# Properties

 Label 7105.2.a.o.1.1 Level $7105$ Weight $2$ Character 7105.1 Self dual yes Analytic conductor $56.734$ 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] = [7105,2,Mod(1,7105)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7105, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("7105.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$7105 = 5 \cdot 7^{2} \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7105.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: $$56.7337106361$$ 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: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 145) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-1.48119$$ of defining polynomial Character $$\chi$$ $$=$$ 7105.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.48119 q^{2} -0.806063 q^{3} +0.193937 q^{4} -1.00000 q^{5} +1.19394 q^{6} +2.67513 q^{8} -2.35026 q^{9} +O(q^{10})$$ $$q-1.48119 q^{2} -0.806063 q^{3} +0.193937 q^{4} -1.00000 q^{5} +1.19394 q^{6} +2.67513 q^{8} -2.35026 q^{9} +1.48119 q^{10} +4.15633 q^{11} -0.156325 q^{12} -2.96239 q^{13} +0.806063 q^{15} -4.35026 q^{16} -5.50659 q^{17} +3.48119 q^{18} +3.19394 q^{19} -0.193937 q^{20} -6.15633 q^{22} +1.84367 q^{23} -2.15633 q^{24} +1.00000 q^{25} +4.38787 q^{26} +4.31265 q^{27} -1.00000 q^{29} -1.19394 q^{30} +4.80606 q^{31} +1.09332 q^{32} -3.35026 q^{33} +8.15633 q^{34} -0.455802 q^{36} -9.50659 q^{37} -4.73084 q^{38} +2.38787 q^{39} -2.67513 q^{40} +11.2750 q^{41} -0.0303172 q^{43} +0.806063 q^{44} +2.35026 q^{45} -2.73084 q^{46} -4.80606 q^{47} +3.50659 q^{48} -1.48119 q^{50} +4.43866 q^{51} -0.574515 q^{52} -1.35026 q^{53} -6.38787 q^{54} -4.15633 q^{55} -2.57452 q^{57} +1.48119 q^{58} -13.2750 q^{59} +0.156325 q^{60} -8.88717 q^{61} -7.11871 q^{62} +7.08110 q^{64} +2.96239 q^{65} +4.96239 q^{66} +5.84367 q^{67} -1.06793 q^{68} -1.48612 q^{69} -1.27504 q^{71} -6.28726 q^{72} +15.2447 q^{73} +14.0811 q^{74} -0.806063 q^{75} +0.619421 q^{76} -3.53690 q^{78} -4.93207 q^{79} +4.35026 q^{80} +3.57452 q^{81} -16.7005 q^{82} -4.41819 q^{83} +5.50659 q^{85} +0.0449056 q^{86} +0.806063 q^{87} +11.1187 q^{88} +3.61213 q^{89} -3.48119 q^{90} +0.357556 q^{92} -3.87399 q^{93} +7.11871 q^{94} -3.19394 q^{95} -0.881286 q^{96} +1.38058 q^{97} -9.76845 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + q^{2} - 2 q^{3} + q^{4} - 3 q^{5} + 4 q^{6} + 3 q^{8} + 3 q^{9}+O(q^{10})$$ 3 * q + q^2 - 2 * q^3 + q^4 - 3 * q^5 + 4 * q^6 + 3 * q^8 + 3 * q^9 $$3 q + q^{2} - 2 q^{3} + q^{4} - 3 q^{5} + 4 q^{6} + 3 q^{8} + 3 q^{9} - q^{10} + 2 q^{11} + 10 q^{12} + 2 q^{13} + 2 q^{15} - 3 q^{16} + 4 q^{17} + 5 q^{18} + 10 q^{19} - q^{20} - 8 q^{22} + 16 q^{23} + 4 q^{24} + 3 q^{25} + 14 q^{26} - 8 q^{27} - 3 q^{29} - 4 q^{30} + 14 q^{31} - 3 q^{32} + 14 q^{34} - 11 q^{36} - 8 q^{37} + 8 q^{38} + 8 q^{39} - 3 q^{40} + 2 q^{41} + 2 q^{43} + 2 q^{44} - 3 q^{45} + 14 q^{46} - 14 q^{47} - 10 q^{48} + q^{50} - 16 q^{51} + 10 q^{52} + 6 q^{53} - 20 q^{54} - 2 q^{55} + 4 q^{57} - q^{58} - 8 q^{59} - 10 q^{60} + 6 q^{61} - 11 q^{64} - 2 q^{65} + 4 q^{66} + 28 q^{67} - 12 q^{68} - 12 q^{69} + 28 q^{71} - 13 q^{72} + 16 q^{73} + 10 q^{74} - 2 q^{75} + 14 q^{76} + 12 q^{78} - 6 q^{79} + 3 q^{80} - q^{81} - 30 q^{82} - 12 q^{83} - 4 q^{85} + 24 q^{86} + 2 q^{87} + 12 q^{88} + 10 q^{89} - 5 q^{90} + 4 q^{92} - 20 q^{93} - 10 q^{95} - 24 q^{96} - 8 q^{97} - 18 q^{99}+O(q^{100})$$ 3 * q + q^2 - 2 * q^3 + q^4 - 3 * q^5 + 4 * q^6 + 3 * q^8 + 3 * q^9 - q^10 + 2 * q^11 + 10 * q^12 + 2 * q^13 + 2 * q^15 - 3 * q^16 + 4 * q^17 + 5 * q^18 + 10 * q^19 - q^20 - 8 * q^22 + 16 * q^23 + 4 * q^24 + 3 * q^25 + 14 * q^26 - 8 * q^27 - 3 * q^29 - 4 * q^30 + 14 * q^31 - 3 * q^32 + 14 * q^34 - 11 * q^36 - 8 * q^37 + 8 * q^38 + 8 * q^39 - 3 * q^40 + 2 * q^41 + 2 * q^43 + 2 * q^44 - 3 * q^45 + 14 * q^46 - 14 * q^47 - 10 * q^48 + q^50 - 16 * q^51 + 10 * q^52 + 6 * q^53 - 20 * q^54 - 2 * q^55 + 4 * q^57 - q^58 - 8 * q^59 - 10 * q^60 + 6 * q^61 - 11 * q^64 - 2 * q^65 + 4 * q^66 + 28 * q^67 - 12 * q^68 - 12 * q^69 + 28 * q^71 - 13 * q^72 + 16 * q^73 + 10 * q^74 - 2 * q^75 + 14 * q^76 + 12 * q^78 - 6 * q^79 + 3 * q^80 - q^81 - 30 * q^82 - 12 * q^83 - 4 * q^85 + 24 * q^86 + 2 * q^87 + 12 * q^88 + 10 * q^89 - 5 * q^90 + 4 * q^92 - 20 * q^93 - 10 * q^95 - 24 * q^96 - 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.48119 −1.04736 −0.523681 0.851914i $$-0.675442\pi$$
−0.523681 + 0.851914i $$0.675442\pi$$
$$3$$ −0.806063 −0.465381 −0.232690 0.972551i $$-0.574753\pi$$
−0.232690 + 0.972551i $$0.574753\pi$$
$$4$$ 0.193937 0.0969683
$$5$$ −1.00000 −0.447214
$$6$$ 1.19394 0.487423
$$7$$ 0 0
$$8$$ 2.67513 0.945802
$$9$$ −2.35026 −0.783421
$$10$$ 1.48119 0.468395
$$11$$ 4.15633 1.25318 0.626590 0.779349i $$-0.284450\pi$$
0.626590 + 0.779349i $$0.284450\pi$$
$$12$$ −0.156325 −0.0451272
$$13$$ −2.96239 −0.821619 −0.410809 0.911721i $$-0.634754\pi$$
−0.410809 + 0.911721i $$0.634754\pi$$
$$14$$ 0 0
$$15$$ 0.806063 0.208125
$$16$$ −4.35026 −1.08757
$$17$$ −5.50659 −1.33554 −0.667772 0.744366i $$-0.732752\pi$$
−0.667772 + 0.744366i $$0.732752\pi$$
$$18$$ 3.48119 0.820525
$$19$$ 3.19394 0.732739 0.366370 0.930469i $$-0.380601\pi$$
0.366370 + 0.930469i $$0.380601\pi$$
$$20$$ −0.193937 −0.0433655
$$21$$ 0 0
$$22$$ −6.15633 −1.31253
$$23$$ 1.84367 0.384433 0.192216 0.981353i $$-0.438432\pi$$
0.192216 + 0.981353i $$0.438432\pi$$
$$24$$ −2.15633 −0.440158
$$25$$ 1.00000 0.200000
$$26$$ 4.38787 0.860533
$$27$$ 4.31265 0.829970
$$28$$ 0 0
$$29$$ −1.00000 −0.185695
$$30$$ −1.19394 −0.217982
$$31$$ 4.80606 0.863194 0.431597 0.902066i $$-0.357950\pi$$
0.431597 + 0.902066i $$0.357950\pi$$
$$32$$ 1.09332 0.193274
$$33$$ −3.35026 −0.583206
$$34$$ 8.15633 1.39880
$$35$$ 0 0
$$36$$ −0.455802 −0.0759669
$$37$$ −9.50659 −1.56287 −0.781437 0.623985i $$-0.785513\pi$$
−0.781437 + 0.623985i $$0.785513\pi$$
$$38$$ −4.73084 −0.767444
$$39$$ 2.38787 0.382366
$$40$$ −2.67513 −0.422975
$$41$$ 11.2750 1.76087 0.880433 0.474171i $$-0.157252\pi$$
0.880433 + 0.474171i $$0.157252\pi$$
$$42$$ 0 0
$$43$$ −0.0303172 −0.00462332 −0.00231166 0.999997i $$-0.500736\pi$$
−0.00231166 + 0.999997i $$0.500736\pi$$
$$44$$ 0.806063 0.121519
$$45$$ 2.35026 0.350356
$$46$$ −2.73084 −0.402640
$$47$$ −4.80606 −0.701036 −0.350518 0.936556i $$-0.613995\pi$$
−0.350518 + 0.936556i $$0.613995\pi$$
$$48$$ 3.50659 0.506132
$$49$$ 0 0
$$50$$ −1.48119 −0.209473
$$51$$ 4.43866 0.621536
$$52$$ −0.574515 −0.0796710
$$53$$ −1.35026 −0.185473 −0.0927364 0.995691i $$-0.529561\pi$$
−0.0927364 + 0.995691i $$0.529561\pi$$
$$54$$ −6.38787 −0.869279
$$55$$ −4.15633 −0.560439
$$56$$ 0 0
$$57$$ −2.57452 −0.341003
$$58$$ 1.48119 0.194490
$$59$$ −13.2750 −1.72826 −0.864131 0.503266i $$-0.832132\pi$$
−0.864131 + 0.503266i $$0.832132\pi$$
$$60$$ 0.156325 0.0201815
$$61$$ −8.88717 −1.13788 −0.568942 0.822377i $$-0.692647\pi$$
−0.568942 + 0.822377i $$0.692647\pi$$
$$62$$ −7.11871 −0.904078
$$63$$ 0 0
$$64$$ 7.08110 0.885138
$$65$$ 2.96239 0.367439
$$66$$ 4.96239 0.610828
$$67$$ 5.84367 0.713919 0.356959 0.934120i $$-0.383813\pi$$
0.356959 + 0.934120i $$0.383813\pi$$
$$68$$ −1.06793 −0.129505
$$69$$ −1.48612 −0.178908
$$70$$ 0 0
$$71$$ −1.27504 −0.151319 −0.0756596 0.997134i $$-0.524106\pi$$
−0.0756596 + 0.997134i $$0.524106\pi$$
$$72$$ −6.28726 −0.740960
$$73$$ 15.2447 1.78426 0.892130 0.451779i $$-0.149210\pi$$
0.892130 + 0.451779i $$0.149210\pi$$
$$74$$ 14.0811 1.63689
$$75$$ −0.806063 −0.0930762
$$76$$ 0.619421 0.0710525
$$77$$ 0 0
$$78$$ −3.53690 −0.400476
$$79$$ −4.93207 −0.554901 −0.277451 0.960740i $$-0.589490\pi$$
−0.277451 + 0.960740i $$0.589490\pi$$
$$80$$ 4.35026 0.486374
$$81$$ 3.57452 0.397168
$$82$$ −16.7005 −1.84426
$$83$$ −4.41819 −0.484959 −0.242480 0.970156i $$-0.577961\pi$$
−0.242480 + 0.970156i $$0.577961\pi$$
$$84$$ 0 0
$$85$$ 5.50659 0.597273
$$86$$ 0.0449056 0.00484230
$$87$$ 0.806063 0.0864191
$$88$$ 11.1187 1.18526
$$89$$ 3.61213 0.382885 0.191442 0.981504i $$-0.438684\pi$$
0.191442 + 0.981504i $$0.438684\pi$$
$$90$$ −3.48119 −0.366950
$$91$$ 0 0
$$92$$ 0.357556 0.0372778
$$93$$ −3.87399 −0.401714
$$94$$ 7.11871 0.734239
$$95$$ −3.19394 −0.327691
$$96$$ −0.881286 −0.0899459
$$97$$ 1.38058 0.140177 0.0700883 0.997541i $$-0.477672\pi$$
0.0700883 + 0.997541i $$0.477672\pi$$
$$98$$ 0 0
$$99$$ −9.76845 −0.981766
$$100$$ 0.193937 0.0193937
$$101$$ 13.0132 1.29486 0.647430 0.762125i $$-0.275844\pi$$
0.647430 + 0.762125i $$0.275844\pi$$
$$102$$ −6.57452 −0.650974
$$103$$ −5.31994 −0.524190 −0.262095 0.965042i $$-0.584413\pi$$
−0.262095 + 0.965042i $$0.584413\pi$$
$$104$$ −7.92478 −0.777088
$$105$$ 0 0
$$106$$ 2.00000 0.194257
$$107$$ −13.8192 −1.33596 −0.667978 0.744181i $$-0.732840\pi$$
−0.667978 + 0.744181i $$0.732840\pi$$
$$108$$ 0.836381 0.0804808
$$109$$ −1.87399 −0.179496 −0.0897479 0.995965i $$-0.528606\pi$$
−0.0897479 + 0.995965i $$0.528606\pi$$
$$110$$ 6.15633 0.586983
$$111$$ 7.66291 0.727331
$$112$$ 0 0
$$113$$ −11.7685 −1.10708 −0.553541 0.832822i $$-0.686724\pi$$
−0.553541 + 0.832822i $$0.686724\pi$$
$$114$$ 3.81336 0.357154
$$115$$ −1.84367 −0.171924
$$116$$ −0.193937 −0.0180066
$$117$$ 6.96239 0.643673
$$118$$ 19.6629 1.81012
$$119$$ 0 0
$$120$$ 2.15633 0.196845
$$121$$ 6.27504 0.570458
$$122$$ 13.1636 1.19178
$$123$$ −9.08840 −0.819473
$$124$$ 0.932071 0.0837025
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ 14.2677 1.26606 0.633029 0.774128i $$-0.281811\pi$$
0.633029 + 0.774128i $$0.281811\pi$$
$$128$$ −12.6751 −1.12033
$$129$$ 0.0244376 0.00215161
$$130$$ −4.38787 −0.384842
$$131$$ −5.89446 −0.515001 −0.257501 0.966278i $$-0.582899\pi$$
−0.257501 + 0.966278i $$0.582899\pi$$
$$132$$ −0.649738 −0.0565525
$$133$$ 0 0
$$134$$ −8.65562 −0.747731
$$135$$ −4.31265 −0.371174
$$136$$ −14.7308 −1.26316
$$137$$ 18.2823 1.56197 0.780983 0.624553i $$-0.214719\pi$$
0.780983 + 0.624553i $$0.214719\pi$$
$$138$$ 2.20123 0.187381
$$139$$ 11.5369 0.978547 0.489274 0.872130i $$-0.337262\pi$$
0.489274 + 0.872130i $$0.337262\pi$$
$$140$$ 0 0
$$141$$ 3.87399 0.326249
$$142$$ 1.88858 0.158486
$$143$$ −12.3127 −1.02964
$$144$$ 10.2243 0.852021
$$145$$ 1.00000 0.0830455
$$146$$ −22.5804 −1.86877
$$147$$ 0 0
$$148$$ −1.84367 −0.151549
$$149$$ 2.77575 0.227398 0.113699 0.993515i $$-0.463730\pi$$
0.113699 + 0.993515i $$0.463730\pi$$
$$150$$ 1.19394 0.0974845
$$151$$ 1.79877 0.146382 0.0731909 0.997318i $$-0.476682\pi$$
0.0731909 + 0.997318i $$0.476682\pi$$
$$152$$ 8.54420 0.693026
$$153$$ 12.9419 1.04629
$$154$$ 0 0
$$155$$ −4.80606 −0.386032
$$156$$ 0.463096 0.0370773
$$157$$ −3.76845 −0.300755 −0.150378 0.988629i $$-0.548049\pi$$
−0.150378 + 0.988629i $$0.548049\pi$$
$$158$$ 7.30536 0.581183
$$159$$ 1.08840 0.0863155
$$160$$ −1.09332 −0.0864346
$$161$$ 0 0
$$162$$ −5.29455 −0.415979
$$163$$ 1.64244 0.128646 0.0643231 0.997929i $$-0.479511\pi$$
0.0643231 + 0.997929i $$0.479511\pi$$
$$164$$ 2.18664 0.170748
$$165$$ 3.35026 0.260818
$$166$$ 6.54420 0.507928
$$167$$ −8.08110 −0.625334 −0.312667 0.949863i $$-0.601222\pi$$
−0.312667 + 0.949863i $$0.601222\pi$$
$$168$$ 0 0
$$169$$ −4.22425 −0.324943
$$170$$ −8.15633 −0.625562
$$171$$ −7.50659 −0.574043
$$172$$ −0.00587961 −0.000448316 0
$$173$$ 7.73813 0.588320 0.294160 0.955756i $$-0.404960\pi$$
0.294160 + 0.955756i $$0.404960\pi$$
$$174$$ −1.19394 −0.0905121
$$175$$ 0 0
$$176$$ −18.0811 −1.36291
$$177$$ 10.7005 0.804301
$$178$$ −5.35026 −0.401019
$$179$$ −21.4010 −1.59959 −0.799795 0.600274i $$-0.795058\pi$$
−0.799795 + 0.600274i $$0.795058\pi$$
$$180$$ 0.455802 0.0339735
$$181$$ −15.2750 −1.13538 −0.567692 0.823241i $$-0.692164\pi$$
−0.567692 + 0.823241i $$0.692164\pi$$
$$182$$ 0 0
$$183$$ 7.16362 0.529550
$$184$$ 4.93207 0.363597
$$185$$ 9.50659 0.698938
$$186$$ 5.73813 0.420740
$$187$$ −22.8872 −1.67368
$$188$$ −0.932071 −0.0679783
$$189$$ 0 0
$$190$$ 4.73084 0.343211
$$191$$ 3.31994 0.240223 0.120111 0.992760i $$-0.461675\pi$$
0.120111 + 0.992760i $$0.461675\pi$$
$$192$$ −5.70782 −0.411926
$$193$$ −4.88129 −0.351363 −0.175681 0.984447i $$-0.556213\pi$$
−0.175681 + 0.984447i $$0.556213\pi$$
$$194$$ −2.04491 −0.146816
$$195$$ −2.38787 −0.170999
$$196$$ 0 0
$$197$$ −24.2374 −1.72685 −0.863423 0.504481i $$-0.831684\pi$$
−0.863423 + 0.504481i $$0.831684\pi$$
$$198$$ 14.4690 1.02827
$$199$$ −16.7513 −1.18747 −0.593734 0.804661i $$-0.702347\pi$$
−0.593734 + 0.804661i $$0.702347\pi$$
$$200$$ 2.67513 0.189160
$$201$$ −4.71037 −0.332244
$$202$$ −19.2750 −1.35619
$$203$$ 0 0
$$204$$ 0.860818 0.0602693
$$205$$ −11.2750 −0.787483
$$206$$ 7.87987 0.549017
$$207$$ −4.33312 −0.301173
$$208$$ 12.8872 0.893564
$$209$$ 13.2750 0.918254
$$210$$ 0 0
$$211$$ −25.3054 −1.74209 −0.871046 0.491201i $$-0.836558\pi$$
−0.871046 + 0.491201i $$0.836558\pi$$
$$212$$ −0.261865 −0.0179850
$$213$$ 1.02776 0.0704211
$$214$$ 20.4690 1.39923
$$215$$ 0.0303172 0.00206761
$$216$$ 11.5369 0.784987
$$217$$ 0 0
$$218$$ 2.77575 0.187997
$$219$$ −12.2882 −0.830360
$$220$$ −0.806063 −0.0543448
$$221$$ 16.3127 1.09731
$$222$$ −11.3503 −0.761780
$$223$$ −17.6932 −1.18483 −0.592413 0.805634i $$-0.701825\pi$$
−0.592413 + 0.805634i $$0.701825\pi$$
$$224$$ 0 0
$$225$$ −2.35026 −0.156684
$$226$$ 17.4314 1.15952
$$227$$ −26.8423 −1.78158 −0.890792 0.454412i $$-0.849849\pi$$
−0.890792 + 0.454412i $$0.849849\pi$$
$$228$$ −0.499293 −0.0330665
$$229$$ 17.2243 1.13821 0.569105 0.822265i $$-0.307290\pi$$
0.569105 + 0.822265i $$0.307290\pi$$
$$230$$ 2.73084 0.180066
$$231$$ 0 0
$$232$$ −2.67513 −0.175631
$$233$$ −9.07381 −0.594445 −0.297222 0.954808i $$-0.596060\pi$$
−0.297222 + 0.954808i $$0.596060\pi$$
$$234$$ −10.3127 −0.674159
$$235$$ 4.80606 0.313513
$$236$$ −2.57452 −0.167587
$$237$$ 3.97556 0.258241
$$238$$ 0 0
$$239$$ 20.4993 1.32599 0.662995 0.748624i $$-0.269285\pi$$
0.662995 + 0.748624i $$0.269285\pi$$
$$240$$ −3.50659 −0.226349
$$241$$ −5.47627 −0.352758 −0.176379 0.984322i $$-0.556438\pi$$
−0.176379 + 0.984322i $$0.556438\pi$$
$$242$$ −9.29455 −0.597476
$$243$$ −15.8192 −1.01480
$$244$$ −1.72355 −0.110339
$$245$$ 0 0
$$246$$ 13.4617 0.858285
$$247$$ −9.46168 −0.602032
$$248$$ 12.8568 0.816411
$$249$$ 3.56134 0.225691
$$250$$ 1.48119 0.0936790
$$251$$ 29.6180 1.86947 0.934736 0.355343i $$-0.115636\pi$$
0.934736 + 0.355343i $$0.115636\pi$$
$$252$$ 0 0
$$253$$ 7.66291 0.481763
$$254$$ −21.1333 −1.32602
$$255$$ −4.43866 −0.277960
$$256$$ 4.61213 0.288258
$$257$$ −17.6629 −1.10178 −0.550891 0.834577i $$-0.685712\pi$$
−0.550891 + 0.834577i $$0.685712\pi$$
$$258$$ −0.0361968 −0.00225351
$$259$$ 0 0
$$260$$ 0.574515 0.0356299
$$261$$ 2.35026 0.145478
$$262$$ 8.73084 0.539393
$$263$$ 27.3561 1.68685 0.843426 0.537245i $$-0.180535\pi$$
0.843426 + 0.537245i $$0.180535\pi$$
$$264$$ −8.96239 −0.551597
$$265$$ 1.35026 0.0829459
$$266$$ 0 0
$$267$$ −2.91160 −0.178187
$$268$$ 1.13330 0.0692275
$$269$$ −10.4993 −0.640153 −0.320077 0.947392i $$-0.603709\pi$$
−0.320077 + 0.947392i $$0.603709\pi$$
$$270$$ 6.38787 0.388754
$$271$$ 9.61801 0.584252 0.292126 0.956380i $$-0.405637\pi$$
0.292126 + 0.956380i $$0.405637\pi$$
$$272$$ 23.9551 1.45249
$$273$$ 0 0
$$274$$ −27.0797 −1.63594
$$275$$ 4.15633 0.250636
$$276$$ −0.288213 −0.0173484
$$277$$ 13.3503 0.802139 0.401070 0.916048i $$-0.368638\pi$$
0.401070 + 0.916048i $$0.368638\pi$$
$$278$$ −17.0884 −1.02489
$$279$$ −11.2955 −0.676244
$$280$$ 0 0
$$281$$ 20.4241 1.21840 0.609199 0.793017i $$-0.291491\pi$$
0.609199 + 0.793017i $$0.291491\pi$$
$$282$$ −5.73813 −0.341701
$$283$$ 8.02047 0.476767 0.238384 0.971171i $$-0.423382\pi$$
0.238384 + 0.971171i $$0.423382\pi$$
$$284$$ −0.247277 −0.0146732
$$285$$ 2.57452 0.152501
$$286$$ 18.2374 1.07840
$$287$$ 0 0
$$288$$ −2.56959 −0.151415
$$289$$ 13.3225 0.783676
$$290$$ −1.48119 −0.0869787
$$291$$ −1.11283 −0.0652355
$$292$$ 2.95651 0.173017
$$293$$ 23.3054 1.36151 0.680757 0.732510i $$-0.261651\pi$$
0.680757 + 0.732510i $$0.261651\pi$$
$$294$$ 0 0
$$295$$ 13.2750 0.772903
$$296$$ −25.4314 −1.47817
$$297$$ 17.9248 1.04010
$$298$$ −4.11142 −0.238168
$$299$$ −5.46168 −0.315857
$$300$$ −0.156325 −0.00902544
$$301$$ 0 0
$$302$$ −2.66433 −0.153315
$$303$$ −10.4894 −0.602603
$$304$$ −13.8945 −0.796902
$$305$$ 8.88717 0.508878
$$306$$ −19.1695 −1.09585
$$307$$ 6.73084 0.384149 0.192075 0.981380i $$-0.438478\pi$$
0.192075 + 0.981380i $$0.438478\pi$$
$$308$$ 0 0
$$309$$ 4.28821 0.243948
$$310$$ 7.11871 0.404316
$$311$$ 22.0567 1.25072 0.625359 0.780337i $$-0.284952\pi$$
0.625359 + 0.780337i $$0.284952\pi$$
$$312$$ 6.38787 0.361642
$$313$$ −5.03761 −0.284743 −0.142371 0.989813i $$-0.545473\pi$$
−0.142371 + 0.989813i $$0.545473\pi$$
$$314$$ 5.58181 0.315000
$$315$$ 0 0
$$316$$ −0.956509 −0.0538078
$$317$$ 34.2941 1.92615 0.963074 0.269237i $$-0.0867714\pi$$
0.963074 + 0.269237i $$0.0867714\pi$$
$$318$$ −1.61213 −0.0904036
$$319$$ −4.15633 −0.232710
$$320$$ −7.08110 −0.395846
$$321$$ 11.1392 0.621729
$$322$$ 0 0
$$323$$ −17.5877 −0.978605
$$324$$ 0.693229 0.0385127
$$325$$ −2.96239 −0.164324
$$326$$ −2.43278 −0.134739
$$327$$ 1.51056 0.0835340
$$328$$ 30.1622 1.66543
$$329$$ 0 0
$$330$$ −4.96239 −0.273171
$$331$$ −34.8324 −1.91456 −0.957281 0.289159i $$-0.906625\pi$$
−0.957281 + 0.289159i $$0.906625\pi$$
$$332$$ −0.856849 −0.0470257
$$333$$ 22.3430 1.22439
$$334$$ 11.9697 0.654952
$$335$$ −5.84367 −0.319274
$$336$$ 0 0
$$337$$ −17.6326 −0.960509 −0.480254 0.877129i $$-0.659456\pi$$
−0.480254 + 0.877129i $$0.659456\pi$$
$$338$$ 6.25694 0.340333
$$339$$ 9.48612 0.515215
$$340$$ 1.06793 0.0579166
$$341$$ 19.9756 1.08174
$$342$$ 11.1187 0.601231
$$343$$ 0 0
$$344$$ −0.0811024 −0.00437275
$$345$$ 1.48612 0.0800099
$$346$$ −11.4617 −0.616184
$$347$$ −3.11871 −0.167421 −0.0837107 0.996490i $$-0.526677\pi$$
−0.0837107 + 0.996490i $$0.526677\pi$$
$$348$$ 0.156325 0.00837991
$$349$$ 13.0738 0.699825 0.349912 0.936782i $$-0.386211\pi$$
0.349912 + 0.936782i $$0.386211\pi$$
$$350$$ 0 0
$$351$$ −12.7757 −0.681919
$$352$$ 4.54420 0.242207
$$353$$ −5.19982 −0.276758 −0.138379 0.990379i $$-0.544189\pi$$
−0.138379 + 0.990379i $$0.544189\pi$$
$$354$$ −15.8496 −0.842394
$$355$$ 1.27504 0.0676720
$$356$$ 0.700523 0.0371277
$$357$$ 0 0
$$358$$ 31.6991 1.67535
$$359$$ 30.4182 1.60541 0.802705 0.596376i $$-0.203393\pi$$
0.802705 + 0.596376i $$0.203393\pi$$
$$360$$ 6.28726 0.331368
$$361$$ −8.79877 −0.463093
$$362$$ 22.6253 1.18916
$$363$$ −5.05808 −0.265480
$$364$$ 0 0
$$365$$ −15.2447 −0.797945
$$366$$ −10.6107 −0.554631
$$367$$ −20.6556 −1.07821 −0.539107 0.842237i $$-0.681238\pi$$
−0.539107 + 0.842237i $$0.681238\pi$$
$$368$$ −8.02047 −0.418096
$$369$$ −26.4993 −1.37950
$$370$$ −14.0811 −0.732042
$$371$$ 0 0
$$372$$ −0.751309 −0.0389535
$$373$$ 11.0884 0.574135 0.287068 0.957910i $$-0.407320\pi$$
0.287068 + 0.957910i $$0.407320\pi$$
$$374$$ 33.9003 1.75294
$$375$$ 0.806063 0.0416249
$$376$$ −12.8568 −0.663041
$$377$$ 2.96239 0.152571
$$378$$ 0 0
$$379$$ 10.0811 0.517831 0.258916 0.965900i $$-0.416635\pi$$
0.258916 + 0.965900i $$0.416635\pi$$
$$380$$ −0.619421 −0.0317756
$$381$$ −11.5007 −0.589199
$$382$$ −4.91748 −0.251600
$$383$$ −16.3576 −0.835832 −0.417916 0.908486i $$-0.637239\pi$$
−0.417916 + 0.908486i $$0.637239\pi$$
$$384$$ 10.2170 0.521382
$$385$$ 0 0
$$386$$ 7.23013 0.368004
$$387$$ 0.0712533 0.00362201
$$388$$ 0.267745 0.0135927
$$389$$ 31.9003 1.61741 0.808706 0.588213i $$-0.200169\pi$$
0.808706 + 0.588213i $$0.200169\pi$$
$$390$$ 3.53690 0.179098
$$391$$ −10.1524 −0.513427
$$392$$ 0 0
$$393$$ 4.75131 0.239672
$$394$$ 35.9003 1.80863
$$395$$ 4.93207 0.248159
$$396$$ −1.89446 −0.0952002
$$397$$ −2.98683 −0.149905 −0.0749523 0.997187i $$-0.523880\pi$$
−0.0749523 + 0.997187i $$0.523880\pi$$
$$398$$ 24.8119 1.24371
$$399$$ 0 0
$$400$$ −4.35026 −0.217513
$$401$$ −21.9756 −1.09741 −0.548704 0.836017i $$-0.684878\pi$$
−0.548704 + 0.836017i $$0.684878\pi$$
$$402$$ 6.97698 0.347980
$$403$$ −14.2374 −0.709217
$$404$$ 2.52373 0.125560
$$405$$ −3.57452 −0.177619
$$406$$ 0 0
$$407$$ −39.5125 −1.95856
$$408$$ 11.8740 0.587850
$$409$$ 22.4387 1.10952 0.554760 0.832010i $$-0.312810\pi$$
0.554760 + 0.832010i $$0.312810\pi$$
$$410$$ 16.7005 0.824780
$$411$$ −14.7367 −0.726909
$$412$$ −1.03173 −0.0508298
$$413$$ 0 0
$$414$$ 6.41819 0.315437
$$415$$ 4.41819 0.216880
$$416$$ −3.23884 −0.158797
$$417$$ −9.29948 −0.455397
$$418$$ −19.6629 −0.961744
$$419$$ −10.3634 −0.506287 −0.253143 0.967429i $$-0.581464\pi$$
−0.253143 + 0.967429i $$0.581464\pi$$
$$420$$ 0 0
$$421$$ 34.0362 1.65882 0.829411 0.558638i $$-0.188676\pi$$
0.829411 + 0.558638i $$0.188676\pi$$
$$422$$ 37.4821 1.82460
$$423$$ 11.2955 0.549206
$$424$$ −3.61213 −0.175420
$$425$$ −5.50659 −0.267109
$$426$$ −1.52232 −0.0737564
$$427$$ 0 0
$$428$$ −2.68006 −0.129545
$$429$$ 9.92478 0.479173
$$430$$ −0.0449056 −0.00216554
$$431$$ 25.7743 1.24151 0.620753 0.784006i $$-0.286827\pi$$
0.620753 + 0.784006i $$0.286827\pi$$
$$432$$ −18.7612 −0.902647
$$433$$ −2.18076 −0.104801 −0.0524004 0.998626i $$-0.516687\pi$$
−0.0524004 + 0.998626i $$0.516687\pi$$
$$434$$ 0 0
$$435$$ −0.806063 −0.0386478
$$436$$ −0.363436 −0.0174054
$$437$$ 5.88858 0.281689
$$438$$ 18.2012 0.869688
$$439$$ 35.5125 1.69492 0.847459 0.530861i $$-0.178131\pi$$
0.847459 + 0.530861i $$0.178131\pi$$
$$440$$ −11.1187 −0.530064
$$441$$ 0 0
$$442$$ −24.1622 −1.14928
$$443$$ −4.34297 −0.206341 −0.103170 0.994664i $$-0.532899\pi$$
−0.103170 + 0.994664i $$0.532899\pi$$
$$444$$ 1.48612 0.0705281
$$445$$ −3.61213 −0.171231
$$446$$ 26.2071 1.24094
$$447$$ −2.23743 −0.105827
$$448$$ 0 0
$$449$$ 31.3357 1.47882 0.739411 0.673254i $$-0.235104\pi$$
0.739411 + 0.673254i $$0.235104\pi$$
$$450$$ 3.48119 0.164105
$$451$$ 46.8627 2.20668
$$452$$ −2.28233 −0.107352
$$453$$ −1.44992 −0.0681233
$$454$$ 39.7586 1.86596
$$455$$ 0 0
$$456$$ −6.88717 −0.322521
$$457$$ 34.3488 1.60677 0.803386 0.595459i $$-0.203030\pi$$
0.803386 + 0.595459i $$0.203030\pi$$
$$458$$ −25.5125 −1.19212
$$459$$ −23.7480 −1.10846
$$460$$ −0.357556 −0.0166711
$$461$$ −11.8641 −0.552568 −0.276284 0.961076i $$-0.589103\pi$$
−0.276284 + 0.961076i $$0.589103\pi$$
$$462$$ 0 0
$$463$$ 40.4953 1.88198 0.940989 0.338438i $$-0.109899\pi$$
0.940989 + 0.338438i $$0.109899\pi$$
$$464$$ 4.35026 0.201956
$$465$$ 3.87399 0.179652
$$466$$ 13.4401 0.622599
$$467$$ 30.2071 1.39782 0.698909 0.715210i $$-0.253669\pi$$
0.698909 + 0.715210i $$0.253669\pi$$
$$468$$ 1.35026 0.0624159
$$469$$ 0 0
$$470$$ −7.11871 −0.328362
$$471$$ 3.03761 0.139966
$$472$$ −35.5125 −1.63459
$$473$$ −0.126008 −0.00579385
$$474$$ −5.88858 −0.270471
$$475$$ 3.19394 0.146548
$$476$$ 0 0
$$477$$ 3.17347 0.145303
$$478$$ −30.3634 −1.38879
$$479$$ −0.0547547 −0.00250181 −0.00125090 0.999999i $$-0.500398\pi$$
−0.00125090 + 0.999999i $$0.500398\pi$$
$$480$$ 0.881286 0.0402250
$$481$$ 28.1622 1.28409
$$482$$ 8.11142 0.369465
$$483$$ 0 0
$$484$$ 1.21696 0.0553163
$$485$$ −1.38058 −0.0626889
$$486$$ 23.4314 1.06287
$$487$$ −0.881286 −0.0399349 −0.0199674 0.999801i $$-0.506356\pi$$
−0.0199674 + 0.999801i $$0.506356\pi$$
$$488$$ −23.7743 −1.07621
$$489$$ −1.32391 −0.0598695
$$490$$ 0 0
$$491$$ 41.0698 1.85346 0.926728 0.375733i $$-0.122609\pi$$
0.926728 + 0.375733i $$0.122609\pi$$
$$492$$ −1.76257 −0.0794629
$$493$$ 5.50659 0.248004
$$494$$ 14.0146 0.630546
$$495$$ 9.76845 0.439059
$$496$$ −20.9076 −0.938780
$$497$$ 0 0
$$498$$ −5.27504 −0.236380
$$499$$ 12.3733 0.553904 0.276952 0.960884i $$-0.410676\pi$$
0.276952 + 0.960884i $$0.410676\pi$$
$$500$$ −0.193937 −0.00867311
$$501$$ 6.51388 0.291019
$$502$$ −43.8700 −1.95801
$$503$$ 2.26774 0.101114 0.0505569 0.998721i $$-0.483900\pi$$
0.0505569 + 0.998721i $$0.483900\pi$$
$$504$$ 0 0
$$505$$ −13.0132 −0.579079
$$506$$ −11.3503 −0.504581
$$507$$ 3.40502 0.151222
$$508$$ 2.76704 0.122767
$$509$$ 10.9018 0.483212 0.241606 0.970374i $$-0.422326\pi$$
0.241606 + 0.970374i $$0.422326\pi$$
$$510$$ 6.57452 0.291124
$$511$$ 0 0
$$512$$ 18.5188 0.818423
$$513$$ 13.7743 0.608152
$$514$$ 26.1622 1.15397
$$515$$ 5.31994 0.234425
$$516$$ 0.00473934 0.000208638 0
$$517$$ −19.9756 −0.878524
$$518$$ 0 0
$$519$$ −6.23743 −0.273793
$$520$$ 7.92478 0.347524
$$521$$ 4.72496 0.207004 0.103502 0.994629i $$-0.466995\pi$$
0.103502 + 0.994629i $$0.466995\pi$$
$$522$$ −3.48119 −0.152368
$$523$$ 1.06793 0.0466973 0.0233486 0.999727i $$-0.492567\pi$$
0.0233486 + 0.999727i $$0.492567\pi$$
$$524$$ −1.14315 −0.0499388
$$525$$ 0 0
$$526$$ −40.5198 −1.76675
$$527$$ −26.4650 −1.15283
$$528$$ 14.5745 0.634274
$$529$$ −19.6009 −0.852211
$$530$$ −2.00000 −0.0868744
$$531$$ 31.1998 1.35396
$$532$$ 0 0
$$533$$ −33.4010 −1.44676
$$534$$ 4.31265 0.186627
$$535$$ 13.8192 0.597458
$$536$$ 15.6326 0.675225
$$537$$ 17.2506 0.744418
$$538$$ 15.5515 0.670472
$$539$$ 0 0
$$540$$ −0.836381 −0.0359921
$$541$$ −7.46168 −0.320803 −0.160401 0.987052i $$-0.551279\pi$$
−0.160401 + 0.987052i $$0.551279\pi$$
$$542$$ −14.2461 −0.611924
$$543$$ 12.3127 0.528386
$$544$$ −6.02047 −0.258125
$$545$$ 1.87399 0.0802730
$$546$$ 0 0
$$547$$ −38.9683 −1.66616 −0.833081 0.553150i $$-0.813425\pi$$
−0.833081 + 0.553150i $$0.813425\pi$$
$$548$$ 3.54561 0.151461
$$549$$ 20.8872 0.891443
$$550$$ −6.15633 −0.262507
$$551$$ −3.19394 −0.136066
$$552$$ −3.97556 −0.169211
$$553$$ 0 0
$$554$$ −19.7743 −0.840131
$$555$$ −7.66291 −0.325273
$$556$$ 2.23743 0.0948881
$$557$$ −22.9986 −0.974481 −0.487241 0.873268i $$-0.661997\pi$$
−0.487241 + 0.873268i $$0.661997\pi$$
$$558$$ 16.7308 0.708273
$$559$$ 0.0898112 0.00379861
$$560$$ 0 0
$$561$$ 18.4485 0.778897
$$562$$ −30.2520 −1.27610
$$563$$ −11.6688 −0.491781 −0.245890 0.969298i $$-0.579080\pi$$
−0.245890 + 0.969298i $$0.579080\pi$$
$$564$$ 0.751309 0.0316358
$$565$$ 11.7685 0.495102
$$566$$ −11.8799 −0.499348
$$567$$ 0 0
$$568$$ −3.41090 −0.143118
$$569$$ 11.3357 0.475216 0.237608 0.971361i $$-0.423637\pi$$
0.237608 + 0.971361i $$0.423637\pi$$
$$570$$ −3.81336 −0.159724
$$571$$ 27.1754 1.13725 0.568627 0.822595i $$-0.307475\pi$$
0.568627 + 0.822595i $$0.307475\pi$$
$$572$$ −2.38787 −0.0998420
$$573$$ −2.67609 −0.111795
$$574$$ 0 0
$$575$$ 1.84367 0.0768866
$$576$$ −16.6424 −0.693435
$$577$$ −22.5950 −0.940641 −0.470321 0.882496i $$-0.655862\pi$$
−0.470321 + 0.882496i $$0.655862\pi$$
$$578$$ −19.7332 −0.820793
$$579$$ 3.93463 0.163517
$$580$$ 0.193937 0.00805278
$$581$$ 0 0
$$582$$ 1.64832 0.0683252
$$583$$ −5.61213 −0.232431
$$584$$ 40.7816 1.68756
$$585$$ −6.96239 −0.287859
$$586$$ −34.5198 −1.42600
$$587$$ −9.31994 −0.384675 −0.192338 0.981329i $$-0.561607\pi$$
−0.192338 + 0.981329i $$0.561607\pi$$
$$588$$ 0 0
$$589$$ 15.3503 0.632497
$$590$$ −19.6629 −0.809509
$$591$$ 19.5369 0.803641
$$592$$ 41.3561 1.69973
$$593$$ 15.1246 0.621093 0.310546 0.950558i $$-0.399488\pi$$
0.310546 + 0.950558i $$0.399488\pi$$
$$594$$ −26.5501 −1.08936
$$595$$ 0 0
$$596$$ 0.538319 0.0220504
$$597$$ 13.5026 0.552625
$$598$$ 8.08981 0.330817
$$599$$ 4.09569 0.167345 0.0836727 0.996493i $$-0.473335\pi$$
0.0836727 + 0.996493i $$0.473335\pi$$
$$600$$ −2.15633 −0.0880316
$$601$$ −22.2276 −0.906682 −0.453341 0.891337i $$-0.649768\pi$$
−0.453341 + 0.891337i $$0.649768\pi$$
$$602$$ 0 0
$$603$$ −13.7342 −0.559298
$$604$$ 0.348847 0.0141944
$$605$$ −6.27504 −0.255117
$$606$$ 15.5369 0.631144
$$607$$ 48.2941 1.96020 0.980098 0.198512i $$-0.0636110\pi$$
0.980098 + 0.198512i $$0.0636110\pi$$
$$608$$ 3.49200 0.141619
$$609$$ 0 0
$$610$$ −13.1636 −0.532979
$$611$$ 14.2374 0.575985
$$612$$ 2.50991 0.101457
$$613$$ −9.74798 −0.393717 −0.196859 0.980432i $$-0.563074\pi$$
−0.196859 + 0.980432i $$0.563074\pi$$
$$614$$ −9.96968 −0.402344
$$615$$ 9.08840 0.366480
$$616$$ 0 0
$$617$$ 18.2170 0.733387 0.366694 0.930342i $$-0.380490\pi$$
0.366694 + 0.930342i $$0.380490\pi$$
$$618$$ −6.35168 −0.255502
$$619$$ −25.0943 −1.00862 −0.504312 0.863521i $$-0.668254\pi$$
−0.504312 + 0.863521i $$0.668254\pi$$
$$620$$ −0.932071 −0.0374329
$$621$$ 7.95112 0.319068
$$622$$ −32.6702 −1.30996
$$623$$ 0 0
$$624$$ −10.3879 −0.415848
$$625$$ 1.00000 0.0400000
$$626$$ 7.46168 0.298229
$$627$$ −10.7005 −0.427338
$$628$$ −0.730841 −0.0291637
$$629$$ 52.3488 2.08729
$$630$$ 0 0
$$631$$ 21.4617 0.854376 0.427188 0.904163i $$-0.359504\pi$$
0.427188 + 0.904163i $$0.359504\pi$$
$$632$$ −13.1939 −0.524827
$$633$$ 20.3977 0.810737
$$634$$ −50.7962 −2.01738
$$635$$ −14.2677 −0.566198
$$636$$ 0.211080 0.00836986
$$637$$ 0 0
$$638$$ 6.15633 0.243731
$$639$$ 2.99668 0.118547
$$640$$ 12.6751 0.501029
$$641$$ 3.17347 0.125344 0.0626722 0.998034i $$-0.480038\pi$$
0.0626722 + 0.998034i $$0.480038\pi$$
$$642$$ −16.4993 −0.651175
$$643$$ 2.74069 0.108082 0.0540411 0.998539i $$-0.482790\pi$$
0.0540411 + 0.998539i $$0.482790\pi$$
$$644$$ 0 0
$$645$$ −0.0244376 −0.000962228 0
$$646$$ 26.0508 1.02495
$$647$$ −6.34297 −0.249368 −0.124684 0.992197i $$-0.539792\pi$$
−0.124684 + 0.992197i $$0.539792\pi$$
$$648$$ 9.56230 0.375642
$$649$$ −55.1754 −2.16582
$$650$$ 4.38787 0.172107
$$651$$ 0 0
$$652$$ 0.318530 0.0124746
$$653$$ 4.08110 0.159706 0.0798529 0.996807i $$-0.474555\pi$$
0.0798529 + 0.996807i $$0.474555\pi$$
$$654$$ −2.23743 −0.0874903
$$655$$ 5.89446 0.230316
$$656$$ −49.0494 −1.91506
$$657$$ −35.8291 −1.39783
$$658$$ 0 0
$$659$$ 9.58181 0.373254 0.186627 0.982431i $$-0.440244\pi$$
0.186627 + 0.982431i $$0.440244\pi$$
$$660$$ 0.649738 0.0252910
$$661$$ 27.5271 1.07068 0.535339 0.844637i $$-0.320184\pi$$
0.535339 + 0.844637i $$0.320184\pi$$
$$662$$ 51.5936 2.00524
$$663$$ −13.1490 −0.510666
$$664$$ −11.8192 −0.458675
$$665$$ 0 0
$$666$$ −33.0943 −1.28238
$$667$$ −1.84367 −0.0713874
$$668$$ −1.56722 −0.0606376
$$669$$ 14.2619 0.551396
$$670$$ 8.65562 0.334396
$$671$$ −36.9380 −1.42597
$$672$$ 0 0
$$673$$ 3.13727 0.120933 0.0604665 0.998170i $$-0.480741\pi$$
0.0604665 + 0.998170i $$0.480741\pi$$
$$674$$ 26.1173 1.00600
$$675$$ 4.31265 0.165994
$$676$$ −0.819237 −0.0315091
$$677$$ 46.2579 1.77784 0.888918 0.458067i $$-0.151458\pi$$
0.888918 + 0.458067i $$0.151458\pi$$
$$678$$ −14.0508 −0.539617
$$679$$ 0 0
$$680$$ 14.7308 0.564902
$$681$$ 21.6366 0.829115
$$682$$ −29.5877 −1.13297
$$683$$ −9.01905 −0.345104 −0.172552 0.985000i $$-0.555201\pi$$
−0.172552 + 0.985000i $$0.555201\pi$$
$$684$$ −1.45580 −0.0556640
$$685$$ −18.2823 −0.698532
$$686$$ 0 0
$$687$$ −13.8838 −0.529702
$$688$$ 0.131888 0.00502817
$$689$$ 4.00000 0.152388
$$690$$ −2.20123 −0.0837994
$$691$$ 50.0625 1.90447 0.952234 0.305368i $$-0.0987794\pi$$
0.952234 + 0.305368i $$0.0987794\pi$$
$$692$$ 1.50071 0.0570483
$$693$$ 0 0
$$694$$ 4.61942 0.175351
$$695$$ −11.5369 −0.437620
$$696$$ 2.15633 0.0817353
$$697$$ −62.0870 −2.35171
$$698$$ −19.3649 −0.732970
$$699$$ 7.31406 0.276643
$$700$$ 0 0
$$701$$ 45.3014 1.71101 0.855505 0.517795i $$-0.173247\pi$$
0.855505 + 0.517795i $$0.173247\pi$$
$$702$$ 18.9234 0.714216
$$703$$ −30.3634 −1.14518
$$704$$ 29.4314 1.10924
$$705$$ −3.87399 −0.145903
$$706$$ 7.70194 0.289866
$$707$$ 0 0
$$708$$ 2.07522 0.0779916
$$709$$ −3.27504 −0.122997 −0.0614983 0.998107i $$-0.519588\pi$$
−0.0614983 + 0.998107i $$0.519588\pi$$
$$710$$ −1.88858 −0.0708772
$$711$$ 11.5917 0.434721
$$712$$ 9.66291 0.362133
$$713$$ 8.86082 0.331840
$$714$$ 0 0
$$715$$ 12.3127 0.460467
$$716$$ −4.15045 −0.155109
$$717$$ −16.5237 −0.617090
$$718$$ −45.0553 −1.68145
$$719$$ −27.7235 −1.03391 −0.516957 0.856011i $$-0.672935\pi$$
−0.516957 + 0.856011i $$0.672935\pi$$
$$720$$ −10.2243 −0.381035
$$721$$ 0 0
$$722$$ 13.0327 0.485026
$$723$$ 4.41422 0.164167
$$724$$ −2.96239 −0.110096
$$725$$ −1.00000 −0.0371391
$$726$$ 7.49200 0.278054
$$727$$ 26.8930 0.997408 0.498704 0.866772i $$-0.333810\pi$$
0.498704 + 0.866772i $$0.333810\pi$$
$$728$$ 0 0
$$729$$ 2.02776 0.0751023
$$730$$ 22.5804 0.835738
$$731$$ 0.166944 0.00617465
$$732$$ 1.38929 0.0513496
$$733$$ −3.17935 −0.117432 −0.0587160 0.998275i $$-0.518701\pi$$
−0.0587160 + 0.998275i $$0.518701\pi$$
$$734$$ 30.5950 1.12928
$$735$$ 0 0
$$736$$ 2.01573 0.0743007
$$737$$ 24.2882 0.894668
$$738$$ 39.2506 1.44483
$$739$$ −29.7440 −1.09415 −0.547076 0.837083i $$-0.684259\pi$$
−0.547076 + 0.837083i $$0.684259\pi$$
$$740$$ 1.84367 0.0677748
$$741$$ 7.62672 0.280174
$$742$$ 0 0
$$743$$ −4.34297 −0.159328 −0.0796640 0.996822i $$-0.525385\pi$$
−0.0796640 + 0.996822i $$0.525385\pi$$
$$744$$ −10.3634 −0.379942
$$745$$ −2.77575 −0.101695
$$746$$ −16.4241 −0.601328
$$747$$ 10.3839 0.379927
$$748$$ −4.43866 −0.162293
$$749$$ 0 0
$$750$$ −1.19394 −0.0435964
$$751$$ 22.5804 0.823970 0.411985 0.911191i $$-0.364836\pi$$
0.411985 + 0.911191i $$0.364836\pi$$
$$752$$ 20.9076 0.762423
$$753$$ −23.8740 −0.870017
$$754$$ −4.38787 −0.159797
$$755$$ −1.79877 −0.0654639
$$756$$ 0 0
$$757$$ 9.88461 0.359262 0.179631 0.983734i $$-0.442510\pi$$
0.179631 + 0.983734i $$0.442510\pi$$
$$758$$ −14.9321 −0.542357
$$759$$ −6.17679 −0.224203
$$760$$ −8.54420 −0.309931
$$761$$ −13.6991 −0.496592 −0.248296 0.968684i $$-0.579871\pi$$
−0.248296 + 0.968684i $$0.579871\pi$$
$$762$$ 17.0348 0.617105
$$763$$ 0 0
$$764$$ 0.643859 0.0232940
$$765$$ −12.9419 −0.467916
$$766$$ 24.2287 0.875419
$$767$$ 39.3258 1.41997
$$768$$ −3.71767 −0.134150
$$769$$ −25.0132 −0.901998 −0.450999 0.892524i $$-0.648932\pi$$
−0.450999 + 0.892524i $$0.648932\pi$$
$$770$$ 0 0
$$771$$ 14.2374 0.512748
$$772$$ −0.946660 −0.0340710
$$773$$ 35.9062 1.29146 0.645728 0.763567i $$-0.276554\pi$$
0.645728 + 0.763567i $$0.276554\pi$$
$$774$$ −0.105540 −0.00379356
$$775$$ 4.80606 0.172639
$$776$$ 3.69323 0.132579
$$777$$ 0 0
$$778$$ −47.2506 −1.69402
$$779$$ 36.0118 1.29026
$$780$$ −0.463096 −0.0165815
$$781$$ −5.29948 −0.189630
$$782$$ 15.0376 0.537744
$$783$$ −4.31265 −0.154122
$$784$$ 0 0
$$785$$ 3.76845 0.134502
$$786$$ −7.03761 −0.251023
$$787$$ −50.3839 −1.79599 −0.897996 0.440003i $$-0.854977\pi$$
−0.897996 + 0.440003i $$0.854977\pi$$
$$788$$ −4.70052 −0.167449
$$789$$ −22.0508 −0.785029
$$790$$ −7.30536 −0.259913
$$791$$ 0 0
$$792$$ −26.1319 −0.928556
$$793$$ 26.3272 0.934908
$$794$$ 4.42407 0.157004
$$795$$ −1.08840 −0.0386014
$$796$$ −3.24869 −0.115147
$$797$$ 5.69323 0.201665 0.100832 0.994903i $$-0.467849\pi$$
0.100832 + 0.994903i $$0.467849\pi$$
$$798$$ 0 0
$$799$$ 26.4650 0.936265
$$800$$ 1.09332 0.0386547
$$801$$ −8.48944 −0.299960
$$802$$ 32.5501 1.14938
$$803$$ 63.3620 2.23600
$$804$$ −0.913513 −0.0322171
$$805$$ 0 0
$$806$$ 21.0884 0.742807
$$807$$ 8.46310 0.297915
$$808$$ 34.8119 1.22468
$$809$$ −7.76257 −0.272918 −0.136459 0.990646i $$-0.543572\pi$$
−0.136459 + 0.990646i $$0.543572\pi$$
$$810$$ 5.29455 0.186032
$$811$$ 26.4894 0.930170 0.465085 0.885266i $$-0.346024\pi$$
0.465085 + 0.885266i $$0.346024\pi$$
$$812$$ 0 0
$$813$$ −7.75272 −0.271900
$$814$$ 58.5256 2.05132
$$815$$ −1.64244 −0.0575323
$$816$$ −19.3093 −0.675962
$$817$$ −0.0968311 −0.00338769
$$818$$ −33.2360 −1.16207
$$819$$ 0 0
$$820$$ −2.18664 −0.0763609
$$821$$ 25.4763 0.889128 0.444564 0.895747i $$-0.353359\pi$$
0.444564 + 0.895747i $$0.353359\pi$$
$$822$$ 21.8279 0.761337
$$823$$ −9.22028 −0.321399 −0.160699 0.987003i $$-0.551375\pi$$
−0.160699 + 0.987003i $$0.551375\pi$$
$$824$$ −14.2315 −0.495779
$$825$$ −3.35026 −0.116641
$$826$$ 0 0
$$827$$ 24.5343 0.853143 0.426571 0.904454i $$-0.359721\pi$$
0.426571 + 0.904454i $$0.359721\pi$$
$$828$$ −0.840350 −0.0292042
$$829$$ −0.201231 −0.00698903 −0.00349452 0.999994i $$-0.501112\pi$$
−0.00349452 + 0.999994i $$0.501112\pi$$
$$830$$ −6.54420 −0.227152
$$831$$ −10.7612 −0.373300
$$832$$ −20.9770 −0.727246
$$833$$ 0 0
$$834$$ 13.7743 0.476966
$$835$$ 8.08110 0.279658
$$836$$ 2.57452 0.0890415
$$837$$ 20.7269 0.716425
$$838$$ 15.3503 0.530266
$$839$$ −1.45580 −0.0502599 −0.0251299 0.999684i $$-0.508000\pi$$
−0.0251299 + 0.999684i $$0.508000\pi$$
$$840$$ 0 0
$$841$$ 1.00000 0.0344828
$$842$$ −50.4142 −1.73739
$$843$$ −16.4631 −0.567019
$$844$$ −4.90763 −0.168928
$$845$$ 4.22425 0.145319
$$846$$ −16.7308 −0.575218
$$847$$ 0 0
$$848$$ 5.87399 0.201714
$$849$$ −6.46501 −0.221878
$$850$$ 8.15633 0.279760
$$851$$ −17.5271 −0.600820
$$852$$ 0.199321 0.00682861
$$853$$ −43.1793 −1.47843 −0.739216 0.673468i $$-0.764804\pi$$
−0.739216 + 0.673468i $$0.764804\pi$$
$$854$$ 0 0
$$855$$ 7.50659 0.256720
$$856$$ −36.9683 −1.26355
$$857$$ 20.9887 0.716962 0.358481 0.933537i $$-0.383295\pi$$
0.358481 + 0.933537i $$0.383295\pi$$
$$858$$ −14.7005 −0.501868
$$859$$ 49.4069 1.68574 0.842871 0.538115i $$-0.180863\pi$$
0.842871 + 0.538115i $$0.180863\pi$$
$$860$$ 0.00587961 0.000200493 0
$$861$$ 0 0
$$862$$ −38.1768 −1.30031
$$863$$ 56.6820 1.92948 0.964738 0.263211i $$-0.0847816\pi$$
0.964738 + 0.263211i $$0.0847816\pi$$
$$864$$ 4.71511 0.160411
$$865$$ −7.73813 −0.263104
$$866$$ 3.23013 0.109764
$$867$$ −10.7388 −0.364708
$$868$$ 0 0
$$869$$ −20.4993 −0.695391
$$870$$ 1.19394 0.0404782
$$871$$ −17.3112 −0.586569
$$872$$ −5.01317 −0.169767
$$873$$ −3.24472 −0.109817
$$874$$ −8.72213 −0.295031
$$875$$ 0 0
$$876$$ −2.38313 −0.0805186
$$877$$ −13.1998 −0.445726 −0.222863 0.974850i $$-0.571540\pi$$
−0.222863 + 0.974850i $$0.571540\pi$$
$$878$$ −52.6009 −1.77519
$$879$$ −18.7856 −0.633622
$$880$$ 18.0811 0.609514
$$881$$ −6.37802 −0.214881 −0.107441 0.994212i $$-0.534266\pi$$
−0.107441 + 0.994212i $$0.534266\pi$$
$$882$$ 0 0
$$883$$ 48.6213 1.63624 0.818119 0.575049i $$-0.195017\pi$$
0.818119 + 0.575049i $$0.195017\pi$$
$$884$$ 3.16362 0.106404
$$885$$ −10.7005 −0.359694
$$886$$ 6.43278 0.216113
$$887$$ 15.0317 0.504716 0.252358 0.967634i $$-0.418794\pi$$
0.252358 + 0.967634i $$0.418794\pi$$
$$888$$ 20.4993 0.687911
$$889$$ 0 0
$$890$$ 5.35026 0.179341
$$891$$ 14.8568 0.497723
$$892$$ −3.43136 −0.114891
$$893$$ −15.3503 −0.513677
$$894$$ 3.31406 0.110839
$$895$$ 21.4010 0.715358
$$896$$ 0 0
$$897$$ 4.40246 0.146994
$$898$$ −46.4142 −1.54886
$$899$$ −4.80606 −0.160291
$$900$$ −0.455802 −0.0151934
$$901$$ 7.43533 0.247707
$$902$$ −69.4128 −2.31119
$$903$$ 0 0
$$904$$ −31.4821 −1.04708
$$905$$ 15.2750 0.507759
$$906$$ 2.14762 0.0713498
$$907$$ −0.342968 −0.0113880 −0.00569402 0.999984i $$-0.501812\pi$$
−0.00569402 + 0.999984i $$0.501812\pi$$
$$908$$ −5.20570 −0.172757
$$909$$ −30.5844 −1.01442
$$910$$ 0 0
$$911$$ 20.9076 0.692701 0.346350 0.938105i $$-0.387421\pi$$
0.346350 + 0.938105i $$0.387421\pi$$
$$912$$ 11.1998 0.370863
$$913$$ −18.3634 −0.607741
$$914$$ −50.8773 −1.68287
$$915$$ −7.16362 −0.236822
$$916$$ 3.34041 0.110370
$$917$$ 0 0
$$918$$ 35.1754 1.16096
$$919$$ −1.90034 −0.0626864 −0.0313432 0.999509i $$-0.509978\pi$$
−0.0313432 + 0.999509i $$0.509978\pi$$
$$920$$ −4.93207 −0.162606
$$921$$ −5.42548 −0.178776
$$922$$ 17.5731 0.578739
$$923$$ 3.77716 0.124327
$$924$$ 0 0
$$925$$ −9.50659 −0.312575
$$926$$ −59.9814 −1.97111
$$927$$ 12.5033 0.410661
$$928$$ −1.09332 −0.0358900
$$929$$ −39.3522 −1.29110 −0.645551 0.763717i $$-0.723372\pi$$
−0.645551 + 0.763717i $$0.723372\pi$$
$$930$$ −5.73813 −0.188161
$$931$$ 0 0
$$932$$ −1.75974 −0.0576423
$$933$$ −17.7791 −0.582061
$$934$$ −44.7426 −1.46402
$$935$$ 22.8872 0.748490
$$936$$ 18.6253 0.608787
$$937$$ 6.37802 0.208361 0.104180 0.994558i $$-0.466778\pi$$
0.104180 + 0.994558i $$0.466778\pi$$
$$938$$ 0 0
$$939$$ 4.06063 0.132514
$$940$$ 0.932071 0.0304008
$$941$$ 26.6253 0.867960 0.433980 0.900923i $$-0.357109\pi$$
0.433980 + 0.900923i $$0.357109\pi$$
$$942$$ −4.49929 −0.146595
$$943$$ 20.7875 0.676934
$$944$$ 57.7499 1.87960
$$945$$ 0 0
$$946$$ 0.186642 0.00606827
$$947$$ −12.2823 −0.399122 −0.199561 0.979885i $$-0.563952\pi$$
−0.199561 + 0.979885i $$0.563952\pi$$
$$948$$ 0.771007 0.0250411
$$949$$ −45.1608 −1.46598
$$950$$ −4.73084 −0.153489
$$951$$ −27.6432 −0.896393
$$952$$ 0 0
$$953$$ 0.821792 0.0266205 0.0133102 0.999911i $$-0.495763\pi$$
0.0133102 + 0.999911i $$0.495763\pi$$
$$954$$ −4.70052 −0.152185
$$955$$ −3.31994 −0.107431
$$956$$ 3.97556 0.128579
$$957$$ 3.35026 0.108299
$$958$$ 0.0811024 0.00262030
$$959$$ 0 0
$$960$$ 5.70782 0.184219
$$961$$ −7.90175 −0.254895
$$962$$ −41.7137 −1.34490
$$963$$ 32.4788 1.04662
$$964$$ −1.06205 −0.0342063
$$965$$ 4.88129 0.157134
$$966$$ 0 0
$$967$$ −37.4314 −1.20371 −0.601856 0.798605i $$-0.705572\pi$$
−0.601856 + 0.798605i $$0.705572\pi$$
$$968$$ 16.7866 0.539540
$$969$$ 14.1768 0.455424
$$970$$ 2.04491 0.0656580
$$971$$ 8.71625 0.279718 0.139859 0.990171i $$-0.455335\pi$$
0.139859 + 0.990171i $$0.455335\pi$$
$$972$$ −3.06793 −0.0984039
$$973$$ 0 0
$$974$$ 1.30536 0.0418263
$$975$$ 2.38787 0.0764731
$$976$$ 38.6615 1.23752
$$977$$ −33.7645 −1.08022 −0.540111 0.841594i $$-0.681618\pi$$
−0.540111 + 0.841594i $$0.681618\pi$$
$$978$$ 1.96097 0.0627050
$$979$$ 15.0132 0.479823
$$980$$ 0 0
$$981$$ 4.40437 0.140621
$$982$$ −60.8324 −1.94124
$$983$$ −43.6082 −1.39088 −0.695442 0.718582i $$-0.744791\pi$$
−0.695442 + 0.718582i $$0.744791\pi$$
$$984$$ −24.3127 −0.775059
$$985$$ 24.2374 0.772269
$$986$$ −8.15633 −0.259750
$$987$$ 0 0
$$988$$ −1.83497 −0.0583780
$$989$$ −0.0558950 −0.00177736
$$990$$ −14.4690 −0.459854
$$991$$ −52.9741 −1.68278 −0.841390 0.540429i $$-0.818262\pi$$
−0.841390 + 0.540429i $$0.818262\pi$$
$$992$$ 5.25457 0.166833
$$993$$ 28.0771 0.891001
$$994$$ 0 0
$$995$$ 16.7513 0.531052
$$996$$ 0.690674 0.0218849
$$997$$ −13.6326 −0.431749 −0.215874 0.976421i $$-0.569260\pi$$
−0.215874 + 0.976421i $$0.569260\pi$$
$$998$$ −18.3272 −0.580139
$$999$$ −40.9986 −1.29714
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 7105.2.a.o.1.1 3
7.6 odd 2 145.2.a.c.1.1 3
21.20 even 2 1305.2.a.p.1.3 3
28.27 even 2 2320.2.a.n.1.2 3
35.13 even 4 725.2.b.e.349.5 6
35.27 even 4 725.2.b.e.349.2 6
35.34 odd 2 725.2.a.e.1.3 3
56.13 odd 2 9280.2.a.bj.1.2 3
56.27 even 2 9280.2.a.br.1.2 3
105.104 even 2 6525.2.a.be.1.1 3
203.202 odd 2 4205.2.a.f.1.3 3

By twisted newform
Twist Min Dim Char Parity Ord Type
145.2.a.c.1.1 3 7.6 odd 2
725.2.a.e.1.3 3 35.34 odd 2
725.2.b.e.349.2 6 35.27 even 4
725.2.b.e.349.5 6 35.13 even 4
1305.2.a.p.1.3 3 21.20 even 2
2320.2.a.n.1.2 3 28.27 even 2
4205.2.a.f.1.3 3 203.202 odd 2
6525.2.a.be.1.1 3 105.104 even 2
7105.2.a.o.1.1 3 1.1 even 1 trivial
9280.2.a.bj.1.2 3 56.13 odd 2
9280.2.a.br.1.2 3 56.27 even 2