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Properties

Label 91.2.e.c.53.1
Level $91$
Weight $2$
Character 91.53
Analytic conductor $0.727$
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] = [91,2,Mod(53,91)] 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("91.53"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(91, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 91 = 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 91.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.726638658394\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 53.1
Root \(-0.862625 - 1.49411i\) of defining polynomial
Character \(\chi\) \(=\) 91.53
Dual form 91.2.e.c.79.1

$q$-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+(-1.36263 - 2.36014i) q^{2} +(0.673208 - 1.16603i) q^{3} +(-2.71349 + 4.69991i) q^{4} +(-1.09358 - 1.89414i) q^{5} -3.66932 q^{6} +(-2.19729 - 1.47375i) q^{7} +9.33940 q^{8} +(0.593582 + 1.02811i) q^{9} +(-2.98028 + 5.16200i) q^{10} +(0.524077 - 0.907729i) q^{11} +(3.65349 + 6.32803i) q^{12} +1.00000 q^{13} +(-0.484172 + 7.19406i) q^{14} -2.94483 q^{15} +(-7.29912 - 12.6424i) q^{16} +(2.64562 - 4.58236i) q^{17} +(1.61766 - 2.80187i) q^{18} +(-0.378453 - 0.655500i) q^{19} +11.8697 q^{20} +(-3.19767 + 1.56996i) q^{21} -2.85648 q^{22} +(-0.326792 - 0.566020i) q^{23} +(6.28736 - 10.8900i) q^{24} +(0.108157 - 0.187333i) q^{25} +(-1.36263 - 2.36014i) q^{26} +5.63766 q^{27} +(12.8888 - 6.32803i) q^{28} -3.10408 q^{29} +(4.01270 + 6.95021i) q^{30} +(-0.513956 + 0.890198i) q^{31} +(-10.5525 + 18.2775i) q^{32} +(-0.705626 - 1.22218i) q^{33} -14.4200 q^{34} +(-0.388575 + 5.77363i) q^{35} -6.44273 q^{36} +(5.44661 + 9.43381i) q^{37} +(-1.03138 + 1.78640i) q^{38} +(0.673208 - 1.16603i) q^{39} +(-10.2134 - 17.6901i) q^{40} +7.32040 q^{41} +(8.06254 + 5.40766i) q^{42} +0.887771 q^{43} +(2.84416 + 4.92623i) q^{44} +(1.29826 - 2.24865i) q^{45} +(-0.890590 + 1.54255i) q^{46} +(-1.16875 - 2.02434i) q^{47} -19.6553 q^{48} +(2.65613 + 6.47650i) q^{49} -0.589510 q^{50} +(-3.56211 - 6.16976i) q^{51} +(-2.71349 + 4.69991i) q^{52} +(-2.44407 + 4.23325i) q^{53} +(-7.68202 - 13.3057i) q^{54} -2.29249 q^{55} +(-20.5213 - 13.7639i) q^{56} -1.01911 q^{57} +(4.22970 + 7.32606i) q^{58} +(0.524077 - 0.907729i) q^{59} +(7.99079 - 13.8404i) q^{60} +(6.24989 + 10.8251i) q^{61} +2.80132 q^{62} +(0.210913 - 3.13385i) q^{63} +28.3200 q^{64} +(-1.09358 - 1.89414i) q^{65} +(-1.92301 + 3.33075i) q^{66} +(-2.23944 + 3.87883i) q^{67} +(14.3578 + 24.8684i) q^{68} -0.879996 q^{69} +(14.1560 - 6.95021i) q^{70} -6.60274 q^{71} +(5.54370 + 9.60197i) q^{72} +(4.14174 - 7.17370i) q^{73} +(14.8434 - 25.7095i) q^{74} +(-0.145624 - 0.252229i) q^{75} +4.10772 q^{76} +(-2.48931 + 1.22218i) q^{77} -3.66932 q^{78} +(-1.07007 - 1.85342i) q^{79} +(-15.9644 + 27.6511i) q^{80} +(2.01457 - 3.48935i) q^{81} +(-9.97496 - 17.2771i) q^{82} -6.66558 q^{83} +(1.29817 - 19.2888i) q^{84} -11.5728 q^{85} +(-1.20970 - 2.09526i) q^{86} +(-2.08969 + 3.61946i) q^{87} +(4.89457 - 8.47765i) q^{88} +(2.88388 + 4.99503i) q^{89} -7.07617 q^{90} +(-2.19729 - 1.47375i) q^{91} +3.54699 q^{92} +(0.691998 + 1.19858i) q^{93} +(-3.18515 + 5.51684i) q^{94} +(-0.827739 + 1.43369i) q^{95} +(14.2081 + 24.6091i) q^{96} -2.88777 q^{97} +(11.6661 - 15.0939i) q^{98} +1.24433 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 4 q^{2} - 8 q^{4} - 2 q^{5} - 10 q^{6} + q^{7} + 18 q^{8} - 3 q^{9} + 5 q^{10} - 11 q^{11} - 5 q^{12} + 10 q^{13} + 10 q^{14} - 10 q^{16} + 5 q^{17} - 9 q^{18} - 9 q^{19} + 2 q^{20} + 2 q^{21} + 16 q^{22}+ \cdots + 22 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/91\mathbb{Z}\right)^\times\).

\(n\) \(15\) \(66\)
\(\chi(n)\) \(1\) \(e\left(\frac{2}{3}\right)\)

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 −0.713536 − 0.700619i \(-0.752907\pi\)
−0.249986 − 0.968250i \(-0.580426\pi\)
\(3\) 0.673208 − 1.16603i 0.388677 − 0.673208i −0.603595 − 0.797291i \(-0.706266\pi\)
0.992272 + 0.124083i \(0.0395989\pi\)
\(4\) −2.71349 + 4.69991i −1.35675 + 2.34996i
\(5\) −1.09358 − 1.89414i −0.489065 − 0.847085i 0.510856 − 0.859666i \(-0.329328\pi\)
−0.999921 + 0.0125813i \(0.995995\pi\)
\(6\) −3.66932 −1.49799
\(7\) −2.19729 − 1.47375i −0.830496 − 0.557025i
\(8\) 9.33940 3.30198
\(9\) 0.593582 + 1.02811i 0.197861 + 0.342705i
\(10\) −2.98028 + 5.16200i −0.942449 + 1.63237i
\(11\) 0.524077 − 0.907729i 0.158015 − 0.273691i −0.776138 − 0.630564i \(-0.782824\pi\)
0.934153 + 0.356873i \(0.116157\pi\)
\(12\) 3.65349 + 6.32803i 1.05467 + 1.82675i
\(13\) 1.00000 0.277350
\(14\) −0.484172 + 7.19406i −0.129400 + 1.92269i
\(15\) −2.94483 −0.760352
\(16\) −7.29912 − 12.6424i −1.82478 − 3.16061i
\(17\) 2.64562 − 4.58236i 0.641658 − 1.11138i −0.343404 − 0.939188i \(-0.611580\pi\)
0.985063 − 0.172197i \(-0.0550865\pi\)
\(18\) 1.61766 − 2.80187i 0.381286 − 0.660407i
\(19\) −0.378453 − 0.655500i −0.0868231 − 0.150382i 0.819344 − 0.573303i \(-0.194338\pi\)
−0.906167 + 0.422921i \(0.861005\pi\)
\(20\) 11.8697 2.65415
\(21\) −3.19767 + 1.56996i −0.697788 + 0.342594i
\(22\) −2.85648 −0.609005
\(23\) −0.326792 − 0.566020i −0.0681408 − 0.118023i 0.829942 − 0.557850i \(-0.188373\pi\)
−0.898083 + 0.439826i \(0.855040\pi\)
\(24\) 6.28736 − 10.8900i 1.28340 − 2.22292i
\(25\) 0.108157 − 0.187333i 0.0216314 − 0.0374667i
\(26\) −1.36263 − 2.36014i −0.267233 − 0.462861i
\(27\) 5.63766 1.08497
\(28\) 12.8888 − 6.32803i 2.43576 − 1.19589i
\(29\) −3.10408 −0.576414 −0.288207 − 0.957568i \(-0.593059\pi\)
−0.288207 + 0.957568i \(0.593059\pi\)
\(30\) 4.01270 + 6.95021i 0.732616 + 1.26893i
\(31\) −0.513956 + 0.890198i −0.0923092 + 0.159884i −0.908482 − 0.417923i \(-0.862758\pi\)
0.816173 + 0.577807i \(0.196091\pi\)
\(32\) −10.5525 + 18.2775i −1.86544 + 3.23104i
\(33\) −0.705626 − 1.22218i −0.122834 − 0.212754i
\(34\) −14.4200 −2.47301
\(35\) −0.388575 + 5.77363i −0.0656811 + 0.975922i
\(36\) −6.44273 −1.07379
\(37\) 5.44661 + 9.43381i 0.895418 + 1.55091i 0.833287 + 0.552841i \(0.186456\pi\)
0.0621309 + 0.998068i \(0.480210\pi\)
\(38\) −1.03138 + 1.78640i −0.167312 + 0.289793i
\(39\) 0.673208 − 1.16603i 0.107800 − 0.186714i
\(40\) −10.2134 − 17.6901i −1.61488 − 2.79706i
\(41\) 7.32040 1.14325 0.571627 − 0.820514i \(-0.306312\pi\)
0.571627 + 0.820514i \(0.306312\pi\)
\(42\) 8.06254 + 5.40766i 1.24408 + 0.834420i
\(43\) 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
\(45\) 1.29826 − 2.24865i 0.193533 − 0.335210i
\(46\) −0.890590 + 1.54255i −0.131310 + 0.227436i
\(47\) −1.16875 − 2.02434i −0.170480 − 0.295281i 0.768108 − 0.640321i \(-0.221199\pi\)
−0.938588 + 0.345040i \(0.887865\pi\)
\(48\) −19.6553 −2.83700
\(49\) 2.65613 + 6.47650i 0.379447 + 0.925214i
\(50\) −0.589510 −0.0833692
\(51\) −3.56211 − 6.16976i −0.498795 − 0.863939i
\(52\) −2.71349 + 4.69991i −0.376294 + 0.651760i
\(53\) −2.44407 + 4.23325i −0.335719 + 0.581482i −0.983623 − 0.180240i \(-0.942313\pi\)
0.647904 + 0.761722i \(0.275646\pi\)
\(54\) −7.68202 − 13.3057i −1.04539 − 1.81067i
\(55\) −2.29249 −0.309119
\(56\) −20.5213 − 13.7639i −2.74228 − 1.83928i
\(57\) −1.01911 −0.134985
\(58\) 4.22970 + 7.32606i 0.555387 + 0.961959i
\(59\) 0.524077 − 0.907729i 0.0682291 − 0.118176i −0.829893 − 0.557923i \(-0.811598\pi\)
0.898122 + 0.439747i \(0.144932\pi\)
\(60\) 7.99079 − 13.8404i 1.03161 − 1.78679i
\(61\) 6.24989 + 10.8251i 0.800217 + 1.38602i 0.919473 + 0.393153i \(0.128616\pi\)
−0.119256 + 0.992864i \(0.538051\pi\)
\(62\) 2.80132 0.355768
\(63\) 0.210913 − 3.13385i 0.0265726 − 0.394828i
\(64\) 28.3200 3.54000
\(65\) −1.09358 − 1.89414i −0.135642 − 0.234939i
\(66\) −1.92301 + 3.33075i −0.236706 + 0.409987i
\(67\) −2.23944 + 3.87883i −0.273592 + 0.473875i −0.969779 − 0.243986i \(-0.921545\pi\)
0.696187 + 0.717860i \(0.254878\pi\)
\(68\) 14.3578 + 24.8684i 1.74114 + 3.01574i
\(69\) −0.879996 −0.105939
\(70\) 14.1560 − 6.95021i 1.69197 − 0.830708i
\(71\) −6.60274 −0.783601 −0.391801 − 0.920050i \(-0.628148\pi\)
−0.391801 + 0.920050i \(0.628148\pi\)
\(72\) 5.54370 + 9.60197i 0.653331 + 1.13160i
\(73\) 4.14174 − 7.17370i 0.484754 − 0.839618i −0.515093 − 0.857134i \(-0.672243\pi\)
0.999847 + 0.0175164i \(0.00557593\pi\)
\(74\) 14.8434 − 25.7095i 1.72551 − 2.98867i
\(75\) −0.145624 − 0.252229i −0.0168152 − 0.0291249i
\(76\) 4.10772 0.471188
\(77\) −2.48931 + 1.22218i −0.283683 + 0.139280i
\(78\) −3.66932 −0.415469
\(79\) −1.07007 − 1.85342i −0.120392 − 0.208526i 0.799530 − 0.600626i \(-0.205082\pi\)
−0.919922 + 0.392100i \(0.871749\pi\)
\(80\) −15.9644 + 27.6511i −1.78487 + 3.09149i
\(81\) 2.01457 − 3.48935i 0.223842 − 0.387705i
\(82\) −9.97496 − 17.2771i −1.10155 − 1.90794i
\(83\) −6.66558 −0.731642 −0.365821 − 0.930685i \(-0.619212\pi\)
−0.365821 + 0.930685i \(0.619212\pi\)
\(84\) 1.29817 − 19.2888i 0.141642 − 2.10458i
\(85\) −11.5728 −1.25525
\(86\) −1.20970 − 2.09526i −0.130445 − 0.225938i
\(87\) −2.08969 + 3.61946i −0.224039 + 0.388047i
\(88\) 4.89457 − 8.47765i 0.521763 − 0.903720i
\(89\) 2.88388 + 4.99503i 0.305691 + 0.529472i 0.977415 − 0.211329i \(-0.0677792\pi\)
−0.671724 + 0.740802i \(0.734446\pi\)
\(90\) −7.07617 −0.745894
\(91\) −2.19729 − 1.47375i −0.230338 − 0.154491i
\(92\) 3.54699 0.369800
\(93\) 0.691998 + 1.19858i 0.0717569 + 0.124287i
\(94\) −3.18515 + 5.51684i −0.328523 + 0.569019i
\(95\) −0.827739 + 1.43369i −0.0849242 + 0.147093i
\(96\) 14.2081 + 24.6091i 1.45011 + 2.51166i
\(97\) −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
\(99\) 1.24433 0.125060
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 91.2.e.c.53.1 ✓ 10
3.2 odd 2 819.2.j.h.235.5 10
4.3 odd 2 1456.2.r.p.417.2 10
7.2 even 3 inner 91.2.e.c.79.1 yes 10
7.3 odd 6 637.2.a.k.1.5 5
7.4 even 3 637.2.a.l.1.5 5
7.5 odd 6 637.2.e.m.79.1 10
7.6 odd 2 637.2.e.m.508.1 10
13.12 even 2 1183.2.e.f.508.5 10
21.2 odd 6 819.2.j.h.352.5 10
21.11 odd 6 5733.2.a.bl.1.1 5
21.17 even 6 5733.2.a.bm.1.1 5
28.23 odd 6 1456.2.r.p.625.2 10
91.25 even 6 8281.2.a.bw.1.1 5
91.38 odd 6 8281.2.a.bx.1.1 5
91.51 even 6 1183.2.e.f.170.5 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
91.2.e.c.53.1 ✓ 10 1.1 even 1 trivial
91.2.e.c.79.1 yes 10 7.2 even 3 inner
637.2.a.k.1.5 5 7.3 odd 6
637.2.a.l.1.5 5 7.4 even 3
637.2.e.m.79.1 10 7.5 odd 6
637.2.e.m.508.1 10 7.6 odd 2
819.2.j.h.235.5 10 3.2 odd 2
819.2.j.h.352.5 10 21.2 odd 6
1183.2.e.f.170.5 10 91.51 even 6
1183.2.e.f.508.5 10 13.12 even 2
1456.2.r.p.417.2 10 4.3 odd 2
1456.2.r.p.625.2 10 28.23 odd 6
5733.2.a.bl.1.1 5 21.11 odd 6
5733.2.a.bm.1.1 5 21.17 even 6
8281.2.a.bw.1.1 5 91.25 even 6
8281.2.a.bx.1.1 5 91.38 odd 6