# 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

# 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

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