# Properties

 Label 3192.2.a.s.1.2 Level $3192$ Weight $2$ Character 3192.1 Self dual yes Analytic conductor $25.488$ Analytic rank $1$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\zeta_{12})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 3$$ x^2 - 3 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$1.73205$$ 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)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{3} +0.732051 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{3} +0.732051 q^{5} +1.00000 q^{7} +1.00000 q^{9} -4.00000 q^{11} -1.46410 q^{13} +0.732051 q^{15} -3.26795 q^{17} +1.00000 q^{19} +1.00000 q^{21} -6.92820 q^{23} -4.46410 q^{25} +1.00000 q^{27} -3.26795 q^{29} -2.00000 q^{31} -4.00000 q^{33} +0.732051 q^{35} -4.92820 q^{37} -1.46410 q^{39} -8.92820 q^{41} +4.92820 q^{43} +0.732051 q^{45} +0.196152 q^{47} +1.00000 q^{49} -3.26795 q^{51} +7.66025 q^{53} -2.92820 q^{55} +1.00000 q^{57} -10.9282 q^{59} +0.928203 q^{61} +1.00000 q^{63} -1.07180 q^{65} +5.46410 q^{67} -6.92820 q^{69} +4.19615 q^{71} -10.3923 q^{73} -4.46410 q^{75} -4.00000 q^{77} +2.92820 q^{79} +1.00000 q^{81} +8.19615 q^{83} -2.39230 q^{85} -3.26795 q^{87} -6.00000 q^{89} -1.46410 q^{91} -2.00000 q^{93} +0.732051 q^{95} +11.8564 q^{97} -4.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{3} - 2 q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^3 - 2 * q^5 + 2 * q^7 + 2 * q^9 $$2 q + 2 q^{3} - 2 q^{5} + 2 q^{7} + 2 q^{9} - 8 q^{11} + 4 q^{13} - 2 q^{15} - 10 q^{17} + 2 q^{19} + 2 q^{21} - 2 q^{25} + 2 q^{27} - 10 q^{29} - 4 q^{31} - 8 q^{33} - 2 q^{35} + 4 q^{37} + 4 q^{39} - 4 q^{41} - 4 q^{43} - 2 q^{45} - 10 q^{47} + 2 q^{49} - 10 q^{51} - 2 q^{53} + 8 q^{55} + 2 q^{57} - 8 q^{59} - 12 q^{61} + 2 q^{63} - 16 q^{65} + 4 q^{67} - 2 q^{71} - 2 q^{75} - 8 q^{77} - 8 q^{79} + 2 q^{81} + 6 q^{83} + 16 q^{85} - 10 q^{87} - 12 q^{89} + 4 q^{91} - 4 q^{93} - 2 q^{95} - 4 q^{97} - 8 q^{99}+O(q^{100})$$ 2 * q + 2 * q^3 - 2 * q^5 + 2 * q^7 + 2 * q^9 - 8 * q^11 + 4 * q^13 - 2 * q^15 - 10 * q^17 + 2 * q^19 + 2 * q^21 - 2 * q^25 + 2 * q^27 - 10 * q^29 - 4 * q^31 - 8 * q^33 - 2 * q^35 + 4 * q^37 + 4 * q^39 - 4 * q^41 - 4 * q^43 - 2 * q^45 - 10 * q^47 + 2 * q^49 - 10 * q^51 - 2 * q^53 + 8 * q^55 + 2 * q^57 - 8 * q^59 - 12 * q^61 + 2 * q^63 - 16 * q^65 + 4 * q^67 - 2 * q^71 - 2 * q^75 - 8 * q^77 - 8 * q^79 + 2 * q^81 + 6 * q^83 + 16 * q^85 - 10 * q^87 - 12 * q^89 + 4 * q^91 - 4 * q^93 - 2 * q^95 - 4 * q^97 - 8 * 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
$$5$$ 0.732051 0.327383 0.163692 0.986512i $$-0.447660\pi$$
0.163692 + 0.986512i $$0.447660\pi$$
$$6$$ 0 0
$$7$$ 1.00000 0.377964
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ −4.00000 −1.20605 −0.603023 0.797724i $$-0.706037\pi$$
−0.603023 + 0.797724i $$0.706037\pi$$
$$12$$ 0 0
$$13$$ −1.46410 −0.406069 −0.203034 0.979172i $$-0.565080\pi$$
−0.203034 + 0.979172i $$0.565080\pi$$
$$14$$ 0 0
$$15$$ 0.732051 0.189015
$$16$$ 0 0
$$17$$ −3.26795 −0.792594 −0.396297 0.918122i $$-0.629705\pi$$
−0.396297 + 0.918122i $$0.629705\pi$$
$$18$$ 0 0
$$19$$ 1.00000 0.229416
$$20$$ 0 0
$$21$$ 1.00000 0.218218
$$22$$ 0 0
$$23$$ −6.92820 −1.44463 −0.722315 0.691564i $$-0.756922\pi$$
−0.722315 + 0.691564i $$0.756922\pi$$
$$24$$ 0 0
$$25$$ −4.46410 −0.892820
$$26$$ 0 0
$$27$$ 1.00000 0.192450
$$28$$ 0 0
$$29$$ −3.26795 −0.606843 −0.303421 0.952856i $$-0.598129\pi$$
−0.303421 + 0.952856i $$0.598129\pi$$
$$30$$ 0 0
$$31$$ −2.00000 −0.359211 −0.179605 0.983739i $$-0.557482\pi$$
−0.179605 + 0.983739i $$0.557482\pi$$
$$32$$ 0 0
$$33$$ −4.00000 −0.696311
$$34$$ 0 0
$$35$$ 0.732051 0.123739
$$36$$ 0 0
$$37$$ −4.92820 −0.810192 −0.405096 0.914274i $$-0.632762\pi$$
−0.405096 + 0.914274i $$0.632762\pi$$
$$38$$ 0 0
$$39$$ −1.46410 −0.234444
$$40$$ 0 0
$$41$$ −8.92820 −1.39435 −0.697176 0.716900i $$-0.745560\pi$$
−0.697176 + 0.716900i $$0.745560\pi$$
$$42$$ 0 0
$$43$$ 4.92820 0.751544 0.375772 0.926712i $$-0.377378\pi$$
0.375772 + 0.926712i $$0.377378\pi$$
$$44$$ 0 0
$$45$$ 0.732051 0.109128
$$46$$ 0 0
$$47$$ 0.196152 0.0286118 0.0143059 0.999898i $$-0.495446\pi$$
0.0143059 + 0.999898i $$0.495446\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ −3.26795 −0.457604
$$52$$ 0 0
$$53$$ 7.66025 1.05222 0.526108 0.850418i $$-0.323651\pi$$
0.526108 + 0.850418i $$0.323651\pi$$
$$54$$ 0 0
$$55$$ −2.92820 −0.394839
$$56$$ 0 0
$$57$$ 1.00000 0.132453
$$58$$ 0 0
$$59$$ −10.9282 −1.42273 −0.711365 0.702822i $$-0.751923\pi$$
−0.711365 + 0.702822i $$0.751923\pi$$
$$60$$ 0 0
$$61$$ 0.928203 0.118844 0.0594221 0.998233i $$-0.481074\pi$$
0.0594221 + 0.998233i $$0.481074\pi$$
$$62$$ 0 0
$$63$$ 1.00000 0.125988
$$64$$ 0 0
$$65$$ −1.07180 −0.132940
$$66$$ 0 0
$$67$$ 5.46410 0.667546 0.333773 0.942653i $$-0.391678\pi$$
0.333773 + 0.942653i $$0.391678\pi$$
$$68$$ 0 0
$$69$$ −6.92820 −0.834058
$$70$$ 0 0
$$71$$ 4.19615 0.497992 0.248996 0.968505i $$-0.419899\pi$$
0.248996 + 0.968505i $$0.419899\pi$$
$$72$$ 0 0
$$73$$ −10.3923 −1.21633 −0.608164 0.793812i $$-0.708094\pi$$
−0.608164 + 0.793812i $$0.708094\pi$$
$$74$$ 0 0
$$75$$ −4.46410 −0.515470
$$76$$ 0 0
$$77$$ −4.00000 −0.455842
$$78$$ 0 0
$$79$$ 2.92820 0.329449 0.164724 0.986340i $$-0.447327\pi$$
0.164724 + 0.986340i $$0.447327\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ 8.19615 0.899645 0.449822 0.893118i $$-0.351487\pi$$
0.449822 + 0.893118i $$0.351487\pi$$
$$84$$ 0 0
$$85$$ −2.39230 −0.259482
$$86$$ 0 0
$$87$$ −3.26795 −0.350361
$$88$$ 0 0
$$89$$ −6.00000 −0.635999 −0.317999 0.948091i $$-0.603011\pi$$
−0.317999 + 0.948091i $$0.603011\pi$$
$$90$$ 0 0
$$91$$ −1.46410 −0.153480
$$92$$ 0 0
$$93$$ −2.00000 −0.207390
$$94$$ 0 0
$$95$$ 0.732051 0.0751068
$$96$$ 0 0
$$97$$ 11.8564 1.20384 0.601918 0.798558i $$-0.294403\pi$$
0.601918 + 0.798558i $$0.294403\pi$$
$$98$$ 0 0
$$99$$ −4.00000 −0.402015
$$100$$ 0 0
$$101$$ −4.73205 −0.470857 −0.235428 0.971892i $$-0.575649\pi$$
−0.235428 + 0.971892i $$0.575649\pi$$
$$102$$ 0 0
$$103$$ −2.92820 −0.288524 −0.144262 0.989539i $$-0.546081\pi$$
−0.144262 + 0.989539i $$0.546081\pi$$
$$104$$ 0 0
$$105$$ 0.732051 0.0714408
$$106$$ 0 0
$$107$$ −12.1962 −1.17905 −0.589523 0.807751i $$-0.700684\pi$$
−0.589523 + 0.807751i $$0.700684\pi$$
$$108$$ 0 0
$$109$$ −4.53590 −0.434460 −0.217230 0.976120i $$-0.569702\pi$$
−0.217230 + 0.976120i $$0.569702\pi$$
$$110$$ 0 0
$$111$$ −4.92820 −0.467764
$$112$$ 0 0
$$113$$ 16.7321 1.57402 0.787009 0.616941i $$-0.211628\pi$$
0.787009 + 0.616941i $$0.211628\pi$$
$$114$$ 0 0
$$115$$ −5.07180 −0.472947
$$116$$ 0 0
$$117$$ −1.46410 −0.135356
$$118$$ 0 0
$$119$$ −3.26795 −0.299572
$$120$$ 0 0
$$121$$ 5.00000 0.454545
$$122$$ 0 0
$$123$$ −8.92820 −0.805029
$$124$$ 0 0
$$125$$ −6.92820 −0.619677
$$126$$ 0 0
$$127$$ −6.53590 −0.579967 −0.289984 0.957032i $$-0.593650\pi$$
−0.289984 + 0.957032i $$0.593650\pi$$
$$128$$ 0 0
$$129$$ 4.92820 0.433904
$$130$$ 0 0
$$131$$ 10.7321 0.937664 0.468832 0.883287i $$-0.344675\pi$$
0.468832 + 0.883287i $$0.344675\pi$$
$$132$$ 0 0
$$133$$ 1.00000 0.0867110
$$134$$ 0 0
$$135$$ 0.732051 0.0630049
$$136$$ 0 0
$$137$$ 4.92820 0.421045 0.210522 0.977589i $$-0.432484\pi$$
0.210522 + 0.977589i $$0.432484\pi$$
$$138$$ 0 0
$$139$$ 19.3205 1.63874 0.819372 0.573262i $$-0.194322\pi$$
0.819372 + 0.573262i $$0.194322\pi$$
$$140$$ 0 0
$$141$$ 0.196152 0.0165190
$$142$$ 0 0
$$143$$ 5.85641 0.489737
$$144$$ 0 0
$$145$$ −2.39230 −0.198670
$$146$$ 0 0
$$147$$ 1.00000 0.0824786
$$148$$ 0 0
$$149$$ −3.46410 −0.283790 −0.141895 0.989882i $$-0.545320\pi$$
−0.141895 + 0.989882i $$0.545320\pi$$
$$150$$ 0 0
$$151$$ 13.4641 1.09569 0.547847 0.836579i $$-0.315448\pi$$
0.547847 + 0.836579i $$0.315448\pi$$
$$152$$ 0 0
$$153$$ −3.26795 −0.264198
$$154$$ 0 0
$$155$$ −1.46410 −0.117599
$$156$$ 0 0
$$157$$ −3.07180 −0.245156 −0.122578 0.992459i $$-0.539116\pi$$
−0.122578 + 0.992459i $$0.539116\pi$$
$$158$$ 0 0
$$159$$ 7.66025 0.607498
$$160$$ 0 0
$$161$$ −6.92820 −0.546019
$$162$$ 0 0
$$163$$ −12.9282 −1.01262 −0.506308 0.862353i $$-0.668990\pi$$
−0.506308 + 0.862353i $$0.668990\pi$$
$$164$$ 0 0
$$165$$ −2.92820 −0.227960
$$166$$ 0 0
$$167$$ 9.46410 0.732354 0.366177 0.930545i $$-0.380666\pi$$
0.366177 + 0.930545i $$0.380666\pi$$
$$168$$ 0 0
$$169$$ −10.8564 −0.835108
$$170$$ 0 0
$$171$$ 1.00000 0.0764719
$$172$$ 0 0
$$173$$ −11.8564 −0.901426 −0.450713 0.892669i $$-0.648830\pi$$
−0.450713 + 0.892669i $$0.648830\pi$$
$$174$$ 0 0
$$175$$ −4.46410 −0.337454
$$176$$ 0 0
$$177$$ −10.9282 −0.821414
$$178$$ 0 0
$$179$$ −14.7321 −1.10113 −0.550563 0.834794i $$-0.685587\pi$$
−0.550563 + 0.834794i $$0.685587\pi$$
$$180$$ 0 0
$$181$$ −7.07180 −0.525643 −0.262821 0.964845i $$-0.584653\pi$$
−0.262821 + 0.964845i $$0.584653\pi$$
$$182$$ 0 0
$$183$$ 0.928203 0.0686148
$$184$$ 0 0
$$185$$ −3.60770 −0.265243
$$186$$ 0 0
$$187$$ 13.0718 0.955904
$$188$$ 0 0
$$189$$ 1.00000 0.0727393
$$190$$ 0 0
$$191$$ 18.9282 1.36960 0.684798 0.728733i $$-0.259890\pi$$
0.684798 + 0.728733i $$0.259890\pi$$
$$192$$ 0 0
$$193$$ 8.92820 0.642666 0.321333 0.946966i $$-0.395869\pi$$
0.321333 + 0.946966i $$0.395869\pi$$
$$194$$ 0 0
$$195$$ −1.07180 −0.0767530
$$196$$ 0 0
$$197$$ 0.928203 0.0661317 0.0330659 0.999453i $$-0.489473\pi$$
0.0330659 + 0.999453i $$0.489473\pi$$
$$198$$ 0 0
$$199$$ 19.3205 1.36959 0.684797 0.728734i $$-0.259891\pi$$
0.684797 + 0.728734i $$0.259891\pi$$
$$200$$ 0 0
$$201$$ 5.46410 0.385408
$$202$$ 0 0
$$203$$ −3.26795 −0.229365
$$204$$ 0 0
$$205$$ −6.53590 −0.456487
$$206$$ 0 0
$$207$$ −6.92820 −0.481543
$$208$$ 0 0
$$209$$ −4.00000 −0.276686
$$210$$ 0 0
$$211$$ −8.00000 −0.550743 −0.275371 0.961338i $$-0.588801\pi$$
−0.275371 + 0.961338i $$0.588801\pi$$
$$212$$ 0 0
$$213$$ 4.19615 0.287516
$$214$$ 0 0
$$215$$ 3.60770 0.246043
$$216$$ 0 0
$$217$$ −2.00000 −0.135769
$$218$$ 0 0
$$219$$ −10.3923 −0.702247
$$220$$ 0 0
$$221$$ 4.78461 0.321848
$$222$$ 0 0
$$223$$ −22.7846 −1.52577 −0.762885 0.646534i $$-0.776218\pi$$
−0.762885 + 0.646534i $$0.776218\pi$$
$$224$$ 0 0
$$225$$ −4.46410 −0.297607
$$226$$ 0 0
$$227$$ −7.32051 −0.485879 −0.242940 0.970041i $$-0.578112\pi$$
−0.242940 + 0.970041i $$0.578112\pi$$
$$228$$ 0 0
$$229$$ 2.39230 0.158088 0.0790440 0.996871i $$-0.474813\pi$$
0.0790440 + 0.996871i $$0.474813\pi$$
$$230$$ 0 0
$$231$$ −4.00000 −0.263181
$$232$$ 0 0
$$233$$ −17.3205 −1.13470 −0.567352 0.823475i $$-0.692032\pi$$
−0.567352 + 0.823475i $$0.692032\pi$$
$$234$$ 0 0
$$235$$ 0.143594 0.00936701
$$236$$ 0 0
$$237$$ 2.92820 0.190207
$$238$$ 0 0
$$239$$ −9.46410 −0.612182 −0.306091 0.952002i $$-0.599021\pi$$
−0.306091 + 0.952002i $$0.599021\pi$$
$$240$$ 0 0
$$241$$ 10.7846 0.694698 0.347349 0.937736i $$-0.387082\pi$$
0.347349 + 0.937736i $$0.387082\pi$$
$$242$$ 0 0
$$243$$ 1.00000 0.0641500
$$244$$ 0 0
$$245$$ 0.732051 0.0467690
$$246$$ 0 0
$$247$$ −1.46410 −0.0931586
$$248$$ 0 0
$$249$$ 8.19615 0.519410
$$250$$ 0 0
$$251$$ −1.66025 −0.104794 −0.0523972 0.998626i $$-0.516686\pi$$
−0.0523972 + 0.998626i $$0.516686\pi$$
$$252$$ 0 0
$$253$$ 27.7128 1.74229
$$254$$ 0 0
$$255$$ −2.39230 −0.149812
$$256$$ 0 0
$$257$$ 2.39230 0.149228 0.0746139 0.997212i $$-0.476228\pi$$
0.0746139 + 0.997212i $$0.476228\pi$$
$$258$$ 0 0
$$259$$ −4.92820 −0.306224
$$260$$ 0 0
$$261$$ −3.26795 −0.202281
$$262$$ 0 0
$$263$$ 2.53590 0.156370 0.0781851 0.996939i $$-0.475087\pi$$
0.0781851 + 0.996939i $$0.475087\pi$$
$$264$$ 0 0
$$265$$ 5.60770 0.344478
$$266$$ 0 0
$$267$$ −6.00000 −0.367194
$$268$$ 0 0
$$269$$ −27.4641 −1.67452 −0.837258 0.546808i $$-0.815843\pi$$
−0.837258 + 0.546808i $$0.815843\pi$$
$$270$$ 0 0
$$271$$ 20.0000 1.21491 0.607457 0.794353i $$-0.292190\pi$$
0.607457 + 0.794353i $$0.292190\pi$$
$$272$$ 0 0
$$273$$ −1.46410 −0.0886115
$$274$$ 0 0
$$275$$ 17.8564 1.07678
$$276$$ 0 0
$$277$$ −24.3923 −1.46559 −0.732796 0.680449i $$-0.761785\pi$$
−0.732796 + 0.680449i $$0.761785\pi$$
$$278$$ 0 0
$$279$$ −2.00000 −0.119737
$$280$$ 0 0
$$281$$ −4.73205 −0.282290 −0.141145 0.989989i $$-0.545078\pi$$
−0.141145 + 0.989989i $$0.545078\pi$$
$$282$$ 0 0
$$283$$ −8.39230 −0.498871 −0.249435 0.968391i $$-0.580245\pi$$
−0.249435 + 0.968391i $$0.580245\pi$$
$$284$$ 0 0
$$285$$ 0.732051 0.0433629
$$286$$ 0 0
$$287$$ −8.92820 −0.527015
$$288$$ 0 0
$$289$$ −6.32051 −0.371795
$$290$$ 0 0
$$291$$ 11.8564 0.695035
$$292$$ 0 0
$$293$$ −27.8564 −1.62739 −0.813694 0.581293i $$-0.802547\pi$$
−0.813694 + 0.581293i $$0.802547\pi$$
$$294$$ 0 0
$$295$$ −8.00000 −0.465778
$$296$$ 0 0
$$297$$ −4.00000 −0.232104
$$298$$ 0 0
$$299$$ 10.1436 0.586619
$$300$$ 0 0
$$301$$ 4.92820 0.284057
$$302$$ 0 0
$$303$$ −4.73205 −0.271849
$$304$$ 0 0
$$305$$ 0.679492 0.0389076
$$306$$ 0 0
$$307$$ 16.9282 0.966144 0.483072 0.875581i $$-0.339521\pi$$
0.483072 + 0.875581i $$0.339521\pi$$
$$308$$ 0 0
$$309$$ −2.92820 −0.166580
$$310$$ 0 0
$$311$$ −7.12436 −0.403985 −0.201993 0.979387i $$-0.564742\pi$$
−0.201993 + 0.979387i $$0.564742\pi$$
$$312$$ 0 0
$$313$$ 19.0718 1.07800 0.539001 0.842305i $$-0.318802\pi$$
0.539001 + 0.842305i $$0.318802\pi$$
$$314$$ 0 0
$$315$$ 0.732051 0.0412464
$$316$$ 0 0
$$317$$ −19.2679 −1.08220 −0.541098 0.840960i $$-0.681991\pi$$
−0.541098 + 0.840960i $$0.681991\pi$$
$$318$$ 0 0
$$319$$ 13.0718 0.731880
$$320$$ 0 0
$$321$$ −12.1962 −0.680723
$$322$$ 0 0
$$323$$ −3.26795 −0.181834
$$324$$ 0 0
$$325$$ 6.53590 0.362546
$$326$$ 0 0
$$327$$ −4.53590 −0.250836
$$328$$ 0 0
$$329$$ 0.196152 0.0108142
$$330$$ 0 0
$$331$$ 8.39230 0.461283 0.230641 0.973039i $$-0.425918\pi$$
0.230641 + 0.973039i $$0.425918\pi$$
$$332$$ 0 0
$$333$$ −4.92820 −0.270064
$$334$$ 0 0
$$335$$ 4.00000 0.218543
$$336$$ 0 0
$$337$$ 1.60770 0.0875767 0.0437884 0.999041i $$-0.486057\pi$$
0.0437884 + 0.999041i $$0.486057\pi$$
$$338$$ 0 0
$$339$$ 16.7321 0.908760
$$340$$ 0 0
$$341$$ 8.00000 0.433224
$$342$$ 0 0
$$343$$ 1.00000 0.0539949
$$344$$ 0 0
$$345$$ −5.07180 −0.273056
$$346$$ 0 0
$$347$$ 19.3205 1.03718 0.518590 0.855023i $$-0.326457\pi$$
0.518590 + 0.855023i $$0.326457\pi$$
$$348$$ 0 0
$$349$$ −22.7846 −1.21963 −0.609816 0.792543i $$-0.708757\pi$$
−0.609816 + 0.792543i $$0.708757\pi$$
$$350$$ 0 0
$$351$$ −1.46410 −0.0781480
$$352$$ 0 0
$$353$$ −25.5167 −1.35811 −0.679057 0.734085i $$-0.737611\pi$$
−0.679057 + 0.734085i $$0.737611\pi$$
$$354$$ 0 0
$$355$$ 3.07180 0.163034
$$356$$ 0 0
$$357$$ −3.26795 −0.172958
$$358$$ 0 0
$$359$$ 29.8564 1.57576 0.787880 0.615828i $$-0.211178\pi$$
0.787880 + 0.615828i $$0.211178\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 0 0
$$363$$ 5.00000 0.262432
$$364$$ 0 0
$$365$$ −7.60770 −0.398205
$$366$$ 0 0
$$367$$ −15.3205 −0.799724 −0.399862 0.916575i $$-0.630942\pi$$
−0.399862 + 0.916575i $$0.630942\pi$$
$$368$$ 0 0
$$369$$ −8.92820 −0.464784
$$370$$ 0 0
$$371$$ 7.66025 0.397701
$$372$$ 0 0
$$373$$ 30.7846 1.59397 0.796983 0.604001i $$-0.206428\pi$$
0.796983 + 0.604001i $$0.206428\pi$$
$$374$$ 0 0
$$375$$ −6.92820 −0.357771
$$376$$ 0 0
$$377$$ 4.78461 0.246420
$$378$$ 0 0
$$379$$ 20.3923 1.04748 0.523741 0.851877i $$-0.324536\pi$$
0.523741 + 0.851877i $$0.324536\pi$$
$$380$$ 0 0
$$381$$ −6.53590 −0.334844
$$382$$ 0 0
$$383$$ 0.679492 0.0347204 0.0173602 0.999849i $$-0.494474\pi$$
0.0173602 + 0.999849i $$0.494474\pi$$
$$384$$ 0 0
$$385$$ −2.92820 −0.149235
$$386$$ 0 0
$$387$$ 4.92820 0.250515
$$388$$ 0 0
$$389$$ −13.6077 −0.689938 −0.344969 0.938614i $$-0.612110\pi$$
−0.344969 + 0.938614i $$0.612110\pi$$
$$390$$ 0 0
$$391$$ 22.6410 1.14501
$$392$$ 0 0
$$393$$ 10.7321 0.541360
$$394$$ 0 0
$$395$$ 2.14359 0.107856
$$396$$ 0 0
$$397$$ 6.00000 0.301131 0.150566 0.988600i $$-0.451890\pi$$
0.150566 + 0.988600i $$0.451890\pi$$
$$398$$ 0 0
$$399$$ 1.00000 0.0500626
$$400$$ 0 0
$$401$$ 6.58846 0.329012 0.164506 0.986376i $$-0.447397\pi$$
0.164506 + 0.986376i $$0.447397\pi$$
$$402$$ 0 0
$$403$$ 2.92820 0.145864
$$404$$ 0 0
$$405$$ 0.732051 0.0363759
$$406$$ 0 0
$$407$$ 19.7128 0.977128
$$408$$ 0 0
$$409$$ −2.53590 −0.125392 −0.0626961 0.998033i $$-0.519970\pi$$
−0.0626961 + 0.998033i $$0.519970\pi$$
$$410$$ 0 0
$$411$$ 4.92820 0.243090
$$412$$ 0 0
$$413$$ −10.9282 −0.537742
$$414$$ 0 0
$$415$$ 6.00000 0.294528
$$416$$ 0 0
$$417$$ 19.3205 0.946129
$$418$$ 0 0
$$419$$ 8.87564 0.433604 0.216802 0.976216i $$-0.430437\pi$$
0.216802 + 0.976216i $$0.430437\pi$$
$$420$$ 0 0
$$421$$ −16.5359 −0.805910 −0.402955 0.915220i $$-0.632017\pi$$
−0.402955 + 0.915220i $$0.632017\pi$$
$$422$$ 0 0
$$423$$ 0.196152 0.00953726
$$424$$ 0 0
$$425$$ 14.5885 0.707644
$$426$$ 0 0
$$427$$ 0.928203 0.0449189
$$428$$ 0 0
$$429$$ 5.85641 0.282750
$$430$$ 0 0
$$431$$ −40.9808 −1.97397 −0.986987 0.160801i $$-0.948592\pi$$
−0.986987 + 0.160801i $$0.948592\pi$$
$$432$$ 0 0
$$433$$ 23.8564 1.14647 0.573233 0.819393i $$-0.305689\pi$$
0.573233 + 0.819393i $$0.305689\pi$$
$$434$$ 0 0
$$435$$ −2.39230 −0.114702
$$436$$ 0 0
$$437$$ −6.92820 −0.331421
$$438$$ 0 0
$$439$$ −3.85641 −0.184056 −0.0920281 0.995756i $$-0.529335\pi$$
−0.0920281 + 0.995756i $$0.529335\pi$$
$$440$$ 0 0
$$441$$ 1.00000 0.0476190
$$442$$ 0 0
$$443$$ 16.3923 0.778822 0.389411 0.921064i $$-0.372679\pi$$
0.389411 + 0.921064i $$0.372679\pi$$
$$444$$ 0 0
$$445$$ −4.39230 −0.208215
$$446$$ 0 0
$$447$$ −3.46410 −0.163846
$$448$$ 0 0
$$449$$ 19.6603 0.927825 0.463912 0.885881i $$-0.346445\pi$$
0.463912 + 0.885881i $$0.346445\pi$$
$$450$$ 0 0
$$451$$ 35.7128 1.68165
$$452$$ 0 0
$$453$$ 13.4641 0.632599
$$454$$ 0 0
$$455$$ −1.07180 −0.0502466
$$456$$ 0 0
$$457$$ −11.3205 −0.529551 −0.264776 0.964310i $$-0.585298\pi$$
−0.264776 + 0.964310i $$0.585298\pi$$
$$458$$ 0 0
$$459$$ −3.26795 −0.152535
$$460$$ 0 0
$$461$$ −9.80385 −0.456611 −0.228305 0.973590i $$-0.573318\pi$$
−0.228305 + 0.973590i $$0.573318\pi$$
$$462$$ 0 0
$$463$$ −16.0000 −0.743583 −0.371792 0.928316i $$-0.621256\pi$$
−0.371792 + 0.928316i $$0.621256\pi$$
$$464$$ 0 0
$$465$$ −1.46410 −0.0678961
$$466$$ 0 0
$$467$$ −8.19615 −0.379273 −0.189636 0.981854i $$-0.560731\pi$$
−0.189636 + 0.981854i $$0.560731\pi$$
$$468$$ 0 0
$$469$$ 5.46410 0.252309
$$470$$ 0 0
$$471$$ −3.07180 −0.141541
$$472$$ 0 0
$$473$$ −19.7128 −0.906396
$$474$$ 0 0
$$475$$ −4.46410 −0.204827
$$476$$ 0 0
$$477$$ 7.66025 0.350739
$$478$$ 0 0
$$479$$ 19.5167 0.891739 0.445869 0.895098i $$-0.352895\pi$$
0.445869 + 0.895098i $$0.352895\pi$$
$$480$$ 0 0
$$481$$ 7.21539 0.328993
$$482$$ 0 0
$$483$$ −6.92820 −0.315244
$$484$$ 0 0
$$485$$ 8.67949 0.394115
$$486$$ 0 0
$$487$$ 5.85641 0.265379 0.132690 0.991158i $$-0.457639\pi$$
0.132690 + 0.991158i $$0.457639\pi$$
$$488$$ 0 0
$$489$$ −12.9282 −0.584634
$$490$$ 0 0
$$491$$ −6.24871 −0.282000 −0.141000 0.990010i $$-0.545032\pi$$
−0.141000 + 0.990010i $$0.545032\pi$$
$$492$$ 0 0
$$493$$ 10.6795 0.480980
$$494$$ 0 0
$$495$$ −2.92820 −0.131613
$$496$$ 0 0
$$497$$ 4.19615 0.188223
$$498$$ 0 0
$$499$$ −1.85641 −0.0831042 −0.0415521 0.999136i $$-0.513230\pi$$
−0.0415521 + 0.999136i $$0.513230\pi$$
$$500$$ 0 0
$$501$$ 9.46410 0.422825
$$502$$ 0 0
$$503$$ −25.2679 −1.12664 −0.563321 0.826238i $$-0.690477\pi$$
−0.563321 + 0.826238i $$0.690477\pi$$
$$504$$ 0 0
$$505$$ −3.46410 −0.154150
$$506$$ 0 0
$$507$$ −10.8564 −0.482150
$$508$$ 0 0
$$509$$ 5.32051 0.235827 0.117914 0.993024i $$-0.462379\pi$$
0.117914 + 0.993024i $$0.462379\pi$$
$$510$$ 0 0
$$511$$ −10.3923 −0.459728
$$512$$ 0 0
$$513$$ 1.00000 0.0441511
$$514$$ 0 0
$$515$$ −2.14359 −0.0944580
$$516$$ 0 0
$$517$$ −0.784610 −0.0345071
$$518$$ 0 0
$$519$$ −11.8564 −0.520438
$$520$$ 0 0
$$521$$ −36.2487 −1.58808 −0.794042 0.607862i $$-0.792027\pi$$
−0.794042 + 0.607862i $$0.792027\pi$$
$$522$$ 0 0
$$523$$ 38.0000 1.66162 0.830812 0.556553i $$-0.187876\pi$$
0.830812 + 0.556553i $$0.187876\pi$$
$$524$$ 0 0
$$525$$ −4.46410 −0.194829
$$526$$ 0 0
$$527$$ 6.53590 0.284708
$$528$$ 0 0
$$529$$ 25.0000 1.08696
$$530$$ 0 0
$$531$$ −10.9282 −0.474244
$$532$$ 0 0
$$533$$ 13.0718 0.566202
$$534$$ 0 0
$$535$$ −8.92820 −0.386000
$$536$$ 0 0
$$537$$ −14.7321 −0.635735
$$538$$ 0 0
$$539$$ −4.00000 −0.172292
$$540$$ 0 0
$$541$$ −22.0000 −0.945854 −0.472927 0.881102i $$-0.656803\pi$$
−0.472927 + 0.881102i $$0.656803\pi$$
$$542$$ 0 0
$$543$$ −7.07180 −0.303480
$$544$$ 0 0
$$545$$ −3.32051 −0.142235
$$546$$ 0 0
$$547$$ 30.9282 1.32239 0.661197 0.750212i $$-0.270049\pi$$
0.661197 + 0.750212i $$0.270049\pi$$
$$548$$ 0 0
$$549$$ 0.928203 0.0396147
$$550$$ 0 0
$$551$$ −3.26795 −0.139219
$$552$$ 0 0
$$553$$ 2.92820 0.124520
$$554$$ 0 0
$$555$$ −3.60770 −0.153138
$$556$$ 0 0
$$557$$ −38.1051 −1.61457 −0.807283 0.590165i $$-0.799063\pi$$
−0.807283 + 0.590165i $$0.799063\pi$$
$$558$$ 0 0
$$559$$ −7.21539 −0.305178
$$560$$ 0 0
$$561$$ 13.0718 0.551892
$$562$$ 0 0
$$563$$ 35.3205 1.48858 0.744291 0.667855i $$-0.232788\pi$$
0.744291 + 0.667855i $$0.232788\pi$$
$$564$$ 0 0
$$565$$ 12.2487 0.515307
$$566$$ 0 0
$$567$$ 1.00000 0.0419961
$$568$$ 0 0
$$569$$ −29.5167 −1.23740 −0.618701 0.785626i $$-0.712341\pi$$
−0.618701 + 0.785626i $$0.712341\pi$$
$$570$$ 0 0
$$571$$ −4.00000 −0.167395 −0.0836974 0.996491i $$-0.526673\pi$$
−0.0836974 + 0.996491i $$0.526673\pi$$
$$572$$ 0 0
$$573$$ 18.9282 0.790737
$$574$$ 0 0
$$575$$ 30.9282 1.28980
$$576$$ 0 0
$$577$$ 14.7846 0.615491 0.307746 0.951469i $$-0.400425\pi$$
0.307746 + 0.951469i $$0.400425\pi$$
$$578$$ 0 0
$$579$$ 8.92820 0.371043
$$580$$ 0 0
$$581$$ 8.19615 0.340034
$$582$$ 0 0
$$583$$ −30.6410 −1.26902
$$584$$ 0 0
$$585$$ −1.07180 −0.0443133
$$586$$ 0 0
$$587$$ 20.1962 0.833584 0.416792 0.909002i $$-0.363154\pi$$
0.416792 + 0.909002i $$0.363154\pi$$
$$588$$ 0 0
$$589$$ −2.00000 −0.0824086
$$590$$ 0 0
$$591$$ 0.928203 0.0381812
$$592$$ 0 0
$$593$$ 9.80385 0.402596 0.201298 0.979530i $$-0.435484\pi$$
0.201298 + 0.979530i $$0.435484\pi$$
$$594$$ 0 0
$$595$$ −2.39230 −0.0980749
$$596$$ 0 0
$$597$$ 19.3205 0.790736
$$598$$ 0 0
$$599$$ −7.80385 −0.318857 −0.159428 0.987210i $$-0.550965\pi$$
−0.159428 + 0.987210i $$0.550965\pi$$
$$600$$ 0 0
$$601$$ 16.1436 0.658511 0.329255 0.944241i $$-0.393202\pi$$
0.329255 + 0.944241i $$0.393202\pi$$
$$602$$ 0 0
$$603$$ 5.46410 0.222515
$$604$$ 0 0
$$605$$ 3.66025 0.148810
$$606$$ 0 0
$$607$$ 16.7846 0.681266 0.340633 0.940196i $$-0.389359\pi$$
0.340633 + 0.940196i $$0.389359\pi$$
$$608$$ 0 0
$$609$$ −3.26795 −0.132424
$$610$$ 0 0
$$611$$ −0.287187 −0.0116183
$$612$$ 0 0
$$613$$ −23.6077 −0.953506 −0.476753 0.879037i $$-0.658186\pi$$
−0.476753 + 0.879037i $$0.658186\pi$$
$$614$$ 0 0
$$615$$ −6.53590 −0.263553
$$616$$ 0 0
$$617$$ 22.7846 0.917274 0.458637 0.888624i $$-0.348338\pi$$
0.458637 + 0.888624i $$0.348338\pi$$
$$618$$ 0 0
$$619$$ −18.5359 −0.745021 −0.372510 0.928028i $$-0.621503\pi$$
−0.372510 + 0.928028i $$0.621503\pi$$
$$620$$ 0 0
$$621$$ −6.92820 −0.278019
$$622$$ 0 0
$$623$$ −6.00000 −0.240385
$$624$$ 0 0
$$625$$ 17.2487 0.689948
$$626$$ 0 0
$$627$$ −4.00000 −0.159745
$$628$$ 0 0
$$629$$ 16.1051 0.642153
$$630$$ 0 0
$$631$$ −34.7846 −1.38475 −0.692377 0.721536i $$-0.743436\pi$$
−0.692377 + 0.721536i $$0.743436\pi$$
$$632$$ 0 0
$$633$$ −8.00000 −0.317971
$$634$$ 0 0
$$635$$ −4.78461 −0.189871
$$636$$ 0 0
$$637$$ −1.46410 −0.0580098
$$638$$ 0 0
$$639$$ 4.19615 0.165997
$$640$$ 0 0
$$641$$ 3.26795 0.129076 0.0645381 0.997915i $$-0.479443\pi$$
0.0645381 + 0.997915i $$0.479443\pi$$
$$642$$ 0 0
$$643$$ 0.392305 0.0154710 0.00773550 0.999970i $$-0.497538\pi$$
0.00773550 + 0.999970i $$0.497538\pi$$
$$644$$ 0 0
$$645$$ 3.60770 0.142053
$$646$$ 0 0
$$647$$ 10.3397 0.406497 0.203249 0.979127i $$-0.434850\pi$$
0.203249 + 0.979127i $$0.434850\pi$$
$$648$$ 0 0
$$649$$ 43.7128 1.71588
$$650$$ 0 0
$$651$$ −2.00000 −0.0783862
$$652$$ 0 0
$$653$$ −45.7128 −1.78888 −0.894440 0.447187i $$-0.852426\pi$$
−0.894440 + 0.447187i $$0.852426\pi$$
$$654$$ 0 0
$$655$$ 7.85641 0.306975
$$656$$ 0 0
$$657$$ −10.3923 −0.405442
$$658$$ 0 0
$$659$$ −11.1244 −0.433343 −0.216672 0.976245i $$-0.569520\pi$$
−0.216672 + 0.976245i $$0.569520\pi$$
$$660$$ 0 0
$$661$$ 5.46410 0.212529 0.106264 0.994338i $$-0.466111\pi$$
0.106264 + 0.994338i $$0.466111\pi$$
$$662$$ 0 0
$$663$$ 4.78461 0.185819
$$664$$ 0 0
$$665$$ 0.732051 0.0283877
$$666$$ 0 0
$$667$$ 22.6410 0.876664
$$668$$ 0 0
$$669$$ −22.7846 −0.880904
$$670$$ 0 0
$$671$$ −3.71281 −0.143332
$$672$$ 0 0
$$673$$ 43.8564 1.69054 0.845270 0.534339i $$-0.179440\pi$$
0.845270 + 0.534339i $$0.179440\pi$$
$$674$$ 0 0
$$675$$ −4.46410 −0.171823
$$676$$ 0 0
$$677$$ 0.535898 0.0205962 0.0102981 0.999947i $$-0.496722\pi$$
0.0102981 + 0.999947i $$0.496722\pi$$
$$678$$ 0 0
$$679$$ 11.8564 0.455007
$$680$$ 0 0
$$681$$ −7.32051 −0.280522
$$682$$ 0 0
$$683$$ −27.8038 −1.06388 −0.531942 0.846781i $$-0.678538\pi$$
−0.531942 + 0.846781i $$0.678538\pi$$
$$684$$ 0 0
$$685$$ 3.60770 0.137843
$$686$$ 0 0
$$687$$ 2.39230 0.0912721
$$688$$ 0 0
$$689$$ −11.2154 −0.427272
$$690$$ 0 0
$$691$$ 29.8564 1.13579 0.567896 0.823101i $$-0.307758\pi$$
0.567896 + 0.823101i $$0.307758\pi$$
$$692$$ 0 0
$$693$$ −4.00000 −0.151947
$$694$$ 0 0
$$695$$ 14.1436 0.536497
$$696$$ 0 0
$$697$$ 29.1769 1.10515
$$698$$ 0 0
$$699$$ −17.3205 −0.655122
$$700$$ 0 0
$$701$$ −5.21539 −0.196983 −0.0984913 0.995138i $$-0.531402\pi$$
−0.0984913 + 0.995138i $$0.531402\pi$$
$$702$$ 0 0
$$703$$ −4.92820 −0.185871
$$704$$ 0 0
$$705$$ 0.143594 0.00540805
$$706$$ 0 0
$$707$$ −4.73205 −0.177967
$$708$$ 0 0
$$709$$ 39.3205 1.47671 0.738356 0.674411i $$-0.235602\pi$$
0.738356 + 0.674411i $$0.235602\pi$$
$$710$$ 0 0
$$711$$ 2.92820 0.109816
$$712$$ 0 0
$$713$$ 13.8564 0.518927
$$714$$ 0 0
$$715$$ 4.28719 0.160332
$$716$$ 0 0
$$717$$ −9.46410 −0.353443
$$718$$ 0 0
$$719$$ 11.4115 0.425579 0.212789 0.977098i $$-0.431745\pi$$
0.212789 + 0.977098i $$0.431745\pi$$
$$720$$ 0 0
$$721$$ −2.92820 −0.109052
$$722$$ 0 0
$$723$$ 10.7846 0.401084
$$724$$ 0 0
$$725$$ 14.5885 0.541802
$$726$$ 0 0
$$727$$ −35.7128 −1.32451 −0.662257 0.749276i $$-0.730401\pi$$
−0.662257 + 0.749276i $$0.730401\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ −16.1051 −0.595669
$$732$$ 0 0
$$733$$ 37.3205 1.37846 0.689232 0.724541i $$-0.257948\pi$$
0.689232 + 0.724541i $$0.257948\pi$$
$$734$$ 0 0
$$735$$ 0.732051 0.0270021
$$736$$ 0 0
$$737$$ −21.8564 −0.805091
$$738$$ 0 0
$$739$$ −47.8564 −1.76043 −0.880213 0.474578i $$-0.842601\pi$$
−0.880213 + 0.474578i $$0.842601\pi$$
$$740$$ 0 0
$$741$$ −1.46410 −0.0537851
$$742$$ 0 0
$$743$$ 5.66025 0.207655 0.103827 0.994595i $$-0.466891\pi$$
0.103827 + 0.994595i $$0.466891\pi$$
$$744$$ 0 0
$$745$$ −2.53590 −0.0929081
$$746$$ 0 0
$$747$$ 8.19615 0.299882
$$748$$ 0 0
$$749$$ −12.1962 −0.445638
$$750$$ 0 0
$$751$$ 24.3923 0.890088 0.445044 0.895509i $$-0.353188\pi$$
0.445044 + 0.895509i $$0.353188\pi$$
$$752$$ 0 0
$$753$$ −1.66025 −0.0605030
$$754$$ 0 0
$$755$$ 9.85641 0.358711
$$756$$ 0 0
$$757$$ −14.7846 −0.537356 −0.268678 0.963230i $$-0.586587\pi$$
−0.268678 + 0.963230i $$0.586587\pi$$
$$758$$ 0 0
$$759$$ 27.7128 1.00591
$$760$$ 0 0
$$761$$ −28.0526 −1.01690 −0.508452 0.861090i $$-0.669782\pi$$
−0.508452 + 0.861090i $$0.669782\pi$$
$$762$$ 0 0
$$763$$ −4.53590 −0.164211
$$764$$ 0 0
$$765$$ −2.39230 −0.0864940
$$766$$ 0 0
$$767$$ 16.0000 0.577727
$$768$$ 0 0
$$769$$ 8.92820 0.321959 0.160980 0.986958i $$-0.448535\pi$$
0.160980 + 0.986958i $$0.448535\pi$$
$$770$$ 0 0
$$771$$ 2.39230 0.0861568
$$772$$ 0 0
$$773$$ 42.7846 1.53886 0.769428 0.638734i $$-0.220542\pi$$
0.769428 + 0.638734i $$0.220542\pi$$
$$774$$ 0 0
$$775$$ 8.92820 0.320711
$$776$$ 0 0
$$777$$ −4.92820 −0.176798
$$778$$ 0 0
$$779$$ −8.92820 −0.319886
$$780$$ 0 0
$$781$$ −16.7846 −0.600601
$$782$$ 0 0
$$783$$ −3.26795 −0.116787
$$784$$ 0 0
$$785$$ −2.24871 −0.0802599
$$786$$ 0 0
$$787$$ 4.14359 0.147703 0.0738516 0.997269i $$-0.476471\pi$$
0.0738516 + 0.997269i $$0.476471\pi$$
$$788$$ 0 0
$$789$$ 2.53590 0.0902804
$$790$$ 0 0
$$791$$ 16.7321 0.594923
$$792$$ 0 0
$$793$$ −1.35898 −0.0482589
$$794$$ 0 0
$$795$$ 5.60770 0.198884
$$796$$ 0 0
$$797$$ 14.7846 0.523698 0.261849 0.965109i $$-0.415668\pi$$
0.261849 + 0.965109i $$0.415668\pi$$
$$798$$ 0 0
$$799$$ −0.641016 −0.0226775
$$800$$ 0 0
$$801$$ −6.00000 −0.212000
$$802$$ 0 0
$$803$$ 41.5692 1.46695
$$804$$ 0 0
$$805$$ −5.07180 −0.178757
$$806$$ 0 0
$$807$$ −27.4641 −0.966782
$$808$$ 0 0
$$809$$ 49.7128 1.74781 0.873905 0.486097i $$-0.161580\pi$$
0.873905 + 0.486097i $$0.161580\pi$$
$$810$$ 0 0
$$811$$ 36.0000 1.26413 0.632065 0.774915i $$-0.282207\pi$$
0.632065 + 0.774915i $$0.282207\pi$$
$$812$$ 0 0
$$813$$ 20.0000 0.701431
$$814$$ 0 0
$$815$$ −9.46410 −0.331513
$$816$$ 0 0
$$817$$ 4.92820 0.172416
$$818$$ 0 0
$$819$$ −1.46410 −0.0511599
$$820$$ 0 0
$$821$$ −41.3205 −1.44210 −0.721048 0.692885i $$-0.756339\pi$$
−0.721048 + 0.692885i $$0.756339\pi$$
$$822$$ 0 0
$$823$$ 22.7846 0.794222 0.397111 0.917771i $$-0.370013\pi$$
0.397111 + 0.917771i $$0.370013\pi$$
$$824$$ 0 0
$$825$$ 17.8564 0.621680
$$826$$ 0 0
$$827$$ 16.9808 0.590479 0.295239 0.955423i $$-0.404601\pi$$
0.295239 + 0.955423i $$0.404601\pi$$
$$828$$ 0 0
$$829$$ −9.71281 −0.337340 −0.168670 0.985673i $$-0.553947\pi$$
−0.168670 + 0.985673i $$0.553947\pi$$
$$830$$ 0 0
$$831$$ −24.3923 −0.846160
$$832$$ 0 0
$$833$$ −3.26795 −0.113228
$$834$$ 0 0
$$835$$ 6.92820 0.239760
$$836$$ 0 0
$$837$$ −2.00000 −0.0691301
$$838$$ 0 0
$$839$$ 42.6410 1.47213 0.736066 0.676910i $$-0.236681\pi$$
0.736066 + 0.676910i $$0.236681\pi$$
$$840$$ 0 0
$$841$$ −18.3205 −0.631742
$$842$$ 0 0
$$843$$ −4.73205 −0.162980
$$844$$ 0 0
$$845$$ −7.94744 −0.273400
$$846$$ 0 0
$$847$$ 5.00000 0.171802
$$848$$ 0 0
$$849$$ −8.39230 −0.288023
$$850$$ 0 0
$$851$$ 34.1436 1.17043
$$852$$ 0 0
$$853$$ 34.0000 1.16414 0.582069 0.813139i $$-0.302243\pi$$
0.582069 + 0.813139i $$0.302243\pi$$
$$854$$ 0 0
$$855$$ 0.732051 0.0250356
$$856$$ 0 0
$$857$$ −7.85641 −0.268370 −0.134185 0.990956i $$-0.542842\pi$$
−0.134185 + 0.990956i $$0.542842\pi$$
$$858$$ 0 0
$$859$$ 37.8564 1.29164 0.645822 0.763488i $$-0.276515\pi$$
0.645822 + 0.763488i $$0.276515\pi$$
$$860$$ 0 0
$$861$$ −8.92820 −0.304272
$$862$$ 0 0
$$863$$ −41.3731 −1.40836 −0.704178 0.710024i $$-0.748684\pi$$
−0.704178 + 0.710024i $$0.748684\pi$$
$$864$$ 0 0
$$865$$ −8.67949 −0.295112
$$866$$ 0 0
$$867$$ −6.32051 −0.214656
$$868$$ 0 0
$$869$$ −11.7128 −0.397330
$$870$$ 0 0
$$871$$ −8.00000 −0.271070
$$872$$ 0 0
$$873$$ 11.8564 0.401279
$$874$$ 0 0
$$875$$ −6.92820 −0.234216
$$876$$ 0 0
$$877$$ −35.5692 −1.20109 −0.600544 0.799592i $$-0.705049\pi$$
−0.600544 + 0.799592i $$0.705049\pi$$
$$878$$ 0 0
$$879$$ −27.8564 −0.939573
$$880$$ 0 0
$$881$$ 23.2679 0.783917 0.391959 0.919983i $$-0.371798\pi$$
0.391959 + 0.919983i $$0.371798\pi$$
$$882$$ 0 0
$$883$$ −28.7846 −0.968679 −0.484340 0.874880i $$-0.660940\pi$$
−0.484340 + 0.874880i $$0.660940\pi$$
$$884$$ 0 0
$$885$$ −8.00000 −0.268917
$$886$$ 0 0
$$887$$ 41.1769 1.38259 0.691293 0.722575i $$-0.257042\pi$$
0.691293 + 0.722575i $$0.257042\pi$$
$$888$$ 0 0
$$889$$ −6.53590 −0.219207
$$890$$ 0 0
$$891$$ −4.00000 −0.134005
$$892$$ 0 0
$$893$$ 0.196152 0.00656399
$$894$$ 0 0
$$895$$ −10.7846 −0.360490
$$896$$ 0 0
$$897$$ 10.1436 0.338685
$$898$$ 0 0
$$899$$ 6.53590 0.217984
$$900$$ 0 0
$$901$$ −25.0333 −0.833981
$$902$$ 0 0
$$903$$ 4.92820 0.164000
$$904$$ 0 0
$$905$$ −5.17691 −0.172086
$$906$$ 0 0
$$907$$ −34.6410 −1.15024 −0.575118 0.818070i $$-0.695044\pi$$
−0.575118 + 0.818070i $$0.695044\pi$$
$$908$$ 0 0
$$909$$ −4.73205 −0.156952
$$910$$ 0 0
$$911$$ 47.5167 1.57430 0.787149 0.616763i $$-0.211556\pi$$
0.787149 + 0.616763i $$0.211556\pi$$
$$912$$ 0 0
$$913$$ −32.7846 −1.08501
$$914$$ 0 0
$$915$$ 0.679492 0.0224633
$$916$$ 0 0
$$917$$ 10.7321 0.354404
$$918$$ 0 0
$$919$$ −3.71281 −0.122474 −0.0612372 0.998123i $$-0.519505\pi$$
−0.0612372 + 0.998123i $$0.519505\pi$$
$$920$$ 0 0
$$921$$ 16.9282 0.557803
$$922$$ 0 0
$$923$$ −6.14359 −0.202219
$$924$$ 0 0
$$925$$ 22.0000 0.723356
$$926$$ 0 0
$$927$$ −2.92820 −0.0961748
$$928$$ 0 0
$$929$$ 26.5885 0.872339 0.436169 0.899865i $$-0.356335\pi$$
0.436169 + 0.899865i $$0.356335\pi$$
$$930$$ 0 0
$$931$$ 1.00000 0.0327737
$$932$$ 0 0
$$933$$ −7.12436 −0.233241
$$934$$ 0 0
$$935$$ 9.56922 0.312947
$$936$$ 0 0
$$937$$ 52.9282 1.72909 0.864545 0.502556i $$-0.167607\pi$$
0.864545 + 0.502556i $$0.167607\pi$$
$$938$$ 0 0
$$939$$ 19.0718 0.622385
$$940$$ 0 0
$$941$$ 30.3923 0.990761 0.495380 0.868676i $$-0.335029\pi$$
0.495380 + 0.868676i $$0.335029\pi$$
$$942$$ 0 0
$$943$$ 61.8564 2.01432
$$944$$ 0 0
$$945$$ 0.732051 0.0238136
$$946$$ 0 0
$$947$$ 10.1436 0.329622 0.164811 0.986325i $$-0.447299\pi$$
0.164811 + 0.986325i $$0.447299\pi$$
$$948$$ 0 0
$$949$$ 15.2154 0.493912
$$950$$ 0 0
$$951$$ −19.2679 −0.624806
$$952$$ 0 0
$$953$$ −1.41154 −0.0457244 −0.0228622 0.999739i $$-0.507278\pi$$
−0.0228622 + 0.999739i $$0.507278\pi$$
$$954$$ 0 0
$$955$$ 13.8564 0.448383
$$956$$ 0 0
$$957$$ 13.0718 0.422551
$$958$$ 0 0
$$959$$ 4.92820 0.159140
$$960$$ 0 0
$$961$$ −27.0000 −0.870968
$$962$$ 0 0
$$963$$ −12.1962 −0.393016
$$964$$ 0 0
$$965$$ 6.53590 0.210398
$$966$$ 0 0
$$967$$ −30.7846 −0.989966 −0.494983 0.868903i $$-0.664826\pi$$
−0.494983 + 0.868903i $$0.664826\pi$$
$$968$$ 0 0
$$969$$ −3.26795 −0.104982
$$970$$ 0 0
$$971$$ −15.2154 −0.488285 −0.244143 0.969739i $$-0.578506\pi$$
−0.244143 + 0.969739i $$0.578506\pi$$
$$972$$ 0 0
$$973$$ 19.3205 0.619387
$$974$$ 0 0
$$975$$ 6.53590 0.209316
$$976$$ 0 0
$$977$$ 25.5167 0.816350 0.408175 0.912904i $$-0.366165\pi$$
0.408175 + 0.912904i $$0.366165\pi$$
$$978$$ 0 0
$$979$$ 24.0000 0.767043
$$980$$ 0 0
$$981$$ −4.53590 −0.144820
$$982$$ 0 0
$$983$$ 8.78461 0.280186 0.140093 0.990138i $$-0.455260\pi$$
0.140093 + 0.990138i $$0.455260\pi$$
$$984$$ 0 0
$$985$$ 0.679492 0.0216504
$$986$$ 0 0
$$987$$ 0.196152 0.00624360
$$988$$ 0 0
$$989$$ −34.1436 −1.08570
$$990$$ 0 0
$$991$$ −30.6410 −0.973344 −0.486672 0.873585i $$-0.661789\pi$$
−0.486672 + 0.873585i $$0.661789\pi$$
$$992$$ 0 0
$$993$$ 8.39230 0.266322
$$994$$ 0 0
$$995$$ 14.1436 0.448382
$$996$$ 0 0
$$997$$ −43.0718 −1.36410 −0.682049 0.731307i $$-0.738911\pi$$
−0.682049 + 0.731307i $$0.738911\pi$$
$$998$$ 0 0
$$999$$ −4.92820 −0.155921
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.s.1.2 2
3.2 odd 2 9576.2.a.bw.1.1 2
4.3 odd 2 6384.2.a.bi.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
3192.2.a.s.1.2 2 1.1 even 1 trivial
6384.2.a.bi.1.2 2 4.3 odd 2
9576.2.a.bw.1.1 2 3.2 odd 2