Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [342,2,Mod(55,342)] 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("342.55"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(342, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([0, 10])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 342 = 2 \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 342.u (of order \(9\), degree \(6\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,0,0,-3,0,-3,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(8)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.73088374913\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{18})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{3} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 114)
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label 289.1
Root \(0.939693 + 0.342020i\) of defining polynomial
Character \(\chi\) \(=\) 342.289
Dual form 342.2.u.b.271.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.939693 + 0.342020i) q^{2} +(0.766044 + 0.642788i) q^{4} +(0.0923963 - 0.0775297i) q^{5} +(2.14543 - 3.71599i) q^{7} +(0.500000 + 0.866025i) q^{8} +(0.113341 - 0.0412527i) q^{10} +(1.28699 + 2.22913i) q^{11} +(0.141559 - 0.802823i) q^{13} +(3.28699 - 2.75811i) q^{14} +(0.173648 + 0.984808i) q^{16} +(-0.439693 - 0.160035i) q^{17} +(3.16637 + 2.99568i) q^{19} +0.120615 q^{20} +(0.446967 + 2.53487i) q^{22} +(-4.25490 - 3.57029i) q^{23} +(-0.865715 + 4.90971i) q^{25} +(0.407604 - 0.705990i) q^{26} +(4.03209 - 1.46756i) q^{28} +(2.20574 - 0.802823i) q^{29} +(-2.67365 + 4.63089i) q^{31} +(-0.173648 + 0.984808i) q^{32} +(-0.358441 - 0.300767i) q^{34} +(-0.0898700 - 0.509678i) q^{35} -8.51754 q^{37} +(1.95084 + 3.89798i) q^{38} +(0.113341 + 0.0412527i) q^{40} +(-0.666374 - 3.77920i) q^{41} +(7.14930 - 5.99898i) q^{43} +(-0.446967 + 2.53487i) q^{44} +(-2.77719 - 4.81023i) q^{46} +(-8.90420 + 3.24086i) q^{47} +(-5.70574 - 9.88263i) q^{49} +(-2.49273 + 4.31753i) q^{50} +(0.624485 - 0.524005i) q^{52} +(-9.77379 - 8.20118i) q^{53} +(0.291737 + 0.106183i) q^{55} +4.29086 q^{56} +2.34730 q^{58} +(14.1420 + 5.14728i) q^{59} +(-1.31521 - 1.10359i) q^{61} +(-4.09627 + 3.43718i) q^{62} +(-0.500000 + 0.866025i) q^{64} +(-0.0491630 - 0.0851529i) q^{65} +(-10.7652 + 3.91820i) q^{67} +(-0.233956 - 0.405223i) q^{68} +(0.0898700 - 0.509678i) q^{70} +(-10.2135 + 8.57013i) q^{71} +(0.396459 + 2.24843i) q^{73} +(-8.00387 - 2.91317i) q^{74} +(0.500000 + 4.33013i) q^{76} +11.0446 q^{77} +(0.843426 + 4.78331i) q^{79} +(0.0923963 + 0.0775297i) q^{80} +(0.666374 - 3.77920i) q^{82} +(1.62449 - 2.81369i) q^{83} +(-0.0530334 + 0.0193026i) q^{85} +(8.76991 - 3.19199i) q^{86} +(-1.28699 + 2.22913i) q^{88} +(-0.595800 + 3.37895i) q^{89} +(-2.67958 - 2.24843i) q^{91} +(-0.964508 - 5.46999i) q^{92} -9.47565 q^{94} +(0.524815 + 0.0313013i) q^{95} +(-2.91400 - 1.06061i) q^{97} +(-1.98158 - 11.2381i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 3 q^{5} - 3 q^{7} + 3 q^{8} - 6 q^{10} + 9 q^{13} + 12 q^{14} + 3 q^{17} + 12 q^{20} + 15 q^{22} - 27 q^{23} - 15 q^{25} + 6 q^{26} + 15 q^{28} + 3 q^{29} - 15 q^{31} + 6 q^{34} - 12 q^{35} - 6 q^{37}+ \cdots + 18 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/342\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(325\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{9}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.939693 + 0.342020i 0.664463 + 0.241845i
\(3\) 0 0
\(4\) 0.766044 + 0.642788i 0.383022 + 0.321394i
\(5\) 0.0923963 − 0.0775297i 0.0413209 − 0.0346723i −0.621893 − 0.783102i \(-0.713636\pi\)
0.663214 + 0.748430i \(0.269192\pi\)
\(6\) 0 0
\(7\) 2.14543 − 3.71599i 0.810896 − 1.40451i −0.101341 − 0.994852i \(-0.532313\pi\)
0.912238 − 0.409662i \(-0.134353\pi\)
\(8\) 0.500000 + 0.866025i 0.176777 + 0.306186i
\(9\) 0 0
\(10\) 0.113341 − 0.0412527i 0.0358415 − 0.0130452i
\(11\) 1.28699 + 2.22913i 0.388042 + 0.672108i 0.992186 − 0.124767i \(-0.0398183\pi\)
−0.604144 + 0.796875i \(0.706485\pi\)
\(12\) 0 0
\(13\) 0.141559 − 0.802823i 0.0392615 − 0.222663i −0.958864 − 0.283866i \(-0.908383\pi\)
0.998125 + 0.0612035i \(0.0194939\pi\)
\(14\) 3.28699 − 2.75811i 0.878485 − 0.737136i
\(15\) 0 0
\(16\) 0.173648 + 0.984808i 0.0434120 + 0.246202i
\(17\) −0.439693 − 0.160035i −0.106641 − 0.0388142i 0.288149 − 0.957586i \(-0.406960\pi\)
−0.394790 + 0.918772i \(0.629183\pi\)
\(18\) 0 0
\(19\) 3.16637 + 2.99568i 0.726416 + 0.687255i
\(20\) 0.120615 0.0269703
\(21\) 0 0
\(22\) 0.446967 + 2.53487i 0.0952936 + 0.540437i
\(23\) −4.25490 − 3.57029i −0.887208 − 0.744456i 0.0804401 − 0.996759i \(-0.474367\pi\)
−0.967648 + 0.252304i \(0.918812\pi\)
\(24\) 0 0
\(25\) −0.865715 + 4.90971i −0.173143 + 0.981942i
\(26\) 0.407604 − 0.705990i 0.0799377 − 0.138456i
\(27\) 0 0
\(28\) 4.03209 − 1.46756i 0.761993 − 0.277343i
\(29\) 2.20574 − 0.802823i 0.409595 − 0.149080i −0.129001 − 0.991644i \(-0.541177\pi\)
0.538596 + 0.842564i \(0.318955\pi\)
\(30\) 0 0
\(31\) −2.67365 + 4.63089i −0.480201 + 0.831733i −0.999742 − 0.0227125i \(-0.992770\pi\)
0.519541 + 0.854446i \(0.326103\pi\)
\(32\) −0.173648 + 0.984808i −0.0306970 + 0.174091i
\(33\) 0 0
\(34\) −0.358441 − 0.300767i −0.0614721 − 0.0515812i
\(35\) −0.0898700 − 0.509678i −0.0151908 − 0.0861514i
\(36\) 0 0
\(37\) −8.51754 −1.40028 −0.700138 − 0.714008i \(-0.746878\pi\)
−0.700138 + 0.714008i \(0.746878\pi\)
\(38\) 1.95084 + 3.89798i 0.316468 + 0.632336i
\(39\) 0 0
\(40\) 0.113341 + 0.0412527i 0.0179208 + 0.00652262i
\(41\) −0.666374 − 3.77920i −0.104070 − 0.590211i −0.991588 − 0.129437i \(-0.958683\pi\)
0.887517 − 0.460774i \(-0.152428\pi\)
\(42\) 0 0
\(43\) 7.14930 − 5.99898i 1.09026 − 0.914835i 0.0935262 − 0.995617i \(-0.470186\pi\)
0.996732 + 0.0807817i \(0.0257417\pi\)
\(44\) −0.446967 + 2.53487i −0.0673827 + 0.382147i
\(45\) 0 0
\(46\) −2.77719 − 4.81023i −0.409474 − 0.709230i
\(47\) −8.90420 + 3.24086i −1.29881 + 0.472729i −0.896609 − 0.442822i \(-0.853977\pi\)
−0.402202 + 0.915551i \(0.631755\pi\)
\(48\) 0 0
\(49\) −5.70574 − 9.88263i −0.815105 − 1.41180i
\(50\) −2.49273 + 4.31753i −0.352525 + 0.610591i
\(51\) 0 0
\(52\) 0.624485 − 0.524005i 0.0866005 − 0.0726665i
\(53\) −9.77379 − 8.20118i −1.34253 − 1.12652i −0.980968 − 0.194172i \(-0.937798\pi\)
−0.361565 − 0.932347i \(-0.617757\pi\)
\(54\) 0 0
\(55\) 0.291737 + 0.106183i 0.0393378 + 0.0143178i
\(56\) 4.29086 0.573390
\(57\) 0 0
\(58\) 2.34730 0.308215
\(59\) 14.1420 + 5.14728i 1.84113 + 0.670118i 0.989220 + 0.146439i \(0.0467811\pi\)
0.851915 + 0.523680i \(0.175441\pi\)
\(60\) 0 0
\(61\) −1.31521 − 1.10359i −0.168395 − 0.141300i 0.554696 − 0.832053i \(-0.312835\pi\)
−0.723091 + 0.690753i \(0.757279\pi\)
\(62\) −4.09627 + 3.43718i −0.520226 + 0.436522i
\(63\) 0 0
\(64\) −0.500000 + 0.866025i −0.0625000 + 0.108253i
\(65\) −0.0491630 − 0.0851529i −0.00609792 − 0.0105619i
\(66\) 0 0
\(67\) −10.7652 + 3.91820i −1.31517 + 0.478684i −0.901909 − 0.431926i \(-0.857834\pi\)
−0.413266 + 0.910611i \(0.635612\pi\)
\(68\) −0.233956 − 0.405223i −0.0283713 − 0.0491405i
\(69\) 0 0
\(70\) 0.0898700 − 0.509678i 0.0107415 − 0.0609182i
\(71\) −10.2135 + 8.57013i −1.21212 + 1.01709i −0.212918 + 0.977070i \(0.568297\pi\)
−0.999199 + 0.0400167i \(0.987259\pi\)
\(72\) 0 0
\(73\) 0.396459 + 2.24843i 0.0464021 + 0.263159i 0.999179 − 0.0405117i \(-0.0128988\pi\)
−0.952777 + 0.303671i \(0.901788\pi\)
\(74\) −8.00387 − 2.91317i −0.930431 − 0.338649i
\(75\) 0 0
\(76\) 0.500000 + 4.33013i 0.0573539 + 0.496700i
\(77\) 11.0446 1.25865
\(78\) 0 0
\(79\) 0.843426 + 4.78331i 0.0948928 + 0.538164i 0.994780 + 0.102042i \(0.0325375\pi\)
−0.899887 + 0.436122i \(0.856351\pi\)
\(80\) 0.0923963 + 0.0775297i 0.0103302 + 0.00866808i
\(81\) 0 0
\(82\) 0.666374 − 3.77920i 0.0735887 − 0.417342i
\(83\) 1.62449 − 2.81369i 0.178310 − 0.308843i −0.762992 − 0.646408i \(-0.776270\pi\)
0.941302 + 0.337566i \(0.109604\pi\)
\(84\) 0 0
\(85\) −0.0530334 + 0.0193026i −0.00575228 + 0.00209366i
\(86\) 8.76991 − 3.19199i 0.945684 − 0.344201i
\(87\) 0 0
\(88\) −1.28699 + 2.22913i −0.137193 + 0.237626i
\(89\) −0.595800 + 3.37895i −0.0631547 + 0.358168i 0.936811 + 0.349837i \(0.113763\pi\)
−0.999965 + 0.00833100i \(0.997348\pi\)
\(90\) 0 0
\(91\) −2.67958 − 2.24843i −0.280896 − 0.235700i
\(92\) −0.964508 − 5.46999i −0.100557 − 0.570286i
\(93\) 0 0
\(94\) −9.47565 −0.977339
\(95\) 0.524815 + 0.0313013i 0.0538449 + 0.00321145i
\(96\) 0 0
\(97\) −2.91400 − 1.06061i −0.295872 − 0.107689i 0.189819 − 0.981819i \(-0.439210\pi\)
−0.485691 + 0.874130i \(0.661432\pi\)
\(98\) −1.98158 − 11.2381i −0.200170 − 1.13522i
\(99\) 0 0
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 342.2.u.b.289.1 6
3.2 odd 2 114.2.i.c.61.1 yes 6
12.11 even 2 912.2.bo.d.289.1 6
19.5 even 9 inner 342.2.u.b.271.1 6
19.9 even 9 6498.2.a.bp.1.1 3
19.10 odd 18 6498.2.a.bu.1.1 3
57.5 odd 18 114.2.i.c.43.1 ✓ 6
57.29 even 18 2166.2.a.p.1.3 3
57.47 odd 18 2166.2.a.r.1.3 3
228.119 even 18 912.2.bo.d.385.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
114.2.i.c.43.1 ✓ 6 57.5 odd 18
114.2.i.c.61.1 yes 6 3.2 odd 2
342.2.u.b.271.1 6 19.5 even 9 inner
342.2.u.b.289.1 6 1.1 even 1 trivial
912.2.bo.d.289.1 6 12.11 even 2
912.2.bo.d.385.1 6 228.119 even 18
2166.2.a.p.1.3 3 57.29 even 18
2166.2.a.r.1.3 3 57.47 odd 18
6498.2.a.bp.1.1 3 19.9 even 9
6498.2.a.bu.1.1 3 19.10 odd 18