Properties

Label 400.10.c.l
Level 400
Weight 10
Character orbit 400.c
Analytic conductor 206.014
Analytic rank 0
Dimension 4
CM no
Inner twists 2

Related objects

Downloads

Learn more about

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: \(4\)
Coefficient field: \(\Q(i, \sqrt{79})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{12}\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 20)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -13 \beta_{1} + \beta_{3} ) q^{3} + ( 19 \beta_{1} - 69 \beta_{3} ) q^{7} + ( -17441 + 26 \beta_{2} ) q^{9} +O(q^{10})\) \( q + ( -13 \beta_{1} + \beta_{3} ) q^{3} + ( 19 \beta_{1} - 69 \beta_{3} ) q^{7} + ( -17441 + 26 \beta_{2} ) q^{9} + ( -51360 - 3 \beta_{2} ) q^{11} + ( -8957 \beta_{1} - 564 \beta_{3} ) q^{13} + ( 15801 \beta_{1} + 348 \beta_{3} ) q^{17} + ( 68636 + 516 \beta_{2} ) q^{19} + ( 1420156 - 916 \beta_{2} ) q^{21} + ( -33273 \beta_{1} - 6393 \beta_{3} ) q^{23} + ( 496678 \beta_{1} - 31558 \beta_{3} ) q^{27} + ( 3446874 + 588 \beta_{2} ) q^{29} + ( -145916 + 2625 \beta_{2} ) q^{31} + ( 607008 \beta_{1} - 47460 \beta_{3} ) q^{33} + ( 563069 \beta_{1} - 59976 \beta_{3} ) q^{37} + ( -237764 + 1625 \beta_{2} ) q^{39} + ( 14886726 + 7218 \beta_{2} ) q^{41} + ( -585409 \beta_{1} - 99615 \beta_{3} ) q^{43} + ( -3124665 \beta_{1} - 42573 \beta_{3} ) q^{47} + ( -55968957 + 2622 \beta_{2} ) q^{49} + ( 13503348 - 11277 \beta_{2} ) q^{51} + ( -470889 \beta_{1} + 212532 \beta_{3} ) q^{53} + ( 9543316 \beta_{1} - 602164 \beta_{3} ) q^{57} + ( -46465428 - 37398 \beta_{2} ) q^{59} + ( 97836962 + 72144 \beta_{2} ) q^{61} + ( -36613235 \beta_{1} + 1252829 \beta_{3} ) q^{63} + ( 10988371 \beta_{1} - 203139 \beta_{3} ) q^{67} + ( 86037132 - 49836 \beta_{2} ) q^{69} + ( -155603508 + 139761 \beta_{2} ) q^{71} + ( 4961203 \beta_{1} + 2047716 \beta_{3} ) q^{73} + ( 3210528 \beta_{1} + 3538140 \beta_{3} ) q^{77} + ( 271130888 + 7002 \beta_{2} ) q^{79} + ( 940619189 - 395174 \beta_{2} ) q^{81} + ( -62845785 \beta_{1} + 192957 \beta_{3} ) q^{83} + ( -32917650 \beta_{1} + 2682474 \beta_{3} ) q^{87} + ( 231145926 - 542196 \beta_{2} ) q^{89} + ( -770018884 - 607317 \beta_{2} ) q^{91} + ( 54984908 \beta_{1} - 3558416 \beta_{3} ) q^{93} + ( 83585837 \beta_{1} - 2902956 \beta_{3} ) q^{97} + ( 738022560 - 1283037 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 69764q^{9} + O(q^{10}) \) \( 4q - 69764q^{9} - 205440q^{11} + 274544q^{19} + 5680624q^{21} + 13787496q^{29} - 583664q^{31} - 951056q^{39} + 59546904q^{41} - 223875828q^{49} + 54013392q^{51} - 185861712q^{59} + 391347848q^{61} + 344148528q^{69} - 622414032q^{71} + 1084523552q^{79} + 3762476756q^{81} + 924583704q^{89} - 3080075536q^{91} + 2952090240q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 39 x^{2} + 400\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} - 19 \nu \)\()/2\)
\(\beta_{2}\)\(=\)\( -8 \nu^{3} + 472 \nu \)
\(\beta_{3}\)\(=\)\( 32 \nu^{2} - 624 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} + 16 \beta_{1}\)\()/320\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} + 624\)\()/32\)
\(\nu^{3}\)\(=\)\((\)\(19 \beta_{2} + 944 \beta_{1}\)\()/320\)

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
−4.44410 + 0.500000i
4.44410 0.500000i
4.44410 + 0.500000i
−4.44410 0.500000i
0 272.211i 0 0 0 10002.6i 0 −54415.9 0
49.2 0 12.2111i 0 0 0 9622.57i 0 19533.9 0
49.3 0 12.2111i 0 0 0 9622.57i 0 19533.9 0
49.4 0 272.211i 0 0 0 10002.6i 0 −54415.9 0
\(n\): e.g. 2-40 or 990-1000
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.l 4
4.b odd 2 1 100.10.c.c 4
5.b even 2 1 inner 400.10.c.l 4
5.c odd 4 1 80.10.a.j 2
5.c odd 4 1 400.10.a.l 2
20.d odd 2 1 100.10.c.c 4
20.e even 4 1 20.10.a.b 2
20.e even 4 1 100.10.a.c 2
40.i odd 4 1 320.10.a.l 2
40.k even 4 1 320.10.a.t 2
60.l odd 4 1 180.10.a.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.10.a.b 2 20.e even 4 1
80.10.a.j 2 5.c odd 4 1
100.10.a.c 2 20.e even 4 1
100.10.c.c 4 4.b odd 2 1
100.10.c.c 4 20.d odd 2 1
180.10.a.e 2 60.l odd 4 1
320.10.a.l 2 40.i odd 4 1
320.10.a.t 2 40.k even 4 1
400.10.a.l 2 5.c odd 4 1
400.10.c.l 4 1.a even 1 1 trivial
400.10.c.l 4 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 74248 T_{3}^{2} + 11048976 \) acting on \(S_{10}^{\mathrm{new}}(400, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - 4484 T^{2} - 587274858 T^{4} - 1737193472676 T^{6} + 150094635296999121 T^{8} \)
$5$ 1
$7$ \( 1 + 31230700 T^{2} + 3486762586041798 T^{4} + \)\(50\!\cdots\!00\)\( T^{6} + \)\(26\!\cdots\!01\)\( T^{8} \)
$11$ \( ( 1 + 102720 T + 7335543382 T^{2} + 242208386819520 T^{3} + 5559917313492231481 T^{4} )^{2} \)
$13$ \( 1 - 13506080684 T^{2} + 64066498776709104822 T^{4} - \)\(15\!\cdots\!36\)\( T^{6} + \)\(12\!\cdots\!41\)\( T^{8} \)
$17$ \( 1 - 419518771196 T^{2} + \)\(71\!\cdots\!22\)\( T^{4} - \)\(58\!\cdots\!64\)\( T^{6} + \)\(19\!\cdots\!81\)\( T^{8} \)
$19$ \( ( 1 - 137272 T + 111610161654 T^{2} - 44295985649518888 T^{3} + \)\(10\!\cdots\!41\)\( T^{4} )^{2} \)
$23$ \( 1 - 5330064218900 T^{2} + \)\(13\!\cdots\!38\)\( T^{4} - \)\(17\!\cdots\!00\)\( T^{6} + \)\(10\!\cdots\!61\)\( T^{8} \)
$29$ \( ( 1 - 6893748 T + 40195999658014 T^{2} - \)\(10\!\cdots\!12\)\( T^{3} + \)\(21\!\cdots\!61\)\( T^{4} )^{2} \)
$31$ \( ( 1 + 291832 T + 38964935800398 T^{2} + 7715927814392939272 T^{3} + \)\(69\!\cdots\!41\)\( T^{4} )^{2} \)
$37$ \( 1 - 310941286370060 T^{2} + \)\(48\!\cdots\!58\)\( T^{4} - \)\(52\!\cdots\!40\)\( T^{6} + \)\(28\!\cdots\!41\)\( T^{8} \)
$41$ \( ( 1 - 29773452 T + 771012402449398 T^{2} - \)\(97\!\cdots\!72\)\( T^{3} + \)\(10\!\cdots\!21\)\( T^{4} )^{2} \)
$43$ \( 1 - 1540458208886372 T^{2} + \)\(10\!\cdots\!94\)\( T^{4} - \)\(38\!\cdots\!28\)\( T^{6} + \)\(63\!\cdots\!01\)\( T^{8} \)
$47$ \( 1 - 2450505224578676 T^{2} + \)\(38\!\cdots\!22\)\( T^{4} - \)\(30\!\cdots\!64\)\( T^{6} + \)\(15\!\cdots\!21\)\( T^{8} \)
$53$ \( 1 - 11327676942925580 T^{2} + \)\(53\!\cdots\!78\)\( T^{4} - \)\(12\!\cdots\!20\)\( T^{6} + \)\(11\!\cdots\!21\)\( T^{8} \)
$59$ \( ( 1 + 92930856 T + 16656477955483462 T^{2} + \)\(80\!\cdots\!84\)\( T^{3} + \)\(75\!\cdots\!21\)\( T^{4} )^{2} \)
$61$ \( ( 1 - 195673924 T + 22434263296171326 T^{2} - \)\(22\!\cdots\!84\)\( T^{3} + \)\(13\!\cdots\!81\)\( T^{4} )^{2} \)
$67$ \( 1 - 83008173082523780 T^{2} + \)\(31\!\cdots\!18\)\( T^{4} - \)\(61\!\cdots\!20\)\( T^{6} + \)\(54\!\cdots\!81\)\( T^{8} \)
$71$ \( ( 1 + 311207016 T + 76405636625293726 T^{2} + \)\(14\!\cdots\!96\)\( T^{3} + \)\(21\!\cdots\!61\)\( T^{4} )^{2} \)
$73$ \( 1 - 60959480039527964 T^{2} + \)\(70\!\cdots\!62\)\( T^{4} - \)\(21\!\cdots\!16\)\( T^{6} + \)\(12\!\cdots\!61\)\( T^{8} \)
$79$ \( ( 1 - 542261776 T + 313115996157615582 T^{2} - \)\(64\!\cdots\!44\)\( T^{3} + \)\(14\!\cdots\!61\)\( T^{4} )^{2} \)
$83$ \( 1 + 43663493653967740 T^{2} + \)\(69\!\cdots\!18\)\( T^{4} + \)\(15\!\cdots\!60\)\( T^{6} + \)\(12\!\cdots\!81\)\( T^{8} \)
$89$ \( ( 1 - 462291852 T + 159603168035249494 T^{2} - \)\(16\!\cdots\!68\)\( T^{3} + \)\(12\!\cdots\!81\)\( T^{4} )^{2} \)
$97$ \( 1 - 1302744298917711740 T^{2} + \)\(11\!\cdots\!78\)\( T^{4} - \)\(75\!\cdots\!60\)\( T^{6} + \)\(33\!\cdots\!21\)\( T^{8} \)
show more
show less