# Properties

 Label 7524.2.a.r.1.6 Level $7524$ Weight $2$ Character 7524.1 Self dual yes Analytic conductor $60.079$ Analytic rank $0$ Dimension $6$ CM no Inner twists $1$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7524,2,Mod(1,7524)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7524, 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("7524.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$7524 = 2^{2} \cdot 3^{2} \cdot 11 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7524.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: $$60.0794424808$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: $$\mathbb{Q}[x]/(x^{6} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - 2x^{5} - 12x^{4} + 28x^{3} + 16x^{2} - 60x + 27$$ x^6 - 2*x^5 - 12*x^4 + 28*x^3 + 16*x^2 - 60*x + 27 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 836) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.6 Root $$2.18868$$ of defining polynomial Character $$\chi$$ $$=$$ 7524.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+3.62873 q^{5} -4.31127 q^{7} +O(q^{10})$$ $$q+3.62873 q^{5} -4.31127 q^{7} -1.00000 q^{11} -6.10370 q^{13} +0.694815 q^{17} +1.00000 q^{19} +7.79640 q^{23} +8.16767 q^{25} -6.02709 q^{29} -6.26529 q^{31} -15.6444 q^{35} +10.1139 q^{37} +2.04380 q^{41} -2.16767 q^{43} +10.2074 q^{47} +11.5870 q^{49} +1.95798 q^{53} -3.62873 q^{55} -5.89831 q^{59} +6.84803 q^{61} -22.1487 q^{65} -8.35725 q^{67} -0.566034 q^{71} +7.29948 q^{73} +4.31127 q^{77} -13.9103 q^{79} +7.05772 q^{83} +2.52129 q^{85} +14.3590 q^{89} +26.3147 q^{91} +3.62873 q^{95} -2.89842 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 2 q^{7}+O(q^{10})$$ 6 * q + 2 * q^7 $$6 q + 2 q^{7} - 6 q^{11} + 2 q^{13} - 12 q^{17} + 6 q^{19} + 2 q^{23} + 26 q^{25} - 4 q^{29} - 20 q^{31} - 20 q^{35} + 22 q^{37} + 2 q^{41} + 10 q^{43} - 16 q^{47} + 48 q^{49} - 12 q^{53} + 14 q^{59} + 12 q^{61} - 10 q^{65} - 12 q^{67} + 30 q^{71} + 24 q^{73} - 2 q^{77} + 14 q^{83} + 12 q^{85} + 14 q^{89} + 20 q^{91} - 46 q^{97}+O(q^{100})$$ 6 * q + 2 * q^7 - 6 * q^11 + 2 * q^13 - 12 * q^17 + 6 * q^19 + 2 * q^23 + 26 * q^25 - 4 * q^29 - 20 * q^31 - 20 * q^35 + 22 * q^37 + 2 * q^41 + 10 * q^43 - 16 * q^47 + 48 * q^49 - 12 * q^53 + 14 * q^59 + 12 * q^61 - 10 * q^65 - 12 * q^67 + 30 * q^71 + 24 * q^73 - 2 * q^77 + 14 * q^83 + 12 * q^85 + 14 * q^89 + 20 * q^91 - 46 * 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
$$3$$ 0 0
$$4$$ 0 0
$$5$$ 3.62873 1.62282 0.811408 0.584480i $$-0.198701\pi$$
0.811408 + 0.584480i $$0.198701\pi$$
$$6$$ 0 0
$$7$$ −4.31127 −1.62951 −0.814753 0.579808i $$-0.803128\pi$$
−0.814753 + 0.579808i $$0.803128\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −1.00000 −0.301511
$$12$$ 0 0
$$13$$ −6.10370 −1.69286 −0.846431 0.532498i $$-0.821253\pi$$
−0.846431 + 0.532498i $$0.821253\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 0.694815 0.168517 0.0842587 0.996444i $$-0.473148\pi$$
0.0842587 + 0.996444i $$0.473148\pi$$
$$18$$ 0 0
$$19$$ 1.00000 0.229416
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 7.79640 1.62566 0.812830 0.582500i $$-0.197926\pi$$
0.812830 + 0.582500i $$0.197926\pi$$
$$24$$ 0 0
$$25$$ 8.16767 1.63353
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −6.02709 −1.11920 −0.559602 0.828762i $$-0.689046\pi$$
−0.559602 + 0.828762i $$0.689046\pi$$
$$30$$ 0 0
$$31$$ −6.26529 −1.12528 −0.562639 0.826703i $$-0.690214\pi$$
−0.562639 + 0.826703i $$0.690214\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −15.6444 −2.64439
$$36$$ 0 0
$$37$$ 10.1139 1.66271 0.831354 0.555744i $$-0.187566\pi$$
0.831354 + 0.555744i $$0.187566\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 2.04380 0.319189 0.159594 0.987183i $$-0.448981\pi$$
0.159594 + 0.987183i $$0.448981\pi$$
$$42$$ 0 0
$$43$$ −2.16767 −0.330566 −0.165283 0.986246i $$-0.552854\pi$$
−0.165283 + 0.986246i $$0.552854\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 10.2074 1.48890 0.744451 0.667677i $$-0.232711\pi$$
0.744451 + 0.667677i $$0.232711\pi$$
$$48$$ 0 0
$$49$$ 11.5870 1.65529
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 1.95798 0.268949 0.134475 0.990917i $$-0.457065\pi$$
0.134475 + 0.990917i $$0.457065\pi$$
$$54$$ 0 0
$$55$$ −3.62873 −0.489298
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ −5.89831 −0.767895 −0.383948 0.923355i $$-0.625436\pi$$
−0.383948 + 0.923355i $$0.625436\pi$$
$$60$$ 0 0
$$61$$ 6.84803 0.876800 0.438400 0.898780i $$-0.355545\pi$$
0.438400 + 0.898780i $$0.355545\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ −22.1487 −2.74720
$$66$$ 0 0
$$67$$ −8.35725 −1.02100 −0.510501 0.859877i $$-0.670540\pi$$
−0.510501 + 0.859877i $$0.670540\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −0.566034 −0.0671759 −0.0335880 0.999436i $$-0.510693\pi$$
−0.0335880 + 0.999436i $$0.510693\pi$$
$$72$$ 0 0
$$73$$ 7.29948 0.854339 0.427170 0.904172i $$-0.359511\pi$$
0.427170 + 0.904172i $$0.359511\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 4.31127 0.491315
$$78$$ 0 0
$$79$$ −13.9103 −1.56503 −0.782513 0.622635i $$-0.786062\pi$$
−0.782513 + 0.622635i $$0.786062\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 7.05772 0.774685 0.387343 0.921936i $$-0.373393\pi$$
0.387343 + 0.921936i $$0.373393\pi$$
$$84$$ 0 0
$$85$$ 2.52129 0.273473
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 14.3590 1.52205 0.761027 0.648720i $$-0.224695\pi$$
0.761027 + 0.648720i $$0.224695\pi$$
$$90$$ 0 0
$$91$$ 26.3147 2.75853
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 3.62873 0.372300
$$96$$ 0 0
$$97$$ −2.89842 −0.294290 −0.147145 0.989115i $$-0.547008\pi$$
−0.147145 + 0.989115i $$0.547008\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 7.57491 0.753732 0.376866 0.926268i $$-0.377002\pi$$
0.376866 + 0.926268i $$0.377002\pi$$
$$102$$ 0 0
$$103$$ 3.29026 0.324199 0.162099 0.986774i $$-0.448173\pi$$
0.162099 + 0.986774i $$0.448173\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −10.9079 −1.05451 −0.527255 0.849707i $$-0.676779\pi$$
−0.527255 + 0.849707i $$0.676779\pi$$
$$108$$ 0 0
$$109$$ 5.42741 0.519852 0.259926 0.965629i $$-0.416302\pi$$
0.259926 + 0.965629i $$0.416302\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −11.9439 −1.12359 −0.561794 0.827277i $$-0.689889\pi$$
−0.561794 + 0.827277i $$0.689889\pi$$
$$114$$ 0 0
$$115$$ 28.2910 2.63815
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ −2.99553 −0.274600
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 11.4946 1.02811
$$126$$ 0 0
$$127$$ 7.72813 0.685761 0.342880 0.939379i $$-0.388597\pi$$
0.342880 + 0.939379i $$0.388597\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 10.5909 0.925335 0.462668 0.886532i $$-0.346892\pi$$
0.462668 + 0.886532i $$0.346892\pi$$
$$132$$ 0 0
$$133$$ −4.31127 −0.373834
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 8.74829 0.747417 0.373708 0.927546i $$-0.378086\pi$$
0.373708 + 0.927546i $$0.378086\pi$$
$$138$$ 0 0
$$139$$ 10.2696 0.871055 0.435527 0.900175i $$-0.356562\pi$$
0.435527 + 0.900175i $$0.356562\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 6.10370 0.510417
$$144$$ 0 0
$$145$$ −21.8707 −1.81626
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 21.5725 1.76729 0.883643 0.468161i $$-0.155083\pi$$
0.883643 + 0.468161i $$0.155083\pi$$
$$150$$ 0 0
$$151$$ −0.461397 −0.0375479 −0.0187740 0.999824i $$-0.505976\pi$$
−0.0187740 + 0.999824i $$0.505976\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −22.7350 −1.82612
$$156$$ 0 0
$$157$$ −4.52449 −0.361093 −0.180547 0.983566i $$-0.557787\pi$$
−0.180547 + 0.983566i $$0.557787\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −33.6124 −2.64903
$$162$$ 0 0
$$163$$ 11.9622 0.936953 0.468477 0.883476i $$-0.344803\pi$$
0.468477 + 0.883476i $$0.344803\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 9.83576 0.761114 0.380557 0.924758i $$-0.375732\pi$$
0.380557 + 0.924758i $$0.375732\pi$$
$$168$$ 0 0
$$169$$ 24.2552 1.86578
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 12.7818 0.971783 0.485891 0.874019i $$-0.338495\pi$$
0.485891 + 0.874019i $$0.338495\pi$$
$$174$$ 0 0
$$175$$ −35.2130 −2.66185
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 8.13311 0.607897 0.303949 0.952688i $$-0.401695\pi$$
0.303949 + 0.952688i $$0.401695\pi$$
$$180$$ 0 0
$$181$$ 18.5543 1.37913 0.689564 0.724225i $$-0.257802\pi$$
0.689564 + 0.724225i $$0.257802\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 36.7004 2.69827
$$186$$ 0 0
$$187$$ −0.694815 −0.0508099
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 10.6265 0.768910 0.384455 0.923144i $$-0.374389\pi$$
0.384455 + 0.923144i $$0.374389\pi$$
$$192$$ 0 0
$$193$$ −0.759689 −0.0546836 −0.0273418 0.999626i $$-0.508704\pi$$
−0.0273418 + 0.999626i $$0.508704\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −1.68678 −0.120178 −0.0600891 0.998193i $$-0.519138\pi$$
−0.0600891 + 0.998193i $$0.519138\pi$$
$$198$$ 0 0
$$199$$ 24.2153 1.71658 0.858290 0.513166i $$-0.171527\pi$$
0.858290 + 0.513166i $$0.171527\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 25.9844 1.82375
$$204$$ 0 0
$$205$$ 7.41641 0.517984
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ −1.00000 −0.0691714
$$210$$ 0 0
$$211$$ −28.4049 −1.95547 −0.977735 0.209841i $$-0.932705\pi$$
−0.977735 + 0.209841i $$0.932705\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ −7.86588 −0.536448
$$216$$ 0 0
$$217$$ 27.0113 1.83365
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −4.24094 −0.285277
$$222$$ 0 0
$$223$$ −8.82043 −0.590660 −0.295330 0.955395i $$-0.595430\pi$$
−0.295330 + 0.955395i $$0.595430\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 17.2796 1.14689 0.573443 0.819246i $$-0.305607\pi$$
0.573443 + 0.819246i $$0.305607\pi$$
$$228$$ 0 0
$$229$$ 2.15539 0.142432 0.0712162 0.997461i $$-0.477312\pi$$
0.0712162 + 0.997461i $$0.477312\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −11.9733 −0.784398 −0.392199 0.919880i $$-0.628285\pi$$
−0.392199 + 0.919880i $$0.628285\pi$$
$$234$$ 0 0
$$235$$ 37.0399 2.41622
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −18.7692 −1.21408 −0.607038 0.794673i $$-0.707643\pi$$
−0.607038 + 0.794673i $$0.707643\pi$$
$$240$$ 0 0
$$241$$ 14.9468 0.962807 0.481404 0.876499i $$-0.340127\pi$$
0.481404 + 0.876499i $$0.340127\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 42.0462 2.68624
$$246$$ 0 0
$$247$$ −6.10370 −0.388369
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 12.1955 0.769771 0.384885 0.922964i $$-0.374241\pi$$
0.384885 + 0.922964i $$0.374241\pi$$
$$252$$ 0 0
$$253$$ −7.79640 −0.490155
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 6.96669 0.434570 0.217285 0.976108i $$-0.430280\pi$$
0.217285 + 0.976108i $$0.430280\pi$$
$$258$$ 0 0
$$259$$ −43.6036 −2.70939
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ −25.0396 −1.54401 −0.772005 0.635617i $$-0.780746\pi$$
−0.772005 + 0.635617i $$0.780746\pi$$
$$264$$ 0 0
$$265$$ 7.10498 0.436455
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ −1.58130 −0.0964134 −0.0482067 0.998837i $$-0.515351\pi$$
−0.0482067 + 0.998837i $$0.515351\pi$$
$$270$$ 0 0
$$271$$ 5.57867 0.338880 0.169440 0.985540i $$-0.445804\pi$$
0.169440 + 0.985540i $$0.445804\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −8.16767 −0.492529
$$276$$ 0 0
$$277$$ 30.5385 1.83488 0.917440 0.397874i $$-0.130252\pi$$
0.917440 + 0.397874i $$0.130252\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 15.0369 0.897029 0.448514 0.893776i $$-0.351953\pi$$
0.448514 + 0.893776i $$0.351953\pi$$
$$282$$ 0 0
$$283$$ 29.1158 1.73076 0.865378 0.501120i $$-0.167079\pi$$
0.865378 + 0.501120i $$0.167079\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −8.81139 −0.520120
$$288$$ 0 0
$$289$$ −16.5172 −0.971602
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ −5.88816 −0.343990 −0.171995 0.985098i $$-0.555021\pi$$
−0.171995 + 0.985098i $$0.555021\pi$$
$$294$$ 0 0
$$295$$ −21.4034 −1.24615
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ −47.5869 −2.75202
$$300$$ 0 0
$$301$$ 9.34540 0.538660
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 24.8496 1.42289
$$306$$ 0 0
$$307$$ 19.2140 1.09660 0.548300 0.836282i $$-0.315275\pi$$
0.548300 + 0.836282i $$0.315275\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −26.7459 −1.51662 −0.758310 0.651894i $$-0.773975\pi$$
−0.758310 + 0.651894i $$0.773975\pi$$
$$312$$ 0 0
$$313$$ −16.0435 −0.906834 −0.453417 0.891298i $$-0.649795\pi$$
−0.453417 + 0.891298i $$0.649795\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −18.5664 −1.04280 −0.521398 0.853314i $$-0.674589\pi$$
−0.521398 + 0.853314i $$0.674589\pi$$
$$318$$ 0 0
$$319$$ 6.02709 0.337453
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 0.694815 0.0386605
$$324$$ 0 0
$$325$$ −49.8530 −2.76535
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ −44.0069 −2.42618
$$330$$ 0 0
$$331$$ −34.6196 −1.90286 −0.951431 0.307861i $$-0.900387\pi$$
−0.951431 + 0.307861i $$0.900387\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ −30.3262 −1.65690
$$336$$ 0 0
$$337$$ 2.84625 0.155045 0.0775224 0.996991i $$-0.475299\pi$$
0.0775224 + 0.996991i $$0.475299\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 6.26529 0.339284
$$342$$ 0 0
$$343$$ −19.7760 −1.06780
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −10.0153 −0.537651 −0.268826 0.963189i $$-0.586636\pi$$
−0.268826 + 0.963189i $$0.586636\pi$$
$$348$$ 0 0
$$349$$ −0.712288 −0.0381279 −0.0190640 0.999818i $$-0.506069\pi$$
−0.0190640 + 0.999818i $$0.506069\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −32.4528 −1.72729 −0.863645 0.504101i $$-0.831824\pi$$
−0.863645 + 0.504101i $$0.831824\pi$$
$$354$$ 0 0
$$355$$ −2.05398 −0.109014
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −19.6749 −1.03840 −0.519200 0.854653i $$-0.673770\pi$$
−0.519200 + 0.854653i $$0.673770\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 26.4878 1.38644
$$366$$ 0 0
$$367$$ −31.5887 −1.64891 −0.824457 0.565924i $$-0.808520\pi$$
−0.824457 + 0.565924i $$0.808520\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −8.44138 −0.438255
$$372$$ 0 0
$$373$$ 20.4557 1.05916 0.529579 0.848260i $$-0.322350\pi$$
0.529579 + 0.848260i $$0.322350\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 36.7876 1.89466
$$378$$ 0 0
$$379$$ 6.22508 0.319761 0.159880 0.987136i $$-0.448889\pi$$
0.159880 + 0.987136i $$0.448889\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 0.592813 0.0302913 0.0151457 0.999885i $$-0.495179\pi$$
0.0151457 + 0.999885i $$0.495179\pi$$
$$384$$ 0 0
$$385$$ 15.6444 0.797314
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ −32.2424 −1.63475 −0.817377 0.576103i $$-0.804573\pi$$
−0.817377 + 0.576103i $$0.804573\pi$$
$$390$$ 0 0
$$391$$ 5.41705 0.273952
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ −50.4765 −2.53975
$$396$$ 0 0
$$397$$ −25.1889 −1.26420 −0.632098 0.774888i $$-0.717806\pi$$
−0.632098 + 0.774888i $$0.717806\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 22.3272 1.11497 0.557484 0.830188i $$-0.311767\pi$$
0.557484 + 0.830188i $$0.311767\pi$$
$$402$$ 0 0
$$403$$ 38.2414 1.90494
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −10.1139 −0.501325
$$408$$ 0 0
$$409$$ 24.1005 1.19169 0.595847 0.803098i $$-0.296817\pi$$
0.595847 + 0.803098i $$0.296817\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 25.4292 1.25129
$$414$$ 0 0
$$415$$ 25.6105 1.25717
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 17.0480 0.832851 0.416425 0.909170i $$-0.363283\pi$$
0.416425 + 0.909170i $$0.363283\pi$$
$$420$$ 0 0
$$421$$ 4.12963 0.201266 0.100633 0.994924i $$-0.467913\pi$$
0.100633 + 0.994924i $$0.467913\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 5.67502 0.275279
$$426$$ 0 0
$$427$$ −29.5237 −1.42875
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 5.86783 0.282643 0.141322 0.989964i $$-0.454865\pi$$
0.141322 + 0.989964i $$0.454865\pi$$
$$432$$ 0 0
$$433$$ 15.5840 0.748920 0.374460 0.927243i $$-0.377828\pi$$
0.374460 + 0.927243i $$0.377828\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 7.79640 0.372952
$$438$$ 0 0
$$439$$ 13.1937 0.629699 0.314850 0.949142i $$-0.398046\pi$$
0.314850 + 0.949142i $$0.398046\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 4.14411 0.196893 0.0984463 0.995142i $$-0.468613\pi$$
0.0984463 + 0.995142i $$0.468613\pi$$
$$444$$ 0 0
$$445$$ 52.1050 2.47002
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 35.9971 1.69881 0.849405 0.527742i $$-0.176961\pi$$
0.849405 + 0.527742i $$0.176961\pi$$
$$450$$ 0 0
$$451$$ −2.04380 −0.0962390
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 95.4889 4.47659
$$456$$ 0 0
$$457$$ 7.94870 0.371825 0.185912 0.982566i $$-0.440476\pi$$
0.185912 + 0.982566i $$0.440476\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 9.33968 0.434992 0.217496 0.976061i $$-0.430211\pi$$
0.217496 + 0.976061i $$0.430211\pi$$
$$462$$ 0 0
$$463$$ 13.0328 0.605686 0.302843 0.953040i $$-0.402064\pi$$
0.302843 + 0.953040i $$0.402064\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −17.1169 −0.792076 −0.396038 0.918234i $$-0.629615\pi$$
−0.396038 + 0.918234i $$0.629615\pi$$
$$468$$ 0 0
$$469$$ 36.0304 1.66373
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 2.16767 0.0996695
$$474$$ 0 0
$$475$$ 8.16767 0.374758
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ −39.8917 −1.82270 −0.911349 0.411635i $$-0.864958\pi$$
−0.911349 + 0.411635i $$0.864958\pi$$
$$480$$ 0 0
$$481$$ −61.7319 −2.81473
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −10.5176 −0.477578
$$486$$ 0 0
$$487$$ −21.3504 −0.967478 −0.483739 0.875212i $$-0.660722\pi$$
−0.483739 + 0.875212i $$0.660722\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 17.0293 0.768524 0.384262 0.923224i $$-0.374456\pi$$
0.384262 + 0.923224i $$0.374456\pi$$
$$492$$ 0 0
$$493$$ −4.18771 −0.188605
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 2.44033 0.109464
$$498$$ 0 0
$$499$$ −4.94667 −0.221443 −0.110722 0.993851i $$-0.535316\pi$$
−0.110722 + 0.993851i $$0.535316\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 3.22196 0.143660 0.0718301 0.997417i $$-0.477116\pi$$
0.0718301 + 0.997417i $$0.477116\pi$$
$$504$$ 0 0
$$505$$ 27.4873 1.22317
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ −10.1751 −0.451003 −0.225502 0.974243i $$-0.572402\pi$$
−0.225502 + 0.974243i $$0.572402\pi$$
$$510$$ 0 0
$$511$$ −31.4700 −1.39215
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 11.9395 0.526115
$$516$$ 0 0
$$517$$ −10.2074 −0.448921
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 33.3792 1.46237 0.731186 0.682179i $$-0.238967\pi$$
0.731186 + 0.682179i $$0.238967\pi$$
$$522$$ 0 0
$$523$$ −2.17534 −0.0951208 −0.0475604 0.998868i $$-0.515145\pi$$
−0.0475604 + 0.998868i $$0.515145\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −4.35321 −0.189629
$$528$$ 0 0
$$529$$ 37.7838 1.64277
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −12.4748 −0.540342
$$534$$ 0 0
$$535$$ −39.5819 −1.71127
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ −11.5870 −0.499089
$$540$$ 0 0
$$541$$ 24.6958 1.06176 0.530878 0.847448i $$-0.321862\pi$$
0.530878 + 0.847448i $$0.321862\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 19.6946 0.843624
$$546$$ 0 0
$$547$$ −9.80330 −0.419159 −0.209579 0.977792i $$-0.567209\pi$$
−0.209579 + 0.977792i $$0.567209\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −6.02709 −0.256763
$$552$$ 0 0
$$553$$ 59.9708 2.55022
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 12.1796 0.516065 0.258033 0.966136i $$-0.416926\pi$$
0.258033 + 0.966136i $$0.416926\pi$$
$$558$$ 0 0
$$559$$ 13.2308 0.559603
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ −10.6703 −0.449698 −0.224849 0.974394i $$-0.572189\pi$$
−0.224849 + 0.974394i $$0.572189\pi$$
$$564$$ 0 0
$$565$$ −43.3412 −1.82338
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 20.8260 0.873072 0.436536 0.899687i $$-0.356205\pi$$
0.436536 + 0.899687i $$0.356205\pi$$
$$570$$ 0 0
$$571$$ 21.5755 0.902907 0.451454 0.892295i $$-0.350906\pi$$
0.451454 + 0.892295i $$0.350906\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 63.6784 2.65557
$$576$$ 0 0
$$577$$ 15.5212 0.646157 0.323079 0.946372i $$-0.395282\pi$$
0.323079 + 0.946372i $$0.395282\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −30.4277 −1.26235
$$582$$ 0 0
$$583$$ −1.95798 −0.0810912
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −4.08383 −0.168558 −0.0842789 0.996442i $$-0.526859\pi$$
−0.0842789 + 0.996442i $$0.526859\pi$$
$$588$$ 0 0
$$589$$ −6.26529 −0.258157
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 45.1039 1.85220 0.926098 0.377283i $$-0.123142\pi$$
0.926098 + 0.377283i $$0.123142\pi$$
$$594$$ 0 0
$$595$$ −10.8700 −0.445626
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 1.29026 0.0527186 0.0263593 0.999653i $$-0.491609\pi$$
0.0263593 + 0.999653i $$0.491609\pi$$
$$600$$ 0 0
$$601$$ −29.0011 −1.18298 −0.591489 0.806313i $$-0.701460\pi$$
−0.591489 + 0.806313i $$0.701460\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 3.62873 0.147529
$$606$$ 0 0
$$607$$ −11.0119 −0.446958 −0.223479 0.974709i $$-0.571741\pi$$
−0.223479 + 0.974709i $$0.571741\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −62.3029 −2.52051
$$612$$ 0 0
$$613$$ 1.94010 0.0783600 0.0391800 0.999232i $$-0.487525\pi$$
0.0391800 + 0.999232i $$0.487525\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −15.1760 −0.610964 −0.305482 0.952198i $$-0.598818\pi$$
−0.305482 + 0.952198i $$0.598818\pi$$
$$618$$ 0 0
$$619$$ 35.8548 1.44113 0.720564 0.693389i $$-0.243883\pi$$
0.720564 + 0.693389i $$0.243883\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ −61.9057 −2.48020
$$624$$ 0 0
$$625$$ 0.872460 0.0348984
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 7.02726 0.280195
$$630$$ 0 0
$$631$$ −1.43092 −0.0569640 −0.0284820 0.999594i $$-0.509067\pi$$
−0.0284820 + 0.999594i $$0.509067\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 28.0433 1.11286
$$636$$ 0 0
$$637$$ −70.7239 −2.80218
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 27.8685 1.10074 0.550369 0.834922i $$-0.314487\pi$$
0.550369 + 0.834922i $$0.314487\pi$$
$$642$$ 0 0
$$643$$ −1.90753 −0.0752256 −0.0376128 0.999292i $$-0.511975\pi$$
−0.0376128 + 0.999292i $$0.511975\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 44.9164 1.76584 0.882922 0.469519i $$-0.155573\pi$$
0.882922 + 0.469519i $$0.155573\pi$$
$$648$$ 0 0
$$649$$ 5.89831 0.231529
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −23.6282 −0.924642 −0.462321 0.886713i $$-0.652983\pi$$
−0.462321 + 0.886713i $$0.652983\pi$$
$$654$$ 0 0
$$655$$ 38.4317 1.50165
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 10.7391 0.418334 0.209167 0.977880i $$-0.432925\pi$$
0.209167 + 0.977880i $$0.432925\pi$$
$$660$$ 0 0
$$661$$ −46.0837 −1.79245 −0.896224 0.443603i $$-0.853700\pi$$
−0.896224 + 0.443603i $$0.853700\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ −15.6444 −0.606665
$$666$$ 0 0
$$667$$ −46.9896 −1.81945
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −6.84803 −0.264365
$$672$$ 0 0
$$673$$ −22.6538 −0.873242 −0.436621 0.899646i $$-0.643825\pi$$
−0.436621 + 0.899646i $$0.643825\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −24.3654 −0.936438 −0.468219 0.883613i $$-0.655104\pi$$
−0.468219 + 0.883613i $$0.655104\pi$$
$$678$$ 0 0
$$679$$ 12.4959 0.479547
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 26.6471 1.01962 0.509811 0.860286i $$-0.329715\pi$$
0.509811 + 0.860286i $$0.329715\pi$$
$$684$$ 0 0
$$685$$ 31.7452 1.21292
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ −11.9509 −0.455294
$$690$$ 0 0
$$691$$ 39.6176 1.50713 0.753563 0.657376i $$-0.228334\pi$$
0.753563 + 0.657376i $$0.228334\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 37.2655 1.41356
$$696$$ 0 0
$$697$$ 1.42007 0.0537888
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −1.92410 −0.0726722 −0.0363361 0.999340i $$-0.511569\pi$$
−0.0363361 + 0.999340i $$0.511569\pi$$
$$702$$ 0 0
$$703$$ 10.1139 0.381451
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −32.6575 −1.22821
$$708$$ 0 0
$$709$$ −6.91616 −0.259742 −0.129871 0.991531i $$-0.541456\pi$$
−0.129871 + 0.991531i $$0.541456\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ −48.8466 −1.82932
$$714$$ 0 0
$$715$$ 22.1487 0.828313
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 12.3035 0.458843 0.229422 0.973327i $$-0.426317\pi$$
0.229422 + 0.973327i $$0.426317\pi$$
$$720$$ 0 0
$$721$$ −14.1852 −0.528284
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ −49.2273 −1.82826
$$726$$ 0 0
$$727$$ −27.6098 −1.02399 −0.511995 0.858989i $$-0.671093\pi$$
−0.511995 + 0.858989i $$0.671093\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −1.50613 −0.0557062
$$732$$ 0 0
$$733$$ −45.0293 −1.66319 −0.831597 0.555380i $$-0.812573\pi$$
−0.831597 + 0.555380i $$0.812573\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 8.35725 0.307843
$$738$$ 0 0
$$739$$ 37.8963 1.39404 0.697018 0.717053i $$-0.254510\pi$$
0.697018 + 0.717053i $$0.254510\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −23.3103 −0.855172 −0.427586 0.903975i $$-0.640636\pi$$
−0.427586 + 0.903975i $$0.640636\pi$$
$$744$$ 0 0
$$745$$ 78.2807 2.86798
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 47.0270 1.71833
$$750$$ 0 0
$$751$$ −14.9643 −0.546056 −0.273028 0.962006i $$-0.588025\pi$$
−0.273028 + 0.962006i $$0.588025\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ −1.67428 −0.0609334
$$756$$ 0 0
$$757$$ −13.3436 −0.484980 −0.242490 0.970154i $$-0.577964\pi$$
−0.242490 + 0.970154i $$0.577964\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 11.6962 0.423986 0.211993 0.977271i $$-0.432005\pi$$
0.211993 + 0.977271i $$0.432005\pi$$
$$762$$ 0 0
$$763$$ −23.3990 −0.847102
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 36.0015 1.29994
$$768$$ 0 0
$$769$$ 41.2234 1.48655 0.743276 0.668985i $$-0.233271\pi$$
0.743276 + 0.668985i $$0.233271\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ 12.9999 0.467574 0.233787 0.972288i $$-0.424888\pi$$
0.233787 + 0.972288i $$0.424888\pi$$
$$774$$ 0 0
$$775$$ −51.1728 −1.83818
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 2.04380 0.0732269
$$780$$ 0 0
$$781$$ 0.566034 0.0202543
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ −16.4181 −0.585988
$$786$$ 0 0
$$787$$ 4.63820 0.165334 0.0826669 0.996577i $$-0.473656\pi$$
0.0826669 + 0.996577i $$0.473656\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 51.4934 1.83089
$$792$$ 0 0
$$793$$ −41.7983 −1.48430
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 11.0271 0.390600 0.195300 0.980744i $$-0.437432\pi$$
0.195300 + 0.980744i $$0.437432\pi$$
$$798$$ 0 0
$$799$$ 7.09226 0.250906
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ −7.29948 −0.257593
$$804$$ 0 0
$$805$$ −121.970 −4.29888
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 18.8659 0.663290 0.331645 0.943404i $$-0.392396\pi$$
0.331645 + 0.943404i $$0.392396\pi$$
$$810$$ 0 0
$$811$$ −13.9642 −0.490348 −0.245174 0.969479i $$-0.578845\pi$$
−0.245174 + 0.969479i $$0.578845\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 43.4076 1.52050
$$816$$ 0 0
$$817$$ −2.16767 −0.0758371
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 30.8485 1.07662 0.538310 0.842747i $$-0.319063\pi$$
0.538310 + 0.842747i $$0.319063\pi$$
$$822$$ 0 0
$$823$$ −9.48137 −0.330500 −0.165250 0.986252i $$-0.552843\pi$$
−0.165250 + 0.986252i $$0.552843\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −16.7454 −0.582296 −0.291148 0.956678i $$-0.594037\pi$$
−0.291148 + 0.956678i $$0.594037\pi$$
$$828$$ 0 0
$$829$$ 1.59130 0.0552682 0.0276341 0.999618i $$-0.491203\pi$$
0.0276341 + 0.999618i $$0.491203\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 8.05085 0.278945
$$834$$ 0 0
$$835$$ 35.6913 1.23515
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 5.12222 0.176839 0.0884193 0.996083i $$-0.471818\pi$$
0.0884193 + 0.996083i $$0.471818\pi$$
$$840$$ 0 0
$$841$$ 7.32587 0.252616
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ 88.0154 3.02782
$$846$$ 0 0
$$847$$ −4.31127 −0.148137
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 78.8516 2.70300
$$852$$ 0 0
$$853$$ −51.2020 −1.75312 −0.876561 0.481290i $$-0.840168\pi$$
−0.876561 + 0.481290i $$0.840168\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 7.92412 0.270683 0.135341 0.990799i $$-0.456787\pi$$
0.135341 + 0.990799i $$0.456787\pi$$
$$858$$ 0 0
$$859$$ −39.4668 −1.34659 −0.673296 0.739373i $$-0.735122\pi$$
−0.673296 + 0.739373i $$0.735122\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 52.1231 1.77429 0.887145 0.461491i $$-0.152685\pi$$
0.887145 + 0.461491i $$0.152685\pi$$
$$864$$ 0 0
$$865$$ 46.3817 1.57703
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 13.9103 0.471873
$$870$$ 0 0
$$871$$ 51.0102 1.72841
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ −49.5563 −1.67531
$$876$$ 0 0
$$877$$ 20.6344 0.696773 0.348386 0.937351i $$-0.386730\pi$$
0.348386 + 0.937351i $$0.386730\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −42.3008 −1.42515 −0.712574 0.701597i $$-0.752471\pi$$
−0.712574 + 0.701597i $$0.752471\pi$$
$$882$$ 0 0
$$883$$ −47.1397 −1.58638 −0.793188 0.608977i $$-0.791580\pi$$
−0.793188 + 0.608977i $$0.791580\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −5.41860 −0.181939 −0.0909694 0.995854i $$-0.528997\pi$$
−0.0909694 + 0.995854i $$0.528997\pi$$
$$888$$ 0 0
$$889$$ −33.3180 −1.11745
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 10.2074 0.341578
$$894$$ 0 0
$$895$$ 29.5129 0.986506
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 37.7615 1.25942
$$900$$ 0 0
$$901$$ 1.36043 0.0453226
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 67.3284 2.23807
$$906$$ 0 0
$$907$$ 22.1148 0.734309 0.367154 0.930160i $$-0.380332\pi$$
0.367154 + 0.930160i $$0.380332\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −18.0321 −0.597429 −0.298715 0.954343i $$-0.596558\pi$$
−0.298715 + 0.954343i $$0.596558\pi$$
$$912$$ 0 0
$$913$$ −7.05772 −0.233576
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ −45.6604 −1.50784
$$918$$ 0 0
$$919$$ −36.9395 −1.21852 −0.609261 0.792970i $$-0.708534\pi$$
−0.609261 + 0.792970i $$0.708534\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 3.45490 0.113720
$$924$$ 0 0
$$925$$ 82.6066 2.71609
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ −10.2792 −0.337250 −0.168625 0.985680i $$-0.553933\pi$$
−0.168625 + 0.985680i $$0.553933\pi$$
$$930$$ 0 0
$$931$$ 11.5870 0.379750
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ −2.52129 −0.0824551
$$936$$ 0 0
$$937$$ 1.42527 0.0465615 0.0232807 0.999729i $$-0.492589\pi$$
0.0232807 + 0.999729i $$0.492589\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −32.0967 −1.04632 −0.523161 0.852234i $$-0.675247\pi$$
−0.523161 + 0.852234i $$0.675247\pi$$
$$942$$ 0 0
$$943$$ 15.9343 0.518892
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 8.61405 0.279919 0.139959 0.990157i $$-0.455303\pi$$
0.139959 + 0.990157i $$0.455303\pi$$
$$948$$ 0 0
$$949$$ −44.5538 −1.44628
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −6.48103 −0.209941 −0.104971 0.994475i $$-0.533475\pi$$
−0.104971 + 0.994475i $$0.533475\pi$$
$$954$$ 0 0
$$955$$ 38.5609 1.24780
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −37.7162 −1.21792
$$960$$ 0 0
$$961$$ 8.25380 0.266252
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ −2.75671 −0.0887415
$$966$$ 0 0
$$967$$ 55.2454 1.77657 0.888286 0.459292i $$-0.151897\pi$$
0.888286 + 0.459292i $$0.151897\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −30.1518 −0.967618 −0.483809 0.875174i $$-0.660747\pi$$
−0.483809 + 0.875174i $$0.660747\pi$$
$$972$$ 0 0
$$973$$ −44.2750 −1.41939
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −22.6608 −0.724984 −0.362492 0.931987i $$-0.618074\pi$$
−0.362492 + 0.931987i $$0.618074\pi$$
$$978$$ 0 0
$$979$$ −14.3590 −0.458917
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ −17.3274 −0.552658 −0.276329 0.961063i $$-0.589118\pi$$
−0.276329 + 0.961063i $$0.589118\pi$$
$$984$$ 0 0
$$985$$ −6.12087 −0.195027
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −16.9000 −0.537389
$$990$$ 0 0
$$991$$ 27.1931 0.863818 0.431909 0.901917i $$-0.357840\pi$$
0.431909 + 0.901917i $$0.357840\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 87.8709 2.78569
$$996$$ 0 0
$$997$$ 21.0570 0.666881 0.333441 0.942771i $$-0.391790\pi$$
0.333441 + 0.942771i $$0.391790\pi$$
$$998$$ 0 0
$$999$$ 0 0
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 7524.2.a.r.1.6 6
3.2 odd 2 836.2.a.d.1.2 6
12.11 even 2 3344.2.a.x.1.5 6
33.32 even 2 9196.2.a.k.1.2 6

By twisted newform
Twist Min Dim Char Parity Ord Type
836.2.a.d.1.2 6 3.2 odd 2
3344.2.a.x.1.5 6 12.11 even 2
7524.2.a.r.1.6 6 1.1 even 1 trivial
9196.2.a.k.1.2 6 33.32 even 2