Newspace parameters
| Level: | \( N \) | \(=\) | \( 605 = 5 \cdot 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 605.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(97.0322109869\) |
| Analytic rank: | \(0\) |
| Dimension: | \(18\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{18} - \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{18} - 481 x^{16} + 94807 x^{14} - 9871083 x^{12} + 583599384 x^{10} - 19602093264 x^{8} + \cdots - 4321382510592 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{13}\cdot 3\cdot 11^{6} \) |
| 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 \(1,\beta_1,\ldots,\beta_{17}\) 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
\( x^{18} - 481 x^{16} + 94807 x^{14} - 9871083 x^{12} + 583599384 x^{10} - 19602093264 x^{8} + \cdots - 4321382510592 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 22\!\cdots\!43 \nu^{16} + \cdots - 12\!\cdots\!96 ) / 10\!\cdots\!80 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 22\!\cdots\!43 \nu^{16} + \cdots + 44\!\cdots\!44 ) / 10\!\cdots\!80 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 25\!\cdots\!53 \nu^{16} + \cdots - 91\!\cdots\!44 ) / 10\!\cdots\!80 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 32\!\cdots\!61 \nu^{16} + \cdots - 41\!\cdots\!44 ) / 10\!\cdots\!80 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 50\!\cdots\!51 \nu^{16} + \cdots - 18\!\cdots\!96 ) / 10\!\cdots\!80 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 22\!\cdots\!11 \nu^{16} + \cdots - 59\!\cdots\!20 ) / 13\!\cdots\!60 \)
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| \(\beta_{8}\) | \(=\) |
\( ( - 37\!\cdots\!81 \nu^{16} + \cdots + 74\!\cdots\!68 ) / 21\!\cdots\!76 \)
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| \(\beta_{9}\) | \(=\) |
\( ( 75\!\cdots\!81 \nu^{16} + \cdots + 54\!\cdots\!40 ) / 10\!\cdots\!80 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 188009014745 \nu^{17} - 85359084526949 \nu^{15} + \cdots - 15\!\cdots\!48 \nu ) / 53\!\cdots\!60 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 54\!\cdots\!25 \nu^{17} + \cdots + 35\!\cdots\!96 \nu ) / 39\!\cdots\!60 \)
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| \(\beta_{12}\) | \(=\) |
\( ( 19\!\cdots\!79 \nu^{17} + \cdots - 80\!\cdots\!16 \nu ) / 13\!\cdots\!60 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 22\!\cdots\!43 \nu^{17} + \cdots + 48\!\cdots\!64 \nu ) / 10\!\cdots\!80 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 28\!\cdots\!75 \nu^{17} + \cdots - 17\!\cdots\!68 \nu ) / 13\!\cdots\!60 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - 31\!\cdots\!93 \nu^{17} + \cdots - 35\!\cdots\!04 \nu ) / 66\!\cdots\!80 \)
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| \(\beta_{16}\) | \(=\) |
\( ( - 86\!\cdots\!15 \nu^{17} + \cdots - 22\!\cdots\!72 \nu ) / 10\!\cdots\!80 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( 32\!\cdots\!97 \nu^{17} + \cdots + 21\!\cdots\!40 \nu ) / 22\!\cdots\!60 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{3} + \beta_{2} + 53 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{14} - \beta_{13} - \beta_{12} + \beta_{11} + 2\beta_{10} + 86\beta_1 \)
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| \(\nu^{4}\) | \(=\) |
\( 2\beta_{9} + 5\beta_{8} + \beta_{6} - 2\beta_{5} + \beta_{4} - 141\beta_{3} + 114\beta_{2} + 4589 \)
|
| \(\nu^{5}\) | \(=\) |
\( 5 \beta_{17} - 8 \beta_{16} + 4 \beta_{15} + 130 \beta_{14} - 179 \beta_{13} - 168 \beta_{12} + \cdots + 8391 \beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( 370 \beta_{9} + 798 \beta_{8} + 2 \beta_{7} + 150 \beta_{6} - 450 \beta_{5} + 74 \beta_{4} + \cdots + 449431 \)
|
| \(\nu^{7}\) | \(=\) |
\( 516 \beta_{17} - 1756 \beta_{16} + 804 \beta_{15} + 14579 \beta_{14} - 25127 \beta_{13} + \cdots + 865408 \beta_1 \)
|
| \(\nu^{8}\) | \(=\) |
\( 52814 \beta_{9} + 103833 \beta_{8} + 1258 \beta_{7} + 15577 \beta_{6} - 70636 \beta_{5} + \cdots + 46530639 \)
|
| \(\nu^{9}\) | \(=\) |
\( 17141 \beta_{17} - 276260 \beta_{16} + 128752 \beta_{15} + 1584712 \beta_{14} - 3241193 \beta_{13} + \cdots + 92076281 \beta_1 \)
|
| \(\nu^{10}\) | \(=\) |
\( 6887398 \beta_{9} + 12752654 \beta_{8} + 360748 \beta_{7} + 1421434 \beta_{6} - 9871140 \beta_{5} + \cdots + 4972446533 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 4440784 \beta_{17} - 38156728 \beta_{16} + 19029856 \beta_{15} + 171555377 \beta_{14} + \cdots + 9998219782 \beta_1 \)
|
| \(\nu^{12}\) | \(=\) |
\( 861423578 \beta_{9} + 1533349217 \beta_{8} + 73518780 \beta_{7} + 120569109 \beta_{6} + \cdots + 542556350657 \)
|
| \(\nu^{13}\) | \(=\) |
\( - 1296503391 \beta_{17} - 4938383792 \beta_{16} + 2679012732 \beta_{15} + 18652547914 \beta_{14} + \cdots + 1101972136031 \beta_1 \)
|
| \(\nu^{14}\) | \(=\) |
\( 105379876858 \beta_{9} + 182772458034 \beta_{8} + 12589919462 \beta_{7} + 9450604274 \beta_{6} + \cdots + 60104608843035 \)
|
| \(\nu^{15}\) | \(=\) |
\( - 238646307896 \beta_{17} - 616910346772 \beta_{16} + 363921406604 \beta_{15} + \cdots + 122892873941944 \beta_1 \)
|
| \(\nu^{16}\) | \(=\) |
\( 12730225629974 \beta_{9} + 21709030334173 \beta_{8} + 1949795752790 \beta_{7} + 639371461397 \beta_{6} + \cdots + 67\!\cdots\!71 \)
|
| \(\nu^{17}\) | \(=\) |
\( - 37359448909463 \beta_{17} - 75557073640460 \beta_{16} + 48107196350408 \beta_{15} + \cdots + 13\!\cdots\!33 \beta_1 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−10.9368 | 26.9053 | 87.6133 | 25.0000 | −294.257 | −22.0537 | −608.231 | 480.894 | −273.420 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −10.1466 | 19.7939 | 70.9544 | 25.0000 | −200.842 | −106.032 | −395.257 | 148.800 | −253.666 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −9.70936 | −15.1249 | 62.2717 | 25.0000 | 146.853 | 82.2514 | −293.919 | −14.2387 | −242.734 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −8.31004 | −17.3654 | 37.0567 | 25.0000 | 144.307 | −159.925 | −42.0217 | 58.5560 | −207.751 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −6.61246 | 2.44105 | 11.7246 | 25.0000 | −16.1414 | 221.647 | 134.070 | −237.041 | −165.311 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −5.74694 | 12.0494 | 1.02729 | 25.0000 | −69.2472 | −129.982 | 177.998 | −97.8118 | −143.673 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −3.50235 | −3.85877 | −19.7336 | 25.0000 | 13.5147 | 185.291 | 181.189 | −228.110 | −87.5587 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | −2.35283 | 25.1485 | −26.4642 | 25.0000 | −59.1702 | −82.5944 | 137.556 | 389.449 | −58.8207 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | −0.741407 | −22.9892 | −31.4503 | 25.0000 | 17.0444 | 50.1534 | 47.0425 | 285.504 | −18.5352 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 0.741407 | −22.9892 | −31.4503 | 25.0000 | −17.0444 | −50.1534 | −47.0425 | 285.504 | 18.5352 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 2.35283 | 25.1485 | −26.4642 | 25.0000 | 59.1702 | 82.5944 | −137.556 | 389.449 | 58.8207 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 3.50235 | −3.85877 | −19.7336 | 25.0000 | −13.5147 | −185.291 | −181.189 | −228.110 | 87.5587 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.13 | 5.74694 | 12.0494 | 1.02729 | 25.0000 | 69.2472 | 129.982 | −177.998 | −97.8118 | 143.673 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.14 | 6.61246 | 2.44105 | 11.7246 | 25.0000 | 16.1414 | −221.647 | −134.070 | −237.041 | 165.311 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.15 | 8.31004 | −17.3654 | 37.0567 | 25.0000 | −144.307 | 159.925 | 42.0217 | 58.5560 | 207.751 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.16 | 9.70936 | −15.1249 | 62.2717 | 25.0000 | −146.853 | −82.2514 | 293.919 | −14.2387 | 242.734 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.17 | 10.1466 | 19.7939 | 70.9544 | 25.0000 | 200.842 | 106.032 | 395.257 | 148.800 | 253.666 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.18 | 10.9368 | 26.9053 | 87.6133 | 25.0000 | 294.257 | 22.0537 | 608.231 | 480.894 | 273.420 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \( -1 \) |
| \(11\) | \( +1 \) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 11.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 605.6.a.m | ✓ | 18 |
| 11.b | odd | 2 | 1 | inner | 605.6.a.m | ✓ | 18 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 605.6.a.m | ✓ | 18 | 1.a | even | 1 | 1 | trivial |
| 605.6.a.m | ✓ | 18 | 11.b | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{18} - 481 T_{2}^{16} + 94807 T_{2}^{14} - 9871083 T_{2}^{12} + 583599384 T_{2}^{10} + \cdots - 4321382510592 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(605))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{18} + \cdots - 4321382510592 \)
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| $3$ |
\( (T^{9} - 27 T^{8} + \cdots - 9178529283)^{2} \)
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| $5$ |
\( (T - 25)^{18} \)
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| $7$ |
\( T^{18} + \cdots - 46\!\cdots\!47 \)
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| $11$ |
\( T^{18} \)
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| $13$ |
\( T^{18} + \cdots - 11\!\cdots\!72 \)
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| $17$ |
\( T^{18} + \cdots - 22\!\cdots\!72 \)
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| $19$ |
\( T^{18} + \cdots - 60\!\cdots\!92 \)
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| $23$ |
\( (T^{9} + \cdots + 51\!\cdots\!16)^{2} \)
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| $29$ |
\( T^{18} + \cdots - 29\!\cdots\!28 \)
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| $31$ |
\( (T^{9} + \cdots - 54\!\cdots\!56)^{2} \)
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| $37$ |
\( (T^{9} + \cdots - 15\!\cdots\!56)^{2} \)
|
| $41$ |
\( T^{18} + \cdots - 33\!\cdots\!47 \)
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| $43$ |
\( T^{18} + \cdots - 39\!\cdots\!87 \)
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| $47$ |
\( (T^{9} + \cdots - 13\!\cdots\!33)^{2} \)
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| $53$ |
\( (T^{9} + \cdots - 22\!\cdots\!48)^{2} \)
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| $59$ |
\( (T^{9} + \cdots - 84\!\cdots\!68)^{2} \)
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| $61$ |
\( T^{18} + \cdots - 70\!\cdots\!07 \)
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| $67$ |
\( (T^{9} + \cdots - 60\!\cdots\!63)^{2} \)
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| $71$ |
\( (T^{9} + \cdots - 19\!\cdots\!48)^{2} \)
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| $73$ |
\( T^{18} + \cdots - 43\!\cdots\!48 \)
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| $79$ |
\( T^{18} + \cdots - 28\!\cdots\!32 \)
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| $83$ |
\( T^{18} + \cdots - 20\!\cdots\!32 \)
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| $89$ |
\( (T^{9} + \cdots + 23\!\cdots\!53)^{2} \)
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| $97$ |
\( (T^{9} + \cdots - 21\!\cdots\!36)^{2} \)
|
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