# Properties

 Label 605.2.j.f Level $605$ Weight $2$ Character orbit 605.j Analytic conductor $4.831$ Analytic rank $0$ Dimension $16$ CM no Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$605 = 5 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 605.j (of order $$10$$, degree $$4$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.83094932229$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$4$$ over $$\Q(\zeta_{10})$$ Coefficient field: 16.0.6879707136000000000000.7 Defining polynomial: $$x^{16} - 4x^{14} + 15x^{12} - 56x^{10} + 209x^{8} - 56x^{6} + 15x^{4} - 4x^{2} + 1$$ x^16 - 4*x^14 + 15*x^12 - 56*x^10 + 209*x^8 - 56*x^6 + 15*x^4 - 4*x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{12} q^{2} - \beta_{3} q^{3} - \beta_{10} q^{4} + (\beta_{14} + \beta_{10} + \beta_{7} - \beta_{6} + \beta_{5} - \beta_{2}) q^{5} - \beta_{13} q^{6} + (2 \beta_{15} + \beta_{12} + 2 \beta_{11} + \beta_{9} - 2 \beta_{7} - \beta_{5} - 2 \beta_{3} + \beta_1) q^{7} - \beta_{3} q^{8} + ( - \beta_{6} - \beta_{4}) q^{9}+O(q^{10})$$ q + b12 * q^2 - b3 * q^3 - b10 * q^4 + (b14 + b10 + b7 - b6 + b5 - b2) * q^5 - b13 * q^6 + (2*b15 + b12 + 2*b11 + b9 - 2*b7 - b5 - 2*b3 + b1) * q^7 - b3 * q^8 + (-b6 - b4) * q^9 $$q + \beta_{12} q^{2} - \beta_{3} q^{3} - \beta_{10} q^{4} + (\beta_{14} + \beta_{10} + \beta_{7} - \beta_{6} + \beta_{5} - \beta_{2}) q^{5} - \beta_{13} q^{6} + (2 \beta_{15} + \beta_{12} + 2 \beta_{11} + \beta_{9} - 2 \beta_{7} - \beta_{5} - 2 \beta_{3} + \beta_1) q^{7} - \beta_{3} q^{8} + ( - \beta_{6} - \beta_{4}) q^{9} + (\beta_{11} - \beta_{9} + \beta_{2} - 1) q^{10} + (2 \beta_{11} + \beta_{9}) q^{12} - 3 \beta_{15} q^{13} - \beta_{14} q^{14} + ( - 2 \beta_{15} - \beta_{12} - 2 \beta_{11} - \beta_{10} - \beta_{9} - \beta_{8} + 2 \beta_{7} + \beta_{5} + \cdots - \beta_1) q^{15}+ \cdots + ( - 3 \beta_{11} + 4 \beta_{9}) q^{98}+O(q^{100})$$ q + b12 * q^2 - b3 * q^3 - b10 * q^4 + (b14 + b10 + b7 - b6 + b5 - b2) * q^5 - b13 * q^6 + (2*b15 + b12 + 2*b11 + b9 - 2*b7 - b5 - 2*b3 + b1) * q^7 - b3 * q^8 + (-b6 - b4) * q^9 + (b11 - b9 + b2 - 1) * q^10 + (2*b11 + b9) * q^12 - 3*b15 * q^13 - b14 * q^14 + (-2*b15 - b12 - 2*b11 - b10 - b9 - b8 + 2*b7 + b5 + 2*b3 - b1) * q^15 + (2*b14 - b13 + 2*b10 - 2*b6 - 2*b2) * q^16 - 2*b7 * q^17 + (b15 - 2*b12 + b11 - 2*b9 - b7 - 3*b5 - b3 + 3*b1) * q^18 + (-3*b14 + b13 - b8 - b4 + 1) * q^19 + (-b15 - 2*b12 - 3*b4) * q^20 + (2*b2 + 3) * q^21 + (-b11 - 4*b9) * q^23 + (-b6 + 2*b4) * q^24 + (-b13 + b8 + b4 + 2*b3 + 2*b1 - 1) * q^25 + (-3*b10 + 3*b8) * q^26 + (-2*b7 + b5) * q^27 + 3*b7 * q^28 + 4*b10 * q^29 + (b14 - b3 + b1) * q^30 + (-b6 + 7*b4) * q^31 + (4*b11 - b9) * q^32 - 2 * q^34 + (-3*b15 - 3*b12 - b6 + 3*b4) * q^35 + (-b14 + 3*b13 - 3*b8 - 3*b4 + 3) * q^36 + (-2*b15 + 2*b12 - 2*b11 + 2*b9 + 2*b7 + 4*b5 + 2*b3 - 4*b1) * q^37 + (-3*b7 - 7*b5) * q^38 + (3*b14 + 3*b13 + 3*b10 - 3*b6 - 3*b2) * q^39 + (-2*b15 - b12 - 2*b11 - b10 - b9 - b8 + 2*b7 + b5 + 2*b3 - b1) * q^40 - b14 * q^41 + (2*b15 + b12) * q^42 + (2*b11 + 7*b9) * q^43 + (2*b9 + b2 - 3) * q^45 + (-3*b6 - 7*b4) * q^46 + (-7*b3 + 4*b1) * q^47 + (-3*b15 - b12 - 3*b11 - b9 + 3*b7 + 2*b5 + 3*b3 - 2*b1) * q^48 + (-3*b14 + b13 - 3*b10 + 3*b6 + 3*b2) * q^49 + (-2*b14 + 6*b13 - 2*b10 + 2*b6 + b5 + 2*b2) * q^50 + (2*b10 + 4*b8) * q^51 + (3*b3 + 3*b1) * q^52 + (-3*b15 + 4*b12) * q^53 + (b2 - 4) * q^54 + (2*b2 + 3) * q^56 + (5*b15 + 2*b12) * q^57 + (4*b3 - 8*b1) * q^58 + (b10 - 3*b8) * q^59 + (b14 + 3*b13 + b10 - 3*b7 - b6 - b2) * q^60 + (-b14 - 10*b13 - b10 + b6 + b2) * q^61 + (b15 + 6*b12 + b11 + 6*b9 - b7 + 5*b5 - b3 - 5*b1) * q^62 + (-b3 - b1) * q^63 + (-b6 - 4*b4) * q^64 + (-3*b11 - 6*b9 - 6) * q^65 + (-b11 + 7*b9) * q^67 + (4*b15 + 2*b12) * q^68 + (-b14 - 5*b13 + 5*b8 + 5*b4 - 5) * q^69 + (b15 + 2*b12 + b11 + 2*b9 - 3*b8 - b7 + b5 - b3 - b1) * q^70 + (-3*b14 - 3*b13 - 3*b10 + 3*b6 + 3*b2) * q^71 + (b7 + b5) * q^72 + (-2*b15 - 4*b12 - 2*b11 - 4*b9 + 2*b7 - 2*b5 + 2*b3 + 2*b1) * q^73 + (4*b14 - 6*b13 + 6*b8 + 6*b4 - 6) * q^74 + (b15 + b12 + 2*b6 - 6*b4) * q^75 + (-b2 + 9) * q^76 + 3*b11 * q^78 + (-2*b6 - 2*b4) * q^79 + (b14 - 6*b13 + 6*b8 + 6*b4 + b3 + 3*b1 - 6) * q^80 + (5*b10 + 2*b8) * q^81 + (-b7 - 2*b5) * q^82 + (-7*b7 - 7*b5) * q^83 + (-3*b10 - 6*b8) * q^84 + (-2*b14 - 2*b13 + 2*b8 + 2*b4 - 4*b3 + 2*b1 - 2) * q^85 + (5*b6 + 12*b4) * q^86 + (-8*b11 - 4*b9) * q^87 + (-2*b2 + 3) * q^89 + (b15 - 4*b12 + 2*b6 + 4*b4) * q^90 + (3*b14 + 9*b13 - 9*b8 - 9*b4 + 9) * q^91 + (5*b15 - 2*b12 + 5*b11 - 2*b9 - 5*b7 - 7*b5 - 5*b3 + 7*b1) * q^92 + (9*b7 + b5) * q^93 + (-4*b14 + b13 - 4*b10 + 4*b6 + 4*b2) * q^94 + (2*b15 + 6*b12 + 2*b11 + b10 + 6*b9 - 9*b8 - 2*b7 + 4*b5 - 2*b3 - 4*b1) * q^95 + (4*b14 + 3*b13 - 3*b8 - 3*b4 + 3) * q^96 + (-5*b15 - 4*b12) * q^97 + (-3*b11 + 4*b9) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q + 4 q^{6} - 4 q^{9}+O(q^{10})$$ 16 * q + 4 * q^6 - 4 * q^9 $$16 q + 4 q^{6} - 4 q^{9} - 16 q^{10} - 4 q^{15} + 4 q^{16} + 4 q^{19} - 12 q^{20} + 48 q^{21} + 8 q^{24} - 4 q^{25} + 12 q^{26} + 28 q^{31} - 32 q^{34} + 12 q^{35} + 12 q^{36} - 12 q^{39} - 4 q^{40} - 48 q^{45} - 28 q^{46} - 4 q^{49} - 24 q^{50} + 16 q^{51} - 64 q^{54} + 48 q^{56} - 12 q^{59} - 12 q^{60} + 40 q^{61} - 16 q^{64} - 96 q^{65} - 20 q^{69} - 12 q^{70} + 12 q^{71} - 24 q^{74} - 24 q^{75} + 144 q^{76} - 8 q^{79} - 24 q^{80} + 8 q^{81} - 24 q^{84} - 8 q^{85} + 48 q^{86} + 48 q^{89} + 16 q^{90} + 36 q^{91} - 4 q^{94} - 36 q^{95} + 12 q^{96}+O(q^{100})$$ 16 * q + 4 * q^6 - 4 * q^9 - 16 * q^10 - 4 * q^15 + 4 * q^16 + 4 * q^19 - 12 * q^20 + 48 * q^21 + 8 * q^24 - 4 * q^25 + 12 * q^26 + 28 * q^31 - 32 * q^34 + 12 * q^35 + 12 * q^36 - 12 * q^39 - 4 * q^40 - 48 * q^45 - 28 * q^46 - 4 * q^49 - 24 * q^50 + 16 * q^51 - 64 * q^54 + 48 * q^56 - 12 * q^59 - 12 * q^60 + 40 * q^61 - 16 * q^64 - 96 * q^65 - 20 * q^69 - 12 * q^70 + 12 * q^71 - 24 * q^74 - 24 * q^75 + 144 * q^76 - 8 * q^79 - 24 * q^80 + 8 * q^81 - 24 * q^84 - 8 * q^85 + 48 * q^86 + 48 * q^89 + 16 * q^90 + 36 * q^91 - 4 * q^94 - 36 * q^95 + 12 * q^96

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 4x^{14} + 15x^{12} - 56x^{10} + 209x^{8} - 56x^{6} + 15x^{4} - 4x^{2} + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{10} + 362 ) / 209$$ (v^10 + 362) / 209 $$\beta_{3}$$ $$=$$ $$( \nu^{11} + 780\nu ) / 209$$ (v^11 + 780*v) / 209 $$\beta_{4}$$ $$=$$ $$( \nu^{12} + 780\nu^{2} ) / 209$$ (v^12 + 780*v^2) / 209 $$\beta_{5}$$ $$=$$ $$( \nu^{13} + 780\nu^{3} ) / 209$$ (v^13 + 780*v^3) / 209 $$\beta_{6}$$ $$=$$ $$( -2\nu^{12} - 1351\nu^{2} ) / 209$$ (-2*v^12 - 1351*v^2) / 209 $$\beta_{7}$$ $$=$$ $$( -4\nu^{13} - 2911\nu^{3} ) / 209$$ (-4*v^13 - 2911*v^3) / 209 $$\beta_{8}$$ $$=$$ $$( -4\nu^{14} - 2911\nu^{4} ) / 209$$ (-4*v^14 - 2911*v^4) / 209 $$\beta_{9}$$ $$=$$ $$( -4\nu^{15} - 2911\nu^{5} ) / 209$$ (-4*v^15 - 2911*v^5) / 209 $$\beta_{10}$$ $$=$$ $$( -7\nu^{14} - 5042\nu^{4} ) / 209$$ (-7*v^14 - 5042*v^4) / 209 $$\beta_{11}$$ $$=$$ $$( -\nu^{15} - 723\nu^{5} ) / 19$$ (-v^15 - 723*v^5) / 19 $$\beta_{12}$$ $$=$$ $$( 60\nu^{15} - 225\nu^{13} + 840\nu^{11} - 3135\nu^{9} + 11704\nu^{7} - 225\nu^{5} + 60\nu^{3} - 15\nu ) / 209$$ (60*v^15 - 225*v^13 + 840*v^11 - 3135*v^9 + 11704*v^7 - 225*v^5 + 60*v^3 - 15*v) / 209 $$\beta_{13}$$ $$=$$ $$( 56\nu^{14} - 224\nu^{12} + 840\nu^{10} - 3135\nu^{8} + 11704\nu^{6} - 3136\nu^{4} + 840\nu^{2} - 224 ) / 209$$ (56*v^14 - 224*v^12 + 840*v^10 - 3135*v^8 + 11704*v^6 - 3136*v^4 + 840*v^2 - 224) / 209 $$\beta_{14}$$ $$=$$ $$( 104\nu^{14} - 390\nu^{12} + 1456\nu^{10} - 5434\nu^{8} + 20273\nu^{6} - 390\nu^{4} + 104\nu^{2} - 26 ) / 209$$ (104*v^14 - 390*v^12 + 1456*v^10 - 5434*v^8 + 20273*v^6 - 390*v^4 + 104*v^2 - 26) / 209 $$\beta_{15}$$ $$=$$ $$( 164\nu^{15} - 615\nu^{13} + 2296\nu^{11} - 8569\nu^{9} + 31977\nu^{7} - 615\nu^{5} + 164\nu^{3} - 41\nu ) / 209$$ (164*v^15 - 615*v^13 + 2296*v^11 - 8569*v^9 + 31977*v^7 - 615*v^5 + 164*v^3 - 41*v) / 209
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{6} + 2\beta_{4}$$ b6 + 2*b4 $$\nu^{3}$$ $$=$$ $$\beta_{7} + 4\beta_{5}$$ b7 + 4*b5 $$\nu^{4}$$ $$=$$ $$4\beta_{10} - 7\beta_{8}$$ 4*b10 - 7*b8 $$\nu^{5}$$ $$=$$ $$4\beta_{11} - 11\beta_{9}$$ 4*b11 - 11*b9 $$\nu^{6}$$ $$=$$ $$-15\beta_{14} + 26\beta_{13} - 26\beta_{8} - 26\beta_{4} + 26$$ -15*b14 + 26*b13 - 26*b8 - 26*b4 + 26 $$\nu^{7}$$ $$=$$ $$-15\beta_{15} + 41\beta_{12}$$ -15*b15 + 41*b12 $$\nu^{8}$$ $$=$$ $$-56\beta_{14} + 97\beta_{13} - 56\beta_{10} + 56\beta_{6} + 56\beta_{2}$$ -56*b14 + 97*b13 - 56*b10 + 56*b6 + 56*b2 $$\nu^{9}$$ $$=$$ $$-56\beta_{15} + 153\beta_{12} - 56\beta_{11} + 153\beta_{9} + 56\beta_{7} + 209\beta_{5} + 56\beta_{3} - 209\beta_1$$ -56*b15 + 153*b12 - 56*b11 + 153*b9 + 56*b7 + 209*b5 + 56*b3 - 209*b1 $$\nu^{10}$$ $$=$$ $$209\beta_{2} - 362$$ 209*b2 - 362 $$\nu^{11}$$ $$=$$ $$209\beta_{3} - 780\beta_1$$ 209*b3 - 780*b1 $$\nu^{12}$$ $$=$$ $$-780\beta_{6} - 1351\beta_{4}$$ -780*b6 - 1351*b4 $$\nu^{13}$$ $$=$$ $$-780\beta_{7} - 2911\beta_{5}$$ -780*b7 - 2911*b5 $$\nu^{14}$$ $$=$$ $$-2911\beta_{10} + 5042\beta_{8}$$ -2911*b10 + 5042*b8 $$\nu^{15}$$ $$=$$ $$-2911\beta_{11} + 7953\beta_{9}$$ -2911*b11 + 7953*b9

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/605\mathbb{Z}\right)^\times$$.

 $$n$$ $$122$$ $$486$$ $$\chi(n)$$ $$-1$$ $$-\beta_{8}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
9.1
 −1.13551 − 1.56290i −0.304260 − 0.418778i 0.304260 + 0.418778i 1.13551 + 1.56290i 1.83730 − 0.596975i 0.492303 − 0.159959i −0.492303 + 0.159959i −1.83730 + 0.596975i −1.13551 + 1.56290i −0.304260 + 0.418778i 0.304260 − 0.418778i 1.13551 − 1.56290i 1.83730 + 0.596975i 0.492303 + 0.159959i −0.492303 − 0.159959i −1.83730 − 0.596975i
−1.83730 0.596975i 0.304260 + 0.418778i 1.40126 + 1.01807i 0.809764 2.08429i −0.309017 0.951057i −0.526994 + 0.725345i 0.304260 + 0.418778i 0.844250 2.59833i −2.73205 + 3.34607i
9.2 −0.492303 0.159959i 1.13551 + 1.56290i −1.40126 1.01807i −0.809764 + 2.08429i −0.309017 0.951057i 1.96677 2.70702i 1.13551 + 1.56290i −0.226216 + 0.696222i 0.732051 0.896575i
9.3 0.492303 + 0.159959i −1.13551 1.56290i −1.40126 1.01807i 1.88023 + 1.21026i −0.309017 0.951057i −1.96677 + 2.70702i −1.13551 1.56290i −0.226216 + 0.696222i 0.732051 + 0.896575i
9.4 1.83730 + 0.596975i −0.304260 0.418778i 1.40126 + 1.01807i −1.88023 1.21026i −0.309017 0.951057i 0.526994 0.725345i −0.304260 0.418778i 0.844250 2.59833i −2.73205 3.34607i
124.1 −1.13551 1.56290i −0.492303 + 0.159959i −0.535233 + 1.64728i 2.23251 0.126049i 0.809017 + 0.587785i 0.852694 + 0.277057i −0.492303 + 0.159959i −2.21028 + 1.60586i −2.73205 3.34607i
124.2 −0.304260 0.418778i −1.83730 + 0.596975i 0.535233 1.64728i −2.23251 + 0.126049i 0.809017 + 0.587785i −3.18230 1.03399i −1.83730 + 0.596975i 0.592242 0.430289i 0.732051 + 0.896575i
124.3 0.304260 + 0.418778i 1.83730 0.596975i 0.535233 1.64728i −0.570005 2.16220i 0.809017 + 0.587785i 3.18230 + 1.03399i 1.83730 0.596975i 0.592242 0.430289i 0.732051 0.896575i
124.4 1.13551 + 1.56290i 0.492303 0.159959i −0.535233 + 1.64728i 0.570005 + 2.16220i 0.809017 + 0.587785i −0.852694 0.277057i 0.492303 0.159959i −2.21028 + 1.60586i −2.73205 + 3.34607i
269.1 −1.83730 + 0.596975i 0.304260 0.418778i 1.40126 1.01807i 0.809764 + 2.08429i −0.309017 + 0.951057i −0.526994 0.725345i 0.304260 0.418778i 0.844250 + 2.59833i −2.73205 3.34607i
269.2 −0.492303 + 0.159959i 1.13551 1.56290i −1.40126 + 1.01807i −0.809764 2.08429i −0.309017 + 0.951057i 1.96677 + 2.70702i 1.13551 1.56290i −0.226216 0.696222i 0.732051 + 0.896575i
269.3 0.492303 0.159959i −1.13551 + 1.56290i −1.40126 + 1.01807i 1.88023 1.21026i −0.309017 + 0.951057i −1.96677 2.70702i −1.13551 + 1.56290i −0.226216 0.696222i 0.732051 0.896575i
269.4 1.83730 0.596975i −0.304260 + 0.418778i 1.40126 1.01807i −1.88023 + 1.21026i −0.309017 + 0.951057i 0.526994 + 0.725345i −0.304260 + 0.418778i 0.844250 + 2.59833i −2.73205 + 3.34607i
444.1 −1.13551 + 1.56290i −0.492303 0.159959i −0.535233 1.64728i 2.23251 + 0.126049i 0.809017 0.587785i 0.852694 0.277057i −0.492303 0.159959i −2.21028 1.60586i −2.73205 + 3.34607i
444.2 −0.304260 + 0.418778i −1.83730 0.596975i 0.535233 + 1.64728i −2.23251 0.126049i 0.809017 0.587785i −3.18230 + 1.03399i −1.83730 0.596975i 0.592242 + 0.430289i 0.732051 0.896575i
444.3 0.304260 0.418778i 1.83730 + 0.596975i 0.535233 + 1.64728i −0.570005 + 2.16220i 0.809017 0.587785i 3.18230 1.03399i 1.83730 + 0.596975i 0.592242 + 0.430289i 0.732051 + 0.896575i
444.4 1.13551 1.56290i 0.492303 + 0.159959i −0.535233 1.64728i 0.570005 2.16220i 0.809017 0.587785i −0.852694 + 0.277057i 0.492303 + 0.159959i −2.21028 1.60586i −2.73205 3.34607i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 444.4 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
11.c even 5 3 inner
55.j even 10 3 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 605.2.j.f 16
5.b even 2 1 inner 605.2.j.f 16
11.b odd 2 1 605.2.j.e 16
11.c even 5 1 605.2.b.d 4
11.c even 5 3 inner 605.2.j.f 16
11.d odd 10 1 605.2.b.e yes 4
11.d odd 10 3 605.2.j.e 16
55.d odd 2 1 605.2.j.e 16
55.h odd 10 1 605.2.b.e yes 4
55.h odd 10 3 605.2.j.e 16
55.j even 10 1 605.2.b.d 4
55.j even 10 3 inner 605.2.j.f 16
55.k odd 20 2 3025.2.a.y 4
55.l even 20 2 3025.2.a.z 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
605.2.b.d 4 11.c even 5 1
605.2.b.d 4 55.j even 10 1
605.2.b.e yes 4 11.d odd 10 1
605.2.b.e yes 4 55.h odd 10 1
605.2.j.e 16 11.b odd 2 1
605.2.j.e 16 11.d odd 10 3
605.2.j.e 16 55.d odd 2 1
605.2.j.e 16 55.h odd 10 3
605.2.j.f 16 1.a even 1 1 trivial
605.2.j.f 16 5.b even 2 1 inner
605.2.j.f 16 11.c even 5 3 inner
605.2.j.f 16 55.j even 10 3 inner
3025.2.a.y 4 55.k odd 20 2
3025.2.a.z 4 55.l even 20 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(605, [\chi])$$:

 $$T_{2}^{16} - 4T_{2}^{14} + 15T_{2}^{12} - 56T_{2}^{10} + 209T_{2}^{8} - 56T_{2}^{6} + 15T_{2}^{4} - 4T_{2}^{2} + 1$$ T2^16 - 4*T2^14 + 15*T2^12 - 56*T2^10 + 209*T2^8 - 56*T2^6 + 15*T2^4 - 4*T2^2 + 1 $$T_{19}^{8} - 2T_{19}^{7} + 30T_{19}^{6} - 112T_{19}^{5} + 1004T_{19}^{4} + 2912T_{19}^{3} + 20280T_{19}^{2} + 35152T_{19} + 456976$$ T19^8 - 2*T19^7 + 30*T19^6 - 112*T19^5 + 1004*T19^4 + 2912*T19^3 + 20280*T19^2 + 35152*T19 + 456976

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16} - 4 T^{14} + 15 T^{12} - 56 T^{10} + \cdots + 1$$
$3$ $$T^{16} - 4 T^{14} + 15 T^{12} - 56 T^{10} + \cdots + 1$$
$5$ $$T^{16} + 2 T^{14} - 21 T^{12} + \cdots + 390625$$
$7$ $$T^{16} - 12 T^{14} + 135 T^{12} + \cdots + 6561$$
$11$ $$T^{16}$$
$13$ $$(T^{8} - 18 T^{6} + 324 T^{4} + \cdots + 104976)^{2}$$
$17$ $$T^{16} - 16 T^{14} + 240 T^{12} + \cdots + 65536$$
$19$ $$(T^{8} - 2 T^{7} + 30 T^{6} - 112 T^{5} + \cdots + 456976)^{2}$$
$23$ $$(T^{4} + 52 T^{2} + 484)^{4}$$
$29$ $$(T^{8} + 48 T^{6} + 2304 T^{4} + \cdots + 5308416)^{2}$$
$31$ $$(T^{8} - 14 T^{7} + 150 T^{6} + \cdots + 4477456)^{2}$$
$37$ $$T^{16} - 48 T^{14} + \cdots + 429981696$$
$41$ $$(T^{8} + 3 T^{6} + 9 T^{4} + 27 T^{2} + \cdots + 81)^{2}$$
$43$ $$(T^{4} + 156 T^{2} + 4761)^{4}$$
$47$ $$T^{16} - 148 T^{14} + \cdots + 23811286661761$$
$53$ $$T^{16} - 148 T^{14} + \cdots + 208827064576$$
$59$ $$(T^{8} + 6 T^{7} + 30 T^{6} + 144 T^{5} + \cdots + 1296)^{2}$$
$61$ $$(T^{8} - 20 T^{7} + 303 T^{6} + \cdots + 88529281)^{2}$$
$67$ $$(T^{4} + 228 T^{2} + 1089)^{4}$$
$71$ $$(T^{8} - 6 T^{7} + 54 T^{6} - 432 T^{5} + \cdots + 104976)^{2}$$
$73$ $$(T^{8} - 24 T^{6} + 576 T^{4} + \cdots + 331776)^{2}$$
$79$ $$(T^{8} + 4 T^{7} + 24 T^{6} + 128 T^{5} + \cdots + 4096)^{2}$$
$83$ $$(T^{8} - 98 T^{6} + 9604 T^{4} + \cdots + 92236816)^{2}$$
$89$ $$(T^{2} - 6 T - 3)^{8}$$
$97$ $$T^{16} - 84 T^{14} + 7020 T^{12} + \cdots + 1679616$$