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Properties

Label 637.2.e.m.79.1
Level $637$
Weight $2$
Character 637.79
Analytic conductor $5.086$
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] = [637,2,Mod(79,637)] 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("637.79"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(637, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([2, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 637 = 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 637.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [10,-4,0,-8,2] 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: \(5.08647060876\)
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: no (minimal twist has level 91)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 79.1
Root \(-0.862625 + 1.49411i\) of defining polynomial
Character \(\chi\) \(=\) 637.79
Dual form 637.2.e.m.508.1

$q$-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} +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} -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} -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} -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} -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} +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} -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} -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} +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} -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} -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} -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} +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} +1.24433 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 4 q^{2} - 8 q^{4} + 2 q^{5} + 10 q^{6} + 18 q^{8} - 3 q^{9} - 5 q^{10} - 11 q^{11} + 5 q^{12} - 10 q^{13} - 10 q^{16} - 5 q^{17} - 9 q^{18} + 9 q^{19} - 2 q^{20} + 16 q^{22} - 10 q^{23} - 9 q^{25}+ \cdots + 22 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/637\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(248\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{3}\right)\)

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 \(0.580426\pi\)
−0.713536 + 0.700619i \(0.752907\pi\)
\(3\) −0.673208 − 1.16603i −0.388677 − 0.673208i 0.603595 − 0.797291i \(-0.293734\pi\)
−0.992272 + 0.124083i \(0.960401\pi\)
\(4\) −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 \(0.00400485\pi\)
\(6\) 3.66932 1.49799
\(7\) 0 0
\(8\) 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 0.934153 − 0.356873i \(-0.116157\pi\)
−0.776138 + 0.630564i \(0.782824\pi\)
\(12\) −3.65349 + 6.32803i −1.05467 + 1.82675i
\(13\) −1.00000 −0.277350
\(14\) 0 0
\(15\) −2.94483 −0.760352
\(16\) −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 \(-0.388420\pi\)
\(18\) 1.61766 + 2.80187i 0.381286 + 0.660407i
\(19\) 0.378453 − 0.655500i 0.0868231 − 0.150382i −0.819344 − 0.573303i \(-0.805662\pi\)
0.906167 + 0.422921i \(0.138995\pi\)
\(20\) −11.8697 −2.65415
\(21\) 0 0
\(22\) −2.85648 −0.609005
\(23\) −0.326792 + 0.566020i −0.0681408 + 0.118023i −0.898083 − 0.439826i \(-0.855040\pi\)
0.829942 + 0.557850i \(0.188373\pi\)
\(24\) −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
\(28\) 0 0
\(29\) −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 \(-0.137242\pi\)
−0.816173 + 0.577807i \(0.803909\pi\)
\(32\) −10.5525 − 18.2775i −1.86544 − 3.23104i
\(33\) 0.705626 − 1.22218i 0.122834 − 0.212754i
\(34\) 14.4200 2.47301
\(35\) 0 0
\(36\) −6.44273 −1.07379
\(37\) 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
\(39\) 0.673208 + 1.16603i 0.107800 + 0.186714i
\(40\) 10.2134 − 17.6901i 1.61488 − 2.79706i
\(41\) −7.32040 −1.14325 −0.571627 − 0.820514i \(-0.693688\pi\)
−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
\(45\) −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 \(-0.778801\pi\)
0.938588 + 0.345040i \(0.112135\pi\)
\(48\) 19.6553 2.83700
\(49\) 0 0
\(50\) −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 0.647904 − 0.761722i \(-0.275646\pi\)
−0.983623 + 0.180240i \(0.942313\pi\)
\(54\) 7.68202 − 13.3057i 1.04539 − 1.81067i
\(55\) 2.29249 0.309119
\(56\) 0 0
\(57\) −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 \(-0.188402\pi\)
−0.898122 + 0.439747i \(0.855068\pi\)
\(60\) 7.99079 + 13.8404i 1.03161 + 1.78679i
\(61\) −6.24989 + 10.8251i −0.800217 + 1.38602i 0.119256 + 0.992864i \(0.461949\pi\)
−0.919473 + 0.393153i \(0.871384\pi\)
\(62\) −2.80132 −0.355768
\(63\) 0 0
\(64\) 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 0.696187 − 0.717860i \(-0.254878\pi\)
−0.969779 + 0.243986i \(0.921545\pi\)
\(68\) −14.3578 + 24.8684i −1.74114 + 3.01574i
\(69\) 0.879996 0.105939
\(70\) 0 0
\(71\) −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 \(-0.327757\pi\)
−0.999847 + 0.0175164i \(0.994424\pi\)
\(74\) 14.8434 + 25.7095i 1.72551 + 2.98867i
\(75\) 0.145624 − 0.252229i 0.0168152 − 0.0291249i
\(76\) −4.10772 −0.471188
\(77\) 0 0
\(78\) −3.66932 −0.415469
\(79\) −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
\(81\) 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 \(-0.380788\pi\)
0.365821 + 0.930685i \(0.380788\pi\)
\(84\) 0 0
\(85\) −11.5728 −1.25525
\(86\) −1.20970 + 2.09526i −0.130445 + 0.225938i
\(87\) 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 \(-0.932221\pi\)
0.671724 + 0.740802i \(0.265554\pi\)
\(90\) 7.07617 0.745894
\(91\) 0 0
\(92\) 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 \(-0.453166\pi\)
0.146604 + 0.989195i \(0.453166\pi\)
\(98\) 0 0
\(99\) 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 637.2.e.m.79.1 10
7.2 even 3 637.2.a.k.1.5 5
7.3 odd 6 91.2.e.c.53.1 ✓ 10
7.4 even 3 inner 637.2.e.m.508.1 10
7.5 odd 6 637.2.a.l.1.5 5
7.6 odd 2 91.2.e.c.79.1 yes 10
21.2 odd 6 5733.2.a.bm.1.1 5
21.5 even 6 5733.2.a.bl.1.1 5
21.17 even 6 819.2.j.h.235.5 10
21.20 even 2 819.2.j.h.352.5 10
28.3 even 6 1456.2.r.p.417.2 10
28.27 even 2 1456.2.r.p.625.2 10
91.12 odd 6 8281.2.a.bw.1.1 5
91.38 odd 6 1183.2.e.f.508.5 10
91.51 even 6 8281.2.a.bx.1.1 5
91.90 odd 2 1183.2.e.f.170.5 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
91.2.e.c.53.1 ✓ 10 7.3 odd 6
91.2.e.c.79.1 yes 10 7.6 odd 2
637.2.a.k.1.5 5 7.2 even 3
637.2.a.l.1.5 5 7.5 odd 6
637.2.e.m.79.1 10 1.1 even 1 trivial
637.2.e.m.508.1 10 7.4 even 3 inner
819.2.j.h.235.5 10 21.17 even 6
819.2.j.h.352.5 10 21.20 even 2
1183.2.e.f.170.5 10 91.90 odd 2
1183.2.e.f.508.5 10 91.38 odd 6
1456.2.r.p.417.2 10 28.3 even 6
1456.2.r.p.625.2 10 28.27 even 2
5733.2.a.bl.1.1 5 21.5 even 6
5733.2.a.bm.1.1 5 21.2 odd 6
8281.2.a.bw.1.1 5 91.12 odd 6
8281.2.a.bx.1.1 5 91.51 even 6