Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [171,2,Mod(106,171)] 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("171.106"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(171, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 171 = 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 171.g (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,1,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.36544187456\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 106.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 171.106
Dual form 171.2.g.a.121.1

$q$-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+(0.500000 + 0.866025i) q^{2} +(-1.50000 + 0.866025i) q^{3} +(0.500000 - 0.866025i) q^{4} +3.00000 q^{5} +(-1.50000 - 0.866025i) q^{6} +(-0.500000 + 0.866025i) q^{7} +3.00000 q^{8} +(1.50000 - 2.59808i) q^{9} +(1.50000 + 2.59808i) q^{10} +(-2.50000 + 4.33013i) q^{11} +1.73205i q^{12} +(-1.00000 + 1.73205i) q^{13} -1.00000 q^{14} +(-4.50000 + 2.59808i) q^{15} +(0.500000 + 0.866025i) q^{16} +(2.50000 - 4.33013i) q^{17} +3.00000 q^{18} +(-4.00000 + 1.73205i) q^{19} +(1.50000 - 2.59808i) q^{20} -1.73205i q^{21} -5.00000 q^{22} +(4.00000 - 6.92820i) q^{23} +(-4.50000 + 2.59808i) q^{24} +4.00000 q^{25} -2.00000 q^{26} +5.19615i q^{27} +(0.500000 + 0.866025i) q^{28} -1.00000 q^{29} +(-4.50000 - 2.59808i) q^{30} +(-1.50000 - 2.59808i) q^{31} +(2.50000 - 4.33013i) q^{32} -8.66025i q^{33} +5.00000 q^{34} +(-1.50000 + 2.59808i) q^{35} +(-1.50000 - 2.59808i) q^{36} -6.00000 q^{37} +(-3.50000 - 2.59808i) q^{38} -3.46410i q^{39} +9.00000 q^{40} -9.00000 q^{41} +(1.50000 - 0.866025i) q^{42} +(-4.00000 - 6.92820i) q^{43} +(2.50000 + 4.33013i) q^{44} +(4.50000 - 7.79423i) q^{45} +8.00000 q^{46} +3.00000 q^{47} +(-1.50000 - 0.866025i) q^{48} +(3.00000 + 5.19615i) q^{49} +(2.00000 + 3.46410i) q^{50} +8.66025i q^{51} +(1.00000 + 1.73205i) q^{52} +(0.500000 + 0.866025i) q^{53} +(-4.50000 + 2.59808i) q^{54} +(-7.50000 + 12.9904i) q^{55} +(-1.50000 + 2.59808i) q^{56} +(4.50000 - 6.06218i) q^{57} +(-0.500000 - 0.866025i) q^{58} +5.00000 q^{59} +5.19615i q^{60} -13.0000 q^{61} +(1.50000 - 2.59808i) q^{62} +(1.50000 + 2.59808i) q^{63} +7.00000 q^{64} +(-3.00000 + 5.19615i) q^{65} +(7.50000 - 4.33013i) q^{66} +(2.00000 - 3.46410i) q^{67} +(-2.50000 - 4.33013i) q^{68} +13.8564i q^{69} -3.00000 q^{70} +(-1.50000 + 2.59808i) q^{71} +(4.50000 - 7.79423i) q^{72} +(2.50000 - 4.33013i) q^{73} +(-3.00000 - 5.19615i) q^{74} +(-6.00000 + 3.46410i) q^{75} +(-0.500000 + 4.33013i) q^{76} +(-2.50000 - 4.33013i) q^{77} +(3.00000 - 1.73205i) q^{78} +(-2.00000 - 3.46410i) q^{79} +(1.50000 + 2.59808i) q^{80} +(-4.50000 - 7.79423i) q^{81} +(-4.50000 - 7.79423i) q^{82} +(-4.50000 + 7.79423i) q^{83} +(-1.50000 - 0.866025i) q^{84} +(7.50000 - 12.9904i) q^{85} +(4.00000 - 6.92820i) q^{86} +(1.50000 - 0.866025i) q^{87} +(-7.50000 + 12.9904i) q^{88} +(4.50000 + 7.79423i) q^{89} +9.00000 q^{90} +(-1.00000 - 1.73205i) q^{91} +(-4.00000 - 6.92820i) q^{92} +(4.50000 + 2.59808i) q^{93} +(1.50000 + 2.59808i) q^{94} +(-12.0000 + 5.19615i) q^{95} +8.66025i q^{96} +(5.00000 + 8.66025i) q^{97} +(-3.00000 + 5.19615i) q^{98} +(7.50000 + 12.9904i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} - 3 q^{3} + q^{4} + 6 q^{5} - 3 q^{6} - q^{7} + 6 q^{8} + 3 q^{9} + 3 q^{10} - 5 q^{11} - 2 q^{13} - 2 q^{14} - 9 q^{15} + q^{16} + 5 q^{17} + 6 q^{18} - 8 q^{19} + 3 q^{20} - 10 q^{22}+ \cdots + 15 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/171\mathbb{Z}\right)^\times\).

\(n\) \(20\) \(154\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{2}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.500000 + 0.866025i 0.353553 + 0.612372i 0.986869 − 0.161521i \(-0.0516399\pi\)
−0.633316 + 0.773893i \(0.718307\pi\)
\(3\) −1.50000 + 0.866025i −0.866025 + 0.500000i
\(4\) 0.500000 − 0.866025i 0.250000 − 0.433013i
\(5\) 3.00000 1.34164 0.670820 − 0.741620i \(-0.265942\pi\)
0.670820 + 0.741620i \(0.265942\pi\)
\(6\) −1.50000 − 0.866025i −0.612372 − 0.353553i
\(7\) −0.500000 + 0.866025i −0.188982 + 0.327327i −0.944911 − 0.327327i \(-0.893852\pi\)
0.755929 + 0.654654i \(0.227186\pi\)
\(8\) 3.00000 1.06066
\(9\) 1.50000 − 2.59808i 0.500000 − 0.866025i
\(10\) 1.50000 + 2.59808i 0.474342 + 0.821584i
\(11\) −2.50000 + 4.33013i −0.753778 + 1.30558i 0.192201 + 0.981356i \(0.438437\pi\)
−0.945979 + 0.324227i \(0.894896\pi\)
\(12\) 1.73205i 0.500000i
\(13\) −1.00000 + 1.73205i −0.277350 + 0.480384i −0.970725 − 0.240192i \(-0.922790\pi\)
0.693375 + 0.720577i \(0.256123\pi\)
\(14\) −1.00000 −0.267261
\(15\) −4.50000 + 2.59808i −1.16190 + 0.670820i
\(16\) 0.500000 + 0.866025i 0.125000 + 0.216506i
\(17\) 2.50000 − 4.33013i 0.606339 − 1.05021i −0.385499 − 0.922708i \(-0.625971\pi\)
0.991838 − 0.127502i \(-0.0406959\pi\)
\(18\) 3.00000 0.707107
\(19\) −4.00000 + 1.73205i −0.917663 + 0.397360i
\(20\) 1.50000 − 2.59808i 0.335410 − 0.580948i
\(21\) − 1.73205i − 0.377964i
\(22\) −5.00000 −1.06600
\(23\) 4.00000 − 6.92820i 0.834058 − 1.44463i −0.0607377 − 0.998154i \(-0.519345\pi\)
0.894795 − 0.446476i \(-0.147321\pi\)
\(24\) −4.50000 + 2.59808i −0.918559 + 0.530330i
\(25\) 4.00000 0.800000
\(26\) −2.00000 −0.392232
\(27\) 5.19615i 1.00000i
\(28\) 0.500000 + 0.866025i 0.0944911 + 0.163663i
\(29\) −1.00000 −0.185695 −0.0928477 − 0.995680i \(-0.529597\pi\)
−0.0928477 + 0.995680i \(0.529597\pi\)
\(30\) −4.50000 − 2.59808i −0.821584 − 0.474342i
\(31\) −1.50000 − 2.59808i −0.269408 − 0.466628i 0.699301 − 0.714827i \(-0.253495\pi\)
−0.968709 + 0.248199i \(0.920161\pi\)
\(32\) 2.50000 − 4.33013i 0.441942 − 0.765466i
\(33\) − 8.66025i − 1.50756i
\(34\) 5.00000 0.857493
\(35\) −1.50000 + 2.59808i −0.253546 + 0.439155i
\(36\) −1.50000 − 2.59808i −0.250000 − 0.433013i
\(37\) −6.00000 −0.986394 −0.493197 − 0.869918i \(-0.664172\pi\)
−0.493197 + 0.869918i \(0.664172\pi\)
\(38\) −3.50000 − 2.59808i −0.567775 − 0.421464i
\(39\) − 3.46410i − 0.554700i
\(40\) 9.00000 1.42302
\(41\) −9.00000 −1.40556 −0.702782 − 0.711405i \(-0.748059\pi\)
−0.702782 + 0.711405i \(0.748059\pi\)
\(42\) 1.50000 − 0.866025i 0.231455 − 0.133631i
\(43\) −4.00000 − 6.92820i −0.609994 − 1.05654i −0.991241 − 0.132068i \(-0.957838\pi\)
0.381246 − 0.924473i \(-0.375495\pi\)
\(44\) 2.50000 + 4.33013i 0.376889 + 0.652791i
\(45\) 4.50000 − 7.79423i 0.670820 − 1.16190i
\(46\) 8.00000 1.17954
\(47\) 3.00000 0.437595 0.218797 − 0.975770i \(-0.429787\pi\)
0.218797 + 0.975770i \(0.429787\pi\)
\(48\) −1.50000 − 0.866025i −0.216506 − 0.125000i
\(49\) 3.00000 + 5.19615i 0.428571 + 0.742307i
\(50\) 2.00000 + 3.46410i 0.282843 + 0.489898i
\(51\) 8.66025i 1.21268i
\(52\) 1.00000 + 1.73205i 0.138675 + 0.240192i
\(53\) 0.500000 + 0.866025i 0.0686803 + 0.118958i 0.898321 − 0.439340i \(-0.144788\pi\)
−0.829640 + 0.558298i \(0.811454\pi\)
\(54\) −4.50000 + 2.59808i −0.612372 + 0.353553i
\(55\) −7.50000 + 12.9904i −1.01130 + 1.75162i
\(56\) −1.50000 + 2.59808i −0.200446 + 0.347183i
\(57\) 4.50000 − 6.06218i 0.596040 − 0.802955i
\(58\) −0.500000 − 0.866025i −0.0656532 − 0.113715i
\(59\) 5.00000 0.650945 0.325472 − 0.945552i \(-0.394477\pi\)
0.325472 + 0.945552i \(0.394477\pi\)
\(60\) 5.19615i 0.670820i
\(61\) −13.0000 −1.66448 −0.832240 − 0.554416i \(-0.812942\pi\)
−0.832240 + 0.554416i \(0.812942\pi\)
\(62\) 1.50000 − 2.59808i 0.190500 − 0.329956i
\(63\) 1.50000 + 2.59808i 0.188982 + 0.327327i
\(64\) 7.00000 0.875000
\(65\) −3.00000 + 5.19615i −0.372104 + 0.644503i
\(66\) 7.50000 − 4.33013i 0.923186 − 0.533002i
\(67\) 2.00000 − 3.46410i 0.244339 − 0.423207i −0.717607 − 0.696449i \(-0.754762\pi\)
0.961946 + 0.273241i \(0.0880957\pi\)
\(68\) −2.50000 − 4.33013i −0.303170 − 0.525105i
\(69\) 13.8564i 1.66812i
\(70\) −3.00000 −0.358569
\(71\) −1.50000 + 2.59808i −0.178017 + 0.308335i −0.941201 − 0.337846i \(-0.890302\pi\)
0.763184 + 0.646181i \(0.223635\pi\)
\(72\) 4.50000 − 7.79423i 0.530330 − 0.918559i
\(73\) 2.50000 − 4.33013i 0.292603 − 0.506803i −0.681822 − 0.731519i \(-0.738812\pi\)
0.974424 + 0.224716i \(0.0721453\pi\)
\(74\) −3.00000 − 5.19615i −0.348743 − 0.604040i
\(75\) −6.00000 + 3.46410i −0.692820 + 0.400000i
\(76\) −0.500000 + 4.33013i −0.0573539 + 0.496700i
\(77\) −2.50000 − 4.33013i −0.284901 − 0.493464i
\(78\) 3.00000 − 1.73205i 0.339683 − 0.196116i
\(79\) −2.00000 − 3.46410i −0.225018 − 0.389742i 0.731307 − 0.682048i \(-0.238911\pi\)
−0.956325 + 0.292306i \(0.905577\pi\)
\(80\) 1.50000 + 2.59808i 0.167705 + 0.290474i
\(81\) −4.50000 − 7.79423i −0.500000 − 0.866025i
\(82\) −4.50000 − 7.79423i −0.496942 − 0.860729i
\(83\) −4.50000 + 7.79423i −0.493939 + 0.855528i −0.999976 − 0.00698436i \(-0.997777\pi\)
0.506036 + 0.862512i \(0.331110\pi\)
\(84\) −1.50000 − 0.866025i −0.163663 − 0.0944911i
\(85\) 7.50000 − 12.9904i 0.813489 − 1.40900i
\(86\) 4.00000 − 6.92820i 0.431331 − 0.747087i
\(87\) 1.50000 − 0.866025i 0.160817 − 0.0928477i
\(88\) −7.50000 + 12.9904i −0.799503 + 1.38478i
\(89\) 4.50000 + 7.79423i 0.476999 + 0.826187i 0.999653 − 0.0263586i \(-0.00839118\pi\)
−0.522654 + 0.852545i \(0.675058\pi\)
\(90\) 9.00000 0.948683
\(91\) −1.00000 − 1.73205i −0.104828 − 0.181568i
\(92\) −4.00000 − 6.92820i −0.417029 − 0.722315i
\(93\) 4.50000 + 2.59808i 0.466628 + 0.269408i
\(94\) 1.50000 + 2.59808i 0.154713 + 0.267971i
\(95\) −12.0000 + 5.19615i −1.23117 + 0.533114i
\(96\) 8.66025i 0.883883i
\(97\) 5.00000 + 8.66025i 0.507673 + 0.879316i 0.999961 + 0.00888289i \(0.00282755\pi\)
−0.492287 + 0.870433i \(0.663839\pi\)
\(98\) −3.00000 + 5.19615i −0.303046 + 0.524891i
\(99\) 7.50000 + 12.9904i 0.753778 + 1.30558i
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 171.2.g.a.106.1 ✓ 2
3.2 odd 2 513.2.g.a.505.1 2
9.4 even 3 171.2.h.a.49.1 yes 2
9.5 odd 6 513.2.h.b.334.1 2
19.7 even 3 171.2.h.a.7.1 yes 2
57.26 odd 6 513.2.h.b.235.1 2
171.121 even 3 inner 171.2.g.a.121.1 yes 2
171.140 odd 6 513.2.g.a.64.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
171.2.g.a.106.1 ✓ 2 1.1 even 1 trivial
171.2.g.a.121.1 yes 2 171.121 even 3 inner
171.2.h.a.7.1 yes 2 19.7 even 3
171.2.h.a.49.1 yes 2 9.4 even 3
513.2.g.a.64.1 2 171.140 odd 6
513.2.g.a.505.1 2 3.2 odd 2
513.2.h.b.235.1 2 57.26 odd 6
513.2.h.b.334.1 2 9.5 odd 6