# Properties

 Label 400.6.a.z Level 400 Weight 6 Character orbit 400.a Self dual yes Analytic conductor 64.154 Analytic rank 0 Dimension 4 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$400 = 2^{4} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 400.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$64.1535279252$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.1595208.1 Defining polynomial: $$x^{4} - x^{3} - 20 x^{2} + 33 x - 3$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{8}\cdot 5^{2}$$ Twist minimal: no (minimal twist has level 40) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( -1 - \beta_{1} ) q^{3} + ( -37 - \beta_{1} + \beta_{3} ) q^{7} + ( 125 + 7 \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{9} +O(q^{10})$$ $$q + ( -1 - \beta_{1} ) q^{3} + ( -37 - \beta_{1} + \beta_{3} ) q^{7} + ( 125 + 7 \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{9} + ( 92 - 18 \beta_{1} - 2 \beta_{2} ) q^{11} + ( 110 - 2 \beta_{2} + 3 \beta_{3} ) q^{13} + ( -168 - 44 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} ) q^{17} + ( 172 - 46 \beta_{1} + 2 \beta_{2} - 8 \beta_{3} ) q^{19} + ( 248 + 149 \beta_{1} - 3 \beta_{2} + 6 \beta_{3} ) q^{21} + ( -1123 - 23 \beta_{1} - 8 \beta_{2} + 5 \beta_{3} ) q^{23} + ( -2038 - 222 \beta_{1} + 16 \beta_{2} + 2 \beta_{3} ) q^{27} + ( -734 - 12 \beta_{1} + 20 \beta_{2} ) q^{29} + ( -528 - 84 \beta_{1} + 12 \beta_{2} + 24 \beta_{3} ) q^{31} + ( 6716 - 156 \beta_{1} - 8 \beta_{2} - 28 \beta_{3} ) q^{33} + ( 2198 - 428 \beta_{1} + 2 \beta_{2} + 17 \beta_{3} ) q^{37} + ( -376 + 36 \beta_{1} + 4 \beta_{2} + 32 \beta_{3} ) q^{39} + ( 2950 - 491 \beta_{1} - 3 \beta_{2} + 2 \beta_{3} ) q^{41} + ( -12069 - 189 \beta_{1} - 54 \beta_{3} ) q^{43} + ( -3681 - 485 \beta_{1} - 56 \beta_{2} - 19 \beta_{3} ) q^{47} + ( 5625 - 285 \beta_{1} - 69 \beta_{2} + 6 \beta_{3} ) q^{49} + ( 15600 + 988 \beta_{1} - 68 \beta_{2} - 88 \beta_{3} ) q^{51} + ( 21074 + 64 \beta_{1} + 34 \beta_{2} + 53 \beta_{3} ) q^{53} + ( 17756 - 572 \beta_{1} - 40 \beta_{2} - 164 \beta_{3} ) q^{57} + ( 11460 + 598 \beta_{1} - 26 \beta_{2} - 88 \beta_{3} ) q^{59} + ( 15482 - 683 \beta_{1} + 61 \beta_{2} - 82 \beta_{3} ) q^{61} + ( -46573 - 521 \beta_{1} + 152 \beta_{2} + 115 \beta_{3} ) q^{63} + ( -18175 + 73 \beta_{1} + 32 \beta_{2} + 26 \beta_{3} ) q^{67} + ( 9592 + 1103 \beta_{1} + 7 \beta_{2} + 26 \beta_{3} ) q^{69} + ( 15704 - 1560 \beta_{1} + 232 \beta_{2} - 24 \beta_{3} ) q^{71} + ( 33268 - 1212 \beta_{1} + 80 \beta_{2} + 48 \beta_{3} ) q^{73} + ( 2860 + 1972 \beta_{1} - 56 \beta_{2} + 440 \beta_{3} ) q^{77} + ( 5408 - 196 \beta_{1} - 100 \beta_{2} + 328 \beta_{3} ) q^{79} + ( 51209 + 3257 \beta_{1} - 63 \beta_{2} - 6 \beta_{3} ) q^{81} + ( -18665 + 1679 \beta_{1} - 352 \beta_{2} + 166 \beta_{3} ) q^{83} + ( 3118 + 2526 \beta_{1} - 112 \beta_{2} - 104 \beta_{3} ) q^{87} + ( 5238 - 564 \beta_{1} + 300 \beta_{2} + 96 \beta_{3} ) q^{89} + ( 60952 - 2120 \beta_{1} - 200 \beta_{2} + 440 \beta_{3} ) q^{91} + ( 26400 + 4608 \beta_{1} - 192 \beta_{2} - 24 \beta_{3} ) q^{93} + ( 14864 + 1860 \beta_{1} + 172 \beta_{2} - 402 \beta_{3} ) q^{97} + ( 33356 - 5062 \beta_{1} + 426 \beta_{2} - 504 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{3} - 148q^{7} + 500q^{9} + O(q^{10})$$ $$4q - 4q^{3} - 148q^{7} + 500q^{9} + 368q^{11} + 440q^{13} - 672q^{17} + 688q^{19} + 992q^{21} - 4492q^{23} - 8152q^{27} - 2936q^{29} - 2112q^{31} + 26864q^{33} + 8792q^{37} - 1504q^{39} + 11800q^{41} - 48276q^{43} - 14724q^{47} + 22500q^{49} + 62400q^{51} + 84296q^{53} + 71024q^{57} + 45840q^{59} + 61928q^{61} - 186292q^{63} - 72700q^{67} + 38368q^{69} + 62816q^{71} + 133072q^{73} + 11440q^{77} + 21632q^{79} + 204836q^{81} - 74660q^{83} + 12472q^{87} + 20952q^{89} + 243808q^{91} + 105600q^{93} + 59456q^{97} + 133424q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 20 x^{2} + 33 x - 3$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$-2 \nu^{3} + 40 \nu - 29$$ $$\beta_{2}$$ $$=$$ $$($$$$22 \nu^{3} + 40 \nu^{2} - 240 \nu - 141$$$$)/3$$ $$\beta_{3}$$ $$=$$ $$($$$$-8 \nu^{3} + 40 \nu^{2} + 120 \nu - 516$$$$)/3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{3} + \beta_{2} + 5 \beta_{1} + 20$$$$)/80$$ $$\nu^{2}$$ $$=$$ $$($$$$5 \beta_{3} + \beta_{2} - 3 \beta_{1} + 820$$$$)/80$$ $$\nu^{3}$$ $$=$$ $$($$$$-\beta_{3} + \beta_{2} + 3 \beta_{1} - 38$$$$)/4$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 1.64654 3.98753 −4.73066 0.0965878
0 −28.9338 0 0 0 −146.828 0 594.165 0
1.2 0 −4.69449 0 0 0 −10.2635 0 −220.962 0
1.3 0 5.49000 0 0 0 188.968 0 −212.860 0
1.4 0 24.1383 0 0 0 −179.876 0 339.657 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 400.6.a.z 4
4.b odd 2 1 200.6.a.k 4
5.b even 2 1 400.6.a.ba 4
5.c odd 4 2 80.6.c.d 8
15.e even 4 2 720.6.f.n 8
20.d odd 2 1 200.6.a.j 4
20.e even 4 2 40.6.c.a 8
40.i odd 4 2 320.6.c.i 8
40.k even 4 2 320.6.c.j 8
60.l odd 4 2 360.6.f.b 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.6.c.a 8 20.e even 4 2
80.6.c.d 8 5.c odd 4 2
200.6.a.j 4 20.d odd 2 1
200.6.a.k 4 4.b odd 2 1
320.6.c.i 8 40.i odd 4 2
320.6.c.j 8 40.k even 4 2
360.6.f.b 8 60.l odd 4 2
400.6.a.z 4 1.a even 1 1 trivial
400.6.a.ba 4 5.b even 2 1
720.6.f.n 8 15.e even 4 2

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$5$$ $$-1$$

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} + 4 T_{3}^{3} - 728 T_{3}^{2} + 432 T_{3} + 18000$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(400))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 + 4 T + 244 T^{2} + 3348 T^{3} + 18486 T^{4} + 813564 T^{5} + 14407956 T^{6} + 57395628 T^{7} + 3486784401 T^{8}$$
$5$ 1
$7$ $$1 + 148 T + 33316 T^{2} + 2108948 T^{3} + 503710694 T^{4} + 35445089036 T^{5} + 9410945395684 T^{6} + 702639103471564 T^{7} + 79792266297612001 T^{8}$$
$11$ $$1 - 368 T + 176204 T^{2} - 51158896 T^{3} + 42277949270 T^{4} - 8239191359696 T^{5} + 4570277964394604 T^{6} - 1537227326344959568 T^{7} +$$$$67\!\cdots\!01$$$$T^{8}$$
$13$ $$1 - 440 T + 886036 T^{2} - 564452872 T^{3} + 392923998710 T^{4} - 209577400203496 T^{5} + 122147586683920564 T^{6} - 22521792926199933080 T^{7} +$$$$19\!\cdots\!01$$$$T^{8}$$
$17$ $$1 + 672 T + 3356228 T^{2} + 3540729312 T^{3} + 5520858237894 T^{4} + 5027329298748384 T^{5} + 6766135176516146372 T^{6} +$$$$19\!\cdots\!96$$$$T^{7} +$$$$40\!\cdots\!01$$$$T^{8}$$
$19$ $$1 - 688 T + 5308396 T^{2} - 6058136368 T^{3} + 15069081422710 T^{4} - 15000545402668432 T^{5} + 32546127598645797196 T^{6} -$$$$10\!\cdots\!12$$$$T^{7} +$$$$37\!\cdots\!01$$$$T^{8}$$
$23$ $$1 + 4492 T + 27426980 T^{2} + 79192657228 T^{3} + 264562508116390 T^{4} + 509711105000837204 T^{5} +$$$$11\!\cdots\!20$$$$T^{6} +$$$$11\!\cdots\!44$$$$T^{7} +$$$$17\!\cdots\!01$$$$T^{8}$$
$29$ $$1 + 2936 T + 58625996 T^{2} + 77951973928 T^{3} + 1466411094282230 T^{4} + 1598884552081323272 T^{5} +$$$$24\!\cdots\!96$$$$T^{6} +$$$$25\!\cdots\!64$$$$T^{7} +$$$$17\!\cdots\!01$$$$T^{8}$$
$31$ $$1 + 2112 T + 80187004 T^{2} + 163080265536 T^{3} + 3196344720873606 T^{4} + 4668849547150239936 T^{5} +$$$$65\!\cdots\!04$$$$T^{6} +$$$$49\!\cdots\!12$$$$T^{7} +$$$$67\!\cdots\!01$$$$T^{8}$$
$37$ $$1 - 8792 T + 164536948 T^{2} - 653354012200 T^{3} + 11523689002604438 T^{4} - 45306152527774275400 T^{5} +$$$$79\!\cdots\!52$$$$T^{6} -$$$$29\!\cdots\!56$$$$T^{7} +$$$$23\!\cdots\!01$$$$T^{8}$$
$41$ $$1 - 11800 T + 337909340 T^{2} - 2943020124776 T^{3} + 51155654972384870 T^{4} -$$$$34\!\cdots\!76$$$$T^{5} +$$$$45\!\cdots\!40$$$$T^{6} -$$$$18\!\cdots\!00$$$$T^{7} +$$$$18\!\cdots\!01$$$$T^{8}$$
$43$ $$1 + 48276 T + 1306891924 T^{2} + 24102642735684 T^{3} + 333606713965182294 T^{4} +$$$$35\!\cdots\!12$$$$T^{5} +$$$$28\!\cdots\!76$$$$T^{6} +$$$$15\!\cdots\!32$$$$T^{7} +$$$$46\!\cdots\!01$$$$T^{8}$$
$47$ $$1 + 14724 T + 590074244 T^{2} + 7755349482468 T^{3} + 177012391425566214 T^{4} +$$$$17\!\cdots\!76$$$$T^{5} +$$$$31\!\cdots\!56$$$$T^{6} +$$$$17\!\cdots\!32$$$$T^{7} +$$$$27\!\cdots\!01$$$$T^{8}$$
$53$ $$1 - 84296 T + 4144600628 T^{2} - 135070817073080 T^{3} + 3234464962764166486 T^{4} -$$$$56\!\cdots\!40$$$$T^{5} +$$$$72\!\cdots\!72$$$$T^{6} -$$$$61\!\cdots\!72$$$$T^{7} +$$$$30\!\cdots\!01$$$$T^{8}$$
$59$ $$1 - 45840 T + 3064286732 T^{2} - 94721285480976 T^{3} + 3348109683185502486 T^{4} -$$$$67\!\cdots\!24$$$$T^{5} +$$$$15\!\cdots\!32$$$$T^{6} -$$$$16\!\cdots\!60$$$$T^{7} +$$$$26\!\cdots\!01$$$$T^{8}$$
$61$ $$1 - 61928 T + 3903014764 T^{2} - 145287706763384 T^{3} + 5198153942066716726 T^{4} -$$$$12\!\cdots\!84$$$$T^{5} +$$$$27\!\cdots\!64$$$$T^{6} -$$$$37\!\cdots\!28$$$$T^{7} +$$$$50\!\cdots\!01$$$$T^{8}$$
$67$ $$1 + 72700 T + 7283604532 T^{2} + 314621051775212 T^{3} + 16093744308113184182 T^{4} +$$$$42\!\cdots\!84$$$$T^{5} +$$$$13\!\cdots\!68$$$$T^{6} +$$$$17\!\cdots\!00$$$$T^{7} +$$$$33\!\cdots\!01$$$$T^{8}$$
$71$ $$1 - 62816 T + 3398787356 T^{2} - 184024084124896 T^{3} + 8353296562609817510 T^{4} -$$$$33\!\cdots\!96$$$$T^{5} +$$$$11\!\cdots\!56$$$$T^{6} -$$$$36\!\cdots\!16$$$$T^{7} +$$$$10\!\cdots\!01$$$$T^{8}$$
$73$ $$1 - 133072 T + 13424418340 T^{2} - 846379227250928 T^{3} + 45561026057566462310 T^{4} -$$$$17\!\cdots\!04$$$$T^{5} +$$$$57\!\cdots\!60$$$$T^{6} -$$$$11\!\cdots\!04$$$$T^{7} +$$$$18\!\cdots\!01$$$$T^{8}$$
$79$ $$1 - 21632 T + 7152876604 T^{2} - 332616618908288 T^{3} + 24121899620797566790 T^{4} -$$$$10\!\cdots\!12$$$$T^{5} +$$$$67\!\cdots\!04$$$$T^{6} -$$$$63\!\cdots\!68$$$$T^{7} +$$$$89\!\cdots\!01$$$$T^{8}$$
$83$ $$1 + 74660 T + 6167006708 T^{2} + 554766838355444 T^{3} + 42874466663031633142 T^{4} +$$$$21\!\cdots\!92$$$$T^{5} +$$$$95\!\cdots\!92$$$$T^{6} +$$$$45\!\cdots\!20$$$$T^{7} +$$$$24\!\cdots\!01$$$$T^{8}$$
$89$ $$1 - 20952 T + 16118164796 T^{2} - 497915996461992 T^{3} +$$$$11\!\cdots\!30$$$$T^{4} -$$$$27\!\cdots\!08$$$$T^{5} +$$$$50\!\cdots\!96$$$$T^{6} -$$$$36\!\cdots\!48$$$$T^{7} +$$$$97\!\cdots\!01$$$$T^{8}$$
$97$ $$1 - 59456 T + 24399465604 T^{2} - 684112850575552 T^{3} +$$$$25\!\cdots\!74$$$$T^{4} -$$$$58\!\cdots\!64$$$$T^{5} +$$$$17\!\cdots\!96$$$$T^{6} -$$$$37\!\cdots\!08$$$$T^{7} +$$$$54\!\cdots\!01$$$$T^{8}$$