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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [171,2,Mod(14,171)] 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("171.14"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(171, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([15, 7])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 171 = 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 171.x (of order \(18\), degree \(6\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.36544187456\)
Analytic rank: \(0\)
Dimension: \(108\)
Relative dimension: \(18\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label 110.12
Character \(\chi\) \(=\) 171.110
Dual form 171.2.x.a.14.12

$q$-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+(0.469442 - 0.393909i) q^{2} +(1.72885 - 0.105209i) q^{3} +(-0.282084 + 1.59978i) q^{4} +(-0.821139 + 2.25606i) q^{5} +(0.770154 - 0.730400i) q^{6} +(-0.676894 - 1.17241i) q^{7} +(1.11056 + 1.92354i) q^{8} +(2.97786 - 0.363781i) q^{9} +(0.503205 + 1.38254i) q^{10} +0.212949i q^{11} +(-0.319372 + 2.79546i) q^{12} +(-2.01801 - 5.54444i) q^{13} +(-0.779587 - 0.283747i) q^{14} +(-1.18227 + 3.98679i) q^{15} +(-1.77394 - 0.645662i) q^{16} +(0.897444 - 2.46571i) q^{17} +(1.25464 - 1.34378i) q^{18} +(-0.154798 - 4.35615i) q^{19} +(-3.37757 - 1.95004i) q^{20} +(-1.29360 - 1.95572i) q^{21} +(0.0838824 + 0.0999672i) q^{22} +(1.61574 + 0.284898i) q^{23} +(2.12237 + 3.20868i) q^{24} +(-0.585319 - 0.491141i) q^{25} +(-3.13134 - 1.80788i) q^{26} +(5.11001 - 0.942220i) q^{27} +(2.06655 - 0.752162i) q^{28} +(-0.914202 + 5.18470i) q^{29} +(1.01542 + 2.33727i) q^{30} -3.57339i q^{31} +(-5.26143 + 1.91500i) q^{32} +(0.0224041 + 0.368157i) q^{33} +(-0.549965 - 1.51102i) q^{34} +(3.20086 - 0.564398i) q^{35} +(-0.258040 + 4.86654i) q^{36} +8.01979i q^{37} +(-1.78859 - 1.98398i) q^{38} +(-4.07217 - 9.37321i) q^{39} +(-5.25156 + 0.925991i) q^{40} +(-6.91883 + 5.80559i) q^{41} +(-1.37764 - 0.408537i) q^{42} +(-0.233045 - 1.32166i) q^{43} +(-0.340671 - 0.0600696i) q^{44} +(-1.62453 + 7.01695i) q^{45} +(0.870720 - 0.502711i) q^{46} +(2.80328 + 0.494293i) q^{47} +(-3.13481 - 0.929620i) q^{48} +(2.58363 - 4.47498i) q^{49} -0.468238 q^{50} +(1.29213 - 4.35726i) q^{51} +(9.43914 - 1.66438i) q^{52} +(-7.65410 - 6.42256i) q^{53} +(2.02771 - 2.45520i) q^{54} +(-0.480426 - 0.174861i) q^{55} +(1.50346 - 2.60407i) q^{56} +(-0.725928 - 7.51485i) q^{57} +(1.61313 + 2.79403i) q^{58} +(1.30385 + 7.39448i) q^{59} +(-6.04448 - 3.01599i) q^{60} +(-4.70028 + 1.71076i) q^{61} +(-1.40759 - 1.67750i) q^{62} +(-2.44220 - 3.24505i) q^{63} +(0.172188 - 0.298239i) q^{64} +14.1657 q^{65} +(0.155538 + 0.164003i) q^{66} +(-6.37567 + 7.59823i) q^{67} +(3.69143 + 2.13125i) q^{68} +(2.82335 + 0.322558i) q^{69} +(1.28030 - 1.52580i) q^{70} +(-7.93377 + 6.65722i) q^{71} +(4.00684 + 5.32405i) q^{72} +(-1.33247 - 7.55679i) q^{73} +(3.15906 + 3.76483i) q^{74} +(-1.06360 - 0.787530i) q^{75} +(7.01255 + 0.981159i) q^{76} +(0.249664 - 0.144144i) q^{77} +(-5.60384 - 2.79612i) q^{78} +(-1.92202 + 5.28069i) q^{79} +(2.91331 - 3.47194i) q^{80} +(8.73533 - 2.16658i) q^{81} +(-0.961119 + 5.45078i) q^{82} +(14.1234 - 8.15413i) q^{83} +(3.49362 - 1.51780i) q^{84} +(4.82586 + 4.04937i) q^{85} +(-0.630016 - 0.528646i) q^{86} +(-1.03505 + 9.05976i) q^{87} +(-0.409617 + 0.236492i) q^{88} +(-0.680140 + 3.85727i) q^{89} +(2.00142 + 3.93397i) q^{90} +(-5.13440 + 6.11895i) q^{91} +(-0.911550 + 2.50446i) q^{92} +(-0.375951 - 6.17786i) q^{93} +(1.51068 - 0.872193i) q^{94} +(9.95485 + 3.22777i) q^{95} +(-8.89476 + 3.86431i) q^{96} +(4.16363 + 4.96202i) q^{97} +(-0.549868 - 3.11846i) q^{98} +(0.0774666 + 0.634132i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 108 q - 9 q^{2} - 3 q^{4} - 9 q^{5} + 3 q^{7} - 24 q^{9} - 12 q^{10} - 9 q^{12} - 6 q^{13} - 9 q^{14} - 36 q^{15} - 9 q^{16} + 27 q^{17} + 36 q^{18} - 15 q^{19} - 18 q^{20} + 3 q^{21} + 30 q^{22} - 45 q^{23}+ \cdots + 39 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(20\) \(154\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{11}{18}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.469442 − 0.393909i 0.331946 − 0.278536i −0.461546 − 0.887116i \(-0.652705\pi\)
0.793492 + 0.608580i \(0.208261\pi\)
\(3\) 1.72885 − 0.105209i 0.998153 − 0.0607422i
\(4\) −0.282084 + 1.59978i −0.141042 + 0.799890i
\(5\) −0.821139 + 2.25606i −0.367225 + 1.00894i 0.609188 + 0.793026i \(0.291496\pi\)
−0.976412 + 0.215915i \(0.930727\pi\)
\(6\) 0.770154 − 0.730400i 0.314414 − 0.298184i
\(7\) −0.676894 − 1.17241i −0.255842 − 0.443131i 0.709282 − 0.704925i \(-0.249019\pi\)
−0.965124 + 0.261794i \(0.915686\pi\)
\(8\) 1.11056 + 1.92354i 0.392642 + 0.680076i
\(9\) 2.97786 − 0.363781i 0.992621 − 0.121260i
\(10\) 0.503205 + 1.38254i 0.159127 + 0.437199i
\(11\) 0.212949i 0.0642065i 0.999485 + 0.0321032i \(0.0102205\pi\)
−0.999485 + 0.0321032i \(0.989779\pi\)
\(12\) −0.319372 + 2.79546i −0.0921947 + 0.806980i
\(13\) −2.01801 − 5.54444i −0.559696 − 1.53775i −0.820081 − 0.572247i \(-0.806072\pi\)
0.260386 − 0.965505i \(-0.416150\pi\)
\(14\) −0.779587 − 0.283747i −0.208353 − 0.0758344i
\(15\) −1.18227 + 3.98679i −0.305261 + 1.02938i
\(16\) −1.77394 − 0.645662i −0.443485 − 0.161415i
\(17\) 0.897444 − 2.46571i 0.217662 − 0.598022i −0.782020 − 0.623254i \(-0.785810\pi\)
0.999682 + 0.0252323i \(0.00803255\pi\)
\(18\) 1.25464 − 1.34378i 0.295721 − 0.316732i
\(19\) −0.154798 − 4.35615i −0.0355132 − 0.999369i
\(20\) −3.37757 − 1.95004i −0.755248 − 0.436043i
\(21\) −1.29360 − 1.95572i −0.282286 − 0.426772i
\(22\) 0.0838824 + 0.0999672i 0.0178838 + 0.0213131i
\(23\) 1.61574 + 0.284898i 0.336905 + 0.0594054i 0.339541 − 0.940591i \(-0.389728\pi\)
−0.00263654 + 0.999997i \(0.500839\pi\)
\(24\) 2.12237 + 3.20868i 0.433226 + 0.654970i
\(25\) −0.585319 − 0.491141i −0.117064 − 0.0982282i
\(26\) −3.13134 − 1.80788i −0.614107 − 0.354555i
\(27\) 5.11001 − 0.942220i 0.983422 − 0.181330i
\(28\) 2.06655 − 0.752162i 0.390541 − 0.142145i
\(29\) −0.914202 + 5.18470i −0.169763 + 0.962774i 0.774253 + 0.632876i \(0.218126\pi\)
−0.944016 + 0.329899i \(0.892985\pi\)
\(30\) 1.01542 + 2.33727i 0.185390 + 0.426726i
\(31\) − 3.57339i − 0.641800i −0.947113 − 0.320900i \(-0.896015\pi\)
0.947113 − 0.320900i \(-0.103985\pi\)
\(32\) −5.26143 + 1.91500i −0.930098 + 0.338528i
\(33\) 0.0224041 + 0.368157i 0.00390005 + 0.0640879i
\(34\) −0.549965 − 1.51102i −0.0943183 − 0.259137i
\(35\) 3.20086 − 0.564398i 0.541045 − 0.0954008i
\(36\) −0.258040 + 4.86654i −0.0430066 + 0.811091i
\(37\) 8.01979i 1.31844i 0.751948 + 0.659222i \(0.229114\pi\)
−0.751948 + 0.659222i \(0.770886\pi\)
\(38\) −1.78859 − 1.98398i −0.290148 − 0.321845i
\(39\) −4.07217 − 9.37321i −0.652069 − 1.50091i
\(40\) −5.25156 + 0.925991i −0.830344 + 0.146412i
\(41\) −6.91883 + 5.80559i −1.08054 + 0.906681i −0.995966 − 0.0897363i \(-0.971398\pi\)
−0.0845746 + 0.996417i \(0.526953\pi\)
\(42\) −1.37764 − 0.408537i −0.212575 − 0.0630386i
\(43\) −0.233045 − 1.32166i −0.0355390 − 0.201552i 0.961868 − 0.273512i \(-0.0881855\pi\)
−0.997407 + 0.0719608i \(0.977074\pi\)
\(44\) −0.340671 − 0.0600696i −0.0513582 − 0.00905583i
\(45\) −1.62453 + 7.01695i −0.242170 + 1.04603i
\(46\) 0.870720 − 0.502711i 0.128381 − 0.0741206i
\(47\) 2.80328 + 0.494293i 0.408900 + 0.0721000i 0.374315 − 0.927302i \(-0.377878\pi\)
0.0345849 + 0.999402i \(0.488989\pi\)
\(48\) −3.13481 − 0.929620i −0.452471 − 0.134179i
\(49\) 2.58363 − 4.47498i 0.369090 − 0.639282i
\(50\) −0.468238 −0.0662189
\(51\) 1.29213 − 4.35726i 0.180935 − 0.610139i
\(52\) 9.43914 − 1.66438i 1.30897 − 0.230807i
\(53\) −7.65410 − 6.42256i −1.05137 − 0.882206i −0.0581349 − 0.998309i \(-0.518515\pi\)
−0.993237 + 0.116103i \(0.962960\pi\)
\(54\) 2.02771 − 2.45520i 0.275936 − 0.334110i
\(55\) −0.480426 − 0.174861i −0.0647806 − 0.0235782i
\(56\) 1.50346 − 2.60407i 0.200908 − 0.347984i
\(57\) −0.725928 − 7.51485i −0.0961515 − 0.995367i
\(58\) 1.61313 + 2.79403i 0.211815 + 0.366874i
\(59\) 1.30385 + 7.39448i 0.169746 + 0.962679i 0.944035 + 0.329846i \(0.106997\pi\)
−0.774288 + 0.632833i \(0.781892\pi\)
\(60\) −6.04448 − 3.01599i −0.780340 − 0.389362i
\(61\) −4.70028 + 1.71076i −0.601810 + 0.219041i −0.624916 − 0.780692i \(-0.714867\pi\)
0.0231061 + 0.999733i \(0.492644\pi\)
\(62\) −1.40759 − 1.67750i −0.178764 − 0.213043i
\(63\) −2.44220 − 3.24505i −0.307688 − 0.408838i
\(64\) 0.172188 − 0.298239i 0.0215235 − 0.0372798i
\(65\) 14.1657 1.75703
\(66\) 0.155538 + 0.164003i 0.0191454 + 0.0201874i
\(67\) −6.37567 + 7.59823i −0.778912 + 0.928271i −0.998884 − 0.0472352i \(-0.984959\pi\)
0.219972 + 0.975506i \(0.429403\pi\)
\(68\) 3.69143 + 2.13125i 0.447652 + 0.258452i
\(69\) 2.82335 + 0.322558i 0.339891 + 0.0388314i
\(70\) 1.28030 − 1.52580i 0.153025 − 0.182368i
\(71\) −7.93377 + 6.65722i −0.941565 + 0.790067i −0.977857 − 0.209274i \(-0.932890\pi\)
0.0362918 + 0.999341i \(0.488445\pi\)
\(72\) 4.00684 + 5.32405i 0.472210 + 0.627445i
\(73\) −1.33247 − 7.55679i −0.155953 − 0.884455i −0.957909 − 0.287072i \(-0.907318\pi\)
0.801956 − 0.597383i \(-0.203793\pi\)
\(74\) 3.15906 + 3.76483i 0.367234 + 0.437652i
\(75\) −1.06360 − 0.787530i −0.122814 − 0.0909361i
\(76\) 7.01255 + 0.981159i 0.804395 + 0.112547i
\(77\) 0.249664 − 0.144144i 0.0284519 − 0.0164267i
\(78\) −5.60384 − 2.79612i −0.634510 − 0.316598i
\(79\) −1.92202 + 5.28069i −0.216244 + 0.594125i −0.999624 − 0.0274180i \(-0.991271\pi\)
0.783380 + 0.621543i \(0.213494\pi\)
\(80\) 2.91331 − 3.47194i 0.325717 − 0.388175i
\(81\) 8.73533 − 2.16658i 0.970592 − 0.240731i
\(82\) −0.961119 + 5.45078i −0.106138 + 0.601938i
\(83\) 14.1234 − 8.15413i 1.55024 − 0.895032i 0.552119 − 0.833765i \(-0.313819\pi\)
0.998121 − 0.0612667i \(-0.0195140\pi\)
\(84\) 3.49362 − 1.51780i 0.381185 − 0.165605i
\(85\) 4.82586 + 4.04937i 0.523438 + 0.439216i
\(86\) −0.630016 − 0.528646i −0.0679363 − 0.0570054i
\(87\) −1.03505 + 9.05976i −0.110969 + 0.971308i
\(88\) −0.409617 + 0.236492i −0.0436653 + 0.0252102i
\(89\) −0.680140 + 3.85727i −0.0720947 + 0.408869i 0.927308 + 0.374300i \(0.122117\pi\)
−0.999402 + 0.0345692i \(0.988994\pi\)
\(90\) 2.00142 + 3.93397i 0.210968 + 0.414677i
\(91\) −5.13440 + 6.11895i −0.538232 + 0.641440i
\(92\) −0.911550 + 2.50446i −0.0950356 + 0.261108i
\(93\) −0.375951 − 6.17786i −0.0389844 − 0.640615i
\(94\) 1.51068 − 0.872193i 0.155815 − 0.0899598i
\(95\) 9.95485 + 3.22777i 1.02135 + 0.331162i
\(96\) −8.89476 + 3.86431i −0.907818 + 0.394399i
\(97\) 4.16363 + 4.96202i 0.422753 + 0.503817i 0.934817 − 0.355130i \(-0.115563\pi\)
−0.512064 + 0.858947i \(0.671119\pi\)
\(98\) −0.549868 − 3.11846i −0.0555451 − 0.315012i
\(99\) 0.0774666 + 0.634132i 0.00778569 + 0.0637327i
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.x.a.110.12 yes 108
3.2 odd 2 513.2.bo.a.224.7 108
9.4 even 3 513.2.cd.a.395.12 108
9.5 odd 6 171.2.bd.a.167.7 yes 108
19.14 odd 18 171.2.bd.a.128.7 yes 108
57.14 even 18 513.2.cd.a.413.12 108
171.14 even 18 inner 171.2.x.a.14.12 ✓ 108
171.166 odd 18 513.2.bo.a.71.7 108
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
171.2.x.a.14.12 ✓ 108 171.14 even 18 inner
171.2.x.a.110.12 yes 108 1.1 even 1 trivial
171.2.bd.a.128.7 yes 108 19.14 odd 18
171.2.bd.a.167.7 yes 108 9.5 odd 6
513.2.bo.a.71.7 108 171.166 odd 18
513.2.bo.a.224.7 108 3.2 odd 2
513.2.cd.a.395.12 108 9.4 even 3
513.2.cd.a.413.12 108 57.14 even 18