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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [650,2,Mod(101,650)] 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("650.101"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(650, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 5])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 650 = 2 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 650.m (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,-2,4,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.19027613138\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.22581504.2
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} + 5x^{6} + 2x^{5} - 11x^{4} + 4x^{3} + 20x^{2} - 32x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 101.1
Root \(0.665665 + 1.24775i\) of defining polynomial
Character \(\chi\) \(=\) 650.101
Dual form 650.2.m.b.251.1

$q$-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+(-0.866025 + 0.500000i) q^{2} +(-1.24775 - 2.16117i) q^{3} +(0.500000 - 0.866025i) q^{4} +(2.16117 + 1.24775i) q^{6} +(-0.286941 - 0.165665i) q^{7} +1.00000i q^{8} +(-1.61378 + 2.79515i) q^{9} +(3.60108 - 2.07908i) q^{11} -2.49551 q^{12} +(-0.448114 - 3.57760i) q^{13} +0.331331 q^{14} +(-0.500000 - 0.866025i) q^{16} +(1.21306 - 2.10108i) q^{17} -3.22756i q^{18} +(1.71006 + 0.987301i) q^{19} +0.826838i q^{21} +(-2.07908 + 3.60108i) q^{22} +(-0.263457 - 0.456321i) q^{23} +(2.16117 - 1.24775i) q^{24} +(2.17688 + 2.87423i) q^{26} +0.567874 q^{27} +(-0.286941 + 0.165665i) q^{28} +(-4.79066 - 8.29766i) q^{29} +4.45512i q^{31} +(0.866025 + 0.500000i) q^{32} +(-8.98652 - 5.18837i) q^{33} +2.42612i q^{34} +(1.61378 + 2.79515i) q^{36} +(3.74326 - 2.16117i) q^{37} -1.97460 q^{38} +(-7.17267 + 5.43241i) q^{39} +(-9.98052 + 5.76225i) q^{41} +(-0.413419 - 0.716063i) q^{42} +(-4.02720 + 6.97531i) q^{43} -4.15817i q^{44} +(0.456321 + 0.263457i) q^{46} +2.12619i q^{47} +(-1.24775 + 2.16117i) q^{48} +(-3.44511 - 5.96711i) q^{49} -6.05440 q^{51} +(-3.32235 - 1.40072i) q^{52} -10.0140 q^{53} +(-0.491793 + 0.283937i) q^{54} +(0.165665 - 0.286941i) q^{56} -4.92763i q^{57} +(8.29766 + 4.79066i) q^{58} +(-2.33833 - 1.35004i) q^{59} +(4.77046 - 8.26268i) q^{61} +(-2.22756 - 3.85824i) q^{62} +(0.926118 - 0.534695i) q^{63} -1.00000 q^{64} +10.3767 q^{66} +(8.77647 - 5.06710i) q^{67} +(-1.21306 - 2.10108i) q^{68} +(-0.657459 + 1.13875i) q^{69} +(-12.9670 - 7.48652i) q^{71} +(-2.79515 - 1.61378i) q^{72} +0.328354i q^{73} +(-2.16117 + 3.74326i) q^{74} +(1.71006 - 0.987301i) q^{76} -1.37773 q^{77} +(3.49551 - 8.29094i) q^{78} +0.321494 q^{79} +(4.13277 + 7.15817i) q^{81} +(5.76225 - 9.98052i) q^{82} +3.29992i q^{83} +(0.716063 + 0.413419i) q^{84} -8.05440i q^{86} +(-11.9551 + 20.7069i) q^{87} +(2.07908 + 3.60108i) q^{88} +(16.0240 - 9.25147i) q^{89} +(-0.464102 + 1.10080i) q^{91} -0.526914 q^{92} +(9.62828 - 5.55889i) q^{93} +(-1.06310 - 1.84134i) q^{94} -2.49551i q^{96} +(-6.12485 - 3.53618i) q^{97} +(5.96711 + 3.44511i) q^{98} +13.4207i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{3} + 4 q^{4} - 6 q^{7} + 2 q^{9} + 12 q^{11} - 4 q^{12} + 10 q^{13} - 4 q^{16} + 6 q^{17} + 6 q^{19} - 6 q^{22} + 6 q^{26} + 4 q^{27} - 6 q^{28} - 12 q^{29} - 24 q^{33} - 2 q^{36} + 6 q^{37}+ \cdots + 24 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/650\mathbb{Z}\right)^\times\).

\(n\) \(27\) \(301\)
\(\chi(n)\) \(1\) \(e\left(\frac{5}{6}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.866025 + 0.500000i −0.612372 + 0.353553i
\(3\) −1.24775 − 2.16117i −0.720391 − 1.24775i −0.960843 − 0.277093i \(-0.910629\pi\)
0.240452 − 0.970661i \(-0.422704\pi\)
\(4\) 0.500000 − 0.866025i 0.250000 − 0.433013i
\(5\) 0 0
\(6\) 2.16117 + 1.24775i 0.882295 + 0.509393i
\(7\) −0.286941 − 0.165665i −0.108453 − 0.0626156i 0.444792 − 0.895634i \(-0.353278\pi\)
−0.553246 + 0.833018i \(0.686611\pi\)
\(8\) 1.00000i 0.353553i
\(9\) −1.61378 + 2.79515i −0.537926 + 0.931716i
\(10\) 0 0
\(11\) 3.60108 − 2.07908i 1.08577 − 0.626868i 0.153320 − 0.988177i \(-0.451003\pi\)
0.932446 + 0.361309i \(0.117670\pi\)
\(12\) −2.49551 −0.720391
\(13\) −0.448114 − 3.57760i −0.124284 − 0.992247i
\(14\) 0.331331 0.0885519
\(15\) 0 0
\(16\) −0.500000 − 0.866025i −0.125000 − 0.216506i
\(17\) 1.21306 − 2.10108i 0.294210 − 0.509587i −0.680591 − 0.732664i \(-0.738277\pi\)
0.974801 + 0.223077i \(0.0716101\pi\)
\(18\) − 3.22756i − 0.760743i
\(19\) 1.71006 + 0.987301i 0.392314 + 0.226502i 0.683162 − 0.730267i \(-0.260604\pi\)
−0.290849 + 0.956769i \(0.593938\pi\)
\(20\) 0 0
\(21\) 0.826838i 0.180431i
\(22\) −2.07908 + 3.60108i −0.443262 + 0.767753i
\(23\) −0.263457 − 0.456321i −0.0549346 − 0.0951494i 0.837250 − 0.546820i \(-0.184162\pi\)
−0.892185 + 0.451670i \(0.850828\pi\)
\(24\) 2.16117 − 1.24775i 0.441148 − 0.254697i
\(25\) 0 0
\(26\) 2.17688 + 2.87423i 0.426921 + 0.563683i
\(27\) 0.567874 0.109287
\(28\) −0.286941 + 0.165665i −0.0542267 + 0.0313078i
\(29\) −4.79066 − 8.29766i −0.889602 − 1.54084i −0.840346 − 0.542050i \(-0.817648\pi\)
−0.0492563 − 0.998786i \(-0.515685\pi\)
\(30\) 0 0
\(31\) 4.45512i 0.800163i 0.916480 + 0.400081i \(0.131018\pi\)
−0.916480 + 0.400081i \(0.868982\pi\)
\(32\) 0.866025 + 0.500000i 0.153093 + 0.0883883i
\(33\) −8.98652 − 5.18837i −1.56435 − 0.903180i
\(34\) 2.42612i 0.416076i
\(35\) 0 0
\(36\) 1.61378 + 2.79515i 0.268963 + 0.465858i
\(37\) 3.74326 − 2.16117i 0.615388 − 0.355295i −0.159683 − 0.987168i \(-0.551047\pi\)
0.775071 + 0.631874i \(0.217714\pi\)
\(38\) −1.97460 −0.320323
\(39\) −7.17267 + 5.43241i −1.14855 + 0.869882i
\(40\) 0 0
\(41\) −9.98052 + 5.76225i −1.55869 + 0.899913i −0.561312 + 0.827604i \(0.689703\pi\)
−0.997382 + 0.0723087i \(0.976963\pi\)
\(42\) −0.413419 − 0.716063i −0.0637920 − 0.110491i
\(43\) −4.02720 + 6.97531i −0.614142 + 1.06373i 0.376392 + 0.926460i \(0.377164\pi\)
−0.990534 + 0.137265i \(0.956169\pi\)
\(44\) − 4.15817i − 0.626868i
\(45\) 0 0
\(46\) 0.456321 + 0.263457i 0.0672808 + 0.0388446i
\(47\) 2.12619i 0.310137i 0.987904 + 0.155069i \(0.0495599\pi\)
−0.987904 + 0.155069i \(0.950440\pi\)
\(48\) −1.24775 + 2.16117i −0.180098 + 0.311938i
\(49\) −3.44511 − 5.96711i −0.492159 − 0.852444i
\(50\) 0 0
\(51\) −6.05440 −0.847785
\(52\) −3.32235 − 1.40072i −0.460727 − 0.194245i
\(53\) −10.0140 −1.37553 −0.687765 − 0.725934i \(-0.741408\pi\)
−0.687765 + 0.725934i \(0.741408\pi\)
\(54\) −0.491793 + 0.283937i −0.0669246 + 0.0386389i
\(55\) 0 0
\(56\) 0.165665 − 0.286941i 0.0221380 − 0.0383441i
\(57\) − 4.92763i − 0.652681i
\(58\) 8.29766 + 4.79066i 1.08954 + 0.629044i
\(59\) −2.33833 − 1.35004i −0.304425 − 0.175760i 0.340004 − 0.940424i \(-0.389572\pi\)
−0.644429 + 0.764664i \(0.722905\pi\)
\(60\) 0 0
\(61\) 4.77046 − 8.26268i 0.610795 − 1.05793i −0.380312 − 0.924858i \(-0.624183\pi\)
0.991107 − 0.133069i \(-0.0424833\pi\)
\(62\) −2.22756 − 3.85824i −0.282900 − 0.489998i
\(63\) 0.926118 − 0.534695i 0.116680 − 0.0673652i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 10.3767 1.27729
\(67\) 8.77647 − 5.06710i 1.07222 − 0.619044i 0.143430 − 0.989660i \(-0.454187\pi\)
0.928786 + 0.370616i \(0.120853\pi\)
\(68\) −1.21306 − 2.10108i −0.147105 − 0.254793i
\(69\) −0.657459 + 1.13875i −0.0791487 + 0.137090i
\(70\) 0 0
\(71\) −12.9670 − 7.48652i −1.53890 − 0.888487i −0.998903 − 0.0468225i \(-0.985090\pi\)
−0.540001 − 0.841664i \(-0.681576\pi\)
\(72\) −2.79515 − 1.61378i −0.329411 − 0.190186i
\(73\) 0.328354i 0.0384309i 0.999815 + 0.0192155i \(0.00611685\pi\)
−0.999815 + 0.0192155i \(0.993883\pi\)
\(74\) −2.16117 + 3.74326i −0.251231 + 0.435145i
\(75\) 0 0
\(76\) 1.71006 − 0.987301i 0.196157 − 0.113251i
\(77\) −1.37773 −0.157007
\(78\) 3.49551 − 8.29094i 0.395788 − 0.938764i
\(79\) 0.321494 0.0361709 0.0180855 − 0.999836i \(-0.494243\pi\)
0.0180855 + 0.999836i \(0.494243\pi\)
\(80\) 0 0
\(81\) 4.13277 + 7.15817i 0.459197 + 0.795352i
\(82\) 5.76225 − 9.98052i 0.636334 − 1.10216i
\(83\) 3.29992i 0.362214i 0.983463 + 0.181107i \(0.0579680\pi\)
−0.983463 + 0.181107i \(0.942032\pi\)
\(84\) 0.716063 + 0.413419i 0.0781289 + 0.0451077i
\(85\) 0 0
\(86\) − 8.05440i − 0.868528i
\(87\) −11.9551 + 20.7069i −1.28172 + 2.22001i
\(88\) 2.07908 + 3.60108i 0.221631 + 0.383876i
\(89\) 16.0240 − 9.25147i 1.69854 − 0.980654i 0.751396 − 0.659852i \(-0.229381\pi\)
0.947146 − 0.320802i \(-0.103952\pi\)
\(90\) 0 0
\(91\) −0.464102 + 1.10080i −0.0486511 + 0.115395i
\(92\) −0.526914 −0.0549346
\(93\) 9.62828 − 5.55889i 0.998406 − 0.576430i
\(94\) −1.06310 − 1.84134i −0.109650 − 0.189919i
\(95\) 0 0
\(96\) − 2.49551i − 0.254697i
\(97\) −6.12485 − 3.53618i −0.621884 − 0.359045i 0.155718 − 0.987802i \(-0.450231\pi\)
−0.777602 + 0.628757i \(0.783564\pi\)
\(98\) 5.96711 + 3.44511i 0.602769 + 0.348009i
\(99\) 13.4207i 1.34883i
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 650.2.m.b.101.1 ✓ 8
5.2 odd 4 650.2.n.c.49.1 8
5.3 odd 4 650.2.n.f.49.4 8
5.4 even 2 650.2.m.d.101.4 yes 8
13.2 odd 12 8450.2.a.ck.1.4 4
13.4 even 6 inner 650.2.m.b.251.1 yes 8
13.11 odd 12 8450.2.a.co.1.4 4
65.4 even 6 650.2.m.d.251.4 yes 8
65.17 odd 12 650.2.n.f.199.4 8
65.24 odd 12 8450.2.a.ch.1.1 4
65.43 odd 12 650.2.n.c.199.1 8
65.54 odd 12 8450.2.a.cl.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
650.2.m.b.101.1 ✓ 8 1.1 even 1 trivial
650.2.m.b.251.1 yes 8 13.4 even 6 inner
650.2.m.d.101.4 yes 8 5.4 even 2
650.2.m.d.251.4 yes 8 65.4 even 6
650.2.n.c.49.1 8 5.2 odd 4
650.2.n.c.199.1 8 65.43 odd 12
650.2.n.f.49.4 8 5.3 odd 4
650.2.n.f.199.4 8 65.17 odd 12
8450.2.a.ch.1.1 4 65.24 odd 12
8450.2.a.ck.1.4 4 13.2 odd 12
8450.2.a.cl.1.1 4 65.54 odd 12
8450.2.a.co.1.4 4 13.11 odd 12