Newspace parameters
Level: | \( N \) | \(=\) | \( 1274 = 2 \cdot 7^{2} \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 1274.o (of order \(6\), degree \(2\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(10.1729412175\) |
Analytic rank: | \(0\) |
Dimension: | \(12\) |
Relative dimension: | \(6\) over \(\Q(\zeta_{6})\) |
Coefficient field: | \(\mathbb{Q}[x]/(x^{12} - \cdots)\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{12} - 6 x^{11} + 39 x^{10} - 140 x^{9} + 460 x^{8} - 1066 x^{7} + 2127 x^{6} - 3172 x^{5} + 3842 x^{4} - 3394 x^{3} + 2141 x^{2} - 832 x + 169 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 182) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 6 x^{11} + 39 x^{10} - 140 x^{9} + 460 x^{8} - 1066 x^{7} + 2127 x^{6} - 3172 x^{5} + 3842 x^{4} - 3394 x^{3} + 2141 x^{2} - 832 x + 169 \) :
\(\beta_{1}\) | \(=\) | \( \nu \) |
\(\beta_{2}\) | \(=\) | \( ( 24 \nu^{10} - 120 \nu^{9} + 751 \nu^{8} - 2284 \nu^{7} + 6728 \nu^{6} - 12694 \nu^{5} + 20323 \nu^{4} - 21914 \nu^{3} + 17046 \nu^{2} - 7860 \nu + 2041 ) / 286 \) |
\(\beta_{3}\) | \(=\) | \( ( - 31 \nu^{10} + 155 \nu^{9} - 976 \nu^{8} + 2974 \nu^{7} - 8881 \nu^{6} + 16885 \nu^{5} - 28044 \nu^{4} + 31106 \nu^{3} - 27273 \nu^{2} + 14085 \nu - 5252 ) / 286 \) |
\(\beta_{4}\) | \(=\) | \( ( 162 \nu^{11} + 186 \nu^{10} - 51 \nu^{9} + 15887 \nu^{8} - 51264 \nu^{7} + 192268 \nu^{6} - 390377 \nu^{5} + 670405 \nu^{4} - 762542 \nu^{3} + 592138 \nu^{2} - 270385 \nu + 66227 ) / 7898 \) |
\(\beta_{5}\) | \(=\) | \( ( 2323 \nu^{11} - 22649 \nu^{10} + 132943 \nu^{9} - 590285 \nu^{8} + 1852053 \nu^{7} - 4762085 \nu^{6} + 9077085 \nu^{5} - 13790463 \nu^{4} + 15194803 \nu^{3} + \cdots - 1517893 ) / 102674 \) |
\(\beta_{6}\) | \(=\) | \( ( - 2323 \nu^{11} + 2904 \nu^{10} - 34218 \nu^{9} - 29708 \nu^{8} + 35569 \nu^{7} - 841546 \nu^{6} + 1541776 \nu^{5} - 3573290 \nu^{4} + 3839377 \nu^{3} + \cdots - 689598 ) / 102674 \) |
\(\beta_{7}\) | \(=\) | \( ( - 324 \nu^{11} + 1782 \nu^{10} - 10668 \nu^{9} + 34641 \nu^{8} - 98512 \nu^{7} + 195608 \nu^{6} - 301272 \nu^{5} + 336079 \nu^{4} - 238324 \nu^{3} + 98790 \nu^{2} + \cdots - 3573 ) / 7898 \) |
\(\beta_{8}\) | \(=\) | \( ( - 5977 \nu^{11} + 33053 \nu^{10} - 223218 \nu^{9} + 741326 \nu^{8} - 2485739 \nu^{7} + 5177975 \nu^{6} - 10201456 \nu^{5} + 12482628 \nu^{4} - 13546965 \nu^{3} + \cdots - 527956 ) / 102674 \) |
\(\beta_{9}\) | \(=\) | \( ( 5977 \nu^{11} - 32694 \nu^{10} + 221423 \nu^{9} - 766456 \nu^{8} + 2597029 \nu^{7} - 6035626 \nu^{6} + 12377355 \nu^{5} - 19119820 \nu^{4} + 23328279 \nu^{3} + \cdots - 3597672 ) / 102674 \) |
\(\beta_{10}\) | \(=\) | \( ( - 6388 \nu^{11} + 30826 \nu^{10} - 187767 \nu^{9} + 543572 \nu^{8} - 1501380 \nu^{7} + 2562258 \nu^{6} - 3406537 \nu^{5} + 2547170 \nu^{4} - 212336 \nu^{3} + \cdots - 435708 ) / 102674 \) |
\(\beta_{11}\) | \(=\) | \( ( - 9240 \nu^{11} + 55128 \nu^{10} - 344643 \nu^{9} + 1207618 \nu^{8} - 3759676 \nu^{7} + 8280896 \nu^{6} - 15228345 \nu^{5} + 20588490 \nu^{4} + \cdots + 1667016 ) / 102674 \) |
\(\nu\) | \(=\) | \( \beta_1 \) |
\(\nu^{2}\) | \(=\) | \( \beta_{6} + \beta_{5} - \beta_{3} + \beta_{2} + \beta _1 - 4 \) |
\(\nu^{3}\) | \(=\) | \( - \beta_{11} + \beta_{10} + \beta_{9} - \beta_{8} - 2 \beta_{7} + 6 \beta_{6} - 4 \beta_{5} - \beta_{3} + 2 \beta_{2} - 5 \beta _1 - 2 \) |
\(\nu^{4}\) | \(=\) | \( - 2 \beta_{11} + 2 \beta_{10} + 3 \beta_{9} - \beta_{8} - 6 \beta_{7} - \beta_{6} - 21 \beta_{5} - 4 \beta_{4} + 6 \beta_{3} - 9 \beta_{2} - 9 \beta _1 + 29 \) |
\(\nu^{5}\) | \(=\) | \( 8 \beta_{11} - 15 \beta_{10} - 8 \beta_{9} + 13 \beta_{8} + 25 \beta_{7} - 73 \beta_{6} + 26 \beta_{5} - 10 \beta_{4} + 11 \beta_{3} - 35 \beta_{2} + 33 \beta _1 + 34 \) |
\(\nu^{6}\) | \(=\) | \( 29 \beta_{11} - 50 \beta_{10} - 48 \beta_{9} + 25 \beta_{8} + 124 \beta_{7} - 81 \beta_{6} + 266 \beta_{5} + 48 \beta_{4} - 41 \beta_{3} + 42 \beta_{2} + 88 \beta _1 - 247 \) |
\(\nu^{7}\) | \(=\) | \( - 60 \beta_{11} + 135 \beta_{10} + 17 \beta_{9} - 115 \beta_{8} - 145 \beta_{7} + 668 \beta_{6} + 14 \beta_{5} + 203 \beta_{4} - 117 \beta_{3} + 400 \beta_{2} - 244 \beta _1 - 526 \) |
\(\nu^{8}\) | \(=\) | \( - 380 \beta_{11} + 778 \beta_{10} + 515 \beta_{9} - 363 \beta_{8} - 1632 \beta_{7} + 1517 \beta_{6} - 2765 \beta_{5} - 340 \beta_{4} + 304 \beta_{3} + 91 \beta_{2} - 949 \beta _1 + 2008 \) |
\(\nu^{9}\) | \(=\) | \( 264 \beta_{11} - 673 \beta_{10} + 454 \beta_{9} + 839 \beta_{8} - 287 \beta_{7} - 5265 \beta_{6} - 3066 \beta_{5} - 2790 \beta_{4} + 1329 \beta_{3} - 3791 \beta_{2} + 1710 \beta _1 + 7116 \) |
\(\nu^{10}\) | \(=\) | \( 4383 \beta_{11} - 9560 \beta_{10} - 4630 \beta_{9} + 4411 \beta_{8} + 17550 \beta_{7} - 20512 \beta_{6} + 25127 \beta_{5} + 650 \beta_{4} - 1974 \beta_{3} - 5273 \beta_{2} + 10483 \beta _1 - 13713 \) |
\(\nu^{11}\) | \(=\) | \( 1759 \beta_{11} - 3186 \beta_{10} - 9684 \beta_{9} - 4506 \beta_{8} + 22413 \beta_{7} + 33458 \beta_{6} + 57148 \beta_{5} + 31493 \beta_{4} - 14866 \beta_{3} + 31330 \beta_{2} - 8841 \beta _1 - 85518 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1274\mathbb{Z}\right)^\times\).
\(n\) | \(197\) | \(885\) |
\(\chi(n)\) | \(1 - \beta_{7}\) | \(-1 + \beta_{7}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
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459.1 |
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− | 1.00000i | −1.27815 | − | 2.21381i | −1.00000 | −3.02030 | + | 1.74377i | −2.21381 | + | 1.27815i | 0 | 1.00000i | −1.76732 | + | 3.06108i | 1.74377 | + | 3.02030i | |||||||||||||||||||||||||||||||||||||||||||
459.2 | − | 1.00000i | −0.233273 | − | 0.404040i | −1.00000 | 2.93529 | − | 1.69469i | −0.404040 | + | 0.233273i | 0 | 1.00000i | 1.39117 | − | 2.40957i | −1.69469 | − | 2.93529i | ||||||||||||||||||||||||||||||||||||||||||||
459.3 | − | 1.00000i | 1.14539 | + | 1.98388i | −1.00000 | −0.781015 | + | 0.450919i | 1.98388 | − | 1.14539i | 0 | 1.00000i | −1.12385 | + | 1.94657i | 0.450919 | + | 0.781015i | ||||||||||||||||||||||||||||||||||||||||||||
459.4 | 1.00000i | −0.432757 | − | 0.749558i | −1.00000 | 3.21409 | − | 1.85566i | 0.749558 | − | 0.432757i | 0 | − | 1.00000i | 1.12544 | − | 1.94932i | 1.85566 | + | 3.21409i | ||||||||||||||||||||||||||||||||||||||||||||
459.5 | 1.00000i | 0.126439 | + | 0.218999i | −1.00000 | −0.993985 | + | 0.573878i | −0.218999 | + | 0.126439i | 0 | − | 1.00000i | 1.46803 | − | 2.54270i | −0.573878 | − | 0.993985i | ||||||||||||||||||||||||||||||||||||||||||||
459.6 | 1.00000i | 1.67234 | + | 2.89658i | −1.00000 | −1.35408 | + | 0.781779i | −2.89658 | + | 1.67234i | 0 | − | 1.00000i | −4.09347 | + | 7.09010i | −0.781779 | − | 1.35408i | ||||||||||||||||||||||||||||||||||||||||||||
569.1 | − | 1.00000i | −0.432757 | + | 0.749558i | −1.00000 | 3.21409 | + | 1.85566i | 0.749558 | + | 0.432757i | 0 | 1.00000i | 1.12544 | + | 1.94932i | 1.85566 | − | 3.21409i | ||||||||||||||||||||||||||||||||||||||||||||
569.2 | − | 1.00000i | 0.126439 | − | 0.218999i | −1.00000 | −0.993985 | − | 0.573878i | −0.218999 | − | 0.126439i | 0 | 1.00000i | 1.46803 | + | 2.54270i | −0.573878 | + | 0.993985i | ||||||||||||||||||||||||||||||||||||||||||||
569.3 | − | 1.00000i | 1.67234 | − | 2.89658i | −1.00000 | −1.35408 | − | 0.781779i | −2.89658 | − | 1.67234i | 0 | 1.00000i | −4.09347 | − | 7.09010i | −0.781779 | + | 1.35408i | ||||||||||||||||||||||||||||||||||||||||||||
569.4 | 1.00000i | −1.27815 | + | 2.21381i | −1.00000 | −3.02030 | − | 1.74377i | −2.21381 | − | 1.27815i | 0 | − | 1.00000i | −1.76732 | − | 3.06108i | 1.74377 | − | 3.02030i | ||||||||||||||||||||||||||||||||||||||||||||
569.5 | 1.00000i | −0.233273 | + | 0.404040i | −1.00000 | 2.93529 | + | 1.69469i | −0.404040 | − | 0.233273i | 0 | − | 1.00000i | 1.39117 | + | 2.40957i | −1.69469 | + | 2.93529i | ||||||||||||||||||||||||||||||||||||||||||||
569.6 | 1.00000i | 1.14539 | − | 1.98388i | −1.00000 | −0.781015 | − | 0.450919i | 1.98388 | + | 1.14539i | 0 | − | 1.00000i | −1.12385 | − | 1.94657i | 0.450919 | − | 0.781015i | ||||||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
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1.a | even | 1 | 1 | trivial |
91.k | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
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Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 1274.2.o.e | 12 | |
7.b | odd | 2 | 1 | 1274.2.o.d | 12 | ||
7.c | even | 3 | 1 | 1274.2.m.c | 12 | ||
7.c | even | 3 | 1 | 1274.2.v.d | 12 | ||
7.d | odd | 6 | 1 | 182.2.m.b | ✓ | 12 | |
7.d | odd | 6 | 1 | 1274.2.v.e | 12 | ||
13.e | even | 6 | 1 | 1274.2.v.d | 12 | ||
21.g | even | 6 | 1 | 1638.2.bj.g | 12 | ||
28.f | even | 6 | 1 | 1456.2.cc.d | 12 | ||
91.k | even | 6 | 1 | inner | 1274.2.o.e | 12 | |
91.l | odd | 6 | 1 | 1274.2.o.d | 12 | ||
91.l | odd | 6 | 1 | 2366.2.d.r | 12 | ||
91.p | odd | 6 | 1 | 182.2.m.b | ✓ | 12 | |
91.t | odd | 6 | 1 | 1274.2.v.e | 12 | ||
91.u | even | 6 | 1 | 1274.2.m.c | 12 | ||
91.v | odd | 6 | 1 | 2366.2.d.r | 12 | ||
91.ba | even | 12 | 1 | 2366.2.a.bf | 6 | ||
91.ba | even | 12 | 1 | 2366.2.a.bh | 6 | ||
273.y | even | 6 | 1 | 1638.2.bj.g | 12 | ||
364.bp | even | 6 | 1 | 1456.2.cc.d | 12 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
182.2.m.b | ✓ | 12 | 7.d | odd | 6 | 1 | |
182.2.m.b | ✓ | 12 | 91.p | odd | 6 | 1 | |
1274.2.m.c | 12 | 7.c | even | 3 | 1 | ||
1274.2.m.c | 12 | 91.u | even | 6 | 1 | ||
1274.2.o.d | 12 | 7.b | odd | 2 | 1 | ||
1274.2.o.d | 12 | 91.l | odd | 6 | 1 | ||
1274.2.o.e | 12 | 1.a | even | 1 | 1 | trivial | |
1274.2.o.e | 12 | 91.k | even | 6 | 1 | inner | |
1274.2.v.d | 12 | 7.c | even | 3 | 1 | ||
1274.2.v.d | 12 | 13.e | even | 6 | 1 | ||
1274.2.v.e | 12 | 7.d | odd | 6 | 1 | ||
1274.2.v.e | 12 | 91.t | odd | 6 | 1 | ||
1456.2.cc.d | 12 | 28.f | even | 6 | 1 | ||
1456.2.cc.d | 12 | 364.bp | even | 6 | 1 | ||
1638.2.bj.g | 12 | 21.g | even | 6 | 1 | ||
1638.2.bj.g | 12 | 273.y | even | 6 | 1 | ||
2366.2.a.bf | 6 | 91.ba | even | 12 | 1 | ||
2366.2.a.bh | 6 | 91.ba | even | 12 | 1 | ||
2366.2.d.r | 12 | 91.l | odd | 6 | 1 | ||
2366.2.d.r | 12 | 91.v | odd | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{12} - 2 T_{3}^{11} + 14 T_{3}^{10} - 4 T_{3}^{9} + 103 T_{3}^{8} - 34 T_{3}^{7} + 354 T_{3}^{6} + 296 T_{3}^{5} + 397 T_{3}^{4} + 90 T_{3}^{3} + 46 T_{3}^{2} - 4 T_{3} + 4 \)
acting on \(S_{2}^{\mathrm{new}}(1274, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{2} + 1)^{6} \)
$3$
\( T^{12} - 2 T^{11} + 14 T^{10} - 4 T^{9} + \cdots + 4 \)
$5$
\( T^{12} - 21 T^{10} + 340 T^{8} + \cdots + 5041 \)
$7$
\( T^{12} \)
$11$
\( T^{12} - 18 T^{11} + 118 T^{10} + \cdots + 495616 \)
$13$
\( T^{12} - 8 T^{11} + 15 T^{10} + \cdots + 4826809 \)
$17$
\( (T^{6} - 4 T^{5} - 39 T^{4} + 168 T^{3} + \cdots - 176)^{2} \)
$19$
\( T^{12} - 12 T^{11} + \cdots + 369869824 \)
$23$
\( (T^{6} - 6 T^{5} - 68 T^{4} + 488 T^{3} + \cdots + 3142)^{2} \)
$29$
\( T^{12} + 10 T^{11} + 139 T^{10} + \cdots + 135424 \)
$31$
\( T^{12} + 18 T^{11} + 118 T^{10} + \cdots + 1024 \)
$37$
\( T^{12} + 310 T^{10} + 35529 T^{8} + \cdots + 1024 \)
$41$
\( T^{12} - 24 T^{11} + 175 T^{10} + \cdots + 1024 \)
$43$
\( T^{12} - 26 T^{11} + 462 T^{10} + \cdots + 8667136 \)
$47$
\( T^{12} + 48 T^{11} + 1016 T^{10} + \cdots + 31719424 \)
$53$
\( T^{12} + 18 T^{11} + \cdots + 2018525184 \)
$59$
\( T^{12} + 168 T^{10} + 7294 T^{8} + \cdots + 4 \)
$61$
\( T^{12} - 28 T^{11} + \cdots + 80364879169 \)
$67$
\( T^{12} - 42 T^{11} + 742 T^{10} + \cdots + 1024 \)
$71$
\( T^{12} - 48 T^{11} + \cdots + 750321664 \)
$73$
\( T^{12} - 48 T^{11} + \cdots + 2708994304 \)
$79$
\( T^{12} + 22 T^{11} + 422 T^{10} + \cdots + 64513024 \)
$83$
\( T^{12} + 464 T^{10} + \cdots + 879478336 \)
$89$
\( T^{12} + 416 T^{10} + \cdots + 10303062016 \)
$97$
\( T^{12} + 60 T^{11} + \cdots + 6400000000 \)
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