Properties

Label 1805.2.a.w
Level $1805$
Weight $2$
Character orbit 1805.a
Self dual yes
Analytic conductor $14.413$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1805 = 5 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1805.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(14.4129975648\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \( x^{16} - 29x^{14} + 339x^{12} - 2038x^{10} + 6639x^{8} - 11261x^{6} + 8701x^{4} - 2592x^{2} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{12} q^{3} + (\beta_{6} - \beta_{5} + 2) q^{4} + q^{5} + (\beta_{5} - \beta_{3} - \beta_{2} - 1) q^{6} + ( - \beta_{3} + 1) q^{7} + (\beta_{10} - \beta_{9} + 2 \beta_1) q^{8} + (\beta_{11} + \beta_{5} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + \beta_{12} q^{3} + (\beta_{6} - \beta_{5} + 2) q^{4} + q^{5} + (\beta_{5} - \beta_{3} - \beta_{2} - 1) q^{6} + ( - \beta_{3} + 1) q^{7} + (\beta_{10} - \beta_{9} + 2 \beta_1) q^{8} + (\beta_{11} + \beta_{5} + 1) q^{9} + \beta_1 q^{10} + ( - \beta_{8} - \beta_{6} + \beta_{5} - \beta_{4}) q^{11} + (2 \beta_{14} + \beta_{9} - 2 \beta_{7}) q^{12} + (\beta_{15} - \beta_{12} + \beta_{9} - \beta_{7} - \beta_1) q^{13} + (\beta_{14} + \beta_{12} + \beta_{10} + \beta_{7} + 2 \beta_1) q^{14} + \beta_{12} q^{15} + (\beta_{11} + \beta_{6} - 3 \beta_{5} - \beta_{3} + \beta_{2} + 4) q^{16} + ( - \beta_{8} - \beta_{4} + 1) q^{17} + ( - \beta_{15} - \beta_{9} + \beta_{7} + \beta_1) q^{18} + (\beta_{6} - \beta_{5} + 2) q^{20} + ( - \beta_{13} + \beta_{12} - \beta_{10} + \beta_1) q^{21} + ( - 2 \beta_{14} + 2 \beta_{13} - \beta_{10} + \beta_{9} - \beta_1) q^{22} + (\beta_{8} + \beta_{3} - \beta_{2} + 3) q^{23} + ( - \beta_{11} + 2 \beta_{8} + \beta_{5} + 2 \beta_{4} - 2 \beta_{2} - 1) q^{24} + q^{25} + ( - 2 \beta_{11} - \beta_{8} - \beta_{6} + 3 \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} - 3) q^{26} + ( - 2 \beta_{14} - 2 \beta_{13} + \beta_{12} - \beta_{9} - \beta_1) q^{27} + (\beta_{8} + 3 \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} + 3) q^{28} + ( - \beta_{14} - 2 \beta_{13} + \beta_{12} + \beta_{10} - 2 \beta_{9} - \beta_{7}) q^{29} + (\beta_{5} - \beta_{3} - \beta_{2} - 1) q^{30} + (\beta_{15} + 2 \beta_{13} - \beta_{12} - \beta_{10} - \beta_{7}) q^{31} + ( - \beta_{15} + \beta_{10} - 3 \beta_{9} + 5 \beta_{7} + 3 \beta_1) q^{32} + ( - \beta_{15} - \beta_{14} + \beta_{12} - \beta_{10} + 2 \beta_{9} + 3 \beta_{7} - \beta_1) q^{33} + ( - 2 \beta_{14} + 2 \beta_{13} + 2 \beta_1) q^{34} + ( - \beta_{3} + 1) q^{35} + (\beta_{8} + \beta_{6} - 6 \beta_{5} - \beta_{4} + 2) q^{36} + (\beta_{15} + \beta_{12} - \beta_{10} - 2 \beta_{9} + \beta_{7} - 2 \beta_1) q^{37} + ( - 2 \beta_{11} + 2 \beta_{8} + \beta_{6} - 3 \beta_{5} + 2 \beta_{3} + 2 \beta_{2} + \cdots - 1) q^{39}+ \cdots + (3 \beta_{11} - 5 \beta_{8} - \beta_{6} + 7 \beta_{5} - \beta_{4} - 2 \beta_{3} - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 26 q^{4} + 16 q^{5} - 2 q^{6} + 22 q^{7} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 26 q^{4} + 16 q^{5} - 2 q^{6} + 22 q^{7} + 18 q^{9} + 12 q^{11} + 42 q^{16} + 22 q^{17} + 26 q^{20} + 42 q^{23} - 14 q^{24} + 16 q^{25} - 26 q^{26} + 46 q^{28} - 2 q^{30} + 22 q^{35} - 8 q^{36} - 38 q^{39} + 74 q^{42} + 88 q^{43} - 48 q^{44} + 18 q^{45} + 32 q^{47} + 30 q^{49} - 22 q^{54} + 12 q^{55} - 2 q^{58} + 20 q^{61} + 6 q^{62} - 6 q^{63} + 24 q^{64} - 24 q^{66} + 84 q^{68} + 44 q^{73} - 122 q^{74} + 4 q^{77} + 42 q^{80} - 36 q^{81} - 50 q^{82} + 56 q^{83} + 22 q^{85} + 34 q^{87} + 6 q^{92} - 58 q^{93} - 96 q^{96} + 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 29x^{14} + 339x^{12} - 2038x^{10} + 6639x^{8} - 11261x^{6} + 8701x^{4} - 2592x^{2} + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{12} - 22\nu^{10} + 181\nu^{8} - 687\nu^{6} + 1190\nu^{4} - 791\nu^{2} + 144 ) / 16 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 3\nu^{14} - 83\nu^{12} + 913\nu^{10} - 5070\nu^{8} + 14881\nu^{6} - 21999\nu^{4} + 13979\nu^{2} - 2448 ) / 64 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{14} - 27\nu^{12} + 289\nu^{10} - 1558\nu^{8} + 4433\nu^{6} - 6343\nu^{4} + 3869\nu^{2} - 640 ) / 16 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3\nu^{14} - 87\nu^{12} + 1017\nu^{10} - 6098\nu^{8} + 19613\nu^{6} - 31799\nu^{4} + 20999\nu^{2} - 3472 ) / 64 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 3\nu^{14} - 87\nu^{12} + 1017\nu^{10} - 6098\nu^{8} + 19613\nu^{6} - 31799\nu^{4} + 21063\nu^{2} - 3728 ) / 64 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 9 \nu^{15} + 261 \nu^{13} - 3035 \nu^{11} + 17990 \nu^{9} - 56855 \nu^{7} + 90357 \nu^{5} - 59269 \nu^{3} + 10672 \nu ) / 256 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -5\nu^{14} + 137\nu^{12} - 1495\nu^{10} + 8254\nu^{8} - 24099\nu^{6} + 35097\nu^{4} - 20993\nu^{2} + 3376 ) / 64 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 3\nu^{15} - 87\nu^{13} + 1017\nu^{11} - 6098\nu^{9} + 19613\nu^{7} - 31799\nu^{5} + 20999\nu^{3} - 3472\nu ) / 64 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 3\nu^{15} - 87\nu^{13} + 1017\nu^{11} - 6098\nu^{9} + 19613\nu^{7} - 31799\nu^{5} + 21063\nu^{3} - 3856\nu ) / 64 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 9\nu^{14} - 261\nu^{12} + 3035\nu^{10} - 17990\nu^{8} + 56855\nu^{6} - 90293\nu^{4} + 58693\nu^{2} - 9712 ) / 64 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 13 \nu^{15} - 377 \nu^{13} + 4391 \nu^{11} - 26110 \nu^{9} + 82803 \nu^{7} - 131433 \nu^{5} + 83993 \nu^{3} - 13328 \nu ) / 128 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 39 \nu^{15} + 1115 \nu^{13} - 12789 \nu^{11} + 74858 \nu^{9} - 233993 \nu^{7} + 368251 \nu^{5} - 237851 \nu^{3} + 39184 \nu ) / 256 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 41 \nu^{15} + 1173 \nu^{13} - 13467 \nu^{11} + 78902 \nu^{9} - 246727 \nu^{7} + 387701 \nu^{5} - 248885 \nu^{3} + 40944 \nu ) / 256 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 69 \nu^{15} + 2001 \nu^{13} - 23311 \nu^{11} + 138734 \nu^{9} - 441179 \nu^{7} + 705921 \nu^{5} - 462033 \nu^{3} + 77296 \nu ) / 256 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} - \beta_{5} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{10} - \beta_{9} + 6\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{11} + 7\beta_{6} - 9\beta_{5} - \beta_{3} + \beta_{2} + 24 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{15} + 9\beta_{10} - 11\beta_{9} + 5\beta_{7} + 39\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 12\beta_{11} + \beta_{8} + 48\beta_{6} - 72\beta_{5} - \beta_{4} - 9\beta_{3} + 13\beta_{2} + 157 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -14\beta_{15} - 4\beta_{14} + 4\beta_{13} - 4\beta_{12} + 70\beta_{10} - 96\beta_{9} + 62\beta_{7} + 263\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 110\beta_{11} + 10\beta_{8} + 333\beta_{6} - 555\beta_{5} - 22\beta_{4} - 62\beta_{3} + 126\beta_{2} + 1068 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 142 \beta_{15} - 76 \beta_{14} + 76 \beta_{13} - 64 \beta_{12} + 521 \beta_{10} - 775 \beta_{9} + 582 \beta_{7} + 1818 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 917\beta_{11} + 66\beta_{8} + 2339\beta_{6} - 4207\beta_{5} - 294\beta_{4} - 381\beta_{3} + 1101\beta_{2} + 7448 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 1277 \beta_{15} - 948 \beta_{14} + 948 \beta_{13} - 720 \beta_{12} + 3821 \beta_{10} - 6041 \beta_{9} + 4961 \beta_{7} + 12801 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 7318 \beta_{11} + 329 \beta_{8} + 16622 \beta_{6} - 31644 \beta_{5} - 3173 \beta_{4} - 2153 \beta_{3} + 9173 \beta_{2} + 52867 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 10820 \beta_{15} - 9864 \beta_{14} + 9848 \beta_{13} - 7020 \beta_{12} + 27948 \beta_{10} - 46280 \beta_{9} + 40492 \beta_{7} + 91437 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 57100 \beta_{11} + 956 \beta_{8} + 119385 \beta_{6} - 237297 \beta_{5} - 30532 \beta_{4} - 11064 \beta_{3} + 74504 \beta_{2} + 380296 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 88588 \beta_{15} - 93016 \beta_{14} + 92552 \beta_{13} - 63440 \beta_{12} + 204953 \beta_{10} - 351497 \beta_{9} + 323164 \beta_{7} + 660662 \beta_1 \) 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
−2.76439
−2.51130
−2.40747
−2.01896
−1.91539
−1.01007
−0.503538
−0.486735
0.486735
0.503538
1.01007
1.91539
2.01896
2.40747
2.51130
2.76439
−2.76439 2.29146 5.64183 1.00000 −6.33447 1.54174 −10.0674 2.25078 −2.76439
1.2 −2.51130 −2.10371 4.30663 1.00000 5.28304 3.12429 −5.79265 1.42558 −2.51130
1.3 −2.40747 −0.306883 3.79590 1.00000 0.738810 3.95092 −4.32356 −2.90582 −2.40747
1.4 −2.01896 −1.69608 2.07621 1.00000 3.42432 1.43286 −0.153862 −0.123308 −2.01896
1.5 −1.91539 1.67190 1.66872 1.00000 −3.20235 −4.22760 0.634526 −0.204741 −1.91539
1.6 −1.01007 2.79667 −0.979758 1.00000 −2.82483 −1.25546 3.00976 4.82135 −1.01007
1.7 −0.503538 −3.01030 −1.74645 1.00000 1.51580 2.48971 1.88648 6.06188 −0.503538
1.8 −0.486735 −0.821147 −1.76309 1.00000 0.399681 3.94354 1.83163 −2.32572 −0.486735
1.9 0.486735 0.821147 −1.76309 1.00000 0.399681 3.94354 −1.83163 −2.32572 0.486735
1.10 0.503538 3.01030 −1.74645 1.00000 1.51580 2.48971 −1.88648 6.06188 0.503538
1.11 1.01007 −2.79667 −0.979758 1.00000 −2.82483 −1.25546 −3.00976 4.82135 1.01007
1.12 1.91539 −1.67190 1.66872 1.00000 −3.20235 −4.22760 −0.634526 −0.204741 1.91539
1.13 2.01896 1.69608 2.07621 1.00000 3.42432 1.43286 0.153862 −0.123308 2.01896
1.14 2.40747 0.306883 3.79590 1.00000 0.738810 3.95092 4.32356 −2.90582 2.40747
1.15 2.51130 2.10371 4.30663 1.00000 5.28304 3.12429 5.79265 1.42558 2.51130
1.16 2.76439 −2.29146 5.64183 1.00000 −6.33447 1.54174 10.0674 2.25078 2.76439
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.16
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1805.2.a.w 16
5.b even 2 1 9025.2.a.cm 16
19.b odd 2 1 inner 1805.2.a.w 16
95.d odd 2 1 9025.2.a.cm 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1805.2.a.w 16 1.a even 1 1 trivial
1805.2.a.w 16 19.b odd 2 1 inner
9025.2.a.cm 16 5.b even 2 1
9025.2.a.cm 16 95.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1805))\):

\( T_{2}^{16} - 29T_{2}^{14} + 339T_{2}^{12} - 2038T_{2}^{10} + 6639T_{2}^{8} - 11261T_{2}^{6} + 8701T_{2}^{4} - 2592T_{2}^{2} + 256 \) Copy content Toggle raw display
\( T_{3}^{16} - 33T_{3}^{14} + 441T_{3}^{12} - 3074T_{3}^{10} + 11974T_{3}^{8} - 25742T_{3}^{6} + 27709T_{3}^{4} - 11321T_{3}^{2} + 841 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} - 29 T^{14} + 339 T^{12} + \cdots + 256 \) Copy content Toggle raw display
$3$ \( T^{16} - 33 T^{14} + 441 T^{12} + \cdots + 841 \) Copy content Toggle raw display
$5$ \( (T - 1)^{16} \) Copy content Toggle raw display
$7$ \( (T^{8} - 11 T^{7} + 25 T^{6} + 136 T^{5} + \cdots + 1421)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} - 6 T^{7} - 33 T^{6} + 242 T^{5} + \cdots - 6704)^{2} \) Copy content Toggle raw display
$13$ \( T^{16} - 140 T^{14} + 7462 T^{12} + \cdots + 3041536 \) Copy content Toggle raw display
$17$ \( (T^{8} - 11 T^{7} + 9 T^{6} + 278 T^{5} + \cdots + 1216)^{2} \) Copy content Toggle raw display
$19$ \( T^{16} \) Copy content Toggle raw display
$23$ \( (T^{8} - 21 T^{7} + 85 T^{6} + 786 T^{5} + \cdots + 2131)^{2} \) Copy content Toggle raw display
$29$ \( T^{16} - 235 T^{14} + \cdots + 5696626576 \) Copy content Toggle raw display
$31$ \( T^{16} - 207 T^{14} + 15676 T^{12} + \cdots + 952576 \) Copy content Toggle raw display
$37$ \( T^{16} - 383 T^{14} + \cdots + 65364080896 \) Copy content Toggle raw display
$41$ \( T^{16} - 390 T^{14} + \cdots + 11497200625 \) Copy content Toggle raw display
$43$ \( (T^{8} - 44 T^{7} + 685 T^{6} + \cdots + 2989936)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} - 16 T^{7} + 49 T^{6} + 468 T^{5} + \cdots + 54071)^{2} \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 203344862089216 \) Copy content Toggle raw display
$59$ \( T^{16} - 358 T^{14} + \cdots + 276743332096 \) Copy content Toggle raw display
$61$ \( (T^{8} - 10 T^{7} - 232 T^{6} + \cdots - 55364)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} - 589 T^{14} + \cdots + 55644563881 \) Copy content Toggle raw display
$71$ \( T^{16} - 276 T^{14} + \cdots + 66324736 \) Copy content Toggle raw display
$73$ \( (T^{8} - 22 T^{7} - 180 T^{6} + \cdots + 22241776)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} - 598 T^{14} + \cdots + 2600298151936 \) Copy content Toggle raw display
$83$ \( (T^{8} - 28 T^{7} - 79 T^{6} + \cdots + 2983616)^{2} \) Copy content Toggle raw display
$89$ \( T^{16} - 649 T^{14} + \cdots + 34781877001 \) Copy content Toggle raw display
$97$ \( T^{16} - 795 T^{14} + \cdots + 29064864921856 \) Copy content Toggle raw display
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