Newspace parameters
Level: | \( N \) | \(=\) | \( 175 = 5^{2} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 175.f (of order \(4\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(10.3253342510\) |
Analytic rank: | \(0\) |
Dimension: | \(16\) |
Relative dimension: | \(8\) over \(\Q(i)\) |
Coefficient field: | \(\mathbb{Q}[x]/(x^{16} + \cdots)\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{16} + 10654x^{12} + 22102125x^{8} + 5700572500x^{4} + 44626562500 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
Coefficient ring index: | \( 2^{3}\cdot 5^{2} \) |
Twist minimal: | no (minimal twist has level 35) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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.
Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 10654x^{12} + 22102125x^{8} + 5700572500x^{4} + 44626562500 \) :
\(\beta_{1}\) | \(=\) | \( \nu \) |
\(\beta_{2}\) | \(=\) | \( ( 13637\nu^{12} + 141541523\nu^{8} + 268260815950\nu^{4} + 33643813420000 ) / 2595065895000 \) |
\(\beta_{3}\) | \(=\) | \( ( -72908\nu^{13} - 790876257\nu^{9} - 1748160601575\nu^{5} - 620163391231250\nu ) / 50603784952500 \) |
\(\beta_{4}\) | \(=\) | \( ( 103709\nu^{14} + 1108253436\nu^{10} + 2329722745125\nu^{6} + 699405434108750\nu^{2} ) / 1934850601125000 \) |
\(\beta_{5}\) | \(=\) | \( ( 922187 \nu^{14} + 16951545 \nu^{12} + 10180112673 \nu^{10} + 177015939555 \nu^{8} + 24362198808750 \nu^{6} + \cdots - 45\!\cdots\!00 ) / 13\!\cdots\!00 \) |
\(\beta_{6}\) | \(=\) | \( ( - 2712347 \nu^{14} + 28252575 \nu^{12} - 29527060063 \nu^{10} + 295026565925 \nu^{8} - 67007189126000 \nu^{6} + \cdots - 54\!\cdots\!00 ) / 21\!\cdots\!00 \) |
\(\beta_{7}\) | \(=\) | \( ( 26023741 \nu^{14} + 53408225 \nu^{12} + 277997050989 \nu^{10} + 614189471025 \nu^{8} + 580713324166500 \nu^{6} + \cdots + 41\!\cdots\!00 ) / 13\!\cdots\!00 \) |
\(\beta_{8}\) | \(=\) | \( ( - 26023741 \nu^{14} + 53408225 \nu^{12} - 277997050989 \nu^{10} + 614189471025 \nu^{8} - 580713324166500 \nu^{6} + \cdots + 41\!\cdots\!00 ) / 13\!\cdots\!00 \) |
\(\beta_{9}\) | \(=\) | \( ( 21675877 \nu^{15} + 273281450 \nu^{13} + 237181688433 \nu^{11} + 2814563615175 \nu^{9} + 544908018727500 \nu^{7} + \cdots - 65\!\cdots\!50 \nu ) / 42\!\cdots\!00 \) |
\(\beta_{10}\) | \(=\) | \( ( 21675877 \nu^{15} - 273281450 \nu^{13} + 237181688433 \nu^{11} - 2814563615175 \nu^{9} + 544908018727500 \nu^{7} + \cdots + 65\!\cdots\!50 \nu ) / 42\!\cdots\!00 \) |
\(\beta_{11}\) | \(=\) | \( ( 103709\nu^{15} + 1108253436\nu^{11} + 2329722745125\nu^{7} + 699405434108750\nu^{3} ) / 1934850601125000 \) |
\(\beta_{12}\) | \(=\) | \( ( - 11171152 \nu^{14} - 118324616783 \nu^{10} - 240090289254625 \nu^{6} - 52\!\cdots\!50 \nu^{2} ) / 10\!\cdots\!00 \) |
\(\beta_{13}\) | \(=\) | \( ( - 753130203 \nu^{15} - 2863109275 \nu^{13} - 8000341557387 \nu^{11} - 30005408823975 \nu^{9} + \cdots - 11\!\cdots\!00 \nu ) / 17\!\cdots\!00 \) |
\(\beta_{14}\) | \(=\) | \( ( - 753130203 \nu^{15} + 2863109275 \nu^{13} - 8000341557387 \nu^{11} + 30005408823975 \nu^{9} + \cdots + 11\!\cdots\!00 \nu ) / 17\!\cdots\!00 \) |
\(\beta_{15}\) | \(=\) | \( ( 4232737\nu^{15} + 45142420038\nu^{11} + 94008451007535\nu^{7} + 24778718008842250\nu^{3} ) / 5701359771315000 \) |
\(\nu\) | \(=\) | \( \beta_1 \) |
\(\nu^{2}\) | \(=\) | \( \beta_{12} + 2\beta_{8} - 2\beta_{7} + 2\beta_{6} - 2\beta_{5} + 41\beta_{4} - 2 \) |
\(\nu^{3}\) | \(=\) | \( \beta_{15} + 4\beta_{14} + 4\beta_{13} + 67\beta_{11} - 8\beta_{10} - 8\beta_{9} \) |
\(\nu^{4}\) | \(=\) | \( 36\beta_{8} + 36\beta_{7} - 201\beta_{6} - 201\beta_{5} - 201\beta_{4} + 93\beta_{2} - 2635 \) |
\(\nu^{5}\) | \(=\) | \( 552\beta_{14} - 552\beta_{13} + 969\beta_{10} - 969\beta_{9} + 423\beta_{3} - 5011\beta_1 \) |
\(\nu^{6}\) | \(=\) | \( -9439\beta_{12} + 4672\beta_{8} - 4672\beta_{7} - 18173\beta_{6} + 18173\beta_{5} - 210644\beta_{4} + 18173 \) |
\(\nu^{7}\) | \(=\) | \( -55129\beta_{15} - 59896\beta_{14} - 59896\beta_{13} - 401203\beta_{11} + 95537\beta_{10} + 95537\beta_{9} \) |
\(\nu^{8}\) | \(=\) | \( 808956 \beta_{8} + 808956 \beta_{7} + 1619979 \beta_{6} + 1619979 \beta_{5} + 1619979 \beta_{4} - 919347 \beta_{2} + 16461715 \) |
\(\nu^{9}\) | \(=\) | \( - 5887608 \beta_{14} + 5887608 \beta_{13} - 8908851 \beta_{10} + 8908851 \beta_{9} - 5777217 \beta_{3} + 33582919 \beta_1 \) |
\(\nu^{10}\) | \(=\) | \( 86195731 \beta_{12} - 91446988 \beta_{8} + 91446988 \beta_{7} + 144279767 \beta_{6} - 144279767 \beta_{5} + 1485183926 \beta_{4} - 144279767 \) |
\(\nu^{11}\) | \(=\) | \( 557649241 \beta_{15} + 552397984 \beta_{14} + 552397984 \beta_{13} + 2889367387 \beta_{11} - 812845823 \beta_{10} - 812845823 \beta_{9} \) |
\(\nu^{12}\) | \(=\) | \( - 9104513724 \beta_{8} - 9104513724 \beta_{7} - 12860150391 \beta_{6} - 12860150391 \beta_{5} - 12860150391 \beta_{4} + 7902954063 \beta_{2} + \cdots - 121492467235 \) |
\(\nu^{13}\) | \(=\) | \( 50630722632 \beta_{14} - 50630722632 \beta_{13} + 73405265679 \beta_{10} - 73405265679 \beta_{9} + 51832282293 \beta_{3} - 252648048151 \beta_1 \) |
\(\nu^{14}\) | \(=\) | \( - 715809307699 \beta_{12} + 858779499652 \beta_{8} - 858779499652 \beta_{7} - 1147049021243 \beta_{6} + 1147049021243 \beta_{5} + \cdots + 1147049021243 \) |
\(\nu^{15}\) | \(=\) | \( - 4727466349489 \beta_{15} - 4584496157536 \beta_{14} - 4584496157536 \beta_{13} - 22296858454723 \beta_{11} + 6594024605867 \beta_{10} + 6594024605867 \beta_{9} \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/175\mathbb{Z}\right)^\times\).
\(n\) | \(101\) | \(127\) |
\(\chi(n)\) | \(-1\) | \(-\beta_{4}\) |
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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118.1 |
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−3.08000 | + | 3.08000i | −1.19219 | + | 1.19219i | − | 10.9728i | 0 | − | 7.34390i | 13.5707 | + | 12.6030i | 9.15614 | + | 9.15614i | 24.1574i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
118.2 | −3.08000 | + | 3.08000i | 1.19219 | − | 1.19219i | − | 10.9728i | 0 | 7.34390i | −12.6030 | − | 13.5707i | 9.15614 | + | 9.15614i | 24.1574i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
118.3 | −1.31781 | + | 1.31781i | −6.68128 | + | 6.68128i | 4.52674i | 0 | − | 17.6093i | 6.92319 | − | 17.1776i | −16.5079 | − | 16.5079i | − | 62.2789i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
118.4 | −1.31781 | + | 1.31781i | 6.68128 | − | 6.68128i | 4.52674i | 0 | 17.6093i | 17.1776 | − | 6.92319i | −16.5079 | − | 16.5079i | − | 62.2789i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
118.5 | −0.172516 | + | 0.172516i | −2.91939 | + | 2.91939i | 7.94048i | 0 | − | 1.00728i | −3.56610 | + | 18.1737i | −2.74998 | − | 2.74998i | 9.95432i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
118.6 | −0.172516 | + | 0.172516i | 2.91939 | − | 2.91939i | 7.94048i | 0 | 1.00728i | −18.1737 | + | 3.56610i | −2.74998 | − | 2.74998i | 9.95432i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
118.7 | 3.57033 | − | 3.57033i | −4.94129 | + | 4.94129i | − | 17.4944i | 0 | 35.2840i | −18.5018 | + | 0.826888i | −33.8983 | − | 33.8983i | − | 21.8328i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
118.8 | 3.57033 | − | 3.57033i | 4.94129 | − | 4.94129i | − | 17.4944i | 0 | − | 35.2840i | −0.826888 | + | 18.5018i | −33.8983 | − | 33.8983i | − | 21.8328i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
132.1 | −3.08000 | − | 3.08000i | −1.19219 | − | 1.19219i | 10.9728i | 0 | 7.34390i | 13.5707 | − | 12.6030i | 9.15614 | − | 9.15614i | − | 24.1574i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
132.2 | −3.08000 | − | 3.08000i | 1.19219 | + | 1.19219i | 10.9728i | 0 | − | 7.34390i | −12.6030 | + | 13.5707i | 9.15614 | − | 9.15614i | − | 24.1574i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
132.3 | −1.31781 | − | 1.31781i | −6.68128 | − | 6.68128i | − | 4.52674i | 0 | 17.6093i | 6.92319 | + | 17.1776i | −16.5079 | + | 16.5079i | 62.2789i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
132.4 | −1.31781 | − | 1.31781i | 6.68128 | + | 6.68128i | − | 4.52674i | 0 | − | 17.6093i | 17.1776 | + | 6.92319i | −16.5079 | + | 16.5079i | 62.2789i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
132.5 | −0.172516 | − | 0.172516i | −2.91939 | − | 2.91939i | − | 7.94048i | 0 | 1.00728i | −3.56610 | − | 18.1737i | −2.74998 | + | 2.74998i | − | 9.95432i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
132.6 | −0.172516 | − | 0.172516i | 2.91939 | + | 2.91939i | − | 7.94048i | 0 | − | 1.00728i | −18.1737 | − | 3.56610i | −2.74998 | + | 2.74998i | − | 9.95432i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
132.7 | 3.57033 | + | 3.57033i | −4.94129 | − | 4.94129i | 17.4944i | 0 | − | 35.2840i | −18.5018 | − | 0.826888i | −33.8983 | + | 33.8983i | 21.8328i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
132.8 | 3.57033 | + | 3.57033i | 4.94129 | + | 4.94129i | 17.4944i | 0 | 35.2840i | −0.826888 | − | 18.5018i | −33.8983 | + | 33.8983i | 21.8328i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.c | odd | 4 | 1 | inner |
7.b | odd | 2 | 1 | inner |
35.f | even | 4 | 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 | 175.4.f.g | 16 | |
5.b | even | 2 | 1 | 35.4.f.b | ✓ | 16 | |
5.c | odd | 4 | 1 | 35.4.f.b | ✓ | 16 | |
5.c | odd | 4 | 1 | inner | 175.4.f.g | 16 | |
7.b | odd | 2 | 1 | inner | 175.4.f.g | 16 | |
35.c | odd | 2 | 1 | 35.4.f.b | ✓ | 16 | |
35.f | even | 4 | 1 | 35.4.f.b | ✓ | 16 | |
35.f | even | 4 | 1 | inner | 175.4.f.g | 16 | |
35.i | odd | 6 | 2 | 245.4.l.b | 32 | ||
35.j | even | 6 | 2 | 245.4.l.b | 32 | ||
35.k | even | 12 | 2 | 245.4.l.b | 32 | ||
35.l | odd | 12 | 2 | 245.4.l.b | 32 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
35.4.f.b | ✓ | 16 | 5.b | even | 2 | 1 | |
35.4.f.b | ✓ | 16 | 5.c | odd | 4 | 1 | |
35.4.f.b | ✓ | 16 | 35.c | odd | 2 | 1 | |
35.4.f.b | ✓ | 16 | 35.f | even | 4 | 1 | |
175.4.f.g | 16 | 1.a | even | 1 | 1 | trivial | |
175.4.f.g | 16 | 5.c | odd | 4 | 1 | inner | |
175.4.f.g | 16 | 7.b | odd | 2 | 1 | inner | |
175.4.f.g | 16 | 35.f | even | 4 | 1 | inner | |
245.4.l.b | 32 | 35.i | odd | 6 | 2 | ||
245.4.l.b | 32 | 35.j | even | 6 | 2 | ||
245.4.l.b | 32 | 35.k | even | 12 | 2 | ||
245.4.l.b | 32 | 35.l | odd | 12 | 2 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} + 2T_{2}^{7} + 2T_{2}^{6} + 20T_{2}^{5} + 549T_{2}^{4} + 1538T_{2}^{3} + 2178T_{2}^{2} + 660T_{2} + 100 \)
acting on \(S_{4}^{\mathrm{new}}(175, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{8} + 2 T^{7} + 2 T^{6} + 20 T^{5} + \cdots + 100)^{2} \)
$3$
\( T^{16} + 10654 T^{12} + \cdots + 44626562500 \)
$5$
\( T^{16} \)
$7$
\( T^{16} + 32 T^{15} + \cdots + 19\!\cdots\!01 \)
$11$
\( (T^{4} + 38 T^{3} - 323 T^{2} + \cdots - 199336)^{4} \)
$13$
\( T^{16} + 22406254 T^{12} + \cdots + 68\!\cdots\!00 \)
$17$
\( T^{16} + 23886114 T^{12} + \cdots + 30\!\cdots\!00 \)
$19$
\( (T^{8} - 20760 T^{6} + \cdots + 132690195125000)^{2} \)
$23$
\( (T^{8} - 36 T^{7} + \cdots + 1537004857600)^{2} \)
$29$
\( (T^{8} + 82730 T^{6} + \cdots + 12\!\cdots\!76)^{2} \)
$31$
\( (T^{8} + 106580 T^{6} + \cdots + 353301362000000)^{2} \)
$37$
\( (T^{8} - 128 T^{7} + \cdots + 13\!\cdots\!00)^{2} \)
$41$
\( (T^{8} + 400420 T^{6} + \cdots + 37\!\cdots\!00)^{2} \)
$43$
\( (T^{8} - 156 T^{7} + \cdots + 12\!\cdots\!00)^{2} \)
$47$
\( T^{16} + 112461415314 T^{12} + \cdots + 12\!\cdots\!00 \)
$53$
\( (T^{8} + 884 T^{7} + \cdots + 31\!\cdots\!00)^{2} \)
$59$
\( (T^{8} - 1078040 T^{6} + \cdots + 18\!\cdots\!00)^{2} \)
$61$
\( (T^{8} + 560420 T^{6} + \cdots + 35\!\cdots\!00)^{2} \)
$67$
\( (T^{8} + 2252 T^{7} + \cdots + 56\!\cdots\!00)^{2} \)
$71$
\( (T^{4} - 1592 T^{3} + \cdots - 101976596376)^{4} \)
$73$
\( T^{16} + 1359643216064 T^{12} + \cdots + 16\!\cdots\!00 \)
$79$
\( (T^{8} + 635270 T^{6} + \cdots + 89\!\cdots\!16)^{2} \)
$83$
\( T^{16} + 4530921507244 T^{12} + \cdots + 33\!\cdots\!00 \)
$89$
\( (T^{8} - 2894640 T^{6} + \cdots + 19\!\cdots\!00)^{2} \)
$97$
\( T^{16} + 7476920645954 T^{12} + \cdots + 80\!\cdots\!00 \)
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