} + 68q^{16} + 72q^{26} + 128q^{29} + 184q^{34} + 32q^{41} + 344q^{44} + 304q^{46} + 112q^{49} - 232q^{56} - 352q^{61} + 220q^{64} + 264q^{74} - 48q^{76} + 400q^{86} + 160q^{89} + 192q^{94} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} + 5 x^{14} + 12 x^{12} + 25 x^{10} + 53 x^{8} + 100 x^{6} + 192 x^{4} + 320 x^{2} + 256$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$11 \nu^{14} + 115 \nu^{12} + 96 \nu^{10} + 467 \nu^{8} + 35 \nu^{6} + 1512 \nu^{4} + 1184 \nu^{2} + 2240$$$$)/2432$$ $$\beta_{2}$$ $$=$$ $$($$$$-47 \nu^{14} - 139 \nu^{12} - 196 \nu^{10} - 807 \nu^{8} - 1179 \nu^{6} - 2156 \nu^{4} - 5584 \nu^{2} - 6144$$$$)/1216$$ $$\beta_{3}$$ $$=$$ $$($$$$-127 \nu^{14} - 471 \nu^{12} - 1136 \nu^{10} - 2407 \nu^{8} - 5959 \nu^{6} - 11736 \nu^{4} - 20192 \nu^{2} - 28736$$$$)/2432$$ $$\beta_{4}$$ $$=$$ $$($$$$-35 \nu^{14} - 93 \nu^{12} - 74 \nu^{10} - 339 \nu^{8} - 861 \nu^{6} - 1650 \nu^{4} - 2040 \nu^{2} - 2816$$$$)/608$$ $$\beta_{5}$$ $$=$$ $$($$$$-179 \nu^{14} - 835 \nu^{12} - 1576 \nu^{10} - 3675 \nu^{8} - 7603 \nu^{6} - 12016 \nu^{4} - 24960 \nu^{2} - 36672$$$$)/2432$$ $$\beta_{6}$$ $$=$$ $$($$$$-165 \nu^{15} - 661 \nu^{13} - 984 \nu^{11} - 2749 \nu^{9} - 5541 \nu^{7} - 10064 \nu^{5} - 17152 \nu^{3} - 19008 \nu$$$$)/9728$$ $$\beta_{7}$$ $$=$$ $$($$$$-101 \nu^{15} - 213 \nu^{13} - 536 \nu^{11} - 1469 \nu^{9} - 2021 \nu^{7} - 7568 \nu^{5} - 10816 \nu^{3} - 17472 \nu$$$$)/4864$$ $$\beta_{8}$$ $$=$$ $$($$$$-59 \nu^{14} - 185 \nu^{12} - 318 \nu^{10} - 667 \nu^{8} - 1497 \nu^{6} - 2662 \nu^{4} - 6088 \nu^{2} - 7040$$$$)/608$$ $$\beta_{9}$$ $$=$$ $$($$$$71 \nu^{15} + 231 \nu^{13} + 440 \nu^{11} + 1135 \nu^{9} + 1815 \nu^{7} + 3168 \nu^{5} + 7200 \nu^{3} + 9152 \nu$$$$)/2432$$ $$\beta_{10}$$ $$=$$ $$($$$$211 \nu^{14} + 603 \nu^{12} + 1344 \nu^{10} + 3099 \nu^{8} + 5259 \nu^{6} + 10376 \nu^{4} + 20832 \nu^{2} + 24064$$$$)/1216$$ $$\beta_{11}$$ $$=$$ $$($$$$431 \nu^{15} + 927 \nu^{13} + 680 \nu^{11} + 3319 \nu^{9} + 7631 \nu^{7} + 6416 \nu^{5} + 17152 \nu^{3} + 36032 \nu$$$$)/9728$$ $$\beta_{12}$$ $$=$$ $$($$$$-495 \nu^{15} - 767 \nu^{13} - 1736 \nu^{11} - 3383 \nu^{9} - 5679 \nu^{7} - 9520 \nu^{5} - 17408 \nu^{3} + 1344 \nu$$$$)/9728$$ $$\beta_{13}$$ $$=$$ $$($$$$-621 \nu^{15} - 2333 \nu^{13} - 3416 \nu^{11} - 6853 \nu^{9} - 14509 \nu^{7} - 23440 \nu^{5} - 63360 \nu^{3} - 87104 \nu$$$$)/9728$$ $$\beta_{14}$$ $$=$$ $$($$$$-977 \nu^{15} - 2849 \nu^{13} - 5528 \nu^{11} - 12681 \nu^{9} - 24817 \nu^{7} - 44848 \nu^{5} - 100352 \nu^{3} - 104768 \nu$$$$)/9728$$ $$\beta_{15}$$ $$=$$ $$($$$$525 \nu^{15} + 1661 \nu^{13} + 3352 \nu^{11} + 6757 \nu^{9} + 14093 \nu^{7} + 25168 \nu^{5} + 50816 \nu^{3} + 65344 \nu$$$$)/4864$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{15} + \beta_{14} + \beta_{12} + \beta_{11} + \beta_{9} - \beta_{7} + 3 \beta_{6}$$$$)/8$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{10} - 2 \beta_{8} + \beta_{5} + 2 \beta_{4} - \beta_{3} - 3 \beta_{2} - 6$$$$)/8$$ $$\nu^{3}$$ $$=$$ $$($$$$-3 \beta_{15} - 5 \beta_{14} - \beta_{13} + \beta_{12} - \beta_{11} - 2 \beta_{9} + \beta_{7} + 4 \beta_{6}$$$$)/8$$ $$\nu^{4}$$ $$=$$ $$($$$$3 \beta_{5} - \beta_{4} - 4 \beta_{3} + 2 \beta_{2} + 7 \beta_{1} - 3$$$$)/8$$ $$\nu^{5}$$ $$=$$ $$($$$$2 \beta_{14} - \beta_{13} - 2 \beta_{12} - 2 \beta_{11} - 9 \beta_{9} - 8 \beta_{7} - 13 \beta_{6}$$$$)/8$$ $$\nu^{6}$$ $$=$$ $$($$$$-\beta_{10} + 4 \beta_{8} - 6 \beta_{5} - 5 \beta_{4} + \beta_{3} + \beta_{2} - 17 \beta_{1} - 15$$$$)/8$$ $$\nu^{7}$$ $$=$$ $$($$$$9 \beta_{15} - 7 \beta_{14} + 12 \beta_{13} + 9 \beta_{12} + 9 \beta_{11} - 19 \beta_{9} + 15 \beta_{7} - \beta_{6}$$$$)/8$$ $$\nu^{8}$$ $$=$$ $$($$$$5 \beta_{10} + 18 \beta_{8} - 5 \beta_{5} - 2 \beta_{4} + 5 \beta_{3} - 17 \beta_{2} - 2$$$$)/8$$ $$\nu^{9}$$ $$=$$ $$($$$$-13 \beta_{15} + 5 \beta_{14} + 9 \beta_{13} - \beta_{12} + \beta_{11} + 74 \beta_{9} - \beta_{7} - 12 \beta_{6}$$$$)/8$$ $$\nu^{10}$$ $$=$$ $$($$$$16 \beta_{10} - 3 \beta_{5} + 17 \beta_{4} + 4 \beta_{3} + 46 \beta_{2} - 7 \beta_{1} + 3$$$$)/8$$ $$\nu^{11}$$ $$=$$ $$($$$$48 \beta_{15} + 30 \beta_{14} + 9 \beta_{13} - 14 \beta_{12} - 30 \beta_{11} + 33 \beta_{9} + 24 \beta_{7} + 85 \beta_{6}$$$$)/8$$ $$\nu^{12}$$ $$=$$ $$($$$$-31 \beta_{10} - 52 \beta_{8} - 34 \beta_{5} - 3 \beta_{4} + 31 \beta_{3} + 31 \beta_{2} + 105 \beta_{1} - 89$$$$)/8$$ $$\nu^{13}$$ $$=$$ $$($$$$-41 \beta_{15} + 71 \beta_{14} - 108 \beta_{13} - 41 \beta_{12} - 41 \beta_{11} + 3 \beta_{9} + \beta_{7} - 255 \beta_{6}$$$$)/8$$ $$\nu^{14}$$ $$=$$ $$($$$$83 \beta_{10} - 18 \beta_{8} + 93 \beta_{5} - 94 \beta_{4} + 83 \beta_{3} + 41 \beta_{2} - 176 \beta_{1} + 466$$$$)/8$$ $$\nu^{15}$$ $$=$$ $$($$$$-11 \beta_{15} - 29 \beta_{14} - 9 \beta_{13} - 135 \beta_{12} + 135 \beta_{11} - 162 \beta_{9} - 135 \beta_{7} + 308 \beta_{6}$$$$)/8$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/900\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$451$$ $$577$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
199.1
 1.28061 + 0.600040i 1.28061 − 0.600040i −0.120653 − 1.40906i −0.120653 + 1.40906i 0.422403 + 1.34966i 0.422403 − 1.34966i −0.957636 + 1.04064i −0.957636 − 1.04064i 0.957636 + 1.04064i 0.957636 − 1.04064i −0.422403 + 1.34966i −0.422403 − 1.34966i 0.120653 − 1.40906i 0.120653 + 1.40906i −1.28061 + 0.600040i −1.28061 − 0.600040i
−1.95141 0.438172i 0 3.61601 + 1.71011i 0 0 6.33166 −6.30701 4.92155i 0 0
199.2 −1.95141 + 0.438172i 0 3.61601 1.71011i 0 0 6.33166 −6.30701 + 4.92155i 0 0
199.3 −1.08539 1.67986i 0 −1.64388 + 3.64660i 0 0 −0.596540 7.91002 1.19648i 0 0
199.4 −1.08539 + 1.67986i 0 −1.64388 3.64660i 0 0 −0.596540 7.91002 + 1.19648i 0 0
199.5 −0.696577 1.87477i 0 −3.02956 + 2.61185i 0 0 −5.46770 7.00695 + 3.86039i 0 0
199.6 −0.696577 + 1.87477i 0 −3.02956 2.61185i 0 0 −5.46770 7.00695 3.86039i 0 0
199.7 −0.169449 1.99281i 0 −3.94257 + 0.675358i 0 0 12.3959 2.01392 + 7.74236i 0 0
199.8 −0.169449 + 1.99281i 0 −3.94257 0.675358i 0 0 12.3959 2.01392 7.74236i 0 0
199.9 0.169449 1.99281i 0 −3.94257 0.675358i 0 0 −12.3959 −2.01392 + 7.74236i 0 0
199.10 0.169449 + 1.99281i 0 −3.94257 + 0.675358i 0 0 −12.3959 −2.01392 7.74236i 0 0
199.11 0.696577 1.87477i 0 −3.02956 2.61185i 0 0 5.46770 −7.00695 + 3.86039i 0 0
199.12 0.696577 + 1.87477i 0 −3.02956 + 2.61185i 0 0 5.46770 −7.00695 3.86039i 0 0
199.13 1.08539 1.67986i 0 −1.64388 3.64660i 0 0 0.596540 −7.91002 1.19648i 0 0
199.14 1.08539 + 1.67986i 0 −1.64388 + 3.64660i 0 0 0.596540 −7.91002 + 1.19648i 0 0
199.15 1.95141 0.438172i 0 3.61601 1.71011i 0 0 −6.33166 6.30701 4.92155i 0 0
199.16 1.95141 + 0.438172i 0 3.61601 + 1.71011i 0 0 −6.33166 6.30701 + 4.92155i 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 199.16 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.b even 2 1 inner
20.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 900.3.f.f 16
3.b odd 2 1 300.3.f.b 16
4.b odd 2 1 inner 900.3.f.f 16
5.b even 2 1 inner 900.3.f.f 16
5.c odd 4 1 180.3.c.b 8
5.c odd 4 1 900.3.c.u 8
12.b even 2 1 300.3.f.b 16
15.d odd 2 1 300.3.f.b 16
15.e even 4 1 60.3.c.a 8
15.e even 4 1 300.3.c.d 8
20.d odd 2 1 inner 900.3.f.f 16
20.e even 4 1 180.3.c.b 8
20.e even 4 1 900.3.c.u 8
40.i odd 4 1 2880.3.e.j 8
40.k even 4 1 2880.3.e.j 8
60.h even 2 1 300.3.f.b 16
60.l odd 4 1 60.3.c.a 8
60.l odd 4 1 300.3.c.d 8
120.q odd 4 1 960.3.e.c 8
120.w even 4 1 960.3.e.c 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.3.c.a 8 15.e even 4 1
60.3.c.a 8 60.l odd 4 1
180.3.c.b 8 5.c odd 4 1
180.3.c.b 8 20.e even 4 1
300.3.c.d 8 15.e even 4 1
300.3.c.d 8 60.l odd 4 1
300.3.f.b 16 3.b odd 2 1
300.3.f.b 16 12.b even 2 1
300.3.f.b 16 15.d odd 2 1
300.3.f.b 16 60.h even 2 1
900.3.c.u 8 5.c odd 4 1
900.3.c.u 8 20.e even 4 1
900.3.f.f 16 1.a even 1 1 trivial
900.3.f.f 16 4.b odd 2 1 inner
900.3.f.f 16 5.b even 2 1 inner
900.3.f.f 16 20.d odd 2 1 inner
960.3.e.c 8 120.q odd 4 1
960.3.e.c 8 120.w even 4 1
2880.3.e.j 8 40.i odd 4 1
2880.3.e.j 8 40.k even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(900, [\chi])$$:

 $$T_{7}^{8} - 224 T_{7}^{6} + 12032 T_{7}^{4} - 188416 T_{7}^{2} + 65536$$ $$T_{29}^{4} - 32 T_{29}^{3} - 2152 T_{29}^{2} + 34688 T_{29} + 1334416$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$65536 + 40960 T^{2} + 8448 T^{4} - 640 T^{6} - 592 T^{8} - 40 T^{10} + 33 T^{12} + 10 T^{14} + T^{16}$$
$3$ $$T^{16}$$
$5$ $$T^{16}$$
$7$ $$( 65536 - 188416 T^{2} + 12032 T^{4} - 224 T^{6} + T^{8} )^{2}$$
$11$ $$( 10496 + 208 T^{2} + T^{4} )^{4}$$
$13$ $$( 155351296 + 21781248 T^{2} + 290528 T^{4} + 1008 T^{6} + T^{8} )^{2}$$
$17$ $$( 77721856 + 7269632 T^{2} + 162144 T^{4} + 848 T^{6} + T^{8} )^{2}$$
$19$ $$( 6544162816 + 173686784 T^{2} + 925952 T^{4} + 1696 T^{6} + T^{8} )^{2}$$
$23$ $$( 101419319296 - 1884176384 T^{2} + 4397312 T^{4} - 3616 T^{6} + T^{8} )^{2}$$
$29$ $$( 1334416 + 34688 T - 2152 T^{2} - 32 T^{3} + T^{4} )^{4}$$
$31$ $$( 59895709696 + 2731491328 T^{2} + 7432448 T^{4} + 5408 T^{6} + T^{8} )^{2}$$
$37$ $$( 59919206656 + 2291918592 T^{2} + 8020448 T^{4} + 6192 T^{6} + T^{8} )^{2}$$
$41$ $$( 87184 + 7264 T - 1800 T^{2} - 8 T^{3} + T^{4} )^{4}$$
$43$ $$( 33624411406336 - 62108155904 T^{2} + 40259072 T^{4} - 10816 T^{6} + T^{8} )^{2}$$
$47$ $$( 1056981385216 - 9701752832 T^{2} + 15726848 T^{4} - 8032 T^{6} + T^{8} )^{2}$$
$53$ $$( 228545188096 + 35356046592 T^{2} + 37352288 T^{4} + 11472 T^{6} + T^{8} )^{2}$$
$59$ $$( 173909016576 + 2174459904 T^{2} + 6273792 T^{4} + 4896 T^{6} + T^{8} )^{2}$$
$61$ $$( -2142704 - 273568 T - 2536 T^{2} + 88 T^{3} + T^{4} )^{4}$$
$67$ $$( 281086590976 - 15044755456 T^{2} + 57554432 T^{4} - 16064 T^{6} + T^{8} )^{2}$$
$71$ $$( 16079971680256 + 101402017792 T^{2} + 64237568 T^{4} + 13952 T^{6} + T^{8} )^{2}$$
$73$ $$( 24622079140096 + 121864272128 T^{2} + 90628704 T^{4} + 17552 T^{6} + T^{8} )^{2}$$
$79$ $$( 3198642669223936 + 2420601929728 T^{2} + 550899968 T^{4} + 41888 T^{6} + T^{8} )^{2}$$
$83$ $$( 4284940379815936 - 2381453017088 T^{2} + 464465408 T^{4} - 36928 T^{6} + T^{8} )^{2}$$
$89$ $$( 70652944 + 757600 T - 20584 T^{2} - 40 T^{3} + T^{4} )^{4}$$
$97$ $$( 3514328965714176 + 2030236372224 T^{2} + 414794592 T^{4} + 34896 T^{6} + T^{8} )^{2}$$
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