# Properties

 Label 725.2.a.e.1.2 Level $725$ Weight $2$ Character 725.1 Self dual yes Analytic conductor $5.789$ 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] = [725,2,Mod(1,725)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(725, 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("725.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$725 = 5^{2} \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 725.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: $$5.78915414654$$ 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 $$\chi$$ $$=$$ 725.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-0.311108 q^{2} -2.90321 q^{3} -1.90321 q^{4} +0.903212 q^{6} +0.903212 q^{7} +1.21432 q^{8} +5.42864 q^{9} +O(q^{10})$$ $$q-0.311108 q^{2} -2.90321 q^{3} -1.90321 q^{4} +0.903212 q^{6} +0.903212 q^{7} +1.21432 q^{8} +5.42864 q^{9} -1.52543 q^{11} +5.52543 q^{12} +0.622216 q^{13} -0.280996 q^{14} +3.42864 q^{16} +7.95407 q^{17} -1.68889 q^{18} -1.09679 q^{19} -2.62222 q^{21} +0.474572 q^{22} -7.52543 q^{23} -3.52543 q^{24} -0.193576 q^{26} -7.05086 q^{27} -1.71900 q^{28} -1.00000 q^{29} -6.90321 q^{31} -3.49532 q^{32} +4.42864 q^{33} -2.47457 q^{34} -10.3319 q^{36} -3.95407 q^{37} +0.341219 q^{38} -1.80642 q^{39} +3.67307 q^{41} +0.815792 q^{42} +10.5161 q^{43} +2.90321 q^{44} +2.34122 q^{46} -6.90321 q^{47} -9.95407 q^{48} -6.18421 q^{49} -23.0923 q^{51} -1.18421 q^{52} -6.42864 q^{53} +2.19358 q^{54} +1.09679 q^{56} +3.18421 q^{57} +0.311108 q^{58} -1.67307 q^{59} -1.86665 q^{61} +2.14764 q^{62} +4.90321 q^{63} -5.76986 q^{64} -1.37778 q^{66} -11.5254 q^{67} -15.1383 q^{68} +21.8479 q^{69} +13.6731 q^{71} +6.59210 q^{72} -10.1891 q^{73} +1.23014 q^{74} +2.08742 q^{76} -1.37778 q^{77} +0.561993 q^{78} +9.13828 q^{79} +4.18421 q^{81} -1.14272 q^{82} -10.7096 q^{83} +4.99063 q^{84} -3.27163 q^{86} +2.90321 q^{87} -1.85236 q^{88} -7.80642 q^{89} +0.561993 q^{91} +14.3225 q^{92} +20.0415 q^{93} +2.14764 q^{94} +10.1476 q^{96} +4.08742 q^{97} +1.92396 q^{98} -8.28100 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - q^{2} - 2 q^{3} + q^{4} - 4 q^{6} - 4 q^{7} - 3 q^{8} + 3 q^{9}+O(q^{10})$$ 3 * q - q^2 - 2 * q^3 + q^4 - 4 * q^6 - 4 * q^7 - 3 * q^8 + 3 * q^9 $$3 q - q^{2} - 2 q^{3} + q^{4} - 4 q^{6} - 4 q^{7} - 3 q^{8} + 3 q^{9} + 2 q^{11} + 10 q^{12} + 2 q^{13} + 6 q^{14} - 3 q^{16} + 4 q^{17} - 5 q^{18} - 10 q^{19} - 8 q^{21} + 8 q^{22} - 16 q^{23} - 4 q^{24} - 14 q^{26} - 8 q^{27} - 12 q^{28} - 3 q^{29} - 14 q^{31} + 3 q^{32} - 14 q^{34} - 11 q^{36} + 8 q^{37} + 8 q^{38} + 8 q^{39} - 2 q^{41} + 16 q^{42} - 2 q^{43} + 2 q^{44} + 14 q^{46} - 14 q^{47} - 10 q^{48} - 5 q^{49} - 16 q^{51} + 10 q^{52} - 6 q^{53} + 20 q^{54} + 10 q^{56} - 4 q^{57} + q^{58} + 8 q^{59} - 6 q^{61} + 8 q^{63} - 11 q^{64} - 4 q^{66} - 28 q^{67} - 12 q^{68} + 12 q^{69} + 28 q^{71} + 13 q^{72} + 16 q^{73} + 10 q^{74} - 14 q^{76} - 4 q^{77} - 12 q^{78} - 6 q^{79} - q^{81} - 30 q^{82} - 12 q^{83} - 12 q^{84} + 24 q^{86} + 2 q^{87} - 12 q^{88} - 10 q^{89} - 12 q^{91} - 4 q^{92} + 20 q^{93} + 24 q^{96} - 8 q^{97} - 21 q^{98} - 18 q^{99}+O(q^{100})$$ 3 * q - q^2 - 2 * q^3 + q^4 - 4 * q^6 - 4 * q^7 - 3 * q^8 + 3 * q^9 + 2 * q^11 + 10 * q^12 + 2 * q^13 + 6 * q^14 - 3 * q^16 + 4 * q^17 - 5 * q^18 - 10 * q^19 - 8 * q^21 + 8 * q^22 - 16 * q^23 - 4 * q^24 - 14 * q^26 - 8 * q^27 - 12 * q^28 - 3 * q^29 - 14 * q^31 + 3 * q^32 - 14 * q^34 - 11 * q^36 + 8 * q^37 + 8 * q^38 + 8 * q^39 - 2 * q^41 + 16 * q^42 - 2 * q^43 + 2 * q^44 + 14 * q^46 - 14 * q^47 - 10 * q^48 - 5 * q^49 - 16 * q^51 + 10 * q^52 - 6 * q^53 + 20 * q^54 + 10 * q^56 - 4 * q^57 + q^58 + 8 * q^59 - 6 * q^61 + 8 * q^63 - 11 * q^64 - 4 * q^66 - 28 * q^67 - 12 * q^68 + 12 * q^69 + 28 * q^71 + 13 * q^72 + 16 * q^73 + 10 * q^74 - 14 * q^76 - 4 * q^77 - 12 * q^78 - 6 * q^79 - q^81 - 30 * q^82 - 12 * q^83 - 12 * q^84 + 24 * q^86 + 2 * q^87 - 12 * q^88 - 10 * q^89 - 12 * q^91 - 4 * q^92 + 20 * q^93 + 24 * q^96 - 8 * q^97 - 21 * q^98 - 18 * q^99

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −0.311108 −0.219986 −0.109993 0.993932i $$-0.535083\pi$$
−0.109993 + 0.993932i $$0.535083\pi$$
$$3$$ −2.90321 −1.67617 −0.838085 0.545540i $$-0.816325\pi$$
−0.838085 + 0.545540i $$0.816325\pi$$
$$4$$ −1.90321 −0.951606
$$5$$ 0 0
$$6$$ 0.903212 0.368735
$$7$$ 0.903212 0.341382 0.170691 0.985325i $$-0.445400\pi$$
0.170691 + 0.985325i $$0.445400\pi$$
$$8$$ 1.21432 0.429327
$$9$$ 5.42864 1.80955
$$10$$ 0 0
$$11$$ −1.52543 −0.459934 −0.229967 0.973198i $$-0.573862\pi$$
−0.229967 + 0.973198i $$0.573862\pi$$
$$12$$ 5.52543 1.59505
$$13$$ 0.622216 0.172572 0.0862858 0.996270i $$-0.472500\pi$$
0.0862858 + 0.996270i $$0.472500\pi$$
$$14$$ −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 $$0.0849820\pi$$
$$18$$ −1.68889 −0.398076
$$19$$ −1.09679 −0.251620 −0.125810 0.992054i $$-0.540153\pi$$
−0.125810 + 0.992054i $$0.540153\pi$$
$$20$$ 0 0
$$21$$ −2.62222 −0.572214
$$22$$ 0.474572 0.101179
$$23$$ −7.52543 −1.56916 −0.784580 0.620028i $$-0.787121\pi$$
−0.784580 + 0.620028i $$0.787121\pi$$
$$24$$ −3.52543 −0.719625
$$25$$ 0 0
$$26$$ −0.193576 −0.0379634
$$27$$ −7.05086 −1.35694
$$28$$ −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
$$33$$ 4.42864 0.770927
$$34$$ −2.47457 −0.424386
$$35$$ 0 0
$$36$$ −10.3319 −1.72198
$$37$$ −3.95407 −0.650045 −0.325022 0.945706i $$-0.605372\pi$$
−0.325022 + 0.945706i $$0.605372\pi$$
$$38$$ 0.341219 0.0553531
$$39$$ −1.80642 −0.289259
$$40$$ 0 0
$$41$$ 3.67307 0.573637 0.286819 0.957985i $$-0.407402\pi$$
0.286819 + 0.957985i $$0.407402\pi$$
$$42$$ 0.815792 0.125879
$$43$$ 10.5161 1.60368 0.801842 0.597536i $$-0.203854\pi$$
0.801842 + 0.597536i $$0.203854\pi$$
$$44$$ 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 $$0.667943\pi$$
$$48$$ −9.95407 −1.43675
$$49$$ −6.18421 −0.883458
$$50$$ 0 0
$$51$$ −23.0923 −3.23357
$$52$$ −1.18421 −0.164220
$$53$$ −6.42864 −0.883042 −0.441521 0.897251i $$-0.645561\pi$$
−0.441521 + 0.897251i $$0.645561\pi$$
$$54$$ 2.19358 0.298508
$$55$$ 0 0
$$56$$ 1.09679 0.146564
$$57$$ 3.18421 0.421759
$$58$$ 0.311108 0.0408505
$$59$$ −1.67307 −0.217815 −0.108908 0.994052i $$-0.534735\pi$$
−0.108908 + 0.994052i $$0.534735\pi$$
$$60$$ 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
$$63$$ 4.90321 0.617747
$$64$$ −5.76986 −0.721232
$$65$$ 0 0
$$66$$ −1.37778 −0.169594
$$67$$ −11.5254 −1.40806 −0.704028 0.710173i $$-0.748617\pi$$
−0.704028 + 0.710173i $$0.748617\pi$$
$$68$$ −15.1383 −1.83579
$$69$$ 21.8479 2.63018
$$70$$ 0 0
$$71$$ 13.6731 1.62269 0.811347 0.584564i $$-0.198734\pi$$
0.811347 + 0.584564i $$0.198734\pi$$
$$72$$ 6.59210 0.776887
$$73$$ −10.1891 −1.19255 −0.596274 0.802781i $$-0.703353\pi$$
−0.596274 + 0.802781i $$0.703353\pi$$
$$74$$ 1.23014 0.143001
$$75$$ 0 0
$$76$$ 2.08742 0.239444
$$77$$ −1.37778 −0.157013
$$78$$ 0.561993 0.0636331
$$79$$ 9.13828 1.02814 0.514068 0.857749i $$-0.328138\pi$$
0.514068 + 0.857749i $$0.328138\pi$$
$$80$$ 0 0
$$81$$ 4.18421 0.464912
$$82$$ −1.14272 −0.126192
$$83$$ −10.7096 −1.17554 −0.587768 0.809030i $$-0.699993\pi$$
−0.587768 + 0.809030i $$0.699993\pi$$
$$84$$ 4.99063 0.544523
$$85$$ 0 0
$$86$$ −3.27163 −0.352789
$$87$$ 2.90321 0.311257
$$88$$ −1.85236 −0.197462
$$89$$ −7.80642 −0.827479 −0.413740 0.910395i $$-0.635778\pi$$
−0.413740 + 0.910395i $$0.635778\pi$$
$$90$$ 0 0
$$91$$ 0.561993 0.0589128
$$92$$ 14.3225 1.49322
$$93$$ 20.0415 2.07821
$$94$$ 2.14764 0.221512
$$95$$ 0 0
$$96$$ 10.1476 1.03569
$$97$$ 4.08742 0.415015 0.207507 0.978233i $$-0.433465\pi$$
0.207507 + 0.978233i $$0.433465\pi$$
$$98$$ 1.92396 0.194349
$$99$$ −8.28100 −0.832271
$$100$$ 0 0
$$101$$ 13.9081 1.38391 0.691956 0.721940i $$-0.256749\pi$$
0.691956 + 0.721940i $$0.256749\pi$$
$$102$$ 7.18421 0.711343
$$103$$ 12.9447 1.27548 0.637740 0.770252i $$-0.279870\pi$$
0.637740 + 0.770252i $$0.279870\pi$$
$$104$$ 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 $$0.678538\pi$$
$$108$$ 13.4193 1.29127
$$109$$ −18.0415 −1.72806 −0.864031 0.503439i $$-0.832068\pi$$
−0.864031 + 0.503439i $$0.832068\pi$$
$$110$$ 0 0
$$111$$ 11.4795 1.08959
$$112$$ 3.09679 0.292619
$$113$$ 10.2810 0.967155 0.483577 0.875302i $$-0.339337\pi$$
0.483577 + 0.875302i $$0.339337\pi$$
$$114$$ −0.990632 −0.0927812
$$115$$ 0 0
$$116$$ 1.90321 0.176709
$$117$$ 3.37778 0.312276
$$118$$ 0.520505 0.0479164
$$119$$ 7.18421 0.658575
$$120$$ 0 0
$$121$$ −8.67307 −0.788461
$$122$$ 0.580728 0.0525767
$$123$$ −10.6637 −0.961514
$$124$$ 13.1383 1.17985
$$125$$ 0 0
$$126$$ −1.52543 −0.135896
$$127$$ −6.22077 −0.552004 −0.276002 0.961157i $$-0.589010\pi$$
−0.276002 + 0.961157i $$0.589010\pi$$
$$128$$ 8.78568 0.776552
$$129$$ −30.5303 −2.68805
$$130$$ 0 0
$$131$$ −11.7605 −1.02752 −0.513759 0.857934i $$-0.671748\pi$$
−0.513759 + 0.857934i $$0.671748\pi$$
$$132$$ −8.42864 −0.733619
$$133$$ −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 $$0.451309\pi$$
$$138$$ −6.79706 −0.578604
$$139$$ −8.56199 −0.726219 −0.363109 0.931747i $$-0.618285\pi$$
−0.363109 + 0.931747i $$0.618285\pi$$
$$140$$ 0 0
$$141$$ 20.0415 1.68780
$$142$$ −4.25380 −0.356971
$$143$$ −0.949145 −0.0793715
$$144$$ 18.6128 1.55107
$$145$$ 0 0
$$146$$ 3.16992 0.262344
$$147$$ 17.9541 1.48083
$$148$$ 7.52543 0.618586
$$149$$ −5.61285 −0.459822 −0.229911 0.973212i $$-0.573844\pi$$
−0.229911 + 0.973212i $$0.573844\pi$$
$$150$$ 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
$$153$$ 43.1798 3.49088
$$154$$ 0.428639 0.0345408
$$155$$ 0 0
$$156$$ 3.43801 0.275261
$$157$$ −2.28100 −0.182043 −0.0910217 0.995849i $$-0.529013\pi$$
−0.0910217 + 0.995849i $$0.529013\pi$$
$$158$$ −2.84299 −0.226176
$$159$$ 18.6637 1.48013
$$160$$ 0 0
$$161$$ −6.79706 −0.535683
$$162$$ −1.30174 −0.102274
$$163$$ −16.3225 −1.27848 −0.639238 0.769009i $$-0.720750\pi$$
−0.639238 + 0.769009i $$0.720750\pi$$
$$164$$ −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 $$0.440917\pi$$
$$168$$ −3.18421 −0.245667
$$169$$ −12.6128 −0.970219
$$170$$ 0 0
$$171$$ −5.95407 −0.455319
$$172$$ −20.0143 −1.52608
$$173$$ −4.23506 −0.321986 −0.160993 0.986956i $$-0.551470\pi$$
−0.160993 + 0.986956i $$0.551470\pi$$
$$174$$ −0.903212 −0.0684723
$$175$$ 0 0
$$176$$ −5.23014 −0.394237
$$177$$ 4.85728 0.365095
$$178$$ 2.42864 0.182034
$$179$$ 9.71456 0.726100 0.363050 0.931770i $$-0.381735\pi$$
0.363050 + 0.931770i $$0.381735\pi$$
$$180$$ 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
$$183$$ 5.41927 0.400604
$$184$$ −9.13828 −0.673683
$$185$$ 0 0
$$186$$ −6.23506 −0.457177
$$187$$ −12.1334 −0.887279
$$188$$ 13.1383 0.958207
$$189$$ −6.36842 −0.463234
$$190$$ 0 0
$$191$$ −14.9447 −1.08136 −0.540680 0.841228i $$-0.681833\pi$$
−0.540680 + 0.841228i $$0.681833\pi$$
$$192$$ 16.7511 1.20891
$$193$$ 14.1476 1.01837 0.509185 0.860657i $$-0.329947\pi$$
0.509185 + 0.860657i $$0.329947\pi$$
$$194$$ −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 $$0.434859\pi$$
$$198$$ 2.57628 0.183088
$$199$$ −22.1432 −1.56969 −0.784845 0.619692i $$-0.787257\pi$$
−0.784845 + 0.619692i $$0.787257\pi$$
$$200$$ 0 0
$$201$$ 33.4608 2.36014
$$202$$ −4.32693 −0.304442
$$203$$ −0.903212 −0.0633930
$$204$$ 43.9496 3.07709
$$205$$ 0 0
$$206$$ −4.02720 −0.280588
$$207$$ −40.8528 −2.83947
$$208$$ 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
$$213$$ −39.6958 −2.71991
$$214$$ 3.42372 0.234040
$$215$$ 0 0
$$216$$ −8.56199 −0.582570
$$217$$ −6.23506 −0.423264
$$218$$ 5.61285 0.380150
$$219$$ 29.5812 1.99891
$$220$$ 0 0
$$221$$ 4.94914 0.332916
$$222$$ −3.57136 −0.239694
$$223$$ −9.03657 −0.605133 −0.302567 0.953128i $$-0.597843\pi$$
−0.302567 + 0.953128i $$0.597843\pi$$
$$224$$ −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 $$0.722717\pi$$
$$228$$ −6.06022 −0.401348
$$229$$ −25.6128 −1.69254 −0.846272 0.532751i $$-0.821158\pi$$
−0.846272 + 0.532751i $$0.821158\pi$$
$$230$$ 0 0
$$231$$ 4.00000 0.263181
$$232$$ −1.21432 −0.0797240
$$233$$ 3.12399 0.204659 0.102330 0.994751i $$-0.467370\pi$$
0.102330 + 0.994751i $$0.467370\pi$$
$$234$$ −1.05086 −0.0686965
$$235$$ 0 0
$$236$$ 3.18421 0.207274
$$237$$ −26.5303 −1.72333
$$238$$ −2.23506 −0.144878
$$239$$ 13.9398 0.901689 0.450845 0.892602i $$-0.351123\pi$$
0.450845 + 0.892602i $$0.351123\pi$$
$$240$$ 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
$$243$$ 9.00492 0.577666
$$244$$ 3.55262 0.227433
$$245$$ 0 0
$$246$$ 3.31756 0.211520
$$247$$ −0.682439 −0.0434225
$$248$$ −8.38271 −0.532302
$$249$$ 31.0923 1.97040
$$250$$ 0 0
$$251$$ −13.7921 −0.870552 −0.435276 0.900297i $$-0.643349\pi$$
−0.435276 + 0.900297i $$0.643349\pi$$
$$252$$ −9.33185 −0.587851
$$253$$ 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 $$0.485307\pi$$
$$258$$ 9.49823 0.591334
$$259$$ −3.57136 −0.221914
$$260$$ 0 0
$$261$$ −5.42864 −0.336024
$$262$$ 3.65878 0.226040
$$263$$ 0.442930 0.0273122 0.0136561 0.999907i $$-0.495653\pi$$
0.0136561 + 0.999907i $$0.495653\pi$$
$$264$$ 5.37778 0.330980
$$265$$ 0 0
$$266$$ 0.308193 0.0188965
$$267$$ 22.6637 1.38700
$$268$$ 21.9353 1.33991
$$269$$ 3.93978 0.240212 0.120106 0.992761i $$-0.461676\pi$$
0.120106 + 0.992761i $$0.461676\pi$$
$$270$$ 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
$$273$$ −1.63158 −0.0987479
$$274$$ −1.10970 −0.0670391
$$275$$ 0 0
$$276$$ −41.5812 −2.50289
$$277$$ −5.57136 −0.334751 −0.167375 0.985893i $$-0.553529\pi$$
−0.167375 + 0.985893i $$0.553529\pi$$
$$278$$ 2.66370 0.159758
$$279$$ −37.4750 −2.24357
$$280$$ 0 0
$$281$$ 6.69535 0.399411 0.199705 0.979856i $$-0.436001\pi$$
0.199705 + 0.979856i $$0.436001\pi$$
$$282$$ −6.23506 −0.371293
$$283$$ −25.8020 −1.53377 −0.766884 0.641785i $$-0.778194\pi$$
−0.766884 + 0.641785i $$0.778194\pi$$
$$284$$ −26.0228 −1.54417
$$285$$ 0 0
$$286$$ 0.295286 0.0174607
$$287$$ 3.31756 0.195829
$$288$$ −18.9748 −1.11810
$$289$$ 46.2672 2.72160
$$290$$ 0 0
$$291$$ −11.8666 −0.695635
$$292$$ 19.3921 1.13484
$$293$$ 18.8430 1.10082 0.550410 0.834895i $$-0.314472\pi$$
0.550410 + 0.834895i $$0.314472\pi$$
$$294$$ −5.58565 −0.325762
$$295$$ 0 0
$$296$$ −4.80150 −0.279082
$$297$$ 10.7556 0.624101
$$298$$ 1.74620 0.101155
$$299$$ −4.68244 −0.270792
$$300$$ 0 0
$$301$$ 9.49823 0.547469
$$302$$ −3.35905 −0.193292
$$303$$ −40.3783 −2.31967
$$304$$ −3.76049 −0.215679
$$305$$ 0 0
$$306$$ −13.4336 −0.767946
$$307$$ 1.65878 0.0946716 0.0473358 0.998879i $$-0.484927\pi$$
0.0473358 + 0.998879i $$0.484927\pi$$
$$308$$ 2.62222 0.149415
$$309$$ −37.5812 −2.13792
$$310$$ 0 0
$$311$$ 21.3002 1.20782 0.603912 0.797051i $$-0.293608\pi$$
0.603912 + 0.797051i $$0.293608\pi$$
$$312$$ −2.19358 −0.124187
$$313$$ −8.62222 −0.487356 −0.243678 0.969856i $$-0.578354\pi$$
−0.243678 + 0.969856i $$0.578354\pi$$
$$314$$ 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 $$0.217771\pi$$
$$318$$ −5.80642 −0.325608
$$319$$ 1.52543 0.0854075
$$320$$ 0 0
$$321$$ 31.9496 1.78325
$$322$$ 2.11462 0.117843
$$323$$ −8.72393 −0.485412
$$324$$ −7.96343 −0.442413
$$325$$ 0 0
$$326$$ 5.07805 0.281247
$$327$$ 52.3783 2.89652
$$328$$ 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
$$333$$ −21.4652 −1.17629
$$334$$ −1.48394 −0.0811976
$$335$$ 0 0
$$336$$ −8.99063 −0.490479
$$337$$ −11.9956 −0.653439 −0.326720 0.945121i $$-0.605943\pi$$
−0.326720 + 0.945121i $$0.605943\pi$$
$$338$$ 3.92396 0.213435
$$339$$ −29.8479 −1.62112
$$340$$ 0 0
$$341$$ 10.5303 0.570250
$$342$$ 1.85236 0.100164
$$343$$ −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 $$0.552766\pi$$
$$348$$ −5.52543 −0.296194
$$349$$ −7.12399 −0.381338 −0.190669 0.981654i $$-0.561066\pi$$
−0.190669 + 0.981654i $$0.561066\pi$$
$$350$$ 0 0
$$351$$ −4.38715 −0.234169
$$352$$ 5.33185 0.284189
$$353$$ 16.9175 0.900428 0.450214 0.892921i $$-0.351348\pi$$
0.450214 + 0.892921i $$0.351348\pi$$
$$354$$ −1.51114 −0.0803160
$$355$$ 0 0
$$356$$ 14.8573 0.787434
$$357$$ −20.8573 −1.10388
$$358$$ −3.02227 −0.159732
$$359$$ 36.7096 1.93746 0.968730 0.248116i $$-0.0798115\pi$$
0.968730 + 0.248116i $$0.0798115\pi$$
$$360$$ 0 0
$$361$$ −17.7971 −0.936687
$$362$$ −0.101710 −0.00534577
$$363$$ 25.1798 1.32159
$$364$$ −1.06959 −0.0560618
$$365$$ 0 0
$$366$$ −1.68598 −0.0881275
$$367$$ −8.41435 −0.439225 −0.219613 0.975587i $$-0.570479\pi$$
−0.219613 + 0.975587i $$0.570479\pi$$
$$368$$ −25.8020 −1.34502
$$369$$ 19.9398 1.03802
$$370$$ 0 0
$$371$$ −5.80642 −0.301455
$$372$$ −38.1432 −1.97763
$$373$$ 8.66370 0.448590 0.224295 0.974521i $$-0.427992\pi$$
0.224295 + 0.974521i $$0.427992\pi$$
$$374$$ 3.77478 0.195189
$$375$$ 0 0
$$376$$ −8.38271 −0.432305
$$377$$ −0.622216 −0.0320457
$$378$$ 1.98126 0.101905
$$379$$ −2.76986 −0.142278 −0.0711390 0.997466i $$-0.522663\pi$$
−0.0711390 + 0.997466i $$0.522663\pi$$
$$380$$ 0 0
$$381$$ 18.0602 0.925253
$$382$$ 4.64941 0.237885
$$383$$ −1.67752 −0.0857171 −0.0428585 0.999081i $$-0.513646\pi$$
−0.0428585 + 0.999081i $$0.513646\pi$$
$$384$$ −25.5067 −1.30163
$$385$$ 0 0
$$386$$ −4.40144 −0.224028
$$387$$ 57.0879 2.90194
$$388$$ −7.77923 −0.394930
$$389$$ −5.77478 −0.292793 −0.146397 0.989226i $$-0.546768\pi$$
−0.146397 + 0.989226i $$0.546768\pi$$
$$390$$ 0 0
$$391$$ −59.8578 −3.02714
$$392$$ −7.50961 −0.379292
$$393$$ 34.1432 1.72230
$$394$$ −1.77478 −0.0894122
$$395$$ 0 0
$$396$$ 15.7605 0.791994
$$397$$ −29.9081 −1.50105 −0.750523 0.660844i $$-0.770198\pi$$
−0.750523 + 0.660844i $$0.770198\pi$$
$$398$$ 6.88892 0.345310
$$399$$ 2.87601 0.143981
$$400$$ 0 0
$$401$$ 8.53035 0.425985 0.212993 0.977054i $$-0.431679\pi$$
0.212993 + 0.977054i $$0.431679\pi$$
$$402$$ −10.4099 −0.519199
$$403$$ −4.29529 −0.213963
$$404$$ −26.4701 −1.31694
$$405$$ 0 0
$$406$$ 0.280996 0.0139456
$$407$$ 6.03164 0.298977
$$408$$ −28.0415 −1.38826
$$409$$ 5.09234 0.251800 0.125900 0.992043i $$-0.459818\pi$$
0.125900 + 0.992043i $$0.459818\pi$$
$$410$$ 0 0
$$411$$ −10.3555 −0.510800
$$412$$ −24.6365 −1.21375
$$413$$ −1.51114 −0.0743582
$$414$$ 12.7096 0.624645
$$415$$ 0 0
$$416$$ −2.17484 −0.106630
$$417$$ 24.8573 1.21727
$$418$$ −0.520505 −0.0254588
$$419$$ −24.3368 −1.18893 −0.594465 0.804122i $$-0.702636\pi$$
−0.594465 + 0.804122i $$0.702636\pi$$
$$420$$ 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
$$423$$ −37.4750 −1.82210
$$424$$ −7.80642 −0.379113
$$425$$ 0 0
$$426$$ 12.3497 0.598344
$$427$$ −1.68598 −0.0815902
$$428$$ 20.9447 1.01240
$$429$$ 2.75557 0.133040
$$430$$ 0 0
$$431$$ 4.26671 0.205520 0.102760 0.994706i $$-0.467233\pi$$
0.102760 + 0.994706i $$0.467233\pi$$
$$432$$ −24.1748 −1.16311
$$433$$ −27.0049 −1.29777 −0.648887 0.760885i $$-0.724765\pi$$
−0.648887 + 0.760885i $$0.724765\pi$$
$$434$$ 1.93978 0.0931123
$$435$$ 0 0
$$436$$ 34.3368 1.64443
$$437$$ 8.25380 0.394833
$$438$$ −9.20294 −0.439734
$$439$$ −2.03164 −0.0969650 −0.0484825 0.998824i $$-0.515439\pi$$
−0.0484825 + 0.998824i $$0.515439\pi$$
$$440$$ 0 0
$$441$$ −33.5718 −1.59866
$$442$$ −1.53972 −0.0732369
$$443$$ 3.46520 0.164637 0.0823184 0.996606i $$-0.473768\pi$$
0.0823184 + 0.996606i $$0.473768\pi$$
$$444$$ −21.8479 −1.03686
$$445$$ 0 0
$$446$$ 2.81135 0.133121
$$447$$ 16.2953 0.770741
$$448$$ −5.21141 −0.246216
$$449$$ 37.3590 1.76308 0.881541 0.472107i $$-0.156506\pi$$
0.881541 + 0.472107i $$0.156506\pi$$
$$450$$ 0 0
$$451$$ −5.60300 −0.263835
$$452$$ −19.5669 −0.920350
$$453$$ −31.3461 −1.47277
$$454$$ 6.03704 0.283332
$$455$$ 0 0
$$456$$ 3.86665 0.181072
$$457$$ −13.4509 −0.629207 −0.314604 0.949223i $$-0.601872\pi$$
−0.314604 + 0.949223i $$0.601872\pi$$
$$458$$ 7.96836 0.372337
$$459$$ −56.0830 −2.61773
$$460$$ 0 0
$$461$$ −16.2766 −0.758075 −0.379037 0.925381i $$-0.623745\pi$$
−0.379037 + 0.925381i $$0.623745\pi$$
$$462$$ −1.24443 −0.0578962
$$463$$ 30.3926 1.41246 0.706231 0.707982i $$-0.250394\pi$$
0.706231 + 0.707982i $$0.250394\pi$$
$$464$$ −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 $$0.491245\pi$$
$$468$$ −6.42864 −0.297164
$$469$$ −10.4099 −0.480685
$$470$$ 0 0
$$471$$ 6.62222 0.305136
$$472$$ −2.03164 −0.0935139
$$473$$ −16.0415 −0.737588
$$474$$ 8.25380 0.379110
$$475$$ 0 0
$$476$$ −13.6731 −0.626704
$$477$$ −34.8988 −1.59790
$$478$$ −4.33677 −0.198359
$$479$$ 41.0464 1.87546 0.937729 0.347367i $$-0.112924\pi$$
0.937729 + 0.347367i $$0.112924\pi$$
$$480$$ 0 0
$$481$$ −2.46028 −0.112179
$$482$$ 5.74620 0.261732
$$483$$ 19.7333 0.897896
$$484$$ 16.5067 0.750304
$$485$$ 0 0
$$486$$ −2.80150 −0.127079
$$487$$ 10.1476 0.459834 0.229917 0.973210i $$-0.426155\pi$$
0.229917 + 0.973210i $$0.426155\pi$$
$$488$$ −2.26671 −0.102609
$$489$$ 47.3876 2.14294
$$490$$ 0 0
$$491$$ −29.2083 −1.31815 −0.659077 0.752075i $$-0.729053\pi$$
−0.659077 + 0.752075i $$0.729053\pi$$
$$492$$ 20.2953 0.914982
$$493$$ −7.95407 −0.358233
$$494$$ 0.212312 0.00955237
$$495$$ 0 0
$$496$$ −23.6686 −1.06275
$$497$$ 12.3497 0.553959
$$498$$ −9.67307 −0.433461
$$499$$ 21.9813 0.984017 0.492008 0.870591i $$-0.336263\pi$$
0.492008 + 0.870591i $$0.336263\pi$$
$$500$$ 0 0
$$501$$ −13.8479 −0.618679
$$502$$ 4.29084 0.191510
$$503$$ −5.77923 −0.257683 −0.128841 0.991665i $$-0.541126\pi$$
−0.128841 + 0.991665i $$0.541126\pi$$
$$504$$ 5.95407 0.265215
$$505$$ 0 0
$$506$$ −3.57136 −0.158766
$$507$$ 36.6178 1.62625
$$508$$ 11.8394 0.525291
$$509$$ 13.6543 0.605218 0.302609 0.953115i $$-0.402142\pi$$
0.302609 + 0.953115i $$0.402142\pi$$
$$510$$ 0 0
$$511$$ −9.20294 −0.407114
$$512$$ −20.3111 −0.897633
$$513$$ 7.73329 0.341433
$$514$$ −0.460282 −0.0203022
$$515$$ 0 0
$$516$$ 58.1057 2.55796
$$517$$ 10.5303 0.463124
$$518$$ 1.11108 0.0488180
$$519$$ 12.2953 0.539703
$$520$$ 0 0
$$521$$ −19.6731 −0.861893 −0.430946 0.902378i $$-0.641820\pi$$
−0.430946 + 0.902378i $$0.641820\pi$$
$$522$$ 1.68889 0.0739208
$$523$$ 15.1383 0.661951 0.330975 0.943639i $$-0.392622\pi$$
0.330975 + 0.943639i $$0.392622\pi$$
$$524$$ 22.3827 0.977793
$$525$$ 0 0
$$526$$ −0.137799 −0.00600832
$$527$$ −54.9086 −2.39186
$$528$$ 15.1842 0.660808
$$529$$ 33.6321 1.46226
$$530$$ 0 0
$$531$$ −9.08250 −0.394147
$$532$$ 1.88538 0.0817417
$$533$$ 2.28544 0.0989935
$$534$$ −7.05086 −0.305120
$$535$$ 0 0
$$536$$ −13.9956 −0.604516
$$537$$ −28.2034 −1.21707
$$538$$ −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
$$543$$ −0.949145 −0.0407317
$$544$$ −27.8020 −1.19200
$$545$$ 0 0
$$546$$ 0.507598 0.0217232
$$547$$ 15.3635 0.656896 0.328448 0.944522i $$-0.393475\pi$$
0.328448 + 0.944522i $$0.393475\pi$$
$$548$$ −6.78859 −0.289994
$$549$$ −10.1334 −0.432481
$$550$$ 0 0
$$551$$ 1.09679 0.0467247
$$552$$ 26.5303 1.12921
$$553$$ 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 $$0.432880\pi$$
$$558$$ 11.6588 0.493556
$$559$$ 6.54326 0.276750
$$560$$ 0 0
$$561$$ 35.2257 1.48723
$$562$$ −2.08297 −0.0878650
$$563$$ 27.4938 1.15872 0.579362 0.815070i $$-0.303302\pi$$
0.579362 + 0.815070i $$0.303302\pi$$
$$564$$ −38.1432 −1.60612
$$565$$ 0 0
$$566$$ 8.02720 0.337408
$$567$$ 3.77923 0.158713
$$568$$ 16.6035 0.696667
$$569$$ 17.3590 0.727729 0.363865 0.931452i $$-0.381457\pi$$
0.363865 + 0.931452i $$0.381457\pi$$
$$570$$ 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
$$573$$ 43.3876 1.81254
$$574$$ −1.03212 −0.0430798
$$575$$ 0 0
$$576$$ −31.3225 −1.30510
$$577$$ 10.6178 0.442024 0.221012 0.975271i $$-0.429064\pi$$
0.221012 + 0.975271i $$0.429064\pi$$
$$578$$ −14.3941 −0.598715
$$579$$ −41.0736 −1.70696
$$580$$ 0 0
$$581$$ −9.67307 −0.401307
$$582$$ 3.69181 0.153030
$$583$$ 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 $$0.440903\pi$$
$$588$$ −34.1704 −1.40916
$$589$$ 7.57136 0.311972
$$590$$ 0 0
$$591$$ −16.5620 −0.681269
$$592$$ −13.5571 −0.557192
$$593$$ −14.1619 −0.581561 −0.290780 0.956790i $$-0.593915\pi$$
−0.290780 + 0.956790i $$0.593915\pi$$
$$594$$ −3.34614 −0.137294
$$595$$ 0 0
$$596$$ 10.6824 0.437570
$$597$$ 64.2864 2.63107
$$598$$ 1.45674 0.0595707
$$599$$ −22.5575 −0.921676 −0.460838 0.887484i $$-0.652451\pi$$
−0.460838 + 0.887484i $$0.652451\pi$$
$$600$$ 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
$$603$$ −62.5674 −2.54794
$$604$$ −20.5491 −0.836130
$$605$$ 0 0
$$606$$ 12.5620 0.510296
$$607$$ −13.5955 −0.551824 −0.275912 0.961183i $$-0.588980\pi$$
−0.275912 + 0.961183i $$0.588980\pi$$
$$608$$ 3.83362 0.155474
$$609$$ 2.62222 0.106258
$$610$$ 0 0
$$611$$ −4.29529 −0.173769
$$612$$ −82.1802 −3.32194
$$613$$ 42.0830 1.69972 0.849858 0.527012i $$-0.176688\pi$$
0.849858 + 0.527012i $$0.176688\pi$$
$$614$$ −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 $$0.735625\pi$$
$$618$$ 11.6918 0.470313
$$619$$ −14.6780 −0.589958 −0.294979 0.955504i $$-0.595313\pi$$
−0.294979 + 0.955504i $$0.595313\pi$$
$$620$$ 0 0
$$621$$ 53.0607 2.12925
$$622$$ −6.62666 −0.265705
$$623$$ −7.05086 −0.282487
$$624$$ −6.19358 −0.247941
$$625$$ 0 0
$$626$$ 2.68244 0.107212
$$627$$ −4.85728 −0.193981
$$628$$ 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
$$633$$ 60.5116 2.40512
$$634$$ −8.58517 −0.340961
$$635$$ 0 0
$$636$$ −35.5210 −1.40850
$$637$$ −3.84791 −0.152460
$$638$$ −0.474572 −0.0187885
$$639$$ 74.2262 2.93634
$$640$$ 0 0
$$641$$ 34.8988 1.37842 0.689209 0.724562i $$-0.257958\pi$$
0.689209 + 0.724562i $$0.257958\pi$$
$$642$$ −9.93978 −0.392292
$$643$$ 41.9768 1.65540 0.827702 0.561168i $$-0.189648\pi$$
0.827702 + 0.561168i $$0.189648\pi$$
$$644$$ 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 $$0.534262\pi$$
$$648$$ 5.08097 0.199599
$$649$$ 2.55215 0.100181
$$650$$ 0 0
$$651$$ 18.1017 0.709462
$$652$$ 31.0651 1.21660
$$653$$ 8.76986 0.343191 0.171596 0.985167i $$-0.445108\pi$$
0.171596 + 0.985167i $$0.445108\pi$$
$$654$$ −16.2953 −0.637196
$$655$$ 0 0
$$656$$ 12.5936 0.491699
$$657$$ −55.3131 −2.15797
$$658$$ 1.93978 0.0756204
$$659$$ 3.29036 0.128174 0.0640872 0.997944i $$-0.479586\pi$$
0.0640872 + 0.997944i $$0.479586\pi$$
$$660$$ 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
$$663$$ −14.3684 −0.558023
$$664$$ −13.0049 −0.504689
$$665$$ 0 0
$$666$$ 6.67799 0.258767
$$667$$ 7.52543 0.291386
$$668$$ −9.07805 −0.351240
$$669$$ 26.2351 1.01431
$$670$$ 0 0
$$671$$ 2.84743 0.109924
$$672$$ 9.16547 0.353566
$$673$$ −44.3970 −1.71138 −0.855689 0.517490i $$-0.826866\pi$$
−0.855689 + 0.517490i $$0.826866\pi$$
$$674$$ 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 $$0.537382\pi$$
$$678$$ 9.28592 0.356624
$$679$$ 3.69181 0.141679
$$680$$ 0 0
$$681$$ 56.3368 2.15883
$$682$$ −3.27607 −0.125447
$$683$$ −37.9224 −1.45106 −0.725531 0.688190i $$-0.758406\pi$$
−0.725531 + 0.688190i $$0.758406\pi$$
$$684$$ 11.3319 0.433284
$$685$$ 0 0
$$686$$ 3.70471 0.141447
$$687$$ 74.3595 2.83699
$$688$$ 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
$$693$$ −7.47949 −0.284123
$$694$$ 1.91258 0.0726005
$$695$$ 0 0
$$696$$ 3.52543 0.133631
$$697$$ 29.2159 1.10663
$$698$$ 2.21633 0.0838892
$$699$$ −9.06959 −0.343043
$$700$$ 0 0
$$701$$ −23.4893 −0.887180 −0.443590 0.896230i $$-0.646295\pi$$
−0.443590 + 0.896230i $$0.646295\pi$$
$$702$$ 1.36488 0.0515140
$$703$$ 4.33677 0.163565
$$704$$ 8.80150 0.331719
$$705$$ 0 0
$$706$$ −5.26317 −0.198082
$$707$$ 12.5620 0.472442
$$708$$ −9.24443 −0.347427
$$709$$ 11.6731 0.438391 0.219196 0.975681i $$-0.429657\pi$$
0.219196 + 0.975681i $$0.429657\pi$$
$$710$$ 0 0
$$711$$ 49.6084 1.86046
$$712$$ −9.47949 −0.355259
$$713$$ 51.9496 1.94553
$$714$$ 6.48886 0.242840
$$715$$ 0 0
$$716$$ −18.4889 −0.690961
$$717$$ −40.4701 −1.51138
$$718$$ −11.4207 −0.426215
$$719$$ 29.5526 1.10213 0.551063 0.834463i $$-0.314222\pi$$
0.551063 + 0.834463i $$0.314222\pi$$
$$720$$ 0 0
$$721$$ 11.6918 0.435426
$$722$$ 5.53680 0.206058
$$723$$ 53.6227 1.99425
$$724$$ −0.622216 −0.0231245
$$725$$ 0 0
$$726$$ −7.83362 −0.290733
$$727$$ −3.88094 −0.143936 −0.0719680 0.997407i $$-0.522928\pi$$
−0.0719680 + 0.997407i $$0.522928\pi$$
$$728$$ 0.682439 0.0252929
$$729$$ −38.6958 −1.43318
$$730$$ 0 0
$$731$$ 83.6454 3.09374
$$732$$ −10.3140 −0.381217
$$733$$ −14.8845 −0.549771 −0.274885 0.961477i $$-0.588640\pi$$
−0.274885 + 0.961477i $$0.588640\pi$$
$$734$$ 2.61777 0.0966236
$$735$$ 0 0
$$736$$ 26.3037 0.969569
$$737$$ 17.5812 0.647612
$$738$$ −6.20342 −0.228351
$$739$$ 2.24935 0.0827438 0.0413719 0.999144i $$-0.486827\pi$$
0.0413719 + 0.999144i $$0.486827\pi$$
$$740$$ 0 0
$$741$$ 1.98126 0.0727836
$$742$$ 1.80642 0.0663159
$$743$$ 3.46520 0.127126 0.0635630 0.997978i $$-0.479754\pi$$
0.0635630 + 0.997978i $$0.479754\pi$$
$$744$$ 24.3368 0.892229
$$745$$ 0 0
$$746$$ −2.69535 −0.0986836
$$747$$ −58.1388 −2.12719
$$748$$ 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
$$753$$ 40.0415 1.45919
$$754$$ 0.193576 0.00704963
$$755$$ 0 0
$$756$$ 12.1204 0.440816
$$757$$ 52.0785 1.89283 0.946413 0.322958i $$-0.104677\pi$$
0.946413 + 0.322958i $$0.104677\pi$$
$$758$$ 0.861725 0.0312993
$$759$$ −33.3274 −1.20971
$$760$$ 0 0
$$761$$ −14.9777 −0.542942 −0.271471 0.962447i $$-0.587510\pi$$
−0.271471 + 0.962447i $$0.587510\pi$$
$$762$$ −5.61868 −0.203543
$$763$$ −16.2953 −0.589929
$$764$$ 28.4429 1.02903
$$765$$ 0 0
$$766$$ 0.521889 0.0188566
$$767$$ −1.04101 −0.0375887
$$768$$ −25.5669 −0.922567
$$769$$ −1.90813 −0.0688091 −0.0344045 0.999408i $$-0.510953\pi$$
−0.0344045 + 0.999408i $$0.510953\pi$$
$$770$$ 0 0
$$771$$ −4.29529 −0.154691
$$772$$ −26.9260 −0.969087
$$773$$ −21.7891 −0.783698 −0.391849 0.920029i $$-0.628165\pi$$
−0.391849 + 0.920029i $$0.628165\pi$$
$$774$$ −17.7605 −0.638388
$$775$$ 0 0
$$776$$ 4.96343 0.178177
$$777$$ 10.3684 0.371965
$$778$$ 1.79658 0.0644105
$$779$$ −4.02858 −0.144339
$$780$$ 0 0
$$781$$ −20.8573 −0.746332
$$782$$ 18.6222 0.665929
$$783$$ 7.05086 0.251977
$$784$$ −21.2034 −0.757265
$$785$$ 0 0
$$786$$ −10.6222 −0.378882
$$787$$ 18.1388 0.646577 0.323288 0.946301i $$-0.395212\pi$$
0.323288 + 0.946301i $$0.395212\pi$$
$$788$$ −10.8573 −0.386775
$$789$$ −1.28592 −0.0457799
$$790$$ 0 0
$$791$$ 9.28592 0.330169
$$792$$ −10.0558 −0.357316
$$793$$ −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 $$0.516714\pi$$
$$798$$ −0.894751 −0.0316738
$$799$$ −54.9086 −1.94253
$$800$$ 0 0
$$801$$ −42.3783 −1.49736
$$802$$ −2.65386 −0.0937110
$$803$$ 15.5428 0.548493
$$804$$ −63.6829 −2.24592
$$805$$ 0 0
$$806$$ 1.33630 0.0470691
$$807$$ −11.4380 −0.402637
$$808$$ 16.8889 0.594150
$$809$$ −26.2953 −0.924493 −0.462247 0.886751i $$-0.652956\pi$$
−0.462247 + 0.886751i $$0.652956\pi$$
$$810$$ 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
$$813$$ −18.0228 −0.632085
$$814$$ −1.87649 −0.0657710
$$815$$ 0 0
$$816$$ −79.1753 −2.77169
$$817$$ −11.5339 −0.403520
$$818$$ −1.58427 −0.0553926
$$819$$ 3.05086 0.106606
$$820$$ 0 0
$$821$$ 1.52987 0.0533929 0.0266965 0.999644i $$-0.491501\pi$$
0.0266965 + 0.999644i $$0.491501\pi$$
$$822$$ 3.22168 0.112369
$$823$$ −46.7195 −1.62854 −0.814269 0.580487i $$-0.802862\pi$$
−0.814269 + 0.580487i $$0.802862\pi$$
$$824$$ 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 $$0.327601\pi$$
$$828$$ 77.7516 2.70205
$$829$$ −8.79706 −0.305534 −0.152767 0.988262i $$-0.548818\pi$$
−0.152767 + 0.988262i $$0.548818\pi$$
$$830$$ 0 0
$$831$$ 16.1748 0.561099
$$832$$ −3.59010 −0.124464
$$833$$ −49.1896 −1.70432
$$834$$ −7.73329 −0.267782
$$835$$ 0 0
$$836$$ −3.18421 −0.110128
$$837$$ 48.6735 1.68240
$$838$$ 7.57136 0.261548
$$839$$ 11.3319 0.391219 0.195609 0.980682i $$-0.437332\pi$$
0.195609 + 0.980682i $$0.437332\pi$$
$$840$$ 0 0
$$841$$ 1.00000 0.0344828
$$842$$ −7.62269 −0.262695
$$843$$ −19.4380 −0.669481
$$844$$ 39.6686 1.36545
$$845$$ 0 0
$$846$$ 11.6588 0.400837
$$847$$ −7.83362 −0.269166
$$848$$ −22.0415 −0.756908
$$849$$ 74.9086 2.57086
$$850$$ 0 0
$$851$$ 29.7560 1.02002
$$852$$ 75.5496 2.58829
$$853$$ −54.8845 −1.87921 −0.939604 0.342263i $$-0.888807\pi$$
−0.939604 + 0.342263i $$0.888807\pi$$
$$854$$ 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 $$0.713825\pi$$
$$858$$ −0.857279 −0.0292670
$$859$$ 1.72885 0.0589875 0.0294938 0.999565i $$-0.490610\pi$$
0.0294938 + 0.999565i $$0.490610\pi$$
$$860$$ 0 0
$$861$$ −9.63158 −0.328243
$$862$$ −1.32741 −0.0452116
$$863$$ 9.40192 0.320045 0.160023 0.987113i $$-0.448843\pi$$
0.160023 + 0.987113i $$0.448843\pi$$
$$864$$ 24.6450 0.838439
$$865$$ 0 0
$$866$$ 8.40144 0.285493
$$867$$ −134.323 −4.56186
$$868$$ 11.8666 0.402780
$$869$$ −13.9398 −0.472875
$$870$$ 0 0
$$871$$ −7.17130 −0.242990
$$872$$ −21.9081 −0.741903
$$873$$ 22.1891 0.750988
$$874$$ −2.56782 −0.0868579
$$875$$ 0 0
$$876$$ −56.2993 −1.90218
$$877$$ −8.91750 −0.301123 −0.150561 0.988601i $$-0.548108\pi$$
−0.150561 + 0.988601i $$0.548108\pi$$
$$878$$ 0.632060 0.0213310
$$879$$ −54.7052 −1.84516
$$880$$ 0 0
$$881$$ −42.1245 −1.41921 −0.709605 0.704600i $$-0.751126\pi$$
−0.709605 + 0.704600i $$0.751126\pi$$
$$882$$ 10.4445 0.351683
$$883$$ 38.4340 1.29341 0.646704 0.762741i $$-0.276147\pi$$
0.646704 + 0.762741i $$0.276147\pi$$
$$884$$ −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 $$0.275337\pi$$
$$888$$ 13.9398 0.467788
$$889$$ −5.61868 −0.188444
$$890$$ 0 0
$$891$$ −6.38271 −0.213829
$$892$$ 17.1985 0.575848
$$893$$ 7.57136 0.253366
$$894$$ −5.06959 −0.169552
$$895$$ 0 0
$$896$$ 7.93533 0.265101
$$897$$ 13.5941 0.453894
$$898$$ −11.6227 −0.387854
$$899$$ 6.90321 0.230235
$$900$$ 0 0
$$901$$ −51.1338 −1.70351
$$902$$ 1.74314 0.0580402
$$903$$ −27.5754 −0.917651
$$904$$ 12.4844 0.415226
$$905$$ 0 0
$$906$$ 9.75203 0.323989
$$907$$ −0.534795 −0.0177576 −0.00887880 0.999961i $$-0.502826\pi$$
−0.00887880 + 0.999961i $$0.502826\pi$$
$$908$$ 36.9318 1.22562
$$909$$ 75.5022 2.50425
$$910$$ 0 0
$$911$$ −23.6686 −0.784177 −0.392088 0.919928i $$-0.628247\pi$$
−0.392088 + 0.919928i $$0.628247\pi$$
$$912$$ 10.9175 0.361515
$$913$$ 16.3368 0.540668
$$914$$ 4.18468 0.138417
$$915$$ 0 0
$$916$$ 48.7467 1.61064
$$917$$ −10.6222 −0.350776
$$918$$ 17.4479 0.575865
$$919$$ 35.7748 1.18010 0.590051 0.807366i $$-0.299108\pi$$
0.590051 + 0.807366i $$0.299108\pi$$
$$920$$ 0 0
$$921$$ −4.81579 −0.158686
$$922$$ 5.06376 0.166766
$$923$$ 8.50760 0.280031
$$924$$ −7.61285 −0.250444
$$925$$ 0 0
$$926$$ −9.45536 −0.310722
$$927$$ 70.2721 2.30804
$$928$$ 3.49532 0.114739
$$929$$ −52.7753 −1.73150 −0.865750 0.500477i $$-0.833158\pi$$
−0.865750 + 0.500477i $$0.833158\pi$$
$$930$$ 0 0
$$931$$ 6.78277 0.222296
$$932$$ −5.94561 −0.194755
$$933$$ −61.8390 −2.02452
$$934$$ −0.369800 −0.0121002
$$935$$ 0 0
$$936$$ 4.10171 0.134069
$$937$$ −42.1245 −1.37615 −0.688073 0.725641i $$-0.741543\pi$$
−0.688073 + 0.725641i $$0.741543\pi$$
$$938$$ 3.23860 0.105744
$$939$$ 25.0321 0.816892
$$940$$ 0 0
$$941$$ −3.89829 −0.127081 −0.0635403 0.997979i $$-0.520239\pi$$
−0.0635403 + 0.997979i $$0.520239\pi$$
$$942$$ −2.06022 −0.0671257
$$943$$ −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 $$0.549680\pi$$
$$948$$ 50.4929 1.63993
$$949$$ −6.33984 −0.205800
$$950$$ 0 0
$$951$$ −80.1156 −2.59793
$$952$$ 8.72393 0.282744
$$953$$ −27.2070 −0.881320 −0.440660 0.897674i $$-0.645256\pi$$
−0.440660 + 0.897674i $$0.645256\pi$$
$$954$$ 10.8573 0.351517
$$955$$ 0 0
$$956$$ −26.5303 −0.858053
$$957$$ −4.42864 −0.143158
$$958$$ −12.7699 −0.412575
$$959$$ 3.22168 0.104033
$$960$$ 0 0
$$961$$ 16.6543 0.537237
$$962$$ 0.765413 0.0246779
$$963$$ −59.7418 −1.92515
$$964$$ 35.1526 1.13219
$$965$$ 0 0
$$966$$ −6.13918 −0.197525
$$967$$ 16.8015 0.540300 0.270150 0.962818i $$-0.412927\pi$$
0.270150 + 0.962818i $$0.412927\pi$$
$$968$$ −10.5319 −0.338507
$$969$$ 25.3274 0.813633
$$970$$ 0 0
$$971$$ −17.4465 −0.559884 −0.279942 0.960017i $$-0.590315\pi$$
−0.279942 + 0.960017i $$0.590315\pi$$
$$972$$ −17.1383 −0.549710
$$973$$ −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 $$0.671357\pi$$
$$978$$ −14.7427 −0.471418
$$979$$ 11.9081 0.380586
$$980$$ 0 0
$$981$$ −97.9407 −3.12701
$$982$$ 9.08694 0.289976
$$983$$ 16.5259 0.527094 0.263547 0.964646i $$-0.415108\pi$$
0.263547 + 0.964646i $$0.415108\pi$$
$$984$$ −12.9491 −0.412804
$$985$$ 0 0
$$986$$ 2.47457 0.0788064
$$987$$ 18.1017 0.576184
$$988$$ 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
$$993$$ −49.1022 −1.55821
$$994$$ −3.84208 −0.121863
$$995$$ 0 0
$$996$$ −59.1753 −1.87504
$$997$$ 15.9956 0.506584 0.253292 0.967390i $$-0.418487\pi$$
0.253292 + 0.967390i $$0.418487\pi$$
$$998$$ −6.83854 −0.216470
$$999$$ 27.8796 0.882070
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 725.2.a.e.1.2 3
3.2 odd 2 6525.2.a.be.1.2 3
5.2 odd 4 725.2.b.e.349.3 6
5.3 odd 4 725.2.b.e.349.4 6
5.4 even 2 145.2.a.c.1.2 3
15.14 odd 2 1305.2.a.p.1.2 3
20.19 odd 2 2320.2.a.n.1.1 3
35.34 odd 2 7105.2.a.o.1.2 3
40.19 odd 2 9280.2.a.br.1.3 3
40.29 even 2 9280.2.a.bj.1.1 3
145.144 even 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 5.4 even 2
725.2.a.e.1.2 3 1.1 even 1 trivial
725.2.b.e.349.3 6 5.2 odd 4
725.2.b.e.349.4 6 5.3 odd 4
1305.2.a.p.1.2 3 15.14 odd 2
2320.2.a.n.1.1 3 20.19 odd 2
4205.2.a.f.1.2 3 145.144 even 2
6525.2.a.be.1.2 3 3.2 odd 2
7105.2.a.o.1.2 3 35.34 odd 2
9280.2.a.bj.1.1 3 40.29 even 2
9280.2.a.br.1.3 3 40.19 odd 2