Properties

Label 4140.2.n.a
Level $4140$
Weight $2$
Character orbit 4140.n
Analytic conductor $33.058$
Analytic rank $0$
Dimension $16$
CM discriminant -115
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4140.n (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(33.0580664368\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \( x^{16} - 8 x^{15} + 60 x^{14} - 280 x^{13} + 1352 x^{12} - 4836 x^{11} + 18782 x^{10} - 55300 x^{9} + 177369 x^{8} - 421148 x^{7} + 1135954 x^{6} - 2123100 x^{5} + \cdots + 11064600 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{8} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{5} - \beta_{12} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{5} - \beta_{12} q^{7} + ( - \beta_{7} + \beta_{5}) q^{17} - \beta_{7} q^{23} - 5 q^{25} + \beta_{4} q^{29} - \beta_{13} q^{31} + \beta_{6} q^{35} + \beta_{11} q^{37} + (\beta_{9} + \beta_{6} + \beta_{3}) q^{41} + \beta_{14} q^{43} + ( - \beta_{13} - 2 \beta_1 + 7) q^{49} - \beta_{10} q^{53} + (\beta_{9} + \beta_{4} - \beta_{3}) q^{59} + (\beta_{12} - \beta_{11} + \beta_{8}) q^{67} + ( - \beta_{9} + \beta_{4} + \beta_{3}) q^{71} + (\beta_{10} - 2 \beta_{2}) q^{83} + ( - \beta_{13} + 2 \beta_1) q^{85} + ( - \beta_{14} + \beta_{12} - \beta_{8}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 80 q^{25} + 112 q^{49}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 8 x^{15} + 60 x^{14} - 280 x^{13} + 1352 x^{12} - 4836 x^{11} + 18782 x^{10} - 55300 x^{9} + 177369 x^{8} - 421148 x^{7} + 1135954 x^{6} - 2123100 x^{5} + \cdots + 11064600 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 20064 \nu^{14} + 140448 \nu^{13} - 937481 \nu^{12} + 3799062 \nu^{11} - 15860063 \nu^{10} + 47822924 \nu^{9} - 162235379 \nu^{8} + \cdots + 149448683340 ) / 26830596240 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 133504 \nu^{14} - 934528 \nu^{13} + 6695037 \nu^{12} - 28021358 \nu^{11} + 129715595 \nu^{10} - 413988444 \nu^{9} + 1596821759 \nu^{8} + \cdots + 465366701700 ) / 106742684880 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 106258 \nu^{15} + 796935 \nu^{14} + 13163751 \nu^{13} - 97651229 \nu^{12} + 812724807 \nu^{11} - 3369231855 \nu^{10} + \cdots - 13243903781760 ) / 5766153719760 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 117109791629 \nu^{15} + 16540767916323 \nu^{14} - 116447710149513 \nu^{13} + 824327028518237 \nu^{12} + \cdots + 62\!\cdots\!40 ) / 30\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2749228324702 \nu^{15} - 62799868141707 \nu^{14} + 451913481511650 \nu^{13} + \cdots - 22\!\cdots\!85 ) / 15\!\cdots\!45 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 539182748555 \nu^{15} + 1009273286571 \nu^{14} - 18554881253337 \nu^{13} + 61298504121473 \nu^{12} + \cdots - 60\!\cdots\!00 ) / 30\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 5498456649404 \nu^{15} - 41238424870530 \nu^{14} + 313297783133112 \nu^{13} + \cdots - 40\!\cdots\!45 ) / 15\!\cdots\!45 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 337968148003 \nu^{15} - 3877090878811 \nu^{14} + 27999579998889 \nu^{13} - 143607177111377 \nu^{12} + 634435421888731 \nu^{11} + \cdots + 53\!\cdots\!00 ) / 77\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 17560214 \nu^{15} - 131701605 \nu^{14} + 1153006347 \nu^{13} - 5497066913 \nu^{12} + 27510455715 \nu^{11} - 95234213943 \nu^{10} + \cdots - 152815016235024 ) / 39283216337016 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 5993238410 \nu^{15} - 50665996107 \nu^{14} + 436312620864 \nu^{13} - 2180875782881 \nu^{12} + 11480401717314 \nu^{11} + \cdots - 84\!\cdots\!80 ) / 91\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 4777460923925 \nu^{15} - 41098626203829 \nu^{14} + 288080795609703 \nu^{13} + \cdots + 21\!\cdots\!60 ) / 69\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 4952697701597 \nu^{15} + 38690862132291 \nu^{14} - 278731694424105 \nu^{13} + \cdots + 17\!\cdots\!20 ) / 69\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 36507237422 \nu^{15} - 266071926057 \nu^{14} + 1892888815392 \nu^{13} - 8141610531515 \nu^{12} + 38054733576438 \nu^{11} + \cdots - 11\!\cdots\!80 ) / 20\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 131055918584 \nu^{15} + 871225968975 \nu^{14} - 6354895784841 \nu^{13} + 26033823037319 \nu^{12} - 126381765343269 \nu^{11} + \cdots + 52\!\cdots\!40 ) / 57\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 451984 \nu^{15} + 3389880 \nu^{14} - 25081536 \nu^{13} + 111616804 \nu^{12} - 534548844 \nu^{11} + 1825342794 \nu^{10} + \cdots + 638989166880 ) / 46311655365 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{12} - 2\beta_{11} - \beta_{9} + \beta_{8} + 6\beta_{7} - \beta_{3} + 6 ) / 12 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{12} - \beta_{9} + \beta_{8} + \beta_{7} - 2\beta_{6} - \beta_{3} - \beta_{2} - \beta _1 - 7 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 2 \beta_{14} - 9 \beta_{13} - 5 \beta_{12} + 15 \beta_{11} - 9 \beta_{10} + 3 \beta_{9} - \beta_{8} + 3 \beta_{7} - 9 \beta_{6} + 7 \beta_{3} - 9 \beta_{2} - 9 \beta _1 - 33 ) / 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 2 \beta_{14} - 3 \beta_{13} - 2 \beta_{12} + 4 \beta_{11} - 3 \beta_{10} + 2 \beta_{9} - 6 \beta_{8} + \beta_{7} - \beta_{5} - 4 \beta_{4} + 4 \beta_{3} + 13 \beta_{2} + 13 \beta _1 - 32 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 15 \beta_{15} + 85 \beta_{14} + 90 \beta_{13} + 100 \beta_{12} + 15 \beta_{11} + 90 \beta_{10} + 12 \beta_{9} - 190 \beta_{8} - 93 \beta_{7} + 15 \beta_{6} - 15 \beta_{5} - 60 \beta_{4} - 167 \beta_{3} + 270 \beta_{2} + 270 \beta _1 - 432 ) / 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 15 \beta_{15} + 144 \beta_{14} + 105 \beta_{13} + 10 \beta_{12} - 6 \beta_{11} + 105 \beta_{10} + 124 \beta_{9} - 64 \beta_{8} - 179 \beta_{7} + 266 \beta_{6} + 153 \beta_{5} + 170 \beta_{4} - 100 \beta_{3} - 16 \beta_{2} + \cdots - 166 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 399 \beta_{15} - 713 \beta_{14} + 252 \beta_{13} - 1997 \beta_{12} - 456 \beta_{11} + 588 \beta_{10} + 1410 \beta_{9} + 2096 \beta_{8} - 2649 \beta_{7} + 2730 \beta_{6} + 1659 \beta_{5} + 1995 \beta_{4} + 1117 \beta_{3} + \cdots - 18 ) / 6 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 301 \beta_{15} - 784 \beta_{14} - 84 \beta_{13} - 1752 \beta_{12} - 412 \beta_{11} + 140 \beta_{10} + 20 \beta_{9} + 1268 \beta_{8} - 560 \beta_{7} - 52 \beta_{6} - 826 \beta_{5} - 444 \beta_{4} + 580 \beta_{3} - 64 \beta_{2} + \cdots + 6174 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 567 \beta_{15} + 4249 \beta_{14} + 8496 \beta_{13} - 8261 \beta_{12} - 20061 \beta_{11} - 8208 \beta_{10} - 18801 \beta_{9} + 5939 \beta_{8} + 29271 \beta_{7} - 17703 \beta_{6} - 32319 \beta_{5} + \cdots + 161940 ) / 6 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 5565 \beta_{15} + 11732 \beta_{14} + 16185 \beta_{13} + 43016 \beta_{12} - 21682 \beta_{11} - 15015 \beta_{10} - 30326 \beta_{9} - 11646 \beta_{8} + 43109 \beta_{7} - 31110 \beta_{6} - 14765 \beta_{5} + \cdots + 53228 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 48939 \beta_{15} - 59081 \beta_{14} - 218394 \beta_{13} + 388459 \beta_{12} + 159072 \beta_{11} - 86922 \beta_{10} - 224340 \beta_{9} - 185554 \beta_{8} + 322935 \beta_{7} - 321222 \beta_{6} + \cdots - 585252 ) / 6 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 13200 \beta_{15} - 114708 \beta_{14} - 310002 \beta_{13} + 87956 \beta_{12} + 276474 \beta_{11} - 3234 \beta_{10} - 23470 \beta_{9} - 41030 \beta_{8} + 103433 \beta_{7} - 40226 \beta_{6} + \cdots - 1127816 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 111111 \beta_{15} - 931853 \beta_{14} - 2890368 \beta_{13} + 2872747 \beta_{12} + 4327959 \beta_{11} + 792480 \beta_{10} + 3355593 \beta_{9} + 183905 \beta_{8} - 3382077 \beta_{7} + \cdots - 32635080 ) / 6 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 89089 \beta_{15} + 2533492 \beta_{14} + 3222219 \beta_{13} + 275988 \beta_{12} - 564014 \beta_{11} + 1437163 \beta_{10} + 7859602 \beta_{9} - 1507742 \beta_{8} - 10540867 \beta_{7} + \cdots - 8754816 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 5565105 \beta_{15} + 30575089 \beta_{14} + 67696470 \beta_{13} - 76646303 \beta_{12} - 56057274 \beta_{11} + 17247366 \beta_{10} + 58581378 \beta_{9} - 17853112 \beta_{8} + \cdots + 332141952 ) / 6 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4140\mathbb{Z}\right)^\times\).

\(n\) \(461\) \(1657\) \(2071\) \(3961\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2069.1
2.65988 2.13910i
−1.65988 + 2.13910i
1.07874 1.43199i
1.07874 + 3.36384i
−0.0787370 + 1.43199i
−0.0787370 3.36384i
2.65988 + 2.65673i
−1.65988 2.65673i
2.65988 + 2.13910i
−1.65988 2.13910i
1.07874 + 1.43199i
1.07874 3.36384i
−0.0787370 1.43199i
−0.0787370 + 3.36384i
2.65988 2.65673i
−1.65988 + 2.65673i
0 0 0 2.23607i 0 −5.21115 0 0 0
2069.2 0 0 0 2.23607i 0 −4.05368 0 0 0
2069.3 0 0 0 2.23607i 0 −3.40113 0 0 0
2069.4 0 0 0 2.23607i 0 −0.918623 0 0 0
2069.5 0 0 0 2.23607i 0 0.918623 0 0 0
2069.6 0 0 0 2.23607i 0 3.40113 0 0 0
2069.7 0 0 0 2.23607i 0 4.05368 0 0 0
2069.8 0 0 0 2.23607i 0 5.21115 0 0 0
2069.9 0 0 0 2.23607i 0 −5.21115 0 0 0
2069.10 0 0 0 2.23607i 0 −4.05368 0 0 0
2069.11 0 0 0 2.23607i 0 −3.40113 0 0 0
2069.12 0 0 0 2.23607i 0 −0.918623 0 0 0
2069.13 0 0 0 2.23607i 0 0.918623 0 0 0
2069.14 0 0 0 2.23607i 0 3.40113 0 0 0
2069.15 0 0 0 2.23607i 0 4.05368 0 0 0
2069.16 0 0 0 2.23607i 0 5.21115 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2069.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
115.c odd 2 1 CM by \(\Q(\sqrt{-115}) \)
3.b odd 2 1 inner
5.b even 2 1 inner
15.d odd 2 1 inner
23.b odd 2 1 inner
69.c even 2 1 inner
345.h even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4140.2.n.a 16
3.b odd 2 1 inner 4140.2.n.a 16
5.b even 2 1 inner 4140.2.n.a 16
15.d odd 2 1 inner 4140.2.n.a 16
23.b odd 2 1 inner 4140.2.n.a 16
69.c even 2 1 inner 4140.2.n.a 16
115.c odd 2 1 CM 4140.2.n.a 16
345.h even 2 1 inner 4140.2.n.a 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4140.2.n.a 16 1.a even 1 1 trivial
4140.2.n.a 16 3.b odd 2 1 inner
4140.2.n.a 16 5.b even 2 1 inner
4140.2.n.a 16 15.d odd 2 1 inner
4140.2.n.a 16 23.b odd 2 1 inner
4140.2.n.a 16 69.c even 2 1 inner
4140.2.n.a 16 115.c odd 2 1 CM
4140.2.n.a 16 345.h even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{8} - 56T_{7}^{6} + 997T_{7}^{4} - 5964T_{7}^{2} + 4356 \) acting on \(S_{2}^{\mathrm{new}}(4140, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{2} + 5)^{8} \) Copy content Toggle raw display
$7$ \( (T^{8} - 56 T^{6} + 997 T^{4} - 5964 T^{2} + \cdots + 4356)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} \) Copy content Toggle raw display
$13$ \( T^{16} \) Copy content Toggle raw display
$17$ \( (T^{4} + 79 T^{2} + 784)^{4} \) Copy content Toggle raw display
$19$ \( T^{16} \) Copy content Toggle raw display
$23$ \( (T^{2} + 23)^{8} \) Copy content Toggle raw display
$29$ \( (T^{8} + 232 T^{6} + 15253 T^{4} + \cdots + 527076)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} - 71 T^{2} + 484)^{4} \) Copy content Toggle raw display
$37$ \( (T^{8} - 296 T^{6} + 27517 T^{4} + \cdots + 2268036)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + 328 T^{6} + 35893 T^{4} + \cdots + 15634116)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} - 172 T^{2} + 36)^{4} \) Copy content Toggle raw display
$47$ \( T^{16} \) Copy content Toggle raw display
$53$ \( (T^{4} + 313 T^{2} + 23716)^{4} \) Copy content Toggle raw display
$59$ \( (T^{8} + 472 T^{6} + 76573 T^{4} + \cdots + 108868356)^{2} \) Copy content Toggle raw display
$61$ \( T^{16} \) Copy content Toggle raw display
$67$ \( (T^{8} - 536 T^{6} + 86437 T^{4} + \cdots + 1313316)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + 568 T^{6} + 110173 T^{4} + \cdots + 207187236)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} \) Copy content Toggle raw display
$79$ \( T^{16} \) Copy content Toggle raw display
$83$ \( (T^{4} + 373 T^{2} + 15376)^{4} \) Copy content Toggle raw display
$89$ \( T^{16} \) Copy content Toggle raw display
$97$ \( (T^{4} - 388 T^{2} + 30276)^{4} \) Copy content Toggle raw display
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