# Properties

 Label 9680.2.a.ca.1.1 Level $9680$ Weight $2$ Character 9680.1 Self dual yes Analytic conductor $77.295$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9680,2,Mod(1,9680)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9680, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("9680.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$9680 = 2^{4} \cdot 5 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9680.a (trivial)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$77.2951891566$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.404.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 5x - 1$$ x^3 - x^2 - 5*x - 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 4840) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$2.86620$$ of defining polynomial Character $$\chi$$ $$=$$ 9680.1

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-2.34889 q^{3} -1.00000 q^{5} -2.86620 q^{7} +2.51730 q^{9} +O(q^{10})$$ $$q-2.34889 q^{3} -1.00000 q^{5} -2.86620 q^{7} +2.51730 q^{9} +6.24970 q^{13} +2.34889 q^{15} -3.73240 q^{17} +8.24970 q^{19} +6.73240 q^{21} -8.94749 q^{23} +1.00000 q^{25} +1.13380 q^{27} +8.69779 q^{29} +5.21509 q^{31} +2.86620 q^{35} -2.69779 q^{37} -14.6799 q^{39} +8.21509 q^{41} -7.13380 q^{43} -2.51730 q^{45} -1.31429 q^{47} +1.21509 q^{49} +8.76700 q^{51} -0.180484 q^{53} -19.3777 q^{57} -3.55191 q^{59} +5.94749 q^{61} -7.21509 q^{63} -6.24970 q^{65} -2.27968 q^{67} +21.0167 q^{69} +2.51730 q^{71} +8.36097 q^{73} -2.34889 q^{75} -15.1280 q^{79} -10.2151 q^{81} +3.55191 q^{83} +3.73240 q^{85} -20.4302 q^{87} +0.732397 q^{89} -17.9129 q^{91} -12.2497 q^{93} -8.24970 q^{95} +1.75030 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - q^{3} - 3 q^{5} - q^{7} + 6 q^{9}+O(q^{10})$$ 3 * q - q^3 - 3 * q^5 - q^7 + 6 * q^9 $$3 q - q^{3} - 3 q^{5} - q^{7} + 6 q^{9} + 2 q^{13} + q^{15} + 4 q^{17} + 8 q^{19} + 5 q^{21} + 2 q^{23} + 3 q^{25} + 11 q^{27} + 14 q^{29} + 2 q^{31} + q^{35} + 4 q^{37} + 11 q^{41} - 29 q^{43} - 6 q^{45} - q^{47} - 10 q^{49} + 8 q^{51} + 10 q^{53} - 2 q^{57} - 6 q^{59} - 11 q^{61} - 8 q^{63} - 2 q^{65} - 7 q^{67} + 28 q^{69} + 6 q^{71} + 4 q^{73} - q^{75} - 6 q^{79} - 17 q^{81} + 6 q^{83} - 4 q^{85} - 34 q^{87} - 13 q^{89} - 28 q^{91} - 20 q^{93} - 8 q^{95} + 22 q^{97}+O(q^{100})$$ 3 * q - q^3 - 3 * q^5 - q^7 + 6 * q^9 + 2 * q^13 + q^15 + 4 * q^17 + 8 * q^19 + 5 * q^21 + 2 * q^23 + 3 * q^25 + 11 * q^27 + 14 * q^29 + 2 * q^31 + q^35 + 4 * q^37 + 11 * q^41 - 29 * q^43 - 6 * q^45 - q^47 - 10 * q^49 + 8 * q^51 + 10 * q^53 - 2 * q^57 - 6 * q^59 - 11 * q^61 - 8 * q^63 - 2 * q^65 - 7 * q^67 + 28 * q^69 + 6 * q^71 + 4 * q^73 - q^75 - 6 * q^79 - 17 * q^81 + 6 * q^83 - 4 * q^85 - 34 * q^87 - 13 * q^89 - 28 * q^91 - 20 * q^93 - 8 * q^95 + 22 * q^97

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ −2.34889 −1.35613 −0.678067 0.735000i $$-0.737182\pi$$
−0.678067 + 0.735000i $$0.737182\pi$$
$$4$$ 0 0
$$5$$ −1.00000 −0.447214
$$6$$ 0 0
$$7$$ −2.86620 −1.08332 −0.541661 0.840597i $$-0.682204\pi$$
−0.541661 + 0.840597i $$0.682204\pi$$
$$8$$ 0 0
$$9$$ 2.51730 0.839101
$$10$$ 0 0
$$11$$ 0 0
$$12$$ 0 0
$$13$$ 6.24970 1.73336 0.866678 0.498869i $$-0.166251\pi$$
0.866678 + 0.498869i $$0.166251\pi$$
$$14$$ 0 0
$$15$$ 2.34889 0.606482
$$16$$ 0 0
$$17$$ −3.73240 −0.905239 −0.452620 0.891704i $$-0.649510\pi$$
−0.452620 + 0.891704i $$0.649510\pi$$
$$18$$ 0 0
$$19$$ 8.24970 1.89261 0.946306 0.323274i $$-0.104783\pi$$
0.946306 + 0.323274i $$0.104783\pi$$
$$20$$ 0 0
$$21$$ 6.73240 1.46913
$$22$$ 0 0
$$23$$ −8.94749 −1.86568 −0.932840 0.360290i $$-0.882678\pi$$
−0.932840 + 0.360290i $$0.882678\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ 1.13380 0.218200
$$28$$ 0 0
$$29$$ 8.69779 1.61514 0.807569 0.589773i $$-0.200783\pi$$
0.807569 + 0.589773i $$0.200783\pi$$
$$30$$ 0 0
$$31$$ 5.21509 0.936658 0.468329 0.883554i $$-0.344856\pi$$
0.468329 + 0.883554i $$0.344856\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 2.86620 0.484476
$$36$$ 0 0
$$37$$ −2.69779 −0.443514 −0.221757 0.975102i $$-0.571179\pi$$
−0.221757 + 0.975102i $$0.571179\pi$$
$$38$$ 0 0
$$39$$ −14.6799 −2.35066
$$40$$ 0 0
$$41$$ 8.21509 1.28298 0.641491 0.767131i $$-0.278316\pi$$
0.641491 + 0.767131i $$0.278316\pi$$
$$42$$ 0 0
$$43$$ −7.13380 −1.08789 −0.543947 0.839119i $$-0.683071\pi$$
−0.543947 + 0.839119i $$0.683071\pi$$
$$44$$ 0 0
$$45$$ −2.51730 −0.375258
$$46$$ 0 0
$$47$$ −1.31429 −0.191708 −0.0958542 0.995395i $$-0.530558\pi$$
−0.0958542 + 0.995395i $$0.530558\pi$$
$$48$$ 0 0
$$49$$ 1.21509 0.173585
$$50$$ 0 0
$$51$$ 8.76700 1.22763
$$52$$ 0 0
$$53$$ −0.180484 −0.0247914 −0.0123957 0.999923i $$-0.503946\pi$$
−0.0123957 + 0.999923i $$0.503946\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ −19.3777 −2.56664
$$58$$ 0 0
$$59$$ −3.55191 −0.462420 −0.231210 0.972904i $$-0.574268\pi$$
−0.231210 + 0.972904i $$0.574268\pi$$
$$60$$ 0 0
$$61$$ 5.94749 0.761498 0.380749 0.924678i $$-0.375666\pi$$
0.380749 + 0.924678i $$0.375666\pi$$
$$62$$ 0 0
$$63$$ −7.21509 −0.909016
$$64$$ 0 0
$$65$$ −6.24970 −0.775180
$$66$$ 0 0
$$67$$ −2.27968 −0.278507 −0.139253 0.990257i $$-0.544470\pi$$
−0.139253 + 0.990257i $$0.544470\pi$$
$$68$$ 0 0
$$69$$ 21.0167 2.53011
$$70$$ 0 0
$$71$$ 2.51730 0.298749 0.149375 0.988781i $$-0.452274\pi$$
0.149375 + 0.988781i $$0.452274\pi$$
$$72$$ 0 0
$$73$$ 8.36097 0.978577 0.489289 0.872122i $$-0.337256\pi$$
0.489289 + 0.872122i $$0.337256\pi$$
$$74$$ 0 0
$$75$$ −2.34889 −0.271227
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −15.1280 −1.70203 −0.851015 0.525141i $$-0.824012\pi$$
−0.851015 + 0.525141i $$0.824012\pi$$
$$80$$ 0 0
$$81$$ −10.2151 −1.13501
$$82$$ 0 0
$$83$$ 3.55191 0.389873 0.194937 0.980816i $$-0.437550\pi$$
0.194937 + 0.980816i $$0.437550\pi$$
$$84$$ 0 0
$$85$$ 3.73240 0.404835
$$86$$ 0 0
$$87$$ −20.4302 −2.19035
$$88$$ 0 0
$$89$$ 0.732397 0.0776339 0.0388169 0.999246i $$-0.487641\pi$$
0.0388169 + 0.999246i $$0.487641\pi$$
$$90$$ 0 0
$$91$$ −17.9129 −1.87778
$$92$$ 0 0
$$93$$ −12.2497 −1.27023
$$94$$ 0 0
$$95$$ −8.24970 −0.846401
$$96$$ 0 0
$$97$$ 1.75030 0.177716 0.0888580 0.996044i $$-0.471678\pi$$
0.0888580 + 0.996044i $$0.471678\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −5.69779 −0.566951 −0.283476 0.958979i $$-0.591487\pi$$
−0.283476 + 0.958979i $$0.591487\pi$$
$$102$$ 0 0
$$103$$ 19.0167 1.87377 0.936886 0.349635i $$-0.113695\pi$$
0.936886 + 0.349635i $$0.113695\pi$$
$$104$$ 0 0
$$105$$ −6.73240 −0.657015
$$106$$ 0 0
$$107$$ −7.45272 −0.720481 −0.360241 0.932859i $$-0.617305\pi$$
−0.360241 + 0.932859i $$0.617305\pi$$
$$108$$ 0 0
$$109$$ −0.551912 −0.0528636 −0.0264318 0.999651i $$-0.508414\pi$$
−0.0264318 + 0.999651i $$0.508414\pi$$
$$110$$ 0 0
$$111$$ 6.33682 0.601464
$$112$$ 0 0
$$113$$ −5.46479 −0.514084 −0.257042 0.966400i $$-0.582748\pi$$
−0.257042 + 0.966400i $$0.582748\pi$$
$$114$$ 0 0
$$115$$ 8.94749 0.834358
$$116$$ 0 0
$$117$$ 15.7324 1.45446
$$118$$ 0 0
$$119$$ 10.6978 0.980665
$$120$$ 0 0
$$121$$ 0 0
$$122$$ 0 0
$$123$$ −19.2964 −1.73990
$$124$$ 0 0
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ 9.38350 0.832651 0.416326 0.909216i $$-0.363318\pi$$
0.416326 + 0.909216i $$0.363318\pi$$
$$128$$ 0 0
$$129$$ 16.7565 1.47533
$$130$$ 0 0
$$131$$ −11.2151 −0.979867 −0.489934 0.871760i $$-0.662979\pi$$
−0.489934 + 0.871760i $$0.662979\pi$$
$$132$$ 0 0
$$133$$ −23.6453 −2.05031
$$134$$ 0 0
$$135$$ −1.13380 −0.0975821
$$136$$ 0 0
$$137$$ −21.3085 −1.82050 −0.910252 0.414054i $$-0.864112\pi$$
−0.910252 + 0.414054i $$0.864112\pi$$
$$138$$ 0 0
$$139$$ −4.18048 −0.354584 −0.177292 0.984158i $$-0.556734\pi$$
−0.177292 + 0.984158i $$0.556734\pi$$
$$140$$ 0 0
$$141$$ 3.08712 0.259982
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ −8.69779 −0.722312
$$146$$ 0 0
$$147$$ −2.85412 −0.235404
$$148$$ 0 0
$$149$$ −15.7491 −1.29022 −0.645108 0.764091i $$-0.723188\pi$$
−0.645108 + 0.764091i $$0.723188\pi$$
$$150$$ 0 0
$$151$$ −0.0871191 −0.00708965 −0.00354483 0.999994i $$-0.501128\pi$$
−0.00354483 + 0.999994i $$0.501128\pi$$
$$152$$ 0 0
$$153$$ −9.39558 −0.759587
$$154$$ 0 0
$$155$$ −5.21509 −0.418886
$$156$$ 0 0
$$157$$ −17.8950 −1.42817 −0.714087 0.700057i $$-0.753158\pi$$
−0.714087 + 0.700057i $$0.753158\pi$$
$$158$$ 0 0
$$159$$ 0.423939 0.0336205
$$160$$ 0 0
$$161$$ 25.6453 2.02113
$$162$$ 0 0
$$163$$ −1.20302 −0.0942276 −0.0471138 0.998890i $$-0.515002\pi$$
−0.0471138 + 0.998890i $$0.515002\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −1.73822 −0.134508 −0.0672539 0.997736i $$-0.521424\pi$$
−0.0672539 + 0.997736i $$0.521424\pi$$
$$168$$ 0 0
$$169$$ 26.0588 2.00452
$$170$$ 0 0
$$171$$ 20.7670 1.58809
$$172$$ 0 0
$$173$$ −3.64528 −0.277145 −0.138573 0.990352i $$-0.544251\pi$$
−0.138573 + 0.990352i $$0.544251\pi$$
$$174$$ 0 0
$$175$$ −2.86620 −0.216664
$$176$$ 0 0
$$177$$ 8.34307 0.627103
$$178$$ 0 0
$$179$$ −4.96539 −0.371131 −0.185565 0.982632i $$-0.559412\pi$$
−0.185565 + 0.982632i $$0.559412\pi$$
$$180$$ 0 0
$$181$$ 22.2664 1.65505 0.827524 0.561430i $$-0.189748\pi$$
0.827524 + 0.561430i $$0.189748\pi$$
$$182$$ 0 0
$$183$$ −13.9700 −1.03269
$$184$$ 0 0
$$185$$ 2.69779 0.198345
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ −3.24970 −0.236381
$$190$$ 0 0
$$191$$ −13.3777 −0.967975 −0.483987 0.875075i $$-0.660812\pi$$
−0.483987 + 0.875075i $$0.660812\pi$$
$$192$$ 0 0
$$193$$ 16.8362 1.21190 0.605949 0.795504i $$-0.292794\pi$$
0.605949 + 0.795504i $$0.292794\pi$$
$$194$$ 0 0
$$195$$ 14.6799 1.05125
$$196$$ 0 0
$$197$$ 16.6978 1.18967 0.594834 0.803849i $$-0.297218\pi$$
0.594834 + 0.803849i $$0.297218\pi$$
$$198$$ 0 0
$$199$$ 14.3610 1.01802 0.509011 0.860760i $$-0.330011\pi$$
0.509011 + 0.860760i $$0.330011\pi$$
$$200$$ 0 0
$$201$$ 5.35472 0.377693
$$202$$ 0 0
$$203$$ −24.9296 −1.74971
$$204$$ 0 0
$$205$$ −8.21509 −0.573767
$$206$$ 0 0
$$207$$ −22.5236 −1.56549
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 10.7491 0.739999 0.369999 0.929032i $$-0.379358\pi$$
0.369999 + 0.929032i $$0.379358\pi$$
$$212$$ 0 0
$$213$$ −5.91288 −0.405144
$$214$$ 0 0
$$215$$ 7.13380 0.486521
$$216$$ 0 0
$$217$$ −14.9475 −1.01470
$$218$$ 0 0
$$219$$ −19.6390 −1.32708
$$220$$ 0 0
$$221$$ −23.3264 −1.56910
$$222$$ 0 0
$$223$$ −17.7266 −1.18706 −0.593529 0.804812i $$-0.702266\pi$$
−0.593529 + 0.804812i $$0.702266\pi$$
$$224$$ 0 0
$$225$$ 2.51730 0.167820
$$226$$ 0 0
$$227$$ 22.6920 1.50612 0.753059 0.657953i $$-0.228577\pi$$
0.753059 + 0.657953i $$0.228577\pi$$
$$228$$ 0 0
$$229$$ −17.5928 −1.16256 −0.581281 0.813703i $$-0.697448\pi$$
−0.581281 + 0.813703i $$0.697448\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 14.8783 0.974708 0.487354 0.873204i $$-0.337962\pi$$
0.487354 + 0.873204i $$0.337962\pi$$
$$234$$ 0 0
$$235$$ 1.31429 0.0857346
$$236$$ 0 0
$$237$$ 35.5340 2.30818
$$238$$ 0 0
$$239$$ −4.24970 −0.274890 −0.137445 0.990509i $$-0.543889\pi$$
−0.137445 + 0.990509i $$0.543889\pi$$
$$240$$ 0 0
$$241$$ 4.88873 0.314911 0.157455 0.987526i $$-0.449671\pi$$
0.157455 + 0.987526i $$0.449671\pi$$
$$242$$ 0 0
$$243$$ 20.5928 1.32103
$$244$$ 0 0
$$245$$ −1.21509 −0.0776294
$$246$$ 0 0
$$247$$ 51.5582 3.28057
$$248$$ 0 0
$$249$$ −8.34307 −0.528720
$$250$$ 0 0
$$251$$ −0.947489 −0.0598050 −0.0299025 0.999553i $$-0.509520\pi$$
−0.0299025 + 0.999553i $$0.509520\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ −8.76700 −0.549011
$$256$$ 0 0
$$257$$ 22.6799 1.41473 0.707366 0.706847i $$-0.249883\pi$$
0.707366 + 0.706847i $$0.249883\pi$$
$$258$$ 0 0
$$259$$ 7.73240 0.480468
$$260$$ 0 0
$$261$$ 21.8950 1.35527
$$262$$ 0 0
$$263$$ −1.75030 −0.107928 −0.0539640 0.998543i $$-0.517186\pi$$
−0.0539640 + 0.998543i $$0.517186\pi$$
$$264$$ 0 0
$$265$$ 0.180484 0.0110871
$$266$$ 0 0
$$267$$ −1.72032 −0.105282
$$268$$ 0 0
$$269$$ 15.2738 0.931263 0.465632 0.884979i $$-0.345827\pi$$
0.465632 + 0.884979i $$0.345827\pi$$
$$270$$ 0 0
$$271$$ 20.0934 1.22059 0.610293 0.792176i $$-0.291052\pi$$
0.610293 + 0.792176i $$0.291052\pi$$
$$272$$ 0 0
$$273$$ 42.0755 2.54652
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −4.55936 −0.273945 −0.136973 0.990575i $$-0.543737\pi$$
−0.136973 + 0.990575i $$0.543737\pi$$
$$278$$ 0 0
$$279$$ 13.1280 0.785951
$$280$$ 0 0
$$281$$ −24.7912 −1.47892 −0.739458 0.673203i $$-0.764918\pi$$
−0.739458 + 0.673203i $$0.764918\pi$$
$$282$$ 0 0
$$283$$ 1.91871 0.114055 0.0570277 0.998373i $$-0.481838\pi$$
0.0570277 + 0.998373i $$0.481838\pi$$
$$284$$ 0 0
$$285$$ 19.3777 1.14783
$$286$$ 0 0
$$287$$ −23.5461 −1.38988
$$288$$ 0 0
$$289$$ −3.06922 −0.180542
$$290$$ 0 0
$$291$$ −4.11127 −0.241007
$$292$$ 0 0
$$293$$ 17.9129 1.04648 0.523241 0.852185i $$-0.324723\pi$$
0.523241 + 0.852185i $$0.324723\pi$$
$$294$$ 0 0
$$295$$ 3.55191 0.206800
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ −55.9191 −3.23389
$$300$$ 0 0
$$301$$ 20.4469 1.17854
$$302$$ 0 0
$$303$$ 13.3835 0.768862
$$304$$ 0 0
$$305$$ −5.94749 −0.340552
$$306$$ 0 0
$$307$$ −0.448088 −0.0255737 −0.0127869 0.999918i $$-0.504070\pi$$
−0.0127869 + 0.999918i $$0.504070\pi$$
$$308$$ 0 0
$$309$$ −44.6682 −2.54109
$$310$$ 0 0
$$311$$ −19.7145 −1.11791 −0.558953 0.829199i $$-0.688797\pi$$
−0.558953 + 0.829199i $$0.688797\pi$$
$$312$$ 0 0
$$313$$ −34.4815 −1.94901 −0.974505 0.224367i $$-0.927969\pi$$
−0.974505 + 0.224367i $$0.927969\pi$$
$$314$$ 0 0
$$315$$ 7.21509 0.406524
$$316$$ 0 0
$$317$$ 32.2559 1.81167 0.905837 0.423626i $$-0.139243\pi$$
0.905837 + 0.423626i $$0.139243\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 17.5056 0.977070
$$322$$ 0 0
$$323$$ −30.7912 −1.71327
$$324$$ 0 0
$$325$$ 6.24970 0.346671
$$326$$ 0 0
$$327$$ 1.29638 0.0716902
$$328$$ 0 0
$$329$$ 3.76700 0.207682
$$330$$ 0 0
$$331$$ −6.05131 −0.332610 −0.166305 0.986074i $$-0.553184\pi$$
−0.166305 + 0.986074i $$0.553184\pi$$
$$332$$ 0 0
$$333$$ −6.79115 −0.372153
$$334$$ 0 0
$$335$$ 2.27968 0.124552
$$336$$ 0 0
$$337$$ −17.0409 −0.928274 −0.464137 0.885763i $$-0.653636\pi$$
−0.464137 + 0.885763i $$0.653636\pi$$
$$338$$ 0 0
$$339$$ 12.8362 0.697168
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 16.5807 0.895273
$$344$$ 0 0
$$345$$ −21.0167 −1.13150
$$346$$ 0 0
$$347$$ −20.7370 −1.11322 −0.556611 0.830773i $$-0.687899\pi$$
−0.556611 + 0.830773i $$0.687899\pi$$
$$348$$ 0 0
$$349$$ 4.33682 0.232145 0.116072 0.993241i $$-0.462970\pi$$
0.116072 + 0.993241i $$0.462970\pi$$
$$350$$ 0 0
$$351$$ 7.08592 0.378219
$$352$$ 0 0
$$353$$ −3.48270 −0.185365 −0.0926826 0.995696i $$-0.529544\pi$$
−0.0926826 + 0.995696i $$0.529544\pi$$
$$354$$ 0 0
$$355$$ −2.51730 −0.133605
$$356$$ 0 0
$$357$$ −25.1280 −1.32991
$$358$$ 0 0
$$359$$ −17.9129 −0.945406 −0.472703 0.881222i $$-0.656722\pi$$
−0.472703 + 0.881222i $$0.656722\pi$$
$$360$$ 0 0
$$361$$ 49.0576 2.58198
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −8.36097 −0.437633
$$366$$ 0 0
$$367$$ 6.91751 0.361091 0.180546 0.983567i $$-0.442214\pi$$
0.180546 + 0.983567i $$0.442214\pi$$
$$368$$ 0 0
$$369$$ 20.6799 1.07655
$$370$$ 0 0
$$371$$ 0.517304 0.0268571
$$372$$ 0 0
$$373$$ −22.4123 −1.16046 −0.580232 0.814451i $$-0.697038\pi$$
−0.580232 + 0.814451i $$0.697038\pi$$
$$374$$ 0 0
$$375$$ 2.34889 0.121296
$$376$$ 0 0
$$377$$ 54.3586 2.79961
$$378$$ 0 0
$$379$$ 20.1626 1.03568 0.517841 0.855477i $$-0.326736\pi$$
0.517841 + 0.855477i $$0.326736\pi$$
$$380$$ 0 0
$$381$$ −22.0409 −1.12919
$$382$$ 0 0
$$383$$ 33.0409 1.68831 0.844154 0.536100i $$-0.180103\pi$$
0.844154 + 0.536100i $$0.180103\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −17.9579 −0.912854
$$388$$ 0 0
$$389$$ 28.9370 1.46717 0.733583 0.679600i $$-0.237847\pi$$
0.733583 + 0.679600i $$0.237847\pi$$
$$390$$ 0 0
$$391$$ 33.3956 1.68889
$$392$$ 0 0
$$393$$ 26.3431 1.32883
$$394$$ 0 0
$$395$$ 15.1280 0.761171
$$396$$ 0 0
$$397$$ −2.96539 −0.148829 −0.0744144 0.997227i $$-0.523709\pi$$
−0.0744144 + 0.997227i $$0.523709\pi$$
$$398$$ 0 0
$$399$$ 55.5403 2.78049
$$400$$ 0 0
$$401$$ 0.0704139 0.00351630 0.00175815 0.999998i $$-0.499440\pi$$
0.00175815 + 0.999998i $$0.499440\pi$$
$$402$$ 0 0
$$403$$ 32.5928 1.62356
$$404$$ 0 0
$$405$$ 10.2151 0.507592
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ 4.14588 0.205000 0.102500 0.994733i $$-0.467316\pi$$
0.102500 + 0.994733i $$0.467316\pi$$
$$410$$ 0 0
$$411$$ 50.0513 2.46885
$$412$$ 0 0
$$413$$ 10.1805 0.500949
$$414$$ 0 0
$$415$$ −3.55191 −0.174357
$$416$$ 0 0
$$417$$ 9.81952 0.480864
$$418$$ 0 0
$$419$$ 26.2738 1.28356 0.641781 0.766888i $$-0.278196\pi$$
0.641781 + 0.766888i $$0.278196\pi$$
$$420$$ 0 0
$$421$$ 14.7503 0.718886 0.359443 0.933167i $$-0.382967\pi$$
0.359443 + 0.933167i $$0.382967\pi$$
$$422$$ 0 0
$$423$$ −3.30846 −0.160863
$$424$$ 0 0
$$425$$ −3.73240 −0.181048
$$426$$ 0 0
$$427$$ −17.0467 −0.824947
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −7.26641 −0.350010 −0.175005 0.984568i $$-0.555994\pi$$
−0.175005 + 0.984568i $$0.555994\pi$$
$$432$$ 0 0
$$433$$ 16.2497 0.780911 0.390455 0.920622i $$-0.372318\pi$$
0.390455 + 0.920622i $$0.372318\pi$$
$$434$$ 0 0
$$435$$ 20.4302 0.979552
$$436$$ 0 0
$$437$$ −73.8141 −3.53101
$$438$$ 0 0
$$439$$ 20.4543 0.976232 0.488116 0.872779i $$-0.337684\pi$$
0.488116 + 0.872779i $$0.337684\pi$$
$$440$$ 0 0
$$441$$ 3.05876 0.145655
$$442$$ 0 0
$$443$$ 16.5628 0.786922 0.393461 0.919341i $$-0.371278\pi$$
0.393461 + 0.919341i $$0.371278\pi$$
$$444$$ 0 0
$$445$$ −0.732397 −0.0347189
$$446$$ 0 0
$$447$$ 36.9930 1.74971
$$448$$ 0 0
$$449$$ −0.308458 −0.0145570 −0.00727851 0.999974i $$-0.502317\pi$$
−0.00727851 + 0.999974i $$0.502317\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ 0.204634 0.00961452
$$454$$ 0 0
$$455$$ 17.9129 0.839769
$$456$$ 0 0
$$457$$ −32.0397 −1.49875 −0.749376 0.662145i $$-0.769646\pi$$
−0.749376 + 0.662145i $$0.769646\pi$$
$$458$$ 0 0
$$459$$ −4.23180 −0.197523
$$460$$ 0 0
$$461$$ −34.7324 −1.61765 −0.808824 0.588050i $$-0.799896\pi$$
−0.808824 + 0.588050i $$0.799896\pi$$
$$462$$ 0 0
$$463$$ 34.0634 1.58306 0.791530 0.611130i $$-0.209285\pi$$
0.791530 + 0.611130i $$0.209285\pi$$
$$464$$ 0 0
$$465$$ 12.2497 0.568066
$$466$$ 0 0
$$467$$ −22.0213 −1.01903 −0.509513 0.860463i $$-0.670174\pi$$
−0.509513 + 0.860463i $$0.670174\pi$$
$$468$$ 0 0
$$469$$ 6.53401 0.301713
$$470$$ 0 0
$$471$$ 42.0334 1.93680
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 8.24970 0.378522
$$476$$ 0 0
$$477$$ −0.454334 −0.0208025
$$478$$ 0 0
$$479$$ 11.6274 0.531268 0.265634 0.964074i $$-0.414419\pi$$
0.265634 + 0.964074i $$0.414419\pi$$
$$480$$ 0 0
$$481$$ −16.8604 −0.768767
$$482$$ 0 0
$$483$$ −60.2380 −2.74093
$$484$$ 0 0
$$485$$ −1.75030 −0.0794770
$$486$$ 0 0
$$487$$ 35.9129 1.62737 0.813684 0.581307i $$-0.197459\pi$$
0.813684 + 0.581307i $$0.197459\pi$$
$$488$$ 0 0
$$489$$ 2.82576 0.127785
$$490$$ 0 0
$$491$$ −6.23180 −0.281237 −0.140619 0.990064i $$-0.544909\pi$$
−0.140619 + 0.990064i $$0.544909\pi$$
$$492$$ 0 0
$$493$$ −32.4636 −1.46209
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −7.21509 −0.323641
$$498$$ 0 0
$$499$$ 9.13963 0.409146 0.204573 0.978851i $$-0.434419\pi$$
0.204573 + 0.978851i $$0.434419\pi$$
$$500$$ 0 0
$$501$$ 4.08291 0.182411
$$502$$ 0 0
$$503$$ 26.7191 1.19135 0.595673 0.803227i $$-0.296885\pi$$
0.595673 + 0.803227i $$0.296885\pi$$
$$504$$ 0 0
$$505$$ 5.69779 0.253548
$$506$$ 0 0
$$507$$ −61.2093 −2.71840
$$508$$ 0 0
$$509$$ −0.870829 −0.0385988 −0.0192994 0.999814i $$-0.506144\pi$$
−0.0192994 + 0.999814i $$0.506144\pi$$
$$510$$ 0 0
$$511$$ −23.9642 −1.06011
$$512$$ 0 0
$$513$$ 9.35352 0.412968
$$514$$ 0 0
$$515$$ −19.0167 −0.837976
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 8.56237 0.375846
$$520$$ 0 0
$$521$$ −28.6966 −1.25722 −0.628610 0.777720i $$-0.716376\pi$$
−0.628610 + 0.777720i $$0.716376\pi$$
$$522$$ 0 0
$$523$$ 0.354723 0.0155109 0.00775547 0.999970i $$-0.497531\pi$$
0.00775547 + 0.999970i $$0.497531\pi$$
$$524$$ 0 0
$$525$$ 6.73240 0.293826
$$526$$ 0 0
$$527$$ −19.4648 −0.847900
$$528$$ 0 0
$$529$$ 57.0576 2.48076
$$530$$ 0 0
$$531$$ −8.94124 −0.388017
$$532$$ 0 0
$$533$$ 51.3419 2.22386
$$534$$ 0 0
$$535$$ 7.45272 0.322209
$$536$$ 0 0
$$537$$ 11.6632 0.503303
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −41.9296 −1.80269 −0.901347 0.433098i $$-0.857420\pi$$
−0.901347 + 0.433098i $$0.857420\pi$$
$$542$$ 0 0
$$543$$ −52.3014 −2.24447
$$544$$ 0 0
$$545$$ 0.551912 0.0236413
$$546$$ 0 0
$$547$$ 14.5173 0.620715 0.310358 0.950620i $$-0.399551\pi$$
0.310358 + 0.950620i $$0.399551\pi$$
$$548$$ 0 0
$$549$$ 14.9716 0.638974
$$550$$ 0 0
$$551$$ 71.7542 3.05683
$$552$$ 0 0
$$553$$ 43.3598 1.84385
$$554$$ 0 0
$$555$$ −6.33682 −0.268983
$$556$$ 0 0
$$557$$ 18.1626 0.769573 0.384787 0.923006i $$-0.374275\pi$$
0.384787 + 0.923006i $$0.374275\pi$$
$$558$$ 0 0
$$559$$ −44.5841 −1.88571
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 6.62274 0.279115 0.139558 0.990214i $$-0.455432\pi$$
0.139558 + 0.990214i $$0.455432\pi$$
$$564$$ 0 0
$$565$$ 5.46479 0.229906
$$566$$ 0 0
$$567$$ 29.2785 1.22958
$$568$$ 0 0
$$569$$ 37.9475 1.59084 0.795421 0.606058i $$-0.207250\pi$$
0.795421 + 0.606058i $$0.207250\pi$$
$$570$$ 0 0
$$571$$ −11.2664 −0.471484 −0.235742 0.971816i $$-0.575752\pi$$
−0.235742 + 0.971816i $$0.575752\pi$$
$$572$$ 0 0
$$573$$ 31.4227 1.31270
$$574$$ 0 0
$$575$$ −8.94749 −0.373136
$$576$$ 0 0
$$577$$ 1.69034 0.0703700 0.0351850 0.999381i $$-0.488798\pi$$
0.0351850 + 0.999381i $$0.488798\pi$$
$$578$$ 0 0
$$579$$ −39.5465 −1.64350
$$580$$ 0 0
$$581$$ −10.1805 −0.422358
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ −15.7324 −0.650455
$$586$$ 0 0
$$587$$ 8.76118 0.361612 0.180806 0.983519i $$-0.442129\pi$$
0.180806 + 0.983519i $$0.442129\pi$$
$$588$$ 0 0
$$589$$ 43.0230 1.77273
$$590$$ 0 0
$$591$$ −39.2213 −1.61335
$$592$$ 0 0
$$593$$ −16.0421 −0.658768 −0.329384 0.944196i $$-0.606841\pi$$
−0.329384 + 0.944196i $$0.606841\pi$$
$$594$$ 0 0
$$595$$ −10.6978 −0.438567
$$596$$ 0 0
$$597$$ −33.7324 −1.38058
$$598$$ 0 0
$$599$$ 8.97164 0.366571 0.183286 0.983060i $$-0.441327\pi$$
0.183286 + 0.983060i $$0.441327\pi$$
$$600$$ 0 0
$$601$$ −6.94124 −0.283139 −0.141570 0.989928i $$-0.545215\pi$$
−0.141570 + 0.989928i $$0.545215\pi$$
$$602$$ 0 0
$$603$$ −5.73864 −0.233696
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 34.9358 1.41800 0.709001 0.705208i $$-0.249146\pi$$
0.709001 + 0.705208i $$0.249146\pi$$
$$608$$ 0 0
$$609$$ 58.5570 2.37285
$$610$$ 0 0
$$611$$ −8.21389 −0.332299
$$612$$ 0 0
$$613$$ −7.52475 −0.303922 −0.151961 0.988387i $$-0.548559\pi$$
−0.151961 + 0.988387i $$0.548559\pi$$
$$614$$ 0 0
$$615$$ 19.2964 0.778105
$$616$$ 0 0
$$617$$ −17.1551 −0.690640 −0.345320 0.938485i $$-0.612230\pi$$
−0.345320 + 0.938485i $$0.612230\pi$$
$$618$$ 0 0
$$619$$ 4.03581 0.162213 0.0811064 0.996705i $$-0.474155\pi$$
0.0811064 + 0.996705i $$0.474155\pi$$
$$620$$ 0 0
$$621$$ −10.1447 −0.407092
$$622$$ 0 0
$$623$$ −2.09919 −0.0841024
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 10.0692 0.401486
$$630$$ 0 0
$$631$$ −27.4290 −1.09193 −0.545965 0.837808i $$-0.683837\pi$$
−0.545965 + 0.837808i $$0.683837\pi$$
$$632$$ 0 0
$$633$$ −25.2485 −1.00354
$$634$$ 0 0
$$635$$ −9.38350 −0.372373
$$636$$ 0 0
$$637$$ 7.59396 0.300884
$$638$$ 0 0
$$639$$ 6.33682 0.250681
$$640$$ 0 0
$$641$$ 2.89618 0.114392 0.0571960 0.998363i $$-0.481784\pi$$
0.0571960 + 0.998363i $$0.481784\pi$$
$$642$$ 0 0
$$643$$ 39.6666 1.56430 0.782149 0.623091i $$-0.214123\pi$$
0.782149 + 0.623091i $$0.214123\pi$$
$$644$$ 0 0
$$645$$ −16.7565 −0.659788
$$646$$ 0 0
$$647$$ 21.4014 0.841376 0.420688 0.907205i $$-0.361789\pi$$
0.420688 + 0.907205i $$0.361789\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 35.1101 1.37607
$$652$$ 0 0
$$653$$ −29.9191 −1.17083 −0.585413 0.810735i $$-0.699068\pi$$
−0.585413 + 0.810735i $$0.699068\pi$$
$$654$$ 0 0
$$655$$ 11.2151 0.438210
$$656$$ 0 0
$$657$$ 21.0471 0.821126
$$658$$ 0 0
$$659$$ 19.9883 0.778635 0.389318 0.921104i $$-0.372711\pi$$
0.389318 + 0.921104i $$0.372711\pi$$
$$660$$ 0 0
$$661$$ −17.6107 −0.684976 −0.342488 0.939522i $$-0.611270\pi$$
−0.342488 + 0.939522i $$0.611270\pi$$
$$662$$ 0 0
$$663$$ 54.7912 2.12791
$$664$$ 0 0
$$665$$ 23.6453 0.916925
$$666$$ 0 0
$$667$$ −77.8234 −3.01333
$$668$$ 0 0
$$669$$ 41.6378 1.60981
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ 24.4364 0.941955 0.470978 0.882145i $$-0.343901\pi$$
0.470978 + 0.882145i $$0.343901\pi$$
$$674$$ 0 0
$$675$$ 1.13380 0.0436400
$$676$$ 0 0
$$677$$ 45.3956 1.74469 0.872347 0.488887i $$-0.162597\pi$$
0.872347 + 0.488887i $$0.162597\pi$$
$$678$$ 0 0
$$679$$ −5.01671 −0.192523
$$680$$ 0 0
$$681$$ −53.3010 −2.04250
$$682$$ 0 0
$$683$$ −31.1914 −1.19350 −0.596752 0.802426i $$-0.703542\pi$$
−0.596752 + 0.802426i $$0.703542\pi$$
$$684$$ 0 0
$$685$$ 21.3085 0.814154
$$686$$ 0 0
$$687$$ 41.3235 1.57659
$$688$$ 0 0
$$689$$ −1.12797 −0.0429724
$$690$$ 0 0
$$691$$ 47.6336 1.81207 0.906034 0.423205i $$-0.139095\pi$$
0.906034 + 0.423205i $$0.139095\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 4.18048 0.158575
$$696$$ 0 0
$$697$$ −30.6620 −1.16141
$$698$$ 0 0
$$699$$ −34.9475 −1.32184
$$700$$ 0 0
$$701$$ −6.59277 −0.249005 −0.124503 0.992219i $$-0.539734\pi$$
−0.124503 + 0.992219i $$0.539734\pi$$
$$702$$ 0 0
$$703$$ −22.2559 −0.839399
$$704$$ 0 0
$$705$$ −3.08712 −0.116268
$$706$$ 0 0
$$707$$ 16.3310 0.614190
$$708$$ 0 0
$$709$$ 40.2213 1.51054 0.755272 0.655411i $$-0.227505\pi$$
0.755272 + 0.655411i $$0.227505\pi$$
$$710$$ 0 0
$$711$$ −38.0817 −1.42818
$$712$$ 0 0
$$713$$ −46.6620 −1.74750
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 9.98210 0.372788
$$718$$ 0 0
$$719$$ 19.2727 0.718749 0.359374 0.933193i $$-0.382990\pi$$
0.359374 + 0.933193i $$0.382990\pi$$
$$720$$ 0 0
$$721$$ −54.5056 −2.02990
$$722$$ 0 0
$$723$$ −11.4831 −0.427062
$$724$$ 0 0
$$725$$ 8.69779 0.323028
$$726$$ 0 0
$$727$$ −26.8662 −0.996412 −0.498206 0.867059i $$-0.666008\pi$$
−0.498206 + 0.867059i $$0.666008\pi$$
$$728$$ 0 0
$$729$$ −17.7250 −0.656480
$$730$$ 0 0
$$731$$ 26.6262 0.984805
$$732$$ 0 0
$$733$$ 3.45553 0.127633 0.0638165 0.997962i $$-0.479673\pi$$
0.0638165 + 0.997962i $$0.479673\pi$$
$$734$$ 0 0
$$735$$ 2.85412 0.105276
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ −21.8708 −0.804531 −0.402266 0.915523i $$-0.631777\pi$$
−0.402266 + 0.915523i $$0.631777\pi$$
$$740$$ 0 0
$$741$$ −121.105 −4.44889
$$742$$ 0 0
$$743$$ −21.6511 −0.794302 −0.397151 0.917753i $$-0.630001\pi$$
−0.397151 + 0.917753i $$0.630001\pi$$
$$744$$ 0 0
$$745$$ 15.7491 0.577002
$$746$$ 0 0
$$747$$ 8.94124 0.327143
$$748$$ 0 0
$$749$$ 21.3610 0.780513
$$750$$ 0 0
$$751$$ 22.9988 0.839238 0.419619 0.907700i $$-0.362164\pi$$
0.419619 + 0.907700i $$0.362164\pi$$
$$752$$ 0 0
$$753$$ 2.22555 0.0811036
$$754$$ 0 0
$$755$$ 0.0871191 0.00317059
$$756$$ 0 0
$$757$$ −0.947489 −0.0344371 −0.0172185 0.999852i $$-0.505481\pi$$
−0.0172185 + 0.999852i $$0.505481\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 17.3264 0.628080 0.314040 0.949410i $$-0.398317\pi$$
0.314040 + 0.949410i $$0.398317\pi$$
$$762$$ 0 0
$$763$$ 1.58189 0.0572682
$$764$$ 0 0
$$765$$ 9.39558 0.339698
$$766$$ 0 0
$$767$$ −22.1984 −0.801537
$$768$$ 0 0
$$769$$ −35.2664 −1.27174 −0.635870 0.771797i $$-0.719358\pi$$
−0.635870 + 0.771797i $$0.719358\pi$$
$$770$$ 0 0
$$771$$ −53.2727 −1.91857
$$772$$ 0 0
$$773$$ −3.65693 −0.131531 −0.0657654 0.997835i $$-0.520949\pi$$
−0.0657654 + 0.997835i $$0.520949\pi$$
$$774$$ 0 0
$$775$$ 5.21509 0.187332
$$776$$ 0 0
$$777$$ −18.1626 −0.651579
$$778$$ 0 0
$$779$$ 67.7721 2.42819
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 9.86157 0.352424
$$784$$ 0 0
$$785$$ 17.8950 0.638699
$$786$$ 0 0
$$787$$ −20.3489 −0.725360 −0.362680 0.931914i $$-0.618138\pi$$
−0.362680 + 0.931914i $$0.618138\pi$$
$$788$$ 0 0
$$789$$ 4.11127 0.146365
$$790$$ 0 0
$$791$$ 15.6632 0.556919
$$792$$ 0 0
$$793$$ 37.1700 1.31995
$$794$$ 0 0
$$795$$ −0.423939 −0.0150356
$$796$$ 0 0
$$797$$ −1.32938 −0.0470889 −0.0235445 0.999723i $$-0.507495\pi$$
−0.0235445 + 0.999723i $$0.507495\pi$$
$$798$$ 0 0
$$799$$ 4.90544 0.173542
$$800$$ 0 0
$$801$$ 1.84366 0.0651427
$$802$$ 0 0
$$803$$ 0 0
$$804$$ 0 0
$$805$$ −25.6453 −0.903877
$$806$$ 0 0
$$807$$ −35.8767 −1.26292
$$808$$ 0 0
$$809$$ 49.8350 1.75211 0.876053 0.482215i $$-0.160167\pi$$
0.876053 + 0.482215i $$0.160167\pi$$
$$810$$ 0 0
$$811$$ 48.8183 1.71424 0.857121 0.515114i $$-0.172251\pi$$
0.857121 + 0.515114i $$0.172251\pi$$
$$812$$ 0 0
$$813$$ −47.1972 −1.65528
$$814$$ 0 0
$$815$$ 1.20302 0.0421399
$$816$$ 0 0
$$817$$ −58.8517 −2.05896
$$818$$ 0 0
$$819$$ −45.0922 −1.57565
$$820$$ 0 0
$$821$$ −35.6441 −1.24399 −0.621993 0.783022i $$-0.713677\pi$$
−0.621993 + 0.783022i $$0.713677\pi$$
$$822$$ 0 0
$$823$$ 27.5702 0.961038 0.480519 0.876984i $$-0.340448\pi$$
0.480519 + 0.876984i $$0.340448\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −29.1517 −1.01370 −0.506852 0.862033i $$-0.669191\pi$$
−0.506852 + 0.862033i $$0.669191\pi$$
$$828$$ 0 0
$$829$$ 7.21629 0.250632 0.125316 0.992117i $$-0.460006\pi$$
0.125316 + 0.992117i $$0.460006\pi$$
$$830$$ 0 0
$$831$$ 10.7094 0.371507
$$832$$ 0 0
$$833$$ −4.53521 −0.157136
$$834$$ 0 0
$$835$$ 1.73822 0.0601538
$$836$$ 0 0
$$837$$ 5.91288 0.204379
$$838$$ 0 0
$$839$$ −2.63783 −0.0910681 −0.0455341 0.998963i $$-0.514499\pi$$
−0.0455341 + 0.998963i $$0.514499\pi$$
$$840$$ 0 0
$$841$$ 46.6515 1.60867
$$842$$ 0 0
$$843$$ 58.2318 2.00561
$$844$$ 0 0
$$845$$ −26.0588 −0.896449
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ −4.50685 −0.154675
$$850$$ 0 0
$$851$$ 24.1384 0.827455
$$852$$ 0 0
$$853$$ 3.32636 0.113892 0.0569462 0.998377i $$-0.481864\pi$$
0.0569462 + 0.998377i $$0.481864\pi$$
$$854$$ 0 0
$$855$$ −20.7670 −0.710217
$$856$$ 0 0
$$857$$ −26.0934 −0.891332 −0.445666 0.895199i $$-0.647033\pi$$
−0.445666 + 0.895199i $$0.647033\pi$$
$$858$$ 0 0
$$859$$ 15.2727 0.521096 0.260548 0.965461i $$-0.416097\pi$$
0.260548 + 0.965461i $$0.416097\pi$$
$$860$$ 0 0
$$861$$ 55.3073 1.88487
$$862$$ 0 0
$$863$$ −1.41931 −0.0483138 −0.0241569 0.999708i $$-0.507690\pi$$
−0.0241569 + 0.999708i $$0.507690\pi$$
$$864$$ 0 0
$$865$$ 3.64528 0.123943
$$866$$ 0 0
$$867$$ 7.20926 0.244839
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ −14.2473 −0.482752
$$872$$ 0 0
$$873$$ 4.40604 0.149122
$$874$$ 0 0
$$875$$ 2.86620 0.0968952
$$876$$ 0 0
$$877$$ −32.5656 −1.09966 −0.549831 0.835276i $$-0.685308\pi$$
−0.549831 + 0.835276i $$0.685308\pi$$
$$878$$ 0 0
$$879$$ −42.0755 −1.41917
$$880$$ 0 0
$$881$$ −1.76076 −0.0593215 −0.0296607 0.999560i $$-0.509443\pi$$
−0.0296607 + 0.999560i $$0.509443\pi$$
$$882$$ 0 0
$$883$$ −37.2727 −1.25432 −0.627162 0.778889i $$-0.715784\pi$$
−0.627162 + 0.778889i $$0.715784\pi$$
$$884$$ 0 0
$$885$$ −8.34307 −0.280449
$$886$$ 0 0
$$887$$ −15.5881 −0.523398 −0.261699 0.965149i $$-0.584283\pi$$
−0.261699 + 0.965149i $$0.584283\pi$$
$$888$$ 0 0
$$889$$ −26.8950 −0.902029
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ −10.8425 −0.362829
$$894$$ 0 0
$$895$$ 4.96539 0.165975
$$896$$ 0 0
$$897$$ 131.348 4.38559
$$898$$ 0 0
$$899$$ 45.3598 1.51283
$$900$$ 0 0
$$901$$ 0.673639 0.0224422
$$902$$ 0 0
$$903$$ −48.0276 −1.59826
$$904$$ 0 0
$$905$$ −22.2664 −0.740160
$$906$$ 0 0
$$907$$ 41.3897 1.37432 0.687162 0.726504i $$-0.258856\pi$$
0.687162 + 0.726504i $$0.258856\pi$$
$$908$$ 0 0
$$909$$ −14.3431 −0.475729
$$910$$ 0 0
$$911$$ 25.5940 0.847966 0.423983 0.905670i $$-0.360632\pi$$
0.423983 + 0.905670i $$0.360632\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 0 0
$$915$$ 13.9700 0.461835
$$916$$ 0 0
$$917$$ 32.1447 1.06151
$$918$$ 0 0
$$919$$ −25.0167 −0.825225 −0.412612 0.910907i $$-0.635384\pi$$
−0.412612 + 0.910907i $$0.635384\pi$$
$$920$$ 0 0
$$921$$ 1.05251 0.0346814
$$922$$ 0 0
$$923$$ 15.7324 0.517838
$$924$$ 0 0
$$925$$ −2.69779 −0.0887027
$$926$$ 0 0
$$927$$ 47.8708 1.57228
$$928$$ 0 0
$$929$$ 1.50060 0.0492331 0.0246165 0.999697i $$-0.492164\pi$$
0.0246165 + 0.999697i $$0.492164\pi$$
$$930$$ 0 0
$$931$$ 10.0241 0.328528
$$932$$ 0 0
$$933$$ 46.3073 1.51603
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −9.01046 −0.294359 −0.147179 0.989110i $$-0.547019\pi$$
−0.147179 + 0.989110i $$0.547019\pi$$
$$938$$ 0 0
$$939$$ 80.9934 2.64312
$$940$$ 0 0
$$941$$ −5.96539 −0.194466 −0.0972331 0.995262i $$-0.530999\pi$$
−0.0972331 + 0.995262i $$0.530999\pi$$
$$942$$ 0 0
$$943$$ −73.5044 −2.39363
$$944$$ 0 0
$$945$$ 3.24970 0.105713
$$946$$ 0 0
$$947$$ 1.17929 0.0383217 0.0191608 0.999816i $$-0.493901\pi$$
0.0191608 + 0.999816i $$0.493901\pi$$
$$948$$ 0 0
$$949$$ 52.2536 1.69622
$$950$$ 0 0
$$951$$ −75.7658 −2.45687
$$952$$ 0 0
$$953$$ 49.9883 1.61928 0.809641 0.586926i $$-0.199662\pi$$
0.809641 + 0.586926i $$0.199662\pi$$
$$954$$ 0 0
$$955$$ 13.3777 0.432891
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 61.0743 1.97219
$$960$$ 0 0
$$961$$ −3.80281 −0.122671
$$962$$ 0 0
$$963$$ −18.7608 −0.604557
$$964$$ 0 0
$$965$$ −16.8362 −0.541977
$$966$$ 0 0
$$967$$ 26.2046 0.842684 0.421342 0.906902i $$-0.361559\pi$$
0.421342 + 0.906902i $$0.361559\pi$$
$$968$$ 0 0
$$969$$ 72.3252 2.32342
$$970$$ 0 0
$$971$$ −15.5940 −0.500434 −0.250217 0.968190i $$-0.580502\pi$$
−0.250217 + 0.968190i $$0.580502\pi$$
$$972$$ 0 0
$$973$$ 11.9821 0.384128
$$974$$ 0 0
$$975$$ −14.6799 −0.470133
$$976$$ 0 0
$$977$$ −37.6632 −1.20495 −0.602476 0.798137i $$-0.705819\pi$$
−0.602476 + 0.798137i $$0.705819\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ −1.38933 −0.0443579
$$982$$ 0 0
$$983$$ −30.3823 −0.969045 −0.484523 0.874779i $$-0.661007\pi$$
−0.484523 + 0.874779i $$0.661007\pi$$
$$984$$ 0 0
$$985$$ −16.6978 −0.532036
$$986$$ 0 0
$$987$$ −8.84830 −0.281644
$$988$$ 0 0
$$989$$ 63.8296 2.02966
$$990$$ 0 0
$$991$$ −12.2139 −0.387987 −0.193994 0.981003i $$-0.562144\pi$$
−0.193994 + 0.981003i $$0.562144\pi$$
$$992$$ 0 0
$$993$$ 14.2139 0.451064
$$994$$ 0 0
$$995$$ −14.3610 −0.455273
$$996$$ 0 0
$$997$$ 27.2034 0.861541 0.430771 0.902461i $$-0.358242\pi$$
0.430771 + 0.902461i $$0.358242\pi$$
$$998$$ 0 0
$$999$$ −3.05876 −0.0967748
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 9680.2.a.ca.1.1 3
4.3 odd 2 4840.2.a.u.1.3 yes 3
11.10 odd 2 9680.2.a.cc.1.1 3
44.43 even 2 4840.2.a.t.1.3 3

By twisted newform
Twist Min Dim Char Parity Ord Type
4840.2.a.t.1.3 3 44.43 even 2
4840.2.a.u.1.3 yes 3 4.3 odd 2
9680.2.a.ca.1.1 3 1.1 even 1 trivial
9680.2.a.cc.1.1 3 11.10 odd 2