# Properties

 Label 9280.2.a.bj.1.3 Level $9280$ Weight $2$ Character 9280.1 Self dual yes Analytic conductor $74.101$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9280,2,Mod(1,9280)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9280, 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("9280.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$9280 = 2^{6} \cdot 5 \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9280.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: $$74.1011730757$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.148.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 3x + 1$$ x^3 - x^2 - 3*x + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 145) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.3 Root $$2.17009$$ of defining polynomial Character $$\chi$$ $$=$$ 9280.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.70928 q^{3} -1.00000 q^{5} +3.70928 q^{7} -0.0783777 q^{9} +O(q^{10})$$ $$q+1.70928 q^{3} -1.00000 q^{5} +3.70928 q^{7} -0.0783777 q^{9} +0.630898 q^{11} +4.34017 q^{13} -1.70928 q^{15} -1.55252 q^{17} +5.70928 q^{19} +6.34017 q^{21} +6.63090 q^{23} +1.00000 q^{25} -5.26180 q^{27} +1.00000 q^{29} -2.29072 q^{31} +1.07838 q^{33} -3.70928 q^{35} +2.44748 q^{37} +7.41855 q^{39} +5.60197 q^{41} -12.5464 q^{43} +0.0783777 q^{45} +2.29072 q^{47} +6.75872 q^{49} -2.65368 q^{51} -0.921622 q^{53} -0.630898 q^{55} +9.75872 q^{57} +3.60197 q^{59} +13.0205 q^{61} -0.290725 q^{63} -4.34017 q^{65} -10.6309 q^{67} +11.3340 q^{69} +15.6020 q^{71} -10.9444 q^{73} +1.70928 q^{75} +2.34017 q^{77} -10.2062 q^{79} -8.75872 q^{81} +3.12783 q^{83} +1.55252 q^{85} +1.70928 q^{87} +1.41855 q^{89} +16.0989 q^{91} -3.91548 q^{93} -5.70928 q^{95} +13.4680 q^{97} -0.0494483 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 2 q^{3} - 3 q^{5} + 4 q^{7} + 3 q^{9}+O(q^{10})$$ 3 * q - 2 * q^3 - 3 * q^5 + 4 * q^7 + 3 * q^9 $$3 q - 2 q^{3} - 3 q^{5} + 4 q^{7} + 3 q^{9} - 2 q^{11} + 2 q^{13} + 2 q^{15} - 4 q^{17} + 10 q^{19} + 8 q^{21} + 16 q^{23} + 3 q^{25} - 8 q^{27} + 3 q^{29} - 14 q^{31} - 4 q^{35} + 8 q^{37} + 8 q^{39} - 2 q^{41} - 2 q^{43} - 3 q^{45} + 14 q^{47} - 5 q^{49} + 16 q^{51} - 6 q^{53} + 2 q^{55} + 4 q^{57} - 8 q^{59} + 6 q^{61} - 8 q^{63} - 2 q^{65} - 28 q^{67} - 12 q^{69} + 28 q^{71} - 16 q^{73} - 2 q^{75} - 4 q^{77} - 6 q^{79} - q^{81} - 12 q^{83} + 4 q^{85} - 2 q^{87} - 10 q^{89} + 12 q^{91} + 20 q^{93} - 10 q^{95} + 8 q^{97} + 18 q^{99}+O(q^{100})$$ 3 * q - 2 * q^3 - 3 * q^5 + 4 * q^7 + 3 * q^9 - 2 * q^11 + 2 * q^13 + 2 * q^15 - 4 * q^17 + 10 * q^19 + 8 * q^21 + 16 * q^23 + 3 * q^25 - 8 * q^27 + 3 * q^29 - 14 * q^31 - 4 * q^35 + 8 * q^37 + 8 * q^39 - 2 * q^41 - 2 * q^43 - 3 * q^45 + 14 * q^47 - 5 * q^49 + 16 * q^51 - 6 * q^53 + 2 * q^55 + 4 * q^57 - 8 * q^59 + 6 * q^61 - 8 * q^63 - 2 * q^65 - 28 * q^67 - 12 * q^69 + 28 * q^71 - 16 * q^73 - 2 * q^75 - 4 * q^77 - 6 * q^79 - q^81 - 12 * q^83 + 4 * q^85 - 2 * q^87 - 10 * q^89 + 12 * q^91 + 20 * q^93 - 10 * q^95 + 8 * q^97 + 18 * 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
$$3$$ 1.70928 0.986851 0.493425 0.869788i $$-0.335745\pi$$
0.493425 + 0.869788i $$0.335745\pi$$
$$4$$ 0 0
$$5$$ −1.00000 −0.447214
$$6$$ 0 0
$$7$$ 3.70928 1.40197 0.700987 0.713174i $$-0.252743\pi$$
0.700987 + 0.713174i $$0.252743\pi$$
$$8$$ 0 0
$$9$$ −0.0783777 −0.0261259
$$10$$ 0 0
$$11$$ 0.630898 0.190223 0.0951114 0.995467i $$-0.469679\pi$$
0.0951114 + 0.995467i $$0.469679\pi$$
$$12$$ 0 0
$$13$$ 4.34017 1.20375 0.601874 0.798591i $$-0.294421\pi$$
0.601874 + 0.798591i $$0.294421\pi$$
$$14$$ 0 0
$$15$$ −1.70928 −0.441333
$$16$$ 0 0
$$17$$ −1.55252 −0.376541 −0.188271 0.982117i $$-0.560288\pi$$
−0.188271 + 0.982117i $$0.560288\pi$$
$$18$$ 0 0
$$19$$ 5.70928 1.30980 0.654899 0.755717i $$-0.272711\pi$$
0.654899 + 0.755717i $$0.272711\pi$$
$$20$$ 0 0
$$21$$ 6.34017 1.38354
$$22$$ 0 0
$$23$$ 6.63090 1.38264 0.691319 0.722550i $$-0.257030\pi$$
0.691319 + 0.722550i $$0.257030\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ −5.26180 −1.01263
$$28$$ 0 0
$$29$$ 1.00000 0.185695
$$30$$ 0 0
$$31$$ −2.29072 −0.411426 −0.205713 0.978612i $$-0.565951\pi$$
−0.205713 + 0.978612i $$0.565951\pi$$
$$32$$ 0 0
$$33$$ 1.07838 0.187721
$$34$$ 0 0
$$35$$ −3.70928 −0.626982
$$36$$ 0 0
$$37$$ 2.44748 0.402363 0.201182 0.979554i $$-0.435522\pi$$
0.201182 + 0.979554i $$0.435522\pi$$
$$38$$ 0 0
$$39$$ 7.41855 1.18792
$$40$$ 0 0
$$41$$ 5.60197 0.874880 0.437440 0.899247i $$-0.355885\pi$$
0.437440 + 0.899247i $$0.355885\pi$$
$$42$$ 0 0
$$43$$ −12.5464 −1.91330 −0.956652 0.291233i $$-0.905935\pi$$
−0.956652 + 0.291233i $$0.905935\pi$$
$$44$$ 0 0
$$45$$ 0.0783777 0.0116839
$$46$$ 0 0
$$47$$ 2.29072 0.334137 0.167068 0.985945i $$-0.446570\pi$$
0.167068 + 0.985945i $$0.446570\pi$$
$$48$$ 0 0
$$49$$ 6.75872 0.965532
$$50$$ 0 0
$$51$$ −2.65368 −0.371590
$$52$$ 0 0
$$53$$ −0.921622 −0.126595 −0.0632973 0.997995i $$-0.520162\pi$$
−0.0632973 + 0.997995i $$0.520162\pi$$
$$54$$ 0 0
$$55$$ −0.630898 −0.0850702
$$56$$ 0 0
$$57$$ 9.75872 1.29257
$$58$$ 0 0
$$59$$ 3.60197 0.468936 0.234468 0.972124i $$-0.424665\pi$$
0.234468 + 0.972124i $$0.424665\pi$$
$$60$$ 0 0
$$61$$ 13.0205 1.66711 0.833553 0.552439i $$-0.186303\pi$$
0.833553 + 0.552439i $$0.186303\pi$$
$$62$$ 0 0
$$63$$ −0.290725 −0.0366279
$$64$$ 0 0
$$65$$ −4.34017 −0.538332
$$66$$ 0 0
$$67$$ −10.6309 −1.29877 −0.649385 0.760459i $$-0.724974\pi$$
−0.649385 + 0.760459i $$0.724974\pi$$
$$68$$ 0 0
$$69$$ 11.3340 1.36446
$$70$$ 0 0
$$71$$ 15.6020 1.85161 0.925806 0.377998i $$-0.123387\pi$$
0.925806 + 0.377998i $$0.123387\pi$$
$$72$$ 0 0
$$73$$ −10.9444 −1.28095 −0.640473 0.767981i $$-0.721262\pi$$
−0.640473 + 0.767981i $$0.721262\pi$$
$$74$$ 0 0
$$75$$ 1.70928 0.197370
$$76$$ 0 0
$$77$$ 2.34017 0.266687
$$78$$ 0 0
$$79$$ −10.2062 −1.14829 −0.574144 0.818754i $$-0.694665\pi$$
−0.574144 + 0.818754i $$0.694665\pi$$
$$80$$ 0 0
$$81$$ −8.75872 −0.973192
$$82$$ 0 0
$$83$$ 3.12783 0.343324 0.171662 0.985156i $$-0.445086\pi$$
0.171662 + 0.985156i $$0.445086\pi$$
$$84$$ 0 0
$$85$$ 1.55252 0.168394
$$86$$ 0 0
$$87$$ 1.70928 0.183254
$$88$$ 0 0
$$89$$ 1.41855 0.150366 0.0751830 0.997170i $$-0.476046\pi$$
0.0751830 + 0.997170i $$0.476046\pi$$
$$90$$ 0 0
$$91$$ 16.0989 1.68762
$$92$$ 0 0
$$93$$ −3.91548 −0.406016
$$94$$ 0 0
$$95$$ −5.70928 −0.585759
$$96$$ 0 0
$$97$$ 13.4680 1.36747 0.683734 0.729731i $$-0.260355\pi$$
0.683734 + 0.729731i $$0.260355\pi$$
$$98$$ 0 0
$$99$$ −0.0494483 −0.00496974
$$100$$ 0 0
$$101$$ −1.10504 −0.109956 −0.0549778 0.998488i $$-0.517509\pi$$
−0.0549778 + 0.998488i $$0.517509\pi$$
$$102$$ 0 0
$$103$$ 15.6248 1.53955 0.769776 0.638314i $$-0.220368\pi$$
0.769776 + 0.638314i $$0.220368\pi$$
$$104$$ 0 0
$$105$$ −6.34017 −0.618738
$$106$$ 0 0
$$107$$ −2.81432 −0.272070 −0.136035 0.990704i $$-0.543436\pi$$
−0.136035 + 0.990704i $$0.543436\pi$$
$$108$$ 0 0
$$109$$ −5.91548 −0.566600 −0.283300 0.959031i $$-0.591429\pi$$
−0.283300 + 0.959031i $$0.591429\pi$$
$$110$$ 0 0
$$111$$ 4.18342 0.397072
$$112$$ 0 0
$$113$$ −1.95055 −0.183492 −0.0917462 0.995782i $$-0.529245\pi$$
−0.0917462 + 0.995782i $$0.529245\pi$$
$$114$$ 0 0
$$115$$ −6.63090 −0.618334
$$116$$ 0 0
$$117$$ −0.340173 −0.0314490
$$118$$ 0 0
$$119$$ −5.75872 −0.527901
$$120$$ 0 0
$$121$$ −10.6020 −0.963815
$$122$$ 0 0
$$123$$ 9.57531 0.863376
$$124$$ 0 0
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ −22.4885 −1.99553 −0.997767 0.0667962i $$-0.978722\pi$$
−0.997767 + 0.0667962i $$0.978722\pi$$
$$128$$ 0 0
$$129$$ −21.4452 −1.88815
$$130$$ 0 0
$$131$$ −3.86603 −0.337777 −0.168888 0.985635i $$-0.554018\pi$$
−0.168888 + 0.985635i $$0.554018\pi$$
$$132$$ 0 0
$$133$$ 21.1773 1.83630
$$134$$ 0 0
$$135$$ 5.26180 0.452863
$$136$$ 0 0
$$137$$ 21.2846 1.81846 0.909232 0.416289i $$-0.136670\pi$$
0.909232 + 0.416289i $$0.136670\pi$$
$$138$$ 0 0
$$139$$ −8.09890 −0.686939 −0.343470 0.939164i $$-0.611602\pi$$
−0.343470 + 0.939164i $$0.611602\pi$$
$$140$$ 0 0
$$141$$ 3.91548 0.329743
$$142$$ 0 0
$$143$$ 2.73820 0.228980
$$144$$ 0 0
$$145$$ −1.00000 −0.0830455
$$146$$ 0 0
$$147$$ 11.5525 0.952836
$$148$$ 0 0
$$149$$ −12.8371 −1.05166 −0.525828 0.850591i $$-0.676245\pi$$
−0.525828 + 0.850591i $$0.676245\pi$$
$$150$$ 0 0
$$151$$ −20.5958 −1.67606 −0.838032 0.545621i $$-0.816294\pi$$
−0.838032 + 0.545621i $$0.816294\pi$$
$$152$$ 0 0
$$153$$ 0.121683 0.00983749
$$154$$ 0 0
$$155$$ 2.29072 0.183995
$$156$$ 0 0
$$157$$ 6.04945 0.482799 0.241399 0.970426i $$-0.422394\pi$$
0.241399 + 0.970426i $$0.422394\pi$$
$$158$$ 0 0
$$159$$ −1.57531 −0.124930
$$160$$ 0 0
$$161$$ 24.5958 1.93842
$$162$$ 0 0
$$163$$ 15.9649 1.25047 0.625235 0.780437i $$-0.285003\pi$$
0.625235 + 0.780437i $$0.285003\pi$$
$$164$$ 0 0
$$165$$ −1.07838 −0.0839516
$$166$$ 0 0
$$167$$ −11.3112 −0.875290 −0.437645 0.899148i $$-0.644187\pi$$
−0.437645 + 0.899148i $$0.644187\pi$$
$$168$$ 0 0
$$169$$ 5.83710 0.449008
$$170$$ 0 0
$$171$$ −0.447480 −0.0342197
$$172$$ 0 0
$$173$$ 10.4969 0.798067 0.399033 0.916936i $$-0.369346\pi$$
0.399033 + 0.916936i $$0.369346\pi$$
$$174$$ 0 0
$$175$$ 3.70928 0.280395
$$176$$ 0 0
$$177$$ 6.15676 0.462770
$$178$$ 0 0
$$179$$ 12.3135 0.920355 0.460178 0.887827i $$-0.347786\pi$$
0.460178 + 0.887827i $$0.347786\pi$$
$$180$$ 0 0
$$181$$ 1.60197 0.119073 0.0595367 0.998226i $$-0.481038\pi$$
0.0595367 + 0.998226i $$0.481038\pi$$
$$182$$ 0 0
$$183$$ 22.2557 1.64519
$$184$$ 0 0
$$185$$ −2.44748 −0.179942
$$186$$ 0 0
$$187$$ −0.979481 −0.0716268
$$188$$ 0 0
$$189$$ −19.5174 −1.41969
$$190$$ 0 0
$$191$$ 13.6248 0.985853 0.492926 0.870071i $$-0.335927\pi$$
0.492926 + 0.870071i $$0.335927\pi$$
$$192$$ 0 0
$$193$$ −16.9711 −1.22160 −0.610802 0.791783i $$-0.709153\pi$$
−0.610802 + 0.791783i $$0.709153\pi$$
$$194$$ 0 0
$$195$$ −7.41855 −0.531253
$$196$$ 0 0
$$197$$ 0.0578588 0.00412227 0.00206114 0.999998i $$-0.499344\pi$$
0.00206114 + 0.999998i $$0.499344\pi$$
$$198$$ 0 0
$$199$$ 5.39189 0.382221 0.191110 0.981569i $$-0.438791\pi$$
0.191110 + 0.981569i $$0.438791\pi$$
$$200$$ 0 0
$$201$$ −18.1711 −1.28169
$$202$$ 0 0
$$203$$ 3.70928 0.260340
$$204$$ 0 0
$$205$$ −5.60197 −0.391258
$$206$$ 0 0
$$207$$ −0.519715 −0.0361227
$$208$$ 0 0
$$209$$ 3.60197 0.249153
$$210$$ 0 0
$$211$$ −4.14834 −0.285584 −0.142792 0.989753i $$-0.545608\pi$$
−0.142792 + 0.989753i $$0.545608\pi$$
$$212$$ 0 0
$$213$$ 26.6681 1.82727
$$214$$ 0 0
$$215$$ 12.5464 0.855656
$$216$$ 0 0
$$217$$ −8.49693 −0.576809
$$218$$ 0 0
$$219$$ −18.7070 −1.26410
$$220$$ 0 0
$$221$$ −6.73820 −0.453261
$$222$$ 0 0
$$223$$ −6.72979 −0.450660 −0.225330 0.974282i $$-0.572346\pi$$
−0.225330 + 0.974282i $$0.572346\pi$$
$$224$$ 0 0
$$225$$ −0.0783777 −0.00522518
$$226$$ 0 0
$$227$$ 22.2472 1.47660 0.738301 0.674472i $$-0.235629\pi$$
0.738301 + 0.674472i $$0.235629\pi$$
$$228$$ 0 0
$$229$$ 7.16290 0.473338 0.236669 0.971590i $$-0.423944\pi$$
0.236669 + 0.971590i $$0.423944\pi$$
$$230$$ 0 0
$$231$$ 4.00000 0.263181
$$232$$ 0 0
$$233$$ 30.1978 1.97832 0.989162 0.146831i $$-0.0469073\pi$$
0.989162 + 0.146831i $$0.0469073\pi$$
$$234$$ 0 0
$$235$$ −2.29072 −0.149430
$$236$$ 0 0
$$237$$ −17.4452 −1.13319
$$238$$ 0 0
$$239$$ −6.43907 −0.416509 −0.208254 0.978075i $$-0.566778\pi$$
−0.208254 + 0.978075i $$0.566778\pi$$
$$240$$ 0 0
$$241$$ 10.9939 0.708177 0.354088 0.935212i $$-0.384791\pi$$
0.354088 + 0.935212i $$0.384791\pi$$
$$242$$ 0 0
$$243$$ 0.814315 0.0522383
$$244$$ 0 0
$$245$$ −6.75872 −0.431799
$$246$$ 0 0
$$247$$ 24.7792 1.57667
$$248$$ 0 0
$$249$$ 5.34632 0.338809
$$250$$ 0 0
$$251$$ −9.41014 −0.593963 −0.296981 0.954883i $$-0.595980\pi$$
−0.296981 + 0.954883i $$0.595980\pi$$
$$252$$ 0 0
$$253$$ 4.18342 0.263009
$$254$$ 0 0
$$255$$ 2.65368 0.166180
$$256$$ 0 0
$$257$$ 5.81658 0.362828 0.181414 0.983407i $$-0.441933\pi$$
0.181414 + 0.983407i $$0.441933\pi$$
$$258$$ 0 0
$$259$$ 9.07838 0.564103
$$260$$ 0 0
$$261$$ −0.0783777 −0.00485146
$$262$$ 0 0
$$263$$ −8.91321 −0.549612 −0.274806 0.961500i $$-0.588614\pi$$
−0.274806 + 0.961500i $$0.588614\pi$$
$$264$$ 0 0
$$265$$ 0.921622 0.0566148
$$266$$ 0 0
$$267$$ 2.42469 0.148389
$$268$$ 0 0
$$269$$ 16.4391 1.00231 0.501154 0.865358i $$-0.332909\pi$$
0.501154 + 0.865358i $$0.332909\pi$$
$$270$$ 0 0
$$271$$ 29.4101 1.78654 0.893269 0.449522i $$-0.148406\pi$$
0.893269 + 0.449522i $$0.148406\pi$$
$$272$$ 0 0
$$273$$ 27.5174 1.66543
$$274$$ 0 0
$$275$$ 0.630898 0.0380446
$$276$$ 0 0
$$277$$ −11.0784 −0.665635 −0.332818 0.942991i $$-0.607999\pi$$
−0.332818 + 0.942991i $$0.607999\pi$$
$$278$$ 0 0
$$279$$ 0.179542 0.0107489
$$280$$ 0 0
$$281$$ −21.1194 −1.25988 −0.629939 0.776644i $$-0.716920\pi$$
−0.629939 + 0.776644i $$0.716920\pi$$
$$282$$ 0 0
$$283$$ 13.7815 0.819226 0.409613 0.912259i $$-0.365664\pi$$
0.409613 + 0.912259i $$0.365664\pi$$
$$284$$ 0 0
$$285$$ −9.75872 −0.578057
$$286$$ 0 0
$$287$$ 20.7792 1.22656
$$288$$ 0 0
$$289$$ −14.5897 −0.858217
$$290$$ 0 0
$$291$$ 23.0205 1.34949
$$292$$ 0 0
$$293$$ −6.14834 −0.359190 −0.179595 0.983741i $$-0.557479\pi$$
−0.179595 + 0.983741i $$0.557479\pi$$
$$294$$ 0 0
$$295$$ −3.60197 −0.209715
$$296$$ 0 0
$$297$$ −3.31965 −0.192626
$$298$$ 0 0
$$299$$ 28.7792 1.66435
$$300$$ 0 0
$$301$$ −46.5380 −2.68240
$$302$$ 0 0
$$303$$ −1.88882 −0.108510
$$304$$ 0 0
$$305$$ −13.0205 −0.745553
$$306$$ 0 0
$$307$$ −10.3896 −0.592967 −0.296484 0.955038i $$-0.595814\pi$$
−0.296484 + 0.955038i $$0.595814\pi$$
$$308$$ 0 0
$$309$$ 26.7070 1.51931
$$310$$ 0 0
$$311$$ 18.7565 1.06358 0.531791 0.846876i $$-0.321519\pi$$
0.531791 + 0.846876i $$0.321519\pi$$
$$312$$ 0 0
$$313$$ 12.3402 0.697508 0.348754 0.937214i $$-0.386605\pi$$
0.348754 + 0.937214i $$0.386605\pi$$
$$314$$ 0 0
$$315$$ 0.290725 0.0163805
$$316$$ 0 0
$$317$$ 30.6986 1.72421 0.862103 0.506734i $$-0.169147\pi$$
0.862103 + 0.506734i $$0.169147\pi$$
$$318$$ 0 0
$$319$$ 0.630898 0.0353235
$$320$$ 0 0
$$321$$ −4.81044 −0.268493
$$322$$ 0 0
$$323$$ −8.86376 −0.493193
$$324$$ 0 0
$$325$$ 4.34017 0.240749
$$326$$ 0 0
$$327$$ −10.1112 −0.559150
$$328$$ 0 0
$$329$$ 8.49693 0.468451
$$330$$ 0 0
$$331$$ 4.08065 0.224293 0.112146 0.993692i $$-0.464227\pi$$
0.112146 + 0.993692i $$0.464227\pi$$
$$332$$ 0 0
$$333$$ −0.191828 −0.0105121
$$334$$ 0 0
$$335$$ 10.6309 0.580828
$$336$$ 0 0
$$337$$ −18.3630 −1.00029 −0.500147 0.865940i $$-0.666721\pi$$
−0.500147 + 0.865940i $$0.666721\pi$$
$$338$$ 0 0
$$339$$ −3.33403 −0.181080
$$340$$ 0 0
$$341$$ −1.44521 −0.0782627
$$342$$ 0 0
$$343$$ −0.894960 −0.0483233
$$344$$ 0 0
$$345$$ −11.3340 −0.610204
$$346$$ 0 0
$$347$$ −8.97107 −0.481592 −0.240796 0.970576i $$-0.577409\pi$$
−0.240796 + 0.970576i $$0.577409\pi$$
$$348$$ 0 0
$$349$$ −26.1978 −1.40234 −0.701168 0.712996i $$-0.747338\pi$$
−0.701168 + 0.712996i $$0.747338\pi$$
$$350$$ 0 0
$$351$$ −22.8371 −1.21895
$$352$$ 0 0
$$353$$ −26.2823 −1.39887 −0.699433 0.714698i $$-0.746564\pi$$
−0.699433 + 0.714698i $$0.746564\pi$$
$$354$$ 0 0
$$355$$ −15.6020 −0.828066
$$356$$ 0 0
$$357$$ −9.84324 −0.520960
$$358$$ 0 0
$$359$$ 22.8722 1.20715 0.603574 0.797307i $$-0.293743\pi$$
0.603574 + 0.797307i $$0.293743\pi$$
$$360$$ 0 0
$$361$$ 13.5958 0.715570
$$362$$ 0 0
$$363$$ −18.1217 −0.951142
$$364$$ 0 0
$$365$$ 10.9444 0.572857
$$366$$ 0 0
$$367$$ −11.0700 −0.577848 −0.288924 0.957352i $$-0.593297\pi$$
−0.288924 + 0.957352i $$0.593297\pi$$
$$368$$ 0 0
$$369$$ −0.439070 −0.0228571
$$370$$ 0 0
$$371$$ −3.41855 −0.177482
$$372$$ 0 0
$$373$$ −11.5753 −0.599347 −0.299673 0.954042i $$-0.596878\pi$$
−0.299673 + 0.954042i $$0.596878\pi$$
$$374$$ 0 0
$$375$$ −1.70928 −0.0882666
$$376$$ 0 0
$$377$$ 4.34017 0.223530
$$378$$ 0 0
$$379$$ 9.31124 0.478286 0.239143 0.970984i $$-0.423133\pi$$
0.239143 + 0.970984i $$0.423133\pi$$
$$380$$ 0 0
$$381$$ −38.4391 −1.96929
$$382$$ 0 0
$$383$$ 33.9649 1.73553 0.867763 0.496978i $$-0.165557\pi$$
0.867763 + 0.496978i $$0.165557\pi$$
$$384$$ 0 0
$$385$$ −2.34017 −0.119266
$$386$$ 0 0
$$387$$ 0.983357 0.0499868
$$388$$ 0 0
$$389$$ 4.12556 0.209174 0.104587 0.994516i $$-0.466648\pi$$
0.104587 + 0.994516i $$0.466648\pi$$
$$390$$ 0 0
$$391$$ −10.2946 −0.520620
$$392$$ 0 0
$$393$$ −6.60811 −0.333335
$$394$$ 0 0
$$395$$ 10.2062 0.513530
$$396$$ 0 0
$$397$$ −17.1050 −0.858477 −0.429239 0.903191i $$-0.641218\pi$$
−0.429239 + 0.903191i $$0.641218\pi$$
$$398$$ 0 0
$$399$$ 36.1978 1.81216
$$400$$ 0 0
$$401$$ −0.554787 −0.0277048 −0.0138524 0.999904i $$-0.504409\pi$$
−0.0138524 + 0.999904i $$0.504409\pi$$
$$402$$ 0 0
$$403$$ −9.94214 −0.495253
$$404$$ 0 0
$$405$$ 8.75872 0.435224
$$406$$ 0 0
$$407$$ 1.54411 0.0765387
$$408$$ 0 0
$$409$$ −20.6537 −1.02126 −0.510629 0.859801i $$-0.670588\pi$$
−0.510629 + 0.859801i $$0.670588\pi$$
$$410$$ 0 0
$$411$$ 36.3812 1.79455
$$412$$ 0 0
$$413$$ 13.3607 0.657437
$$414$$ 0 0
$$415$$ −3.12783 −0.153539
$$416$$ 0 0
$$417$$ −13.8432 −0.677907
$$418$$ 0 0
$$419$$ 6.02666 0.294422 0.147211 0.989105i $$-0.452970\pi$$
0.147211 + 0.989105i $$0.452970\pi$$
$$420$$ 0 0
$$421$$ 12.5380 0.611063 0.305532 0.952182i $$-0.401166\pi$$
0.305532 + 0.952182i $$0.401166\pi$$
$$422$$ 0 0
$$423$$ −0.179542 −0.00872962
$$424$$ 0 0
$$425$$ −1.55252 −0.0753083
$$426$$ 0 0
$$427$$ 48.2967 2.33724
$$428$$ 0 0
$$429$$ 4.68035 0.225969
$$430$$ 0 0
$$431$$ −18.0410 −0.869006 −0.434503 0.900670i $$-0.643076\pi$$
−0.434503 + 0.900670i $$0.643076\pi$$
$$432$$ 0 0
$$433$$ 18.8143 0.904158 0.452079 0.891978i $$-0.350682\pi$$
0.452079 + 0.891978i $$0.350682\pi$$
$$434$$ 0 0
$$435$$ −1.70928 −0.0819535
$$436$$ 0 0
$$437$$ 37.8576 1.81098
$$438$$ 0 0
$$439$$ 5.54411 0.264606 0.132303 0.991209i $$-0.457763\pi$$
0.132303 + 0.991209i $$0.457763\pi$$
$$440$$ 0 0
$$441$$ −0.529734 −0.0252254
$$442$$ 0 0
$$443$$ −17.8082 −0.846092 −0.423046 0.906108i $$-0.639039\pi$$
−0.423046 + 0.906108i $$0.639039\pi$$
$$444$$ 0 0
$$445$$ −1.41855 −0.0672458
$$446$$ 0 0
$$447$$ −21.9421 −1.03783
$$448$$ 0 0
$$449$$ −10.6947 −0.504715 −0.252358 0.967634i $$-0.581206\pi$$
−0.252358 + 0.967634i $$0.581206\pi$$
$$450$$ 0 0
$$451$$ 3.53427 0.166422
$$452$$ 0 0
$$453$$ −35.2039 −1.65403
$$454$$ 0 0
$$455$$ −16.0989 −0.754728
$$456$$ 0 0
$$457$$ −21.7998 −1.01975 −0.509875 0.860249i $$-0.670308\pi$$
−0.509875 + 0.860249i $$0.670308\pi$$
$$458$$ 0 0
$$459$$ 8.16904 0.381298
$$460$$ 0 0
$$461$$ −22.4124 −1.04385 −0.521925 0.852991i $$-0.674786\pi$$
−0.521925 + 0.852991i $$0.674786\pi$$
$$462$$ 0 0
$$463$$ −2.10277 −0.0977241 −0.0488621 0.998806i $$-0.515559\pi$$
−0.0488621 + 0.998806i $$0.515559\pi$$
$$464$$ 0 0
$$465$$ 3.91548 0.181576
$$466$$ 0 0
$$467$$ 18.6042 0.860901 0.430451 0.902614i $$-0.358355\pi$$
0.430451 + 0.902614i $$0.358355\pi$$
$$468$$ 0 0
$$469$$ −39.4329 −1.82084
$$470$$ 0 0
$$471$$ 10.3402 0.476450
$$472$$ 0 0
$$473$$ −7.91548 −0.363954
$$474$$ 0 0
$$475$$ 5.70928 0.261960
$$476$$ 0 0
$$477$$ 0.0722347 0.00330740
$$478$$ 0 0
$$479$$ 8.89884 0.406598 0.203299 0.979117i $$-0.434834\pi$$
0.203299 + 0.979117i $$0.434834\pi$$
$$480$$ 0 0
$$481$$ 10.6225 0.484344
$$482$$ 0 0
$$483$$ 42.0410 1.91293
$$484$$ 0 0
$$485$$ −13.4680 −0.611550
$$486$$ 0 0
$$487$$ −12.9711 −0.587775 −0.293888 0.955840i $$-0.594949\pi$$
−0.293888 + 0.955840i $$0.594949\pi$$
$$488$$ 0 0
$$489$$ 27.2885 1.23403
$$490$$ 0 0
$$491$$ 13.8615 0.625561 0.312780 0.949826i $$-0.398740\pi$$
0.312780 + 0.949826i $$0.398740\pi$$
$$492$$ 0 0
$$493$$ −1.55252 −0.0699220
$$494$$ 0 0
$$495$$ 0.0494483 0.00222254
$$496$$ 0 0
$$497$$ 57.8720 2.59591
$$498$$ 0 0
$$499$$ 22.3545 1.00073 0.500364 0.865815i $$-0.333200\pi$$
0.500364 + 0.865815i $$0.333200\pi$$
$$500$$ 0 0
$$501$$ −19.3340 −0.863781
$$502$$ 0 0
$$503$$ 34.4885 1.53777 0.768884 0.639389i $$-0.220813\pi$$
0.768884 + 0.639389i $$0.220813\pi$$
$$504$$ 0 0
$$505$$ 1.10504 0.0491736
$$506$$ 0 0
$$507$$ 9.97721 0.443104
$$508$$ 0 0
$$509$$ 28.7526 1.27444 0.637218 0.770684i $$-0.280085\pi$$
0.637218 + 0.770684i $$0.280085\pi$$
$$510$$ 0 0
$$511$$ −40.5958 −1.79585
$$512$$ 0 0
$$513$$ −30.0410 −1.32634
$$514$$ 0 0
$$515$$ −15.6248 −0.688509
$$516$$ 0 0
$$517$$ 1.44521 0.0635604
$$518$$ 0 0
$$519$$ 17.9421 0.787573
$$520$$ 0 0
$$521$$ −21.6020 −0.946399 −0.473200 0.880955i $$-0.656901\pi$$
−0.473200 + 0.880955i $$0.656901\pi$$
$$522$$ 0 0
$$523$$ −4.20620 −0.183924 −0.0919622 0.995762i $$-0.529314\pi$$
−0.0919622 + 0.995762i $$0.529314\pi$$
$$524$$ 0 0
$$525$$ 6.34017 0.276708
$$526$$ 0 0
$$527$$ 3.55640 0.154919
$$528$$ 0 0
$$529$$ 20.9688 0.911687
$$530$$ 0 0
$$531$$ −0.282314 −0.0122514
$$532$$ 0 0
$$533$$ 24.3135 1.05314
$$534$$ 0 0
$$535$$ 2.81432 0.121673
$$536$$ 0 0
$$537$$ 21.0472 0.908253
$$538$$ 0 0
$$539$$ 4.26406 0.183666
$$540$$ 0 0
$$541$$ −26.7792 −1.15133 −0.575665 0.817686i $$-0.695257\pi$$
−0.575665 + 0.817686i $$0.695257\pi$$
$$542$$ 0 0
$$543$$ 2.73820 0.117508
$$544$$ 0 0
$$545$$ 5.91548 0.253391
$$546$$ 0 0
$$547$$ −2.33176 −0.0996990 −0.0498495 0.998757i $$-0.515874\pi$$
−0.0498495 + 0.998757i $$0.515874\pi$$
$$548$$ 0 0
$$549$$ −1.02052 −0.0435547
$$550$$ 0 0
$$551$$ 5.70928 0.243223
$$552$$ 0 0
$$553$$ −37.8576 −1.60987
$$554$$ 0 0
$$555$$ −4.18342 −0.177576
$$556$$ 0 0
$$557$$ −30.8781 −1.30835 −0.654174 0.756344i $$-0.726984\pi$$
−0.654174 + 0.756344i $$0.726984\pi$$
$$558$$ 0 0
$$559$$ −54.4534 −2.30314
$$560$$ 0 0
$$561$$ −1.67420 −0.0706849
$$562$$ 0 0
$$563$$ 34.1750 1.44030 0.720152 0.693816i $$-0.244072\pi$$
0.720152 + 0.693816i $$0.244072\pi$$
$$564$$ 0 0
$$565$$ 1.95055 0.0820603
$$566$$ 0 0
$$567$$ −32.4885 −1.36439
$$568$$ 0 0
$$569$$ −30.6947 −1.28679 −0.643395 0.765535i $$-0.722475\pi$$
−0.643395 + 0.765535i $$0.722475\pi$$
$$570$$ 0 0
$$571$$ 25.7275 1.07666 0.538332 0.842733i $$-0.319055\pi$$
0.538332 + 0.842733i $$0.319055\pi$$
$$572$$ 0 0
$$573$$ 23.2885 0.972889
$$574$$ 0 0
$$575$$ 6.63090 0.276528
$$576$$ 0 0
$$577$$ 16.0228 0.667037 0.333519 0.942743i $$-0.391764\pi$$
0.333519 + 0.942743i $$0.391764\pi$$
$$578$$ 0 0
$$579$$ −29.0082 −1.20554
$$580$$ 0 0
$$581$$ 11.6020 0.481331
$$582$$ 0 0
$$583$$ −0.581449 −0.0240812
$$584$$ 0 0
$$585$$ 0.340173 0.0140644
$$586$$ 0 0
$$587$$ −19.6248 −0.810000 −0.405000 0.914317i $$-0.632729\pi$$
−0.405000 + 0.914317i $$0.632729\pi$$
$$588$$ 0 0
$$589$$ −13.0784 −0.538885
$$590$$ 0 0
$$591$$ 0.0988967 0.00406807
$$592$$ 0 0
$$593$$ 30.9627 1.27148 0.635742 0.771902i $$-0.280694\pi$$
0.635742 + 0.771902i $$0.280694\pi$$
$$594$$ 0 0
$$595$$ 5.75872 0.236085
$$596$$ 0 0
$$597$$ 9.21622 0.377195
$$598$$ 0 0
$$599$$ 24.4619 0.999484 0.499742 0.866174i $$-0.333428\pi$$
0.499742 + 0.866174i $$0.333428\pi$$
$$600$$ 0 0
$$601$$ 16.3857 0.668389 0.334194 0.942504i $$-0.391536\pi$$
0.334194 + 0.942504i $$0.391536\pi$$
$$602$$ 0 0
$$603$$ 0.833226 0.0339316
$$604$$ 0 0
$$605$$ 10.6020 0.431031
$$606$$ 0 0
$$607$$ 16.6986 0.677775 0.338888 0.940827i $$-0.389949\pi$$
0.338888 + 0.940827i $$0.389949\pi$$
$$608$$ 0 0
$$609$$ 6.34017 0.256917
$$610$$ 0 0
$$611$$ 9.94214 0.402216
$$612$$ 0 0
$$613$$ −5.83096 −0.235510 −0.117755 0.993043i $$-0.537570\pi$$
−0.117755 + 0.993043i $$0.537570\pi$$
$$614$$ 0 0
$$615$$ −9.57531 −0.386114
$$616$$ 0 0
$$617$$ −11.7237 −0.471976 −0.235988 0.971756i $$-0.575833\pi$$
−0.235988 + 0.971756i $$0.575833\pi$$
$$618$$ 0 0
$$619$$ 8.41628 0.338279 0.169139 0.985592i $$-0.445901\pi$$
0.169139 + 0.985592i $$0.445901\pi$$
$$620$$ 0 0
$$621$$ −34.8904 −1.40010
$$622$$ 0 0
$$623$$ 5.26180 0.210809
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 6.15676 0.245877
$$628$$ 0 0
$$629$$ −3.79976 −0.151506
$$630$$ 0 0
$$631$$ −12.7792 −0.508734 −0.254367 0.967108i $$-0.581867\pi$$
−0.254367 + 0.967108i $$0.581867\pi$$
$$632$$ 0 0
$$633$$ −7.09066 −0.281829
$$634$$ 0 0
$$635$$ 22.4885 0.892430
$$636$$ 0 0
$$637$$ 29.3340 1.16226
$$638$$ 0 0
$$639$$ −1.22285 −0.0483751
$$640$$ 0 0
$$641$$ −0.0722347 −0.00285310 −0.00142655 0.999999i $$-0.500454\pi$$
−0.00142655 + 0.999999i $$0.500454\pi$$
$$642$$ 0 0
$$643$$ −32.7175 −1.29025 −0.645126 0.764076i $$-0.723195\pi$$
−0.645126 + 0.764076i $$0.723195\pi$$
$$644$$ 0 0
$$645$$ 21.4452 0.844404
$$646$$ 0 0
$$647$$ −15.8082 −0.621483 −0.310742 0.950494i $$-0.600577\pi$$
−0.310742 + 0.950494i $$0.600577\pi$$
$$648$$ 0 0
$$649$$ 2.27247 0.0892024
$$650$$ 0 0
$$651$$ −14.5236 −0.569224
$$652$$ 0 0
$$653$$ 15.3112 0.599175 0.299588 0.954069i $$-0.403151\pi$$
0.299588 + 0.954069i $$0.403151\pi$$
$$654$$ 0 0
$$655$$ 3.86603 0.151058
$$656$$ 0 0
$$657$$ 0.857798 0.0334659
$$658$$ 0 0
$$659$$ −17.1278 −0.667205 −0.333603 0.942714i $$-0.608264\pi$$
−0.333603 + 0.942714i $$0.608264\pi$$
$$660$$ 0 0
$$661$$ 26.2290 1.02019 0.510095 0.860118i $$-0.329610\pi$$
0.510095 + 0.860118i $$0.329610\pi$$
$$662$$ 0 0
$$663$$ −11.5174 −0.447301
$$664$$ 0 0
$$665$$ −21.1773 −0.821219
$$666$$ 0 0
$$667$$ 6.63090 0.256749
$$668$$ 0 0
$$669$$ −11.5031 −0.444734
$$670$$ 0 0
$$671$$ 8.21461 0.317122
$$672$$ 0 0
$$673$$ 46.4657 1.79112 0.895561 0.444938i $$-0.146774\pi$$
0.895561 + 0.444938i $$0.146774\pi$$
$$674$$ 0 0
$$675$$ −5.26180 −0.202527
$$676$$ 0 0
$$677$$ 27.8394 1.06995 0.534977 0.844867i $$-0.320320\pi$$
0.534977 + 0.844867i $$0.320320\pi$$
$$678$$ 0 0
$$679$$ 49.9565 1.91716
$$680$$ 0 0
$$681$$ 38.0267 1.45718
$$682$$ 0 0
$$683$$ −39.0966 −1.49599 −0.747995 0.663704i $$-0.768984\pi$$
−0.747995 + 0.663704i $$0.768984\pi$$
$$684$$ 0 0
$$685$$ −21.2846 −0.813242
$$686$$ 0 0
$$687$$ 12.2434 0.467114
$$688$$ 0 0
$$689$$ −4.00000 −0.152388
$$690$$ 0 0
$$691$$ −24.7480 −0.941460 −0.470730 0.882277i $$-0.656009\pi$$
−0.470730 + 0.882277i $$0.656009\pi$$
$$692$$ 0 0
$$693$$ −0.183417 −0.00696745
$$694$$ 0 0
$$695$$ 8.09890 0.307209
$$696$$ 0 0
$$697$$ −8.69717 −0.329429
$$698$$ 0 0
$$699$$ 51.6163 1.95231
$$700$$ 0 0
$$701$$ −0.187952 −0.00709886 −0.00354943 0.999994i $$-0.501130\pi$$
−0.00354943 + 0.999994i $$0.501130\pi$$
$$702$$ 0 0
$$703$$ 13.9733 0.527014
$$704$$ 0 0
$$705$$ −3.91548 −0.147465
$$706$$ 0 0
$$707$$ −4.09890 −0.154155
$$708$$ 0 0
$$709$$ −13.6020 −0.510833 −0.255416 0.966831i $$-0.582213\pi$$
−0.255416 + 0.966831i $$0.582213\pi$$
$$710$$ 0 0
$$711$$ 0.799939 0.0300001
$$712$$ 0 0
$$713$$ −15.1896 −0.568854
$$714$$ 0 0
$$715$$ −2.73820 −0.102403
$$716$$ 0 0
$$717$$ −11.0061 −0.411032
$$718$$ 0 0
$$719$$ −9.27617 −0.345943 −0.172971 0.984927i $$-0.555337\pi$$
−0.172971 + 0.984927i $$0.555337\pi$$
$$720$$ 0 0
$$721$$ 57.9565 2.15841
$$722$$ 0 0
$$723$$ 18.7915 0.698864
$$724$$ 0 0
$$725$$ 1.00000 0.0371391
$$726$$ 0 0
$$727$$ 29.0121 1.07600 0.538000 0.842945i $$-0.319180\pi$$
0.538000 + 0.842945i $$0.319180\pi$$
$$728$$ 0 0
$$729$$ 27.6681 1.02474
$$730$$ 0 0
$$731$$ 19.4785 0.720438
$$732$$ 0 0
$$733$$ 34.0638 1.25818 0.629088 0.777334i $$-0.283428\pi$$
0.629088 + 0.777334i $$0.283428\pi$$
$$734$$ 0 0
$$735$$ −11.5525 −0.426121
$$736$$ 0 0
$$737$$ −6.70701 −0.247056
$$738$$ 0 0
$$739$$ −1.49466 −0.0549820 −0.0274910 0.999622i $$-0.508752\pi$$
−0.0274910 + 0.999622i $$0.508752\pi$$
$$740$$ 0 0
$$741$$ 42.3545 1.55593
$$742$$ 0 0
$$743$$ 17.8082 0.653318 0.326659 0.945142i $$-0.394077\pi$$
0.326659 + 0.945142i $$0.394077\pi$$
$$744$$ 0 0
$$745$$ 12.8371 0.470315
$$746$$ 0 0
$$747$$ −0.245152 −0.00896964
$$748$$ 0 0
$$749$$ −10.4391 −0.381435
$$750$$ 0 0
$$751$$ −23.7503 −0.866661 −0.433331 0.901235i $$-0.642662\pi$$
−0.433331 + 0.901235i $$0.642662\pi$$
$$752$$ 0 0
$$753$$ −16.0845 −0.586153
$$754$$ 0 0
$$755$$ 20.5958 0.749559
$$756$$ 0 0
$$757$$ −26.1939 −0.952034 −0.476017 0.879436i $$-0.657920\pi$$
−0.476017 + 0.879436i $$0.657920\pi$$
$$758$$ 0 0
$$759$$ 7.15061 0.259551
$$760$$ 0 0
$$761$$ −44.7214 −1.62115 −0.810574 0.585636i $$-0.800845\pi$$
−0.810574 + 0.585636i $$0.800845\pi$$
$$762$$ 0 0
$$763$$ −21.9421 −0.794359
$$764$$ 0 0
$$765$$ −0.121683 −0.00439946
$$766$$ 0 0
$$767$$ 15.6332 0.564481
$$768$$ 0 0
$$769$$ 10.8950 0.392882 0.196441 0.980516i $$-0.437062\pi$$
0.196441 + 0.980516i $$0.437062\pi$$
$$770$$ 0 0
$$771$$ 9.94214 0.358057
$$772$$ 0 0
$$773$$ −34.1171 −1.22711 −0.613554 0.789653i $$-0.710261\pi$$
−0.613554 + 0.789653i $$0.710261\pi$$
$$774$$ 0 0
$$775$$ −2.29072 −0.0822853
$$776$$ 0 0
$$777$$ 15.5174 0.556685
$$778$$ 0 0
$$779$$ 31.9832 1.14592
$$780$$ 0 0
$$781$$ 9.84324 0.352219
$$782$$ 0 0
$$783$$ −5.26180 −0.188041
$$784$$ 0 0
$$785$$ −6.04945 −0.215914
$$786$$ 0 0
$$787$$ −39.7548 −1.41711 −0.708554 0.705657i $$-0.750652\pi$$
−0.708554 + 0.705657i $$0.750652\pi$$
$$788$$ 0 0
$$789$$ −15.2351 −0.542385
$$790$$ 0 0
$$791$$ −7.23513 −0.257252
$$792$$ 0 0
$$793$$ 56.5113 2.00678
$$794$$ 0 0
$$795$$ 1.57531 0.0558704
$$796$$ 0 0
$$797$$ −18.7298 −0.663443 −0.331722 0.943377i $$-0.607629\pi$$
−0.331722 + 0.943377i $$0.607629\pi$$
$$798$$ 0 0
$$799$$ −3.55640 −0.125816
$$800$$ 0 0
$$801$$ −0.111183 −0.00392845
$$802$$ 0 0
$$803$$ −6.90480 −0.243665
$$804$$ 0 0
$$805$$ −24.5958 −0.866889
$$806$$ 0 0
$$807$$ 28.0989 0.989128
$$808$$ 0 0
$$809$$ −31.9421 −1.12303 −0.561513 0.827468i $$-0.689781\pi$$
−0.561513 + 0.827468i $$0.689781\pi$$
$$810$$ 0 0
$$811$$ 17.8888 0.628161 0.314081 0.949396i $$-0.398304\pi$$
0.314081 + 0.949396i $$0.398304\pi$$
$$812$$ 0 0
$$813$$ 50.2700 1.76305
$$814$$ 0 0
$$815$$ −15.9649 −0.559227
$$816$$ 0 0
$$817$$ −71.6307 −2.50604
$$818$$ 0 0
$$819$$ −1.26180 −0.0440907
$$820$$ 0 0
$$821$$ −30.9939 −1.08169 −0.540847 0.841121i $$-0.681896\pi$$
−0.540847 + 0.841121i $$0.681896\pi$$
$$822$$ 0 0
$$823$$ 16.5008 0.575182 0.287591 0.957753i $$-0.407146\pi$$
0.287591 + 0.957753i $$0.407146\pi$$
$$824$$ 0 0
$$825$$ 1.07838 0.0375443
$$826$$ 0 0
$$827$$ −43.1155 −1.49927 −0.749637 0.661849i $$-0.769772\pi$$
−0.749637 + 0.661849i $$0.769772\pi$$
$$828$$ 0 0
$$829$$ −22.5958 −0.784785 −0.392393 0.919798i $$-0.628353\pi$$
−0.392393 + 0.919798i $$0.628353\pi$$
$$830$$ 0 0
$$831$$ −18.9360 −0.656882
$$832$$ 0 0
$$833$$ −10.4931 −0.363563
$$834$$ 0 0
$$835$$ 11.3112 0.391442
$$836$$ 0 0
$$837$$ 12.0533 0.416624
$$838$$ 0 0
$$839$$ 1.21235 0.0418549 0.0209274 0.999781i $$-0.493338\pi$$
0.0209274 + 0.999781i $$0.493338\pi$$
$$840$$ 0 0
$$841$$ 1.00000 0.0344828
$$842$$ 0 0
$$843$$ −36.0989 −1.24331
$$844$$ 0 0
$$845$$ −5.83710 −0.200802
$$846$$ 0 0
$$847$$ −39.3256 −1.35124
$$848$$ 0 0
$$849$$ 23.5564 0.808453
$$850$$ 0 0
$$851$$ 16.2290 0.556323
$$852$$ 0 0
$$853$$ −5.93618 −0.203251 −0.101625 0.994823i $$-0.532404\pi$$
−0.101625 + 0.994823i $$0.532404\pi$$
$$854$$ 0 0
$$855$$ 0.447480 0.0153035
$$856$$ 0 0
$$857$$ 14.5503 0.497027 0.248514 0.968628i $$-0.420058\pi$$
0.248514 + 0.968628i $$0.420058\pi$$
$$858$$ 0 0
$$859$$ 6.32192 0.215701 0.107851 0.994167i $$-0.465603\pi$$
0.107851 + 0.994167i $$0.465603\pi$$
$$860$$ 0 0
$$861$$ 35.5174 1.21043
$$862$$ 0 0
$$863$$ −3.28005 −0.111654 −0.0558270 0.998440i $$-0.517780\pi$$
−0.0558270 + 0.998440i $$0.517780\pi$$
$$864$$ 0 0
$$865$$ −10.4969 −0.356906
$$866$$ 0 0
$$867$$ −24.9378 −0.846932
$$868$$ 0 0
$$869$$ −6.43907 −0.218430
$$870$$ 0 0
$$871$$ −46.1399 −1.56339
$$872$$ 0 0
$$873$$ −1.05559 −0.0357264
$$874$$ 0 0
$$875$$ −3.70928 −0.125396
$$876$$ 0 0
$$877$$ −18.2823 −0.617350 −0.308675 0.951168i $$-0.599886\pi$$
−0.308675 + 0.951168i $$0.599886\pi$$
$$878$$ 0 0
$$879$$ −10.5092 −0.354467
$$880$$ 0 0
$$881$$ 29.7464 1.00218 0.501091 0.865394i $$-0.332932\pi$$
0.501091 + 0.865394i $$0.332932\pi$$
$$882$$ 0 0
$$883$$ −13.8127 −0.464835 −0.232417 0.972616i $$-0.574664\pi$$
−0.232417 + 0.972616i $$0.574664\pi$$
$$884$$ 0 0
$$885$$ −6.15676 −0.206957
$$886$$ 0 0
$$887$$ −56.3318 −1.89144 −0.945718 0.324989i $$-0.894639\pi$$
−0.945718 + 0.324989i $$0.894639\pi$$
$$888$$ 0 0
$$889$$ −83.4161 −2.79769
$$890$$ 0 0
$$891$$ −5.52586 −0.185123
$$892$$ 0 0
$$893$$ 13.0784 0.437651
$$894$$ 0 0
$$895$$ −12.3135 −0.411595
$$896$$ 0 0
$$897$$ 49.1917 1.64246
$$898$$ 0 0
$$899$$ −2.29072 −0.0763999
$$900$$ 0 0
$$901$$ 1.43084 0.0476681
$$902$$ 0 0
$$903$$ −79.5462 −2.64713
$$904$$ 0 0
$$905$$ −1.60197 −0.0532512
$$906$$ 0 0
$$907$$ −21.8082 −0.724128 −0.362064 0.932153i $$-0.617928\pi$$
−0.362064 + 0.932153i $$0.617928\pi$$
$$908$$ 0 0
$$909$$ 0.0866105 0.00287269
$$910$$ 0 0
$$911$$ 4.76099 0.157739 0.0788693 0.996885i $$-0.474869\pi$$
0.0788693 + 0.996885i $$0.474869\pi$$
$$912$$ 0 0
$$913$$ 1.97334 0.0653080
$$914$$ 0 0
$$915$$ −22.2557 −0.735749
$$916$$ 0 0
$$917$$ −14.3402 −0.473554
$$918$$ 0 0
$$919$$ 34.1256 1.12570 0.562849 0.826560i $$-0.309705\pi$$
0.562849 + 0.826560i $$0.309705\pi$$
$$920$$ 0 0
$$921$$ −17.7587 −0.585170
$$922$$ 0 0
$$923$$ 67.7152 2.22887
$$924$$ 0 0
$$925$$ 2.44748 0.0804727
$$926$$ 0 0
$$927$$ −1.22463 −0.0402222
$$928$$ 0 0
$$929$$ −12.5769 −0.412635 −0.206318 0.978485i $$-0.566148\pi$$
−0.206318 + 0.978485i $$0.566148\pi$$
$$930$$ 0 0
$$931$$ 38.5874 1.26465
$$932$$ 0 0
$$933$$ 32.0599 1.04960
$$934$$ 0 0
$$935$$ 0.979481 0.0320325
$$936$$ 0 0
$$937$$ −29.7464 −0.971774 −0.485887 0.874022i $$-0.661503\pi$$
−0.485887 + 0.874022i $$0.661503\pi$$
$$938$$ 0 0
$$939$$ 21.0928 0.688336
$$940$$ 0 0
$$941$$ 7.47641 0.243724 0.121862 0.992547i $$-0.461113\pi$$
0.121862 + 0.992547i $$0.461113\pi$$
$$942$$ 0 0
$$943$$ 37.1461 1.20964
$$944$$ 0 0
$$945$$ 19.5174 0.634903
$$946$$ 0 0
$$947$$ 15.2846 0.496682 0.248341 0.968673i $$-0.420115\pi$$
0.248341 + 0.968673i $$0.420115\pi$$
$$948$$ 0 0
$$949$$ −47.5006 −1.54194
$$950$$ 0 0
$$951$$ 52.4724 1.70153
$$952$$ 0 0
$$953$$ −54.0288 −1.75016 −0.875081 0.483976i $$-0.839192\pi$$
−0.875081 + 0.483976i $$0.839192\pi$$
$$954$$ 0 0
$$955$$ −13.6248 −0.440887
$$956$$ 0 0
$$957$$ 1.07838 0.0348590
$$958$$ 0 0
$$959$$ 78.9504 2.54944
$$960$$ 0 0
$$961$$ −25.7526 −0.830728
$$962$$ 0 0
$$963$$ 0.220580 0.00710808
$$964$$ 0 0
$$965$$ 16.9711 0.546318
$$966$$ 0 0
$$967$$ −15.7671 −0.507037 −0.253518 0.967331i $$-0.581588\pi$$
−0.253518 + 0.967331i $$0.581588\pi$$
$$968$$ 0 0
$$969$$ −15.1506 −0.486708
$$970$$ 0 0
$$971$$ −48.1627 −1.54562 −0.772808 0.634640i $$-0.781148\pi$$
−0.772808 + 0.634640i $$0.781148\pi$$
$$972$$ 0 0
$$973$$ −30.0410 −0.963071
$$974$$ 0 0
$$975$$ 7.41855 0.237584
$$976$$ 0 0
$$977$$ −8.28685 −0.265120 −0.132560 0.991175i $$-0.542320\pi$$
−0.132560 + 0.991175i $$0.542320\pi$$
$$978$$ 0 0
$$979$$ 0.894960 0.0286031
$$980$$ 0 0
$$981$$ 0.463642 0.0148029
$$982$$ 0 0
$$983$$ 22.9177 0.730963 0.365481 0.930819i $$-0.380904\pi$$
0.365481 + 0.930819i $$0.380904\pi$$
$$984$$ 0 0
$$985$$ −0.0578588 −0.00184354
$$986$$ 0 0
$$987$$ 14.5236 0.462291
$$988$$ 0 0
$$989$$ −83.1937 −2.64541
$$990$$ 0 0
$$991$$ 22.3234 0.709125 0.354562 0.935032i $$-0.384630\pi$$
0.354562 + 0.935032i $$0.384630\pi$$
$$992$$ 0 0
$$993$$ 6.97495 0.221343
$$994$$ 0 0
$$995$$ −5.39189 −0.170934
$$996$$ 0 0
$$997$$ −14.3630 −0.454879 −0.227440 0.973792i $$-0.573035\pi$$
−0.227440 + 0.973792i $$0.573035\pi$$
$$998$$ 0 0
$$999$$ −12.8781 −0.407446
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 9280.2.a.bj.1.3 3
4.3 odd 2 9280.2.a.br.1.1 3
8.3 odd 2 2320.2.a.n.1.3 3
8.5 even 2 145.2.a.c.1.3 3
24.5 odd 2 1305.2.a.p.1.1 3
40.13 odd 4 725.2.b.e.349.1 6
40.29 even 2 725.2.a.e.1.1 3
40.37 odd 4 725.2.b.e.349.6 6
56.13 odd 2 7105.2.a.o.1.3 3
120.29 odd 2 6525.2.a.be.1.3 3
232.173 even 2 4205.2.a.f.1.1 3

By twisted newform
Twist Min Dim Char Parity Ord Type
145.2.a.c.1.3 3 8.5 even 2
725.2.a.e.1.1 3 40.29 even 2
725.2.b.e.349.1 6 40.13 odd 4
725.2.b.e.349.6 6 40.37 odd 4
1305.2.a.p.1.1 3 24.5 odd 2
2320.2.a.n.1.3 3 8.3 odd 2
4205.2.a.f.1.1 3 232.173 even 2
6525.2.a.be.1.3 3 120.29 odd 2
7105.2.a.o.1.3 3 56.13 odd 2
9280.2.a.bj.1.3 3 1.1 even 1 trivial
9280.2.a.br.1.1 3 4.3 odd 2