# Properties

 Label 725.2.b.e.349.4 Level $725$ Weight $2$ Character 725.349 Analytic conductor $5.789$ Analytic rank $0$ Dimension $6$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [725,2,Mod(349,725)]

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([1, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("725.349");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$725 = 5^{2} \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 725.b (of order $$2$$, degree $$1$$, not minimal)

## 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: no Analytic conductor: $$5.78915414654$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.350464.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2$$ x^6 - 2*x^5 + 2*x^4 + 2*x^3 + 4*x^2 - 4*x + 2 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: no (minimal twist has level 145) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 349.4 Root $$1.45161 - 1.45161i$$ of defining polynomial Character $$\chi$$ $$=$$ 725.349 Dual form 725.2.b.e.349.3

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+0.311108i q^{2} -2.90321i q^{3} +1.90321 q^{4} +0.903212 q^{6} -0.903212i q^{7} +1.21432i q^{8} -5.42864 q^{9} +O(q^{10})$$ $$q+0.311108i q^{2} -2.90321i q^{3} +1.90321 q^{4} +0.903212 q^{6} -0.903212i q^{7} +1.21432i q^{8} -5.42864 q^{9} -1.52543 q^{11} -5.52543i q^{12} +0.622216i q^{13} +0.280996 q^{14} +3.42864 q^{16} -7.95407i q^{17} -1.68889i q^{18} +1.09679 q^{19} -2.62222 q^{21} -0.474572i q^{22} -7.52543i q^{23} +3.52543 q^{24} -0.193576 q^{26} +7.05086i q^{27} -1.71900i q^{28} +1.00000 q^{29} -6.90321 q^{31} +3.49532i q^{32} +4.42864i q^{33} +2.47457 q^{34} -10.3319 q^{36} +3.95407i q^{37} +0.341219i q^{38} +1.80642 q^{39} +3.67307 q^{41} -0.815792i q^{42} +10.5161i q^{43} -2.90321 q^{44} +2.34122 q^{46} +6.90321i q^{47} -9.95407i q^{48} +6.18421 q^{49} -23.0923 q^{51} +1.18421i q^{52} -6.42864i q^{53} -2.19358 q^{54} +1.09679 q^{56} -3.18421i q^{57} +0.311108i q^{58} +1.67307 q^{59} -1.86665 q^{61} -2.14764i q^{62} +4.90321i q^{63} +5.76986 q^{64} -1.37778 q^{66} +11.5254i q^{67} -15.1383i q^{68} -21.8479 q^{69} +13.6731 q^{71} -6.59210i q^{72} -10.1891i q^{73} -1.23014 q^{74} +2.08742 q^{76} +1.37778i q^{77} +0.561993i q^{78} -9.13828 q^{79} +4.18421 q^{81} +1.14272i q^{82} -10.7096i q^{83} -4.99063 q^{84} -3.27163 q^{86} -2.90321i q^{87} -1.85236i q^{88} +7.80642 q^{89} +0.561993 q^{91} -14.3225i q^{92} +20.0415i q^{93} -2.14764 q^{94} +10.1476 q^{96} -4.08742i q^{97} +1.92396i q^{98} +8.28100 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 2 q^{4} - 8 q^{6} - 6 q^{9}+O(q^{10})$$ 6 * q - 2 * q^4 - 8 * q^6 - 6 * q^9 $$6 q - 2 q^{4} - 8 q^{6} - 6 q^{9} + 4 q^{11} - 12 q^{14} - 6 q^{16} + 20 q^{19} - 16 q^{21} + 8 q^{24} - 28 q^{26} + 6 q^{29} - 28 q^{31} + 28 q^{34} - 22 q^{36} - 16 q^{39} - 4 q^{41} - 4 q^{44} + 28 q^{46} + 10 q^{49} - 32 q^{51} - 40 q^{54} + 20 q^{56} - 16 q^{59} - 12 q^{61} + 22 q^{64} - 8 q^{66} - 24 q^{69} + 56 q^{71} - 20 q^{74} - 28 q^{76} + 12 q^{79} - 2 q^{81} + 24 q^{84} + 48 q^{86} + 20 q^{89} - 24 q^{91} + 48 q^{96} + 36 q^{99}+O(q^{100})$$ 6 * q - 2 * q^4 - 8 * q^6 - 6 * q^9 + 4 * q^11 - 12 * q^14 - 6 * q^16 + 20 * q^19 - 16 * q^21 + 8 * q^24 - 28 * q^26 + 6 * q^29 - 28 * q^31 + 28 * q^34 - 22 * q^36 - 16 * q^39 - 4 * q^41 - 4 * q^44 + 28 * q^46 + 10 * q^49 - 32 * q^51 - 40 * q^54 + 20 * q^56 - 16 * q^59 - 12 * q^61 + 22 * q^64 - 8 * q^66 - 24 * q^69 + 56 * q^71 - 20 * q^74 - 28 * q^76 + 12 * q^79 - 2 * q^81 + 24 * q^84 + 48 * q^86 + 20 * q^89 - 24 * q^91 + 48 * q^96 + 36 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/725\mathbb{Z}\right)^\times$$.

 $$n$$ $$176$$ $$552$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0.311108i 0.219986i 0.993932 + 0.109993i $$0.0350829\pi$$
−0.993932 + 0.109993i $$0.964917\pi$$
$$3$$ − 2.90321i − 1.67617i −0.545540 0.838085i $$-0.683675\pi$$
0.545540 0.838085i $$-0.316325\pi$$
$$4$$ 1.90321 0.951606
$$5$$ 0 0
$$6$$ 0.903212 0.368735
$$7$$ − 0.903212i − 0.341382i −0.985325 0.170691i $$-0.945400\pi$$
0.985325 0.170691i $$-0.0546000\pi$$
$$8$$ 1.21432i 0.429327i
$$9$$ −5.42864 −1.80955
$$10$$ 0 0
$$11$$ −1.52543 −0.459934 −0.229967 0.973198i $$-0.573862\pi$$
−0.229967 + 0.973198i $$0.573862\pi$$
$$12$$ − 5.52543i − 1.59505i
$$13$$ 0.622216i 0.172572i 0.996270 + 0.0862858i $$0.0274998\pi$$
−0.996270 + 0.0862858i $$0.972500\pi$$
$$14$$ 0.280996 0.0750994
$$15$$ 0 0
$$16$$ 3.42864 0.857160
$$17$$ − 7.95407i − 1.92914i −0.263819 0.964572i $$-0.584982\pi$$
0.263819 0.964572i $$-0.415018\pi$$
$$18$$ − 1.68889i − 0.398076i
$$19$$ 1.09679 0.251620 0.125810 0.992054i $$-0.459847\pi$$
0.125810 + 0.992054i $$0.459847\pi$$
$$20$$ 0 0
$$21$$ −2.62222 −0.572214
$$22$$ − 0.474572i − 0.101179i
$$23$$ − 7.52543i − 1.56916i −0.620028 0.784580i $$-0.712879\pi$$
0.620028 0.784580i $$-0.287121\pi$$
$$24$$ 3.52543 0.719625
$$25$$ 0 0
$$26$$ −0.193576 −0.0379634
$$27$$ 7.05086i 1.35694i
$$28$$ − 1.71900i − 0.324861i
$$29$$ 1.00000 0.185695
$$30$$ 0 0
$$31$$ −6.90321 −1.23985 −0.619927 0.784660i $$-0.712838\pi$$
−0.619927 + 0.784660i $$0.712838\pi$$
$$32$$ 3.49532i 0.617890i
$$33$$ 4.42864i 0.770927i
$$34$$ 2.47457 0.424386
$$35$$ 0 0
$$36$$ −10.3319 −1.72198
$$37$$ 3.95407i 0.650045i 0.945706 + 0.325022i $$0.105372\pi$$
−0.945706 + 0.325022i $$0.894628\pi$$
$$38$$ 0.341219i 0.0553531i
$$39$$ 1.80642 0.289259
$$40$$ 0 0
$$41$$ 3.67307 0.573637 0.286819 0.957985i $$-0.407402\pi$$
0.286819 + 0.957985i $$0.407402\pi$$
$$42$$ − 0.815792i − 0.125879i
$$43$$ 10.5161i 1.60368i 0.597536 + 0.801842i $$0.296146\pi$$
−0.597536 + 0.801842i $$0.703854\pi$$
$$44$$ −2.90321 −0.437676
$$45$$ 0 0
$$46$$ 2.34122 0.345194
$$47$$ 6.90321i 1.00694i 0.864014 + 0.503468i $$0.167943\pi$$
−0.864014 + 0.503468i $$0.832057\pi$$
$$48$$ − 9.95407i − 1.43675i
$$49$$ 6.18421 0.883458
$$50$$ 0 0
$$51$$ −23.0923 −3.23357
$$52$$ 1.18421i 0.164220i
$$53$$ − 6.42864i − 0.883042i −0.897251 0.441521i $$-0.854439\pi$$
0.897251 0.441521i $$-0.145561\pi$$
$$54$$ −2.19358 −0.298508
$$55$$ 0 0
$$56$$ 1.09679 0.146564
$$57$$ − 3.18421i − 0.421759i
$$58$$ 0.311108i 0.0408505i
$$59$$ 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 $$0.538129\pi$$
$$62$$ − 2.14764i − 0.272751i
$$63$$ 4.90321i 0.617747i
$$64$$ 5.76986 0.721232
$$65$$ 0 0
$$66$$ −1.37778 −0.169594
$$67$$ 11.5254i 1.40806i 0.710173 + 0.704028i $$0.248617\pi$$
−0.710173 + 0.704028i $$0.751383\pi$$
$$68$$ − 15.1383i − 1.83579i
$$69$$ −21.8479 −2.63018
$$70$$ 0 0
$$71$$ 13.6731 1.62269 0.811347 0.584564i $$-0.198734\pi$$
0.811347 + 0.584564i $$0.198734\pi$$
$$72$$ − 6.59210i − 0.776887i
$$73$$ − 10.1891i − 1.19255i −0.802781 0.596274i $$-0.796647\pi$$
0.802781 0.596274i $$-0.203353\pi$$
$$74$$ −1.23014 −0.143001
$$75$$ 0 0
$$76$$ 2.08742 0.239444
$$77$$ 1.37778i 0.157013i
$$78$$ 0.561993i 0.0636331i
$$79$$ −9.13828 −1.02814 −0.514068 0.857749i $$-0.671862\pi$$
−0.514068 + 0.857749i $$0.671862\pi$$
$$80$$ 0 0
$$81$$ 4.18421 0.464912
$$82$$ 1.14272i 0.126192i
$$83$$ − 10.7096i − 1.17554i −0.809030 0.587768i $$-0.800007\pi$$
0.809030 0.587768i $$-0.199993\pi$$
$$84$$ −4.99063 −0.544523
$$85$$ 0 0
$$86$$ −3.27163 −0.352789
$$87$$ − 2.90321i − 0.311257i
$$88$$ − 1.85236i − 0.197462i
$$89$$ 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
$$92$$ − 14.3225i − 1.49322i
$$93$$ 20.0415i 2.07821i
$$94$$ −2.14764 −0.221512
$$95$$ 0 0
$$96$$ 10.1476 1.03569
$$97$$ − 4.08742i − 0.415015i −0.978233 0.207507i $$-0.933465\pi$$
0.978233 0.207507i $$-0.0665351\pi$$
$$98$$ 1.92396i 0.194349i
$$99$$ 8.28100 0.832271
$$100$$ 0 0
$$101$$ 13.9081 1.38391 0.691956 0.721940i $$-0.256749\pi$$
0.691956 + 0.721940i $$0.256749\pi$$
$$102$$ − 7.18421i − 0.711343i
$$103$$ 12.9447i 1.27548i 0.770252 + 0.637740i $$0.220130\pi$$
−0.770252 + 0.637740i $$0.779870\pi$$
$$104$$ −0.755569 −0.0740896
$$105$$ 0 0
$$106$$ 2.00000 0.194257
$$107$$ 11.0049i 1.06389i 0.846780 + 0.531943i $$0.178538\pi$$
−0.846780 + 0.531943i $$0.821462\pi$$
$$108$$ 13.4193i 1.29127i
$$109$$ 18.0415 1.72806 0.864031 0.503439i $$-0.167932\pi$$
0.864031 + 0.503439i $$0.167932\pi$$
$$110$$ 0 0
$$111$$ 11.4795 1.08959
$$112$$ − 3.09679i − 0.292619i
$$113$$ 10.2810i 0.967155i 0.875302 + 0.483577i $$0.160663\pi$$
−0.875302 + 0.483577i $$0.839337\pi$$
$$114$$ 0.990632 0.0927812
$$115$$ 0 0
$$116$$ 1.90321 0.176709
$$117$$ − 3.37778i − 0.312276i
$$118$$ 0.520505i 0.0479164i
$$119$$ −7.18421 −0.658575
$$120$$ 0 0
$$121$$ −8.67307 −0.788461
$$122$$ − 0.580728i − 0.0525767i
$$123$$ − 10.6637i − 0.961514i
$$124$$ −13.1383 −1.17985
$$125$$ 0 0
$$126$$ −1.52543 −0.135896
$$127$$ 6.22077i 0.552004i 0.961157 + 0.276002i $$0.0890097\pi$$
−0.961157 + 0.276002i $$0.910990\pi$$
$$128$$ 8.78568i 0.776552i
$$129$$ 30.5303 2.68805
$$130$$ 0 0
$$131$$ −11.7605 −1.02752 −0.513759 0.857934i $$-0.671748\pi$$
−0.513759 + 0.857934i $$0.671748\pi$$
$$132$$ 8.42864i 0.733619i
$$133$$ − 0.990632i − 0.0858987i
$$134$$ −3.58565 −0.309753
$$135$$ 0 0
$$136$$ 9.65878 0.828234
$$137$$ − 3.56691i − 0.304742i −0.988323 0.152371i $$-0.951309\pi$$
0.988323 0.152371i $$-0.0486909\pi$$
$$138$$ − 6.79706i − 0.578604i
$$139$$ 8.56199 0.726219 0.363109 0.931747i $$-0.381715\pi$$
0.363109 + 0.931747i $$0.381715\pi$$
$$140$$ 0 0
$$141$$ 20.0415 1.68780
$$142$$ 4.25380i 0.356971i
$$143$$ − 0.949145i − 0.0793715i
$$144$$ −18.6128 −1.55107
$$145$$ 0 0
$$146$$ 3.16992 0.262344
$$147$$ − 17.9541i − 1.48083i
$$148$$ 7.52543i 0.618586i
$$149$$ 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 $$0.355217\pi$$
$$152$$ 1.33185i 0.108027i
$$153$$ 43.1798i 3.49088i
$$154$$ −0.428639 −0.0345408
$$155$$ 0 0
$$156$$ 3.43801 0.275261
$$157$$ 2.28100i 0.182043i 0.995849 + 0.0910217i $$0.0290133\pi$$
−0.995849 + 0.0910217i $$0.970987\pi$$
$$158$$ − 2.84299i − 0.226176i
$$159$$ −18.6637 −1.48013
$$160$$ 0 0
$$161$$ −6.79706 −0.535683
$$162$$ 1.30174i 0.102274i
$$163$$ − 16.3225i − 1.27848i −0.769009 0.639238i $$-0.779250\pi$$
0.769009 0.639238i $$-0.220750\pi$$
$$164$$ 6.99063 0.545877
$$165$$ 0 0
$$166$$ 3.33185 0.258602
$$167$$ − 4.76986i − 0.369103i −0.982823 0.184551i $$-0.940917\pi$$
0.982823 0.184551i $$-0.0590832\pi$$
$$168$$ − 3.18421i − 0.245667i
$$169$$ 12.6128 0.970219
$$170$$ 0 0
$$171$$ −5.95407 −0.455319
$$172$$ 20.0143i 1.52608i
$$173$$ − 4.23506i − 0.321986i −0.986956 0.160993i $$-0.948530\pi$$
0.986956 0.160993i $$-0.0514696\pi$$
$$174$$ 0.903212 0.0684723
$$175$$ 0 0
$$176$$ −5.23014 −0.394237
$$177$$ − 4.85728i − 0.365095i
$$178$$ 2.42864i 0.182034i
$$179$$ −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 $$0.496132\pi$$
$$182$$ 0.174840i 0.0129600i
$$183$$ 5.41927i 0.400604i
$$184$$ 9.13828 0.673683
$$185$$ 0 0
$$186$$ −6.23506 −0.457177
$$187$$ 12.1334i 0.887279i
$$188$$ 13.1383i 0.958207i
$$189$$ 6.36842 0.463234
$$190$$ 0 0
$$191$$ −14.9447 −1.08136 −0.540680 0.841228i $$-0.681833\pi$$
−0.540680 + 0.841228i $$0.681833\pi$$
$$192$$ − 16.7511i − 1.20891i
$$193$$ 14.1476i 1.01837i 0.860657 + 0.509185i $$0.170053\pi$$
−0.860657 + 0.509185i $$0.829947\pi$$
$$194$$ 1.27163 0.0912976
$$195$$ 0 0
$$196$$ 11.7699 0.840704
$$197$$ − 5.70471i − 0.406444i −0.979133 0.203222i $$-0.934859\pi$$
0.979133 0.203222i $$-0.0651413\pi$$
$$198$$ 2.57628i 0.183088i
$$199$$ 22.1432 1.56969 0.784845 0.619692i $$-0.212743\pi$$
0.784845 + 0.619692i $$0.212743\pi$$
$$200$$ 0 0
$$201$$ 33.4608 2.36014
$$202$$ 4.32693i 0.304442i
$$203$$ − 0.903212i − 0.0633930i
$$204$$ −43.9496 −3.07709
$$205$$ 0 0
$$206$$ −4.02720 −0.280588
$$207$$ 40.8528i 2.83947i
$$208$$ 2.13335i 0.147921i
$$209$$ −1.67307 −0.115729
$$210$$ 0 0
$$211$$ −20.8430 −1.43489 −0.717445 0.696615i $$-0.754689\pi$$
−0.717445 + 0.696615i $$0.754689\pi$$
$$212$$ − 12.2351i − 0.840308i
$$213$$ − 39.6958i − 2.71991i
$$214$$ −3.42372 −0.234040
$$215$$ 0 0
$$216$$ −8.56199 −0.582570
$$217$$ 6.23506i 0.423264i
$$218$$ 5.61285i 0.380150i
$$219$$ −29.5812 −1.99891
$$220$$ 0 0
$$221$$ 4.94914 0.332916
$$222$$ 3.57136i 0.239694i
$$223$$ − 9.03657i − 0.605133i −0.953128 0.302567i $$-0.902157\pi$$
0.953128 0.302567i $$-0.0978435\pi$$
$$224$$ 3.15701 0.210937
$$225$$ 0 0
$$226$$ −3.19850 −0.212761
$$227$$ 19.4050i 1.28795i 0.765045 + 0.643977i $$0.222717\pi$$
−0.765045 + 0.643977i $$0.777283\pi$$
$$228$$ − 6.06022i − 0.401348i
$$229$$ 25.6128 1.69254 0.846272 0.532751i $$-0.178842\pi$$
0.846272 + 0.532751i $$0.178842\pi$$
$$230$$ 0 0
$$231$$ 4.00000 0.263181
$$232$$ 1.21432i 0.0797240i
$$233$$ 3.12399i 0.204659i 0.994751 + 0.102330i $$0.0326296\pi$$
−0.994751 + 0.102330i $$0.967370\pi$$
$$234$$ 1.05086 0.0686965
$$235$$ 0 0
$$236$$ 3.18421 0.207274
$$237$$ 26.5303i 1.72333i
$$238$$ − 2.23506i − 0.144878i
$$239$$ −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 $$0.702802\pi$$
$$242$$ − 2.69826i − 0.173451i
$$243$$ 9.00492i 0.577666i
$$244$$ −3.55262 −0.227433
$$245$$ 0 0
$$246$$ 3.31756 0.211520
$$247$$ 0.682439i 0.0434225i
$$248$$ − 8.38271i − 0.532302i
$$249$$ −31.0923 −1.97040
$$250$$ 0 0
$$251$$ −13.7921 −0.870552 −0.435276 0.900297i $$-0.643349\pi$$
−0.435276 + 0.900297i $$0.643349\pi$$
$$252$$ 9.33185i 0.587851i
$$253$$ 11.4795i 0.721710i
$$254$$ −1.93533 −0.121433
$$255$$ 0 0
$$256$$ 8.80642 0.550401
$$257$$ − 1.47949i − 0.0922883i −0.998935 0.0461442i $$-0.985307\pi$$
0.998935 0.0461442i $$-0.0146934\pi$$
$$258$$ 9.49823i 0.591334i
$$259$$ 3.57136 0.221914
$$260$$ 0 0
$$261$$ −5.42864 −0.336024
$$262$$ − 3.65878i − 0.226040i
$$263$$ 0.442930i 0.0273122i 0.999907 + 0.0136561i $$0.00434701\pi$$
−0.999907 + 0.0136561i $$0.995653\pi$$
$$264$$ −5.37778 −0.330980
$$265$$ 0 0
$$266$$ 0.308193 0.0188965
$$267$$ − 22.6637i − 1.38700i
$$268$$ 21.9353i 1.33991i
$$269$$ −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 $$0.439621\pi$$
$$272$$ − 27.2716i − 1.65359i
$$273$$ − 1.63158i − 0.0987479i
$$274$$ 1.10970 0.0670391
$$275$$ 0 0
$$276$$ −41.5812 −2.50289
$$277$$ 5.57136i 0.334751i 0.985893 + 0.167375i $$0.0535292\pi$$
−0.985893 + 0.167375i $$0.946471\pi$$
$$278$$ 2.66370i 0.159758i
$$279$$ 37.4750 2.24357
$$280$$ 0 0
$$281$$ 6.69535 0.399411 0.199705 0.979856i $$-0.436001\pi$$
0.199705 + 0.979856i $$0.436001\pi$$
$$282$$ 6.23506i 0.371293i
$$283$$ − 25.8020i − 1.53377i −0.641785 0.766884i $$-0.721806\pi$$
0.641785 0.766884i $$-0.278194\pi$$
$$284$$ 26.0228 1.54417
$$285$$ 0 0
$$286$$ 0.295286 0.0174607
$$287$$ − 3.31756i − 0.195829i
$$288$$ − 18.9748i − 1.11810i
$$289$$ −46.2672 −2.72160
$$290$$ 0 0
$$291$$ −11.8666 −0.695635
$$292$$ − 19.3921i − 1.13484i
$$293$$ 18.8430i 1.10082i 0.834895 + 0.550410i $$0.185528\pi$$
−0.834895 + 0.550410i $$0.814472\pi$$
$$294$$ 5.58565 0.325762
$$295$$ 0 0
$$296$$ −4.80150 −0.279082
$$297$$ − 10.7556i − 0.624101i
$$298$$ 1.74620i 0.101155i
$$299$$ 4.68244 0.270792
$$300$$ 0 0
$$301$$ 9.49823 0.547469
$$302$$ 3.35905i 0.193292i
$$303$$ − 40.3783i − 2.31967i
$$304$$ 3.76049 0.215679
$$305$$ 0 0
$$306$$ −13.4336 −0.767946
$$307$$ − 1.65878i − 0.0946716i −0.998879 0.0473358i $$-0.984927\pi$$
0.998879 0.0473358i $$-0.0150731\pi$$
$$308$$ 2.62222i 0.149415i
$$309$$ 37.5812 2.13792
$$310$$ 0 0
$$311$$ 21.3002 1.20782 0.603912 0.797051i $$-0.293608\pi$$
0.603912 + 0.797051i $$0.293608\pi$$
$$312$$ 2.19358i 0.124187i
$$313$$ − 8.62222i − 0.487356i −0.969856 0.243678i $$-0.921646\pi$$
0.969856 0.243678i $$-0.0783541\pi$$
$$314$$ −0.709636 −0.0400471
$$315$$ 0 0
$$316$$ −17.3921 −0.978381
$$317$$ − 27.5955i − 1.54992i −0.632012 0.774959i $$-0.717771\pi$$
0.632012 0.774959i $$-0.282229\pi$$
$$318$$ − 5.80642i − 0.325608i
$$319$$ −1.52543 −0.0854075
$$320$$ 0 0
$$321$$ 31.9496 1.78325
$$322$$ − 2.11462i − 0.117843i
$$323$$ − 8.72393i − 0.485412i
$$324$$ 7.96343 0.442413
$$325$$ 0 0
$$326$$ 5.07805 0.281247
$$327$$ − 52.3783i − 2.89652i
$$328$$ 4.46028i 0.246278i
$$329$$ 6.23506 0.343750
$$330$$ 0 0
$$331$$ 16.9131 0.929626 0.464813 0.885409i $$-0.346122\pi$$
0.464813 + 0.885409i $$0.346122\pi$$
$$332$$ − 20.3827i − 1.11865i
$$333$$ − 21.4652i − 1.17629i
$$334$$ 1.48394 0.0811976
$$335$$ 0 0
$$336$$ −8.99063 −0.490479
$$337$$ 11.9956i 0.653439i 0.945121 + 0.326720i $$0.105943\pi$$
−0.945121 + 0.326720i $$0.894057\pi$$
$$338$$ 3.92396i 0.213435i
$$339$$ 29.8479 1.62112
$$340$$ 0 0
$$341$$ 10.5303 0.570250
$$342$$ − 1.85236i − 0.100164i
$$343$$ − 11.9081i − 0.642979i
$$344$$ −12.7699 −0.688505
$$345$$ 0 0
$$346$$ 1.31756 0.0708325
$$347$$ 6.14764i 0.330023i 0.986292 + 0.165011i $$0.0527661\pi$$
−0.986292 + 0.165011i $$0.947234\pi$$
$$348$$ − 5.52543i − 0.296194i
$$349$$ 7.12399 0.381338 0.190669 0.981654i $$-0.438934\pi$$
0.190669 + 0.981654i $$0.438934\pi$$
$$350$$ 0 0
$$351$$ −4.38715 −0.234169
$$352$$ − 5.33185i − 0.284189i
$$353$$ 16.9175i 0.900428i 0.892921 + 0.450214i $$0.148652\pi$$
−0.892921 + 0.450214i $$0.851348\pi$$
$$354$$ 1.51114 0.0803160
$$355$$ 0 0
$$356$$ 14.8573 0.787434
$$357$$ 20.8573i 1.10388i
$$358$$ − 3.02227i − 0.159732i
$$359$$ −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
$$362$$ 0.101710i 0.00534577i
$$363$$ 25.1798i 1.32159i
$$364$$ 1.06959 0.0560618
$$365$$ 0 0
$$366$$ −1.68598 −0.0881275
$$367$$ 8.41435i 0.439225i 0.975587 + 0.219613i $$0.0704794\pi$$
−0.975587 + 0.219613i $$0.929521\pi$$
$$368$$ − 25.8020i − 1.34502i
$$369$$ −19.9398 −1.03802
$$370$$ 0 0
$$371$$ −5.80642 −0.301455
$$372$$ 38.1432i 1.97763i
$$373$$ 8.66370i 0.448590i 0.974521 + 0.224295i $$0.0720078\pi$$
−0.974521 + 0.224295i $$0.927992\pi$$
$$374$$ −3.77478 −0.195189
$$375$$ 0 0
$$376$$ −8.38271 −0.432305
$$377$$ 0.622216i 0.0320457i
$$378$$ 1.98126i 0.101905i
$$379$$ 2.76986 0.142278 0.0711390 0.997466i $$-0.477337\pi$$
0.0711390 + 0.997466i $$0.477337\pi$$
$$380$$ 0 0
$$381$$ 18.0602 0.925253
$$382$$ − 4.64941i − 0.237885i
$$383$$ − 1.67752i − 0.0857171i −0.999081 0.0428585i $$-0.986354\pi$$
0.999081 0.0428585i $$-0.0136465\pi$$
$$384$$ 25.5067 1.30163
$$385$$ 0 0
$$386$$ −4.40144 −0.224028
$$387$$ − 57.0879i − 2.90194i
$$388$$ − 7.77923i − 0.394930i
$$389$$ 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
$$392$$ 7.50961i 0.379292i
$$393$$ 34.1432i 1.72230i
$$394$$ 1.77478 0.0894122
$$395$$ 0 0
$$396$$ 15.7605 0.791994
$$397$$ 29.9081i 1.50105i 0.660844 + 0.750523i $$0.270198\pi$$
−0.660844 + 0.750523i $$0.729802\pi$$
$$398$$ 6.88892i 0.345310i
$$399$$ −2.87601 −0.143981
$$400$$ 0 0
$$401$$ 8.53035 0.425985 0.212993 0.977054i $$-0.431679\pi$$
0.212993 + 0.977054i $$0.431679\pi$$
$$402$$ 10.4099i 0.519199i
$$403$$ − 4.29529i − 0.213963i
$$404$$ 26.4701 1.31694
$$405$$ 0 0
$$406$$ 0.280996 0.0139456
$$407$$ − 6.03164i − 0.298977i
$$408$$ − 28.0415i − 1.38826i
$$409$$ −5.09234 −0.251800 −0.125900 0.992043i $$-0.540182\pi$$
−0.125900 + 0.992043i $$0.540182\pi$$
$$410$$ 0 0
$$411$$ −10.3555 −0.510800
$$412$$ 24.6365i 1.21375i
$$413$$ − 1.51114i − 0.0743582i
$$414$$ −12.7096 −0.624645
$$415$$ 0 0
$$416$$ −2.17484 −0.106630
$$417$$ − 24.8573i − 1.21727i
$$418$$ − 0.520505i − 0.0254588i
$$419$$ 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 $$0.296331\pi$$
$$422$$ − 6.48442i − 0.315656i
$$423$$ − 37.4750i − 1.82210i
$$424$$ 7.80642 0.379113
$$425$$ 0 0
$$426$$ 12.3497 0.598344
$$427$$ 1.68598i 0.0815902i
$$428$$ 20.9447i 1.01240i
$$429$$ −2.75557 −0.133040
$$430$$ 0 0
$$431$$ 4.26671 0.205520 0.102760 0.994706i $$-0.467233\pi$$
0.102760 + 0.994706i $$0.467233\pi$$
$$432$$ 24.1748i 1.16311i
$$433$$ − 27.0049i − 1.29777i −0.760885 0.648887i $$-0.775235\pi$$
0.760885 0.648887i $$-0.224765\pi$$
$$434$$ −1.93978 −0.0931123
$$435$$ 0 0
$$436$$ 34.3368 1.64443
$$437$$ − 8.25380i − 0.394833i
$$438$$ − 9.20294i − 0.439734i
$$439$$ 2.03164 0.0969650 0.0484825 0.998824i $$-0.484561\pi$$
0.0484825 + 0.998824i $$0.484561\pi$$
$$440$$ 0 0
$$441$$ −33.5718 −1.59866
$$442$$ 1.53972i 0.0732369i
$$443$$ 3.46520i 0.164637i 0.996606 + 0.0823184i $$0.0262324\pi$$
−0.996606 + 0.0823184i $$0.973768\pi$$
$$444$$ 21.8479 1.03686
$$445$$ 0 0
$$446$$ 2.81135 0.133121
$$447$$ − 16.2953i − 0.770741i
$$448$$ − 5.21141i − 0.246216i
$$449$$ −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
$$452$$ 19.5669i 0.920350i
$$453$$ − 31.3461i − 1.47277i
$$454$$ −6.03704 −0.283332
$$455$$ 0 0
$$456$$ 3.86665 0.181072
$$457$$ 13.4509i 0.629207i 0.949223 + 0.314604i $$0.101872\pi$$
−0.949223 + 0.314604i $$0.898128\pi$$
$$458$$ 7.96836i 0.372337i
$$459$$ 56.0830 2.61773
$$460$$ 0 0
$$461$$ −16.2766 −0.758075 −0.379037 0.925381i $$-0.623745\pi$$
−0.379037 + 0.925381i $$0.623745\pi$$
$$462$$ 1.24443i 0.0578962i
$$463$$ 30.3926i 1.41246i 0.707982 + 0.706231i $$0.249606\pi$$
−0.707982 + 0.706231i $$0.750394\pi$$
$$464$$ 3.42864 0.159171
$$465$$ 0 0
$$466$$ −0.971896 −0.0450222
$$467$$ − 1.18865i − 0.0550043i −0.999622 0.0275022i $$-0.991245\pi$$
0.999622 0.0275022i $$-0.00875532\pi$$
$$468$$ − 6.42864i − 0.297164i
$$469$$ 10.4099 0.480685
$$470$$ 0 0
$$471$$ 6.62222 0.305136
$$472$$ 2.03164i 0.0935139i
$$473$$ − 16.0415i − 0.737588i
$$474$$ −8.25380 −0.379110
$$475$$ 0 0
$$476$$ −13.6731 −0.626704
$$477$$ 34.8988i 1.59790i
$$478$$ − 4.33677i − 0.198359i
$$479$$ −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
$$482$$ − 5.74620i − 0.261732i
$$483$$ 19.7333i 0.897896i
$$484$$ −16.5067 −0.750304
$$485$$ 0 0
$$486$$ −2.80150 −0.127079
$$487$$ − 10.1476i − 0.459834i −0.973210 0.229917i $$-0.926155\pi$$
0.973210 0.229917i $$-0.0738454\pi$$
$$488$$ − 2.26671i − 0.102609i
$$489$$ −47.3876 −2.14294
$$490$$ 0 0
$$491$$ −29.2083 −1.31815 −0.659077 0.752075i $$-0.729053\pi$$
−0.659077 + 0.752075i $$0.729053\pi$$
$$492$$ − 20.2953i − 0.914982i
$$493$$ − 7.95407i − 0.358233i
$$494$$ −0.212312 −0.00955237
$$495$$ 0 0
$$496$$ −23.6686 −1.06275
$$497$$ − 12.3497i − 0.553959i
$$498$$ − 9.67307i − 0.433461i
$$499$$ −21.9813 −0.984017 −0.492008 0.870591i $$-0.663737\pi$$
−0.492008 + 0.870591i $$0.663737\pi$$
$$500$$ 0 0
$$501$$ −13.8479 −0.618679
$$502$$ − 4.29084i − 0.191510i
$$503$$ − 5.77923i − 0.257683i −0.991665 0.128841i $$-0.958874\pi$$
0.991665 0.128841i $$-0.0411258\pi$$
$$504$$ −5.95407 −0.265215
$$505$$ 0 0
$$506$$ −3.57136 −0.158766
$$507$$ − 36.6178i − 1.62625i
$$508$$ 11.8394i 0.525291i
$$509$$ −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
$$512$$ 20.3111i 0.897633i
$$513$$ 7.73329i 0.341433i
$$514$$ 0.460282 0.0203022
$$515$$ 0 0
$$516$$ 58.1057 2.55796
$$517$$ − 10.5303i − 0.463124i
$$518$$ 1.11108i 0.0488180i
$$519$$ −12.2953 −0.539703
$$520$$ 0 0
$$521$$ −19.6731 −0.861893 −0.430946 0.902378i $$-0.641820\pi$$
−0.430946 + 0.902378i $$0.641820\pi$$
$$522$$ − 1.68889i − 0.0739208i
$$523$$ 15.1383i 0.661951i 0.943639 + 0.330975i $$0.107378\pi$$
−0.943639 + 0.330975i $$0.892622\pi$$
$$524$$ −22.3827 −0.977793
$$525$$ 0 0
$$526$$ −0.137799 −0.00600832
$$527$$ 54.9086i 2.39186i
$$528$$ 15.1842i 0.660808i
$$529$$ −33.6321 −1.46226
$$530$$ 0 0
$$531$$ −9.08250 −0.394147
$$532$$ − 1.88538i − 0.0817417i
$$533$$ 2.28544i 0.0989935i
$$534$$ 7.05086 0.305120
$$535$$ 0 0
$$536$$ −13.9956 −0.604516
$$537$$ 28.2034i 1.21707i
$$538$$ − 1.22570i − 0.0528435i
$$539$$ −9.43356 −0.406332
$$540$$ 0 0
$$541$$ 2.68244 0.115327 0.0576635 0.998336i $$-0.481635\pi$$
0.0576635 + 0.998336i $$0.481635\pi$$
$$542$$ 1.93132i 0.0829571i
$$543$$ − 0.949145i − 0.0407317i
$$544$$ 27.8020 1.19200
$$545$$ 0 0
$$546$$ 0.507598 0.0217232
$$547$$ − 15.3635i − 0.656896i −0.944522 0.328448i $$-0.893475\pi$$
0.944522 0.328448i $$-0.106525\pi$$
$$548$$ − 6.78859i − 0.289994i
$$549$$ 10.1334 0.432481
$$550$$ 0 0
$$551$$ 1.09679 0.0467247
$$552$$ − 26.5303i − 1.12921i
$$553$$ 8.25380i 0.350987i
$$554$$ −1.73329 −0.0736406
$$555$$ 0 0
$$556$$ 16.2953 0.691074
$$557$$ − 9.87955i − 0.418610i −0.977850 0.209305i $$-0.932880\pi$$
0.977850 0.209305i $$-0.0671202\pi$$
$$558$$ 11.6588i 0.493556i
$$559$$ −6.54326 −0.276750
$$560$$ 0 0
$$561$$ 35.2257 1.48723
$$562$$ 2.08297i 0.0878650i
$$563$$ 27.4938i 1.15872i 0.815070 + 0.579362i $$0.196698\pi$$
−0.815070 + 0.579362i $$0.803302\pi$$
$$564$$ 38.1432 1.60612
$$565$$ 0 0
$$566$$ 8.02720 0.337408
$$567$$ − 3.77923i − 0.158713i
$$568$$ 16.6035i 0.696667i
$$569$$ −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 $$0.678740\pi$$
$$572$$ − 1.80642i − 0.0755304i
$$573$$ 43.3876i 1.81254i
$$574$$ 1.03212 0.0430798
$$575$$ 0 0
$$576$$ −31.3225 −1.30510
$$577$$ − 10.6178i − 0.442024i −0.975271 0.221012i $$-0.929064\pi$$
0.975271 0.221012i $$-0.0709359\pi$$
$$578$$ − 14.3941i − 0.598715i
$$579$$ 41.0736 1.70696
$$580$$ 0 0
$$581$$ −9.67307 −0.401307
$$582$$ − 3.69181i − 0.153030i
$$583$$ 9.80642i 0.406141i
$$584$$ 12.3729 0.511993
$$585$$ 0 0
$$586$$ −5.86220 −0.242165
$$587$$ − 8.94470i − 0.369187i −0.982815 0.184594i $$-0.940903\pi$$
0.982815 0.184594i $$-0.0590969\pi$$
$$588$$ − 34.1704i − 1.40916i
$$589$$ −7.57136 −0.311972
$$590$$ 0 0
$$591$$ −16.5620 −0.681269
$$592$$ 13.5571i 0.557192i
$$593$$ − 14.1619i − 0.581561i −0.956790 0.290780i $$-0.906085\pi$$
0.956790 0.290780i $$-0.0939149\pi$$
$$594$$ 3.34614 0.137294
$$595$$ 0 0
$$596$$ 10.6824 0.437570
$$597$$ − 64.2864i − 2.63107i
$$598$$ 1.45674i 0.0595707i
$$599$$ 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 $$0.810706\pi$$
$$602$$ 2.95497i 0.120436i
$$603$$ − 62.5674i − 2.54794i
$$604$$ 20.5491 0.836130
$$605$$ 0 0
$$606$$ 12.5620 0.510296
$$607$$ 13.5955i 0.551824i 0.961183 + 0.275912i $$0.0889799\pi$$
−0.961183 + 0.275912i $$0.911020\pi$$
$$608$$ 3.83362i 0.155474i
$$609$$ −2.62222 −0.106258
$$610$$ 0 0
$$611$$ −4.29529 −0.173769
$$612$$ 82.1802i 3.32194i
$$613$$ 42.0830i 1.69972i 0.527012 + 0.849858i $$0.323312\pi$$
−0.527012 + 0.849858i $$0.676688\pi$$
$$614$$ 0.516060 0.0208265
$$615$$ 0 0
$$616$$ −1.67307 −0.0674099
$$617$$ 33.5067i 1.34893i 0.738307 + 0.674464i $$0.235625\pi$$
−0.738307 + 0.674464i $$0.764375\pi$$
$$618$$ 11.6918i 0.470313i
$$619$$ 14.6780 0.589958 0.294979 0.955504i $$-0.404687\pi$$
0.294979 + 0.955504i $$0.404687\pi$$
$$620$$ 0 0
$$621$$ 53.0607 2.12925
$$622$$ 6.62666i 0.265705i
$$623$$ − 7.05086i − 0.282487i
$$624$$ 6.19358 0.247941
$$625$$ 0 0
$$626$$ 2.68244 0.107212
$$627$$ 4.85728i 0.193981i
$$628$$ 4.34122i 0.173234i
$$629$$ 31.4509 1.25403
$$630$$ 0 0
$$631$$ 11.3176 0.450545 0.225273 0.974296i $$-0.427673\pi$$
0.225273 + 0.974296i $$0.427673\pi$$
$$632$$ − 11.0968i − 0.441407i
$$633$$ 60.5116i 2.40512i
$$634$$ 8.58517 0.340961
$$635$$ 0 0
$$636$$ −35.5210 −1.40850
$$637$$ 3.84791i 0.152460i
$$638$$ − 0.474572i − 0.0187885i
$$639$$ −74.2262 −2.93634
$$640$$ 0 0
$$641$$ 34.8988 1.37842 0.689209 0.724562i $$-0.257958\pi$$
0.689209 + 0.724562i $$0.257958\pi$$
$$642$$ 9.93978i 0.392292i
$$643$$ 41.9768i 1.65540i 0.561168 + 0.827702i $$0.310352\pi$$
−0.561168 + 0.827702i $$0.689648\pi$$
$$644$$ −12.9362 −0.509759
$$645$$ 0 0
$$646$$ 2.71408 0.106784
$$647$$ 5.46520i 0.214859i 0.994213 + 0.107430i $$0.0342620\pi$$
−0.994213 + 0.107430i $$0.965738\pi$$
$$648$$ 5.08097i 0.199599i
$$649$$ −2.55215 −0.100181
$$650$$ 0 0
$$651$$ 18.1017 0.709462
$$652$$ − 31.0651i − 1.21660i
$$653$$ 8.76986i 0.343191i 0.985167 + 0.171596i $$0.0548922\pi$$
−0.985167 + 0.171596i $$0.945108\pi$$
$$654$$ 16.2953 0.637196
$$655$$ 0 0
$$656$$ 12.5936 0.491699
$$657$$ 55.3131i 2.15797i
$$658$$ 1.93978i 0.0756204i
$$659$$ −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 $$0.374474\pi$$
$$662$$ 5.26178i 0.204505i
$$663$$ − 14.3684i − 0.558023i
$$664$$ 13.0049 0.504689
$$665$$ 0 0
$$666$$ 6.67799 0.258767
$$667$$ − 7.52543i − 0.291386i
$$668$$ − 9.07805i − 0.351240i
$$669$$ −26.2351 −1.01431
$$670$$ 0 0
$$671$$ 2.84743 0.109924
$$672$$ − 9.16547i − 0.353566i
$$673$$ − 44.3970i − 1.71138i −0.517490 0.855689i $$-0.673134\pi$$
0.517490 0.855689i $$-0.326866\pi$$
$$674$$ −3.73191 −0.143748
$$675$$ 0 0
$$676$$ 24.0049 0.923266
$$677$$ 6.09726i 0.234337i 0.993112 + 0.117168i $$0.0373817\pi$$
−0.993112 + 0.117168i $$0.962618\pi$$
$$678$$ 9.28592i 0.356624i
$$679$$ −3.69181 −0.141679
$$680$$ 0 0
$$681$$ 56.3368 2.15883
$$682$$ 3.27607i 0.125447i
$$683$$ − 37.9224i − 1.45106i −0.688190 0.725531i $$-0.741594\pi$$
0.688190 0.725531i $$-0.258406\pi$$
$$684$$ −11.3319 −0.433284
$$685$$ 0 0
$$686$$ 3.70471 0.141447
$$687$$ − 74.3595i − 2.83699i
$$688$$ 36.0558i 1.37461i
$$689$$ 4.00000 0.152388
$$690$$ 0 0
$$691$$ 13.3145 0.506507 0.253254 0.967400i $$-0.418499\pi$$
0.253254 + 0.967400i $$0.418499\pi$$
$$692$$ − 8.06022i − 0.306404i
$$693$$ − 7.47949i − 0.284123i
$$694$$ −1.91258 −0.0726005
$$695$$ 0 0
$$696$$ 3.52543 0.133631
$$697$$ − 29.2159i − 1.10663i
$$698$$ 2.21633i 0.0838892i
$$699$$ 9.06959 0.343043
$$700$$ 0 0
$$701$$ −23.4893 −0.887180 −0.443590 0.896230i $$-0.646295\pi$$
−0.443590 + 0.896230i $$0.646295\pi$$
$$702$$ − 1.36488i − 0.0515140i
$$703$$ 4.33677i 0.163565i
$$704$$ −8.80150 −0.331719
$$705$$ 0 0
$$706$$ −5.26317 −0.198082
$$707$$ − 12.5620i − 0.472442i
$$708$$ − 9.24443i − 0.347427i
$$709$$ −11.6731 −0.438391 −0.219196 0.975681i $$-0.570343\pi$$
−0.219196 + 0.975681i $$0.570343\pi$$
$$710$$ 0 0
$$711$$ 49.6084 1.86046
$$712$$ 9.47949i 0.355259i
$$713$$ 51.9496i 1.94553i
$$714$$ −6.48886 −0.242840
$$715$$ 0 0
$$716$$ −18.4889 −0.690961
$$717$$ 40.4701i 1.51138i
$$718$$ − 11.4207i − 0.426215i
$$719$$ −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
$$722$$ − 5.53680i − 0.206058i
$$723$$ 53.6227i 1.99425i
$$724$$ 0.622216 0.0231245
$$725$$ 0 0
$$726$$ −7.83362 −0.290733
$$727$$ 3.88094i 0.143936i 0.997407 + 0.0719680i $$0.0229279\pi$$
−0.997407 + 0.0719680i $$0.977072\pi$$
$$728$$ 0.682439i 0.0252929i
$$729$$ 38.6958 1.43318
$$730$$ 0 0
$$731$$ 83.6454 3.09374
$$732$$ 10.3140i 0.381217i
$$733$$ − 14.8845i − 0.549771i −0.961477 0.274885i $$-0.911360\pi$$
0.961477 0.274885i $$-0.0886399\pi$$
$$734$$ −2.61777 −0.0966236
$$735$$ 0 0
$$736$$ 26.3037 0.969569
$$737$$ − 17.5812i − 0.647612i
$$738$$ − 6.20342i − 0.228351i
$$739$$ −2.24935 −0.0827438 −0.0413719 0.999144i $$-0.513173\pi$$
−0.0413719 + 0.999144i $$0.513173\pi$$
$$740$$ 0 0
$$741$$ 1.98126 0.0727836
$$742$$ − 1.80642i − 0.0663159i
$$743$$ 3.46520i 0.127126i 0.997978 + 0.0635630i $$0.0202464\pi$$
−0.997978 + 0.0635630i $$0.979754\pi$$
$$744$$ −24.3368 −0.892229
$$745$$ 0 0
$$746$$ −2.69535 −0.0986836
$$747$$ 58.1388i 2.12719i
$$748$$ 23.0923i 0.844340i
$$749$$ 9.93978 0.363192
$$750$$ 0 0
$$751$$ 3.16992 0.115672 0.0578360 0.998326i $$-0.481580\pi$$
0.0578360 + 0.998326i $$0.481580\pi$$
$$752$$ 23.6686i 0.863106i
$$753$$ 40.0415i 1.45919i
$$754$$ −0.193576 −0.00704963
$$755$$ 0 0
$$756$$ 12.1204 0.440816
$$757$$ − 52.0785i − 1.89283i −0.322958 0.946413i $$-0.604677\pi$$
0.322958 0.946413i $$-0.395323\pi$$
$$758$$ 0.861725i 0.0312993i
$$759$$ 33.3274 1.20971
$$760$$ 0 0
$$761$$ −14.9777 −0.542942 −0.271471 0.962447i $$-0.587510\pi$$
−0.271471 + 0.962447i $$0.587510\pi$$
$$762$$ 5.61868i 0.203543i
$$763$$ − 16.2953i − 0.589929i
$$764$$ −28.4429 −1.02903
$$765$$ 0 0
$$766$$ 0.521889 0.0188566
$$767$$ 1.04101i 0.0375887i
$$768$$ − 25.5669i − 0.922567i
$$769$$ 1.90813 0.0688091 0.0344045 0.999408i $$-0.489047\pi$$
0.0344045 + 0.999408i $$0.489047\pi$$
$$770$$ 0 0
$$771$$ −4.29529 −0.154691
$$772$$ 26.9260i 0.969087i
$$773$$ − 21.7891i − 0.783698i −0.920029 0.391849i $$-0.871835\pi$$
0.920029 0.391849i $$-0.128165\pi$$
$$774$$ 17.7605 0.638388
$$775$$ 0 0
$$776$$ 4.96343 0.178177
$$777$$ − 10.3684i − 0.371965i
$$778$$ 1.79658i 0.0644105i
$$779$$ 4.02858 0.144339
$$780$$ 0 0
$$781$$ −20.8573 −0.746332
$$782$$ − 18.6222i − 0.665929i
$$783$$ 7.05086i 0.251977i
$$784$$ 21.2034 0.757265
$$785$$ 0 0
$$786$$ −10.6222 −0.378882
$$787$$ − 18.1388i − 0.646577i −0.946301 0.323288i $$-0.895212\pi$$
0.946301 0.323288i $$-0.104788\pi$$
$$788$$ − 10.8573i − 0.386775i
$$789$$ 1.28592 0.0457799
$$790$$ 0 0
$$791$$ 9.28592 0.330169
$$792$$ 10.0558i 0.357316i
$$793$$ − 1.16146i − 0.0412445i
$$794$$ −9.30465 −0.330210
$$795$$ 0 0
$$796$$ 42.1432 1.49373
$$797$$ 2.96343i 0.104970i 0.998622 + 0.0524851i $$0.0167142\pi$$
−0.998622 + 0.0524851i $$0.983286\pi$$
$$798$$ − 0.894751i − 0.0316738i
$$799$$ 54.9086 1.94253
$$800$$ 0 0
$$801$$ −42.3783 −1.49736
$$802$$ 2.65386i 0.0937110i
$$803$$ 15.5428i 0.548493i
$$804$$ 63.6829 2.24592
$$805$$ 0 0
$$806$$ 1.33630 0.0470691
$$807$$ 11.4380i 0.402637i
$$808$$ 16.8889i 0.594150i
$$809$$ 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 $$0.359212\pi$$
$$812$$ − 1.71900i − 0.0603252i
$$813$$ − 18.0228i − 0.632085i
$$814$$ 1.87649 0.0657710
$$815$$ 0 0
$$816$$ −79.1753 −2.77169
$$817$$ 11.5339i 0.403520i
$$818$$ − 1.58427i − 0.0553926i
$$819$$ −3.05086 −0.106606
$$820$$ 0 0
$$821$$ 1.52987 0.0533929 0.0266965 0.999644i $$-0.491501\pi$$
0.0266965 + 0.999644i $$0.491501\pi$$
$$822$$ − 3.22168i − 0.112369i
$$823$$ − 46.7195i − 1.62854i −0.580487 0.814269i $$-0.697138\pi$$
0.580487 0.814269i $$-0.302862\pi$$
$$824$$ −15.7190 −0.547597
$$825$$ 0 0
$$826$$ 0.470127 0.0163578
$$827$$ − 29.6499i − 1.03103i −0.856881 0.515514i $$-0.827601\pi$$
0.856881 0.515514i $$-0.172399\pi$$
$$828$$ 77.7516i 2.70205i
$$829$$ 8.79706 0.305534 0.152767 0.988262i $$-0.451182\pi$$
0.152767 + 0.988262i $$0.451182\pi$$
$$830$$ 0 0
$$831$$ 16.1748 0.561099
$$832$$ 3.59010i 0.124464i
$$833$$ − 49.1896i − 1.70432i
$$834$$ 7.73329 0.267782
$$835$$ 0 0
$$836$$ −3.18421 −0.110128
$$837$$ − 48.6735i − 1.68240i
$$838$$ 7.57136i 0.261548i
$$839$$ −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
$$842$$ 7.62269i 0.262695i
$$843$$ − 19.4380i − 0.669481i
$$844$$ −39.6686 −1.36545
$$845$$ 0 0
$$846$$ 11.6588 0.400837
$$847$$ 7.83362i 0.269166i
$$848$$ − 22.0415i − 0.756908i
$$849$$ −74.9086 −2.57086
$$850$$ 0 0
$$851$$ 29.7560 1.02002
$$852$$ − 75.5496i − 2.58829i
$$853$$ − 54.8845i − 1.87921i −0.342263 0.939604i $$-0.611193\pi$$
0.342263 0.939604i $$-0.388807\pi$$
$$854$$ −0.524521 −0.0179487
$$855$$ 0 0
$$856$$ −13.3635 −0.456755
$$857$$ 36.4385i 1.24471i 0.782733 + 0.622357i $$0.213825\pi$$
−0.782733 + 0.622357i $$0.786175\pi$$
$$858$$ − 0.857279i − 0.0292670i
$$859$$ −1.72885 −0.0589875 −0.0294938 0.999565i $$-0.509390\pi$$
−0.0294938 + 0.999565i $$0.509390\pi$$
$$860$$ 0 0
$$861$$ −9.63158 −0.328243
$$862$$ 1.32741i 0.0452116i
$$863$$ 9.40192i 0.320045i 0.987113 + 0.160023i $$0.0511567\pi$$
−0.987113 + 0.160023i $$0.948843\pi$$
$$864$$ −24.6450 −0.838439
$$865$$ 0 0
$$866$$ 8.40144 0.285493
$$867$$ 134.323i 4.56186i
$$868$$ 11.8666i 0.402780i
$$869$$ 13.9398 0.472875
$$870$$ 0 0
$$871$$ −7.17130 −0.242990
$$872$$ 21.9081i 0.741903i
$$873$$ 22.1891i 0.750988i
$$874$$ 2.56782 0.0868579
$$875$$ 0 0
$$876$$ −56.2993 −1.90218
$$877$$ 8.91750i 0.301123i 0.988601 + 0.150561i $$0.0481081\pi$$
−0.988601 + 0.150561i $$0.951892\pi$$
$$878$$ 0.632060i 0.0213310i
$$879$$ 54.7052 1.84516
$$880$$ 0 0
$$881$$ −42.1245 −1.41921 −0.709605 0.704600i $$-0.751126\pi$$
−0.709605 + 0.704600i $$0.751126\pi$$
$$882$$ − 10.4445i − 0.351683i
$$883$$ 38.4340i 1.29341i 0.762741 + 0.646704i $$0.223853\pi$$
−0.762741 + 0.646704i $$0.776147\pi$$
$$884$$ 9.41927 0.316804
$$885$$ 0 0
$$886$$ −1.07805 −0.0362179
$$887$$ − 38.6365i − 1.29729i −0.761092 0.648643i $$-0.775337\pi$$
0.761092 0.648643i $$-0.224663\pi$$
$$888$$ 13.9398i 0.467788i
$$889$$ 5.61868 0.188444
$$890$$ 0 0
$$891$$ −6.38271 −0.213829
$$892$$ − 17.1985i − 0.575848i
$$893$$ 7.57136i 0.253366i
$$894$$ 5.06959 0.169552
$$895$$ 0 0
$$896$$ 7.93533 0.265101
$$897$$ − 13.5941i − 0.453894i
$$898$$ − 11.6227i − 0.387854i
$$899$$ −6.90321 −0.230235
$$900$$ 0 0
$$901$$ −51.1338 −1.70351
$$902$$ − 1.74314i − 0.0580402i
$$903$$ − 27.5754i − 0.917651i
$$904$$ −12.4844 −0.415226
$$905$$ 0 0
$$906$$ 9.75203 0.323989
$$907$$ 0.534795i 0.0177576i 0.999961 + 0.00887880i $$0.00282625\pi$$
−0.999961 + 0.00887880i $$0.997174\pi$$
$$908$$ 36.9318i 1.22562i
$$909$$ −75.5022 −2.50425
$$910$$ 0 0
$$911$$ −23.6686 −0.784177 −0.392088 0.919928i $$-0.628247\pi$$
−0.392088 + 0.919928i $$0.628247\pi$$
$$912$$ − 10.9175i − 0.361515i
$$913$$ 16.3368i 0.540668i
$$914$$ −4.18468 −0.138417
$$915$$ 0 0
$$916$$ 48.7467 1.61064
$$917$$ 10.6222i 0.350776i
$$918$$ 17.4479i 0.575865i
$$919$$ −35.7748 −1.18010 −0.590051 0.807366i $$-0.700892\pi$$
−0.590051 + 0.807366i $$0.700892\pi$$
$$920$$ 0 0
$$921$$ −4.81579 −0.158686
$$922$$ − 5.06376i − 0.166766i
$$923$$ 8.50760i 0.280031i
$$924$$ 7.61285 0.250444
$$925$$ 0 0
$$926$$ −9.45536 −0.310722
$$927$$ − 70.2721i − 2.30804i
$$928$$ 3.49532i 0.114739i
$$929$$ 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
$$932$$ 5.94561i 0.194755i
$$933$$ − 61.8390i − 2.02452i
$$934$$ 0.369800 0.0121002
$$935$$ 0 0
$$936$$ 4.10171 0.134069
$$937$$ 42.1245i 1.37615i 0.725641 + 0.688073i $$0.241543\pi$$
−0.725641 + 0.688073i $$0.758457\pi$$
$$938$$ 3.23860i 0.105744i
$$939$$ −25.0321 −0.816892
$$940$$ 0 0
$$941$$ −3.89829 −0.127081 −0.0635403 0.997979i $$-0.520239\pi$$
−0.0635403 + 0.997979i $$0.520239\pi$$
$$942$$ 2.06022i 0.0671257i
$$943$$ − 27.6414i − 0.900129i
$$944$$ 5.73636 0.186703
$$945$$ 0 0
$$946$$ 4.99063 0.162259
$$947$$ 9.56691i 0.310883i 0.987845 + 0.155441i $$0.0496800\pi$$
−0.987845 + 0.155441i $$0.950320\pi$$
$$948$$ 50.4929i 1.63993i
$$949$$ 6.33984 0.205800
$$950$$ 0 0
$$951$$ −80.1156 −2.59793
$$952$$ − 8.72393i − 0.282744i
$$953$$ − 27.2070i − 0.881320i −0.897674 0.440660i $$-0.854744\pi$$
0.897674 0.440660i $$-0.145256\pi$$
$$954$$ −10.8573 −0.351517
$$955$$ 0 0
$$956$$ −26.5303 −0.858053
$$957$$ 4.42864i 0.143158i
$$958$$ − 12.7699i − 0.412575i
$$959$$ −3.22168 −0.104033
$$960$$ 0 0
$$961$$ 16.6543 0.537237
$$962$$ − 0.765413i − 0.0246779i
$$963$$ − 59.7418i − 1.92515i
$$964$$ −35.1526 −1.13219
$$965$$ 0 0
$$966$$ −6.13918 −0.197525
$$967$$ − 16.8015i − 0.540300i −0.962818 0.270150i $$-0.912927\pi$$
0.962818 0.270150i $$-0.0870733\pi$$
$$968$$ − 10.5319i − 0.338507i
$$969$$ −25.3274 −0.813633
$$970$$ 0 0
$$971$$ −17.4465 −0.559884 −0.279942 0.960017i $$-0.590315\pi$$
−0.279942 + 0.960017i $$0.590315\pi$$
$$972$$ 17.1383i 0.549710i
$$973$$ − 7.73329i − 0.247918i
$$974$$ 3.15701 0.101157
$$975$$ 0 0
$$976$$ −6.40006 −0.204861
$$977$$ 32.0513i 1.02541i 0.858564 + 0.512706i $$0.171357\pi$$
−0.858564 + 0.512706i $$0.828643\pi$$
$$978$$ − 14.7427i − 0.471418i
$$979$$ −11.9081 −0.380586
$$980$$ 0 0
$$981$$ −97.9407 −3.12701
$$982$$ − 9.08694i − 0.289976i
$$983$$ 16.5259i 0.527094i 0.964646 + 0.263547i $$0.0848925\pi$$
−0.964646 + 0.263547i $$0.915108\pi$$
$$984$$ 12.9491 0.412804
$$985$$ 0 0
$$986$$ 2.47457 0.0788064
$$987$$ − 18.1017i − 0.576184i
$$988$$ 1.29883i 0.0413211i
$$989$$ 79.1378 2.51644
$$990$$ 0 0
$$991$$ −9.34920 −0.296987 −0.148494 0.988913i $$-0.547442\pi$$
−0.148494 + 0.988913i $$0.547442\pi$$
$$992$$ − 24.1289i − 0.766094i
$$993$$ − 49.1022i − 1.55821i
$$994$$ 3.84208 0.121863
$$995$$ 0 0
$$996$$ −59.1753 −1.87504
$$997$$ − 15.9956i − 0.506584i −0.967390 0.253292i $$-0.918487\pi$$
0.967390 0.253292i $$-0.0815134\pi$$
$$998$$ − 6.83854i − 0.216470i
$$999$$ −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.b.e.349.4 6
5.2 odd 4 725.2.a.e.1.2 3
5.3 odd 4 145.2.a.c.1.2 3
5.4 even 2 inner 725.2.b.e.349.3 6
15.2 even 4 6525.2.a.be.1.2 3
15.8 even 4 1305.2.a.p.1.2 3
20.3 even 4 2320.2.a.n.1.1 3
35.13 even 4 7105.2.a.o.1.2 3
40.3 even 4 9280.2.a.br.1.3 3
40.13 odd 4 9280.2.a.bj.1.1 3
145.28 odd 4 4205.2.a.f.1.2 3

By twisted newform
Twist Min Dim Char Parity Ord Type
145.2.a.c.1.2 3 5.3 odd 4
725.2.a.e.1.2 3 5.2 odd 4
725.2.b.e.349.3 6 5.4 even 2 inner
725.2.b.e.349.4 6 1.1 even 1 trivial
1305.2.a.p.1.2 3 15.8 even 4
2320.2.a.n.1.1 3 20.3 even 4
4205.2.a.f.1.2 3 145.28 odd 4
6525.2.a.be.1.2 3 15.2 even 4
7105.2.a.o.1.2 3 35.13 even 4
9280.2.a.bj.1.1 3 40.13 odd 4
9280.2.a.br.1.3 3 40.3 even 4