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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [465,2,Mod(119,465)] 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("465.119"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(465, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 3, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 465 = 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 465.t (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [104,0] 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: \(3.71304369399\)
Analytic rank: \(0\)
Dimension: \(104\)
Relative dimension: \(52\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 119.40
Character \(\chi\) \(=\) 465.119
Dual form 465.2.t.d.254.40

$q$-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+1.36470 q^{2} +(-0.0687450 - 1.73069i) q^{3} -0.137607 q^{4} +(-1.77591 - 1.35873i) q^{5} +(-0.0938159 - 2.36186i) q^{6} +(-1.46427 - 0.845397i) q^{7} -2.91718 q^{8} +(-2.99055 + 0.237952i) q^{9} +(-2.42357 - 1.85426i) q^{10} +(2.40661 + 4.16838i) q^{11} +(0.00945982 + 0.238155i) q^{12} +(-0.307395 - 0.532425i) q^{13} +(-1.99828 - 1.15371i) q^{14} +(-2.22946 + 3.16694i) q^{15} -3.70585 q^{16} +(-3.77148 - 2.17746i) q^{17} +(-4.08119 + 0.324732i) q^{18} +(2.90894 - 5.03844i) q^{19} +(0.244378 + 0.186972i) q^{20} +(-1.36246 + 2.59231i) q^{21} +(3.28429 + 5.68857i) q^{22} -4.43060i q^{23} +(0.200542 + 5.04873i) q^{24} +(1.30768 + 4.82597i) q^{25} +(-0.419501 - 0.726597i) q^{26} +(0.617405 + 5.15934i) q^{27} +(0.201495 + 0.116333i) q^{28} -8.46868 q^{29} +(-3.04253 + 4.32191i) q^{30} +(-0.0385902 - 5.56763i) q^{31} +0.777011 q^{32} +(7.04871 - 4.45165i) q^{33} +(-5.14692 - 2.97157i) q^{34} +(1.45174 + 3.49090i) q^{35} +(0.411522 - 0.0327440i) q^{36} +(2.81390 - 4.87382i) q^{37} +(3.96982 - 6.87593i) q^{38} +(-0.900328 + 0.568607i) q^{39} +(5.18064 + 3.96367i) q^{40} +(8.07624 - 4.66282i) q^{41} +(-1.85934 + 3.53771i) q^{42} +(2.53588 - 4.39228i) q^{43} +(-0.331168 - 0.573600i) q^{44} +(5.63425 + 3.64078i) q^{45} -6.04641i q^{46} +5.15494 q^{47} +(0.254759 + 6.41366i) q^{48} +(-2.07061 - 3.58640i) q^{49} +(1.78459 + 6.58597i) q^{50} +(-3.50924 + 6.67693i) q^{51} +(0.0422999 + 0.0732656i) q^{52} +(-5.16045 + 2.97938i) q^{53} +(0.842570 + 7.04093i) q^{54} +(1.38980 - 10.6726i) q^{55} +(4.27155 + 2.46618i) q^{56} +(-8.91993 - 4.68810i) q^{57} -11.5572 q^{58} +(-4.93527 - 2.84938i) q^{59} +(0.306790 - 0.435795i) q^{60} +9.69632i q^{61} +(-0.0526639 - 7.59812i) q^{62} +(4.58014 + 2.17978i) q^{63} +8.47208 q^{64} +(-0.177518 + 1.36320i) q^{65} +(9.61934 - 6.07514i) q^{66} +(-4.16359 + 2.40385i) q^{67} +(0.518984 + 0.299635i) q^{68} +(-7.66797 + 0.304581i) q^{69} +(1.98118 + 4.76402i) q^{70} +(-6.23532 + 3.59996i) q^{71} +(8.72397 - 0.694149i) q^{72} +(0.805479 + 1.39513i) q^{73} +(3.84011 - 6.65127i) q^{74} +(8.26234 - 2.59495i) q^{75} +(-0.400292 + 0.693327i) q^{76} -8.13818i q^{77} +(-1.22867 + 0.775974i) q^{78} +(0.206366 + 0.119146i) q^{79} +(6.58124 + 5.03526i) q^{80} +(8.88676 - 1.42321i) q^{81} +(11.0216 - 6.36332i) q^{82} +(-1.80720 + 1.04339i) q^{83} +(0.187484 - 0.356721i) q^{84} +(3.73920 + 8.99141i) q^{85} +(3.46071 - 5.99412i) q^{86} +(0.582179 + 14.6566i) q^{87} +(-7.02053 - 12.1599i) q^{88} +14.1440 q^{89} +(7.68903 + 4.96855i) q^{90} +1.03949i q^{91} +0.609683i q^{92} +(-9.63317 + 0.449534i) q^{93} +7.03493 q^{94} +(-12.0119 + 4.99531i) q^{95} +(-0.0534156 - 1.34476i) q^{96} +10.2539i q^{97} +(-2.82575 - 4.89434i) q^{98} +(-8.18897 - 11.8931i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 104 q + 56 q^{4} - 12 q^{6} - 2 q^{9} - 20 q^{10} - 24 q^{16} + 12 q^{19} - 6 q^{21} - 48 q^{24} + 8 q^{25} - 40 q^{31} + 108 q^{34} + 36 q^{36} + 52 q^{39} - 76 q^{40} - 4 q^{45} + 80 q^{49} - 20 q^{51}+ \cdots + 66 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/465\mathbb{Z}\right)^\times\).

\(n\) \(187\) \(311\) \(406\)
\(\chi(n)\) \(-1\) \(-1\) \(e\left(\frac{1}{6}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.36470 0.964985 0.482493 − 0.875900i \(-0.339732\pi\)
0.482493 + 0.875900i \(0.339732\pi\)
\(3\) −0.0687450 − 1.73069i −0.0396899 − 0.999212i
\(4\) −0.137607 −0.0688037
\(5\) −1.77591 − 1.35873i −0.794209 − 0.607644i
\(6\) −0.0938159 − 2.36186i −0.0383002 − 0.964225i
\(7\) −1.46427 − 0.845397i −0.553443 − 0.319530i 0.197067 − 0.980390i \(-0.436858\pi\)
−0.750509 + 0.660860i \(0.770192\pi\)
\(8\) −2.91718 −1.03138
\(9\) −2.99055 + 0.237952i −0.996849 + 0.0793173i
\(10\) −2.42357 − 1.85426i −0.766400 − 0.586368i
\(11\) 2.40661 + 4.16838i 0.725622 + 1.25681i 0.958718 + 0.284360i \(0.0917810\pi\)
−0.233096 + 0.972454i \(0.574886\pi\)
\(12\) 0.00945982 + 0.238155i 0.00273082 + 0.0687495i
\(13\) −0.307395 − 0.532425i −0.0852562 − 0.147668i 0.820244 − 0.572014i \(-0.193838\pi\)
−0.905500 + 0.424346i \(0.860504\pi\)
\(14\) −1.99828 − 1.15371i −0.534064 − 0.308342i
\(15\) −2.22946 + 3.16694i −0.575643 + 0.817701i
\(16\) −3.70585 −0.926462
\(17\) −3.77148 − 2.17746i −0.914718 − 0.528112i −0.0327716 − 0.999463i \(-0.510433\pi\)
−0.881946 + 0.471350i \(0.843767\pi\)
\(18\) −4.08119 + 0.324732i −0.961945 + 0.0765400i
\(19\) 2.90894 − 5.03844i 0.667357 − 1.15590i −0.311283 − 0.950317i \(-0.600759\pi\)
0.978640 − 0.205579i \(-0.0659079\pi\)
\(20\) 0.244378 + 0.186972i 0.0546446 + 0.0418082i
\(21\) −1.36246 + 2.59231i −0.297312 + 0.565689i
\(22\) 3.28429 + 5.68857i 0.700214 + 1.21281i
\(23\) − 4.43060i − 0.923843i −0.886921 − 0.461922i \(-0.847160\pi\)
0.886921 − 0.461922i \(-0.152840\pi\)
\(24\) 0.200542 + 5.04873i 0.0409354 + 1.03057i
\(25\) 1.30768 + 4.82597i 0.261537 + 0.965194i
\(26\) −0.419501 − 0.726597i −0.0822709 − 0.142497i
\(27\) 0.617405 + 5.15934i 0.118820 + 0.992916i
\(28\) 0.201495 + 0.116333i 0.0380789 + 0.0219849i
\(29\) −8.46868 −1.57260 −0.786298 − 0.617848i \(-0.788005\pi\)
−0.786298 + 0.617848i \(0.788005\pi\)
\(30\) −3.04253 + 4.32191i −0.555487 + 0.789069i
\(31\) −0.0385902 − 5.56763i −0.00693101 − 0.999976i
\(32\) 0.777011 0.137357
\(33\) 7.04871 − 4.45165i 1.22702 − 0.774933i
\(34\) −5.14692 − 2.97157i −0.882689 − 0.509621i
\(35\) 1.45174 + 3.49090i 0.245388 + 0.590070i
\(36\) 0.411522 − 0.0327440i 0.0685870 − 0.00545733i
\(37\) 2.81390 − 4.87382i 0.462602 − 0.801250i −0.536488 − 0.843908i \(-0.680249\pi\)
0.999090 + 0.0426578i \(0.0135825\pi\)
\(38\) 3.96982 − 6.87593i 0.643990 − 1.11542i
\(39\) −0.900328 + 0.568607i −0.144168 + 0.0910499i
\(40\) 5.18064 + 3.96367i 0.819131 + 0.626712i
\(41\) 8.07624 − 4.66282i 1.26130 − 0.728210i 0.287971 − 0.957639i \(-0.407019\pi\)
0.973325 + 0.229429i \(0.0736860\pi\)
\(42\) −1.85934 + 3.53771i −0.286902 + 0.545881i
\(43\) 2.53588 − 4.39228i 0.386718 − 0.669816i −0.605288 − 0.796007i \(-0.706942\pi\)
0.992006 + 0.126191i \(0.0402752\pi\)
\(44\) −0.331168 − 0.573600i −0.0499255 − 0.0864735i
\(45\) 5.63425 + 3.64078i 0.839904 + 0.542735i
\(46\) − 6.04641i − 0.891495i
\(47\) 5.15494 0.751926 0.375963 − 0.926635i \(-0.377312\pi\)
0.375963 + 0.926635i \(0.377312\pi\)
\(48\) 0.254759 + 6.41366i 0.0367712 + 0.925732i
\(49\) −2.07061 − 3.58640i −0.295801 − 0.512342i
\(50\) 1.78459 + 6.58597i 0.252379 + 0.931397i
\(51\) −3.50924 + 6.67693i −0.491391 + 0.934958i
\(52\) 0.0422999 + 0.0732656i 0.00586594 + 0.0101601i
\(53\) −5.16045 + 2.97938i −0.708842 + 0.409250i −0.810632 − 0.585556i \(-0.800876\pi\)
0.101790 + 0.994806i \(0.467543\pi\)
\(54\) 0.842570 + 7.04093i 0.114659 + 0.958149i
\(55\) 1.38980 − 10.6726i 0.187400 − 1.43909i
\(56\) 4.27155 + 2.46618i 0.570809 + 0.329557i
\(57\) −8.91993 − 4.68810i −1.18147 − 0.620954i
\(58\) −11.5572 −1.51753
\(59\) −4.93527 − 2.84938i −0.642518 − 0.370958i 0.143066 − 0.989713i \(-0.454304\pi\)
−0.785584 + 0.618755i \(0.787637\pi\)
\(60\) 0.306790 − 0.435795i 0.0396064 − 0.0562609i
\(61\) 9.69632i 1.24149i 0.784014 + 0.620743i \(0.213169\pi\)
−0.784014 + 0.620743i \(0.786831\pi\)
\(62\) −0.0526639 − 7.59812i −0.00668832 − 0.964962i
\(63\) 4.58014 + 2.17978i 0.577043 + 0.274626i
\(64\) 8.47208 1.05901
\(65\) −0.177518 + 1.36320i −0.0220184 + 0.169085i
\(66\) 9.61934 − 6.07514i 1.18406 − 0.747798i
\(67\) −4.16359 + 2.40385i −0.508664 + 0.293677i −0.732284 − 0.680999i \(-0.761546\pi\)
0.223620 + 0.974676i \(0.428212\pi\)
\(68\) 0.518984 + 0.299635i 0.0629360 + 0.0363361i
\(69\) −7.66797 + 0.304581i −0.923115 + 0.0366673i
\(70\) 1.98118 + 4.76402i 0.236796 + 0.569409i
\(71\) −6.23532 + 3.59996i −0.739996 + 0.427237i −0.822068 − 0.569389i \(-0.807180\pi\)
0.0820718 + 0.996626i \(0.473846\pi\)
\(72\) 8.72397 − 0.694149i 1.02813 − 0.0818063i
\(73\) 0.805479 + 1.39513i 0.0942742 + 0.163288i 0.909305 − 0.416129i \(-0.136614\pi\)
−0.815031 + 0.579417i \(0.803280\pi\)
\(74\) 3.84011 − 6.65127i 0.446404 − 0.773195i
\(75\) 8.26234 − 2.59495i 0.954053 − 0.299639i
\(76\) −0.400292 + 0.693327i −0.0459167 + 0.0795300i
\(77\) − 8.13818i − 0.927432i
\(78\) −1.22867 + 0.775974i −0.139120 + 0.0878618i
\(79\) 0.206366 + 0.119146i 0.0232180 + 0.0134049i 0.511564 − 0.859245i \(-0.329066\pi\)
−0.488346 + 0.872650i \(0.662400\pi\)
\(80\) 6.58124 + 5.03526i 0.735805 + 0.562960i
\(81\) 8.88676 − 1.42321i 0.987418 − 0.158135i
\(82\) 11.0216 − 6.36332i 1.21713 − 0.702712i
\(83\) −1.80720 + 1.04339i −0.198366 + 0.114527i −0.595893 − 0.803064i \(-0.703202\pi\)
0.397527 + 0.917590i \(0.369868\pi\)
\(84\) 0.187484 − 0.356721i 0.0204562 − 0.0389215i
\(85\) 3.73920 + 8.99141i 0.405573 + 0.975255i
\(86\) 3.46071 − 5.99412i 0.373177 − 0.646362i
\(87\) 0.582179 + 14.6566i 0.0624162 + 1.57136i
\(88\) −7.02053 − 12.1599i −0.748391 − 1.29625i
\(89\) 14.1440 1.49926 0.749631 − 0.661856i \(-0.230231\pi\)
0.749631 + 0.661856i \(0.230231\pi\)
\(90\) 7.68903 + 4.96855i 0.810495 + 0.523732i
\(91\) 1.03949i 0.108968i
\(92\) 0.609683i 0.0635639i
\(93\) −9.63317 + 0.449534i −0.998913 + 0.0466145i
\(94\) 7.03493 0.725597
\(95\) −12.0119 + 4.99531i −1.23240 + 0.512508i
\(96\) −0.0534156 − 1.34476i −0.00545170 − 0.137249i
\(97\) 10.2539i 1.04112i 0.853824 + 0.520561i \(0.174277\pi\)
−0.853824 + 0.520561i \(0.825723\pi\)
\(98\) −2.82575 − 4.89434i −0.285443 − 0.494403i
\(99\) −8.18897 − 11.8931i −0.823022 − 1.19530i
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 465.2.t.d.119.40 yes 104
3.2 odd 2 inner 465.2.t.d.119.14 yes 104
5.4 even 2 inner 465.2.t.d.119.13 ✓ 104
15.14 odd 2 inner 465.2.t.d.119.39 yes 104
31.6 odd 6 inner 465.2.t.d.254.39 yes 104
93.68 even 6 inner 465.2.t.d.254.13 yes 104
155.99 odd 6 inner 465.2.t.d.254.14 yes 104
465.254 even 6 inner 465.2.t.d.254.40 yes 104
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
465.2.t.d.119.13 ✓ 104 5.4 even 2 inner
465.2.t.d.119.14 yes 104 3.2 odd 2 inner
465.2.t.d.119.39 yes 104 15.14 odd 2 inner
465.2.t.d.119.40 yes 104 1.1 even 1 trivial
465.2.t.d.254.13 yes 104 93.68 even 6 inner
465.2.t.d.254.14 yes 104 155.99 odd 6 inner
465.2.t.d.254.39 yes 104 31.6 odd 6 inner
465.2.t.d.254.40 yes 104 465.254 even 6 inner