Properties

Label 2100.2.x.d
Level 2100
Weight 2
Character orbit 2100.x
Analytic conductor 16.769
Analytic rank 0
Dimension 16
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) \(=\) \( 2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2100.x (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(16.7685844245\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 8 x^{14} + 8 x^{12} - 8 x^{10} + 212 x^{8} + 248 x^{6} + 368 x^{4} + 32 x^{2} + 100\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{9} \)
Twist minimal: no (minimal twist has level 420)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{8} q^{3} + ( -\beta_{3} - \beta_{4} + \beta_{10} ) q^{7} -\beta_{5} q^{9} +O(q^{10})\) \( q + \beta_{8} q^{3} + ( -\beta_{3} - \beta_{4} + \beta_{10} ) q^{7} -\beta_{5} q^{9} + ( 1 - \beta_{2} + \beta_{5} + \beta_{7} ) q^{11} + ( -\beta_{4} - \beta_{6} - \beta_{10} - 2 \beta_{11} + \beta_{14} + \beta_{15} ) q^{13} + ( -1 + 2 \beta_{5} + \beta_{7} + \beta_{9} + \beta_{10} + \beta_{12} - \beta_{14} + \beta_{15} ) q^{17} + ( -\beta_{3} + \beta_{4} + \beta_{6} + \beta_{10} + \beta_{11} + \beta_{12} ) q^{19} + ( 1 - \beta_{5} - \beta_{9} + \beta_{14} ) q^{21} + ( -1 - 2 \beta_{1} + 2 \beta_{2} - \beta_{4} - 2 \beta_{5} - \beta_{6} - \beta_{7} - \beta_{10} ) q^{23} + \beta_{3} q^{27} + ( 1 - 2 \beta_{5} - 2 \beta_{6} - \beta_{7} - \beta_{9} - 2 \beta_{10} ) q^{29} + ( -1 + \beta_{3} + 2 \beta_{5} + \beta_{7} + \beta_{9} - 2 \beta_{10} + \beta_{11} - \beta_{12} - 2 \beta_{13} - 2 \beta_{14} ) q^{31} + ( \beta_{8} + \beta_{11} - \beta_{13} ) q^{33} + ( 1 + 2 \beta_{1} + 2 \beta_{2} + \beta_{4} - 3 \beta_{5} - \beta_{6} - 2 \beta_{7} - 4 \beta_{9} - \beta_{10} ) q^{37} + ( 1 + 2 \beta_{1} + \beta_{6} - \beta_{7} - \beta_{9} + \beta_{10} ) q^{39} + ( -1 - 2 \beta_{3} + 2 \beta_{5} + \beta_{7} - 2 \beta_{8} + \beta_{9} + \beta_{10} + \beta_{13} - 2 \beta_{14} ) q^{41} + ( -2 + 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{5} ) q^{43} + ( -2 - \beta_{4} + 4 \beta_{5} - \beta_{6} + 2 \beta_{7} + 2 \beta_{9} - \beta_{10} - 2 \beta_{12} - 2 \beta_{14} + 2 \beta_{15} ) q^{47} + ( -1 - 2 \beta_{1} - 2 \beta_{3} + 3 \beta_{5} - 2 \beta_{6} + \beta_{7} + 2 \beta_{8} + \beta_{9} - 2 \beta_{10} + 2 \beta_{15} ) q^{49} + ( -1 - \beta_{2} - \beta_{4} + \beta_{5} + \beta_{7} ) q^{51} + ( 4 + \beta_{1} - \beta_{2} + \beta_{4} + 2 \beta_{5} + \beta_{6} - 2 \beta_{7} + \beta_{10} ) q^{53} + ( -\beta_{1} - \beta_{2} + \beta_{5} + \beta_{7} + \beta_{9} ) q^{57} + ( 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{6} - 2 \beta_{8} - 2 \beta_{13} ) q^{59} + ( -1 + 4 \beta_{3} + 2 \beta_{5} + \beta_{7} + 4 \beta_{8} + \beta_{9} - 2 \beta_{14} ) q^{61} + ( \beta_{8} - \beta_{13} ) q^{63} + ( 4 - 2 \beta_{1} - 2 \beta_{2} + \beta_{4} - 2 \beta_{5} - \beta_{6} + 2 \beta_{7} - 2 \beta_{9} - \beta_{10} ) q^{67} + ( -1 + 2 \beta_{3} - \beta_{4} + 2 \beta_{5} - \beta_{6} + \beta_{7} + \beta_{9} - 2 \beta_{10} - 2 \beta_{11} - 2 \beta_{12} + \beta_{13} + 2 \beta_{15} ) q^{69} + ( -2 + 3 \beta_{2} + 2 \beta_{4} - 3 \beta_{5} - 3 \beta_{9} ) q^{71} + ( \beta_{4} + \beta_{6} - 4 \beta_{8} + \beta_{10} + 2 \beta_{11} + 3 \beta_{14} + 3 \beta_{15} ) q^{73} + ( -\beta_{1} - \beta_{2} - 4 \beta_{3} + 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} + 2 \beta_{9} + \beta_{10} + \beta_{12} - \beta_{14} + \beta_{15} ) q^{77} + ( -2 + 4 \beta_{6} + 2 \beta_{7} + 2 \beta_{9} + 4 \beta_{10} ) q^{79} - q^{81} + ( 3 \beta_{4} + 3 \beta_{6} - 4 \beta_{8} + 3 \beta_{10} + 4 \beta_{11} + 2 \beta_{13} ) q^{83} + ( -2 + 2 \beta_{3} + \beta_{4} + 4 \beta_{5} + \beta_{6} + 2 \beta_{7} + 2 \beta_{9} - \beta_{10} - 2 \beta_{14} + 2 \beta_{15} ) q^{87} + ( -2 - 3 \beta_{4} + 4 \beta_{5} - 3 \beta_{6} + 2 \beta_{7} + 3 \beta_{8} + 2 \beta_{9} - 3 \beta_{10} - 3 \beta_{11} - 3 \beta_{12} + 4 \beta_{15} ) q^{89} + ( 4 \beta_{2} + 2 \beta_{3} - \beta_{4} + 3 \beta_{8} - \beta_{10} - \beta_{11} + \beta_{12} - \beta_{13} - 4 \beta_{14} ) q^{91} + ( 1 - \beta_{1} + \beta_{2} - \beta_{4} - \beta_{5} - \beta_{6} - 2 \beta_{7} - \beta_{10} ) q^{93} + ( -5 + 2 \beta_{3} + \beta_{4} + 10 \beta_{5} + \beta_{6} + 5 \beta_{7} + 5 \beta_{9} - 3 \beta_{10} - 2 \beta_{12} - 5 \beta_{14} + 5 \beta_{15} ) q^{97} + ( -\beta_{1} - \beta_{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + O(q^{10}) \) \( 16q + 16q^{11} + 8q^{21} - 8q^{23} - 16q^{37} - 48q^{43} - 16q^{51} + 40q^{53} + 8q^{57} + 48q^{67} - 32q^{71} + 24q^{77} - 16q^{81} + 32q^{91} + 8q^{93} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 8 x^{14} + 8 x^{12} - 8 x^{10} + 212 x^{8} + 248 x^{6} + 368 x^{4} + 32 x^{2} + 100\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 168 \nu^{14} + 313 \nu^{12} - 16892 \nu^{10} + 53080 \nu^{8} - 40938 \nu^{6} + 450802 \nu^{4} - 403288 \nu^{2} + 148090 \)\()/255270\)
\(\beta_{2}\)\(=\)\((\)\( -926 \nu^{14} + 35451 \nu^{12} - 243201 \nu^{10} + 298601 \nu^{8} - 346280 \nu^{6} + 5518342 \nu^{4} + 5075426 \nu^{2} + 5183600 \)\()/1276350\)
\(\beta_{3}\)\(=\)\((\)\( -570 \nu^{15} + 4712 \nu^{13} - 4682 \nu^{11} - 3835 \nu^{9} - 108472 \nu^{7} - 107896 \nu^{5} + 14152 \nu^{3} + 194682 \nu \)\()/170180\)
\(\beta_{4}\)\(=\)\((\)\( -5309 \nu^{14} + 52964 \nu^{12} - 133744 \nu^{10} + 152449 \nu^{8} - 925640 \nu^{6} + 42048 \nu^{4} - 235896 \nu^{2} - 1778150 \)\()/1276350\)
\(\beta_{5}\)\(=\)\((\)\( 2329 \nu^{14} - 16984 \nu^{12} + 3874 \nu^{10} + 8131 \nu^{8} + 466620 \nu^{6} + 946682 \nu^{4} + 792956 \nu^{2} + 235200 \)\()/425450\)
\(\beta_{6}\)\(=\)\((\)\(-11316 \nu^{15} - 8064 \nu^{14} + 98651 \nu^{13} + 44539 \nu^{12} - 144631 \nu^{11} + 130096 \nu^{10} + 68966 \nu^{9} - 369536 \nu^{8} - 2351580 \nu^{7} - 1404540 \nu^{6} - 1175218 \nu^{5} - 5420342 \nu^{4} + 166486 \nu^{3} - 2935756 \nu^{2} + 4605860 \nu + 379600\)\()/2552700\)
\(\beta_{7}\)\(=\)\((\)\( 7589 \nu^{14} - 54794 \nu^{12} + 7819 \nu^{10} + 33071 \nu^{8} + 1427600 \nu^{6} + 3316632 \nu^{4} + 3429726 \nu^{2} + 3909500 \)\()/1276350\)
\(\beta_{8}\)\(=\)\((\)\( 7803 \nu^{15} - 62803 \nu^{13} + 63288 \nu^{11} - 43563 \nu^{9} + 1587570 \nu^{7} + 2005314 \nu^{5} + 2316332 \nu^{3} + 415970 \nu \)\()/850900\)
\(\beta_{9}\)\(=\)\((\)\( 12236 \nu^{14} - 94911 \nu^{12} + 60231 \nu^{10} + 17089 \nu^{8} + 2540690 \nu^{6} + 3712778 \nu^{4} + 3670474 \nu^{2} - 942350 \)\()/1276350\)
\(\beta_{10}\)\(=\)\((\)\(11316 \nu^{15} - 18682 \nu^{14} - 98651 \nu^{13} + 150467 \nu^{12} + 144631 \nu^{11} - 137392 \nu^{10} - 68966 \nu^{9} - 64638 \nu^{8} + 2351580 \nu^{7} - 3255820 \nu^{6} + 1175218 \nu^{5} - 5336246 \nu^{4} - 166486 \nu^{3} - 3407548 \nu^{2} - 4605860 \nu - 3176700\)\()/2552700\)
\(\beta_{11}\)\(=\)\((\)\(-16334 \nu^{15} + 18682 \nu^{14} + 101559 \nu^{13} - 150467 \nu^{12} + 102621 \nu^{11} + 137392 \nu^{10} - 54961 \nu^{9} + 64638 \nu^{8} - 3695480 \nu^{7} + 3255820 \nu^{6} - 9322322 \nu^{5} + 5336246 \nu^{4} - 13804966 \nu^{3} + 3407548 \nu^{2} - 1360210 \nu + 3176700\)\()/2552700\)
\(\beta_{12}\)\(=\)\((\)\(-23638 \nu^{15} + 18682 \nu^{14} + 230578 \nu^{13} - 150467 \nu^{12} - 532523 \nu^{11} + 137392 \nu^{10} + 615878 \nu^{9} + 64638 \nu^{8} - 5443240 \nu^{7} + 3255820 \nu^{6} + 2884356 \nu^{5} + 5336246 \nu^{4} - 445002 \nu^{3} + 3407548 \nu^{2} + 11472260 \nu + 3176700\)\()/2552700\)
\(\beta_{13}\)\(=\)\((\)\(-29273 \nu^{15} + 18682 \nu^{14} + 228943 \nu^{13} - 150467 \nu^{12} - 185753 \nu^{11} + 137392 \nu^{10} + 126018 \nu^{9} + 64638 \nu^{8} - 6059090 \nu^{7} + 3255820 \nu^{6} - 8361154 \nu^{5} + 5336246 \nu^{4} - 9991622 \nu^{3} + 3407548 \nu^{2} - 1660020 \nu + 3176700\)\()/2552700\)
\(\beta_{14}\)\(=\)\((\)\(15299 \nu^{15} + 33799 \nu^{14} - 127039 \nu^{13} - 251609 \nu^{12} + 162509 \nu^{11} + 91294 \nu^{10} - 174804 \nu^{9} + 98946 \nu^{8} + 3259370 \nu^{7} + 6768010 \nu^{6} + 2681062 \nu^{5} + 12709502 \nu^{4} + 5233886 \nu^{3} + 11857936 \nu^{2} + 2801520 \nu + 3102000\)\()/2552700\)
\(\beta_{15}\)\(=\)\((\)\(25290 \nu^{15} - 33799 \nu^{14} - 200555 \nu^{13} + 251609 \nu^{12} + 167875 \nu^{11} - 91294 \nu^{10} - 20180 \nu^{9} - 98946 \nu^{8} + 5151300 \nu^{7} - 6768010 \nu^{6} + 6855310 \nu^{5} - 12709502 \nu^{4} + 4591250 \nu^{3} - 11857936 \nu^{2} - 641960 \nu - 3102000\)\()/2552700\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{15} + \beta_{14} + \beta_{13} + \beta_{6} + \beta_{4}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{10} + \beta_{9} - 2 \beta_{7} - \beta_{6} - 2 \beta_{5} - \beta_{4} + \beta_{2} - \beta_{1} + 2\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(3 \beta_{13} - \beta_{12} - \beta_{11} - 2 \beta_{10} + 3 \beta_{8} + 3 \beta_{6} + 3 \beta_{4} - 4 \beta_{3}\)\()/2\)
\(\nu^{4}\)\(=\)\(2 \beta_{9} - 6 \beta_{7} + \beta_{5} - 3 \beta_{4} + 2 \beta_{2} + 7\)
\(\nu^{5}\)\(=\)\((\)\(14 \beta_{15} - 12 \beta_{14} + 12 \beta_{13} - 8 \beta_{12} - 2 \beta_{11} - 17 \beta_{10} + 13 \beta_{9} + 12 \beta_{8} + 13 \beta_{7} + 19 \beta_{6} + 26 \beta_{5} + 19 \beta_{4} - 10 \beta_{3} - 13\)\()/2\)
\(\nu^{6}\)\(=\)\(-4 \beta_{10} + 7 \beta_{9} - 30 \beta_{7} - 4 \beta_{6} + 8 \beta_{5} - 8 \beta_{4} + 7 \beta_{2} - 4 \beta_{1} + 51\)
\(\nu^{7}\)\(=\)\(41 \beta_{15} - 19 \beta_{14} + 24 \beta_{13} - 23 \beta_{12} - 6 \beta_{11} - 54 \beta_{10} + 30 \beta_{9} + 7 \beta_{8} + 30 \beta_{7} + 49 \beta_{6} + 60 \beta_{5} + 49 \beta_{4} - 37 \beta_{3} - 30\)
\(\nu^{8}\)\(=\)\(-50 \beta_{10} + 16 \beta_{9} - 178 \beta_{7} - 50 \beta_{6} + 40 \beta_{5} - 52 \beta_{4} + 42 \beta_{2} - 36 \beta_{1} + 258\)
\(\nu^{9}\)\(=\)\(155 \beta_{15} - 141 \beta_{14} + 25 \beta_{13} - 122 \beta_{12} - 6 \beta_{11} - 340 \beta_{10} + 148 \beta_{9} + 14 \beta_{8} + 148 \beta_{7} + 237 \beta_{6} + 296 \beta_{5} + 237 \beta_{4} - 256 \beta_{3} - 148\)
\(\nu^{10}\)\(=\)\(-279 \beta_{10} - 29 \beta_{9} - 980 \beta_{7} - 279 \beta_{6} + 422 \beta_{5} - 281 \beta_{4} + 211 \beta_{2} - 219 \beta_{1} + 1320\)
\(\nu^{11}\)\(=\)\(768 \beta_{15} - 1056 \beta_{14} - 421 \beta_{13} - 709 \beta_{12} + 99 \beta_{11} - 2148 \beta_{10} + 912 \beta_{9} - 205 \beta_{8} + 912 \beta_{7} + 1117 \beta_{6} + 1824 \beta_{5} + 1117 \beta_{4} - 1288 \beta_{3} - 912\)
\(\nu^{12}\)\(=\)\(-1752 \beta_{10} - 878 \beta_{9} - 5128 \beta_{7} - 1752 \beta_{6} + 3166 \beta_{5} - 1154 \beta_{4} + 914 \beta_{2} - 1394 \beta_{1} + 6878\)
\(\nu^{13}\)\(=\)\(3562 \beta_{15} - 6356 \beta_{14} - 5216 \beta_{13} - 3856 \beta_{12} + 1118 \beta_{11} - 12839 \beta_{10} + 4959 \beta_{9} - 3304 \beta_{8} + 4959 \beta_{7} + 4885 \beta_{6} + 9918 \beta_{5} + 4885 \beta_{4} - 6958 \beta_{3} - 4959\)
\(\nu^{14}\)\(=\)\(-11166 \beta_{10} - 8796 \beta_{9} - 26354 \beta_{7} - 11166 \beta_{6} + 20760 \beta_{5} - 4774 \beta_{4} + 3782 \beta_{2} - 8704 \beta_{1} + 33556\)
\(\nu^{15}\)\(=\)\(13138 \beta_{15} - 37998 \beta_{14} - 45324 \beta_{13} - 20110 \beta_{12} + 9788 \beta_{11} - 74344 \beta_{10} + 25568 \beta_{9} - 28154 \beta_{8} + 25568 \beta_{7} + 18698 \beta_{6} + 51136 \beta_{5} + 18698 \beta_{4} - 37578 \beta_{3} - 25568\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2100\mathbb{Z}\right)^\times\).

\(n\) \(701\) \(1051\) \(1177\) \(1501\)
\(\chi(n)\) \(1\) \(1\) \(-\beta_{5}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1357.1
0.919224 + 1.30296i
0.550947 0.483398i
−0.522506 1.01508i
−2.36188 + 0.195512i
2.36188 0.195512i
−0.919224 1.30296i
−0.550947 + 0.483398i
0.522506 + 1.01508i
0.919224 1.30296i
0.550947 + 0.483398i
−0.522506 + 1.01508i
−2.36188 0.195512i
2.36188 + 0.195512i
−0.919224 + 1.30296i
−0.550947 0.483398i
0.522506 1.01508i
0 −0.707107 0.707107i 0 0 0 −2.16604 1.51930i 0 1.00000i 0
1357.2 0 −0.707107 0.707107i 0 0 0 −1.05782 + 2.42508i 0 1.00000i 0
1357.3 0 −0.707107 0.707107i 0 0 0 0.235858 + 2.63522i 0 1.00000i 0
1357.4 0 −0.707107 0.707107i 0 0 0 1.57379 2.12678i 0 1.00000i 0
1357.5 0 0.707107 + 0.707107i 0 0 0 −2.12678 + 1.57379i 0 1.00000i 0
1357.6 0 0.707107 + 0.707107i 0 0 0 −1.51930 2.16604i 0 1.00000i 0
1357.7 0 0.707107 + 0.707107i 0 0 0 2.42508 1.05782i 0 1.00000i 0
1357.8 0 0.707107 + 0.707107i 0 0 0 2.63522 + 0.235858i 0 1.00000i 0
1693.1 0 −0.707107 + 0.707107i 0 0 0 −2.16604 + 1.51930i 0 1.00000i 0
1693.2 0 −0.707107 + 0.707107i 0 0 0 −1.05782 2.42508i 0 1.00000i 0
1693.3 0 −0.707107 + 0.707107i 0 0 0 0.235858 2.63522i 0 1.00000i 0
1693.4 0 −0.707107 + 0.707107i 0 0 0 1.57379 + 2.12678i 0 1.00000i 0
1693.5 0 0.707107 0.707107i 0 0 0 −2.12678 1.57379i 0 1.00000i 0
1693.6 0 0.707107 0.707107i 0 0 0 −1.51930 + 2.16604i 0 1.00000i 0
1693.7 0 0.707107 0.707107i 0 0 0 2.42508 + 1.05782i 0 1.00000i 0
1693.8 0 0.707107 0.707107i 0 0 0 2.63522 0.235858i 0 1.00000i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1693.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
7.b odd 2 1 inner
35.f even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2100.2.x.d 16
5.b even 2 1 420.2.x.a 16
5.c odd 4 1 420.2.x.a 16
5.c odd 4 1 inner 2100.2.x.d 16
7.b odd 2 1 inner 2100.2.x.d 16
15.d odd 2 1 1260.2.ba.b 16
15.e even 4 1 1260.2.ba.b 16
20.d odd 2 1 1680.2.cz.c 16
20.e even 4 1 1680.2.cz.c 16
35.c odd 2 1 420.2.x.a 16
35.f even 4 1 420.2.x.a 16
35.f even 4 1 inner 2100.2.x.d 16
105.g even 2 1 1260.2.ba.b 16
105.k odd 4 1 1260.2.ba.b 16
140.c even 2 1 1680.2.cz.c 16
140.j odd 4 1 1680.2.cz.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
420.2.x.a 16 5.b even 2 1
420.2.x.a 16 5.c odd 4 1
420.2.x.a 16 35.c odd 2 1
420.2.x.a 16 35.f even 4 1
1260.2.ba.b 16 15.d odd 2 1
1260.2.ba.b 16 15.e even 4 1
1260.2.ba.b 16 105.g even 2 1
1260.2.ba.b 16 105.k odd 4 1
1680.2.cz.c 16 20.d odd 2 1
1680.2.cz.c 16 20.e even 4 1
1680.2.cz.c 16 140.c even 2 1
1680.2.cz.c 16 140.j odd 4 1
2100.2.x.d 16 1.a even 1 1 trivial
2100.2.x.d 16 5.c odd 4 1 inner
2100.2.x.d 16 7.b odd 2 1 inner
2100.2.x.d 16 35.f even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{4} - 4 T_{11}^{3} - 8 T_{11}^{2} + 16 T_{11} - 4 \) acting on \(S_{2}^{\mathrm{new}}(2100, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( ( 1 + T^{4} )^{4} \)
$5$ 1
$7$ \( 1 - 32 T^{3} + 36 T^{4} - 96 T^{5} + 512 T^{6} - 1216 T^{7} + 3334 T^{8} - 8512 T^{9} + 25088 T^{10} - 32928 T^{11} + 86436 T^{12} - 537824 T^{13} + 5764801 T^{16} \)
$11$ \( ( 1 - 4 T + 36 T^{2} - 116 T^{3} + 546 T^{4} - 1276 T^{5} + 4356 T^{6} - 5324 T^{7} + 14641 T^{8} )^{4} \)
$13$ \( 1 - 536 T^{4} + 97308 T^{8} - 1310248 T^{12} - 1567451002 T^{16} - 37421993128 T^{20} + 79377124999068 T^{24} - 12487773625649816 T^{28} + 665416609183179841 T^{32} \)
$17$ \( 1 - 904 T^{4} + 210332 T^{8} + 67692360 T^{12} - 44568692154 T^{16} + 5653733599560 T^{20} + 1467225014080412 T^{24} - 526690502455703944 T^{28} + 48661191875666868481 T^{32} \)
$19$ \( ( 1 + 120 T^{2} + 6692 T^{4} + 228264 T^{6} + 5227942 T^{8} + 82403304 T^{10} + 872108132 T^{12} + 5645505720 T^{14} + 16983563041 T^{16} )^{2} \)
$23$ \( ( 1 + 4 T + 8 T^{2} - 4 T^{3} - 1548 T^{4} - 3956 T^{5} - 3432 T^{6} + 53556 T^{7} + 1130086 T^{8} + 1231788 T^{9} - 1815528 T^{10} - 48132652 T^{11} - 433193868 T^{12} - 25745372 T^{13} + 1184287112 T^{14} + 13619301788 T^{15} + 78310985281 T^{16} )^{2} \)
$29$ \( ( 1 - 136 T^{2} + 8924 T^{4} - 381688 T^{6} + 12377958 T^{8} - 320999608 T^{10} + 6311775644 T^{12} - 80895971656 T^{14} + 500246412961 T^{16} )^{2} \)
$31$ \( ( 1 - 136 T^{2} + 8612 T^{4} - 354840 T^{6} + 11712390 T^{8} - 341001240 T^{10} + 7953362852 T^{12} - 120700500616 T^{14} + 852891037441 T^{16} )^{2} \)
$37$ \( ( 1 + 8 T + 32 T^{2} + 56 T^{3} - 2980 T^{4} - 15992 T^{5} - 31008 T^{6} + 263416 T^{7} + 5332518 T^{8} + 9746392 T^{9} - 42449952 T^{10} - 810042776 T^{11} - 5584999780 T^{12} + 3883261592 T^{13} + 82103245088 T^{14} + 759455017064 T^{15} + 3512479453921 T^{16} )^{2} \)
$41$ \( ( 1 - 200 T^{2} + 20668 T^{4} - 1400376 T^{6} + 67532294 T^{8} - 2354032056 T^{10} + 58402828348 T^{12} - 950020848200 T^{14} + 7984925229121 T^{16} )^{2} \)
$43$ \( ( 1 + 24 T + 288 T^{2} + 2504 T^{3} + 19620 T^{4} + 145992 T^{5} + 988256 T^{6} + 5831128 T^{7} + 34926502 T^{8} + 250738504 T^{9} + 1827285344 T^{10} + 11607385944 T^{11} + 67076875620 T^{12} + 368109141272 T^{13} + 1820552558112 T^{14} + 6523646666568 T^{15} + 11688200277601 T^{16} )^{2} \)
$47$ \( 1 + 2312 T^{4} - 7152868 T^{8} - 8327985864 T^{12} + 35781048692550 T^{16} - 40637914388829384 T^{20} - \)\(17\!\cdots\!48\)\( T^{24} + \)\(26\!\cdots\!92\)\( T^{28} + \)\(56\!\cdots\!21\)\( T^{32} \)
$53$ \( ( 1 - 20 T + 200 T^{2} - 1284 T^{3} + 2996 T^{4} + 21028 T^{5} - 195432 T^{6} + 604308 T^{7} - 304634 T^{8} + 32028324 T^{9} - 548968488 T^{10} + 3130585556 T^{11} + 23639881076 T^{12} - 536963013012 T^{13} + 4432872225800 T^{14} - 23494222796740 T^{15} + 62259690411361 T^{16} )^{2} \)
$59$ \( ( 1 + 280 T^{2} + 33212 T^{4} + 2343784 T^{6} + 136126182 T^{8} + 8158712104 T^{10} + 402441793532 T^{12} + 11810549419480 T^{14} + 146830437604321 T^{16} )^{2} \)
$61$ \( ( 1 - 312 T^{2} + 47324 T^{4} - 4644168 T^{6} + 328520806 T^{8} - 17280949128 T^{10} + 655240579484 T^{12} - 16074356800632 T^{14} + 191707312997281 T^{16} )^{2} \)
$67$ \( ( 1 - 24 T + 288 T^{2} - 2984 T^{3} + 22788 T^{4} - 56616 T^{5} - 752032 T^{6} + 14609960 T^{7} - 156156506 T^{8} + 978867320 T^{9} - 3375871648 T^{10} - 17027998008 T^{11} + 459203745348 T^{12} - 4028773319288 T^{13} + 26052014064672 T^{14} - 145457078527752 T^{15} + 406067677556641 T^{16} )^{2} \)
$71$ \( ( 1 + 8 T + 140 T^{2} + 1024 T^{3} + 14962 T^{4} + 72704 T^{5} + 705740 T^{6} + 2863288 T^{7} + 25411681 T^{8} )^{4} \)
$73$ \( 1 - 9816 T^{4} + 48230108 T^{8} - 385344995944 T^{12} + 2907384584774982 T^{16} - 10943120062961734504 T^{20} + \)\(38\!\cdots\!48\)\( T^{24} - \)\(22\!\cdots\!36\)\( T^{28} + \)\(65\!\cdots\!61\)\( T^{32} \)
$79$ \( ( 1 - 248 T^{2} + 39836 T^{4} - 4744648 T^{6} + 419330758 T^{8} - 29611348168 T^{10} + 1551615426716 T^{12} - 60285688969208 T^{14} + 1517108809906561 T^{16} )^{2} \)
$83$ \( 1 + 11592 T^{4} + 91463132 T^{8} + 885997919352 T^{12} + 6666213559678726 T^{16} + 42047973661939327992 T^{20} + \)\(20\!\cdots\!12\)\( T^{24} + \)\(12\!\cdots\!12\)\( T^{28} + \)\(50\!\cdots\!81\)\( T^{32} \)
$89$ \( ( 1 + 328 T^{2} + 42716 T^{4} + 2964280 T^{6} + 188103750 T^{8} + 23480061880 T^{10} + 2680097566556 T^{12} + 163009863435208 T^{14} + 3936588805702081 T^{16} )^{2} \)
$97$ \( 1 - 1624 T^{4} + 150045276 T^{8} - 609733450088 T^{12} + 17981575892543942 T^{16} - 53979263937940026728 T^{20} + \)\(11\!\cdots\!36\)\( T^{24} - \)\(11\!\cdots\!84\)\( T^{28} + \)\(61\!\cdots\!21\)\( T^{32} \)
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