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Properties

Label 322.2.k.b.83.4
Level $322$
Weight $2$
Character 322.83
Analytic conductor $2.571$
Analytic rank $0$
Dimension $80$
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [322,2,Mod(83,322)] 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("322.83"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(322, base_ring=CyclotomicField(22)) chi = DirichletCharacter(H, H._module([11, 21])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 322 = 2 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 322.k (of order \(22\), degree \(10\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [80,8] 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: \(2.57118294509\)
Analytic rank: \(0\)
Dimension: \(80\)
Relative dimension: \(8\) over \(\Q(\zeta_{22})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label 83.4
Character \(\chi\) \(=\) 322.83
Dual form 322.2.k.b.97.4

$q$-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+(0.959493 - 0.281733i) q^{2} +(-0.668109 + 0.578920i) q^{3} +(0.841254 - 0.540641i) q^{4} +(0.308243 + 2.14388i) q^{5} +(-0.477945 + 0.743697i) q^{6} +(-1.85023 + 1.89121i) q^{7} +(0.654861 - 0.755750i) q^{8} +(-0.315723 + 2.19590i) q^{9} +(0.899756 + 1.97019i) q^{10} +(-0.279879 + 0.953181i) q^{11} +(-0.249061 + 0.848225i) q^{12} +(2.08290 - 0.951227i) q^{13} +(-1.24246 + 2.33587i) q^{14} +(-1.44707 - 1.25389i) q^{15} +(0.415415 - 0.909632i) q^{16} +(0.147156 + 0.0945714i) q^{17} +(0.315723 + 2.19590i) q^{18} +(-0.158784 + 0.102044i) q^{19} +(1.41838 + 1.63689i) q^{20} +(0.141295 - 2.33466i) q^{21} +0.993421i q^{22} +(4.79388 + 0.136919i) q^{23} +0.884035i q^{24} +(0.296274 - 0.0869940i) q^{25} +(1.73053 - 1.49952i) q^{26} +(-2.49415 - 3.88097i) q^{27} +(-0.534044 + 2.59129i) q^{28} +(1.17704 + 0.756436i) q^{29} +(-1.74172 - 0.795416i) q^{30} +(-4.70460 - 4.07656i) q^{31} +(0.142315 - 0.989821i) q^{32} +(-0.364825 - 0.798856i) q^{33} +(0.167839 + 0.0492820i) q^{34} +(-4.62483 - 3.38370i) q^{35} +(0.921591 + 2.01800i) q^{36} +(7.69386 + 1.10621i) q^{37} +(-0.123603 + 0.142645i) q^{38} +(-0.840918 + 1.84135i) q^{39} +(1.82209 + 1.17099i) q^{40} +(5.21177 - 0.749340i) q^{41} +(-0.522180 - 2.27990i) q^{42} +(-4.89515 + 4.24168i) q^{43} +(0.279879 + 0.953181i) q^{44} -4.80506 q^{45} +(4.63827 - 1.21922i) q^{46} -1.93506i q^{47} +(0.249061 + 0.848225i) q^{48} +(-0.153333 - 6.99832i) q^{49} +(0.259764 - 0.166940i) q^{50} +(-0.153065 + 0.0220075i) q^{51} +(1.23797 - 1.92632i) q^{52} +(-7.78039 - 3.55319i) q^{53} +(-3.48651 - 3.02108i) q^{54} +(-2.12977 - 0.306215i) q^{55} +(0.217640 + 2.63678i) q^{56} +(0.0470095 - 0.160100i) q^{57} +(1.34247 + 0.394185i) q^{58} +(7.09551 - 3.24041i) q^{59} +(-1.89526 - 0.272497i) q^{60} +(0.153531 - 0.177185i) q^{61} +(-5.66253 - 2.58599i) q^{62} +(-3.56875 - 4.66001i) q^{63} +(-0.142315 - 0.989821i) q^{64} +(2.68135 + 4.17226i) q^{65} +(-0.575111 - 0.663713i) q^{66} +(-1.39179 - 4.74001i) q^{67} +0.174925 q^{68} +(-3.28210 + 2.68379i) q^{69} +(-5.39079 - 1.94367i) q^{70} +(-4.30653 + 1.26451i) q^{71} +(1.45280 + 1.67662i) q^{72} +(-3.31556 - 5.15912i) q^{73} +(7.69386 - 1.10621i) q^{74} +(-0.147581 + 0.229640i) q^{75} +(-0.0784082 + 0.171690i) q^{76} +(-1.28482 - 2.29291i) q^{77} +(-0.288086 + 2.00368i) q^{78} +(4.80556 - 2.19463i) q^{79} +(2.07819 + 0.610211i) q^{80} +(-2.47272 - 0.726057i) q^{81} +(4.78955 - 2.18731i) q^{82} +(1.69002 - 11.7543i) q^{83} +(-1.14335 - 2.04043i) q^{84} +(-0.157390 + 0.344635i) q^{85} +(-3.50185 + 5.44898i) q^{86} +(-1.22430 + 0.176028i) q^{87} +(0.537084 + 0.835719i) q^{88} +(9.01116 + 10.3994i) q^{89} +(-4.61042 + 1.35374i) q^{90} +(-2.05486 + 5.69918i) q^{91} +(4.10689 - 2.47658i) q^{92} +5.50319 q^{93} +(-0.545168 - 1.85667i) q^{94} +(-0.267714 - 0.308958i) q^{95} +(0.477945 + 0.743697i) q^{96} +(1.74345 + 12.1259i) q^{97} +(-2.11878 - 6.67164i) q^{98} +(-2.00473 - 0.915528i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 80 q + 8 q^{2} - 8 q^{4} + 11 q^{7} + 8 q^{8} + 14 q^{9} - 11 q^{14} - 8 q^{16} - 14 q^{18} - 33 q^{21} + 18 q^{23} + 6 q^{25} - 11 q^{28} + 8 q^{29} + 22 q^{30} + 8 q^{32} - 33 q^{35} - 8 q^{36} - 22 q^{37}+ \cdots + 132 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/322\mathbb{Z}\right)^\times\).

\(n\) \(185\) \(281\)
\(\chi(n)\) \(-1\) \(e\left(\frac{21}{22}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.959493 − 0.281733i 0.678464 − 0.199215i
\(3\) −0.668109 + 0.578920i −0.385733 + 0.334239i −0.826044 − 0.563606i \(-0.809414\pi\)
0.440311 + 0.897845i \(0.354868\pi\)
\(4\) 0.841254 − 0.540641i 0.420627 − 0.270320i
\(5\) 0.308243 + 2.14388i 0.137850 + 0.958770i 0.934914 + 0.354875i \(0.115477\pi\)
−0.797063 + 0.603896i \(0.793614\pi\)
\(6\) −0.477945 + 0.743697i −0.195120 + 0.303613i
\(7\) −1.85023 + 1.89121i −0.699319 + 0.714809i
\(8\) 0.654861 − 0.755750i 0.231528 − 0.267198i
\(9\) −0.315723 + 2.19590i −0.105241 + 0.731967i
\(10\) 0.899756 + 1.97019i 0.284528 + 0.623029i
\(11\) −0.279879 + 0.953181i −0.0843867 + 0.287395i −0.990865 − 0.134858i \(-0.956942\pi\)
0.906478 + 0.422252i \(0.138760\pi\)
\(12\) −0.249061 + 0.848225i −0.0718978 + 0.244861i
\(13\) 2.08290 − 0.951227i 0.577692 − 0.263823i −0.105069 − 0.994465i \(-0.533506\pi\)
0.682761 + 0.730642i \(0.260779\pi\)
\(14\) −1.24246 + 2.33587i −0.332062 + 0.624287i
\(15\) −1.44707 − 1.25389i −0.373632 − 0.323754i
\(16\) 0.415415 − 0.909632i 0.103854 − 0.227408i
\(17\) 0.147156 + 0.0945714i 0.0356906 + 0.0229369i 0.558364 − 0.829596i \(-0.311429\pi\)
−0.522674 + 0.852533i \(0.675065\pi\)
\(18\) 0.315723 + 2.19590i 0.0744166 + 0.517579i
\(19\) −0.158784 + 0.102044i −0.0364275 + 0.0234105i −0.558728 − 0.829351i \(-0.688710\pi\)
0.522300 + 0.852762i \(0.325074\pi\)
\(20\) 1.41838 + 1.63689i 0.317159 + 0.366021i
\(21\) 0.141295 − 2.33466i 0.0308330 − 0.509465i
\(22\) 0.993421i 0.211798i
\(23\) 4.79388 + 0.136919i 0.999592 + 0.0285495i
\(24\) 0.884035i 0.180453i
\(25\) 0.296274 − 0.0869940i 0.0592549 − 0.0173988i
\(26\) 1.73053 − 1.49952i 0.339385 − 0.294079i
\(27\) −2.49415 − 3.88097i −0.479999 − 0.746893i
\(28\) −0.534044 + 2.59129i −0.100925 + 0.489708i
\(29\) 1.17704 + 0.756436i 0.218570 + 0.140467i 0.645345 − 0.763892i \(-0.276714\pi\)
−0.426774 + 0.904358i \(0.640350\pi\)
\(30\) −1.74172 − 0.795416i −0.317993 − 0.145222i
\(31\) −4.70460 − 4.07656i −0.844971 − 0.732172i 0.120491 − 0.992714i \(-0.461553\pi\)
−0.965462 + 0.260543i \(0.916099\pi\)
\(32\) 0.142315 − 0.989821i 0.0251579 − 0.174977i
\(33\) −0.364825 − 0.798856i −0.0635079 − 0.139063i
\(34\) 0.167839 + 0.0492820i 0.0287841 + 0.00845179i
\(35\) −4.62483 − 3.38370i −0.781740 − 0.571950i
\(36\) 0.921591 + 2.01800i 0.153599 + 0.336334i
\(37\) 7.69386 + 1.10621i 1.26486 + 0.181860i 0.741909 − 0.670501i \(-0.233921\pi\)
0.522954 + 0.852361i \(0.324830\pi\)
\(38\) −0.123603 + 0.142645i −0.0200510 + 0.0231401i
\(39\) −0.840918 + 1.84135i −0.134655 + 0.294852i
\(40\) 1.82209 + 1.17099i 0.288098 + 0.185149i
\(41\) 5.21177 − 0.749340i 0.813942 − 0.117027i 0.277242 − 0.960800i \(-0.410580\pi\)
0.536700 + 0.843773i \(0.319671\pi\)
\(42\) −0.522180 − 2.27990i −0.0805741 − 0.351796i
\(43\) −4.89515 + 4.24168i −0.746504 + 0.646849i −0.942672 − 0.333721i \(-0.891696\pi\)
0.196168 + 0.980570i \(0.437150\pi\)
\(44\) 0.279879 + 0.953181i 0.0421934 + 0.143697i
\(45\) −4.80506 −0.716296
\(46\) 4.63827 − 1.21922i 0.683875 − 0.179764i
\(47\) − 1.93506i − 0.282257i −0.989991 − 0.141128i \(-0.954927\pi\)
0.989991 − 0.141128i \(-0.0450730\pi\)
\(48\) 0.249061 + 0.848225i 0.0359489 + 0.122431i
\(49\) −0.153333 − 6.99832i −0.0219047 − 0.999760i
\(50\) 0.259764 − 0.166940i 0.0367362 − 0.0236089i
\(51\) −0.153065 + 0.0220075i −0.0214334 + 0.00308166i
\(52\) 1.23797 − 1.92632i 0.171676 − 0.267133i
\(53\) −7.78039 − 3.55319i −1.06872 − 0.488068i −0.198179 − 0.980166i \(-0.563503\pi\)
−0.870540 + 0.492098i \(0.836230\pi\)
\(54\) −3.48651 − 3.02108i −0.474454 − 0.411117i
\(55\) −2.12977 − 0.306215i −0.287178 − 0.0412900i
\(56\) 0.217640 + 2.63678i 0.0290833 + 0.352355i
\(57\) 0.0470095 − 0.160100i 0.00622656 − 0.0212057i
\(58\) 1.34247 + 0.394185i 0.176275 + 0.0517591i
\(59\) 7.09551 − 3.24041i 0.923756 − 0.421865i 0.104000 − 0.994577i \(-0.466836\pi\)
0.819757 + 0.572712i \(0.194109\pi\)
\(60\) −1.89526 − 0.272497i −0.244677 − 0.0351793i
\(61\) 0.153531 − 0.177185i 0.0196577 − 0.0226862i −0.745836 − 0.666130i \(-0.767950\pi\)
0.765493 + 0.643444i \(0.222495\pi\)
\(62\) −5.66253 − 2.58599i −0.719142 − 0.328421i
\(63\) −3.56875 − 4.66001i −0.449620 − 0.587106i
\(64\) −0.142315 − 0.989821i −0.0177894 − 0.123728i
\(65\) 2.68135 + 4.17226i 0.332581 + 0.517506i
\(66\) −0.575111 − 0.663713i −0.0707912 − 0.0816975i
\(67\) −1.39179 − 4.74001i −0.170034 − 0.579084i −0.999781 − 0.0209433i \(-0.993333\pi\)
0.829746 − 0.558141i \(-0.188485\pi\)
\(68\) 0.174925 0.0212127
\(69\) −3.28210 + 2.68379i −0.395118 + 0.323091i
\(70\) −5.39079 − 1.94367i −0.644323 − 0.232313i
\(71\) −4.30653 + 1.26451i −0.511091 + 0.150070i −0.527102 − 0.849802i \(-0.676722\pi\)
0.0160109 + 0.999872i \(0.494903\pi\)
\(72\) 1.45280 + 1.67662i 0.171214 + 0.197591i
\(73\) −3.31556 − 5.15912i −0.388057 − 0.603829i 0.591182 − 0.806539i \(-0.298662\pi\)
−0.979239 + 0.202709i \(0.935025\pi\)
\(74\) 7.69386 − 1.10621i 0.894393 − 0.128594i
\(75\) −0.147581 + 0.229640i −0.0170412 + 0.0265166i
\(76\) −0.0784082 + 0.171690i −0.00899404 + 0.0196942i
\(77\) −1.28482 − 2.29291i −0.146419 − 0.261301i
\(78\) −0.288086 + 2.00368i −0.0326193 + 0.226872i
\(79\) 4.80556 − 2.19463i 0.540668 − 0.246915i −0.126312 − 0.991991i \(-0.540314\pi\)
0.666980 + 0.745076i \(0.267587\pi\)
\(80\) 2.07819 + 0.610211i 0.232348 + 0.0682236i
\(81\) −2.47272 − 0.726057i −0.274747 − 0.0806730i
\(82\) 4.78955 − 2.18731i 0.528917 − 0.241548i
\(83\) 1.69002 − 11.7543i 0.185504 − 1.29021i −0.657973 − 0.753041i \(-0.728586\pi\)
0.843477 − 0.537165i \(-0.180505\pi\)
\(84\) −1.14335 − 2.04043i −0.124750 − 0.222630i
\(85\) −0.157390 + 0.344635i −0.0170713 + 0.0373809i
\(86\) −3.50185 + 5.44898i −0.377614 + 0.587579i
\(87\) −1.22430 + 0.176028i −0.131259 + 0.0188722i
\(88\) 0.537084 + 0.835719i 0.0572533 + 0.0890879i
\(89\) 9.01116 + 10.3994i 0.955181 + 1.10234i 0.994669 + 0.103117i \(0.0328817\pi\)
−0.0394884 + 0.999220i \(0.512573\pi\)
\(90\) −4.61042 + 1.35374i −0.485981 + 0.142697i
\(91\) −2.05486 + 5.69918i −0.215408 + 0.597436i
\(92\) 4.10689 − 2.47658i 0.428173 − 0.258202i
\(93\) 5.50319 0.570654
\(94\) −0.545168 − 1.85667i −0.0562298 − 0.191501i
\(95\) −0.267714 − 0.308958i −0.0274669 − 0.0316985i
\(96\) 0.477945 + 0.743697i 0.0487801 + 0.0759033i
\(97\) 1.74345 + 12.1259i 0.177020 + 1.23120i 0.863611 + 0.504158i \(0.168197\pi\)
−0.686591 + 0.727044i \(0.740894\pi\)
\(98\) −2.11878 − 6.67164i −0.214029 − 0.673937i
\(99\) −2.00473 − 0.915528i −0.201483 − 0.0920140i
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 322.2.k.b.83.4 ✓ 80
7.6 odd 2 inner 322.2.k.b.83.5 yes 80
23.5 odd 22 inner 322.2.k.b.97.5 yes 80
161.97 even 22 inner 322.2.k.b.97.4 yes 80
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
322.2.k.b.83.4 ✓ 80 1.1 even 1 trivial
322.2.k.b.83.5 yes 80 7.6 odd 2 inner
322.2.k.b.97.4 yes 80 161.97 even 22 inner
322.2.k.b.97.5 yes 80 23.5 odd 22 inner