Properties

Label 7.10.c.a
Level $7$
Weight $10$
Character orbit 7.c
Analytic conductor $3.605$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 7 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 7.c (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.60525085315\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Defining polynomial: \( x^{10} - x^{9} + 430 x^{8} + 61 x^{7} + 146753 x^{6} + 23608 x^{5} + 16136944 x^{4} + 30575648 x^{3} + 1399072384 x^{2} + 1034227200 x + 761760000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{3}\cdot 7^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 4 \beta_{3} + \beta_{2} - \beta_1) q^{2} + ( - \beta_{7} - \beta_{5} - 33 \beta_{3} - \beta_1 + 33) q^{3} + ( - \beta_{8} + \beta_{7} + \beta_{5} + \beta_{4} + 191 \beta_{3} + 7 \beta_1 - 191) q^{4} + ( - \beta_{9} + 307 \beta_{3}) q^{5} + ( - 2 \beta_{6} + 11 \beta_{5} + 7 \beta_{4} + 60 \beta_{2} - 897) q^{6} + ( - \beta_{9} - 10 \beta_{8} - 10 \beta_{7} + 3 \beta_{6} - 9 \beta_{5} + \cdots - 797) q^{7}+ \cdots + (7 \beta_{9} + 16 \beta_{8} - 86 \beta_{7} - 7284 \beta_{3} + 258 \beta_{2} - 258 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 4 \beta_{3} + \beta_{2} - \beta_1) q^{2} + ( - \beta_{7} - \beta_{5} - 33 \beta_{3} - \beta_1 + 33) q^{3} + ( - \beta_{8} + \beta_{7} + \beta_{5} + \beta_{4} + 191 \beta_{3} + 7 \beta_1 - 191) q^{4} + ( - \beta_{9} + 307 \beta_{3}) q^{5} + ( - 2 \beta_{6} + 11 \beta_{5} + 7 \beta_{4} + 60 \beta_{2} - 897) q^{6} + ( - \beta_{9} - 10 \beta_{8} - 10 \beta_{7} + 3 \beta_{6} - 9 \beta_{5} + \cdots - 797) q^{7}+ \cdots + ( - 309295 \beta_{6} + 1130938 \beta_{5} + \cdots - 200734674) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 18 q^{2} + 161 q^{3} - 940 q^{4} + 1533 q^{5} - 8708 q^{6} - 1036 q^{7} + 34272 q^{8} - 35734 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 18 q^{2} + 161 q^{3} - 940 q^{4} + 1533 q^{5} - 8708 q^{6} - 1036 q^{7} + 34272 q^{8} - 35734 q^{9} + 4298 q^{10} + 42213 q^{11} + 135604 q^{12} - 319676 q^{13} - 39522 q^{14} + 151394 q^{15} + 322064 q^{16} + 324681 q^{17} - 1012868 q^{18} - 16121 q^{19} - 350616 q^{20} - 1557857 q^{21} - 62692 q^{22} + 2638863 q^{23} + 8449728 q^{24} - 1304092 q^{25} + 4179252 q^{26} - 18331558 q^{27} - 22156316 q^{28} + 15292500 q^{29} + 20557942 q^{30} + 19179237 q^{31} - 6263520 q^{32} + 1689359 q^{33} - 62909700 q^{34} - 43746759 q^{35} + 71476528 q^{36} + 39566985 q^{37} + 67365270 q^{38} - 44299486 q^{39} + 5721744 q^{40} - 53436852 q^{41} - 183129856 q^{42} + 101835992 q^{43} + 99704916 q^{44} + 85098230 q^{45} - 14489202 q^{46} + 32509659 q^{47} - 185141600 q^{48} - 49024598 q^{49} + 3328464 q^{50} + 44168403 q^{51} + 103893272 q^{52} - 25714707 q^{53} + 51200926 q^{54} - 144695222 q^{55} + 115352832 q^{56} - 121710346 q^{57} - 46645516 q^{58} + 46776513 q^{59} + 132391756 q^{60} - 113075039 q^{61} + 467465628 q^{62} + 318071530 q^{63} - 192008960 q^{64} - 338113566 q^{65} - 836682602 q^{66} - 126707879 q^{67} + 32262636 q^{68} + 1323616182 q^{69} + 697712470 q^{70} - 1188736032 q^{71} - 950557728 q^{72} - 859257651 q^{73} + 591757530 q^{74} - 169061732 q^{75} + 1101475592 q^{76} + 1911891891 q^{77} + 519432424 q^{78} - 527065417 q^{79} - 1257352656 q^{80} + 551662715 q^{81} - 1341703076 q^{82} - 144863208 q^{83} + 486452204 q^{84} - 1197360222 q^{85} - 678648216 q^{86} - 340781350 q^{87} + 903700608 q^{88} + 1661554797 q^{89} + 1967758744 q^{90} + 726641384 q^{91} - 1301840952 q^{92} - 423057489 q^{93} - 272580882 q^{94} - 1197123495 q^{95} - 1441922272 q^{96} + 869770188 q^{97} - 2404833858 q^{98} - 1900777180 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - x^{9} + 430 x^{8} + 61 x^{7} + 146753 x^{6} + 23608 x^{5} + 16136944 x^{4} + 30575648 x^{3} + 1399072384 x^{2} + 1034227200 x + 761760000 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 174976544647 \nu^{9} + 543280294119 \nu^{8} - 2080323777608 \nu^{7} + 306747495137243 \nu^{6} - 527825025841115 \nu^{5} + \cdots - 72\!\cdots\!00 ) / 49\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 439915122044593 \nu^{9} - 359425911506973 \nu^{8} + \cdots + 45\!\cdots\!00 ) / 45\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 9108600924689 \nu^{9} + 744234087958953 \nu^{8} + \cdots + 14\!\cdots\!20 ) / 23\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 44987322634457 \nu^{9} - 60129676436289 \nu^{8} + 230247989821048 \nu^{7} + \cdots + 26\!\cdots\!80 ) / 69\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 10228081391315 \nu^{9} + \cdots + 13\!\cdots\!84 ) / 69\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 24\!\cdots\!57 \nu^{9} + \cdots - 60\!\cdots\!00 ) / 15\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 20\!\cdots\!03 \nu^{9} + \cdots + 20\!\cdots\!00 ) / 31\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 10\!\cdots\!87 \nu^{9} + \cdots + 10\!\cdots\!00 ) / 53\!\cdots\!00 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{8} - \beta_{7} - 687\beta_{3} - \beta_{2} + \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -27\beta_{5} + \beta_{4} - 507\beta_{2} - 375 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 14 \beta_{9} - 169 \beta_{8} + 155 \beta_{7} - 14 \beta_{6} + 155 \beta_{5} + 169 \beta_{4} + 87639 \beta_{3} - 134 \beta _1 - 87639 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 28\beta_{9} - 583\beta_{8} - 11457\beta_{7} + 99345\beta_{3} + 142643\beta_{2} - 142643\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 12040\beta_{6} - 89029\beta_{5} - 107929\beta_{4} + 71655\beta_{2} + 49481343 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 9450 \beta_{9} + 115386 \beta_{8} + 1949460 \beta_{7} + 9450 \beta_{6} + 1949460 \beta_{5} - 115386 \beta_{4} - 13417998 \beta_{3} + 21095143 \beta _1 + 13417998 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 4129692 \beta_{9} + 33731905 \beta_{8} - 25531489 \beta_{7} - 14657685447 \beta_{3} - 22648273 \beta_{2} + 22648273 \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( -8200416\beta_{6} - 1242725823\beta_{5} + 81699181\beta_{4} - 12756581151\beta_{2} - 8505431979 ) / 4 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/7\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(-\beta_{3}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1
−8.71912 15.1020i
−5.11725 8.86334i
−0.371984 0.644295i
5.89912 + 10.2176i
8.80924 + 15.2580i
−8.71912 + 15.1020i
−5.11725 + 8.86334i
−0.371984 + 0.644295i
5.89912 10.2176i
8.80924 15.2580i
−19.4382 + 33.6680i 113.728 + 196.982i −499.691 865.489i −162.760 + 281.909i −8842.67 5234.95 3598.46i 18947.7 −16026.5 + 27758.7i −6327.54 10959.6i
2.2 −12.2345 + 21.1908i −79.7348 138.105i −43.3662 75.1124i 1014.15 1756.56i 3902.06 −4235.51 4734.35i −10405.9 −2873.78 + 4977.54i 24815.2 + 42981.2i
2.3 −2.74397 + 4.75269i −1.70307 2.94981i 240.941 + 417.323i −828.924 + 1435.74i 18.6927 2822.68 + 5690.88i −5454.36 9835.70 17035.9i −4549.08 7879.24i
2.4 9.79824 16.9710i 104.977 + 181.826i 63.9892 + 110.832i 983.791 1703.98i 4114.37 −5768.52 + 2660.41i 12541.3 −12198.9 + 21129.2i −19278.8 33391.9i
2.5 15.6185 27.0520i −56.7670 98.3234i −231.874 401.617i −239.755 + 415.269i −3546.46 1428.40 6189.77i 1507.26 3396.51 5882.92i 7489.23 + 12971.7i
4.1 −19.4382 33.6680i 113.728 196.982i −499.691 + 865.489i −162.760 281.909i −8842.67 5234.95 + 3598.46i 18947.7 −16026.5 27758.7i −6327.54 + 10959.6i
4.2 −12.2345 21.1908i −79.7348 + 138.105i −43.3662 + 75.1124i 1014.15 + 1756.56i 3902.06 −4235.51 + 4734.35i −10405.9 −2873.78 4977.54i 24815.2 42981.2i
4.3 −2.74397 4.75269i −1.70307 + 2.94981i 240.941 417.323i −828.924 1435.74i 18.6927 2822.68 5690.88i −5454.36 9835.70 + 17035.9i −4549.08 + 7879.24i
4.4 9.79824 + 16.9710i 104.977 181.826i 63.9892 110.832i 983.791 + 1703.98i 4114.37 −5768.52 2660.41i 12541.3 −12198.9 21129.2i −19278.8 + 33391.9i
4.5 15.6185 + 27.0520i −56.7670 + 98.3234i −231.874 + 401.617i −239.755 415.269i −3546.46 1428.40 + 6189.77i 1507.26 3396.51 + 5882.92i 7489.23 12971.7i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 7.10.c.a 10
3.b odd 2 1 63.10.e.b 10
4.b odd 2 1 112.10.i.c 10
7.b odd 2 1 49.10.c.g 10
7.c even 3 1 inner 7.10.c.a 10
7.c even 3 1 49.10.a.e 5
7.d odd 6 1 49.10.a.f 5
7.d odd 6 1 49.10.c.g 10
21.h odd 6 1 63.10.e.b 10
28.g odd 6 1 112.10.i.c 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.10.c.a 10 1.a even 1 1 trivial
7.10.c.a 10 7.c even 3 1 inner
49.10.a.e 5 7.c even 3 1
49.10.a.f 5 7.d odd 6 1
49.10.c.g 10 7.b odd 2 1
49.10.c.g 10 7.d odd 6 1
63.10.e.b 10 3.b odd 2 1
63.10.e.b 10 21.h odd 6 1
112.10.i.c 10 4.b odd 2 1
112.10.i.c 10 28.g odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{10}^{\mathrm{new}}(7, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + 18 T^{9} + \cdots + 10212166139904 \) Copy content Toggle raw display
$3$ \( T^{10} - 161 T^{9} + \cdots + 86\!\cdots\!89 \) Copy content Toggle raw display
$5$ \( T^{10} - 1533 T^{9} + \cdots + 10\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{10} + 1036 T^{9} + \cdots + 10\!\cdots\!07 \) Copy content Toggle raw display
$11$ \( T^{10} - 42213 T^{9} + \cdots + 91\!\cdots\!25 \) Copy content Toggle raw display
$13$ \( (T^{5} + 159838 T^{4} + \cdots + 79\!\cdots\!92)^{2} \) Copy content Toggle raw display
$17$ \( T^{10} - 324681 T^{9} + \cdots + 29\!\cdots\!81 \) Copy content Toggle raw display
$19$ \( T^{10} + 16121 T^{9} + \cdots + 39\!\cdots\!01 \) Copy content Toggle raw display
$23$ \( T^{10} - 2638863 T^{9} + \cdots + 42\!\cdots\!01 \) Copy content Toggle raw display
$29$ \( (T^{5} - 7646250 T^{4} + \cdots + 73\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( T^{10} - 19179237 T^{9} + \cdots + 14\!\cdots\!69 \) Copy content Toggle raw display
$37$ \( T^{10} - 39566985 T^{9} + \cdots + 47\!\cdots\!25 \) Copy content Toggle raw display
$41$ \( (T^{5} + 26718426 T^{4} + \cdots - 20\!\cdots\!40)^{2} \) Copy content Toggle raw display
$43$ \( (T^{5} - 50917996 T^{4} + \cdots + 72\!\cdots\!36)^{2} \) Copy content Toggle raw display
$47$ \( T^{10} - 32509659 T^{9} + \cdots + 34\!\cdots\!25 \) Copy content Toggle raw display
$53$ \( T^{10} + 25714707 T^{9} + \cdots + 47\!\cdots\!21 \) Copy content Toggle raw display
$59$ \( T^{10} - 46776513 T^{9} + \cdots + 79\!\cdots\!25 \) Copy content Toggle raw display
$61$ \( T^{10} + 113075039 T^{9} + \cdots + 27\!\cdots\!41 \) Copy content Toggle raw display
$67$ \( T^{10} + 126707879 T^{9} + \cdots + 20\!\cdots\!25 \) Copy content Toggle raw display
$71$ \( (T^{5} + 594368016 T^{4} + \cdots + 25\!\cdots\!92)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + 859257651 T^{9} + \cdots + 87\!\cdots\!29 \) Copy content Toggle raw display
$79$ \( T^{10} + 527065417 T^{9} + \cdots + 26\!\cdots\!21 \) Copy content Toggle raw display
$83$ \( (T^{5} + 72431604 T^{4} + \cdots + 39\!\cdots\!36)^{2} \) Copy content Toggle raw display
$89$ \( T^{10} - 1661554797 T^{9} + \cdots + 21\!\cdots\!41 \) Copy content Toggle raw display
$97$ \( (T^{5} - 434885094 T^{4} + \cdots - 84\!\cdots\!56)^{2} \) Copy content Toggle raw display
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