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