# 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 $2$

# Related objects

## 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: 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.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

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