Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [74,2,Mod(3,74)] 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("74.3"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(74, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([13])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 74 = 2 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 74.h (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: \(0.590892974957\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(2\) over \(\Q(\zeta_{18})\)
Coefficient field: \(\Q(\zeta_{36})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - x^{6} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label 25.1
Root \(-0.642788 - 0.766044i\) of defining polynomial
Character \(\chi\) \(=\) 74.25
Dual form 74.2.h.a.3.1

$q$-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+(-0.642788 - 0.766044i) q^{2} +(1.43969 + 1.20805i) q^{3} +(-0.173648 + 0.984808i) q^{4} +(0.273629 + 0.751790i) q^{5} -1.87939i q^{6} +(-0.138449 + 0.0503913i) q^{7} +(0.866025 - 0.500000i) q^{8} +(0.0923963 + 0.524005i) q^{9} +(0.400019 - 0.692853i) q^{10} +(-2.40570 - 4.16679i) q^{11} +(-1.43969 + 1.20805i) q^{12} +(1.91858 + 0.338298i) q^{13} +(0.127595 + 0.0736672i) q^{14} +(-0.514255 + 1.41290i) q^{15} +(-0.939693 - 0.342020i) q^{16} +(-3.43779 + 0.606175i) q^{17} +(0.342020 - 0.407604i) q^{18} +(-4.33625 + 5.16775i) q^{19} +(-0.787884 + 0.138925i) q^{20} +(-0.260199 - 0.0947047i) q^{21} +(-1.64560 + 4.52124i) q^{22} +(-3.61610 - 2.08776i) q^{23} +(1.85083 + 0.326352i) q^{24} +(3.33991 - 2.80251i) q^{25} +(-0.974090 - 1.68717i) q^{26} +(2.31908 - 4.01676i) q^{27} +(-0.0255844 - 0.145096i) q^{28} +(2.63134 - 1.51921i) q^{29} +(1.41290 - 0.514255i) q^{30} +8.13740i q^{31} +(0.342020 + 0.939693i) q^{32} +(1.57021 - 8.90509i) q^{33} +(2.67412 + 2.24386i) q^{34} +(-0.0757674 - 0.0902961i) q^{35} -0.532089 q^{36} +(6.08227 + 0.0772535i) q^{37} +6.74601 q^{38} +(2.35349 + 2.80478i) q^{39} +(0.612865 + 0.514255i) q^{40} +(0.676822 - 3.83845i) q^{41} +(0.0947047 + 0.260199i) q^{42} -8.10852i q^{43} +(4.52124 - 1.64560i) q^{44} +(-0.368660 + 0.212846i) q^{45} +(0.725070 + 4.11208i) q^{46} +(-4.16911 + 7.22111i) q^{47} +(-0.939693 - 1.62760i) q^{48} +(-5.34568 + 4.48556i) q^{49} +(-4.29370 - 0.757095i) q^{50} +(-5.68164 - 3.28030i) q^{51} +(-0.666317 + 1.83069i) q^{52} +(10.2327 + 3.72440i) q^{53} +(-4.56769 + 0.805407i) q^{54} +(2.47428 - 2.94874i) q^{55} +(-0.0947047 + 0.112865i) q^{56} +(-12.4857 + 2.20157i) q^{57} +(-2.85517 - 1.03920i) q^{58} +(-2.96569 + 8.14816i) q^{59} +(-1.30214 - 0.751790i) q^{60} +(-0.346344 - 0.0610698i) q^{61} +(6.23361 - 5.23062i) q^{62} +(-0.0391975 - 0.0678921i) q^{63} +(0.500000 - 0.866025i) q^{64} +(0.270651 + 1.53494i) q^{65} +(-7.83101 + 4.52124i) q^{66} +(-2.25471 + 0.820647i) q^{67} -3.49082i q^{68} +(-2.68397 - 7.37414i) q^{69} +(-0.0204685 + 0.116082i) q^{70} +(5.34953 + 4.48879i) q^{71} +(0.342020 + 0.407604i) q^{72} +1.13399 q^{73} +(-3.85043 - 4.70895i) q^{74} +8.19401 q^{75} +(-4.33625 - 5.16775i) q^{76} +(0.543037 + 0.455662i) q^{77} +(0.635792 - 3.60576i) q^{78} +(0.646510 + 1.77627i) q^{79} -0.800038i q^{80} +(9.69119 - 3.52730i) q^{81} +(-3.37547 + 1.94883i) q^{82} +(-1.71997 - 9.75442i) q^{83} +(0.138449 - 0.239801i) q^{84} +(-1.39639 - 2.41863i) q^{85} +(-6.21149 + 5.21206i) q^{86} +(5.62359 + 0.991591i) q^{87} +(-4.16679 - 2.40570i) q^{88} +(-3.45924 + 9.50418i) q^{89} +(0.400019 + 0.145595i) q^{90} +(-0.282673 + 0.0498429i) q^{91} +(2.68397 - 3.19863i) q^{92} +(-9.83035 + 11.7154i) q^{93} +(8.21154 - 1.44792i) q^{94} +(-5.07158 - 1.84591i) q^{95} +(-0.642788 + 1.76604i) q^{96} +(5.08812 + 2.93763i) q^{97} +(6.87228 + 1.21177i) q^{98} +(1.96114 - 1.64560i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 6 q^{3} - 12 q^{7} - 6 q^{9} + 6 q^{10} - 6 q^{11} - 6 q^{12} + 6 q^{13} - 18 q^{14} - 18 q^{19} - 6 q^{21} - 18 q^{25} + 12 q^{26} - 6 q^{27} - 6 q^{28} + 18 q^{29} + 24 q^{30} - 6 q^{33} + 12 q^{34}+ \cdots + 30 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/74\mathbb{Z}\right)^\times\).

\(n\) \(39\)
\(\chi(n)\) \(e\left(\frac{5}{18}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.642788 − 0.766044i −0.454519 − 0.541675i
\(3\) 1.43969 + 1.20805i 0.831207 + 0.697465i 0.955568 − 0.294772i \(-0.0952436\pi\)
−0.124361 + 0.992237i \(0.539688\pi\)
\(4\) −0.173648 + 0.984808i −0.0868241 + 0.492404i
\(5\) 0.273629 + 0.751790i 0.122371 + 0.336211i 0.985719 − 0.168397i \(-0.0538592\pi\)
−0.863349 + 0.504608i \(0.831637\pi\)
\(6\) − 1.87939i − 0.767256i
\(7\) −0.138449 + 0.0503913i −0.0523288 + 0.0190461i −0.368052 − 0.929805i \(-0.619975\pi\)
0.315723 + 0.948851i \(0.397753\pi\)
\(8\) 0.866025 − 0.500000i 0.306186 − 0.176777i
\(9\) 0.0923963 + 0.524005i 0.0307988 + 0.174668i
\(10\) 0.400019 − 0.692853i 0.126497 − 0.219099i
\(11\) −2.40570 − 4.16679i −0.725346 − 1.25634i −0.958832 − 0.283975i \(-0.908347\pi\)
0.233486 − 0.972360i \(-0.424987\pi\)
\(12\) −1.43969 + 1.20805i −0.415603 + 0.348733i
\(13\) 1.91858 + 0.338298i 0.532119 + 0.0938270i 0.433251 − 0.901274i \(-0.357367\pi\)
0.0988686 + 0.995100i \(0.468478\pi\)
\(14\) 0.127595 + 0.0736672i 0.0341013 + 0.0196884i
\(15\) −0.514255 + 1.41290i −0.132780 + 0.364810i
\(16\) −0.939693 − 0.342020i −0.234923 − 0.0855050i
\(17\) −3.43779 + 0.606175i −0.833786 + 0.147019i −0.574214 − 0.818705i \(-0.694692\pi\)
−0.259572 + 0.965724i \(0.583581\pi\)
\(18\) 0.342020 − 0.407604i 0.0806149 − 0.0960731i
\(19\) −4.33625 + 5.16775i −0.994805 + 1.18556i −0.0121861 + 0.999926i \(0.503879\pi\)
−0.982619 + 0.185636i \(0.940565\pi\)
\(20\) −0.787884 + 0.138925i −0.176176 + 0.0310646i
\(21\) −0.260199 − 0.0947047i −0.0567801 − 0.0206663i
\(22\) −1.64560 + 4.52124i −0.350842 + 0.963931i
\(23\) −3.61610 − 2.08776i −0.754009 − 0.435327i 0.0731316 − 0.997322i \(-0.476701\pi\)
−0.827141 + 0.561995i \(0.810034\pi\)
\(24\) 1.85083 + 0.326352i 0.377800 + 0.0666163i
\(25\) 3.33991 − 2.80251i 0.667981 − 0.560503i
\(26\) −0.974090 − 1.68717i −0.191035 − 0.330882i
\(27\) 2.31908 − 4.01676i 0.446307 − 0.773026i
\(28\) −0.0255844 − 0.145096i −0.00483499 − 0.0274206i
\(29\) 2.63134 − 1.51921i 0.488628 − 0.282109i −0.235377 − 0.971904i \(-0.575633\pi\)
0.724005 + 0.689795i \(0.242299\pi\)
\(30\) 1.41290 − 0.514255i 0.257960 − 0.0938896i
\(31\) 8.13740i 1.46152i 0.682634 + 0.730760i \(0.260834\pi\)
−0.682634 + 0.730760i \(0.739166\pi\)
\(32\) 0.342020 + 0.939693i 0.0604612 + 0.166116i
\(33\) 1.57021 − 8.90509i 0.273338 − 1.55018i
\(34\) 2.67412 + 2.24386i 0.458609 + 0.384818i
\(35\) −0.0757674 − 0.0902961i −0.0128070 − 0.0152628i
\(36\) −0.532089 −0.0886815
\(37\) 6.08227 + 0.0772535i 0.999919 + 0.0127004i
\(38\) 6.74601 1.09435
\(39\) 2.35349 + 2.80478i 0.376860 + 0.449124i
\(40\) 0.612865 + 0.514255i 0.0969024 + 0.0813108i
\(41\) 0.676822 − 3.83845i 0.105702 − 0.599465i −0.885236 − 0.465142i \(-0.846003\pi\)
0.990938 − 0.134322i \(-0.0428858\pi\)
\(42\) 0.0947047 + 0.260199i 0.0146133 + 0.0401496i
\(43\) − 8.10852i − 1.23654i −0.785966 − 0.618269i \(-0.787834\pi\)
0.785966 − 0.618269i \(-0.212166\pi\)
\(44\) 4.52124 − 1.64560i 0.681602 − 0.248083i
\(45\) −0.368660 + 0.212846i −0.0549565 + 0.0317292i
\(46\) 0.725070 + 4.11208i 0.106906 + 0.606293i
\(47\) −4.16911 + 7.22111i −0.608127 + 1.05331i 0.383422 + 0.923573i \(0.374746\pi\)
−0.991549 + 0.129734i \(0.958588\pi\)
\(48\) −0.939693 − 1.62760i −0.135633 − 0.234923i
\(49\) −5.34568 + 4.48556i −0.763669 + 0.640794i
\(50\) −4.29370 − 0.757095i −0.607221 − 0.107069i
\(51\) −5.68164 − 3.28030i −0.795589 − 0.459334i
\(52\) −0.666317 + 1.83069i −0.0924015 + 0.253871i
\(53\) 10.2327 + 3.72440i 1.40557 + 0.511586i 0.929827 − 0.367998i \(-0.119957\pi\)
0.475744 + 0.879584i \(0.342179\pi\)
\(54\) −4.56769 + 0.805407i −0.621584 + 0.109602i
\(55\) 2.47428 − 2.94874i 0.333632 − 0.397607i
\(56\) −0.0947047 + 0.112865i −0.0126555 + 0.0150822i
\(57\) −12.4857 + 2.20157i −1.65378 + 0.291606i
\(58\) −2.85517 − 1.03920i −0.374902 − 0.136453i
\(59\) −2.96569 + 8.14816i −0.386100 + 1.06080i 0.582642 + 0.812729i \(0.302019\pi\)
−0.968742 + 0.248072i \(0.920203\pi\)
\(60\) −1.30214 − 0.751790i −0.168105 − 0.0970557i
\(61\) −0.346344 − 0.0610698i −0.0443448 − 0.00781919i 0.151432 − 0.988468i \(-0.451612\pi\)
−0.195777 + 0.980649i \(0.562723\pi\)
\(62\) 6.23361 − 5.23062i 0.791670 − 0.664290i
\(63\) −0.0391975 − 0.0678921i −0.00493842 − 0.00855360i
\(64\) 0.500000 − 0.866025i 0.0625000 − 0.108253i
\(65\) 0.270651 + 1.53494i 0.0335702 + 0.190386i
\(66\) −7.83101 + 4.52124i −0.963931 + 0.556526i
\(67\) −2.25471 + 0.820647i −0.275457 + 0.100258i −0.476055 − 0.879416i \(-0.657934\pi\)
0.200598 + 0.979674i \(0.435711\pi\)
\(68\) − 3.49082i − 0.423324i
\(69\) −2.68397 − 7.37414i −0.323112 − 0.887742i
\(70\) −0.0204685 + 0.116082i −0.00244645 + 0.0138745i
\(71\) 5.34953 + 4.48879i 0.634873 + 0.532721i 0.902439 − 0.430818i \(-0.141775\pi\)
−0.267566 + 0.963539i \(0.586219\pi\)
\(72\) 0.342020 + 0.407604i 0.0403075 + 0.0480366i
\(73\) 1.13399 0.132724 0.0663618 − 0.997796i \(-0.478861\pi\)
0.0663618 + 0.997796i \(0.478861\pi\)
\(74\) −3.85043 − 4.70895i −0.447603 − 0.547404i
\(75\) 8.19401 0.946162
\(76\) −4.33625 − 5.16775i −0.497402 − 0.592781i
\(77\) 0.543037 + 0.455662i 0.0618848 + 0.0519275i
\(78\) 0.635792 − 3.60576i 0.0719893 − 0.408271i
\(79\) 0.646510 + 1.77627i 0.0727380 + 0.199846i 0.970734 − 0.240158i \(-0.0771992\pi\)
−0.897996 + 0.440004i \(0.854977\pi\)
\(80\) − 0.800038i − 0.0894470i
\(81\) 9.69119 − 3.52730i 1.07680 − 0.391923i
\(82\) −3.37547 + 1.94883i −0.372759 + 0.215212i
\(83\) −1.71997 − 9.75442i −0.188791 − 1.07069i −0.920987 − 0.389593i \(-0.872616\pi\)
0.732196 − 0.681094i \(-0.238495\pi\)
\(84\) 0.138449 − 0.239801i 0.0151060 − 0.0261644i
\(85\) −1.39639 − 2.41863i −0.151460 − 0.262337i
\(86\) −6.21149 + 5.21206i −0.669802 + 0.562031i
\(87\) 5.62359 + 0.991591i 0.602912 + 0.106310i
\(88\) −4.16679 − 2.40570i −0.444182 − 0.256448i
\(89\) −3.45924 + 9.50418i −0.366679 + 1.00744i 0.609937 + 0.792450i \(0.291195\pi\)
−0.976616 + 0.214992i \(0.931028\pi\)
\(90\) 0.400019 + 0.145595i 0.0421657 + 0.0153471i
\(91\) −0.282673 + 0.0498429i −0.0296322 + 0.00522496i
\(92\) 2.68397 − 3.19863i 0.279823 − 0.333480i
\(93\) −9.83035 + 11.7154i −1.01936 + 1.21483i
\(94\) 8.21154 − 1.44792i 0.846956 − 0.149341i
\(95\) −5.07158 − 1.84591i −0.520333 − 0.189386i
\(96\) −0.642788 + 1.76604i −0.0656042 + 0.180246i
\(97\) 5.08812 + 2.93763i 0.516620 + 0.298271i 0.735551 − 0.677470i \(-0.236923\pi\)
−0.218930 + 0.975740i \(0.570257\pi\)
\(98\) 6.87228 + 1.21177i 0.694205 + 0.122407i
\(99\) 1.96114 − 1.64560i 0.197102 − 0.165389i
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 74.2.h.a.25.1 yes 12
3.2 odd 2 666.2.bj.c.469.2 12
4.3 odd 2 592.2.bq.b.321.2 12
37.3 even 18 inner 74.2.h.a.3.1 ✓ 12
37.15 odd 36 2738.2.a.s.1.2 6
37.22 odd 36 2738.2.a.r.1.1 6
111.77 odd 18 666.2.bj.c.595.2 12
148.3 odd 18 592.2.bq.b.225.2 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
74.2.h.a.3.1 ✓ 12 37.3 even 18 inner
74.2.h.a.25.1 yes 12 1.1 even 1 trivial
592.2.bq.b.225.2 12 148.3 odd 18
592.2.bq.b.321.2 12 4.3 odd 2
666.2.bj.c.469.2 12 3.2 odd 2
666.2.bj.c.595.2 12 111.77 odd 18
2738.2.a.r.1.1 6 37.22 odd 36
2738.2.a.s.1.2 6 37.15 odd 36