Newspace parameters
| Level: | \( N \) | \(=\) | \( 200 = 2^{3} \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 200.d (of order \(2\), degree \(1\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(11.8003820011\) |
| Analytic rank: | \(0\) |
| Dimension: | \(12\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{12} - \cdots)\) |
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| Defining polynomial: |
\( x^{12} - 4x^{11} + 7x^{10} - 12x^{9} + 21x^{8} - 68x^{6} + 336x^{4} - 768x^{3} + 1792x^{2} - 4096x + 4096 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{15}\cdot 5^{4} \) |
| Twist minimal: | no (minimal twist has level 40) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) 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
\( x^{12} - 4x^{11} + 7x^{10} - 12x^{9} + 21x^{8} - 68x^{6} + 336x^{4} - 768x^{3} + 1792x^{2} - 4096x + 4096 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 2 \nu^{11} + 5 \nu^{10} - 10 \nu^{9} + 27 \nu^{8} - 30 \nu^{7} - 55 \nu^{6} + 64 \nu^{5} + \cdots + 5888 ) / 960 \)
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| \(\beta_{2}\) | \(=\) |
\( ( 7 \nu^{11} - 22 \nu^{10} + 65 \nu^{9} - 186 \nu^{8} + 99 \nu^{7} + 206 \nu^{6} - 404 \nu^{5} + \cdots - 31360 ) / 1920 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 93 \nu^{11} - 204 \nu^{10} + 379 \nu^{9} - 388 \nu^{8} + 817 \nu^{7} + 1672 \nu^{6} - 4708 \nu^{5} + \cdots - 191488 ) / 15360 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 23 \nu^{11} - 12 \nu^{10} + 95 \nu^{9} - 36 \nu^{8} + 29 \nu^{7} - 424 \nu^{6} - 644 \nu^{5} + \cdots - 1280 ) / 3840 \)
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| \(\beta_{5}\) | \(=\) |
\( ( - 97 \nu^{11} + 220 \nu^{10} + 105 \nu^{9} + 52 \nu^{8} + 635 \nu^{7} - 1800 \nu^{6} + 3444 \nu^{5} + \cdots + 2048 ) / 15360 \)
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| \(\beta_{6}\) | \(=\) |
\( ( - 3 \nu^{11} + 4 \nu^{10} + 3 \nu^{9} + 12 \nu^{8} - 55 \nu^{7} - 72 \nu^{6} + 132 \nu^{5} + \cdots + 7712 ) / 480 \)
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| \(\beta_{7}\) | \(=\) |
\( ( - 99 \nu^{11} + 260 \nu^{10} - 325 \nu^{9} + 364 \nu^{8} - 15 \nu^{7} - 1960 \nu^{6} + 2908 \nu^{5} + \cdots + 106496 ) / 7680 \)
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| \(\beta_{8}\) | \(=\) |
\( ( 49 \nu^{11} - 156 \nu^{10} - 9 \nu^{9} + 236 \nu^{8} - 27 \nu^{7} + 168 \nu^{6} - 1476 \nu^{5} + \cdots - 19200 ) / 3840 \)
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| \(\beta_{9}\) | \(=\) |
\( ( 235 \nu^{11} - 332 \nu^{10} + 125 \nu^{9} - 804 \nu^{8} + 2359 \nu^{7} + 416 \nu^{6} - 1820 \nu^{5} + \cdots - 340992 ) / 15360 \)
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| \(\beta_{10}\) | \(=\) |
\( ( - 59 \nu^{11} + 134 \nu^{10} - 189 \nu^{9} + 314 \nu^{8} - 407 \nu^{7} - 1182 \nu^{6} + \cdots + 109312 ) / 3840 \)
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| \(\beta_{11}\) | \(=\) |
\( ( - 173 \nu^{11} + 352 \nu^{10} - 459 \nu^{9} + 1424 \nu^{8} - 1633 \nu^{7} - 1956 \nu^{6} + \cdots + 421376 ) / 7680 \)
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| \(\nu\) | \(=\) |
\( ( -\beta_{7} - \beta_{6} + \beta_{5} - \beta_{3} + \beta_{2} + 5\beta _1 + 3 ) / 10 \)
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| \(\nu^{2}\) | \(=\) |
\( ( 2\beta_{10} - \beta_{7} - \beta_{6} + 2\beta_{3} + \beta_{2} + 2\beta _1 + 2 ) / 10 \)
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| \(\nu^{3}\) | \(=\) |
\( ( - 3 \beta_{11} + 3 \beta_{10} - 4 \beta_{9} + 3 \beta_{8} - 2 \beta_{7} + 8 \beta_{5} - 5 \beta_{4} + \cdots + 30 ) / 20 \)
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| \(\nu^{4}\) | \(=\) |
\( ( - \beta_{11} + \beta_{10} - \beta_{8} - 4 \beta_{7} - 2 \beta_{6} + 2 \beta_{5} + \beta_{4} - 4 \beta_{3} + \cdots + 2 ) / 4 \)
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| \(\nu^{5}\) | \(=\) |
\( ( 7 \beta_{11} + 3 \beta_{10} + 6 \beta_{9} + 3 \beta_{8} - 14 \beta_{7} - 3 \beta_{6} + 16 \beta_{5} + \cdots - 91 ) / 10 \)
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| \(\nu^{6}\) | \(=\) |
\( ( 17 \beta_{11} + 23 \beta_{10} - 44 \beta_{9} - 7 \beta_{8} - 42 \beta_{7} - 64 \beta_{6} + 2 \beta_{5} + \cdots + 22 ) / 20 \)
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| \(\nu^{7}\) | \(=\) |
\( ( 33 \beta_{11} - 37 \beta_{10} + 24 \beta_{9} + 27 \beta_{8} + 62 \beta_{7} - 86 \beta_{6} + 80 \beta_{5} + \cdots + 1540 ) / 20 \)
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| \(\nu^{8}\) | \(=\) |
\( ( 11 \beta_{11} - 7 \beta_{10} + 2 \beta_{9} + 3 \beta_{8} - 7 \beta_{7} - 15 \beta_{6} + 16 \beta_{5} + \cdots - 114 ) / 2 \)
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| \(\nu^{9}\) | \(=\) |
\( ( 139 \beta_{11} + 237 \beta_{10} + 12 \beta_{9} + 21 \beta_{8} + 290 \beta_{7} + 544 \beta_{6} + \cdots + 610 ) / 20 \)
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| \(\nu^{10}\) | \(=\) |
\( ( - 387 \beta_{11} - 109 \beta_{10} - 416 \beta_{9} - 123 \beta_{8} + 796 \beta_{7} + 970 \beta_{6} + \cdots + 3198 ) / 20 \)
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| \(\nu^{11}\) | \(=\) |
\( ( - 223 \beta_{11} - 99 \beta_{10} + 586 \beta_{9} - 467 \beta_{8} - 86 \beta_{7} + 859 \beta_{6} + \cdots + 3559 ) / 10 \)
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Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/200\mathbb{Z}\right)^\times\).
| \(n\) | \(101\) | \(151\) | \(177\) |
| \(\chi(n)\) | \(-1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 101.1 |
|
−2.74276 | − | 0.690860i | − | 4.24443i | 7.04543 | + | 3.78972i | 0 | −2.93231 | + | 11.6414i | 14.6308 | −16.7057 | − | 15.2617i | 8.98481 | 0 | |||||||||||||||||||||||||||||||||||||||||||||
| 101.2 | −2.74276 | + | 0.690860i | 4.24443i | 7.04543 | − | 3.78972i | 0 | −2.93231 | − | 11.6414i | 14.6308 | −16.7057 | + | 15.2617i | 8.98481 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 101.3 | −2.20360 | − | 1.77318i | 9.57890i | 1.71169 | + | 7.81474i | 0 | 16.9851 | − | 21.1081i | −21.5703 | 10.0850 | − | 20.2557i | −64.7554 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 101.4 | −2.20360 | + | 1.77318i | − | 9.57890i | 1.71169 | − | 7.81474i | 0 | 16.9851 | + | 21.1081i | −21.5703 | 10.0850 | + | 20.2557i | −64.7554 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 101.5 | −1.52528 | − | 2.38191i | − | 1.51777i | −3.34703 | + | 7.26618i | 0 | −3.61520 | + | 2.31503i | −5.13620 | 22.4126 | − | 3.11063i | 24.6964 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 101.6 | −1.52528 | + | 2.38191i | 1.51777i | −3.34703 | − | 7.26618i | 0 | −3.61520 | − | 2.31503i | −5.13620 | 22.4126 | + | 3.11063i | 24.6964 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 101.7 | 0.337480 | − | 2.80822i | − | 7.99849i | −7.77221 | − | 1.89544i | 0 | −22.4615 | − | 2.69933i | −9.93501 | −7.94578 | + | 21.1864i | −36.9759 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 101.8 | 0.337480 | + | 2.80822i | 7.99849i | −7.77221 | + | 1.89544i | 0 | −22.4615 | + | 2.69933i | −9.93501 | −7.94578 | − | 21.1864i | −36.9759 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 101.9 | 2.54175 | − | 1.24077i | 0.888401i | 4.92097 | − | 6.30746i | 0 | 1.10230 | + | 2.25809i | −26.6173 | 4.68175 | − | 22.1378i | 26.2107 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 101.10 | 2.54175 | + | 1.24077i | − | 0.888401i | 4.92097 | + | 6.30746i | 0 | 1.10230 | − | 2.25809i | −26.6173 | 4.68175 | + | 22.1378i | 26.2107 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 101.11 | 2.59241 | − | 1.13111i | − | 6.25785i | 5.44116 | − | 5.86462i | 0 | −7.07834 | − | 16.2229i | 34.6280 | 7.47214 | − | 21.3581i | −12.1606 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 101.12 | 2.59241 | + | 1.13111i | 6.25785i | 5.44116 | + | 5.86462i | 0 | −7.07834 | + | 16.2229i | 34.6280 | 7.47214 | + | 21.3581i | −12.1606 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 8.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 200.4.d.b | 12 | |
| 4.b | odd | 2 | 1 | 800.4.d.d | 12 | ||
| 5.b | even | 2 | 1 | 40.4.d.a | ✓ | 12 | |
| 5.c | odd | 4 | 1 | 200.4.f.b | 12 | ||
| 5.c | odd | 4 | 1 | 200.4.f.c | 12 | ||
| 8.b | even | 2 | 1 | inner | 200.4.d.b | 12 | |
| 8.d | odd | 2 | 1 | 800.4.d.d | 12 | ||
| 15.d | odd | 2 | 1 | 360.4.k.c | 12 | ||
| 20.d | odd | 2 | 1 | 160.4.d.a | 12 | ||
| 20.e | even | 4 | 1 | 800.4.f.b | 12 | ||
| 20.e | even | 4 | 1 | 800.4.f.c | 12 | ||
| 40.e | odd | 2 | 1 | 160.4.d.a | 12 | ||
| 40.f | even | 2 | 1 | 40.4.d.a | ✓ | 12 | |
| 40.i | odd | 4 | 1 | 200.4.f.b | 12 | ||
| 40.i | odd | 4 | 1 | 200.4.f.c | 12 | ||
| 40.k | even | 4 | 1 | 800.4.f.b | 12 | ||
| 40.k | even | 4 | 1 | 800.4.f.c | 12 | ||
| 60.h | even | 2 | 1 | 1440.4.k.c | 12 | ||
| 80.k | odd | 4 | 1 | 1280.4.a.ba | 6 | ||
| 80.k | odd | 4 | 1 | 1280.4.a.bd | 6 | ||
| 80.q | even | 4 | 1 | 1280.4.a.bb | 6 | ||
| 80.q | even | 4 | 1 | 1280.4.a.bc | 6 | ||
| 120.i | odd | 2 | 1 | 360.4.k.c | 12 | ||
| 120.m | even | 2 | 1 | 1440.4.k.c | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 40.4.d.a | ✓ | 12 | 5.b | even | 2 | 1 | |
| 40.4.d.a | ✓ | 12 | 40.f | even | 2 | 1 | |
| 160.4.d.a | 12 | 20.d | odd | 2 | 1 | ||
| 160.4.d.a | 12 | 40.e | odd | 2 | 1 | ||
| 200.4.d.b | 12 | 1.a | even | 1 | 1 | trivial | |
| 200.4.d.b | 12 | 8.b | even | 2 | 1 | inner | |
| 200.4.f.b | 12 | 5.c | odd | 4 | 1 | ||
| 200.4.f.b | 12 | 40.i | odd | 4 | 1 | ||
| 200.4.f.c | 12 | 5.c | odd | 4 | 1 | ||
| 200.4.f.c | 12 | 40.i | odd | 4 | 1 | ||
| 360.4.k.c | 12 | 15.d | odd | 2 | 1 | ||
| 360.4.k.c | 12 | 120.i | odd | 2 | 1 | ||
| 800.4.d.d | 12 | 4.b | odd | 2 | 1 | ||
| 800.4.d.d | 12 | 8.d | odd | 2 | 1 | ||
| 800.4.f.b | 12 | 20.e | even | 4 | 1 | ||
| 800.4.f.b | 12 | 40.k | even | 4 | 1 | ||
| 800.4.f.c | 12 | 20.e | even | 4 | 1 | ||
| 800.4.f.c | 12 | 40.k | even | 4 | 1 | ||
| 1280.4.a.ba | 6 | 80.k | odd | 4 | 1 | ||
| 1280.4.a.bb | 6 | 80.q | even | 4 | 1 | ||
| 1280.4.a.bc | 6 | 80.q | even | 4 | 1 | ||
| 1280.4.a.bd | 6 | 80.k | odd | 4 | 1 | ||
| 1440.4.k.c | 12 | 60.h | even | 2 | 1 | ||
| 1440.4.k.c | 12 | 120.m | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(200, [\chi])\):
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\( T_{3}^{12} + 216T_{3}^{10} + 16140T_{3}^{8} + 493760T_{3}^{6} + 5547312T_{3}^{4} + 13618560T_{3}^{2} + 7529536 \)
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\( T_{7}^{6} + 14T_{7}^{5} - 1258T_{7}^{4} - 23408T_{7}^{3} + 166612T_{7}^{2} + 4186552T_{7} + 14843128 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} + 2 T^{11} + \cdots + 262144 \)
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| $3$ |
\( T^{12} + 216 T^{10} + \cdots + 7529536 \)
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| $5$ |
\( T^{12} \)
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| $7$ |
\( (T^{6} + 14 T^{5} + \cdots + 14843128)^{2} \)
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| $11$ |
\( T^{12} + \cdots + 26\!\cdots\!00 \)
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| $13$ |
\( T^{12} + \cdots + 24\!\cdots\!00 \)
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| $17$ |
\( (T^{6} - 14500 T^{4} + \cdots - 7473839808)^{2} \)
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| $19$ |
\( T^{12} + \cdots + 17\!\cdots\!24 \)
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| $23$ |
\( (T^{6} + 302 T^{5} + \cdots + 881168216)^{2} \)
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| $29$ |
\( T^{12} + \cdots + 11\!\cdots\!00 \)
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| $31$ |
\( (T^{6} + \cdots - 1437816300032)^{2} \)
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| $37$ |
\( T^{12} + \cdots + 21\!\cdots\!04 \)
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| $41$ |
\( (T^{6} + \cdots - 71667547865600)^{2} \)
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| $43$ |
\( T^{12} + \cdots + 25\!\cdots\!04 \)
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| $47$ |
\( (T^{6} + \cdots - 72048375466472)^{2} \)
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| $53$ |
\( T^{12} + \cdots + 86\!\cdots\!76 \)
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| $59$ |
\( T^{12} + \cdots + 10\!\cdots\!24 \)
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| $61$ |
\( T^{12} + \cdots + 61\!\cdots\!00 \)
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| $67$ |
\( T^{12} + \cdots + 87\!\cdots\!36 \)
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| $71$ |
\( (T^{6} + \cdots - 36\!\cdots\!48)^{2} \)
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| $73$ |
\( (T^{6} + \cdots - 378730163491776)^{2} \)
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| $79$ |
\( (T^{6} + \cdots - 44\!\cdots\!00)^{2} \)
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| $83$ |
\( T^{12} + \cdots + 78\!\cdots\!24 \)
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| $89$ |
\( (T^{6} + \cdots - 62\!\cdots\!00)^{2} \)
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| $97$ |
\( (T^{6} + \cdots - 33\!\cdots\!28)^{2} \)
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