Properties

Label 1200.3.c.m
Level $1200$
Weight $3$
Character orbit 1200.c
Analytic conductor $32.698$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1200.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(32.6976317232\)
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 \(\left(\mathbb{Z}/1200\mathbb{Z}\right)^\times\).

\(n\) \(401\) \(577\) \(751\) \(901\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(1\)

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:

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

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1200, [\chi])\):

\( T_{7}^{8} + 228 T_{7}^{6} + 16788 T_{7}^{4} + 441568 T_{7}^{2} + 3779136 \)
\( T_{11}^{8} + 888 T_{11}^{6} + 226128 T_{11}^{4} + 14951168 T_{11}^{2} + 232989696 \)

Hecke characteristic polynomials

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