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Properties

Label 322.2.k.b.97.5
Level $322$
Weight $2$
Character 322.97
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 97.5
Character \(\chi\) \(=\) 322.97
Dual form 322.2.k.b.83.5

$q$-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} +(0.217640 + 2.63678i) 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} +(-0.534044 + 2.59129i) 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} +(-1.38108 + 1.88765i) 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} +(-1.24246 + 2.33587i) 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} +(-5.72002 - 0.346178i) 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} +(-1.85695 + 1.42210i) 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} +(-6.90527 + 1.14774i) 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} +(-1.85023 + 1.89121i) 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} +(5.72141 - 1.31041i) 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} +(2.45242 - 0.945431i) 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} +(-2.18238 + 0.841328i) 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} +(-6.94891 - 0.844192i) 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 \(\left(\mathbb{Z}/322\mathbb{Z}\right)^\times\).

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

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 \(-0.190586\pi\)
−0.440311 + 0.897845i \(0.645132\pi\)
\(4\) 0.841254 + 0.540641i 0.420627 + 0.270320i
\(5\) −0.308243 + 2.14388i −0.137850 + 0.958770i 0.797063 + 0.603896i \(0.206386\pi\)
−0.934914 + 0.354875i \(0.884523\pi\)
\(6\) 0.477945 + 0.743697i 0.195120 + 0.303613i
\(7\) 0.217640 + 2.63678i 0.0822600 + 0.996611i
\(8\) 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 0.906478 − 0.422252i \(-0.138760\pi\)
−0.990865 + 0.134858i \(0.956942\pi\)
\(12\) 0.249061 + 0.848225i 0.0718978 + 0.244861i
\(13\) −2.08290 − 0.951227i −0.577692 − 0.263823i 0.105069 − 0.994465i \(-0.466494\pi\)
−0.682761 + 0.730642i \(0.739221\pi\)
\(14\) −0.534044 + 2.59129i −0.142729 + 0.692552i
\(15\) −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 \(-0.688571\pi\)
0.522674 + 0.852533i \(0.324935\pi\)
\(18\) 0.315723 − 2.19590i 0.0744166 − 0.517579i
\(19\) 0.158784 + 0.102044i 0.0364275 + 0.0234105i 0.558728 − 0.829351i \(-0.311290\pi\)
−0.522300 + 0.852762i \(0.674926\pi\)
\(20\) −1.41838 + 1.63689i −0.317159 + 0.366021i
\(21\) −1.38108 + 1.88765i −0.301376 + 0.411920i
\(22\) − 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
\(28\) −1.24246 + 2.33587i −0.234803 + 0.441438i
\(29\) 1.17704 − 0.756436i 0.218570 − 0.140467i −0.426774 − 0.904358i \(-0.640350\pi\)
0.645345 + 0.763892i \(0.276714\pi\)
\(30\) −1.74172 + 0.795416i −0.317993 + 0.145222i
\(31\) 4.70460 − 4.07656i 0.844971 − 0.732172i −0.120491 − 0.992714i \(-0.538447\pi\)
0.965462 + 0.260543i \(0.0839014\pi\)
\(32\) 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
\(35\) −5.72002 − 0.346178i −0.966861 − 0.0585147i
\(36\) 0.921591 − 2.01800i 0.153599 − 0.336334i
\(37\) 7.69386 − 1.10621i 1.26486 − 0.181860i 0.522954 − 0.852361i \(-0.324830\pi\)
0.741909 + 0.670501i \(0.233921\pi\)
\(38\) 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 \(-0.589420\pi\)
−0.536700 + 0.843773i \(0.680329\pi\)
\(42\) −1.85695 + 1.42210i −0.286534 + 0.219434i
\(43\) −4.89515 − 4.24168i −0.746504 − 0.646849i 0.196168 − 0.980570i \(-0.437150\pi\)
−0.942672 + 0.333721i \(0.891696\pi\)
\(44\) 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
\(49\) −6.90527 + 1.14774i −0.986467 + 0.163962i
\(50\) 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 −0.870540 − 0.492098i \(-0.836230\pi\)
−0.198179 + 0.980166i \(0.563503\pi\)
\(54\) 3.48651 − 3.02108i 0.474454 − 0.411117i
\(55\) 2.12977 − 0.306215i 0.287178 − 0.0412900i
\(56\) −1.85023 + 1.89121i −0.247247 + 0.252723i
\(57\) 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 \(-0.533164\pi\)
−0.819757 + 0.572712i \(0.805891\pi\)
\(60\) −1.89526 + 0.272497i −0.244677 + 0.0351793i
\(61\) −0.153531 − 0.177185i −0.0196577 − 0.0226862i 0.745836 − 0.666130i \(-0.232050\pi\)
−0.765493 + 0.643444i \(0.777505\pi\)
\(62\) 5.66253 − 2.58599i 0.719142 − 0.328421i
\(63\) 5.72141 − 1.31041i 0.720829 − 0.165096i
\(64\) −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 0.829746 + 0.558141i \(0.188485\pi\)
−0.999781 + 0.0209433i \(0.993333\pi\)
\(68\) −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 0.0160109 − 0.999872i \(-0.494903\pi\)
−0.527102 + 0.849802i \(0.676722\pi\)
\(72\) 1.45280 − 1.67662i 0.171214 − 0.197591i
\(73\) 3.31556 − 5.15912i 0.388057 − 0.603829i −0.591182 − 0.806539i \(-0.701338\pi\)
0.979239 + 0.202709i \(0.0649747\pi\)
\(74\) 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
\(77\) 2.45242 − 0.945431i 0.279479 − 0.107742i
\(78\) −0.288086 − 2.00368i −0.0326193 − 0.226872i
\(79\) 4.80556 + 2.19463i 0.540668 + 0.246915i 0.666980 − 0.745076i \(-0.267587\pi\)
−0.126312 + 0.991991i \(0.540314\pi\)
\(80\) −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 −0.843477 − 0.537165i \(-0.819495\pi\)
0.657973 − 0.753041i \(-0.271414\pi\)
\(84\) −2.18238 + 0.841328i −0.238117 + 0.0917964i
\(85\) −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 0.0394884 + 0.999220i \(0.487427\pi\)
−0.994669 + 0.103117i \(0.967118\pi\)
\(90\) 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 0.686591 + 0.727044i \(0.259106\pi\)
−0.863611 + 0.504158i \(0.831803\pi\)
\(98\) −6.94891 − 0.844192i −0.701946 − 0.0852763i
\(99\) −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.97.5 yes 80
7.6 odd 2 inner 322.2.k.b.97.4 yes 80
23.14 odd 22 inner 322.2.k.b.83.4 ✓ 80
161.83 even 22 inner 322.2.k.b.83.5 yes 80
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
322.2.k.b.83.4 ✓ 80 23.14 odd 22 inner
322.2.k.b.83.5 yes 80 161.83 even 22 inner
322.2.k.b.97.4 yes 80 7.6 odd 2 inner
322.2.k.b.97.5 yes 80 1.1 even 1 trivial