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Properties

Label 2070.3.c.a
Level $2070$
Weight $3$
Character orbit 2070.c
Analytic conductor $56.403$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 2070 = 2 \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2070.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(56.4034147226\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Defining polynomial: \( x^{16} + 78x^{14} + 2165x^{12} + 28310x^{10} + 184804x^{8} + 569634x^{6} + 696037x^{4} + 285578x^{2} + 529 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2^{7} \)
Twist minimal: no (minimal twist has level 230)
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.

\(f(q)\) \(=\) \( q + \beta_{6} q^{2} + 2 q^{4} - \beta_{2} q^{5} + (\beta_{7} + \beta_1) q^{7} + 2 \beta_{6} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{6} q^{2} + 2 q^{4} - \beta_{2} q^{5} + (\beta_{7} + \beta_1) q^{7} + 2 \beta_{6} q^{8} - \beta_{8} q^{10} + (\beta_{9} - \beta_{7} - \beta_{4} - \beta_{2} - \beta_1) q^{11} + (\beta_{15} - \beta_{14} + \beta_{13} - \beta_{12} + \beta_{11} + \beta_{10} - \beta_{7} - 3 \beta_{6} - \beta_{5} + \beta_{4} + \cdots + 1) q^{13}+ \cdots + (5 \beta_{15} - \beta_{14} + 3 \beta_{13} - \beta_{12} + 3 \beta_{11} + \beta_{10} - \beta_{7} - \beta_{6} + \cdots - 17) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 32 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 32 q^{4} + 24 q^{13} + 64 q^{16} - 4 q^{23} - 80 q^{25} - 96 q^{26} + 108 q^{29} - 116 q^{31} - 60 q^{35} + 156 q^{41} - 124 q^{46} + 128 q^{47} - 28 q^{49} + 48 q^{52} + 160 q^{58} - 204 q^{59} - 64 q^{62} + 128 q^{64} - 120 q^{70} - 236 q^{71} - 112 q^{73} + 936 q^{77} - 64 q^{82} + 60 q^{85} - 8 q^{92} - 216 q^{94} + 160 q^{95} - 256 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 78x^{14} + 2165x^{12} + 28310x^{10} + 184804x^{8} + 569634x^{6} + 696037x^{4} + 285578x^{2} + 529 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 249266 \nu^{15} + 19450521 \nu^{13} + 540300528 \nu^{11} + 7075107694 \nu^{9} + 46294879486 \nu^{7} + 143284317912 \nu^{5} + \cdots + 76734676377 \nu ) / 1473983980 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 3655930587 \nu^{15} - 279912880683 \nu^{13} - 7519929098840 \nu^{11} - 93142815057018 \nu^{9} - 550737169153146 \nu^{7} + \cdots + 192163143584185 \nu ) / 13607820103360 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 2028139777 \nu^{15} - 159386627629 \nu^{13} - 4483504013264 \nu^{11} - 59956611391678 \nu^{9} - 407018725514790 \nu^{7} + \cdots - 890813001228721 \nu ) / 3401955025840 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3266665611 \nu^{14} + 251231034675 \nu^{12} + 6803101750504 \nu^{10} + 85389448480554 \nu^{8} + 517393068635482 \nu^{6} + \cdots + 38708131230591 ) / 6803910051680 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 328071 \nu^{14} + 25254583 \nu^{12} + 684448696 \nu^{10} + 8586649314 \nu^{8} + 51832164194 \nu^{6} + 133890163508 \nu^{4} + 92129883455 \nu^{2} + \cdots + 885659459 ) / 505359680 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 651909559 \nu^{15} + 49847669703 \nu^{13} + 1334253632328 \nu^{11} + 16362993734546 \nu^{9} + 94189913828650 \nu^{7} + \cdots - 93749086185413 \nu ) / 485993575120 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 191987899 \nu^{15} + 14808207659 \nu^{13} + 402812007992 \nu^{11} + 5087217548666 \nu^{9} + 31116072823834 \nu^{7} + \cdots + 8504894308903 \nu ) / 114351429440 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 229620557 \nu^{15} - 17699955559 \nu^{13} - 480902133464 \nu^{11} - 6059544792898 \nu^{9} - 36877647447520 \nu^{7} + \cdots - 4166619657821 \nu ) / 121498393780 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 14365389855 \nu^{14} + 1105876709199 \nu^{12} + 29972322851736 \nu^{10} + 375976473404450 \nu^{8} + \cdots - 44834433467221 ) / 6803910051680 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 45221309975 \nu^{15} - 21002456926 \nu^{14} + 3479923444575 \nu^{13} - 1617708410646 \nu^{12} + 94261526625960 \nu^{11} + \cdots - 82437933075262 ) / 13607820103360 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 33188908451 \nu^{15} + 2553066418411 \nu^{13} + 69108994984088 \nu^{11} + 865212237602154 \nu^{9} + \cdots - 392465157363513 \nu ) / 6803910051680 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 45221309975 \nu^{15} - 28278344419 \nu^{14} + 3479923444575 \nu^{13} - 2174316072571 \nu^{12} + 94261526625960 \nu^{11} + \cdots + 28012979344985 ) / 13607820103360 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 45221309975 \nu^{15} + 40620036785 \nu^{14} - 3479923444575 \nu^{13} + 3124787270393 \nu^{12} - 94261526625960 \nu^{11} + \cdots + 6504254102973 ) / 13607820103360 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 45221309975 \nu^{15} + 68821388859 \nu^{14} - 3479923444575 \nu^{13} + 5299632441843 \nu^{12} - 94261526625960 \nu^{11} + \cdots + 78181554830047 ) / 13607820103360 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{14} + \beta_{12} - \beta_{11} - 3\beta_{10} + \beta_{7} + 5\beta_{6} - 3\beta_{5} - \beta_{4} - \beta_{2} - 18 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 7\beta_{12} + 5\beta_{9} - 19\beta_{8} + 2\beta_{7} - 7\beta_{4} + 5\beta_{3} + 10\beta_{2} - 25\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 14 \beta_{15} - 47 \beta_{14} - 14 \beta_{13} - 31 \beta_{12} + 43 \beta_{11} + 79 \beta_{10} - 31 \beta_{7} - 179 \beta_{6} + 117 \beta_{5} + 31 \beta_{4} + 31 \beta_{2} + 400 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 349 \beta_{12} - 193 \beta_{9} + 929 \beta_{8} - 28 \beta_{7} + 313 \beta_{4} - 223 \beta_{3} - 342 \beta_{2} + 834 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 359 \beta_{15} + 898 \beta_{14} + 325 \beta_{13} + 543 \beta_{12} - 872 \beta_{11} - 1235 \beta_{10} + 543 \beta_{7} + 3226 \beta_{6} - 2129 \beta_{5} - 543 \beta_{4} - 543 \beta_{2} - 5998 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 13910 \beta_{12} + 6966 \beta_{9} - 36654 \beta_{8} + 52 \beta_{7} - 11798 \beta_{4} + 8514 \beta_{3} + 12044 \beta_{2} - 29577 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 28822 \beta_{15} - 66115 \beta_{14} - 25026 \beta_{13} - 39353 \beta_{12} + 66439 \beta_{11} + 84671 \beta_{10} - 39353 \beta_{7} - 232271 \beta_{6} + 154427 \beta_{5} + 39353 \beta_{4} + \cdots + 406476 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 520505 \beta_{12} - 250339 \beta_{9} + 1366801 \beta_{8} + 12974 \beta_{7} + 430885 \beta_{4} - 313151 \beta_{3} - 434946 \beta_{2} + 1065837 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 1081278 \beta_{15} + 2410481 \beta_{14} + 923762 \beta_{13} + 1431797 \beta_{12} - 2458153 \beta_{11} - 3010165 \beta_{10} + 1431797 \beta_{7} + 8384705 \beta_{6} + \cdots - 14401944 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 19083917 \beta_{12} + 9029821 \beta_{9} - 50060453 \beta_{8} - 679532 \beta_{7} - 15643385 \beta_{4} + 11400387 \beta_{3} + 15800198 \beta_{2} - 38587470 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 19844966 \beta_{15} - 43809734 \beta_{14} - 16848718 \beta_{13} - 26024800 \beta_{12} + 44933550 \beta_{11} + 54218408 \beta_{10} - 26024800 \beta_{7} - 151719058 \beta_{6} + \cdots + 259136963 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 694927048 \beta_{12} - 326698624 \beta_{9} + 1822403440 \beta_{8} + 27487912 \beta_{7} + 567460000 \beta_{4} - 413915040 \beta_{3} - 574139472 \beta_{2} + 1399069645 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 1446079476 \beta_{15} + 3181503425 \beta_{14} + 1224795972 \beta_{13} + 1890378909 \beta_{12} - 3270129841 \beta_{11} - 3924573439 \beta_{10} + \cdots - 18751391662 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 25246432839 \beta_{12} + 11839142477 \beta_{9} - 66202882547 \beta_{8} - 1035217254 \beta_{7} - 20585412727 \beta_{4} + 15018968005 \beta_{3} + \cdots - 50751416961 \beta_1 ) / 2 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2070\mathbb{Z}\right)^\times\).

\(n\) \(461\) \(1657\) \(1891\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
91.1
− 6.02373i
1.00527i
3.68124i
2.26343i
− 2.26343i
− 3.68124i
− 1.00527i
6.02373i
− 2.98291i
0.0431371i
− 3.47734i
1.01877i
− 1.01877i
3.47734i
− 0.0431371i
2.98291i
−1.41421 0 2.00000 − 2.23607i 0 − 7.10180i −2.82843 0 3.16228i
91.2 −1.41421 0 2.00000 − 2.23607i 0 − 1.47532i −2.82843 0 3.16228i
91.3 −1.41421 0 2.00000 − 2.23607i 0 1.16919i −2.82843 0 3.16228i
91.4 −1.41421 0 2.00000 − 2.23607i 0 10.1866i −2.82843 0 3.16228i
91.5 −1.41421 0 2.00000 2.23607i 0 − 10.1866i −2.82843 0 − 3.16228i
91.6 −1.41421 0 2.00000 2.23607i 0 − 1.16919i −2.82843 0 − 3.16228i
91.7 −1.41421 0 2.00000 2.23607i 0 1.47532i −2.82843 0 − 3.16228i
91.8 −1.41421 0 2.00000 2.23607i 0 7.10180i −2.82843 0 − 3.16228i
91.9 1.41421 0 2.00000 − 2.23607i 0 − 8.51262i 2.82843 0 − 3.16228i
91.10 1.41421 0 2.00000 − 2.23607i 0 − 8.24199i 2.82843 0 − 3.16228i
91.11 1.41421 0 2.00000 − 2.23607i 0 − 7.05858i 2.82843 0 − 3.16228i
91.12 1.41421 0 2.00000 − 2.23607i 0 7.61815i 2.82843 0 − 3.16228i
91.13 1.41421 0 2.00000 2.23607i 0 − 7.61815i 2.82843 0 3.16228i
91.14 1.41421 0 2.00000 2.23607i 0 7.05858i 2.82843 0 3.16228i
91.15 1.41421 0 2.00000 2.23607i 0 8.24199i 2.82843 0 3.16228i
91.16 1.41421 0 2.00000 2.23607i 0 8.51262i 2.82843 0 3.16228i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 91.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2070.3.c.a 16
3.b odd 2 1 230.3.d.a ✓ 16
12.b even 2 1 1840.3.k.d 16
15.d odd 2 1 1150.3.d.b 16
15.e even 4 2 1150.3.c.c 32
23.b odd 2 1 inner 2070.3.c.a 16
69.c even 2 1 230.3.d.a ✓ 16
276.h odd 2 1 1840.3.k.d 16
345.h even 2 1 1150.3.d.b 16
345.l odd 4 2 1150.3.c.c 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.3.d.a ✓ 16 3.b odd 2 1
230.3.d.a ✓ 16 69.c even 2 1
1150.3.c.c 32 15.e even 4 2
1150.3.c.c 32 345.l odd 4 2
1150.3.d.b 16 15.d odd 2 1
1150.3.d.b 16 345.h even 2 1
1840.3.k.d 16 12.b even 2 1
1840.3.k.d 16 276.h odd 2 1
2070.3.c.a 16 1.a even 1 1 trivial
2070.3.c.a 16 23.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{16} + 406 T_{7}^{14} + 67901 T_{7}^{12} + 6012916 T_{7}^{10} + 299518572 T_{7}^{8} + 8103516608 T_{7}^{6} + 100475819056 T_{7}^{4} + 285091956800 T_{7}^{2} + \cdots + 221645107264 \) acting on \(S_{3}^{\mathrm{new}}(2070, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 2)^{8} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{2} + 5)^{8} \) Copy content Toggle raw display
$7$ \( T^{16} + 406 T^{14} + \cdots + 221645107264 \) Copy content Toggle raw display
$11$ \( T^{16} + 1016 T^{14} + \cdots + 21\!\cdots\!44 \) Copy content Toggle raw display
$13$ \( (T^{8} - 12 T^{7} - 898 T^{6} + \cdots - 343464224)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + 1858 T^{14} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{16} + 4184 T^{14} + \cdots + 11\!\cdots\!24 \) Copy content Toggle raw display
$23$ \( T^{16} + 4 T^{15} + \cdots + 61\!\cdots\!61 \) Copy content Toggle raw display
$29$ \( (T^{8} - 54 T^{7} - 2309 T^{6} + \cdots - 61767459836)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + 58 T^{7} + \cdots + 229759835104)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} + 14482 T^{14} + \cdots + 27\!\cdots\!04 \) Copy content Toggle raw display
$41$ \( (T^{8} - 78 T^{7} + \cdots - 212194449184)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} + 13412 T^{14} + \cdots + 18\!\cdots\!24 \) Copy content Toggle raw display
$47$ \( (T^{8} - 64 T^{7} - 5764 T^{6} + \cdots + 13232824136)^{2} \) Copy content Toggle raw display
$53$ \( T^{16} + 21250 T^{14} + \cdots + 16\!\cdots\!64 \) Copy content Toggle raw display
$59$ \( (T^{8} + 102 T^{7} + \cdots - 42922529206784)^{2} \) Copy content Toggle raw display
$61$ \( T^{16} + 28128 T^{14} + \cdots + 54\!\cdots\!84 \) Copy content Toggle raw display
$67$ \( T^{16} + 52678 T^{14} + \cdots + 20\!\cdots\!04 \) Copy content Toggle raw display
$71$ \( (T^{8} + 118 T^{7} + \cdots - 24390990617024)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + 56 T^{7} + \cdots + 1317400530416)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} + 82216 T^{14} + \cdots + 18\!\cdots\!24 \) Copy content Toggle raw display
$83$ \( T^{16} + 69862 T^{14} + \cdots + 39\!\cdots\!64 \) Copy content Toggle raw display
$89$ \( T^{16} + 67928 T^{14} + \cdots + 54\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{16} + 83856 T^{14} + \cdots + 37\!\cdots\!04 \) Copy content Toggle raw display
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