Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [605,2,Mod(9,605)] 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("605.9"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(605, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([5, 6])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 605 = 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 605.j (of order \(10\), degree \(4\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,0,0,6,-2,12] 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: \(4.83094932229\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 7x^{14} + 25x^{12} - 57x^{10} + 194x^{8} - 303x^{6} + 235x^{4} - 33x^{2} + 121 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 55)
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label 9.4
Root \(1.92464 - 0.625353i\) of defining polynomial
Character \(\chi\) \(=\) 605.9
Dual form 605.2.j.h.269.4

$q$-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+(1.92464 + 0.625353i) q^{2} +(1.54035 + 2.12011i) q^{3} +(1.69513 + 1.23158i) q^{4} +(2.01689 - 0.965471i) q^{5} +(1.63880 + 5.04369i) q^{6} +(-0.567697 + 0.781367i) q^{7} +(0.113351 + 0.156015i) q^{8} +(-1.19513 + 3.67823i) q^{9} +(4.48555 - 0.596911i) q^{10} +5.49092i q^{12} +(-4.30362 - 1.39833i) q^{13} +(-1.58124 + 1.14884i) q^{14} +(5.15362 + 2.78887i) q^{15} +(-1.17437 - 3.61433i) q^{16} +(-3.17338 + 1.03109i) q^{17} +(-4.60038 + 6.33188i) q^{18} +(2.65163 - 1.92652i) q^{19} +(4.60795 + 0.847376i) q^{20} -2.53103 q^{21} -3.36643i q^{23} +(-0.156167 + 0.480634i) q^{24} +(3.13573 - 3.89451i) q^{25} +(-7.40846 - 5.38256i) q^{26} +(-2.16214 + 0.702522i) q^{27} +(-1.92464 + 0.625353i) q^{28} +(3.97470 + 2.88779i) q^{29} +(8.17482 + 8.59039i) q^{30} +(-0.129282 + 0.397889i) q^{31} -8.07636i q^{32} -6.75241 q^{34} +(-0.390597 + 2.12403i) q^{35} +(-6.55594 + 4.76317i) q^{36} +(-3.72512 + 5.12719i) q^{37} +(6.30817 - 2.04965i) q^{38} +(-3.66446 - 11.2780i) q^{39} +(0.379246 + 0.205228i) q^{40} +(-4.68068 + 3.40071i) q^{41} +(-4.87132 - 1.58279i) q^{42} -2.26205i q^{43} +(1.14077 + 8.57246i) q^{45} +(2.10520 - 6.47915i) q^{46} +(2.54173 + 3.49838i) q^{47} +(5.85383 - 8.05710i) q^{48} +(1.87486 + 5.77024i) q^{49} +(8.47058 - 5.53458i) q^{50} +(-7.07414 - 5.13966i) q^{51} +(-5.57303 - 7.67061i) q^{52} +(-2.53047 - 0.822201i) q^{53} -4.60066 q^{54} -0.186254 q^{56} +(8.16885 + 2.65422i) q^{57} +(5.84397 + 8.04353i) q^{58} +(-8.19153 - 5.95150i) q^{59} +(5.30132 + 11.0746i) q^{60} +(-0.763537 - 2.34993i) q^{61} +(-0.497642 + 0.684945i) q^{62} +(-2.19558 - 3.02195i) q^{63} +(2.70184 - 8.31540i) q^{64} +(-10.0300 + 1.33473i) q^{65} +9.60059i q^{67} +(-6.64917 - 2.16045i) q^{68} +(7.13718 - 5.18546i) q^{69} +(-2.08002 + 3.84373i) q^{70} +(-1.68510 - 5.18621i) q^{71} +(-0.709327 + 0.230474i) q^{72} +(0.843791 - 1.16138i) q^{73} +(-10.3758 + 7.53847i) q^{74} +(13.0869 + 0.649186i) q^{75} +6.86752 q^{76} -23.9977i q^{78} +(-0.310240 + 0.954820i) q^{79} +(-5.85811 - 6.15591i) q^{80} +(4.56679 + 3.31797i) q^{81} +(-11.1353 + 3.61806i) q^{82} +(-7.03346 + 2.28531i) q^{83} +(-4.29042 - 3.11717i) q^{84} +(-5.40489 + 5.14342i) q^{85} +(1.41458 - 4.35363i) q^{86} +12.8750i q^{87} +12.1964 q^{89} +(-3.16523 + 17.2123i) q^{90} +(3.53576 - 2.56888i) q^{91} +(4.14603 - 5.70652i) q^{92} +(-1.04271 + 0.338796i) q^{93} +(2.70418 + 8.32259i) q^{94} +(3.48805 - 6.44566i) q^{95} +(17.1227 - 12.4404i) q^{96} +(2.87147 + 0.932997i) q^{97} +12.2781i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 6 q^{4} - 2 q^{5} + 12 q^{6} + 2 q^{9} + 8 q^{14} + 24 q^{15} + 6 q^{16} + 6 q^{19} + 12 q^{20} + 8 q^{21} - 4 q^{24} + 24 q^{25} - 50 q^{26} + 22 q^{29} - 4 q^{30} - 22 q^{31} - 16 q^{34} - 8 q^{35}+ \cdots + 94 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/605\mathbb{Z}\right)^\times\).

\(n\) \(122\) \(486\)
\(\chi(n)\) \(-1\) \(e\left(\frac{3}{5}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.92464 + 0.625353i 1.36092 + 0.442191i 0.896352 − 0.443344i \(-0.146208\pi\)
0.464572 + 0.885535i \(0.346208\pi\)
\(3\) 1.54035 + 2.12011i 0.889320 + 1.22404i 0.973751 + 0.227614i \(0.0730925\pi\)
−0.0844316 + 0.996429i \(0.526907\pi\)
\(4\) 1.69513 + 1.23158i 0.847564 + 0.615791i
\(5\) 2.01689 − 0.965471i 0.901983 − 0.431772i
\(6\) 1.63880 + 5.04369i 0.669035 + 2.05908i
\(7\) −0.567697 + 0.781367i −0.214569 + 0.295329i −0.902711 − 0.430247i \(-0.858427\pi\)
0.688142 + 0.725576i \(0.258427\pi\)
\(8\) 0.113351 + 0.156015i 0.0400758 + 0.0551595i
\(9\) −1.19513 + 3.67823i −0.398376 + 1.22608i
\(10\) 4.48555 − 0.596911i 1.41846 − 0.188760i
\(11\) 0 0
\(12\) 5.49092i 1.58509i
\(13\) −4.30362 − 1.39833i −1.19361 − 0.387827i −0.356203 − 0.934408i \(-0.615929\pi\)
−0.837406 + 0.546581i \(0.815929\pi\)
\(14\) −1.58124 + 1.14884i −0.422604 + 0.307040i
\(15\) 5.15362 + 2.78887i 1.33066 + 0.720083i
\(16\) −1.17437 − 3.61433i −0.293592 − 0.903582i
\(17\) −3.17338 + 1.03109i −0.769658 + 0.250077i −0.667419 − 0.744683i \(-0.732601\pi\)
−0.102239 + 0.994760i \(0.532601\pi\)
\(18\) −4.60038 + 6.33188i −1.08432 + 1.49244i
\(19\) 2.65163 − 1.92652i 0.608325 − 0.441974i −0.240499 − 0.970649i \(-0.577311\pi\)
0.848824 + 0.528675i \(0.177311\pi\)
\(20\) 4.60795 + 0.847376i 1.03037 + 0.189479i
\(21\) −2.53103 −0.552316
\(22\) 0 0
\(23\) − 3.36643i − 0.701948i −0.936385 − 0.350974i \(-0.885851\pi\)
0.936385 − 0.350974i \(-0.114149\pi\)
\(24\) −0.156167 + 0.480634i −0.0318775 + 0.0981089i
\(25\) 3.13573 − 3.89451i 0.627146 − 0.778902i
\(26\) −7.40846 − 5.38256i −1.45292 − 1.05561i
\(27\) −2.16214 + 0.702522i −0.416104 + 0.135200i
\(28\) −1.92464 + 0.625353i −0.363722 + 0.118181i
\(29\) 3.97470 + 2.88779i 0.738083 + 0.536249i 0.892110 − 0.451818i \(-0.149224\pi\)
−0.154027 + 0.988067i \(0.549224\pi\)
\(30\) 8.17482 + 8.59039i 1.49251 + 1.56838i
\(31\) −0.129282 + 0.397889i −0.0232197 + 0.0714629i −0.961995 − 0.273067i \(-0.911962\pi\)
0.938775 + 0.344530i \(0.111962\pi\)
\(32\) − 8.07636i − 1.42771i
\(33\) 0 0
\(34\) −6.75241 −1.15803
\(35\) −0.390597 + 2.12403i −0.0660229 + 0.359027i
\(36\) −6.55594 + 4.76317i −1.09266 + 0.793861i
\(37\) −3.72512 + 5.12719i −0.612406 + 0.842905i −0.996773 − 0.0802758i \(-0.974420\pi\)
0.384367 + 0.923181i \(0.374420\pi\)
\(38\) 6.30817 − 2.04965i 1.02332 − 0.332497i
\(39\) −3.66446 − 11.2780i −0.586783 − 1.80593i
\(40\) 0.379246 + 0.205228i 0.0599640 + 0.0324494i
\(41\) −4.68068 + 3.40071i −0.730999 + 0.531102i −0.889879 − 0.456196i \(-0.849212\pi\)
0.158880 + 0.987298i \(0.449212\pi\)
\(42\) −4.87132 − 1.58279i −0.751660 − 0.244229i
\(43\) − 2.26205i − 0.344960i −0.985013 − 0.172480i \(-0.944822\pi\)
0.985013 − 0.172480i \(-0.0551780\pi\)
\(44\) 0 0
\(45\) 1.14077 + 8.57246i 0.170057 + 1.27791i
\(46\) 2.10520 − 6.47915i 0.310395 − 0.955298i
\(47\) 2.54173 + 3.49838i 0.370749 + 0.510292i 0.953104 − 0.302642i \(-0.0978688\pi\)
−0.582355 + 0.812934i \(0.697869\pi\)
\(48\) 5.85383 − 8.05710i 0.844927 − 1.16294i
\(49\) 1.87486 + 5.77024i 0.267838 + 0.824320i
\(50\) 8.47058 − 5.53458i 1.19792 − 0.782708i
\(51\) −7.07414 − 5.13966i −0.990577 − 0.719697i
\(52\) −5.57303 − 7.67061i −0.772840 − 1.06372i
\(53\) −2.53047 − 0.822201i −0.347587 − 0.112938i 0.130020 − 0.991511i \(-0.458496\pi\)
−0.477607 + 0.878573i \(0.658496\pi\)
\(54\) −4.60066 −0.626071
\(55\) 0 0
\(56\) −0.186254 −0.0248892
\(57\) 8.16885 + 2.65422i 1.08199 + 0.351560i
\(58\) 5.84397 + 8.04353i 0.767350 + 1.05617i
\(59\) −8.19153 − 5.95150i −1.06645 − 0.774819i −0.0911765 − 0.995835i \(-0.529063\pi\)
−0.975270 + 0.221016i \(0.929063\pi\)
\(60\) 5.30132 + 11.0746i 0.684398 + 1.42972i
\(61\) −0.763537 − 2.34993i −0.0977609 − 0.300877i 0.890202 − 0.455565i \(-0.150563\pi\)
−0.987963 + 0.154688i \(0.950563\pi\)
\(62\) −0.497642 + 0.684945i −0.0632005 + 0.0869881i
\(63\) −2.19558 − 3.02195i −0.276617 − 0.380730i
\(64\) 2.70184 − 8.31540i 0.337729 − 1.03942i
\(65\) −10.0300 + 1.33473i −1.24407 + 0.165553i
\(66\) 0 0
\(67\) 9.60059i 1.17290i 0.809986 + 0.586449i \(0.199475\pi\)
−0.809986 + 0.586449i \(0.800525\pi\)
\(68\) −6.64917 − 2.16045i −0.806330 − 0.261992i
\(69\) 7.13718 − 5.18546i 0.859215 − 0.624256i
\(70\) −2.08002 + 3.84373i −0.248611 + 0.459413i
\(71\) −1.68510 − 5.18621i −0.199985 − 0.615490i −0.999882 − 0.0153533i \(-0.995113\pi\)
0.799897 − 0.600137i \(-0.204887\pi\)
\(72\) −0.709327 + 0.230474i −0.0835950 + 0.0271617i
\(73\) 0.843791 − 1.16138i 0.0987582 − 0.135929i −0.756778 − 0.653672i \(-0.773228\pi\)
0.855536 + 0.517743i \(0.173228\pi\)
\(74\) −10.3758 + 7.53847i −1.20616 + 0.876329i
\(75\) 13.0869 + 0.649186i 1.51114 + 0.0749616i
\(76\) 6.86752 0.787758
\(77\) 0 0
\(78\) − 23.9977i − 2.71721i
\(79\) −0.310240 + 0.954820i −0.0349047 + 0.107426i −0.966991 − 0.254811i \(-0.917987\pi\)
0.932086 + 0.362237i \(0.117987\pi\)
\(80\) −5.85811 − 6.15591i −0.654956 − 0.688251i
\(81\) 4.56679 + 3.31797i 0.507421 + 0.368663i
\(82\) −11.1353 + 3.61806i −1.22968 + 0.399548i
\(83\) −7.03346 + 2.28531i −0.772022 + 0.250845i −0.668430 − 0.743775i \(-0.733034\pi\)
−0.103592 + 0.994620i \(0.533034\pi\)
\(84\) −4.29042 − 3.11717i −0.468123 − 0.340112i
\(85\) −5.40489 + 5.14342i −0.586242 + 0.557882i
\(86\) 1.41458 − 4.35363i 0.152538 − 0.469464i
\(87\) 12.8750i 1.38034i
\(88\) 0 0
\(89\) 12.1964 1.29282 0.646410 − 0.762991i \(-0.276270\pi\)
0.646410 + 0.762991i \(0.276270\pi\)
\(90\) −3.16523 + 17.2123i −0.333645 + 1.81433i
\(91\) 3.53576 − 2.56888i 0.370648 − 0.269292i
\(92\) 4.14603 − 5.70652i 0.432254 − 0.594946i
\(93\) −1.04271 + 0.338796i −0.108124 + 0.0351315i
\(94\) 2.70418 + 8.32259i 0.278914 + 0.858410i
\(95\) 3.48805 − 6.44566i 0.357867 − 0.661310i
\(96\) 17.1227 − 12.4404i 1.74758 − 1.26969i
\(97\) 2.87147 + 0.932997i 0.291553 + 0.0947315i 0.451142 − 0.892452i \(-0.351017\pi\)
−0.159589 + 0.987184i \(0.551017\pi\)
\(98\) 12.2781i 1.24027i
\(99\) 0 0
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 605.2.j.h.9.4 16
5.4 even 2 inner 605.2.j.h.9.1 16
11.2 odd 10 605.2.j.d.124.4 16
11.3 even 5 55.2.j.a.4.4 yes 16
11.4 even 5 605.2.b.g.364.8 8
11.5 even 5 inner 605.2.j.h.269.1 16
11.6 odd 10 605.2.j.g.269.4 16
11.7 odd 10 605.2.b.f.364.1 8
11.8 odd 10 605.2.j.d.444.1 16
11.9 even 5 55.2.j.a.14.1 yes 16
11.10 odd 2 605.2.j.g.9.1 16
33.14 odd 10 495.2.ba.a.334.1 16
33.20 odd 10 495.2.ba.a.289.4 16
44.3 odd 10 880.2.cd.c.609.4 16
44.31 odd 10 880.2.cd.c.289.1 16
55.3 odd 20 275.2.h.d.26.1 16
55.4 even 10 605.2.b.g.364.1 8
55.7 even 20 3025.2.a.bk.1.8 8
55.9 even 10 55.2.j.a.14.4 yes 16
55.14 even 10 55.2.j.a.4.1 ✓ 16
55.18 even 20 3025.2.a.bk.1.1 8
55.19 odd 10 605.2.j.d.444.4 16
55.24 odd 10 605.2.j.d.124.1 16
55.29 odd 10 605.2.b.f.364.8 8
55.37 odd 20 3025.2.a.bl.1.1 8
55.39 odd 10 605.2.j.g.269.1 16
55.42 odd 20 275.2.h.d.201.4 16
55.47 odd 20 275.2.h.d.26.4 16
55.48 odd 20 3025.2.a.bl.1.8 8
55.49 even 10 inner 605.2.j.h.269.4 16
55.53 odd 20 275.2.h.d.201.1 16
55.54 odd 2 605.2.j.g.9.4 16
165.14 odd 10 495.2.ba.a.334.4 16
165.119 odd 10 495.2.ba.a.289.1 16
220.119 odd 10 880.2.cd.c.289.4 16
220.179 odd 10 880.2.cd.c.609.1 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
55.2.j.a.4.1 ✓ 16 55.14 even 10
55.2.j.a.4.4 yes 16 11.3 even 5
55.2.j.a.14.1 yes 16 11.9 even 5
55.2.j.a.14.4 yes 16 55.9 even 10
275.2.h.d.26.1 16 55.3 odd 20
275.2.h.d.26.4 16 55.47 odd 20
275.2.h.d.201.1 16 55.53 odd 20
275.2.h.d.201.4 16 55.42 odd 20
495.2.ba.a.289.1 16 165.119 odd 10
495.2.ba.a.289.4 16 33.20 odd 10
495.2.ba.a.334.1 16 33.14 odd 10
495.2.ba.a.334.4 16 165.14 odd 10
605.2.b.f.364.1 8 11.7 odd 10
605.2.b.f.364.8 8 55.29 odd 10
605.2.b.g.364.1 8 55.4 even 10
605.2.b.g.364.8 8 11.4 even 5
605.2.j.d.124.1 16 55.24 odd 10
605.2.j.d.124.4 16 11.2 odd 10
605.2.j.d.444.1 16 11.8 odd 10
605.2.j.d.444.4 16 55.19 odd 10
605.2.j.g.9.1 16 11.10 odd 2
605.2.j.g.9.4 16 55.54 odd 2
605.2.j.g.269.1 16 55.39 odd 10
605.2.j.g.269.4 16 11.6 odd 10
605.2.j.h.9.1 16 5.4 even 2 inner
605.2.j.h.9.4 16 1.1 even 1 trivial
605.2.j.h.269.1 16 11.5 even 5 inner
605.2.j.h.269.4 16 55.49 even 10 inner
880.2.cd.c.289.1 16 44.31 odd 10
880.2.cd.c.289.4 16 220.119 odd 10
880.2.cd.c.609.1 16 220.179 odd 10
880.2.cd.c.609.4 16 44.3 odd 10
3025.2.a.bk.1.1 8 55.18 even 20
3025.2.a.bk.1.8 8 55.7 even 20
3025.2.a.bl.1.1 8 55.37 odd 20
3025.2.a.bl.1.8 8 55.48 odd 20