Newspace parameters
Level: | \( N \) | \(=\) | \( 1900 = 2^{2} \cdot 5^{2} \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 1900.s (of order \(6\), degree \(2\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(15.1715763840\) |
Analytic rank: | \(0\) |
Dimension: | \(16\) |
Relative dimension: | \(8\) over \(\Q(\zeta_{6})\) |
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: | \( x^{16} - 17x^{14} + 215x^{12} - 1176x^{10} + 4775x^{8} - 2898x^{6} + 1385x^{4} - 164x^{2} + 16 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
Coefficient ring index: | \( 2^{2}\cdot 3^{2} \) |
Twist minimal: | no (minimal twist has level 380) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) 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 \( x^{16} - 17x^{14} + 215x^{12} - 1176x^{10} + 4775x^{8} - 2898x^{6} + 1385x^{4} - 164x^{2} + 16 \) :
\(\beta_{1}\) | \(=\) | \( \nu \) |
\(\beta_{2}\) | \(=\) | \( ( 26971964 \nu^{14} - 444227322 \nu^{12} + 5546383462 \nu^{10} - 28530870622 \nu^{8} + 110759128493 \nu^{6} - 8522047166 \nu^{4} + \cdots - 60023019862 ) / 17067637587 \) |
\(\beta_{3}\) | \(=\) | \( ( - 41610718 \nu^{14} + 684671526 \nu^{12} - 8556625619 \nu^{10} + 44015705039 \nu^{8} - 171758222896 \nu^{6} + 13147299967 \nu^{4} + \cdots - 6161981722 ) / 17067637587 \) |
\(\beta_{4}\) | \(=\) | \( ( 7345 \nu^{14} - 127233 \nu^{12} + 1618437 \nu^{10} - 9124664 \nu^{8} + 37577239 \nu^{6} - 30836000 \nu^{4} + 10921017 \nu^{2} - 1293284 ) / 2504606 \) |
\(\beta_{5}\) | \(=\) | \( ( 40106707 \nu^{14} - 677309787 \nu^{12} + 8548959173 \nu^{10} - 46239259076 \nu^{8} + 186744961489 \nu^{6} - 97460284762 \nu^{4} + \cdots - 6408773852 ) / 5251580796 \) |
\(\beta_{6}\) | \(=\) | \( ( - 40106707 \nu^{15} + 677309787 \nu^{13} - 8548959173 \nu^{11} + 46239259076 \nu^{9} - 186744961489 \nu^{7} + 97460284762 \nu^{5} + \cdots + 6408773852 \nu ) / 5251580796 \) |
\(\beta_{7}\) | \(=\) | \( ( - 96592348 \nu^{14} + 1588531329 \nu^{12} - 19862780534 \nu^{10} + 102175124654 \nu^{8} - 399408961640 \nu^{6} + 30519266062 \nu^{4} + \cdots - 24794252808 ) / 5689212529 \) |
\(\beta_{8}\) | \(=\) | \( ( - 304415798 \nu^{15} + 5006038191 \nu^{13} - 62598583759 \nu^{11} + 322010208379 \nu^{9} - 1259225979323 \nu^{7} + \cdots - 208838310356 \nu ) / 17067637587 \) |
\(\beta_{9}\) | \(=\) | \( ( - 331387762 \nu^{15} + 5450265513 \nu^{13} - 68144967221 \nu^{11} + 350541079001 \nu^{9} - 1369985107816 \nu^{7} + \cdots - 97612377733 \nu ) / 17067637587 \) |
\(\beta_{10}\) | \(=\) | \( ( 913391869 \nu^{14} - 15395107497 \nu^{12} + 194194776311 \nu^{10} - 1046891021486 \nu^{8} + 4221230619577 \nu^{6} + \cdots - 144830854988 ) / 34135275174 \) |
\(\beta_{11}\) | \(=\) | \( ( 589848415 \nu^{14} - 10109146509 \nu^{12} + 128159645582 \nu^{10} - 710466949547 \nu^{8} + 2902973030392 \nu^{6} + \cdots - 99796456472 ) / 17067637587 \) |
\(\beta_{12}\) | \(=\) | \( ( 621164806 \nu^{15} - 10215859500 \nu^{13} + 127733308823 \nu^{11} - 657066452963 \nu^{9} + 2568211992736 \nu^{7} + \cdots + 171995136157 \nu ) / 17067637587 \) |
\(\beta_{13}\) | \(=\) | \( ( - 3080990861 \nu^{15} + 52709730381 \nu^{13} - 667890407323 \nu^{11} + 3691698257488 \nu^{9} - 15063857001587 \nu^{7} + \cdots + 517752221236 \nu ) / 68270550348 \) |
\(\beta_{14}\) | \(=\) | \( ( - 640799947 \nu^{15} + 10828264986 \nu^{13} - 136673641894 \nu^{11} + 739737686495 \nu^{9} - 2985522971342 \nu^{7} + \cdots + 18500239168 \nu ) / 5689212529 \) |
\(\beta_{15}\) | \(=\) | \( ( 7925043745 \nu^{15} - 134009754417 \nu^{13} + 1691462224943 \nu^{11} - 9161831867528 \nu^{9} + 36948597094099 \nu^{7} + \cdots - 228957502496 \nu ) / 68270550348 \) |
\(\nu\) | \(=\) | \( \beta_1 \) |
\(\nu^{2}\) | \(=\) | \( -\beta_{10} + 4\beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + 4 \) |
\(\nu^{3}\) | \(=\) | \( \beta_{15} + \beta_{14} + \beta_{13} + \beta_{9} - 8\beta_{6} + 8\beta_1 \) |
\(\nu^{4}\) | \(=\) | \( \beta_{11} - 9\beta_{10} + 32\beta_{5} - 13\beta_{4} \) |
\(\nu^{5}\) | \(=\) | \( 14\beta_{15} + 9\beta_{14} + 18\beta_{13} - 14\beta_{12} - 9\beta_{8} - 72\beta_{6} \) |
\(\nu^{6}\) | \(=\) | \( 14\beta_{7} - 150\beta_{3} - 81\beta_{2} - 278 \) |
\(\nu^{7}\) | \(=\) | \( -164\beta_{12} - 233\beta_{9} - 81\beta_{8} - 671\beta_1 \) |
\(\nu^{8}\) | \(=\) | \( - 164 \beta_{11} + 752 \beta_{10} + 164 \beta_{7} - 2518 \beta_{5} + 1629 \beta_{4} - 1629 \beta_{3} - 752 \beta_{2} - 2518 \) |
\(\nu^{9}\) | \(=\) | \( -1793\beta_{15} - 752\beta_{14} - 2670\beta_{13} - 2670\beta_{9} + 6403\beta_{6} - 6403\beta_1 \) |
\(\nu^{10}\) | \(=\) | \( -1793\beta_{11} + 7155\beta_{10} - 23530\beta_{5} + 17122\beta_{4} \) |
\(\nu^{11}\) | \(=\) | \( -18915\beta_{15} - 7155\beta_{14} - 28882\beta_{13} + 18915\beta_{12} + 7155\beta_{8} + 62117\beta_{6} \) |
\(\nu^{12}\) | \(=\) | \( -18915\beta_{7} + 176626\beta_{3} + 69272\beta_{2} + 224948 \) |
\(\nu^{13}\) | \(=\) | \( 195541\beta_{12} + 302895\beta_{9} + 69272\beta_{8} + 609390\beta_1 \) |
\(\nu^{14}\) | \(=\) | \( 195541 \beta_{11} - 678662 \beta_{10} - 195541 \beta_{7} + 2185022 \beta_{5} - 1801803 \beta_{4} + 1801803 \beta_{3} + 678662 \beta_{2} + 2185022 \) |
\(\nu^{15}\) | \(=\) | \( 1997344\beta_{15} + 678662\beta_{14} + 3120485\beta_{13} + 3120485\beta_{9} - 6022811\beta_{6} + 6022811\beta_1 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1900\mathbb{Z}\right)^\times\).
\(n\) | \(77\) | \(401\) | \(951\) |
\(\chi(n)\) | \(-1\) | \(\beta_{5}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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49.1 |
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0 | −2.74101 | − | 1.58253i | 0 | 0 | 0 | 1.53315i | 0 | 3.50877 | + | 6.07738i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
49.2 | 0 | −2.18309 | − | 1.26041i | 0 | 0 | 0 | 2.72743i | 0 | 1.67727 | + | 2.90511i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
49.3 | 0 | −0.614201 | − | 0.354609i | 0 | 0 | 0 | − | 3.11079i | 0 | −1.24850 | − | 2.16247i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
49.4 | 0 | −0.306096 | − | 0.176725i | 0 | 0 | 0 | − | 4.30507i | 0 | −1.43754 | − | 2.48989i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
49.5 | 0 | 0.306096 | + | 0.176725i | 0 | 0 | 0 | 4.30507i | 0 | −1.43754 | − | 2.48989i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
49.6 | 0 | 0.614201 | + | 0.354609i | 0 | 0 | 0 | 3.11079i | 0 | −1.24850 | − | 2.16247i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
49.7 | 0 | 2.18309 | + | 1.26041i | 0 | 0 | 0 | − | 2.72743i | 0 | 1.67727 | + | 2.90511i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
49.8 | 0 | 2.74101 | + | 1.58253i | 0 | 0 | 0 | − | 1.53315i | 0 | 3.50877 | + | 6.07738i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
349.1 | 0 | −2.74101 | + | 1.58253i | 0 | 0 | 0 | − | 1.53315i | 0 | 3.50877 | − | 6.07738i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
349.2 | 0 | −2.18309 | + | 1.26041i | 0 | 0 | 0 | − | 2.72743i | 0 | 1.67727 | − | 2.90511i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
349.3 | 0 | −0.614201 | + | 0.354609i | 0 | 0 | 0 | 3.11079i | 0 | −1.24850 | + | 2.16247i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
349.4 | 0 | −0.306096 | + | 0.176725i | 0 | 0 | 0 | 4.30507i | 0 | −1.43754 | + | 2.48989i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
349.5 | 0 | 0.306096 | − | 0.176725i | 0 | 0 | 0 | − | 4.30507i | 0 | −1.43754 | + | 2.48989i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
349.6 | 0 | 0.614201 | − | 0.354609i | 0 | 0 | 0 | − | 3.11079i | 0 | −1.24850 | + | 2.16247i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
349.7 | 0 | 2.18309 | − | 1.26041i | 0 | 0 | 0 | 2.72743i | 0 | 1.67727 | − | 2.90511i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
349.8 | 0 | 2.74101 | − | 1.58253i | 0 | 0 | 0 | 1.53315i | 0 | 3.50877 | − | 6.07738i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.b | even | 2 | 1 | inner |
19.c | even | 3 | 1 | inner |
95.i | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 1900.2.s.d | 16 | |
5.b | even | 2 | 1 | inner | 1900.2.s.d | 16 | |
5.c | odd | 4 | 1 | 380.2.i.c | ✓ | 8 | |
5.c | odd | 4 | 1 | 1900.2.i.d | 8 | ||
15.e | even | 4 | 1 | 3420.2.t.w | 8 | ||
19.c | even | 3 | 1 | inner | 1900.2.s.d | 16 | |
20.e | even | 4 | 1 | 1520.2.q.m | 8 | ||
95.i | even | 6 | 1 | inner | 1900.2.s.d | 16 | |
95.l | even | 12 | 1 | 7220.2.a.p | 4 | ||
95.m | odd | 12 | 1 | 380.2.i.c | ✓ | 8 | |
95.m | odd | 12 | 1 | 1900.2.i.d | 8 | ||
95.m | odd | 12 | 1 | 7220.2.a.r | 4 | ||
285.v | even | 12 | 1 | 3420.2.t.w | 8 | ||
380.v | even | 12 | 1 | 1520.2.q.m | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
380.2.i.c | ✓ | 8 | 5.c | odd | 4 | 1 | |
380.2.i.c | ✓ | 8 | 95.m | odd | 12 | 1 | |
1520.2.q.m | 8 | 20.e | even | 4 | 1 | ||
1520.2.q.m | 8 | 380.v | even | 12 | 1 | ||
1900.2.i.d | 8 | 5.c | odd | 4 | 1 | ||
1900.2.i.d | 8 | 95.m | odd | 12 | 1 | ||
1900.2.s.d | 16 | 1.a | even | 1 | 1 | trivial | |
1900.2.s.d | 16 | 5.b | even | 2 | 1 | inner | |
1900.2.s.d | 16 | 19.c | even | 3 | 1 | inner | |
1900.2.s.d | 16 | 95.i | even | 6 | 1 | inner | |
3420.2.t.w | 8 | 15.e | even | 4 | 1 | ||
3420.2.t.w | 8 | 285.v | even | 12 | 1 | ||
7220.2.a.p | 4 | 95.l | even | 12 | 1 | ||
7220.2.a.r | 4 | 95.m | odd | 12 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{16} - 17T_{3}^{14} + 215T_{3}^{12} - 1176T_{3}^{10} + 4775T_{3}^{8} - 2898T_{3}^{6} + 1385T_{3}^{4} - 164T_{3}^{2} + 16 \)
acting on \(S_{2}^{\mathrm{new}}(1900, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{16} \)
$3$
\( T^{16} - 17 T^{14} + 215 T^{12} + \cdots + 16 \)
$5$
\( T^{16} \)
$7$
\( (T^{8} + 38 T^{6} + 473 T^{4} + 2249 T^{2} + \cdots + 3136)^{2} \)
$11$
\( (T^{4} - 2 T^{3} - 35 T^{2} + 21 T + 267)^{4} \)
$13$
\( T^{16} - 91 T^{14} + 5830 T^{12} + \cdots + 7311616 \)
$17$
\( T^{16} - 69 T^{14} + \cdots + 1358954496 \)
$19$
\( (T^{8} + 3 T^{7} - 43 T^{6} - 21 T^{5} + \cdots + 130321)^{2} \)
$23$
\( T^{16} - 78 T^{14} + 4671 T^{12} + \cdots + 8503056 \)
$29$
\( (T^{8} + 5 T^{7} + 84 T^{6} + 341 T^{5} + \cdots + 74529)^{2} \)
$31$
\( (T^{4} + 10 T^{3} - 29 T^{2} - 303 T - 277)^{4} \)
$37$
\( (T^{8} + 242 T^{6} + 16249 T^{4} + \cdots + 171396)^{2} \)
$41$
\( (T^{8} + 8 T^{7} + 59 T^{6} + 106 T^{5} + \cdots + 324)^{2} \)
$43$
\( T^{16} - 113 T^{14} + \cdots + 1003875856 \)
$47$
\( T^{16} - 306 T^{14} + \cdots + 1912622616576 \)
$53$
\( T^{16} - 261 T^{14} + \cdots + 2200843458576 \)
$59$
\( (T^{8} + 11 T^{7} + 176 T^{6} + \cdots + 2518569)^{2} \)
$61$
\( (T^{8} - 12 T^{7} + 166 T^{6} - 608 T^{5} + \cdots + 841)^{2} \)
$67$
\( T^{16} + \cdots + 118018152595456 \)
$71$
\( (T^{8} - 14 T^{7} + 197 T^{6} + \cdots + 5049009)^{2} \)
$73$
\( T^{16} - 290 T^{14} + \cdots + 96947540496 \)
$79$
\( (T^{8} + 13 T^{7} + 297 T^{6} + \cdots + 9096256)^{2} \)
$83$
\( (T^{8} + 189 T^{6} + 10810 T^{4} + \cdots + 1695204)^{2} \)
$89$
\( (T^{8} + 5 T^{7} + 147 T^{6} + \cdots + 1382976)^{2} \)
$97$
\( T^{16} + \cdots + 178268320144656 \)
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