Properties

Label 400.10.c.q
Level $400$
Weight $10$
Character orbit 400.c
Analytic conductor $206.014$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 400.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(206.014334466\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} + \cdots)\)
Defining polynomial: \(x^{6} + 1305 x^{4} + 433104 x^{2} + 16000000\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{8}\cdot 5^{8} \)
Twist minimal: no (minimal twist has level 25)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{2} q^{3} + ( -27 \beta_{1} - 26 \beta_{2} - 23 \beta_{5} ) q^{7} + ( -19409 + 15 \beta_{3} + 8 \beta_{4} ) q^{9} +O(q^{10})\) \( q -\beta_{2} q^{3} + ( -27 \beta_{1} - 26 \beta_{2} - 23 \beta_{5} ) q^{7} + ( -19409 + 15 \beta_{3} + 8 \beta_{4} ) q^{9} + ( 18227 - \beta_{3} + 17 \beta_{4} ) q^{11} + ( -623 \beta_{1} - 460 \beta_{2} + 209 \beta_{5} ) q^{13} + ( 686 \beta_{1} + 168 \beta_{2} - 835 \beta_{5} ) q^{17} + ( 272886 + 164 \beta_{3} + 351 \beta_{4} ) q^{19} + ( -791931 + 21 \beta_{3} + 420 \beta_{4} ) q^{21} + ( 9822 \beta_{1} + 3426 \beta_{2} - 1338 \beta_{5} ) q^{23} + ( -17409 \beta_{1} + 18163 \beta_{2} - 13617 \beta_{5} ) q^{27} + ( -724835 + 1483 \beta_{3} + 508 \beta_{4} ) q^{29} + ( -1424710 - 598 \beta_{3} - 534 \beta_{4} ) q^{31} + ( 64491 \beta_{1} - 7288 \beta_{2} + 24420 \beta_{5} ) q^{33} + ( -19697 \beta_{1} + 37572 \beta_{2} - 3831 \beta_{5} ) q^{37} + ( -16805627 + 5043 \beta_{3} + 7832 \beta_{4} ) q^{39} + ( 1977665 + 11588 \beta_{3} + 4904 \beta_{4} ) q^{41} + ( -58561 \beta_{1} - 2732 \beta_{2} + 24439 \beta_{5} ) q^{43} + ( -205331 \beta_{1} - 9384 \beta_{2} - 83831 \beta_{5} ) q^{47} + ( 2144411 - 2792 \beta_{3} + 17904 \beta_{4} ) q^{49} + ( 8544697 - 5919 \beta_{3} - 5311 \beta_{4} ) q^{51} + ( -216242 \beta_{1} + 314856 \beta_{2} + 211780 \beta_{5} ) q^{53} + ( 761544 \beta_{1} + 111584 \beta_{2} + 204519 \beta_{5} ) q^{57} + ( 1902007 + 13381 \beta_{3} - 21680 \beta_{4} ) q^{59} + ( 41757807 - 56485 \beta_{3} - 22980 \beta_{4} ) q^{61} + ( 920772 \beta_{1} + 589692 \beta_{2} + 76428 \beta_{5} ) q^{63} + ( 153369 \beta_{1} + 322747 \beta_{2} + 1059625 \beta_{5} ) q^{67} + ( 104788422 - 4500 \beta_{3} - 94824 \beta_{4} ) q^{69} + ( -99224327 - 74345 \beta_{3} + 48040 \beta_{4} ) q^{71} + ( -876806 \beta_{1} + 56864 \beta_{2} - 1374247 \beta_{5} ) q^{73} + ( 1683837 \beta_{1} + 701052 \beta_{2} + 70551 \beta_{5} ) q^{77} + ( -103381487 - 194525 \beta_{3} - 75234 \beta_{4} ) q^{79} + ( 466165172 - 204207 \beta_{3} + 147640 \beta_{4} ) q^{81} + ( -2207037 \beta_{1} + 165897 \beta_{2} - 1002837 \beta_{5} ) q^{83} + ( -2743125 \beta_{1} + 2351284 \beta_{2} - 1725681 \beta_{5} ) q^{87} + ( 368000190 + 136359 \beta_{3} - 4176 \beta_{4} ) q^{89} + ( -394503662 + 124222 \beta_{3} + 447552 \beta_{4} ) q^{91} + ( -82782 \beta_{1} + 540432 \beta_{2} + 254460 \beta_{5} ) q^{93} + ( -2129372 \beta_{1} - 62616 \beta_{2} - 4150494 \beta_{5} ) q^{97} + ( -297151739 + 696363 \beta_{3} - 82942 \beta_{4} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 116468q^{9} + O(q^{10}) \) \( 6q - 116468q^{9} + 109398q^{11} + 1637690q^{19} - 4750788q^{21} - 4350960q^{29} - 8548132q^{31} - 100828184q^{39} + 11852622q^{41} + 12907858q^{49} + 51269398q^{51} + 11341920q^{59} + 250613852q^{61} + 628549884q^{69} - 595101192q^{71} - 620050340q^{79} + 2797694726q^{81} + 2207720070q^{89} - 2366375312q^{91} - 1784469044q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} + 1305 x^{4} + 433104 x^{2} + 16000000\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -\nu^{5} + 2695 \nu^{3} + 2178896 \nu \)\()/107136\)
\(\beta_{2}\)\(=\)\((\)\( -187 \nu^{5} - 240035 \nu^{3} - 30762448 \nu \)\()/4464000\)
\(\beta_{3}\)\(=\)\((\)\( -35 \nu^{4} - 17275 \nu^{2} + 2247682 \)\()/1674\)
\(\beta_{4}\)\(=\)\((\)\( 85 \nu^{4} + 68525 \nu^{2} + 6099304 \)\()/2232\)
\(\beta_{5}\)\(=\)\((\)\( -2339 \nu^{5} - 2624395 \nu^{3} - 697834256 \nu \)\()/13392000\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-25 \beta_{5} + 100 \beta_{2} + 19 \beta_{1}\)\()/1000\)
\(\nu^{2}\)\(=\)\((\)\(84 \beta_{4} + 153 \beta_{3} - 434977\)\()/1000\)
\(\nu^{3}\)\(=\)\((\)\(14725 \beta_{5} - 64900 \beta_{2} + 15737 \beta_{1}\)\()/1000\)
\(\nu^{4}\)\(=\)\((\)\(-8292 \beta_{4} - 24669 \beta_{3} + 55782341\)\()/200\)
\(\nu^{5}\)\(=\)\((\)\(-14788525 \beta_{5} + 42984100 \beta_{2} - 23325761 \beta_{1}\)\()/1000\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/400\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(177\) \(351\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
27.7229i
22.2334i
6.48955i
6.48955i
22.2334i
27.7229i
0 268.664i 0 0 0 637.237i 0 −52497.3 0
49.2 0 210.171i 0 0 0 9905.49i 0 −24489.0 0
49.3 0 30.5073i 0 0 0 4010.25i 0 18752.3 0
49.4 0 30.5073i 0 0 0 4010.25i 0 18752.3 0
49.5 0 210.171i 0 0 0 9905.49i 0 −24489.0 0
49.6 0 268.664i 0 0 0 637.237i 0 −52497.3 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 400.10.c.q 6
4.b odd 2 1 25.10.b.c 6
5.b even 2 1 inner 400.10.c.q 6
5.c odd 4 1 400.10.a.u 3
5.c odd 4 1 400.10.a.y 3
12.b even 2 1 225.10.b.m 6
20.d odd 2 1 25.10.b.c 6
20.e even 4 1 25.10.a.c 3
20.e even 4 1 25.10.a.d yes 3
60.h even 2 1 225.10.b.m 6
60.l odd 4 1 225.10.a.m 3
60.l odd 4 1 225.10.a.p 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.10.a.c 3 20.e even 4 1
25.10.a.d yes 3 20.e even 4 1
25.10.b.c 6 4.b odd 2 1
25.10.b.c 6 20.d odd 2 1
225.10.a.m 3 60.l odd 4 1
225.10.a.p 3 60.l odd 4 1
225.10.b.m 6 12.b even 2 1
225.10.b.m 6 60.h even 2 1
400.10.a.u 3 5.c odd 4 1
400.10.a.y 3 5.c odd 4 1
400.10.c.q 6 1.a even 1 1 trivial
400.10.c.q 6 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{6} + 117283 T_{3}^{4} + 3296636163 T_{3}^{2} + \)\(29\!\cdots\!81\)\( \) acting on \(S_{10}^{\mathrm{new}}(400, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \)
$3$ \( 2967381766881 + 3296636163 T^{2} + 117283 T^{4} + T^{6} \)
$5$ \( T^{6} \)
$7$ \( \)\(64\!\cdots\!44\)\( + 1624330022979888 T^{2} + 114606892 T^{4} + T^{6} \)
$11$ \( ( 75283351667163 - 1257655633 T - 54699 T^{2} + T^{3} )^{2} \)
$13$ \( \)\(22\!\cdots\!56\)\( + \)\(64\!\cdots\!88\)\( T^{2} + 47782585008 T^{4} + T^{6} \)
$17$ \( \)\(31\!\cdots\!49\)\( + \)\(11\!\cdots\!03\)\( T^{2} + 113679545147 T^{4} + T^{6} \)
$19$ \( ( 226123024842854125 - 499889436625 T - 818845 T^{2} + T^{3} )^{2} \)
$23$ \( \)\(14\!\cdots\!56\)\( + \)\(81\!\cdots\!88\)\( T^{2} + 5844421656108 T^{4} + T^{6} \)
$29$ \( ( -3964526545895424000 - 11063731072000 T + 2175480 T^{2} + T^{3} )^{2} \)
$31$ \( ( 817098195664566648 + 3532627327452 T + 4274066 T^{2} + T^{3} )^{2} \)
$37$ \( \)\(29\!\cdots\!84\)\( + \)\(59\!\cdots\!08\)\( T^{2} + 204279936810732 T^{4} + T^{6} \)
$41$ \( ( \)\(15\!\cdots\!47\)\( - 750102076057093 T - 5926311 T^{2} + T^{3} )^{2} \)
$43$ \( \)\(10\!\cdots\!56\)\( + \)\(12\!\cdots\!88\)\( T^{2} + 260277885634608 T^{4} + T^{6} \)
$47$ \( \)\(21\!\cdots\!64\)\( + \)\(11\!\cdots\!48\)\( T^{2} + 2150337413915312 T^{4} + T^{6} \)
$53$ \( \)\(16\!\cdots\!56\)\( + \)\(10\!\cdots\!88\)\( T^{2} + 18968285309086508 T^{4} + T^{6} \)
$59$ \( ( -\)\(49\!\cdots\!00\)\( - 6651292273432000 T - 5670960 T^{2} + T^{3} )^{2} \)
$61$ \( ( \)\(62\!\cdots\!12\)\( - 12890308075143508 T - 125306926 T^{2} + T^{3} )^{2} \)
$67$ \( \)\(77\!\cdots\!49\)\( + \)\(21\!\cdots\!03\)\( T^{2} + 110005113588027547 T^{4} + T^{6} \)
$71$ \( ( -\)\(11\!\cdots\!32\)\( - 50509406137014928 T + 297550596 T^{2} + T^{3} )^{2} \)
$73$ \( \)\(19\!\cdots\!81\)\( + \)\(11\!\cdots\!63\)\( T^{2} + 197376584603920683 T^{4} + T^{6} \)
$79$ \( ( -\)\(26\!\cdots\!00\)\( - 183462234827962500 T + 310025170 T^{2} + T^{3} )^{2} \)
$83$ \( \)\(70\!\cdots\!81\)\( + \)\(16\!\cdots\!63\)\( T^{2} + 285265670458184883 T^{4} + T^{6} \)
$89$ \( ( \)\(19\!\cdots\!75\)\( + 269595002285863875 T - 1103860035 T^{2} + T^{3} )^{2} \)
$97$ \( \)\(11\!\cdots\!64\)\( + \)\(85\!\cdots\!48\)\( T^{2} + 1696277618616206412 T^{4} + T^{6} \)
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