# Properties

 Label 1305.2.a.p.1.2 Level $1305$ Weight $2$ Character 1305.1 Self dual yes Analytic conductor $10.420$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1305,2,Mod(1,1305)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1305, 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("1305.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1305 = 3^{2} \cdot 5 \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1305.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: $$10.4204774638$$ Analytic rank: $$1$$ 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 1.2 Root $$0.311108$$ of defining polynomial Character $$\chi$$ $$=$$ 1305.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-0.311108 q^{2} -1.90321 q^{4} -1.00000 q^{5} -0.903212 q^{7} +1.21432 q^{8} +O(q^{10})$$ $$q-0.311108 q^{2} -1.90321 q^{4} -1.00000 q^{5} -0.903212 q^{7} +1.21432 q^{8} +0.311108 q^{10} +1.52543 q^{11} -0.622216 q^{13} +0.280996 q^{14} +3.42864 q^{16} +7.95407 q^{17} -1.09679 q^{19} +1.90321 q^{20} -0.474572 q^{22} -7.52543 q^{23} +1.00000 q^{25} +0.193576 q^{26} +1.71900 q^{28} +1.00000 q^{29} -6.90321 q^{31} -3.49532 q^{32} -2.47457 q^{34} +0.903212 q^{35} +3.95407 q^{37} +0.341219 q^{38} -1.21432 q^{40} -3.67307 q^{41} -10.5161 q^{43} -2.90321 q^{44} +2.34122 q^{46} -6.90321 q^{47} -6.18421 q^{49} -0.311108 q^{50} +1.18421 q^{52} -6.42864 q^{53} -1.52543 q^{55} -1.09679 q^{56} -0.311108 q^{58} +1.67307 q^{59} -1.86665 q^{61} +2.14764 q^{62} -5.76986 q^{64} +0.622216 q^{65} +11.5254 q^{67} -15.1383 q^{68} -0.280996 q^{70} -13.6731 q^{71} +10.1891 q^{73} -1.23014 q^{74} +2.08742 q^{76} -1.37778 q^{77} +9.13828 q^{79} -3.42864 q^{80} +1.14272 q^{82} -10.7096 q^{83} -7.95407 q^{85} +3.27163 q^{86} +1.85236 q^{88} +7.80642 q^{89} +0.561993 q^{91} +14.3225 q^{92} +2.14764 q^{94} +1.09679 q^{95} -4.08742 q^{97} +1.92396 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - q^{2} + q^{4} - 3 q^{5} + 4 q^{7} - 3 q^{8}+O(q^{10})$$ 3 * q - q^2 + q^4 - 3 * q^5 + 4 * q^7 - 3 * q^8 $$3 q - q^{2} + q^{4} - 3 q^{5} + 4 q^{7} - 3 q^{8} + q^{10} - 2 q^{11} - 2 q^{13} - 6 q^{14} - 3 q^{16} + 4 q^{17} - 10 q^{19} - q^{20} - 8 q^{22} - 16 q^{23} + 3 q^{25} + 14 q^{26} + 12 q^{28} + 3 q^{29} - 14 q^{31} + 3 q^{32} - 14 q^{34} - 4 q^{35} - 8 q^{37} + 8 q^{38} + 3 q^{40} + 2 q^{41} + 2 q^{43} - 2 q^{44} + 14 q^{46} - 14 q^{47} - 5 q^{49} - q^{50} - 10 q^{52} - 6 q^{53} + 2 q^{55} - 10 q^{56} - q^{58} - 8 q^{59} - 6 q^{61} - 11 q^{64} + 2 q^{65} + 28 q^{67} - 12 q^{68} + 6 q^{70} - 28 q^{71} - 16 q^{73} - 10 q^{74} - 14 q^{76} - 4 q^{77} - 6 q^{79} + 3 q^{80} + 30 q^{82} - 12 q^{83} - 4 q^{85} - 24 q^{86} + 12 q^{88} + 10 q^{89} - 12 q^{91} - 4 q^{92} + 10 q^{95} + 8 q^{97} - 21 q^{98}+O(q^{100})$$ 3 * q - q^2 + q^4 - 3 * q^5 + 4 * q^7 - 3 * q^8 + q^10 - 2 * q^11 - 2 * q^13 - 6 * q^14 - 3 * q^16 + 4 * q^17 - 10 * q^19 - q^20 - 8 * q^22 - 16 * q^23 + 3 * q^25 + 14 * q^26 + 12 * q^28 + 3 * q^29 - 14 * q^31 + 3 * q^32 - 14 * q^34 - 4 * q^35 - 8 * q^37 + 8 * q^38 + 3 * q^40 + 2 * q^41 + 2 * q^43 - 2 * q^44 + 14 * q^46 - 14 * q^47 - 5 * q^49 - q^50 - 10 * q^52 - 6 * q^53 + 2 * q^55 - 10 * q^56 - q^58 - 8 * q^59 - 6 * q^61 - 11 * q^64 + 2 * q^65 + 28 * q^67 - 12 * q^68 + 6 * q^70 - 28 * q^71 - 16 * q^73 - 10 * q^74 - 14 * q^76 - 4 * q^77 - 6 * q^79 + 3 * q^80 + 30 * q^82 - 12 * q^83 - 4 * q^85 - 24 * q^86 + 12 * q^88 + 10 * q^89 - 12 * q^91 - 4 * q^92 + 10 * q^95 + 8 * q^97 - 21 * q^98

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −0.311108 −0.219986 −0.109993 0.993932i $$-0.535083\pi$$
−0.109993 + 0.993932i $$0.535083\pi$$
$$3$$ 0 0
$$4$$ −1.90321 −0.951606
$$5$$ −1.00000 −0.447214
$$6$$ 0 0
$$7$$ −0.903212 −0.341382 −0.170691 0.985325i $$-0.554600\pi$$
−0.170691 + 0.985325i $$0.554600\pi$$
$$8$$ 1.21432 0.429327
$$9$$ 0 0
$$10$$ 0.311108 0.0983809
$$11$$ 1.52543 0.459934 0.229967 0.973198i $$-0.426138\pi$$
0.229967 + 0.973198i $$0.426138\pi$$
$$12$$ 0 0
$$13$$ −0.622216 −0.172572 −0.0862858 0.996270i $$-0.527500\pi$$
−0.0862858 + 0.996270i $$0.527500\pi$$
$$14$$ 0.280996 0.0750994
$$15$$ 0 0
$$16$$ 3.42864 0.857160
$$17$$ 7.95407 1.92914 0.964572 0.263819i $$-0.0849820\pi$$
0.964572 + 0.263819i $$0.0849820\pi$$
$$18$$ 0 0
$$19$$ −1.09679 −0.251620 −0.125810 0.992054i $$-0.540153\pi$$
−0.125810 + 0.992054i $$0.540153\pi$$
$$20$$ 1.90321 0.425571
$$21$$ 0 0
$$22$$ −0.474572 −0.101179
$$23$$ −7.52543 −1.56916 −0.784580 0.620028i $$-0.787121\pi$$
−0.784580 + 0.620028i $$0.787121\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0.193576 0.0379634
$$27$$ 0 0
$$28$$ 1.71900 0.324861
$$29$$ 1.00000 0.185695
$$30$$ 0 0
$$31$$ −6.90321 −1.23985 −0.619927 0.784660i $$-0.712838\pi$$
−0.619927 + 0.784660i $$0.712838\pi$$
$$32$$ −3.49532 −0.617890
$$33$$ 0 0
$$34$$ −2.47457 −0.424386
$$35$$ 0.903212 0.152671
$$36$$ 0 0
$$37$$ 3.95407 0.650045 0.325022 0.945706i $$-0.394628\pi$$
0.325022 + 0.945706i $$0.394628\pi$$
$$38$$ 0.341219 0.0553531
$$39$$ 0 0
$$40$$ −1.21432 −0.192001
$$41$$ −3.67307 −0.573637 −0.286819 0.957985i $$-0.592598\pi$$
−0.286819 + 0.957985i $$0.592598\pi$$
$$42$$ 0 0
$$43$$ −10.5161 −1.60368 −0.801842 0.597536i $$-0.796146\pi$$
−0.801842 + 0.597536i $$0.796146\pi$$
$$44$$ −2.90321 −0.437676
$$45$$ 0 0
$$46$$ 2.34122 0.345194
$$47$$ −6.90321 −1.00694 −0.503468 0.864014i $$-0.667943\pi$$
−0.503468 + 0.864014i $$0.667943\pi$$
$$48$$ 0 0
$$49$$ −6.18421 −0.883458
$$50$$ −0.311108 −0.0439973
$$51$$ 0 0
$$52$$ 1.18421 0.164220
$$53$$ −6.42864 −0.883042 −0.441521 0.897251i $$-0.645561\pi$$
−0.441521 + 0.897251i $$0.645561\pi$$
$$54$$ 0 0
$$55$$ −1.52543 −0.205689
$$56$$ −1.09679 −0.146564
$$57$$ 0 0
$$58$$ −0.311108 −0.0408505
$$59$$ 1.67307 0.217815 0.108908 0.994052i $$-0.465265\pi$$
0.108908 + 0.994052i $$0.465265\pi$$
$$60$$ 0 0
$$61$$ −1.86665 −0.239000 −0.119500 0.992834i $$-0.538129\pi$$
−0.119500 + 0.992834i $$0.538129\pi$$
$$62$$ 2.14764 0.272751
$$63$$ 0 0
$$64$$ −5.76986 −0.721232
$$65$$ 0.622216 0.0771764
$$66$$ 0 0
$$67$$ 11.5254 1.40806 0.704028 0.710173i $$-0.251383\pi$$
0.704028 + 0.710173i $$0.251383\pi$$
$$68$$ −15.1383 −1.83579
$$69$$ 0 0
$$70$$ −0.280996 −0.0335855
$$71$$ −13.6731 −1.62269 −0.811347 0.584564i $$-0.801266\pi$$
−0.811347 + 0.584564i $$0.801266\pi$$
$$72$$ 0 0
$$73$$ 10.1891 1.19255 0.596274 0.802781i $$-0.296647\pi$$
0.596274 + 0.802781i $$0.296647\pi$$
$$74$$ −1.23014 −0.143001
$$75$$ 0 0
$$76$$ 2.08742 0.239444
$$77$$ −1.37778 −0.157013
$$78$$ 0 0
$$79$$ 9.13828 1.02814 0.514068 0.857749i $$-0.328138\pi$$
0.514068 + 0.857749i $$0.328138\pi$$
$$80$$ −3.42864 −0.383334
$$81$$ 0 0
$$82$$ 1.14272 0.126192
$$83$$ −10.7096 −1.17554 −0.587768 0.809030i $$-0.699993\pi$$
−0.587768 + 0.809030i $$0.699993\pi$$
$$84$$ 0 0
$$85$$ −7.95407 −0.862740
$$86$$ 3.27163 0.352789
$$87$$ 0 0
$$88$$ 1.85236 0.197462
$$89$$ 7.80642 0.827479 0.413740 0.910395i $$-0.364222\pi$$
0.413740 + 0.910395i $$0.364222\pi$$
$$90$$ 0 0
$$91$$ 0.561993 0.0589128
$$92$$ 14.3225 1.49322
$$93$$ 0 0
$$94$$ 2.14764 0.221512
$$95$$ 1.09679 0.112528
$$96$$ 0 0
$$97$$ −4.08742 −0.415015 −0.207507 0.978233i $$-0.566535\pi$$
−0.207507 + 0.978233i $$0.566535\pi$$
$$98$$ 1.92396 0.194349
$$99$$ 0 0
$$100$$ −1.90321 −0.190321
$$101$$ −13.9081 −1.38391 −0.691956 0.721940i $$-0.743251\pi$$
−0.691956 + 0.721940i $$0.743251\pi$$
$$102$$ 0 0
$$103$$ −12.9447 −1.27548 −0.637740 0.770252i $$-0.720130\pi$$
−0.637740 + 0.770252i $$0.720130\pi$$
$$104$$ −0.755569 −0.0740896
$$105$$ 0 0
$$106$$ 2.00000 0.194257
$$107$$ −11.0049 −1.06389 −0.531943 0.846780i $$-0.678538\pi$$
−0.531943 + 0.846780i $$0.678538\pi$$
$$108$$ 0 0
$$109$$ −18.0415 −1.72806 −0.864031 0.503439i $$-0.832068\pi$$
−0.864031 + 0.503439i $$0.832068\pi$$
$$110$$ 0.474572 0.0452487
$$111$$ 0 0
$$112$$ −3.09679 −0.292619
$$113$$ 10.2810 0.967155 0.483577 0.875302i $$-0.339337\pi$$
0.483577 + 0.875302i $$0.339337\pi$$
$$114$$ 0 0
$$115$$ 7.52543 0.701750
$$116$$ −1.90321 −0.176709
$$117$$ 0 0
$$118$$ −0.520505 −0.0479164
$$119$$ −7.18421 −0.658575
$$120$$ 0 0
$$121$$ −8.67307 −0.788461
$$122$$ 0.580728 0.0525767
$$123$$ 0 0
$$124$$ 13.1383 1.17985
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ 6.22077 0.552004 0.276002 0.961157i $$-0.410990\pi$$
0.276002 + 0.961157i $$0.410990\pi$$
$$128$$ 8.78568 0.776552
$$129$$ 0 0
$$130$$ −0.193576 −0.0169778
$$131$$ 11.7605 1.02752 0.513759 0.857934i $$-0.328252\pi$$
0.513759 + 0.857934i $$0.328252\pi$$
$$132$$ 0 0
$$133$$ 0.990632 0.0858987
$$134$$ −3.58565 −0.309753
$$135$$ 0 0
$$136$$ 9.65878 0.828234
$$137$$ 3.56691 0.304742 0.152371 0.988323i $$-0.451309\pi$$
0.152371 + 0.988323i $$0.451309\pi$$
$$138$$ 0 0
$$139$$ −8.56199 −0.726219 −0.363109 0.931747i $$-0.618285\pi$$
−0.363109 + 0.931747i $$0.618285\pi$$
$$140$$ −1.71900 −0.145282
$$141$$ 0 0
$$142$$ 4.25380 0.356971
$$143$$ −0.949145 −0.0793715
$$144$$ 0 0
$$145$$ −1.00000 −0.0830455
$$146$$ −3.16992 −0.262344
$$147$$ 0 0
$$148$$ −7.52543 −0.618586
$$149$$ 5.61285 0.459822 0.229911 0.973212i $$-0.426156\pi$$
0.229911 + 0.973212i $$0.426156\pi$$
$$150$$ 0 0
$$151$$ 10.7971 0.878652 0.439326 0.898328i $$-0.355217\pi$$
0.439326 + 0.898328i $$0.355217\pi$$
$$152$$ −1.33185 −0.108027
$$153$$ 0 0
$$154$$ 0.428639 0.0345408
$$155$$ 6.90321 0.554479
$$156$$ 0 0
$$157$$ 2.28100 0.182043 0.0910217 0.995849i $$-0.470987\pi$$
0.0910217 + 0.995849i $$0.470987\pi$$
$$158$$ −2.84299 −0.226176
$$159$$ 0 0
$$160$$ 3.49532 0.276329
$$161$$ 6.79706 0.535683
$$162$$ 0 0
$$163$$ 16.3225 1.27848 0.639238 0.769009i $$-0.279250\pi$$
0.639238 + 0.769009i $$0.279250\pi$$
$$164$$ 6.99063 0.545877
$$165$$ 0 0
$$166$$ 3.33185 0.258602
$$167$$ 4.76986 0.369103 0.184551 0.982823i $$-0.440917\pi$$
0.184551 + 0.982823i $$0.440917\pi$$
$$168$$ 0 0
$$169$$ −12.6128 −0.970219
$$170$$ 2.47457 0.189791
$$171$$ 0 0
$$172$$ 20.0143 1.52608
$$173$$ −4.23506 −0.321986 −0.160993 0.986956i $$-0.551470\pi$$
−0.160993 + 0.986956i $$0.551470\pi$$
$$174$$ 0 0
$$175$$ −0.903212 −0.0682764
$$176$$ 5.23014 0.394237
$$177$$ 0 0
$$178$$ −2.42864 −0.182034
$$179$$ −9.71456 −0.726100 −0.363050 0.931770i $$-0.618265\pi$$
−0.363050 + 0.931770i $$0.618265\pi$$
$$180$$ 0 0
$$181$$ 0.326929 0.0243005 0.0121502 0.999926i $$-0.496132\pi$$
0.0121502 + 0.999926i $$0.496132\pi$$
$$182$$ −0.174840 −0.0129600
$$183$$ 0 0
$$184$$ −9.13828 −0.673683
$$185$$ −3.95407 −0.290709
$$186$$ 0 0
$$187$$ 12.1334 0.887279
$$188$$ 13.1383 0.958207
$$189$$ 0 0
$$190$$ −0.341219 −0.0247547
$$191$$ 14.9447 1.08136 0.540680 0.841228i $$-0.318167\pi$$
0.540680 + 0.841228i $$0.318167\pi$$
$$192$$ 0 0
$$193$$ −14.1476 −1.01837 −0.509185 0.860657i $$-0.670053\pi$$
−0.509185 + 0.860657i $$0.670053\pi$$
$$194$$ 1.27163 0.0912976
$$195$$ 0 0
$$196$$ 11.7699 0.840704
$$197$$ 5.70471 0.406444 0.203222 0.979133i $$-0.434859\pi$$
0.203222 + 0.979133i $$0.434859\pi$$
$$198$$ 0 0
$$199$$ −22.1432 −1.56969 −0.784845 0.619692i $$-0.787257\pi$$
−0.784845 + 0.619692i $$0.787257\pi$$
$$200$$ 1.21432 0.0858654
$$201$$ 0 0
$$202$$ 4.32693 0.304442
$$203$$ −0.903212 −0.0633930
$$204$$ 0 0
$$205$$ 3.67307 0.256538
$$206$$ 4.02720 0.280588
$$207$$ 0 0
$$208$$ −2.13335 −0.147921
$$209$$ −1.67307 −0.115729
$$210$$ 0 0
$$211$$ −20.8430 −1.43489 −0.717445 0.696615i $$-0.754689\pi$$
−0.717445 + 0.696615i $$0.754689\pi$$
$$212$$ 12.2351 0.840308
$$213$$ 0 0
$$214$$ 3.42372 0.234040
$$215$$ 10.5161 0.717189
$$216$$ 0 0
$$217$$ 6.23506 0.423264
$$218$$ 5.61285 0.380150
$$219$$ 0 0
$$220$$ 2.90321 0.195735
$$221$$ −4.94914 −0.332916
$$222$$ 0 0
$$223$$ 9.03657 0.605133 0.302567 0.953128i $$-0.402157\pi$$
0.302567 + 0.953128i $$0.402157\pi$$
$$224$$ 3.15701 0.210937
$$225$$ 0 0
$$226$$ −3.19850 −0.212761
$$227$$ −19.4050 −1.28795 −0.643977 0.765045i $$-0.722717\pi$$
−0.643977 + 0.765045i $$0.722717\pi$$
$$228$$ 0 0
$$229$$ −25.6128 −1.69254 −0.846272 0.532751i $$-0.821158\pi$$
−0.846272 + 0.532751i $$0.821158\pi$$
$$230$$ −2.34122 −0.154375
$$231$$ 0 0
$$232$$ 1.21432 0.0797240
$$233$$ 3.12399 0.204659 0.102330 0.994751i $$-0.467370\pi$$
0.102330 + 0.994751i $$0.467370\pi$$
$$234$$ 0 0
$$235$$ 6.90321 0.450316
$$236$$ −3.18421 −0.207274
$$237$$ 0 0
$$238$$ 2.23506 0.144878
$$239$$ −13.9398 −0.901689 −0.450845 0.892602i $$-0.648877\pi$$
−0.450845 + 0.892602i $$0.648877\pi$$
$$240$$ 0 0
$$241$$ −18.4701 −1.18977 −0.594883 0.803813i $$-0.702802\pi$$
−0.594883 + 0.803813i $$0.702802\pi$$
$$242$$ 2.69826 0.173451
$$243$$ 0 0
$$244$$ 3.55262 0.227433
$$245$$ 6.18421 0.395095
$$246$$ 0 0
$$247$$ 0.682439 0.0434225
$$248$$ −8.38271 −0.532302
$$249$$ 0 0
$$250$$ 0.311108 0.0196762
$$251$$ 13.7921 0.870552 0.435276 0.900297i $$-0.356651\pi$$
0.435276 + 0.900297i $$0.356651\pi$$
$$252$$ 0 0
$$253$$ −11.4795 −0.721710
$$254$$ −1.93533 −0.121433
$$255$$ 0 0
$$256$$ 8.80642 0.550401
$$257$$ 1.47949 0.0922883 0.0461442 0.998935i $$-0.485307\pi$$
0.0461442 + 0.998935i $$0.485307\pi$$
$$258$$ 0 0
$$259$$ −3.57136 −0.221914
$$260$$ −1.18421 −0.0734415
$$261$$ 0 0
$$262$$ −3.65878 −0.226040
$$263$$ 0.442930 0.0273122 0.0136561 0.999907i $$-0.495653\pi$$
0.0136561 + 0.999907i $$0.495653\pi$$
$$264$$ 0 0
$$265$$ 6.42864 0.394908
$$266$$ −0.308193 −0.0188965
$$267$$ 0 0
$$268$$ −21.9353 −1.33991
$$269$$ −3.93978 −0.240212 −0.120106 0.992761i $$-0.538324\pi$$
−0.120106 + 0.992761i $$0.538324\pi$$
$$270$$ 0 0
$$271$$ 6.20787 0.377101 0.188551 0.982063i $$-0.439621\pi$$
0.188551 + 0.982063i $$0.439621\pi$$
$$272$$ 27.2716 1.65359
$$273$$ 0 0
$$274$$ −1.10970 −0.0670391
$$275$$ 1.52543 0.0919867
$$276$$ 0 0
$$277$$ 5.57136 0.334751 0.167375 0.985893i $$-0.446471\pi$$
0.167375 + 0.985893i $$0.446471\pi$$
$$278$$ 2.66370 0.159758
$$279$$ 0 0
$$280$$ 1.09679 0.0655456
$$281$$ −6.69535 −0.399411 −0.199705 0.979856i $$-0.563999\pi$$
−0.199705 + 0.979856i $$0.563999\pi$$
$$282$$ 0 0
$$283$$ 25.8020 1.53377 0.766884 0.641785i $$-0.221806\pi$$
0.766884 + 0.641785i $$0.221806\pi$$
$$284$$ 26.0228 1.54417
$$285$$ 0 0
$$286$$ 0.295286 0.0174607
$$287$$ 3.31756 0.195829
$$288$$ 0 0
$$289$$ 46.2672 2.72160
$$290$$ 0.311108 0.0182689
$$291$$ 0 0
$$292$$ −19.3921 −1.13484
$$293$$ 18.8430 1.10082 0.550410 0.834895i $$-0.314472\pi$$
0.550410 + 0.834895i $$0.314472\pi$$
$$294$$ 0 0
$$295$$ −1.67307 −0.0974099
$$296$$ 4.80150 0.279082
$$297$$ 0 0
$$298$$ −1.74620 −0.101155
$$299$$ 4.68244 0.270792
$$300$$ 0 0
$$301$$ 9.49823 0.547469
$$302$$ −3.35905 −0.193292
$$303$$ 0 0
$$304$$ −3.76049 −0.215679
$$305$$ 1.86665 0.106884
$$306$$ 0 0
$$307$$ −1.65878 −0.0946716 −0.0473358 0.998879i $$-0.515073\pi$$
−0.0473358 + 0.998879i $$0.515073\pi$$
$$308$$ 2.62222 0.149415
$$309$$ 0 0
$$310$$ −2.14764 −0.121978
$$311$$ −21.3002 −1.20782 −0.603912 0.797051i $$-0.706392\pi$$
−0.603912 + 0.797051i $$0.706392\pi$$
$$312$$ 0 0
$$313$$ 8.62222 0.487356 0.243678 0.969856i $$-0.421646\pi$$
0.243678 + 0.969856i $$0.421646\pi$$
$$314$$ −0.709636 −0.0400471
$$315$$ 0 0
$$316$$ −17.3921 −0.978381
$$317$$ 27.5955 1.54992 0.774959 0.632012i $$-0.217771\pi$$
0.774959 + 0.632012i $$0.217771\pi$$
$$318$$ 0 0
$$319$$ 1.52543 0.0854075
$$320$$ 5.76986 0.322545
$$321$$ 0 0
$$322$$ −2.11462 −0.117843
$$323$$ −8.72393 −0.485412
$$324$$ 0 0
$$325$$ −0.622216 −0.0345143
$$326$$ −5.07805 −0.281247
$$327$$ 0 0
$$328$$ −4.46028 −0.246278
$$329$$ 6.23506 0.343750
$$330$$ 0 0
$$331$$ 16.9131 0.929626 0.464813 0.885409i $$-0.346122\pi$$
0.464813 + 0.885409i $$0.346122\pi$$
$$332$$ 20.3827 1.11865
$$333$$ 0 0
$$334$$ −1.48394 −0.0811976
$$335$$ −11.5254 −0.629701
$$336$$ 0 0
$$337$$ 11.9956 0.653439 0.326720 0.945121i $$-0.394057\pi$$
0.326720 + 0.945121i $$0.394057\pi$$
$$338$$ 3.92396 0.213435
$$339$$ 0 0
$$340$$ 15.1383 0.820988
$$341$$ −10.5303 −0.570250
$$342$$ 0 0
$$343$$ 11.9081 0.642979
$$344$$ −12.7699 −0.688505
$$345$$ 0 0
$$346$$ 1.31756 0.0708325
$$347$$ −6.14764 −0.330023 −0.165011 0.986292i $$-0.552766\pi$$
−0.165011 + 0.986292i $$0.552766\pi$$
$$348$$ 0 0
$$349$$ −7.12399 −0.381338 −0.190669 0.981654i $$-0.561066\pi$$
−0.190669 + 0.981654i $$0.561066\pi$$
$$350$$ 0.280996 0.0150199
$$351$$ 0 0
$$352$$ −5.33185 −0.284189
$$353$$ 16.9175 0.900428 0.450214 0.892921i $$-0.351348\pi$$
0.450214 + 0.892921i $$0.351348\pi$$
$$354$$ 0 0
$$355$$ 13.6731 0.725691
$$356$$ −14.8573 −0.787434
$$357$$ 0 0
$$358$$ 3.02227 0.159732
$$359$$ −36.7096 −1.93746 −0.968730 0.248116i $$-0.920188\pi$$
−0.968730 + 0.248116i $$0.920188\pi$$
$$360$$ 0 0
$$361$$ −17.7971 −0.936687
$$362$$ −0.101710 −0.00534577
$$363$$ 0 0
$$364$$ −1.06959 −0.0560618
$$365$$ −10.1891 −0.533323
$$366$$ 0 0
$$367$$ 8.41435 0.439225 0.219613 0.975587i $$-0.429521\pi$$
0.219613 + 0.975587i $$0.429521\pi$$
$$368$$ −25.8020 −1.34502
$$369$$ 0 0
$$370$$ 1.23014 0.0639520
$$371$$ 5.80642 0.301455
$$372$$ 0 0
$$373$$ −8.66370 −0.448590 −0.224295 0.974521i $$-0.572008\pi$$
−0.224295 + 0.974521i $$0.572008\pi$$
$$374$$ −3.77478 −0.195189
$$375$$ 0 0
$$376$$ −8.38271 −0.432305
$$377$$ −0.622216 −0.0320457
$$378$$ 0 0
$$379$$ −2.76986 −0.142278 −0.0711390 0.997466i $$-0.522663\pi$$
−0.0711390 + 0.997466i $$0.522663\pi$$
$$380$$ −2.08742 −0.107082
$$381$$ 0 0
$$382$$ −4.64941 −0.237885
$$383$$ −1.67752 −0.0857171 −0.0428585 0.999081i $$-0.513646\pi$$
−0.0428585 + 0.999081i $$0.513646\pi$$
$$384$$ 0 0
$$385$$ 1.37778 0.0702184
$$386$$ 4.40144 0.224028
$$387$$ 0 0
$$388$$ 7.77923 0.394930
$$389$$ 5.77478 0.292793 0.146397 0.989226i $$-0.453232\pi$$
0.146397 + 0.989226i $$0.453232\pi$$
$$390$$ 0 0
$$391$$ −59.8578 −3.02714
$$392$$ −7.50961 −0.379292
$$393$$ 0 0
$$394$$ −1.77478 −0.0894122
$$395$$ −9.13828 −0.459797
$$396$$ 0 0
$$397$$ 29.9081 1.50105 0.750523 0.660844i $$-0.229802\pi$$
0.750523 + 0.660844i $$0.229802\pi$$
$$398$$ 6.88892 0.345310
$$399$$ 0 0
$$400$$ 3.42864 0.171432
$$401$$ −8.53035 −0.425985 −0.212993 0.977054i $$-0.568321\pi$$
−0.212993 + 0.977054i $$0.568321\pi$$
$$402$$ 0 0
$$403$$ 4.29529 0.213963
$$404$$ 26.4701 1.31694
$$405$$ 0 0
$$406$$ 0.280996 0.0139456
$$407$$ 6.03164 0.298977
$$408$$ 0 0
$$409$$ 5.09234 0.251800 0.125900 0.992043i $$-0.459818\pi$$
0.125900 + 0.992043i $$0.459818\pi$$
$$410$$ −1.14272 −0.0564350
$$411$$ 0 0
$$412$$ 24.6365 1.21375
$$413$$ −1.51114 −0.0743582
$$414$$ 0 0
$$415$$ 10.7096 0.525715
$$416$$ 2.17484 0.106630
$$417$$ 0 0
$$418$$ 0.520505 0.0254588
$$419$$ 24.3368 1.18893 0.594465 0.804122i $$-0.297364\pi$$
0.594465 + 0.804122i $$0.297364\pi$$
$$420$$ 0 0
$$421$$ 24.5018 1.19414 0.597072 0.802188i $$-0.296331\pi$$
0.597072 + 0.802188i $$0.296331\pi$$
$$422$$ 6.48442 0.315656
$$423$$ 0 0
$$424$$ −7.80642 −0.379113
$$425$$ 7.95407 0.385829
$$426$$ 0 0
$$427$$ 1.68598 0.0815902
$$428$$ 20.9447 1.01240
$$429$$ 0 0
$$430$$ −3.27163 −0.157772
$$431$$ −4.26671 −0.205520 −0.102760 0.994706i $$-0.532767\pi$$
−0.102760 + 0.994706i $$0.532767\pi$$
$$432$$ 0 0
$$433$$ 27.0049 1.29777 0.648887 0.760885i $$-0.275235\pi$$
0.648887 + 0.760885i $$0.275235\pi$$
$$434$$ −1.93978 −0.0931123
$$435$$ 0 0
$$436$$ 34.3368 1.64443
$$437$$ 8.25380 0.394833
$$438$$ 0 0
$$439$$ −2.03164 −0.0969650 −0.0484825 0.998824i $$-0.515439\pi$$
−0.0484825 + 0.998824i $$0.515439\pi$$
$$440$$ −1.85236 −0.0883076
$$441$$ 0 0
$$442$$ 1.53972 0.0732369
$$443$$ 3.46520 0.164637 0.0823184 0.996606i $$-0.473768\pi$$
0.0823184 + 0.996606i $$0.473768\pi$$
$$444$$ 0 0
$$445$$ −7.80642 −0.370060
$$446$$ −2.81135 −0.133121
$$447$$ 0 0
$$448$$ 5.21141 0.246216
$$449$$ −37.3590 −1.76308 −0.881541 0.472107i $$-0.843494\pi$$
−0.881541 + 0.472107i $$0.843494\pi$$
$$450$$ 0 0
$$451$$ −5.60300 −0.263835
$$452$$ −19.5669 −0.920350
$$453$$ 0 0
$$454$$ 6.03704 0.283332
$$455$$ −0.561993 −0.0263466
$$456$$ 0 0
$$457$$ 13.4509 0.629207 0.314604 0.949223i $$-0.398128\pi$$
0.314604 + 0.949223i $$0.398128\pi$$
$$458$$ 7.96836 0.372337
$$459$$ 0 0
$$460$$ −14.3225 −0.667789
$$461$$ 16.2766 0.758075 0.379037 0.925381i $$-0.376255\pi$$
0.379037 + 0.925381i $$0.376255\pi$$
$$462$$ 0 0
$$463$$ −30.3926 −1.41246 −0.706231 0.707982i $$-0.749606\pi$$
−0.706231 + 0.707982i $$0.749606\pi$$
$$464$$ 3.42864 0.159171
$$465$$ 0 0
$$466$$ −0.971896 −0.0450222
$$467$$ 1.18865 0.0550043 0.0275022 0.999622i $$-0.491245\pi$$
0.0275022 + 0.999622i $$0.491245\pi$$
$$468$$ 0 0
$$469$$ −10.4099 −0.480685
$$470$$ −2.14764 −0.0990634
$$471$$ 0 0
$$472$$ 2.03164 0.0935139
$$473$$ −16.0415 −0.737588
$$474$$ 0 0
$$475$$ −1.09679 −0.0503241
$$476$$ 13.6731 0.626704
$$477$$ 0 0
$$478$$ 4.33677 0.198359
$$479$$ −41.0464 −1.87546 −0.937729 0.347367i $$-0.887076\pi$$
−0.937729 + 0.347367i $$0.887076\pi$$
$$480$$ 0 0
$$481$$ −2.46028 −0.112179
$$482$$ 5.74620 0.261732
$$483$$ 0 0
$$484$$ 16.5067 0.750304
$$485$$ 4.08742 0.185600
$$486$$ 0 0
$$487$$ −10.1476 −0.459834 −0.229917 0.973210i $$-0.573845\pi$$
−0.229917 + 0.973210i $$0.573845\pi$$
$$488$$ −2.26671 −0.102609
$$489$$ 0 0
$$490$$ −1.92396 −0.0869155
$$491$$ 29.2083 1.31815 0.659077 0.752075i $$-0.270947\pi$$
0.659077 + 0.752075i $$0.270947\pi$$
$$492$$ 0 0
$$493$$ 7.95407 0.358233
$$494$$ −0.212312 −0.00955237
$$495$$ 0 0
$$496$$ −23.6686 −1.06275
$$497$$ 12.3497 0.553959
$$498$$ 0 0
$$499$$ 21.9813 0.984017 0.492008 0.870591i $$-0.336263\pi$$
0.492008 + 0.870591i $$0.336263\pi$$
$$500$$ 1.90321 0.0851142
$$501$$ 0 0
$$502$$ −4.29084 −0.191510
$$503$$ −5.77923 −0.257683 −0.128841 0.991665i $$-0.541126\pi$$
−0.128841 + 0.991665i $$0.541126\pi$$
$$504$$ 0 0
$$505$$ 13.9081 0.618904
$$506$$ 3.57136 0.158766
$$507$$ 0 0
$$508$$ −11.8394 −0.525291
$$509$$ −13.6543 −0.605218 −0.302609 0.953115i $$-0.597858\pi$$
−0.302609 + 0.953115i $$0.597858\pi$$
$$510$$ 0 0
$$511$$ −9.20294 −0.407114
$$512$$ −20.3111 −0.897633
$$513$$ 0 0
$$514$$ −0.460282 −0.0203022
$$515$$ 12.9447 0.570412
$$516$$ 0 0
$$517$$ −10.5303 −0.463124
$$518$$ 1.11108 0.0488180
$$519$$ 0 0
$$520$$ 0.755569 0.0331339
$$521$$ 19.6731 0.861893 0.430946 0.902378i $$-0.358180\pi$$
0.430946 + 0.902378i $$0.358180\pi$$
$$522$$ 0 0
$$523$$ −15.1383 −0.661951 −0.330975 0.943639i $$-0.607378\pi$$
−0.330975 + 0.943639i $$0.607378\pi$$
$$524$$ −22.3827 −0.977793
$$525$$ 0 0
$$526$$ −0.137799 −0.00600832
$$527$$ −54.9086 −2.39186
$$528$$ 0 0
$$529$$ 33.6321 1.46226
$$530$$ −2.00000 −0.0868744
$$531$$ 0 0
$$532$$ −1.88538 −0.0817417
$$533$$ 2.28544 0.0989935
$$534$$ 0 0
$$535$$ 11.0049 0.475784
$$536$$ 13.9956 0.604516
$$537$$ 0 0
$$538$$ 1.22570 0.0528435
$$539$$ −9.43356 −0.406332
$$540$$ 0 0
$$541$$ 2.68244 0.115327 0.0576635 0.998336i $$-0.481635\pi$$
0.0576635 + 0.998336i $$0.481635\pi$$
$$542$$ −1.93132 −0.0829571
$$543$$ 0 0
$$544$$ −27.8020 −1.19200
$$545$$ 18.0415 0.772812
$$546$$ 0 0
$$547$$ −15.3635 −0.656896 −0.328448 0.944522i $$-0.606525\pi$$
−0.328448 + 0.944522i $$0.606525\pi$$
$$548$$ −6.78859 −0.289994
$$549$$ 0 0
$$550$$ −0.474572 −0.0202358
$$551$$ −1.09679 −0.0467247
$$552$$ 0 0
$$553$$ −8.25380 −0.350987
$$554$$ −1.73329 −0.0736406
$$555$$ 0 0
$$556$$ 16.2953 0.691074
$$557$$ 9.87955 0.418610 0.209305 0.977850i $$-0.432880\pi$$
0.209305 + 0.977850i $$0.432880\pi$$
$$558$$ 0 0
$$559$$ 6.54326 0.276750
$$560$$ 3.09679 0.130863
$$561$$ 0 0
$$562$$ 2.08297 0.0878650
$$563$$ 27.4938 1.15872 0.579362 0.815070i $$-0.303302\pi$$
0.579362 + 0.815070i $$0.303302\pi$$
$$564$$ 0 0
$$565$$ −10.2810 −0.432525
$$566$$ −8.02720 −0.337408
$$567$$ 0 0
$$568$$ −16.6035 −0.696667
$$569$$ −17.3590 −0.727729 −0.363865 0.931452i $$-0.618543\pi$$
−0.363865 + 0.931452i $$0.618543\pi$$
$$570$$ 0 0
$$571$$ −25.4479 −1.06496 −0.532480 0.846443i $$-0.678740\pi$$
−0.532480 + 0.846443i $$0.678740\pi$$
$$572$$ 1.80642 0.0755304
$$573$$ 0 0
$$574$$ −1.03212 −0.0430798
$$575$$ −7.52543 −0.313832
$$576$$ 0 0
$$577$$ −10.6178 −0.442024 −0.221012 0.975271i $$-0.570936\pi$$
−0.221012 + 0.975271i $$0.570936\pi$$
$$578$$ −14.3941 −0.598715
$$579$$ 0 0
$$580$$ 1.90321 0.0790266
$$581$$ 9.67307 0.401307
$$582$$ 0 0
$$583$$ −9.80642 −0.406141
$$584$$ 12.3729 0.511993
$$585$$ 0 0
$$586$$ −5.86220 −0.242165
$$587$$ 8.94470 0.369187 0.184594 0.982815i $$-0.440903\pi$$
0.184594 + 0.982815i $$0.440903\pi$$
$$588$$ 0 0
$$589$$ 7.57136 0.311972
$$590$$ 0.520505 0.0214289
$$591$$ 0 0
$$592$$ 13.5571 0.557192
$$593$$ −14.1619 −0.581561 −0.290780 0.956790i $$-0.593915\pi$$
−0.290780 + 0.956790i $$0.593915\pi$$
$$594$$ 0 0
$$595$$ 7.18421 0.294524
$$596$$ −10.6824 −0.437570
$$597$$ 0 0
$$598$$ −1.45674 −0.0595707
$$599$$ 22.5575 0.921676 0.460838 0.887484i $$-0.347549\pi$$
0.460838 + 0.887484i $$0.347549\pi$$
$$600$$ 0 0
$$601$$ −40.6133 −1.65665 −0.828326 0.560246i $$-0.810706\pi$$
−0.828326 + 0.560246i $$0.810706\pi$$
$$602$$ −2.95497 −0.120436
$$603$$ 0 0
$$604$$ −20.5491 −0.836130
$$605$$ 8.67307 0.352610
$$606$$ 0 0
$$607$$ 13.5955 0.551824 0.275912 0.961183i $$-0.411020\pi$$
0.275912 + 0.961183i $$0.411020\pi$$
$$608$$ 3.83362 0.155474
$$609$$ 0 0
$$610$$ −0.580728 −0.0235130
$$611$$ 4.29529 0.173769
$$612$$ 0 0
$$613$$ −42.0830 −1.69972 −0.849858 0.527012i $$-0.823312\pi$$
−0.849858 + 0.527012i $$0.823312\pi$$
$$614$$ 0.516060 0.0208265
$$615$$ 0 0
$$616$$ −1.67307 −0.0674099
$$617$$ −33.5067 −1.34893 −0.674464 0.738307i $$-0.735625\pi$$
−0.674464 + 0.738307i $$0.735625\pi$$
$$618$$ 0 0
$$619$$ −14.6780 −0.589958 −0.294979 0.955504i $$-0.595313\pi$$
−0.294979 + 0.955504i $$0.595313\pi$$
$$620$$ −13.1383 −0.527646
$$621$$ 0 0
$$622$$ 6.62666 0.265705
$$623$$ −7.05086 −0.282487
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ −2.68244 −0.107212
$$627$$ 0 0
$$628$$ −4.34122 −0.173234
$$629$$ 31.4509 1.25403
$$630$$ 0 0
$$631$$ 11.3176 0.450545 0.225273 0.974296i $$-0.427673\pi$$
0.225273 + 0.974296i $$0.427673\pi$$
$$632$$ 11.0968 0.441407
$$633$$ 0 0
$$634$$ −8.58517 −0.340961
$$635$$ −6.22077 −0.246864
$$636$$ 0 0
$$637$$ 3.84791 0.152460
$$638$$ −0.474572 −0.0187885
$$639$$ 0 0
$$640$$ −8.78568 −0.347285
$$641$$ −34.8988 −1.37842 −0.689209 0.724562i $$-0.742042\pi$$
−0.689209 + 0.724562i $$0.742042\pi$$
$$642$$ 0 0
$$643$$ −41.9768 −1.65540 −0.827702 0.561168i $$-0.810352\pi$$
−0.827702 + 0.561168i $$0.810352\pi$$
$$644$$ −12.9362 −0.509759
$$645$$ 0 0
$$646$$ 2.71408 0.106784
$$647$$ −5.46520 −0.214859 −0.107430 0.994213i $$-0.534262\pi$$
−0.107430 + 0.994213i $$0.534262\pi$$
$$648$$ 0 0
$$649$$ 2.55215 0.100181
$$650$$ 0.193576 0.00759268
$$651$$ 0 0
$$652$$ −31.0651 −1.21660
$$653$$ 8.76986 0.343191 0.171596 0.985167i $$-0.445108\pi$$
0.171596 + 0.985167i $$0.445108\pi$$
$$654$$ 0 0
$$655$$ −11.7605 −0.459520
$$656$$ −12.5936 −0.491699
$$657$$ 0 0
$$658$$ −1.93978 −0.0756204
$$659$$ −3.29036 −0.128174 −0.0640872 0.997944i $$-0.520414\pi$$
−0.0640872 + 0.997944i $$0.520414\pi$$
$$660$$ 0 0
$$661$$ 19.7560 0.768421 0.384211 0.923246i $$-0.374474\pi$$
0.384211 + 0.923246i $$0.374474\pi$$
$$662$$ −5.26178 −0.204505
$$663$$ 0 0
$$664$$ −13.0049 −0.504689
$$665$$ −0.990632 −0.0384151
$$666$$ 0 0
$$667$$ −7.52543 −0.291386
$$668$$ −9.07805 −0.351240
$$669$$ 0 0
$$670$$ 3.58565 0.138526
$$671$$ −2.84743 −0.109924
$$672$$ 0 0
$$673$$ 44.3970 1.71138 0.855689 0.517490i $$-0.173134\pi$$
0.855689 + 0.517490i $$0.173134\pi$$
$$674$$ −3.73191 −0.143748
$$675$$ 0 0
$$676$$ 24.0049 0.923266
$$677$$ −6.09726 −0.234337 −0.117168 0.993112i $$-0.537382\pi$$
−0.117168 + 0.993112i $$0.537382\pi$$
$$678$$ 0 0
$$679$$ 3.69181 0.141679
$$680$$ −9.65878 −0.370397
$$681$$ 0 0
$$682$$ 3.27607 0.125447
$$683$$ −37.9224 −1.45106 −0.725531 0.688190i $$-0.758406\pi$$
−0.725531 + 0.688190i $$0.758406\pi$$
$$684$$ 0 0
$$685$$ −3.56691 −0.136285
$$686$$ −3.70471 −0.141447
$$687$$ 0 0
$$688$$ −36.0558 −1.37461
$$689$$ 4.00000 0.152388
$$690$$ 0 0
$$691$$ 13.3145 0.506507 0.253254 0.967400i $$-0.418499\pi$$
0.253254 + 0.967400i $$0.418499\pi$$
$$692$$ 8.06022 0.306404
$$693$$ 0 0
$$694$$ 1.91258 0.0726005
$$695$$ 8.56199 0.324775
$$696$$ 0 0
$$697$$ −29.2159 −1.10663
$$698$$ 2.21633 0.0838892
$$699$$ 0 0
$$700$$ 1.71900 0.0649722
$$701$$ 23.4893 0.887180 0.443590 0.896230i $$-0.353705\pi$$
0.443590 + 0.896230i $$0.353705\pi$$
$$702$$ 0 0
$$703$$ −4.33677 −0.163565
$$704$$ −8.80150 −0.331719
$$705$$ 0 0
$$706$$ −5.26317 −0.198082
$$707$$ 12.5620 0.472442
$$708$$ 0 0
$$709$$ 11.6731 0.438391 0.219196 0.975681i $$-0.429657\pi$$
0.219196 + 0.975681i $$0.429657\pi$$
$$710$$ −4.25380 −0.159642
$$711$$ 0 0
$$712$$ 9.47949 0.355259
$$713$$ 51.9496 1.94553
$$714$$ 0 0
$$715$$ 0.949145 0.0354960
$$716$$ 18.4889 0.690961
$$717$$ 0 0
$$718$$ 11.4207 0.426215
$$719$$ −29.5526 −1.10213 −0.551063 0.834463i $$-0.685778\pi$$
−0.551063 + 0.834463i $$0.685778\pi$$
$$720$$ 0 0
$$721$$ 11.6918 0.435426
$$722$$ 5.53680 0.206058
$$723$$ 0 0
$$724$$ −0.622216 −0.0231245
$$725$$ 1.00000 0.0371391
$$726$$ 0 0
$$727$$ 3.88094 0.143936 0.0719680 0.997407i $$-0.477072\pi$$
0.0719680 + 0.997407i $$0.477072\pi$$
$$728$$ 0.682439 0.0252929
$$729$$ 0 0
$$730$$ 3.16992 0.117324
$$731$$ −83.6454 −3.09374
$$732$$ 0 0
$$733$$ 14.8845 0.549771 0.274885 0.961477i $$-0.411360\pi$$
0.274885 + 0.961477i $$0.411360\pi$$
$$734$$ −2.61777 −0.0966236
$$735$$ 0 0
$$736$$ 26.3037 0.969569
$$737$$ 17.5812 0.647612
$$738$$ 0 0
$$739$$ 2.24935 0.0827438 0.0413719 0.999144i $$-0.486827\pi$$
0.0413719 + 0.999144i $$0.486827\pi$$
$$740$$ 7.52543 0.276640
$$741$$ 0 0
$$742$$ −1.80642 −0.0663159
$$743$$ 3.46520 0.127126 0.0635630 0.997978i $$-0.479754\pi$$
0.0635630 + 0.997978i $$0.479754\pi$$
$$744$$ 0 0
$$745$$ −5.61285 −0.205639
$$746$$ 2.69535 0.0986836
$$747$$ 0 0
$$748$$ −23.0923 −0.844340
$$749$$ 9.93978 0.363192
$$750$$ 0 0
$$751$$ 3.16992 0.115672 0.0578360 0.998326i $$-0.481580\pi$$
0.0578360 + 0.998326i $$0.481580\pi$$
$$752$$ −23.6686 −0.863106
$$753$$ 0 0
$$754$$ 0.193576 0.00704963
$$755$$ −10.7971 −0.392945
$$756$$ 0 0
$$757$$ −52.0785 −1.89283 −0.946413 0.322958i $$-0.895323\pi$$
−0.946413 + 0.322958i $$0.895323\pi$$
$$758$$ 0.861725 0.0312993
$$759$$ 0 0
$$760$$ 1.33185 0.0483113
$$761$$ 14.9777 0.542942 0.271471 0.962447i $$-0.412490\pi$$
0.271471 + 0.962447i $$0.412490\pi$$
$$762$$ 0 0
$$763$$ 16.2953 0.589929
$$764$$ −28.4429 −1.02903
$$765$$ 0 0
$$766$$ 0.521889 0.0188566
$$767$$ −1.04101 −0.0375887
$$768$$ 0 0
$$769$$ −1.90813 −0.0688091 −0.0344045 0.999408i $$-0.510953\pi$$
−0.0344045 + 0.999408i $$0.510953\pi$$
$$770$$ −0.428639 −0.0154471
$$771$$ 0 0
$$772$$ 26.9260 0.969087
$$773$$ −21.7891 −0.783698 −0.391849 0.920029i $$-0.628165\pi$$
−0.391849 + 0.920029i $$0.628165\pi$$
$$774$$ 0 0
$$775$$ −6.90321 −0.247971
$$776$$ −4.96343 −0.178177
$$777$$ 0 0
$$778$$ −1.79658 −0.0644105
$$779$$ 4.02858 0.144339
$$780$$ 0 0
$$781$$ −20.8573 −0.746332
$$782$$ 18.6222 0.665929
$$783$$ 0 0
$$784$$ −21.2034 −0.757265
$$785$$ −2.28100 −0.0814122
$$786$$ 0 0
$$787$$ −18.1388 −0.646577 −0.323288 0.946301i $$-0.604788\pi$$
−0.323288 + 0.946301i $$0.604788\pi$$
$$788$$ −10.8573 −0.386775
$$789$$ 0 0
$$790$$ 2.84299 0.101149
$$791$$ −9.28592 −0.330169
$$792$$ 0 0
$$793$$ 1.16146 0.0412445
$$794$$ −9.30465 −0.330210
$$795$$ 0 0
$$796$$ 42.1432 1.49373
$$797$$ −2.96343 −0.104970 −0.0524851 0.998622i $$-0.516714\pi$$
−0.0524851 + 0.998622i $$0.516714\pi$$
$$798$$ 0 0
$$799$$ −54.9086 −1.94253
$$800$$ −3.49532 −0.123578
$$801$$ 0 0
$$802$$ 2.65386 0.0937110
$$803$$ 15.5428 0.548493
$$804$$ 0 0
$$805$$ −6.79706 −0.239565
$$806$$ −1.33630 −0.0470691
$$807$$ 0 0
$$808$$ −16.8889 −0.594150
$$809$$ 26.2953 0.924493 0.462247 0.886751i $$-0.347044\pi$$
0.462247 + 0.886751i $$0.347044\pi$$
$$810$$ 0 0
$$811$$ 24.3783 0.856037 0.428018 0.903770i $$-0.359212\pi$$
0.428018 + 0.903770i $$0.359212\pi$$
$$812$$ 1.71900 0.0603252
$$813$$ 0 0
$$814$$ −1.87649 −0.0657710
$$815$$ −16.3225 −0.571752
$$816$$ 0 0
$$817$$ 11.5339 0.403520
$$818$$ −1.58427 −0.0553926
$$819$$ 0 0
$$820$$ −6.99063 −0.244123
$$821$$ −1.52987 −0.0533929 −0.0266965 0.999644i $$-0.508499\pi$$
−0.0266965 + 0.999644i $$0.508499\pi$$
$$822$$ 0 0
$$823$$ 46.7195 1.62854 0.814269 0.580487i $$-0.197138\pi$$
0.814269 + 0.580487i $$0.197138\pi$$
$$824$$ −15.7190 −0.547597
$$825$$ 0 0
$$826$$ 0.470127 0.0163578
$$827$$ 29.6499 1.03103 0.515514 0.856881i $$-0.327601\pi$$
0.515514 + 0.856881i $$0.327601\pi$$
$$828$$ 0 0
$$829$$ −8.79706 −0.305534 −0.152767 0.988262i $$-0.548818\pi$$
−0.152767 + 0.988262i $$0.548818\pi$$
$$830$$ −3.33185 −0.115650
$$831$$ 0 0
$$832$$ 3.59010 0.124464
$$833$$ −49.1896 −1.70432
$$834$$ 0 0
$$835$$ −4.76986 −0.165068
$$836$$ 3.18421 0.110128
$$837$$ 0 0
$$838$$ −7.57136 −0.261548
$$839$$ −11.3319 −0.391219 −0.195609 0.980682i $$-0.562668\pi$$
−0.195609 + 0.980682i $$0.562668\pi$$
$$840$$ 0 0
$$841$$ 1.00000 0.0344828
$$842$$ −7.62269 −0.262695
$$843$$ 0 0
$$844$$ 39.6686 1.36545
$$845$$ 12.6128 0.433895
$$846$$ 0 0
$$847$$ 7.83362 0.269166
$$848$$ −22.0415 −0.756908
$$849$$ 0 0
$$850$$ −2.47457 −0.0848771
$$851$$ −29.7560 −1.02002
$$852$$ 0 0
$$853$$ 54.8845 1.87921 0.939604 0.342263i $$-0.111193\pi$$
0.939604 + 0.342263i $$0.111193\pi$$
$$854$$ −0.524521 −0.0179487
$$855$$ 0 0
$$856$$ −13.3635 −0.456755
$$857$$ −36.4385 −1.24471 −0.622357 0.782733i $$-0.713825\pi$$
−0.622357 + 0.782733i $$0.713825\pi$$
$$858$$ 0 0
$$859$$ 1.72885 0.0589875 0.0294938 0.999565i $$-0.490610\pi$$
0.0294938 + 0.999565i $$0.490610\pi$$
$$860$$ −20.0143 −0.682482
$$861$$ 0 0
$$862$$ 1.32741 0.0452116
$$863$$ 9.40192 0.320045 0.160023 0.987113i $$-0.448843\pi$$
0.160023 + 0.987113i $$0.448843\pi$$
$$864$$ 0 0
$$865$$ 4.23506 0.143996
$$866$$ −8.40144 −0.285493
$$867$$ 0 0
$$868$$ −11.8666 −0.402780
$$869$$ 13.9398 0.472875
$$870$$ 0 0
$$871$$ −7.17130 −0.242990
$$872$$ −21.9081 −0.741903
$$873$$ 0 0
$$874$$ −2.56782 −0.0868579
$$875$$ 0.903212 0.0305341
$$876$$ 0 0
$$877$$ 8.91750 0.301123 0.150561 0.988601i $$-0.451892\pi$$
0.150561 + 0.988601i $$0.451892\pi$$
$$878$$ 0.632060 0.0213310
$$879$$ 0 0
$$880$$ −5.23014 −0.176308
$$881$$ 42.1245 1.41921 0.709605 0.704600i $$-0.248874\pi$$
0.709605 + 0.704600i $$0.248874\pi$$
$$882$$ 0 0
$$883$$ −38.4340 −1.29341 −0.646704 0.762741i $$-0.723853\pi$$
−0.646704 + 0.762741i $$0.723853\pi$$
$$884$$ 9.41927 0.316804
$$885$$ 0 0
$$886$$ −1.07805 −0.0362179
$$887$$ 38.6365 1.29729 0.648643 0.761092i $$-0.275337\pi$$
0.648643 + 0.761092i $$0.275337\pi$$
$$888$$ 0 0
$$889$$ −5.61868 −0.188444
$$890$$ 2.42864 0.0814082
$$891$$ 0 0
$$892$$ −17.1985 −0.575848
$$893$$ 7.57136 0.253366
$$894$$ 0 0
$$895$$ 9.71456 0.324722
$$896$$ −7.93533 −0.265101
$$897$$ 0 0
$$898$$ 11.6227 0.387854
$$899$$ −6.90321 −0.230235
$$900$$ 0 0
$$901$$ −51.1338 −1.70351
$$902$$ 1.74314 0.0580402
$$903$$ 0 0
$$904$$ 12.4844 0.415226
$$905$$ −0.326929 −0.0108675
$$906$$ 0 0
$$907$$ 0.534795 0.0177576 0.00887880 0.999961i $$-0.497174\pi$$
0.00887880 + 0.999961i $$0.497174\pi$$
$$908$$ 36.9318 1.22562
$$909$$ 0 0
$$910$$ 0.174840 0.00579590
$$911$$ 23.6686 0.784177 0.392088 0.919928i $$-0.371753\pi$$
0.392088 + 0.919928i $$0.371753\pi$$
$$912$$ 0 0
$$913$$ −16.3368 −0.540668
$$914$$ −4.18468 −0.138417
$$915$$ 0 0
$$916$$ 48.7467 1.61064
$$917$$ −10.6222 −0.350776
$$918$$ 0 0
$$919$$ 35.7748 1.18010 0.590051 0.807366i $$-0.299108\pi$$
0.590051 + 0.807366i $$0.299108\pi$$
$$920$$ 9.13828 0.301280
$$921$$ 0 0
$$922$$ −5.06376 −0.166766
$$923$$ 8.50760 0.280031
$$924$$ 0 0
$$925$$ 3.95407 0.130009
$$926$$ 9.45536 0.310722
$$927$$ 0 0
$$928$$ −3.49532 −0.114739
$$929$$ 52.7753 1.73150 0.865750 0.500477i $$-0.166842\pi$$
0.865750 + 0.500477i $$0.166842\pi$$
$$930$$ 0 0
$$931$$ 6.78277 0.222296
$$932$$ −5.94561 −0.194755
$$933$$ 0 0
$$934$$ −0.369800 −0.0121002
$$935$$ −12.1334 −0.396803
$$936$$ 0 0
$$937$$ 42.1245 1.37615 0.688073 0.725641i $$-0.258457\pi$$
0.688073 + 0.725641i $$0.258457\pi$$
$$938$$ 3.23860 0.105744
$$939$$ 0 0
$$940$$ −13.1383 −0.428523
$$941$$ 3.89829 0.127081 0.0635403 0.997979i $$-0.479761\pi$$
0.0635403 + 0.997979i $$0.479761\pi$$
$$942$$ 0 0
$$943$$ 27.6414 0.900129
$$944$$ 5.73636 0.186703
$$945$$ 0 0
$$946$$ 4.99063 0.162259
$$947$$ −9.56691 −0.310883 −0.155441 0.987845i $$-0.549680\pi$$
−0.155441 + 0.987845i $$0.549680\pi$$
$$948$$ 0 0
$$949$$ −6.33984 −0.205800
$$950$$ 0.341219 0.0110706
$$951$$ 0 0
$$952$$ −8.72393 −0.282744
$$953$$ −27.2070 −0.881320 −0.440660 0.897674i $$-0.645256\pi$$
−0.440660 + 0.897674i $$0.645256\pi$$
$$954$$ 0 0
$$955$$ −14.9447 −0.483599
$$956$$ 26.5303 0.858053
$$957$$ 0 0
$$958$$ 12.7699 0.412575
$$959$$ −3.22168 −0.104033
$$960$$ 0 0
$$961$$ 16.6543 0.537237
$$962$$ 0.765413 0.0246779
$$963$$ 0 0
$$964$$ 35.1526 1.13219
$$965$$ 14.1476 0.455429
$$966$$ 0 0
$$967$$ −16.8015 −0.540300 −0.270150 0.962818i $$-0.587073\pi$$
−0.270150 + 0.962818i $$0.587073\pi$$
$$968$$ −10.5319 −0.338507
$$969$$ 0 0
$$970$$ −1.27163 −0.0408295
$$971$$ 17.4465 0.559884 0.279942 0.960017i $$-0.409685\pi$$
0.279942 + 0.960017i $$0.409685\pi$$
$$972$$ 0 0
$$973$$ 7.73329 0.247918
$$974$$ 3.15701 0.101157
$$975$$ 0 0
$$976$$ −6.40006 −0.204861
$$977$$ −32.0513 −1.02541 −0.512706 0.858564i $$-0.671357\pi$$
−0.512706 + 0.858564i $$0.671357\pi$$
$$978$$ 0 0
$$979$$ 11.9081 0.380586
$$980$$ −11.7699 −0.375974
$$981$$ 0 0
$$982$$ −9.08694 −0.289976
$$983$$ 16.5259 0.527094 0.263547 0.964646i $$-0.415108\pi$$
0.263547 + 0.964646i $$0.415108\pi$$
$$984$$ 0 0
$$985$$ −5.70471 −0.181767
$$986$$ −2.47457 −0.0788064
$$987$$ 0 0
$$988$$ −1.29883 −0.0413211
$$989$$ 79.1378 2.51644
$$990$$ 0 0
$$991$$ −9.34920 −0.296987 −0.148494 0.988913i $$-0.547442\pi$$
−0.148494 + 0.988913i $$0.547442\pi$$
$$992$$ 24.1289 0.766094
$$993$$ 0 0
$$994$$ −3.84208 −0.121863
$$995$$ 22.1432 0.701987
$$996$$ 0 0
$$997$$ −15.9956 −0.506584 −0.253292 0.967390i $$-0.581513\pi$$
−0.253292 + 0.967390i $$0.581513\pi$$
$$998$$ −6.83854 −0.216470
$$999$$ 0 0
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 1305.2.a.p.1.2 3
3.2 odd 2 145.2.a.c.1.2 3
5.4 even 2 6525.2.a.be.1.2 3
12.11 even 2 2320.2.a.n.1.1 3
15.2 even 4 725.2.b.e.349.4 6
15.8 even 4 725.2.b.e.349.3 6
15.14 odd 2 725.2.a.e.1.2 3
21.20 even 2 7105.2.a.o.1.2 3
24.5 odd 2 9280.2.a.bj.1.1 3
24.11 even 2 9280.2.a.br.1.3 3
87.86 odd 2 4205.2.a.f.1.2 3

By twisted newform
Twist Min Dim Char Parity Ord Type
145.2.a.c.1.2 3 3.2 odd 2
725.2.a.e.1.2 3 15.14 odd 2
725.2.b.e.349.3 6 15.8 even 4
725.2.b.e.349.4 6 15.2 even 4
1305.2.a.p.1.2 3 1.1 even 1 trivial
2320.2.a.n.1.1 3 12.11 even 2
4205.2.a.f.1.2 3 87.86 odd 2
6525.2.a.be.1.2 3 5.4 even 2
7105.2.a.o.1.2 3 21.20 even 2
9280.2.a.bj.1.1 3 24.5 odd 2
9280.2.a.br.1.3 3 24.11 even 2