Properties

Label 1045.4.a.c
Level $1045$
Weight $4$
Character orbit 1045.a
Self dual yes
Analytic conductor $61.657$
Analytic rank $1$
Dimension $20$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1045 = 5 \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1045.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(61.6569959560\)
Analytic rank: \(1\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Defining polynomial: \( x^{20} - x^{19} - 105 x^{18} + 103 x^{17} + 4500 x^{16} - 4345 x^{15} - 101844 x^{14} + 95592 x^{13} + 1317797 x^{12} - 1160501 x^{11} - 9914845 x^{10} + 7570653 x^{9} + \cdots + 150528 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + \beta_{5} q^{3} + (\beta_{2} + 3) q^{4} - 5 q^{5} + (\beta_{6} + \beta_1 - 3) q^{6} + (\beta_{7} + 2) q^{7} + ( - \beta_{3} - 3 \beta_1 + 1) q^{8} + ( - \beta_{10} - \beta_{5} + 7) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + \beta_{5} q^{3} + (\beta_{2} + 3) q^{4} - 5 q^{5} + (\beta_{6} + \beta_1 - 3) q^{6} + (\beta_{7} + 2) q^{7} + ( - \beta_{3} - 3 \beta_1 + 1) q^{8} + ( - \beta_{10} - \beta_{5} + 7) q^{9} + 5 \beta_1 q^{10} - 11 q^{11} + (\beta_{17} + \beta_{10} - \beta_{7} - \beta_{6} + 3 \beta_{5} + \beta_{3} - \beta_{2} + 3 \beta_1 - 2) q^{12} + (\beta_{18} - \beta_{5} - \beta_1 + 3) q^{13} + ( - \beta_{12} - \beta_{7} - 2 \beta_{2} - 2 \beta_1 - 5) q^{14} - 5 \beta_{5} q^{15} + ( - \beta_{18} - \beta_{17} + \beta_{12} + \beta_{10} + \beta_{9} - 2 \beta_{6} + \beta_{3} + 3 \beta_{2} + \cdots + 15) q^{16}+ \cdots + (11 \beta_{10} + 11 \beta_{5} - 77) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - q^{2} - 8 q^{3} + 51 q^{4} - 100 q^{5} - 54 q^{6} + 49 q^{7} + 9 q^{8} + 146 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - q^{2} - 8 q^{3} + 51 q^{4} - 100 q^{5} - 54 q^{6} + 49 q^{7} + 9 q^{8} + 146 q^{9} + 5 q^{10} - 220 q^{11} - 59 q^{12} + 60 q^{13} - 89 q^{14} + 40 q^{15} + 275 q^{16} - 155 q^{17} + 45 q^{18} + 380 q^{19} - 255 q^{20} + 105 q^{21} + 11 q^{22} - 154 q^{23} - 397 q^{24} + 500 q^{25} + 176 q^{26} - 206 q^{27} + 155 q^{28} - 305 q^{29} + 270 q^{30} - 759 q^{31} - 254 q^{32} + 88 q^{33} - 565 q^{34} - 245 q^{35} + 705 q^{36} + 698 q^{37} - 19 q^{38} - 758 q^{39} - 45 q^{40} + 547 q^{41} + 106 q^{42} - 925 q^{43} - 561 q^{44} - 730 q^{45} - 254 q^{46} - 681 q^{47} - 540 q^{48} + 213 q^{49} - 25 q^{50} - 899 q^{51} + 889 q^{52} - 419 q^{53} - 2241 q^{54} + 1100 q^{55} - 2473 q^{56} - 152 q^{57} - 1440 q^{58} - 2829 q^{59} + 295 q^{60} - 959 q^{61} + 1575 q^{62} - 426 q^{63} + 93 q^{64} - 300 q^{65} + 594 q^{66} - 1020 q^{67} - 4218 q^{68} - 572 q^{69} + 445 q^{70} + 106 q^{71} + 210 q^{72} + 558 q^{73} - 3439 q^{74} - 200 q^{75} + 969 q^{76} - 539 q^{77} - 3599 q^{78} + 536 q^{79} - 1375 q^{80} - 2128 q^{81} - 1255 q^{82} - 4179 q^{83} - 2024 q^{84} + 775 q^{85} - 1119 q^{86} - 557 q^{87} - 99 q^{88} - 4120 q^{89} - 225 q^{90} - 111 q^{91} - 2831 q^{92} + 801 q^{93} + 1213 q^{94} - 1900 q^{95} - 6147 q^{96} + 1414 q^{97} - 7869 q^{98} - 1606 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - x^{19} - 105 x^{18} + 103 x^{17} + 4500 x^{16} - 4345 x^{15} - 101844 x^{14} + 95592 x^{13} + 1317797 x^{12} - 1160501 x^{11} - 9914845 x^{10} + 7570653 x^{9} + \cdots + 150528 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 11 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 19\nu + 1 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 17\!\cdots\!19 \nu^{19} + \cdots - 27\!\cdots\!72 ) / 54\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 33\!\cdots\!79 \nu^{19} + \cdots - 18\!\cdots\!84 ) / 37\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 48\!\cdots\!48 \nu^{19} + \cdots + 89\!\cdots\!08 ) / 53\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 18\!\cdots\!01 \nu^{19} + \cdots - 22\!\cdots\!32 ) / 10\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 19\!\cdots\!41 \nu^{19} + \cdots + 10\!\cdots\!48 ) / 10\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 19\!\cdots\!57 \nu^{19} + \cdots + 80\!\cdots\!48 ) / 10\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 22\!\cdots\!35 \nu^{19} + \cdots - 14\!\cdots\!28 ) / 97\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 27\!\cdots\!89 \nu^{19} + \cdots + 62\!\cdots\!84 ) / 10\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 51\!\cdots\!07 \nu^{19} + \cdots + 48\!\cdots\!36 ) / 13\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 15\!\cdots\!86 \nu^{19} + \cdots - 20\!\cdots\!96 ) / 27\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 69\!\cdots\!63 \nu^{19} + \cdots - 87\!\cdots\!92 ) / 10\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 89\!\cdots\!99 \nu^{19} + \cdots - 25\!\cdots\!64 ) / 10\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 11\!\cdots\!93 \nu^{19} + \cdots + 10\!\cdots\!72 ) / 13\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 12\!\cdots\!71 \nu^{19} + \cdots + 43\!\cdots\!48 ) / 10\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 37\!\cdots\!79 \nu^{19} + \cdots - 81\!\cdots\!76 ) / 27\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 51\!\cdots\!03 \nu^{19} + \cdots + 13\!\cdots\!12 ) / 27\!\cdots\!88 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 11 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 19\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{18} - \beta_{17} + \beta_{12} + \beta_{10} + \beta_{9} - 2\beta_{6} + \beta_{3} + 27\beta_{2} + 215 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 3 \beta_{19} - 3 \beta_{18} - 3 \beta_{17} + 3 \beta_{15} - \beta_{14} + 2 \beta_{13} + \beta_{12} + 6 \beta_{10} + 2 \beta_{9} + 2 \beta_{8} + \beta_{7} - 5 \beta_{6} + 12 \beta_{5} + 2 \beta_{4} + 38 \beta_{3} + 7 \beta_{2} + 422 \beta _1 - 12 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 46 \beta_{18} - 40 \beta_{17} - \beta_{16} + 4 \beta_{15} + 2 \beta_{14} + 4 \beta_{13} + 46 \beta_{12} + \beta_{11} + 51 \beta_{10} + 40 \beta_{9} - \beta_{8} + 2 \beta_{7} - 94 \beta_{6} + 23 \beta_{5} + 2 \beta_{4} + 50 \beta_{3} + 705 \beta_{2} + 43 \beta _1 + 4902 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 130 \beta_{19} - 159 \beta_{18} - 149 \beta_{17} - 6 \beta_{16} + 147 \beta_{15} - 46 \beta_{14} + 93 \beta_{13} + 59 \beta_{12} + 13 \beta_{11} + 313 \beta_{10} + 104 \beta_{9} + 95 \beta_{8} + 72 \beta_{7} - 301 \beta_{6} + 562 \beta_{5} + \cdots + 579 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 11 \beta_{19} - 1661 \beta_{18} - 1292 \beta_{17} - 54 \beta_{16} + 236 \beta_{15} + 84 \beta_{14} + 223 \beta_{13} + 1607 \beta_{12} + 87 \beta_{11} + 1982 \beta_{10} + 1308 \beta_{9} - 22 \beta_{8} + 95 \beta_{7} - 3408 \beta_{6} + \cdots + 120340 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 4271 \beta_{19} - 6297 \beta_{18} - 5567 \beta_{17} - 340 \beta_{16} + 5316 \beta_{15} - 1575 \beta_{14} + 3305 \beta_{13} + 2549 \beta_{12} + 795 \beta_{11} + 12086 \beta_{10} + 4017 \beta_{9} + 3441 \beta_{8} + \cdots + 45091 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 846 \beta_{19} - 55036 \beta_{18} - 39243 \beta_{17} - 2185 \beta_{16} + 9967 \beta_{15} + 2533 \beta_{14} + 8997 \beta_{13} + 50974 \beta_{12} + 4270 \beta_{11} + 69417 \beta_{10} + 40328 \beta_{9} + \cdots + 3094240 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 128035 \beta_{19} - 223257 \beta_{18} - 187549 \beta_{17} - 13615 \beta_{16} + 172773 \beta_{15} - 48817 \beta_{14} + 108006 \beta_{13} + 96255 \beta_{12} + 33885 \beta_{11} + 416453 \beta_{10} + \cdots + 2118774 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 44522 \beta_{19} - 1750196 \beta_{18} - 1168530 \beta_{17} - 79459 \beta_{16} + 370735 \beta_{15} + 66442 \beta_{14} + 320437 \beta_{13} + 1550768 \beta_{12} + 168694 \beta_{11} + \cdots + 82177470 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 3699520 \beta_{19} - 7480110 \beta_{18} - 6013437 \beta_{17} - 476114 \beta_{16} + 5353570 \beta_{15} - 1450529 \beta_{14} + 3410026 \beta_{13} + 3383330 \beta_{12} + 1248054 \beta_{11} + \cdots + 84526989 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 1979265 \beta_{19} - 54415322 \beta_{18} - 34629199 \beta_{17} - 2731437 \beta_{16} + 12932089 \beta_{15} + 1583656 \beta_{14} + 10731240 \beta_{13} + 46273786 \beta_{12} + \cdots + 2234952776 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 105330882 \beta_{19} - 242415793 \beta_{18} - 187736926 \beta_{17} - 15577910 \beta_{16} + 162009101 \beta_{15} - 42255623 \beta_{14} + 105779221 \beta_{13} + \cdots + 3105704128 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 79748010 \beta_{19} - 1669451069 \beta_{18} - 1027307523 \beta_{17} - 90564337 \beta_{16} + 434268389 \beta_{15} + 34046648 \beta_{14} + 347412829 \beta_{13} + \cdots + 61887295479 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 2985292839 \beta_{19} - 7692052264 \beta_{18} - 5772398627 \beta_{17} - 491796238 \beta_{16} + 4843493623 \beta_{15} - 1219174744 \beta_{14} + 3245700414 \beta_{13} + \cdots + 108549113772 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 3010521868 \beta_{19} - 50805181552 \beta_{18} - 30570060429 \beta_{17} - 2927121960 \beta_{16} + 14224935444 \beta_{15} + 618279097 \beta_{14} + \cdots + 1737872315738 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 84647898327 \beta_{19} - 240671770704 \beta_{18} - 175897812579 \beta_{17} - 15221997041 \beta_{16} + 143922218163 \beta_{15} - 35017992312 \beta_{14} + \cdots + 3668728755334 \) Copy content Toggle raw display

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
5.49185
4.63319
4.18815
3.91893
2.81607
2.35645
2.10393
2.01989
0.783853
0.0251982
−0.130697
−1.09490
−1.13531
−1.56126
−1.78835
−3.55600
−3.59432
−4.38362
−4.96274
−5.13031
−5.49185 3.98529 22.1605 −5.00000 −21.8866 23.4740 −77.7672 −11.1175 27.4593
1.2 −4.63319 −7.91113 13.4664 −5.00000 36.6537 11.8333 −25.3269 35.5860 23.1659
1.3 −4.18815 8.40510 9.54062 −5.00000 −35.2018 −15.6625 −6.45234 43.6456 20.9408
1.4 −3.91893 −3.32066 7.35803 −5.00000 13.0134 −28.1182 2.51584 −15.9732 19.5947
1.5 −2.81607 3.33662 −0.0697562 −5.00000 −9.39614 35.3821 22.7250 −15.8670 14.0803
1.6 −2.35645 −6.45217 −2.44713 −5.00000 15.2042 27.1536 24.6182 14.6305 11.7823
1.7 −2.10393 −1.35130 −3.57349 −5.00000 2.84303 −13.9401 24.3498 −25.1740 10.5196
1.8 −2.01989 7.09682 −3.92003 −5.00000 −14.3348 −8.46588 24.0772 23.3649 10.0995
1.9 −0.783853 0.0603595 −7.38557 −5.00000 −0.0473130 −1.09978 12.0600 −26.9964 3.91926
1.10 −0.0251982 −6.38249 −7.99937 −5.00000 0.160827 −27.7864 0.403155 13.7361 0.125991
1.11 0.130697 8.99140 −7.98292 −5.00000 1.17515 14.2539 −2.08892 53.8454 −0.653486
1.12 1.09490 3.72657 −6.80118 −5.00000 4.08024 3.04995 −16.2059 −13.1127 −5.47452
1.13 1.13531 −7.07177 −6.71106 −5.00000 −8.02868 20.7485 −16.7017 23.0099 −5.67657
1.14 1.56126 −1.47971 −5.56246 −5.00000 −2.31022 22.4493 −21.1746 −24.8104 −7.80632
1.15 1.78835 −7.45188 −4.80180 −5.00000 −13.3266 −20.9424 −22.8941 28.5305 −8.94175
1.16 3.55600 6.62353 4.64511 −5.00000 23.5532 −2.03165 −11.9300 16.8711 −17.7800
1.17 3.59432 4.02265 4.91912 −5.00000 14.4587 10.5448 −11.0737 −10.8183 −17.9716
1.18 4.38362 −4.80730 11.2161 −5.00000 −21.0733 18.1824 14.0981 −3.88988 −21.9181
1.19 4.96274 −9.59476 16.6288 −5.00000 −47.6163 −3.47238 42.8227 65.0593 −24.8137
1.20 5.13031 1.57482 18.3201 −5.00000 8.07930 −16.5525 52.9453 −24.5200 −25.6516
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.20
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(1\)
\(11\) \(1\)
\(19\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1045.4.a.c 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1045.4.a.c 20 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{20} + T_{2}^{19} - 105 T_{2}^{18} - 103 T_{2}^{17} + 4500 T_{2}^{16} + 4345 T_{2}^{15} - 101844 T_{2}^{14} - 95592 T_{2}^{13} + 1317797 T_{2}^{12} + 1160501 T_{2}^{11} - 9914845 T_{2}^{10} - 7570653 T_{2}^{9} + \cdots + 150528 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1045))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} + T^{19} - 105 T^{18} + \cdots + 150528 \) Copy content Toggle raw display
$3$ \( T^{20} + 8 T^{19} + \cdots + 353918232320 \) Copy content Toggle raw display
$5$ \( (T + 5)^{20} \) Copy content Toggle raw display
$7$ \( T^{20} - 49 T^{19} + \cdots + 40\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( (T + 11)^{20} \) Copy content Toggle raw display
$13$ \( T^{20} - 60 T^{19} + \cdots + 30\!\cdots\!92 \) Copy content Toggle raw display
$17$ \( T^{20} + 155 T^{19} + \cdots - 11\!\cdots\!56 \) Copy content Toggle raw display
$19$ \( (T - 19)^{20} \) Copy content Toggle raw display
$23$ \( T^{20} + 154 T^{19} + \cdots - 79\!\cdots\!36 \) Copy content Toggle raw display
$29$ \( T^{20} + 305 T^{19} + \cdots - 23\!\cdots\!04 \) Copy content Toggle raw display
$31$ \( T^{20} + 759 T^{19} + \cdots - 22\!\cdots\!16 \) Copy content Toggle raw display
$37$ \( T^{20} - 698 T^{19} + \cdots + 15\!\cdots\!52 \) Copy content Toggle raw display
$41$ \( T^{20} - 547 T^{19} + \cdots + 26\!\cdots\!80 \) Copy content Toggle raw display
$43$ \( T^{20} + 925 T^{19} + \cdots + 57\!\cdots\!88 \) Copy content Toggle raw display
$47$ \( T^{20} + 681 T^{19} + \cdots - 30\!\cdots\!20 \) Copy content Toggle raw display
$53$ \( T^{20} + 419 T^{19} + \cdots - 58\!\cdots\!96 \) Copy content Toggle raw display
$59$ \( T^{20} + 2829 T^{19} + \cdots + 39\!\cdots\!56 \) Copy content Toggle raw display
$61$ \( T^{20} + 959 T^{19} + \cdots + 88\!\cdots\!88 \) Copy content Toggle raw display
$67$ \( T^{20} + 1020 T^{19} + \cdots - 57\!\cdots\!64 \) Copy content Toggle raw display
$71$ \( T^{20} - 106 T^{19} + \cdots - 52\!\cdots\!64 \) Copy content Toggle raw display
$73$ \( T^{20} - 558 T^{19} + \cdots - 20\!\cdots\!12 \) Copy content Toggle raw display
$79$ \( T^{20} - 536 T^{19} + \cdots + 37\!\cdots\!72 \) Copy content Toggle raw display
$83$ \( T^{20} + 4179 T^{19} + \cdots - 79\!\cdots\!76 \) Copy content Toggle raw display
$89$ \( T^{20} + 4120 T^{19} + \cdots - 28\!\cdots\!80 \) Copy content Toggle raw display
$97$ \( T^{20} - 1414 T^{19} + \cdots - 18\!\cdots\!40 \) Copy content Toggle raw display
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