# Properties

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

## $q$-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 $$-0.964917\pi$$
0.993932 0.109993i $$-0.0350829\pi$$
$$3$$ 2.90321i 1.67617i 0.545540 + 0.838085i $$0.316325\pi$$
−0.545540 + 0.838085i $$0.683675\pi$$
$$4$$ 1.90321 0.951606
$$5$$ 0 0
$$6$$ 0.903212 0.368735
$$7$$ 0.903212i 0.341382i 0.985325 + 0.170691i $$0.0546000\pi$$
−0.985325 + 0.170691i $$0.945400\pi$$
$$8$$ − 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 $$-0.972500\pi$$
0.996270 0.0862858i $$-0.0274998\pi$$
$$14$$ 0.280996 0.0750994
$$15$$ 0 0
$$16$$ 3.42864 0.857160
$$17$$ 7.95407i 1.92914i 0.263819 + 0.964572i $$0.415018\pi$$
−0.263819 + 0.964572i $$0.584982\pi$$
$$18$$ 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 $$0.287121\pi$$
−0.620028 + 0.784580i $$0.712879\pi$$
$$24$$ 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 $$-0.894628\pi$$
0.945706 0.325022i $$-0.105372\pi$$
$$38$$ − 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 $$-0.703854\pi$$
0.597536 0.801842i $$-0.296146\pi$$
$$44$$ −2.90321 −0.437676
$$45$$ 0 0
$$46$$ 2.34122 0.345194
$$47$$ − 6.90321i − 1.00694i −0.864014 0.503468i $$-0.832057\pi$$
0.864014 0.503468i $$-0.167943\pi$$
$$48$$ 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 $$0.145561\pi$$
−0.897251 + 0.441521i $$0.854439\pi$$
$$54$$ −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 $$-0.751383\pi$$
0.710173 0.704028i $$-0.248617\pi$$
$$68$$ 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 $$0.203353\pi$$
−0.802781 + 0.596274i $$0.796647\pi$$
$$74$$ −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 $$0.199993\pi$$
−0.809030 + 0.587768i $$0.800007\pi$$
$$84$$ −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 $$0.0665351\pi$$
−0.978233 + 0.207507i $$0.933465\pi$$
$$98$$ − 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 $$-0.779870\pi$$
0.770252 0.637740i $$-0.220130\pi$$
$$104$$ −0.755569 −0.0740896
$$105$$ 0 0
$$106$$ 2.00000 0.194257
$$107$$ − 11.0049i − 1.06389i −0.846780 0.531943i $$-0.821462\pi$$
0.846780 0.531943i $$-0.178538\pi$$
$$108$$ − 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 $$-0.839337\pi$$
0.875302 0.483577i $$-0.160663\pi$$
$$114$$ 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 $$-0.910990\pi$$
0.961157 0.276002i $$-0.0890097\pi$$
$$128$$ − 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 $$0.0486909\pi$$
−0.988323 + 0.152371i $$0.951309\pi$$
$$138$$ 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 $$-0.970987\pi$$
0.995849 0.0910217i $$-0.0290133\pi$$
$$158$$ 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 $$0.220750\pi$$
−0.769009 + 0.639238i $$0.779250\pi$$
$$164$$ 6.99063 0.545877
$$165$$ 0 0
$$166$$ 3.33185 0.258602
$$167$$ 4.76986i 0.369103i 0.982823 + 0.184551i $$0.0590832\pi$$
−0.982823 + 0.184551i $$0.940917\pi$$
$$168$$ 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 $$0.0514696\pi$$
−0.986956 + 0.160993i $$0.948530\pi$$
$$174$$ 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 $$-0.829947\pi$$
0.860657 0.509185i $$-0.170053\pi$$
$$194$$ 1.27163 0.0912976
$$195$$ 0 0
$$196$$ 11.7699 0.840704
$$197$$ 5.70471i 0.406444i 0.979133 + 0.203222i $$0.0651413\pi$$
−0.979133 + 0.203222i $$0.934859\pi$$
$$198$$ − 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 $$0.0978435\pi$$
−0.953128 + 0.302567i $$0.902157\pi$$
$$224$$ 3.15701 0.210937
$$225$$ 0 0
$$226$$ −3.19850 −0.212761
$$227$$ − 19.4050i − 1.28795i −0.765045 0.643977i $$-0.777283\pi$$
0.765045 0.643977i $$-0.222717\pi$$
$$228$$ 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 $$-0.967370\pi$$
0.994751 0.102330i $$-0.0326296\pi$$
$$234$$ 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 $$0.0146934\pi$$
−0.998935 + 0.0461442i $$0.985307\pi$$
$$258$$ − 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 $$-0.995653\pi$$
0.999907 0.0136561i $$-0.00434701\pi$$
$$264$$ −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 $$-0.946471\pi$$
0.985893 0.167375i $$-0.0535292\pi$$
$$278$$ − 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 $$0.278194\pi$$
−0.641785 + 0.766884i $$0.721806\pi$$
$$284$$ 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 $$-0.814472\pi$$
0.834895 0.550410i $$-0.185528\pi$$
$$294$$ 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 $$0.0150731\pi$$
−0.998879 + 0.0473358i $$0.984927\pi$$
$$308$$ − 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 $$0.0783541\pi$$
−0.969856 + 0.243678i $$0.921646\pi$$
$$314$$ −0.709636 −0.0400471
$$315$$ 0 0
$$316$$ −17.3921 −0.978381
$$317$$ 27.5955i 1.54992i 0.632012 + 0.774959i $$0.282229\pi$$
−0.632012 + 0.774959i $$0.717771\pi$$
$$318$$ 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 $$-0.894057\pi$$
0.945121 0.326720i $$-0.105943\pi$$
$$338$$ − 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 $$-0.947234\pi$$
0.986292 0.165011i $$-0.0527661\pi$$
$$348$$ 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 $$-0.851348\pi$$
0.892921 0.450214i $$-0.148652\pi$$
$$354$$ 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 $$-0.929521\pi$$
0.975587 0.219613i $$-0.0704794\pi$$
$$368$$ 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 $$-0.927992\pi$$
0.974521 0.224295i $$-0.0720078\pi$$
$$374$$ −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 $$0.0136465\pi$$
−0.999081 + 0.0428585i $$0.986354\pi$$
$$384$$ 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 $$-0.729802\pi$$
0.660844 0.750523i $$-0.270198\pi$$
$$398$$ − 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 $$0.224765\pi$$
−0.760885 + 0.648887i $$0.775235\pi$$
$$434$$ −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 $$-0.973768\pi$$
0.996606 0.0823184i $$-0.0262324\pi$$
$$444$$ 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 $$-0.898128\pi$$
0.949223 0.314604i $$-0.101872\pi$$
$$458$$ − 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 $$-0.750394\pi$$
0.707982 0.706231i $$-0.249606\pi$$
$$464$$ 3.42864 0.159171
$$465$$ 0 0
$$466$$ −0.971896 −0.0450222
$$467$$ 1.18865i 0.0550043i 0.999622 + 0.0275022i $$0.00875532\pi$$
−0.999622 + 0.0275022i $$0.991245\pi$$
$$468$$ 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 $$0.0738454\pi$$
−0.973210 + 0.229917i $$0.926155\pi$$
$$488$$ 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 $$0.0411258\pi$$
−0.991665 + 0.128841i $$0.958874\pi$$
$$504$$ −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 $$-0.892622\pi$$
0.943639 0.330975i $$-0.107378\pi$$
$$524$$ −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 $$0.106525\pi$$
−0.944522 + 0.328448i $$0.893475\pi$$
$$548$$ 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 $$0.0671202\pi$$
−0.977850 + 0.209305i $$0.932880\pi$$
$$558$$ − 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 $$-0.803302\pi$$
0.815070 0.579362i $$-0.196698\pi$$
$$564$$ 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 $$0.0709359\pi$$
−0.975271 + 0.221012i $$0.929064\pi$$
$$578$$ 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 $$0.0590969\pi$$
−0.982815 + 0.184594i $$0.940903\pi$$
$$588$$ 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 $$0.0939149\pi$$
−0.956790 + 0.290780i $$0.906085\pi$$
$$594$$ 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 $$-0.911020\pi$$
0.961183 0.275912i $$-0.0889799\pi$$
$$608$$ − 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 $$-0.676688\pi$$
0.527012 0.849858i $$-0.323312\pi$$
$$614$$ 0.516060 0.0208265
$$615$$ 0 0
$$616$$ −1.67307 −0.0674099
$$617$$ − 33.5067i − 1.34893i −0.738307 0.674464i $$-0.764375\pi$$
0.738307 0.674464i $$-0.235625\pi$$
$$618$$ − 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 $$-0.689648\pi$$
0.561168 0.827702i $$-0.310352\pi$$
$$644$$ −12.9362 −0.509759
$$645$$ 0 0
$$646$$ 2.71408 0.106784
$$647$$ − 5.46520i − 0.214859i −0.994213 0.107430i $$-0.965738\pi$$
0.994213 0.107430i $$-0.0342620\pi$$
$$648$$ − 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 $$-0.945108\pi$$
0.985167 0.171596i $$-0.0548922\pi$$
$$654$$ 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 $$0.326866\pi$$
−0.517490 + 0.855689i $$0.673134\pi$$
$$674$$ −3.73191 −0.143748
$$675$$ 0 0
$$676$$ 24.0049 0.923266
$$677$$ − 6.09726i − 0.234337i −0.993112 0.117168i $$-0.962618\pi$$
0.993112 0.117168i $$-0.0373817\pi$$
$$678$$ − 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 $$0.258406\pi$$
−0.688190 + 0.725531i $$0.741594\pi$$
$$684$$ −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 $$-0.977072\pi$$
0.997407 0.0719680i $$-0.0229279\pi$$
$$728$$ − 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 $$0.0886399\pi$$
−0.961477 + 0.274885i $$0.911360\pi$$
$$734$$ −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 $$-0.979754\pi$$
0.997978 0.0635630i $$-0.0202464\pi$$
$$744$$ −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 $$0.395323\pi$$
−0.322958 + 0.946413i $$0.604677\pi$$
$$758$$ − 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 $$0.128165\pi$$
−0.920029 + 0.391849i $$0.871835\pi$$
$$774$$ 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 $$0.104788\pi$$
−0.946301 + 0.323288i $$0.895212\pi$$
$$788$$ 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 $$-0.983286\pi$$
0.998622 0.0524851i $$-0.0167142\pi$$
$$798$$ 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 $$0.302862\pi$$
−0.580487 + 0.814269i $$0.697138\pi$$
$$824$$ −15.7190 −0.547597
$$825$$ 0 0
$$826$$ 0.470127 0.0163578
$$827$$ 29.6499i 1.03103i 0.856881 + 0.515514i $$0.172399\pi$$
−0.856881 + 0.515514i $$0.827601\pi$$
$$828$$ − 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 $$0.388807\pi$$
−0.342263 + 0.939604i $$0.611193\pi$$
$$854$$ −0.524521 −0.0179487
$$855$$ 0 0
$$856$$ −13.3635 −0.456755
$$857$$ − 36.4385i − 1.24471i −0.782733 0.622357i $$-0.786175\pi$$
0.782733 0.622357i $$-0.213825\pi$$
$$858$$ 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 $$-0.948843\pi$$
0.987113 0.160023i $$-0.0511567\pi$$
$$864$$ −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 $$-0.951892\pi$$
0.988601 0.150561i $$-0.0481081\pi$$
$$878$$ − 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 $$-0.776147\pi$$
0.762741 0.646704i $$-0.223853\pi$$
$$884$$ 9.41927 0.316804
$$885$$ 0 0
$$886$$ −1.07805 −0.0362179
$$887$$ 38.6365i 1.29729i 0.761092 + 0.648643i $$0.224663\pi$$
−0.761092 + 0.648643i $$0.775337\pi$$
$$888$$ − 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 $$-0.997174\pi$$
0.999961 0.00887880i $$-0.00282625\pi$$
$$908$$ − 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 $$-0.758457\pi$$
0.725641 0.688073i $$-0.241543\pi$$
$$938$$ − 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 $$-0.950320\pi$$
0.987845 0.155441i $$-0.0496800\pi$$
$$948$$ − 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 $$0.145256\pi$$
−0.897674 + 0.440660i $$0.854744\pi$$
$$954$$ −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 $$0.0870733\pi$$
−0.962818 + 0.270150i $$0.912927\pi$$
$$968$$ 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 $$-0.828643\pi$$
0.858564 0.512706i $$-0.171357\pi$$
$$978$$ 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 $$-0.915108\pi$$
0.964646 0.263547i $$-0.0848925\pi$$
$$984$$ 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 $$0.0815134\pi$$
−0.967390 + 0.253292i $$0.918487\pi$$
$$998$$ 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.349.3 6
5.2 odd 4 145.2.a.c.1.2 3
5.3 odd 4 725.2.a.e.1.2 3
5.4 even 2 inner 725.2.b.e.349.4 6
15.2 even 4 1305.2.a.p.1.2 3
15.8 even 4 6525.2.a.be.1.2 3
20.7 even 4 2320.2.a.n.1.1 3
35.27 even 4 7105.2.a.o.1.2 3
40.27 even 4 9280.2.a.br.1.3 3
40.37 odd 4 9280.2.a.bj.1.1 3
145.57 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 5.2 odd 4
725.2.a.e.1.2 3 5.3 odd 4
725.2.b.e.349.3 6 1.1 even 1 trivial
725.2.b.e.349.4 6 5.4 even 2 inner
1305.2.a.p.1.2 3 15.2 even 4
2320.2.a.n.1.1 3 20.7 even 4
4205.2.a.f.1.2 3 145.57 odd 4
6525.2.a.be.1.2 3 15.8 even 4
7105.2.a.o.1.2 3 35.27 even 4
9280.2.a.bj.1.1 3 40.37 odd 4
9280.2.a.br.1.3 3 40.27 even 4