Properties

Label 1078.2.c.b
Level $1078$
Weight $2$
Character orbit 1078.c
Analytic conductor $8.608$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1078 = 2 \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1078.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(8.60787333789\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \( x^{16} - 6 x^{15} + 18 x^{14} - 36 x^{13} + 34 x^{12} + 18 x^{11} - 72 x^{10} + 132 x^{9} - 93 x^{8} - 102 x^{7} + 144 x^{6} - 432 x^{5} + 502 x^{4} + 288 x^{3} + 72 x^{2} + 12 x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{12} \)
Twist minimal: no (minimal twist has level 154)
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.

\(f(q)\) \(=\) \( q + \beta_{9} q^{2} + \beta_{14} q^{3} - q^{4} + (\beta_{11} - \beta_{8}) q^{5} - \beta_{13} q^{6} - \beta_{9} q^{8} + ( - \beta_{2} - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{9} q^{2} + \beta_{14} q^{3} - q^{4} + (\beta_{11} - \beta_{8}) q^{5} - \beta_{13} q^{6} - \beta_{9} q^{8} + ( - \beta_{2} - 2) q^{9} + (\beta_{10} + \beta_{6}) q^{10} + ( - \beta_{12} + \beta_{9} - \beta_{3} - 1) q^{11} - \beta_{14} q^{12} + ( - \beta_{10} - \beta_{4}) q^{13} + ( - 3 \beta_1 - 2) q^{15} + q^{16} + ( - \beta_{13} - 2 \beta_{10} - \beta_{6} + 2 \beta_{4}) q^{17} + ( - \beta_{15} - 2 \beta_{9}) q^{18} + ( - \beta_{13} + \beta_{10} + 2 \beta_{6} - \beta_{4}) q^{19} + ( - \beta_{11} + \beta_{8}) q^{20} + ( - \beta_{9} + \beta_{7} + \beta_1) q^{22} + ( - \beta_{3} + \beta_{2} - 2) q^{23} + \beta_{13} q^{24} + ( - \beta_{2} - 2 \beta_1 - 1) q^{25} + (\beta_{11} + \beta_{5}) q^{26} + ( - \beta_{14} + 4 \beta_{11} - 2 \beta_{8} - 3 \beta_{5}) q^{27} + (\beta_{12} - 2 \beta_{9} + \beta_{7}) q^{29} + ( - 3 \beta_{12} + \beta_{9}) q^{30} + ( - \beta_{14} + 2 \beta_{11} - 3 \beta_{8} - 2 \beta_{5}) q^{31} + \beta_{9} q^{32} + ( - \beta_{14} - \beta_{13} + \beta_{11} - 3 \beta_{10} + \beta_{8} - 2 \beta_{6} - \beta_{5}) q^{33} + ( - \beta_{14} + 2 \beta_{11} - \beta_{8} - 2 \beta_{5}) q^{34} + (\beta_{2} + 2) q^{36} + ( - \beta_{3} + 2) q^{37} + ( - \beta_{14} - \beta_{11} + 2 \beta_{8} + \beta_{5}) q^{38} + (\beta_{15} + 4 \beta_{12} - \beta_{9} + \beta_{7}) q^{39} + ( - \beta_{10} - \beta_{6}) q^{40} + ( - 3 \beta_{13} + \beta_{10} - 2 \beta_{4}) q^{41} + (\beta_{15} + 3 \beta_{12} + \beta_{9} + 3 \beta_{7}) q^{43} + (\beta_{12} - \beta_{9} + \beta_{3} + 1) q^{44} + (\beta_{14} - 6 \beta_{11} + 3 \beta_{8}) q^{45} + (\beta_{15} - 2 \beta_{9} + \beta_{7}) q^{46} - 2 \beta_{5} q^{47} + \beta_{14} q^{48} + ( - \beta_{15} - 2 \beta_{12} + \beta_{9}) q^{50} + ( - 3 \beta_{15} + \beta_{12} - 7 \beta_{9} + \beta_{7}) q^{51} + (\beta_{10} + \beta_{4}) q^{52} + ( - 2 \beta_{3} + \beta_{2} + \beta_1 + 4) q^{53} + (\beta_{13} + 4 \beta_{10} + 2 \beta_{6} - 3 \beta_{4}) q^{54} + ( - \beta_{14} + 2 \beta_{13} - 3 \beta_{11} + 2 \beta_{8} + \beta_{5} - \beta_{4}) q^{55} + ( - 2 \beta_{12} - 4 \beta_{9} + \beta_{7}) q^{57} + (\beta_{3} - \beta_1 + 1) q^{58} + (3 \beta_{14} + 2 \beta_{11} + \beta_{5}) q^{59} + (3 \beta_1 + 2) q^{60} + ( - \beta_{13} - \beta_{10} - 3 \beta_{6} - 2 \beta_{4}) q^{61} + (\beta_{13} + 2 \beta_{10} + 3 \beta_{6} - 2 \beta_{4}) q^{62} - q^{64} + (2 \beta_{15} + 4 \beta_{12} + 2 \beta_{9} + \beta_{7}) q^{65} + ( - \beta_{14} + \beta_{13} + 3 \beta_{11} + \beta_{10} - 2 \beta_{8} - \beta_{6} - \beta_{4}) q^{66} + (2 \beta_{3} + \beta_1 - 1) q^{67} + (\beta_{13} + 2 \beta_{10} + \beta_{6} - 2 \beta_{4}) q^{68} + ( - 3 \beta_{11} + 3 \beta_{8} + 2 \beta_{5}) q^{69} + ( - 2 \beta_{3} - 2 \beta_{2} - \beta_1) q^{71} + (\beta_{15} + 2 \beta_{9}) q^{72} + ( - 5 \beta_{10} - \beta_{6} + 2 \beta_{4}) q^{73} + (2 \beta_{9} + \beta_{7}) q^{74} + ( - \beta_{14} - 2 \beta_{11} + 2 \beta_{8} - 3 \beta_{5}) q^{75} + (\beta_{13} - \beta_{10} - 2 \beta_{6} + \beta_{4}) q^{76} + (\beta_{3} - \beta_{2} - 4 \beta_1 - 3) q^{78} + (3 \beta_{15} + 2 \beta_{12} - 3 \beta_{9} + \beta_{7}) q^{79} + (\beta_{11} - \beta_{8}) q^{80} + (2 \beta_{3} + \beta_{2} - 4 \beta_1 - 1) q^{81} + ( - 3 \beta_{14} - \beta_{11} + 2 \beta_{5}) q^{82} + ( - 3 \beta_{13} + 2 \beta_{10} - \beta_{6} - 2 \beta_{4}) q^{83} + ( - 6 \beta_{12} + 4 \beta_{9} - 2 \beta_{7}) q^{85} + (3 \beta_{3} - \beta_{2} - 3 \beta_1 - 4) q^{86} + (2 \beta_{13} + 4 \beta_{10} + \beta_{6} - \beta_{4}) q^{87} + (\beta_{9} - \beta_{7} - \beta_1) q^{88} + (3 \beta_{14} - 4 \beta_{11} - \beta_{8} + \beta_{5}) q^{89} + ( - \beta_{13} - 6 \beta_{10} - 3 \beta_{6}) q^{90} + (\beta_{3} - \beta_{2} + 2) q^{92} + ( - \beta_{3} + 3 \beta_{2} - 3 \beta_1 + 4) q^{93} - 2 \beta_{4} q^{94} + ( - 3 \beta_{12} - 5 \beta_{9} + \beta_{7}) q^{95} - \beta_{13} q^{96} + ( - 2 \beta_{14} + 3 \beta_{11} - 2 \beta_{8} + 4 \beta_{5}) q^{97} + ( - \beta_{15} + 5 \beta_{12} - 5 \beta_{9} + \beta_{7} - \beta_{3} + 2 \beta_{2} + \beta_1 + 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 16 q^{4} - 32 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 16 q^{4} - 32 q^{9} - 16 q^{11} - 8 q^{15} + 16 q^{16} - 8 q^{22} - 32 q^{23} + 32 q^{36} + 32 q^{37} + 16 q^{44} + 56 q^{53} + 24 q^{58} + 8 q^{60} - 16 q^{64} - 24 q^{67} + 8 q^{71} - 16 q^{78} + 16 q^{81} - 40 q^{86} + 8 q^{88} + 32 q^{92} + 88 q^{93} + 56 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 6 x^{15} + 18 x^{14} - 36 x^{13} + 34 x^{12} + 18 x^{11} - 72 x^{10} + 132 x^{9} - 93 x^{8} - 102 x^{7} + 144 x^{6} - 432 x^{5} + 502 x^{4} + 288 x^{3} + 72 x^{2} + 12 x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 73568941 \nu^{15} - 2828429898 \nu^{14} + 10706428236 \nu^{13} - 20127265496 \nu^{12} + 35741021050 \nu^{11} + \cdots - 10982429182739 ) / 3707507912227 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 973484628 \nu^{15} + 16798064640 \nu^{14} - 57412057192 \nu^{13} + 106920971791 \nu^{12} - 159815035576 \nu^{11} + \cdots + 17707406591301 ) / 3707507912227 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 4604477275 \nu^{15} + 26065356324 \nu^{14} - 75543186530 \nu^{13} + 146897647868 \nu^{12} - 132924225826 \nu^{11} + \cdots + 1817177370827 ) / 3707507912227 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 275511061668 \nu^{15} + 1657750582996 \nu^{14} - 4984224043414 \nu^{13} + 9981450357924 \nu^{12} - 9469500994272 \nu^{11} + \cdots - 1555184012514 ) / 3707507912227 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 464283294072 \nu^{15} + 2897231896860 \nu^{14} - 9050241140038 \nu^{13} + 18865423115472 \nu^{12} - 20243277302808 \nu^{11} + \cdots - 1395125725490 ) / 3707507912227 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 476129408728 \nu^{15} - 2861932186210 \nu^{14} + 8584036352598 \nu^{13} - 17121557060007 \nu^{12} + 16021649705480 \nu^{11} + \cdots + 2687120456379 ) / 3707507912227 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 770100281810 \nu^{15} - 4733263012640 \nu^{14} + 14562049553051 \nu^{13} - 29920520825502 \nu^{12} + 30799143245438 \nu^{11} + \cdots + 4887214330899 ) / 3707507912227 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 805007695745 \nu^{15} + 5029032860754 \nu^{14} - 15724330059984 \nu^{13} + 32793740003767 \nu^{12} - 35212340782612 \nu^{11} + \cdots - 2408034273855 ) / 3707507912227 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 1616321538 \nu^{15} + 9938916556 \nu^{14} - 30594542475 \nu^{13} + 62871582510 \nu^{12} - 64714538406 \nu^{11} - 18641341380 \nu^{10} + \cdots - 10233974547 ) / 3810388399 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 854458 \nu^{15} - 5149288 \nu^{14} + 15489103 \nu^{13} - 31002870 \nu^{12} + 29353426 \nu^{11} + 15677868 \nu^{10} - 63062803 \nu^{9} + 114195864 \nu^{8} + \cdots + 4877403 ) / 1828267 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 1570148876 \nu^{15} - 9814610316 \nu^{14} + 30686832232 \nu^{13} - 63994473524 \nu^{12} + 68737309508 \nu^{11} + 12449599248 \nu^{10} + \cdots + 4714210963 ) / 1973128213 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 3087253540240 \nu^{15} - 18982048227910 \nu^{14} + 58429841322972 \nu^{13} - 120068214463500 \nu^{12} + \cdots + 19545417290592 ) / 3707507912227 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 3131859052620 \nu^{15} - 18867452624394 \nu^{14} + 56728752984694 \nu^{13} - 113478926578056 \nu^{12} + \cdots + 17856504555702 ) / 3707507912227 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 5329561831437 \nu^{15} + 33315128327418 \nu^{14} - 104171995568578 \nu^{13} + 217250160715424 \nu^{12} + \cdots - 15992012287118 ) / 3707507912227 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 7516121976528 \nu^{15} + 46209561787616 \nu^{14} - 142236306189628 \nu^{13} + 292274046112479 \nu^{12} + \cdots - 47582417802291 ) / 3707507912227 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{12} + \beta_{9} - \beta_{7} + \beta_{5} + \beta_{4} + \beta_{3} + \beta _1 + 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{13} + \beta_{12} + 2\beta_{10} + 2\beta_{9} + \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{14} - \beta_{13} - 2\beta_{11} + 2\beta_{10} + \beta_{8} + \beta_{6} - 3\beta_{5} + 3\beta_{4} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -5\beta_{14} - 9\beta_{11} + 3\beta_{8} - 5\beta_{5} + \beta_{2} + 6\beta _1 + 13 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 8 \beta_{15} - 8 \beta_{14} + 8 \beta_{13} + 35 \beta_{12} - 14 \beta_{11} - 14 \beta_{10} + 29 \beta_{9} + 8 \beta_{8} - 3 \beta_{7} - 8 \beta_{6} - 11 \beta_{5} - 11 \beta_{4} + 3 \beta_{3} + 8 \beta_{2} + 35 \beta _1 + 64 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 8\beta_{15} + 34\beta_{12} + 29\beta_{9} + \beta_{7} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 51 \beta_{15} - 49 \beta_{14} - 49 \beta_{13} + 196 \beta_{12} - 82 \beta_{11} + 82 \beta_{10} + 142 \beta_{9} + 51 \beta_{8} + 2 \beta_{7} + 51 \beta_{6} - 47 \beta_{5} + 47 \beta_{4} + 2 \beta_{3} - 51 \beta_{2} - 196 \beta _1 - 338 ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -148\beta_{14} - 245\beta_{11} + 150\beta_{8} - 118\beta_{5} + 10\beta_{3} - 50\beta_{2} - 188\beta _1 - 323 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 278 \beta_{14} + 278 \beta_{13} - 456 \beta_{11} - 456 \beta_{10} + 298 \beta_{8} - 298 \beta_{6} - 223 \beta_{5} - 223 \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 288 \beta_{15} + 809 \beta_{13} + 1027 \beta_{12} - 1320 \beta_{10} + 674 \beta_{9} + 70 \beta_{7} - 864 \beta_{6} - 599 \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 1673 \beta_{15} + 1533 \beta_{14} + 1533 \beta_{13} + 5864 \beta_{12} + 2490 \beta_{11} - 2490 \beta_{10} + 3716 \beta_{9} - 1673 \beta_{8} + 408 \beta_{7} - 1673 \beta_{6} + 1125 \beta_{5} - 1125 \beta_{4} + \cdots - 9580 ) / 4 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 428\beta_{3} - 1603\beta_{2} - 5576\beta _1 - 9071 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 9210 \beta_{15} - 8354 \beta_{14} + 8354 \beta_{13} - 31783 \beta_{12} - 13502 \beta_{11} - 13502 \beta_{10} - 19597 \beta_{9} + 9210 \beta_{8} - 2489 \beta_{7} - 9210 \beta_{6} - 5865 \beta_{5} + \cdots - 51380 ) / 4 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 8782 \beta_{15} + 23857 \beta_{13} - 30180 \beta_{12} - 38504 \beta_{10} - 18481 \beta_{9} - 2459 \beta_{7} - 26346 \beta_{6} - 16480 \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 45285 \beta_{14} + 45285 \beta_{13} + 73006 \beta_{11} - 73006 \beta_{10} - 50203 \beta_{8} - 50203 \beta_{6} + 31097 \beta_{5} - 31097 \beta_{4} ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1078\mathbb{Z}\right)^\times\).

\(n\) \(199\) \(981\)
\(\chi(n)\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1077.1
−0.186243 + 0.0499037i
0.601150 2.24352i
−0.347596 + 1.29724i
1.60599 0.430324i
0.430324 1.60599i
−1.29724 + 0.347596i
2.24352 0.601150i
−0.0499037 + 0.186243i
−0.0499037 0.186243i
2.24352 + 0.601150i
−1.29724 0.347596i
0.430324 + 1.60599i
1.60599 + 0.430324i
−0.347596 1.29724i
0.601150 + 2.24352i
−0.186243 0.0499037i
1.00000i 3.12703i −1.00000 2.20288i −3.12703 0 1.00000i −6.77832 2.20288
1077.2 1.00000i 2.59518i −1.00000 3.53900i −2.59518 0 1.00000i −3.73495 −3.53900
1077.3 1.00000i 1.55924i −1.00000 1.01909i −1.55924 0 1.00000i 0.568783 1.01909
1077.4 1.00000i 1.02738i −1.00000 1.25868i −1.02738 0 1.00000i 1.94448 −1.25868
1077.5 1.00000i 1.02738i −1.00000 1.25868i 1.02738 0 1.00000i 1.94448 1.25868
1077.6 1.00000i 1.55924i −1.00000 1.01909i 1.55924 0 1.00000i 0.568783 −1.01909
1077.7 1.00000i 2.59518i −1.00000 3.53900i 2.59518 0 1.00000i −3.73495 3.53900
1077.8 1.00000i 3.12703i −1.00000 2.20288i 3.12703 0 1.00000i −6.77832 −2.20288
1077.9 1.00000i 3.12703i −1.00000 2.20288i 3.12703 0 1.00000i −6.77832 −2.20288
1077.10 1.00000i 2.59518i −1.00000 3.53900i 2.59518 0 1.00000i −3.73495 3.53900
1077.11 1.00000i 1.55924i −1.00000 1.01909i 1.55924 0 1.00000i 0.568783 −1.01909
1077.12 1.00000i 1.02738i −1.00000 1.25868i 1.02738 0 1.00000i 1.94448 1.25868
1077.13 1.00000i 1.02738i −1.00000 1.25868i −1.02738 0 1.00000i 1.94448 −1.25868
1077.14 1.00000i 1.55924i −1.00000 1.01909i −1.55924 0 1.00000i 0.568783 1.01909
1077.15 1.00000i 2.59518i −1.00000 3.53900i −2.59518 0 1.00000i −3.73495 −3.53900
1077.16 1.00000i 3.12703i −1.00000 2.20288i −3.12703 0 1.00000i −6.77832 2.20288
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1077.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
11.b odd 2 1 inner
77.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1078.2.c.b 16
7.b odd 2 1 inner 1078.2.c.b 16
7.c even 3 1 154.2.i.a 16
7.c even 3 1 1078.2.i.c 16
7.d odd 6 1 154.2.i.a 16
7.d odd 6 1 1078.2.i.c 16
11.b odd 2 1 inner 1078.2.c.b 16
21.g even 6 1 1386.2.bk.c 16
21.h odd 6 1 1386.2.bk.c 16
28.f even 6 1 1232.2.bn.b 16
28.g odd 6 1 1232.2.bn.b 16
77.b even 2 1 inner 1078.2.c.b 16
77.h odd 6 1 154.2.i.a 16
77.h odd 6 1 1078.2.i.c 16
77.i even 6 1 154.2.i.a 16
77.i even 6 1 1078.2.i.c 16
231.k odd 6 1 1386.2.bk.c 16
231.l even 6 1 1386.2.bk.c 16
308.m odd 6 1 1232.2.bn.b 16
308.n even 6 1 1232.2.bn.b 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.2.i.a 16 7.c even 3 1
154.2.i.a 16 7.d odd 6 1
154.2.i.a 16 77.h odd 6 1
154.2.i.a 16 77.i even 6 1
1078.2.c.b 16 1.a even 1 1 trivial
1078.2.c.b 16 7.b odd 2 1 inner
1078.2.c.b 16 11.b odd 2 1 inner
1078.2.c.b 16 77.b even 2 1 inner
1078.2.i.c 16 7.c even 3 1
1078.2.i.c 16 7.d odd 6 1
1078.2.i.c 16 77.h odd 6 1
1078.2.i.c 16 77.i even 6 1
1232.2.bn.b 16 28.f even 6 1
1232.2.bn.b 16 28.g odd 6 1
1232.2.bn.b 16 308.m odd 6 1
1232.2.bn.b 16 308.n even 6 1
1386.2.bk.c 16 21.g even 6 1
1386.2.bk.c 16 21.h odd 6 1
1386.2.bk.c 16 231.k odd 6 1
1386.2.bk.c 16 231.l even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + 20T_{3}^{6} + 126T_{3}^{4} + 272T_{3}^{2} + 169 \) acting on \(S_{2}^{\mathrm{new}}(1078, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{8} \) Copy content Toggle raw display
$3$ \( (T^{8} + 20 T^{6} + 126 T^{4} + 272 T^{2} + \cdots + 169)^{2} \) Copy content Toggle raw display
$5$ \( (T^{8} + 20 T^{6} + 108 T^{4} + 188 T^{2} + \cdots + 100)^{2} \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( (T^{8} + 8 T^{7} + 20 T^{6} - 40 T^{5} + \cdots + 14641)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} - 44 T^{6} + 510 T^{4} - 908 T^{2} + \cdots + 25)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} - 92 T^{6} + 2352 T^{4} + \cdots + 78400)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} - 80 T^{6} + 1272 T^{4} - 164 T^{2} + \cdots + 4)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 8 T^{3} - 6 T^{2} - 34 T - 14)^{4} \) Copy content Toggle raw display
$29$ \( (T^{8} + 36 T^{6} + 366 T^{4} + 900 T^{2} + \cdots + 9)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + 164 T^{6} + 7200 T^{4} + \cdots + 3136)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 8 T^{3} + 12 T^{2} - 2 T - 2)^{4} \) Copy content Toggle raw display
$41$ \( (T^{8} - 200 T^{6} + 12588 T^{4} + \cdots + 1674436)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + 244 T^{6} + 19536 T^{4} + \cdots + 1201216)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + 80 T^{6} + 1920 T^{4} + \cdots + 4096)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} - 14 T^{3} + 24 T^{2} + 322 T - 1082)^{4} \) Copy content Toggle raw display
$59$ \( (T^{8} + 296 T^{6} + 19974 T^{4} + \cdots + 2356225)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} - 300 T^{6} + 26838 T^{4} + \cdots + 1306449)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 6 T^{3} - 54 T^{2} - 48 T + 417)^{4} \) Copy content Toggle raw display
$71$ \( (T^{4} - 2 T^{3} - 174 T^{2} - 338 T + 3262)^{4} \) Copy content Toggle raw display
$73$ \( (T^{8} - 204 T^{6} + 11868 T^{4} + \cdots + 125316)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + 400 T^{6} + 46182 T^{4} + \cdots + 33953929)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} - 260 T^{6} + 9888 T^{4} + \cdots + 107584)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + 456 T^{6} + 62088 T^{4} + \cdots + 49617936)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + 644 T^{6} + 145902 T^{4} + \cdots + 496532089)^{2} \) Copy content Toggle raw display
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