# Properties

 Label 600.3.c.d Level $600$ Weight $3$ Character orbit 600.c Analytic conductor $16.349$ Analytic rank $0$ Dimension $16$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$600 = 2^{3} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 600.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$16.3488158616$$ Analytic rank: $$0$$ Dimension: $$16$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ Defining polynomial: $$x^{16} + 138 x^{12} + 3393 x^{8} + 15208 x^{4} + 1296$$ Coefficient ring: $$\Z[a_1, \ldots, a_{37}]$$ Coefficient ring index: $$2^{26}\cdot 3^{2}$$ Twist minimal: no (minimal twist has level 120) 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.

 $$f(q)$$ $$=$$ $$q -\beta_{3} q^{3} + ( -\beta_{1} + \beta_{2} + \beta_{3} ) q^{7} + ( -3 + \beta_{12} + \beta_{14} ) q^{9} +O(q^{10})$$ $$q -\beta_{3} q^{3} + ( -\beta_{1} + \beta_{2} + \beta_{3} ) q^{7} + ( -3 + \beta_{12} + \beta_{14} ) q^{9} + ( -1 - 2 \beta_{4} + \beta_{7} + \beta_{8} - 2 \beta_{11} - 2 \beta_{13} ) q^{11} + ( -\beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{5} - \beta_{6} + \beta_{10} - \beta_{15} ) q^{13} + ( -\beta_{3} + \beta_{9} - \beta_{10} - 2 \beta_{15} ) q^{17} + ( 1 - \beta_{7} - \beta_{11} - 2 \beta_{12} - 2 \beta_{13} ) q^{19} + ( 2 - 3 \beta_{4} - \beta_{7} + 2 \beta_{8} - 3 \beta_{11} - 3 \beta_{13} ) q^{21} + ( 2 \beta_{3} + 3 \beta_{6} + 4 \beta_{9} - \beta_{15} ) q^{23} + ( -3 \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{5} - \beta_{6} + 4 \beta_{9} + \beta_{10} ) q^{27} + ( 4 \beta_{8} - 2 \beta_{11} - \beta_{14} ) q^{29} + ( 12 + 2 \beta_{4} + 3 \beta_{8} + 4 \beta_{12} - 2 \beta_{13} + 3 \beta_{14} ) q^{31} + ( -6 \beta_{1} + \beta_{2} - \beta_{3} - 3 \beta_{5} + 2 \beta_{6} + 5 \beta_{9} + 5 \beta_{10} - \beta_{15} ) q^{33} + ( 3 \beta_{1} - 2 \beta_{2} - 9 \beta_{3} - 5 \beta_{5} - \beta_{6} + \beta_{10} - \beta_{15} ) q^{37} + ( 11 + 2 \beta_{4} + 3 \beta_{7} + 6 \beta_{11} - \beta_{12} + 3 \beta_{13} - \beta_{14} ) q^{39} + ( -9 \beta_{8} - 4 \beta_{11} - 5 \beta_{14} ) q^{41} + ( -22 \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{6} + \beta_{10} - \beta_{15} ) q^{43} + ( -4 \beta_{3} + \beta_{6} + 4 \beta_{10} - \beta_{15} ) q^{47} + ( -7 - 4 \beta_{4} - \beta_{8} + 2 \beta_{12} + 4 \beta_{13} - \beta_{14} ) q^{49} + ( 6 - \beta_{4} + \beta_{7} - 8 \beta_{8} + 2 \beta_{11} - 4 \beta_{13} - \beta_{14} ) q^{51} + ( 4 \beta_{6} + 3 \beta_{9} - 2 \beta_{10} - 6 \beta_{15} ) q^{53} + ( -3 \beta_{1} - 4 \beta_{2} - \beta_{3} - \beta_{5} + 5 \beta_{6} - 5 \beta_{9} - \beta_{10} + \beta_{15} ) q^{57} + ( -8 \beta_{8} - 5 \beta_{11} + 3 \beta_{14} ) q^{59} + ( 2 - 8 \beta_{4} - \beta_{8} + 6 \beta_{12} + 8 \beta_{13} - \beta_{14} ) q^{61} + ( 6 \beta_{1} - 6 \beta_{2} - 13 \beta_{3} - 5 \beta_{5} + 10 \beta_{10} - 5 \beta_{15} ) q^{63} + ( -16 \beta_{1} - 4 \beta_{2} + 9 \beta_{3} + 2 \beta_{5} - 3 \beta_{6} + 3 \beta_{10} - 3 \beta_{15} ) q^{67} + ( -12 + 3 \beta_{4} - 3 \beta_{7} - 6 \beta_{8} + 11 \beta_{11} - 2 \beta_{12} - \beta_{13} - 3 \beta_{14} ) q^{69} + ( -3 - 6 \beta_{4} + 3 \beta_{7} - \beta_{8} - \beta_{11} - 6 \beta_{13} - 7 \beta_{14} ) q^{71} + ( 3 \beta_{1} - 6 \beta_{2} - 8 \beta_{3} - 4 \beta_{5} - 2 \beta_{6} + 2 \beta_{10} - 2 \beta_{15} ) q^{73} + ( 20 \beta_{3} + 12 \beta_{6} + 10 \beta_{9} + 4 \beta_{10} + 12 \beta_{15} ) q^{77} + ( -13 + 4 \beta_{4} + \beta_{7} + 2 \beta_{8} + \beta_{11} + 2 \beta_{12} - 2 \beta_{13} + 2 \beta_{14} ) q^{79} + ( 33 - 3 \beta_{8} + 6 \beta_{11} + 2 \beta_{12} + 12 \beta_{13} - 7 \beta_{14} ) q^{81} + ( 13 \beta_{3} + 3 \beta_{6} + 20 \beta_{9} - 7 \beta_{10} + 3 \beta_{15} ) q^{83} + ( 9 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} - 2 \beta_{5} - \beta_{6} + 2 \beta_{10} - 9 \beta_{15} ) q^{87} + ( 4 + 8 \beta_{4} - 4 \beta_{7} - 2 \beta_{8} - 12 \beta_{11} + 8 \beta_{13} + 2 \beta_{14} ) q^{89} + ( 71 + 12 \beta_{4} + 3 \beta_{7} - \beta_{8} + 3 \beta_{11} - 8 \beta_{12} - 6 \beta_{13} - \beta_{14} ) q^{91} + ( -27 \beta_{1} - 3 \beta_{2} - 18 \beta_{3} + 5 \beta_{5} - 6 \beta_{6} + 6 \beta_{9} + 8 \beta_{10} - 7 \beta_{15} ) q^{93} + ( -5 \beta_{1} + 4 \beta_{2} - 4 \beta_{3} + 8 \beta_{5} + 8 \beta_{6} - 8 \beta_{10} + 8 \beta_{15} ) q^{97} + ( -9 - 4 \beta_{4} - 5 \beta_{7} + 19 \beta_{8} + 12 \beta_{11} + 4 \beta_{12} - 6 \beta_{13} + 4 \beta_{14} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q - 40q^{9} + O(q^{10})$$ $$16q - 40q^{9} + 16q^{19} + 56q^{21} + 240q^{31} + 144q^{39} - 128q^{49} + 128q^{51} + 16q^{61} - 200q^{69} - 176q^{79} + 448q^{81} + 1120q^{91} - 64q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} + 138 x^{12} + 3393 x^{8} + 15208 x^{4} + 1296$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-763 \nu^{14} - 105618 \nu^{10} - 2636235 \nu^{6} - 13306756 \nu^{2}$$$$)/1927656$$ $$\beta_{2}$$ $$=$$ $$($$$$11905 \nu^{15} + 35058 \nu^{14} - 9198 \nu^{13} + 1627734 \nu^{11} + 4519404 \nu^{10} - 1237860 \nu^{9} + 38391705 \nu^{7} + 78790914 \nu^{6} - 27464814 \nu^{5} + 135441508 \nu^{3} - 73584312 \nu^{2} - 41323320 \nu$$$$)/92527488$$ $$\beta_{3}$$ $$=$$ $$($$$$-11905 \nu^{15} + 71346 \nu^{14} + 9198 \nu^{13} - 1627734 \nu^{11} + 9825516 \nu^{10} + 1237860 \nu^{9} - 38391705 \nu^{7} + 238690242 \nu^{6} + 27464814 \nu^{5} - 135441508 \nu^{3} + 956837064 \nu^{2} + 41323320 \nu$$$$)/92527488$$ $$\beta_{4}$$ $$=$$ $$($$$$11905 \nu^{15} + 9198 \nu^{13} - 32652 \nu^{12} + 1627734 \nu^{11} + 1237860 \nu^{9} - 4560264 \nu^{8} + 38391705 \nu^{7} + 27464814 \nu^{5} - 114442668 \nu^{4} + 135441508 \nu^{3} + 41323320 \nu - 332141040$$$$)/46263744$$ $$\beta_{5}$$ $$=$$ $$($$$$11905 \nu^{15} + 73248 \nu^{14} - 9198 \nu^{13} + 1627734 \nu^{11} + 10139328 \nu^{10} - 1237860 \nu^{9} + 38391705 \nu^{7} + 253078560 \nu^{6} - 27464814 \nu^{5} + 135441508 \nu^{3} + 1184921088 \nu^{2} - 41323320 \nu$$$$)/46263744$$ $$\beta_{6}$$ $$=$$ $$($$$$-103381 \nu^{15} - 71346 \nu^{14} - 5562 \nu^{13} - 14361006 \nu^{11} - 9825516 \nu^{10} - 492012 \nu^{9} - 363079917 \nu^{7} - 238690242 \nu^{6} + 17991846 \nu^{5} - 1783912756 \nu^{3} - 956837064 \nu^{2} + 518165784 \nu$$$$)/92527488$$ $$\beta_{7}$$ $$=$$ $$($$$$30217 \nu^{15} + 9198 \nu^{13} + 18648 \nu^{12} + 4162566 \nu^{11} + 1237860 \nu^{9} + 2298384 \nu^{8} + 101661345 \nu^{7} + 27464814 \nu^{5} + 29909592 \nu^{4} + 454803652 \nu^{3} + 133850808 \nu - 95466816$$$$)/23131872$$ $$\beta_{8}$$ $$=$$ $$($$$$32945 \nu^{15} + 12762 \nu^{13} + 4537662 \nu^{11} + 1758996 \nu^{9} + 110503233 \nu^{7} + 42343074 \nu^{5} + 466611092 \nu^{3} + 138394728 \nu$$$$)/23131872$$ $$\beta_{9}$$ $$=$$ $$($$$$-32945 \nu^{15} + 12762 \nu^{13} - 4537662 \nu^{11} + 1758996 \nu^{9} - 110503233 \nu^{7} + 42343074 \nu^{5} - 466611092 \nu^{3} + 138394728 \nu$$$$)/23131872$$ $$\beta_{10}$$ $$=$$ $$($$$$156583 \nu^{15} + 71346 \nu^{14} - 64386 \nu^{13} + 21533466 \nu^{11} + 9825516 \nu^{10} - 8665020 \nu^{9} + 521820495 \nu^{7} + 238690242 \nu^{6} - 192253698 \nu^{5} + 2225539132 \nu^{3} + 956837064 \nu^{2} - 659373192 \nu$$$$)/92527488$$ $$\beta_{11}$$ $$=$$ $$($$$$-21061 \nu^{15} - 9198 \nu^{13} - 2895150 \nu^{11} - 1237860 \nu^{9} - 70026525 \nu^{7} - 27464814 \nu^{5} - 295122580 \nu^{3} - 87587064 \nu$$$$)/11565936$$ $$\beta_{12}$$ $$=$$ $$($$$$-9559 \nu^{15} - 2030 \nu^{13} - 2936 \nu^{12} - 1320394 \nu^{11} - 284932 \nu^{9} - 381712 \nu^{8} - 32592959 \nu^{7} - 7493294 \nu^{5} - 7864760 \nu^{4} - 148170284 \nu^{3} - 43495128 \nu - 30449184$$$$)/5140416$$ $$\beta_{13}$$ $$=$$ $$($$$$24484 \nu^{15} + 6867 \nu^{13} + 12825 \nu^{12} + 3377820 \nu^{11} + 950562 \nu^{9} + 1714662 \nu^{8} + 82932084 \nu^{7} + 23726115 \nu^{5} + 36088065 \nu^{4} + 367243516 \nu^{3} + 108194868 \nu + 53385588$$$$)/11565936$$ $$\beta_{14}$$ $$=$$ $$($$$$26543 \nu^{15} + 2754 \nu^{13} + 3672942 \nu^{11} + 402696 \nu^{9} + 91416699 \nu^{7} + 12548286 \nu^{5} + 433460732 \nu^{3} + 126530712 \nu$$$$)/11565936$$ $$\beta_{15}$$ $$=$$ $$($$$$224249 \nu^{15} - 71346 \nu^{14} - 31230 \nu^{13} + 31011270 \nu^{11} - 9825516 \nu^{10} - 4459428 \nu^{9} + 769725297 \nu^{7} - 238690242 \nu^{6} - 127851102 \nu^{5} + 3603127364 \nu^{3} - 956837064 \nu^{2} - 1053569016 \nu$$$$)/92527488$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-4 \beta_{15} + 2 \beta_{14} - 4 \beta_{13} - \beta_{11} + \beta_{10} + 2 \beta_{8} + 2 \beta_{7} - 4 \beta_{6} - 4 \beta_{4} - 9 \beta_{3} - 2$$$$)/24$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{15} + \beta_{10} - \beta_{6} - 3 \beta_{5} - 3 \beta_{3} - 12 \beta_{1}$$$$)/6$$ $$\nu^{3}$$ $$=$$ $$($$$$8 \beta_{15} + 2 \beta_{14} - 12 \beta_{13} - \beta_{11} - 5 \beta_{10} - 4 \beta_{9} + 10 \beta_{8} + 6 \beta_{7} + 12 \beta_{6} - 12 \beta_{4} + 25 \beta_{3} - 6$$$$)/8$$ $$\nu^{4}$$ $$=$$ $$($$$$-11 \beta_{14} - 14 \beta_{13} - 36 \beta_{12} - 8 \beta_{11} - 11 \beta_{8} - 8 \beta_{7} - 2 \beta_{4} - 196$$$$)/6$$ $$\nu^{5}$$ $$=$$ $$($$$$200 \beta_{15} - 34 \beta_{14} + 332 \beta_{13} - 19 \beta_{11} - 185 \beta_{10} - 204 \beta_{9} - 370 \beta_{8} - 166 \beta_{7} + 332 \beta_{6} + 332 \beta_{4} + 717 \beta_{3} + 166$$$$)/24$$ $$\nu^{6}$$ $$=$$ $$($$$$69 \beta_{15} - 69 \beta_{10} + 69 \beta_{6} + 131 \beta_{5} + 75 \beta_{3} + 20 \beta_{2} + 286 \beta_{1}$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$-1888 \beta_{15} - 278 \beta_{14} + 3220 \beta_{13} - 509 \beta_{11} + 2119 \beta_{10} + 2484 \beta_{9} - 4094 \beta_{8} - 1610 \beta_{7} - 3220 \beta_{6} + 3220 \beta_{4} - 7227 \beta_{3} + 1610$$$$)/24$$ $$\nu^{8}$$ $$=$$ $$($$$$1339 \beta_{14} + 1492 \beta_{13} + 4170 \beta_{12} + 538 \beta_{11} + 1339 \beta_{8} + 538 \beta_{7} - 416 \beta_{4} + 17048$$$$)/6$$ $$\nu^{9}$$ $$=$$ $$($$$$-6264 \beta_{15} + 886 \beta_{14} - 10756 \beta_{13} + 2329 \beta_{11} + 7707 \beta_{10} + 9172 \beta_{9} + 14550 \beta_{8} + 5378 \beta_{7} - 10756 \beta_{6} - 10756 \beta_{4} - 24727 \beta_{3} - 5378$$$$)/8$$ $$\nu^{10}$$ $$=$$ $$($$$$-24503 \beta_{15} + 24503 \beta_{10} - 24503 \beta_{6} - 43731 \beta_{5} - 23235 \beta_{3} - 9282 \beta_{2} - 85872 \beta_{1}$$$$)/6$$ $$\nu^{11}$$ $$=$$ $$($$$$191408 \beta_{15} + 26806 \beta_{14} - 329204 \beta_{13} + 81349 \beta_{11} - 245951 \beta_{10} - 294036 \beta_{9} + 458638 \beta_{8} + 164602 \beta_{7} + 329204 \beta_{6} - 329204 \beta_{4} + 766563 \beta_{3} - 164602$$$$)/24$$ $$\nu^{12}$$ $$=$$ $$($$$$-49957 \beta_{14} - 52158 \beta_{13} - 152072 \beta_{12} - 16172 \beta_{11} - 49957 \beta_{8} - 16172 \beta_{7} + 19814 \beta_{4} - 584540$$$$)/2$$ $$\nu^{13}$$ $$=$$ $$($$$$1969912 \beta_{15} - 275234 \beta_{14} + 3389356 \beta_{13} - 889139 \beta_{11} - 2583817 \beta_{10} - 3093948 \beta_{9} - 4788626 \beta_{8} - 1694678 \beta_{7} + 3389356 \beta_{6} + 3389356 \beta_{4} + 7943085 \beta_{3} + 1694678$$$$)/24$$ $$\nu^{14}$$ $$=$$ $$($$$$2694055 \beta_{15} - 2694055 \beta_{10} + 2694055 \beta_{6} + 4747917 \beta_{5} + 2491221 \beta_{3} + 1077552 \beta_{2} + 9116454 \beta_{1}$$$$)/6$$ $$\nu^{15}$$ $$=$$ $$($$$$-6790240 \beta_{15} - 948210 \beta_{14} + 11684060 \beta_{13} - 3151599 \beta_{11} + 8993629 \beta_{10} + 10776220 \beta_{9} - 16618250 \beta_{8} - 5842030 \beta_{7} - 11684060 \beta_{6} + 11684060 \beta_{4} - 27467929 \beta_{3} + 5842030$$$$)/8$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/600\mathbb{Z}\right)^\times$$.

 $$n$$ $$151$$ $$301$$ $$401$$ $$577$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
449.1
 −2.27869 − 2.27869i −2.27869 + 2.27869i −1.09102 + 1.09102i −1.09102 − 1.09102i −1.57158 + 1.57158i −1.57158 − 1.57158i 0.383916 + 0.383916i 0.383916 − 0.383916i −0.383916 − 0.383916i −0.383916 + 0.383916i 1.57158 − 1.57158i 1.57158 + 1.57158i 1.09102 − 1.09102i 1.09102 + 1.09102i 2.27869 + 2.27869i 2.27869 − 2.27869i
0 −2.98579 0.291610i 0 0 0 4.46268i 0 8.82993 + 1.74137i 0
449.2 0 −2.98579 + 0.291610i 0 0 0 4.46268i 0 8.82993 1.74137i 0
449.3 0 −1.79813 2.40140i 0 0 0 10.2132i 0 −2.53346 + 8.63606i 0
449.4 0 −1.79813 + 2.40140i 0 0 0 10.2132i 0 −2.53346 8.63606i 0
449.5 0 −0.864473 2.87275i 0 0 0 9.02416i 0 −7.50537 + 4.96683i 0
449.6 0 −0.864473 + 2.87275i 0 0 0 9.02416i 0 −7.50537 4.96683i 0
449.7 0 −0.323191 2.98254i 0 0 0 4.72640i 0 −8.79110 + 1.92786i 0
449.8 0 −0.323191 + 2.98254i 0 0 0 4.72640i 0 −8.79110 1.92786i 0
449.9 0 0.323191 2.98254i 0 0 0 4.72640i 0 −8.79110 1.92786i 0
449.10 0 0.323191 + 2.98254i 0 0 0 4.72640i 0 −8.79110 + 1.92786i 0
449.11 0 0.864473 2.87275i 0 0 0 9.02416i 0 −7.50537 4.96683i 0
449.12 0 0.864473 + 2.87275i 0 0 0 9.02416i 0 −7.50537 + 4.96683i 0
449.13 0 1.79813 2.40140i 0 0 0 10.2132i 0 −2.53346 8.63606i 0
449.14 0 1.79813 + 2.40140i 0 0 0 10.2132i 0 −2.53346 + 8.63606i 0
449.15 0 2.98579 0.291610i 0 0 0 4.46268i 0 8.82993 1.74137i 0
449.16 0 2.98579 + 0.291610i 0 0 0 4.46268i 0 8.82993 + 1.74137i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 449.16 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 600.3.c.d 16
3.b odd 2 1 inner 600.3.c.d 16
4.b odd 2 1 1200.3.c.m 16
5.b even 2 1 inner 600.3.c.d 16
5.c odd 4 1 120.3.l.a 8
5.c odd 4 1 600.3.l.f 8
12.b even 2 1 1200.3.c.m 16
15.d odd 2 1 inner 600.3.c.d 16
15.e even 4 1 120.3.l.a 8
15.e even 4 1 600.3.l.f 8
20.d odd 2 1 1200.3.c.m 16
20.e even 4 1 240.3.l.d 8
20.e even 4 1 1200.3.l.x 8
40.i odd 4 1 960.3.l.h 8
40.k even 4 1 960.3.l.g 8
60.h even 2 1 1200.3.c.m 16
60.l odd 4 1 240.3.l.d 8
60.l odd 4 1 1200.3.l.x 8
120.q odd 4 1 960.3.l.g 8
120.w even 4 1 960.3.l.h 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.3.l.a 8 5.c odd 4 1
120.3.l.a 8 15.e even 4 1
240.3.l.d 8 20.e even 4 1
240.3.l.d 8 60.l odd 4 1
600.3.c.d 16 1.a even 1 1 trivial
600.3.c.d 16 3.b odd 2 1 inner
600.3.c.d 16 5.b even 2 1 inner
600.3.c.d 16 15.d odd 2 1 inner
600.3.l.f 8 5.c odd 4 1
600.3.l.f 8 15.e even 4 1
960.3.l.g 8 40.k even 4 1
960.3.l.g 8 120.q odd 4 1
960.3.l.h 8 40.i odd 4 1
960.3.l.h 8 120.w even 4 1
1200.3.c.m 16 4.b odd 2 1
1200.3.c.m 16 12.b even 2 1
1200.3.c.m 16 20.d odd 2 1
1200.3.c.m 16 60.h even 2 1
1200.3.l.x 8 20.e even 4 1
1200.3.l.x 8 60.l odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{8} + 228 T_{7}^{6} + 16788 T_{7}^{4} + 441568 T_{7}^{2} + 3779136$$ acting on $$S_{3}^{\mathrm{new}}(600, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16}$$
$3$ $$43046721 + 10628820 T^{2} + 577368 T^{4} - 111780 T^{6} - 22482 T^{8} - 1380 T^{10} + 88 T^{12} + 20 T^{14} + T^{16}$$
$5$ $$T^{16}$$
$7$ $$( 3779136 + 441568 T^{2} + 16788 T^{4} + 228 T^{6} + T^{8} )^{2}$$
$11$ $$( 232989696 + 14951168 T^{2} + 226128 T^{4} + 888 T^{6} + T^{8} )^{2}$$
$13$ $$( 11943936 + 10681344 T^{2} + 166672 T^{4} + 776 T^{6} + T^{8} )^{2}$$
$17$ $$( 15872256 - 25369856 T^{2} + 312672 T^{4} - 1104 T^{6} + T^{8} )^{2}$$
$19$ $$( 16736 + 3424 T - 708 T^{2} - 4 T^{3} + T^{4} )^{4}$$
$23$ $$( 93650688576 - 885144416 T^{2} + 2697012 T^{4} - 2964 T^{6} + T^{8} )^{2}$$
$29$ $$( 3474395136 + 145293824 T^{2} + 1160592 T^{4} + 2376 T^{6} + T^{8} )^{2}$$
$31$ $$( -151296 + 71360 T - 924 T^{2} - 60 T^{3} + T^{4} )^{4}$$
$37$ $$( 967458816 + 1378778112 T^{2} + 10675728 T^{4} + 6472 T^{6} + T^{8} )^{2}$$
$41$ $$( 43961355472896 + 77260718592 T^{2} + 46587024 T^{4} + 11528 T^{6} + T^{8} )^{2}$$
$43$ $$( 2504800014336 + 18640283392 T^{2} + 20724708 T^{4} + 7932 T^{6} + T^{8} )^{2}$$
$47$ $$( 13517317696 - 266492768 T^{2} + 1541364 T^{4} - 2612 T^{6} + T^{8} )^{2}$$
$53$ $$( 6801580544256 - 88193961216 T^{2} + 82087776 T^{4} - 16976 T^{6} + T^{8} )^{2}$$
$59$ $$( 15563214360576 + 64836527616 T^{2} + 49411216 T^{4} + 12616 T^{6} + T^{8} )^{2}$$
$61$ $$( 30631296 + 12544 T - 12508 T^{2} - 4 T^{3} + T^{4} )^{4}$$
$67$ $$( 446523314176 + 49314659584 T^{2} + 58130916 T^{4} + 15484 T^{6} + T^{8} )^{2}$$
$71$ $$( 35499479924736 + 145459224576 T^{2} + 83806272 T^{4} + 16304 T^{6} + T^{8} )^{2}$$
$73$ $$( 8637109698816 + 21599799552 T^{2} + 19264608 T^{4} + 7312 T^{6} + T^{8} )^{2}$$
$79$ $$( -12384 + 8256 T - 1268 T^{2} + 44 T^{3} + T^{4} )^{4}$$
$83$ $$( 336130569170496 - 2057217800544 T^{2} + 532895092 T^{4} - 41716 T^{6} + T^{8} )^{2}$$
$89$ $$( 1334603390386176 + 2220134105088 T^{2} + 649648384 T^{4} + 49312 T^{6} + T^{8} )^{2}$$
$97$ $$( 105474871849216 + 499769625856 T^{2} + 205298016 T^{4} + 25936 T^{6} + T^{8} )^{2}$$