# Properties

 Label 3192.2.a.v.1.3 Level $3192$ Weight $2$ Character 3192.1 Self dual yes Analytic conductor $25.488$ Analytic rank $1$ 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] = [3192,2,Mod(1,3192)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3192, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("3192.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3192 = 2^{3} \cdot 3 \cdot 7 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3192.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: $$25.4882483252$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.316.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 4x + 2$$ x^3 - x^2 - 4*x + 2 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.3 Root $$2.34292$$ of defining polynomial Character $$\chi$$ $$=$$ 3192.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{3} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{3} -1.00000 q^{7} +1.00000 q^{9} +2.68585 q^{11} -6.97858 q^{13} +2.29273 q^{17} -1.00000 q^{19} -1.00000 q^{21} -4.39312 q^{23} -5.00000 q^{25} +1.00000 q^{27} +0.978577 q^{29} -7.66442 q^{31} +2.68585 q^{33} +5.66442 q^{37} -6.97858 q^{39} +7.37169 q^{41} -9.37169 q^{43} -4.68585 q^{47} +1.00000 q^{49} +2.29273 q^{51} -12.9786 q^{53} -1.00000 q^{57} -0.585462 q^{59} -6.58546 q^{61} -1.00000 q^{63} +12.0575 q^{67} -4.39312 q^{69} -1.60688 q^{71} -7.37169 q^{73} -5.00000 q^{75} -2.68585 q^{77} +14.3503 q^{79} +1.00000 q^{81} +8.35027 q^{83} +0.978577 q^{87} -15.9572 q^{89} +6.97858 q^{91} -7.66442 q^{93} +5.27131 q^{97} +2.68585 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 3 q^{3} - 3 q^{7} + 3 q^{9}+O(q^{10})$$ 3 * q + 3 * q^3 - 3 * q^7 + 3 * q^9 $$3 q + 3 q^{3} - 3 q^{7} + 3 q^{9} - 4 q^{11} - 6 q^{13} + 4 q^{17} - 3 q^{19} - 3 q^{21} - 4 q^{23} - 15 q^{25} + 3 q^{27} - 12 q^{29} + 4 q^{31} - 4 q^{33} - 10 q^{37} - 6 q^{39} - 2 q^{41} - 4 q^{43} - 2 q^{47} + 3 q^{49} + 4 q^{51} - 24 q^{53} - 3 q^{57} + 4 q^{59} - 14 q^{61} - 3 q^{63} - 4 q^{69} - 14 q^{71} + 2 q^{73} - 15 q^{75} + 4 q^{77} + 4 q^{79} + 3 q^{81} - 14 q^{83} - 12 q^{87} - 18 q^{89} + 6 q^{91} + 4 q^{93} - 2 q^{97} - 4 q^{99}+O(q^{100})$$ 3 * q + 3 * q^3 - 3 * q^7 + 3 * q^9 - 4 * q^11 - 6 * q^13 + 4 * q^17 - 3 * q^19 - 3 * q^21 - 4 * q^23 - 15 * q^25 + 3 * q^27 - 12 * q^29 + 4 * q^31 - 4 * q^33 - 10 * q^37 - 6 * q^39 - 2 * q^41 - 4 * q^43 - 2 * q^47 + 3 * q^49 + 4 * q^51 - 24 * q^53 - 3 * q^57 + 4 * q^59 - 14 * q^61 - 3 * q^63 - 4 * q^69 - 14 * q^71 + 2 * q^73 - 15 * q^75 + 4 * q^77 + 4 * q^79 + 3 * q^81 - 14 * q^83 - 12 * q^87 - 18 * q^89 + 6 * q^91 + 4 * q^93 - 2 * q^97 - 4 * q^99

## 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
$$3$$ 1.00000 0.577350
$$4$$ 0 0
$$5$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$6$$ 0 0
$$7$$ −1.00000 −0.377964
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 2.68585 0.809813 0.404907 0.914358i $$-0.367304\pi$$
0.404907 + 0.914358i $$0.367304\pi$$
$$12$$ 0 0
$$13$$ −6.97858 −1.93551 −0.967755 0.251895i $$-0.918946\pi$$
−0.967755 + 0.251895i $$0.918946\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 2.29273 0.556069 0.278034 0.960571i $$-0.410317\pi$$
0.278034 + 0.960571i $$0.410317\pi$$
$$18$$ 0 0
$$19$$ −1.00000 −0.229416
$$20$$ 0 0
$$21$$ −1.00000 −0.218218
$$22$$ 0 0
$$23$$ −4.39312 −0.916028 −0.458014 0.888945i $$-0.651439\pi$$
−0.458014 + 0.888945i $$0.651439\pi$$
$$24$$ 0 0
$$25$$ −5.00000 −1.00000
$$26$$ 0 0
$$27$$ 1.00000 0.192450
$$28$$ 0 0
$$29$$ 0.978577 0.181717 0.0908586 0.995864i $$-0.471039\pi$$
0.0908586 + 0.995864i $$0.471039\pi$$
$$30$$ 0 0
$$31$$ −7.66442 −1.37657 −0.688286 0.725440i $$-0.741636\pi$$
−0.688286 + 0.725440i $$0.741636\pi$$
$$32$$ 0 0
$$33$$ 2.68585 0.467546
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 5.66442 0.931225 0.465613 0.884989i $$-0.345834\pi$$
0.465613 + 0.884989i $$0.345834\pi$$
$$38$$ 0 0
$$39$$ −6.97858 −1.11747
$$40$$ 0 0
$$41$$ 7.37169 1.15126 0.575632 0.817709i $$-0.304756\pi$$
0.575632 + 0.817709i $$0.304756\pi$$
$$42$$ 0 0
$$43$$ −9.37169 −1.42917 −0.714585 0.699549i $$-0.753384\pi$$
−0.714585 + 0.699549i $$0.753384\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −4.68585 −0.683501 −0.341750 0.939791i $$-0.611020\pi$$
−0.341750 + 0.939791i $$0.611020\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ 2.29273 0.321047
$$52$$ 0 0
$$53$$ −12.9786 −1.78274 −0.891372 0.453272i $$-0.850257\pi$$
−0.891372 + 0.453272i $$0.850257\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ −1.00000 −0.132453
$$58$$ 0 0
$$59$$ −0.585462 −0.0762207 −0.0381103 0.999274i $$-0.512134\pi$$
−0.0381103 + 0.999274i $$0.512134\pi$$
$$60$$ 0 0
$$61$$ −6.58546 −0.843182 −0.421591 0.906786i $$-0.638528\pi$$
−0.421591 + 0.906786i $$0.638528\pi$$
$$62$$ 0 0
$$63$$ −1.00000 −0.125988
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 12.0575 1.47306 0.736531 0.676403i $$-0.236462\pi$$
0.736531 + 0.676403i $$0.236462\pi$$
$$68$$ 0 0
$$69$$ −4.39312 −0.528869
$$70$$ 0 0
$$71$$ −1.60688 −0.190702 −0.0953511 0.995444i $$-0.530397\pi$$
−0.0953511 + 0.995444i $$0.530397\pi$$
$$72$$ 0 0
$$73$$ −7.37169 −0.862791 −0.431396 0.902163i $$-0.641979\pi$$
−0.431396 + 0.902163i $$0.641979\pi$$
$$74$$ 0 0
$$75$$ −5.00000 −0.577350
$$76$$ 0 0
$$77$$ −2.68585 −0.306081
$$78$$ 0 0
$$79$$ 14.3503 1.61453 0.807266 0.590188i $$-0.200946\pi$$
0.807266 + 0.590188i $$0.200946\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ 8.35027 0.916561 0.458281 0.888808i $$-0.348465\pi$$
0.458281 + 0.888808i $$0.348465\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 0.978577 0.104914
$$88$$ 0 0
$$89$$ −15.9572 −1.69145 −0.845727 0.533615i $$-0.820833\pi$$
−0.845727 + 0.533615i $$0.820833\pi$$
$$90$$ 0 0
$$91$$ 6.97858 0.731554
$$92$$ 0 0
$$93$$ −7.66442 −0.794764
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 5.27131 0.535220 0.267610 0.963527i $$-0.413766\pi$$
0.267610 + 0.963527i $$0.413766\pi$$
$$98$$ 0 0
$$99$$ 2.68585 0.269938
$$100$$ 0 0
$$101$$ −8.00000 −0.796030 −0.398015 0.917379i $$-0.630301\pi$$
−0.398015 + 0.917379i $$0.630301\pi$$
$$102$$ 0 0
$$103$$ −11.0790 −1.09164 −0.545821 0.837902i $$-0.683782\pi$$
−0.545821 + 0.837902i $$0.683782\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −7.56404 −0.731243 −0.365622 0.930764i $$-0.619144\pi$$
−0.365622 + 0.930764i $$0.619144\pi$$
$$108$$ 0 0
$$109$$ −4.87819 −0.467246 −0.233623 0.972327i $$-0.575058\pi$$
−0.233623 + 0.972327i $$0.575058\pi$$
$$110$$ 0 0
$$111$$ 5.66442 0.537643
$$112$$ 0 0
$$113$$ 10.3503 0.973671 0.486836 0.873494i $$-0.338151\pi$$
0.486836 + 0.873494i $$0.338151\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ −6.97858 −0.645170
$$118$$ 0 0
$$119$$ −2.29273 −0.210174
$$120$$ 0 0
$$121$$ −3.78623 −0.344203
$$122$$ 0 0
$$123$$ 7.37169 0.664683
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −0.393115 −0.0348833 −0.0174417 0.999848i $$-0.505552\pi$$
−0.0174417 + 0.999848i $$0.505552\pi$$
$$128$$ 0 0
$$129$$ −9.37169 −0.825132
$$130$$ 0 0
$$131$$ −1.80765 −0.157935 −0.0789677 0.996877i $$-0.525162\pi$$
−0.0789677 + 0.996877i $$0.525162\pi$$
$$132$$ 0 0
$$133$$ 1.00000 0.0867110
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −14.7862 −1.26327 −0.631636 0.775265i $$-0.717616\pi$$
−0.631636 + 0.775265i $$0.717616\pi$$
$$138$$ 0 0
$$139$$ −13.3717 −1.13417 −0.567086 0.823659i $$-0.691929\pi$$
−0.567086 + 0.823659i $$0.691929\pi$$
$$140$$ 0 0
$$141$$ −4.68585 −0.394619
$$142$$ 0 0
$$143$$ −18.7434 −1.56740
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 1.00000 0.0824786
$$148$$ 0 0
$$149$$ 5.07896 0.416085 0.208042 0.978120i $$-0.433291\pi$$
0.208042 + 0.978120i $$0.433291\pi$$
$$150$$ 0 0
$$151$$ 9.56404 0.778310 0.389155 0.921172i $$-0.372767\pi$$
0.389155 + 0.921172i $$0.372767\pi$$
$$152$$ 0 0
$$153$$ 2.29273 0.185356
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 15.9572 1.27352 0.636760 0.771062i $$-0.280274\pi$$
0.636760 + 0.771062i $$0.280274\pi$$
$$158$$ 0 0
$$159$$ −12.9786 −1.02927
$$160$$ 0 0
$$161$$ 4.39312 0.346226
$$162$$ 0 0
$$163$$ −1.37169 −0.107439 −0.0537196 0.998556i $$-0.517108\pi$$
−0.0537196 + 0.998556i $$0.517108\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 9.37169 0.725203 0.362602 0.931944i $$-0.381889\pi$$
0.362602 + 0.931944i $$0.381889\pi$$
$$168$$ 0 0
$$169$$ 35.7005 2.74620
$$170$$ 0 0
$$171$$ −1.00000 −0.0764719
$$172$$ 0 0
$$173$$ −18.7862 −1.42829 −0.714145 0.699997i $$-0.753184\pi$$
−0.714145 + 0.699997i $$0.753184\pi$$
$$174$$ 0 0
$$175$$ 5.00000 0.377964
$$176$$ 0 0
$$177$$ −0.585462 −0.0440060
$$178$$ 0 0
$$179$$ −4.35027 −0.325154 −0.162577 0.986696i $$-0.551981\pi$$
−0.162577 + 0.986696i $$0.551981\pi$$
$$180$$ 0 0
$$181$$ −10.9786 −0.816031 −0.408016 0.912975i $$-0.633779\pi$$
−0.408016 + 0.912975i $$0.633779\pi$$
$$182$$ 0 0
$$183$$ −6.58546 −0.486811
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 6.15792 0.450312
$$188$$ 0 0
$$189$$ −1.00000 −0.0727393
$$190$$ 0 0
$$191$$ 14.3503 1.03835 0.519175 0.854668i $$-0.326239\pi$$
0.519175 + 0.854668i $$0.326239\pi$$
$$192$$ 0 0
$$193$$ −13.3288 −0.959431 −0.479716 0.877424i $$-0.659260\pi$$
−0.479716 + 0.877424i $$0.659260\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −26.4507 −1.88453 −0.942266 0.334867i $$-0.891309\pi$$
−0.942266 + 0.334867i $$0.891309\pi$$
$$198$$ 0 0
$$199$$ 4.78623 0.339287 0.169643 0.985506i $$-0.445738\pi$$
0.169643 + 0.985506i $$0.445738\pi$$
$$200$$ 0 0
$$201$$ 12.0575 0.850473
$$202$$ 0 0
$$203$$ −0.978577 −0.0686827
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ −4.39312 −0.305343
$$208$$ 0 0
$$209$$ −2.68585 −0.185784
$$210$$ 0 0
$$211$$ 18.6858 1.28639 0.643193 0.765704i $$-0.277609\pi$$
0.643193 + 0.765704i $$0.277609\pi$$
$$212$$ 0 0
$$213$$ −1.60688 −0.110102
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 7.66442 0.520295
$$218$$ 0 0
$$219$$ −7.37169 −0.498133
$$220$$ 0 0
$$221$$ −16.0000 −1.07628
$$222$$ 0 0
$$223$$ −22.2070 −1.48709 −0.743547 0.668684i $$-0.766858\pi$$
−0.743547 + 0.668684i $$0.766858\pi$$
$$224$$ 0 0
$$225$$ −5.00000 −0.333333
$$226$$ 0 0
$$227$$ 2.62831 0.174447 0.0872235 0.996189i $$-0.472201\pi$$
0.0872235 + 0.996189i $$0.472201\pi$$
$$228$$ 0 0
$$229$$ 11.9572 0.790151 0.395075 0.918649i $$-0.370718\pi$$
0.395075 + 0.918649i $$0.370718\pi$$
$$230$$ 0 0
$$231$$ −2.68585 −0.176716
$$232$$ 0 0
$$233$$ −6.58546 −0.431428 −0.215714 0.976457i $$-0.569208\pi$$
−0.215714 + 0.976457i $$0.569208\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 14.3503 0.932150
$$238$$ 0 0
$$239$$ 12.9786 0.839514 0.419757 0.907636i $$-0.362115\pi$$
0.419757 + 0.907636i $$0.362115\pi$$
$$240$$ 0 0
$$241$$ 6.72869 0.433433 0.216717 0.976235i $$-0.430465\pi$$
0.216717 + 0.976235i $$0.430465\pi$$
$$242$$ 0 0
$$243$$ 1.00000 0.0641500
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 6.97858 0.444036
$$248$$ 0 0
$$249$$ 8.35027 0.529177
$$250$$ 0 0
$$251$$ −1.80765 −0.114098 −0.0570490 0.998371i $$-0.518169\pi$$
−0.0570490 + 0.998371i $$0.518169\pi$$
$$252$$ 0 0
$$253$$ −11.7992 −0.741811
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −15.3717 −0.958860 −0.479430 0.877580i $$-0.659157\pi$$
−0.479430 + 0.877580i $$0.659157\pi$$
$$258$$ 0 0
$$259$$ −5.66442 −0.351970
$$260$$ 0 0
$$261$$ 0.978577 0.0605724
$$262$$ 0 0
$$263$$ 18.3503 1.13153 0.565763 0.824568i $$-0.308582\pi$$
0.565763 + 0.824568i $$0.308582\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ −15.9572 −0.976562
$$268$$ 0 0
$$269$$ −10.5855 −0.645407 −0.322704 0.946500i $$-0.604592\pi$$
−0.322704 + 0.946500i $$0.604592\pi$$
$$270$$ 0 0
$$271$$ −8.00000 −0.485965 −0.242983 0.970031i $$-0.578126\pi$$
−0.242983 + 0.970031i $$0.578126\pi$$
$$272$$ 0 0
$$273$$ 6.97858 0.422363
$$274$$ 0 0
$$275$$ −13.4292 −0.809813
$$276$$ 0 0
$$277$$ 1.21377 0.0729283 0.0364642 0.999335i $$-0.488391\pi$$
0.0364642 + 0.999335i $$0.488391\pi$$
$$278$$ 0 0
$$279$$ −7.66442 −0.458857
$$280$$ 0 0
$$281$$ −25.5640 −1.52502 −0.762511 0.646975i $$-0.776034\pi$$
−0.762511 + 0.646975i $$0.776034\pi$$
$$282$$ 0 0
$$283$$ 22.5426 1.34002 0.670010 0.742352i $$-0.266290\pi$$
0.670010 + 0.742352i $$0.266290\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −7.37169 −0.435137
$$288$$ 0 0
$$289$$ −11.7434 −0.690787
$$290$$ 0 0
$$291$$ 5.27131 0.309010
$$292$$ 0 0
$$293$$ −19.2860 −1.12670 −0.563350 0.826218i $$-0.690488\pi$$
−0.563350 + 0.826218i $$0.690488\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 2.68585 0.155849
$$298$$ 0 0
$$299$$ 30.6577 1.77298
$$300$$ 0 0
$$301$$ 9.37169 0.540175
$$302$$ 0 0
$$303$$ −8.00000 −0.459588
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 4.78623 0.273165 0.136582 0.990629i $$-0.456388\pi$$
0.136582 + 0.990629i $$0.456388\pi$$
$$308$$ 0 0
$$309$$ −11.0790 −0.630260
$$310$$ 0 0
$$311$$ 19.4292 1.10173 0.550865 0.834594i $$-0.314298\pi$$
0.550865 + 0.834594i $$0.314298\pi$$
$$312$$ 0 0
$$313$$ 25.3288 1.43167 0.715836 0.698269i $$-0.246046\pi$$
0.715836 + 0.698269i $$0.246046\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 29.6791 1.66695 0.833473 0.552561i $$-0.186349\pi$$
0.833473 + 0.552561i $$0.186349\pi$$
$$318$$ 0 0
$$319$$ 2.62831 0.147157
$$320$$ 0 0
$$321$$ −7.56404 −0.422183
$$322$$ 0 0
$$323$$ −2.29273 −0.127571
$$324$$ 0 0
$$325$$ 34.8929 1.93551
$$326$$ 0 0
$$327$$ −4.87819 −0.269765
$$328$$ 0 0
$$329$$ 4.68585 0.258339
$$330$$ 0 0
$$331$$ −5.22846 −0.287382 −0.143691 0.989623i $$-0.545897\pi$$
−0.143691 + 0.989623i $$0.545897\pi$$
$$332$$ 0 0
$$333$$ 5.66442 0.310408
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −9.32885 −0.508175 −0.254087 0.967181i $$-0.581775\pi$$
−0.254087 + 0.967181i $$0.581775\pi$$
$$338$$ 0 0
$$339$$ 10.3503 0.562149
$$340$$ 0 0
$$341$$ −20.5855 −1.11477
$$342$$ 0 0
$$343$$ −1.00000 −0.0539949
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −19.3864 −1.04072 −0.520358 0.853948i $$-0.674201\pi$$
−0.520358 + 0.853948i $$0.674201\pi$$
$$348$$ 0 0
$$349$$ 7.95715 0.425937 0.212968 0.977059i $$-0.431687\pi$$
0.212968 + 0.977059i $$0.431687\pi$$
$$350$$ 0 0
$$351$$ −6.97858 −0.372489
$$352$$ 0 0
$$353$$ 33.6216 1.78950 0.894748 0.446571i $$-0.147355\pi$$
0.894748 + 0.446571i $$0.147355\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ −2.29273 −0.121344
$$358$$ 0 0
$$359$$ 11.7220 0.618661 0.309331 0.950955i $$-0.399895\pi$$
0.309331 + 0.950955i $$0.399895\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 0 0
$$363$$ −3.78623 −0.198726
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −11.2138 −0.585354 −0.292677 0.956211i $$-0.594546\pi$$
−0.292677 + 0.956211i $$0.594546\pi$$
$$368$$ 0 0
$$369$$ 7.37169 0.383755
$$370$$ 0 0
$$371$$ 12.9786 0.673814
$$372$$ 0 0
$$373$$ −18.4507 −0.955339 −0.477669 0.878540i $$-0.658518\pi$$
−0.477669 + 0.878540i $$0.658518\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −6.82908 −0.351715
$$378$$ 0 0
$$379$$ 11.9425 0.613443 0.306722 0.951799i $$-0.400768\pi$$
0.306722 + 0.951799i $$0.400768\pi$$
$$380$$ 0 0
$$381$$ −0.393115 −0.0201399
$$382$$ 0 0
$$383$$ 3.21377 0.164216 0.0821080 0.996623i $$-0.473835\pi$$
0.0821080 + 0.996623i $$0.473835\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −9.37169 −0.476390
$$388$$ 0 0
$$389$$ 24.2070 1.22735 0.613673 0.789560i $$-0.289691\pi$$
0.613673 + 0.789560i $$0.289691\pi$$
$$390$$ 0 0
$$391$$ −10.0722 −0.509375
$$392$$ 0 0
$$393$$ −1.80765 −0.0911840
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −6.20077 −0.311208 −0.155604 0.987820i $$-0.549732\pi$$
−0.155604 + 0.987820i $$0.549732\pi$$
$$398$$ 0 0
$$399$$ 1.00000 0.0500626
$$400$$ 0 0
$$401$$ 5.17935 0.258644 0.129322 0.991603i $$-0.458720\pi$$
0.129322 + 0.991603i $$0.458720\pi$$
$$402$$ 0 0
$$403$$ 53.4868 2.66437
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 15.2138 0.754119
$$408$$ 0 0
$$409$$ −12.6002 −0.623038 −0.311519 0.950240i $$-0.600838\pi$$
−0.311519 + 0.950240i $$0.600838\pi$$
$$410$$ 0 0
$$411$$ −14.7862 −0.729351
$$412$$ 0 0
$$413$$ 0.585462 0.0288087
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ −13.3717 −0.654815
$$418$$ 0 0
$$419$$ 25.1365 1.22800 0.613999 0.789307i $$-0.289560\pi$$
0.613999 + 0.789307i $$0.289560\pi$$
$$420$$ 0 0
$$421$$ −20.4935 −0.998792 −0.499396 0.866374i $$-0.666445\pi$$
−0.499396 + 0.866374i $$0.666445\pi$$
$$422$$ 0 0
$$423$$ −4.68585 −0.227834
$$424$$ 0 0
$$425$$ −11.4637 −0.556069
$$426$$ 0 0
$$427$$ 6.58546 0.318693
$$428$$ 0 0
$$429$$ −18.7434 −0.904939
$$430$$ 0 0
$$431$$ 34.3074 1.65253 0.826265 0.563281i $$-0.190461\pi$$
0.826265 + 0.563281i $$0.190461\pi$$
$$432$$ 0 0
$$433$$ −34.5573 −1.66072 −0.830359 0.557229i $$-0.811865\pi$$
−0.830359 + 0.557229i $$0.811865\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 4.39312 0.210151
$$438$$ 0 0
$$439$$ −4.92104 −0.234868 −0.117434 0.993081i $$-0.537467\pi$$
−0.117434 + 0.993081i $$0.537467\pi$$
$$440$$ 0 0
$$441$$ 1.00000 0.0476190
$$442$$ 0 0
$$443$$ −12.4422 −0.591148 −0.295574 0.955320i $$-0.595511\pi$$
−0.295574 + 0.955320i $$0.595511\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 5.07896 0.240227
$$448$$ 0 0
$$449$$ −27.6069 −1.30285 −0.651425 0.758713i $$-0.725828\pi$$
−0.651425 + 0.758713i $$0.725828\pi$$
$$450$$ 0 0
$$451$$ 19.7992 0.932309
$$452$$ 0 0
$$453$$ 9.56404 0.449358
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 23.5725 1.10267 0.551337 0.834283i $$-0.314118\pi$$
0.551337 + 0.834283i $$0.314118\pi$$
$$458$$ 0 0
$$459$$ 2.29273 0.107016
$$460$$ 0 0
$$461$$ −3.41454 −0.159031 −0.0795154 0.996834i $$-0.525337\pi$$
−0.0795154 + 0.996834i $$0.525337\pi$$
$$462$$ 0 0
$$463$$ 34.7434 1.61466 0.807331 0.590099i $$-0.200911\pi$$
0.807331 + 0.590099i $$0.200911\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −16.3503 −0.756600 −0.378300 0.925683i $$-0.623491\pi$$
−0.378300 + 0.925683i $$0.623491\pi$$
$$468$$ 0 0
$$469$$ −12.0575 −0.556765
$$470$$ 0 0
$$471$$ 15.9572 0.735267
$$472$$ 0 0
$$473$$ −25.1709 −1.15736
$$474$$ 0 0
$$475$$ 5.00000 0.229416
$$476$$ 0 0
$$477$$ −12.9786 −0.594248
$$478$$ 0 0
$$479$$ −12.6002 −0.575716 −0.287858 0.957673i $$-0.592943\pi$$
−0.287858 + 0.957673i $$0.592943\pi$$
$$480$$ 0 0
$$481$$ −39.5296 −1.80240
$$482$$ 0 0
$$483$$ 4.39312 0.199894
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −25.7648 −1.16751 −0.583757 0.811928i $$-0.698418\pi$$
−0.583757 + 0.811928i $$0.698418\pi$$
$$488$$ 0 0
$$489$$ −1.37169 −0.0620301
$$490$$ 0 0
$$491$$ −20.0575 −0.905184 −0.452592 0.891718i $$-0.649501\pi$$
−0.452592 + 0.891718i $$0.649501\pi$$
$$492$$ 0 0
$$493$$ 2.24361 0.101047
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 1.60688 0.0720786
$$498$$ 0 0
$$499$$ 29.2860 1.31102 0.655511 0.755186i $$-0.272453\pi$$
0.655511 + 0.755186i $$0.272453\pi$$
$$500$$ 0 0
$$501$$ 9.37169 0.418696
$$502$$ 0 0
$$503$$ 19.4292 0.866307 0.433153 0.901320i $$-0.357401\pi$$
0.433153 + 0.901320i $$0.357401\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 35.7005 1.58552
$$508$$ 0 0
$$509$$ 20.0722 0.889686 0.444843 0.895609i $$-0.353259\pi$$
0.444843 + 0.895609i $$0.353259\pi$$
$$510$$ 0 0
$$511$$ 7.37169 0.326104
$$512$$ 0 0
$$513$$ −1.00000 −0.0441511
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −12.5855 −0.553508
$$518$$ 0 0
$$519$$ −18.7862 −0.824624
$$520$$ 0 0
$$521$$ −39.4868 −1.72995 −0.864973 0.501818i $$-0.832665\pi$$
−0.864973 + 0.501818i $$0.832665\pi$$
$$522$$ 0 0
$$523$$ −31.1281 −1.36114 −0.680568 0.732685i $$-0.738267\pi$$
−0.680568 + 0.732685i $$0.738267\pi$$
$$524$$ 0 0
$$525$$ 5.00000 0.218218
$$526$$ 0 0
$$527$$ −17.5725 −0.765468
$$528$$ 0 0
$$529$$ −3.70054 −0.160893
$$530$$ 0 0
$$531$$ −0.585462 −0.0254069
$$532$$ 0 0
$$533$$ −51.4439 −2.22828
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ −4.35027 −0.187728
$$538$$ 0 0
$$539$$ 2.68585 0.115688
$$540$$ 0 0
$$541$$ 22.7862 0.979657 0.489828 0.871819i $$-0.337059\pi$$
0.489828 + 0.871819i $$0.337059\pi$$
$$542$$ 0 0
$$543$$ −10.9786 −0.471136
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −2.30115 −0.0983902 −0.0491951 0.998789i $$-0.515666\pi$$
−0.0491951 + 0.998789i $$0.515666\pi$$
$$548$$ 0 0
$$549$$ −6.58546 −0.281061
$$550$$ 0 0
$$551$$ −0.978577 −0.0416888
$$552$$ 0 0
$$553$$ −14.3503 −0.610236
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 27.7367 1.17524 0.587620 0.809137i $$-0.300065\pi$$
0.587620 + 0.809137i $$0.300065\pi$$
$$558$$ 0 0
$$559$$ 65.4011 2.76617
$$560$$ 0 0
$$561$$ 6.15792 0.259988
$$562$$ 0 0
$$563$$ 22.5426 0.950058 0.475029 0.879970i $$-0.342438\pi$$
0.475029 + 0.879970i $$0.342438\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ −1.00000 −0.0419961
$$568$$ 0 0
$$569$$ −6.43596 −0.269810 −0.134905 0.990859i $$-0.543073\pi$$
−0.134905 + 0.990859i $$0.543073\pi$$
$$570$$ 0 0
$$571$$ −7.91431 −0.331204 −0.165602 0.986193i $$-0.552957\pi$$
−0.165602 + 0.986193i $$0.552957\pi$$
$$572$$ 0 0
$$573$$ 14.3503 0.599491
$$574$$ 0 0
$$575$$ 21.9656 0.916028
$$576$$ 0 0
$$577$$ 28.7434 1.19660 0.598301 0.801271i $$-0.295843\pi$$
0.598301 + 0.801271i $$0.295843\pi$$
$$578$$ 0 0
$$579$$ −13.3288 −0.553928
$$580$$ 0 0
$$581$$ −8.35027 −0.346428
$$582$$ 0 0
$$583$$ −34.8585 −1.44369
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −38.1067 −1.57283 −0.786415 0.617699i $$-0.788065\pi$$
−0.786415 + 0.617699i $$0.788065\pi$$
$$588$$ 0 0
$$589$$ 7.66442 0.315807
$$590$$ 0 0
$$591$$ −26.4507 −1.08803
$$592$$ 0 0
$$593$$ −20.4507 −0.839808 −0.419904 0.907569i $$-0.637936\pi$$
−0.419904 + 0.907569i $$0.637936\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 4.78623 0.195887
$$598$$ 0 0
$$599$$ −2.30742 −0.0942788 −0.0471394 0.998888i $$-0.515010\pi$$
−0.0471394 + 0.998888i $$0.515010\pi$$
$$600$$ 0 0
$$601$$ 39.3435 1.60486 0.802428 0.596749i $$-0.203541\pi$$
0.802428 + 0.596749i $$0.203541\pi$$
$$602$$ 0 0
$$603$$ 12.0575 0.491021
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −29.7073 −1.20578 −0.602890 0.797824i $$-0.705984\pi$$
−0.602890 + 0.797824i $$0.705984\pi$$
$$608$$ 0 0
$$609$$ −0.978577 −0.0396539
$$610$$ 0 0
$$611$$ 32.7005 1.32292
$$612$$ 0 0
$$613$$ 4.62831 0.186936 0.0934678 0.995622i $$-0.470205\pi$$
0.0934678 + 0.995622i $$0.470205\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −31.2860 −1.25953 −0.629763 0.776787i $$-0.716848\pi$$
−0.629763 + 0.776787i $$0.716848\pi$$
$$618$$ 0 0
$$619$$ 40.1151 1.61236 0.806181 0.591670i $$-0.201531\pi$$
0.806181 + 0.591670i $$0.201531\pi$$
$$620$$ 0 0
$$621$$ −4.39312 −0.176290
$$622$$ 0 0
$$623$$ 15.9572 0.639310
$$624$$ 0 0
$$625$$ 25.0000 1.00000
$$626$$ 0 0
$$627$$ −2.68585 −0.107262
$$628$$ 0 0
$$629$$ 12.9870 0.517826
$$630$$ 0 0
$$631$$ 33.9572 1.35181 0.675906 0.736987i $$-0.263752\pi$$
0.675906 + 0.736987i $$0.263752\pi$$
$$632$$ 0 0
$$633$$ 18.6858 0.742696
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −6.97858 −0.276501
$$638$$ 0 0
$$639$$ −1.60688 −0.0635674
$$640$$ 0 0
$$641$$ 47.1365 1.86178 0.930890 0.365300i $$-0.119034\pi$$
0.930890 + 0.365300i $$0.119034\pi$$
$$642$$ 0 0
$$643$$ −7.61531 −0.300318 −0.150159 0.988662i $$-0.547979\pi$$
−0.150159 + 0.988662i $$0.547979\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 35.9290 1.41251 0.706257 0.707955i $$-0.250382\pi$$
0.706257 + 0.707955i $$0.250382\pi$$
$$648$$ 0 0
$$649$$ −1.57246 −0.0617245
$$650$$ 0 0
$$651$$ 7.66442 0.300392
$$652$$ 0 0
$$653$$ −27.7367 −1.08542 −0.542710 0.839920i $$-0.682602\pi$$
−0.542710 + 0.839920i $$0.682602\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ −7.37169 −0.287597
$$658$$ 0 0
$$659$$ 6.39312 0.249040 0.124520 0.992217i $$-0.460261\pi$$
0.124520 + 0.992217i $$0.460261\pi$$
$$660$$ 0 0
$$661$$ 17.1365 0.666533 0.333266 0.942833i $$-0.391849\pi$$
0.333266 + 0.942833i $$0.391849\pi$$
$$662$$ 0 0
$$663$$ −16.0000 −0.621389
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −4.29900 −0.166458
$$668$$ 0 0
$$669$$ −22.2070 −0.858574
$$670$$ 0 0
$$671$$ −17.6875 −0.682820
$$672$$ 0 0
$$673$$ 39.8715 1.53693 0.768466 0.639891i $$-0.221020\pi$$
0.768466 + 0.639891i $$0.221020\pi$$
$$674$$ 0 0
$$675$$ −5.00000 −0.192450
$$676$$ 0 0
$$677$$ −23.9572 −0.920748 −0.460374 0.887725i $$-0.652285\pi$$
−0.460374 + 0.887725i $$0.652285\pi$$
$$678$$ 0 0
$$679$$ −5.27131 −0.202294
$$680$$ 0 0
$$681$$ 2.62831 0.100717
$$682$$ 0 0
$$683$$ −17.7220 −0.678112 −0.339056 0.940766i $$-0.610108\pi$$
−0.339056 + 0.940766i $$0.610108\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 11.9572 0.456194
$$688$$ 0 0
$$689$$ 90.5720 3.45052
$$690$$ 0 0
$$691$$ −1.25662 −0.0478039 −0.0239020 0.999714i $$-0.507609\pi$$
−0.0239020 + 0.999714i $$0.507609\pi$$
$$692$$ 0 0
$$693$$ −2.68585 −0.102027
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 16.9013 0.640183
$$698$$ 0 0
$$699$$ −6.58546 −0.249085
$$700$$ 0 0
$$701$$ −22.9210 −0.865716 −0.432858 0.901462i $$-0.642495\pi$$
−0.432858 + 0.901462i $$0.642495\pi$$
$$702$$ 0 0
$$703$$ −5.66442 −0.213638
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 8.00000 0.300871
$$708$$ 0 0
$$709$$ 7.75639 0.291297 0.145649 0.989336i $$-0.453473\pi$$
0.145649 + 0.989336i $$0.453473\pi$$
$$710$$ 0 0
$$711$$ 14.3503 0.538177
$$712$$ 0 0
$$713$$ 33.6707 1.26098
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 12.9786 0.484694
$$718$$ 0 0
$$719$$ −34.3565 −1.28128 −0.640641 0.767840i $$-0.721331\pi$$
−0.640641 + 0.767840i $$0.721331\pi$$
$$720$$ 0 0
$$721$$ 11.0790 0.412602
$$722$$ 0 0
$$723$$ 6.72869 0.250243
$$724$$ 0 0
$$725$$ −4.89289 −0.181717
$$726$$ 0 0
$$727$$ 16.4998 0.611943 0.305971 0.952041i $$-0.401019\pi$$
0.305971 + 0.952041i $$0.401019\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ −21.4868 −0.794717
$$732$$ 0 0
$$733$$ 3.57246 0.131952 0.0659759 0.997821i $$-0.478984\pi$$
0.0659759 + 0.997821i $$0.478984\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 32.3847 1.19291
$$738$$ 0 0
$$739$$ −6.74338 −0.248059 −0.124030 0.992279i $$-0.539582\pi$$
−0.124030 + 0.992279i $$0.539582\pi$$
$$740$$ 0 0
$$741$$ 6.97858 0.256364
$$742$$ 0 0
$$743$$ 36.2646 1.33042 0.665209 0.746657i $$-0.268342\pi$$
0.665209 + 0.746657i $$0.268342\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 8.35027 0.305520
$$748$$ 0 0
$$749$$ 7.56404 0.276384
$$750$$ 0 0
$$751$$ 45.5934 1.66373 0.831864 0.554980i $$-0.187274\pi$$
0.831864 + 0.554980i $$0.187274\pi$$
$$752$$ 0 0
$$753$$ −1.80765 −0.0658745
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 24.0722 0.874920 0.437460 0.899238i $$-0.355878\pi$$
0.437460 + 0.899238i $$0.355878\pi$$
$$758$$ 0 0
$$759$$ −11.7992 −0.428285
$$760$$ 0 0
$$761$$ 40.3650 1.46323 0.731614 0.681719i $$-0.238767\pi$$
0.731614 + 0.681719i $$0.238767\pi$$
$$762$$ 0 0
$$763$$ 4.87819 0.176602
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 4.08569 0.147526
$$768$$ 0 0
$$769$$ 50.4998 1.82107 0.910534 0.413434i $$-0.135671\pi$$
0.910534 + 0.413434i $$0.135671\pi$$
$$770$$ 0 0
$$771$$ −15.3717 −0.553598
$$772$$ 0 0
$$773$$ −27.2860 −0.981409 −0.490705 0.871326i $$-0.663261\pi$$
−0.490705 + 0.871326i $$0.663261\pi$$
$$774$$ 0 0
$$775$$ 38.3221 1.37657
$$776$$ 0 0
$$777$$ −5.66442 −0.203210
$$778$$ 0 0
$$779$$ −7.37169 −0.264118
$$780$$ 0 0
$$781$$ −4.31585 −0.154433
$$782$$ 0 0
$$783$$ 0.978577 0.0349715
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −13.2860 −0.473595 −0.236797 0.971559i $$-0.576098\pi$$
−0.236797 + 0.971559i $$0.576098\pi$$
$$788$$ 0 0
$$789$$ 18.3503 0.653287
$$790$$ 0 0
$$791$$ −10.3503 −0.368013
$$792$$ 0 0
$$793$$ 45.9572 1.63199
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −26.7862 −0.948817 −0.474408 0.880305i $$-0.657338\pi$$
−0.474408 + 0.880305i $$0.657338\pi$$
$$798$$ 0 0
$$799$$ −10.7434 −0.380074
$$800$$ 0 0
$$801$$ −15.9572 −0.563818
$$802$$ 0 0
$$803$$ −19.7992 −0.698700
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −10.5855 −0.372626
$$808$$ 0 0
$$809$$ 14.5855 0.512798 0.256399 0.966571i $$-0.417464\pi$$
0.256399 + 0.966571i $$0.417464\pi$$
$$810$$ 0 0
$$811$$ 2.04285 0.0717340 0.0358670 0.999357i $$-0.488581\pi$$
0.0358670 + 0.999357i $$0.488581\pi$$
$$812$$ 0 0
$$813$$ −8.00000 −0.280572
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 9.37169 0.327874
$$818$$ 0 0
$$819$$ 6.97858 0.243851
$$820$$ 0 0
$$821$$ 16.0920 0.561613 0.280807 0.959764i $$-0.409398\pi$$
0.280807 + 0.959764i $$0.409398\pi$$
$$822$$ 0 0
$$823$$ 6.82908 0.238047 0.119023 0.992891i $$-0.462024\pi$$
0.119023 + 0.992891i $$0.462024\pi$$
$$824$$ 0 0
$$825$$ −13.4292 −0.467546
$$826$$ 0 0
$$827$$ 33.4355 1.16267 0.581333 0.813666i $$-0.302531\pi$$
0.581333 + 0.813666i $$0.302531\pi$$
$$828$$ 0 0
$$829$$ 31.4783 1.09329 0.546644 0.837365i $$-0.315905\pi$$
0.546644 + 0.837365i $$0.315905\pi$$
$$830$$ 0 0
$$831$$ 1.21377 0.0421052
$$832$$ 0 0
$$833$$ 2.29273 0.0794384
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ −7.66442 −0.264921
$$838$$ 0 0
$$839$$ −18.0722 −0.623923 −0.311961 0.950095i $$-0.600986\pi$$
−0.311961 + 0.950095i $$0.600986\pi$$
$$840$$ 0 0
$$841$$ −28.0424 −0.966979
$$842$$ 0 0
$$843$$ −25.5640 −0.880472
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 3.78623 0.130096
$$848$$ 0 0
$$849$$ 22.5426 0.773661
$$850$$ 0 0
$$851$$ −24.8845 −0.853028
$$852$$ 0 0
$$853$$ −10.4998 −0.359505 −0.179753 0.983712i $$-0.557530\pi$$
−0.179753 + 0.983712i $$0.557530\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −22.6148 −0.772508 −0.386254 0.922392i $$-0.626231\pi$$
−0.386254 + 0.922392i $$0.626231\pi$$
$$858$$ 0 0
$$859$$ −16.9870 −0.579589 −0.289794 0.957089i $$-0.593587\pi$$
−0.289794 + 0.957089i $$0.593587\pi$$
$$860$$ 0 0
$$861$$ −7.37169 −0.251227
$$862$$ 0 0
$$863$$ 35.9656 1.22428 0.612141 0.790748i $$-0.290308\pi$$
0.612141 + 0.790748i $$0.290308\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ −11.7434 −0.398826
$$868$$ 0 0
$$869$$ 38.5426 1.30747
$$870$$ 0 0
$$871$$ −84.1445 −2.85113
$$872$$ 0 0
$$873$$ 5.27131 0.178407
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −23.4208 −0.790865 −0.395432 0.918495i $$-0.629405\pi$$
−0.395432 + 0.918495i $$0.629405\pi$$
$$878$$ 0 0
$$879$$ −19.2860 −0.650501
$$880$$ 0 0
$$881$$ −37.0361 −1.24778 −0.623889 0.781513i $$-0.714448\pi$$
−0.623889 + 0.781513i $$0.714448\pi$$
$$882$$ 0 0
$$883$$ −2.54262 −0.0855658 −0.0427829 0.999084i $$-0.513622\pi$$
−0.0427829 + 0.999084i $$0.513622\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −1.37169 −0.0460569 −0.0230285 0.999735i $$-0.507331\pi$$
−0.0230285 + 0.999735i $$0.507331\pi$$
$$888$$ 0 0
$$889$$ 0.393115 0.0131847
$$890$$ 0 0
$$891$$ 2.68585 0.0899792
$$892$$ 0 0
$$893$$ 4.68585 0.156806
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 30.6577 1.02363
$$898$$ 0 0
$$899$$ −7.50023 −0.250147
$$900$$ 0 0
$$901$$ −29.7564 −0.991329
$$902$$ 0 0
$$903$$ 9.37169 0.311870
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 11.0705 0.367591 0.183796 0.982964i $$-0.441162\pi$$
0.183796 + 0.982964i $$0.441162\pi$$
$$908$$ 0 0
$$909$$ −8.00000 −0.265343
$$910$$ 0 0
$$911$$ −37.0508 −1.22755 −0.613774 0.789482i $$-0.710349\pi$$
−0.613774 + 0.789482i $$0.710349\pi$$
$$912$$ 0 0
$$913$$ 22.4275 0.742243
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 1.80765 0.0596940
$$918$$ 0 0
$$919$$ −29.4868 −0.972679 −0.486339 0.873770i $$-0.661668\pi$$
−0.486339 + 0.873770i $$0.661668\pi$$
$$920$$ 0 0
$$921$$ 4.78623 0.157712
$$922$$ 0 0
$$923$$ 11.2138 0.369106
$$924$$ 0 0
$$925$$ −28.3221 −0.931225
$$926$$ 0 0
$$927$$ −11.0790 −0.363881
$$928$$ 0 0
$$929$$ −37.5359 −1.23151 −0.615756 0.787937i $$-0.711149\pi$$
−0.615756 + 0.787937i $$0.711149\pi$$
$$930$$ 0 0
$$931$$ −1.00000 −0.0327737
$$932$$ 0 0
$$933$$ 19.4292 0.636084
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 2.78623 0.0910222 0.0455111 0.998964i $$-0.485508\pi$$
0.0455111 + 0.998964i $$0.485508\pi$$
$$938$$ 0 0
$$939$$ 25.3288 0.826576
$$940$$ 0 0
$$941$$ −53.3288 −1.73847 −0.869235 0.494399i $$-0.835388\pi$$
−0.869235 + 0.494399i $$0.835388\pi$$
$$942$$ 0 0
$$943$$ −32.3847 −1.05459
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −40.8438 −1.32724 −0.663622 0.748068i $$-0.730982\pi$$
−0.663622 + 0.748068i $$0.730982\pi$$
$$948$$ 0 0
$$949$$ 51.4439 1.66994
$$950$$ 0 0
$$951$$ 29.6791 0.962411
$$952$$ 0 0
$$953$$ 19.0508 0.617116 0.308558 0.951205i $$-0.400154\pi$$
0.308558 + 0.951205i $$0.400154\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 2.62831 0.0849611
$$958$$ 0 0
$$959$$ 14.7862 0.477472
$$960$$ 0 0
$$961$$ 27.7434 0.894948
$$962$$ 0 0
$$963$$ −7.56404 −0.243748
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −0.384694 −0.0123709 −0.00618546 0.999981i $$-0.501969\pi$$
−0.00618546 + 0.999981i $$0.501969\pi$$
$$968$$ 0 0
$$969$$ −2.29273 −0.0736531
$$970$$ 0 0
$$971$$ 4.49977 0.144405 0.0722023 0.997390i $$-0.476997\pi$$
0.0722023 + 0.997390i $$0.476997\pi$$
$$972$$ 0 0
$$973$$ 13.3717 0.428677
$$974$$ 0 0
$$975$$ 34.8929 1.11747
$$976$$ 0 0
$$977$$ 35.4355 1.13368 0.566841 0.823827i $$-0.308165\pi$$
0.566841 + 0.823827i $$0.308165\pi$$
$$978$$ 0 0
$$979$$ −42.8585 −1.36976
$$980$$ 0 0
$$981$$ −4.87819 −0.155749
$$982$$ 0 0
$$983$$ −1.17092 −0.0373467 −0.0186733 0.999826i $$-0.505944\pi$$
−0.0186733 + 0.999826i $$0.505944\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 4.68585 0.149152
$$988$$ 0 0
$$989$$ 41.1709 1.30916
$$990$$ 0 0
$$991$$ 25.7648 0.818446 0.409223 0.912434i $$-0.365800\pi$$
0.409223 + 0.912434i $$0.365800\pi$$
$$992$$ 0 0
$$993$$ −5.22846 −0.165920
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −25.4439 −0.805817 −0.402909 0.915240i $$-0.632001\pi$$
−0.402909 + 0.915240i $$0.632001\pi$$
$$998$$ 0 0
$$999$$ 5.66442 0.179214
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 3192.2.a.v.1.3 3
3.2 odd 2 9576.2.a.cc.1.1 3
4.3 odd 2 6384.2.a.bv.1.1 3

By twisted newform
Twist Min Dim Char Parity Ord Type
3192.2.a.v.1.3 3 1.1 even 1 trivial
6384.2.a.bv.1.1 3 4.3 odd 2
9576.2.a.cc.1.1 3 3.2 odd 2