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(7,74)] 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("74.7"); 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([16])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 74 = 2 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 74.f (of order \(9\), degree \(6\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.590892974957\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{18})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{3} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label 49.1
Root \(0.939693 + 0.342020i\) of defining polynomial
Character \(\chi\) \(=\) 74.49
Dual form 74.2.f.a.71.1

$q$-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+(-0.766044 + 0.642788i) q^{2} +(-1.43969 - 1.20805i) q^{3} +(0.173648 - 0.984808i) q^{4} +(3.31908 - 1.20805i) q^{5} +1.87939 q^{6} +(0.826352 - 0.300767i) q^{7} +(0.500000 + 0.866025i) q^{8} +(0.0923963 + 0.524005i) q^{9} +(-1.76604 + 3.05888i) q^{10} +(-1.67365 - 2.89884i) q^{11} +(-1.43969 + 1.20805i) q^{12} +(-1.11334 + 6.31407i) q^{13} +(-0.439693 + 0.761570i) q^{14} +(-6.23783 - 2.27038i) q^{15} +(-0.939693 - 0.342020i) q^{16} +(-0.520945 - 2.95442i) q^{17} +(-0.407604 - 0.342020i) q^{18} +(3.55303 + 2.98135i) q^{19} +(-0.613341 - 3.47843i) q^{20} +(-1.55303 - 0.565258i) q^{21} +(3.14543 + 1.14484i) q^{22} +(-2.91875 + 5.05542i) q^{23} +(0.326352 - 1.85083i) q^{24} +(5.72668 - 4.80526i) q^{25} +(-3.20574 - 5.55250i) q^{26} +(-2.31908 + 4.01676i) q^{27} +(-0.152704 - 0.866025i) q^{28} +(1.63429 + 2.83067i) q^{29} +(6.23783 - 2.27038i) q^{30} -1.12061 q^{31} +(0.939693 - 0.342020i) q^{32} +(-1.09240 + 6.19529i) q^{33} +(2.29813 + 1.92836i) q^{34} +(2.37939 - 1.99654i) q^{35} +0.532089 q^{36} +(-3.44356 + 5.01417i) q^{37} -4.63816 q^{38} +(9.23055 - 7.74535i) q^{39} +(2.70574 + 2.27038i) q^{40} +(1.49020 - 8.45134i) q^{41} +(1.55303 - 0.565258i) q^{42} +5.61587 q^{43} +(-3.14543 + 1.14484i) q^{44} +(0.939693 + 1.62760i) q^{45} +(-1.01367 - 5.74881i) q^{46} +(2.56418 - 4.44129i) q^{47} +(0.939693 + 1.62760i) q^{48} +(-4.76991 + 4.00243i) q^{49} +(-1.29813 + 7.36208i) q^{50} +(-2.81908 + 4.88279i) q^{51} +(6.02481 + 2.19285i) q^{52} +(-0.252374 - 0.0918566i) q^{53} +(-0.805407 - 4.56769i) q^{54} +(-9.05690 - 7.59964i) q^{55} +(0.673648 + 0.565258i) q^{56} +(-1.51367 - 8.58445i) q^{57} +(-3.07145 - 1.11792i) q^{58} +(-7.15910 - 2.60570i) q^{59} +(-3.31908 + 5.74881i) q^{60} +(0.369585 - 2.09602i) q^{61} +(0.858441 - 0.720317i) q^{62} +(0.233956 + 0.405223i) q^{63} +(-0.500000 + 0.866025i) q^{64} +(3.93242 + 22.3019i) q^{65} +(-3.14543 - 5.44804i) q^{66} +(-8.01754 + 2.91815i) q^{67} -3.00000 q^{68} +(10.3093 - 3.75227i) q^{69} +(-0.539363 + 3.05888i) q^{70} +(10.0719 + 8.45134i) q^{71} +(-0.407604 + 0.342020i) q^{72} -8.57398 q^{73} +(-0.585122 - 6.05455i) q^{74} -14.0496 q^{75} +(3.55303 - 2.98135i) q^{76} +(-2.25490 - 1.89209i) q^{77} +(-2.09240 + 11.8666i) q^{78} +(10.8833 - 3.96118i) q^{79} -3.53209 q^{80} +(9.69119 - 3.52730i) q^{81} +(4.29086 + 7.43199i) q^{82} +(0.724155 + 4.10689i) q^{83} +(-0.826352 + 1.43128i) q^{84} +(-5.29813 - 9.17664i) q^{85} +(-4.30200 + 3.60981i) q^{86} +(1.06670 - 6.04958i) q^{87} +(1.67365 - 2.89884i) q^{88} +(-7.86959 - 2.86429i) q^{89} +(-1.76604 - 0.642788i) q^{90} +(0.979055 + 5.55250i) q^{91} +(4.47178 + 3.75227i) q^{92} +(1.61334 + 1.35375i) q^{93} +(0.890530 + 5.05044i) q^{94} +(15.3944 + 5.60310i) q^{95} +(-1.76604 - 0.642788i) q^{96} +(8.53209 - 14.7780i) q^{97} +(1.08125 - 6.13208i) q^{98} +(1.36437 - 1.14484i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 3 q^{3} + 3 q^{5} + 6 q^{7} + 3 q^{8} - 3 q^{9} - 6 q^{10} - 9 q^{11} - 3 q^{12} + 3 q^{14} - 18 q^{15} - 6 q^{18} + 9 q^{19} + 3 q^{20} + 3 q^{21} + 3 q^{22} - 15 q^{23} + 3 q^{24} + 21 q^{25} - 9 q^{26}+ \cdots + 27 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(39\)
\(\chi(n)\) \(e\left(\frac{7}{9}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.766044 + 0.642788i −0.541675 + 0.454519i
\(3\) −1.43969 − 1.20805i −0.831207 − 0.697465i 0.124361 − 0.992237i \(-0.460312\pi\)
−0.955568 + 0.294772i \(0.904756\pi\)
\(4\) 0.173648 − 0.984808i 0.0868241 − 0.492404i
\(5\) 3.31908 − 1.20805i 1.48434 − 0.540254i 0.532385 − 0.846502i \(-0.321296\pi\)
0.951952 + 0.306248i \(0.0990737\pi\)
\(6\) 1.87939 0.767256
\(7\) 0.826352 − 0.300767i 0.312332 − 0.113679i −0.181098 − 0.983465i \(-0.557965\pi\)
0.493430 + 0.869786i \(0.335743\pi\)
\(8\) 0.500000 + 0.866025i 0.176777 + 0.306186i
\(9\) 0.0923963 + 0.524005i 0.0307988 + 0.174668i
\(10\) −1.76604 + 3.05888i −0.558472 + 0.967302i
\(11\) −1.67365 − 2.89884i −0.504624 − 0.874034i −0.999986 − 0.00534749i \(-0.998298\pi\)
0.495362 − 0.868687i \(-0.335036\pi\)
\(12\) −1.43969 + 1.20805i −0.415603 + 0.348733i
\(13\) −1.11334 + 6.31407i −0.308785 + 1.75121i 0.296345 + 0.955081i \(0.404232\pi\)
−0.605130 + 0.796127i \(0.706879\pi\)
\(14\) −0.439693 + 0.761570i −0.117513 + 0.203538i
\(15\) −6.23783 − 2.27038i −1.61060 − 0.586210i
\(16\) −0.939693 − 0.342020i −0.234923 − 0.0855050i
\(17\) −0.520945 − 2.95442i −0.126348 − 0.716553i −0.980498 − 0.196527i \(-0.937034\pi\)
0.854151 − 0.520026i \(-0.174078\pi\)
\(18\) −0.407604 − 0.342020i −0.0960731 − 0.0806149i
\(19\) 3.55303 + 2.98135i 0.815122 + 0.683968i 0.951824 − 0.306644i \(-0.0992060\pi\)
−0.136703 + 0.990612i \(0.543650\pi\)
\(20\) −0.613341 − 3.47843i −0.137147 − 0.777800i
\(21\) −1.55303 − 0.565258i −0.338900 − 0.123349i
\(22\) 3.14543 + 1.14484i 0.670608 + 0.244081i
\(23\) −2.91875 + 5.05542i −0.608601 + 1.05413i 0.382870 + 0.923802i \(0.374936\pi\)
−0.991471 + 0.130326i \(0.958398\pi\)
\(24\) 0.326352 − 1.85083i 0.0666163 − 0.377800i
\(25\) 5.72668 − 4.80526i 1.14534 − 0.961051i
\(26\) −3.20574 − 5.55250i −0.628697 − 1.08893i
\(27\) −2.31908 + 4.01676i −0.446307 + 0.773026i
\(28\) −0.152704 − 0.866025i −0.0288583 − 0.163663i
\(29\) 1.63429 + 2.83067i 0.303479 + 0.525641i 0.976922 − 0.213598i \(-0.0685184\pi\)
−0.673442 + 0.739240i \(0.735185\pi\)
\(30\) 6.23783 − 2.27038i 1.13887 − 0.414513i
\(31\) −1.12061 −0.201268 −0.100634 − 0.994923i \(-0.532087\pi\)
−0.100634 + 0.994923i \(0.532087\pi\)
\(32\) 0.939693 − 0.342020i 0.166116 − 0.0604612i
\(33\) −1.09240 + 6.19529i −0.190162 + 1.07846i
\(34\) 2.29813 + 1.92836i 0.394127 + 0.330711i
\(35\) 2.37939 − 1.99654i 0.402190 − 0.337477i
\(36\) 0.532089 0.0886815
\(37\) −3.44356 + 5.01417i −0.566118 + 0.824324i
\(38\) −4.63816 −0.752408
\(39\) 9.23055 − 7.74535i 1.47807 − 1.24025i
\(40\) 2.70574 + 2.27038i 0.427815 + 0.358979i
\(41\) 1.49020 − 8.45134i 0.232730 − 1.31988i −0.614612 − 0.788830i \(-0.710687\pi\)
0.847342 − 0.531048i \(-0.178202\pi\)
\(42\) 1.55303 − 0.565258i 0.239638 − 0.0872212i
\(43\) 5.61587 0.856412 0.428206 − 0.903681i \(-0.359146\pi\)
0.428206 + 0.903681i \(0.359146\pi\)
\(44\) −3.14543 + 1.14484i −0.474191 + 0.172592i
\(45\) 0.939693 + 1.62760i 0.140081 + 0.242628i
\(46\) −1.01367 − 5.74881i −0.149458 − 0.847616i
\(47\) 2.56418 − 4.44129i 0.374024 − 0.647828i −0.616157 − 0.787624i \(-0.711311\pi\)
0.990180 + 0.139795i \(0.0446445\pi\)
\(48\) 0.939693 + 1.62760i 0.135633 + 0.234923i
\(49\) −4.76991 + 4.00243i −0.681416 + 0.571776i
\(50\) −1.29813 + 7.36208i −0.183584 + 1.04116i
\(51\) −2.81908 + 4.88279i −0.394750 + 0.683727i
\(52\) 6.02481 + 2.19285i 0.835492 + 0.304094i
\(53\) −0.252374 − 0.0918566i −0.0346662 − 0.0126175i 0.324629 − 0.945841i \(-0.394761\pi\)
−0.359295 + 0.933224i \(0.616983\pi\)
\(54\) −0.805407 − 4.56769i −0.109602 − 0.621584i
\(55\) −9.05690 − 7.59964i −1.22123 − 1.02474i
\(56\) 0.673648 + 0.565258i 0.0900200 + 0.0755358i
\(57\) −1.51367 − 8.58445i −0.200491 − 1.13704i
\(58\) −3.07145 − 1.11792i −0.403301 − 0.146790i
\(59\) −7.15910 − 2.60570i −0.932035 − 0.339233i −0.169019 − 0.985613i \(-0.554060\pi\)
−0.763016 + 0.646380i \(0.776282\pi\)
\(60\) −3.31908 + 5.74881i −0.428491 + 0.742168i
\(61\) 0.369585 − 2.09602i 0.0473205 − 0.268368i −0.951963 − 0.306213i \(-0.900938\pi\)
0.999284 + 0.0378447i \(0.0120492\pi\)
\(62\) 0.858441 − 0.720317i 0.109022 − 0.0914804i
\(63\) 0.233956 + 0.405223i 0.0294756 + 0.0510533i
\(64\) −0.500000 + 0.866025i −0.0625000 + 0.108253i
\(65\) 3.93242 + 22.3019i 0.487757 + 2.76620i
\(66\) −3.14543 − 5.44804i −0.387176 − 0.670608i
\(67\) −8.01754 + 2.91815i −0.979499 + 0.356508i −0.781645 − 0.623723i \(-0.785619\pi\)
−0.197853 + 0.980232i \(0.563397\pi\)
\(68\) −3.00000 −0.363803
\(69\) 10.3093 − 3.75227i 1.24109 − 0.451720i
\(70\) −0.539363 + 3.05888i −0.0644662 + 0.365606i
\(71\) 10.0719 + 8.45134i 1.19532 + 1.00299i 0.999751 + 0.0222993i \(0.00709866\pi\)
0.195565 + 0.980691i \(0.437346\pi\)
\(72\) −0.407604 + 0.342020i −0.0480366 + 0.0403075i
\(73\) −8.57398 −1.00351 −0.501754 − 0.865010i \(-0.667312\pi\)
−0.501754 + 0.865010i \(0.667312\pi\)
\(74\) −0.585122 − 6.05455i −0.0680191 − 0.703828i
\(75\) −14.0496 −1.62231
\(76\) 3.55303 − 2.98135i 0.407561 − 0.341984i
\(77\) −2.25490 − 1.89209i −0.256970 − 0.215623i
\(78\) −2.09240 + 11.8666i −0.236917 + 1.34362i
\(79\) 10.8833 − 3.96118i 1.22446 − 0.445668i 0.352764 − 0.935712i \(-0.385242\pi\)
0.871697 + 0.490044i \(0.163019\pi\)
\(80\) −3.53209 −0.394900
\(81\) 9.69119 − 3.52730i 1.07680 − 0.391923i
\(82\) 4.29086 + 7.43199i 0.473846 + 0.820726i
\(83\) 0.724155 + 4.10689i 0.0794864 + 0.450790i 0.998411 + 0.0563559i \(0.0179481\pi\)
−0.918924 + 0.394434i \(0.870941\pi\)
\(84\) −0.826352 + 1.43128i −0.0901624 + 0.156166i
\(85\) −5.29813 − 9.17664i −0.574663 − 0.995346i
\(86\) −4.30200 + 3.60981i −0.463897 + 0.389256i
\(87\) 1.06670 − 6.04958i 0.114363 − 0.648583i
\(88\) 1.67365 − 2.89884i 0.178411 − 0.309018i
\(89\) −7.86959 − 2.86429i −0.834174 − 0.303615i −0.110603 − 0.993865i \(-0.535278\pi\)
−0.723571 + 0.690250i \(0.757501\pi\)
\(90\) −1.76604 − 0.642788i −0.186157 − 0.0677558i
\(91\) 0.979055 + 5.55250i 0.102633 + 0.582060i
\(92\) 4.47178 + 3.75227i 0.466215 + 0.391201i
\(93\) 1.61334 + 1.35375i 0.167296 + 0.140378i
\(94\) 0.890530 + 5.05044i 0.0918511 + 0.520914i
\(95\) 15.3944 + 5.60310i 1.57943 + 0.574866i
\(96\) −1.76604 − 0.642788i −0.180246 − 0.0656042i
\(97\) 8.53209 − 14.7780i 0.866302 − 1.50048i 0.000554205 − 1.00000i \(-0.499824\pi\)
0.865748 − 0.500480i \(-0.166843\pi\)
\(98\) 1.08125 − 6.13208i 0.109223 − 0.619434i
\(99\) 1.36437 − 1.14484i 0.137124 − 0.115061i
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.f.a.49.1 ✓ 6
3.2 odd 2 666.2.x.c.271.1 6
4.3 odd 2 592.2.bc.b.49.1 6
37.16 even 9 2738.2.a.m.1.1 3
37.21 even 18 2738.2.a.p.1.1 3
37.34 even 9 inner 74.2.f.a.71.1 yes 6
111.71 odd 18 666.2.x.c.145.1 6
148.71 odd 18 592.2.bc.b.145.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
74.2.f.a.49.1 ✓ 6 1.1 even 1 trivial
74.2.f.a.71.1 yes 6 37.34 even 9 inner
592.2.bc.b.49.1 6 4.3 odd 2
592.2.bc.b.145.1 6 148.71 odd 18
666.2.x.c.145.1 6 111.71 odd 18
666.2.x.c.271.1 6 3.2 odd 2
2738.2.a.m.1.1 3 37.16 even 9
2738.2.a.p.1.1 3 37.21 even 18