# Properties

 Label 91.2.a.d.1.1 Level $91$ Weight $2$ Character 91.1 Self dual yes Analytic conductor $0.727$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [91,2,Mod(1,91)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(91, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("91.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$91 = 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 91.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: $$0.726638658394$$ Analytic rank: $$0$$ 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: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-1.81361$$ of defining polynomial Character $$\chi$$ $$=$$ 91.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.81361 q^{2} -3.10278 q^{3} +1.28917 q^{4} +2.81361 q^{5} +5.62721 q^{6} -1.00000 q^{7} +1.28917 q^{8} +6.62721 q^{9} +O(q^{10})$$ $$q-1.81361 q^{2} -3.10278 q^{3} +1.28917 q^{4} +2.81361 q^{5} +5.62721 q^{6} -1.00000 q^{7} +1.28917 q^{8} +6.62721 q^{9} -5.10278 q^{10} +3.10278 q^{11} -4.00000 q^{12} +1.00000 q^{13} +1.81361 q^{14} -8.72999 q^{15} -4.91638 q^{16} -0.524438 q^{17} -12.0192 q^{18} +0.813607 q^{19} +3.62721 q^{20} +3.10278 q^{21} -5.62721 q^{22} +7.33804 q^{23} -4.00000 q^{24} +2.91638 q^{25} -1.81361 q^{26} -11.2544 q^{27} -1.28917 q^{28} +8.28917 q^{29} +15.8328 q^{30} +1.39194 q^{31} +6.33804 q^{32} -9.62721 q^{33} +0.951124 q^{34} -2.81361 q^{35} +8.54359 q^{36} -6.15165 q^{37} -1.47556 q^{38} -3.10278 q^{39} +3.62721 q^{40} -4.20555 q^{41} -5.62721 q^{42} +6.75971 q^{43} +4.00000 q^{44} +18.6464 q^{45} -13.3083 q^{46} -5.97028 q^{47} +15.2544 q^{48} +1.00000 q^{49} -5.28917 q^{50} +1.62721 q^{51} +1.28917 q^{52} -2.49472 q^{53} +20.4111 q^{54} +8.72999 q^{55} -1.28917 q^{56} -2.52444 q^{57} -15.0333 q^{58} -4.47054 q^{59} -11.2544 q^{60} -2.00000 q^{61} -2.52444 q^{62} -6.62721 q^{63} -1.66196 q^{64} +2.81361 q^{65} +17.4600 q^{66} +10.0383 q^{67} -0.676089 q^{68} -22.7683 q^{69} +5.10278 q^{70} -8.72999 q^{71} +8.54359 q^{72} -2.34307 q^{73} +11.1567 q^{74} -9.04888 q^{75} +1.04888 q^{76} -3.10278 q^{77} +5.62721 q^{78} -13.5436 q^{79} -13.8328 q^{80} +15.0383 q^{81} +7.62721 q^{82} +16.4791 q^{83} +4.00000 q^{84} -1.47556 q^{85} -12.2594 q^{86} -25.7194 q^{87} +4.00000 q^{88} -10.6464 q^{89} -33.8172 q^{90} -1.00000 q^{91} +9.45998 q^{92} -4.31889 q^{93} +10.8277 q^{94} +2.28917 q^{95} -19.6655 q^{96} -1.18639 q^{97} -1.81361 q^{98} +20.5628 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + q^{2} - 2 q^{3} + 3 q^{4} + 2 q^{5} + 4 q^{6} - 3 q^{7} + 3 q^{8} + 7 q^{9}+O(q^{10})$$ 3 * q + q^2 - 2 * q^3 + 3 * q^4 + 2 * q^5 + 4 * q^6 - 3 * q^7 + 3 * q^8 + 7 * q^9 $$3 q + q^{2} - 2 q^{3} + 3 q^{4} + 2 q^{5} + 4 q^{6} - 3 q^{7} + 3 q^{8} + 7 q^{9} - 8 q^{10} + 2 q^{11} - 12 q^{12} + 3 q^{13} - q^{14} - 6 q^{15} - q^{16} + 4 q^{17} - 15 q^{18} - 4 q^{19} - 2 q^{20} + 2 q^{21} - 4 q^{22} + 10 q^{23} - 12 q^{24} - 5 q^{25} + q^{26} - 8 q^{27} - 3 q^{28} + 24 q^{29} + 20 q^{30} - 4 q^{31} + 7 q^{32} - 16 q^{33} + 14 q^{34} - 2 q^{35} - q^{36} - 10 q^{38} - 2 q^{39} - 2 q^{40} + 2 q^{41} - 4 q^{42} + 10 q^{43} + 12 q^{44} + 22 q^{45} - 18 q^{46} - 8 q^{47} + 20 q^{48} + 3 q^{49} - 15 q^{50} - 8 q^{51} + 3 q^{52} + 8 q^{53} + 32 q^{54} + 6 q^{55} - 3 q^{56} - 2 q^{57} + 12 q^{58} - 4 q^{59} - 8 q^{60} - 6 q^{61} - 2 q^{62} - 7 q^{63} - 17 q^{64} + 2 q^{65} + 12 q^{66} - 12 q^{67} + 22 q^{68} - 6 q^{69} + 8 q^{70} - 6 q^{71} - q^{72} - 10 q^{73} + 30 q^{74} - 16 q^{75} - 8 q^{76} - 2 q^{77} + 4 q^{78} - 14 q^{79} - 14 q^{80} + 3 q^{81} + 10 q^{82} - 12 q^{83} + 12 q^{84} - 10 q^{85} - 26 q^{86} - 26 q^{87} + 12 q^{88} + 2 q^{89} - 28 q^{90} - 3 q^{91} - 12 q^{92} - 22 q^{93} - 10 q^{94} + 6 q^{95} - 4 q^{96} - 10 q^{97} + q^{98} + 14 q^{99}+O(q^{100})$$ 3 * q + q^2 - 2 * q^3 + 3 * q^4 + 2 * q^5 + 4 * q^6 - 3 * q^7 + 3 * q^8 + 7 * q^9 - 8 * q^10 + 2 * q^11 - 12 * q^12 + 3 * q^13 - q^14 - 6 * q^15 - q^16 + 4 * q^17 - 15 * q^18 - 4 * q^19 - 2 * q^20 + 2 * q^21 - 4 * q^22 + 10 * q^23 - 12 * q^24 - 5 * q^25 + q^26 - 8 * q^27 - 3 * q^28 + 24 * q^29 + 20 * q^30 - 4 * q^31 + 7 * q^32 - 16 * q^33 + 14 * q^34 - 2 * q^35 - q^36 - 10 * q^38 - 2 * q^39 - 2 * q^40 + 2 * q^41 - 4 * q^42 + 10 * q^43 + 12 * q^44 + 22 * q^45 - 18 * q^46 - 8 * q^47 + 20 * q^48 + 3 * q^49 - 15 * q^50 - 8 * q^51 + 3 * q^52 + 8 * q^53 + 32 * q^54 + 6 * q^55 - 3 * q^56 - 2 * q^57 + 12 * q^58 - 4 * q^59 - 8 * q^60 - 6 * q^61 - 2 * q^62 - 7 * q^63 - 17 * q^64 + 2 * q^65 + 12 * q^66 - 12 * q^67 + 22 * q^68 - 6 * q^69 + 8 * q^70 - 6 * q^71 - q^72 - 10 * q^73 + 30 * q^74 - 16 * q^75 - 8 * q^76 - 2 * q^77 + 4 * q^78 - 14 * q^79 - 14 * q^80 + 3 * q^81 + 10 * q^82 - 12 * q^83 + 12 * q^84 - 10 * q^85 - 26 * q^86 - 26 * q^87 + 12 * q^88 + 2 * q^89 - 28 * q^90 - 3 * q^91 - 12 * q^92 - 22 * q^93 - 10 * q^94 + 6 * q^95 - 4 * q^96 - 10 * q^97 + q^98 + 14 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.81361 −1.28241 −0.641207 0.767368i $$-0.721566\pi$$
−0.641207 + 0.767368i $$0.721566\pi$$
$$3$$ −3.10278 −1.79139 −0.895694 0.444671i $$-0.853321\pi$$
−0.895694 + 0.444671i $$0.853321\pi$$
$$4$$ 1.28917 0.644584
$$5$$ 2.81361 1.25828 0.629142 0.777291i $$-0.283407\pi$$
0.629142 + 0.777291i $$0.283407\pi$$
$$6$$ 5.62721 2.29730
$$7$$ −1.00000 −0.377964
$$8$$ 1.28917 0.455790
$$9$$ 6.62721 2.20907
$$10$$ −5.10278 −1.61364
$$11$$ 3.10278 0.935522 0.467761 0.883855i $$-0.345061\pi$$
0.467761 + 0.883855i $$0.345061\pi$$
$$12$$ −4.00000 −1.15470
$$13$$ 1.00000 0.277350
$$14$$ 1.81361 0.484707
$$15$$ −8.72999 −2.25407
$$16$$ −4.91638 −1.22910
$$17$$ −0.524438 −0.127195 −0.0635974 0.997976i $$-0.520257\pi$$
−0.0635974 + 0.997976i $$0.520257\pi$$
$$18$$ −12.0192 −2.83294
$$19$$ 0.813607 0.186654 0.0933271 0.995636i $$-0.470250\pi$$
0.0933271 + 0.995636i $$0.470250\pi$$
$$20$$ 3.62721 0.811069
$$21$$ 3.10278 0.677081
$$22$$ −5.62721 −1.19973
$$23$$ 7.33804 1.53009 0.765044 0.643978i $$-0.222717\pi$$
0.765044 + 0.643978i $$0.222717\pi$$
$$24$$ −4.00000 −0.816497
$$25$$ 2.91638 0.583276
$$26$$ −1.81361 −0.355677
$$27$$ −11.2544 −2.16592
$$28$$ −1.28917 −0.243630
$$29$$ 8.28917 1.53926 0.769630 0.638490i $$-0.220441\pi$$
0.769630 + 0.638490i $$0.220441\pi$$
$$30$$ 15.8328 2.89065
$$31$$ 1.39194 0.250000 0.125000 0.992157i $$-0.460107\pi$$
0.125000 + 0.992157i $$0.460107\pi$$
$$32$$ 6.33804 1.12042
$$33$$ −9.62721 −1.67588
$$34$$ 0.951124 0.163116
$$35$$ −2.81361 −0.475586
$$36$$ 8.54359 1.42393
$$37$$ −6.15165 −1.01133 −0.505663 0.862731i $$-0.668752\pi$$
−0.505663 + 0.862731i $$0.668752\pi$$
$$38$$ −1.47556 −0.239368
$$39$$ −3.10278 −0.496842
$$40$$ 3.62721 0.573513
$$41$$ −4.20555 −0.656797 −0.328398 0.944539i $$-0.606509\pi$$
−0.328398 + 0.944539i $$0.606509\pi$$
$$42$$ −5.62721 −0.868298
$$43$$ 6.75971 1.03085 0.515423 0.856936i $$-0.327635\pi$$
0.515423 + 0.856936i $$0.327635\pi$$
$$44$$ 4.00000 0.603023
$$45$$ 18.6464 2.77964
$$46$$ −13.3083 −1.96221
$$47$$ −5.97028 −0.870855 −0.435427 0.900224i $$-0.643403\pi$$
−0.435427 + 0.900224i $$0.643403\pi$$
$$48$$ 15.2544 2.20179
$$49$$ 1.00000 0.142857
$$50$$ −5.28917 −0.748001
$$51$$ 1.62721 0.227855
$$52$$ 1.28917 0.178776
$$53$$ −2.49472 −0.342676 −0.171338 0.985212i $$-0.554809\pi$$
−0.171338 + 0.985212i $$0.554809\pi$$
$$54$$ 20.4111 2.77760
$$55$$ 8.72999 1.17715
$$56$$ −1.28917 −0.172272
$$57$$ −2.52444 −0.334370
$$58$$ −15.0333 −1.97397
$$59$$ −4.47054 −0.582015 −0.291007 0.956721i $$-0.593990\pi$$
−0.291007 + 0.956721i $$0.593990\pi$$
$$60$$ −11.2544 −1.45294
$$61$$ −2.00000 −0.256074 −0.128037 0.991769i $$-0.540868\pi$$
−0.128037 + 0.991769i $$0.540868\pi$$
$$62$$ −2.52444 −0.320604
$$63$$ −6.62721 −0.834950
$$64$$ −1.66196 −0.207744
$$65$$ 2.81361 0.348985
$$66$$ 17.4600 2.14917
$$67$$ 10.0383 1.22638 0.613188 0.789937i $$-0.289887\pi$$
0.613188 + 0.789937i $$0.289887\pi$$
$$68$$ −0.676089 −0.0819878
$$69$$ −22.7683 −2.74098
$$70$$ 5.10278 0.609898
$$71$$ −8.72999 −1.03606 −0.518029 0.855363i $$-0.673334\pi$$
−0.518029 + 0.855363i $$0.673334\pi$$
$$72$$ 8.54359 1.00687
$$73$$ −2.34307 −0.274235 −0.137118 0.990555i $$-0.543784\pi$$
−0.137118 + 0.990555i $$0.543784\pi$$
$$74$$ 11.1567 1.29694
$$75$$ −9.04888 −1.04487
$$76$$ 1.04888 0.120314
$$77$$ −3.10278 −0.353594
$$78$$ 5.62721 0.637156
$$79$$ −13.5436 −1.52377 −0.761887 0.647710i $$-0.775727\pi$$
−0.761887 + 0.647710i $$0.775727\pi$$
$$80$$ −13.8328 −1.54655
$$81$$ 15.0383 1.67092
$$82$$ 7.62721 0.842285
$$83$$ 16.4791 1.80882 0.904410 0.426665i $$-0.140312\pi$$
0.904410 + 0.426665i $$0.140312\pi$$
$$84$$ 4.00000 0.436436
$$85$$ −1.47556 −0.160047
$$86$$ −12.2594 −1.32197
$$87$$ −25.7194 −2.75741
$$88$$ 4.00000 0.426401
$$89$$ −10.6464 −1.12851 −0.564256 0.825600i $$-0.690837\pi$$
−0.564256 + 0.825600i $$0.690837\pi$$
$$90$$ −33.8172 −3.56464
$$91$$ −1.00000 −0.104828
$$92$$ 9.45998 0.986271
$$93$$ −4.31889 −0.447848
$$94$$ 10.8277 1.11680
$$95$$ 2.28917 0.234864
$$96$$ −19.6655 −2.00710
$$97$$ −1.18639 −0.120460 −0.0602300 0.998185i $$-0.519183\pi$$
−0.0602300 + 0.998185i $$0.519183\pi$$
$$98$$ −1.81361 −0.183202
$$99$$ 20.5628 2.06663
$$100$$ 3.75971 0.375971
$$101$$ 13.1028 1.30377 0.651887 0.758316i $$-0.273977\pi$$
0.651887 + 0.758316i $$0.273977\pi$$
$$102$$ −2.95112 −0.292205
$$103$$ 4.41110 0.434639 0.217319 0.976101i $$-0.430269\pi$$
0.217319 + 0.976101i $$0.430269\pi$$
$$104$$ 1.28917 0.126413
$$105$$ 8.72999 0.851960
$$106$$ 4.52444 0.439452
$$107$$ 0.578337 0.0559100 0.0279550 0.999609i $$-0.491100\pi$$
0.0279550 + 0.999609i $$0.491100\pi$$
$$108$$ −14.5089 −1.39611
$$109$$ 5.57331 0.533827 0.266913 0.963721i $$-0.413996\pi$$
0.266913 + 0.963721i $$0.413996\pi$$
$$110$$ −15.8328 −1.50959
$$111$$ 19.0872 1.81168
$$112$$ 4.91638 0.464554
$$113$$ 5.44584 0.512302 0.256151 0.966637i $$-0.417546\pi$$
0.256151 + 0.966637i $$0.417546\pi$$
$$114$$ 4.57834 0.428801
$$115$$ 20.6464 1.92528
$$116$$ 10.6861 0.992183
$$117$$ 6.62721 0.612686
$$118$$ 8.10780 0.746383
$$119$$ 0.524438 0.0480751
$$120$$ −11.2544 −1.02738
$$121$$ −1.37279 −0.124799
$$122$$ 3.62721 0.328392
$$123$$ 13.0489 1.17658
$$124$$ 1.79445 0.161146
$$125$$ −5.86248 −0.524356
$$126$$ 12.0192 1.07075
$$127$$ −12.8816 −1.14306 −0.571530 0.820581i $$-0.693650\pi$$
−0.571530 + 0.820581i $$0.693650\pi$$
$$128$$ −9.66196 −0.854004
$$129$$ −20.9739 −1.84664
$$130$$ −5.10278 −0.447543
$$131$$ 9.04888 0.790604 0.395302 0.918551i $$-0.370640\pi$$
0.395302 + 0.918551i $$0.370640\pi$$
$$132$$ −12.4111 −1.08025
$$133$$ −0.813607 −0.0705486
$$134$$ −18.2056 −1.57272
$$135$$ −31.6655 −2.72533
$$136$$ −0.676089 −0.0579741
$$137$$ −6.25945 −0.534781 −0.267390 0.963588i $$-0.586161\pi$$
−0.267390 + 0.963588i $$0.586161\pi$$
$$138$$ 41.2927 3.51507
$$139$$ −11.5733 −0.981636 −0.490818 0.871262i $$-0.663302\pi$$
−0.490818 + 0.871262i $$0.663302\pi$$
$$140$$ −3.62721 −0.306555
$$141$$ 18.5244 1.56004
$$142$$ 15.8328 1.32866
$$143$$ 3.10278 0.259467
$$144$$ −32.5819 −2.71516
$$145$$ 23.3225 1.93682
$$146$$ 4.24940 0.351683
$$147$$ −3.10278 −0.255913
$$148$$ −7.93051 −0.651884
$$149$$ 8.52444 0.698349 0.349175 0.937058i $$-0.386462\pi$$
0.349175 + 0.937058i $$0.386462\pi$$
$$150$$ 16.4111 1.33996
$$151$$ −11.9844 −0.975278 −0.487639 0.873045i $$-0.662142\pi$$
−0.487639 + 0.873045i $$0.662142\pi$$
$$152$$ 1.04888 0.0850751
$$153$$ −3.47556 −0.280983
$$154$$ 5.62721 0.453454
$$155$$ 3.91638 0.314571
$$156$$ −4.00000 −0.320256
$$157$$ −12.8277 −1.02377 −0.511883 0.859055i $$-0.671052\pi$$
−0.511883 + 0.859055i $$0.671052\pi$$
$$158$$ 24.5628 1.95411
$$159$$ 7.74055 0.613866
$$160$$ 17.8328 1.40980
$$161$$ −7.33804 −0.578319
$$162$$ −27.2736 −2.14282
$$163$$ −13.4600 −1.05427 −0.527133 0.849783i $$-0.676733\pi$$
−0.527133 + 0.849783i $$0.676733\pi$$
$$164$$ −5.42166 −0.423361
$$165$$ −27.0872 −2.10873
$$166$$ −29.8867 −2.31965
$$167$$ −2.02972 −0.157064 −0.0785322 0.996912i $$-0.525023\pi$$
−0.0785322 + 0.996912i $$0.525023\pi$$
$$168$$ 4.00000 0.308607
$$169$$ 1.00000 0.0769231
$$170$$ 2.67609 0.205247
$$171$$ 5.39194 0.412332
$$172$$ 8.71440 0.664467
$$173$$ 20.2978 1.54321 0.771605 0.636102i $$-0.219454\pi$$
0.771605 + 0.636102i $$0.219454\pi$$
$$174$$ 46.6449 3.53614
$$175$$ −2.91638 −0.220458
$$176$$ −15.2544 −1.14985
$$177$$ 13.8711 1.04261
$$178$$ 19.3083 1.44722
$$179$$ −11.0036 −0.822445 −0.411223 0.911535i $$-0.634898\pi$$
−0.411223 + 0.911535i $$0.634898\pi$$
$$180$$ 24.0383 1.79171
$$181$$ 0.691675 0.0514118 0.0257059 0.999670i $$-0.491817\pi$$
0.0257059 + 0.999670i $$0.491817\pi$$
$$182$$ 1.81361 0.134433
$$183$$ 6.20555 0.458727
$$184$$ 9.45998 0.697399
$$185$$ −17.3083 −1.27253
$$186$$ 7.83276 0.574326
$$187$$ −1.62721 −0.118994
$$188$$ −7.69670 −0.561339
$$189$$ 11.2544 0.818639
$$190$$ −4.15165 −0.301192
$$191$$ 7.83276 0.566759 0.283379 0.959008i $$-0.408544\pi$$
0.283379 + 0.959008i $$0.408544\pi$$
$$192$$ 5.15667 0.372151
$$193$$ −12.2056 −0.878575 −0.439287 0.898347i $$-0.644769\pi$$
−0.439287 + 0.898347i $$0.644769\pi$$
$$194$$ 2.15165 0.154480
$$195$$ −8.72999 −0.625167
$$196$$ 1.28917 0.0920835
$$197$$ −18.8222 −1.34103 −0.670513 0.741898i $$-0.733926\pi$$
−0.670513 + 0.741898i $$0.733926\pi$$
$$198$$ −37.2927 −2.65028
$$199$$ 21.6116 1.53201 0.766004 0.642836i $$-0.222242\pi$$
0.766004 + 0.642836i $$0.222242\pi$$
$$200$$ 3.75971 0.265851
$$201$$ −31.1466 −2.19691
$$202$$ −23.7633 −1.67198
$$203$$ −8.28917 −0.581786
$$204$$ 2.09775 0.146872
$$205$$ −11.8328 −0.826436
$$206$$ −8.00000 −0.557386
$$207$$ 48.6308 3.38007
$$208$$ −4.91638 −0.340890
$$209$$ 2.52444 0.174619
$$210$$ −15.8328 −1.09256
$$211$$ −17.3764 −1.19624 −0.598119 0.801407i $$-0.704085\pi$$
−0.598119 + 0.801407i $$0.704085\pi$$
$$212$$ −3.21611 −0.220884
$$213$$ 27.0872 1.85598
$$214$$ −1.04888 −0.0716997
$$215$$ 19.0192 1.29710
$$216$$ −14.5089 −0.987202
$$217$$ −1.39194 −0.0944913
$$218$$ −10.1078 −0.684586
$$219$$ 7.27001 0.491262
$$220$$ 11.2544 0.758773
$$221$$ −0.524438 −0.0352775
$$222$$ −34.6167 −2.32332
$$223$$ −10.5486 −0.706388 −0.353194 0.935550i $$-0.614904\pi$$
−0.353194 + 0.935550i $$0.614904\pi$$
$$224$$ −6.33804 −0.423478
$$225$$ 19.3275 1.28850
$$226$$ −9.87662 −0.656983
$$227$$ −6.95112 −0.461362 −0.230681 0.973029i $$-0.574095\pi$$
−0.230681 + 0.973029i $$0.574095\pi$$
$$228$$ −3.25443 −0.215530
$$229$$ −21.0872 −1.39348 −0.696740 0.717323i $$-0.745367\pi$$
−0.696740 + 0.717323i $$0.745367\pi$$
$$230$$ −37.4444 −2.46901
$$231$$ 9.62721 0.633424
$$232$$ 10.6861 0.701579
$$233$$ 6.08362 0.398551 0.199276 0.979943i $$-0.436141\pi$$
0.199276 + 0.979943i $$0.436141\pi$$
$$234$$ −12.0192 −0.785717
$$235$$ −16.7980 −1.09578
$$236$$ −5.76328 −0.375157
$$237$$ 42.0227 2.72967
$$238$$ −0.951124 −0.0616522
$$239$$ −14.2056 −0.918881 −0.459440 0.888209i $$-0.651950\pi$$
−0.459440 + 0.888209i $$0.651950\pi$$
$$240$$ 42.9200 2.77047
$$241$$ 8.44082 0.543721 0.271860 0.962337i $$-0.412361\pi$$
0.271860 + 0.962337i $$0.412361\pi$$
$$242$$ 2.48970 0.160044
$$243$$ −12.8972 −0.827357
$$244$$ −2.57834 −0.165061
$$245$$ 2.81361 0.179755
$$246$$ −23.6655 −1.50886
$$247$$ 0.813607 0.0517685
$$248$$ 1.79445 0.113948
$$249$$ −51.1310 −3.24030
$$250$$ 10.6322 0.672442
$$251$$ −23.9844 −1.51388 −0.756941 0.653483i $$-0.773307\pi$$
−0.756941 + 0.653483i $$0.773307\pi$$
$$252$$ −8.54359 −0.538196
$$253$$ 22.7683 1.43143
$$254$$ 23.3622 1.46588
$$255$$ 4.57834 0.286707
$$256$$ 20.8469 1.30293
$$257$$ 15.6116 0.973827 0.486913 0.873450i $$-0.338123\pi$$
0.486913 + 0.873450i $$0.338123\pi$$
$$258$$ 38.0383 2.36816
$$259$$ 6.15165 0.382245
$$260$$ 3.62721 0.224950
$$261$$ 54.9341 3.40033
$$262$$ −16.4111 −1.01388
$$263$$ −15.1708 −0.935472 −0.467736 0.883868i $$-0.654930\pi$$
−0.467736 + 0.883868i $$0.654930\pi$$
$$264$$ −12.4111 −0.763850
$$265$$ −7.01916 −0.431183
$$266$$ 1.47556 0.0904725
$$267$$ 33.0333 2.02160
$$268$$ 12.9411 0.790502
$$269$$ 23.2927 1.42018 0.710092 0.704109i $$-0.248653\pi$$
0.710092 + 0.704109i $$0.248653\pi$$
$$270$$ 57.4288 3.49501
$$271$$ 12.7456 0.774238 0.387119 0.922030i $$-0.373470\pi$$
0.387119 + 0.922030i $$0.373470\pi$$
$$272$$ 2.57834 0.156335
$$273$$ 3.10278 0.187788
$$274$$ 11.3522 0.685810
$$275$$ 9.04888 0.545668
$$276$$ −29.3522 −1.76679
$$277$$ 8.12193 0.488000 0.244000 0.969775i $$-0.421540\pi$$
0.244000 + 0.969775i $$0.421540\pi$$
$$278$$ 20.9894 1.25886
$$279$$ 9.22471 0.552269
$$280$$ −3.62721 −0.216767
$$281$$ 19.0333 1.13543 0.567715 0.823225i $$-0.307827\pi$$
0.567715 + 0.823225i $$0.307827\pi$$
$$282$$ −33.5960 −2.00062
$$283$$ 11.1466 0.662598 0.331299 0.943526i $$-0.392513\pi$$
0.331299 + 0.943526i $$0.392513\pi$$
$$284$$ −11.2544 −0.667827
$$285$$ −7.10278 −0.420732
$$286$$ −5.62721 −0.332744
$$287$$ 4.20555 0.248246
$$288$$ 42.0036 2.47508
$$289$$ −16.7250 −0.983821
$$290$$ −42.2978 −2.48381
$$291$$ 3.68111 0.215791
$$292$$ −3.02061 −0.176768
$$293$$ −14.1758 −0.828161 −0.414080 0.910240i $$-0.635897\pi$$
−0.414080 + 0.910240i $$0.635897\pi$$
$$294$$ 5.62721 0.328186
$$295$$ −12.5783 −0.732339
$$296$$ −7.93051 −0.460952
$$297$$ −34.9200 −2.02626
$$298$$ −15.4600 −0.895572
$$299$$ 7.33804 0.424370
$$300$$ −11.6655 −0.673509
$$301$$ −6.75971 −0.389623
$$302$$ 21.7350 1.25071
$$303$$ −40.6550 −2.33557
$$304$$ −4.00000 −0.229416
$$305$$ −5.62721 −0.322213
$$306$$ 6.30330 0.360336
$$307$$ 13.5592 0.773863 0.386932 0.922108i $$-0.373535\pi$$
0.386932 + 0.922108i $$0.373535\pi$$
$$308$$ −4.00000 −0.227921
$$309$$ −13.6867 −0.778606
$$310$$ −7.10278 −0.403411
$$311$$ 0.426686 0.0241952 0.0120976 0.999927i $$-0.496149\pi$$
0.0120976 + 0.999927i $$0.496149\pi$$
$$312$$ −4.00000 −0.226455
$$313$$ −18.1517 −1.02599 −0.512996 0.858391i $$-0.671464\pi$$
−0.512996 + 0.858391i $$0.671464\pi$$
$$314$$ 23.2645 1.31289
$$315$$ −18.6464 −1.05060
$$316$$ −17.4600 −0.982200
$$317$$ 9.42166 0.529173 0.264587 0.964362i $$-0.414764\pi$$
0.264587 + 0.964362i $$0.414764\pi$$
$$318$$ −14.0383 −0.787230
$$319$$ 25.7194 1.44001
$$320$$ −4.67609 −0.261401
$$321$$ −1.79445 −0.100156
$$322$$ 13.3083 0.741644
$$323$$ −0.426686 −0.0237415
$$324$$ 19.3869 1.07705
$$325$$ 2.91638 0.161772
$$326$$ 24.4111 1.35201
$$327$$ −17.2927 −0.956291
$$328$$ −5.42166 −0.299361
$$329$$ 5.97028 0.329152
$$330$$ 49.1255 2.70427
$$331$$ −17.4005 −0.956420 −0.478210 0.878246i $$-0.658714\pi$$
−0.478210 + 0.878246i $$0.658714\pi$$
$$332$$ 21.2444 1.16594
$$333$$ −40.7683 −2.23409
$$334$$ 3.68111 0.201421
$$335$$ 28.2439 1.54313
$$336$$ −15.2544 −0.832197
$$337$$ 22.0524 1.20127 0.600637 0.799522i $$-0.294914\pi$$
0.600637 + 0.799522i $$0.294914\pi$$
$$338$$ −1.81361 −0.0986472
$$339$$ −16.8972 −0.917731
$$340$$ −1.90225 −0.103164
$$341$$ 4.31889 0.233881
$$342$$ −9.77886 −0.528780
$$343$$ −1.00000 −0.0539949
$$344$$ 8.71440 0.469849
$$345$$ −64.0610 −3.44893
$$346$$ −36.8122 −1.97903
$$347$$ 25.3522 1.36098 0.680488 0.732759i $$-0.261768\pi$$
0.680488 + 0.732759i $$0.261768\pi$$
$$348$$ −33.1567 −1.77738
$$349$$ −5.70529 −0.305397 −0.152699 0.988273i $$-0.548796\pi$$
−0.152699 + 0.988273i $$0.548796\pi$$
$$350$$ 5.28917 0.282718
$$351$$ −11.2544 −0.600717
$$352$$ 19.6655 1.04818
$$353$$ −28.6761 −1.52627 −0.763137 0.646237i $$-0.776342\pi$$
−0.763137 + 0.646237i $$0.776342\pi$$
$$354$$ −25.1567 −1.33706
$$355$$ −24.5628 −1.30366
$$356$$ −13.7250 −0.727422
$$357$$ −1.62721 −0.0861212
$$358$$ 19.9561 1.05472
$$359$$ 11.0433 0.582845 0.291423 0.956594i $$-0.405871\pi$$
0.291423 + 0.956594i $$0.405871\pi$$
$$360$$ 24.0383 1.26693
$$361$$ −18.3380 −0.965160
$$362$$ −1.25443 −0.0659312
$$363$$ 4.25945 0.223563
$$364$$ −1.28917 −0.0675708
$$365$$ −6.59247 −0.345066
$$366$$ −11.2544 −0.588278
$$367$$ 27.3466 1.42748 0.713741 0.700409i $$-0.246999\pi$$
0.713741 + 0.700409i $$0.246999\pi$$
$$368$$ −36.0766 −1.88062
$$369$$ −27.8711 −1.45091
$$370$$ 31.3905 1.63191
$$371$$ 2.49472 0.129519
$$372$$ −5.56777 −0.288676
$$373$$ 16.1461 0.836014 0.418007 0.908444i $$-0.362729\pi$$
0.418007 + 0.908444i $$0.362729\pi$$
$$374$$ 2.95112 0.152599
$$375$$ 18.1900 0.939326
$$376$$ −7.69670 −0.396927
$$377$$ 8.28917 0.426914
$$378$$ −20.4111 −1.04983
$$379$$ 26.1305 1.34223 0.671117 0.741351i $$-0.265815\pi$$
0.671117 + 0.741351i $$0.265815\pi$$
$$380$$ 2.95112 0.151389
$$381$$ 39.9688 2.04767
$$382$$ −14.2056 −0.726819
$$383$$ −21.0489 −1.07555 −0.537774 0.843089i $$-0.680735\pi$$
−0.537774 + 0.843089i $$0.680735\pi$$
$$384$$ 29.9789 1.52985
$$385$$ −8.72999 −0.444921
$$386$$ 22.1361 1.12670
$$387$$ 44.7980 2.27721
$$388$$ −1.52946 −0.0776466
$$389$$ −21.6061 −1.09547 −0.547736 0.836651i $$-0.684510\pi$$
−0.547736 + 0.836651i $$0.684510\pi$$
$$390$$ 15.8328 0.801723
$$391$$ −3.84835 −0.194619
$$392$$ 1.28917 0.0651128
$$393$$ −28.0766 −1.41628
$$394$$ 34.1361 1.71975
$$395$$ −38.1063 −1.91734
$$396$$ 26.5089 1.33212
$$397$$ −27.6952 −1.38998 −0.694992 0.719017i $$-0.744592\pi$$
−0.694992 + 0.719017i $$0.744592\pi$$
$$398$$ −39.1950 −1.96467
$$399$$ 2.52444 0.126380
$$400$$ −14.3380 −0.716902
$$401$$ −2.57834 −0.128756 −0.0643780 0.997926i $$-0.520506\pi$$
−0.0643780 + 0.997926i $$0.520506\pi$$
$$402$$ 56.4877 2.81735
$$403$$ 1.39194 0.0693376
$$404$$ 16.8917 0.840393
$$405$$ 42.3119 2.10250
$$406$$ 15.0333 0.746090
$$407$$ −19.0872 −0.946117
$$408$$ 2.09775 0.103854
$$409$$ 15.1169 0.747483 0.373742 0.927533i $$-0.378075\pi$$
0.373742 + 0.927533i $$0.378075\pi$$
$$410$$ 21.4600 1.05983
$$411$$ 19.4217 0.958000
$$412$$ 5.68665 0.280161
$$413$$ 4.47054 0.219981
$$414$$ −88.1971 −4.33465
$$415$$ 46.3658 2.27601
$$416$$ 6.33804 0.310748
$$417$$ 35.9094 1.75849
$$418$$ −4.57834 −0.223934
$$419$$ −9.99446 −0.488261 −0.244131 0.969742i $$-0.578503\pi$$
−0.244131 + 0.969742i $$0.578503\pi$$
$$420$$ 11.2544 0.549160
$$421$$ −25.9250 −1.26351 −0.631753 0.775170i $$-0.717664\pi$$
−0.631753 + 0.775170i $$0.717664\pi$$
$$422$$ 31.5139 1.53407
$$423$$ −39.5663 −1.92378
$$424$$ −3.21611 −0.156188
$$425$$ −1.52946 −0.0741898
$$426$$ −49.1255 −2.38014
$$427$$ 2.00000 0.0967868
$$428$$ 0.745574 0.0360387
$$429$$ −9.62721 −0.464806
$$430$$ −34.4933 −1.66341
$$431$$ 30.6761 1.47762 0.738808 0.673916i $$-0.235389\pi$$
0.738808 + 0.673916i $$0.235389\pi$$
$$432$$ 55.3311 2.66212
$$433$$ −3.51941 −0.169132 −0.0845661 0.996418i $$-0.526950\pi$$
−0.0845661 + 0.996418i $$0.526950\pi$$
$$434$$ 2.52444 0.121177
$$435$$ −72.3643 −3.46960
$$436$$ 7.18494 0.344096
$$437$$ 5.97028 0.285597
$$438$$ −13.1849 −0.630001
$$439$$ 32.3517 1.54406 0.772030 0.635586i $$-0.219241\pi$$
0.772030 + 0.635586i $$0.219241\pi$$
$$440$$ 11.2544 0.536534
$$441$$ 6.62721 0.315582
$$442$$ 0.951124 0.0452404
$$443$$ 15.4458 0.733854 0.366927 0.930250i $$-0.380410\pi$$
0.366927 + 0.930250i $$0.380410\pi$$
$$444$$ 24.6066 1.16778
$$445$$ −29.9547 −1.41999
$$446$$ 19.1310 0.905881
$$447$$ −26.4494 −1.25101
$$448$$ 1.66196 0.0785200
$$449$$ −14.4705 −0.682907 −0.341453 0.939899i $$-0.610919\pi$$
−0.341453 + 0.939899i $$0.610919\pi$$
$$450$$ −35.0524 −1.65239
$$451$$ −13.0489 −0.614448
$$452$$ 7.02061 0.330222
$$453$$ 37.1849 1.74710
$$454$$ 12.6066 0.591657
$$455$$ −2.81361 −0.131904
$$456$$ −3.25443 −0.152402
$$457$$ −34.6705 −1.62182 −0.810910 0.585171i $$-0.801027\pi$$
−0.810910 + 0.585171i $$0.801027\pi$$
$$458$$ 38.2439 1.78702
$$459$$ 5.90225 0.275493
$$460$$ 26.6167 1.24101
$$461$$ 12.5400 0.584047 0.292024 0.956411i $$-0.405671\pi$$
0.292024 + 0.956411i $$0.405671\pi$$
$$462$$ −17.4600 −0.812312
$$463$$ 12.1517 0.564735 0.282368 0.959306i $$-0.408880\pi$$
0.282368 + 0.959306i $$0.408880\pi$$
$$464$$ −40.7527 −1.89190
$$465$$ −12.1517 −0.563519
$$466$$ −11.0333 −0.511107
$$467$$ −37.0333 −1.71370 −0.856848 0.515569i $$-0.827581\pi$$
−0.856848 + 0.515569i $$0.827581\pi$$
$$468$$ 8.54359 0.394928
$$469$$ −10.0383 −0.463526
$$470$$ 30.4650 1.40525
$$471$$ 39.8016 1.83396
$$472$$ −5.76328 −0.265276
$$473$$ 20.9739 0.964379
$$474$$ −76.2127 −3.50056
$$475$$ 2.37279 0.108871
$$476$$ 0.676089 0.0309885
$$477$$ −16.5330 −0.756996
$$478$$ 25.7633 1.17838
$$479$$ 12.0086 0.548687 0.274343 0.961632i $$-0.411540\pi$$
0.274343 + 0.961632i $$0.411540\pi$$
$$480$$ −55.3311 −2.52551
$$481$$ −6.15165 −0.280491
$$482$$ −15.3083 −0.697275
$$483$$ 22.7683 1.03599
$$484$$ −1.76975 −0.0804434
$$485$$ −3.33804 −0.151573
$$486$$ 23.3905 1.06101
$$487$$ 11.1184 0.503821 0.251911 0.967751i $$-0.418941\pi$$
0.251911 + 0.967751i $$0.418941\pi$$
$$488$$ −2.57834 −0.116716
$$489$$ 41.7633 1.88860
$$490$$ −5.10278 −0.230520
$$491$$ 0.0594386 0.00268243 0.00134121 0.999999i $$-0.499573\pi$$
0.00134121 + 0.999999i $$0.499573\pi$$
$$492$$ 16.8222 0.758403
$$493$$ −4.34715 −0.195786
$$494$$ −1.47556 −0.0663887
$$495$$ 57.8555 2.60041
$$496$$ −6.84333 −0.307274
$$497$$ 8.72999 0.391593
$$498$$ 92.7316 4.15540
$$499$$ 10.2978 0.460991 0.230496 0.973073i $$-0.425965\pi$$
0.230496 + 0.973073i $$0.425965\pi$$
$$500$$ −7.55773 −0.337992
$$501$$ 6.29776 0.281363
$$502$$ 43.4983 1.94142
$$503$$ −9.32391 −0.415733 −0.207866 0.978157i $$-0.566652\pi$$
−0.207866 + 0.978157i $$0.566652\pi$$
$$504$$ −8.54359 −0.380562
$$505$$ 36.8661 1.64052
$$506$$ −41.2927 −1.83569
$$507$$ −3.10278 −0.137799
$$508$$ −16.6066 −0.736799
$$509$$ 39.6952 1.75946 0.879730 0.475473i $$-0.157723\pi$$
0.879730 + 0.475473i $$0.157723\pi$$
$$510$$ −8.30330 −0.367676
$$511$$ 2.34307 0.103651
$$512$$ −18.4842 −0.816892
$$513$$ −9.15667 −0.404277
$$514$$ −28.3133 −1.24885
$$515$$ 12.4111 0.546898
$$516$$ −27.0388 −1.19032
$$517$$ −18.5244 −0.814704
$$518$$ −11.1567 −0.490196
$$519$$ −62.9794 −2.76449
$$520$$ 3.62721 0.159064
$$521$$ 22.3627 0.979729 0.489865 0.871798i $$-0.337046\pi$$
0.489865 + 0.871798i $$0.337046\pi$$
$$522$$ −99.6288 −4.36063
$$523$$ −20.6550 −0.903178 −0.451589 0.892226i $$-0.649143\pi$$
−0.451589 + 0.892226i $$0.649143\pi$$
$$524$$ 11.6655 0.509611
$$525$$ 9.04888 0.394925
$$526$$ 27.5139 1.19966
$$527$$ −0.729988 −0.0317988
$$528$$ 47.3311 2.05982
$$529$$ 30.8469 1.34117
$$530$$ 12.7300 0.552955
$$531$$ −29.6272 −1.28571
$$532$$ −1.04888 −0.0454745
$$533$$ −4.20555 −0.182163
$$534$$ −59.9094 −2.59253
$$535$$ 1.62721 0.0703506
$$536$$ 12.9411 0.558969
$$537$$ 34.1416 1.47332
$$538$$ −42.2439 −1.82126
$$539$$ 3.10278 0.133646
$$540$$ −40.8222 −1.75671
$$541$$ 5.62167 0.241695 0.120847 0.992671i $$-0.461439\pi$$
0.120847 + 0.992671i $$0.461439\pi$$
$$542$$ −23.1155 −0.992894
$$543$$ −2.14611 −0.0920985
$$544$$ −3.32391 −0.142512
$$545$$ 15.6811 0.671705
$$546$$ −5.62721 −0.240822
$$547$$ −10.3970 −0.444542 −0.222271 0.974985i $$-0.571347\pi$$
−0.222271 + 0.974985i $$0.571347\pi$$
$$548$$ −8.06949 −0.344711
$$549$$ −13.2544 −0.565685
$$550$$ −16.4111 −0.699772
$$551$$ 6.74412 0.287309
$$552$$ −29.3522 −1.24931
$$553$$ 13.5436 0.575932
$$554$$ −14.7300 −0.625817
$$555$$ 53.7038 2.27960
$$556$$ −14.9200 −0.632747
$$557$$ 14.6550 0.620951 0.310475 0.950581i $$-0.399512\pi$$
0.310475 + 0.950581i $$0.399512\pi$$
$$558$$ −16.7300 −0.708237
$$559$$ 6.75971 0.285905
$$560$$ 13.8328 0.584541
$$561$$ 5.04888 0.213164
$$562$$ −34.5189 −1.45609
$$563$$ −24.7456 −1.04290 −0.521451 0.853281i $$-0.674609\pi$$
−0.521451 + 0.853281i $$0.674609\pi$$
$$564$$ 23.8811 1.00558
$$565$$ 15.3225 0.644621
$$566$$ −20.2156 −0.849725
$$567$$ −15.0383 −0.631550
$$568$$ −11.2544 −0.472225
$$569$$ −20.5330 −0.860789 −0.430395 0.902641i $$-0.641626\pi$$
−0.430395 + 0.902641i $$0.641626\pi$$
$$570$$ 12.8816 0.539552
$$571$$ −41.8953 −1.75326 −0.876631 0.481163i $$-0.840214\pi$$
−0.876631 + 0.481163i $$0.840214\pi$$
$$572$$ 4.00000 0.167248
$$573$$ −24.3033 −1.01529
$$574$$ −7.62721 −0.318354
$$575$$ 21.4005 0.892464
$$576$$ −11.0141 −0.458922
$$577$$ −20.1744 −0.839870 −0.419935 0.907554i $$-0.637947\pi$$
−0.419935 + 0.907554i $$0.637947\pi$$
$$578$$ 30.3325 1.26167
$$579$$ 37.8711 1.57387
$$580$$ 30.0666 1.24845
$$581$$ −16.4791 −0.683670
$$582$$ −6.67609 −0.276733
$$583$$ −7.74055 −0.320581
$$584$$ −3.02061 −0.124994
$$585$$ 18.6464 0.770933
$$586$$ 25.7094 1.06204
$$587$$ −18.7441 −0.773653 −0.386826 0.922153i $$-0.626429\pi$$
−0.386826 + 0.922153i $$0.626429\pi$$
$$588$$ −4.00000 −0.164957
$$589$$ 1.13249 0.0466636
$$590$$ 22.8122 0.939162
$$591$$ 58.4011 2.40230
$$592$$ 30.2439 1.24302
$$593$$ −2.98084 −0.122409 −0.0612043 0.998125i $$-0.519494\pi$$
−0.0612043 + 0.998125i $$0.519494\pi$$
$$594$$ 63.3311 2.59850
$$595$$ 1.47556 0.0604921
$$596$$ 10.9894 0.450145
$$597$$ −67.0560 −2.74442
$$598$$ −13.3083 −0.544218
$$599$$ 7.47411 0.305384 0.152692 0.988274i $$-0.451206\pi$$
0.152692 + 0.988274i $$0.451206\pi$$
$$600$$ −11.6655 −0.476243
$$601$$ 21.4700 0.875780 0.437890 0.899028i $$-0.355726\pi$$
0.437890 + 0.899028i $$0.355726\pi$$
$$602$$ 12.2594 0.499658
$$603$$ 66.5260 2.70915
$$604$$ −15.4499 −0.628649
$$605$$ −3.86248 −0.157032
$$606$$ 73.7321 2.99516
$$607$$ −22.9044 −0.929660 −0.464830 0.885400i $$-0.653884\pi$$
−0.464830 + 0.885400i $$0.653884\pi$$
$$608$$ 5.15667 0.209131
$$609$$ 25.7194 1.04220
$$610$$ 10.2056 0.413211
$$611$$ −5.97028 −0.241532
$$612$$ −4.48059 −0.181117
$$613$$ −20.1461 −0.813694 −0.406847 0.913496i $$-0.633372\pi$$
−0.406847 + 0.913496i $$0.633372\pi$$
$$614$$ −24.5910 −0.992413
$$615$$ 36.7144 1.48047
$$616$$ −4.00000 −0.161165
$$617$$ −13.7844 −0.554939 −0.277470 0.960734i $$-0.589496\pi$$
−0.277470 + 0.960734i $$0.589496\pi$$
$$618$$ 24.8222 0.998495
$$619$$ −19.6655 −0.790424 −0.395212 0.918590i $$-0.629329\pi$$
−0.395212 + 0.918590i $$0.629329\pi$$
$$620$$ 5.04888 0.202768
$$621$$ −82.5855 −3.31404
$$622$$ −0.773841 −0.0310282
$$623$$ 10.6464 0.426538
$$624$$ 15.2544 0.610666
$$625$$ −31.0766 −1.24307
$$626$$ 32.9200 1.31575
$$627$$ −7.83276 −0.312810
$$628$$ −16.5371 −0.659903
$$629$$ 3.22616 0.128635
$$630$$ 33.8172 1.34731
$$631$$ −16.1672 −0.643608 −0.321804 0.946806i $$-0.604289\pi$$
−0.321804 + 0.946806i $$0.604289\pi$$
$$632$$ −17.4600 −0.694521
$$633$$ 53.9149 2.14293
$$634$$ −17.0872 −0.678619
$$635$$ −36.2439 −1.43829
$$636$$ 9.97887 0.395688
$$637$$ 1.00000 0.0396214
$$638$$ −46.6449 −1.84669
$$639$$ −57.8555 −2.28873
$$640$$ −27.1849 −1.07458
$$641$$ −29.0036 −1.14557 −0.572786 0.819705i $$-0.694137\pi$$
−0.572786 + 0.819705i $$0.694137\pi$$
$$642$$ 3.25443 0.128442
$$643$$ −39.2233 −1.54681 −0.773407 0.633910i $$-0.781449\pi$$
−0.773407 + 0.633910i $$0.781449\pi$$
$$644$$ −9.45998 −0.372775
$$645$$ −59.0122 −2.32360
$$646$$ 0.773841 0.0304464
$$647$$ −11.9844 −0.471156 −0.235578 0.971855i $$-0.575698\pi$$
−0.235578 + 0.971855i $$0.575698\pi$$
$$648$$ 19.3869 0.761590
$$649$$ −13.8711 −0.544487
$$650$$ −5.28917 −0.207458
$$651$$ 4.31889 0.169271
$$652$$ −17.3522 −0.679564
$$653$$ 45.3311 1.77394 0.886971 0.461826i $$-0.152806\pi$$
0.886971 + 0.461826i $$0.152806\pi$$
$$654$$ 31.3622 1.22636
$$655$$ 25.4600 0.994804
$$656$$ 20.6761 0.807266
$$657$$ −15.5280 −0.605805
$$658$$ −10.8277 −0.422109
$$659$$ 6.12193 0.238477 0.119238 0.992866i $$-0.461955\pi$$
0.119238 + 0.992866i $$0.461955\pi$$
$$660$$ −34.9200 −1.35926
$$661$$ −27.5280 −1.07072 −0.535358 0.844625i $$-0.679823\pi$$
−0.535358 + 0.844625i $$0.679823\pi$$
$$662$$ 31.5577 1.22653
$$663$$ 1.62721 0.0631957
$$664$$ 21.2444 0.824442
$$665$$ −2.28917 −0.0887701
$$666$$ 73.9377 2.86503
$$667$$ 60.8263 2.35520
$$668$$ −2.61665 −0.101241
$$669$$ 32.7300 1.26541
$$670$$ −51.2233 −1.97893
$$671$$ −6.20555 −0.239563
$$672$$ 19.6655 0.758614
$$673$$ 27.9547 1.07757 0.538787 0.842442i $$-0.318883\pi$$
0.538787 + 0.842442i $$0.318883\pi$$
$$674$$ −39.9945 −1.54053
$$675$$ −32.8222 −1.26333
$$676$$ 1.28917 0.0495834
$$677$$ −12.6605 −0.486583 −0.243291 0.969953i $$-0.578227\pi$$
−0.243291 + 0.969953i $$0.578227\pi$$
$$678$$ 30.6449 1.17691
$$679$$ 1.18639 0.0455296
$$680$$ −1.90225 −0.0729479
$$681$$ 21.5678 0.826479
$$682$$ −7.83276 −0.299932
$$683$$ −28.3033 −1.08300 −0.541498 0.840702i $$-0.682143\pi$$
−0.541498 + 0.840702i $$0.682143\pi$$
$$684$$ 6.95112 0.265783
$$685$$ −17.6116 −0.672906
$$686$$ 1.81361 0.0692438
$$687$$ 65.4288 2.49626
$$688$$ −33.2333 −1.26701
$$689$$ −2.49472 −0.0950412
$$690$$ 116.182 4.42295
$$691$$ 12.2353 0.465452 0.232726 0.972542i $$-0.425236\pi$$
0.232726 + 0.972542i $$0.425236\pi$$
$$692$$ 26.1672 0.994729
$$693$$ −20.5628 −0.781114
$$694$$ −45.9789 −1.74533
$$695$$ −32.5628 −1.23518
$$696$$ −33.1567 −1.25680
$$697$$ 2.20555 0.0835412
$$698$$ 10.3472 0.391646
$$699$$ −18.8761 −0.713960
$$700$$ −3.75971 −0.142104
$$701$$ 51.0419 1.92783 0.963913 0.266219i $$-0.0857743\pi$$
0.963913 + 0.266219i $$0.0857743\pi$$
$$702$$ 20.4111 0.770367
$$703$$ −5.00502 −0.188768
$$704$$ −5.15667 −0.194349
$$705$$ 52.1205 1.96297
$$706$$ 52.0071 1.95731
$$707$$ −13.1028 −0.492781
$$708$$ 17.8822 0.672053
$$709$$ 42.5910 1.59954 0.799770 0.600307i $$-0.204955\pi$$
0.799770 + 0.600307i $$0.204955\pi$$
$$710$$ 44.5472 1.67183
$$711$$ −89.7563 −3.36612
$$712$$ −13.7250 −0.514365
$$713$$ 10.2141 0.382523
$$714$$ 2.95112 0.110443
$$715$$ 8.72999 0.326483
$$716$$ −14.1855 −0.530135
$$717$$ 44.0766 1.64607
$$718$$ −20.0283 −0.747448
$$719$$ −42.4933 −1.58473 −0.792366 0.610046i $$-0.791151\pi$$
−0.792366 + 0.610046i $$0.791151\pi$$
$$720$$ −91.6727 −3.41644
$$721$$ −4.41110 −0.164278
$$722$$ 33.2580 1.23773
$$723$$ −26.1900 −0.974015
$$724$$ 0.891685 0.0331392
$$725$$ 24.1744 0.897814
$$726$$ −7.72496 −0.286700
$$727$$ −3.75614 −0.139307 −0.0696537 0.997571i $$-0.522189\pi$$
−0.0696537 + 0.997571i $$0.522189\pi$$
$$728$$ −1.28917 −0.0477798
$$729$$ −5.09775 −0.188806
$$730$$ 11.9561 0.442517
$$731$$ −3.54505 −0.131118
$$732$$ 8.00000 0.295689
$$733$$ 45.7819 1.69099 0.845497 0.533980i $$-0.179304\pi$$
0.845497 + 0.533980i $$0.179304\pi$$
$$734$$ −49.5960 −1.83062
$$735$$ −8.72999 −0.322010
$$736$$ 46.5089 1.71434
$$737$$ 31.1466 1.14730
$$738$$ 50.5472 1.86067
$$739$$ −14.0539 −0.516981 −0.258491 0.966014i $$-0.583225\pi$$
−0.258491 + 0.966014i $$0.583225\pi$$
$$740$$ −22.3133 −0.820255
$$741$$ −2.52444 −0.0927375
$$742$$ −4.52444 −0.166097
$$743$$ 4.74557 0.174098 0.0870491 0.996204i $$-0.472256\pi$$
0.0870491 + 0.996204i $$0.472256\pi$$
$$744$$ −5.56777 −0.204125
$$745$$ 23.9844 0.878721
$$746$$ −29.2827 −1.07212
$$747$$ 109.211 3.99581
$$748$$ −2.09775 −0.0767014
$$749$$ −0.578337 −0.0211320
$$750$$ −32.9894 −1.20460
$$751$$ 36.1008 1.31734 0.658669 0.752433i $$-0.271120\pi$$
0.658669 + 0.752433i $$0.271120\pi$$
$$752$$ 29.3522 1.07036
$$753$$ 74.4182 2.71195
$$754$$ −15.0333 −0.547480
$$755$$ −33.7194 −1.22718
$$756$$ 14.5089 0.527682
$$757$$ −1.03474 −0.0376084 −0.0188042 0.999823i $$-0.505986\pi$$
−0.0188042 + 0.999823i $$0.505986\pi$$
$$758$$ −47.3905 −1.72130
$$759$$ −70.6449 −2.56425
$$760$$ 2.95112 0.107049
$$761$$ 29.8414 1.08175 0.540874 0.841104i $$-0.318094\pi$$
0.540874 + 0.841104i $$0.318094\pi$$
$$762$$ −72.4877 −2.62595
$$763$$ −5.57331 −0.201768
$$764$$ 10.0978 0.365324
$$765$$ −9.77886 −0.353556
$$766$$ 38.1744 1.37930
$$767$$ −4.47054 −0.161422
$$768$$ −64.6832 −2.33406
$$769$$ 23.6358 0.852329 0.426164 0.904646i $$-0.359864\pi$$
0.426164 + 0.904646i $$0.359864\pi$$
$$770$$ 15.8328 0.570573
$$771$$ −48.4394 −1.74450
$$772$$ −15.7350 −0.566315
$$773$$ 13.0278 0.468576 0.234288 0.972167i $$-0.424724\pi$$
0.234288 + 0.972167i $$0.424724\pi$$
$$774$$ −81.2460 −2.92033
$$775$$ 4.05944 0.145819
$$776$$ −1.52946 −0.0549045
$$777$$ −19.0872 −0.684749
$$778$$ 39.1849 1.40485
$$779$$ −3.42166 −0.122594
$$780$$ −11.2544 −0.402973
$$781$$ −27.0872 −0.969256
$$782$$ 6.97939 0.249583
$$783$$ −93.2898 −3.33391
$$784$$ −4.91638 −0.175585
$$785$$ −36.0922 −1.28819
$$786$$ 50.9200 1.81625
$$787$$ 46.2141 1.64736 0.823678 0.567058i $$-0.191918\pi$$
0.823678 + 0.567058i $$0.191918\pi$$
$$788$$ −24.2650 −0.864404
$$789$$ 47.0716 1.67579
$$790$$ 69.1099 2.45882
$$791$$ −5.44584 −0.193632
$$792$$ 26.5089 0.941951
$$793$$ −2.00000 −0.0710221
$$794$$ 50.2283 1.78253
$$795$$ 21.7789 0.772417
$$796$$ 27.8610 0.987508
$$797$$ 53.1155 1.88145 0.940723 0.339176i $$-0.110148\pi$$
0.940723 + 0.339176i $$0.110148\pi$$
$$798$$ −4.57834 −0.162071
$$799$$ 3.13104 0.110768
$$800$$ 18.4842 0.653514
$$801$$ −70.5558 −2.49297
$$802$$ 4.67609 0.165118
$$803$$ −7.27001 −0.256553
$$804$$ −40.1533 −1.41610
$$805$$ −20.6464 −0.727689
$$806$$ −2.52444 −0.0889195
$$807$$ −72.2721 −2.54410
$$808$$ 16.8917 0.594247
$$809$$ −54.4635 −1.91484 −0.957418 0.288705i $$-0.906775\pi$$
−0.957418 + 0.288705i $$0.906775\pi$$
$$810$$ −76.7371 −2.69627
$$811$$ 38.0978 1.33779 0.668897 0.743356i $$-0.266767\pi$$
0.668897 + 0.743356i $$0.266767\pi$$
$$812$$ −10.6861 −0.375010
$$813$$ −39.5466 −1.38696
$$814$$ 34.6167 1.21331
$$815$$ −37.8711 −1.32657
$$816$$ −8.00000 −0.280056
$$817$$ 5.49974 0.192412
$$818$$ −27.4161 −0.958582
$$819$$ −6.62721 −0.231574
$$820$$ −15.2544 −0.532708
$$821$$ −2.30330 −0.0803858 −0.0401929 0.999192i $$-0.512797\pi$$
−0.0401929 + 0.999192i $$0.512797\pi$$
$$822$$ −35.2233 −1.22855
$$823$$ 23.6172 0.823243 0.411621 0.911355i $$-0.364963\pi$$
0.411621 + 0.911355i $$0.364963\pi$$
$$824$$ 5.68665 0.198104
$$825$$ −28.0766 −0.977503
$$826$$ −8.10780 −0.282106
$$827$$ −48.1643 −1.67484 −0.837419 0.546562i $$-0.815936\pi$$
−0.837419 + 0.546562i $$0.815936\pi$$
$$828$$ 62.6933 2.17874
$$829$$ 13.0716 0.453996 0.226998 0.973895i $$-0.427109\pi$$
0.226998 + 0.973895i $$0.427109\pi$$
$$830$$ −84.0893 −2.91878
$$831$$ −25.2005 −0.874197
$$832$$ −1.66196 −0.0576179
$$833$$ −0.524438 −0.0181707
$$834$$ −65.1255 −2.25511
$$835$$ −5.71083 −0.197631
$$836$$ 3.25443 0.112557
$$837$$ −15.6655 −0.541480
$$838$$ 18.1260 0.626153
$$839$$ −17.6756 −0.610229 −0.305114 0.952316i $$-0.598695\pi$$
−0.305114 + 0.952316i $$0.598695\pi$$
$$840$$ 11.2544 0.388315
$$841$$ 39.7103 1.36932
$$842$$ 47.0177 1.62034
$$843$$ −59.0560 −2.03400
$$844$$ −22.4011 −0.771076
$$845$$ 2.81361 0.0967910
$$846$$ 71.7577 2.46708
$$847$$ 1.37279 0.0471695
$$848$$ 12.2650 0.421181
$$849$$ −34.5855 −1.18697
$$850$$ 2.77384 0.0951420
$$851$$ −45.1411 −1.54742
$$852$$ 34.9200 1.19634
$$853$$ −5.48970 −0.187964 −0.0939818 0.995574i $$-0.529960\pi$$
−0.0939818 + 0.995574i $$0.529960\pi$$
$$854$$ −3.62721 −0.124121
$$855$$ 15.1708 0.518831
$$856$$ 0.745574 0.0254832
$$857$$ −11.0489 −0.377422 −0.188711 0.982033i $$-0.560431\pi$$
−0.188711 + 0.982033i $$0.560431\pi$$
$$858$$ 17.4600 0.596074
$$859$$ 45.2616 1.54430 0.772152 0.635437i $$-0.219180\pi$$
0.772152 + 0.635437i $$0.219180\pi$$
$$860$$ 24.5189 0.836087
$$861$$ −13.0489 −0.444705
$$862$$ −55.6344 −1.89491
$$863$$ −5.90225 −0.200915 −0.100457 0.994941i $$-0.532031\pi$$
−0.100457 + 0.994941i $$0.532031\pi$$
$$864$$ −71.3311 −2.42673
$$865$$ 57.1099 1.94180
$$866$$ 6.38283 0.216898
$$867$$ 51.8938 1.76241
$$868$$ −1.79445 −0.0609076
$$869$$ −42.0227 −1.42552
$$870$$ 131.240 4.44947
$$871$$ 10.0383 0.340135
$$872$$ 7.18494 0.243313
$$873$$ −7.86248 −0.266105
$$874$$ −10.8277 −0.366254
$$875$$ 5.86248 0.198188
$$876$$ 9.37227 0.316660
$$877$$ −4.90727 −0.165707 −0.0828534 0.996562i $$-0.526403\pi$$
−0.0828534 + 0.996562i $$0.526403\pi$$
$$878$$ −58.6732 −1.98012
$$879$$ 43.9844 1.48356
$$880$$ −42.9200 −1.44683
$$881$$ 44.2822 1.49190 0.745952 0.665999i $$-0.231995\pi$$
0.745952 + 0.665999i $$0.231995\pi$$
$$882$$ −12.0192 −0.404706
$$883$$ −58.8605 −1.98081 −0.990407 0.138181i $$-0.955874\pi$$
−0.990407 + 0.138181i $$0.955874\pi$$
$$884$$ −0.676089 −0.0227393
$$885$$ 39.0278 1.31190
$$886$$ −28.0127 −0.941104
$$887$$ −10.1289 −0.340096 −0.170048 0.985436i $$-0.554392\pi$$
−0.170048 + 0.985436i $$0.554392\pi$$
$$888$$ 24.6066 0.825744
$$889$$ 12.8816 0.432036
$$890$$ 54.3260 1.82101
$$891$$ 46.6605 1.56319
$$892$$ −13.5989 −0.455326
$$893$$ −4.85746 −0.162549
$$894$$ 47.9688 1.60432
$$895$$ −30.9597 −1.03487
$$896$$ 9.66196 0.322783
$$897$$ −22.7683 −0.760211
$$898$$ 26.2439 0.875769
$$899$$ 11.5381 0.384816
$$900$$ 24.9164 0.830546
$$901$$ 1.30833 0.0435866
$$902$$ 23.6655 0.787976
$$903$$ 20.9739 0.697966
$$904$$ 7.02061 0.233502
$$905$$ 1.94610 0.0646906
$$906$$ −67.4389 −2.24051
$$907$$ 37.9547 1.26026 0.630132 0.776488i $$-0.283001\pi$$
0.630132 + 0.776488i $$0.283001\pi$$
$$908$$ −8.96117 −0.297387
$$909$$ 86.8349 2.88013
$$910$$ 5.10278 0.169155
$$911$$ 5.57477 0.184700 0.0923501 0.995727i $$-0.470562\pi$$
0.0923501 + 0.995727i $$0.470562\pi$$
$$912$$ 12.4111 0.410973
$$913$$ 51.1310 1.69219
$$914$$ 62.8787 2.07984
$$915$$ 17.4600 0.577209
$$916$$ −27.1849 −0.898216
$$917$$ −9.04888 −0.298820
$$918$$ −10.7044 −0.353296
$$919$$ 15.7844 0.520679 0.260340 0.965517i $$-0.416165\pi$$
0.260340 + 0.965517i $$0.416165\pi$$
$$920$$ 26.6167 0.877525
$$921$$ −42.0711 −1.38629
$$922$$ −22.7427 −0.748990
$$923$$ −8.72999 −0.287351
$$924$$ 12.4111 0.408295
$$925$$ −17.9406 −0.589882
$$926$$ −22.0383 −0.724224
$$927$$ 29.2333 0.960148
$$928$$ 52.5371 1.72462
$$929$$ −45.2630 −1.48503 −0.742516 0.669829i $$-0.766368\pi$$
−0.742516 + 0.669829i $$0.766368\pi$$
$$930$$ 22.0383 0.722665
$$931$$ 0.813607 0.0266649
$$932$$ 7.84281 0.256900
$$933$$ −1.32391 −0.0433429
$$934$$ 67.1638 2.19767
$$935$$ −4.57834 −0.149728
$$936$$ 8.54359 0.279256
$$937$$ 53.6188 1.75165 0.875824 0.482630i $$-0.160318\pi$$
0.875824 + 0.482630i $$0.160318\pi$$
$$938$$ 18.2056 0.594432
$$939$$ 56.3205 1.83795
$$940$$ −21.6555 −0.706324
$$941$$ 20.7753 0.677255 0.338628 0.940920i $$-0.390037\pi$$
0.338628 + 0.940920i $$0.390037\pi$$
$$942$$ −72.1844 −2.35190
$$943$$ −30.8605 −1.00496
$$944$$ 21.9789 0.715351
$$945$$ 31.6655 1.03008
$$946$$ −38.0383 −1.23673
$$947$$ −10.8605 −0.352919 −0.176460 0.984308i $$-0.556465\pi$$
−0.176460 + 0.984308i $$0.556465\pi$$
$$948$$ 54.1744 1.75950
$$949$$ −2.34307 −0.0760592
$$950$$ −4.30330 −0.139618
$$951$$ −29.2333 −0.947955
$$952$$ 0.676089 0.0219122
$$953$$ −25.7180 −0.833087 −0.416543 0.909116i $$-0.636759\pi$$
−0.416543 + 0.909116i $$0.636759\pi$$
$$954$$ 29.9844 0.970781
$$955$$ 22.0383 0.713143
$$956$$ −18.3133 −0.592296
$$957$$ −79.8016 −2.57962
$$958$$ −21.7789 −0.703643
$$959$$ 6.25945 0.202128
$$960$$ 14.5089 0.468271
$$961$$ −29.0625 −0.937500
$$962$$ 11.1567 0.359706
$$963$$ 3.83276 0.123509
$$964$$ 10.8816 0.350474
$$965$$ −34.3416 −1.10550
$$966$$ −41.2927 −1.32857
$$967$$ 33.5038 1.07741 0.538705 0.842494i $$-0.318914\pi$$
0.538705 + 0.842494i $$0.318914\pi$$
$$968$$ −1.76975 −0.0568820
$$969$$ 1.32391 0.0425302
$$970$$ 6.05390 0.194379
$$971$$ 2.03831 0.0654126 0.0327063 0.999465i $$-0.489587\pi$$
0.0327063 + 0.999465i $$0.489587\pi$$
$$972$$ −16.6267 −0.533302
$$973$$ 11.5733 0.371023
$$974$$ −20.1643 −0.646107
$$975$$ −9.04888 −0.289796
$$976$$ 9.83276 0.314739
$$977$$ 15.1411 0.484406 0.242203 0.970226i $$-0.422130\pi$$
0.242203 + 0.970226i $$0.422130\pi$$
$$978$$ −75.7422 −2.42197
$$979$$ −33.0333 −1.05575
$$980$$ 3.62721 0.115867
$$981$$ 36.9355 1.17926
$$982$$ −0.107798 −0.00343998
$$983$$ 49.3124 1.57282 0.786411 0.617704i $$-0.211937\pi$$
0.786411 + 0.617704i $$0.211937\pi$$
$$984$$ 16.8222 0.536272
$$985$$ −52.9583 −1.68739
$$986$$ 7.88403 0.251079
$$987$$ −18.5244 −0.589639
$$988$$ 1.04888 0.0333692
$$989$$ 49.6030 1.57728
$$990$$ −104.927 −3.33480
$$991$$ 5.43171 0.172544 0.0862720 0.996272i $$-0.472505\pi$$
0.0862720 + 0.996272i $$0.472505\pi$$
$$992$$ 8.82220 0.280105
$$993$$ 53.9900 1.71332
$$994$$ −15.8328 −0.502185
$$995$$ 60.8066 1.92770
$$996$$ −65.9165 −2.08865
$$997$$ −53.6061 −1.69772 −0.848861 0.528616i $$-0.822711\pi$$
−0.848861 + 0.528616i $$0.822711\pi$$
$$998$$ −18.6761 −0.591181
$$999$$ 69.2333 2.19044
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 91.2.a.d.1.1 3
3.2 odd 2 819.2.a.i.1.3 3
4.3 odd 2 1456.2.a.t.1.3 3
5.4 even 2 2275.2.a.m.1.3 3
7.2 even 3 637.2.e.j.508.3 6
7.3 odd 6 637.2.e.i.79.3 6
7.4 even 3 637.2.e.j.79.3 6
7.5 odd 6 637.2.e.i.508.3 6
7.6 odd 2 637.2.a.j.1.1 3
8.3 odd 2 5824.2.a.bs.1.1 3
8.5 even 2 5824.2.a.by.1.3 3
13.5 odd 4 1183.2.c.f.337.5 6
13.8 odd 4 1183.2.c.f.337.2 6
13.12 even 2 1183.2.a.i.1.3 3
21.20 even 2 5733.2.a.x.1.3 3
91.90 odd 2 8281.2.a.bg.1.3 3

By twisted newform
Twist Min Dim Char Parity Ord Type
91.2.a.d.1.1 3 1.1 even 1 trivial
637.2.a.j.1.1 3 7.6 odd 2
637.2.e.i.79.3 6 7.3 odd 6
637.2.e.i.508.3 6 7.5 odd 6
637.2.e.j.79.3 6 7.4 even 3
637.2.e.j.508.3 6 7.2 even 3
819.2.a.i.1.3 3 3.2 odd 2
1183.2.a.i.1.3 3 13.12 even 2
1183.2.c.f.337.2 6 13.8 odd 4
1183.2.c.f.337.5 6 13.5 odd 4
1456.2.a.t.1.3 3 4.3 odd 2
2275.2.a.m.1.3 3 5.4 even 2
5733.2.a.x.1.3 3 21.20 even 2
5824.2.a.bs.1.1 3 8.3 odd 2
5824.2.a.by.1.3 3 8.5 even 2
8281.2.a.bg.1.3 3 91.90 odd 2