Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [441,2,Mod(214,441)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("441.214"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(441, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 441.h (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,-2,2,6,5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.52140272914\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.309123.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 3x^{5} + 10x^{4} - 15x^{3} + 19x^{2} - 12x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 63)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 373.3
Root \(0.500000 + 0.224437i\) of defining polynomial
Character \(\chi\) \(=\) 441.373
Dual form 441.2.h.c.214.3

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.69963 q^{2} +(1.29418 - 1.15113i) q^{3} +0.888736 q^{4} +(1.79418 + 3.10761i) q^{5} +(2.19963 - 1.95649i) q^{6} -1.88874 q^{8} +(0.349814 - 2.97954i) q^{9} +(3.04944 + 5.28179i) q^{10} +(1.40545 - 2.43430i) q^{11} +(1.15019 - 1.02305i) q^{12} +(-0.500000 + 0.866025i) q^{13} +(5.89926 + 1.95649i) q^{15} -4.98762 q^{16} +(2.05563 + 3.56046i) q^{17} +(0.594554 - 5.06410i) q^{18} +(0.444368 - 0.769668i) q^{19} +(1.59455 + 2.76185i) q^{20} +(2.38874 - 4.13741i) q^{22} +(-2.93818 - 5.08907i) q^{23} +(-2.44437 + 2.17417i) q^{24} +(-3.93818 + 6.82112i) q^{25} +(-0.849814 + 1.47192i) q^{26} +(-2.97710 - 4.25874i) q^{27} +(0.849814 + 1.47192i) q^{29} +(10.0265 + 3.32530i) q^{30} -6.98762 q^{31} -4.69963 q^{32} +(-0.983290 - 4.76828i) q^{33} +(3.49381 + 6.05146i) q^{34} +(0.310892 - 2.64802i) q^{36} +(-2.38255 + 4.12669i) q^{37} +(0.755260 - 1.30815i) q^{38} +(0.349814 + 1.69636i) q^{39} +(-3.38874 - 5.86946i) q^{40} +(2.70582 - 4.68661i) q^{41} +(-2.60507 - 4.51212i) q^{43} +(1.24907 - 2.16345i) q^{44} +(9.88688 - 4.25874i) q^{45} +(-4.99381 - 8.64953i) q^{46} -2.66621 q^{47} +(-6.45489 + 5.74138i) q^{48} +(-6.69344 + 11.5934i) q^{50} +(6.75890 + 2.24159i) q^{51} +(-0.444368 + 0.769668i) q^{52} +(0.0618219 + 0.107079i) q^{53} +(-5.05996 - 7.23828i) q^{54} +10.0865 q^{55} +(-0.310892 - 1.50761i) q^{57} +(1.44437 + 2.50172i) q^{58} -8.87636 q^{59} +(5.24288 + 1.73880i) q^{60} +3.87636 q^{61} -11.8764 q^{62} +1.98762 q^{64} -3.58836 q^{65} +(-1.67123 - 8.10430i) q^{66} +12.3090 q^{67} +(1.82691 + 3.16431i) q^{68} +(-9.66071 - 3.20397i) q^{69} -2.87636 q^{71} +(-0.660706 + 5.62755i) q^{72} +(5.32072 + 9.21576i) q^{73} +(-4.04944 + 7.01384i) q^{74} +(2.75526 + 13.3611i) q^{75} +(0.394926 - 0.684031i) q^{76} +(0.594554 + 2.88318i) q^{78} -7.08650 q^{79} +(-8.94870 - 15.4996i) q^{80} +(-8.75526 - 2.08457i) q^{81} +(4.59888 - 7.96550i) q^{82} +(2.05563 + 3.56046i) q^{83} +(-7.37636 + 12.7762i) q^{85} +(-4.42766 - 7.66893i) q^{86} +(2.79418 + 0.926690i) q^{87} +(-2.65452 + 4.59776i) q^{88} +(-4.80470 + 8.32199i) q^{89} +(16.8040 - 7.23828i) q^{90} +(-2.61126 - 4.52284i) q^{92} +(-9.04325 + 8.04364i) q^{93} -4.53156 q^{94} +3.18911 q^{95} +(-6.08217 + 5.40987i) q^{96} +(-3.66071 - 6.34053i) q^{97} +(-6.76145 - 5.03913i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 2 q^{2} + 2 q^{3} + 6 q^{4} + 5 q^{5} + q^{6} - 12 q^{8} - 4 q^{9} + 2 q^{11} + 13 q^{12} - 3 q^{13} + 11 q^{15} + 6 q^{16} + 12 q^{17} + 10 q^{18} + 3 q^{19} + 16 q^{20} + 15 q^{22} - 15 q^{24}+ \cdots - 46 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/441\mathbb{Z}\right)^\times\).

\(n\) \(199\) \(344\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{3}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.69963 1.20182 0.600909 − 0.799317i \(-0.294805\pi\)
0.600909 + 0.799317i \(0.294805\pi\)
\(3\) 1.29418 − 1.15113i 0.747196 − 0.664603i
\(4\) 0.888736 0.444368
\(5\) 1.79418 + 3.10761i 0.802383 + 1.38977i 0.918044 + 0.396479i \(0.129768\pi\)
−0.115661 + 0.993289i \(0.536899\pi\)
\(6\) 2.19963 − 1.95649i 0.897994 − 0.798733i
\(7\) 0 0
\(8\) −1.88874 −0.667769
\(9\) 0.349814 − 2.97954i 0.116605 − 0.993178i
\(10\) 3.04944 + 5.28179i 0.964318 + 1.67025i
\(11\) 1.40545 − 2.43430i 0.423758 − 0.733970i −0.572546 − 0.819873i \(-0.694044\pi\)
0.996304 + 0.0859026i \(0.0273774\pi\)
\(12\) 1.15019 − 1.02305i 0.332030 − 0.295328i
\(13\) −0.500000 + 0.866025i −0.138675 + 0.240192i −0.926995 − 0.375073i \(-0.877618\pi\)
0.788320 + 0.615265i \(0.210951\pi\)
\(14\) 0 0
\(15\) 5.89926 + 1.95649i 1.52318 + 0.505163i
\(16\) −4.98762 −1.24691
\(17\) 2.05563 + 3.56046i 0.498564 + 0.863538i 0.999999 − 0.00165734i \(-0.000527549\pi\)
−0.501435 + 0.865196i \(0.667194\pi\)
\(18\) 0.594554 − 5.06410i 0.140138 − 1.19362i
\(19\) 0.444368 − 0.769668i 0.101945 − 0.176574i −0.810541 − 0.585682i \(-0.800827\pi\)
0.912486 + 0.409108i \(0.134160\pi\)
\(20\) 1.59455 + 2.76185i 0.356553 + 0.617568i
\(21\) 0 0
\(22\) 2.38874 − 4.13741i 0.509280 − 0.882099i
\(23\) −2.93818 − 5.08907i −0.612652 − 1.06115i −0.990792 − 0.135396i \(-0.956769\pi\)
0.378139 − 0.925749i \(-0.376564\pi\)
\(24\) −2.44437 + 2.17417i −0.498955 + 0.443802i
\(25\) −3.93818 + 6.82112i −0.787636 + 1.36422i
\(26\) −0.849814 + 1.47192i −0.166662 + 0.288667i
\(27\) −2.97710 − 4.25874i −0.572943 − 0.819595i
\(28\) 0 0
\(29\) 0.849814 + 1.47192i 0.157807 + 0.273329i 0.934077 − 0.357071i \(-0.116224\pi\)
−0.776271 + 0.630399i \(0.782891\pi\)
\(30\) 10.0265 + 3.32530i 1.83059 + 0.607114i
\(31\) −6.98762 −1.25501 −0.627507 − 0.778611i \(-0.715925\pi\)
−0.627507 + 0.778611i \(0.715925\pi\)
\(32\) −4.69963 −0.830785
\(33\) −0.983290 − 4.76828i −0.171169 − 0.830051i
\(34\) 3.49381 + 6.05146i 0.599183 + 1.03782i
\(35\) 0 0
\(36\) 0.310892 − 2.64802i 0.0518154 − 0.441337i
\(37\) −2.38255 + 4.12669i −0.391688 + 0.678424i −0.992672 − 0.120837i \(-0.961442\pi\)
0.600984 + 0.799261i \(0.294775\pi\)
\(38\) 0.755260 − 1.30815i 0.122519 − 0.212210i
\(39\) 0.349814 + 1.69636i 0.0560151 + 0.271635i
\(40\) −3.38874 − 5.86946i −0.535806 − 0.928044i
\(41\) 2.70582 − 4.68661i 0.422578 − 0.731926i −0.573613 − 0.819126i \(-0.694459\pi\)
0.996191 + 0.0872002i \(0.0277920\pi\)
\(42\) 0 0
\(43\) −2.60507 − 4.51212i −0.397270 − 0.688092i 0.596118 − 0.802897i \(-0.296709\pi\)
−0.993388 + 0.114805i \(0.963376\pi\)
\(44\) 1.24907 − 2.16345i 0.188304 − 0.326153i
\(45\) 9.88688 − 4.25874i 1.47385 − 0.634856i
\(46\) −4.99381 − 8.64953i −0.736297 − 1.27530i
\(47\) −2.66621 −0.388906 −0.194453 − 0.980912i \(-0.562293\pi\)
−0.194453 + 0.980912i \(0.562293\pi\)
\(48\) −6.45489 + 5.74138i −0.931683 + 0.828697i
\(49\) 0 0
\(50\) −6.69344 + 11.5934i −0.946595 + 1.63955i
\(51\) 6.75890 + 2.24159i 0.946436 + 0.313885i
\(52\) −0.444368 + 0.769668i −0.0616227 + 0.106734i
\(53\) 0.0618219 + 0.107079i 0.00849190 + 0.0147084i 0.870240 − 0.492628i \(-0.163964\pi\)
−0.861748 + 0.507336i \(0.830630\pi\)
\(54\) −5.05996 − 7.23828i −0.688574 − 0.985005i
\(55\) 10.0865 1.36006
\(56\) 0 0
\(57\) −0.310892 − 1.50761i −0.0411787 − 0.199688i
\(58\) 1.44437 + 2.50172i 0.189655 + 0.328492i
\(59\) −8.87636 −1.15560 −0.577802 − 0.816177i \(-0.696089\pi\)
−0.577802 + 0.816177i \(0.696089\pi\)
\(60\) 5.24288 + 1.73880i 0.676853 + 0.224478i
\(61\) 3.87636 0.496317 0.248158 − 0.968720i \(-0.420175\pi\)
0.248158 + 0.968720i \(0.420175\pi\)
\(62\) −11.8764 −1.50830
\(63\) 0 0
\(64\) 1.98762 0.248453
\(65\) −3.58836 −0.445082
\(66\) −1.67123 − 8.10430i −0.205714 − 0.997571i
\(67\) 12.3090 1.50379 0.751894 − 0.659284i \(-0.229141\pi\)
0.751894 + 0.659284i \(0.229141\pi\)
\(68\) 1.82691 + 3.16431i 0.221546 + 0.383729i
\(69\) −9.66071 − 3.20397i −1.16301 − 0.385713i
\(70\) 0 0
\(71\) −2.87636 −0.341361 −0.170680 − 0.985326i \(-0.554597\pi\)
−0.170680 + 0.985326i \(0.554597\pi\)
\(72\) −0.660706 + 5.62755i −0.0778650 + 0.663214i
\(73\) 5.32072 + 9.21576i 0.622744 + 1.07862i 0.988973 + 0.148099i \(0.0473154\pi\)
−0.366229 + 0.930525i \(0.619351\pi\)
\(74\) −4.04944 + 7.01384i −0.470738 + 0.815342i
\(75\) 2.75526 + 13.3611i 0.318150 + 1.54281i
\(76\) 0.394926 − 0.684031i 0.0453011 − 0.0784638i
\(77\) 0 0
\(78\) 0.594554 + 2.88318i 0.0673200 + 0.326456i
\(79\) −7.08650 −0.797294 −0.398647 − 0.917104i \(-0.630520\pi\)
−0.398647 + 0.917104i \(0.630520\pi\)
\(80\) −8.94870 − 15.4996i −1.00049 − 1.73291i
\(81\) −8.75526 − 2.08457i −0.972807 − 0.231619i
\(82\) 4.59888 − 7.96550i 0.507862 − 0.879642i
\(83\) 2.05563 + 3.56046i 0.225635 + 0.390811i 0.956510 − 0.291700i \(-0.0942210\pi\)
−0.730875 + 0.682512i \(0.760888\pi\)
\(84\) 0 0
\(85\) −7.37636 + 12.7762i −0.800078 + 1.38578i
\(86\) −4.42766 − 7.66893i −0.477447 − 0.826962i
\(87\) 2.79418 + 0.926690i 0.299568 + 0.0993516i
\(88\) −2.65452 + 4.59776i −0.282972 + 0.490123i
\(89\) −4.80470 + 8.32199i −0.509297 + 0.882129i 0.490645 + 0.871360i \(0.336761\pi\)
−0.999942 + 0.0107692i \(0.996572\pi\)
\(90\) 16.8040 − 7.23828i 1.77130 − 0.762981i
\(91\) 0 0
\(92\) −2.61126 − 4.52284i −0.272243 − 0.471539i
\(93\) −9.04325 + 8.04364i −0.937742 + 0.834086i
\(94\) −4.53156 −0.467395
\(95\) 3.18911 0.327196
\(96\) −6.08217 + 5.40987i −0.620759 + 0.552142i
\(97\) −3.66071 − 6.34053i −0.371688 − 0.643783i 0.618137 − 0.786070i \(-0.287888\pi\)
−0.989825 + 0.142287i \(0.954554\pi\)
\(98\) 0 0
\(99\) −6.76145 − 5.03913i −0.679551 − 0.506452i
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 441.2.h.c.373.3 6
3.2 odd 2 1323.2.h.d.226.1 6
7.2 even 3 63.2.f.b.22.1 ✓ 6
7.3 odd 6 441.2.g.d.67.1 6
7.4 even 3 441.2.g.e.67.1 6
7.5 odd 6 441.2.f.d.148.1 6
7.6 odd 2 441.2.h.b.373.3 6
9.2 odd 6 1323.2.g.c.667.3 6
9.7 even 3 441.2.g.e.79.1 6
21.2 odd 6 189.2.f.a.64.3 6
21.5 even 6 1323.2.f.c.442.3 6
21.11 odd 6 1323.2.g.c.361.3 6
21.17 even 6 1323.2.g.b.361.3 6
21.20 even 2 1323.2.h.e.226.1 6
28.23 odd 6 1008.2.r.k.337.1 6
63.2 odd 6 189.2.f.a.127.3 6
63.5 even 6 3969.2.a.p.1.1 3
63.11 odd 6 1323.2.h.d.802.1 6
63.16 even 3 63.2.f.b.43.1 yes 6
63.20 even 6 1323.2.g.b.667.3 6
63.23 odd 6 567.2.a.g.1.1 3
63.25 even 3 inner 441.2.h.c.214.3 6
63.34 odd 6 441.2.g.d.79.1 6
63.38 even 6 1323.2.h.e.802.1 6
63.40 odd 6 3969.2.a.m.1.3 3
63.47 even 6 1323.2.f.c.883.3 6
63.52 odd 6 441.2.h.b.214.3 6
63.58 even 3 567.2.a.d.1.3 3
63.61 odd 6 441.2.f.d.295.1 6
84.23 even 6 3024.2.r.g.1009.1 6
252.23 even 6 9072.2.a.cd.1.3 3
252.79 odd 6 1008.2.r.k.673.1 6
252.191 even 6 3024.2.r.g.2017.1 6
252.247 odd 6 9072.2.a.bq.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
63.2.f.b.22.1 ✓ 6 7.2 even 3
63.2.f.b.43.1 yes 6 63.16 even 3
189.2.f.a.64.3 6 21.2 odd 6
189.2.f.a.127.3 6 63.2 odd 6
441.2.f.d.148.1 6 7.5 odd 6
441.2.f.d.295.1 6 63.61 odd 6
441.2.g.d.67.1 6 7.3 odd 6
441.2.g.d.79.1 6 63.34 odd 6
441.2.g.e.67.1 6 7.4 even 3
441.2.g.e.79.1 6 9.7 even 3
441.2.h.b.214.3 6 63.52 odd 6
441.2.h.b.373.3 6 7.6 odd 2
441.2.h.c.214.3 6 63.25 even 3 inner
441.2.h.c.373.3 6 1.1 even 1 trivial
567.2.a.d.1.3 3 63.58 even 3
567.2.a.g.1.1 3 63.23 odd 6
1008.2.r.k.337.1 6 28.23 odd 6
1008.2.r.k.673.1 6 252.79 odd 6
1323.2.f.c.442.3 6 21.5 even 6
1323.2.f.c.883.3 6 63.47 even 6
1323.2.g.b.361.3 6 21.17 even 6
1323.2.g.b.667.3 6 63.20 even 6
1323.2.g.c.361.3 6 21.11 odd 6
1323.2.g.c.667.3 6 9.2 odd 6
1323.2.h.d.226.1 6 3.2 odd 2
1323.2.h.d.802.1 6 63.11 odd 6
1323.2.h.e.226.1 6 21.20 even 2
1323.2.h.e.802.1 6 63.38 even 6
3024.2.r.g.1009.1 6 84.23 even 6
3024.2.r.g.2017.1 6 252.191 even 6
3969.2.a.m.1.3 3 63.40 odd 6
3969.2.a.p.1.1 3 63.5 even 6
9072.2.a.bq.1.1 3 252.247 odd 6
9072.2.a.cd.1.3 3 252.23 even 6