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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [345,2,Mod(16,345)] 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("345.16"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(345, base_ring=CyclotomicField(22)) chi = DirichletCharacter(H, H._module([0, 0, 8])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 345 = 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 345.m (of order \(11\), degree \(10\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [50,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.75483886973\)
Analytic rank: \(0\)
Dimension: \(50\)
Relative dimension: \(5\) over \(\Q(\zeta_{11})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

Embedding invariants

Embedding label 196.3
Character \(\chi\) \(=\) 345.196
Dual form 345.2.m.d.301.3

$q$-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+(0.116038 - 0.0745732i) q^{2} +(0.142315 - 0.989821i) q^{3} +(-0.822926 + 1.80196i) q^{4} +(-0.959493 + 0.281733i) q^{5} +(-0.0573002 - 0.125470i) q^{6} +(-2.92236 - 3.37258i) q^{7} +(0.0781472 + 0.543526i) q^{8} +(-0.959493 - 0.281733i) q^{9} +(-0.0903281 + 0.104244i) q^{10} +(-2.33198 - 1.49867i) q^{11} +(1.66650 + 1.07100i) q^{12} +(-0.663696 + 0.765946i) q^{13} +(-0.590609 - 0.173419i) q^{14} +(0.142315 + 0.989821i) q^{15} +(-2.54492 - 2.93700i) q^{16} +(-2.24688 - 4.91998i) q^{17} +(-0.132347 + 0.0388607i) q^{18} +(-1.21091 + 2.65153i) q^{19} +(0.281922 - 1.96081i) q^{20} +(-3.75415 + 2.41265i) q^{21} -0.382360 q^{22} +(4.25233 - 2.21759i) q^{23} +0.549115 q^{24} +(0.841254 - 0.540641i) q^{25} +(-0.0198950 + 0.138373i) q^{26} +(-0.415415 + 0.909632i) q^{27} +(8.48214 - 2.49058i) q^{28} +(-2.01937 - 4.42180i) q^{29} +(0.0903281 + 0.104244i) q^{30} +(1.49803 + 10.4190i) q^{31} +(-1.56807 - 0.460428i) q^{32} +(-1.81530 + 2.09496i) q^{33} +(-0.627623 - 0.403349i) q^{34} +(3.75415 + 2.41265i) q^{35} +(1.29726 - 1.49712i) q^{36} +(-8.75061 - 2.56941i) q^{37} +(0.0572208 + 0.397980i) q^{38} +(0.663696 + 0.765946i) q^{39} +(-0.228111 - 0.499492i) q^{40} +(-6.12462 + 1.79835i) q^{41} +(-0.255706 + 0.559918i) q^{42} +(-0.700120 + 4.86944i) q^{43} +(4.61960 - 2.96883i) q^{44} +1.00000 q^{45} +(0.328060 - 0.574434i) q^{46} +9.95133 q^{47} +(-3.26928 + 2.10104i) q^{48} +(-1.83793 + 12.7831i) q^{49} +(0.0573002 - 0.125470i) q^{50} +(-5.18967 + 1.52382i) q^{51} +(-0.834030 - 1.82627i) q^{52} +(1.42364 + 1.64297i) q^{53} +(0.0196302 + 0.136531i) q^{54} +(2.65975 + 0.780972i) q^{55} +(1.60471 - 1.85194i) q^{56} +(2.45221 + 1.57594i) q^{57} +(-0.564071 - 0.362506i) q^{58} +(7.85509 - 9.06525i) q^{59} +(-1.90073 - 0.558105i) q^{60} +(-1.44488 - 10.0493i) q^{61} +(0.950807 + 1.09729i) q^{62} +(1.85382 + 4.05929i) q^{63} +(7.24128 - 2.12623i) q^{64} +(0.421020 - 0.921905i) q^{65} +(-0.0544155 + 0.378468i) q^{66} +(1.06973 - 0.687471i) q^{67} +10.7146 q^{68} +(-1.58985 - 4.52464i) q^{69} +0.615543 q^{70} +(-1.66823 + 1.07210i) q^{71} +(0.0781472 - 0.543526i) q^{72} +(2.23396 - 4.89170i) q^{73} +(-1.20701 + 0.354411i) q^{74} +(-0.415415 - 0.909632i) q^{75} +(-3.78145 - 4.36402i) q^{76} +(1.76049 + 12.2445i) q^{77} +(0.134133 + 0.0393850i) q^{78} +(-1.79851 + 2.07559i) q^{79} +(3.26928 + 2.10104i) q^{80} +(0.841254 + 0.540641i) q^{81} +(-0.576580 + 0.665409i) q^{82} +(2.05109 + 0.602256i) q^{83} +(-1.25810 - 8.75025i) q^{84} +(3.54199 + 4.08767i) q^{85} +(0.281889 + 0.617251i) q^{86} +(-4.66417 + 1.36953i) q^{87} +(0.632330 - 1.38461i) q^{88} +(0.889889 - 6.18931i) q^{89} +(0.116038 - 0.0745732i) q^{90} +4.52278 q^{91} +(0.496645 + 9.48743i) q^{92} +10.5262 q^{93} +(1.15473 - 0.742102i) q^{94} +(0.414840 - 2.88527i) q^{95} +(-0.678901 + 1.48659i) q^{96} +(-4.42369 + 1.29891i) q^{97} +(0.740004 + 1.62038i) q^{98} +(1.81530 + 2.09496i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 50 q + 2 q^{2} + 5 q^{3} - 4 q^{4} - 5 q^{5} + 9 q^{6} + 3 q^{7} + 6 q^{8} - 5 q^{9} + 2 q^{10} + 19 q^{11} + 4 q^{12} + 15 q^{13} + 43 q^{14} + 5 q^{15} - 76 q^{16} - 23 q^{17} + 2 q^{18} - 7 q^{19} - 4 q^{20}+ \cdots + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/345\mathbb{Z}\right)^\times\).

\(n\) \(116\) \(166\) \(277\)
\(\chi(n)\) \(1\) \(e\left(\frac{10}{11}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.116038 − 0.0745732i 0.0820513 − 0.0527312i −0.498972 − 0.866618i \(-0.666289\pi\)
0.581023 + 0.813887i \(0.302653\pi\)
\(3\) 0.142315 − 0.989821i 0.0821655 − 0.571474i
\(4\) −0.822926 + 1.80196i −0.411463 + 0.900979i
\(5\) −0.959493 + 0.281733i −0.429098 + 0.125995i
\(6\) −0.0573002 − 0.125470i −0.0233927 − 0.0512229i
\(7\) −2.92236 − 3.37258i −1.10455 − 1.27472i −0.958392 − 0.285455i \(-0.907855\pi\)
−0.146156 − 0.989261i \(-0.546690\pi\)
\(8\) 0.0781472 + 0.543526i 0.0276292 + 0.192165i
\(9\) −0.959493 − 0.281733i −0.319831 − 0.0939109i
\(10\) −0.0903281 + 0.104244i −0.0285642 + 0.0329649i
\(11\) −2.33198 − 1.49867i −0.703119 − 0.451867i 0.139609 − 0.990207i \(-0.455415\pi\)
−0.842728 + 0.538339i \(0.819052\pi\)
\(12\) 1.66650 + 1.07100i 0.481078 + 0.309170i
\(13\) −0.663696 + 0.765946i −0.184076 + 0.212435i −0.840286 − 0.542143i \(-0.817613\pi\)
0.656210 + 0.754578i \(0.272159\pi\)
\(14\) −0.590609 − 0.173419i −0.157847 − 0.0463481i
\(15\) 0.142315 + 0.989821i 0.0367455 + 0.255571i
\(16\) −2.54492 − 2.93700i −0.636231 − 0.734250i
\(17\) −2.24688 − 4.91998i −0.544949 − 1.19327i −0.959101 − 0.283064i \(-0.908649\pi\)
0.414152 − 0.910208i \(-0.364078\pi\)
\(18\) −0.132347 + 0.0388607i −0.0311946 + 0.00915956i
\(19\) −1.21091 + 2.65153i −0.277802 + 0.608302i −0.996177 − 0.0873540i \(-0.972159\pi\)
0.718375 + 0.695656i \(0.244886\pi\)
\(20\) 0.281922 − 1.96081i 0.0630397 − 0.438451i
\(21\) −3.75415 + 2.41265i −0.819223 + 0.526483i
\(22\) −0.382360 −0.0815194
\(23\) 4.25233 − 2.21759i 0.886672 − 0.462399i
\(24\) 0.549115 0.112088
\(25\) 0.841254 − 0.540641i 0.168251 − 0.108128i
\(26\) −0.0198950 + 0.138373i −0.00390173 + 0.0271372i
\(27\) −0.415415 + 0.909632i −0.0799467 + 0.175059i
\(28\) 8.48214 − 2.49058i 1.60297 − 0.470676i
\(29\) −2.01937 − 4.42180i −0.374987 − 0.821107i −0.999205 − 0.0398571i \(-0.987310\pi\)
0.624218 − 0.781250i \(-0.285418\pi\)
\(30\) 0.0903281 + 0.104244i 0.0164916 + 0.0190323i
\(31\) 1.49803 + 10.4190i 0.269054 + 1.87131i 0.457475 + 0.889222i \(0.348754\pi\)
−0.188422 + 0.982088i \(0.560337\pi\)
\(32\) −1.56807 − 0.460428i −0.277199 − 0.0813929i
\(33\) −1.81530 + 2.09496i −0.316002 + 0.364686i
\(34\) −0.627623 − 0.403349i −0.107636 − 0.0691737i
\(35\) 3.75415 + 2.41265i 0.634567 + 0.407812i
\(36\) 1.29726 − 1.49712i 0.216210 − 0.249520i
\(37\) −8.75061 − 2.56941i −1.43859 − 0.422408i −0.532839 − 0.846217i \(-0.678875\pi\)
−0.905752 + 0.423808i \(0.860693\pi\)
\(38\) 0.0572208 + 0.397980i 0.00928245 + 0.0645608i
\(39\) 0.663696 + 0.765946i 0.106276 + 0.122650i
\(40\) −0.228111 − 0.499492i −0.0360674 − 0.0789767i
\(41\) −6.12462 + 1.79835i −0.956504 + 0.280855i −0.722493 − 0.691379i \(-0.757004\pi\)
−0.234012 + 0.972234i \(0.575185\pi\)
\(42\) −0.255706 + 0.559918i −0.0394563 + 0.0863972i
\(43\) −0.700120 + 4.86944i −0.106767 + 0.742583i 0.864161 + 0.503215i \(0.167849\pi\)
−0.970929 + 0.239368i \(0.923060\pi\)
\(44\) 4.61960 − 2.96883i 0.696430 − 0.447569i
\(45\) 1.00000 0.149071
\(46\) 0.328060 − 0.574434i 0.0483698 − 0.0846957i
\(47\) 9.95133 1.45155 0.725775 − 0.687932i \(-0.241481\pi\)
0.725775 + 0.687932i \(0.241481\pi\)
\(48\) −3.26928 + 2.10104i −0.471881 + 0.303259i
\(49\) −1.83793 + 12.7831i −0.262561 + 1.82615i
\(50\) 0.0573002 − 0.125470i 0.00810347 − 0.0177441i
\(51\) −5.18967 + 1.52382i −0.726699 + 0.213378i
\(52\) −0.834030 − 1.82627i −0.115659 − 0.253258i
\(53\) 1.42364 + 1.64297i 0.195552 + 0.225679i 0.845054 − 0.534681i \(-0.179568\pi\)
−0.649502 + 0.760360i \(0.725023\pi\)
\(54\) 0.0196302 + 0.136531i 0.00267133 + 0.0185795i
\(55\) 2.65975 + 0.780972i 0.358640 + 0.105306i
\(56\) 1.60471 − 1.85194i 0.214439 − 0.247475i
\(57\) 2.45221 + 1.57594i 0.324803 + 0.208738i
\(58\) −0.564071 − 0.362506i −0.0740661 − 0.0475994i
\(59\) 7.85509 − 9.06525i 1.02265 − 1.18020i 0.0391554 − 0.999233i \(-0.487533\pi\)
0.983490 − 0.180963i \(-0.0579213\pi\)
\(60\) −1.90073 − 0.558105i −0.245383 − 0.0720510i
\(61\) −1.44488 − 10.0493i −0.184998 − 1.28669i −0.844733 − 0.535188i \(-0.820241\pi\)
0.659735 − 0.751498i \(-0.270668\pi\)
\(62\) 0.950807 + 1.09729i 0.120753 + 0.139356i
\(63\) 1.85382 + 4.05929i 0.233559 + 0.511423i
\(64\) 7.24128 − 2.12623i 0.905160 − 0.265779i
\(65\) 0.421020 − 0.921905i 0.0522211 − 0.114348i
\(66\) −0.0544155 + 0.378468i −0.00669808 + 0.0465862i
\(67\) 1.06973 − 0.687471i 0.130688 − 0.0839879i −0.473664 − 0.880706i \(-0.657069\pi\)
0.604351 + 0.796718i \(0.293432\pi\)
\(68\) 10.7146 1.29934
\(69\) −1.58985 − 4.52464i −0.191395 − 0.544703i
\(70\) 0.615543 0.0735715
\(71\) −1.66823 + 1.07210i −0.197982 + 0.127235i −0.635876 − 0.771791i \(-0.719361\pi\)
0.437894 + 0.899027i \(0.355725\pi\)
\(72\) 0.0781472 − 0.543526i 0.00920973 − 0.0640551i
\(73\) 2.23396 − 4.89170i 0.261466 − 0.572530i −0.732681 − 0.680573i \(-0.761731\pi\)
0.994146 + 0.108043i \(0.0344584\pi\)
\(74\) −1.20701 + 0.354411i −0.140312 + 0.0411994i
\(75\) −0.415415 − 0.909632i −0.0479680 − 0.105035i
\(76\) −3.78145 − 4.36402i −0.433762 − 0.500588i
\(77\) 1.76049 + 12.2445i 0.200626 + 1.39539i
\(78\) 0.134133 + 0.0393850i 0.0151876 + 0.00445948i
\(79\) −1.79851 + 2.07559i −0.202349 + 0.233523i −0.847850 − 0.530237i \(-0.822103\pi\)
0.645501 + 0.763759i \(0.276649\pi\)
\(80\) 3.26928 + 2.10104i 0.365517 + 0.234904i
\(81\) 0.841254 + 0.540641i 0.0934726 + 0.0600712i
\(82\) −0.576580 + 0.665409i −0.0636726 + 0.0734822i
\(83\) 2.05109 + 0.602256i 0.225137 + 0.0661062i 0.392355 − 0.919814i \(-0.371660\pi\)
−0.167218 + 0.985920i \(0.553479\pi\)
\(84\) −1.25810 − 8.75025i −0.137270 − 0.954731i
\(85\) 3.54199 + 4.08767i 0.384182 + 0.443370i
\(86\) 0.281889 + 0.617251i 0.0303969 + 0.0665599i
\(87\) −4.66417 + 1.36953i −0.500052 + 0.146829i
\(88\) 0.632330 − 1.38461i 0.0674066 − 0.147600i
\(89\) 0.889889 − 6.18931i 0.0943280 − 0.656066i −0.886721 − 0.462305i \(-0.847022\pi\)
0.981049 − 0.193760i \(-0.0620685\pi\)
\(90\) 0.116038 − 0.0745732i 0.0122315 − 0.00786070i
\(91\) 4.52278 0.474116
\(92\) 0.496645 + 9.48743i 0.0517788 + 0.989133i
\(93\) 10.5262 1.09151
\(94\) 1.15473 − 0.742102i 0.119102 − 0.0765420i
\(95\) 0.414840 − 2.88527i 0.0425617 − 0.296023i
\(96\) −0.678901 + 1.48659i −0.0692901 + 0.151724i
\(97\) −4.42369 + 1.29891i −0.449158 + 0.131885i −0.498485 − 0.866898i \(-0.666110\pi\)
0.0493278 + 0.998783i \(0.484292\pi\)
\(98\) 0.740004 + 1.62038i 0.0747517 + 0.163683i
\(99\) 1.81530 + 2.09496i 0.182444 + 0.210552i
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 345.2.m.d.196.3 ✓ 50
23.2 even 11 inner 345.2.m.d.301.3 yes 50
23.5 odd 22 7935.2.a.bt.1.13 25
23.18 even 11 7935.2.a.bu.1.13 25
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
345.2.m.d.196.3 ✓ 50 1.1 even 1 trivial
345.2.m.d.301.3 yes 50 23.2 even 11 inner
7935.2.a.bt.1.13 25 23.5 odd 22
7935.2.a.bu.1.13 25 23.18 even 11