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Properties

Label 819.2.j.h.352.5
Level $819$
Weight $2$
Character 819.352
Analytic conductor $6.540$
Analytic rank $0$
Dimension $10$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [819,2,Mod(235,819)] 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("819.235"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(819, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 4, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 819 = 3^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 819.j (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [10,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.53974792554\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - x^{9} + 8x^{8} + 7x^{7} + 41x^{6} + 18x^{5} + 58x^{4} + 28x^{3} + 64x^{2} + 16x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 91)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 352.5
Root \(-0.862625 + 1.49411i\) of defining polynomial
Character \(\chi\) \(=\) 819.352
Dual form 819.2.j.h.235.5

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.36263 - 2.36014i) q^{2} +(-2.71349 - 4.69991i) q^{4} +(1.09358 - 1.89414i) q^{5} +(-2.19729 + 1.47375i) q^{7} -9.33940 q^{8} +(-2.98028 - 5.16200i) q^{10} +(-0.524077 - 0.907729i) q^{11} +1.00000 q^{13} +(0.484172 + 7.19406i) q^{14} +(-7.29912 + 12.6424i) q^{16} +(-2.64562 - 4.58236i) q^{17} +(-0.378453 + 0.655500i) q^{19} -11.8697 q^{20} -2.85648 q^{22} +(0.326792 - 0.566020i) q^{23} +(0.108157 + 0.187333i) q^{25} +(1.36263 - 2.36014i) q^{26} +(12.8888 + 6.32803i) q^{28} +3.10408 q^{29} +(-0.513956 - 0.890198i) q^{31} +(10.5525 + 18.2775i) q^{32} -14.4200 q^{34} +(0.388575 + 5.77363i) q^{35} +(5.44661 - 9.43381i) q^{37} +(1.03138 + 1.78640i) q^{38} +(-10.2134 + 17.6901i) q^{40} -7.32040 q^{41} +0.887771 q^{43} +(-2.84416 + 4.92623i) q^{44} +(-0.890590 - 1.54255i) q^{46} +(1.16875 - 2.02434i) q^{47} +(2.65613 - 6.47650i) q^{49} +0.589510 q^{50} +(-2.71349 - 4.69991i) q^{52} +(2.44407 + 4.23325i) q^{53} -2.29249 q^{55} +(20.5213 - 13.7639i) q^{56} +(4.22970 - 7.32606i) q^{58} +(-0.524077 - 0.907729i) q^{59} +(6.24989 - 10.8251i) q^{61} -2.80132 q^{62} +28.3200 q^{64} +(1.09358 - 1.89414i) q^{65} +(-2.23944 - 3.87883i) q^{67} +(-14.3578 + 24.8684i) q^{68} +(14.1560 + 6.95021i) q^{70} +6.60274 q^{71} +(4.14174 + 7.17370i) q^{73} +(-14.8434 - 25.7095i) q^{74} +4.10772 q^{76} +(2.48931 + 1.22218i) q^{77} +(-1.07007 + 1.85342i) q^{79} +(15.9644 + 27.6511i) q^{80} +(-9.97496 + 17.2771i) q^{82} +6.66558 q^{83} -11.5728 q^{85} +(1.20970 - 2.09526i) q^{86} +(4.89457 + 8.47765i) q^{88} +(-2.88388 + 4.99503i) q^{89} +(-2.19729 + 1.47375i) q^{91} -3.54699 q^{92} +(-3.18515 - 5.51684i) q^{94} +(0.827739 + 1.43369i) q^{95} -2.88777 q^{97} +(-11.6661 - 15.0939i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 4 q^{2} - 8 q^{4} + 2 q^{5} + q^{7} - 18 q^{8} + 5 q^{10} + 11 q^{11} + 10 q^{13} - 10 q^{14} - 10 q^{16} - 5 q^{17} - 9 q^{19} - 2 q^{20} + 16 q^{22} + 10 q^{23} - 9 q^{25} + 4 q^{26} + 37 q^{28}+ \cdots + 7 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/819\mathbb{Z}\right)^\times\).

\(n\) \(92\) \(379\) \(703\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{1}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.36263 − 2.36014i 0.963521 − 1.66887i 0.249986 − 0.968250i \(-0.419574\pi\)
0.713536 − 0.700619i \(-0.247093\pi\)
\(3\) 0 0
\(4\) −2.71349 − 4.69991i −1.35675 − 2.34996i
\(5\) 1.09358 − 1.89414i 0.489065 − 0.847085i −0.510856 − 0.859666i \(-0.670672\pi\)
0.999921 + 0.0125813i \(0.00400485\pi\)
\(6\) 0 0
\(7\) −2.19729 + 1.47375i −0.830496 + 0.557025i
\(8\) −9.33940 −3.30198
\(9\) 0 0
\(10\) −2.98028 − 5.16200i −0.942449 − 1.63237i
\(11\) −0.524077 − 0.907729i −0.158015 − 0.273691i 0.776138 − 0.630564i \(-0.217176\pi\)
−0.934153 + 0.356873i \(0.883843\pi\)
\(12\) 0 0
\(13\) 1.00000 0.277350
\(14\) 0.484172 + 7.19406i 0.129400 + 1.92269i
\(15\) 0 0
\(16\) −7.29912 + 12.6424i −1.82478 + 3.16061i
\(17\) −2.64562 − 4.58236i −0.641658 − 1.11138i −0.985063 − 0.172197i \(-0.944913\pi\)
0.343404 − 0.939188i \(-0.388420\pi\)
\(18\) 0 0
\(19\) −0.378453 + 0.655500i −0.0868231 + 0.150382i −0.906167 − 0.422921i \(-0.861005\pi\)
0.819344 + 0.573303i \(0.194338\pi\)
\(20\) −11.8697 −2.65415
\(21\) 0 0
\(22\) −2.85648 −0.609005
\(23\) 0.326792 − 0.566020i 0.0681408 − 0.118023i −0.829942 − 0.557850i \(-0.811627\pi\)
0.898083 + 0.439826i \(0.144960\pi\)
\(24\) 0 0
\(25\) 0.108157 + 0.187333i 0.0216314 + 0.0374667i
\(26\) 1.36263 − 2.36014i 0.267233 − 0.462861i
\(27\) 0 0
\(28\) 12.8888 + 6.32803i 2.43576 + 1.19589i
\(29\) 3.10408 0.576414 0.288207 − 0.957568i \(-0.406941\pi\)
0.288207 + 0.957568i \(0.406941\pi\)
\(30\) 0 0
\(31\) −0.513956 − 0.890198i −0.0923092 − 0.159884i 0.816173 − 0.577807i \(-0.196091\pi\)
−0.908482 + 0.417923i \(0.862758\pi\)
\(32\) 10.5525 + 18.2775i 1.86544 + 3.23104i
\(33\) 0 0
\(34\) −14.4200 −2.47301
\(35\) 0.388575 + 5.77363i 0.0656811 + 0.975922i
\(36\) 0 0
\(37\) 5.44661 − 9.43381i 0.895418 − 1.55091i 0.0621309 − 0.998068i \(-0.480210\pi\)
0.833287 − 0.552841i \(-0.186456\pi\)
\(38\) 1.03138 + 1.78640i 0.167312 + 0.289793i
\(39\) 0 0
\(40\) −10.2134 + 17.6901i −1.61488 + 2.79706i
\(41\) −7.32040 −1.14325 −0.571627 − 0.820514i \(-0.693688\pi\)
−0.571627 + 0.820514i \(0.693688\pi\)
\(42\) 0 0
\(43\) 0.887771 0.135384 0.0676919 − 0.997706i \(-0.478437\pi\)
0.0676919 + 0.997706i \(0.478437\pi\)
\(44\) −2.84416 + 4.92623i −0.428774 + 0.742658i
\(45\) 0 0
\(46\) −0.890590 − 1.54255i −0.131310 − 0.227436i
\(47\) 1.16875 − 2.02434i 0.170480 − 0.295281i −0.768108 − 0.640321i \(-0.778801\pi\)
0.938588 + 0.345040i \(0.112135\pi\)
\(48\) 0 0
\(49\) 2.65613 − 6.47650i 0.379447 − 0.925214i
\(50\) 0.589510 0.0833692
\(51\) 0 0
\(52\) −2.71349 − 4.69991i −0.376294 − 0.651760i
\(53\) 2.44407 + 4.23325i 0.335719 + 0.581482i 0.983623 − 0.180240i \(-0.0576875\pi\)
−0.647904 + 0.761722i \(0.724354\pi\)
\(54\) 0 0
\(55\) −2.29249 −0.309119
\(56\) 20.5213 − 13.7639i 2.74228 − 1.83928i
\(57\) 0 0
\(58\) 4.22970 − 7.32606i 0.555387 − 0.961959i
\(59\) −0.524077 − 0.907729i −0.0682291 − 0.118176i 0.829893 − 0.557923i \(-0.188402\pi\)
−0.898122 + 0.439747i \(0.855068\pi\)
\(60\) 0 0
\(61\) 6.24989 − 10.8251i 0.800217 − 1.38602i −0.119256 − 0.992864i \(-0.538051\pi\)
0.919473 − 0.393153i \(-0.128616\pi\)
\(62\) −2.80132 −0.355768
\(63\) 0 0
\(64\) 28.3200 3.54000
\(65\) 1.09358 − 1.89414i 0.135642 − 0.234939i
\(66\) 0 0
\(67\) −2.23944 − 3.87883i −0.273592 − 0.473875i 0.696187 − 0.717860i \(-0.254878\pi\)
−0.969779 + 0.243986i \(0.921545\pi\)
\(68\) −14.3578 + 24.8684i −1.74114 + 3.01574i
\(69\) 0 0
\(70\) 14.1560 + 6.95021i 1.69197 + 0.830708i
\(71\) 6.60274 0.783601 0.391801 − 0.920050i \(-0.371852\pi\)
0.391801 + 0.920050i \(0.371852\pi\)
\(72\) 0 0
\(73\) 4.14174 + 7.17370i 0.484754 + 0.839618i 0.999847 − 0.0175164i \(-0.00557593\pi\)
−0.515093 + 0.857134i \(0.672243\pi\)
\(74\) −14.8434 − 25.7095i −1.72551 − 2.98867i
\(75\) 0 0
\(76\) 4.10772 0.471188
\(77\) 2.48931 + 1.22218i 0.283683 + 0.139280i
\(78\) 0 0
\(79\) −1.07007 + 1.85342i −0.120392 + 0.208526i −0.919922 − 0.392100i \(-0.871749\pi\)
0.799530 + 0.600626i \(0.205082\pi\)
\(80\) 15.9644 + 27.6511i 1.78487 + 3.09149i
\(81\) 0 0
\(82\) −9.97496 + 17.2771i −1.10155 + 1.90794i
\(83\) 6.66558 0.731642 0.365821 − 0.930685i \(-0.380788\pi\)
0.365821 + 0.930685i \(0.380788\pi\)
\(84\) 0 0
\(85\) −11.5728 −1.25525
\(86\) 1.20970 − 2.09526i 0.130445 − 0.225938i
\(87\) 0 0
\(88\) 4.89457 + 8.47765i 0.521763 + 0.903720i
\(89\) −2.88388 + 4.99503i −0.305691 + 0.529472i −0.977415 − 0.211329i \(-0.932221\pi\)
0.671724 + 0.740802i \(0.265554\pi\)
\(90\) 0 0
\(91\) −2.19729 + 1.47375i −0.230338 + 0.154491i
\(92\) −3.54699 −0.369800
\(93\) 0 0
\(94\) −3.18515 − 5.51684i −0.328523 − 0.569019i
\(95\) 0.827739 + 1.43369i 0.0849242 + 0.147093i
\(96\) 0 0
\(97\) −2.88777 −0.293209 −0.146604 − 0.989195i \(-0.546834\pi\)
−0.146604 + 0.989195i \(0.546834\pi\)
\(98\) −11.6661 − 15.0939i −1.17845 − 1.52471i
\(99\) 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 819.2.j.h.352.5 10
3.2 odd 2 91.2.e.c.79.1 yes 10
7.2 even 3 5733.2.a.bl.1.1 5
7.4 even 3 inner 819.2.j.h.235.5 10
7.5 odd 6 5733.2.a.bm.1.1 5
12.11 even 2 1456.2.r.p.625.2 10
21.2 odd 6 637.2.a.l.1.5 5
21.5 even 6 637.2.a.k.1.5 5
21.11 odd 6 91.2.e.c.53.1 ✓ 10
21.17 even 6 637.2.e.m.508.1 10
21.20 even 2 637.2.e.m.79.1 10
39.38 odd 2 1183.2.e.f.170.5 10
84.11 even 6 1456.2.r.p.417.2 10
273.116 odd 6 1183.2.e.f.508.5 10
273.194 even 6 8281.2.a.bx.1.1 5
273.233 odd 6 8281.2.a.bw.1.1 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
91.2.e.c.53.1 ✓ 10 21.11 odd 6
91.2.e.c.79.1 yes 10 3.2 odd 2
637.2.a.k.1.5 5 21.5 even 6
637.2.a.l.1.5 5 21.2 odd 6
637.2.e.m.79.1 10 21.20 even 2
637.2.e.m.508.1 10 21.17 even 6
819.2.j.h.235.5 10 7.4 even 3 inner
819.2.j.h.352.5 10 1.1 even 1 trivial
1183.2.e.f.170.5 10 39.38 odd 2
1183.2.e.f.508.5 10 273.116 odd 6
1456.2.r.p.417.2 10 84.11 even 6
1456.2.r.p.625.2 10 12.11 even 2
5733.2.a.bl.1.1 5 7.2 even 3
5733.2.a.bm.1.1 5 7.5 odd 6
8281.2.a.bw.1.1 5 273.233 odd 6
8281.2.a.bx.1.1 5 273.194 even 6